CATS Proceedings Printout - Graz University of Technology

th
The 15 International IGTE Symposium
on Numerical Field Calculation in Electrical Engineering
Institute for Fundamentals and Theory
in Electrical Engineering - IGTE
Proceedings
Sept. 17 - 19, 2012
Hotel Novapark, Graz, Austria
ISBN: 978-3-85125-258-3
Verlag der Technischen Universität Graz
www.ub.tugraz.at/Verlag
Graz University
of Technology

The 15th International IGTE Symposium on Numerical Field Calculation in Electrical
Engineering is sponsored and supported by:

.
- I - 15th IGTE Symposium 2012
Table of Contents
Multi Domain Multi Scale Problems in the High Frequency Finite 1
Element Method (FEM)
Istvan Bardi, Kezhong Zhao, Rickard Petersson, John Silvestro, Nancy Lambert
A parallel-TLM algorithm. Modelling the Earth-ionosphere waveguide 7
Sergio Toledo-Redondo, Alfonso Salinas, Jesús Fornieles, Jorge Portí,
Bruno Besser, Herbert I. M. Lichtenegger
A Novel Parametric Model Order Reduction Approach with Applications 13
to Geometrically Parameterized Microwave Devices
Stefan Burgard, Ortwin Farle, Romanus Dyczij-Edlinger
Efficient Finite-Element Computation of Far-Fields of Phased Arrays by 19
Order Reduction
Alexander Sommer, Ortwin Farle, Romanus Dyczij-Edlinger
Nanoparticle device for biomedical and optoelectronics applications 25
Renato Iovine, Luigi La Spada, Lucio Vegni
Validation of measurements with conjugate heat transfer models 31
Maximilian Schrittwieser, Oszkár Bíró, Ernst Farnleitner, Gebhard Kastner
Computing the shielding effectiveness of waveguides using FE-mesh 37
truncation by surface operator implementation
Christian Tuerk, Werner Renhart, Christian Magele
Heat Transfer Analysis on End Windings of a Hydro Generator using a 41
Stator-Slot-Sector Model
Stephan Klomberg, Ernst Farnleitner, Gebhard Kastner, Oszkár Bíró
Numerical Investigation of Linear Systems Obtained by Extended 47
Element-Free Galerkin Method
Taku Itoh, Soichiro Ikuno, Atsushi Kamitani
Electromagnetic Wave Propagation Simulation in Corrugated 53
Waveguide using Meshless Time Domain Method
Soichiro Ikuno, Yoshihisa Fujita, Taku Itoh, Susumu Nakata, Atsushi Kamitani
Optimization of Permanent Magnet Linear Actuator for Braille Screen 59
Ivan Yatchev, Iosko Balabozov, Krastio Hinov, Vultchan Gueorgiev,
Dimitar Karastoyanov
3D Finite Element Analysis of Induction Heating System for High 63
Frequency Welding
Ilona Iatcheva, Georgi Gigov, Georgi Kunov, Rumena Stancheva
Optimization Algorithms in the View of State Space Concepts 67
Markus Neumayer, Daniel Watzenig, Gerald Steiner, Bernhard Brandstätter
Quasi TEM Analysis of 2D Symmetrically Coupled Strip Lines with Finite 73
Grounded Plane using HBEM
Saša Ilić, Mirjana Perić, Slavoljub Aleksić, Nebojsa Raicevic

- II - 15th IGTE Symposium 2012
Design Approach for a Line-Start Internal Permanent Magnet 78
Synchronous Motor
Vera Elistratova, Michel Hecquet, Pascal Brochet, Darius Vizireanu,
Maxime Dessoude
Speed-up of Nonlinear Electromagnetic Field Analysis using Fixed-Point 84
Method
Norio Takahashi, Kouske Shimoyama, Daisuke Miyagi, Hiroyuki Kaimori
Software agent based domain decomposition method 89
Matthias Jüttner, André Buchau, Michael Rauscher, Wolfgang M. Rucker,
Peter Göhner
Stochastic Jiles-Atherton model accounting for soft magnetic material 95
variability
Rindra Ramarotafika, Abdelkader Benabou, Stéphane Clénet
Human exposure to the magnetic field produced by MFDC spot welding 101
systems
Davide Bavastro, Aldo Canova, Luca Giaccone, Michele Manca, Marco Simioli
A Circuital Approach for Eddy Currents Fast Evaluation in Beam-like 108
Structures
Alessandro Formisano, Raffaele Martone
Convergence Characteristics of Preconditioned MRTR Method with 113
Eisenstat’s Technique in Real Symmetric Sparse Matrix
Yoshifumi Okamoto, Tomonori Tsuburaya, Koji Fujiwara, Shuji Sato
High Frequency Mixing Rule Based Effective Medium Theory of 119
Metamaterials
Zsolt Szabo
Enhancement of Maximum Starting Torque and Efficiency in Permanent 125
Magnet Synchronous Motors
Jawad Faiz, Vahid Ghorbanian, Bashir Mahdi Ebrahimi
Core Losses Estimation Techniques in Electrical Machines with 131
Different Supplies-A Review
Jawad Faiz, Amir Masoud Takbash, Bashir Mahdi Ebrahimi
Fast Computation of Inductances and Capacitances of High Voltage 137
Power Transformer Windings
Tomislav Župan, Željko Štih, Bojan Trkulja
Numerical and Experimental Investigations of the Structural 144
Characteristics of Stator Core Stacks
Mathias Mair, Bernhard Weilharter, Siegfried Rainer, Katrin Ellermann,
Oszkár Bíró
Proper Location of the Regulating Coil in Transformers from 154
Short-circuit Forces Point of View
Oluş Sonmez, Bilal Düzgün, Güven Kömürgöz
Robust Design of IPM motors using Co-Evolutionary Algorithms 160
Min Li, Andre Ruela, Frederico Guimaraes, Jaime Ramirez, David Lowther

- III - 15th IGTE Symposium 2012
Free-form optimization for magnetic design 167
Zoran Andjelic, Salih Sadovic
Optimization for ECT treatment planning 171
Paolo Di Barba, Luca Giovanni Campana, Fabrizio Dughiero, Carlo Riccardo Rossi,
Elisabetta Sieni
Investigation of the Electroporation Effect in a Singel Cell 175
Jaime Ramirez, William Figueiredo, Joao Francisco Vale, Isabela Metzker,
Rafael Santos, Elizabeth Silva, David Lowther
Anisotropic Model for the Numerical Computation of Magnetostriction 181
in Steel Sheets
Manfred Kaltenbacher, Adrian Volk, Michael Ertl
Analytic Approximation Solution for the Schwarz-Christoffel Parameter 186
Problem
Norbert Eidenberger, Bernhard G. Zagar
Additional Eddy Current Losses in Induction Machines Due to 190
Interlaminar Short Circuits
Paul Handgruber, Andrej Stermecki, Oszkár Bíró, Georg Ofner
Evaluating the influence of manufacturing tolerances in permanent 198
magnet synchronous machines
Isabel Coenen, Thomas Herold, Christelle Piantsop Mboo, Kay Hameyer
Eddy current analysis of a PWM controlled induction machine 204
Hai Van Jorks, Erion Gjonaj, Thomas Weiland
Computation of end-winding inductances of rotating electrical 208
machinery through three-dimensional magnetostatic integral FEM formulation
Flavio Calvano, Giorgio Dal Mut, Fabrizio Ferraioli, Alessandro Formisano,
Fabrizio Marignetti, Raffaele Martone, Guglielmo Rubinacci,
Antonello Tamburrino, Salvatore Ventre
Magnetomechanical Coupled FE Simulations of Rotating Electrical 214
Machines
Anouar Belahcen, Katarzyna Fonteyn, Reijo Kouhia, Paavo Rasilo, Antero Arkkio
Saturable Model of Squirrel-cage Induction Motors under Stator 220
Inter-turn Fault
Jawad Faiz, Mansour Ojaghi, Mahdi Sabouri
Accurate Magnetostatic Simulation of Step-Lap Joints in Transformer 226
Cores Using Anisotropic Higher Order FEM
Andreas Hauck, Michael Ertl, Joachim Schöberl, Manfred Kaltenbacher
Finite Element Based Modeling of Wound Rotor Induction Machines 232
Martin Mohr, Oszkár Bíró, Andrej Stermecki, Franz Diwoky
Post Insulator Optimization Based on Dynamic Population Size 238
Peter Kitak, Arnel Glotic, Igor Ticar
Simulation of the Absorbing Clamp Method for Optimizing the 242
Shielding of Power Cables
Szabolcs Gyimóthy, József Pávó, Péter Kiss, Tomoaki Toratani, Ryuichi Katsumi,
Gábor Varga

- IV - 15th IGTE Symposium 2012
A Neural Network Approach to Sizing an Electrical Machine 248
Steven Bielby, David Lowther
Exploring and Exploiting Parallelism in the Finite Element Method on 254
Multi-core Processors: an Overview
Hussein Moghnieh, David Lowther
Diagnosis of real cracks from the three spatial components of the eddy 262
current testing signals
Milan Smetana, Ladislav Janousek, Mihai Rebican, Tatiana Strapacova,
Anton Duca, Gabriel Preda
Adaptive Galaxy-Based Search Approach Applied to Loney’s Solenoid 267
Benchmark Problem
Leandro dos Santos Coelho, Teodoro Cardoso Bora, Piergiorgio Alotto
Implementation of a 3D magnetic circuit model for automotive 271
applications
Ioannis Anastasiadis, Andreas Buchinger, Tobias Werth, Lukas Bellwald,
Kurt Preis
Robust Optimization of Passive RFID Antennas Loaded by Non-linear 276
Circuits
Yuta Watanabe, Hajime Igarashi
Mixed Order Edge-based Finite Element Analysis by Means of 282
Nonconforming Connection
Yoshifumi Okamoto, Shuji Sato
Topology Optimization Using Parallel Search Strategy for Magnetic 288
Devices
Takumi Nagano, Shogo Yasukawa, Shinji Wakao, Yoshifumi Okamoto
Modeling of the Road Influence on the Grounding System in its Vicinity 294
Dragan Vuckovic, Nenad Cvetkovic, Dejan Krstic, Miodrag Stojanovic
Interaction Magnetic Force Calculation of Axial Passive Magnetic 300
Bearing Using Magnetization Charges and Discretization Technique
Saša Ilić, Ana Vuckovic, Slavoljub Aleksić
Consideration of erroneous magnets in the electromagnetic field 305
simulation
Peter Offermann, Isabel Coenen, Kay Hameyer
Potential of Spheroids in a Homogeneous Magnetic Field in Cartesian 310
Coordinates
Markus Kraiger, Bernhard Schnizer
Application of Signal Processing Tools for Fault Diagnosis in Induction 315
Motors-A Review
Jawad Faiz, Amir Masoud Takbash, Bashir Mahdi Ebrahimi, Subhasis Nandi
Experimental Calibration of Numerical Model of Thermoelastic Actuator 321
Lukas Voracek, Vaclav Kotlan, Bohus Ulrych

- V - 15th IGTE Symposium 2012
Scattering Calculations of Passive UHF-RFID Transponders 327
Thomas Bauernfeind, Gergely Koczka, Kurt Preis, Oszkár Bíró
Simulation of a high speed Reluctance Machine including hysteresis 331
and eddy current losses
Bernhard Schweighofer, Hannes Wegleiter, Manes Recheis, Paul Fulmek
An Iterative Domain Decomposition Method for Solving Wave 337
Propagation Problems
Koczka Gergely, Thomas Bauernfeind, Kurt Preis, Oszkár Bíró
On Effectiveness of Model Reduction for Computational 340
Electromagnetism
Yuki Sato, Hajime Igarashi
Calculation of eddy-current probe signal for a volumetric defect using 346
global series expansion
Sandor Bilicz, József Pávó, Szabolcs Gyimóthy
Bodies motion computation using eddy-current integral equation 352
Mihai Maricaru, Ioan R. Ciric, Horia Gavrila, George-Marian Vasilescu,
Florea I. Hantila
Adaptive Inductance Computation on GPU’s 357
Andrea Gaetano Chiariello, Alessandro Formisano, Raffaele Martone
The reduced-basis method applied to transport equations of a 362
lithium-ion battery
Stefan Volkwein, Andrea Wesche
Surrogate Parameter Optimization based on Space Mapping for 368
Lithium-Ion Cell Models
Matthias Scharrer, Bettina Suhr, Daniel Watzenig
Large Scale Energy Storage with Redox Flow Batteries 374
Piergiorgio Alotto, Massimo Guarnieri, Federico Moro, Andrea Stella
Model Order Reduction for a Lithium-Ion Cell 380
Bettina Suhr, Jelena Rubesa
Automatic domain detection for a meshfree post-processing in 386
boundary element methods
André Buchau, Matthias Jüttner, Wolfgang M. Rucker
Efficient modeling of coil filament losses in 2D 392
Leena Lehti, Janne Keränen, Saku Suuriniemi, Timo Tarhasaari, Lauri Kettunen
Optimization of Energy Storage Usage 398
Arnel Glotic, Peter Kitak, Igor Ticar, Adnan Glotic
Adaptive Surrogate Approach for Bayesian Inference in Inverse 403
Problems
Markus Neumayer, Helcio R.B. Orlande, Marcello J. Colaco, Daniel Watzenig,
Gerald Steiner, Bernhard Brandstätter, George S. Dulikravich

- 1 - 15th IGTE Symposium 2012
Multi-Domain Multi-Scale Problems in High
Frequency Finite Element Methods
Istvan Bardi, Kezhong Zhao, Rickard Petersson, John Silvestro and Nancy Lambert
ANSYS Inc., 225 W Station Square Drive, Pittsburgh, PA 15219, U.S.A.
E-mail: steve.bardi@ansys.com
Abstract—This paper presents Domain Decomposition Methods to overcome the challenges posed by multi-domain, multi-scale
high frequency problems. By decomposing large electromagnetic regions into smaller domains, the Finite Element Method can
cope with the simulation of electrically large problems. A hybrid Finite Element and Boundary Integral procedure is also
presented that allows for domains to employ different solution methods in different subdomains. The Robin Transmission
Condition (RTC) is applied to link the domains and preserve field continuity on interfaces. Real life examples demonstrate the
accuracy and efficiency of the new method.
Index Terms—FEM, hybrid FEM and boundary element method, multi scale problems, Robin Transmission Condition
I. INTRODUCTION
The finite element method (FEM) is a powerful tool
for simulating high frequency structures. There are
several features of the method that have become expected
elements of a successful commercial simulator. These
elements include spurious mode free hierarchical, higher
order vector basis functions, curvilinear elements,
automatic/adaptive meshing, transfinite elements, mesh
truncation methods, broad band frequency sweeps,
parameterization and preconditioned iterative solvers.
However, new challenges have emerged in recent years:
the simulation tools need to cope with multi-scale
problems that start on the chip level, couple to the
package and board levels, and encompass the platform
and antenna levels. Each component in a multi-scale
problem can require millions of unknowns to simulate.
Chip complexity rises to billions of circuit elements;
packages involve large numbers of ports; printed circuit
boards (PCBs) often contain thousands of traces on many
layers; and platforms and antennas often involve
dimensions of hundreds of wavelengths.
While recent advances in High Performance
Computing (HPC) hardware greatly accelerate numerical
computations, new algorithms are needed to exploit the
new HPC environment. In particular, efficient and
effective physics-based parallelization is required to
address the challenges of multi-scale and multi-domain
simulation. This paper presents an overview of domain
decomposition methods that exploits the physical nature
of multi-scale, multi-domain problems to tackle
heretofore impossibly complex high frequency problems.
II. BASICS OF DOMAIN DECOMPOSITION METHOD
A basic characteristic of HPC is the use of multiple
processors to perform computations in parallel. An
algebraic approach to using HPC is to partition large
matrices into smaller sub-matrices. In some cases, this is
inefficient even when iterative solvers are employed.
Physics-based domain decomposition is typically more
efficient because the subdomains exploit the geometry
and the field. In this case, both the solution domain and
the mesh are partitioned into smaller subdomains and
meshes. The mesh of the sub-domains can be
overlapping, conformal touching, non-conformal
touching or even non-touching [1-3].
Another important advantage for DDM is the ability to
link differing solution methods and physics. For instance,
the finite element method is better at simulating complex
geometries, while the boundary element method copes
better with electrically large but simple, smooth
structures. This paper presents hybrid finite element–
boundary integral (FE-BI) methods that allows nonconformal
touching domains and disjoint regions.
Figure 1: Incident surface electric and magnetic current
densities impinging on an FEM domain
A. Single FE domain with surface electric and
magnetic current excitations
Consider a computational domain where an incident
field impinges on a section of a boundary as illustrated in
Figure 1. The wave equation to be solved is
2
imp
1 r
E1
ko1rE1 jkoJ
1 (1)
imp
where J1 is the impressed current density. The total
field description is used inside the domain, while the
scattered field description is used outside
sc inc
sc inc
E1 E1
E1
; H1 H1
H1
(2)
For the sake of simplicity, a Dirichlet boundary condition
is used on 1 \ 12
n 1 E 0
(3)
, both the electric and the magnetic field jump
On 12

inc
inc
with n1 E1
and n2 H2
, respectively [4].
Consequently, an absorbing boundary condition is
sc
sc
required for E and H . Assume we know an operator
that provides perfect absorption:
sc
sc
n1 H1
ABC ( n1
n1
E1
)
(4)
Since, the total field description is used in the
computational domain, the scattered field variables can
be eliminated using (2). Then, the Neumann boundary
condition for the total magnetic field is
n1
1r
E1
(5)
inc inc
jo ( ABC ( e1)
ABC ( e1
) J1
/ )
where the J electric and the magnetic current densities e
is introduced as
J n
H and e n
n
E
(6)
where is the wave impedance. Introducing the first
order ABC generates the simple form
inc inc
n1 1r
E1
jko( e1
e1
J1
)
(7)
Equations (5) and (7) are Robin transmission boundary
conditions. They generalize the Neumann boundary
condition to include the incident fields. Thus, to excite
the computational domain by an external incident field,
the electric and magnetic surface current densities inc
J
inc
and e need to be specified. The transmission
conditions are first order when the first order ABC is
used and higher order when higher order ABC’s are used.
B. Multiple FE domains with surface electric and
magnetic current coupling
Now consider a computational domain that is
subdivided into two subdomains (Fig. 2).
Figure 2: Decomposition into two non-overlapping
subdomains
The boundary value problem (BVP) for the first domain
is
2
imp
1 r
E1
ko1rE1 jkoJ
1 in 1 (8)
inc inc
n1 1r
E1
jkoJ1 jko(
e1
e1
J1
) on 12
(9)
and similarly for the second domain
2
imp
2r
E2 ko 2rE2
jkoJ
2 in 2 (10)
inc inc
n2 2r
E2
jkoJ 2 jko(
e2
e2
J2
) on 12
(11)
The incident field for the first domain is the field of the
second domain and vice versa
inc
e1 e2
; e2 e1
inc (12)
inc
J J
J J
inc
(13)
1
2 ; 2 1
- 2 - 15th IGTE Symposium 2012
Applying this to Equations (9) and (11), we get
n1 1r
E1
jkoJ1
jko(
e1
e2
J2
) (14)
n2 2r
E2
jkoJ
2 jko(
e2
e1
J1)
(15)
The right hand sides of Eqs. (14) and (15) are the
Neumann Boundary conditions for domain 1 and 2 ,
respectively. They will be included into the finite element
formulation as natural boundary conditions. Since J1 and
J 2 were introduced, Eqs. (14), (15) must be prescribed
explicitly as well
J1 e1
e2
J2
(16)
J2 e2
e1
J1
(17)
It can be proved ([1]) that solution of the differential
equations (8) and (10) are unique applying natural and
essential interface conditions (14), (15) and (16), 17),
respectively. Applying Galerkin’s method, the bilinear
form and the essential boundary condition for domain 1
is the following:
b(
, E ) jk v , e jk v , e jk v , J
v1 1 o 1 1 o 1 2
12
o 1 2
12
12
imp v1J11 jk
(18)
o
jk o(
w1, e1
12
w1,
J1
12
w1,
e2
12
w1,
J2
) 0
12
(19)
The same applies to 2 . Note, that the testing functions
w should be orthogonal to those of v. Discretizing the
scalar products, yields the matrix equation [1]
K1
G 21
where
G12
u1
y1
K
2 u
2
y1
(20)
Ek
bk
u
k
ek
; y
k
0
; k=1,2
J k
0
(21)
A
k
T
K k
Ck
0
Ck
vv
Bk
jkoTkk
wv
jkoTkk
0
0
ww
jk oTkk
(22)
0
G
12 G 21
0
0
0
vv
jkoT12
wv
jkoT12
0
vw
jk
oT12
ww
jk oT12
(23)
, v n v n
vv
;
ww
n
w , n
w
j
12
Tij i Tij i j
12
(24)
vw
Tij n
vi
, n
w j
(25)
12
Matrices A, B, C and b are the same as in the case of
standard FE discretization and can be found in [1] along
with the definitions of the scalar products. The structure
of Eq. (18) shows, that the variables of the FE domains
are coupled just via the surface electric and magnetic
current variables, which are called cement variables.
C. Hybrid FE - BI domains with surface electric and
magnetic current coupling
Fig. 3 shows two separated domains 1 and 2 . The
fields in these domains are coupled via the free space
domain ext . The FEM is used in 1 and 2 , while
Boundary Integral Method (BI) is used in ext .

Figure 3: Decomposition into two FEM and one BI
subdomains
The boundary value problem for the finite element
domains is similar to that in section B
2
imp
ir
Ei ko irEi
jkoJ
i in i (26)
ni 1i
i o i o i i i
E jk J jk ( e e J ) on i (27)
Ji ei
ei
Ji
on i (28)
Eqs. (27) and (28) are the Neumann and the Robin
transmission boundary conditions, respectively.
For the unbounded subdomain ext , the boundary
integral equation representation is used, based on
Stratton-Chu [2]. The boundary integral equation for the
electric and the magnetic current densities are
1
2
inc
ei ei
{ nk
( C(
nk
ek
)) jk onk
( A(
Jk
))
2
k1
1
( jk o)
(
Jk
)} on i (29)
jk
2
o inc
J i Ji
{ jkonk
nk
( C(
Jk
))
2
k1
2
jkonk nk (
A(
nk
ek
))
(
nk
ek
)} on i (30)
and the Robin transmission boundary condition is:
Ji ei
ei
Ji
on i (31)
where
'
'
'
' '
A ( x)
xgds ; (
x ) ( x)
gds ; C (x)
x
gds
(32)
Applying Galerkin’s method again, the matrix equation to
be solved is
K
1 N12x
1
y1
(33)
T
N12
K 2 x
2
y1
where
AII
A 0 0
0
I
A A T D T D
I
T
T
K 0 D T D T
i
T
T
0 T D Q T P
ii
ii
T
T
T
0 D T P Q T
ii
ii
I
0
0 0 0 0 Ei
y i
0 0 0 0 0 e
i 0
;
N12
0
0 0 0 0 x i J
;
i y (34)
i 0
inc
E
0
0 0 Q12
P12
ei
y
i
T
inc
0
0 0 P
12 Q12
J
i
H
y
i
Further details are provided in [1].
This general hybrid finite element-boundary integral
equation method (hybrid FE-BI) is very flexible. The
subdomains can be FEM, BEM or any other numerical
method. If just one FEM subdomain exists, it provides a
perfect absorbing boundary condition (FE-BI).
- 3 - 15th IGTE Symposium 2012
III. SOLVING THE MATRIX EQUATIONS
In this section, the solution of the matrix equations is
presented via a stationary alternating Schwartz algorithm
based on Jacobi Splitting. The idea is to eliminate the
internal variables and solve for the surface current
densities also called cement variables. Performing this
process iteratively is called domain iteration. Partitioning
the variables accordingly
e
k Ek
c k ; u k
J
; (35)
k c
k
Eq. (20) for the k-th domain is
Ek
bk
0
Kk
c j
c
g
k 0
kj
(36)
g kj can be read from Eq.(23) and k and j are the domain
indices. Note, that the internal variables are expressed by
the cement variables. Supposing, the inverse matrix of the
k-th domain is known and also partitioned to internal and
cement variable blocks, we get:
E E
E c
k Ek
k k
P b P
g
k k k k kj
c
j
c
(37)
c
c c
k E
k k
k
P b
Pk
g
k k
kj
where
Ek
Ek
Ek
ck
P
k Pk
K k c
k Ek
ck
ck
Pk
Pk
1
(38)
These equations allow domain iterations to be applied
k E P b P g c
n1
k
E Ek
k
c E
k
k
Ek
ck
k
n
kj j
(39)
n1
k k ck
ck
n
ck P bk
Pk
gkjc
j
(40)
where the superscript n provides the iteration number.
cc
The matrix P k is called the numerical Green’s function
and quantities c k and c j in Eq. (40) are called the
cement variables. The domain iteration works with the
cement variables only, but it needs blocks of the inverse
of the system matrix of the internal variables. Eq. (39)
provides the update for the internal variables, which are
not included in the domain iteration because they do not
needed to be updated unless the right-hand-side changes.
Other, popular methods also can be applied, such as
GMRES, a Krylov Subspace Method. The domain
iteration needs to invert the subdomain matrices in each
iteration. For this purpose, either a multifrontal direction
solver can be used or a p-type multiplicative Schwarz
preconditioner (pMUS) iterative solver. The iteration
matrix
cc
Akj Pk
(41)
is dense but it can be replaced by a sparse matrix using
Adaptive Cross Approximation (ACA)
~ mn
kj
mn
mr
rn
Akj
A Ukj
Vkj
(42)
where m and n are the row and column numbers and r is
the rank of the matrix.
The domain iteration method presented above was
derived for the case when the solution domain is
partitioned into two sub-domains with one coupling
surface interface. If the solution domain is partitioned
into multiple domains with multiple coupling surfaces,
the number of the cement variables increases but the

essence remains the same: the subdomain variables are
expressed in terms of cement variables and the domain
iteration is set up for the cement variables. The same
applies when the subdomains are coupled via BI domains.
The convergence of the domain iteration depends on
the order of the Robin transmission boundary conditions.
For simplicity, first order Robin boundary conditions
were used in the above derivations. Higher order
conditions are also available in [5]. Higher order
transmission condition enforce the requirement that the
eigenvalues of the system matrix be inside the unit circle.
This is a necessary condition for a good domain iteration
convergence. A second order transmission boundary
condition can be realized as in Eqs. (16) and (17)
J j Aje j Bj
e
j Jk
Ake
k Bk
ek
(43)
J A e B
e
J
A e B
e
k k k k k j j j j
(44)
where k and j are the indices of the neighboring domains
and denotes the surface operator. Constants k A , k B , Aj
and B j can be optimized for convergence. Figure 4
shows the improvement in convergence provided by the
second order Robin Transmission Condition (RTC).
Figure 4: Convergence with first and second order RTC
III. REPETITIVE STRUCTURES
If identical substructures exist in the computational
domain, the computational effort of storing and solving
the equations is dramatically reduced. Repeated
identical substructures are called unit cells and have
the same mesh. Only one unit cell is stored physically
in the computer; the other unit cells are virtual. The
physically-stored unit cell is called the parent, while
the virtual ones are called children. A structure can
have multiple parents. In the case of non-conformal
domain decomposition, no constraints are applied to
the mesh. In the case of conformal DDM, the parent
mesh must be constrained so that it matches with the
surface meshes of the children.
For the sake of simplicity, assume that the entire
computational domain consists of just one repeated
structure. This is usually the case when finite antenna
arrays are simulated. Figure 5 shows a single parent
case with an internal block and matrices of repetitive
unit cells. Here there is one system matrix and two
coupling matrices. Thus, only three of the sixteen
- 4 - 15th IGTE Symposium 2012
j
matrix blocks need to be stored and matrix block A
must be factored once instead of 4 times.
Figure 5: Internal blocks and corresponding matrices
of repetitive unit cells
IV. MULTI DOMAIN DDM WITH FE-BI
As it has been shown, DDM is based on a divide-andconquer
philosophy. Instead of tackling a large and
complex problem directly as a whole, the original
problem is partitioned into smaller, possibly repetitive,
and easier to solve sub-domains. In this paper, DDM is
used as an effective FEM preconditioner, where a higher
order Robin’s transmission condition (RTC) is devised to
enforce the continuity of electromagnetic fields between
adjacent sub-domains and accelerates the convergence of
the iterative process. DDM is also employed to provide a
hybrid FEM-BEM approach where the treatment of the
radiation condition is exact. The hybrid finite elementboundary
integral (FE-BI) method allows FEM-domains
to be disconnected with the coupling between disjoint
domains provided via Green’s functions. The advantages
of DDM-based FE-BI compared to traditional FE-BI
include modularity of FEM and BI domains in terms of
mesh and basis functions. This “non-conformal” ability
significantly simplifies the integration of existing stateof-art
FEM and BEM solvers. The continuity
enforcement through Robin’s RTC naturally renders
present FE-BI free of internal resonance issue. Since
domains are allowed to be disjoint, if one or more subdomains
are purely metallic or highly conducting, DDM
can allow the integral equation method to be applied to
these sub-domains directly to reduce memory
consumption.
V. APPLICATIONS
To illustrate the effectiveness and accuracy of DDM,
an array of tapered slot antennas is considered. The
antenna element is of the Vivaldi type. The antenna is
similar to the one described in [8]. The rectangular array
spacing is 34 mm along y and 36mm along x. The εr = 6
substrate is 0.02 λ0 thick and the height and opening size
of the slot is ≈0.5 and 0.45λ0 respectively. To show the
accuracy of the simulation an array of 81 elements (9x9)
was analyzed using DDM. For comparison, a full array
model of the 81 elements with a slightly different edge
treatment was also created and simulated using FEM in a
single domain. The model simulated without using DDM
will be referred to as the explicit model. The two patterns
for the φ=0° cut (perpendicular to the slot faces) for all
elements excited with equal amplitude and 0° phase shift
are shown in Figure 6. Excellent agreement is obtained.
In addition to being able to compute the field patterns, the
full scattering matrix can also be extracted from the DDM
simulation. To verify the accuracy of this computation,
consider the data shown in Figure 7. This plot compares

the refection coefficient of the center element (element
#41) in the array and also the coupling terms (S41,-- dB)
for the coupling between the center element and the next
4 elements along the same row of slots. Again agreement
between the two sets of data is excellent.
Another infinite array simulation was performed using
linked boundary conditions and the active element pattern
was computed [9]. The active element pattern is the
radiation pattern for an infinite array of elements where
only a single element is excited. Finite arrays of 9x9 and
21x21 elements were simulated using DDM. The
radiation pattern with only the center element excited was
computed for each of these arrays. For comparison a
single antenna element on a finite ground plane was also
analyzed. The normalized φ=90° patterns for these 4
antennas is shown in Figure 8. The agreement with the
infinite array active element pattern improves as the array
size increases from 1 to 9x9 to 21x21 elements. This
demonstrates the accuracy of the DDM simulation
procedure for large arrays. As a final test, a 15x15 array
of Vivaldi elements was simulated using DDM. In this
case, the radiation pattern for 0° scan angle was
calculated. The 3D polar of this pattern is shown in
Figure 9.
All simulations were run on a Linux cluster. Each
machine in the cluster had 12 CPUs and 96 GB memory.
The explicit 81 element model was run on a single
machine. For the DDM simulation, the domain
simulations were distributed over several machines and
CPUs. For the 9x9 array, 62 domains were used and
21GB Ram was required; for the 15x15 array, 68
domains were used and the total memory required was
≈28GB. The latter simulation shows the power of this
approach – even though the number of tetrahedra
increased significantly, the memory usage was still less
than 30GB.
Figure 6: Comparison of the φ=0° patterns for all
elements excited with equal phase and magnitude for 9x9
array. The DDM data is the solid black line and the
explicit model data is the dashed red line.
Table I shows a comparison of solver statistics of
different element sizes and methods.
TABLE I
COMPARISON OF MEMORY AND SOLUTION TIME OF
DIFFEREN METHODS/ARRAYS
Time Number
of tets
Memory
Explicit (9 x 9) 190 min 1.7 m 50 GB
DDM (9 x 9) 90 min 1.6 m 21 GB
DDM (15 x 15) 300 min 4.3 m 28 GB
- 5 - 15th IGTE Symposium 2012
Figure 7. S41,-where element 41 is the center element of
the array and elements 42-45 are the remaining elements
along the middle row computed using two different
approaches.
Figure 8: Phi =90 °element patterns calculated using the
infinite array approximation (Element_pattern) and from
a 9x9 and 21x21 element array compared to the pattern
for a single isolated element (iso).
Figure 9: 3D polar plot of the radiation pattern for the
15x15 array where all elements are excited with equal
amplitude and phase
The next example demonstrates the efficiency of FE-
BI. Figure 10 shows an Apache helicopter with a
conformal FE-BI surface and it has been simulated at 900
MHz. Table II shows a comparison with pure FEM and
IE methods. The results show the superiority of FE-BI,
due to its conformal mesh truncation capability.

Figure 10: Apache helicopter with conformal FE-BI
boundary
TABLE II
COMPARISON OF MEMORY AND SOLUTION TIME OF
DIFFEREN METHODS
Number
of cores
Memory Time
FEM (PML box) 12 300 GB 330 min
IE 12 83 GB 328 min
FE-BI
(conformal)
12 21 GB 63 min
VI. CONCLUSION
The proliferation of High Performance Computing
(HPC) has made parallelization a basic requirement for
simulation codes today. Computational tasks can be
distributed on different machines (nodes) or cores
(distributed or shared memory). DDM is an ideal
procedure for achieving high HPC efficiency. The
subdomain solutions are fully independent of each other,
so they can be evaluated in parallel, either by using
distributed or shared memory. Subdomain solvers also
exploit multi-processing and iterative solution methods.
Both the Schwartz or Krylov domain iteration methods
distribute tasks with high parallelism. The standard
Message Passing Interface (MPI) can be used to control
the data exchange between the nodes and cores. As
demonstrated in the examples, the hybridized FE and BI
DDM procedure provides a flexible and efficient tool to
solve multi scale multi domain problems.
- 6 - 15th IGTE Symposium 2012
[1]
REFERENCES
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Method with Nonconformal Meshes for Finite Periodic and
Semi-Periodic Structures," IEEE Transactions on Antennas and
Propagation, vol. 55, pp. 2559 - 2570, September, 2007.
[2] K. Zhao, V. Rawat and J.F Lee, "A Domain Decomposition
Method for Electromagnetic Radiation and Scattering Analysis of
Multi-Target Problems," IEEE Transactions on Antennas and
Propagation, vol. 56, pp. 2211 - 2221, August 2008.
[3] I. Bardi, Zs. Badics and Z. Cendes, "Total and Scattered Field
Formulations in the Transfinite Element Method," IEEE
Transactions on Magnetics, vol. 44, pp. 778-781, June, 2008.
[4] R. F. Harrington, Time–Harmonic Electromagnetic Fields, John
Wiley & Sons, Inc. New York, 2000.
[5] Y. Shao, Z. Peng and J.F Lee, "Full-Wave Real-Life 3-D Package
Signal Integrity Analysis Using Nonconformal Domain
[6]
Decomposition Method," IEEE Transactions on Nicrowave
Theory and Techniques, vol. 59, pp. 230 - 241, February 2011.
W. C. Chew and C.C. Lu, "The use of Huygens’ equivalence
principle for solving the volume integral equation for scattering,"
IEEE Transactions on Antennas and Propagation, vol. 41, pp. 897
- 904, July 1993.
[7] Y.J Li and J.M. Jin, "A New Dual–Primal Domain Decomposition
Approach for Finite Element Simulation of 3-D Large – Scale
Electromagnetic Problems," IEEE Transactions on Antennas and
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[8] L.E. R. Petersson and J-M Jin, “Analysis of periodic structures via
a time-domain finite-element formualiton with a Floquet abc,”
IEEE Trans. AP, pp. 933-944, Mar. 2009.
[9] J. Manges, J. Silvestro and R. Petersson, “Accurate and Efficient
Extraction of Antenna Array Performance from Numerical Unit-
Cell Data,” 2011 European Microwave Conference

- 7 - 15th IGTE Symposium 2012
Parallelization of the Transmission Line Matrix
method. Modelling Schumann Resonances and
Atmospherics
S. Toledo-Redondo∗ ,A.Salinas∗ , J. Fornieles∗ ,J.Portí † ,B.Besser ‡ , and H.I.M. Lichtenegger ‡
∗Department of Electromagnetism and Matter Physics, University of Granada, Spain.
† Department of Applied Physics, University of Granada, Spain.
‡ Space Research Institute, Austrian Academy of Sciences, Graz, Austria
E-mail: sergiotr@ugr.es
Abstract—In this paper, a parallelization of the Transmission-Line Modelling (TLM) method is presented. It is intended to
work efficiently regardless of the spatial topology of the problem, by transforming the initial topology into a one-dimensional
structure. It is designed for shared memory environments, and its implementation is carried out using OpenMP directives.
The algorithm is applied to find the first cut-off frequency of the Earth-ionosphere waveguide by solving two models of
the real system. The performance of the algorithm for the mentioned problem is studied in terms of speedup over two
different platforms. Relative speedups of up to 16 are achieved with the use of 32 CPUs. Finally, the whole Earth-ionosphere
cavity is simulated, with an accuracy of 5 km grid size, leading to an error of less than 1.5% in the Schumann Resonance
frequencies. The spatial resolution achieved also enables for the first time the possibility of using this model to study the
global effects generated by local phenomena in the Earth-ionosphere cavity.
Index Terms—Earth-ionosphere waveguide, Schumann resonances, shared memory, speedup, TLM.
I. INTRODUCTION
Numerical methods are a tool for embracing scientific
and technological problems which are difficult or even
impossible to solve analytically. In addition, simulation is
often an intermediary step between design and construction
of prototypes in industry. High performance computers
are one of the keys of the present importance of these
methods, because they allow simulating more and more
complex situations as the technology evolves. However,
the top speed of processors seems to have reached its top
[1], and the tendency of CPU manufacturers is to ship
multi-core processors instead of building faster single
CPUs [2].
The Transmission-Line Modelling method [3] is employed
for the simulation of electromagnetic problems
since 1971 [4], although it can simulate other problems
as well, such as heat or particle diffusion, acoustic propagation,
deformation in electric solids, waves in fluids,
etc. [5] [6]. It has been used previously, in 2D form,
for the study of atmospheric phenomena, e.g., Schumann
Resonances [7], which is a problem similar to the one
that will be addressed in this paper.
The propagation of atmospherics in the Earthionosphere
waveguide is a complex problem which involves
several natural media (ground, oceans, atmosphere,
ionospheric plasma) as well as the phenomena of
lightning [8], [9]. A parallel-TLM algorithm is employed
to model the propagation of these natural signals and
allows finding the first cut-off frequency under two
different approximated models of the natural waveguide
formed by the ground and the ionosphere.
Programming efficient algorithms with these relatively
new hardware solutions is not straightforward. Different
approaches must be taken into account according to
the kind of hardware used. For instance, the way of
designing a parallel code for a Graphical Processing Unit
(GPU) [10] will be different than for a multi-core system
with shared memory access [11]. Programming shared
memory environments is probably the most similar to
traditional computing, i.e., not parallel, but still there is
a great difference in the way we should conceive the
algorithms [12].
In Section 2, the TLM method is briefly introduced, as
well as the Symmetric Condensed Node (SCN). Section
3 describes the approach employed to parallelize the
method. In Section 4, the Earth-ionosphere waveguide is
introduced, and it is solved by means of the proposed
algorithm. The model is benchmarked and its performance
in terms of speedup is presented. In Section 5,
the model is employed to solve the whole lossless Earthionosphere
cavity, obtaining its Schumann Resonances.
A brief summary as well as the main conclusions of the
paper are detailed in Section 6.
II. THE TLM METHOD
TLM is a numerical method intended for simulation of
propagation problems which are governed by differential
equations. Problems which have to deal with electromagnetics,
heat diffusion, gravity waves, acoustic waves,
etc. are suitable to be modelled with this technique. The

Fig. 1. Scheme of the Symmetric Condensed Node with 12 link lines.
idea behind the method is to build a circuit based on
transmission lines which behaves in analogous form as
the problem we want to implement, i.e., the governing
equations are the same for the circuit and for the physical
problem. In this work, TLM will be used to solve
Maxwell equations and the study of the Earth-ionosphere
waveguide.
The TLM method discretizes both time and space and
sets up an iterative process in which the six components
of the electromagnetic field evolve in time from a known
initial situation. Therefore, the fields radiated and/or
propagated in the space are simulated with arbitrary
accuracy, which is constrained by the size of the space
discretization. A thumbnail rule is that the minimum
wavelength (λ) of interest must be ten times larger than
the size of the cell (Δl), i.e., Δl ≤ λ/10 [13].
Depending on the problem we want to model, each
independent cell will be simulated by a different set up
of transmission lines and node. For 3D electromagnetic
problems, the most used circuit since it was formulated is
the Symmetrical Condensed Node (SCN) [14], together
with its variations. In this paper, the SCN with stubs for
conductivity [15] will be used. In Fig. 1, a scheme of the
transmission lines arrangement is shown. With its 12 link
lines, the node is capable of modelling the behavior of
Maxwell’s equations for a differential of volume, ΔV .
Regardless of the node employed for modelling each
ΔV , the process iteration of TLM is always the same.
For each node and time step n, t = nΔt (where Δt ≤
Δl/2c for a cubic SCN, being c the speed of light in the
medium), there is a set of incident pulses or voltages Vi,
one for each transmission line of the node. During the
time step they travel along the line, and they are either
reflected and or transmitted to other lines, depending on
the node structure. The transmitted/reflected pulses, or
simply the scattered pulses Vr, are related to Vi by the
scattering matrix S:
Vr = S · Vi . (1)
At the next time step, the scattered voltages from
each node are converted into incident voltages of the
- 8 - 15th IGTE Symposium 2012
nearby node, thus propagating the pulses along the entire
network. It is important to fix time synchronism in a
manner that all pulses in the mesh are simultaneously
incident at the center of their respective node at each
time nΔt.
III. PARALLELIZATION OF TLM
TLM method is a time and memory consuming application,
when applied to large problems. In its most
basic form, at least 12 floating point numbers (floats)
are needed in order to store the 12 voltages of each
node. Six more floats become necessary if either nodes
of variable size, permittivity (εr), or permeability (μr) of
the medium are required. Finally, three more floats must
be used for adding electric conductivity. The problems
solved in this work make use of 15 different transmission
lines per node, 12 required by the basic SCN configuration,
plus 3 for modelling the electric conductivity.
The operation which consumes most of the computational
time is reflected in Eq. 1, since this matrix
multiplication must be performed at each node for each
time step. For the case involved in this paper, the matrix
multiplication has been reduced to 18 floating point
multiplications and 36 additions, at the cost of executing
different portion of code for the nodes which are at the
border of the spatial distribution. The number of nodes
for the problems involved in this work are on the order of
10 6 , and all these computations can be done concurrently
for different nodes.
Parallelizing the independent matrix multiplications
of Eq. 1 requires doubling the minimum memory size
required to hold the problem. Since the input voltage
for one node is the output voltage for another node,
Vi and Vr must be stored in different variables. On a
sequential implementation of the model, i.e., not parallel,
the order of execution is known, and the calculated Vr can
overwrite Vi, if its value is previously stored on a local
variable which can be erased when all related Vr have
been calculated. Since the order in which the matrix multiplications
will be performed is not known for a parallel
version, doubling the minimum required memory size is
a non-avoidable penalty of engaging parallelization. In
the described TLM implementation, each node has the
requirement of 30 floating-point variables (120 bytes) for
storing the line voltages.
The algorithm has been designed to be independent
of its spatial topology, and it should provide the same
performance regardless of the arrangement of the nodes
in the space. The motivation for this constraint is to have
a very flexible tool for solving different problems. In
order to work in the same way regardless of the spatial
geometry, the algorithm is split into two different steps:
the pre-processing and the TLM computation itself.
The pre-processing is a fast (when compared to TLM
computation) operation which is in charge of transforming
an arbitrary topology to a common one which can
accommodate any kind of spatial distribution. The idea

is to assign a unique identifier to each node of the initial
topology and to store a vector with the unique identifiers
of the adjacent nodes. In this way, the result can be seen
as a one-dimensional vector of nodes where each one
knows which others are their neighbors. Any complex
distribution of nodes can be simplified to this unified
arrangement, regardless of the arbitrary initial geometry.
On the other hand, abstracting the initial topology to
this new paradigm brings a penalty of 6 integers per
node to store the neighbor identifier at each direction
(for 3D topologies) plus another integer to mark the kind
of medium that the node belongs to. Therefore, the total
amount of RAM memory required for each node is 152
bytes.
The TLM computation loop employed is shown in
high-level pseudo-code below, where the OpenMP directives
have been included:
#pragma omp parallel private(private variables)
{
for(t=0..TotalTime)
{
#pragma omp single
{
//The reflected pulses become the incident at new time step
Vi = Vr;
//system feeding
for(i=0..NumberOfFeeds) V[i]= feeding;
//store the relevant output
for(i=0..NumberOfOutputs) output=V[i];
}
#pragma omp for schedule (static)
for(i=0..NumberOfNodes) Vr[i]=S*Vi[i];
}end for(t)
}end pragma parallel
The main loop of the code is inside a #pragma omp. In
this way, the overhead of creating (and destroying) new
threads needs to be computed only once for all the execution.
It mainly consists of iteration over time steps, which
are not parallelizable, and which need synchronization
of the threads which work inside each iteration. Each
time step iteration is divided into two blocks; a sequential
block and a parallel block. The sequential block performs
three different operations:
• Swap Vi by Vr. As we mentioned before, Vi and
Vr must be stored in separate memory addresses in
order to enable parallelization. At the beginning of
a time step, the reflected pulses from the previous
iteration become the incident pulses on the neighbor
nodes. In this implementation, the pointers of the
vectors Vi and Vr are only exchanged, and the
complexity of neighbor swapping the pulses is done
implicitly in the matrix calculations, avoiding extra
reading and writing to memory, although adding a
small penalty in processing and complexity to the
code.
• System feeding. The initial electromagnetic problem
may have sources on its initial definition. These
sources bring external voltage pulses to the system,
which are added in this part of the code.
• Output storage. Some key nodes are marked as
output and, therefore, the temporal evolution of
their voltages is necessary to reconstruct the fields’
- 9 - 15th IGTE Symposium 2012
evolution afterwards. All the line voltages at each
time step from these output nodes are stored in
memory.
The parallel block is in charge for the matrix multiplication
of each node. It is composed of a parallel for.
Since the Vr calculation can be performed independently
for each node, the OpenMP directive is in charge to
distribute the computations between the available number
of threads. Therefore, each thread will compute a portion
of the total range of i. Since the clause schedule
(static) is present, all the available threads will
iterate the same portion of the i range.
In order to reduce the total time of computation,
several optimizations have been included here, which
make the real code hard to read. One of them reduces
the number of multiplications, by identifying operations
which are performed several times for the calculation of
different reflected pulses. Another, the most complex one,
deals with the neighboring of the nodes. It is implemented
in such a way that the nodes on the edges of the initial
geometry are treated in a different manner than the
internal nodes are. The code is different but the amount
of computation remains similar, except for the nodes
which are edge in more than one of its sides. In this case
the computations are larger. The number of nodes being
edge for more than one side is usually small on most
geometries. This is true for the models considered in this
work due to the fact that scheduling the parallel for
as static improves the performance of the algorithm,
although few nodes require a bit more computation than
others.
IV. MODELLING THE EARTH-IONOSPHERE
WAVEGUIDE
The algorithm described above has been employed to
simulate the Earth-ionosphere waveguide. The surface of
Earth behaves like a good conductor in the Very Low
Frequency range (VLF, i.e., in the order of kHz), with
conductivity ∼10−2 S/m for ground and ∼3.2 S/m for
sea water [16]. There is air above the ground, which is
of dielectric nature. As the altitude increases, the number
of free electrons increases too, the density of neutral
decreases, and the air starts behaving like a conductor.
A typical conductivity profile with altitude dependence
is shown in Fig. 2 (top) [17].
The excitation sources of the waveguide are lightning
strokes [18]. They generate a broadband signal which
differs in orientation, strength and duration depending of
its nature (cloud to ground, cloud to cloud, Q-bursts, etc.).
A typical stroke in positive cloud to ground lightning
is depicted in Fig. 2 (bottom) [19], which has been
employed as excitation in the problem considered.
The signal originated by the stroke travels a certain distance,
guided between the two plates before extinguishing
due to losses. On a first approximation, the system
can be regarded as the infinite parallel-plate waveguide.
According to [20], the cut-off frequencies for a lossless
waveguide of this geometry are located at:

Fig. 2. Conductivity profile of the atmosphere with altitude (top),
extracted from [17], and typical current for cloud to ground lightning
(bottom).
Amplitude [a.u.]
6
5
4
3
2
1
Vertical electric field as a function of frequency
lossy waveguide
lossless waveguide
0
1000 2000 3000 4000 5000 6000 7000 8000
Frequency (Hz)
Fig. 3. Detail of the first and second cut-off frequencies for the lossless
and the lossy Earth-ionosphere waveguide.
fn = nc
(2)
2h
where c is the speed of light in vacuum, h is the
distance between the parallel plates, n is the mode
number, and fn the associated cut-off frequency of the
mode.
The problem has been modelled with the algorithm,
both for a lossless and for a lossy waveguide. For the
lossless waveguide, the conductivity is supposed to be
zero in the dielectric. For the lossy waveguide, the night
conductivity profile from Fig. 2 (top) is applied. The
parallel plates are taken as perfect conductors in both
cases. Details of the first and second cut-off frequencies
are depicted in Fig. 3, which correspond to electric
field in the z-direction, at a distance of 45 km in ydirection
from the source (see Fig. 4 for definition of
the directions). It is interesting to note the effect of
the conductivity, which increases the value of the cutoff
frequencies, being equivalent to having a narrower
waveguide.
A. Algorithm Benchmarking
The total execution time of the waveguide model over
different computers has been measured, using a different
- 10 - 15th IGTE Symposium 2012
Fig. 4. Spatial arrangement of the waveguide model.
total time of execution (s)
2000
1000
500
200
100
Scalability of TLM algorithm
Absolute time of execution Relative speedups
SM32
SM32 round-robin
SM8
SM8 round-robin
0 5 10 15 20 25 30
Number of CPUs
speedup = time 1core / time nCores
18
16
14
12
10
8
6
4
2
SM32
SM32 round-robin
SM8
SM8 round-robin
0
0 5 10 15 20
Number of CPUs
25 30
Fig. 5. Total time of execution (left) and relative speedups (right) for
the different platforms.
number of CPUs, in order to determine the scalability of
our algorithm. Two different computers have been used
in the benchmarking process:
• SuperMicro8 (SM8). Server with 2 AMD opteron
quad-core processors 2.0 GHz and 32 GB RAM, in
Not Uniform Memory Access (NUMA) configuration.
The OS is OpenSUSE 11.4 and the compiler
employed is opencc 4.2.4 (level 2 of optimization).
• SuperMicro32 (SM32). Server with 4 AMD opteron
eight-core processors 2.0 GHz and 96 GB RAM, in
Not Uniform Memory Access (NUMA) configuration.
The OS is OpenSUSE 11.4 and the compiler
employed is opencc 4.2.4 (level 2 of optimization).
The problem benchmarked makes use of symmetry
and the initial grid is two-dimensional. According to
Fig. 4, the symmetry is applied in the x-direction. The
conductivity profile is extended along z, and the output
measured at a certain distance along the y-direction. The
node size is 1.5 km, the time step is 2.5 μs, the number
of time steps is 7,500, and the total number of nodes is
∼106 (67 nodes in z, 15,000 nodes in y). The excitation is
placed next to the ground, at the center of the waveguide.
With this configuration, a total of 7.5·109 step-node
computations must be performed to solve the problem.
The total execution time and relative speedups are shown
in Fig. 5, for both platforms. The relative speedup is
defined as the execution time using n cores divided by
the execution time using 1 core. A maximum speedup of
6 is achieved with SM8, when making use of its 8 CPUs.
On SM32, a maximum speedup of 16 is obtained when
using 30 CPUs.
As it can be observed in Fig. 5, two benchmarks
have been measured for each computer. The aim is to
compare the performance when using or not Round-
Robin memory allocation policy. This policy consists in
requesting the operative system to balance the memory
reservation equally among the different portions of RAM.

This can be accomplished via the numactl tool [21].
If this policy is not enabled, the memory reservation
will be performed sequentially, and only some memory
blocks will be used. Since the computers employed have
NUMA architecture, each processor can access faster to
a certain RAM circuit, while the access to the others is
slower. Moreover, if the Round-Robin policy is not set,
the different CPUs will have to compete to gain access
to the particular RAM circuit, slowing down the overall
computation. This technique is especially effective for a
large number of CPUs (see Fig. 5).
V. MODELLING THE WHOLE EARTH-IONOSPHERE
CAVITY
In this section, the whole cavity is considered in the
simulation, leading to a much more time and RAM memory
consuming model. The cavity has been considered as
the space between two concentric spheres of 6,370 and
6,470 km, with perfect conducting walls at the borders
and no conductivity at the interior. The spherical shell
has been modeled by cubic nodes, in this case of Δl=5
km of size. The total number of nodes is ∼4.14·108 ,and
the amount of RAM required is ∼61.5 GBytes. Around
1.1 GBytes are employed for storing the outputs. For a
spatial grid with a 5 km resolution, the time step required
is 8.34 μs. The number of time iterations calculated was
2.4·105 , and therefore the simulated time length is ∼2
s. A frequency resolution of 0.5 Hz is achieved when
the FFT is computed with these parameters. The total
execution time required when using 32 cores on SM32
(see Section IV-A) is roughly 6.0·105 s, i.e., around seven
days.
The excitation source of the cavity has been located at
θ=0 and r=6,372 km, i.e., the North Pole. The excitation
corresponds to a vertical positive Cloud to Ground (+CG)
lightning, and its current is shown in Fig. 2 (bottom). This
stroke starts at t=0, and lasts for 500 μs.
With this spatial arrangement, the problem has symmetry
over the φ coordinate, and therefore the outputs had
been located all φ=0. A total of 101 nodes are marked as
output, and they are equally spaced along the coordinate
θ, from 0 to π, for r=6,370 km, i.e., at the surface,
because it is the common location for SR measurements.
As SR analytical models state [22] [23], the two
relevant components of the electromagnetic field are Er
and Hφ. In Figure 6, the six components of the output
corresponding to θ=π/4 have been plotted, in order to
show this fact. The other output nodes show similar
results, where the two components mentioned are much
greater than the rest.
In order to corroborate the results from the simulations,
the relationship between the modal amplitude of the six
first SR and the angular distance to the source for the
101 nodes marked as output (θ=0, θ=π/100,..., θ=π) has
been plotted in Figure 7. This result is in agreement
with analytical model results [22] [23], which show the
amplitude dependence of SR modes with the distance to
the source.
- 11 - 15th IGTE Symposium 2012
Fig. 6. Spectra of Electric (left) and Magnetic (right) field components.
The relevance of Er and Hφ can be observed.
Hphi [T/sqrt(Hz)]
6e-09
5e-09
4e-09
3e-09
2e-09
1e-09
Dependence of SR amplitude with distance to the source (θ), lossless cavity
SR1
SR2
SR3
0
0 π/4 π/2
θ [rad]
3π/4 π
Hphi [T/sqrt(Hz)]
1.2e-08
1e-08
8e-09
6e-09
4e-09
2e-09
SR4
SR5
SR6
0
0 π/4 π/2
θ [rad]
3π/4 π
Fig. 7. Dependence of SR modal amplitude with θ, for the lossless
cavity.
The simulation has been repeated changing only the
size of the spatial grid to Δl=10 km. Doubling the size
of the nodes reduces by a factor of eight the number
of nodes, at the cost of a poorer fitting of the spherical
geometry and worse spatial resolution. The maximum
valid frequency is also reduced by a factor of two, but
this is not important for the study of SR, because the
top frequency is still 3 kHz (the condition is λ ≥ 10Δl).
The amount of memory required is reduced to 9.1 GBytes
(with 1.1 GBytes for storing the results). The execution
time, again with 32 cores in SM32, is reduced to 7.6·10 4
s, i.e., roughly 21 hours. The magnetic fields in φ direction
at an angular distance of π/4 of the two simulations
are compared in Figure 8.
The six maxima from each spectra of Hφ have been
extracted and averaged, with the aim of using them as a
proxy of the resonance position. The results are shown
in Table I, for both simulations.
It can be observed that the results for the central frequencies
of the six SR are similar in the two simulations
and with the results from the analytical solution. For the
case of the 10 km size simulation, the errors for the
central frequencies are always under 3%. This error is

Amplitude [a.u.]
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
10 km
5 km
Comparison of H φ at θ=π/4
0
0 10 20 30 40 50
Frequency (Hz)
Fig. 8. Comparison of Hφ at θ=π/4, for the two simulations (5 km
and 10 km).
TABLE I
SR CENTRAL VALUES IN HZ, LOSSLESS CAVITY.
1st SR 2nd SR 3rd SR 4th SR 5th SR 6th SR
10 km 10.24 17.74 24.98 32.35 39.63 46.93
5km 10.47 17.99 25.48 32.96 39.98 47.47
Analytical 10.51 18.20 25.75 33.24 40.71 48.17
reduced to less than 1.5% for the 5 km simulation.
VI. CONCLUSION
The TLM is briefly described and the inherent parallel
areas of the algorithm are pointed out. It has been
parallelized for shared memory architectures, by using
OpenMP. The solution obtained has been employed to
simulate the Earth-ionosphere waveguide, and to observe
the changes produced by the conductivity profile over
the cut-off frequency, by comparing the results with
the lossless waveguide. The effect of this conductivity
is to increase the value of the cut-off frequencies,
being equivalent to having a narrower waveguide. The
algorithm has been run on two different platforms and
benchmarked. The algorithm scales up to a speedup of
16 by using 30 CPUs. In order to obtain the maxima
speedups, it is necessary to set a policy of Round-Robin
memory allocation, in order to minimize the effects of
the NUMA architecture. Finally, a huge simulation (the
whole Earth-ionosphere cavity with a 5 km resolution)
has been performed for validation of the parallel algorithm.
The lossless version of the cavity is solved, and
the electromagnetic fields obtained are consistent with
the analytical solution of the cavity. Since a Cartesian
grid of only 5 km size per cell was employed, the
errors are lower than 1.5% for the Schumann resonance
frequencies. The spatial resolution achieved enables the
possibility of using this model to study the global effects
generated by local phenomena in the Earth-ionosphere
cavity.
Acknowledgments: This work was supported by the
Consejería de Innovación, Ciencia y Empresa of Andalusian
Government and Ministerio de Ciencia e Innovación
- 12 - 15th IGTE Symposium 2012
of Spain under projects with references PO7-FQM-03280
and FIS2010-15170, co-financed with FEDER funds of
the European Union.
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R., Maaskant, R., Lu, Y., Che, Q., Lu, R., Su, Z.: A new direction
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TLM, IEEE Press, Piscataway, N.J. (1995)
[4] Johns, P.B, Beurle, R.L.: Numerical solution of 2-dimensional
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[5] De Cogan, D., Pulko, S.H., O’Connor, W.J.: Transmission-Line
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(1995)
[6] Enders, P., Pulko, S.H., Stubbs, D.M.: TLM for diffusion: consistent
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251–259 (2002)
[7] Morente, J.A., Molina-Cuberos, G.J., Portí, J., Besser, B.P., Salinas,
A., Schwingenschuch, K., Lichtenegger, H.: A numerical simulation
of Earth’s electromagnetic cavity with the Transmission Line
Matrix method: Schumann resonances, J. Geophys, Res., 108(A5),
1195–1205 (2003)
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first cut-off frequency of the Earth-ionosphere waveguide observed
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[9] Cummer, S.A.:Modeling electromagnetic propagation in the Earthionosphere
waveguide, IEEE Trans. ant. Propag., 48(9), 1420–1432
(2000)
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a hands-on approach, Morgan Kaufmann, Burlington, M.A.
(2010)
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Shared Memory Parallel Programming, The MIT Press, Cambridge,
Massachussets (2007)
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to writing parallel applications, O’Reilly, Sebastopol, CA (2009)
[13] Morente, J.A., Jiménez, G., Portí, J., Khalladi, M.: Dispersion
analysis for TLM mesh of symmetrical condensed nodes with
stubs, IEEE Trans. Microwave Theory Tech. 43(2), 452–456 (1995)
[14] Johns, P.B.: A symmetrical condensed node for the TLM method,
IEEE Trans. Microwave Theory Tech. 35(4), 370–377 (1987)
[15] Naylor, P., Desai, R.A.: New three dimensional symmetrical condensed
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by TLM, Electron. Lett., 26(7), 492-494 (1990)
[16] Rycroft, M.J., Harrison, R.G., Nicoll, K.A., Mareev, E.A.: An
overview of Earth’s global electric circuit and atmospheric conductivity,
Space Sci. Rev., 137, 83–105 (2008)
[17] Pechony, O., Price, C.: Schumann resonance parameters calculated
with a partially uniform model on Earth, Venus, Mars, and
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[19] Baba, Y., Rakov, V.A.: Present understanding of the lightning return
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[22] Sentman, D.D., Schumann Resonances, in Handbook of atmospheric electrodynamics,
CRC Press, Boca Raton, Fla, (1995)
[23] Toledo-Redondo, S., Salinas, A., Portí, J., Morente, J.A., Fornieles, J.,
Méndez, A., Galindo-Zaldívar, J., Pedrera, A., Ruiz-Constán, A., and Anahnah,
F., Study of Schumann resonances based on magnetotelluric records
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114, (2010).

- 13 - 15th IGTE Symposium 2012
A Novel Parametric Model Order Reduction
Approach with Applications to Geometrically
Parameterized Microwave Devices
Stefan Burgard∗ , Ortwin Farle∗ , and Romanus Dyczij-Edlinger∗ ∗Chair for Electromagnetic Theory, Saarland University, D-66123 Saarbrücken, Germany
E-mail: edlinger@lte.uni-saarland.de
Abstract—Methods of model-order reduction approximate the transfer behavior of a given high-dimensional system by that
of a low-order one, which is much faster to evaluate. In the parametric case, the system features additional parameters,
such as material properties or geometric design variables. The parametric order-reduction methods available today still
exhibit a number of limitations, particularly with respect to convergence rates and the size of the reduced-order model.
This contribution presents a novel technique based on affine parameter reconstruction and parameter-dependent projection
matrices. It features high rates of convergence, supports local adaptation, and yields reduced-order models that are of very
low dimension and thus fast to evaluate.
Index Terms—Computer-aided engineering, geometric parameters, parametric model order reduction, parametric models.
I. INTRODUCTION
This paper addresses microwave components with linear
time-invariant system properties. Since most practical
structures possess complicated shape and inhomogeneous
material properties, their fields-level analysis requires numerical
methods, such as the finite-element (FE) method.
FE discretization in the frequency domain results in
systems of linear equations which are characterized by
sparse matrices of high dimension. While solving a FE
system at one single operating frequency may not be particularly
demanding on modern computers, the analysis
of broad frequency bands still tends to be very timeconsuming.
The situation gets even worse when multiple
parameters, such as material properties or geometric
design variables, are present, and entire response surfaces
are to be computed.
Methods of model-order reduction (MOR) address this
issue by approximating the behavior of the original system
by a reduced-order model (ROM) that is very cheap
to solve. As long as the frequency is the sole parameter,
powerful single-point [1], [2] or multi-point [3], [4]
algorithms are readily available. The incorporation of
additional parameters, especially those of geometric type,
still poses challenges, with respect to convergence rates,
computing times, and model dimension. One particular
difficulty with geometric parameters is that they enter the
FE matrices in the form of multivariate rational functions
of complicated structure. The authors use the technique
of [5] and [6] for affine geometry approximation.
Present parametric model-order reduction (PMOR)
techniques fall under two categories: The one class comprises
methods [7], [5] that employ one global projection
space for all parameters, including the frequency. It is
characteristic of such entire-domain methods that the
ROM dimension is large and rises quickly with increasing
size of the parameter domain. The other class includes
methods that instantiate frequency-domain ROMs for
a set of sampling points in the domain of geometric
parameters and employ interpolation over sub-domains
to account for geometry variations. Thanks to their local
nature, the resulting sub-domain ROMs are of low dimension.
While existing techniques [8] - [11] interpolate
the frequency-domain ROMs directly, the method proposed
in this paper interpolates projection matrices. One
particular advantage of this approach is to decouple the
approximation of the effects of geometric parameters on
the FE matrices, which may be the dominant source of
error, from the actual PMOR process.
The remainder of the paper is organized as follows:
Section II presents the underlying parameter-dependent
FE system. The treatment of geometric parameters is
reviewed in Section III. The new PMOR approach is
developed in Section IV. This constitutes the main contribution
of the paper. Section V gives numerical examples
that demonstrate the benefits of the suggested approach.
A brief summery in Section VI closes the paper.
II. ORIGINAL SYSTEM
We consider a time-harmonic electromagnetic FE system
of dimension N which possesses Q inputs and outputs,
respectively, and depends on the frequency f ∈ R
and a vector p ∈P⊂RP of P geometric parameters.
The input vector is denoted by u, the generalized state
by x(f,p), and the output by y(f,p). The system is
assumed to be of the form
I
J
φi(f)Ai(p) x(f,p) = θj(f)Bj u, (1a)
i=1
j=1
J
y(f,p) = ηj(f)B
j=1
T
j x(f,p), (1b)
wherein Bj ∈ RN×Q , and the functions φi,θj,ηj : R →
C and Ai : RP → RN×N are continuous. Eq. (1) implies

that the topology of the FE mesh, i.e. the number and
connectivity of the FE nodes, must remain the same over
the whole parameter domain P. Meshes of this kind
can be constructed for a wide class of parameterized
geometries by, e.g., the morphing method of [12].
III. GEOMETRY INTERPOLATION
In many cases, the parameter-dependent matrices
Ai(p) are just multivariate rational functions. Nevertheless,
their explicit representation [13] is quite complex
in practice, because it requires tracking the effects of
all geometric parameters from the solid model through
the mesh generation process to the FE matrix generation
stage. Therefore, the present paper follows the suggestion
of [5] and [6] to approximate Ai(p) by a function of
simpler structure. We set
Ai(p) ≈
Γβ(p)A
β
β
i for p ∈P, (2)
wherein β =[β1,...,βP ] is a multi-index, A β
i ∈ CN×N ,
and Γβ : R P ↦→ R is a suitable interpolation function.
Thus, the approximate system reads:
φi(f)
i
Γβ(p)A
β
m
i x ′ (f,p)=
θj(f)Bju, (3a)
y
j
′
(f,p) =
x ′ (f,p), (3b)
ηj(f)B
j
T j
The interpolation functions Γβ are obtained as follows:
For each parameter p, choose a set Np of interpolation
points ψ k p,
Np = ψ k p ∈ R
k =1,...,Kp , (4)
and associated interpolation functions γ k p : R → R with
γ k p (ψ l p)=δkl for ψ l p ∈Np. (5)
Next construct a tensor-grid G = N1 × ... ×NP for
the domain P. Then the interpolation point pβ and
interpolation function Γβ are given by
pβ =[ψ β1
1 ,...,ψβP P ], (6a)
Γβ(p) =γ β1
1 (ψ1) · ...· γ βP
P (ψP ). (6b)
By (6) and (2), the interpolation matrices A β
i
are given
by the system matrices Ai of the original FE system (1)
at the interpolation point pβ:
A β
i = Ai(pβ). (7)
IV. PARAMETRIC ORDER REDUCTION
We construct the parametric ROM by replacing the
test and trial space, respectively, of the interpolated FE
system (3) by an n dimensional subspace S(p) which
depends continuously on the parameter vector p. For
this purpose, a Galerkin procedure based on a parameterdependent
projection matrix V(p) :RP → RN×n , with
S(p) =range {V(p)} , (8)
- 14 - 15th IGTE Symposium 2012
1 1
1 2
[ , ]
1 2
1 2
[ , ]
1 3
1 2
1,1
1,2
2 1
1 2
[ , ]
2 2
1 2
2,1
2,2
3 1
1 2
[ , ]
3 2
[ , ] [ , ]
1 2
2 3 3 3
[ , ] [ , ] [ , ]
Fig. 1. Hypercube topology
1 2
1 2
is applied to (3). The resulting ROM is of the form
φi(f)
i
Γβ(p)
β
Ãβi
(p)
˜x =
θj(f)
j
˜ Bj(p)u, (9a)
˜y(f,p) = ηj(f) ˜ B T
j (p) ˜x(f,p), (9b)
with
j
à β
i (p) =VT (p)A β
i V(p), (10a)
˜Bj(p) =V T (p)Bj. (10b)
As long as n ≪ N, the frequency response of the ROM
can be computed much more efficiently than the original
one.
A. Parameter dependent projection matrix
We start by computing n dimensional one-parameter
ROMs with respect to frequency at all interpolation
points pβ ∈ G. The resulting projection matrices are
denoted by ˆ Vβ ∈ C N×n .
The interpolation points pβ ∈Gsubdivide the param-
eter domain into hypercubes H β ⊂ R P . Based on the
line segments L k p = ψ k p,ψ k+1
p
H β = L β1
1
,wehave
× ...×LβP P . (11)
Fig. 1 illustrates the setting in the two-dimensional case.
Starting from one-dimensional hat functions ξ k p : R → R,
ξ k p (ψ) =
⎧
ψ k−1
p −ψ
for ψ ∈Lk−1 p ,
⎪⎨ ψ
⎪⎩
k−1
p −ψk p
ψ k+1
p −ψ
ψ k+1
p −ψk for ψ ∈L
p
k p,
0 else,
(12)
we construct piecewise multi-linear interpolation functions
Ξβ : R P → R of compact support:
Ξβ(p) =ξ β1
1 (ψ1) · ...· ξ βP
P (ψP ). (13)
Within a given hypercube Hα , the parameterdependent
projection matrix V(p) is defined by
V(p) =
Ξβ(p) ˆ VβT α β for p ∈H α . (14)
pβ∈H α

Herein, the matrices T α β ∈ Rn×n are provided in order
to conduct state transformations. They are constructed
as follows: Following [8] and [9], a singular value
decomposition [14] is performed,
= U diag σW H , (15)
ˆVβ1 ,..., ˆ Vβ (2P )
to determine a basis Rα ∈ RN×n for the n dimensional
subspace of highest energy over the hypercube Hα , i.e.,
the subspace corresponding to the n largest singular
values:
Rα = U(:, 1:n). (16)
For any relevant state Rα˜x, we require the ROM state
at the interpolation point pβ, ˆ VβTα β ˜x, to be as close as
possible:
!
=min ∀˜x ∈ C n , (17a)
Rα˜x − ˆ VβT α β ˜x2
⇒ ˜x − R H α ˆ VβT α β ˜x = 0 ∀˜x ∈ Cn . (17b)
Thus,
T α β =
R T α ˆ −1 Vβ . (18)
Eq. (18) underlines that interpolating the bases ˆ Vβ directly,
which is equivalent to taking Tα β = I, may cause
gross error.
B. Assembly
Plugging (14) into (10) leads to the following representation
of the reduced matrices within the hypercube Hα :
à β
i (p) =
pγ∈Hα
pδ∈Hα Ξγ(p)Ξδ(p) A β
i,γ,δ , (19a)
˜Bj(p) =
Ξγ(p)Bj,γ, (19b)
wherein
pγ∈H α
A β
i,γ,δ =(Tα γ ) T V T γ A β
i VδT α δ , (20a)
Bj,γ =(T α γ ) T V T γ Bj. (20b)
Note that all the coefficient matrices in (20) are of
reduced size and can be computed in advance. No O(N)
operations are required during the solution process.
V. NUMERICAL EXAMPLES
In the examples below, the single-parameter ROMs
with respect to frequency at the interpolation points are
computed by means of the single-point algorithm of [1].
A. Dielectric Post
Fig. 2 shows the H plane filter of [16]. It consists of
a dielectric post at the center of an air-filled rectangular
waveguide. The model features two parameters: the operating
frequency f ∈ [15, 25] GHz and the geometric
parameter p ∈ [−1.5, 1.5] mm which defines the radius r
of the post according to
r =2.5 mm + p. (21)
Fig. 3 shows instantiations of the parametric mesh [12]
for p ∈{−1.5, 0, 1.5} mm.
- 15 - 15th IGTE Symposium 2012
10
Γ (1)
WG
Ω
10
ɛd
20
μd
Γ (2)
d
ɛr = μr =1
Γ (1)
d
2r
Γ (2)
WG
5
Fig. 2. Structure of rectangular waveguide filter [16]. All dimensions
are in mm. Material properties of rod: relative electric permittivity ɛd =
4, relative magnetic permeability μd =1. Waveguide ports are denoted
by Γ (1)
WG and Γ(2)
WG , respectively.
Fig. 3. Instantiations of the parametric FE mesh for different values
of the geometry parameter: p ∈ {−1.5, 0, 1.5} mm. Note that the
meshes share the same topology.
1) Response Surface and Errors: Fig. 4 presents the
response surface of the magnitude of the reflection coefficient
|S11|. The parametric ROM is based on M =5
interpolation points placed at the locations of the zeros
of the fifth-order Chebyshev polynomial of the first kind.
The expansion frequency for the single-parameter ROMs
is set at the center of the frequency band, f exp =20GHz.
We define the error in S11 by
ES11 (f,p) = S11(f,p) − S11(f,p), (22)
wherein ˜ S11 denotes the PMOR result, and S11 is the
reference solution, which is computed by conventional
FE analysis, using the same mesh. The complete error
surface is given in Fig. 5. It can be seen that errors are
in the order of 10 −3 , which is below the typical level of
the FE discretization error. Note that calculating the error
surface is only possible for very simple structures, like
the present filter, because each of the 101×101 sampling
points in f − p space requires a separate FE run.

Fig. 4. Dielectric post: Response surface of the magnitude of the
reflection coefficient S11 as a function of operating frequency f and
radius variation p.
Fig. 5. Dielectric post: Error surface of |S11|.
Computational data for conventional FE analysis and
two different PMOR models, ROM3 and ROM5, using
M =3and M =5interpolation points, respectively,
are given in Table I. It can be seen that, even though the
dimension of the original FE system is very small, the
larger of the two models, ROM5, is still 150 times faster
to evaluate.
2) Analysis of Suggested Procedure: Our first goal
is to compare the proposed method, which employs
TABLE I
COMPUTATIONAL DATA FOR DIELECTRIC POST.
Model ROM5 ROM3 FE
Number of grid points 5 3 -
Moment-matching order 10 7 -
Dimension 22 16 5616
Model generation (s) ∗ 365.8 177.7 -
Evaluations per s∗ 481.0 757.3 3.2
|Average error in S11| 4.99 · 10−4 5.93 · 10−3 0
∗ MATLAB code on Intel Pentium 4 (3GHz), one thread used.
- 16 - 15th IGTE Symposium 2012
|E |
S11
10 0
10 −1
10 −2
10 −3
10 −4
Direct interpolation of ROM bases
Present method
10
−1.5 −1 −0.5 0 0.5 1 1.5
−5
Radius variation (mm)
Fig. 6. Dielectric post: Magnitude of error in reflection coefficient S11
as a function of radius variation p at f =25GHz. Note the positive
effects of state transformations in the present method.
state transformations, to direct interpolation of the ROM
bases ˆ Vβ. For this purpose, we consider a ROM based
on M =5equidistant sampling points. Fig. 6 presents
the error in S11 (22) as a function of radius variation p
at f =25GHz: The necessity of proper state transformations
is evident.
The next test addresses the rate of convergence of the
proposed method. We start from 3 equidistant interpolation
points at refinement level r =1, and refine the grid
recursively. Thus, the total number of points at refinement
level r is
|G| =2 r +1. (23)
Our measure of error is ĒS11 , the average error in S11
at f =25GHz,
ĒS11
= 1
Ns
Ns
n=1
| S11(pn) − S11(pn)|, (24)
based on Ns = 257 equally spaced sampling points,
pn ∈ [−1.5, 1.5] mm. Fig. 7 presents the magnitude of
the average error as a function of refinement level for
direct ROM interpolation, a variant of the present method
which uses piecewise linear geometry interpolation, and
the suggested approach, employing global polynomial
interpolation for the geometry. The results of Fig. 7
show that the total error is dominated by the effects
of geometry interpolation: The suggested method clearly
outperforms competing approaches.
B. Mitered Microstrip Bend
Our second example, the mitered microstrip bend
shown in Fig. 8, is a truly three-dimensional structure
with more than one million FE unknowns. The
model features two parameters, the operating frequency
f ∈ [1, 10] GHz and a geometric parameter p ∈
[−0.7, 0.7] mm which controls the width t of the mitered

|Average error in S 11 |
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
10 −6
Proposed approach − global polynomial
Proposed approach − piecewise linear
ROM interpolation − piecewise linear
0 1 2
Refinement level
3 4
Fig. 7. Dielectric post: Magnitude of average error versus grid
refinement level r at f = 25 GHz. The proposed method benefits
from interpolating the FE matrices by polynomials of higher-order.
μr,ɛr
2.413
t
60
60 0.794
Fig. 8. Structure of a mitered microstrip bend. Dimensions are in mm.
Material properties of substrate: ɛr =2.2, μr =1.
bend. We have:
t =1.7062 mm + p. (25)
Again, the parametric ROM is based on M = 5
interpolation points placed at the locations of the zeros
of the fifth-order Chebyshev polynomial of the first kind.
The expansion frequency for the single-parameter ROMs
is set to f exp =5GHz.
Fig. 9 shows the response surface of the magnitude of
the reflection coefficient S11, calculated by the proposed
method. Fig. 10 presents |S11| and the corresponding
error |ES11 | (22) with respect to conventional FE simulations
versus frequency for the case p =0.2 mm. The
fact that the error is always more than 25 dB below the
signal level underlines the high quality of the ROM.
Table II provides computational data for conventional
FE simulation and the ROM. It can be seen that it
takes more than 2 hours to build the parametric ROM.
However, once the ROM is available, it can be evaluated
more than 2200 times per second. For comparison, one
- 17 - 15th IGTE Symposium 2012
Fig. 9. Mitered microstrip bend: Response surface of the magnitude
of the reflection coefficient |S11| as a function of operating frequency
f and miter parameter p.
|S 11 | (dB)
−30
−40
−50
−60
−70
−80
−90
−100
Proposed approach
Error of proposed approach
2 4 6
Frequency (GHz)
8 10
Fig. 10. Mitered microstrip bend: |S11| and error |ES11 | versus
frequency. Parameter: p =0.2 mm.
conventional FE solution takes 180 s, which is more than
400 000 times longer!
VI. CONCLUSIONS
This paper has presented a PMOR methodology for
FE models with geometric parameters. It is characteristic
of the new approach that geometry approximation
is separated from the actual ROM generation process.
Moreover, the suggested method incorporates state transformations
that improve the quality of the interpolated
projection matrices. In consequence, the present PMOR
method reaches higher rates of convergence than previous
approaches. Since the resulting parametric models are of
small dimension, they are very fast to evaluate.
TABLE II
COMPUTATIONAL DATA FOR MICROSTRIP BEND.
Model ROM FE
Number of grid points 5 -
Moment-matching order 20 -
Dimension 42 1,175,382
Model generation (s) ∗ 7513.2 -
Evaluations per s ∗ 2267.9 5.56 · 10 −3
|Avr. error in S11| at p =0.2 mm 1.06 · 10 −4 0
∗ MATLAB code on Intel Xeon E5620, one thread used.

REFERENCES
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Reduction of Polynomial Matrix Equation using Single-Point
Well-Conditioned Asymptotic Waveform Evaluation: Dreivations
and Theory,” Int. J. Numer. Meth. Eng., vol. 58, pp. 2325 – 2342,
Dec. 2003.
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[4] A. Schultschik, O. Farle, R. Dyczij-Edlinger, “An Adaptive Multi-
Point Fast Frequency Sweep for Large-Scale Finite Element
Models,” IEEE Trans. Magn., vol. 45, no. 3, pp. 1108 – 1111,
March 2009 .
[5] R. Dyczij-Edlinger and O. Farle, “Finite element analysis of
linear boundary value problems with geometrical parameters,”
COMPEL, vol. 28, no. 4, pp. 779 – 794, 2009.
[6] O. Farle, S. Burgard, and R. Dyczij-Edlinger “Passivity Preserving
Parametric Model-Order Reduction for Non-affine Parameters,”
Math. Comp. Model. Dyn. Sys., vol. 17, no. 3, pp. 279 – 294,
2011.
[7] J.R. Phillips, “Variational interconnect analysis via PMTBR,”
ICCAD, pp. 872 – 879, 7-11 Nov. 2004.
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Nonlinear Models by Superposition of Locally Reduced Models,”
Methoden und Anwendungen der Regelungstechnik, pp. 27 – 36,
Aachen:Shaker-Verlag, 2009.
[9] H. Panzer, J. Mohring, R. Eid, and B. Lohmann, “Parametric
Model Order Reduction by Matrix Interpolation,” at - Automatisierungstechnik,
vol. 58, no. 8, pp. 475 – 484, 2010.
[10] O. Farle and R. Dyczij-Edlinger, “Numerically Stable Moment
Matching for Linear Systems Parameterized by Polynomials in
Multiple Variables with Applications to Finite Element Models of
Microwave Structures,” IEEE Trans. Antennas Propag., vol. 58,
no. 11, pp. 3675 – 3684, Sep. 2010.
[11] D. Amsallem, J. Cortial, K. Carlberg, C. Farhat, “A method
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[12] S. Burgard, Morphing von Finite-Elemente-Netzen, Studienarbeit,
Lehrstuhl für Theoretische Elektrotechnik, Universität des Saarlandes,
2008. In German.
[13] J. Pomplun and F. Schmidt, “Accelerated a Posteriori Error
Estimation for the Reduced Basis Method with Application to
3D Electromagnetic Scattering Problems,” SIAM J. Sci. Comput.,
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[14] G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore:Johns
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- 18 - 15th IGTE Symposium 2012

- 19 - 15th IGTE Symposium 2012
Efficient Finite-Element Computation of
Far-Fields of Phased Arrays by Order Reduction
A. Sommer∗ , O. Farle∗ , and R. Dyczij-Edlinger∗ ∗Chair for Electromagnetic Theory, Saarland University, D-66123 Saarbrücken, Germany
E-mail: edlinger@lte.uni-saarland.de
Abstract—This paper presents an efficient numerical method for computing the far-fields of phased antenna arrays over
broad frequency bands as well as wide ranges of steering and look angles. The suggested approach combines finiteelement
analysis, projection-based model-order reduction, and empirical interpolation. Numerical results demonstrate that
evaluation times are reduced by orders of magnitude, compared to traditional methods.
Index Terms—Empirical interpolation, far field computation, finite-element method, and model order reduction.
I. INTRODUCTION
In many areas of application, such as radar or wireless
communications, phased antenna arrays need to be analyzed
over broad frequency bands as well as wide ranges
of steering and look angles. Finite-element (FE) based
analysis of such structures involves two major steps:
First, the near field is computed as a function of angular
frequency ω and steering angles (θs,φs). Second, a
discrete near-field-to-far-field (NF-FF) operator is applied
to determine the far field as a function of frequency and
look angles (θ,φ). Using conventional approaches, this
procedure tends to be very time-consuming. The reasons
are as follows: Since typical antenna arrays are electrically
large and consist of high numbers of radiators, the
corresponding FE systems are of very large dimension.
Moreover, broadband analysis requires large numbers of
sampling frequencies. At each of them, the large-scale
FE system has to solved, by matrix factorization or some
iterative method. In addition, wide variations in steering
angles imply large numbers of sampling angles (θs,φs).
Each of them leads to separate excitation and right-hand
side (RHS), respectively. Finally, wide variations in look
angles result in large numbers of sampling points (θ,φ).
Each of them requires a separate NF-FF transformation
at each operating point (θs,φs; ω) of the antenna array.
To reduce computational efforts, we propose a twostep
approach: We first construct a reduced-order model
(ROM) for the near fields which is very cheap to solve
at any value of the parameter triple (θs,φs; ω). For
this purpose, a multi-point model-order reduction (MOR)
method with self-adaptive expansion point selection [1],
[2] is applied. The second step utilizes the empirical
interpolation (EI) method [3], [4] to construct an affine
approximation to the NF-FF operator as a function of
frequency and look angles. It, too, is very fast to evaluate
at any value of the parameter triple (θ,φ; ω). Combining
both steps yields a highly efficient numerical model
for computing the far-fields as a function of the five
parameters (θ,φ; θs,φs; ω), which is ideally suited for
fast online evaluation: In Section VI, computing a farfield
pattern based on 4830 look angles takes only 2.4 s,
on a personal computer executing plain MATLAB code.
The construction of the model by MOR and EI is
much more time-consuming and must be performed in
advance, in an offline step. The most expensive procedure
in the MOR algorithm is the FE analysis of the array at a
number of expansion points (θs,φs; ω), which are chosen
adaptively. Note that, as long as a direct solver is used,
changes in the three parameters are not equally expensive:
Since ω affects the FE matrix, each frequency value
requires a new matrix factorization, which is computationally
expensive. On the other hand, the steering angles
(θs,φs) enter the RHS only. Thus, changes in angle just
require additional forward-back substitutions, which are
much cheaper. Therefore, the adaptive point placement
strategy of the MOR method ought to vary ω as rarely
as possible. The new frequency-slicing greedy method
presented in Section IV-B implements this strategy in
a systematic fashion. For the highest-accuracy ROM of
Section VI, it improves computing time by a factor of 7
at the cost of increasing ROM size by 5%, compared to
state-of-the-art methods [1], [2].
The numerical experiments of Section VI indicate that
both MOR and EI feature exponential convergence, in
accordance with theoretical results [5]. Our results for
a real-world example [6] demonstrate that the suggested
two-step approach for the broadband analysis of radiation
patterns of phased arrays achieves high accuracy and reduces
evaluation times by orders of magnitude, compared
to conventional approaches.
II. FAR-FIELD COMPUTATION
By the vector Huygens principle in the frequency
domain [7], the radiation vector F of an arbitrary antenna
array, which is enclosed by a surface S, isgivenby
ω j ˆr·r c F (ˆr,ω)=ˆr × e 0
S
′
J s (r ′ ,ω)dS ′ × ˆr
+ 1
ω j c ˆr·r
e 0
η0 S
′
M s (r ′ ,ω)dS ′ × ˆr. (1)
Here c0 and η0 denote the vacuum speed of light and
characteristic impedance, respectively, and ˆr is the unit
vector in the direction of the observer. The equivalent
electric and magnetic surface current densities, J s and

M s, are given in terms of the electric and magnetic nearfields
E and H by
J s (r ′ ,ω)=ˆn × H (r ′ ,ω) with r ′ ∈ S, (2a)
M s (r ′ ,ω)=−ˆn × E (r ′ ,ω) with r ′ ∈ S, (2b)
wherein ˆn stands for the outward-pointing unit normal
vector on the Huygens surface S. The far-fields EF and
HF are obtained from the radiation vector (1) by
ω
c r
e−j 0
EF (r,ω)=−jμ0ω F (ˆr,ω) , (3a)
4πr
HF (r,ω)=−j ω
ω −j c r
e 0
ˆr × F (ˆr,ω) (3b)
c0 4πr
and the directive gain [8] is given by
2 μ0ω F (ˆr,ω)
D (ˆr,ω)=
8πc0
2
2 , (4)
P (ω)
wherein μ0 describes the vacuum permeability. In (4),
the total radiated power P (ω) is determined from
P (ω) = 1
2 ℜ
ˆn × E (r
S
′ ,ω) · H (r ′ ,ω)dS ′
. (5)
III. FE MODEL AND MULTI-POINT MOR METHOD
FE analysis of the near fields of a phased array of L
antennas leads to a linear system of the form
(A0 + ωA1+ω 2 L
A2) x (p) =ω up (p) bp, (6a)
p=1
y (p) = C0 + ω −1
C1 x (p) , (6b)
P (p) =ω −1 x ∗ (p) Dx (p) . (6c)
Herein, A0, A1, A2 ∈ CN×N are the stiffness, damping,
and mass matrices, respectively, x denotes the solution
vector in terms ofE, p = (θs,φs,ω) ∈ R3 the parameter
vector, and N the dimension of the FE system. The
output vector y ∈ C6H holds the electric and the magnetic
near-field values E and H, respectively, sampled
at H points on the Huygens surface S. In (6b), the
matrices C0 ∈ C6H×N and C1 ∈ C6H×N carry out
the sampling process and magnetic field computation on
S. Furthermore, the Hermitian matrix D ∈ CN×N of the
bilinear form (6c) represents the computation of the total
radiated power (5). It can be seen that the system matrix
of (6a) depends on the angular frequency ω only, while
the RHS also depends on the steering angles θs and φs.
Note that the RHS is constructed by a superposition of
L linearly independent vectors bp ∈ CN with parameterdependent
weights up (p).
To obtain the near-fields vector y and the total radiated
power P , the large-scale system (6a) has to be solved for
each parameter vector p of interest. Our goal is to bypass
this time-consuming procedure. Since the FE system (6)
exhibits affine parameter dependence [1], it is well-suited
for projection-based MOR. The idea is to approximate
the FE solution x (p) in a low dimensional subspace
according to
x (p) ≈ ˆx (p) =V˜x (p) (7)
- 20 - 15th IGTE Symposium 2012
with ˆx ∈ CN , ˜x ∈ Cn , V ∈ CN×n , and n ≪ N for
all p ∈ D. Here, D denotes the considered parameter
domain. For numerical stability, the columns of the trial
matrix V are chosen to be orthogonal. Substituting the
approximation (7) for x (p) in (6a) and testing with V∗ leads to the ROM:
2
ω q L
Ãq ˜x (p) =ω up (p) ˜ bp, (8a)
q=0
ˆy (p) =
p=1
1
ω −r Cr˜x ˜ (p) , (8b)
r=0
ˆP (p) =ω −1˜x ∗ (p) ˜ D˜x (p) , (8c)
wherein the reduced matrices and vectors are given by
Ãq = V ∗ AqV with Ãq ∈ C n×n , (9)
˜bp = V ∗ bp with bp
˜ ∈ C n , (10)
˜Cr = CrV with Cr
˜ ∈ C 6H×n , (11)
˜D = V ∗ DV with D˜ n×n
∈ C . (12)
Using a multi-point (MP) MOR method, the trial matrix
V is constructed from FE solutions on a discrete set
De ⊂Dof expansion points pi ∈ De such that
range V =span{x (p1) ,...,x (pn)} . (13)
As long as n ≪ N, the ROM (8) can be solved much
more efficiently than the original system (6). Thus, the
computational costs for determining both the near-field
values and the total radiated power can be kept very low.
IV. SELF-ADAPTIVE EXPANSION POINT SELECTION
The residual r of ˆx with respect to (6a) takes the form
r (p) =
2
ω q L
(AqV) ˜x (p) − ω up (p) bp. (14)
q=0
Thus, the computation of its 2-norm,
r (p) 2
2 =
2 2
p=1
ω
q1=0 q2=0
q1+q2 ˜x (p) ∗ V ∗ A ∗ q1Aq2V ˜x (p)
− 2ℜ
+ ω 2
L
p=0 q=0
L
p1=0 p2=0
2
ω q+1 b ∗ pAqV
˜x (p)
L
up1(p)up2(p) b ∗ p1bp2
, (15)
just involves matrices and vectors of the ROM dimension
n ≪ N and is therefore very fast. This motivates the use
of a residual-based error indicator ρ(Dds) in the pointplacement
strategy:
ρ(Dds) = max r(p)2 . (16)
p∈Dds
Herein, Dds stands for a dense sampling of the considered
domain.

Algorithm 1 Conventional Greedy Algorithm.
Given: Dds, p1 ∈ Dds, and ɛ.
n =0. {Initialize ROM dimension.}
repeat
n ← n +1.
ωc = ω(pn).
Compute LU factorization of A(ωc).
Determine x (pn) by forward-back substitution.
Construct ROM by (8a).
Compute residual r(p) 2 for all p ∈ Dds.
Place expansion point pn+1 using (17).
until ρn(Dds)

Let P denote the number of far-field look angles for
which (23) is to be evaluated. It can be seen that, although
the solution ˜x (p) of the ROM (8) is used, the computational
effort for merely one single operating-point p ∈ D
is still of complexity O (PH + Hn), i.e., the far-field
computation itself is expensive, too. Our solution to this
problem is to adopt an idea from [2] and employ the
EI method [3], [4] to construct an affine decomposition
of the exponential function (22). The offline part of this
method uses a greedy strategy to determine a set of M
basis functions {qm} M
m=1 , interpolation points {r′ m} M
m=1
and parameter values {d ′ m} M
m=1 such that the interpolant
ê (d, r ′ ) defined by
ê (d, r ′ M
)= αm (d) qm (r ′ ) (24)
m=1
approximates (22) for all (r ′ , d) ∈ S ×M. Having
constructed the interpolation matrix
⎡
⎤
⎢
BM = ⎣
. ..
⎥
⎦ (25)
q1 (r ′ 1)
.
q1 (r ′ M ) ... qM (r ′ M )
offline, the parameter-dependent coefficients
{αm (d)} M
m=1 are obtained online, by solving the
lower triangular system
⎡ ⎤ ⎡
α1 (d) e (r
⎢
[BM ]
. ⎥ ⎢
⎣ . ⎦ = ⎣
αM (d)
′ ⎤
1, d)
. ⎥
. ⎦ . (26)
, d)
e (r ′ M
Substituting the empirical interpolant (24) for e (d, r ′ ) in
(23) results in
Ix (˜x (p) , ê (d, r ′ ) ,ω)=ω −1 M
ΔS αm (d)
H
h=1
m=1
qm (r ′ h) ˆn (r ′ h) × ˜ C1 (r ′ h) ˜x (p) . (27)
Under the precondition that the sampling points on the
Huygens surface S remain constant for all M steps of the
EI method, the online part of (24) can be implemented
such that it takes only O (M) operations. Thus, the
computational efforts for computing P far-field values
by (27) for a given operating-point p ∈ D are only of
order O (PM + Mn). Since, in practice, M ≪ H, the
costs of the far-field computation are greatly reduced.
VI. NUMERICAL RESULTS
In the following, we consider the FE model of a
dual-polarized tapered slot antenna array (TSAA) [6]
consisting of L = 40 antennas, whose geometry is
depicted in Fig. 1. The frequency band is given by
f ∈ [2, 4] GHz, and the scan angles of interest are in
the range of (θs,φs) ∈ 0, π
2
3 × [0, 2π) rad .Weuse
#Dds = 17040 training points in the offline part of
the self-adaptive multi-point method of Section IV and
construct the ROM (8) by both the conventional greedy
algorithm and the new FSG approach of Section IV-B.
- 22 - 15th IGTE Symposium 2012
Fig. 1. Geometry of the TSAA [6]. Dimensions: length l =8cm,
width w = 8 cm, height h = 7 cm, and displacement of adjacent
antennas s =2cm.
Maximum norm: local error indicator
10 0
10 −2
10 −4
10 −6
10 −8
Conventional greedy method
FSG method
50 100 150 200 250 300 350
ROM dimension n
Fig. 2. Normalized error indicator (16) versus ROM dimension n
for the conventional and the new FSG method. Circles mark changes
in expansion-point frequency in the FSG method, requiring matrix
factorization.
A. Properties of FSG method
Fig. 2 presents the behavior of the normalized error
indicator (16) as a function of ROM dimension n. It
can be seen that the standard method achieves nearly
constant rates of convergence, whereas the FSG approach
converges rather slowly during the early stages of the
iteration. This behavior is expected because, early on, the
frequency sampling of the FSG method is very poor. On
the other hand, the standard procedure must factorize the
FE matrix at each iteration, whereas the FSG method requires
factorizations only when the expansion frequency
changes, i.e., at the iterations marked by circles in Fig. 2.
Thus, to compare overall computational efficiency, we
have measured computing times for the same threshold
ɛ of the error indicator. Table I presents the results. It
can be seen that, depending on the threshold level, the
proposed FSG method is 5 to 7.5 times faster.

Relative error e n
TABLE I
TSAA: COMPUTING TIME FOR ROM CONSTRUCTION (8)
Residual Time t ∗ Speed-up Dimension n
threshold ɛ Alg. 1 Alg. 2 factor Alg. 1 Alg. 2
2.7 e−3 92.16 h 18.34 h 5.025 153 285
4.6 e−5 164.98 h 24.25 h 6.803 265 333
1.3 e−6 221.32 h 29.46 h 7.513 346 363
∗ MATLAB code on Intel(R) Xeon(R) E5620 CPU at 2.40 GHz.
10 0
10 −2
10 −4
10 −6
10 −8
Conventional greedy method
FSG method
50 100 150 200 250 300 350
ROM dimension n
Fig. 3. Relative error in near-fields (28) versus ROM dimension n
for the conventional approach and the FSG method. Parameter: p =
( π π
rad, − rad, 3.645 GHz) /∈ Dds.
4 6
B. Error in near-fields
Our measure for the error in the near-fields at a given
parameter vector p is the relative error e(p) defined by
en (p) = x (p) − ˆxn (p)2 .
x (p)2 (28)
To investigate the convergence behavior of the
projection-based MOR method, we choose a representative
parameter vector, p = π π
4 rad, − 6 rad, 3.645 GHz /∈
Dds, and evaluate (28) as a function of ROM dimension
n. Fig. 3 shows that both the conventional approach
and the FSG method exhibit exponential convergence.
C. Error in far-fields
The following tests are based on #Mds = 28380
training points in the offline part of the EI method. The
considered look angles are in the range of (θ,φ) ∈
π
2
0, × [0, 2π) rad .
2
We first investigate the error of the empirical interpolant
êm (d, r ′ ) of (24) with respect to the true value
of the exponential function e (d, r ′ ) of (22). For this
purpose, we choose a representative parameter vector,
d =(3.789 GHz, − π π
4 rad, 5 rad) /∈ Mds and monitor the
relative error em(d),
em (d) = max
, (29)
r ′ ∈Sh
e (d, r ′ ) − êm (d, r ′ )
e (d, r ′ )
as a function of the number of EI coefficients m. The
results shown in Fig. 4 demonstrate that the EI method
leads to exponential convergence.
- 23 - 15th IGTE Symposium 2012
Relative error e m
10 2
10 0
10 −2
10 −4
10 −6
10
0 200 400 600 800
−8
Number of coefficients m
Fig. 4. Relative error in exponential function (29) versus number of
EI coefficients m. Parameter: d = 3.789 GHz, − π
4
rad, π
5 rad .
TABLE II
AVERAGE ERROR IN DIRECTIVE GAIN.
Steering angles
(θs,φs)
Average error eD (30)
2.57 GHz 3.30 GHz 3.95 GHz
( π π
rad, 4 2 rad) 1.2 × 10−3 1.7 × 10−3 2.6 × 10−3 ( π π
rad, 4 4 rad) 1.1 × 10−3 1.8 × 10−3 2.8 × 10−3 ( π
6 rad, 0 rad) 1.3 × 10−3 1.9 × 10−3 3.3 × 10−3 ( π π
rad, − 3 3 rad) 2.2 × 10−3 2.0 × 10−3 3.1 × 10−3 In our final test, we consider the error in radiation
pattern of the phased antenna array, by measuring the
average error in directive gain eD,
eD (p) = 1
P
D
(p, dp) −
P
p=1
ˆ
D (p, dp)
. (30)
D (p, dp)
Table II presents error values for 12 different parameter
vectors p /∈ Dds, corresponding to the far-field plots in
Fig. 5 – Fig. 7. It can be seen that the results of the
suggested MOR approach are in very good agreement
with reference data. Computational parameters for this
test can be found in Table III and Table IV.
D. Overall runtime performance
Fig. 5 – Fig. 7 show three-dimensional radiation patterns
of the TSAA for different operating frequencies
and four steering angles per frequency. Computational
data of the original FE model and the ROM are given in
Table III and Table IV, respectively. Without doubt, the
offline part of the algorithm leads to some one-time costs
for constructing the ROM and the affine approximation
to the NF-FF operator. However, once they are available,
computing time for the near-fields improves by a factor of
68000, compared to conventional FE analysis. Moreover,
post-processing time for one radiation pattern based on
P =4, 830 look angles reduces by a factor of 12. Thus,
the total speed-up factor for computing one near-field
solution plus the corresponding far-field pattern is 910.
VII. CONCLUSIONS
An efficient two-step MOR method for computing
the far-field patterns of phased antenna arrays has been

(a) θs = π
4
rad, φs = π
2
π
π
rad. (b) θs = rad, φs = 4 4 rad.
(c) θs = π
π
π
rad, φs =0rad. (d) θs = rad, φs = − 6 3 3 rad.
Fig. 5. Radiation patterns of the TSAA at f =2.57 GHz, determined
by the two-step MOR method using P =4, 830 look angles.
(a) θs = π
4
rad, φs = π
2
π
π
rad. (b) θs = rad, φs = 4 4 rad.
(c) θs = π
π
π
rad, φs =0rad. (d) θs = rad, φs = − 6 3 3 rad.
Fig. 6. Radiation patterns of the TSAA at f =3.30 GHz, determined
by the two-step MOR method using P =4, 830 look angles.
presented. Thanks to the new FSG technique, evaluation
times for a real-world example [6] improve by a factor
of 5 to 7.5 over earlier MOR approaches, and by a factor
of 910 compared to conventional FE analysis.
REFERENCES
[1] V. de la Rubia, U. Razafison, and Y. Maday, ”Reliable fast
frequency sweep for microwave devices via the reduced-basis
method”, IEEE Trans. Microw. Theory Techn., vol. 57, pp. 2923-
2937, Dec. 2009.
[2] M. Fares, J. S. Hesthaven, Y. Maday, and B. Stamm, ”The reduced
basis method for the electric field integral equation”, J. Comput.
Phys., vol. 230, pp. 5532-5555, 2011.
- 24 - 15th IGTE Symposium 2012
(a) θs = π
4 rad, φs = π
2 rad. (b) θs = π
4 rad, φs = π
4 rad.
(c) θs = π
π
π
rad, φs =0rad. (d) θs = rad, φs = − 6 3 3 rad.
Fig. 7. Radiation patterns of the TSAA at f =3.95 GHz, determined
by the two-step MOR method using P =4, 830 look angles.
TABLE III
COMPUTATIONAL DATA OF ORIGINAL FE MODEL OF TSAA.
Parameters: θs = π
π
rad, φs = rad, f = 2.57 GHz.
4 2
FE dimension N 2, 553, 439
Number of near-field points H 12, 800
Number of look angles P 4, 830
Time for solving FE system (6a) 2192.4 s∗ Time for computing radiation pattern 28.9 s∗ ∗ MATLAB code on Intel(R) Xeon(R) E5620 CPU at 2.40 GHz.
TABLE IV
COMPUTATIONAL DATA OF REDUCED-ORDER MODEL OF TSAA.
Parameters: θs = π
π
rad, φs = rad, f = 2.57 GHz.
4 2
ROM dimension n 300
Number of EI coefficients m 350
Number of look angles P 4, 830
Offline time for generating ROM (8) 20.36 h∗ Offline time for EI method 33.97 h∗ Online time for solving ROM (8a) 0.0321 s∗ Online time for radiation pattern 2.4087 s∗ ∗ MATLAB code on Intel(R) Xeon(R) E5620 CPU at 2.40 GHz.
[3] M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera, ”An
’empirical interpolation’ method: application to efficient reducedbasis
discretisation of partial differential equations”, C. R. Acad.
Sci. Paris, Ser. I 339, pp. 667-672, 2004.
[4] M. A. Grepl, Y. Maday, N. C. Nguyen, A. T. Patera, ”Efficient reduced
basis treatment of nonaffine and nonlinear partial differential
equations”, M2AN Math. Model. Numer. Anal. 41, pp. 575605,
2007.
[5] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P.
Wojtaszczyk, ”Convergence rates for greedy algorithms in reduced
basis methods”, SIAM J. Math. Anal., vol. 43, pp. 1457-1472,
2011.
[6] T.-H. Chio, and D. H. Schaubert, ”Parameter study and design of
wide-band widescan dual-polarized tapered slot antenna arrays”,
IEEE Trans. Antennas Propag., vol. 48, pp. 879-886, June 2000.
[7] E. J. Rothwell and M. J. Cloud, ”Electromagnetics”, CRC Press,
2009
[8] S. J. Orfanidis, ”Electromagnetic Waves and Antennas”,
http://www.ece.rutgers.edu/ orfanidi/ewa.

- 25 - 15th IGTE Symposium 2012
Nanoparticle device for biomedical and
optoelectronics applications
R. Iovine, L. La Spada and L. Vegni
Department of Applied Electronics, University of Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy
E-mail: riovine@uniroma3.it
Abstract—In this contribution a nanoparticle device, operating in the visible regime based on the Localized Surface Plasmon
Resonance (LSPR) phenomenon, is presented. The nanoparticle electromagnetic properties are evaluated by a new analytical
model and compared to the results obtained by numerical analysis. A near-field enhancement is obtained by arranging the
nanoparticles in a linear array. Analytical formulas, describing such enhancement, are presented. The structure can find
application for medical diagnostics and optoelectronics applications.
Index Terms— LSPR, Medical diagnostics, Nanoparticle, Near-field Enhancement, Optoelectronics Applications
I. INTRODUCTION
In the last few years, several researches have paid
attention to gold nanoparticles optical properties relate to
the interaction of these structures with electromagnetic
field at Visible (VIS) and Near Infrared Region (NIR)
[1].
When the electromagnetic field interacts with small metal
particles the conduction electrons start oscillating
collectively. This phenomenon is now well known and
called Localized Surface Plasmon (LSP) [2]. If the
frequency of the incident field matches the natural
frequency oscillation of the electrons cloud the resonance
condition is established with a strong dependence on the
shape, size, composition of the nanoparticles as well as
on the dielectric properties of the background
environment [3].
The analytical closed form electromagnetic solution to
evaluate the electromagnetic behavior of metal
nanoparticles exists only for the spherical shapes [4]. The
possibility to predict the electromagnetic properties of
different kind of shapes is now very important due to the
fact that the progress in nanofabrication technology
allows to realize many shapes of particles [5] suitable for
several application field such as biomedical sensing [6]
and thin film solar cells [7].
For example in [8] the possibility to control the
enhancement of the Surface Enhanced Raman Scattering
(SERS) using gold nanoparticles in the field of diagnostic
oncology is reported. In [9] the possibility to use gold
nanoparticles to produce in an efficient way heat energy
from absorbed light energy that may be employed for
selective PhotoThermal Therapy (PTT) is referred.
The aim of this contribution is to propose the design of a
nanostructure device consisting in a gold linear chain
array of nanocubes, deposited on a silica substrate.
For the cube particle a new analytical quasi static model
describing its resonant behavior in terms of absorption
and scattering cross section is presented. The results
obtained by the analytical model are compared to the
other ones performed through proper full-wave
simulations [10] and by using the boundary integral
method approach [11].
The electromagnetic behavior of the device is evaluated
for different inter-particle distance. In particular, the far
field properties and the near electric field distribution are
numerically obtained and the performances of the
structure are analyzed for possible optoelectronics
applications (design of absorbing layers) and for
biosensing applications (refractive index measurements).
II. QUASI STATIC ANALYTICAL MODEL FOR THE
CUBE PARTICLE
In general the nanoparticles have a size smaller compared
to wavelength (e.g. at optical frequencies) so, it is
possible to assume that all the conduction electrons in a
nanoparticle see the same field at a given time (quasi –
static approximation).
Figure 1: Scheme of interaction between electromagnetic field and
small particles compared to wavelength.
The displacement of the electrons by incident
electromagnetic field induces a dipolar charge separation
(positive nuclei – free electrons) generating a restoring
force which conflicts with incident field. The electron
position is determined by the following equation:
(1)
where is the electron mass, is the electron damping
coefficient and is the restoring coefficient.
The relation (1) is a second order inhomogeneous
differential equation with the following solution for
harmonic excitation:
(2)
where
is the natural frequency of the system.

This model is equivalent to a classical mechanical
oscillator and represents a good physical interpretation to
understand the Localized Surface Plasmon Resonance
(LSPR) phenomenon. The resonance condition is
established for and the denominator of (2) tends
to zero and the coefficient and are very difficult to
evaluate and are implicitly related to
geometry/electromagnetic properties of the particles and
permittivity value of the dielectric environment.
However, exploiting the limit of electrically small
particles it is possible to evaluate the resonant behavior of
the cube nanoparticle in accurate way. In order to study
such electromagnetic properties, in terms of scattering
and absorption cross-section, the following assumptions
will be done:
the particle is homogeneous and the surrounding
material is a homogeneous, isotropic and nonabsorbing
medium.
The impinging plane wave has the electric field E
parallel and the propagation vector k perpendicular
to the nanoparticle principal axis, as depicted in
Figure 2.
Figure 2: Geometrical sketch of the gold nanocube particle.
Under such conditions, we can relate the macroscopic
nanoparticle properties to the polarizability of the
nanoparticle.
It is well known that [12], in case of an arbitrary shaped
particle, its polarizability can be expressed as:
(3)
where is the volume of the particle, the surrounding
dielectric environment permittivity, the inclusion
dielectric permittivity and is the depolarization factor.
The nanoparticle polarizability strongly depends on the
inclusion geometry, its metallic electromagnetic
properties , and the permittivity of the surrounding
dielectric environment . In particular, the factor of
a nanoparticle plays a critical role in the polarizability
resonant behaviour for the LSPR strength.
Starting from [12], it is possible to develop new
analytical closed-form formulas for the scattering and
absorption cross-section of the aforementioned particles.
The general corresponding expressions read, respectively:
(4)
- 26 - 15th IGTE Symposium 2012
where is the wavenumber, is the
wavelength and is the refractive index of the
surrounding dielectric environment. Im stands for
"Imaginary part".
By considering the electric field polarization of the
impinging plane wave, the absorption cross-section reads
[13]:
where is:
III. BOUNDARY ELEMENT METHOD APPROACH
Under quasi - static approximation the electric field can
be expressed through the scalar potential as:
(5)
(6)
(7)
For homogeneous isotropic frequency-dispersive media
can be determined easily from the Laplace equation:
(8)
In fact, by assuming an impulsive source the solution of
(8) is well known through the Green function
as:
(9)
where and are the position vector and source vector,
respectively.
However, if we have an inhomogeneous medium such as
a nanoparticle embedded in a dielectric environment
(Figure 3) the solutions (9) are also valid but need to be
satisfied by appropriate boundary conditions.
Figure 3: Gold nanocube particle embedded in a dielectric environment.

In [11] it is possible to evaluate the scalar potential for
the inhomogeneous medium:
(10)
by adding an artificial charge distribution at the boundary
of discontinuity, determined from the continuity of
the tangential electric field and normal component of the
dielectric displacement [14].
The expression (10) can be converted from boundary
integrals to bounday elements. Following the procedure
reported in [15] it is possible to discretize the particle
boundary into small surface by assuming that surface
charges are located at the center of the surface element.
In this way, it is possible to obtain numerically for a
given external excitation the surface charge density
and, consequently, the near electric field distribution and
the far field properties in terms of absorption, scattering
and extinction cross sections.
IV. RESULTS FOR THE SINGLE PARTICLE
The electromagnetic properties for the cube particle are
evaluated using the quasi static analytical model,
boundary element method (BEM) approach [15] and are
compared to the results obtained with full-wave
numerical simulations [10].
We have assumed that the structure is excited by an
impinging plane wave as shown in Figure 2. In addition:
for the cube particle, experimental values [16] of the
complex permittivity function have been inserted;
the surrounding dielectric medium is vacuum.
Far field properties in terms of absorption and scattering
cross - section are shown in Figure 4 and Figure 5.
Figure 4: Absorption and scattering cross section spectra obtained with
the analytical model (l=50 nm).
- 27 - 15th IGTE Symposium 2012
Figure 5: Absorption and scattering cross section spectra obtained
through full-wave simulations (l=50nm).
There is a good agreement among the results obtained
with the analytical model (Figure 4) and full-wave
simulations (Figure 5).
Full-wave simulations are also compared with the
numerical results obtained with the BEM as shown in
Figure 6.
Figure 6: Comparison between extinction spectra obtained with BEM
and full-wave simulations (l=50nm).
The difference among the results shown in Figure 6 could
be associated to the different discretization of the edge of
the particle with these two approaches.
Near electric field distribution is obtained through fullwave
simulation as depicted in Figure 7.
Figure 7: Near electric field distribution for a single nanocube particle
(l=50nm). The incident electric field amplitude is 1 V/m.
In Figure 7 is clearly shown the dipolar charge repartition
according to the quasi-static approach.
V. LSPR DEVICE
To enhance the mechanism of the LSPR it is possible the
use of inter-coupling among nanoparticles. Such effect

originates from the charge induction among two or more
nanoparticles which interact stronger as they get closer to
each other [17].
To use this enhancement mechanism we propose a
structure consisting in a linear chain of gold nanocubes
deposited on a silica substrate, excited by a plane wave as
depicted in Figure 8.
Figure 8: Linear chain of gold nanocubes on silica substrate with
a=500nm, b=100nm, l=50 nm, l/8

Figure 11: Absorption and scattering cross section spectra obtained with
the full-wave simulations (d=l= 50nm).
VII. BIOSENSING APPLICATION OF THE DEVICE
By using very small inter-particle distance among the
nanoparticles it is possible to obtain high scattering and
low absorption efficiencies (Figure 12, TABLE I). These
properties are very important for biosensing applications.
In fact high absorption efficiency could heat the
biological sample invalidating medical diagnosis.
Figure 12: Absorption and scattering cross section spectra obtained with
the full-wave simulations (d=l/8= 6.25nm).
For biosensing application we suppose that the device
(grey) is in direct contact with the biological sample
under test (green) as depicted in Figure 13. The sensor
behavior is related to the effective refractive index
variation of the overall system "LSPR device - biological
compound".
Once the biological compound is placed on the device,
the system "sensor-biological compound" is illuminated
by an optical electromagnetic field (Figure 13). The
detected signal has a new frequency position and its
magnitude and amplitude width are both dependent on
the different characteristics of the biological compound.
- 29 - 15th IGTE Symposium 2012
Figure 13: The sensing system operation scheme
The biological sample used to test this device is an insilico
replica with values or Refractive Index (RI) taken
from the literature. In particular the RI values of rat
mammary adipose and tumor tissue have been considered
[18]. These data (TABLE II) were acquired using an
interferometric imaging system (Optical Coherence
Tomography - OCT technique).
TABLE II
Tissue type Refractive
index
(mean value)
Tumor 1.39
Adipose 1.467
The data show that a difference exists between the RI of a
adipose tissue and that of tumor tissue.
The electromagnetic sensor response is evaluated in terms
of extinction cross-section through full-wave simulations
[10] as depicted in Figure 14.
Figure 14: Extinction spectra for rat mammary cancer (RI=1.39) and
adipose tissue (RI=1.467).
As shown in Figure 14 the resonant peak shifts from 634
nm for a tumor tissue to 650 nm for a regular adipose
tissue. Sensitivity is evaluated as S=Δλ/Δn expressed in
nm/RIU (Refractive Index Unit). In this case sensitivity
reached 207nm/RIU.

Near electric field distribution obtained for this sensing
platform (Figure 15) is less concentrated compared to the
other one obtained for d=l (Figure 10).
Figure 15: Near electric field distribution for d=l/8= 6.25 nm. The
incident electric field amplitude is 1 V/m.
This result is in accord to the prevailing scattering
phenomenon (TABLE I).
VIII. CONCLUSION
In this paper a nanostructure device operating in the
visible regime was proposed. The device consisting in a
gold linear chain array of nanocubes, deposited on a silica
substrate. In this way a near-field enhancement is
obtained and analytical formulas to describe this
phenomenon are presented.
For the single nanoparticle good agreement among
analytical results and numerical solutions was achieved.
Exploiting electromagnetic properties of the device it was
shown that the proposed structure could be successfully
used as a biomedical sensor or as an optoelectronic
device.
[1]
REFERENCES
A. Moores and F. Goettmann, "The plasmon band in noble metal
nanoparticles: an introduction to theory and applications," New
Journal of Chemistry, vol. 30, pp. 1121-1132, 2006.
[2] E. Hutter and J.H. Fendler, "Exploitation of Localized Surface
Plasmon Resonance," Advanced Materials, vol. 16, pp. 1685-
1706, 2004.
[3] L.J. Sherry, S.-H. Chang, G.C. Schatz and R.P. Van Duyne,
"Localized Surface Plasmon Resonance Spectroscopy of Single
Silver Nanocubes," Nano Lett., vol. 5, pp. 2034–2038, 2005.
[4] G. Mie, "Contributions to the optics of turbid media, particularly
of colloidal metal solutions," Ann. Phys., vol. 25, pp. 377-445,
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[5] M. Tréguer-Delapierre, J. Majimel, S. Mornet, E. Duguet and S.
Ravaine, "Synthesis of non-spherical gold nanoparticles," Gold
Bulletin, vol. 41, pp. 195-207, 2008.
[6] W. Cai, T. Gao, H. Hong and J. Sun, “Application of gold
nanoparticles in cancer nanotechnology,” Nanotechnology,
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Science and Application, vol. 1, pp. 17-32, 2008.
K.R. Catchpole and A. Polman, “Plasmonic solar cells,” Optics
Express, vol. 16, pp. 21793-21800, 2008.
[8] D.S. Grubisha, R.J. Lipert, H.-Y. Park, J. Driskell and M.D.
Porter, "Femtomolar Detection of Prostate-Specific Antigen: An
Immunoassay Based on Surface - Enhanced Raman Scattering and
Immunogold Labels," Anal. Chem., vol. 75, pp. 5936-5943, 2003.
[9] S. Kessentini, D. Barchiesi, T. Grosges and M. Lamy de la
Chapelle, "Selective and Collaborative Optimization Methods for
Plasmonics: A Comparison," PIERS Online, vol. 7, pp. 291-295,
2011.
[10] CST Computer Simulation Technology, www.cst.com
- 30 - 15th IGTE Symposium 2012
[11] U. Hohenester and J. Krenn, "Surface plasmon resonances of
single and coupled metallic nanoparticles: A boundary integral
method approach," Phys. Rev. B, vol. 72, pp.195429, 2005.
[12] A. Sihvola, "Electromagnetic Mixing Formulas and
Applications," The Instution of Engineering and Technology -
London, 2008.
[13] L. La Spada, R. Iovine and L. Vegni, "Nanoparticle
Electromagnetic Properties for Sensing Applications," Advances
in Nanoparticles, vol. 1, pp. 9-14, 2012.
[14] F.J. Garcìa de Abajo, "Retarded field calculation of electron
energy loss in inhomogeneous dielectrics," Physical Review B,
vol. 65, pp. 115418.1-115418.17, 2002.
[15] U. Hohenester and A. Trugler, "MNPBEM- A Matlab toolbox for
the simulation of plasmonic nanoparticles," Computer Physics
Communications, vol. 183, pp. 370-381, 2012.
[16] P.B. Johnson and R.W. Christy, “Optical Constants of the Noble
Metals,” Phys. Rev. B, vol. 6, pp.4370-4379, 1972.
[17] T. Chung, S.-Y. Lee, E.Y. Song, H. Chun and B. Lee, "Plasmonic
Nanostructures for Nano-Scale Bio - Sensing," Sensors, vol. 11,
pp. 10907-10929, 2011.
[18] A.M. Zisk, E.J. Chaney and S.A. Boppart, "Refractive index of
carciogen-induced rat mammary tumours," Phys. Med. Biol., vol.
51, pp. 2165-2177, 2006.

- 31 - 15th IGTE Symposium 2012
Validation of measurements with conjugate heat
transfer models
M. Schrittwieser 1, 2 , O. Bíró 1, 2 , E. Farnleitner 3 , and G. Kastner 3
1 Institute for Fundamentals and Theory in Electrical Engineering, Inffeldgasse 18, A-8010 Graz, Austria
2 Christian Doppler Laboratory for Multiphysical Simulation, Analysis and Design of Electrical Machines,
Innfeldgasse 18 A-8010 Graz, Austria
3 Andritz Hydro GmbH, Dr. Karl- Widdmann- Strasse 5, A-8160 Weiz, Austria
E-mail: schrittwieser@tugraz.at
Abstract— The paper presents a comparison of thermal measurements on three stator duct models of an electrical machine.
These models differ from each other by the slot section components. The measurements show the advantages and
disadvantages of different variations. In order to study the measurement results in detail, a comparison with Computational
Fluid Dynamics (CFD) was conducted, where it was useful to apply the Conjugate Heat Transfer (CHT) method, because it
takes the convection and conduction into account. Therefore the conditions for the numerical heat transfer model can be
determined more realistically, especially for the temperature rise in the solid domains caused by losses.
Index Terms— Fluid Flow, Measurement, Stators, Thermal Analysis
I. INTRODUCTION
Hydro generators located in water power plants
produce electric power in the range of more than 10
MVA. The arising losses lead to a temperature rise in the
electrical machine. The temperature rise is caused by
copper, hysteresis, eddy current and mechanical losses
during the generator operating. The heat has to be
discharged to ensure the operating characteristics and this
is the purpose of the cooling scheme. For designing the
cooling of a generator, thermal and air flow networks are
mostly used. Therefore the used parameters have to be
established theoretically, by measurements or by CFD.
The temperature rise has to be handled by solving the
energy equation with the focus on heat convection and
heat conduction. The convective heat transfer coefficient
(HTC) is one of the most important parameters of these
networks and must be known accurately. Examples of
networks are presented in [1] and [2].
In the last years several investigations have been
carried out on the topic of heat transfer, especially for low
power electrical machines. Two different methods have
emerged to get information about the HTC. One uses
thermal resistances, defined with the aid of temperatures
gained by measurements [3], [4] and [5]. The other
employs CFD calculations combined with measurements
[6], [7] and [8].
The convective HTC has been calculated for large
numerical models with CFD at different parts in [9] and
[10] where the numerical effort is very high due to the
large number of nodes in the model. A special set-up of
boundary conditions has been tried to reduce the section
to be analyzed for comparable results with special
attention payed to the rotor stator interaction. Only the
fluid material properties are significant in these CFD
simulations and the temperatures have been defined at the
walls as a boundary condition from measurements. The
refinement of the mesh near the wall for calculating an
exact heat transfer is very important. An indicator of the
mesh density is the dimensionless wall distance y + which
should be about y 1
+ ≤ [11]. The primary reason for this
is that the HTC is a function of the dimensionless wall
distance [12].
The heat transfer caused by conduction has been
considered in several papers by the finite element method
(FEM) [13], [14] and [15]. The advantage of CFD over
FEM is the consideration of the actual wall heat transfer
coefficient. The disadvantage of the CFD is that the
losses cannot be considered, while FEM is capable of
this. Therefore the copper and iron losses have to be
implemented differently in CFD e.g. using the conjugate
heat transfer (CHT) method. The sources can be defined
in the solid domains and the material properties play an
important role for the CHT solution.
This paper presents a mutual validation of calorimetric
measurements and a numerical calculation. The CHT
method (fluid and solid heat transfer) is applied to a stator
duct model. The losses have been defined as sources in
the solid domains. The main objective is to evaluate the
slot geometries with different winding assemblies. All
three models have been measured at 5 flow rate points to
pinpoint their thermal characteristics.
II. MEASUREMENT
A simplified model of a stator section has been under
experimental investigation at the ANDRITZ Hydro. The
main objective of the measurements has been to find and
compare the thermal characteristics of different winding
assemblies.
Air has been used as cooling fluid for the experimental
set-up.
A. Investigated model
The laboratory model and the cooling scheme are
shown in detail in Fig. 1. The cooling fluid streams from
the Inlet through the measuring nozzle (a) to the
temperature probe (b) and from there through rectangular
channels of wood (c) into the stator duct model (d). After
heat exchange the warm air streams through wood ducts
to an outlet channel, which contains resistance thermo
elements (f). The outer surface of the model has been

insulated (e) for reduction of secondary heat flux.
Fig. 1: Calorimetric measurement and experimental set-up of the stator
laboratory model; (a) measuring nozzle, (b) Pt-100 temperature probe,
(c) wood channels; (d) stator duct model; (e) insulation of the model and
(f) resistance thermo elements
B. Measuring physical parameters
The measuring nozzle defines the volume flow rate Vin
immediately in front of the model inlet.
The fluid temperature T and density has been
measured at the inlet and outlet of the stator duct model.
These calorimetric measurement data allow calculating
the heat flux after reaching steady state. The energy
exchange occurs in the stator duct model. Therefore, it is
important to calculate also the solid temperature and fluid
temperature in the ducts. Fig. 2 shows the positions of the
temperature probes in the iron domain.
Fig. 2: Position of measurement probes in the iron; (a) 1 st stator core, (b)
2 nd stator core and (c) heating rod
The stator model consists of a section including 5 slots
in circumferential direction and 3 ventilation ducts with
distance bars between the laminated iron sheets in axial
direction. The temperature has been measured in two
stator cores. Therefore, fifteen Pt-100 resistance
thermometers with 20 mm probe length have been
positioned at each stator core.
The heat sources have been simulated with heating
rods positioned in the winding bars made of solid copper.
The source has been induced with heating rods positioned
in a hole in the middle of the copper bars, see Fig. 3. The
- 32 - 15th IGTE Symposium 2012
length of the rod has been 100 mm, with a diameter of 6
mm and a constant heat output. The upper and lower bars
have been heated up to reach steady state. The heat output
has been constant during the whole experiment.
Fig. 3: Position of temperature measurement devices for the cooling
fluid; solid temperature positions for measuring (a) copper temperature
and (b) spacer temperature; position of (c) the heating rod
Thereupon the temperatures have been measured in
each copper bar with two NiCrNi thermocouples 60 mm
in length. The temperature in the spacer has been
measured by a Pt-100.
C. Results of measurements
Table I shows the measurement data obtained for the 5
different operating points for each model under
investigation. The temperature differences have been
normalized by the fluid inlet temperature.
Model A
Model B
Model C
TABLE I
CALORIMETRIC MEASUREMENT RESULTS
Vin in Tout − TinTcopper
−Tin
m Tin
Tin
3 /s kg/m 3
T − T
iron in
Tin
0.080 1.130 0.12 1.39 0.27
0.060 1.129 0.16 1.60 0.36
0.040 1.138 0.26 1.99 0.55
0.025 1.134 0.41 2.35 0.81
0.015 1.133 0.68 2.96 1.30
0.079 1.155 0.15 1.80 0.48
0.061 1.154 0.21 2.04 0.60
0.041 1.152 0.31 2.40 0.83
0.025 1.154 0.51 2.94 1.22
0.015 1.152 0.81 3.55 1.80
0.078 1.155 0.15 1.80 0.51
0.060 1.149 0.20 2.03 0.64
0.041 1.147 0.31 2.34 0.85
0.025 1.149 0.51 2.84 1.23
0.015 1.151 0.81 3.39 1.77
III. MODEL GEOMETRIES
The measurement set-up has been implemented in
ANSYS CFX [11].
The whole numerical model is shown in Fig. 4. The
cooling scheme is the same as during the measurements
i.e. the wood channels have also been modeled. Adiabatic
walls have been defined at the top and the bottom of the

numerical model in z-direction.
In addition to this simulation, a pperiodic
boundary
condition has been defined at the surfaaces
normal to the
x-direction. The goal of this is to reduce
the section to be
analyzed (less number of nodes togeether
with smaller
elements).
Fig. 4: CHT model
Fig. 5 visualizes the numerical statoor
model in detail.
For the calculation, one slot section hass
been investigated
due to the inlet condition, which is the same for each slot
section as in the measurement. The wwinding
assemblies
are nearly the same, i.e. copper bars (d) and (e) with
insulation (f), spacer (g) between the wwinding
bars and,
for positioning in radial direction, the sllot
wedge (h). The
iron (b) and (c) is located at the top aand
bottom of the
fluid (a).
The difference in the models is thee
contact between
insulation and the iron.
Fig. 5: Numerical stator model consists of the (a) fluid in the stator duct,
(b) iron teeth, (c) iron yoke, (d) top copper bar, (e) bottom copper bar,
(f) insulation, (g) spacer between bars and (h) slott
wedge
The following models differ from eacch
other in the type
of the winding assembly. There are diffferent
options for
mounting the winding, which will be eexplained
in detail
for each model.
A. Model A
This is a model with an air gap (white) between
insulation and iron teeth, see Fig. 6. TThis
air gap has a
constant length. The cooling fluid caan
stream in axial
direction from one duct to another due tto
the air gap.
Fig. 6: Model A with air gap (white)
- 33 - 15th IGTE Symposium 2012
B. Model B
A ripple spring (white dashhed)
is positioned on one
side between the iron teeth annd
the insulation instead of
the air gap, as shown in Fig. 7. This ripple spring has had
a corrugation in diagonal direcction.
This corrugation has
been smoothed along the surfaace.
The implementation of
this has been done with a thermmal
resistance at the contact
interface. This causes an asymmmetric
energy transport and
the fluxes are higher at the sidee
without a ripple spring.
Fig. 7: Model B with ripple spring (whhite
dashed)
C. Model C
This model is similar to moodel
A, with the difference
that epoxy resin (white dotted) ) is present. This is shown
in Fig. 8. In this case the air caan
stream in axial direction
through the air gap (white), likee
in model A.
Fig. 8: Model C with epoxy resin (whitte
dotted) and air gap (white)
IV. NUMERICAAL
METHOD
The material properties havee
a significant influence on
the numerical solution. In CFDD,
the fluid properties play
the most important role and tthe
solid domains are not
taken into account. Only the coonvection
has an influence
in such calculations and the connduction
is not considered.
The conjugate heat transferr
method differs from the
conventional CFD simulation iin
the consideration of the
heat conduction in the energyy
equation. Therefore, the
thermal conductivity has to bbe
known and defined for
each medium in the CFD code [16].
A. Turbulence Model
Computational Fluid Dynammics
uses the Finite Volume
Method for solving the transporrt
equations:
∂ ρ ∂ρ
u j
+
∂t ∂x
j
= 0
(1)
∂ρu ∂ρuu
i i j ∂ p ∂ρτ
ij
+ = + + ρ fi
∂t ∂xj∂ x xi ∂x
j
(2)
∂ρet ∂ρue i t
+
∂t ∂x ∂uip
∂ ∂uiτij ∂qj
=− + + ρuf
i i + + Q(3)
∂x
∂x ∂x
j j
j j
These equations can be solveed
for laminar flows. If the
velocity and all other parametters
vary in a random and
chaotic way, the regime is calleed
turbulent [17]. For most
problems, it is unnecessary to resolve the detailed
turbulent fluctuations and it is sufficient to calculate the
time averaged properties of the flow. Therefore, the

Reynolds Averaged Navier Stokes (RANS) equations
have been used. The Reynolds Stress Tensor is another
unknown variable and further equations must be defined
for the solution to calculate the unknown parameters [18].
In the present case the Shear Stress Transport (SST)
turbulence model [19] has been used. The advantage of
the SST turbulence model is that it combines the
advantages of the k- turbulence model in the free stream
and the advantage of the k- turbulence model near the
wall [20], [21].
B. Model Configuration and Boundary Conditions
The mass flow rate and the inlet temperature have been
defined as measured, see Table I. The pressure at the
outlet has been defined as ambient pressure.
The heat output from the heating element has been
defined on a length of 100 mm in the middle of the
copper bars with a constant value gained from the
measurement.
C. Fluid Properties
The specific heat capacity cp, the dynamic viscosity
and the thermal conductivity have been defined as the
following constant values in Table II.
TABLE II
AIR IDEAL GAS MATERIAL PARAMETERS
cp
J/kgK Pa s W/mK
1004.4 1.831·10 -5 2.61·10 -5
For an ideal gas, the density is calculated with the ideal
gas equation [16]:
n⋅p ρ =
R ⋅T
0
abs
- 34 - 15th IGTE Symposium 2012
(4)
dh = cp dT
(5)
Here, n is the molecular weight, pabs is the absolute
pressure, T is the temperature, R0 is the universal gas
constant and is the density.
These material properties from Table 2 have been
adapted to implement the temperature dependence of the
streaming fluid. In this case the specific heat capacity cp
is expressed by the zero pressure polynomial [11]
cp
2 3 4
= a1+ a2T + aT 3 + a4T + aT 5 (6)
R
S
with the temperature T in Kelvin and the gas constant for
air Rs = 287.058 J/kgK and the following coefficients:
a1 = 3.57 , a2 = -4.3·10 -4 K -1 ,
a3 = -4.2·10 -8 K -2 , a4 = 3.1·10 -9 K -3 ,
a5 = -2.4·10 -12 K -4 .
The values for the viscosity are approximated by the
Sutherlands formula
nμ
μ T0+ Sμ T
= ,
(7)
μ0
T + SμT0
similarly to the conductivity
λ
λ
T + S T
+
0 λ
=
0 T SλT0 In these formulas, S and S stand for the Sutherland
constant and n and n for the appropriate exponents. For
the reference viscosity and reference conductivity the
following values has been chosen from a material
property table [22] at the reference temperature T0=325 K
which is close to the mean operating temperature of the
cooling fluid.
S=77.80 K , 0=1.97·10 -5 Pa s , n=1.57 ,
S=60.71 K , 0=2.82·10 -3 W/mK , n=1.66 .
The material properties are accurately approximated in
the temperature range from about 260 K to 670 K with
this approach. It is not recommended to use the same
parameters outside this range of temperature [22].
D. Solid Properties
The CHT method solves the following transport
equation in the solid domains:
nλ
∂ρh ∂ρu h ∂ ∂T
+ = λ+ S
∂t ∂x ∂x
∂x
t s t
j E
j j j
The important parameter in this equation is the thermal
conductivity , which has a great influence on the results
of the heat conduction and have to be known exactly.
These parameters have been defined as isotropic for the
copper, insulation, spacer, slot wedge, ripple spring and
epoxy resin and as anisotropic for the iron.
V. COMPARING NUMERICAL RESULTS WITH
MEASUREMENTS
The following figures show a comparison of
measurement data (dashed line) and CHT solution data
(solid line) for the three different parameters. The
temperature values have been normalized in the following
figures.
A. Copper temperature
The temperature difference in Fig. 9 is calculated with
an average value of the copper temperature in the top and
bottom bar and the fluid temperature at the inlet. The
deviation is due to the copper temperature because the
inlet temperature has been defined from the
measurements for the CFD calculation and is exactly the
same like in the measurement.
An average deviation has been calculated with 1.98 %
for model A, 0.95 % K for model B and 1.06 % for model
C. The diagram shows that the differences between the
models become smaller with a higher flow rate for the
measurements contrary to the calculation. The difference
of the results at the highest flow rate is calculated for
model A with 3.54 %, for model B with 1.34 % and for
model C with 1.57 %.
.
.
(8)
(9)

Normalized temperature difference
1,1
1,0
0,9
0,8
0,7
0,6
Measurement A CHT A
Measurement B CHT B
Measurement C CHT C
0,5
0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09
Volume flow rate in m 3 /s
Fig. 9: Normalized temperature difference between copper temperature
and fluid inlet temperature for the three stator duct models
The distribution of the temperature is plotted in Fig. 10
for stator duct model A and B. The surface is defined at
the middle of the first stator core (see Fig. 2) and the
temperature is shown at the whole solid domains.
Fig. 10: Temperature distribution through the middle of stator core 1
with all parts; (a) model A, (b) model B and (c) model C
The highest copper temperatures are found in model A
(a). The asymmetric temperature in model B (b) is also
- 35 - 15th IGTE Symposium 2012
recognizable in the iron; the temperature in the iron is
higher on the opposite side of the ripple spring (bottom
side) caused by the higher heat flux. The epoxy resin in
model C (c) contributes a lower temperature in the
insulation than along the air gap (see detailed in Fig. 10
c). This will have a positive effect on the properties of the
insulation during the aging.
B. Iron temperature
The iron temperature has been calculated as an average
value of all measuring points (Fig. 2). The difference to
the fluid inlet temperature has been plotted as before. The
average deviation is 13.20 % for model A, 6.04 % for
model B and 3.52 % for model C. It is worth noting that
model A has the highest deviation for the iron
temperature, see Fig. 11. The deviation for each winding
assembly decreases with a higher volume flow rate.
Normalized temperature difference
1,2
1,1
1,0
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
Measurement A CHT A
Measurement B CHT B
Measurement C CHT C
0,0
0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09
Volume flow rate in m
Fig. 11: Temperature difference between iron temperature and fluid
inlet temperature for the three stator duct models
3 /s
Normalized temperature difference
C. Fluid temperatures
The fluid temperature rise is shown in Fig. 12.
1,2
1,1
1,0
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
Measurement A CHT A
Measurement B CHT B
Measurement C CHT C
0,0
0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09
Volume flow rate in m
Fig. 12: Difference between fluid inlet and outlet temperature
3 /s

The temperature difference has been calculated with the
inlet and outlet temperature of the air. The difference
decreases with the volume flow rate. The average value
of the difference is 0.97 % for model A, 1.09 % for model
B and 0.99 % for model C. The highest deviation at the
lowest flow rate is about 2.84 % for model A, 3.47 % for
model B and 3.10 % for model C.
VI. CONCLUSION
The paper has described the conjugate heat transfer
method for a stator model example. The advantage of
using CHT is that the heat transfer coefficient is
inherently solved and needs not be defined as constant at
the surfaces.
The comparison of the numerical solution shows a
good agreement with measurements for each stator duct
model. The average deviation of the temperature
difference between copper and fluid inlet temperature has
been less than about 1.4 % for all models. The
temperature difference has been calculated between the
iron and fluid inlet temperature. Therefore, the average
deviation is under 8 %. The heating up of the air has been
calculated with a difference less than 1.5 % and at the
lowest operating point the difference reaches the maximal
deviation of 3.1 % and the slightest deviation with the
highest flow rate of about 0.2 %. This can be explained
by the fact that steady state is reached faster for lower
flow rates than for higher ones. Based on these results,
the conclusion can be drawn that the best agreement is
obtained for model C and the worst for model A. These
investigations provide a determination of proper model
conditions in the slot region, which can be used for
further CHT researches.
VII. ACKNOWLEDGMENT
This work has been supported by the Christian Doppler
Laboratory for Multiphysical Simulation, Analysis and
Design of Electrical Machines (MuSicEl) and ANDRITZ
Hydro GmbH.
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- 37 - 15th IGTE Symposium 2012
Computing the shielding effectiveness of waveguides using FE-mesh
truncation by surface operator implementation
C. Tuerk∗ , W. Renhart † , and C. Magele †
∗Armament and Defence Technology Agency, Ministry of Defence and Sports, Rossauer Laende 1, A-1090 Vienna
† Institute for Fundamentals and Theory in Electrical Engineering, Inffeldgasse 18, A-8010 Graz
E-mail: christian.tuerk@bmlvs.gv.at
Abstract—A plane wave incident perpendicular to one open end of a conductive tube, as part of a honeycomb-structure, is
attenuated on its way through it. In order to calculate its total attenuation for various frequencies the FE-method will be
used. This requires a reflectionless truncation of the FE-mesh for which a Surface Operator Boundary Condition (SOBC)
will be employed. In order to show the accuracy and applicability of the FEM with SOBC, the results will be compared
to entirely analytical solutions as well as to easy-to-use engineering formulae.
Index Terms—Finite Element Method (FEM), Shielding, Surface Operator Boundary Condition (SOBC), Waveguide
I. INTRODUCTION
Previous works i.e. [1] have shown the implementation
of a surface operator boundary condition derived from
an analytical model into the FE-mesh. Honeycombs can
be considered waveguides-beyond-cutoff (WBC) and are
therefore employed as vents for large shielded enclosures,
like shielded rooms, while maintaining a certain degree
of attenuation of a plane wave incident on it.
The resulting attenuation imposed by a single conductive
tube will be calculated under different ratios of
length-to-diameter of the tube and at selected frequencies.
Existing literature like [2] provide engineering rules for
designing waveguides-beyond-cutoff (WBC) as a shielding
component whereas others like [3] present analytical
details on the physics of the transmission of electromagnetic
power in waveguides of various cross-sections. The
results, obtained through numerical computation in the
frequency range of 3GHz to 18GHz for a practical
design of a real life waveguide are then compared to
both approaches and subsequently discussed.
II. MODELLING
A. Surface Operator Boundary Conditions (SOBC)
Fig. 1 shows the setup used for the computation of the
plane wave field strength incident on the tube and exiting
it. The right-hand-side boundary is modelled by means
of SOBCs (Γtr) matching the impedance of free space
between the tube and the termination of the problem area.
A plane wave travelling along the x-axis will experience
a certain degree of attenuation by the waveguide as long
as the waveguide-beyond-cutoff (WBC) condition is met.
A fraction of the initial power of the wave penetrates the
waveguide and is terminated reflectionless at Γtr.
Based on results obtained through [5] and [4] the implementation
of this surface operator boundary condition
for the truncation surface Γtr of the FE-mesh can be
directly derived from the Maxwell equations
∇× E = −jωμ H, ∇× H = jωɛ E. (1)
After splitting the field vectors E and H as well as the
∇-operator into their normal and orthogonal tangential
Fig. 1. Modelling the Waveguide
components as in 2
E = Et + nEn, H = Ht + nHn, ∇ = ∇t + ∂
n (2)
∂n
the Maxwell Equations can be reformulated as follows:
n Hn = − 1
jωμ ∇t × Et
(3)
n En = 1
jωɛ ∇t × Ht. (4)
With these relations the normal components of the field
components En and Hn can be eliminated in equation 1.
A couple of mathematical operations finally yield
∂(n × Et)
= −jωμ
∂n
Ht − 1
jωɛ∇t × (∇t × Ht) (5)
∂(n × Ht)
= jωɛ
∂n
Et + 1
jωμ∇t × (∇t × Et). (6)
These equations are commonly valid, consequently on the
truncation surface (see fig. 1) too. On Γtr the situation
is as shown in fig. 2 in a local coordinate system.
The propagation of the wave can be represented by the
wave vector k. Due to the knowledge of the angle of
incidence on Γtr it can be decomposed into its normal
and tangential components as given in the following set
of equations:
k = kt
+
β, β = ± k2 − kt 2 , k = ω √ μɛ. (7)

Fig. 2. Wave at any point on Γtr
The surface normal n is represented by the local coordinate
ζ. In order to get rid of the ∂
∂n term on the left-handside
of equation 5 and equation 6 an integration along ζ
over the half-space must be performed. Assuming a lossy
media, the field components must decay to zero at infinity
which allows for
∞
Ht0e
ζ=0
−jβζ dζ = 1
jβ Ht0
(8)
∞
Et0e −jβζ dζ = 1
jβ Et0.
ζ=0
Ht0 and
Et0 are the tangential field vectors at ζ =0.
Together with equation 7, relations 5 and 6 can now be
rewritten as
n ×
Et0 =
n ×
Ht0 =
−ωμHt0
k2 − kt 2 + ∇t × (∇t × Ht0)
ωɛ k2 − kt 2
(9)
ωɛEt0
k2 − kt 2 − ∇t × (∇t × Et0)
ωμ k2 − kt 2
. (10)
Transverse components of the outgoing wave may be
transformed into the Fourier domain, only to see, that its
tangential derivatives can be expressed as ∇t = −j kt.
Substitution in equation 9 and 10 leads to
n ×
Et0 =
n ×
Ht0 =
−ωμHt0
k2 − kt 2 − kt × ( kt × Ht0)
ωɛ k2 − kt 2
(11)
ωɛEt0
k2 − kt 2 + kt × ( kt × Et0)
ωμ k2 .
2
− kt
(12)
These relations between the tangential components of
Et0
and Ht0 can now be used to model the so called
surface operator boundary conditions (SOBC) on Γtr.
Equations 11 and 12 allow for any angle of incidence of
the plane wave on a truncating surface Γtr. Since only
perpendicular incidence on the waveguide and on Γtr are
considered, the use of a first-order SOBC is reasonable
- kt =0.
Application of the Galerkin method to the well-known
A, v-formulation makes use of the n × Ht on the Neu-
- 38 - 15th IGTE Symposium 2012
mann Boundary (see [6]).
−
+
Ω
+
ΓH
Ω
∇× Ni · 1
μ ∇× AdΩ
Ni · (n × ( 1
μ ∇× A)) dΓ
n× H
Ni · (σ + jωɛ)jω( A −∇v)dΩ =0. (13)
On the Neumann boundary (ΓH) the underbraced term
in equation 13 is substituted by the Fourier transformed
integral of equation 6 which prescribes the truncation of
the FE-mesh directly.
B. Surface Impedance Boundary Conditions (SIBC)
An increased incident angle results always in a larger
wave vector kt and obviously the curl curl-terms in
equations 11 and 12 become more and more relevance
to achieve accurate boundary conditions. If the wave
propagates perpendicularly to Γtr, the vector kt equals
zero. This is the considered case for all results presented
herein. Hence the second term on the right-hand-side in
equations 11 and 12 equal zero and first order SIBCs
remain:
n ×
−ωμHt0 μ
Et0 = = −
k
ɛ Ht0 = −Z0 Ht0 (14)
n ×
ωɛ
Et0 ɛ
Ht0 = = −
k μ Et0 = 1
Et0
(15)
Z0
The impedance of the mesh-terminating plane Γtr can
now be directly prescribed.
III. SETUP
Fig. 1 shows the setup used for the computation of the
plane wave field strength incident on the tube and exiting
it. The right-hand-side of the problem area is terminated
by means of the introduced SOBC. A plane wave originating
from the stimulus plane penetrates the tube. Only
a fraction of the incident power ”leaks” through it, since
at the frequencies considered it represents a waveguidebeyond-cutoff
(WBC). This small fraction of the incident
wave is terminated reflectionless at Γtr. The detail of the
aluminium tube with a square cross-section and lengths
ranging from 20mm ... 80mm is shown in fig. 3.
The grid shown in figure 3 represents the macro
elements used for modelling only.
IV. RESULTS
A. Finite Element Method with SOBC
Since frequencies above 1GHz are of interest, simulations
at distinct frequencies in the range of 3 ... 18GHz at
a stepwidth of 3GHz are considered. At each frequency
the length of the tube is stepped through by 10mm in
the range between 20mm and 80mm. The cross-section
of the waveguide is kept constant. Fig. 4 shows the
resulting attenuation of a plane wave on its way through

Fig. 3. Details of the waveguide-beyond-cutoff
the WBC. At 15GHz the attenuation of the incident
wave starts to approach zero and the tube becomes a
waveguide as known from RF-applications and has also
been described in [3]. As long as the frequencies are
Fig. 4. Attenuation of a plane wave at distinct lengths and frequencies
below the cutoff-frequency, the attenuation does not only
depend on the ratio between f, the frequency used, and
the cutoff-frequency fc of the structure, but also depends
on the length of the tube. The relationship is non-linear
and therefore clearly contrasting the engineering rulesof-thumb
as provided in the following section.
The following figure (Fig. 5) shows the computation
of the field strengths on either side of the waveguidebeyond-cutoff.
It is operated at 9GHz and the righthand-side
is terminated by means of the SOBC described
before. The colors in the figure refer to the absolute value
of the field strengths of the electrical component of the
plane wave at a particular moment. Due to the necessity
of a fine mesh for the computation of fields along the
waveguide (coloured grey), no field strengths are visible.
Following the general formula of the power density of a
plane wave
S = 1
2 Re( E × H∗ ) (16)
and the impedance of free space of
Z0 =
μ0
ɛ0
- 39 - 15th IGTE Symposium 2012
Fig. 5. A waveguide 30mm in length, operated at 9GHz
the attenuation of the power through the waveguide can
be calculated. With
at =20lg |Emaxin|
|Emaxout|
(17)
the degree of the attenuation (at)[dB] can be determined
based on the field strength of the electrical component
of the plane wave on the left-hand-side of the tube
(|Emaxin|) and on the right-hand-side (|Emaxout|). The
maxima of the respective field strengths are taken from
a line parallel to the x-axis along the centre of the tube.
B. Engineering Rules
For applications using frequencies below approximately
1GHz [2] proposes the use of simple ”design
rules”:
fc = 150
b , fc[GHz], diameter[mm] (18)
at = 27.3
b l, at[dB],
b =
diameter, length[mm] (19)
√ 2a, forsquare cross − section[mm]
f ≤ fc
10 , usablefrequencyf (20)
with fc being the cutoff-frequency in [GHz], at representing
the shielding effectiveness in [dB] and any
dimension given in [mm]. Formulae 18 to 20 show that
the cutoff-frequency only depends on the diameter of the
tube which is, to some degree, in accordance with [3].
It has to be distinguished whether a square, rectangular
or circular waveguide is used. As for the rectangular
cross-sections [3] reads that the larger dimension governs
the cutoff-frequency fc. For circular shapes the diameter
counts. One may also have noticed that the engineering
rules do not account for any matter in the waveguide
but free space. Since the WBC is used as a vent with
shielding properties its cutoff-wavelength follows
λc = c0
fc
≈ 2a. (21)
This is in line with [3] and equation 22 if ɛ = ɛ0 and
μ = μ0. Waveguides filled with dielectric matter for
transmission properties are beyond the scope of this work
since they are neither useful as vents nor as a shielding
component.

As long as the frequency of interest is below the
highest usable frequency as given in equation 20 the
tube yields an attenuation according to equation 19.
Application of this set of formulae to the waveguide
under consideration at 12GHz provides the following
graph (fig. 6):
Fig. 6. Engineering rules applied at 12GHz
Figure 6 shows the application of the engineering rules
at the cutoff-frequency fc = 12GHz. The calculation
of the shielding effectiveness with the engineering rules
(blue dashed line) naturally exceed the limits obtained
by means of the numerical value since equation 20 has
not been considered so far. This equation is obviously a
very rough estimate of the maximum usable frequency.
It requires this waveguide not to be used above 1.2GHz.
This is very conservative, since the green solid line
(the uppermost line) shows the course of the shielding
effectiveness at 9GHz of this particular waveguide. The
engineering rules yield similar results, but on the safe
side. Since it is not clear which limit in terms of shielding
effectiveness underlies this set of easy-to-use engineering
rules, one has to be very careful with its application. Even
if it was possible to adjust equation 20 to this result, the
behaviour of a waveguide may render this unreliable due
to its nonlinear attenutation of a plane wave as fig. 4
clearly shows.
C. Analytical Approach
When considering a waveguide-beyond-cutoff (WBC)
for shielding purposes, the lowest mode of a TE or TMwave
propagating through it is of interest. It represents
the cutoff-frequency fc. For waveguides with a square
cross-section [3] reads for TE10-mode
fc = 1
2 √ 1
(22)
ɛμ a
with a being the length of the edge of the square. For a
waveguide as used for this work, fc =14.99GHz which
matches the result shown in figure 4. With increasing
frequencies the attenuation of the plane waves vanishes
above approximately 15GHz regardless of the length of
it. In other words, illuminating this particular waveguide
at frequencies ≥ 15GHz will render it useless as a shield.
- 40 - 15th IGTE Symposium 2012
Since waveguides are generally used for transmission
of electromagnetic energy there are, apart from the engineering
rules above, no analytical formulations available
to determine the attenuation of a plane wave penetrating a
waveguide below its cutoff-frequency - there is no distinct
mode of energy flow in the waveguide. For the same
reason there are no analytical formulations known for
plane waves penetrating a waveguide at other angles than
perpendicular to the cross-section of it (see section V).
V. CONCLUSION
This paper shows how Surface Operator Boundary
Conditions (SOBC) can be implemented in an A − v
formulation to be used with the Galerkin method. The
SOBC are used to model a Neumann Boundary Condition
which allows for reflectionless termination of a problem
area. The use of the SOBC allows for a significant
speed-up of the computation of the problem because the
absorbing boundary is only a single term which does not
require additional finite elements to be modelled. For the
construction of vents in a shielded room, waveguides below
their cutoff-frequencies are employed. The described
model has been used for the computation of the shielding
effectiveness of waveguides at frequencies exceeding
1GHz and compared and contrasted to an analytical
approach and a set of easy-to-use engineering rules. It
can now clearly be shown, that well known and verified
analytical solutions can be met by numerical models
as far as the cutoff-frequency of square waveguides is
concerned. By the same token, it can be shown that
simple design rules are very conservative i.e. delivering
smaller numbers of shielding attenuation than actually
can be yielded in real designs. It can not be said, that this
set of easy-to-use rules are valid only below ≈ 1GHz.
So far, only plane waves incident perpendicular to
an open end of the waveguide have been modelled and
computed. Future efforts will be put on different angles
of incidence. There exist hints, that stacked arrays of
waveguides (honeycomb structures) suffer a deterioration
of total shielding effectiveness compared to the attenuation
provided by a single tube. This behaviour may also
be investigated in the future.
REFERENCES
[1] W. Renhart, C. Magele and C. Tuerk, ”Thin Layer Transition
Matrix Description Applied to the Finite Element Method”,IEEE
Trans on Magn., Vol. 45, No. 3, 2009, pp. 1638- 1641.
[2] Louis T. Gnecco, ”The Design of Shielded Enclosures”, Newnes
Press, ISBN 0-7506-7270-6
[3] Karoly Simonyi, ”Theoretische Elektrotechnik”, 6. Auflage, VEB
Deutscher Verlag der Wissenschaften, Berlin 1977
[4] W. Renhart, C. Magele and C. Tuerk, ”Improved FE-mesh truncation
by surface operator implementation to speed up antenna
design” (unpublished).
[5] Sergei Tretyakov, Analytical Modeling in Applied Electromagnetics,
1st ed. Artech House, chapters 2, 3, 2003.
[6] O. Biro, ”Edge element formulations of eddy current problems”,
Computer methods in applied mechanics and engineering, vol. 169,
pp. 391-405, 1999.

- 41 - 15th IGTE Symposium 2012
Heat Transfer Analysis on End Windings of a Hydro
Generator using a Stator-Slot-Sector Model
1, 2 Stephan Klomberg, 3 Ernst Farnleitner, 3 Gebhard Kastner, 1, 2 Oszkár Bíró
1 Christian Doppler Laboratory for Multiphysical Simulation, Analysis and Design of Electrical Machines,
Inffeldgasse 18, A-8010 Graz, Austria
2 Institute for Fundamentals and Theory in Electrical Engineering, Inffeldgasse 18, A-8010 Graz, Austria
3 Andritz Hydro GmbH, Dr.-Karl-Widdmann-Strasse 5, A-8160 Weiz, Austria
E-mail: stephan.klomberg@tugraz.at
Abstract — An accurate evaluation of the convective heat transfer coefficient on end windings needs usually large numerical
models. These calculations involve an enormous amount of time and are not feasible for finding correlations between the
convective heat transfer coefficient, massflow, rotational speed and geometry. On the basis of a parameter study this paper
shows that a simplified stator-slot-sector model is equally accurate as a pole-sector-model but more practicable and faster.
Index Terms—Cooling, Electric machines, Fluid dynamics,
Heat transfer.
I. INTRODUCTION
Large hydro generators are working with a high
efficiency nevertheless the losses of these machines can
reach up to several MW’s. These heat losses must be
dissipated from the generator. Designing the cooling of a
generator is nowadays well-engineered. The use of flow
and thermal networks in this subject is state of the art [1].
Flow networks decompose the complex geometry in
discrete network elements to calculate the air flow
through a machine. They include pressure generating
elements (active) like fans and poles or other rotating
components and passive elements like ducts, inlets or
outlets. The fundamentals of these components are
determined theoretically, by measurements or
computational fluid dynamics (CFD). The computation of
the temperature rise in the active parts is handled with
thermal networks where the convective heat transfer and
heat conduction coefficients and a reference air
temperature are major parameters. Examples about flow
and thermal networks are found in [1] and [2].
Strictly speaking, the most important factor, the
convective heat transfer coefficient, has to be calculated
with large numerical generator models. Analyzing these
models needs much time and is inappropriate for
parameter studies. The need of coefficients for the
networks makes the consideration of an equivalent
smaller model enabling a faster calculation rational.
The standard equation for the convective wall heat
transfer coefficient is
q W
=
(1)
TW- Tref According to [3] this equations is applicable for forced
convection if q W is the wall heat flux density, TWall the
wall temperature and Tref a reference temperature in a
properly control surface in the calculating volume.
In the last years, several investigations have been
carried out on the topic of heat transfer on end windings
of electrical machines especially for totally enclosed fan
cooled induction motors. The most development has been
on smaller machines in a power class of a few kVA. One
method obtains the heat transfer coefficients by
measuring temperatures and implements these
temperatures in thermal resistances [4], [5]. A second
kind of approach involves CFD calculations combined
with measurements [6], [7]. The end winding heat
transfer of large hydro generators have not yet been
investigated.
The large hydro generator presented in this paper is air
ventilated by a motor-driven fan in radial direction. A
longitudinal section of the investigated generator is
shown in Figure 1. The fluid enters the machine at the
inlet (a) without a spin, flows through the end winding
bars (d, e) in the pole area (f) and through the stator ducts
(h) to the outlet (j).
b
c
d
e
axis of rotation
a j
Figure 1: Profile of the investigated hydro generator
showing the (a) inlet, (b) bearing support, (c) support
ring, (d) bottom bar, (e) top bar, (f) salient pole, (g) airgap,
(h) stator ducts, (i) outlet area, (j) outlet, (k) shaft
The purpose of this paper is to develop a so called slotsector
model which is smaller and faster to calculate than
k
h
i
f
g

a standard model with all components and an enormous
number of elements. The slot-sector model should be
investigated and optimized for calculating an accurate
heat transfer coefficient.
II. NUMERICAL SIMULATION OF THE HEAT FLUX
Turbulence models are one of the most important parts
in numerical fluid simulation. Therefore the two most
commonly used models, the standard k- model by Jones
and Launder [8] and the shear stress transport model by
Menter [9] have been compared.
The fundamental equation to calculate the heat flux at
the wall is
W= cpu *
T + (TW -T) (2)
where is the density and cp is the specific heat capacity.
It should be pointed out that the two turbulence models
apply different approaches for the dimensionless near
wall velocity u * and the dimensionless temperature at the
wall T + . Vieser et al tested and explained these heat
transfer predictions in [10] for different test cases.
A strong impact on the wall treatment has the density
of the used mesh near walls described by the
dimensionless distance from the wall
y + = u y
(3)
This parameter depends especially on the height of the
first element adjacent to the wall y. The other
parameters are the friction velocity u and the kinematic
viscosity . The smaller y is chosen, the more accurate
the calculated heat transfer coefficient becomes. A
parameter study in section IV will show this correlation.
A short overview of the influence of y + on the
convective heat transfer coefficient is
T + =fy +
=fW W=fT + , u * y + =f(y)
u * =fy + (4)
The commercial software package ANSYS-CFX-13
[11] has been used for the numerical simulations
described in this paper. There are two main calculation
methods for a rotor stator simulation, the transient and the
steady-state approach. A transient calculation needs large
computing resources and takes a long time. Therefore the
steady-state method has been chosen. There are two
variants, the stage method and the frozen rotor method.
These steady-state approaches are only approximations
because the transient terms in the flow equations are
neglected. Nevertheless, their balance of computational
efficiency and accuracy is ideal for parameter studies
with many working points. They differ in the treatment of
the interface between two components. The stage method
averages the fluxes in circumferential direction on bands
and transmits these fluxes to the downstream component.
- 42 - 15th IGTE Symposium 2012
Only one passage per component has to be modeled, and
furthermore, it can be used for large pitch ratios which
highly reduce the number of elements.
The frozen approach works with a frame change at the
interface without averaging the fluxes. Therefore, it needs
to model more passages per component. The conservation
equations for the rotor are solved in a rotating system, the
equations of the stator in a static frame of reference. The
consistency of speed and pressure is combined at the
interface. These relations are illustrated in Figure 2 and
explained in detail in [11].
Figure 2: Steady-state methods: stage and frozen rotor
R1/ R2 and S1/ S2 are rotational periodicities; pR/ pS are
pitch ratios
The standard setting in ANSYS-CFX-13 for ideal gas
is temperature independent, i.e. the thermal conductivity
, the specific heat capacity cp and the dynamic viscosity
are constant. This is a simplification of reality which
may make the results more inaccurate. An ideal gas with
temperature dependence has been modeled as a
consequence, and compared to a temperature independent
ideal gas.
The dynamic viscosity (5) and the thermal conductivity
(6) have been modeled with the Sutherland’s formula
[11]. The reference temperature Tref has been set to 325
K. The reference molecular viscosity o, the reference
molecular conductivity o, the Sutherland constants S/ S
and the temperature exponents n/ n are listed below for
both equations.
0
0
S = 77.8 K
0 = 1.972 10 -5 Pa s
n = 1.574
= Tref + S
T + S T
n
Tref (5)
= Tref + S
T + S T
n (6)
Tref S = 60.7 K
0 = 2.82 10 -2 W / m K
n = 1.676
As illustrated in equation (7), the specific heat capacity
has been calculated with the zero pressure polynomial
[11].
cp RS = t 1 +t 2 T+t 3 T 2 +t 4T 3 +t 5T 4 (7)

RS = 287.058 J / kg K
t1 = 3.574
t2 = -4.2691 10 -4
t3 = -4.1854 10 -8
t4 = 3.0986 10 -9
t5 = -2.3848 10 -12
All physical values have been found by automatically
adjusting them to measured thermodynamic properties of
dry air gathered in [12]. These values are valid in a
temperature range from 260 K to 670 K.
III. EXPLANATION OF THE 3D MODEL
This chapter shows the structure of the reference model
called pole-sector model (PSM). Four different slot-sector
models (SSM) will be explained, too.
The reference model has been reduced to one pole
sector of the whole circumference of the generator. A
rotational periodic condition is given at both
circumferential sides. A symmetry condition in axial
direction further reduces the numerical effort. The model
is shown in Figure 3. It is simulated with the frozen rotor
approach and the number of elements is about 30 million.
The calculation time of the PSM is longer than a week
because of this large amount of elements. Nevertheless,
the mesh of this model is rather coarse over the whole
volume. This fact is especially important near the wall of
the end windings.
c
d
b
a
Figure 3: Pole-sector model: (a) inlet, (b) bearing support,
(c) support ring, (d) bottom bar, (e) top bar, (f) salient
pole, (g) air-gap, (h) stator ducts, (i) outlet area, (j) outlet
Measuring temperatures at walls in a large hydro
generator demands a high effort and a long preparation
time. The measuring sensors must be fixed during the
construction of the components, which makes such
investigations complicated and expensive. Not least due
to these facts, the calculated wall heat transfer
coefficients (WHTC) of the PSM have been taken as
e
j
f
i
h
g
- 43 - 15th IGTE Symposium 2012
reference values. By means of simulating several working
points with different volume flow rates and rotational
speeds, a wide scope has been covered. The volume flow
rate is set as the inlet boundary condition and the static
pressure as the outlet condition. All walls, especially the
end winding walls, of the model have a fixed
temperature. Conduction in solids is not considered.
The aim of the investigations is developing a
simplified model with acceptable computational demands
for a numerical parameter study. The best fitting
computational approach for this issue is the stage model.
The question is how much can the PSM be reduced by
maintaining similar accuracy. To clarify this, four
different simplifications have been modeled.
The first idea was to reduce the model as much as
possible. In order to achieve this, the whole generator has
been reduced to a circumferential section of one slot.
Furthermore, the rotor, the stator ducts and the outlet area
are not considered. The interface between the rotor
domain and the inlet domain as well as the air gap serves
as a simplified outlet. Due to this, it is difficult to find an
appropriate boundary condition at the simplified outlet.
Only one end winding bar is considered and a diffuser
has been put in front of the inlet to get a radial inflow
onto the end winding area. The number of elements is
only about 0.6 million due to all these reductions. This
slot-sector model is named SSM_1.
The next step was extending the model SSM_1 with
the rotor domain to get the second model (SSM_2). The
outlet is moved to the symmetry plane of the rotor. The
interface between the rotor and the stator ducts is
assumed to be a fixed wall. The number of elements is
less than 1 million.
The third model (SSM_3) is enhanced with the stator
ducts and the outlet area. These parts have also a
circumferential extension of one slot only. Because of
this, the same boundary conditions as in the PSM are
possible. The number of elements rises to 1.5 million.
The last remaining problem has been the inflow. A slot
section of the inlet domain doesn’t allow a three
dimensional spreading of the flow onto the end windings.
The consideration of the entire inlet area leads to the last
simplified slot-sector model called SSM_4. The final
model includes all components, but it contains only one
slot with a pitch ratio of 22.5, i.e. one end winding bar
and its surrounding stator ducts are modeled. This model
has the best features for using the steady-state approach
stage and a circumferential averaging of the WHTC is
expected to be appropriate.
The number of elements has been reduced to 2 million.
On the one hand, only one slot section has been modeled,
and on the other hand, the rotor and the inlet domains
have been geometrically simplified and meshed coarser
than the components of the PSM. The meshes of the end
winding bars of the PSM and of all four SSM have the
same structure and mesh density.
An accurate prediction of the heat transfer coefficient
is possible with a fine near wall mesh only. By means of
a parameter study, the influence of the mesh density on
the WHTC has been investigated. The end windings’
domain is meshed starting from an extremely coarse grid

to a very fine one. These various meshes are illustrated in
Table 1.
TABLE I
DIFFERENT MESHES OF THE END WINDING BAR
y 1.element number of
[mm] elements
5,00 45.000
3,00 65.000
2,00 81.000
1,00 146.000
0,50 318.000
0,25 693.000
0,12 989.000
0,06 1.492.000
0,05 1.682.000
The focus has been on a defined height of the first
element at the walls. The rest of the volume is
automatically meshed with a defined ratio of growth and
a Poisson distribution normal to the wall [13].
IV. RESULTS
The evaluation has been carried out by calculating the
WHTC at the end windings as defined in (1). Further
results are normalized to the reference values for a better
overview. The end winding bar is split into 5 zones to get
the variation of the WHTC along the bar. Figure 4 shows
the zones, beginning with T1 and T2 on the top bar. TB3
is the junction from the top to the bottom bar and it is
followed by B2 and B1.
Figure 4: End winding bar with 5 zones
Figure 5 shows the comparison of the WHTCs
obtained by the four slot-sector models along an end
winding bar. As a criterion for an acceptable agreement
between the PSM and the SSMs, a ratio PSM / SSM in
the range of 0.8 - 1.2 has been chosen. The first 3 slotsector
models cannot fulfill this target, especially the area
TB3 at the end of the bar is too inaccurate. The version
SSM_4 is just in the range, except in zone T1. This area
- 44 - 15th IGTE Symposium 2012
of the top bar is located at the beginning of the air gap
and the rotor and is highly influenced by the motion of
the rotor. This can be also seen in Figure 6, detail x and y,
where the velocity is very high.
PSM / SSM
2,0
1,8
1,6
1,4
1,2
1,0
0,8
0,6
0,4
0,2
SSM_1 SSM_2
SSM_3 SSM_4
0,0 T1 T2 TB3 B2 B1
Figure 5: Comparison of the four slot-sector models
Figure 6 shows the comparison of the velocity contours
in the symmetry plane of the inlet for the models SSM_4
and SSM_3. The contours of SSM_1 and SSM_2 are
similar to SSM_3. These pictures essentially show that
the inflow velocity is higher if the whole inlet is used for
the calculations. Hence the ratios of the slot-sector
models 1-3 in the zones TB3, B2 and B1 in Figure 5 are
out of the range of 0.8 - 1.2.
x y
Figure 6: Velocity in the symmetry plane of the inlet
domain of a) SSM_4 and b) SSM_3
The interaction between the rotor and the end windings is
illustrated in Figure 7. The turbulent kinetic energy at the
interfaces between the inlet and the rotor as well as
between the top bar and the inlet is shown. There are
vortices with high energy at the PSM contour seen in
Figure 7a. The computation with the model SSM_4

generates the well known circumferential bands (see
Figure 7b) characteristic of the stage method. In other
words, by averaging the physical values on
circumferential bands it is not possible to calculate local
vortices. Therefore the use of a slot-sector model
underestimates the rotor stator interaction.
Inlet – Top bar
Inlet – Top bar
Inlet - Rotor
Inlet - Rotor
Figure 7: Turbulent kinetic energy on selected interfaces
in a) PSM, b) SSM_4
The graph in Figure 8 shows the heat transfer
coefficient in dependence on the dimensionless distance
from the wall at the top bar. The SST and the k-
turbulence models have been used for this study. The
WHTC increases with decreasing dimensionless distance
from the wall. The coefficient reaches its peak at about y +
= 5 and fluctuates around the maximum value. The factor
y + is very sensitive to varying the near wall velocity due
to a different volume flow rate or rotational speed with
the same mesh and geometry. This mesh refinement study
confirms the investigations of [14].
- 45 - 15th IGTE Symposium 2012
y=min
1,1
1,0
0,9
0,8
0,7
0,6
0 1
y
10 100
+ [-]
Figure 8: Mesh refinement study in the zones T1, T2,
TB3 with the k- and the SST turbulence model
Depending on the previous investigations, a parameter
study with various working points has been carried out.
The slot-sector model SSM_4 has been used with
different operating conditions but the SST turbulence
model has always been applied. The results have been
averaged and a standard deviation has been calculated.
Figure 9 shows the results obtained.
PSM / SSM
1,4
1,2
1,0
0,8
0,6
0,4
0,2
T1 SST T1 k-e
T2 SST T2 k-e
TB3 SST TB3 k-e
averaged ratio (y+ < 30, ideal gas temperature independent)
averaged ratio (y+ < 1, ideal gas temperature independent)
averaged ratio (y+ < 1, ideal gas temperature dependent)
standard deviation
0,0 T1 T2 TB3 B2 B1
Figure 9: Normalized WHTC in dependence of y + and the
type of ideal gas
First, the SSM has been calculated with the same mesh
density near the end winding walls as the PSM. The
results are located in the given range of 0.8 – 1.2. The
second simulation run has been done with the finest mesh
of the end winding domain. The curve is decreasing
nearly parallel to the first one. The reference model has
been only simulated with a coarse mesh near the end
winding bars, hence an estimation of the accuracy is not
possible. These two curves have been calculated with a
temperature independent ideal gas. Therefore, the last
curve has been simulated with an ideal gas with
temperature dependency. The ratio of the heat transfer
coefficients increases about 5% with temperature
independence assumed.
Regarding these findings, it can be stated that a slotsector
model with a very fine mesh near walls and an
adjusted ideal gas leads to sufficiently accurate results.

V. CONCLUSION
The comparison of a pole-sector model with various
slot-sector models shows that a simplification with less
numerical effort is possible. An extreme reduction of the
generator is not recommended, because all components
have to be considered. Due to modeling one slot only, the
stage approach is an adequate and fast calculating method
for this kind of model structure. The averaged deviation
of the wall heat transfer coefficient from the reference
values is about 12%. Possible improvements of the slotsector
model are the use of an adjusted ideal gas and a
fine mesh near walls. Furthermore, the influence of the
dimensionless distance from the wall has been confirmed.
ACKNOWLEDGMENT
This work has been supported by the Christian
Doppler Research Association (CDG) and by the Andritz
Hydro GmbH.
- 46 - 15th IGTE Symposium 2012
REFERENCES
[1] E. Farnleitner and G. Kastner, "Moderne Methoden der
Ventilationsauslegung von Pumpspeichergeneratoren," e&i, vol.
127, pp. 24-29, 2010.
[2] G. Traxler-Samek, R. Zickermann and A. Schwery, "Cooling
airflow, losses, and temperatures in large air-cooled synchronous
machines," IEEE Transactions on Industrial Electronics, vol. 57,
no. 1, pp. 172-180, Jan. 2010.
[3] H. Herwig, "Kritische Anmerkungen zu einem weitverbreiteten
Konzept: der Wärmeübergangskoeffizient a," Forschung im
Ingenieurwesen, vol. 63, pp. 13-17, 1997.
[4] A. Boglietti and A. Cavagnino, "Analysis of the endwinding
cooling effects in TEFC induction iotors," IEEE Transactions on
Industry Applications, vol. 43, no. 5, pp. 1214-1222, Sept.-Oct.
2007.
[5] A. Boglietti, A. Cavagnino, D. Staton and M. Popescu,
"Experimental assessment of end region cooling arrangements in
induction motor endwindings," IET Electric Power Applications,
vol. 5, no. 2, pp. 203-209, Feb. 2011.
[6] C. Micallef, S. Pickering, K. Simmons and K. Bradley, "Improved
cooling in the end region of a strip-wound totally enclosed fancooled
induction electric machine," IEEE Transactions on
Industrial Electronics, vol. 55, no. 10, pp. 3517-3524, Oct. 2008.
[7] M. Hettegger, B. Streibl, O. Bíró and H. Neudorfer, "Identifying
the heat transfer coefficients on the end-winding of an electrical
machine by measurements and simulations," in 19th ICEM, Rome,
2010.
[8] W. P. Jones and B. E. Launder, "The prediction of laminarization
with a two-equation model of turbulence," International Journal of
Heat and Mass Transfer, vol. 15, no. 2, pp. 301-314, Feb. 1972.
[9] F. R. Menter, "Two-equation eddy-viscosity turbulence models for
engineering applications," AIAA Journal, vol. 32, pp. 1598-1605,
1994.
[10] W. Vieser, T. Esch and F. Menter, "Heat transfer predictions using
advanced two-equation turbulence models," CFX Technical
Memorandum, vol. CFX-VAL10/0602, 2002.
[11] "ANSYS 13.0 documentation," ANSYS, Inc., Canonsburg, 2010.
[12] F. Kreith, R. M. Manglik and M. S. Bohn, Principles of Heat
Transfer, 7 ed., Stamford: Cengage Learning, 2011.
[13] "ANSYS ICEM CFD 13.0 documentation," ANSYS, Inc.,
Canonsburg, 2010.
[14] M. Schrittwieser, A. Marn, E. Farnleitner and G. Kastner,
"Numerical analysis of heat transfer and flow of stator duct
models," in 20th ICEM, Marseille, 2012.

- 47 - 15th IGTE Symposium 2012
Numerical Investigation of Linear Systems Obtained
by Extended Element-Free Galerkin Method
Taku Itoh∗ , Soichiro Ikuno∗ , and Atsushi Kamitani †
∗ Tokyo University of Technology, 1404-1 Katakura, Hachioji, Tokyo 192-0982, Japan
† Yamagata University, 4-3-16 Johnan, Yonezawa, Yamagata, 992-8510, Japan
E-mail: taku@m.ieice.org
Abstract—To impose not only the essential boundary condition but also the natural one without any integrations, the
Element-Free Galerkin method (EFG) has been reformulated, and this method is called an eXtended EFG (X-EFG). A
linear system obtained by the X-EFG becomes an asymmetric saddle point problem. Numerical experiments show that,
by using IC-Bi-CGSTAB and IC-GMRES(m), the linear system can be solved more than 9 times faster than the LU
factorization in relatively large problem. However, there are some cases where these iterative methods sometimes do not
converge regardless of the condition number. To avoid these cases, and to stably solve the linear system as fast as possible,
a flow chart for choosing an appropriate solver has been constructed by using the results of the numerical experiments.
Index Terms—Meshless methods, Element-free Galerkin methods, Saddle point problems, Asymmetric linear systems
I. INTRODUCTION
Meshless methods such as the Element-Free Galerkin
method (EFG) [1] and the Meshless Local Petrov-
Galerkin method (MLPG) [2] have widely been applied
to numerical simulations in a lot of fields, including
electromagnetics [3], [4], [5], [6], [7]. In the meshless
methods, elements of a geometrical structure are no
longer necessary.
Especially in the EFG, the Lagrange multiplier [1] is
employed for imposing the essential boundary condition.
Recently, the EFG has been reformulated together with
anewimposing method of the essential boundary condition
[8]. In this EFG, the essential boundary condition
can be satisfied without using any integrations. However,
it must be noted here that, in this EFG, the natural
boundary condition is imposed by evaluating integrations.
Especially for a three-dimensional (3D) problem, surface
integrals must be evaluated for imposing the natural
boundary condition. For this reason, if there exists an
easier method for imposing the natural boundary condition,
the method is helpful for developing a numerical
code based on the EFG.
The purpose of the present study is to reformulate
a 3D EFG so that not only the essential boundary
condition but also the natural one can be imposed without
any integrations. The reformulated method is called an
eXtended EFG (X-EFG). A linear system obtained by
the X-EFG becomes a saddle point problem, and the
coefficient matrix is asymmetric. For the purpose of
stably obtaining a numerical solution as fast as possible,
appropriate solvers for the asymmetric linear system are
also investigated numerically.
II. EXTENDED ELEMENT-FREE GALERKIN METHOD
A. New Reformulation
In this section, we describe a new reformulation of
EFG which is different from that described in [8]. For
simplicity, we consider a 3D Poisson problem:
−Δu = f in V, (1)
u =ū on SD, (2)
∂u
∂n =¯q on SN, (3)
where V is a region bounded by a simple closed surface
∂V that consists of both SD and SN. Here, SD and SN
satisfy SD ∪ SN = ∂V and SD ∩ SN = φ. In addition, ū
and ¯q are known functions on SD and SN, respectively,
and n is an outward unit normal on ∂V . Furthermore,
f(x) is a given function on V , and x =[x, y, z] T .
From (1), the weak form is derived as
∀
w s.t. w∂V : ∇w ·∇u d
=0 3
x = wf d 3 x . (4)
V
where w(x) is a test function. Note that the constraint of
w(x) in (4) is different from that in [8].
To discretize (4), the nodes, x1, x2,...,xN are first
placed both in V and on ∂V , and shape functions
φ1(x),φ2(x),...,φN(x) are determined by using the
Moving Least-Squares (MLS) approximation [1], [2], [7].
Here, N is the number of nodes in V ∪ ∂V . In the
following, M denotes the number of nodes on ∂V .In
addition, the orthonormal system in R N and that in R M
are denoted by {e1, e2,...,eN } and {ē1, ē2,...,ēM },
respectively.
Let us first discretize the weak form (4). To this end,
we assume that u and w can be expanded with φi(x)(i =
1, 2,...,N) as follows:
N
N
u(x) = ûiφi(x), w(x) = ˆwiφi(x). (5)
i=1
By substituting (5) into (4), we obtain
where
i=1
( ˆw,Aû − f) =0, (6)
ˆw =[ˆw1, ˆw2,..., ˆwN ] T , (7)
V

and
û =[û1, û2,...,ûN] T . (8)
In addition, A and f are defined as
N N
A ≡ ∇φi ·∇φj d 3 xeie T j , (9)
f ≡
i=1 j=1
N
V
φif d 3 xei . (10)
i=1 V
Next, the constraint w| ∂V =0 in (4) is discretized. To
this end, the constraint is rewritten as the equivalent
proposition:
∀
β(s, t) :
∂V
β(s, t)w(x(s, t)) dS =0, (11)
and an arbitrary function β(s, t) is assumed to be
contained in span(N1,N2,...,NM ), where N1(s, t),
N2(s, t),...,NM (s, t) are linearly independent functions
on ∂V . Here, s and t are parameters for representing
∂V . By using N1(s, t),N2(s, t),...,NM (s, t), (11) can
be discretized as
( ˆw, ck) =0(k =1, 2,...,M), (12)
where ck (k =1, 2,...,M) are defined as
N
ck ≡ Nk(s, t)φi(x(s, t)) dS ei. (13)
i=1
∂V
Note that (12) indicate ˆw ∈ V ⊥ , where
V = span(c1, c2,...,cM ). (14)
Hence, the weak form (4) can be discretized as
∀ ˆw ∈ V ⊥ :(ˆw,Aû − f) =0. (15)
Since (V ⊥ ) ⊥ = V, (15) can be written as
Aû − f ∈ V. (16)
Therefore, there exists ˆv ∈ R M such that
Aû + C ˆv = f, (17)
where C ≡ [c1, c2,...,cM ] and ˆv ≡ [ˆv1, ˆv2,...,ˆvM ] T .
Finally, the essential boundary condition (2) and
the natural one (3) are simultaneously discretized. By
the similar procedures for discretizing the constraint
w| ∂V =0 , (2) and (3) can be discretized as
D T û = g, (18)
where D ≡ [d1, d2,...,dM ], and
⎧
N
Nk(s, t)φi(x(s, t)) dS ei,
⎪⎨ i=1 ∂V
for xk ∈ SD,
dk ≡ N
Nk(s, t)
⎪⎩ i=1 ∂V
∂φi
(x(s, t)) dS ei,
∂n
for xk ∈ SN
(k =1, 2,...,M). (19)
- 48 - 15th IGTE Symposium 2012
In addition, g ≡ [g1,g2,...,gM ], and
⎧
⎪⎨ Nk(s, t)ū(s, t) dS, for xk ∈ SD,
SD
gk ≡
⎪⎩ Nk(s, t)¯q(s, t) dS, for xk ∈ SN
SN
(k =1, 2,...,M). (20)
Note that, for xk ∈ SD, dk is exactly the same asck.
Equations (17) and (18) can be written in the form,
A C
DT
û f
= . (21)
O ˆv g
Equation (21) is a discretized form of the Poisson problem.
Throughout this subsection, the reformulation of
EFG is finished. In the following, the reformulated EFG
is referred to as an eXtended EFG (X-EFG).
B. Selection of linearly independent functions
As mentioned above, the linearly independent functions
Nk (k =1, 2,...,M) are required for discretizing
the essential and natural boundary conditions. Here, the
δ-functions defined on ∂V are employed as Nk (k =
1, 2,...,M) so that the these boundary conditions may
be satisfied exactly. The explicit form of Nk(s, t) is given
as
Nk(s, t) = δ(s − sk)δ(t − tk)
∂x ∂x
(k =1, 2,...,M).
×
∂s ∂t
(22)
Note that, on ∂V , the kth boundary node xk is represented
by sk and tk, i.e., xk = x(sk,tk). By using (22),
C, dk and gk(k =1, 2,...,M) can be rewritten as
C =
N
M
φi(x(sk,tk))eiē
i=1 k=1
T
k , (23)
⎧
N
⎪⎨ φi(x(sk,tk))ei, for xk ∈ SD,
dk = i=1
N ∂φi
⎪⎩
∂n
i=1
(x(sk,tk))ei,
(24)
for xk ∈ SN,
ū(sk,tk), for xk ∈ SD,
gk =
(25)
¯q(sk,tk), for xk ∈ SN.
It must be noted here that, in the X-EFG, the coefficient
matrix is not symmetric except for the case where
∂V = SD. However, the essential and natural boundary
conditions can easily be imposed, since C, D and g can
be evaluated without any integrations.
III. SOLVING LINEAR SYSTEM (21)
A linear system that has a coefficient matrix of a 2 ×
2 block structure as in (21) are called a saddle point
problem. In this section, we consider solving the saddle
point problem (21). For the following discussion, (21) is
rewritten as
Aˆx = b, (26)

where
A≡
A C
D T O
û
, ˆx ≡
ˆv
f
, and b ≡
g
. (27)
A. Direct solvers
As a direct solver for saddle point problems, there is a
method that utilizes the 2 × 2 structure of A [9]. In this
method, A is decomposed by the Cholesky factorization.
Since A is a main part of A, the computational cost may
be decreased by using this method in comparison with
that of the Gaussian elimination. However, we do not
employ this method for solving (21). This is because
there were some cases that the Cholesky factorization
of A was failed in preliminary numerical experiments.
Thus, we consider that the method in [9] is not stable
for solving (21) in this problem.
It must be noted here that A can be decomposed by the
LU factorization. Hence, we adopt the LU factorization as
a direct solver. In addition, an ordering method is used to
decrease fill-ins before the LU factorization is executed.
B. Iterative Schemes for Solving Saddle Point Problems
As an iterative scheme for solving saddle point problems,
Uzawa’s method [10] is well known. Starting with
initial guesses û0 and ˆv0, Uzawa’s method consists of
the following coupled iteration:
Aûk+1 = f − C ˆvk, (28)
ˆvk+1 = ˆvk + ω(D T ûk+1 − g), (29)
where ω>0 is a relaxation parameter. In (28), a linear
system depending on the size of A must be solved.
Hence, if the size of A is large, the computational cost
for solving (28) may be expensive.
On the other hand, the Arrow-Hurwicz method [10]
is also well known as an inexpensive iterative scheme
in comparison with the Uzawa’s method. Starting with
initial guesses û0 and ˆv0, the Arrow-Hurwicz method
consists of the following coupled iteration:
ûk+1 = ûk + α(f − Aûk − C ˆvk), (30)
ˆvk+1 = ˆvk + ω(D T ûk+1 − g), (31)
where α is also a relaxation parameter. The Arrow-
Hurwicz method is useful for the case where the size of
A is large. This is because a linear system do not exist
in this iteration. This iteration can be written in terms
of a matrix splitting A = P−Q, i.e., as the fixed-point
iteration,
P ˆxk+1 = Qˆxk + b, (32)
where
1
P≡ αI O
DT − 1
ω I
1
, Q≡ αI − A −C
O − 1
ω I
, (33)
and ˆx T k ≡ ûT k ˆvT
k . In 3D problems, the size of A tends
to be large. Thus, we adopt the Arrow-Hurwicz method
as an iterative scheme for solving (21).
- 49 - 15th IGTE Symposium 2012
C. Preconditioned Krylov Subspace Methods
For asymmetric linear systems, the incomplete LU
factorization (ILU) [10] is well known as a preconditioner
for Krylov subspace methods. Although ILU can be applied
to (21), we do not employ ILU. This is because the
coefficient matrix of (21) is almost symmetric. Namely,
we consider utilizing the matrix property.
To utilize “almost symmetric”, we adopt the incomplete
Cholesky factorization (IC) [11] as a preconditioner
for Krylov subspace methods. To this end, we propose
a strategy for generating preconditioned matrices. In
this strategy, preconditioned matrices LDLT of IC are
generated as
A D
DT
LDL
O
T , (34)
where L is a lower triangular matrix, and D is a diagonal
matrix. In (34), we assume
A C
A =
DT
A D
O DT
. (35)
O
As mentioned above, if xk ∈ SD, the kth column ofD
is exactly the same as that of C. In addition, there is
no difference between the matrix A of (21) and that of
(34). Hence, we consider that the assumption (35) can
be acceptable. Note that we adopt an algorithm of IC in
which matrices LDL T are as sparse as the matrix A [11].
Even we use IC as a preconditioner, Krylov subspace
methods for asymmetric linear systems have to be chosen.
As Krylov subspace methods for asymmetric linear
systems, Bi-CGSTAB [12] and GMRES(m) [13] are well
known and these iterative methods have produced a lot
of attractive results [14]. Here, m is some fixed integer
parameter, and GMRES(m) restarts every m steps [13].
For solving (21), we adopt both methods with IC. In the
following, Bi-CGSTAB with IC and GMRES(m) with IC
are referred to as IC-Bi-CGSTAB and IC-GMRES(m),
respectively.
IV. NUMERICAL EXPERIMENTS
In this section, some numerical solvers as chosen
in Section III are applied to a linear system (21). To
generate the linear system (21), a 3D Poisson problem is
discretized by using the X-EFG.
Throughout the present section, the region V is assumed
as V =(−0.5, 0.5) × (−0.5, 0.5) × (−0.5, 0.5).
In addition, the natural boundary condition is imposed
on the surface SN defined as −0.25 ≤ x ≤ 0.25,
−0.25 ≤ y ≤ 0.25 and z = 0.5, and the essential
boundary condition is imposed on SD ≡ ∂V − SN.
Moreover, the functions f(x), ū and ¯q are determined
so that the analytic solution of the 3D Poisson problem
may be u =exp(−x 2 − y 2 − z 2 ).
The boundary nodes x1, x2,...,xM are uniformly
placed on ∂V , and the nodes xM+1, xM+2,...,xN are
also uniformly placed in V . In addition, the exponential

Fig. 1. Dependence of the relative error on the size of coefficient
matrix. In this figure, u e and u n are exact and numerical solutions,
respectively.
weight function [1],
⎧
⎨exp[−(r/c)
w(r) ≡
⎩
2 ] − exp[−(R/c) 2 ]
1 − exp[−(R/c) 2 (r ≤ R),
]
(36)
0 (r>R),
is adopted for the MLS approximation. Here, R denotes a
support radius, and c is a user-specified parameter. We set
R =1.9h and c = h, where h is the minimum distance
between two nodes.
In the MLS approximation, the shape functions
φi(x) (i =1, 2,...,N) can be determined by
φi(x) =p T (x)B −1 (x)bi(x), (37)
where p T (x) =[1xyz]. In addition, the matrix B(x)
and the vector bi(x) are defined as
B(x) =
N
wk(x)p(xk)p T (xk), (38)
k=1
bi(x) =wi(x)p(xi), (39)
where wi(x) =w(|x−xi|). In (9), the partial derivatives
of φi(x) by X(= x, y, and z) can be obtained as
where
φi,X(x) =p T X(x)B −1 (x)bi(x)
+p T (x)[B −1
X (x)bi(x)+B −1 (x)bi,X(x)], (40)
B −1
X (x) =−B−1 (x)BX(x)B −1 (x). (41)
For evaluating (9) and (10), a cubic cell structure being
independent of the nodes is used [1], [7], and the Gauss-
Legendre quadrature is employed. The number NQ of
quadrature points depends on the number m of nodes in
a cell. Throughout this section, NQ is handled on the
similar criterion in [1], i.e., NQ = nQ × nQ × nQ, where
nQ = ⌊ √ m +0.5⌋ +2. In addition, the number NC of
cells is set as NC = mC × mC × mC, where mC =
⌊N 1/3 ⌋.
As a LU factorization, we adopt the sequential SuperLU
[15]. In addition, the Column Approximate Minimum
Degree Ordering (COLAMD) [16] is employed
- 50 - 15th IGTE Symposium 2012
Fig. 2. Dependence of the computational time for solving (21) on the
size of coefficient matrix.
as an ordering method. This ordering method can easily
be used by setting options.ColPerm = COLAMD in
the SuperLU. For the Arrow-Hurwicz method, we set
α =1.5 and ω =0.05. In addition, for IC-GMRES(m),
we set m = 200. For all iterative solvers, an initial guess
of ˆx in (26) is set as ˆx0 = 0.
Computations were performed on a computer equipped
with a 2.66GHz Intel Core i7 920 processor, 24GB RAM,
Ubuntu Linux ver. 12.04, and g++ ver. 4.6.3. Note that we
only used a single core of this processor in the following
experiments. Compiler options were set as “-O3 -Wall
-m64” for all solvers.
A. Determining εtol for Iterative Solvers
For the Arrow-Hurwicz method, the iteration is repeated
until ||ˆxk+1 − ˆxk||/||ˆxk+1|| ≤ εtol is satisfied,
where k is the iteration number and ˆxk is the approximate
solution in the kth iteration. Also, for IC-Bi-
CGSTAB and IC-GMRES(m), the iteration is repeated
until ||rk+1||/||b|| ≤ εtol is satisfied, where rk+1 is the
(k +1)th residual vector that can be obtained in the
algorithm of Krylov subspace methods. Note that the
maximum norm is adopted for the definition of ||·||.
To determine εtol, the dependence of relative error on
the size of coefficient matrix is shown in Fig. 1. Here,
the relative error ε ≡||ue−un ||/||ue ||, where ue and un are the exact and numerical solutions of u, respectively.
In addition, by the first equation of (5), un is evaluated
with û that is determined by the LU factorization. From
Fig. 1, we see ε>10−4 . Hence, we consider that, in this
problem, εtol =10−8is sufficient for obtaining un that
has almost the same accuracy shown in Fig. 1.
B. Performance of Direct and Iterative Solvers
Let us first investigate the performance of the LU factorization,
the Arrow-Hurwicz method, IC-Bi-CGSTAB
and IC-GMRES(m). To this end, the dependence of
the computational time for solving (21) by using these
methods on the size of coefficient matrix is shown in

(a)
(b)
Fig. 3. Histories of the relative residual for IC-Bi-CGSTAB and IC-
GMRES(m), and those of the relative error for the Arrow-Hurwicz
method. (a) and (b) are for N + M = 19083 and 42083, respectively.
Fig. 2. We see from this figure that the computational
time of the LU factorization is less than that of the
Arrow-Hurwicz method. In addition, from this figure,
there is no obvious difference between the computational
time of IC-Bi-CGSTAB and that of IC-GMRES(m), and
the computational time of both methods is less than that
of the LU factorization. Especially for the case where the
size N + M of the coefficient matrix is relatively large,
the computational time can be decreased by using both
methods, e.g., for N + M = 42083, IC-BiCGSTAB and
IC-GMRES(m) are about 9 and 15 times faster than the
LU factorization. From these results, we consider that the
strategy described in (34) works well for solving (21).
It must be noted here that the Arrow-Hurwicz method
does not converge for N +M = 42083. Similarly, IC-Bi-
CGSTAB and IC-GMRES(m) do not converge for N +
M = 19083. Thus, in Fig. 2, there is no data for these 3
cases. Hence, the iterative solvers are not always stable
in this problem.
To investigate behavior of the iterative solvers for
N + M = 19083 and 42083, histories of the relative
residual ||rk+1||/||b|| for Krylov subspace methods, and
those of the relative error ||ˆxk+1 − ˆxk||/||ˆxk+1|| for
- 51 - 15th IGTE Symposium 2012
Fig. 4. Dependence of the condition number of A on the size of
coefficient matrix.
the Arrow-Hurwicz method are shown in Fig. 3. We
see from Fig. 3(a) that IC-BICGSTAB rapidly diverges
and IC-GMRES(m) oscillates. In addition, the Arrow-
Hurwicz method converges though the convergence speed
is slow. Indeed, the iteration number for the Arrow-
Hurwicz method is 35324 when ||ˆxk+1 − ˆxk||/||ˆxk+1||
is satisfied. From Fig. 3(b), we see that IC-Bi-CGSTAB
and IC-GMRES(m) converge rapidly. In addition, the
relative residual of IC-GMRES(m) is stably decreased
until restarting. For N +M = 42083, the Arrow-Hurwicz
method does not converge, even after more than 500000
iterations.
Next, we investigate a property of the coefficient matrix
of (21). To this end, the dependence of the condition
number of A on the size of coefficient matrix is shown
in Fig. 4. We see from this figure that the condition
numbers are not very large, even for N + M = 19083
and 42083. Hence, from the condition numbers, it is
difficult to obtain the reason why the Krylov subspace
methods and the Arrow-Hurwicz method do not converge
for N + M = 19083 and 42083, respectively.
From these results, it is difficult that we recognize an
appropriate solver in advance. Hence, to stably solve (21)
as fast as possible, we suggest that IC-GMRES(m) is
first used in order to choose an appropriate solver. After
ˆn iterations of IC-GMRES(m), if
||rk+1||/||b|| ≤ ˆεtol
(42)
is not satisfied, then it is recognized that IC-GMRES(m)
will not converge. In this case, the iteration of IC-
GMRES(m) is finished, and (21) is solved by the LU
factorization. If (42) is satisfied after ˆn iterations, then it
is recognized that IC-GMRES(m) will converge. Hence,
in this case, the iteration of IC-GMRES(m) is continued.
We consider that ˆεtol =10 −2 and ˆn = max(30, (N +
M)/500) are reasonable choice for this problem.
Although the convergence speed is slow, the Arrow-
Hurwicz method may work for the case where not only
Krylov subspace methods do not converge but also the
LU factorization cannot execute. This may occur when
N + M is too large.

The above suggestions for choosing solvers are summarized
as a flow chart shown in Fig. 5. Note that, in
this flow chart, IC-Bi-CGSTAB can be used instead of
IC-GMRES(m) by setting ˆεtol and ˆn appropriately. This
is because, in Fig. 2, when IC-GMRES(m) converges,
IC-Bi-CGSTAB also converges, and there is no obvious
difference between the computational time of IC-
GMRES(m) and that of IC-Bi-CGSTAB.
V. CONCLUSION
To impose not only the essential boundary condition
but also the natural one without any integrations, the EFG
has been reformulated, and this method is called a X-
EFG. A linear system obtained by the X-EFG becomes
an asymmetric saddle point problem. To investigate appropriate
solvers for this problem, the linear system that
is obtained from a 3D Poisson problem discretized by
the X-EFG has been solved by the LU factorization,
the Arrow-Hurwicz method, IC-Bi-CGSTAB, and IC-
GMRES(m) in the numerical experiments. Conclusions
obtained in the present study are summarized as follows:
1) By using the X-EFG, the essential and natural
boundary conditions can be imposed without any
integrations.
2) Although the linear system obtained by the X-
EFG has the asymmetric coefficient matrix, the
incomplete Cholesky factorization works well as a
preconditioner for Bi-CGSTAB and GMRES(m).
3) By using IC-BiCGSTAB and IC-GMRES(m), the
linear system can be solved faster than the LU
factorization in relatively large problems. However,
these iterative methods sometimes do not converge
regardless of the condition number.
4) To stably solve the linear system as fast as possible,
an appropriate solver can be chosen by the flow
chart shown in Fig. 5.
As future work, the X-EFG will be applied to more
practical problems in various fields, including electromagnetics.
ACKNOWLEDGMENTS
This work was partially supported by JSPS KAKENHI
Grant Numbers 24700053 and 22360042.
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- 52 - 15th IGTE Symposium 2012
✓
Start
✒
✏
✑
❄
Iteration of IC-GMRES(m) is repeated
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❄
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- 53 - 15th IGTE Symposium 2012
Electromagnetic Wave Propagation Simulation
in Corrugated Waveguide using Meshless Time
Domain Method
Soichiro Ikuno∗ , Yoshihisa Fujita∗ , Taku Itoh∗ , Susumu Nakata † and Atsushi Kamitani ‡
∗Tokyo University of Technology, 1404-1 Katakura, Hachioji, Tokyo 192-0982, Japan
† Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan
‡ Yamagata University, 4-3-16 Johnan, Yonezawa, Yamagata, 992-8510, Japan
E-mail: s.ikuno@ieee.org
Abstract—The simulation of the electromagnetic wave propagation in complex shaped corrugated waveguide using Meshless
Time Domain Method (MTDM) based on the Radial Point Interpolation method is numerically investigated. MTDM does
not require finite elements or meshes of a geometrical structure as well as other meshless method. In MTDM, only the
necessary information is the location of nodes, and the arrangement of the node structure of electric fields and magnetic
fields. By using the simulation code for analyzing a magnetic wave propagation phenomenon in a complex shaped waveguide,
the influence of node alignment on a wave propagation is numerically evaluated. Moreover, the influence of frequencies
and pitch of corrugate on the dumping rate is evaluated. The results of computation show that the node alignment based
on the staggered grid that is generally used in standard FDTD is suitable for the numerical calculation. In addition, the
relationship between a pitch of corrugate and a frequency is numerically evaluated.
Index Terms—FDTD, RPIM, wave propagation, corrugated waveguide
I. INTRODUCTION
In the Large Helical Device (LHD), the electron cyclotron
heating device is used for plasma heating. The
electrical power that is made by the gyrotron system
transmits to LHD by using long corrugated waveguide.
However, it is not clear that the shape of curvature of the
waveguide or transmission gain of electromagnetic wave
propagation theoretically.
Finite Difference Time Domain (FDTD) method has
provided the solution of Maxwell’s equation directly,
and the method is applied for electromagnetic wave
propagation simulation frequently [1], [2]. Furthermore,
FDTD method has great advantages in terms of parallelization
and treatment of problems and so on. However,
the numerical domain should be divided into rectangle
meshes if FDTD method is applied for the simulation,
and it is difficult to treat the problem in the complex
domain.
As is well known that the meshless approach does
not require finite elements or meshless of a geometrical
structure. And various meshless approaches such as the
diffuse element method [3], the element-free Galerkin
(EFG) method [4] and the meshless local Petrov-Galerkin
(MLPG) [5] method and the radial point interpolation
method (RPIM) has been developed [6]. And these
methods are applied to a variety of engineering fields
and the fields of computational magnetics [7], [8], [9].
Particularly, meshless approaches based on RPIM are
applied to time dependent problems [10]. Meshless Time
Domain Method (MTDM) [11] does not require finite
elements or meshes of a geometrical structure as well as
other meshless method. In MTDM, only the necessary
information is the location of nodes, and the arrangement
of the node structure of electric fields and magnetic
fields. Thus, MTDM can be easily applied for the time
dependent simulation of the problem in the complex
shaped domain.
The purpose of the present study is to develop the
numerical code for analyzing electromagnetic wave propagation
in corrugated waveguide, and to investigate the
optimal shape of corrugated waveguide.
II. SHAPE FUNCTION BASED ON RPIM
First, we scatter N nodes x1, x2, ··· , xN in the target
domain and the boundary, and assign the Radial Basis
Function (RBF) w1(x),w2(x), ··· ,wN (x) with compact
support to the nodes. Then, the solution u(x) can
be expanded as
u(x) =[w(x) T , p(x) T ]G −1
u
= φ(x)u, (1)
0
where the vector w(x), p(x), u(x) and φ(x) are defined
by
w(x) =[w1(x),w2(x), ··· ,wN (x)] T , (2)
p(x) =[p1(x),p2(x), ··· ,pM(x)] T , (3)
u =[u1,u2, ··· ,uN ] T , (4)
φ(x) =[φ1(x),φ2(x), ··· ,φN(x)] T . (5)
where φi(x) denotes a shape function on i−th node.
The components of the vector p(x) are monomials of
the space variables. For example, p(x) T =[1,x,y] and
p(x) T =[1,x,y,x2 ,xy,y2 ] are monomials for the linear
and the quadratic approximation. Furthermore, the matrix
G is defined by following equation.
W P
G =
P T
, (6)
O

Here, the matrices W and P are defined by following
equations.
W =[w(x1), w(x2), ··· , w(xn)] T , (7)
P =[p(x1), p(x2), ··· , p(xn)] T . (8)
In the present study, following three functions are
adopted for the weight function.
⎧
⎨ e
wi(xj) =
⎩
−(r/c)2 − e−(R/c)2 1.0 − e−(R/c)2 , r < R,
(9)
0, r ≥ R,
r
2 wi(xj) =1.0− 6.0
R
r
3
r
4 +8.0 − 3.0
(10)
R R
r
2
−0.5
wi(xj) = +1.0
(11)
R
Here, R denotes a support radius of the influence domain
and c denotes a constant. Moreover, r is defined by
r = |x − xi|. Under the above assumptions, the shape
function and its derivative can be expressed as
N
M
φk(x) = wi(x)gi,k + pj(x)gN+j,k, (12)
∂φk
∂x =
∂φk
∂y =
N
i=1
N
i=1
i=1
∂wi(x)
∂x gi,k +
∂wi(x)
gi,k +
∂y
j=1
M
j=1
M
j=1
∂pj(x)
∂x gN+j,k, (13)
∂pj(x)
gN+j,k, (14)
∂y
where, gi,j denotes the (i, j) element of matrix G −1 .
Note that the shape function satisfy the Kronecker
delta function property, i.e.
1, i = j,
φi(xj) =
(15)
0, i = j.
From this property the function can be expanded by using
the shape function based on RPIM as follows.
u(xi) =
N
φi(xj)ui = ui. (16)
j=1
In the next section, Meshless Time Domain Method is
formulated by using above shape function.
III. MESHLESS TIME DOMAIN METHOD
In the present study, 2D electromagnetic wave propagation
of TM mode is adopted for the evaluation. The
governing equation of the problem is defined by
ε ∂Ez
∂t = −σEz + ∂Hy
∂x
μ ∂Hx
∂t
μ ∂Hy
∂t
∂Hx
− , (17)
∂y
= −∂Ez , (18)
∂y
∂Ez
= , (19)
∂x
- 54 - 15th IGTE Symposium 2012
where, Hx and Hy denote the magnetic field of x and
y component, and Ez denotes the electric field of z
component. In addition, ε, σ and μ denote permitivity,
permeability and electroconductivity, respectively.
The system is discretized with respect to time by
applying Leap Frog Method, and it is transformed to
following equations.
ε n+1
Ez − E
Δt
n z
+ σE n+ 1
2
z
= ∂Hn+ 1
1
2
2
y ∂Hn+ x
− ,
∂x ∂y
μ
H
Δt
(20)
n+1/2
x − H n−1/2
x = − ∂En z
,
∂y
μ
H
Δt
(21)
n+1/2
y − H n−1/2
y = ∂En z
.
∂x
(22)
As we mentioned above, the shape function of RPIM
has the Kronecker delta function property (15). By using
the shape function and the property, the system can be
discretized with respect to space as follows.
E n+1
ε σ
z,i = α − E
Δt 2
n z,i
⎤
N 1 n+ 2
+ H
N 1 n+ 2 H ⎦ , (23)
H
H
1 n+ 2
x,i
1 n+ 2
y,i
j=1
y,j
1
2 = Hn− x,i
1
2 = Hn− y,i
Here, φ E i and φH i
∂φ H j
∂x −
Δt
−
μ
Δt
+
μ
N
j=1
N
j=1
j=1
x,j
E n ∂φ
z,j
E j
∂y
E n ∂φ
z,j
E j
∂x
∂φ H i
∂y
, (24)
. (25)
denote the shape function for electric
field and magnetic field, and the parameter α is defined
as following equation.
Note that, the average of E n z,i
1
α = ε σ . (26)
+
Δt 2
and En+1 z,i is adopted for
E n+1/2
z,i . By solving (23), (24) and (25) alternately in
each time step, we can obtain the result that describes
the time dependent behavior of the electromagnetic wave
propagation in various shape of wave guide.
In the present study, the Perfectly Matched Layer
(PML) and the Perfect Magnetic Conductor (PMC) are
used for absorbing boundary condition and boundary
condition. The electric field of z component is divided
into
Ez = Ezx + Ezy, (27)
where components are governed by following equations.
jωεEzx + σxEzx = ∂Hy
, (28)
∂x
jωεEzy + σyEzx = − ∂Hx
. (29)
∂x

Here, j denotes a imaginary unit, and ω denotes a angular
frequency. By using (27), the basic governing equation
of PML is written as follows.
ε ∂Ezx
∂t = −σxEzx + ∂Hy
, (30)
∂x
ε ∂Ezy
∂t = −σyEzy − ∂Hx
, (31)
∂x
μ ∂Hx
∂t = −σ∗ yHx − ∂Ez
, (32)
∂y
μ ∂Hy
∂t = −σ∗ yHy + ∂Ez
. (33)
∂x
Here, μ denotes permeability. Taking into account the
delta function property of the shape function based on
RPIM, and discretizing respect to time using the Leap-
Flog method, we can obtain following discretized equations
for PML
E n zx,m =
E n zy,m =
H
H
1 n+ 2
x,m =
1 n+ 2
y,m =
ε
Δt
ε
Δt
σx
− E
2
n−1
zx,m +
N
H
i=1
ε σy
+
Δt 2
σx
− E
2
n−1
zy,m +
N
H
i=1
ε σy
+
Δt 2
μ
Δt − σ∗
1
y n− 2 Hx,m −
2
μ
Δt + σ∗ y
2
μ
Δt − σ∗
1
x n− 2 Hy,m +
2
μ
Δt + σ∗ x
2
N
i=1
N
i=1
n− 1
2
y,i
n− 1
2
x,i
∂φ H i
∂x
∂φ H i
∂y
E n ∂φ
z,i
E i
∂y
E n ∂φ
z,i
E i
∂x
, (34)
, (35)
, (36)
, (37)
where Δt denotes a step size of time and superscript n
denotes number of steps.
In MTDM, nodes for electric field and magnetic field
should be separated, and following four types of node
alignment is adopted for accuracy evaluation as shown
in Fig. 1.
Fist alignment type is based on normal meshless
method that means a node for electric field and magnetic
field is located same position as shown in Fig. 1 (a). The
node for magnetic field located a center of diagonally of
nodes for electric field in second type as shown in Fig.
1 (b). Third type is based on staggered grid which is
generally used in standard FDTD, and fourth type is a
mixed version of second and third type as shown in Fig.
1 (c) and (b), respectively.
IV. INFLUENCE OF NODE ALIGNMENT
As is well known that FDTD is an explicit method.
Thus, the method must be satisfies the Courant condition,
- 55 - 15th IGTE Symposium 2012
(a) (b)
(c) (d)
Fig. 1. The schematic view of four types of node alignment of electric
field and magnetic field.
i.e.,
Δt < 1
v
1
2 1
+
Δx
1
Δy
,
2
(38)
where Δx and Δy denote a division size of x and y
direction, and v denotes a wave speed. On the other
hand, MTDM has not concept of mesh as we mentioned
above. Therefore, following criterion is derived for stable
calculation [11].
min |xi − x|
i
Δt <
. (39)
v
Here, min |xi − x| denotes a distance of neighboring
i
node, and the step size of time Δt is determined so as
to satisfy the criterion (39).
To evaluate the influence of node alignment, value of
the dumping rate RD is introduced.
RD =
Γout
Γin
Pz dl
Pz dl
(40)
Here, Pz denote a pointing vector P = B × E of z
component, and Γin, Γout denote a the source input line
and the observation line, respectively. We can see from
above equation, if the value of RD satisfies RD =1,
the waveguide regards as an ideal zero loss waveguide.
In addition, physical parameters for the calculation are
shown in Table I

TABLE I
PHYSICAL PARAMETERS FOR THE CALCULATION. HERE λ DENOTES
A WAVE LENGTH.
Damping rate, R D
Input Wave Sine wave
Amplitude 1.0 [V/m]
Frequency 1.0, 15.0, 30.0 [GHz]
Wave speed 3.0 × 10 8 [m/s]
Distance of neighboring node 20/λ
Number of layer for PML 16
Dimension of PML 4
Reflectance factor of PML −80 [dB]
1.4
1.2
1
0.8
0.6
0.4
0.2
(a)
(b)
(c)
0
0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.058 0.06
Support radius, R
Fig. 2. The values of dumping rate RD are plotted as a function of
support radius R in case of the first type node alignment as shown in
Fig. 1 (a). Note that lines (a), (b) and (c) are evaluated by using Eq.
(9), (10) and (11), respectively.
In Fig. 2, 3, 4 and 5, we show the influence of
support radius R on dumping rate RD with various
weight functions. Note that a normal line waveguide is
adopted for the evaluation, and same value of support
radius R is adopted for electric field shape function and
magnetic field shape function. In addition, a frequency
of input wave is fixed as 1 [GHz]. We can see from
these figures that the values of dumping rate RD are not
strictly stable in case of spline weight function is adopted
for the shape function construction. On the other hand,
if the Gauss type weight function is adopted for weight
function the value of dumping rate RD generally continue
to be flat around unit value in case of all the types
of node alignment. From this result, Gauss type weight
function (9) is suitable for MTDM weight function, and
for the rest of this Gauss type weight function is adopted
for following calculation. Furthermore, we can see from
these figures that node alignments of third type (see Fig.
1 (c)) lead us stable calculation. Thus, in the following
calculation node alignment of third type is adopted.
V. WAVE PROPAGATION SIMULATION IN
CORRUGATED WAVEGUIDE
Let us first show the distribution of electric field in
curved corrugated waveguide. The schematic view of
the curved corrugated waveguide which is used in the
calculation is shown in Fig. 6 (a). The pitch of the
- 56 - 15th IGTE Symposium 2012
Damping rate, R D
1.4
1.2
1
0.8
0.6
0.4
0.2
(a)
(b)
(c)
0
0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.058 0.06
Support radius, R
Fig. 3. The values of dumping rate RD are plotted as a function of
support radius R in case of the second type node alignment as shown
in Fig. 1 (b). Note that lines (a), (b) and (c) are evaluated by using Eq.
(9), (10) and (11), respectively.
Damping rate, R D
1.4
1.2
1
0.8
0.6
0.4
0.2
(a)
(b)
(c)
0
0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.058 0.06
Support radius, R
Fig. 4. The values of dumping rate RD are plotted as a function of
support radius R in case of the third type node alignment as shown in
Fig. 1 (c). Note that lines (a), (b) and (c) are evaluated by using Eq.
(9), (10) and (11), respectively.
Damping rate, R D
1.4
1.2
1
0.8
0.6
0.4
0.2
(a)
(b)
(c)
0
0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.058 0.06
Support radius, R
Fig. 5. The values of dumping rate RD are plotted as a function of
support radius R in case of the fourth type node alignment as shown
in Fig. 1 (d). Note that lines (a), (b) and (c) are evaluated by using Eq.
(9), (10) and (11), respectively.

y(m)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 0.05 0.1 0.15 0.2
x(m)
(a) (b)
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
E z
Fig. 6. (a): The schematic view of the analytic region and (b): the
distribution of electric field Ez in corrugated waveguide in case of
W =50mm.
(a) (b)
Fig. 7. Analytic models for evaluating dumping rate RD. (a): Line
wave guide, (b): Curved wave guide
corrugate made at regular intervals for straight part, and
unequally-spaced gaps are made on a curved part as
shown in Fig. 6 (a). The distribution of electric field
of z component Ez in case of W = 50 mm is also
shown in Fig. 6 (b). In this figure, the reflected wave
is observed at the curved part of waveguide. Note that
the reflected wave increase as the width of waveguide
W increases. In other words, the damping rate increase
as the value of W increase, and this phenomenon also
relevant to wavelength and curvature of waveguide.
Next, we evaluate the influence of frequencies and
pitch of corrugate on the dumping rate RD. The analytic
models for evaluating dumping rate RD are line corrugated
waveguide (see Fig. 7 (a)) and curved corrugated
waveguide (see Fig. 7 (b)). The pitch of corrugate shape
is Cλ where C denotes a constant, and the pitch of
the corrugate made at regular intervals for straight part
and unequally-spaced gaps are made on a curved part
in curved corrugated waveguide as well as previous
evaluation.
By using the analytic models, the influence of frequen-
-2.5
- 57 - 15th IGTE Symposium 2012
Damping rate, R D
2
1.5
1
0.5
1GHz
15GHz
30GHz
0
0λ 0.2λ 0.4λ 0.6λ 0.8λ 1λ 1.2λ
Pitch of Corrugated waveguide
Fig. 8. The influence of frequencies and pitch of corrugate on dumping
rate RD in the line waveguide.
Damping rate, R D
2.5
2
1.5
1
0.5
1GHz
15GHz
30GHz
0
0λ 0.2λ 0.4λ 0.6λ 0.8λ 1λ 1.2λ
Pitch of Corrugated waveguide
Fig. 9. The influence of frequencies and pitch of corrugate on dumping
rate RD in the curved waveguide.
cies and pitch of corrugate D on dumping rate RD in the
corrugated waveguide is evaluated, and the results are
shown in Fig. 8 and 9. In the line waveguide, the magnetic
wave propagates stationary in case of 0.0λ

• The values of dumping rate RD are not strictly stable
in case of spline weight function is adopted for the
shape function construction.
• On the other hand, if the Gauss type weight function
is adopted for weight function the value of dumping
rate RD generally continue to be flat around unit
value in case of all the types of node alignment.
• The node alignment based on the staggered grid that
is generally used in standard FDTD should be used
for MTDM simulation.
• The reflected wave increase as the width of waveguide
W increases. In other words, the damping rate
increase as the value of W increase, and this phenomenon
also relevant to wavelength and curvature
of waveguide.
• In the line corrugated waveguide, the magnetic wave
propagated stationary in case of 0.0λ

- 59 - 15th IGTE Symposium 2012
Optimization of Permanent Magnet Linear Actuator
for Braille Screen
*Ivan S. Yatchev, *Iosko S. Balabozov, *Krastio L. Hinov, *Vultchan T. Gueorgiev and
**Dimitar N. Karastoyanov
* Faculty of Electrical Engineering, Technical University of Sofia, 8, Kliment Ohridsky Blvd., 1000 Sofia, Bulgaria
** Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bonchev St.,
Block 2, 1113 Sofia, Bulgaria
E-mail: yatchev@tu-sofia.bg
Abstract—Permanent magnet linear actuator intended for driving a needle in Braille screen has been optimized. The mover
of the actuator is a combined one - it consists of permanent magnet and ferromagnetic discs. The optimization is carried out
with respect to minimal magnetomotive force ensuring required minimum electromagnetic force on the mover. The
optimization factors are dimensions of the cores and mover parts under additional constraint for overall dimension of the
actuator. Finite element analysis, response surface methodology and design of experiments have been employed for the
optimization. The obtained optimal solution is verified again by finite element analysis.
Index Terms—actuators, Braille screen, optimization, secondary models.
I. INTRODUCTION
Application of permanent magnets in the constructions
of different actuators has been intensively increased in
recent years. One of the reasons for their application is
the possibility for development of energy efficient
actuators. New constructions of permanent magnet
actuators are employed for different purposes. One such
purpose is the facilitation of perception of images by
visually impaired people using the so called Braille
screens. Recently, different approaches have been utilized
for the actuators used to move Braille dots [1]-[6].
Typical view of a Braille screen is shown in Fig. 1.
Figure 1: Braille screen with needles (dots) driven by
linear actuators.
In the present paper, recently developed permanent
magnet linear actuator for driving a needle (dot) in Braille
screen is optimized using response surface methodology
(RSM) and design of experiments (DoE).
The nature of the main application puts very firm
requirements about the driver of the Braille screen
needles. These requirements can be summarized as
follows:
- firm dimension constraints-especially in radial
direction: outer diameter of the driver 3-6 mm;
- holding force 02-05 N;
- minimum energy consumption.
The minimum energy consumption can be achieved by
polarized construction of the driving electromagnet
actuator because no power will be consumed at steady
state.
II. ACTUATOR CONSTRUCTION
The principal actuator construction is shown in Fig. 2.
The moving part is axially magnetized cylindrical
permanent magnet with two ferromagnetic discs on both
sides.
The two coils are connected in series in such way that
they create magnetic flux of opposite directions in the
region of the permanent magnet. In this way, depending
on the polarity of the power supply, the permanent
magnet will move either up or down. When motion up is
needed, the upper coil should create flux in the air gap
coinciding with the flux of the permanent magnet. Lower
coil at the same time will create opposite flux and the
permanent magnet will move in upper direction. When
motion down is needed, the polarity of the power supply
is reversed. The motion is transferred to the Braille dot
using the non-magnetic shaft.
Figure 2: Principal construction of the studied actuator.
1–upper shaft; 2–upper core; 3–outer core; 4–upper coil; 5-upper disc;
6–magnet; 7–lower disc; 8–lower coil; 9–lower core; 10–lower shaft

The actuator features increased energy efficiency, as
the power supply is needed only during the switching
between the two end positions of the mover. In each end
position, the permanent magnet creates holding force,
which keeps the mover in this position.
III. STATIC FORCE CHARACTERISTICS
Static magnetic field of the actuator is modeled using
the finite element method and the program FEMM [7].
Axisymmetric model is adopted as the actuator features
rotational symmetry. The electromagnetic force acting on
the mover is obtained using the weighted stress tensor
approach.
Typical static force characteristics of the actuator are
shown in Fig. 3. The stroke of the actuator, denoted with
x, is set to zero when the shaft is situated symmetrically
between the upper and lower cores.
c1=-1,c2=1
c1=1,c2=-1
1.2
1
F, N
c1=0,c2=0
0.8
0.6
0.4
0.2
0
-0.6 -0.4 -0.2 -0.2 0 0.2 0.4 0.6
-0.4
-0.6
-0.8
-1
-1.2
x, mm
Figure 3: Typical force-stroke characteristics of the
studied construction.
c1 and c2 show the direction of MMF in both coils of the construction.
c1=-1, c2=1 – upward movement of the shaft;
c1=1, c2=-1 –downward movement of the shaft; c1=0, c2=0 –non
energized coils, the force is due to the permanent magnet only.
The upper and lower curves in Fig. 3 represent the
force when the shaft is moving in upward and downward
direction. The middle curve shows the force when no
current flows in the coils. In that case the force is due to
the magnetic flux of the permanent magnet. The
characteristic is symmetrical towards the origin of the
force-stroke coordinate system and its final values (when
the shaft is close to upper or lower cores) is called
holding force – Fh. This is the only force that keeps the
shaft in both stable position – upper and lower and it
should resist to the force created by the touching fingers
and the mover’s own weight.
The starting force – Fs is the initial force that acts on
the shaft when it is in its final upper position and both
coils are energized in such a manner to create force in
downward direction or the opposite – the shaft is in final
lower position and force is acting upward.
The construction should guarantee overcoming of the
holding force, created by the permanent magnet, when
the coils are properly energized.
The upper coil excites in the upper core magnetic flux
that is equal or bigger than the flux of the permanent
magnet but contrary directed. At the same time, the flux
excited by the lower coil is coincident with the flux of the
- 60 - 15th IGTE Symposium 2012
permanent magnet.
The construction minimizes the requirements towards
the starting force and guarantees that it will start moving
even for small value of the starting force if only it
exceeds the own weight of the shaft.
IV. SECONDARY MODELS
Finite element method, DoE and RSM have been used
for creation of the secondary models. Full factorial design
has been applied.
The fixed geometric parameters are shown in Fig. 4 and
their values are given in Table 1.
Figure 4. Fixed parameters of the actuator.
TABLE I
FIXED GEOMETRIC PARAMETERS
Dimension
Designation
(in Fig. 4)
Value
(in mm)
Outer core diameter D 5
Outer magnet diameter Dm 2
Inner coil diameter Dw1 2.4
Outer coil diameter Dw2 4
Shaft diameter Ds 1
Inner core diameter Dc 1.2
Core thickness hc 2
The varied parameters are:
- The length of the upper and lower cores - hw,
- The axial dimension of the ferromagnetic disks -
hd,
- The length of the permanent magnet - hm,
- The current density in the coils - J.
The varied parameters with geometric representations
are shown in Fig. 5.

Figure 5: Varied geometric parameters of the actuator.
The DoE methodology has been used for varied
parameters to create polynomial secondary models. For
each combination of values of the varied parameters a
family of static force-stroke characteristics was obtained.
Based on them secondary models for holding force Fh,
starting force Fs and ampere-turns of the coils – NI have
been made.
The precision of secondary models has been estimated
by the relative error between the value obtained by te
secondary model and corresponding value obtained by
the FEM model. The difference between secondary and
FEM models for 27 calculation points is given in Fig.6.
relative error, %
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0 5 10 15 20 25 30
number of calculation points
Figure 6: Relative error between secondary and FEM
models.
V. OPTIMIZATION
The objective function is minimal magnetomotive force
of the coils. The optimization parameters are dimensions
of the permanent magnet, ferromagnetic discs and the
cores. As constraints, minimal electromagnetic force
acting on the mover, minimal starting force and overall
outer diameter of the actuator have been set. The
Fs
Fh
- 61 - 15th IGTE Symposium 2012
optimization is carried out using sequential quadratic
programming.
The canonic form of the optimization problem is:
where:
- NI — ampere-turns — minimizing energy
consumption with satisfied force requirements;
- Fh — holding force — mover (shaft) in upper
position, no current in the coils;
- Fs — starting force — mover (shaft) in upper or
lower position and energized coils;
- J — coils current density;
- hw, hm, hd—geometric dimensions according to
the sketch in Fig. 6.
Minimization of magneto-motive force NI is direct
subsequence of the requirement for minimum energy
consumption.
Constraints for Fs and Fh have already been discussed.
The lower bounds for the dimensions are imposed by the
manufacturing limits and the upper bound for the current
density is determined by the thermal balance of the
actuator.
The radial dimensions of the construction are directly
dependent by the outer diameter of the core – D which
fixed value was discussed earlier. The influence of those
parameters on the behavior of the construction have been
studied in previous work [8] that make clear that there is
no need radial dimensions to be included in the set of
optimization parameters.
The optimization is carried out by sequential quadratic
programming. The optimization results are as follows:
The optimal parameters were set as input values to the
FEM model. The force-stroke characteristics of the
optimal actuator is shown in Fig.7 and Fig.8.
Fh, N
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
Fh - holding force in lower position of the shaft
Fh - holding force in upper position of the shaft
-0.4 -0.3 -0.2 -0.1 0
x, mm
0.1 0.2 0.3 0.4
Figure 7: Force-stroke characteristic of the optimal
actuator. The force is created by the permanent magnet
only (no current in the coils).

F, N
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
F - final force (shaft imoved in lower position, coils still energized)
Fs - starting force (shaft in upper position)
Fh - force switches to the holding force when current is ceased
-0.4 -0.3 -0.2 -0.1 0
x, mm
0.1 0.2 0.3 0.4
Figure 8. Force-stroke characteristic of the optimal
actuator. Coils are energized. The shaft is displaced from
final upper to final lower position.
In Figs. 9 and 10, the magnetic field of the optimal
actuator is plotted for two cases.
Figure 9: Magnetic field of the optimal actuator with
shaft in upper position and coils energized to create
downward force.
Figure 10: Magnetic field of the optimal actuator with no
current in the coils.
- 62 - 15th IGTE Symposium 2012
The force constraints for Fs and Fh are active which
can be expected when minimum energy consumption is
required. The active constraint for hw is also expected
because longer upper and lower cores size which
respectively means longer coils will increase the leakage
coil flux and corrupted coil efficiency.
VI. CONCLUSION
The employed approach has confirmed its robustness
for solution to the optimization problem for the actuator.
The obtained optimal solution satisfies the specific
requirements for actuators for Braille screen.
[1]
REFERENCES
Nobels T., F. Allemeersch, K. Hameyer, “Design of a high power
density electromagnetic actuator for a portable Braille display.“
Int. Conf. EPE-PEMC 2002, Dubrovnik & Cavtat, 2002.
[2] Kawaguchi Y., K. Ioi, Y. Ohtsubo, “Design of new Braille display
using inverse principle of tuned mass damper.” Proc.of SICE
annual conference 2010, Taipei, Taiwan, Aug. 18-21, pp. 379-383.
[3] Kwon, H.J., Lee, S.W., Lee, S. Braille code display device with a
PDMS membrane and thermopneumatic actuator. IEEE
[4]
international conference on micro electro mechanical systems
(XXI), MEMS, Tucson, 2008, pp. 527-530.
Chaves, D., Peixoto, I., Lima, A., Vieira, M., de Araujo, C.
Microtuators of SMA for Braille display system. IEEE
international workshop on medical measurements and
[5]
applications, MeMeA Cetraro, Italy, May 20-30, 2009, pp. 64-68.
Hernandez, H., Preza, E., Velazquez, R. Characterization of a
piezoelectric ultrasonic linear motor for braille displays.
Electronics, robotics and automotive mechanics conference
CERMA Cuernavaca, Mexico, Sep. 22-25, 2009, pp. 402-407.
[6] Cho, H.C., Kim, B.S., Park, J.J., Song, J.B. (2006) Development
of a Braille display using piezoelectric linear motors.
[7]
International joint conference SICE-ICASE, 2006, Busan, Korea,
Oct. 18-21, pp. 1917-1921.
D. Meeker, Finite element method magnetics version 3.4, 2005.
[8] Yatchev I., K. Hinov, V. Gueorgiev, D. Karastoyanov, I.
Balabozov, Force characteristics of an electromagnetic actuator
for Braille screen, Proceedings of Thirteenth International
Conference on Electrical Machines, Drives and Power Systems
ELMA 2011, 21-22 October 2011, Varna, Bulgaria, pp. 338-341.

- 63 - 15th IGTE Symposium 2012
3D Finite Element Analysis of Induction Heating
System for High Frequency Welding
*Ilona I. Iatcheva, *Georgi H. Gigov , *Georgi C. Kunov and *Rumena D. Stancheva
*Technical University of Sofia, Kliment Ohridski 8, Sofia 1000, Bulgaria
E-mail: iiach@tu-sofia.bg
Abstract—The aim of the work is investigation of induction heating system used for longitudinal, high frequency pipe
welding. The problem was considered as 3D coupled electromagnetic and temperature field problem and has been solved
using finite element method and COMSOL 4.2 software package. Time harmonic electromagnetic and transient thermal fields
have been studied in order to estimate system efficiency and factors influencing on the quality of the welding process and
required energy.
Index Terms— finite element method, high frequency welding, 3D coupled field analysis.
small scale, carbon steel tubes and pipes. It consists of
spiral inductor, which induced a voltage across the edges
of the moving open pipe material. The induced voltage
causes high frequency currents, concentrated on the
surface layer due to the skin and proximity effects. The
currents flow along the two edges in opposite directions
in so called “V”-zone (Fig.2) to the point where they
meet, causing rapid heating of the metal and surface
melting. The weld squeeze rolls are used to apply
pressure, which forces the heated metal into contact and
forms welding bond.
I. INTRODUCTION
The induction heating is widely used in the heat
treatment of conducting details due to its advantages:
high quality and efficiency of the heating processes, good
accuracy in heating of certain zones in a short time and
clean operating conditions [ 1]-[4].
The aim of the present research is investigation of
induction heating system used for high frequency
longitudinal pipe welding. The main task is to determine
optimal factors and parameters influencing on quality of
the welding process and required energy: welding
frequency, welding speed, ‘vee’ angle, presence of the
ferrite impeder (inner and outer), tube thickness and etc.
The solution of the problem is based on the precise 3Dmodelling
and FEM analysis of the electromagnetic and
thermal processes, taking place in the investigated system.
Detailed determination of the electromagnetic and
temperature field distribution and its dependence on the
mentioned above parameters is important condition for
effective control and management of the welding process.
II. INVESTIGATED INDUCTION HEATING SYSTEM
The principal geometry of the investigated system is
shown in Fig.1.
squeeze
point of roll
closure
direction of
movement
impeder
core
inductor
steel pipe
cooling water
welded
bond
Figure 1: Geometry of the investigated induction system
The system is designed for high frequency welding of
Figure 2: In the “V”-zone HF currents flow along the two edges in
opposite directions.
The system includes also inner ferrite impeder, which
concentrates magnetic flux and improves the welding
efficiency. The cooling water flows inside the inductor
and impeder for system cooling.
As it can be seen from the geometry in Fig.1 the
impeder is located not along the pipe axe, but moved
closer to the welded region - i.e. the system is not
axesymmetric and has to be analysed as three
dimensional.
The system has been investigated and electromagnetic
and thermal processes have been modelled for the
parameters shown in Table I.
TABLE I
PARAMETERS OF THE SYSTEM
Parameter Value
Applied current I 1000 A
Voltage U 500 V
cos 0,1
Frequency f 200kHz 500kHz
End heating temperature 13001450 0 C
Cooling water
temperature
40 0 C

III. MATHEMATICAL MODEL OF THE COUPLED FIELD
PROBLEM
Mathematical modeling of the processes in the
investigated system for high frequency welding are based
on the analysis of coupled – electromagnetic and
temperature field distribution in the considered device.
As it has been already mention the geometry of the object
is a complex, nonsymmetric and electromagnetic and
thermal field have to be studied as three-dimensional.
The present work deals with modeling of the 3D time
harmonic electromagnetic field. The eddy current losses,
obtained in electromagnetic field analysis are field
sources in modeling of the transient thermal field
The electromagnetic field problem has been studied not
only in the system elements, but also in wide buffer zone
around the devise. It helps to define correct boundary
conditions in field modeling. In Fig.3 is shown
investigated region, used in electromagnetic field
modeling. It includes domains: 1- inductor; 2impeder;
3- welded pipe; 4- cooling water; 5- buffer
zone with air.
1
4
Figure 3: Investigation domains
2
5
3
Electromagnetic field distribution can be described with
equations (1) and (2):
-1
A
( A)
J e
(1)
t
E j
A V
(2)
where A is magnetic vector potential , J is current
density, E is electrical strength , V is scalar electric
potential, is electric conductivity and is magnetic
permeability.
The boundary conditions are A 0
for the buffer zone
boundaries.
The time varying electromagnetic field produces eddy
currents:
- 64 - 15th IGTE Symposium 2012
J jA
(3)
and corresponding Joule losses – source of the heating in
the region:
1
*
[ ] JJ
Q
2
(4)
The transient thermal field is modeled by equation:
T
. C (
kT
) Q (5)
t
where k is thermal conductivity , T is temperature, is
density, C is heat capacity and Q is heat source, obtained
in electromagnetic field analysis.
IV. FEM ANALYSIS - 3D COUPLED PROBLEM
Numerical simulation of the coupled - electromagnetic
and thermal fields was carried out using FEM and
COMSOL 4.2 package [4].
In Fig.4 is shown investigated system with the buffer
zone around it and Fig.5 presents FEM mesh, used in
solving the problem.
Figure 4: Investigated system with the buffer zone around it.
Figure 5: FEM mesh.

Some results, obtained in solving the problem for
frequency 300 KHz are shown in Fig. 6, Fig. 7, Fig. 8,
Fig. 9 and Fig. 10.
The analysis of electromagnetic field distribution
indicates that maximal value of the magnetic flux density
is about 0.19T. These values are reached in the “V” zone
and around the inductor. Two different cross sections
illustrate distribution of the magnetic flux density in the
system in Fig.6 and Fig.7.
Figure 6: Distribution of magnetic flux density in the
investigated region, f= 300 KHz.
Figure 7: Distribution of magnetic flux density along the
“V” zone, f= 300 KHz.
The results, obtained for current density distribution in
the entire region are shown in Fig.8. Two specific for the
problem cross sections - around “point of closure” and
spiral inductors are picking out. The maximal value is
- 65 - 15th IGTE Symposium 2012
1,13x10 9 A/m 2 . Current density distribution around the
“point of closure” is shown in Fig.9 and in Fig.10 around
the spiral inductor.
.
Figure 8: Current density distribution in the entire region
Figure 9: Current density distribution around “point of
closure”
V. CONCLUSION
3D-coupled electromagnetic and temperature field
problem and has been solved using finite element method
and COMSOL 4.2 software package in order to
investigate induction heating system for high frequency

pipe welding. The obtained temperature value around the
“point of closure” is about 1400 0 C.
REFERENCES
[1] R.Baumer, Y.Adonyi ”Transient High-Frequency Welding
[2]
Simulations of Dual-Phase Steels”, Welding Journal, October
2009, vol. 88, pp. 193 – 201
D.Kim, T. Kim, Y.Park, K.Sung, M.Kang, C.Kim, I.Lee and
S.Rhee, “Estimation of weld quality in high-frequency electric
resistance welding”, Welding Journal, March 2007, pp. 27 – 31.
[3] A. Shamov, I. Lunin, V. Ivanov, High frequency metal welding,
[4]
Leningrad, ‘Mashinostroenie”1977 (In Russian).
COMSOL Version 4.2 User’s Guide, 2011.
- 66 - 15th IGTE Symposium 2012

- 67 - 15th IGTE Symposium 2012
Optimization Algorithms in the View of State
Space Concepts
M. Neumayer∗ , D. Watzenig∗ , G. Steiner∗ , and B. Brandstätter †
∗Institute of Electrical Measurement and Measurement Signal Processing, Graz University of Technology,
Kopernikusgasse 24/4, A-8010 Graz, Austria, E-mail: neumayer@TUGraz.at
† Elin Motoren GmbH, Elinmotorenstrasse 1, A-8160 Preding/Weiz, Austria
Abstract—The working principles of optimization algorithms offer several characteristics which naturally arise in state
estimation, or more generally when dealing with state space systems. In this paper we will treat similarities between the
two disciplines and show how concepts of state estimation, including the incorporation of model uncertainty information,
can be used in optimization.
Index Terms—optimization, state space methods
I. INTRODUCTION
Numerical optimization is generally referred to as
solving a problem of form [1]
x ∗ = argminΨ(x) (1)
s.t. C(x) ≤ 0, (2)
where Ψ:R N → R 1 is called the objective function
and the vector x ∈ R N contains the variables of interest.
Possible constraints on the vector x are formulated by
the vectorial function C(x) as a set of equalities and
inequalities. By this the space of the feasible solutions
becomes a subspace of R N .
An enormous variety of algorithms and solution
strategies for such problems has been the output of
research activities in the past years. Yet it has to be
mentioned that the presented form of the optimization
problem is only a part of actual problems. I.e. the
discipline of multi objective optimization asks for an
optimal solution given several objective functions Ψi [2].
Such formulations are of importance in multi physical
problem scenarios. Another important class are robust
optimization approaches which aim for a stable solution
under the scenario of uncertainty or tolerances in the
objective function [3]. Thus, existing manufacturing
tolerances can be incorporated to optimization based
design process. A general distinction between the number
of the different (iterative) optimization algorithms used
today can be made by separating them into deterministic
and stochastic methods.
Deterministic methods most often make use of gradient
and curvature information of the objective function Ψ
in order to efficiently detect the minimum. Hereby efficiency
is typically defined by the number of evaluations
of Ψ. Classical first and second order deterministic methods
try to minimize Ψ by determining a descent direction
out of gradient or gradient and Hessian information. I.e.
the classical steepest descent method uses the iteration
xk+1 = xk − sg(xk), (3)
to find x∗ in a step by step approach. Hereby
g(xk) = ∇Ψ is defined as the gradient of Ψ with
respect to the elements of xk. Due to this nature the
result of deterministic methods can be strongly affected
by the starting point x0. Also local minima of Ψ will
result in a termination of the algorithm before the global
minima is found. Stochastic methods rely on some
sort of randomness to explore the parameter space in
search for the minimum. A main difference with respect
to deterministic methods is their ability to overcome
local minima of the objective function. For stochastic
optimization algorithms it has become common to let
them run for a certain time or a number of evaluations of
Ψ. Of course, hybrid algorithms have been proposed to
combine the advantages of the two classes of algorithms.
Although we have only pointed out the basics of some
fundamental concepts of optimization we can observe,
that in all algorithms some kind of evolution from the
vector xk to the vector xk+1 occurs. Modern system
theory uses so called state space models as a unified
framework to describe dynamical systems [4]. The general
form of a discrete time, nonlinear, time-variant state
space model is given by
xk+1 = F k(xk)+Bk(uk)+wk, (4)
yk = Hk(xk)+vk, (5)
where F k : R N → R N presents the system dynamics,
Bk : R L → R N describes the affect onto xk+1 due
to an input term uk ∈ R L , and Hk : R N → R M
describes a measurement process. The terms wk ∈ R N
and vk ∈ R M are referred to as process noise and
measurement noise. We observe some similarities
between the state space concept and the topics discussed
in concern with optimization. Yet we have to say that
state space methods and models follow a quite organized

scheme.
In this paper we will point out similarities between optimization
techniques and state space models and methods.
The paper is structured as follows. In section II we
give a short introduction about the state space concept.
In section II we review optimization techniques in the
sense of state space methods and present similarities
as well as mathematical tools potential for a general
description. Section IV lists several state space techniques
which relate with topics of optimization and thus could
potentially be used to improve optimization. Finally we
present an exemplary hybrid optimization scheme which
we derive from the state space view and demonstrate its
behavior using some of the suggested approaches.
II. THE STATE SPACE CONCEPT IN MORE DETAIL
uk Bk
wk
xk+1
z −1
F k
Hk
Fig. 1. Diagram of the state space model given by equation (4) and (5).
Figure 1 depicts the structure of a state space model
given by equations (4) and (5). The core purpose of a
state space model is to describe the evolution of the
state vector x over the time or the discrete time steps,
respectively. As can be observed by the equations (4)
and (5) or by figure 1, this evolution is determined
by a deterministic drift due to the dynamics of F k
and the input uk and a stochastic diffusion due to
the process noise wk. The system is referred to be an
autonomous system if B is zero. The function Hk
provides a deterministic measure about the internal state.
In addition the measurement noise vk acts as an additive
disturber. We can already observe that the state space
concept is able to provides several aspects which we
pointed out in the introductional part about optimization
algorithms in a natural way.
It should be mentioned that a state space model is
referred to be linear if all system components are matrices.
This class is of large importance as many technical
processes can be described by this.
A. State Space Methods
In this part of section II we want to give a
short introduction about two important disciplines in
association with state space models. These are state
estimation and state control.
State estimation is referred to the task to find an
estimate ˆxk of the state xk using the measurements y k ,
the input uk and the model.
vk
y k
- 68 - 15th IGTE Symposium 2012
State control is a special kind of feedback control
where the system input uk is formed as a function of
the state vector xk. A notable controller out of this class
is the dead beat control system. This approach enables
a control system to reach the steady state within a finite
number of iterations.
III. OPTIMIZATION AND STATE SPACE METHODS
Looking onto all points discussed so far we can consider
a relation between the measurement function H k
and the objective function Ψ. I.e. for a design problem
where one is interested to meet a desired output yd, Ψ
could be of form
Ψ(x) =(H k(x) − y d ) T W (Hk(x) − y d ) . (6)
Hereby the positive definite matrix W presents a weighting
matrix. From a system theoretic point of view we
could consider the function Ψ as a (nonlinear) control
plant of MISO (multiple input single output) type.
A. Classical Deterministic Methods Reviewed
Classical deterministic optimization methods like the
already mentioned steepest descent method (see equation
(3)) take use of local gradient or curvature information
of the function Ψ. While the steepest descent algorithm
just takes use of the gradient information the well known
Gauss-Newton (GN) method defined by
xn+1 = xn + sG −1
k gk, (7)
takes use of the Hessian G matrix which provides curvature
information about Ψ to improve the convergence
behavior. In both schemes, the steepest descent method
and the GN method, the system matrix F isgivenbythe
identity matrix I. For objective functions Ψ of form (6),
the practical realization of the GN method is given by
xn+1 = xn − s(JJ T ) −1 Jr (8)
where J is the Jacobian of the system H with respect
to the state vector x. Herebyr =(y − y d) defines the
residual vector of the output of F with respect to y d .
The gradient g = ∇xΨ of the objective function (6) with
respect to x (to keep the notation short we set the matrix
W to be the identity matrix I) isgivenasg = J(y−y d)
and (JJ T ) approximates the Hessian G [1].
JJ T −1
z −1
I
−sJ
H k
Fig. 2. State space representation of a second order scheme.
y k
−y d
Figure 2 depicts the GN scheme as a control system for
the objective function (plant) Ψ following equation (8).
For the steepest descent method the matrix B is replaced
by the identity matrix I. Note, that all matrices depend

on the iteration index k. A control system with this
property is referred to as a time varying control system.
We observe that neither the steepest descent algorithm
nor the GN method are state space control systems as
these methods do not take use of the state vector x.
However, we can observe a closed loop scheme in figure
2. It is hard to argue whether we see the steepest descent
algorithm as a drive system or as a closed loop control
system as for B = −g and s replacing the input u
(scalar) no closed loop is required. However, with respect
to the different input matrix B the powerfulness of the
GN-method becomes clear from a system theoretic point
of view.
B. Stochastic Methods Reviewed
With the availability of more and more computational
power stochastic optimization methods have become
of increased interest for many practical problems.
Interesting issues for the application of stochastic
methods is their ability to overcome local minima, and
the not given necessity for derivative information. This
is of concern for not differentiable or not continuous
problems
In contrast to deterministic methods, stochastic
methods most often rely on a set of N individual vectors
x N which explore the objective function on their own.
Over the time a mutual exchange of information from
the the different realizations x N is performed which
mixes the individuals. Concepts about the individual
exploration of each individual on Ψ as well as the
exchange of mutual information between the individuals
is often based on concepts of nature like evolution
principles resulting in the class of genetic algorithms
(GA). I.e. certain elements of two arbitrarily selected
vectors xi and xj are exchanged, replaced by a weighted
mean or just individually disturbed by a random variable.
A contrastable aspect with respect to the behavior of
deterministic methods is the fact, that the combining
principles do not automatically remove the weakest
individual (the realization with the highest value of Ψ).
Instead also the strongest individual can be removed by
some random procedure. Exactly this property enables
the behavior that stochastic methods can overcome local
minima. Other well known strategies for stochastic
optimization are particle swarm optimization (PSO),
nitching evolution techniques or differential evolution
(DE) [5]. Also the behavior of an ant colony or bacteria
in a nutrient solution [6] have been used as strategies to
find a solution minimizing Ψ.
The enormous variety of differently labeled stochastic
algorithms [7] makes it often hard to distinguish the
differences between. More important it is hard to
charge the efficiency of the different methods and their
suitability for different applications. In the following we
will provide an approach to present several aspects of
- 69 - 15th IGTE Symposium 2012
stochastic optimization within the unified framework of
state space techniques.
In state space models randomness has the unified
entrance into the system formulated by the process noise
w. By setting the deterministic input vector u to zero the
resulting system becomes an autonomous system. While
different stochastic optimization strategies are originated
by more or less random inspiritments, system theory
takes use of probabilistic methods to describe the behavior
in concern with randomness. Hereby any random
process is described by a probability density function
(pdf) denoted by π(·). The mathematical framework used
to describe stochastic behavior is based on Bayes law
π(x|y) = π(y|x)π(x)
∝ π(y|x)π(x), (9)
π(y)
where π(y|x) is referred to as the likelihood function
and π(x) is referred to as prior. The evidence π(y) has
the role of a normalization constant and can be skipped
leading to the right hand side formula of the posterior
distribution π(x|y). The likelihood function provides a
probability measure for x originating a certain output y.
The prior π(x) gives a probability statement about x.
We can already link this concepts to the optimization
problem given by equation (1) and the constraints
formulated in equation (2), as the likelihood function
obviously is able to express C(x) by becoming zero for
infeasible solutions. However, this concept also enables
the possibility of a continuous measure for the state x,
i.e. we can incorporate ”gray regions” for the solution.
The understanding of the likelihood is maybe not that
obvious. For easier explanation we write the likelihood
corresponding to the objective function (6) as
π(yd |x) ∝ exp − (Hk(x) − yd ) T
W (H k(x) − yd ) .
(10)
The likelihood function is the exponential of the negative
objective function but it states Ψ as a probability measure.
It has to be noted that 0 <
N
exp(−Ψ(x))dx < ∞
has to hold in the Lebesgue sense, to form a likelihood
function from an objective function. Such a formulation
is known from simulated annealing (SA). Hereby a
stochastic algorithm seeks for the modes (maxima) of
the function exp(−Ψ(x)/T ), where T is an artificial
temperature which decreases over time. Note, that due
to the temperature T , SA is different with respect to
Bayesian inference as the likelihood has a physical
meaning where no term like T occurs. Given all these
aspects stochastic optimization can be fully seen in the
context of state estimation and we can work out some
conceptual ideas that are used in state estimation in the
next section.
The exchange of mutual information depends on a so
called resampling scheme which stays outside the state
space model. While different stochastic methods have
brought up a variety of exchange schemes also state

estimation methods have brought up unified methods like
residual, stratified, or systematic resampling, etc. [8]. We
will not focus on these aspects of stochastic optimization
methods at this point, but we will provide a description
about the stochastic diffusion of states in the state space
view of optimization.
While deterministic methods select the state update
from gradient or curvature information in order to decrease
Ψ and thus follow strictly deterministic rules,
the probabilistic change is summarized by means of
pdf’s π(·). State space theorists have developed the
ChapmanKolmogorov equation
π(xk|yd )= π(xk|xk−1)π(xk−1|yd )dxk−1, (11)
R N
to provide a probabilistic measure about the state evolution
given the current state and its posterior. While
equation (11) is not hard to derive using the mathematical
tool of marginalization, it provides two interesting insides
about the update in stochastic optimization methods.
• The state update is described by π(xk|xk−1) and
does not depend on the current value of the objective
function.
• The update probability depends on π(xk−1|y d ),but
there is no guarantee that xk−1 will be changed.
The transition kernel π(xk|xk−1) describes the probability
of the state exchange from state xk−1 to the state xk.
A remarkable point about this formulation is the fact, that
the update is independent from the current value of Ψ or
the posterior. This is an important fact that explains the
powerfulness of stochastic methods. If the kernel would
depend on Ψ, stochastic methods would end with the
same stalling behavior in local minima as deterministic
methods do, as then a deterministic drift is present.
In most cases the kernel π(xk|xk−1) is even reduced
to π(xk). The pdf π(xk−1|y d) in equation (11) induces
another important principle in stochastic optimization
which can be directly connected to the mutual information
exchange. It states, that the update of the state due
to the proposal kernel is not guaranteed. Instead we can
see the result π(xk|y d) only provides a relative number
for the new state π(xk) to be accepted.
IV. STATE SPACE METHODS FOR OPTIMIZATION
In this section we want to discuss some more state
space concepts and their use for stochastic optimization.
We have selected these methods as we see them
to be important with nowadays needs. State estimation
techniques are among the algorithms which have seen
one of the strongest developments in the past decades.
The early origin was given by the Apollo space flight
programm in the 1960’s where the Kalman filter has
seen it’s breakthrough. Since then both, single point
and population-based methods have been developed, to
regain knowledge from the hidden states of a system
given the actually observed function values for an optimal
designed objective function in order to recover x from
- 70 - 15th IGTE Symposium 2012
noisy observations. In this sense we first have to discuss
the meaning of the likelihood function π(x|y d) in more
detail. Following the definition of a multivariate Gaussian
random variable y
y ∝ exp −(y − μ) T Σ −1 (y − μ) , (12)
where μ expresses the mean and Σ is the covariance
matrix, we observe, that the likelihood function has the
mean of a Gaussian distribution expressing uncertainty
about y. In this sense the measurement noise v becomes
relevant for a first as the likelihood function expressed
this noise in terms of a probability measure. This will
lead us directly to the aspects brought in the following
subsection.
The consequent use of this approach brought up powerful
stochastic state estimation algorithms like state observers,
sequential Monte Carlo methods like the already
mentioned Kalman filter or Particle filters, or even more
powerful Markov chain Monte Carlo (MCMC) methods.
A. Enhanced Error Model
A matter of concern with the solution of physical
motivated optimization problems are the computational
costs in concern with the evaluation of the objective
function Ψ. This especially holds if the underlying problem
requires the solution of partial differential equations
(PDE’s) which has to be done by numerical methods
like the finite element method (FEM). Recently the
use of approximation techniques has become popular in
both, state estimation and optimization [9], [10]. Hereby
the computational costly evaluation of H k is replaces
by a cheap approximation or surrogate function H ∗ k.
Subsequently this leads to the cost function Ψ∗ due to
the approximation error
e = H ∗ k − H k. (13)
We can reformulate the relation between H k and H ∗ k to
H ∗ k = Hk +(H ∗ k − H k) =Hk + e. (14)
This is an interesting formulation as we can look on
the approximation error e as an additive error similar to
the measurement noise v depicted in figure 1. Although
the approximation error e depends on the state x, and
thus is a deterministic error, we can think about a
probabilistic description about e in the concept of a
Gaussian distribution. This is an approach often taken
in several fields of state estimation and system theory.
It ends up exactly in the idea covered by the so called
enhanced error model [11]. Although the approximation
error e is of deterministic nature a probabilistic model is
built from samples about the state space R N . Then the
likelihood function π ∗ (y d|x) becomes
π ∗ (y d|x) ∝ exp −(y ∗ − y d + μ e) T Σ −1
e (y ∗ − y d + μ e) ,
(15)
and the optimization can be performed on this
computational less costly function. Given the degree

of accuracy of the approximation H ∗ k the solution can
be seen as good as a solution obtained by Hk, orthe
approximation approach can be used to find a good
initial solution which can be refined in less optimization
steps using the accurate model.
In general the determination of the mean μ e and the
covariance matrix Σe requires a large number of samples.
However, during the setup and model testing phase for the
optimization problem typically enough data is generated
to describe e in the presented way.
B. Hybrid Schemes
Another useful aspect about the use of state space
schemes for optimization is the natural possibility to incorporate
both, deterministic and stochastic methods for
building hybrid optimization schemes. This can be easily
done by enabling the input vector uk and building an
outer feedback system as discussed in subsection III-A.
The natural representation of the interaction between
the deterministic drift and the stochastic interaction is
therefore of interest, as it illustrates the powerfulness
of the combination. I.e. if only some elements of the
gradient g are available because the function Ψ is not
steady with respect to this variables, we can only use
the available gradient information for the input vector
u. The other components of x are updated by the
stochastic algorithm. In addition, an outer resampling
scheme retains the property of a stochastic optimization
scheme to overcome local minima.
C. Robust Schemes
Uncertainty is in many aspects a concerning topic in
state estimation. This is given due to the fact, that models
often do not cover all physical aspects due to reduction.
Also optimization engineers have developed robust target
functions in order to find solutions insensitive with
respect to parameter variations of x [3]. Such robust
objective functions are typically of form
min max Ψ(x, ξ), (16)
x ξ
where ξ describes an immanent given uncertainty in the
parameters. Most often the absolute value of ξ is limited.
Robust state estimation has brought up the H∞ concept
[4], where the estimation error e = x − ˆx is minimized
using an approach of form
min max J (x, ˆx, v, w). (17)
ˆx v,w
Hereby no limitations about the process noise w and
the measurement noise v are assumed. The H∞ filter
seeks for the best estimate under worst case conditions.
Mostly game theoretic approaches are used to formulate
the function J . We pointed this out, as control scientist
have gained a lot of experience in the field and there
might be useful aspects for optimization.
- 71 - 15th IGTE Symposium 2012
V. A NUMERICAL EXAMPLE
To provide a numerical example about the presented
considerations of state space methods for optimization
we want to present a simple optimization problem
consisting of an inverse problem for a resistor network
example. Figure 3(a) depicts the resistor network under
investigation. The black lines illustrate resistors with
a value of R1 = 1Ω. The gray colored lines mark a
circular disk of radius r where resistors with a value of
R2 are placed. Hereby the mapping between the circle
radius and the resistor values is discontinuous by the
way, that the resistor has to be fully placed inside the
circle. It is now aim to find the radius r of the circle
and the resistor value R2 from some electrical boundary
measurements. These measurement built the vector yd. A problem of this kind is a classical inverse problem
where we aim on the determination of the state vector
x = T r R2 from measurements yd.
Figure 3 exemplary depicts a part of the cost function.
Hereby the R2 and r were set to R2 =0.5Ω and r =
0.5m. The corner length was set to r =1mand was
discretized by 40 resistors. A current is injected at the
upper left corner and 5 equidistant measurement points
(ampere meters) are connected to the lower edge.
(a) Resistor network.
0.7
0.6
0
x 10
8
7
6
5
4
3
2
1
−4
Fig. 3. Test example and objective function.
Ψ
0.5
0.4
r (m)
0.55
0.5
0.45
0.3
0.2 0.4
R (Ω)
2
(b) Objective function Ψ.
We now want to apply a hybrid optimization approach
where the corner points about the algorithm can be stated
by the following:
• We use a population based scheme.
• We use gradient information about the resistor
value R2.
• We work on a reduced model (only half the number
of resistor elements per edge).
In state estimation such an algorithm belongs to the class
of sequential Monte Carlo (SMC) methods and is mostly
referred to as Particle filter (PF) [12].
Arguable one of the most interesting points in this list
is the use of a reduced model to solve the optimization
problem. Figure 4(a) depicts the objective function (6)
(W was set to be the identity matrix) when using the
reduced model for solving the optimization problem with
data from the fine model. One can obtain, that the depicted
part of the objective function does not even include
a minima. Figure 4(b) depicts the likelihood of form (15),
using an enhanced error model. As we can see, the point

where the likelihood function has its maxima presents
the true solution. Thus, if our optimization algorithm is
designed to minimize the corresponding objective function
is should be possible to find the solution although
working on the reduced model. Figure 5 depicts the
Ψ *
0.2
0.15
0.1
0.05
0
0.7
0.6
0.5
0.4
0.3
r (m)
0.7
0.6
0.5
0.4
0.3
R (Ω)
2
(a) Objective function Ψ ∗ .
π(r,R 2 )
1
0.8
0.6
0.4
0.2
0
0.7
0.6
0.5
0.4
0.3
r (m)
0.3
0.4
R 2 (Ω)
(b) Posteriori probability π ∗ (r, R2).
Fig. 4. Determination of r and R2 using a reduced model.
behavior and the result of the proposed hybrid scheme
for the given problem using the reduced model for the
solution. Figure 5(a) depicts the state of the population.
As can be seen, the population is clustered around the
correct solution. Hereby the background color depicts
the objective function for the fine model but as stated
the coarse model is used! Figure 5(b) and figure 5(c)
depict the decrease of the objective function and the
increase of the likelihood function, respectively. The dots
illustread the spread of the population. As can be seen
both, the likelihood function and the objective function
can become smaller or larger, respectively. Thus, the
property of stochastic methods is given.
Ψ(r,R 2 )
0.035
0.03
0.025
0.02
0.015
0.01
0.005
r (m)
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0
1 2 3 4 5 6
Iteration
7 8 9 10
(b) Objective function.
0.3
0.3 0.4 0.5 0.6 0.7
R (Ω)
2
(a) Particles.
Fig. 5. Output of the particle filter.
π(r,R 2 )
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1 2 3 4 5 6
Iteration
7 8 9 10
(c) Posteriori probability.
VI. OUTLOOK
In this paper we demonstrated a state space view on
optimization algorithms. Both, deterministic and stochastic
methods were exploited in the content of the unified
state space representation. We demonstrated that
0.5
0.6
0.7
- 72 - 15th IGTE Symposium 2012
deterministic approaches can be considered as standard
feedback systems, whereas stochastic methods can be
directly linked to state estimation. We explored features
of stochastic state estimation using Bayes law and subsequently
demonstrated the usefulness of state estimation
techniques for optimization. All our considerations are
summarized in a hybrid optimization algorithm working
on a reduced model where we demonstrated the natural
interaction of deterministic and stochastic methods using
state space descriptions. Further research will focus in
two directions. First, we would extend the presented
hybrid scheme using some more sophisticated methods.
Second we consider work on a formal description of
different stochastic algorithms using methods from probability
theory.
REFERENCES
[1] R. Fletcher, Practical Methods of Optimization; (2nd Ed.), Wiley-
Interscience, New York, USA, 1987.
[2] L. dos Santos Coelho and P. Alotto, Multiobjective Electromagnetic
Optimization Based on a Nondominated Sorting Genetic Approach
With a Chaotic Crossover Operator, IEEE Transactions on Magnetics,
vol.44, no.6, pp.1078-1081, 2008.
[3] P. Alotto, C. Magele, W. Renhart, A. Weber, G. Steiner Robust
target functions in electromagnetic design, COMPEL: The International
Journal for Computation and Mathematics in Electrical
and Electronic Engineering, Vol. 22 Iss: 3, pp.549 - 560, 2003.
[4] D. Simon, Optimal state estimation, Kalman, H∞ and nonlinear
approaches, Wiley - Interscience, John Wiley & Sons, Inc., New
Jersey, 2006.
[5] R. Storn and K. Price, Differential evolution - a simple and efficient
heuristic for global optimization over continuous spaces, Journal
of Global Optimization 11: pp.341-359, 1997.
[6] L. dos Santos Coelho, C. da Costa Silveira, C.A. Sierakowski, and
P. Alotto, Improved Bacterial Foraging Strategy Applied to TEAM
Workshop Benchmark Problem, IEEE Transactions on Magnetics,
vol.46, no.8, pp.2903-2906, Aug. 2010.
[7] O. Hajji, S. Brisset, and P. Brochet, Comparing stochastic optimization
methods used in electrical engineering, Systems, Man
and Cybernetics, 2002 IEEE International Conference on , vol.7,
no., pp. 6 pp. vol.7, 6-9 Oct. 2002.
[8] R. Douc, O. Cappe, and E. MoulinesComparison of resampling
schemes for particle filtering, In 4th International Symposium on
Image and Signal Processing and Analysis (ISPA), pp.64-69, 2005.
[9] Albunni, M.N.; Rischmuller, V.; Fritzsche, T.; Lohmann, B.; ,
Multiobjective Optimization of the Design of Nonlinear Electromagnetic
Systems Using Parametric Reduced Order Models, IEEE
Transactions on Magnetics, vol.45, no.3, pp.1474-1477, March
2009.
[10] A. I. Forrester, A. Sóbester and A. J. Keane, Engineering Design
via Surrogate Modelling A Practical Guide, Wiley, 2008.
[11] J. P. Kaipio and E. Somersalo, Statistical and computational
inverse problems, New York: Applied Mathematical Sciences,
Springer, 2004.
[12] M.S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp,
A tutorial on particle filters for online nonlinear/non-Gaussian
Bayesian tracking IEEE Transactions on Signal Processing 50 (2),
pp.174188, 2002.

- 73 - 15th IGTE Symposium 2012
Quasi TEM Analysis of 2D Symmetrically Coupled
Strip Lines with Finite Grounded Plane using HBEM
*Saša S. Ilić, *Mirjana T. Perić, *Slavoljub R. Aleksić and *Nebojša B. Raičević
*University of Niš, Faculty of Electronic Engineering of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia
E-mail: sasa.ilic@elfak.ni.ac.rs
Abstract—The hybrid boundary element method (HBEM), based on combination between equivalent electrodes method
(EEM) and boundary element method (BEM), is applied for characteristic parameters determination of symmetrically coupled
strip lines with a finite width grounded plane. Even and odd modes are considered in the paper. All results for the characteristic
impedance and the effective dielectric permittivity are compared with the finite element method (FEM).
Index Terms—Characteristic impedance, Equivalent Electrodes Method (EEM), Finite Element Method (FEM), Hybrid
Boundary Element Method (HBEM).
strip lines parameters, when the strip line is above an infinite-width
grounded plane [14]. The HBEM can be also
applied to analysis of corona effects [15] and metamaterial
structures [16]. A problem of symmetrically
coupled strip line placed above infinite grounded plane is
investigated in [17].
The HBEM is applied, in this paper, to calculate the
even- and odd- mode characteristic impedance of 2D
symmetrically coupled strip lines with finite grounded
plane, shown in Fig. 1. The quasi TEM analysis is used in
this paper.
I. INTRODUCTION
Over the years, many authors have analyzed
symmetrically and asymmetrically coupled or ordinary
strip lines with width-limited dielectric substrate using
numerous numerical and analytical methods [1]-[9]. The
variational method [1], the Garlekin’s method, the method
of moments [2]-[4], the boundary element method [5], the
conformal mapping, the moving perfect electric wall
method [6]-[9] etc. are some of the commonly used
methods. On the other side, the problem of the widthlimited
microstrip grounded plane has not been so often
investigated, although these forms of microstrips are
typical in practice. In [7]-[9] the microstrip line with
finite-width dielectric and grounded plane was analyzed.
A so-called moving perfect electric wall method (MPEW)
in conjunction with the conformal mapping method
(CMM) was applied in those papers.
An application of boundary element method (BEM)
usually contains singular and nearly singular integrals
whose evaluation is difficult although original problems
are not singular. In order to avoid numerical integrations,
it is possible to substitute small boundary segments by
total charges placed at their centres. The Green’s function
for the electric scalar potential of the charges, placed in
the free space at the boundary of two dielectrics, is used
and the proposed method is called the hybrid boundary
element method (HBEM) [10-17].
This method presents a combination of BEM and
equivalent electrodes method (EEM). The basic idea is in
replacing an arbitrary shaped electrode by equivalent
electrodes (EEs), and an arbitrary shaped boundary
surface between any two dielectric layers by discrete
equivalent total charges per unit length placed in the air.
The basic Green’s function for the electric scalar potential
of the charges placed in the free space at the boundary
surface of two dielectrics is used. The method is based on
the EEM, on the point-matching method (PMM) for the
potential of the perfect electric conductor (PEC)
electrodes and for the normal component of the electric
field at the boundary surface between any two dielectric
layers.
The HBEM is applied, until now, to solving
multilayered electromagnetic problems [10], grounding
systems [12], electromagnetic field determination in vicinity
of cable terminations [13], as well as to calculation of
Figure 1: Symmetrically coupled strip line with finite grounded plane.
Symmetrically coupled strip lines can be used as basic
elements for filters, phase shifters, directional couplers,
baluns and combiners [18].
II. THEORETICAL BACKGROUND
The HBEM is applied and corresponding model is formed,
Fig. 2.
Figure 2: Corresponding HBEM model.

Using the existing symmetry, the electric scalar potential
of whole system from Fig. 2 is determined:
(e (e, o)
2
B ln l
4
B ln l
3
B ln l
Ki
A
i 1 k 1
( x
Ki
i 3 k 1
( x
Mi
i 1 m 1
( x
q
0
d
ik
2
q
2
2
dik
a
ik
q
x
x
0
a ik
t
im
x
0
t im
)
)
ln
)
2
2
2
ln
ln
( x
( y
( x
( y
( y
( x
x
y
dik
y
y
dik
x
a ik
)
t im
)
a ik
x
)
)
2
2
)
2
t im
)
2
2
2
,
( y
( y
( y
y
dik
y
)
a ik
y
2
)
t im
where the coefficients A and B have following values:
0,
odd(o)
mode;
A
1,
even(e)
mode.
B
1,
odd(o)
mode;
1,
even(e)
mode .
The electric field is E grad( g ( ) . The total number of
unknowns N tot , will be denoted by:
4
3
N K M A .
tot
i
i 1 i 1
A relation between the normal component of the
electric field and the total surface charges is given with
Eq. (2):
n ˆi ( 0
Eim
)
0
( 0
)
t
im i , t im i
q
t im i
,
l im
(2)
where i M m , , 1 , 3 , 2 , i 1 , nˆ i ( nˆ 1 nnˆ
ˆ2
yyˆ
ˆ , nˆ 3 xxˆ
ˆ )
are unit normal vectors oriented from the layer
the layer 0 .
towards
Using the PMM for the potential of the perfect
conductors given by (1), the PMM for the normal
component of the electric field (2), and the electrical
neutrality condition (3) (only for the even mode!), it is
possible to determine unknown free charges per unit
length on conductors, the total charges per unit length on
the boundary surfaces between two dielectric layers and
the unknown constant 0 .
The electrical neutrality condition is:
2
Ki
4 Ki
q d dik
qa
a ik 0 0.
(3)
i 1 k 1 i 3 k 1
After solving the system of linear equations, it is
possible to calculate the capacitance per unit length of the
i
2
)
2
(1)
- 74 - 15th IGTE Symposium 2012
strip line given by (4):
K1
K3
(e, o) 1
C qd1k
qa
3k
. (4)
U
k 1 k 1
With the developed program code, the characteristic
impedance of the symmetrically coupled strip line is
calculated as
(e, o) (e, o) eff ef (e, o)
Z c Zc
0 / r ,
where
ef eff (e, o) ( (e, o) ( (e, o)
r
C
/C 0
(e, o)
is the effective dielectric permittivity, and Z c0
is the
characteristic impedance of the symmetrically coupled
strip line without dielectric layer (free space), for even (e)
and odd (o) modes, respectively.
In order to verify the obtained numerical results for the
characteristic impedance and the effective dielectric
permittivity, the finite element method (FEM) [19] is
used.
III. RESULTS
The results convergence and computation time for the
even and odd modes can be noticed from Table I, for
parameters: r 3 , d / w1
4 , h / d 0 0.
5 , t 1 / w1
0
. 1 ,
s / w1
1
. 0 , w 2 / w1
6 6.
0 and t 2 / t1
2
. 0 , where N tot
is the total number of unknowns.
N tot
TABLE I
CONVERGENCE OF RESULTS AND CPU TIME
Even mode Odd mode
eff ef
r
Zc
[ ] eff ef
r
Zc
[ ]
t(s)
298 2.119 158.860 1.817 77.293 4.4
370 2.120 158.871 1.823 77.207 6.9
444 2.121 158.901 1.827 77.162 9.7
585 2.123 158.906 1.832 77.093 16.7
655 2.123 158.917 1.833 77.077 20.9
726 2.124 158.905 1.835 77.048 25.7
800 2.124 158.916 1.836 77.039 31.5
872 2.124 158.916 1.837 77.026 38.8
940 2.125 158.905 1.838 77.007 45.4
1014 2.125 158.914 1.839 77.003 51.4
1085 2.125 158.904 1.840 76.987 58.1
1155 2.125 158.911 1.840 76.986 65.4
1225 2.125 158.917 1.840 76.984 74.1
1296 2.126 158.909 1.841 76.973 86.5
1370 2.125 158.915 1.841 76.972 97.6
First, a very good convergence of values of both
parameters is achieved for the both modes. Second, a
computation time was much shorter comparing to the time
required by FEM: we needed up to 97.6 seconds for the
system of 1370 unknowns, while FEM for solving the
same problem took about 15 minutes with a few hundreds
of thousands of finite elements.
Equipotential contours and distributions of polarized

charges per unit length along boundary surface are shown
in Figs. 3-6 (even and odd modes, respectively) for para-
meters:
r 3 , d / w1
4 , h / d 0 0.
5,
t 1 / w1
0 0.
1,
s / w1
1
. 0 , w 2 / w1
6
. 0 and t 2 / t1
2
. 0 .
Figure 3: Equipotential contours (Even mode).
Figure 4: Equipotential contours (Odd mode).
t 2
t1
1
2
3
4
- 75 - 15th IGTE Symposium 2012
Figure 5: Distribution of polarized charges per unit length along
boundary surface (Even mode).
Figure 6: Distribution of polarized charges per unit length along
boundary surface (Odd mode).
TABLE II
COMPARED RESULTS FOR CHARACTERISTIC IMPEDANCE OF STRIP LINE VERSUS 2 1 t t AND h d FOR PARAMETERS:
r 3 , d / w1
4 , t 1/
w1
0 0.
05 , s / w1
1
. 0 AND w 2 / w1
6 6.
0 .
h
d
Even mode Odd mode
HBEM FEM HBEM FEM
eff ef
r
Zc
[ ] eff ef
r
Zc
[ ] eff ef
r
Zc
[ ] eff ef
r
Zc
[ ]
0.2 2.3967 84.950 2.3969 84.864 2.0470 64.218 2.0558 63.942
0.4 2.2182 138.000 2.2185 137.829 1.9009 77.198 1.9120 76.805
0.6 2.0856 182.116 2.0862 181.851 1.8642 81.055 1.8757 80.626
0.8 1.9849 220.428 1.9863 220.066 1.8550 82.450 1.8676 81.983
1.0 1.9059 254.371 1.9076 253.892 1.8533 83.020 1.8666 82.318
0.2 2.3938 84.935 2.3965 84.783 2.0470 64.215 2.0555 63.948
0.4 2.2144 137.901 2.2172 137.624 1.9008 77.192 1.9119 76.801
0.6 2.0817 181.901 2.0844 181.526 1.8642 81.048 1.8757 80.618
0.8 1.9810 220.107 1.9844 219.602 1.8543 82.457 1.8676 81.981
1.0 1.9021 253.950 1.9056 253.368 1.8533 83.016 1.8670 82.545
0.2 2.3923 84.907 2.3957 84.741 2.0469 64.213 2.0553 63.949
0.4 2.2122 137.786 2.2156 137.492 1.9008 77.188 1.9119 76.794
0.6 2.0793 181.675 2.0827 181.265 1.8641 81.043 1.8755 80.620
0.8 1.9785 219.777 1.9826 219.241 1.8543 82.453 1.8676 81.978
1.0 1.8997 253.521 1.9038 252.901 1.8532 83.013 1.8666 82.554
0.2 2.3912 84.878 2.3942 84.739 2.0468 64.212 2.0557 63.939
0.4 2.2105 137.677 2.2139 137.412 1.9007 77.184 1.9118 76.792
0.6 2.0775 181.465 2.0809 181.083 1.8641 81.039 1.8756 80.600
0.8 1.9766 219.468 1.9809 218.918 1.8543 82.449 1.8675 81.973
1.0 1.8978 253.120 1.9021 252.483 1.8532 83.010 1.8666 82.548

- 76 - 15th IGTE Symposium 2012
TABLE III
COMPARED RESULTS FOR CHARACTERISTIC IMPEDANCE OF STRIP LINE VERSUS 2 1 w w FOR PARAMETERS:
r 3 , d / w1
4 , t 1/
w1
0 0.
05 , h / d 0
. 5 , s / w1
1
. 0 AND t 2 / t1
2
. 0 .
Even mode Odd mode
w 2 HBEM FEM HBEM FEM
w1
eff ef
r
Zc
[ ] eff ef
r
Zc
[ ] eff ef
r
Zc
[ ] eff ef
r
Zc
[ ]
4.5 2.2100 168.392 2.2120 168.080 1.8799 80.064 1.8915 79.655
5.0 2.1823 165.314 2.1845 165.004 1.8784 79.885 1.8899 79.479
5.5 2.1605 162.805 2.1630 162.490 1.8772 79.743 1.8888 79.344
6.0 2.1430 160.759 2.1458 160.438 1.8749 79.656 1.8876 79.242
8.0 2.0981 155.589 2.1026 155.229 1.8718 79.427 1.8858 78.982
10.0 2.0753 152.991 2.0805 152.606 1.8670 79.352 1.8848 78.898
15.0 2.0492 150.340 2.0575 149.885 1.8598 79.343 1.8598 79.343
TABLE IV
COMPARED RESULTS FOR CHARACTERISTIC IMPEDANCE OF STRIP LINE VERSUS 1 1 w t FOR PARAMETERS:
r 3 , d / w1
4 , w 2 / w1
6 6.
0 , h / d 0 0.
5 , s / w1
1
. 0 AND t 2 / t1
2
. 0 .
Even mode Odd mode
t1
HBEM FEM HBEM FEM
w1
eff ef
r
Zc
[ ] eff ef
r
Zc
[ ] eff ef
r
Zc
[ ] eff ef
r
Zc
[ ]
0.01 2.1768 162.282 2.1632 162.257 1.9031 82.453 1.9251 81.744
0.02 2.1600 161.998 2.1583 161.804 1.8971 81.628 1.9147 81.052
0.03 2.1526 161.595 2.1537 161.316 1.8899 80.913 1.9051 80.416
0.04 2.1474 161.167 2.1496 160.857 1.8827 80.257 1.8963 79.801
0.05 2.1430 160.759 2.1458 160.438 1.8749 79.656 1.8876 79.242
0.06 2.1388 160.362 2.1422 160.021 1.8679 79.074 1.8800 78.670
0.07 2.1352 159.976 2.1388 159.641 1.8610 78.519 1.8754 78.222
0.08 2.1319 159.605 2.1356 159.271 1.8543 77.985 1.8645 77.640
0.09 2.1288 159.250 2.1264 158.700 1.8447 77.470 1.8577 77.130
0.10 2.1258 158.909 2.1293 158.592 1.8413 76.973 1.8507 76.655
To validate the accuracy of the presented method, a
comparison is made with the finite element method results
obtained using the FEM software [19]. Those results are
shown in Tables II-IV.
As presented in the tables, numerical results for the
effective dielectric permittivity and the characteristic
impedance obtained using the HBEM are obviously in
very good agreement with the FEM values (with few
hundreds of thousands finite elements) with divergence
less than 0.4% for the most of the cases.
Distributions of characteristic impedance versus s / w1
for different values of dielectric permittivity r are
shown in Figs. 7 and 8, for:
d / w1
4 , h / d 0 0.
5,
t 1 / w1
0 0.
1 , w 2 / w1
6 6.
0
and t 2 / t1
2 2.
0 .
Fig.7 shows that increasing the values of parameter
s / w1,
decreasing the characteristic impedance for even
mode. But, for the odd mode, Fig. 8, increasing the
parameter s / w1,
increasing the characteristic impedance
too. The lowest values for the characteristic impedance
are obtained for the highest value of dielectric
permittivity.
The obtained values are compared with the FEM
results, also. A very good results agreement is obtained.
Figure 7: Distribution of characteristic impedance versus s / w1
for
different values of dielectric permittivity (Even mode).
IV. CONCLUSION
A newly developed hybrid boundary element method is
applied to quasi TEM analysis of 2D symmetrically
coupled strip lines with finite grounded plane. Two quasistatic
parameters are calculated: effective dielectric
permittivity and characteristic impedance of the line. We
have compared the values of parameters with those
obtained by the finite element method. A very good
agreement of the results is achieved: maximal relative

error of the characteristic impedance is less than 0.4%.
Figure 8: Distribution of characteristic impedance versus s / w1
for
different values of dielectric permittivity (Odd mode).
All calculations were performed on computer with dual
core INTEL processor 2.8 GHz and 4 GB of RAM.
This method can be successfully applied to static,
stationary and quasi-stationary electromagnetic fields, as
well as to the analysis of the fields in mechanics, fluid
dynamics, conductive heat flow etc.
Acknowledgement
This research was partially supported by funding from
the Serbian Ministry of Education and Science in the
frame of the project TR 33008.
REFERENCES
[1] T. Fukuda, T. Sugie, K. Wakino, Y.-D. Lin, and T. Kitazawa,
“Variational method of coupled strip lines with an inclined
dielectric substrate,” in Asia Pacific Microwave Conference –
APMC 2009, December 7-10, 2009, pp. 866-869.
[2] R. F. Harrington, Field computation by Moment Methods. New
York: Macmillan, 1968.
[3] T. G. Bryant and J. A. Weiss, “Parameters of microstrip
transmission lines and of coupled pairs of microstrip lines,” IEEE
Trans. Microwave Theory Tech., vol. MMT-16, pp. 1021-1027,
Dec. 1968.
[4] A. Farrar and A. T. Adams, “Characteristic impedance of
microstrip by the method of moments,” IEEE Trans. Microwave
Theory Tech., vol. MMT-18, pp. 65-66, Jan. 1970.
[5] K. Li, and Y. Fujii, “Indirect boundary element method of applied
- 77 - 15th IGTE Symposium 2012
to generalized microstrip line analysis with applications to sideproximity
effect in MMICs,” IEEE Trans. Microwave Theory and
Techniques, vol. 40, pp. 237–244, Feb. 1992.
[6] C.E. Smith, and R.S. Chang, “Microstrip transmission line with
finite width dielectric,” IEEE Trans. Microwave Theory and
Techniques, vol. 28, pp. 90–94, Feb. 1980.
[7] J. Svacina, “Analytical models of width-limited microstrip lines,”
Microwave and Optical Technology Letters, vol. 36, pp. 63–65,
Jan. 2003.
[8] J. Svacina, “New method for analysis of microstrip with finitewidth
ground plane”, Microwave and Optical Technology Letters,
Vol. 48, No. 2, pp. 396-399, Feb. 2006.
[9] C.E. Smith, and R.S. Chang, “Microstrip transmission line with
finite width dielectric and ground plane,” IEEE Trans. Microwave
Theory and Techniques, vol. 33, pp. 835–839, Sept. 1985.
[10] N. B. Raičević, S. R. Aleksić and S. S. Ilić, “A hybrid boundary
element method for multilayer electrostatic and magnetostatic
problems,” J. Electromagnetics, No. 30, pp. 507-524, 2010.
[11] N. B. Raičević, S. R. Aleksić, “One method for electric field determination
in the vicinity of infinitely thin electrode shells,”
Journal Engineering Analysis with Boundary Elements, Elsevier,
No. 34, pp. 97-104, 2010.
[12] S. S. Ilić, N. B. Raičević, and S. R. Aleksić, “Application of new
hybrid boundary element method on grounding systems,” in 14th
International IGTE'10 Symp., Graz, Austria, Sept. 19-22, 2010,
pp. 160-165.
[13] N. B. Raičević, S. S. Ilić, and S. R. Aleksić, “Application of new
hybrid boundary element method on the cable terminations,” in
14th International IGTE'10 Symp., Graz, Austria, Sept. 19-22,
2010, pp. 56-61.
[14] S. S. Ilić, S. R. Aleksić, and N. B. Raičević, “TEM analysis of
strip line with finite width of dielectric substrate by using new
hybrid boundary element method,” in 10-th International Conf.
on Applied Electromagnetics ПЕС 2011, Niš, Serbia, September
25-29, Sept. 2011, CD Proc. О8-4.
[15] B. Petković, S. Ilić, S. Aleksić, N. Raičević, and D. Antić, “A
novel approach to the positive DC nonlinear corona design,” J.
Electromagnetics, vol. 31, no. 7, pp. 505-524, Oct. 2011.
[16] N. B. Raicevic, and S. S. Ilic, “One hybrid method application on
complex media strip lines determination,” in 3rd International
Congress on Advanced Electromagnetic Materials in Microwaves
and Optics, METAMATERIALS 2009, London, United Kingdom,
2009, pp. 698-700.
[17] S. S. Ilić, M. T. Perić, S. R. Aleksić, and N. B. Raičević, “Quasi
TEM analysis of 2D symmetrically coupled strip lines with
infinite grounded plane using HBEM,” in Proc. XVII-th
International Symposium on Electrical Apparatus and
Technologies SIELA 2012, Bourgas, Bulgaria, 28–30 May, 2012,
pp.147-155.
[18] A. M. Abbosh, “Analytical closed-form solutions for different
configurations of parallel-coupled microstrip lines”, in IET
Microwaves, Antennas & Propagation, Vol. 3, Iss. 1, pp. 137-
147, 2009.
[19] D. Meeker, FEMM 4.2, Available:
http://www.femm.info/wiki/Download

- 78 - 15th IGTE Symposium 2012
Design Approach for a Line-Start Internal Permanent
Magnet Synchronous Motor
1,2 V. Elistratova, 1 M. Hecquet, 1 P. Brochet, 2 D. Vizireanu and 2 M. Dessoude
1 L2EP, Ecole Centrale de Lille, Cité Scientifique - BP 48 - 59651 Villeneuve d'Ascq, France
2 EDF R&D, 1 avenue du Général de Gaulle, 92141 Clamart Cedex, France
E-mail: vera.elistratova@ec-lille.fr
Abstract—The work described in this paper deals with the analytical design and optimization of a line-start permanent
magnet synchronous motor (LSPM) with radial magnet configuration. The design approach considers a LSPM as an
induction motor (IM) combined with a permanent magnet rotor arrangement and takes into account the characteristics of
both asynchronous and synchronous regimes and the motor thermal behavior.
Index Terms — LSPM, Eco-design, Optimization, Multi-physical model.
I. INTRODUCTION
A large amount of the primary energy resources are
converted into electric energy. As the main portion of
greenhouse gases is produced by fossil fuels, electricity
generation is responsible for the worldwide air pollution
and global warming [1].
Electric motors are one of the main sources of
electricity consumption (Fig.1) and this expense is up to
70% in the industrial processes in Europe [2]. As so, the
electric motors are responsible for a huge share of
emission of CO2. Moreover, there are a total potential of
improving the energy efficiency of applications using
electric motors in the range of 20-30%. The main factors
of such improvements are the use of variable speed drives
and the use of energy efficient motors. Therefore, electric
motor optimization for a better efficiency is essential for
energy saving and the reduction of CO2 emissions.
Figure 1: Distribution of the industrial electric consumption [2]
Until now low- and medium - power induction motors
(IMs) are widely used in many industrial applications,
such as pumps and fans. In spite of the low cost, IMs
normally suffer from relatively poor operational
efficiency and power factor [3]. Although a permanent
magnet synchronous machine (PMSM) can achieve high
operational efficiency and power factor, it lacks the
starting capability of the IM. For last few decades the
line-start permanent magnet synchronous motor (LSPM)
has been designed, constructed and tested. Compared
with an IM, a LSPM has a lot of advantages: synchronous
speed, higher power factor and efficiency, small size, etc.
Besides it has an ability to start when connected directly
to the mains.
II. PERFORMANCE DESIGN
The objective of this research is to find an analytical
model for the LSPM with different magnet
configurations. In the present paper the LSPM with radial
magnet arrangement (Fig.2) is designed. This
configuration has a number of advantages: simple and
robust structure, better protection of buried permanent
magnets (PMs) from demagnetization. Moreover, in
comparison with the other topologies this one has one of
the best asynchronous loading capabilities [4, 10].
Figure 2: LSPM architecture under study
An analytical model of a PMSM with the same rotor
architecture may be found in [6, 11]. In our case this
model can be applied under the assumption that during
the steady-state regime a LSPM operates as a PMSM.
In general, the structure of a LSPM is similar to an IM
but the rotor includes both cage and inserted permanent
magnets. Hence, the LSPM combines IMs and PMSM
structure features: the LSPM will start due to the resultant
of two torque components i.e. the asynchronous torque
and magnet opponent torque (braking torque). If for the
entire speed range during starting the asynchronous
torque is higher than the sum of the braking and load
torque, the motor will reach the synchronous regime [4].
Therefore, the design of a LSPM has to take into account
both types of performances: starting capacity and
efficiency in steady state regime.
To simplify the LSPM design process, the proposed
procedure applied in this article treats separately the
running modes: the motor can be considered as an IM
during its start and synchronization and as a PMSM
during the steady-state regime. Figure 3 shows the
workflow diagram applied for the design approach.

Figure 3: Diagram summarizing the design procedure
At the first stage of the design we enter the data
concerning the power and the stator parameters. At this
stage the same design methodology as for an IM could be
applied. For economic reasons, the stator of the LSPM is
identical to the IM of the same power.
The stage 2 is to choose a squirrel cage that gives the
value of the rotor cage resistance, the level of saturation
in a rotor tooth and the number of rotor bars.
At the next stage a configuration of the permanent
magnets has to be chosen. As soon as we know the
geometry of the rotor and the PMs, the d- and q-axis
reactances Xd and Xq and the no-load EMF could be
computed. Using these parameters, it can be simulated
the start of the LSPM taking into account the braking
torque caused by PMs. If the motor is not able to start, the
designer has to go back either to the Stage 2 in order to
change the rotor squirrel cage to improve the starting
torque or to the Stage 3 to change the configuration of
PMs and to reduce the breaking torque.
Finally, after the successful start of the LSPM (Stage
5), performances and steady-state characteristics are
calculated. If they are acceptable, the design solution is to
be considered as one for the optimization procedure (see
chapters III, IV). Otherwise, the design procedure is to be
repeat starting from the Stage 2.
Each stage of the diagram will be further detailed.
A. Asynchronous Torque
The electromagnetic design of induction motor is a
well-known problem. In this paper the asynchronous part
design is based on the methodology proposed in [8].
The classical expression for the asynchronous torque
can be written as follows:
- 79 - 15th IGTE Symposium 2012
'
2 R2
3pV
Tc
s
'
R2
2 ' 2
2 f ( Rsc ) ( X1 cX2)
s
where V is the phase RMS voltage, p is the number of
poles, c=1+X1σ/Xm, X1σ , X`2σ are the stator and rotor
leakage reactances, Xm is the magnetizing reactance.
B. Braking torque
The braking torque is found as a function of the back-
EMF and the stator resistance [5]:
T
br
2
2 2 2
(1 s)( Rs Xsq(1 s)
)
s
s
2
RsXsd Xsq 2 2
s
3p
ER
2 ( (1 ) )
where Rs is the stator resistance, ωs is synchronous
electrical speed, E is the RMS value of back-EMF, s is
the slip, Xsd, Xsq are the direct and quadrature
synchronous reactances respectively.
The braking torque peaks the maximum at low speed
and declines near synchronous speed.
C. Steady state regime
During the steady state regime the rotor of LSPM
rotates at synchronous speed and its cage has no
influence. In this state the performance of the machine
could be calculated as for PMSM.
As stated in [10] the RMS armature current is a
function of the motor equivalent electrical parameters:
2 2
Ia I ad Iaq,
(3)
where the axis currents are
I
ad
I
V( X cos R sin )
EX
ad
sq s
2
Xsq Xsd Rs
sq
V( R cos X sin )
ER
,
s sd
2
Xsq Xsd Rs
s
where δ is the load angle.
The input power of the motor is
2
in 3[ aq ad aq( sd sq) s a ],
,
(1)
(2)
(4)
(5)
P I EI I X X R I
(6)
Neglecting the stator core losses the electromagnetic
power is
P 3[ I EI I ( X X )].
(7)
elm aq ad aq sd sq
The electromagnetic torque developed by a PMSM is
Pelm
Telm
,
(8)
2
ns
where ns is the synchronous speed of the rotating
magnetic field.

Taking into consideration the Joule losses Pj, the
mechanical losses Pm and the stator core losses Ps, the
efficiency can be expressed as
Pin PjPsPm .
(9)
P
in
D. Calculation of the back-EMF and the direct and
quadrature synchronous reactances
The analytical model of the d-q machine parameters is
interesting as it provides the fast evaluation of LSPM
performances at steady-state regime, the obtainment of all
the characteristics and their integration into the
optimization procedure. For example, it permits us to find
the optimal volume of magnets in terms of the improved
efficiency and reduced braking torque.
To compute the d- and q-axis reactances Xd and Xq and
the back-EMF, the finite element simulation or
experimental testing are usually used [3-5, 9, 12].
However, there are a number of papers where all these
parameters are analytically expressed as function of the
studied machine geometry [6, 10, 11].
According to the model in [6] the flat-topped value of
the flux density in the air gap is
B
ag
2Br
emehhmp
(4eagehhmmp2eagempRh ,
e e R 2 e e pR e e R )
ag m h m h r m h r
(10)
where em is the magnet width, hm is the magnet height, eag
is the air gap length, eh is the hub thickness, Rh is the
external hub radius (in our case as the shaft is made of the
non-magnetic material eh = Rh), Rr is the rotor radius, μ0
is the permeability of vacuum, μm is the magnet relative
permeability, Br is the remanent magnetization,
α=em/(2∙Rh), β=/(2·Rr) are geometrical coefficients.
The first harmonic of the flux density in the air gap is
4 i Bag1 Bag
sin ,
(11)
2
where αi=2/π is the ratio of the average-to-maximum
value of the normal component of the air gap magnetic
flux density.
The first harmonic of the back EMF can be expressed
32RbLact f NskwBremehhmsin( p)
E
,
pe ( (4 eh pe( 2 p) R) ehR)
ag h m m m h h m r
(12)
where Lact is the rotor active length, Ns is the number of
turns in series per phase, kw is the global winding
coefficient.
The direct and quadrature synchronous inductances
could be found as a function of the machine geometry
and winding arrangement:
- 80 - 15th IGTE Symposium 2012
2
4sin( p
)
2p
p
Ld
2 eagempRhemeagRh)
2 2
6Lact0kwNsRb
Lq
2p sin(2 p)
2 2
eag p
2(6eag Rb2(4 eag Rb)cos(
)
(2 eag Rb)(cos(2
) sin(2 )))
.
2
p(2 eag Rb)
eag
2 2
6Lact0kNR w s b sin(2 p) 8eeR
m h r sin( p)
,
2 2
eag p p(4eagehhmpehemRr (13)
(14)
III. OPTIMIZATION PROBLEM
The goal of optimization process consists in finding the
set of optimal configurations R * taking into account
parameters and constraints imposed by the design
specification. Table I presents the specification for the
studied LSPM. In the presented study the dependency
between the efficiency and the magnet braking torque is
analyzed.
Table I. Specification of the designed LSPM machine
Parameter Value/Feasible interval
Power, [kW] 7.5
Voltage LL, [V] 400
Supply frequency f, [Hz] 50
Rated speed, [rpm] 1500
Height of the shaft axe, [mm] 132
Rated Torque, [Nm] 47.75
Overload conditions, Tmax/Trated
≥1.6
Ambient temperature, Tamb [°C] [-10; 40]
Stator winding temperature rise
average, [K]
80
Stator winding hot spot, [K] 90
Load torque, [p.u.]
Linearly from zero to nominal
speed, starting from 0.8pu to 1
p.u.
Power factor, cosφ ≥0.8
Efficiency η, [%] To be maximized
Geometry of permanent magnets Radial magnet configuration
The design vector X =[x1, x2,…, xn] T identifies the set
of design variables. The design variables can be freely
varied by the designer to define a designed object [7].
The permanent magnet geometry is analytically
predetermined form the imposed specification (Table I).
Consequently, the design vector of the studied problem is
composed of 3 variables: x1 – length of the air gap eag; x2
– magnet height hm; x3 – magnet width em. According to
equations (12-14) these 3 parameters are sufficient to
compute the d- and q-axis reactances Xd and Xq and the
back-EMF. Due to manufacturing constraints all of the
components of design vector X are discontinuous and
standardized.

Formally, the problem is expressed as follows:
minimize1
η, Tbr ,
X
(15)
subject to GX ( )= g 1( X), g 2( X),..., g n(
X)
0,n
=2.
Electromagnetic constraints of the problem G(X) are
specified in the Table II.
Table II. Constraints of the optimization problem
Function Constraint level
Power factor cosφ, p.u. ≥0.8
≥1.6
Tmax/Trated
Where {Tmax/Trated, cosϕ} are the feasible domains for
the maximum torque ratio for synchronous operation and
power factor.
Taking into consideration the fact that all the
components are discrete, in order to find R* a lot of
configurations have to be investigated.
IV. OPTIMIZATION TECHNIQUE
The optimization method applied for the considering
problem (15) is the exhaustive enumeration (EE) [7, 13].
It is an exact method with evaluations of all possible
combinations of the PM dimensions and air gap length.
The method doesn’t have any heuristic rules at all.
Because of the presence of several objective functions,
the aim of multi-objective evolutionary algorithms is to
find compromise solutions rather than a single optimal
point as in scalar optimization problems [14].
These tradeoff solutions are usually called Pareto
optimal solutions. The EE was applied in order to obtain
a genuine Pareto-Front. The method is not pretended to
be the best one in terms of total time of calculation, but
on the other hand, it gives reliable results.
Input parameters were:
Design vector:
X =[x1, x2,…, xn] T in our case n = 3.
Objective functions:
F(X) = {f1(X),f2(X),…, fm(X)} in our case m=2
and F(X) = {(1- η), Tbr}.
Constraints:
G(X) = {g1(X), g2(X),…, gk(X)} in our case k =2.
The feasible set Ω= {ω1, ω2,…, ωn},where ωi is the
subset which contains all feasible values for the
component xi of the design vector, for i=1…n. In
our case n = 3. As all of the components of design
vector X are discrete, Ω is a finite set that is
composed of the possible standardized values.
Output parameters:
The set of optimal solutions:
R * = {Xi * X 0 |G (Xi * ) ≤, for i=1,..m}, where
Xi * is the degenerate interval, and each component
*
of X is a Pareto optimal solution. Therefore Xi
has following features:
- 81 - 15th IGTE Symposium 2012
*
fl( ) fl( i) for l 1...
m,
*
f j( ) f j( i)
for at least one index j.
The problem (15) was treated and a total of 360
combinations has been enumerated. Among these 360
combinations there are 160 that belong to the feasible
domain defined by optimization constraints. In Fig.4 the
feasible set of solutions for the EE and the Pareto frontier
are presented.
Figure 4: Pareto front of efficiency versus braking torque
V. DESIGN RESULTS
A boundary point of the maximal efficiency from the
Pareto frontier has been chosen for a deeper investigation.
Based on this optimal solution and solving the set of
equations (1-14) all the characteristics for steady-state
regime and optimal dimensions of permanent magnets
and air gap length have been found (Table III).
Table III. Optimal solution for of the designed LSPM
Parameter Value
Efficiency η, [%] 91.2
Braking torque, Nm 11.66
Power factor cosφ, p.u. 0.983
Air gap length eag, mm 0.7
Magnet height hm, mm 29.0
Magnet width em, mm 15.0
Overload condition, Tmax/Trated
1.783
Table III shows that the efficiency of the designed
LSPM compared with a premium efficiency class
induction motor (PEIM) is greater than 0.8% [20]. The
power factor of the PEIM (0.9 p.u.) is much lower
compared with the designed LSPM (0.983 p.u.). It means
the LSPM can achieve a very high power factor in a wide
output power range. This feature assists in saving energy
when the motor is running at different loads.
Based on the designed data the static and dynamic
characteristics were obtained. Figures 5-7 show that the
designed LSPM is able to start and synchronize even at
85% of the rated voltage.

Figure 5: Torque versus speed curve of the studied motor
with supplied phase voltage equal 231V
Figure 6: Torque versus speed curve of the studied motor
with supplied phase voltage equal 85% * 231V
Figure 7: Motor speed during transient start
supplied with different voltages
VI. THERMAL MODEL
An increase in motor temperature can cause the stator
winding insulation degradation and permanent magnet
material decreased performances. According to the design
specification (Table I), the acceptable heating in the
LSPM doesn’t have to exceed 90K. To predict the motor
transient thermal behavior an analytical model based on
the general cylindrical component [21] was developed
(Fig. 8).
Figure 8: A simplified model of LSPM as the heating body
- 82 - 15th IGTE Symposium 2012
This model corresponds to the system of equations:
dcu
cu 12 ( ) 1
P A cu st A cu C1 ,
dt
dst
P 2 12
st A st A ( cu st ) C2 .
dt
(16)
where ∆θcu and ∆θst are the average heating in the copper
and respectively the stator laminations, A1, A2, A12 are
the heat transfer coefficients, C1 and C2 represent the heat
capacities of stator core and stator winding. In order to
determine the temperature, a simplified equivalent
thermal network (ETN) model of the LSPM is considered
(Fig. 9).
Figure 9: Simplified equivalent thermal network of LSPM
(
cu, a cu, c ) cu -cu, a a, in - cu, c s, st Pcu,
(
cu, a rot, c c, f ) s, st cu, c curot, c rot
c, ff Ps,
st,
(17)
(
rot, a rot, c ) s, st rot, a a, in rot, s s, st Prot
( cu, a rot, a a, f ) a, in cu,
a curot, a rot
a, ff Pa,
in,
( c, fa, f f) fc, fs, sta, fa, in0.
where Δθcu is the heating in the copper winding; Δθs,st -
heating in the steel stator pack, Δθrot - heating in the rotor;
Δθa,in - heating in the air gap; Δθf - heating in the motor
case, Рcu - source of losses in the copper winding, Рs,st -
source of losses in the stator pack, Рrot - source of losses
in the rotor, Рa,in - source of mechanical and additional
losses, Λcu,c - thermal conductivity between the slot
winding and stator core, Λcu,a – thermal conductivity
between the winding and the air gap, Λrot,a – thermal
conductivity between the rotor and the air gap, Λrot,с –
thermal conductivity between the rotor and the stator
core, Λa,f – thermal conductivity between the air inside
the motor and the motor case, Λс,f – thermal conductivity
between the stator core and the motor frame; Λf – thermal
conductivity between the motor frame and the external
air.
The solution of the systems (16, 17) enabled us to
model overheating in the main parts of the designed
motor (Figs. 10, 11).

Figure 10: The increase of winding temperature
Figure 11: The increase of rotor core temperature
According to figures 10, 11 maximal overheating in
winding is 79.4°C, maximal overheating in rotor is about
of 26.3°C that is in compliance with the specification
requirements (Table I).
VII. CONCLUSION
A design method for a LSPM motor considering the
asynchronous starting capacity and the synchronous
steady state performances is proposed in order to find out
an optimal design solution for the given motor topology.
The approach is based on the design of an asynchronous
machine incorporating the effect of magnets. The present
analytical model takes into account the radial magnet
topology and is to be extended for the other LSPM
architectures.
It has been shown that the efficiency and the power
factor of the designed LSPM is greater compared with a
PEIM of the same power.
Thereafter, an analytical thermal model was developed.
The proposed thermal model allows predicting the
overheating in the main parts of the motor. During the
design process the thermal model didn’t take part in
optimization procedure.
In future investigations it might be possible to combine
the electro-magnetic and thermal optimization problems
in order to integrate them into optimization procedure.
The verification of the analytical approach will be
provided by both finite element and experimental models.
- 83 - 15th IGTE Symposium 2012
REFERENCES
[1] Key world energy statistics. International Energy Agency, 2010.
[2] La rentabilité énergétique les entrainements, Mesures 803, Mars
2008, www.mesures.com.
[3] Jian Li and Jungtae Song and Yunhyun Cho. A High-Performance
Line-Start Permanent Magnet Synchronous Motor Amended From
a Small Industrial Three-Phase Induction Motor. In Industrial
Electronics, 2010 IEEE International Symposium, pp. 1308 -1313.
[4] T. Ruan, H. Pan, Y. Xia « Design and Analysis of Two Different
Line-Start PM Synchronous Motors», Artificial Intelligence,
Management Science and Electronic Commerce (AIMSEC), 2011.
[5] Soulard, J.; Nee, H.-P.; , "Study of the synchronization of linestart
permanent magnet synchronous motors," Industry
Applications Conference, 2000. Conference Record of the 2000
IEEE , vol.1, no., pp.424-431 vol.1, 2000.
[6] X. Jannot, J.-C. Vannier, J. Saint-Michel and M. Gabsi, An
Analytical Model for Interior Permanent-Magnet Synchronous
Machine with Circumferential Magnetization Design, IEEE,
10.1109/ELECTROMOTION.2009.5259155, July 2009.
[7] P.Venkataraman, Applied Optimization with
Matlab Programming, A Wiley - Interscience publication, John
Wiley & Sons, New York, 2001.
[8] I.P. Kopylov, Electric Machines: M., Energoatomizdat, 1986.
[9] K. Kurihara, M. Azizur Rahman, High Efficiency Line-Start
Interior Permanent Magnet Synchronous Motors, IEEE Trans.
Industry Applications, Vol. 40 Issue 3, May 2004.
[10] J.F.Gieras, M. Wing, Permanent Magnet Motor Technology,
USA, Marcel Dekker, 2002.
[11] D.Fodorean, A. Miraoui, Dimensionnement rapide des machines
synchrones à aimants permanents (MSAP), Techniques de
l’ingénieur, Nov. 10, 2009.
[12] H-P. Nee, L. Lefevre, P. Thelin, J. Soulard, Determination of d
and q reactances of permanent magnet synchronous motors
without measurements of the rotor position, IEEE Trans. on
Industry Applications, Vol. 36, No. 5, 1330-1335, Oct. 2000.
[13] D. Samarkanov, F. Gillon, P.Brochet, D. Laloy , Optimal design
of induction machine using interval algorithms, COMPEL: The
International Journal for Computation and Mathematics in
Electrical and Electronic Engineering, Vol. 31, N°.5, pages. 1492 -
1502, ISBN. 0332-1649, 8-2012.
[14] P. Alotto, U. Baumgartner, F. Freschi, M. Jaindl, A. Köstinger,
Ch. Magele, W. Renhart, and M. Repetto, SMES Benchmark
Extended: Introducing Pareto Optimal Solutions Into TEAM22,
IEEE Transactions on Magnetics, Vol. 44, No.6, pp. 1066-1069,
2008.
[15] Mellor, P.H.; Roberts, D.; Turner, D.R.; , "Lumped parameter
thermal model for electrical machines of TEFC design," Electric
Power Applications, IEE Proceedings B , vol.138, no.5, pp.205-
218, Sep 1991.
[16] IEC 60034-30, Standard on efficiency classes for low voltage AC
motors, 2008.
[17] D. Stoia, M. Antonoaie, D. Ilea, M. Cernat, Design of Line Start
PM Motors with High Power Factor, Proc. POWERENG 2007,
Setubal, Portugal, 12-14 April, 2007, published on CD-Rom,
IEEE Catalog Number 07EX1654C, ISBN: 1-4244-0895-4, paper
186.
[18] T. Miller, Synchronization of line-start permanent magnet AC
motor, IEEE Trans. Power Apparatus and Systems, vol. PAS-103,
July 1984, pp 1822-1828.
[19] T. Tran, S. Brisset, P. Brochet, A Benchmark for Multi-objective,
Multi-Level and Combinatorial Optimizations of a Safety
Isolating Transformer, COMPUMAG 2007, Aachen, Germany,
6- 2007
[20] X. Feng, L. Liu, J. Kang, Y. Zhang, Super Premium Efficient
Line Start-up Permanent Magnet Synchronous Motor, Proc. Of
XIX International Conference on Electrical Machines, ICEM2010,
Roma, Italy, Sept. 6-8, 2010.
[21] A.I. Borisenko Cooling of industrial electrical machinery,
Energoatomizdat, 1983.

- 84 - 15th IGTE Symposium 2012
Speed-up of Nonlinear Magnetic Field Analysis using a Modified
Fixed-Point Method
Norio Takahashi 1 , Kousuke Shimomura 1 , Daisuke Miyagi 2 and Hiroyuki Kaimori 3
1 Dept. Electrical and Electronic Eng., Okayama University, Okayama 700-8530 Japan
2 Dept. Electrical Eng., Tohoku University, Sendai 980-8579 Japan
3 Science Solutions Int. Lab., Inc., Tokyo 153-0065 Japan
The nonlinear finite element analysis of magnetic fields using the Fixed-Point method (FPM) requires a number of iterations and
long CPU time compared with those using the Newton-Raphson method (NRM). On the other hand, the Fixed-Point method has an
advantage that the convergence can be obtained even for a complicated nonlinear anisotropy problem, of which the convergence is
very difficult using a conventional Newton-Raphson method. Moreover, it has an advantage that a software can be easily obtained by
slightly modifying a linear FEM software. We then achieved the speed-up of the Fixed-Point method by updating the reluctivity at each
iteration (This is called a modified Fixed-Point method). It is shown that the formulation of the Fixed-Point method using the
derivative of reluctivity is almost the same as that of the Newton-Raphson method. The convergence properties of these methods are
compared. It is shown that the modified Fixed-Point method has an advantage that the programming is easy and it has a similar
convergence property to the Newton-Raphson method for an isotropic nonlinear problem.
Index Terms—finite element method, Fixed-Point method, Newton-Raphson method, nonlinear electromagnetic analysis
I. INTRODUCTION
The Fixed-Point method [1,2] has an advantage that the
convergence can be obtained even for a complicated nonlinear
problems [3] such as the analysis considering vector magnetic
properties treating an anisotropic material [4, 5], in which the
convergence is sometimes difficult. In addition, it has an
advantage that the software for nonlinear analysis can be
easily obtained by adding a small change to that for linear
analysis. But, the Fixed-Point method requires a number of
iterations and long CPU time compared with those of the
Newton-Raphson method [6]. It is reported that the CPU time
can be reduced by using a constant reluctivity in the
beginning of nonlinear iterations [7,8 ]. However, nearly ten
times longer CPU time is still necessary compared with the
Newton-Raphson method.
In this paper, a modified Fixed-Point method, which
updates the derivative of reluctivity at each iteration, is
proposed. Furthermore, it is pointed out that the formulation of
the Fixed-Point method using the derivative of reluctivity is
the same as the Newton-Raphson method. The convergence
characteristic of the newly proposed Fixed-Point method is
compared with those of the Newton-Raphson method.
II. FORMULATION OF NRM AND FPM
A. Newton-Raphson Method
There are two kinds of methods which deal with the
nonlinearity in the Newton-Raphson method (NRM). One is
the method A (NRM(B 2 )) which uses ν-B 2 curve. In this
method, the magnetic field strength H is given by
2
H ( B ) B
(1)
B is the flux density. The reluctivity ν is given by
2 H(
B)
( B )
(2)
B
The other is the method B (NRM(B)) which uses the B-H
curve directly. In this method, the magnetic field strength H is
given by
B
H H(
B )
(3)
B
1) Method A (NRM(B 2 )
The static magnetic field equation can be written as follows
in the case of the Newton-Raphson method using the -B 2
curve:
H
(
A)
J
(4)
0
where, A is the magnetic vector potential. J0 is the forced
current density. The Galerkin equation G * i(A (k) ) of (4) is given
by
* ( k )
( k 1)
( k 1)
Gi ( A )
N i ( A ) dV N iJ
0dV
(5)
where, Ni is the interpolation function of the edge element.
The residual Gi(A) at the k-th nonlinear iteration is given by
( k ) * ( k )
( k )
G ( A ) G i ( A ) G
( A )
i
( k )
j
i
( k 1)
( k 1)
N i ( A ) dV N iJ
0dV
( k 1)
( k 1)
N i ( A ) dV A
A
* ( k )
G i ( A ) N (
i
( k 1)
Ν ) dV
( k 1)
( k 1)
( k )
N dV
i
A A
( k )
i
A
j
2
B
B
2B
(7)
2
2
Aj
B
Aj
B
Aj
where, A, ν etc. in (6) and (7) are values at the k-th iteration.
∂ν/∂B 2 is the term which represents nonlinear magnetic
properties. The process of calculation is as follows:
1) The initial value of ν is determined.
2) δA (0) is set to zero.
3) A is updated by A (k) =A (k-1) +δA (k) using δA (k) calculated by
(6).
j
( k )
i
1
(6)

4) ν (k) is calculated using the ν-B 2 curve from B obtained by
A (k) .
5) The process from 3) to 5) is repeated.
6) It is judged to be converged if δB(A (k) ) is less than a
specified small value.
2) Method B (NRM(B))
The static magnetic field equation can be written as follows
in the case of the Newton-Raphson method using the B-H
curve:
H J
(8)
0
The Galerkin equation G * i(A) of (8) is given by
* ( k )
G i ( A )
N i HdV
N iJ
0dV
(9)
The residual Gi(A) at the k-th nonlinear iteration is given by
( k ) * ( k )
( k )
G ( A ) G i ( A ) G
( A )
i
i
( k 1)
( k )
N dV dV dV
i H N iJ
N i H
0
( k 1)
* ( k )
H
( B ) ( k ) (10)
G i ( A ) N
dV
i
Bi
B
( k 1)
* ( k )
H
( B )
( k )
G i ( A ) N i A
dV
B
∂H(B)/∂B is the term which represents nonlinear magnetic
properties.
The process of calculation is as follows:
1) The initial value of ∂H(B (0) )/∂B is determined.
2) δA (0) is set to zero.
3) A is updated by A (k) =A (k-1) +δA (k) using δA (k) calculated by
(10).
4) H (k) is calculated using the B-H curve from B obtained by
A (k) .
5) The process from 3) to 5) is repeated.
6) It is judged to be converged if δB(A (k) ) is less than a
specified small value.
B. Fixed-Point Method
In the Newton-Raphson method, the reluctivity is updated
in each nonlinear iteration as explained above. In the Fixed-
Point method, the reluctivity is fixed at the first step and it is
not changed during the nonlinear iterations.
According to the concept of the Fixed-Point method [1], the
magnetic field strength is given by
H( B)
ν B H
(11)
FP
FP
where, FP is the Fixed-Point reluctivity which is constant
during the nonlinear iterations, HFP is an additional magnetic
field strength.
The static magnetic field equation can be written as follows
in the case of the Fixed-Point method:
FP 0 )
( J H B FP
(12)
where, HFP (k) at the k-th nonlinear iteration can be obtained by
the following equation:
( k )
( k1)
( k1)
H FP H(
B ) νFPB
(13)
where, H(B (k-1) ) is the magnetic field strength vector on the B-
H curve corresponding to the flux density B (k-1) at the (k-1)-th
nonlinear iteration. HFP (k) converges to some value after
iterations. The residual Gi(A) of (12) is given by
- 85 - 15th IGTE Symposium 2012
( k )
Gi
( A)
( k )
N i ( FP
A ) dV N iJ
0dV
(14)
( k )
N i H FP dV
By substituting HFP (k) in (13) into HFP (k) in (14), we obtain
( k ) * ( k )
G ( A ) G ( A ) N H
i
i
)
dV
* ( k )
( k 1)
( k 1)
G ( A ) N ( H(
B )
B ) dV
i
* ( k )
( k )
G ( A ) N ( A
) dV
i
i
i
i
FP
FP
FP
i
i
i
FP
( k
FP
( k )
( k 1)
N ( A ) dV N J dV N H ( B ) dV
( k 1)
N ( A ) dV
i 0
( k 1)
( k 1)
N ( A
) dV N J dV N H(
B ) dV
where, δA (k) =A (k) -A (k-1) .
Gi * (A (k) ) is given by
*
Gi i FP
i 0
i
0
FP
i
i
2
(15)
( k )
( k )
( A )
N (
A ) dV N J dV
(16)
In the actual calculation, (14) is used in the Fixed-Point
method.
The process of calculation is as follows:
1) The initial value of FP is determined.
2) HFP (0) is set to zero.
3) B (k) is obtained from A (k) which is calculated by (14).
4) HFP (k) is obtained by (13).
5) The right hand side of (14) is updated and the process from
3) to 5) is repeated.
6) It is judged to be converged if the change of B (k) is less
than the specified small value.
According to (15), we found that HFP (k) is given by
H
A
H H
(17)
( k )
( k )
( k )
( k1)
FP FP
FP
FP
(17) means that the difference HFP (k) is the same as H in (9)
of the Newton-Raphson method and it can be used as the
judgment of the convergence.
Fig.1 shows the concept of the nonlinear magnetic field
analysis using the Fixed-Point method. A white circle on the B
axis is a convergence target. In this method, the reluctivity FP
shown in Fig.1 is given as an initial value, and FP is not
changed during the iterations. The flux density B (1) is obtained
by the linear magnetic field analysis. Next, the HFP (1) which
corresponds to the flux density B (1) on the B-H curve and
FPB (1) on the line of FP shown in Fig.1(a) is obtained. During
iterations, HFP (k) becomes the same value, which means the
difference HFP (k) becomes almost zero. Then, the converged
result can be obtained.
C. Modified Fixed-Point Method
In the modified Fixed-Point method, the derivative of
reluctivity is updated at each iteration. In this expression, the
HFP (k) at the k-th nonlinear iteration in (13) can be rewritten by
the following equation:
( k 1)
( k )
( k 1)
H(
B ) ( k 1)
H H(
B ) B
FP
(18)
B
The residual Gi(A (k) ) is given by
k
k
k
k
k
Gi
G i i
dV
( 1)
( ) * ( )
( 1)
H(
B ) ( 1)
( A ) ( A ) N H(
B ) B (19)
B
(19) can be written as follows:

H
FPB (1) FPB (1)
HB (1) HB νFP
(1) νFP
H (1)
FP
H
FPB (2) FPB (2)
HB (2) HB (2)
H (1)
FP
H (2)
FP
H
FPB (2) FPB (2)
HB (3) HB (3)
H (2)
FP
H (3)
FP
ν
ν
FP
ν
FP
FP
ν
ν
FP
FP
(a)
B (1) B (1)
B (2) B (2)
(b)
H
B (3) B (3)
H
( 3 )
FP
H
( 2 )
FP
( 1 )
FP
B-H curve
( k 1)
( k ) H
( B )
( k )
G ( A ) NAdV NJdV
i
i
i 0
B
( k 1)
( k 1)
H
( B )
( k 1)
N i H(
B ) dV
N i
A dV
B
(20)
( k 1)
( k 1)
H
( B )
( k )
N H(
B ) dV N J dV
N A
dV
i
i 0
i
B
( k 1)
* ( k ) H
( B )
( k )
G ( A )
N A
dV
i
i
B
(10) and (20) denote that the formulation of the modified
Fixed-Point method is the same as that of the Newton-
Raphson method.
In the actual calculation of the modified Fixed-Point
method, (19) is used.
The process of calculation is as follows:
1) The initial value of ∂H(B (0) )/∂B is determined.
2) HFP (0) is set to zero.
3) B (k) is obtained from A (k) which is calculated by (19).
4) HFP (k) is obtained by (18).
B
B-H curve
B-H curve
(c)
Fig. 1 Conceptual diagram of Fixed-Point method. (a) 1 st step. (b) 2 nd step.
(c) 3 rd step.
B
- 86 - 15th IGTE Symposium 2012
5) The right hand side of (19) is updated and the process from
3) to 5) is repeated.
6) It is judged to be converged if the change of B (k) is less
than the specified small value.
Fig.2 shows the concept of the nonlinear magnetic field
analysis using the modified Fixed-Point method. In this
method, the reluctivity νFP shown in Fig.2 (a) is given as an
initial value, and the derivative ∂H/∂B is updated at each
iteration. At the initial iteration, the linear magnetic field
analysis is carried out using the given ∂H/∂B, and the flux
density B (1) is obtained. Next, H(B (1) ) corresponding to the
flux density B (1) on the B-H curve and
∂H(B (1) )/∂B·B (1) =VFPB (1) on the line ∂H(B (1) )/∂B shown in
Fig.2(a) is obtained. At the first step, HFP (k) =HFP (1) following
the definition of HFP (1) in (17). The iteration is carried out
H(B (1) H(B ) (1) )
H
FPB (1) FPB (1)
H (1)
FP
0
H
H(B (2) H(B ) (2) )
FPB (2) FPB (2)
H (1)
FP
0
H
H(B (3) H(B ) (3) )
FPB (3) FPB (3)
H (2)
FP
0
( 1 )
H ( B )
B
( 2 )
B (1) B (1)
(a)
H ( B )
( 2 )
H FP
B
H
H
( 1 )
FP
( 2 )
FP
B (2) B
( 2)
H(
B )
B
(2)
( 2)
H(
B )
B
(b)
( 3 )
H ( B )
( 3 )
H FP
B
B (3) B (3)
( 3 )
H ( B )
B
H ( B )
B
H
B-H curve
( 1 )
B
B-H curve
B
B-H curve
( 3 )
FP
B
(c)
Fig. 2 Conceptual diagram of Modified Fixed-Point method. (a) 1 st step. (b)
2 nd step. (c) 3 rd step.
3

until δHFP becomes near to zero. H (which corresponds to
HFP in (17)) can be directly obtained by using the Newton-
Raphson method as shown in (10). The modified Fixed-Point
method needs two steps (Eqs. (18) and (19)), but the concept
is the same as that of the Newton- Raphson method.
III. ANALYZED MODEL
The modified Fixed-Point method is applied to the analysis
of the magnetic field in the billet heater model [9] shown in
Fig.3. Analysis domain of the model is 1/8. The material of
the yoke is 35A230(non-oriented electrical steel). The material
of the billet is S45C(carbon steel). The numbers of elements
and nodes are 107632 and 115101, respectively. The ampere
turns of the coil are set as 70000AT (60Hz). The CPU time
and number of iteration of the Fixed-Point method (FPM),
modified Fixed-Point method (MFPM), Newton-Raphson
method using ν-B 2 curve (NRM(B 2 )), and Newton-Raphson
method using B-H curve (NRM(B)) are compared. For
simplicity, only the calculation of the 1st step of the step by
step method for the nonlinear eddy current analysis is carried
out in order to compare the performance of each method. As
the total CPU time is almost equal to the multiple of number
of steps, the comparison of only the 1st step is sufficient for
the comparison of each method.
IV. RESULTS AND DISCUSSION
Fig.4 shows an example of distribution of flux density of
NRM(B) and MFPM. The results of NRM(B 2 ) and FPM are
also the same as Fig.4. The comparison of the CPU time and
the number of iterations are shown in Table I. The
convergence property is shown in Fig.5. The convergence
criterion is B(A) < 2.010 -3 . The convergence criterion
( )
G n
/ G
( 0)
of the ICCG method is chosen as less than 10 -5 .
Intel Core2 Duo E8400@ 3.16GHz, 3GB RAM is used. These
results suggest that the convergence property of MFPM is near
to that of NRM. Especially, MFPM is faster than NRM (B). It
is also clarified that NRM(B 2 ) is faster than NRM(B).
fire-resistant material
billet
y
150
z
x
unit:mm
200
100 100
yoke
15 15 2510 2510 50 50
adiabator
billet
iron core
V. CONCLUSIONS
z
x
unit:mm
The obtained results can be summarized as follows:
(a) The formulation of the modified Fixed-Point method
y
10
200
25
200
300
15
50
(a) (b)
Fig. 3 Analyzed model of billet heater. (a) bird’s eye biew (1/8 region). (b) xy
plane.
- 87 - 15th IGTE Symposium 2012
148
TABLE I
COMPARISON OF CPU TIME AND ITERATIONS
Method CPU Time (sec) Iterations
NRM(B2) 370.08 13
NEM(B) 654.83 28
FPM 3432.24 101
MFPM 781.69 22
PC performance : Intel Core2 Duo E8400@ 3.16GHz, 3GB RAM
Flux density B[T]
3.16
3.24
2.88
2.52
2.16
1.80
1.44
1.08
0.72
0.36
0.00
y
x
(a) (b)
Fig.4 Comparison of numerical results of Flux distribution using NRM(B)
and MFPM. (a) NRM(B). (b) MFPM.
Number of nonconverged
elements
Fig. 5 Convergence Property.
NRM(B2 NRM(B )
NRM(B)
FPM
MFPM
2 )
NRM(B)
FPM
MFPM
Iterations
(MFPM) using the derivative of reluctivity is almost the
same as that of the Newton-Raphson method (NRM).
(b) The modified Fixed-Point method (MFPM) has an
advantage that the CPU time is less than that of the
Newton-Raphson method (NRM) in some condition, or
MFPM has almost the same performance as NRM.
Moreover,the programming is easy compared with NRM.
[1]
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4

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- 88 - 15th IGTE Symposium 2012
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- 89 - 15th IGTE Symposium 2012
Software Agent Based Domain Decomposition Method
1) M. Jüttner, 1) A. Buchau, 2) M. Rauscher, 1) W. M. Rucker, and 2) P. Göhner
1) Institute for Theory of Electrical Engineering, Pfaffenwaldring 47, D-70569 Stuttgart, Germany
2) Institute of Industrial Automation and Software Engineering, Pfaffenwaldring 47, D-70569 Stuttgart, Germany
E-mail: ite@ite.uni-stuttgart.de
Abstract—A workbench is described, able to divide complex coupled three dimensional multiphysics simulations into
smaller parts based on existing domain decomposition techniques. These parts are calculated by software agents allowing to
widely distributes the calculation over multiple distributed computers and even into the cloud to speed up the performance,
to make larger simulations possible and to actively manipulate and control the strategy and the process of solving.
Index Terms— cloud computing, coupled multiphysics problems, domain decomposition, software agents
different resources like idle workstations, laptops or
smartphones and even online resources located within the
cloud. These resources can be used by established and
multifunctional solving methods like FEM or BEM. FEM
is able to solve non-linear and anisotropic material effects
and lead to large systems of non-linear equations. An
alternative to the FEM is the BEM. At BEM only the
surface of a model needs to be discretised. This leads to a
much smaller system of equations represented in a dense
matrix. The calculation time for the matrix gets
acceptable if we use matrix compression. Therefore the
fast multipole method (FMM) and the adaptive cross
approximation (ACA) can be used [7], allowing to
calculate hysteresis effects for magnetic fields [8].
I. INTRODUCTION
Finding a solution for complex three dimensional
coupled field problems more efficiently is the goal of this
approach. Therefore, established methods for numerical
solutions like finite element methods (FEM) and
boundary element methods (BEM) are combined with the
idea of software agents.
Software agents are a way to develop flexible and
efficient software based on the concept of agent oriented
software development [1]. Therefore, the system is
divided into autonomous and self-organized agents.
These agents are independent of each other and capable
to make decisions within there possibilities. Therefore
they are able to interact with each other via messages and
data exchange. Based on this the agents negotiate to reach
their individual goals. The communication between the
agents also allows a dynamic handling of multiple
situations per agent as well as for the global system to
grant dynamic and well fitting agent behaviour. Within
the context of agent based systems a systematic
distribution of the functions, necessary for solving a
problem, to different agents grant a limited coupling
between different agents and results into an even more
flexible and manageable system. This flexibility and
dynamic allow the approach described in section II to
perfectly handle systems with multiple boundary
conditions and to solve weak coupled systems. The
approach of software agents is currently well established
in automation technology and used for example for selfmanagement
in automation systems [2], modelling smart
grids [3], prioritization of test cases [4] or optimising
electromagnetic field problems [5].
Because of the big influence of available computer
resources for solving numerical problems nowadays
workstations including multiple multicore CPUs and a
relatively large RAM could be used. To handle these
resources modern programming languages are available
and grant a quite good usage of all of these resources.
Based on the increase of calculation power the problems
getting larger and coupled effects are considered as well.
Nowadays large simulation problems are mostly solved
on huge computer clusters with identical computers. The
usage of temporary available resources grant, due to
modern operation systems a large performance alternative
[6]. Products like the Microsoft High Performance
Computing Server or the Enterprise Linux Cluster from
Redhat or Suse offer a simple way to spread tasks to
II. COUPLING SOFTWARE AGENTS AND SOLVER
Considering a large coupled simulation, the creation of
equations including all effects is mostly not reasonable.
Splitting the problem into smaller parts that can be solved
iteratively can reduce the total expense of the large nonlinear
problem [9], especially when small changes in the
partial problems can be ignored and do not lead to further
iterations of the calculation. Therefore a so called
coordination agent is created. This agent splits the
coupled problem into partial problems. Examples for
different classes of partial problems are different single
physic problems as well as geometry or material based
partial problems. All functions of the coordination agent
are described in section II.B. Then, the partial problems
are assigned to different calculation agents. Fig. 1
describes the cooperation of software agents.
Fig. 1: Concept of cooperation agents

Each calculation agent solves its problem with an
optimized approach for its partial problem. Due to that a
combination of multiple methods like FEM or BEM for
different partial problems are possible. This allows a
combination of the advantages of FEM and BEM for
multiple different calculation resources. For the
calculation agents there is no need to be within one
system. Different resources can be used if the calculation
agents are distributed to multiple computers or even the
cloud as displayed in Fig. 2. The calculation agents are
described in detail in section II.C. The collection of these
software agents is able to solve coupled problems. The
necessity of this new approach handling multiple physics
and large systems can be seen in [10]. The simulation was
only possible with height effort to reach convergence.
Fig. 2: Distributed Agents
A. Steps to a successful simulation
Setting the approach into its context and describing the
process of creating and solving a complex coupled
problem with this approach is topic of this subsection.
The process is visualized in Fig. 3.
Build a finer
mesh
Modelling the system with a FEM-software
Including mesh and boundary conditions
Export mesh and boundary conditions
Divide mesh into smaller parts
Mapping the boundary conditions
Mesh-management within the agent-system
Calculation Agent 1
Solve partial problem
Parallel and independent
Exchange of boundary
conditions
Exchange of status
Exchange of results
no
status,
boundary
conditions
convergence
yes
Combine solutions
Calculation
Agent
2
...
conceivable
Calculation
Agent
n
Fig. 3: The process for a simulation
- Initially a model estimating the actual problem
needs to be created. This approach does not set any
special requirements to the model itself. So the model can
be created with commonly used CAD-software tools. The
same holds for the creation of the mesh of the model, so
common meshing-tools can be used. For complex
coupled problems it is important to consider all effects
within one single mesh because the calculation is
influenced by the geometry of all physics as well as their
coupling.
- To allow any solver to create a suitable solution, the
boundary conditions for the given mesh including all
coupled physics has to be set. The boundary conditions as
well as the previously created mesh have to be exported
in a way it can be handed over to the coordination agent.
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- The mesh and the boundary conditions are now
handed to a coordination agent. The coordination agent is
responsible for finding a solution for the given problem.
Therefore it splits the problem into smaller parts. The
quantity of smaller parts depends on the number of
available calculation agents or is defined manually. In
case of a resource based splitting the assignment of the
partial problems to the different calculation agents is
intuitive. In case of a manual quantity of partial problems
each partial problem is solved iteratively according to
availability of resources. The partial problems are now
distributed to the available calculation agents.
- The calculation agents now start solving their
allocated partial problem. Because of each calculation
agent running on its own hardware the agent is able to use
all resources of this hardware to solve its partial problem
fast and highly parallel. If a new calculation agent,
including new hardware resources, appears in the system
the coordination agents has to determine if the resource
should used and which agent splits its partial problem. In
case of a drop out of a calculation agent the coordination
agent needs to attach the partial problem to another
calculation agent. This behaviour allows a dynamic
adaption of the system. To do so, information based on
status of different calculation agents has to be distributed.
- To allow the system of agents to solve a coupled
problem, the calculation agents has to exchange results
between each other as soon as they are available. If a
calculation agent is able to understand and interpret the
results of another calculation agent, new boundary
conditions can be derived and integrated into the own
calculation. This process continues until all calculation
agents finished. If some partial problems do not reach
convergence the calculation can be interrupted and
reinitialised with a new set of partial problems without
recalculating successfully solved parts.
- Finally the coordination agent combines all results
depending on the way they were split before and returns
the result to the user.
An example describing the advantages of this approach
is shown in section II.E.
B. The Coordination Agent
The coordination agent is an independent program with a
small set of functions. It’s visible to the in- and outside
and represents the interface between the users and the
calculation agents. The graphical user interface (GUI)
provides the interface to the user. The GUI is controls all
functions described below. The internal interface is
realised via a message system handling different types of
messages received from other agents. In addition to that
the coordination agent offers process variables like the
convergence criteria and the overall progress, so each
problem needs to be assigned to at least one coordination
agent. The different functions of the coordination agent
are summarized in Fig. 4 and described in detail in the
following.
- Via its GUI the coordination agent offers the
interface to load a problem. A second problem can only
be loaded, if the first is solved and the results are either
collected by the user or the actual calculations are
interrupted and possible results are dropped. The GUI is
additionally used to display calculated results.

- Solving a model only gets possible if calculation
agents are available. These agents need to be able to solve
all different classes of the actual problem. Therefore the
coordination agent collects and manages information
about all available calculation agents and their
possibilities to solve problems. In this context economical
aspects can also be considered within the process of
solving by the possibility to weight different agents. In
case of multiple coordination agents working in the same
surrounding it’s necessary to care about the status of
agents to avoid multiple tasks for the same agent.
- To instruct a calculation agent to solve a partial
problem, the partial problem must be created. In case of a
simple problem and a single calculation agent able to
solve the problem, the partial problem can be the problem
and can directly be handed over to the calculation agent.
In all other cases the problem must be split.
An obvious splitting for weak coupled systems is based
on the different types of physics. Further opportunities
for splitting results out of the method of BEM-FEM
coupling (combining the positive effects on both methods
by calculating non-linear equations with FEM while the
linear once are calculated with BEM). An approach
splitting the different physics and considering BEM/FEM
coupling is realised for electromagnetic field problems in
[11]. There it was shown that iterative coupling of BEM
and FEM results in an increase of convergence compared
to a strong coupling for the different physics. So this
segmentation based on the different type of equations and
physics is used.
Another way to decompose different domains is based
on regions solvable with the same numerical method. The
regions are usually segmented by borders of the different
materials. This is especially useful for distributed
calculation and for different discretisation size within one
model. The idea as well as a domain decomposition based
on the number of available resources is realised for FEM
in FETI [12] and for BEM in BETI [13].
Another idea is based on overlapping regions only
considering Dirichlet boundary conditions [14]. This is of
special interest, if we take a look at the amount of data
exchanged between different agents.
All mentioned methods for domain decomposition
have in common, that the decomposition has to be done
before the actual solving is done. A flexible or dynamic
adaption to results or partial solutions gets possible with
this agent based approach. This rapidly increases the
speed of convergence for a complex simulation. So this
approach gets more flexible, more dynamic and more
adapted to available resources compared to existing
domain decomposition algorithms.
- In a next step the partial problems need to be
distributed to the different calculation agents. Therefore
the coordination agent reserves required calculation
agents. Further it shares all necessary information
including the actual partial problem and initialises the
solving process.
If a calculation agent finishes, a notification is received
by the coordination agent. The coordination agent then
updates its progress variables and checks, if all other
agents working on the same problem have finished. In
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this case the overall solution is available. If other agents
are still working, only a partial solution can be offered. In
case of no solution can be found a creation of a finer
mesh and an initialisation of the splitting process are done
in hope to find solvable partial problems. The user finally
gets informed about these circumstances.
Fig. 4: The Coordination agent
C. The Calculation Agent
Calculation agents are independent programs. At their
start up all necessary parameters are set independent from
a usage by a coordination agent. The functions of the
calculation agent are summarized in Fig. 5 and described
in detail in the following.
- Before a calculation agent can offer its service to a
coordination agent, it has to complete its description.
Therefore a unique name must be set for each calculation
agent while it’s initialised. Also the status the agent is
currently in and the problem classes the calculation agent
is capable of solving needs to be specified as well. The
problem classes the calculation agent is able to solve
depend on the specific solver the calculation agent is
connected to. To reach a flexible system and to allow
calculation agents to connect to different solvers, a solver
interface is created capable of handling all data and
information connected to the solver. The solver interface
is described in detail in section II.D. A important
information is the commissioner of the actual tasks the
calculation agent is working for. Therefore the name of
the coordination agent is stored within each agent to send
notifications to the commissioner if this gets necessary.
- To manage the solver interface and to satisfy its
needs is the major task for a running calculation agent.
Therefore the agent provides all information requested by
a solver and pass them onto the solver interface.
Examples are the initial boundary conditions that are
received from the coordination agent and the tolerances
for the solver.
The calculation agent also has to make sure, when ever
another calculation agent reports an available result, the
calculation agent has to check whether the result does
influence the own calculation or not. In case of an
influence, a re-initialisation of the solver process is
necessary. This includes a stop of the actual solver
process, an update of the boundary conditions after
calculation agent has received the result from the other
calculation agent and a start of the new initialized solver.
In case of a successful calculated partial solution the
calculation agent notify all calculation agents connected
to the problem about this result and distribute the result
about the new boundary conditions if they are requested.
Also the coordination agent needs to be informed about

the successful calculation and the availability of the
results. In case of a failure or a not converging solving
algorithm chosen by the calculation agent, the
coordination agent also has to be informed. The same
holds for a drop of available solver resources.
- Whenever a calculation agent is started a GUI
provided by each calculation agent is displayed. This GUI
allows setting the calculation agent parameters as well as
connecting it to a solver interface includes setting
necessary parameters therefor. Examples for these
parameters are the host the solver is running on, the port
this host allows to establish a connection and the problem
classes that could be solved with the given solver. All
functions the GUI provides are needed whenever the
coordination agent instructs a calculation agent to solve a
problem. Therefore the GUI provides functions like
loading a model, starting the calculation, extracting the
results and a possibility to cancel the actual solving
process. These functions can also being used without a
connection to a coordination agent. In this case the
calculation agent solves simple problems on its own.
- To track the actual process of solving a partial
problem and to understand the behaviour of a calculation
agent each calculation agent offers a separate function of
writing a log file and displaying it within the GUI.
To grant the availability for the coordination agent during
the complete process of solving and to allow the
calculation agent to react flexible to information from
other agents at least two threads are created within the
calculation agent. The first thread represents the
functionality of the calculation agent. Further threads are
used by the solver and its interface to solve the partial
problem. Only the realisation with multiple threads
allows the calculation agent to handle requests form the
coordination agent as well as checking for changes in the
boundary setting and interrupt in case of necessity while
the solver is calculating a solution for the problem. A
quick and efficient message exchange allows sending and
receiving as well as processing messages with very little
delays is another important part to grant the flexibility of
the system. Consequences for the solver of received
messages are handled by the solver interface.
Fig. 5: The Calculation Agent
D. The Solver Interface
The solver interface is part of the calculation agent. It is
the bridge between the calculation agent and a solver. It
allows the calculation agent to solve at least on problem
class. The functions of the solver interface are described
in the following and summarize Fig. 6.
- To establish a connection between the calculation
agent and different solvers the solver interface can be
understood as a collection of libraries controlling a
variety of solvers. While starting the calculation agent the
specific type of solver must be selected and parameters
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for the reachability of the solver must be set. Examples
are the host name and the network port the solver can be
reached or the local running solver application. In case of
multiple solver interfaces managed by a single calculation
agent it gets possible to create calculation agents with the
possibility to solve multiclass problems.
- If a connection to a solver is established the
necessary parameters need to be set. Therefore the partial
problem received by the calculation agent must be
translated to a form the solver does understand and
passed to the solver.
- In the next step the solver interface starts the solver.
- While the solver is calculating the major task of the
solver interface is to control the solver and manage its
output. This includes monitoring all information created
by the solver as well as interrupting the solver for
checking possible changes due to results of other agents.
Additional information like the convergence behaviour of
the actual partial problem, the availability of the solver,
its resources and a guess for the remaining calculation
time are collected by the solver interface and passed to
the calculation agent. The analyses of the information
allow a quick reaction from the calculation agent and also
the coordination agent to any changes of the system. An
example is a temporary unavailable solver. Due to that
act the calculation agent gets temporary unable to solve
problems and has to disconnect itself from the
coordination agent and the problem. Then the
coordination agent has to find an alternative for the
calculation agent to successfully solve the problem.
Another example concerns the possibility of convergence.
If the calculation agent recognizes a convergence is
unlikely, the calculation agent has to reconsider the
chosen form of the solver or in the worst case a message
has to be sent to the coordination agent to replace the
actual splitting by a different on.
- After a successful calculation the solver interface
notifies the calculation agent to inform other agents about
the available result. All information connected to the
message exchange between different agents is handled by
the calculation agent. The solver interface only takes care
about the information directly connected to the solver.
If the result is requested, the solver interface extracts the
result and translates it into a form that can be shared with
other agents. In case the result or the solver is no longer
needed the solver interface detach the connection to the
solver, release reserved resources and initialise the
deregistering process of the calculation agent.
Fig. 6: The solver interface
E. Processing Details
The way a problem is solved do significantly depend
on the timing of the agents finishing their calculations
and informing others agents about their results. So in
coupled systems not every effect has the same meaning at
each moment within the process of solving. The time to

calculate a partial result for two physics depends for an
identic mesh on the linearity of the materials for the
different physics as well as on the resources each agent is
able to use. Therefore the dependent partial results are
offered at a different time and the timing issue to the
global system needs to be taken special care of.
As an example the temperature at a circuit board after a
certain time should be calculated like it’s shown in Fig. 7.
The system consists of one coordination agent and two
calculation agents. The first calculation agent calculates
the electric field and as a side effect, the resistive losses.
The second agent takes care of the calculation of the heat
conduction from the transistor. In the described approach
the recalculation of the overall temperature simulation
will automatically be initialised if the result of the electric
simulation including the resistive losses is present and do
significantly change the result of the temperature
simulation. Because of the parallel calculation of the
circuit board it’s only necessary to recalculate some
values of the matrix. Fig. 8 and Fig. 9 show two different
scenarios for handling the different calculation times.
Fig. 7: Coupled Problem
In Fig. 8 a calculation procedure is assumed where the
calculation agent responsible for solving the electric field
problem has finished its calculation first. In that case the
calculation agent responsible for the temperature has to
check whether the heat radiated from the electric current
does significantly change the own result or can be
ignored. In the example the heat has to be considered.
Therefore the temperature calculation agent has to update
its calculation based on the result of the electric
calculation agent. Therefore it adapts its boundary
conditions to the result and recalculates again. If the
temperature calculation agent also finishes and in case of
no more calculation agents working on the problem the
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coordination agent requests all calculated partial results
and combines them to a single result that is finally
offered to the user.
Fig. 8: Solution case I
In Fig. 9 the opposite case is considered. The
calculation agent responsible for solving the temperature
problem finishes first. In this simplified case the
temperature does not have any influence on the electric
conductivity so the electric calculation agent responses a
“not acknowledge” (NACK) to the given information
about a result. This NACK means that there is no
influence expected from the partial result of the
temperature calculation agent to the electric calculation
agent. Then the electric calculation agent passes on and
finishes its calculation regularly. If the temperaturecalculation
agent obtains the result of the electriccalculation
agent it checks its result and recalculates it as
in the previous case. The following continues equally.
Fig. 9: Solution case II

III. IMPLEMENTATION ENVIRONMENT
A. The Agent System
The actual implementation is using the Java Agent
Development Framework (JADE) as a middle-ware to
implement the mentioned agents. JADE is a Java agent
based development framework. It is distributed by
Telecom Italia and currently available under Lesser
General Public License Version 2 (LGPLv2) with the
latest version 4.2.0 and a release on 26 th June 2012. The
implementation in Java, grants a system and operating
independent environment for usage. The minimal system
requirement is a running Java 1.4 runtime-environment
available for mostly every system and even smartphones.
The agent communication is based on a protocol
containing seven layers that ensure the correct
transmission and reception of message from different
types. The complete system is thereby based on a
standard offered by the “Foundation for Intelligent
Physical Agents” (FIPA) that was inherited by IEEE in
2005
B. The Solver
Each calculation agent can connect itself to two different
solvers. As a FEM-software, COMSOL Multiphysics
[15] offers a bench of modern algorithms able to solve
coupled problems. It also includes the creation of a mesh
with multiple mesh types. In this context it is quite useful
that all elements available in the software can be reached
via the Java-API COMSOL Multiphysics offers. It is also
possible to run the software as a server and connect to the
server via an offered jar library for Java-programs. The
library also passes results that can be visualised within a
GUI. The prototype of the GUI for the calculation agent
is based on the example for the usage of the COMSOL
API. Within the API the GUI and the solver are already
realised in parallel threats and notifications are send when
a task starts or finishes. This helps initialising further
events. Necessary functions to control the solver and its
behaviour are also implemented. As a BEM-software the
calculation agents are prepared to connect to FAMU [8].
IV. CONCLUSION
This approach will efficiently solve highly complex
three dimensional coupled field problems based on the
idea of software agents spread onto multiple distributed
computers including the cloud. The distributed computers
run a so called calculation agent, able to solve smaller
problems. The total expenditure for finding a solution is
reduced by the creation of multiple smaller equation
systems. The complex coupled problem is split by a
variation of already established domain decomposition
methods into these smaller problems. The domain
decomposition used, is based on different physics and
different material properties as well as geometrical
aspects. Every partial problem is then assigned to a
calculation agent and handled for its own. Finding a
solution of a coupled problem only gets possible by the
communication and negotiation between the different
agents. The communication also allows to dynamically
adapt the system to new surrounding and to find
convergence in coupled systems
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An additional approach to reduce the effort for finding
a solution is reached by the independent decision agents
are allowed to take. This concerns especially the way the
calculation agents solving the given partial problem. This
includes the decision for a numerical method like FEM or
BEM and the reaction on changed boundary conditions.
Meaning, slightly modified boundary conditions are
skipped for the partial result if no change in the result is
expected instead of initialisation a new calculation cycle.
The domain decomposition and the coordination of the
interworking of different calculation agents are handled
by a so called coordination agent. In this paper the
realisation of a calculation agent as well as the realisation
of a coordination agent with their different functions and
their interworking is described.
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Human exposure to the magnetic field
produced by MFDC spot welding systems
D. Bavastro ∗ , A. Canova ∗ , L. Giaccone ∗ , M. Manca ∗ , M. Simioli †
∗ Politecnico di Torino - Dipartimento Energia, C.so Duca degli abruzzi 24, 10129 Torino, Italy
† KGR S.p.a, via Nicolao Cena 65, 10032 Brandizzo, Italy
E-mail: luca.giaccone@polito.it
Abstract—In this paper the magnetic field emission of Medium Frequency Direct Current (MFDC) spot
welding system is analyzed with reference to the exposure of working population. In the first part of the
paper experimental measurements have been carried out in order to get the magnetic field emission of
a selected MFDC system. The measurement is performed in time domain acquiring the waveform of the
magnetic field. In the second part of the paper the methodologies suggested by the ICNIRP guidelines have
been adopted for the analysis of the waveforms: the equivalent frequency method, the multiple frequency method
and the weighted multiple frequency method. An exhaustive comparison of the possible methodology suggested
by the guidelines is given and contextualized in the regulatory framework. The emission of MFDC spot
welding system has been completely characterized. By means of the spectral analysis it is found that the
overcoming of the limit is mainly due to the 2000 Hz and 4000 Hz components. This result is useful for
manufacturers because, in order to minimize the overall emission, it is possible to think about a mitigation
system that encloses only the related internal components.
Index Terms—pulsed magnetic fields, quasi-static magnetic fields, spot welding, MFDC
I. INTRODUCTION
Protection of the working population against
the possible effects of extremely low frequency
(ELF) electromagnetic fields is a concern of the
European Community, which has published 2004
Directive 2004/40/EC [1]. The Directive refers to
the risk to the health and safety of workers due
to known short-term adverse effects in the human
body caused by the circulation of induced currents,
by energy absorption, and contact currents. One of
the most important points stated in the Directive is
the rationale of exposure at low frequency which
is defined in accordance with ICNIRP 1998 guidelines
[2]. These Guidelines report that in the ELF
frequency range, the risk to the health and safety of
workers is due to known short-term adverse effects
caused by the circulation of induced currents in the
human body.
INCIRP guidelines and directive 2004/40/EC
provide a definition of reference or action values
(i.e., values which can be directly measured like
magnetic flux density) and exposure limit values
(i.e., limit which are based directly on established
health effects and biological considerations like
current density).
In this paper the exposure to the magnetic field
produced by Medium Frequency Direct Current
(MFCD) spot welding devices is analyzed. For this
application the magnetic field waveform is pulsed
and non-sinusoidal. While continuous wave mode
of exposure is strictly defined in ICNIRP guidelines,
the evaluation for pulsed or non-sinusoidal
magnetic field waveforms is still an open question
[3], [4], [5], [6], [7], [8], [9]. In the ICNIRP guidelines
(year 1998), the problem of non-sinusoidal
waveforms was tackled by means of superposition
of harmonic values. This approach, even if possible,
has been highly criticized afterwards because
of an excessive conservative estimates of exposure
levels. Due to the increasing importance of nonsinusoidal
sources of magnetic fields, in 2003
ICNIRP has published a new guideline for pulsed
and complex non-sinusoidal waveforms [10]. This
document is strongly based on the result obtained
from K. Jokela [11]. It addresses the exposure evaluation
in non-sinusoidal conditions by means of
proper weighting factors to be applied to different
harmonic components of the waveform spectrum.
In 2010 the ICNIRP published a new set of
guidelines [13]. There are two main differences
between this document and the older one: 1) the
dosimetric quantity used for ELF electromagnetic
field is E (V/m) instead of J (A/m2 ). 2) The limits
imposed on action values have been increased
as can be observed in Fig. 1.
From the analysis of the current literature, several
papers analyze the same problem by computing
the spectrum of the measured welding current.

Fig. 1. Reference levels for occupational exposure to time
varying magnetic field. Comparison between 1998 and 2010
values.
Afterward, the welder is usually modeled as a
coil and, simulations are performed in frequency
domain for each spectral component of the current
in order to derive the current density inside a
human model or a simplified and standardized
model [8], [9], [12]. Even if these kind of simulations
are not an easy task, the procedure is often
preferred because it is easier to measure the current
in time domain rather than the magnetic field,
excpecially for quasi-rectangular waveform [14].
The drawback of this procedure is that assuming
a spectrum for the current, the generated magnetic
field is characterized by the same spectrum in all
the surrounding point due to the neglection of the
nonlinear electrical devices inside the body of the
welder.
In this paper the different methodologies provided
by the ICNIRP to analyze pulsed and nonsinusoidal
magnetic field will be applied to the
MFCD spot welding devices. The actual waveform
of the magnetic field have been measured taking
care to the possible measurement problem [14].
Finally, in order to compute the exposure level
the limit provided by the ICNIRP 1998 has been
employed due to the fact that currently the Italian
regulation framework refers to those guidelines.
II. MFDC SYSTEMS
In Fig. 2 the conversion chain of the MFDC
system is represented. The supply power, taken
from the three-phase system at 50 Hz, is driven
by means of a rectifier to an IGBT switch. The
waveform at the input/output of the transformer is
characterized by a frequency of 1000 Hz and, after
a full-wave rectification (f = 2000 Hz), is applied
to the welder terminals. The welder terminal can
be considered as a R-L load. The switching of
the IGBT bridge is controlled so that the welding
current reaches a desired (constant) value. The
- 102 - 15th IGTE Symposium 2012
welding process can be performed by means of
a single impulse or with more the one impulse.
In Fig. 3 the waveform of the weld current is
shown. As it can be observed, the actual current
is not perfectly rectangular because the conversion
system is not able to nullify completely a ripple at
2000 Hz (and higher harmonics) that is superposed
to the weld current. The main weld parameter are:
the current peak (Ip) that is usually in the order
of some kA, the weld time that is the duration of
the single pulse and the hold time that is a period
when the current is zero after the welding, but the
electrodes are still applied to the sheet to chill the
weld.
Fig. 2. MFDC spot welding device under analysis. The
reference system for the measurement is placed in the center
of the welding coil.
current
peak
weld time hold time
Fig. 3. Weld current and its main parameters. Weld time =
140 ms, hold time = 40 ms.
In the spot welding sector the MFDC welder are
often preferred to the AC ones because of some
benefit: a shorter weld times is required due to the
the DC output current, hence significant energy
saving is obtained. Moreover, MFDC systems are
very stable in working condition far from the rating
power (useful range: 20-95%). Conversely, AC
systems are unstable and inefficient when used
outside the 70-90% of the rating power.
III. EXPERIMENTAL MEASUREMENT
In this paper the magnetic field emission of
a MFDC welder produced by KGR S.p.A. is
analyzed. In Fig. 4 the layout with dimensions

is shown. The MWG model is a manual welder,
therefore the operator is quite close to the device
during the welding operation. In Fig. 5 a classical
working configuration is reported in frontal and
lateral view.
Several observation points have been defined
in order to evaluate the human exposure in the
working configuration represented in Fig. 5. It
has to be stressed that the considered working
configuration is also the most critical one because
the operator is faced to the welder coil.
The measurements points are summarized in Table
I. The coordinates are related to the reference
system in Fig. 6.
Fig. 4. Weld current and its main parameters. Weld time =
140 ms, hold time = 40 ms.
(a) (b)
Fig. 5. working configuration: front view (a) and side view
(b)
TABLE I
FIELD POINTS
field point x (m) y (m) z (m)
P1 0 0 0.28
P2 0 0 0.5
P3 0.5 0 0.28
P4 0.5 0 0.5
P5 0.8 0 0.28
P6 0.8 0 0.5
Al the measurement will be referred to the
current represented in Fig. 3: current peak (Ip)
equal to 12 kA, wled time equal to 140 ms and
hold time equal to 40 ms.
For each measurement point the waveform of
the magnetic field has been measured along the
three axis (x, y and z). Finally the rms values
- 103 - 15th IGTE Symposium 2012
Fig. 6. Reference system for measurement points definition
has been computed. The total field waveforms for
each observation points are shown in Fig. 7. By
comparing those waveforms it is possible to derive
some considerations: 1) as expected, the maximum
value is observed for the field points P1 and P2
that are faced to the welder coil. 2) P1, P3 and P5
present a higher peak values with respect to P2,
P5 and P6 because they are closer to the welder.
3) by defining two groups of points with the same
distance from the welder [P1, P3, P5] and [P2,
P4, P6] it is possible to note that the peak value
decreases by moving far from the welder coil. 4)
the decreasing low for the peak value is not true
for the medium frequency ripple superposed to the
waveforms. It seems that the observation points
P3 and P4 are characterized by the higher ripple.
This consideration will be better investigated in the
following by analyzing the spectrum components
of all the waveforms. At this stage, a qualitative
explanation can be given considering that P3 and
P4 are located in front of the full wave rectifier
connected to the secondary winding of the medium
frequency transformer. Hence, P3 and P4 are the
points more influenced by the conversion system.
IV. HUMAN EXPOSURE EVALUATION
The magnetic field produced by MFDC spot
welding system is a pulsed magnetic field (see
Fig. 7). With reference to the ICNIRP guidelines
[2], [10], [11], it can be analyzed with three different
approaches: the equivalent frequency method,
the multiple frequency method and the weighted
multiple frequency method.
A. Equivalent frequency method
The equivalent frequency method is introduced
with the note 4 of Table 4 and 6 in the ICNIRP
guidelines [2] . Afterward, it is better detailed in
the guidelines focused on pulsed magnetic fields
[10]. The method simply takes into account the
pulse duration (tp) and maximum value of the
waveform during impulse. Finally it refers to the
equivalent and continuous sinusoidal field with a
frequency feq =1/tp in order to test compliance
with the reference levels.

(a)
(b)
(c)
Fig. 7. P1 and P2 (a) P3 and P4 (c) P5 and P6
Fig. 8. Equivalent frequency method: application to the
measurement point P1
If a single pulse of the magnetic field measured
in the observation point P1 is analyzed by means
of the equivalent frequency method the result of
Fig. 8 is obtained. The single pulse is characterized
- 104 - 15th IGTE Symposium 2012
by tp = 140 ms that leads to a continuous sinusoidal
field with frequency equal to feq =3.5 Hz.
The amplitude of the sinusoidal field is imposed
to the maximum valued observed during the single
pulse, i.e. 6298.3 μT.
It is now simple to identify the reference level
for a 3.5 Hz sinusoidal field from Fig. 1, that is
16327 μT. It must be stressed that reference level
is a rms value, therefore it must be compared with
6298.3/ √ 2 = 4453.5 μT. Finally, by means of the
equivalent frequency method, the welder produces
a magnetic field that is 3.5 times lower than the
applicable reference level in the observation point
with the higher emission.
B. Multiple frequency method
The multiple frequency method is suggested for
non-coherent waveform, i.e. waveform that can not
be measured with repeatability property because of
their intrinsic variation in time.
The procedure is summarized in the following
steps:
• selection of the signal to be analyzed
• perform the Fourier Transform of the signal
• computing the global indicator defined as:
where:
Is =
65 kHz
Bj
BL,j
j=1 Hz
+
10MHz j>65 kHz
Bj
b
(1)
• Bj is the magnetic flux density at frequency
j;
• BL,j is the magnetic flux density reference
level;
• b is 30.7 μT (rms) for occupational exposure
The exposure is compliant with the limit if Is < 1.
In order to test the compliance with this method
a single impulse in each observation point has
been selected in order to perform the Fourier
Transform. Finally, in order to better point out the
rationale of this methodology, the Is representation
is provided:
• graphically: in the x-axis is represented
the frequency and in the y-axis the ration
Bj/BL,j. With this it is possible to understand
what are the frequency components that
exceed the relative limit.
• numerically: computation of Is with (1)
For the sake of brevity the graphical result are
shown just for the measurement points P1 and P3.
In Fig. 9 it is possible to see the spectrum of the P1
and P3 waveforms. It is possible to observe that the
harmonic content is mainly located in the range 0-
100 Hz with significant values of DC component.
The spectrum is also characterized from a 2000 Hz

(a)P1:x=0m,y=0m,z=0.28 m
(b)P3:x=0.5m,y=0m,z=0.28 m
Fig. 9. Spectral analysis of the measured waveform
(a)P1:x=0m,y=0m,z=0.28 m
(b)P3:x=0.5m,y=0m,z=0.28 m
Fig. 10. Computation of the ICNIRP limit by means of the
multiple frequency method
- 105 - 15th IGTE Symposium 2012
component and its multiple. As explained before
those components are generated from the final
full wave rectifier. In fact, the observation points
P3 that is faced to the rectifier, present in its
spectrum the higher harmonic content for this set
of frequencies.
From the analysis of Fig. 10 it comes out that
none of the observation points can be considered
complaint if the multiple frequency method is
adopted because the Is values exceed significantly
the unitary reference level (see also Table II for the
other measurement points). Remembering that the
y-axis of Fig. 10 represents the summation terms
of (1) it is possible to observe that the overcoming
of the limit is mainly due to the 2000 Hz and 4000
Hz components. For point P3 those harmonics are
about 5 times higher than the relative reference
level. It is worth noting that reference levels become
stricter for higher frequencies. Moreover,
above 820 Hz eq. (1) imposes a constant limit
equal to 30.7 μT even if in Fig. 1 the trend is
still decreasing above 65 kHz.
Finally, it must be stressed that this procedure
is based on the assumption that the spectral components
add in phase, i.e., all maxima coincide at
the same time and results in a sharp peak. This is
a realistic assumption when the number of spectral
components is limited and their phases are not
coherent, i.e., they vary randomly. For fixed coherent
phases the assumption may be unnecessarily
conservative [2], [11], [13].
C. Weighted multiple frequency method
The weighted multiple frequency method is proposed
for coherent waveform, i.e. waveform that
can be measured with repeatability property. For
these signals the multiple frequency method always
leads to a conservative exposure assessments.
This modification provides an alternative method
based on weighted peak that more closely recognizes
the nature of biological interactions.
The procedure is summarized in the following
steps:
• selection of the signal to be analyzed
• perform the Fourier Transform of the signal
• computing the global indicator defined as:
Iw = max
(WF) j
Bj cos (2πfjt + θj + ϕj)
j
(2)
where:
• Bj is the amplitude of the j-th frequency
component;

• WFj is the weighting function where the
magnitude is equal with the inverse of the
peak reference level at j-th frequency
• θj is the phase of the i-th component of B
• ϕj is the phases of the weighting function
of the i-th component. It should satisfy the
conditions:
– ϕ(f) =π/2 if ffc
• fc = 820 Hz for occupational exposure
The exposure is compliant with the limit if Iw < 1.
Conversely from the multiple frequency method
here the spectrum of the waveform is weighted by
a complex function. It is possible to observe that
the ICNIRP definition of the weighting function
for magnetic flux density waveform is a high-pass
filter with cut-off frequency equal to fc = 820 Hz
[10], [11].
In order to better point out the rationale of this
methodology, the Iw representation is provided:
• graphically: it is provided the
graph of the function of time:
j (WF) j Bj cos (2πfjt + θj + ϕj)
• numerically: computation of Iw with (2)
From the analysis of the results presented in
Fig. 11 it is possible to observe that even applying
the weighting methods none of the observation
points can be considered compliant with
the unitary reference level (see also Table II for
the other measurement points). In spite of this
result, the weighting procedure allows a significant
reduction for some observation points (see P1
and P3 values). This is in accordance to the fact
that for pulsed fields associated to spot welding
machine the multiple frequency method is a too
conservative assessment procedure.
V. CONCLUSIONS
In this paper the magnetic field generated from
MFDC spot welding systems have been analyzed.
All the possible methodology suggested in the
international guidelines provided by the ICNIRP
have been employed: the equivalent frequency
method, the multiple frequency method and the
weighted multiple frequency method.
The main conclusion is that the three methodologies
lead to contrasting result. By means of
equivalent frequency method the emission is compliant
with the reference level even in the maximum
emission point (3.5 times lower than the
limit). On the other hand, the other two approaches
provide opposite results, i.e., none of the surrounding
points is compliant with the limit.
In Table II the results for weighted and nonweighted
multiple frequency method are summarized.
For most of the observation points, a high
- 106 - 15th IGTE Symposium 2012
(a)P1:x=0m,y=0m,z=0.28 m
(b)P3:x=0.5m,y=0m,z=0.28 m
Fig. 11. Computation of the ICNIRP limit by means of the
wighted multiple frequency method
difference in the result is observed. In our opinion
for the spot welding process the multiple frequency
method is too conservative because the waveform
is clearly coherent due to the supply control system.
However, result for weighted multiple frequency
method are still quite far from the unity
limit.
The definition of exposure limits for pulsed
magnetic fields is still an open problem. Therefore
no Standard and guidelines define precisely
what is the constraint and the procedure to be
observed for spot welding machine as well as other
technologies characterized by similar emission of
pulsed magnetic field (e.g MRI devices). One of
the consequence is that the Directive 2004/40/EC
has been modified extending the deadline for the
application of the reference levels [15], [16] that
now is fixed for October 31th 2013. Concerning
the EU directives, it must be stressed that reference
levels are always related to acute effects and not to
possible long term effect. In the literature, as well
as in the real life applications, there is no evidence
of acute effects. For possible long term effects, the
epidemiological studies recognizes that it is hard
to estimate the exposure because may arise other
type of exposure in the same workplace which are
correlated to the same EMF exposure and which
may affect the health of workers. An example
concerns exposure to welding fumes which may

increase lung cancer risks among welders [17],
[18].
TABLE II
COMPARISON BETWEEN WEIGHTED AND NON-WEIGHTED
MULTIPLE FREQUENCY METHOD
field point Is Iw
P1 99.68 16.77
P2 13.60 4.23
P3 125.28 36.41
P4 18.23 7.51
P5 23.64 7.80
P6 6.38 4.61
From the technical point of view it is quite impossible
to apply a mitigation system to the welder
coil for (obvious) operating reason. The analysis of
Table II together with Fig. 9 allows to understand
that the overcoming of the limit is mainly due to
the 2000 Hz and 4000 Hz components. For point
P3 those harmonics are about 5 times higher than
the relative reference level. Therefore, the future
development of this work is to design a shielding
case for the transformer and the full wave rectifier.
Obviously the mitigation system will not reduce
the maximum value of the magnetic field (that is
generated from the welder coil). The aim is just to
reduce as much as possible the medium frequency
components that seem to play a significant role in
equations (1) and (2).
REFERENCES
[1] Directive of the european parliament and of the council
of 29 april 2004 on the minimum health and safety
requirements regarding the exposure of workers to the
risks arising from physical agents (electromagnetic fields)
european parliament and council.
[2] ICNIRP. Guidelines for limiting exposure to time varying
electric, magnetic and electromagnetic fields (up to 300
GHz). Health Phys, 74(4):494–522, 1998.
[3] H. Heinrich. Assessment of non-sinusoidal, pulsed or
intermittent exposure to low frequency electric and magnetic
fields. Health Phys, 96(6):541–546, 2007.
[4] R. Scorretti, N. Burais, A. Fabregue, and O. Nicolas, and
L. Nicolas. Computation of the induced current density
into the human body due to relative LF magnetic field
generated by realistic devices. IEEE Transactions on
Magnetics, 40(2):643–646, 2004.
[5] D. Desideri and A. Maschio. Magnetic field emissions
up to 400 kHz from a welding equipment. In Proc.
Int. Symp. Electromagnetic Compatibility, pages 151–
156, Barcelona, 2006.
[6] D. Desideri, A. Maschio, and P. Mattavelli. Human
exposure topulsed current waveforms below 100 kHz.
In 391-396, editor, Proc. Int. Symp. Electromagnetic
Compatibility, Hamburg, Sep. 8–12, 2008.
[7] G. Kang and O. Gandhi. Comparison of various safety
guidelines for electronic article surveillance devices with
pulsed magnetic fields. IEEE Transactions on Biomedical
Engineering, 50(1), 2003.
[8] A. Canova, F. Freschi, and M. Repetto. Evaluation of
workers expo- sure to magnetic fields. The European
Physical Journal Applied Physics, 52(2), 2010.
- 107 - 15th IGTE Symposium 2012
[9] A. Canova, F. Freschi, L. Giaccone, and M. Repetto. Exposure
of working population to pulsed magnetic fields.
IEEE Transaction on Magnetics, 46(8):2819–2822, 2010.
[10] ICNIRP. Guidance on determining compliance of exposure
to pulsed and complex non-sinusoidal waveform
below 100 kHz with icnirp guidelines. Health Phys,
84(3):383–387, 2003.
[11] K. Jokela. Restricting exposure to pulsed and broadband
magnetic fields. Health Phys, 79(4):373–388, 2000.
[12] F. Dughiero, M. Forzan, and E. Sieni. A numerical
evaluation on electromagnetic fields exposure on real
human body models until 100 khz. COMPEL, 29:1552–
1561, 2010.
[13] ICNIRP. Guidelines for limiting exposure to time-varying
electric and magnetic fields (1 Hz to 100 kHz). Health
Phys, 99(6):818–836, 2010.
[14] G. Crotti and D. Giordano. Problems in the detection
of quasi-rectangular magnetic flux density waveforms. In
18th Symposium IMEKO TC4, Natal (Brasil), September
2001.
[15] Directive of the european parliament and of the council of
23 april 2008 amending directive 2004/40/ec on minimum
health and safety requirements regarding the exposure
of workers to the risks arising from physical agents
(electromagnetic fields) (18th individual directive within
the meaning of article 16(1) of directive 89/391/eec).
[16] Directive of the european parliament and of the council of
19 april 2012 amending directive 2004/40/ec on minimum
health and safety requirements regarding the exposure
of workers to the risks arising from physical agents
(electromagnetic fields) (18th individual directive within
the meaning of article 16(1) of directive 89/391/eec).
[17] R.M. Sterns. Cancer incidence among welders: possible
effects of exposure to extremely low frequency
electromagnetic radiation (elf) and to welding fumes.
Environmental Health Perspectives, 76:221–229, 1987.
[18] Review of the scientific evidence for limiting exposure to
electromagnetic fields (0-300 ghz). Technical Report Vol.
15 N.3, National Radiological Protection Board, 2004.

- 108 - 15th IGTE Symposium 2012
A Circuital Approach for Eddy Currents Fast
Evaluation in Beam-like Structures
A. Formisano
Dipartimento di Ingegneria Industriale e dell’Informazione
Seconda Università di Napoli, via Roma 29, I-81031 Aversa (CE), Italy
E-mail: Alessandro.Formisano@unina2.it
Abstract — The electromagnetic analysis of mechanic or civil structures composed by an interconnection of beam-like
elements, mechanically interconnected to create a structural mesh, can be formulated in terms of an equivalent lumped
elements electric network. This is the case of the eddy currents evaluation in a truss bridge or a bridge crane exposed to a
time varying electromagnetic field or in the reinforcement of buildings concrete. In such cases, if compatible with the
accuracy needs, the network approach can be very effective thanks to the strong reduction of the model complexity. The
paper proposes such a kind of formulation, based on concept of partial inductance. Advantage is taken from automated treebuilding
algorithms for electric networks, and on minimum order formulations based on loop currents to further reduce
computational complexity.
Index Terms— Eddy Currents, Electric Circuit Theory, Filamentary Structures
I. INTRODUCTION
The use of metallic materials in mechanical and civil
engineering is a common practice, due to the extremely
favorable behavior of such materials with respect to
stresses and mechanical solicitations. On the other hand,
when exposed to time varying electromagnetic fields,
metallic structures react by generating a field due to
induced currents in their volume. The effect of such fields
may reveal critical in some particular applications, such
as when forces on the structures must be taken under
control, or when aging phenomena are facilitated by
electric currents in the metal, or when the electromagnetic
field map must be strictly controlled in critical regions
not far from the structures (e.g. to reduce interference on
electronic devices or to avoid impact on physical
phenomena requiring controlled field maps, such as
Nuclear Magnetic Resonance, or finally to limit human
exposure to electromagnetic energy).
In these cases, the possible interactions with
surrounding electromagnetic field sources must be
considered in the design phase. Unfortunately, fully 3D
numerical analysis would usually be required, since no
particular symmetry or simplification can be expected to
reduce model complexity. On the other hand, such a
model would require a quite large computational effort,
although just an estimate of the electromagnetic effects
due to the mechanical structure are often enough in the
design step.
In the particular cases when the mechanical structures
can be modeled using interconnects of beam-like
elements (e.g. truss bridges, bridge cranes, iron rebar in
reinforced concrete, etc.), the electromagnetic analysis, in
the magneto-quasi-static limit and assuming linear
behavior of all the materials, can be formulated in terms
of an equivalent electric network, composed by lumped
elements. The current in each branch of the network is
related to the current density in the corresponding beamlike
element of the structure thanks to a filamentary
current element approximation of the actual beam. Each
current element, or current stick, is defined by stick tips,
and a (scalar) stick currents.
The interconnection of sticks is defined through a
suitable incidence matrix, defined from the actual 3D
topology. The lumped network is created by associating
to each single beam of the original structure a lumped
parameter model, falling in the typical circuit classes. It
follows that a resistive parameter has to be used to model
the Ohmic behavior of the metallic element. In addition, a
set of inductances should be used to represent the
induction phenomena among sticks and their capability to
accumulate magnetic energy. Finally, an electromotive
force (typical of the voltage sources in the circuits) can be
used to represent the induction phenomena from assigned
external currents.
In principle, also capacitive parameters should be
considered to take into account the capability to
accumulate electric energy, but in the range of
frequencies here considered the impact of capacitive
phenomena can be neglected.
The mathematical tools able to face such a class of
systems are the well known Kirchhoff laws, replacing the
most general Maxwell ones, significantly reducing the
complexity of the model.
Although the lumped network approach to treat similar
structures is quite diffused in the electromagnetic analysis
of mechanical structures [1-4], the network analysis is
usually performed using standard computer codes, not
necessarily guaranteeing the minimum computational
effort. In this paper, an automated fundamental loop
method is proposed to achieve a minimum complexity
resolution of the network.
Once currents in each branch are known, simple closed
formulas can be used to estimate the field produced by
each stick, forces on the structure elements, and Ohmic
losses due to induction phenomena. This approach is
particularly useful when a quick, yet not accurate
estimation of the impact of metallic structures, is

equired.
II. MATHEMATICAL MODELING
Let's consider a structure composed of Nb metallic
beams, connected in a general way at their tips in Nn
nodes.
The equivalent lumped network will be composed of
Nb branches, with the same topology as the mechanical
structure. According to the geometrical characteristics of
the beam and, in addition, to the electromagnetic
characteristics of the materials, each branches is endowed
with a suitable set of circuit elements, including a
resistance, an inductance, some mutual inductances, and a
suited number of voltage sources.
A very simple example is reported in Fig. 1(a), while
in Fig. 1(b) the equivalent electric network is sketched.
Since the assembly is immersed in a time varying field,
we will assume that each stick, arbitrarily oriented,
carries a current ik, k=1, 2...Nb, that represents the
unknown to be determined. The currents are induced by
an external field, but their value depends also on the
structure itself, through material properties and
geometrical relationships.
voltage
source
+ -
(a)
branch
resistance
branch
current ik
(b)
self inductance
and mutual with
all other branches
Figure 1: (a) An example of mechanical interconnect of beam like
metallic elements (the structure of a truss bridge); (b) its representation
as an electric network.
Each stick can be characterized by a resistance Rk.
depending from its resistivity k, length Lk and cross
section Sk. If assuming that the penetration depth at the
highest frequency involved is smaller than the transverse
dimension of the beam modeled by stick, and that both
- 109 - 15th IGTE Symposium 2012
resistivity and cross section are uniform along the beam,
the resistance associated the k-th stick can be computed
as [5]:
Rk=k*Lk/Sk
Of course, if any of the above exposed hypotheses
falls, the more general expression using the line integral
along the beam axis of k(l)/Sk(l) can be used.
The resistances are then assembled into a diagonal
resistance matrix R.
In addition, if assuming a linear magnetic behavior, the
sticks assembly is characterized also by an inductance
matrix Mb, whose elements describe the mutual
inductance between sticks or, on the diagonal, their self
inductance. Under the same assumptions used for (1)
about skin depth, the self inductance Mkk of the k-th stick
can be computed using [6]:
4 2Lk
Mkk 210 Lkln
1 r
Lk
where r is the geometric mean distance and is the
arithmetic mean distance on the corresponding k-th beam
cross section. (2) provides self inductance in Henry is Lk
is in meters.
The mutual inductance Mjk between the j-th and k-th
stick can be computed using formulas from [6]; as a
possible alternative, assuming a limited dimension of the
cross section, the mutual inductance can be evaluated also
by line integrating (numerically) the vector potential
Ak(x) of stick k on the axis of stick j:
ˆ
M A x tˆdl
jk k j dl j
j
where j is the centerline along the j-th beam, xj is a
generic point along j, and ˆt is the centerline tangent unit
vector. The following expression has been used in this
study for Ak [5]:
j
where a, b and c are defined in Fig. 2, â is the unit
vector along the k-th stick, and suitable countermeasures
have been taken to avoid singularities when sticks are
aligned [7].
(1)
(2)
(3)
I
k 0 ˆ
cba A x a ln
4
cba
(4)
c
Ak b
xj
Figure 2: Basic elements form computation of vector potential using
(3).
Note that eq. (3) can be easily generalized to massive
a

conductors, if the thin beam approximation may reveal
too crude for the analysis, while this is not the case for
closed form expressions found in [6].
The structure is supposed to be immersed in the timevarying
magnetic field produced by another set of Ne
external currents ie, linked with the sticks by means of a
mutual inductance matrix Me.
Elements of Me can be easily evaluated using
expressions based on (3). For the particular shape of field
source, suitable analytical (possibly approximate)
expression are available; e.g., the mutual inductance of a
stick and a power line can be easily computed using
formulas from [6]. Of course, the more general procedure
based on suitable decomposition in elementary sticks and
a numerical evaluation of (3) can be used.
If assuming that external sources are given (because
not influenced by eddy currents induced in the structure
or some other factor) in each branch, the induced emf can
be circuitally described as a voltage source. The set of the
voltages is given by
e= Me die/dt + d Me /dt ie
where the last term vanishes in case of time invariance of
the matrix Me.
Within these hypotheses, the system can be regarded as
a R-L circuit, where sticks play the role of branches and
Nn nodes represent the stick tips.
The network topology is described by the incidence
matrix (Nn rows and Nb column) providing, for each
branch, the couple of starting-ending nodes. The
incidence matrix can be easily recovered from CAD
schemes for the mechanical assembly, where available, or
by survey of the drawings.
It is well known that the rank of incidence matrix is
lower than the number of nodes and its value, for a
connected network is Nn–1; consequently, often the
“reduced” incidence matrix A is used, as it will be done in
the following. The graph theory guarantees that
independent columns in an incidence matrix do not form
loops. It follows that a basis of the columns set defines a
set of branches able to connect all the nodes of the
network, i.e. a tree of the network
A fast and effective way to search for independent
columns is to determine the echelon form of A [8] and
select the branch corresponding to the leading
coefficients Of course, in general several trees can be
defined for an assigned network; the choice of the
extracted tree among all the possible ones can be
controlled by ordering the columns of the incidence
matrix in such a way that column corresponding to
favourite branches are the leftmost ones.
In order to simplify a number of automatic treatment of
the network topology, it is recommended to rearrange the
branch numbering of the matrix in such a way to include
in the first NT positions (Nn-1 in case of connected
networks), the columns of the tree branches. Then, the
incidence matrix A can be partitioned as:
(5)
A =(AT; AC) (6)
- 110 - 15th IGTE Symposium 2012
where, AT is the NTxNT non singular matrix corresponding
to tree branches, and AC the NTxNb matrix corresponding
to co-tree branches.
Several effective methods can be used to face with the
analysis of this network.
One of the most effective and popular is the nodal
technique whose unknowns are the nodes potential set vn.
Here, for simplicity just the formulation in case of linear,
memory free, voltage controlled components is
highlighted:
Yn vn = Jn
where Yn and Jn are the nodal matrix and the nodal drive
equivalent current vector, respectively. Both can be easily
evaluated by the branch parameters. In particular
Yn = A Yb A T , where Yb is the NbxNb matrix with the
conductances (self or mutual) of the branches. The
method can be extended to circuits with linear current
controlled components or linear dynamical components
and, in addition, also in presence of non linear
components. Of course, in any case, the existence and the
uniqueness of solution has to be assessed.
Here the classical dual formulation is proposed, whose
unknowns are the principal loop currents IL [9]. The
model, for general dynamic systems, can be stated in time
domain; here, taking advantage from linearity, the more
compact formulation in the Laplace space is used:
M
s s s
L M
(7)
Z I E (8)
where IL and EM are the arrays of the principal loop
currents and driven voltages, respectively, and s is the
complex frequency.
In (8) for simplicity, the hypotheses of linear, voltage
controlled components has been assumed.
T
The loop impedance matrix BZ
M b B
Z can be
deduced from the branch impedance matrix Zb = (Rb+sMb)
and from the topological NLxNb matrix B of the principal
loops related to the tree [8], where NL = Nb-NT is the
maximum number of independent loops. Similar
expressions hold for loop voltage sources EM, driven by
external currents.
It should be noticed that the partitioned form of B
B = B ; 1 includes an identity matrix for the co-tree
T L
columns. In addition, the first partition B can be easily
T
deduced by the reduced incidence matrix:
T -1
T T L
B =- A A
(9)
Once loop currents are known, currents in each branch
are easily computed as I = B IL.
From branch currents, estimates of the other electrical
quantities can be easily recovered using closed form
expression for stick currents. This is the case, for
example, of the flux density produced in any points of

space [5], or the total Ohmic power, or, finally, the net
force acting on the structure.
III. NUMERICAL EXAMPLES
In this section, firstly a simple example is discussed to
illustrate the various steps of the proposed method; then a
more complex case is presented to show the effectiveness
of the approach.
1. As a first example, the field produced by a reinforced
concrete beam near a power line is considered. The
beam is 2.4 m long, with a transverse dimension of 30
cm, a reinforcement diameter of 2 cm, a resistivity of
5x10 -5 m and is 5 m away from the line. The line
carries 100 A of current at 50 Hz, and is assumed 20
m long.
The networks has 16 nodes; the 15 tree branches are
branches 1-15 in Fig. 3, and the non-trivial part of the
fundamental loop matrix BT is reported in table I.
The currents in each branch can easily be computed
using a linear system with 13 equations, and then
post-processed to obtain estimates of the desired
quantities.
The highest currents are in the “longitudinal”
branches #17 and #19 (1.4 mA), and #21 and #23 (1.5
mA).
A 2D FEM model, neglecting connecting elements in
the mesh, and correcting conductivity to take into
account the finite length of actual geometry, provides
a current of 1.5 mA in the four “longitudinal” beams.
The "disturbance" magnetic field produced by the
beam is 1.72 nT at a point 10 cm away from the
power line. Note that in the 2D FEM model, the
reinforcement contribution was hidden by the
numerical errors.
2. The second example compares computational
complexity in the case of a fully 3D geometry either
using a commercial FEM package (COMSOL
Multiphysics Ver. 4.2a, [10]) or using the proposed
approach.
The aim is to estimate total Ohmic losses in the
metallic structure of the truss beam depicted in Fig. 1.
The bridge is 16 m long, 4 m large and 4 m high.
Each beam is a square with a 0.4 m side.
The bridge is made of non magnetic structural steel,
with a conductivity of 4x10 6 S/m.
The excitation field is provided by a circular coil
radius 5 m, hanging 10 m above the bridge. Of course
such excitation is not realistic, but has been adopted
for its ease of modelling with FEM package.
The FEM model is solved with a mesh composed of
34,000 2 nd order tetrahedral elements, giving
300,000 unknowns, and requires 165 s to be solved
on a i7-based PC, with 4GB RAM.
The total Ohmic losses are 4 mW using FEM model
and 3.5 mW using the proposed method. A map of
losses density is reported in Fig.4.
- 111 - 15th IGTE Symposium 2012
0.3 m
Figure 3: Graph of the equivalent network for case 1: capital letters
indicate nodes, number indicate branches, dashed lines are co-tree
branches
Loops
C
3
2
B
1
16
A
19
D
18
2.4 m
G
7
6
4
F
5
17
23
H
22
20
E
TABLE I
NON TRIVIAL PART OF THE FUNDAMENTAL LOOP MATRIX
Branches
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0
2 -1 -1 -1 -1 1 1 1 0 0 0 0 0 0 0 0
3 0 -1 -1 -1 0 1 1 0 0 0 0 0 0 0 0
4 0 0 -1 -1 0 0 1 0 0 0 0 0 0 0 0
5 0 0 0 0 -1 -1 -1 0 0 0 0 0 0 0 0
6 0 0 0 0 -1 -1 -1 -1 1 1 1 0 0 0 0
7 0 0 0 0 0 -1 -1 -1 0 1 1 0 0 0 0
8 0 0 0 0 0 0 -1 -1 0 0 1 0 0 0 0
9 0 0 0 0 0 0 0 0 -1 -1 -1 0 0 0 0
10 0 0 0 0 0 0 0 0 -1 -1 -1 -1 1 1 1
11 0 0 0 0 0 0 0 0 0 -1 -1 -1 0 1 1
12 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 1
13 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1
Figure 4: Ohmic Losses density plot for FEM solution of example 2.
IV. CONCLUSIONS
A prompt method to assess the effect of mechanical
structures composed by interconnected conducting
beams, exposed to low frequency magnetic fields, has
been presented. The method is based on equivalent
electric network analysis, and adopts a minimum order
formulation to solve the network.
ACKNOWLEDGEMENTS
11
L 26
O
15
14
P
12
N 28
13 M
The author wishes to thank prof. R. Martone for
fruitful discussions and for his support.
This work has been partly supported by Seconda
Università di Napoli, Italy, under PRIST grant
“Generazione distribuita di energia da fonti tradizionali e
8
21
K
10
J
9
27
24
I
25

innovabili: aspetti ingegneristici e giuridici-economiciambientali”
REFERENCES
[1] A. Ruehli, "Equivalent Circuit Models for Three-Dimensional
Multiconductor Systems", IEEE Trans. on Microwave Th. and
Tech., vol. MTT-22, pp. 216-221, 1974.
[2] W. Pinello, A. Ruehli, “Time Domain Solutions for Coupled
Problems using PEEC Models with Waveform Relaxation”,
Proceedings of Antennas and Propagation Society International
Symposium AP-S. Digest, pp. 2118-2121, 1996.
[3] A.Y. Wu, K.S. Sun, “Formulation and implementation of the
current filament method for the analysis of current diffusion and
heating in railguns and homopolar generators”, IEEE Trans. on.
Mag., vol. 25, pp. 610-615, 1989.
[4] B. Azzerboni, E. Cardelli, M. Raugi, ”Network mesh model for
flux compression generators analysis, IEEE Trans. on Magn., vol.
27, pp. 3951-3954, 1991.
[5] H. A. Haus, J. R. Melcher, Electromagnetic Fields and Energy,
Englewood Cliffs, NJ: Prentice Hall, 1989.
[6] F. Grover, Inductance Calculation, New York: Van Nostrand,
1946.
[7] J. D. Hanson, S. P. Hirshman, “Compact expressions for the Biot–
Savart fields of a filamentary segment”, Physics of Plasmas, vol.
9, pp. 4410-4412, 2002.
[8] L. Chua, I. Lin, Computer-Aided Analysis of Electronic Circuits,
3rd ed., vol. 2. Oxford: Clarendon Press, 1982.
[9] J. Nilsson, S. Riedel. Electric Circuits, Englewood Cliffs, NJ:
Prentice Hall, 2010.
[10] www.Comsol.com, last visited on Sept., 4 th 2012.
- 112 - 15th IGTE Symposium 2012

- 113 - 15th IGTE Symposium 2012
Effectiveness of the Preconditioned
MRTR Method Supported by Eisenstat’s Technique
in Real Symmetric Sparse Matrices
*Yoshifumi Okamoto, *Tomonori Tsuburaya, † Koji Fujiwara, and *Shuji Sato
*Department of Electrical and Electronic Systems Engineering, Utsunomiya University
7-1-2 Yoto, Utsunomiya, Tochigi 321-8585, Japan
† Department of Electrical Engineering, Doshisha University
1-3 Tataramiyakodani, Kyotanabe, Kyoto 610-0321, Japan
E-mail: okamotoy@cc.utsunomiya-u.ac.jp
Abstract—The Incomplete Cholesky Conjugate Gradient (ICCG) method is widely used to solve the indefinite algebraic
equations obtained from the edge-based finite element method. However, when a linear solver based on the minimum
residual without residual oscillations is used, there is a possibility of the elapsed time being shortened. This paper shows the
effectiveness of the preconditioned Minimized Residual method based on the Three-term Recurrence formula of the CG-type
(MRTR) method with Eisenstat’s technique by making a comparison with ICCG method for real symmetric sparse matrices.
Index Terms— Eisenstat’s technique, ICCG method, MRTR method, split preconditioning.
A x b,
(1)
I. INTRODUCTION
where A is a large sparse n-by-n matrix, x is a solution n-
The ICCG method [1] is widely used as a solver for a vector, and b is a n-vector. Now, suppose that the
real symmetric indefinite linear equation derived from the diagonal scaling has already been applied to (1).
edge-based finite element method. While the behavior of The recurrence formula for x in MRTR method is
a residual in CG iterations is oscillatory, a monotonic designed using the expression
decrease in the residual is mathematically ensured in
MRTR method [2], which is an algorithm identical to that
of Orthomin(2) [3] and which is based on the minimum
residual. Therefore, there is a possibility that MRTR can
solve linear equations faster than the CG method.
IC factorization is widely recognized as a powerful
preconditioner for a symmetric linear system. However, it
may not necessarily be powerful in various magnetic
field problems. Some of the split preconditioners such as
symmetric Gauss-Seidel (SGS) and diagonal IC
factorization (DIC) might be capable of improving the
convergence characteristics of linear solvers.
SGS and DIC preconditioners, in which the offdiagonal
components in the original linear system are
directly used in the preconditioned matrix, can utilize the
Eisenstat’s technique [4], in which the matrix-vector
product can be replaced by forward and backward
substitutions. Therefore, there is a possibility that SGS
and DIC preconditioners supported by Eisenstat’s
technique can reduce the elapsed time by reducing the
computational cost of an iterative process.
This paper shows the effectiveness of preconditioned
MRTR method supported by Eisenstat’s technique for the
xk 1 x0
zk
1,
zk
1
K k ( A;
r0
) ,
(2)
where xk+1 is the solution vector in step (k + 1) in the
iterative process and Kk(A;r0) is the Krylov subspace
spanned by A and the initial residual vector r0. The
residual vector rk+1 is comprehended in Kk+1(A;r0) as
follows:
rk 1
b Axk 1
r0
Azk
1
K k 1(
A;
r0
) . (3)
Furthermore, the approximate solution (x0 + z) in step (k
+ 1) satisfies the minimization condition as follows:
min || b A(
x0
z)
|| 2 min || r0
Az
|| 2 , (4)
zS
k
zS
k
where Sk is a subspace comprehended by Kk(A;r0). Using
a three-term recurrence formula involving Lanczos
polynomials, the algorithm of MRTR method is given as
follows:
Algorithm 1 (MRTR method). Let x0 be an initial guess,
and put r0 = b – Ax0. Set y0 = – r0 and 0 = (y0, y0).
For k = 0, 1, 2, …, repeat the following steps until the
condition ||rk||2/||b||2 < MR holds:
(
Ark
, rk
) / ( Ark
, Ark
)
( k 0)
k ( Ark
, r )
k
k
,
( k 1)
k ( Ark
, Ark
) ( yk
, Ark
)( Ark
, yk
)
several linear systems derived from the edge-based finite
element method in the magnetic field analysis.
Comparisons have been made with other well-known
preconditioned CG methods.
0
( yk
, Ark
)( Ark
, r )
k
k
k ( Ark
, Ark
) ( yk
, Ark
)( Ark
, yk
)
k 1
k ( Ark , rk
),
( k 0)
,
( k 1)
II. PRECONDITIONED MRTR METHOD
A. MRTR method
The symmetric sparse linear system can be defined as
follows:
k 1
pk rk
k
pk
1,
k
xk 1
xk
k pk
,
yk 1
k yk
k Ark
,

k 1
rk
yk
1,
where (u, v) denotes the inner product of vectors u and v.
The above algorithm is mathematically equivalent to the
conjugate residual (CR) method [5].
TABLE I shows the computational cost of MRTR
method, along with a comparison with the CG method.
Au, (u, v), (u + v), and u denote a matrix-vector
product, the inner product, the sum of vectors, and a
scalar-vector product, respectively. The computational
cost of MRTR method is nearly identical to that of the
CG method owing to the same number of computations
for the matrix-vector product.
TABLE I
COMPUTATIONAL COST OF LINEAR SOLVERS
linear solver Au (u, v) u + v u
CG 1 2 3 3
MRTR 1 4 4 4
B. Preconditioning
MRTR method can be combined with split
preconditioning techniques as long as the preconditioner
M, which can be written in the form M = CC T (with C : a
lower triangular matrix), is used. The preconditioned
T
matrix Aˆ
1
C AC retains the symmetry of A. Here,
we utilize M and C derived using shifted IC factorization
[6], DIC [1], and SGS preconditioning [7].
IC preconditioner
IC factorization is performed as follows:
ˆ ˆ ˆ T
, ˆ ˆ 1/
2
M LDL
C LD
,
(5)
i1
2
aii
li
k d k k ( i j),
k 1
lij j1
(6)
aij
li
kl
j kd
k k ( i j),
k 1
dii 1 / lii
,
(7)
where lij and dii are components of Lˆ and Dˆ and is
the shifted parameter. is determined by performing the
following steps: 1. Set = 1.05. 2. Perform IC
factorization. 3. If all diagonal components become
positive, shifted IC factorization is stopped. Otherwise,
return to step 1, add 0.05 to , and iterate steps 1-3.
If (5) is used for forward and backward substitutions,
there is a possibility of cache miss in backward
substitution. Hence, M is modified as follows:
ˆ ˆ ˆ 1
( ) ( ˆ ˆ T
M LD
D LD)
.
(8)
Therefore, the process of forward and backward
substitutions to compute the unknown vector u can be
described as follows:
ˆ ˆ ˆ 1 ( ) ( ˆ ˆ T
L D D LD)
u v,
(9)
where v is a known vector and the diagonal components
of LDˆ ˆ become 1.0. Forward and backward substitutions
is performed by following a two-step procedure
consisting of
ˆ ˆ
ˆ 1
( ) ,
( ˆ ˆ T
LD
y v y D LD)
u,
(10)
- 114 - 15th IGTE Symposium 2012
ˆ ˆ T
( LD)
u Dˆ
y.
(11)
Consequently, the computational cost can be reduced by
using the forward substitution (10) and backward
T
substitution (11) instead of the expression M Lˆ
Dˆ
Lˆ
.
DIC preconditioner
The large sparse matrix A can be split into three terms
as follows:
T
A L I L ,
(12)
where L is the strictly lower triangular part of A and I is a
unit matrix. The diagonal matrix Dˆ obtained using
shifted IC factorization (see (6) and (7)) is utilized. Thus,
M can be defined as follows:
ˆ ˆ 1
ˆ T
( ) ( ) , ( ˆ ) ˆ 1/
2
M L D D L D C L D D . (13)
Similar to the case of the IC preconditioner, the
procedure for forward and backward substitution should
be attentively schemed. The forward and backward
substitution for the DIC preconditioner is designed to
make the diagonal component 1.0:
ˆ 1 ( ) ˆ ( ˆ 1
T
L D I D LD
I)
u v,
(14)
ˆ 1
( ) , ˆ ( ˆ 1
T
LD I y v y D LD
I ) u,
(15)
( LDˆ
1
T
I ) u Dˆ
1
y.
(16)
SGS preconditioner
Using (12), M for the SGS preconditioner can be
defined as follows:
T
M ( L I ) ( L I ) , C L I.
(17)
Then, the algorithm of the preconditioned CG method is
as follows:
Algorithm 2 (Preconditioned CG method). Let x0 be M –1
b, and put r0 = b – Ax0. Set p0 = M –1 r0 and u0 = p0.
For k = 0, 1, 2, …, repeat the following steps until the
condition ||rk||2/||b||2 < CG holds:
Apk
,
( rk , uk
) / ( pk
, ),
x x
p ,
k
k 1
k k k
rk rk
k,
1
uk
M r
,
1
1 k 1
k ( rk 1,
uk
1)
/ ( rk
, uk
),
pk 1
uk
1
k pk
.
The algorithm of preconditioned MRTR method [8] is as
follows:
Algorithm 3 (Preconditioned MRTR method). Let x0 be
M –1 b, and put r0 = b – Ax0. Set u0 = M –1 r0, y0 = – r0, and
z0 = M –1 y0.
For k = 0, 1, 2, …, repeat the following steps until the
condition ||rk||2/||b||2 < MR holds:
1
AM rk
Auk
,
w
1
1
1
M AM rk
M
,
(
w,
rk
) / ( ,
w)
k ( w,
r )
k
k
k ( ,
w)
( yk
, w)(
w,
y
k
)
( k 0)
,
( k 1)

0
( yk
, w)(
w,
r )
k
k
k ( ,
w)
( yk
, w)(
w,
y
( w,
r ),
k 1
k k
p
k 1
k uk
k pk
1
k
x x p
k 1
k k k
yk 1
k yk
k,
rk
1
rk
yk
1,
zk 1
k zk
k w,
u u z
k 1
k k 1.
,
,
k
)
( k 0)
,
( k 1)
C. Eisenstat’s technique
Here, Eisenstat’s approach, in which the preconditioned
matrix and vectors are mainly utilized, is applied to the
preconditioned linear solvers in order to reduce the
computational cost for the matrix-vector product.
First, we apply Eisenstat’s technique to the DIC
preconditioner using the expression ˆ 1
( ) ˆ 1/
2
C LD
I D ,
and the preconditioned matrix-vector product Apˆ ˆ
k can be
transformed into
Aˆ
pˆ
k
ˆ 1/
2 ˆ 1
1
T
D ( LD
I)
( L I L )
ˆ 1
T
ˆ 1/
2
( LD
I)
D pˆ
k
ˆ 1/
2
( ˆ 1
1
) {( ˆ 1
D LD
I LD
I)
Dˆ
( 2Dˆ
I)
ˆ ( ˆ 1
T
) }( ˆ 1
T
) ˆ 1/
2
D LD
I LD
I D pˆ
k
ˆ 1/
2
( ˆ 1
T
) ˆ 1/
2 ˆ ˆ 1/
2
( ˆ 1
1
D LD
I D pk
D LD
I)
{ ˆ 1/
2 ˆ ( 2 ˆ )( ˆ 1
T
) ˆ 1/
2
D p
ˆ
k D I LD
I D pk
}
ˆ 1/
2 T
1
ˆ { ˆ 1/
2 ˆ ( 2 ˆ T
D C p
) ˆ
k C D pk
D I C pk
} .
(18)
It is shown that Apˆ ˆ
k can be replaced by one backward
substitution
T
C pˆ k and one forward substitution
1
ˆ 1/
2
{ ˆ ( 2 ˆ T
C D p ) ˆ
k D I C pk
} . On the other hand, the
formulation of Apˆ ˆ
k with the SGS preconditioner is as
follows:
Aˆ
pˆ
k
1
T
T
( L I)
( L I L ) ( L I)
pˆ
k
1
T
T
( L I)
{( L I)
I ( L I)
} ( L I)
pˆ
k
(19)
T
1
T
( L
I)
pˆ
( ) { ˆ ( ) ˆ
k L I pk
L I pk
}
T
C pˆ
1
C ( pˆ
T
C pˆ
) .
k
k
The applicable scope of Eisenstat’s technique is restricted
to the preconditioned matrix, in which the lower
triangular part of the original equation is used as it is. The
preconditioned CG method supported by Eisenstat’s
technique is as follows:
Algorithm 4 (Preconditioned CG method supported by
Eisenstat’s technique). Set
( LDˆ
C
L I
1
I)
Dˆ
1/
2
k
( DIC)
.
( SGS)
ˆ0 0
Let x0 be M –1 1
b, and put r0 = b – Ax0. Set r C r ,
pˆ rˆ
.
0
0
- 115 - 15th IGTE Symposium 2012
For k = 0, 1, 2, …, repeat the following steps until the
condition ||rk||2/||b||2 < CG holds:
T
u C pˆ
k ,
ˆ 1/
2 1
{ ˆ 1
ˆ ˆ
D u C D
A pk
1
u C ( pˆ
k u)
( ˆ , ˆ ) / ( ˆ , ˆ ˆ
k rk
rk
pk
Apk
),
xk 1
xk
ku,
r rˆ
Ap ˆ ˆ ,
ˆk 1
k k k
r ˆ k 1 C rk
1,
( ˆ , ˆ ) / ( ˆ , ˆ
k rk 1
rk
1
rk
rk
pˆ ˆ ˆ
k 1
rk
1
k pk
.
/ 2
),
pˆ
( 2Dˆ
I)
u}
k
( DIC)
,
( SGS)
Preconditioned MRTR method supported by Eisenstat’s
technique is as follows:
Algorithm 5 (Preconditioned MRTR method supported
by Eisenstat’s technique). Set
( LDˆ
C
L I
1
I)
Dˆ
1/
2
( DIC)
.
( SGS)
Let x0 be M –1 1
b and put r0 = b – Ax0. Set rˆ
0 C r0
,
yˆ ˆ 0 r0
.
For k = 0, 1, 2, …, repeat the following steps until the
condition ||rk||2/||b||2 < MR holds:
T
u C rˆ
k ,
ˆ 1/
2 1 ˆ 1/
2
{ ˆ ( 2 ˆ
ˆ
) }
ˆ
D u C D rk
D I u
Ark
1
u C ( rˆ
k u)
(
Aˆ
rˆ
ˆ ˆ ˆ ˆ ˆ
k , rk
) / ( Ark
, Ark
)
( ˆ
k ˆ , ˆ
k Ark
rk
)
( ˆ ˆ , ˆ ˆ ) ( ˆ , ˆ ˆ )( ˆ
A A A Aˆ
, ˆ
k rk
rk
yk
rk
rk
yk
)
0
ˆ ˆ
( yˆ
ˆ ˆ ˆ
k , Ark
)( Ark
, r )
k
k
( ˆ ˆ , ˆ ˆ ) ( ˆ , ˆ ˆ )( ˆ
A A A Aˆ
, ˆ
k rk
rk
yk
rk
rk
y
( ˆ ˆ , ˆ
k 1
k Ark rk
),
k 1
pk u k
pk
1,
x x p
k 1
k k k
y
yˆ
Ar ˆ ˆ ,
ˆ k 1
k k k k
rˆ
ˆ ˆ
k 1
rk
yk
1,
r ˆ k 1 C rk
1.
k
,
k
)
( DIC)
,
( SGS)
( k 0)
,
( k 1)
( k 0)
( k 1)
In the linear solver using Eisenstat’s technique, the lower
triangular matrix-vector product C r ˆk 1
is computed to
evaluate the residual rk+1.
D. Computational cost of preconditioned linear solvers
TABLE II shows the computational cost of
preconditioned linear solvers. Au, Lu, L -1 u, and L -T u
denote the matrix-vector product, the lower triangular
matrix-vector product, forward substitution, and
backward substitution, respectively. The abbreviations
EDIC and ESGS represent the DIC and SGS
,

preconditioners using Eisenstat’s technique, respectively.
The computational cost of the preconditioned linear
solvers using Eisenstat’s technique (EDIC and ESGS) is
lower than that of the other preconditioned solvers by
10 %. The reason why the computational cost does not
reduce significantly when Eisenstat’s technique is used is
the additional computation of the lower triangular matrixvector
product Cr ˆk 1
, whose cost is approximately equal
to that of forward or backward substitution.
TABLE II
COMPUTATIONAL COST OF PRECONDITIONED LINEAR SOLVERS
linear
solver precond. Au Lu L-1u L -T u
app. costs per one ite.
(Au + Lu + L -1 u + L -T u)
IC 1 0 1 1 1.0
DIC 1 0 1 1 1.0
CG EDIC 0 1 1 1 0.9
SGS 1 0 1 1 1.0
ESGS 0 1 1 1 0.9
MRTR
IC 1 0 1 1 1.0
DIC 1 0 1 1 1.0
EDIC 0 1 1 1 0.9
SGS 1 0 1 1 1.0
ESGS 0 1 1 1 0.9
approximate computational costs:
Au = 0.4, Lu = 0.3, L -1 u = 0.3, L -T u = 0.3
III. ANALYSIS MODEL
Figure 1 shows finite element meshes of model
problems used for performing a magnetic field analysis.
The unknown numbers in all meshes are determined by
the absolute edge number based on the nodal number.
Figure 1 (a) shows a box shield model [9] in which the
magnetic shielding part is composed of four-layer finite
elements in the thickness direction; the shielding
thickness is 1 mm. Magnetostatic and eddy current
analyses are carried out by considering the magnetic
nonlinearity of SS400.
Figures 1 (b) and (c) show the permanent-magnet-type
MRI model [10]. For this model, magnetostatic field
analysis is performed by considering the magnetic
nonlinearity of the pole piece, yoke, and props. The 2ndorder
hexahedral elements are of the Serendipity type.
Finally, Figure 1 (d) shows the IPM motor (D-model)
proposed by the IEEJ committee. A strongly coupled
analysis is performed between the magnetic field and
AC-driven three-phase circuit. The stator and overhung
rotor are considered to be magnetically nonlinear, and the
conductivity of the magnet is set to be 6.944 × 10 5 S/m.
The number of revolutions per minute is set to 1500, and
the pitch of the mechanical angle is 1°. The total number
of time steps is set to 360.
- 116 - 15th IGTE Symposium 2012
TABLE III lists the analyzed conditions. The Newton-
Raphson (NR) method, along with the line search
technique based on functional minimization (0, 1.0) [11],
is used as the nonlinear analysis method. GEAR’s
implicit scheme [12], [13] of 2nd order is used for the
discretization of the time domain based on the A-
formulation.
IV. NUMERICAL RESULTS
A. Verification of computational accuracy
The computational accuracy of preconditioned MRTR
method is verified for the box shield model. Figure 2
shows the analysis results. The magnetic flux density Bz
in the z-direction on z-axis is shown in Figure 2 (a). The
characteristic of the ICCG method coincide with that of
ESGS-MRTR. The relative error for two characteristics is
less than 10 -4 % at points A and B. Similarly, the relative
error for eddy current loss PJe as shown in Figure 2 (b) is
less than 10 -4 % at these points. It is to be noted that other
preconditioned linear solvers have similar characteristics.
20
240
y
75
20d
y
(unit:mm) magnet: Br = 1.2 T
z
y z
yoke: SS400
z
16
100
(unit: mm)
coil:2 kAT
magnetic shielding
(SS400)
x
(a) Box shield model
prop:
SS400
x gradientcoil
polepiece:
SS400
13
(unit:mm) magnet: Br = 1.2 T
z
240 20
y
75
yoke: SS400
13
prop: SS400
x
gradientcoil
polepiece
: SS400
(b) MRI (1st order tetra.) (c) MRI (2nd order hexa.)
(unit : mm)
45
32.5
TABLE III
ANALYZED CONDITIONS
x
v
w
u
v
w
u
rotor core
(50A350)
stator core
(50A350)
30
shaft
(S45C)
(d) IPM motor
Figure 1: Finite element meshes.
analysis model formul. discret. no. of nodes no. of elements DoF nonlinear circuit field
box shield
A
A
1st-hexa 72,900 67,980
magnet
enlarged view
CG, MR || B
|| 2
197,472 static 10-3 10-2
206,427 time domain 10-3 10-2
MRI
A 1st-tetra 279,090 49,813 323,965 static 10-3 10-3 A 2nd-hexa 93,879 87,120 1,014,600 static 10-3 10-3 IPM motor 1st-hexa 381,197 352,980 1,030,156 time domain 10-3 10-2
A

B z [T]
ESGS-MRTR
0.025
0.020
0.015
0.010
0.005
ICCG
point A
point B
0 0.05 0.10 0.15 0.20
z [m]
2.0
1.5
1.0
P Je [W] ESGS-MRTR
0.5
0 0.01 0.02 0.03 0.04
t [s]
point B
point A
ICCG
(a) (b)
Figure 2: Some characteristics of the box shield model:
(a) the Bz distribution in the z-axis direction and (b) the
distribution of eddy current loss.
log 10(||r (k) || 2 / ||b|| 2)
log 10(||r (k) || 2 / ||b|| 2)
1
0 DIC-CG
-1
-2
EDIC-CG
DIC-MRTR
EDIC-MRTR
SGS-CG
-3
ICCG
ESGS-CG
-4
IC-MRTR
-5
SGS-MRTR
-6 ESGS-MRTR
-7
0 200 400 600
iteration number k
800 1000
1
0
-1
-2
-3
(a)
DIC-CG
EDIC-CG
DIC-MRTR
EDIC-MRTR
SGS-CG
ESGS-CG
-4 IC-MRTR
-5
SGS-MRTR
ESGS-MRTR
-6
ICCG
-7
0 150 300 450 600
iteration number k
750 900
(b)
Figure 3: Convergence characteristics of preconditioned
linear solvers for the box shield model. (a) Magnetostatic
field analysis and (b) eddy current analysis.
B. Convergence characteristics and elapsed time
Figure 3 shows the convergence characteristics of the
box shield model, obtained by the magnetostatic and
eddy current analyses. The characteristics are normalized
by the initial norm of the residual in the 1st NR iteration.
In MRTR method, the monotonic decrease in the residual
has been mathematically proved; nevertheless, there are
some noise spikes in the characteristics in the case of
preconditioned MRTR method. The generation of noise
is likely to be caused by changes in the NR iteration.
Noise generation is also observed for the preconditioned
CG method. The characteristics of preconditioned MRTR
method are superior to those of the preconditioned CG
method because the monotonic decrease in the residual is
mathematically guaranteed in the former method. The
characteristics of the ESGS and EDIC preconditioners
- 117 - 15th IGTE Symposium 2012
linear
solver
CG
MRTR
linear
solver
CG
MRTR
TABLE IV
ANALYSIS RESULTS FOR BOX SHIELD MODEL
(a) MAGNETOSTATIC FIELD ANALYSIS
precond. total linear ite. NR ite. time for precond. [s] elapsed time [s]
IC 544 (1.00) 5 1.07 8.2 (1.00)
DIC 979 (1.80) 5 1.06 13.8 (1.68)
EDIC 979 (1.80) 5 1.07 13.0 (1.59)
SGS 653 (1.20) 5 0.03 8.3 (1.01)
ESGS 653 (1.20) 5 0.03 7.9 (0.96)
IC 448 (0.82) 5 1.01 7.4 (0.90)
DIC 812 (1.49) 5 1.09 12.4 (1.51)
EDIC 812 (1.49) 5 1.06 11.5 (1.40)
SGS 552 (1.01) 5 0.03 7.7 (0.94)
ESGS 552 (1.01) 5 0.03 7.0 (0.85)
(b) EDDY CURRENT ANALYSIS IN TIME DOMAIN (1ST TIME STEP)
precond. total linear ite. NR ite. time for precond. [s] elapsed time [s]
IC 503 (1.00) 4 1.07 8.6 (1.00)
DIC 844 (1.68) 4 1.05 13.6 (1.58)
EDIC 844 (1.68) 4 1.06 12.8 (1.49)
SGS 595 (1.18) 4 0.03 8.7 (1.01)
ESGS 595 (1.18) 4 0.03 8.2 (0.95)
IC 393 (0.78) 4 1.00 7.4 (0.86)
DIC 694 (1.38) 4 1.07 12.1 (1.41)
EDIC 694 (1.38) 4 1.03 11.2 (1.30)
SGS 470 (0.93) 4 0.03 7.4 (0.86)
ESGS 470 (0.93) 4 0.03 6.8 (0.79)
are consistent with those of SGS and DIC, respectively,
and the characteristics of the DIC preconditioner are
inferior to those of other preconditioned solvers. The IC-
MRTR characteristics are the best among all
preconditioned solvers. However, the elapsed time of
ESGS-MRTR is the shortest among all solvers, as can be
seen in TABLE IV. The reason for this is the reduction in
the computational cost when Eisenstat’s technique is
used. All results are obtained by using a PC (CPU: Intel
Core i7 2600K/4.2 GHz; memory: 16 GB). Following all
problems are solved with the same hardware.
Figure 4 shows the convergence characteristics of MRI
models. The convergence characteristics of
preconditioned MRTR are superior to those of the
preconditioned CG method. The DIC preconditioner is
not very effective in improving the convergence
characteristics. While the IC preconditioner is successful
in the case of a tetrahedron, the SGS preconditioner is the
most effective for a 2nd-order hexahedron. The
effectiveness of the preconditioner depends on the target
problem. TABLE V shows the analysis results for the
MRI model. The elapsed time of ESGS-MRTR is the
shortest among all preconditioned solvers.
TABLE VI shows the analysis results for an IPM
motor. The number of NR iterations is different for all
solvers owing to the slight discrepancy in the converged
solution in every time step. The elapsed time of ESGS-
MRTR is the shortest among all linear solvers.
V. CONCLUSION
This paper shows the suitability of preconditioned
MRTR method for solving an algebraic equation derived
from the edge-based finite element method in a magnetic
field. There is a possibility of reducing the elapsed time
in the case of MRTR method by using the symmetric

Gauss-Seidel preconditioner supported by Eisenstat’s
technique.
log 10(||r (k) || 2 / ||b|| 2)
log 10(||r (k) || 2 / ||b|| 2)
1
0
-1
DIC-CG
EDIC-CG DIC-MRTR
EDIC-MRTR
-2
ICCG
-3
SGS-CG
-4
-5
IC-MRTR
ESGS-CG
-6 SGS-MRTR
-7
-8
ESGS-MRTR
-9
0 200 400 600 800
iteration number k
1000
1
0
-1
-2
-3
(a)
ICCG
SGS-CG
ESGS-CG
DIC-MRTR
EDIC-MRTR
DIC-CG
-4
EDIC-CG
-5
IC-MRTR
-6 SGS-MRTR
-7 ESGS-MRTR
-8
0 900 1800 2700 3600
iteration number k
4500
(b)
Figure 4: Convergence characteristics of preconditioned
linear solvers for the MRI model. (a) Tetrahedron and (b)
2nd-order hexahedron.
linear
solver
CG
MRTR
linear
solver
CG
MRTR
TABLE V
ANALYSIS RESULTS FOR THE MRI MODEL
(a) 1ST ORDER TETRAHEDRON
precond. total linear ite. NR ite. time for precond. [s] elapsed time [s]
IC 572 (1.00) 7 0.71 11.5 (1.00)
DIC 958 (1.67) 7 0.73 18.6 (1.62)
EDIC 958 (1.67) 7 0.72 17.6 (1.53)
SGS 649 (1.14) 7 0.03 11.9 (1.03)
ESGS 649 (1.14) 7 0.03 11.3 (0.98)
IC 481 (0.84) 7 0.73 10.8 (0.94)
DIC 777 (1.36) 7 0.72 16.7 (1.45)
EDIC 777 (1.36) 7 0.74 15.3 (1.33)
SGS 542 (0.95) 7 0.03 11.0 (0.96)
ESGS 542 (0.95) 7 0.03 10.1 (0.88)
(b) 2ND ORDER HEXAHEDRON
precond. total linear ite. NR ite. time for precond. [s] elapsed time [s]
IC 3,795 (1.00) 8 25.6 570.7 (1.00)
DIC 4,305 (1.13) 8 25.2 643.5 (1.13)
EDIC 4,219 (1.11) 8 25.3 585.4 (1.03)
SGS 2,590 (0.68) 8 0.50 370.1 (0.65)
ESGS 2,590 (0.68) 8 0.50 342.8 (0.60)
IC 2,785 (0.73) 8 23.9 443.3 (0.78)
DIC 3,357 (0.88) 8 25.7 530.9 (0.93)
EDIC 3,357 (0.88) 8 23.3 470.3 (0.82)
SGS 2,112 (0.56) 8 0.50 316.1 (0.55)
ESGS 2,112 (0.56) 8 0.50 287.6 (0.50)
- 118 - 15th IGTE Symposium 2012
linear
solver
CG
MRTR
TABLE VI
ANALYSIS RESULTS FOR THE IPM MOTOR
precond. total linear ite. total NR ite. time for precond. [h] elapsed time [h]
IC 1,666,584 (1.00) 4,070 (0.80) 4.84 39.1 (1.00)
DIC 1,799,885 (1.08) 5,065 (1.00) 4.81 43.6 (1.12)
EDIC 1,779,282 (1.07) 4,999 (0.99) 4.82 40.9 (1.05)
SGS 1,400,867 (0.84) 4,996 (0.99) 0.03 29.7 (0.76)
ESGS 1,406,236 (0.84) 4,993 (0.99) 0.03 28.6 (0.73)
IC 931,375 (0.56) 4,450 (0.88) 5.24 26.6 (0.68)
DIC 1,088,578 (0.65) 4,716 (0.93) 4.95 29.7 (0.76)
EDIC 1,067,788 (0.64) 5,242 (1.03) 4.96 27.6 (0.71)
SGS 866,637 (0.52) 5,178 (1.02) 0.04 19.6 (0.50)
ESGS 864,545 (0.52) 5,162 (1.02) 0.04 18.2 (0.47)
ACKNOWLEDGMENT
The authors would like to thank Dr. K. Abe and Dr. Y.
Takahashi for their advice and helpful comments. This
work was supported by a Japan Society for the Promotion
of Science (JSPS) Grant-in-Aid for Young Scientists (B)
(Grant Number: 23760252).
REFERENCES
[1] J. A. Meijerink and H. A. van der Vorst, “An iterative solution
method for linear systems of which the coefficient matrix is a
symmetric M-matrix,” Mathematics of Computation, Vol. 31, No.
137, pp. 148-162, Jan. 1977.
[2] K. Abe, S.-L. Zhang, and T. Mitsui, “MRTR method: an iterative
method based on the three-term recurrence formula of CG-type for
nonsymmetric matrix,” The Japan Society for Industrial and
Applied Mathematics, Vol. 7, No. 1, pp. 37-50, Mar. 1997. (in
Japanese)
[3] K. Abe and S.-L. Zhang, “A variant algorithm of the Orthomin(m)
method for solving linear systems,” Appl. Math. Comput., Vol. 206,
No. 1, pp. 42-49, Dec. 2008.
[4] S. C. Eisenstat, “Efficient implementation of a class of
preconditioned conjugate gradient methods,” SIAM J. Sci. Stat.
Comput., Vol. 2, No. 1, pp. 1-4, Mar. 1981.
[5] S. C. Eisenstat, H. C. Elman, and M. H. Schultz, “Variational
iterative methods for nonsymmetric systems of linear equations,”
SIAM J. Numer. Anal., Vol. 20, No. 2, pp. 345-357, Apr. 1983.
[6] K. Fujiwara, T. Nakata, and H. Fusayasu, “Acceleration of
convergence characteristic of the ICCG method,” IEEE Trans.
Magn., Vol. 29, No. 2, pp. 1958-1961, Mar. 1993.
[7] O. Axelsson, “A generalized SSOR method,” BIT Numerical
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positive definite matrices of MRTR method,” Trans. JSCES, No.
20060007, pp. 231-237, Feb. 2006. (in Japanese)
[9] A. Kameari, “Improvement of ICCG convergence for thin
elements in magnetic field analyses using the finite-element
method,” IEEE Trans. Magn., Vol. 44, No. 6, pp. 1178-1181, Jun.
2008.
[10] K. Miyata, K. Ohashi, A. Muraoka, and N. Takahashi, “3-D
magnetic field analysis of permanent-magnet type of MRI taking
account of minor loop,” IEEE Trans. Magn., Vol. 42, No. 4, pp.
1451-1454, Apr. 2006.
[11] Y. Okamoto, K. Fujiwara, and R. Himeno, “Exact minimization of
energy functional for NR method with line-search technique,” IEEE
Trans. Magn., Vol. 45, No. 3, pp. 1288-1291, Mar. 2009.
[12] C. W. GEAR, Numerical Initial Value Problems in Ordinary
Differential Equations. Englewood Cliffs, NJ: Prentice-Hall, Inc.,
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[13] Y. Okamoto, K. Fujiwara, and Y. Ishihara, “Effectiveness of
higher order time integration in time-domain finite-element
analysis,” IEEE Trans. Magn., Vol. 46, No. 8, pp. 3321-3324, Aug.
2010.

- 119 - 15th IGTE Symposium 2012
High Frequency Mixing Rule Based Effective
Medium Theory of Metamaterials
Zsolt Szabó
Department of Broadband Infocommunications and Electromagnetic Theory,
Budapest University of Technology and Economics, Egry József 18, 1111 Budapest, Hungary,
E-mail: szabo@evt.bme.hu
Abstract— The electromagnetic response of metamaterials is governed by the collective behavior of engineered electric and
magnetic dipoles. Therefore metamaterials may be replaced by hypothetical composites of spherical particles embedded in a
host material. The effective electric permittivity and magnetic permeability of such systems can be computed with high
frequency extension of the Maxwell-Garnett mixing rule. The validity of this assumption is discussed and as a benchmark the
effective electromagnetic material parameters of a deep subwavelength spherical composite are calculated in three different
ways: with the Maxwell-Garnett mixing rule, high frequency mixing rule and directly extracted from transmission reflection
data. The developed theory is applied to find the parameters of a composite with similar magnetic response as a metamaterial
built of split ring resonator.
Index Terms—metamaterials, effective medium theory, Maxwell-Garnett mixing, Mie theory.
I. INTRODUCTION
Recently metamaterials are in focus of very intensive
research and due to their unique properties are promising
over the full electromagnetic spectrum [1-3]. The research
of metamaterials has started with the goal of producing
materials with negative refractive index i.e. simultaneous
negative electric permittivity and magnetic permeability
for imaging applications below the diffraction limit [4].
However, the ultimate goal of the metamaterial research
is to fabricate materials with arbitrarily configurable
electric and magnetic properties. With an advance in
micro- and nano-manufacturing techniques there are
possibilities to produce subwavelength structures that can
support symmetric and anti-symmetric modes. The
associated current flow produces electric and magnetic
dipole moments. A metamaterial with a customized
optical response can be built as a superposition of such
nano-elements. A very common design of an artificial
material with tailored negative permittivity is the wire
medium [5]. The most common designs to produce
artificial magnetism are the variations of the split ring
resonators [6] or pairs of nanorods [7]. The superposition
of subwavelength structures with negative electric
permittivity and magnetic permeability can lead to
negative refractive index [8] even at optical frequencies
[9]. However the losses and the finite size of the unit cell
results in a cutoff frequency, limiting the applicability of
metamaterials. In addition toward optical frequencies it is
increasingly challenging to fabricate the meta-structures,
especially the negative magnetic response.
The design of devices with metamaterials often
requires the application of the effective medium theory.
However robust effective metamaterial parameter
extraction and homogenization are unsolved theoretical
challenges of the metamaterial research. In spite of
considerable progress, researchers are still debating the
fundamental issues and question the validity of the
effective medium concept, which is considered by many
as the Achilles-heel of this research field.
In this paper it is argued that metamaterials can be
homogenized when their electromagnetic response is
governed by the excitation of electric and magnetic
dipoles. The electromagnetic response of spherical
particles can be replaced with static dipoles when the
sphere is very small compared to the optical wavelength
of the incident electromagnetic wave and by radiating
dipoles, when the size is larger. The analytical formulas
of the Mie theory explain precisely the scattering
mechanism. Metamaterials may be equivalent to a
properly chosen hypothetical composite of spherical
particles embedded in a host material. Therefore well
developed effective medium theories of composite
materials can be applied to metamaterials. The validity of
this assumption is discussed and as a benchmark the
effective electromagnetic material parameters of a deep
subwavelength composite of spherical particles are
calculated in three different ways: with the Maxwell-
Garnett mixing rule, high frequency mixing rule and
extracted directly from transmission reflection data. The
developed theory is applied to find the parameters of a
composite with similar magnetic response as a
metamaterial built up of split ring resonator.
II. EFFECTIVE MEDIUM THEORIES OF METAMATERIALS
Several effective medium theories of metamaterials
have been developed. In Fig. 1 two models of
metamaterial homogenization are presented. The effective
metamaterial parameters can be extracted by replacing the
electromagnetic response of the metamaterials with the
electromagnetic response of a homogeneous isotropic slab
is it is shown in Fig. 1.b. The model of Fig. 1.c replaces
the metamaterial with the hypothetical composite of
spherical particles embedded in a host material. In both
cases the electromagnetic properties can be determined in
such a way that the metamaterial slab and the slab with
the homogenized material parameters have the same
reflection S 11 and transmission S 21 parameters.
When the metamaterial is replaced with homogeneous
slab, from the Fresnel relations the effective metamaterial
parameters can be expressed. However the extracted wave
impedance is exact only in the quasi static limit [10] and

the unique extraction of the refractive index is
cumbersome due to the branching problem of the
refractive index; that is the calculation of the refractive
index involves the evaluation of a complex logarithm that
is a multi-valued function. To remove this ambiguity, the
Kramers–Kronig relation can be applied to estimate the
refractive index from the extinction coefficient [11]. The
physically realistic exact values of the refractive index are
determined by selecting those branches of the logarithmic
function which are closest to those predicted by the
Kramers–Kronig relation. Finally from the wave
impedance and from the refractive index the electric
permittivity and the magnetic permeability can be
calculated.
(a)
- 120 - 15th IGTE Symposium 2012
x = ε μ ωr
c , where ω is the angular frequency of
h h
r r 0
the incident radiation and c 0 is the speed of light in
vacuum, provides the guideline for the validity of the
Maxwell Garnett mixing rule, with the necessary
condition x 1.
However the limits of the Mixing-
Garnett mixing rule can be extended. The Mie theory
explains precisely the scattering mechanism of standalone
spherical particles of any size and offer analytic solution
in form of infinite series [14]. When the magnetic
permeability of the host material and of the spherical
particle is equal, the Mie coefficients are
mΨn( mx) Ψ′ n( x) −Ψn( x) Ψ′
n(
mx)
an
=
,
mΨ mx ξ′ x −ξ x Ψ′
mx
b
n( ) n( ) n( ) n(
)
( mx) ′ ( x) m ( x) ′ ( mx)
( mx) ξ′ ( x) mξ ( x) ′ ( mx)
Ψ Ψ − Ψ Ψ
=
, (4)
n n n n
n
Ψn n − n Ψn
where m =
i i
ε r μr h h
ε r μr
is the contrast of the
refractive index and n Ψ and ξ n are the Riccati-Bessel
functions. The radiating electric and magnetic dipole
polarizabilities correspond to the first terms of the
expansion and can be expressed with the Mie scattering
coefficients as
3
3
3r
3r
α e = i a1,
α 3 m = i b1.
3
2x
2x
(5)
(b) (c)
Figure 1: Homogenization models of metamaterials
Substituting (5) in the Clausius-Mossotti relation leads to
the expressions of the effective electric permittivity and
with a similar argument to the expression of the effective
magnetic permeability [15, 16]
The Maxwell-Garnett mixing rule [10, 12, 13] can
provide the effective electric permittivity of dilute, two
component mixtures and it is derived with the assumption
that the spherical inclusions can be replaced by static
electric dipoles with polarizability
3
eff h x + 3iζa1(
mx)
εr = εr
3
x − 3 iζa1( mx
2 )
3
eff h x + 3iζb1(
mx)
μr = μr
3
x − 3 iζb1( mx
2 )
,
. (6)
i h
εr − εr
3
αe
= r , i h
εr + 2εr
(1)
where 1
h
where ε r is the electric permittivity of the host material,
i
ε r is the electric permittivity and r is the radius of the
spherical inclusions. The connection between the
eff
polarizability and the effective electric permittivity ε r is
given by the Clausius-Mossotti relation [13]
eff h
εr − εr ζ
= α
eff h 3 e , (2)
εr + 2εr
r
where ζ is the filling factor of the spherical inclusion.
When (1) is substituted in the Clausius-Mossotti relation
it results in the Maxwell-Garnett mixing formula
eff h i h
εr −εr εr −εr
= ζ . (3)
eff h i h
εr + 2εr εr + 2εr
In this relation, the size of the spherical inclusions is not
appearing in a direct way; the filling factor ζ is the only
geometry factor in the Maxwell-Garnett formula. The
static dipole approximation is valid only for spheres,
which are very small compared to the optical wavelength
of the incident electromagnetic wave. The size parameter
a and b 1 are the first terms of the Mie scattering
coefficients and i = − 1 is the imaginary unit. The
evaluation of a 1 and b1 is trivial, because in (5) for
n = 1 , the Riccati-Bessel functions and the derivatives
can be expressed with simple expression of trigonometric
functions as
sin ρ
Ψ 1 ( ρ) = − cos ρ ,
ρ
1 cosρ
Ψ ′ 1 ( ρ) = sin ρ1−
2 +
,
ρ ρ
cos ρ
ξ1( ρ) =Ψ1( ρ) − i + sin ρ
ρ ,
1
ξ′ 1( ρ) =Ψ ′ 1( ρ) + i Ψ 1( ρ) + cos ρ 2
ρ .
When the size of the spherical inclusions is not small
enough to be replaced with static dipoles, but it is small
enough to disregard all higher order modes of (4) then the
high frequency mixing formulas (6) are applicable. Note
that the resonance based magnetic metamaterials are
working under similar conditions [1]. Metamaterials has

finite unit cell sizes, and especially magnetic
metamaterials has unit cells, which are not deep
subwavelength. The strength of the resonance decreases
with the size of the unit cell and the resonance is not
strong enough to produce negative permeability for
structures with deep sub-wavelength elements.
Metamaterials with larger unit cell can support higher
order modes at frequencies, which are just slightly
different than the frequency region where the double
negative behavior occurs. Special care must be taken
when metamaterial parameters are extracted directly from
transmission reflection data, and it is not sufficient to
enforce the continuity of the refractive index, because we
may extract erroneous effective metamaterial parameters
for frequency regions where they do not even exist. The
high frequency mixing, which is based on the Mie theory
provides estimate for the limits of the homogenization.
III. EFFECTIVE MATERIAL PARAMETERS OF COMPOSITE
WITH SPHERICAL METALLIC INCLUSIONS
In this section the effective parameters of the
composite material with the unit cell illustrated in Fig. 2.a
are calculated. This composite serves as benchmark to
compare the effective material parameters calculated with
the Maxwell Garnett mixing rule, the high frequency
mixing rule and directly extracted from transmission
reflection data. The geometry and the composition are
selected such that the size parameter of the spheres at
optical frequencies satisfies the condition x 1.
The
length of the cubic unit cell is 15 nm and the radius of the
sphere is 3 nm. The spherical inclusions are made of Ag
and are embedded in SiO2 host and the calculations take
into account the frequency dispersion of the materials
parameters. Fig. 2.b presents the electric permittivity of
the Ag inclusions, and Fig. 2.c plots the electric
permittivity of the SiO2 host [17, 18]. The composite is
considered infinitely large in the x and y directions (see
Fig. 2.a) and only-one-unit-cell thick in the z direction.
The Maxwell-Garnett type mixing rules do not require
cubic unit cells; the requirement is that the inclusions are
separated. For periodically arranged spherical inclusions,
when the filling factor is high, the Maxwell-Garnett
mixing rule has to be modified [12]. On the other hand
disorder and inaccuracy of shapes destroys the collective
effects and extends the limits of the theory.
The aim of the calculation is to determine the effective
parameters of this composite in the frequency range from
0.4 to 1 PHz. The calculations of the transmission
reflection data (S-parameters) of this paper are performed
with the frequency-domain solver of the commercial
software CST Microwave Studio [19]. Due to periodicity,
one unit cell with perfect electric conducting and perfect
magnetic conducting boundary conditions in the x and the
y directions is sufficient to calculate the S-parameters
below the frequencies where diffraction occurs. In the z
direction additional air regions are added to the
computational space by positioning waveguide ports at
one-unit-cell distance from the surface of the composite.
The fundamental mode of the waveguide ports is excited
to launch a plane wave, which is propagating along the z
direction; at the same time the waveguide ports act as
- 121 - 15th IGTE Symposium 2012
absorbing boundary condition and permits the automatic
calculation of the S-parameters [19]. The online algorithm
[20] is applied to extract the electromagnetic parameters
from the S parameters. To get a good estimate for the
Kramers–Kronig integral, the simulations cover the 0.25–
1.25 PHz frequency interval. When this frequency
interval is even larger, the accuracy of the Kramers–
Kronig approximation does not change noticeably in the
frequency range of interest.
(a)
(b)
(c)
Figure 2: The geometry of the composite material is
shown in (a), the electric permittivity of the spherical
inclusions made of Ag is presented in (b) and the electric
permittivity of the SiO2 host materials is plotted in (c).
Fig. 3.a and 3.b presents the magnitude and phase of
2 2
the S-parameters. The absorption A = 1− S11
− S21
in
function of frequency is plotted as well, showing a
resonant peek at f 1 = 0.7192 PHz.
The effective electric permittivity of the composite
material, which is calculated in three different ways, with
the high frequency mixing rule, with the Maxwell-Garnett
mixing and extracted from the S-parameters, are
presented in Fig. 4. The magnetic permeability is obtained
from the high frequency mixing rule and it is extracted
from the S-parameters as well, and it has values close to
one over the frequency range of interest. Comparing the
real and imaginary parts of the electric permittivity
obtained with the three different methods, a very good
agreement can be observed. The peak in the imaginary
part of the electric permittivity corresponds to the
absorption peek of Fig. 3.a, which reveals that it is
electric resonance. The electric permittivity has Lorentz
shape and can be successfully fitted with a single
oscillator model

( − )
2
εrs εr∞ω0 εr ( ω) = εr∞+
. (7)
2 2
ω0+ iδω−ω
where the static electric permittivity ε rs = 2.379 , the
electric permittivity at very high frequencies ε r∞
= 2.23 ,
the resonant frequency ω0= 2π ⋅ 0.7228 rad/fs and
damping constant δ = 0.33 1/fs.
(a)
(b)
Figure 3: The S-parameters of the one-unit-cell thick
composite material. In (a), the magnitude, and in (b) the
phase of the S-parameters is plotted. Note the absorption
peek at f 1 = 0.7192 PHz.
Figure 4: The effective electric permittivity of the
composite material calculated with the high frequency
mixing rule, Maxwell-Garnett mixing and extracted from
the S-parameters.
The electric permittivity at the frequency of the
absorption peek is ε = 2.5319 + 2.0465i
and the
r
- 122 - 15th IGTE Symposium 2012
corresponding optical wavelength is
λ opt = c0 ( n f1)
= 245.04 nm, which is much larger than
any characteristic dimension of the composite (the size of
the unit cell is 15 nm), showing that the resonant behavior
is related to the composition rather than structuring.
IV. EQUIVALENT COMPOSITES OF METAMATERIALS
DESIGNED WITH THE HIGH FREQUENCY MIXING RULE
In this section the equivalent composite of a magnetic
metamaterial is determined. The geometry of the
metamaterial is the well studied split ring resonator [1, 2,
3, 8] as it is shown in Fig. 5. The dimensions and the
material parameters are the same as in [8]. The size of the
cubic unit cell is 5 mm, the split ring resonators are made
of copper, the outer length of the exterior split ring
resonators is 3 mm, the width of both split rings is
0.25 mm, the thickness is 0.02 mm, the size of the gaps
and the distance between the split ring resonators is
0.5 mm. The substrate is made of dielectric with
ε r = 3.84 and the thickness of the substrate is 0.25 mm.
The metamaterial is periodic in the direction
perpendicular to the propagation of the electromagnetic
wave (z direction), the electric field is polarized in y
direction, which means that the magnetic field is
perpendicular to the plane of the split ring resonators. The
metamaterial of [8] was designed to experimentally
demonstrate the negative refraction. The role of the splitring
resonators is to provide the negative magnetic
response, while additional copper wires placed on the
back side of the substrate are responsible for producing
the negative electric permittivity, leading to a negative
refractive index at a frequency of 10 GHz. In our
numerical simulations the metamaterial is only-one-unitcell
thick.
Figure 5: The unit cell of the magnetic metamaterial slab
is composed of metallic split-ring resonators.
The reflection, transmission and absorption spectrum
of the double negative metamaterial [8] is presented in
Fig. 8.a, while in Fig. 8.b the electromagnetic response of
the split ring resonators is shown. The simulations reveal
that the position of the resonant peek at 10 GHz is not
changed by removing the wires; nevertheless the shape of
the transmission and reflection curves is greatly affected.

The effective parameters of the magnetic metamaterial
built of split ring resonators are extracted from the Sparameters
with [20] and are shown in Fig. 7.a. The
magnetic permeability has Lorentz shape with negative
values and it is similar to the magnetic permeability of
[8]. The effective electric permittivity has a shape of antiresonance,
which may be an artifact caused by the
replacement of the anisotropic metamaterial structure with
the homogenized model of isotropic slab.
(a)
(b)
Figure 6: In (a) the reflection, transmission and
absorption spectrum of the double negative metamaterial
is presented, while in (b) the electromagnetic response of
the split ring resonators is shown.
Spectral fitting is carried out to find the parameters of
the composite, which is magnetically equivalent to the
metamaterial, built of split ring resonators. The
parameters of the high frequency mixing rule, the radius
and the electric permittivity of the spherical inclusions,
the filling factor and the electric permittivity of the host
material are determined by minimizing the mean square
error,
2 2
N HF TR HF TR
Re( μri ) Re( μri ) Im ( μri) Im ( μ
− − ri )
+
TR TR
i= 1 Re( μri ) Im(
μri
)
Ω=
2N
where N is the number of data points in the spectra, Re()
and Im() return the real and imaginary parts of the
magnetic permeability
μ extracted from the S-
TR
r
parameters or HF
μ r calculated with the high frequency
- 123 - 15th IGTE Symposium 2012
mixing rule. The minimization is performed with the
differential evolution algorithm [21]. The minimization
provides r = 2.31 mm for the radius of the spherical
inclusions, the electric permittivity of the inclusions is
i
ε r = 37.67 , the filling factor is ζ = 0.13 and the electric
h
permittivity of the host material is ε r = 1.
Note that the
results of this optimization are implementable; several
materials exist at microwave frequencies with even higher
electric permittivity and are available as powder or
suspension [7].
(a)
(b)
Figure 7: In (a) the effective magnetic permeability and
electric permittivity of the metamaterial made of split-ring
resonators extracted from S-parameters is shown. In (b)
the material parameters of the equivalent composite are
presented.
The real and imaginary parts of the magnetic
permeability and the electric permittivity of the equivalent
composite are plotted in Fig. 7. b. As it can be seen there
is a good agreement between the effective magnetic
permeability of the metamaterial and the permeability of
the composite. Comparing the real part of the electric
permittivities it can be observed that they are comparable
TR
at low frequencies, for example at 5 GHz ε = 1.57 and
ε = 1.43 , even though there is no optimization goal
HF
r
formulated for permittivity in the mean square error of the
minimization procedure. On the other hand there is no
anti-resonant behavior in the electric permittivity of the
composite in the frequency region of the magnetic
resonance. The magnetic resonance of the composite is
followed by electric resonance, which appears at the
r

upper end of the investigated frequency region. To move
the electric resonance outside of this frequency region,
the bounds of the optimization parameters were changed
and several minimization runs were performed. As a
result it can be observed that the model does not provide
enough freedom to maintain the strength and the position
of the magnetic resonance and at the same time to change
the position of the electric resonance to higher
frequencies. The extension of the high frequency model to
ellipsoidal particles may solve this issue.
In Fig. 8 the magnitudes of the S-parameters for the
metamaterial built of split ring resonator and those for the
equivalent composite are presented. The difference
between the curves is due to the difference in electric
permittivities. The correspondence may be improved by
considering frequency dependent material parameters.
Figure 8: Comparison between the magnitudes of the
S-parameters of the metamaterial built of split ring
resonator and the S-parameters of the equivalent
composite.
V. CONCLUSIONS
High frequency mixing rule, which is based on the
Clausius-Mossotti relation and the first terms of the Mie
expansion corresponding to radiating dipoles has been
applied to characterize composites and metamaterials. To
validate the model the effective electric permittivity and
magnetic permeability of a deep subwavelength
composite were calculated and it was shown that similar
results are produced by the high frequency mixing rule,
the Maxwell-Garnett mixing rule or extracted directly
from the S-parameters.
The developed high frequency model can open
alternative ways to engineer required electromagnetic
properties. It was shown that equivalent composite, which
has similar effective magnetic permeability, can be
assigned to the magnetic metamaterial built of split ring
resonators.
VI. ACKNOWLEDGEMENT
This work has been supported by the János Bolyai
Research Fellowship of the Hungarian Academy of
Sciences and OTKA 105996.
- 124 - 15th IGTE Symposium 2012
[1]
REFERENCES
L. Solymár and E. Shamonina, Waves in Metamaterials, Oxford,
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[3] N. Engheta, R. W. Ziolkowski, Metamaterials Physics and
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Phenomena, IEEE Transactions on Microwave Theory and
Techniques, vol. 47, 2075, 1999.
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Composite Medium with Simultaneously Negative Permeability
and Permittivity, Phys. Rev. Lett., vol. 84, p. 4184, 2000.
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Negative-index metamaterial at 780 nm wavelength, Opt. Let.,
vol. 32, no. 1, pp. 53-55, 2007.
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Comission, 2010.
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extraction of metamaterial parameters based on Kramers-Kronig
relationship,” IEEE Trans. Microwave Theory Tech., vol. 58, no.
10, pp. 2646-2653, 2010.
[12] A. Sihvola, Electromagnetic Mixing Formulas and Applications,
The Institution of Electrical Engineers, London, United Kingdom,
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by Small Particles, Wiley-VCH, 2004.
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Optics Communications, vol 182, pp. 273–279, 2000.
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spectra of granular materials, Phys. Rev. B, vol. 43, pp. 10780–
10788, 1991.
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Constants of Solids, Academic Press, New York, 1997.
[18] [Online] http://www.sspectra.com/sopra.html
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[21] K. V. Price, R. M. Storn, J. A. Lampinen, Differential Evolution,
A Practical Approach to Global Optimization, Springer, 2005.

- 125 - 15th IGTE Symposium 2012
Enhancement of Maximum Starting Torque and
Efficiency in Permanent Magnet Synchronous Motors
Jawad Faiz, Vahid Ghorbanian and Bashir Mahdi Ebrahimi
Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer
Engineering, College of Engineering, University of Tehran, Tehran 1439957131, Iran
(e-mail: jfaiz@ut.ac.ir)
Abstract— This paper presents a new algorithm for enhancement of maximum starting torque and steady-state efficiency in
permanent magnet (PM) motors. This algorithm includes two strategies which are used to raise starting torque and decrease
losses in PM motors. Therefore, transient and steady-state operations of the PM motor are improved. It is essential to model
the core in the efficiency estimation of losses of the PM motor. Appropriate control coefficients based on the introduced
algorithm are set for two aforementioned goals. Simulation results are presented the competency of the proposed algorithm.
Index Terms— PM motor, control strategy, starting torque, losses minimization, efficiency.
I. INTRODUCTION
Application of permanent magnet (PM) motors is
increasing due to the advanced technology in PM
manufacturing, and its high power density, improved
power factor and high efficiency. Some applications such
as electric and hybrid vehicles with huge start-stop
actions need high starting torque to quickly accelerate the
vehicles. Enhancement of the starting torque involves the
increase of the stator windings currents leading to
windings temperature rise. Therefore, the stator current
must be limited. Since the performance improvement of
these motors has considerable effects upon the electrical
power consumption over long time applications, the
instantaneous optimal control and appropriate operating
point over different loads and speeds should be
considered.
Nowadays, a wide range of the motor speeds and
torques are achieved by application of vector control
methods in the motors [1]-[4]. These control methods
provide stability and precise required speed and accurate
response because of the feedback in the motor which can
control the flux and torque independently [5]. The
controllable quantities are id and iq currents. By
controlling current id, the motor flux and consequently
speed is adjusted and by controlling current iq, the steadystate
output torque is regulated [6]. In [7], [8], a method
has been introduced to control the torque of the PM
machine by limiting the windings currents. This machine
has been used as a generator of a wind turbine. When the
wind speed is higher than that of the rated speed, torque
rises and the windings insulation may fail. In some
methods [7], there is no need to have a mechanical sensor.
Since magnetic saturation has considerable impact on the
motor behavior at high currents; the saturation effects are
approximately modeled in this paper. Meanwhile, strategy
of maximum torque control is normally applied to the
motor in the non-starting case [3].
Different strategies have been so far applied to improve
the efficiency of the PM motor. In [9], efficiency of the
motor has been improved through introducing a new teeth
and slot structure. In [7], [12], a method based on the dq
model of the motor has been proposed in which the core
losses have been taken into account by a resistance in the
equivalent circuit of the motor. This technique is called
the “loss model control”, in which copper and core losses
are evaluated as an analytical function of equivalent
circuit parameters and id and iq currents of the motor. By
obtaining the optimal currents, the motor losses can be
minimized and efficiency maximized. In interior PM
(IPM) motors Lq and Ld are unequal and it is difficult to
obtain the optimal operating point at different speeds and
torques analytically. To overcome this problem, normally
id and iq are expressed as functions of the motor speed and
its coefficients are stored in a lookup table. The objection
of this method is that by increasing the motor speed and
torque ranges, the tables will be larger and it must be
updated by change of the motor parameters. In [1], [5],
[10], motor efficiency has been improved by id =0
method. In the PM motor model, motor reluctance torque
appears as coefficient of id and by putting id =0, the
reluctance torque as an opposed torque is eliminated and
consequently its output power increases for a fixed speed.
Applying id=0 to IPM motors needs a high power
inverter. Therefore, this method is normally applied to
surface-mounted PM (SPM) motor [10]. The unity power
factor method can develop less maximum torque
compared to other methods [10]. So, it is not suitable in
the high torque applications. Flux-linkage control method
presents a better performance in IPM.
On contrary to the loss model control which depends
on the motor model accuracy, methods presented in [2],
[3], [13] is called search control method which is
independent of the motor and drive parameters. In this
method, attempt has been made to reduce the input power
and this is normally done through the control of voltage
or dc link current of the inverter. Application of this
control method may produce undesirable oscillations in
the torque and speed of the motor which leads to
instability [14]. In this method the use of a frequency
stabilizer is necessary.
Previous papers have not taken into account both high
starting torque and steady-state efficiency improvement in
PM motor. This paper investigates the control strategy of
the maximum torque and efficiency enhancement in the
steady-state operation of the motor. By application of this
method, torque raises up to 4 pu and current up to 2 pu.

Distinction of this paper and [10] is the design of
intelligent system for applying the limitation on id and iq,
So, at any instant sensitivity of torque against each current
components is measured and a component that has less
effect in of the torque development is limited. To improve
the steady-state efficiency of the motor, loss model
control algorithm of [5] is used. In section II the motor
model is introduced. In section III control strategy is
described. Section IV and V present the simulation
method and results respectively. Finally section VI
concludes the paper.
II. MODEL OF MOTOR
The proposed control strategies in this paper are based
on the analytical equations of the motor model. Since
enhancement of the maximum starting torque and steadystate
efficiency of the motor are carried out through the
control of id and iq current vectors, the Park’s model
converts three-phase abc equations of the motor into twophase
dq equations in which id and iq currents of the
motor are available. Figure 1 shows the IPM motor
model where Ra is the stator resistance, Ld is the d-axis
inductance and Lq is the q-axis inductance.
Figure 1: Two-axes model of IPM motor
These two inductances are not equal in the IPM motors
and they develop a considerable reluctance torque. Rc is
the iron losses equivalent resistance. The iron losses
consist of the hysteresis and eddy current losses, and are
modeled by equivalent resistance Rc, which depends on
the temperature and frequency. To simplify the
computations, this resistance is evaluated at the rated
conditions. In [5], leakage and magnetizing inductances
are separated and inserted in different branches. Since Rc
is very larger than that of the other impedances, the
current of the losses branch is small and the current of the
left hand side and right hand side have no considerable
difference. Therefore, sum of leakage and magnetizing
inductances are used in the model.
The governing equations of the motor model are as
follows:
diod
vd
Raid
Lqioq
Ld
(1)
dt
- 126 - 15th IGTE Symposium 2012
dioq
vq
Raiq
Ldiod
a Lq
dt
(2)
icd id
iod
(3)
icq iq
ioq
(4)
diod
( Lqioq
Ld
)
i
dt
cd
Rc
(5)
dioq
( (
Ldiod
a ) Lq
)
i
dt
cq
RC
The developed electromagnetic torque of the motor is:
(6)
3P
Te ( )[ aLq
( Ld
Lq
) iodioq
]
2
The dynamic equation of the motor is as follows:
(7)
dr
Te
Tm
c sign(
r ) Fr
J
dt
(8)
III. CONTROL STRATEGY
A. Enhancement of Maximum Torque
Eqn. (7) indicates that the motor torque depends on id
and iq components of currents. At the starting, stator
current does not so much depend on the load, and
normally is 2 to 2.5 times the rated value. The starting
torque of the motor can be up to 2.5 times the rated
torque and generally there is no need to reduce it.
However, in some applications such as ABS brake of cars
the motor must have very short declaration time (about
fractional of ms), and the starting torque, about 2.5 times
the rated torque, cannot response quickly in the no control
mode. Therefore, it is necessary to apply the high starting
torque in the case of no control case over longer time in
order to provide an appropriate declaration time. In the
previous studies some control methods have been
proposed to increase the starting torque; however a
limited starting current has not been considered. Here a
novel technique is introduced that optimally limits the
stator current under vector control and also enhanced the
maximum starting torque. By applying this method, the
starting torque rises up to 4 times the rated torque. The
basis of this method is the use of id and iq components of
current. Since the equivalent resistance of the iron losses
is almost infinite:
i i
q
oq
id iod
(10)
Suppose the stator current is
starting period, iq is as follows:
is constant during the
2 2 2 2 2 2
(11)
i i i i i i
q
d
s
q
s
d
Combining (7) and (11) leads to:
3P
2 2
(12)
Te ( ) is
id
[ a ( Ld
Lq
) id
]
2
where is can be taken as 2 to 2.5 times the rated current.
In practice, the back-emf increases by acceleration of the
motor and this decreases the stator current. However, to
simplify the equations it was taken to be constant. The
(9)

optimal id is obtained by putting the derivative of the
maximum torque versus id equal to zero:
2
dTe
a
a 2 is
0 id
( )
2
2
did
4(
Ld
Lq
) 4(
Ld
Lq
) 2 (13)
For positive id, the reluctance torque increases and the
total torque of the motor reduces. So, only the negative
sign is acceptable.
2
a
a 2 is
id
( )
2
2
4(
Ld
Lq
) 4(
Ld
Lq
) 2
(14)
By applying id from (14) to the reference point id,
torque rises. But the stator current becomes larger than
the permissible current by applying the obtained
components of the current. Therefore, the torque
sensitivity versus the current components is calculated
and the current that has less influence the torque
development is limited:
T Te
3P
e
(15)
Si | i cte ( )( Ld
Lq
) iq
d
q
id
2
T Te
3P
e Si | i cte ( )( a ( Ld
Lq
) iq
)
q
d
(16)
iq
2
By applying this limit, the stator current does not rise
further than 2 times the rated current.
B. Improvement of Steady-state Efficiency
The major factor in the efficiency reduction of the
motor is the increase of the copper and iron losses. In the
vector controlled motor supplied by an inverter, high
order harmonics generates additional losses. In spite of
this, the major part of the losses allocated to the
fundamental harmonic. Copper losses directly and iron
losses indirectly is proportional with the motor current.
The copper and iron losses arising from the fundamental
harmonic is optimized by vector control of the stator
currents. The high order harmonic losses are
uncontrollable. The basis of the losses control is the
estimation of the motor losses using the presented model
and its optimization versus the motor currents. Since
efficiency is defined in steady-state, the time derivatives
of the dynamic equations are set equal to zero and losses
are calculated as follows:
Lqioq
2
(
iod
)
3R
2 2 3R
Rc
Wcu
( iod
, ioq
, ) ( )( id
iq
) ( )
2
2 ioq
(
a Ldiod
) 2
(
)
Rc
(17)
2
2
3R
( )
c 2 2 3
Lqioq
W fe(
iod
, ioq,
) ( )( icd
icq)
( )
2
2R
2 (18)
c ( a Ldiod
)
where Wcu and Wfe are the copper losses and iron losses
respectively. The motor losses are function of iod, ioq and
. In these equations, the influence of temperature rise
and higher harmonics on the resistances and magnetic
saturation upon the inductances have been ignored and
taken to be constant. However, the losses increase
nonlinearly due to the high harmonics and considerable
rise of the magnetizing current because of the saturation.
The magnetic saturation occurs normally at starting, and
at the steady-state mode the motor operates at the knee of
the magnetization characteristic; therefore, neglecting the
- 127 - 15th IGTE Symposium 2012
saturation is acceptable. By combining (7), (17), (18), the
following equation is obtained:
Wc W fe(
iod
, Te
, ) Wcu
( iod
, Te
, )
Wc
( iod
, Te
, )
(19)
As indicated in (19), the total losses of the motor depend
on function of the operating point and iod current. At the
operating point with fixed speed and torque, the optimal
iod is obtained for losses reduction using the analytical
derivative of (19).However, in the IPM motor, Ld and Lq
are not identical, therefore the equations are complicated
and use of analytical derivative is difficult. Sometimes,
the currents of the motor are expressed as a polynomial
versus each other where its polynomial coefficients
depending on the speed of the motor. The coefficients of
the polynomials versus the motor operating point are
stored in a look-up table. However, this method is quick
but interpolation over different operating points leads to a
highly approximated method. Meanwhile, the tables over
wide range of speed and torque become large, and these
tables must be updated by change of the motor type.
Flow-chart reported in [5] has optimized the motor losses
without using analytical derivative and also lookup table.
The presented algorithm is an iterative one and it
normally converges to an appropriate solution after 14
iterations. In the present paper, the section related to the
response of the final conditional expression reported in
[5] is modified and therefore the optimal response point is
achieved with lower number of iterations at any operating
point. This algorithm increases the developed
electromagnetic torque of the motor at a constant of
operating point and consequently efficiency of the motor
improves. Since iq is the torque component of the current,
its value depends largely on the load of the motor and its
large change will lose the stable operating point.
Therefore, the reluctance torque value is controllable
by change of id. In the traditional methods such as id=0,
the reluctance torque is almost zero and demagnetization
effect of the stator current diminishes. The idea used in
the new method is to make negative id which makes the
reluctance torque positive and improves the efficiency
Figure 2: Loss minimization algorithm

Figure 3: Motor and control system
of the motor at fixed speed. idmax is generally taken to be a
small positive value and idmin a large negative value. d
defines the step variations of id and x the mean value of id
in every step. To achieve an appropriate response the step
number depends on the value of id which is fixed at an
optimal value. The simulation results show the
improvement of the motor efficiency by applying the
presented control method compared to id =0 method.
IV. SIMULATION METHOD
The above-mentioned control strategies are applied to a
PM motor under vector control. The output of these
control methods provides the reference values of the drive
current. Since the motor supply under vector control is
PWM type, the high order odd harmonics are injected to
the motor. Amplitude of these harmonics varies with
changing the operating point. Figure. 3 shows the
complete system of the motor and drive. The LMA block
is for efficiency improvement strategy in steady-state and
T/A block is for the maximum starting torque
enhancement. Limitation block limits the motor currents.
The reference current values are transformed into the twophase
reference voltages by Vqd_ref. Then the motor
reference voltages are formed by transforming the twophase
to three-phase voltages and applying to the PWM
block. The maximum starting torque algorithm during
transient and efficiency improvement algorithm during
the steady-state periods are applied to the motor. In the
motor model, a low-pass filter is used for eliminating
high-order harmonics from the control process.
Specifications of the simulated motor have been
summarized in Table I.
V. SIMULATION RESULTS
The results have been obtained by simulation of the
motor under control using Simulink. First, the results of
applying T/A algorithm during the transient mode to the
motor is considered. By using motor parameters and
- 128 - 15th IGTE Symposium 2012
TABLE I
NAMEPLATES AND PARAMETERS OF IPM
MOTOR
Number of poles 6
Rated Torque (Nm) 1.8
Rated rms current (A) 3.6
Rated speed (rpm) 4000
Stator winding resistance Ra ( 2.21
Core loss equivalent resistance Rc ( 840
Direct axis inductance (mH) 9.77
Quadrature axis inductance (mH) 14.94
Permanent magnet flux a (Wb) 0.0844
Mechanical losses (Nm) 0.04
considering constant maximum current of the motor, id
component of the stator current is calculated by T/A and
applied to the input reference of the drive. Since a high
torque is necessary at the starting, at the first instant the
current id increases in the negative direction considerably
(Figure. 4a). Negative current id leads to the positive
reluctance torque and increases the total torque of the
motor. By applying T/A, value of iq also increases and the
motor current rises over permissible limit.
Therefore, currents are limited. Figure. 4b exhibits the
variations of the normalized electromagnetic torque of the
motor (based on the rated values) in which the torque
raises up to 3.8 pu due to T/A applications. This torque is
very larger than the case in which the motor is able to
develop with no control strategy. In order to study the
performance of the stator current limiter system, threephase
currents of the motor are shown in figure. 5 which
indicates that the phase current of the motor raises up to
1.8 pu. This current rises up to 2.5 pu when current
limiter is not used. The reason for a constant torque
during the transient mode is that the tolerable peak
current by the stator winding is assumed constant. If this
current as a function of the motor emf is applied to the
model, the motor torque will decrease by time. After
completion of the transient period, the control algorithm

iabc(pu)
Speed(rpm)
id(A)
torque(pu)
-5
0 0.01 0.02 0.03 0.04 0.05
time(s)
2
1
0
-1
0
-1
-2
-3
-4
4
3.5
3
2.5
2
1.5
1
0.5
0 0.01 0.02 0.03 0.04 0.05
time(s)
-2
0 0.01 0.02 0.03 0.04 0.05
time(s)
5000
4000
3000
2000
1000
(a)
(b)
Figure 4: (a) id and (b) motor torque at starting period
is converted into LMA. Change of the control strategy is
also visible in the three-phase currents of the motor which
creates some irregularities in the currents. In addition to
limiting the current, reducing overshoot
Figure 5: Three-phase currents of controlled motor
with T/A control
Without T/A control
0
0 0.01 0.02 0.03 0.04
time(s)
Figure 6: Time variations of motor speed
and shortening settling time of the motor speed are other
advantages of applying T/A. Figure. 6 compares the
motor speed variations without T/A application and
- 129 - 15th IGTE Symposium 2012
controlling cases. Settling time of the motor from 0 to the
rated speed decreases from 0.025 in the no control case to
0.01 in the application of T/A and without overshoot. In
fact, by applying the maximum torque control, the motor
becomes more stable. It is noted that the torque jump due
to switching from T/A to LMA is not present in the speed
signal. The reasons are the high inertia and long
mechanical time constant of the motor compared to its
electrical time constant. The LMA attempts to find the
optimal id for the efficiency improvement. Normally, id is
chosen negative values by applying this algorithm. The
influence of the negative id is the enhancement of the
torque and decrease of losses in the motor. Simulation of
PM motor under LMA control over a wide range of the
speed and torque has been carried out and the effect of
this algorithm on the motor variables and system
efficiency has been investigated. Meanwhile, the outputs
of LMA with id=0 are compared and advantage of this
method over conventional methods is given.
Figure. 7 shows the variations of the copper and iron
losses of the motor versus speed and torque. By raising
the speed at the rated load, the iron losses increase and in
this case the losses reduction algorithm shows its
dominant effects. Also at fixed speed and high loads,
reduction of the total iron and copper losses is
considerable. Meanwhile, by increasing the negative
value of id, demagnetization effect of PM decreases and
for a fixed output power, supply voltage reduces. This
means the efficiency improvement. Difference between
the motor losses in two cases id=0 and LMA causes the
motor efficiency change.
Figure. 8 shows the efficiency versus speed and torque
of the motor. Efficiency of the motor has been compared
in two id=0 and LMA cases. According to figure. 8a,
efficiency of the motor under LMA control over different
speeds, shows a relative increase by id=0 method. By
increasing the speed of the motor, the rate of efficiency
improvement also rises. It means that whatever the motor
approaches more to the rated operating point, its
efficiency improves. Figure. 8b shows the efficiency
versus load torque at the rated speed, and it emphasizes
the efficiency improvement of the LMA in the motor
compared to that of the conventional methods. The
impact of this method is higher for higher loads. As
shown in figure. 8a, there is no much difference between
LMA and id=0 at low load levels. So, efficiency will not
be considerably changed. This is not true over the low
speeds. Generally, electrical machines operate in the knee
of the magnetization characteristic where they have peak
power density; therefore they have the maximum
efficiency at the rated operating point. By applying LMA
at the rated operating point, a 3% rise of the efficiency
occurs. In the references, LMA algorithm has been
applied to the PM motor experimentally. The difference
between the simulation and experimental results is due to
the approximations included in the simulation. The most
important factor is ignoring the magnetic saturation.

Total loss(W)
82
80
78
76
74
72
70
LMA
id=0
68
500 1000 1500 2000 2500 3000 3500 4000
Speed(rpm)
Total losses(W)
120
100
80
60
40
20
LMA
id=0
0
0 0.5 1 1.5 2
torque(Nm)
Comparision of the motor efficiencies at rated load (1.8 N.m)
- 130 - versus the angular 15th speed in the case IGTE of LMA and id=0 controls. Symposium 2012
Efficiencies [%]
Efficiencies [%]
100
90
80
70
60
50
40
80
60
40
20
0
0 1000 2000 3000 4000
Angular speed [rpm]
(a) (a)
VI. CONCLUSION
Two algorithms for enhancing the maximum starting
torque and steady-state efficiency of a PM motor were
investigated. In both methods, stator current components
have been used, transient and steady-state performance of
a PMSM have been improved by closed-loop control
strategy and application of the two algorithms. These
algorithms are independent of the PM motor type. The
simulation results shown that by applying two algorithms,
performance of the motor is considerably improved
compared to that of the conventional control methods. By
applying the stator current limiting method during the
starting period, the stator winding insulation is prevented
against the damage due to high current. The LMA method
improves the steady-state efficiency of the motor up to
3% at the rated load and T/A method causes the increase
of the starting torque up to 4 times of the rated torque.
Therefore, by applying the proposed control methods in
addition to providing a high starting torques without the
risk of the short circuit of the windings; the extra losses
arising from the imprecise control of electrical motors can
be prevented.
AKNOWLEGEMENT
We sincerely thank the Iran’s National Elites
Foundation (INEF) for financial support of the project.
VII. REFRENCES
[1] S.Morimoto, Y.Tong, Y.Takeda, and T. Hirasa, “Loss minimization
control of permanent magnet synchronous motor drives,”IEEE
Transactions on Industrial Electronics, vol. 41, no. 5, pp. 511-517,
Oct 1994.
[2] C.Mademlis, L.Xypteras, and N.Margaris, “Loss minimization in
surface permanent magnet synchronous motor drives",IEEE
Transactions on Industrial Electronics,vol. 47, no. 1, pp. 115-122,
Feb 2000.
Comparision of the motor efficiencies at rated speed (4000)
versus the load torque in the case of LMA and of id=0 controls
id=0
LMA
id=0
LMA
30
0 0.5 1 1.5 2
Torque(N.m)
(b) (b)
Figure 7: Losses of motor versus (a) speed and (b) torque
Figure 8: Efficiency of the motor versus (a)speed (b)
torque
[3] Sadegh Vaez,M.A.Rahman, "Adaptive Loss Minimization Control of
Inverter Fed IPM Motor Drives".IEEE Power Electronics Specialists
Conference, pp. 861-868, vo. 2, 1997.
[4] T.M.Jahns, G.B.Kliman, T.W.Neumann, “Interior permnanent
magnet synchronous motor for adjustable speed drives,” IEEE
Transactions Industry Applications, vol. 22, no. 4, pp. 738-747,
July/August 1986
[5] C.Cavallaro, A.O.Tommaso, R.Miceli, and A.Raciti, “Efficiency
enhansment of permanent magnet synchronous motor drives by
online loss minimization approaches,”IEEE Transactions on
Industry Applications, vol. 52, no. 4, pp. 1153-1160, August 2005.
[6] J.S.Yim, S.K.Sul, B.H.Bae, N.R.Patel, and S.Hiti, “Modified current
control schemes for high-performance permanent-magnet ac drives
with low sampling to operating frequency ratio,” IEEE Transactions
Industry Applications, vol. 45, no. 2, pp. 763-771, March/April
2009.
[7] S.Morimoto, H.Nakayama, and M.Sanada, “Sensorless output
maximization control for variable-speed wind generation system
using IPMSG,”IEEE Transactions on Industry Applications, vol.
41, no. 1, pp. 60-67, Jan/Feb 2005.
[8] T.Nakamura, S.Morimoto, m.sanada, and Y.Takada, “Optimum
control of IPMSG for wind generation system,” IEEE Power
Conversion Conference, Osaka, pp. 1435-1440, 2002.
[9] C.Chris, G.R.Slemon, and R.Bonert, “Minimization of iron loss
of permanent magnet synchronous machines,”IEEE Transactions
on Energy Conversion, vol. 20, no. 1, pp. 121- 127, March 2005.
[10] S.Morimoto, Y.Takeda, and T.Hirasa, “Current phase control
methods for permanent magnet synchronous motors,”IEEE
Transactions on Power Electronics, vol. 5, no. 2, pp. 133, April
1990.
[11] S.Morimoto, Y.Takeda, T.Hirasa, and K.Taniguchi, “Expansion of
operating limits for permanent magnet motor by current vector
control considering inverter capacity,”IEEE Transactions on
Industry Applications, vol. 26, no. 5, pp. 866-871, Sep/Oct 1990.
[12] K.Yamazaki, “Torque and efficiency calculation of an interior
permanent magent motor considering harmonic iron losses of both
the stator and rotor,” IEEE Transactions Magnetics,vol. 39, no. 3,
pp. 1460-1463, May 2003.
[13] R.S.Colby, and D.W.Novotny, “An efficiency-optimizing
permanent magnet synchronous motor drive,”IEEE Transaction on
Industry Applications, vol. 24, no. 3, pp. 462-469,May/June 1988.
[14] A.Kusko, and D.Galler, “Control means for minimization of losses,
in AC and DC motor drives,”IEEE Transactions on Industry
Applications, vol. 19, no. 4, pp. 561-570 ,July/August 1983.

- 131 - 15th IGTE Symposium 2012
Core Losses Estimation Techniques in Electrical
Machines with Different Supplies – A Review
Jawad Faiz, A.M. Takbash and B. M. Ebrahimi
Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer Engineering, College of
Engineering, University of Tehran, Tehran, Iran, Email: jfaiz@ut.ac.ir
Abstract—In this paper, different methods for core losses estimation in ferromagnetic materials with non-sinusoidal supply
are studied. At this end, the origin of the core losses in the aforementioned materials is addressed. Since magnetization
excitation is the most effective factor upon the core losses, different core losses estimation with six general types of excitations
are considered and features of these methods, their advantages and disadvantages are investigated.
Index Terms—Core Loss, Finite Element Method, Hysteresis Loop, Steinmetz Equation.
I. INTRODUCTION
The major role of ferromagnetic materials in electrical
machines leads to wide research towards a better
realization of these materials and their characteristics.
One of the most important features in these materials is
their losses. Core losses are generated due to magnetic
flux residual and eddy current. From physics point of
view, these two factors have an identical origin; they are
movement of magnetic domains walls as well as internal
movement of the magnetic domains. The magnetic
residual flux is a well-known phenomenon. When an
external magnetic field is applied to a ferromagnetic
material, magnetic dipoles try to align with this external
field. Even after removing the external magnetic field,
some magnetic domains preserve their alignments and
such case the material is magnetized. By changing the
magnetic field a small amount of energy is stored in the
material due to existing residual flux. The level of this
stored energy depends on the material type. If a conductor
is imposed on a varying magnetic field or moved
appropriately in the magnetic field, eddy current is
induced in the conductor. Any variation in magnetic field
that causes the movement of the magnetic domains walls
is a factor inducing eddy current. Eddy current generates
heat and electromagnetic forces. For dc excitation, there
are residual and eddy current losses as well. The reason
for the residual losses is the internal movement of the
magnetic domains which itself generates microscopic
currents [1]. Based on these two factors, the core losses
have been classified into two classes [2]. Some categorize
the core losses into three classes [3], in which the third
class is stray losses due to the external factors such as
external magnetic fields. Two factors influence the core
losses. The first factor depends on the magnetic material
alloy, for instance rising Si content in SiFe magnetic alloy
reduces the eddy current losses. The second factor is the
external factor. Magnetic properties of magnetic materials
are affected by cutting, pressing and welding processes.
For instance, cutting or welding process of magnetic
sheets can increase the residual losses of the sheets.
Pressing the sheets causes the increase of the eddy
currents. One of methods for reduction of the eddy
current is using thin sheets. However, this leads to higher
residual losses [4], [5]. Another external factor affecting
the core losses is the magnetizing excitation. In the case
of sinusoidal excitation, the well-known classic Steinmetz
equations can be used for core losses estimation:
P k f B k f B
(1)
x 2 2
ir h s m e s m.
where Pir is the core losses, fs is the supply frequency, Bm
is the magnetic flux density magnitude and kh, ke, xare the
Steinmetz factors. Such equations leads to error in the
case of non-sinusoidal excitation and new equations must
be introduced. Application of different drives with
various switching patterns and also internal faults in
electrical machines leads to non-sinusoidal excitation.
Wide application of inverter-fed electrical machines and
different faults such as rotor broken bars and eccentricity
are important research topics in recent years. Therefore,
core losses estimation in the magnetic core over such noncommon
conditions is important. The trend of core losses
estimation can be classified as shown in Figure 1. In this
classification six general methods has been introduced
which will be discussed in this paper.
II. FINITE-ELEMENT-BASED METHODS
Finite element methods (FEM) are time consuming and
high precision techniques that take into account the
geometry and physics of the machine. There are three
steps in modeling electrical machines. They include
geometrical modeling of motor considering physical
characteristics, modeling motor supply considering
electrical characteristics and finally modeling the motor
load taking into account mechanical features. FEMs
provide magnetic field distributions within induction
motor based on its geometrical and magnetic parameters.
Other quantities such as air gap flux density can be also
estimated using the magnetic field distribution. In FEM,
the coupling between electrical and magnetic fields and
motor rotation can be taken into account. Losses in
different parts of the machine can be estimated having the
magnetic field distribution in various sections of the
motor. A new method has been introduced in [6] for core
losses estimation in a no-load motor with direct-fed and
PWM-fed motor using 2D time stepping FEM in which
the simulation results have been compared to the

Steinmetz
Eq.
Hysteresis
Model
FEM
- 132 - 15th IGTE Symposium 2012
Iron Loss
Calculation
Methods
Equivalent
Circuit
Physical Eq.
Figure 1: Different methods for core losses estimation in non-sinusoidal excitation
experimental results. To include stray losses and
rotational residual losses, a modification factor has been
considered in the losses calculation process. This
modification factor depends on the peak magnetic flux
density and its distortion. In [7], a new model for
laminated core of induction motor has been presented
based on 3D FEM. This modeling method is based on the
reduced magnetic potential equations and used to estimate
the core losses with non-sinusoidal excitation caused by
PWM application. The results are more precise than that
of the 2D FEM. In [8], the authors emphasize the need for
precise data in core losses estimation from magnetic
fields, therefore 2D FE model is applied; impact of nonsinusoidal
supply and its harmonics on the rotor magnetic
field has been investigated and the well-known integral
equations have been then used to estimate the core losses.
Core losses estimation in rotating electrical machines is
more complicated than that of the static machine because
of more complicated structure and rotating magnetic
fields [9]. So, FEM modeling has been modified in order
to include the stray core losses due to magnetic field
rotating vector and its harmonics. At this end, two types
of PM motors with two different structures have been
modeled using a new method and the simulation results
have been compared to the experimental results. In [10],
impact of the stator slot shapes upon the core losses has
been discussed and three different structures of induction
motor for minimizing the core losses have been
investigated. In this case, core losses are estimated using
magnetic flux density and field intensity and integration
of their product. Core losses distribution over core crosssection
and impact of different stator slot shapes has been
considered using FEM. In [11], a model based on eddy
current analysis in the magnetic sheets is presented which
is capable to estimate the stray losses for computation of
high frequency core losses and residual losses in core
sheets of electrical machines. Advantages of this method
is taking into account the magnetic field distribution
along sheets thickness using one-dimensional non-linear
FEM over non-linear 2D elements and also impact of
frequency and magnetic flux density on the quantities
related to the core losses. In [12], a model has been
introduced to study the impact of PWM supply on the
induction motor core losses. Triple losses of the core have
been presented by a combined model using FEM. The
results have been compared to the traditional modeling
and experimental results. This combined model consists
of two static and dynamic models in which the impact of
Control
Strategy
the residual minor loops has been also included. In
addition, the losses distribution over the motor and
separation of different components of the core losses has
been pointed out. In [13], induction motor performance
has been analyzed using 2D FEM. This is one of the few
works in which the impact of internal fault of induction
motor such as rotor broken bars and eccentricity and also
application of the PWM drive upon core and Ohmic
losses have been considered and shown that the rotor
broken bar causes the increase of the losses around the
damaged bar; in addition PWM supply also increases the
core losses density. In [14], Permanent magnet (PM)
motor under three static, dynamic and mixed
eccentricities faults have been analyzed using FEM.
Finally, impact of these faults on core and Ohmic losses
has been investigated. Figure 2 shows the impact of
different eccentricities on the eddy current and residual
losses.
III. HYSTERESIS LOOP MODEL BASED METHODS
Precise mathematical model of hysteresis loop in
magnetic material could be useful in accurate estimation
of the core losses. First, classical hysteresis models such
as [15] have been used to calculate the losses. Recently,
improved hysteresis model such as loss surface model
(LSM) and energy-based hysteresis vector-model have
been employed to estimate the losses which are briefly
described below. The LSM is a numerical and dynamic
model for core losses evaluation which has been applied
to thick magnetic laminations in [16]. This method is
based on the definition of the magnetic field as a surface
function of magnetic flux density and its rate as follows:
dB dB
S H( B, ) Hstat ( B) Hdyn ( B,
). (2)
dt dt
In fact, this method is combination of a static and
dynamic model and is capable to model the static and
dynamic behaviors of the hysteresis loop. The static
behavior is modeled using different hysteresis curves and
dynamic behavior using six parameters which depend on
the magnetic flux density and time variation of its
derivative. Vector magnetic hysteresis model has been
used in [17] to estimate the core losses. In this method,
magnetic field intensity has been evaluated using the
vertical components of the magnetic flux density and then
core losses have been calculated by integration of the
product of the magnetic flux density and field intensity.

- 133 - 15th IGTE Symposium 2012
Figure 2: (a) Hysteresis losses and (b) eddy current losses for different degrees and types of eccentricity [14]
Application of two Preisach and Jill-Atherton models
have been compared in three different magnetic materials
in order to obtain an optimal method for precise modeling
of the magnetic cores using FEM with reasonable
computation time [18]. In mathematical models the full
hysteresis loop and a series of the magnetic parameters of
the proposed material must be available which
complicated its application [19]
IV. STEINMETZ EQUATIONS-BASED METHODS
Purpose of the improved Steinmetz equations is to
estimate core losses for non-sinusoidal magnetic flux
density analytically. This modification is done using
different methods. Improved Steinmetz equations have
been employed to estimate the core losses in switched
reluctance motor (SRM) [20], [21]. First SRM behavior is
analyzed using FEM and then improved core losses
equations for SRM including the impact of the minor
hysteresis loops beside of current harmonics effect are
considered.
ab. B 1 dB
max
2
Pc kcfChfBmax C ( ) .
2 e avg (3)
2
dt
where Kcf is the modification factor that takes into
account the impact of the minor hysteresis loops within
the major loops. Another idea for improving the
Steinmetz classical equations is obtaining an equivalent
frequency for proposed non-sinusoidal signal which have
been used for magnetic sheets [22] and transformer [23]:
2
2
T dB
feq
( ) dt.
2 2
( B 0
max Bmin ) (4)
dt
where Bmax and Bmin are the maximum and minimum of
the magnetic flux density. In the other words, remagnetizing
frequency is substituted by an equivalent
frequency versus magnetic flux density variations. In
addition the impacts of dc upon the core losses have been
considered in [22] and the modified Steinmetz equations
(MSE) have been introduced. Another method of
modifying Steinmetz equations is introducing the
coefficients in order to take into account the nonsinusoidal
magnetic excitation waveform. In this case, the
distorted magnetic flux density waveform is used to
determine these coefficients. For instance, in [24] the
Steinmetz equations have been changed as such that the
hysteresis losses versus the mean value of the rectified
waveform of the magnetic flux density and eddy current
losses versus the rms value of this waveform have been
expressed. Consequently, these coefficients are estimated
when the magnetic flux density waveforms in two directfed
and PWM-fed are known and the core losses in
abnormal operation are expressed versus the core losses
in the normal operation. In [25], the traditional Steinmetz
equations and their modification have been used to
estimate the magnetic sheets losses under PWM-fed; such
that the impact of frequency and magnetic flux density
variations upon the Steinmetz equation have been
included in the coefficients for different materials and
also impact of the magnetic flux density waveform
variations due to drive on the core losses. In order to
consider the frequency and magnetic flux density on the
Steinmetz coefficients, these coefficients are considered
as 3 rd order equations versus magnetic flux density.
V. PHYSICS-DEFINED LOSSES BASED METHODS
A long time ago, there was a procedure for core losses
estimation based on the physics definitions of various
core losses. For instance in [26], a modeling method was
introduced for magnetic domains in material and then
eddy current, its cause and losses were discussed.
Advantage of this model is its capability to use over wide
range of the flux density up to the saturation level and
wide frequency band. In [27], forming the eddy current in
the magnetic sheets and external magnetic fields effect
has been considered. Meanwhile, impacts of the internal
magnetic fields (adjacent magnetic domains walls) on the
eddy current have been included. In [28], core losses as
non-linear function of frequency are expressed as three
types of hysteresis, classic and stray losses. The classic
and stray losses are as follows:
( class)
2 2 2 2
P d
I max
fm/6.
(5)
( exc) 2L
( class)
P (1.63 ) P .
(6)
d
where is the conductivity, d is the magnetic sheet
thickness and L is the magnetic domain dimensions. Core
losses have been categorized into two types: 1: Core
losses constant against frequency (hysteresis losses), and
2: core losses depending on frequency (eddy current
losses and abnormal stray losses) [29]. Each category has
been expressed by equations versus frequency and
magnetic flux density. This method is not a precise
method, the main reason is the classification of the losses
versus dependency and independency on the
frequency. In addition, this method has appropriate results

Figure 4: various losses in the mains-fed and inverter-fed
induction motor [31]
over particular amplitude of the magnetic flux density due
to the simplification of the method. In [30], [31], core
losses have been divided into hysteresis losses, eddy
current losses, and stray eddy current losses. Classic eddy
current losses and stray eddy current losses are as follows:
2
d 1 T dB
Wc( ) dt.
f 12m
T (7)
0 dt
v
1 1 T dB
W GV
S dt.
(8)
c
0
fm 0
v T dt
where mv is the magnetic material density. In the stray
eddy currents losses S is the magnetic sheet cross-section,
G is the dimensions factor and V0 is a parameter that
determines the local fields distribution. Then an
equivalent frequency is defined to estimate the losses
based on physics definition taking into account the
harmonics within the core magnetic flux density. Figure 4
shows various losses in the mains-fed and inverter-fed
induction motor. Also in [32], integrally defined of triple
subdivision of core losses has been used to estimate the
core losses. In this case, losses equations have been
presented versus mmf for different emf (sinusoidal,
triangular and rectangular) waveforms considering
relationship between emf and magnetic flux density and
piecewise-linearized modeling of emf waveform. The
relevant equations due to different losses have been given
for each waveform. In addition, impact of different
parameters such as duty cycle upon various core losses
has been investigated. In [33], the influence of minor
hysteresis loops within the magnetic sheet hysteresis loop
upon core losses estimation has been investigated and
their complicated time variations versus peak magnetic
flux density and magnetic polarization vector for
estimation of the triple core losses have been presented.
Figure 5 shows that how these minor hysteresis loops are
generated. Variations in the magnetic polarization
envelop causes minor hysteresis loops within the major
hysteresis loop. Dividing the core losses into three
different losses and integral definitions versus magnetic
flux density, its derivative lead to complicated
computations. An important point in these methods is
their dependency on the coefficients which depend on the
physical and chemical characteristics of the proposed
material and its molecular structure; however, they are not
1.5
- 134 - 15th IGTE Symposium 2012
Figure 5: Generating minor hysteresis loops within major
hysteresis loop [33].
Figure 6: dq equivalent circuit considering core losses [36].
available and their computations need some tests,
therefore application of these methods is difficult. In
[34], eddy current losses have been evaluated by
application of different double-magnetic-excitation on the
magnetic sheets using Maxwell equations. These
computations have been carried out on a steel sheet and
can be extended to the whole electrical machine. In
addition, computations results have large difference with
the test results and no justification has been given. Also
application of the complicated Maxwell equations is an
important problem particularly in the selection of the
numerical solution method for solution of the equations.
In [35], superposition method has been applied to
estimate the eddy current losses in the PWM-fed motor
and transformer. Since different losses do not vary versus
magnetic flux density, application of the superposition is
not correct.
VI. EQUIVALENT CIRCUIT-BASED METHODS
Equivalent circuit of induction motor has been
frequently used to analyze the motor behavior and find its
core losses. As shown in Figure 6, at this end a simplified
dq model of ac motors can be used which does not lead to
accurate results [36]. In addition, in dq model harmonics
are ignored and this decreases the precision of the
method. In [37], an equivalent circuit with a
harmonic supply has been introduced to estimate core
losses. In this case, dq model has been used and
parameters of the equivalent circuit are calculated using
FEM. In [38], impact of unbalanced supplied induction
motor on the motor efficiency has been presented. The
proposed equivalent circuit of induction motor is not
accurate, so it has been modified by layered magnetic
core. To do so, a resistance with leakage inductances has
been connected in parallel. The major point in the

Method
Steinmetz Eqn.
MSE
Hysteresis model
Physical Eqn.
FEM
Equivalent Circuit
Control Strategy
application of the equivalent circuits for core losses
estimation is its strong dependency on the parameters
which may vary due to the operating conditions.
VII. CONTROL STRATEGY-BASED METHODS
Since efficiency of electrical machine is an important
factor beside its life, one of the major quantities aiming to
reduce is the core losses of drive-fed motor. In [39], [40],
strategies for squirrel-cage induction motor and PM
synchronous motor control have been introduced to
decrease the losses. In these strategies, all equations for
minimizing the losses depend on the motor parameters
and determination of these parameters is themselves
critical. In [41], impact of the core losses of induction
motor on the stator-flux oriented control has been studied
and a control strategy taking into account the induction
motor losses has been introduced. In this method, core
losses have been modeled by a resistance in parallel with
the magnetizing inductance. In [42], a vector controlbased
strategy for PMSM has been presented to maximize
the motor efficiency. At this end, a model has been
suggested for losses. In this method, d-component of the
stator current is determined to maximize the IPM motor
efficiency. In this strategy, losses are divided into Ohmic,
core, mechanical and harmonic losses and they are then
calculated.
This test-based method can be extended to different
types of PM and reluctance motors. However, they
depend strongly on the motor parameters while these
parameters are not in turn constant and vary by changing
the operating point of the motor. Table I summarizes the
core losses estimation for different supplies. In this table,
some factors such as complexity of the methods, need for
magnetic material parameters, response to the nonsinusoidal
excitation waveforms and precision of the
methods have been compared. As seen in Table 1, some
accurate methods are complicated and need huge data
from the magnetic material. Some methods are simple
with low accurate responses. So, a proper method must be
selected based-on the application.
VIII. CONCLUSION
Different methods were proposed for core losses
estimation in magnetic materials for different excitations.
These methods were classified and studied. Some factors
such as complexity of methods for magnetic material
parameters estimation, response to non-sinusoidal
- 135 - 15th IGTE Symposium 2012
TABLE I
COMPARISON OF DIFFERENT CORE LOSSES ESTIMATION METHODS
Complex waveform Complexity Material knowledge
-
Low
Low
+
Low
Low
+
High
High
+
High
High
+
High
Medium
-
Medium
Low
+
Medium
Low
Accuracy
Low
Medium
Good
Good
Good
Low
Medium
excitation waveforms and precision for each method were
investigated and summarized. The methods such as FEM,
hysteresis model, physical equation lead to accurate
results but they are complicated methods and need a huge
data of the magnetic material. MSE method is simple and
no need huge data of the magnetic material with average
accuracy. Control strategies are not used in direct core
losses estimation. Equivalent electrical circuits of motor
parameters also depend on the operating point of the
motor and this is considered a difficulty of these methods.
ACKNOWLEDGMENT
The authors would like to thank Iran’s National Elites
Foundation (INEF) for financial support of the project.
[1] C. D. Graham,
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- 137 - 15th IGTE Symposium 2012
Fast Computation of Inductances and Capacitances of
High Voltage Power Transformer Windings
*Župan, Tomislav, *Štih, Željko and *Trkulja, Bojan
*Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia
E-mail: tomislav.zupan@fer.hr
Abstract— Inductances and capacitances in analysis of fast transients in high voltage transformers are usually calculated on
the basis of simple analytical approximations for applications in lumped circuit models. This paper presents the application
of the boundary element method to calculation of capacitances and inductances of transformer windings with coils of
rectangular cross section. Two-dimensional axially symmetric mathematical model of electric and magnetic field in
transformer, which is based on integral equations, is introduced. The accuracy and applicability of the proposed approach is
illustrated by an example.
Index Terms—boundary element method, capacitance calculation, inductance calculation, rectangular cross section coils.
I. INTRODUCTION
Power transformers are one of the most important
segments of electric power systems. Due to their long
term running requirements they are faced with numerous
overvoltage strikes during their lifetime. Therefore it is of
great importance to know how the power transformer
windings will react on such excitations.
Since most of the overvoltage strikes are characterized
by high frequency impulse signals (due to lightning
discharges, short circuits, etc.), the capacitances between
windings, which are negligible during normal operational
frequency of power grid, become significant. Thus it is
meaningful to have, alongside the strict calculation of
inductances, a method for precise calculation of
capacitances of high voltage power transformer windings.
Distribution of voltage along windings of transformers
due to fast excitations (switching, lightning, and testing)
is one of the most important inputs to insulation design.
Lumped circuit model of winding is most frequently
applied in calculation of such distribution [1]-[3]. Parts of
windings are represented by equivalent capacitances,
inductances and resistances. Depending on the way of the
connection of the conductors, the equivalent diagram is
made for each of the turns or, for the sake of faster
calculation, turns are grouped.
There are numerous papers dealing with the presented
problem by using the numerical approach based on the
finite element method (FEM) [1], [4]. In this paper we
introduce the application of the boundary element method
(BEM) to calculation of capacitances and inductances of
windings of transformers with coils of rectangular cross
section. Main advantage of this approach is ease of use,
because the discretization is done only at material
interfaces and boundaries, thus effectively reducing the
order of the mathematical model by one dimension.
II. COMPUTATION OF CAPACITANCES
Typical arrangement inside the high voltage power
transformer can be seen in Fig. 1 (simplified). Windings
usually consist of low voltage, high voltage and
regulation voltage parts. Conductors are typically of
rectangular cross section insulated by paper insulation
and immersed in oil, grouped in discs with radial and
axial canals for heat transfer purposes.
z
r
LV
HV RV
core
Fig. 1. Windings in high voltage power transformers
Usual approach to calculation of capacitances is based
on simple analytical parallel-plate approximations [2]:
2 2
Rout Rin
CParallel
0
,
(1)
drc
o p
where o and p are relative permittivities of oil and
paper insulation, respectively, drc height of the radial
canal between conductors in z axis direction, width of
insulation between conductors, in R and Rout conductor's
inner and outer radius, respectively,
cylindrical approximations [5]:
or analytical
20
phc CSerial
,
Rav
ln 2
R
av
2
(2)
oil
paper
insulation
conductor

where c h is conductor's height, and Rav average radius
between two conductors.
These analytical methods result in significant errors and
consequently in unsatisfactory accuracy of computation
of voltage distribution along the winding.
Electromagnetic field in power transformers is
transverse and the capacitances may be computed on the
basis of electrostatic analysis. The electric field in a space
composed of conducting regions at known potentials and
regions filled with different dielectrics can be solved by a
pair of coupled integral equations [6]. The potential (A)
at any point A on the surface of conductor is related to the
charge density (B) on total surface of all boundaries
(conductor–dielectric and dielectric–dielectric) by the
equation:
A B G B, A dS 0,
(3)
S
where:
1
GB, A
,
(4)
4 d AB
and dAB is the distance between points A and B. The
charge density (D) at any point D on the dielectric–
dielectric boundary is related to the charge density (B)
on total surface of all boundaries by the equation:
o
i
D2 BDNB, DndS 0.
(5)
o i S
Here, n is the unit vector normal to the surface at the
point D, o is the permittivity of the region in the
direction of the normal unit vector on the surface at the
point D and i is the permittivity of the region in the
opposite direction.
Geometry of the windings can be approximated with
axially symmetry, and the problem becomes twodimensional.
Therefore, we may integrate surface
integrals in (3) and (5) with respect to circumferential
direction and reduce them to line integrals. Kernels of
integral equations are:
1 r
Prr ( , ) kKk 2
r
DN( r, r) DR( r, r) ar DZ( r, r) az
1
DR( r, r)
4 r
3
r k
2 r1k KkEk rr
EkKk 2
k 2r
(6)
1
DZ( r, r)
4r 3
r zz k
E k r 2 2r 1
k
2
k
4rr
.
2 2
rr zz
Here, k is the modulus of the elliptic integrals of the
first and the second kind K(k) and E(k). The vector
r ra r za
z defines the position of the source point
(B), and the vector r rar zaz
defines the position of
the point of interest (A, D).
- 138 - 15th IGTE Symposium 2012
Terms Prr ( , ) , DR( r, r)
and DZ( r, r)
represent the
kernels for electric potential and radial and axial
components of electrical induction vector of uniformly
charged ring with negligible cross section, respectively
[7]. Because the conductors inside power transformers
have rectangular cross section, their surface can be
divided, using BEM approach, into either thin cylinders
or thin discs. As can be seen on Fig.2, both of the cases,
for the sake of generality, can be represented with
truncated cone.
truncated
cone
disc
cylinder
Fig. 2. General representation of the conductor segment
division
The final expressions for Prr ( , b, re)
, after integrating
(6) over l, are:
1 2
l r() t K( k )
Prr ( , b, re)
dt
q
0
2 2
e b e b
l z z r r
rt () ( re rb) trb zt () ( ze zb) tzb, where r(t) and z(t) represent parametric notation of the
general point on segment l, l is the length of the segment
and q and k are:
2
2 2
q r rt () zzt () 2
rrt ()
4 rr( t)
k .
q
From the equation E ,0,
r z
obtain the kernels of electrical induction vector:
1
l
DR( r, rb, re) r( t)
0
2 2
2() rtKk ()(1 k) Ek ()2() rt krrt ()
2 2 3 2
k (1 k ) q
1
l rt () zzt () Ek
( )
DZ( r, rb, re)
dt.
2 3 2
(1 k ) q
0
z
z
ze
zb
The system of integral equations (3) and (5) is solved
by BEM. Boundaries and interfaces between two
dielectrics are divided into finite segments and the
l
rb re r
dt
r
(7)
(8)
we
(9)

unknown distribution of surface charge density on the ith
segment is approximated as linear combination of
predefined basis functions:
N
( r)
t ( r).
(10)
i in in
i1
The simplest approach is to use basis functions which
are constant (N=1) on the finite segment. The application
of (10) to (3) and (5) results in:
N
( r) P( r, r, r) dC ; r C
0 i p k i 0
i1 Ci
N
o i
i( r) 2 i
DN( r, rp, rk) dCi; o
i i1 Ci
r Ci.
(11)
Here, C0 is boundary at known potential 0 and Ci is
interface between two dielectrics. A linear equation
system for unknown coefficients i is derived by
enforcing an exact solution at midpoints of each finite
segment (point-matching [8]). The integrals in (11) are
Ci,
the integrals become singular. In such a case their vicinity
is treated separately and this contribution is calculated
analytically (logarithmic singularities).
After the determination of the surface charge
distribution
calculated by:
on conductors, the capacitances are
Qij
Cij ; i j.
(12)
i j
Here, i and j are potentials of i-th and j-th conductor,
respectively, and Qij is total charge on the j-th conductor
influenced by the charge on the i-th conductor:
j
Q dS S
ij j j kj kj
S
k 1
j
N
,
- 139 - 15th IGTE Symposium 2012
(13)
where kj is surface charge density on k-th segment of jth
conductor, Nj is the number of finite segments on j-th
conductor and Skj is the surface of the k-th segment of jth
conductor.
We calculate the capacitances by setting the potential of
the i-th conductor to 1V and the potential of all other
conductors to zero. Then, we obtain the total charge on
conductors and use (12) to calculate the capacitances.
III. CAPACITANCE CALCULATION -EXAMPLE
The following procedure has been tested on two
examples, first one showing the calculation of turn-toturn
capacitances of high voltage winding in a power
transformer and the second one showing a more
"macroscopic" approach, where the conductors in one
row of high voltage winding are grouped into disc and
then the disc-by-disc capacitances are observed.
The turn-to-turn capacitances were calculated in three
ways:
BEM approach. Total number of unknown
coefficients of surface charge distribution was
896.
FEM approach using Ansoft Maxwell ®
package.
Total number of elements was 36180, and total
number of nodes was 2103, which results in 0.1%
error in calculation of energy.
Analytical approach based on cylindrical
approximation shown in (2) for calculation of
serial capacitance between radially neighboring
conductors or parallel-plate approximation shown
in (1) for calculation of parallel capacitance
between axially neighboring conductors.
Following proposed BEM approach, graphical
depiction of one example where the middle conductor's
potential is set to 1V, illustrating the distribution of the
surface charge density on the observed conductor and the
influenced surface charge densities on neighboring
conductors can be seen on Fig. 3.
Fig. 3. BEM solution for calculation of turn-to-turn
capacitances
The same example was solved using Ansoft Maxwell ®
package, as can be seen on Fig. 4.
Fig. 4. FEM solution for calculation of turn-to-turn
capacitances (Ansoft Maxwell ® )
Comparison of the results for turn-to-turn capacitance
calculation of various approaches is given in Table 1. CiS
and CiP represent the serial and parallel capacitance
between two innermost conductors, CS and CP between

two middlemost conductors, and CoS and CoP between
two outermost conductors.
TABLE I
TURN-TO-TURN CAPACITANCE RESULTS
turn-to- CiS CiP CS CP CoS CoP
turn [nF] [nF] [nF] [nF] [nF] [nF]
® Maxwell 1.70 0.09 1.81 0.03 1.93 0.10
Analytical 1.56 0.04 1.66 0.04 1.76 0.05
BEM 1.68 0.09 1.77 0.03 1.89 0.10
The disc-to-disc example results solved by BEM can be
seen in Fig. 5 and the same example solved using Ansoft
Maxwell ® package can be seen in Fig. 6.
Fig. 5. BEM solution for calculation of disc-to-disc
capacitances
Fig. 6. FEM solution for calculation of disc-to-disc
capacitances (Ansoft Maxwell ® )
Comparison of the results for disc-to-disc capacitance
calculation is given in Table 2.
TABLE II
DISC-TO-DISC CAPACITANCE RESULTS
disc-to- Cbottom Cmid1 Cmid2 Ctop
disc [nF] [nF] [nF] [nF]
®
Maxwell 2.83 2.88 2.90 2.83
Analytical 2.36 2.36 2.36 2.36
BEM 2.81 2.73 2.79 2.75
The cumulative results show significant errors in
analytical approach and justify the necessity of
application of numerical approaches. Even in the case of
very coarse discretization of the BEM approach, the
- 140 - 15th IGTE Symposium 2012
results differ by only a few percents from the results
obtained by FEM.
IV. COMPUTATION OF INDUCTANCES
Using the analogy introduced in capacitance
computation, inductances can be computed on the basis
of magnetostatic analysis. The magnetic field in space
composed of conducting regions with known currents and
regions filled with different magnetic materials can be
solved by a pair of coupled integral equations. Due to the
linearity of the computation, by imposing the constant
magnetic permeability, the magnetic vector potential and
magnetic field strength can be written as:
Ar ( ) AM( r) AS( r)
(14)
Hr ( ) HM( r) HS(
r),
where A
is total magnetic vector potential, AM
is
magnetic vector potential caused by surface
magnetization current density KM
and S A is magnetic
vector potential caused by imposed current density S J .
The same subscripts and definitions are valid for
magnetic field strength.
Magnetic vector potential AA ( ) at any point A on the
surface that restricts the model is related to the surface
magnetization current density K( B)
on total surface of
all boundaries by the equation:
AA ( ) KBGBAdS ( ) ( , ) A(
A),
(15)
S
where GBA ( , ) is written in equation (4). The surface
magnetization current density K( D)
at any point D on
the boundary of two different magnetic materials is
related to the surface magnetization current density
K( B)
on total surface of all the boundaries by the
equation:
o
i
K( D) 2 K( B) HT( B, D) d S
o i
S
(16)
o
i
2 HS( D) n.
o i
Here, n is the unit vector normal to the surface at the
point D, o is the permeability of the region in the
direction of the normal unit vector on the surface at the
point D and i is the permeability of the region in the
opposite direction.
Assuming the same simplification as in the capacitance
calculation, geometry of the windings can be
approximated with axially symmetry, and the problem
becomes two-dimensional so we may integrate surface
integrals in (15) and (16) with respect to circumferential
direction and reduce them to line integrals. Kernels of
integral equations are:
2
1 r 1 k
Grr ( , ) 1 Kk
( ) Ek
( )
rk
2
(17)
HT ( r, r) HR( r, r) a HZ( r, r) a n,
S
r z

where Grr ( , ) is the kernel for magnetic vector potential
and HR( r, r)
and HZ( r, r)
are kernels for radial and
axial components of magnetic field strength:
k zz HR( r, r) K(
k)
4
rr
r
2 2
2
r r zz
Ek ( )
2 2
rr zz
k
HZ( r, r) K
( k)
(18)
4
rr
2 2
2
r r zz
Ek ( )
2 2
rr zz
2 4rr
k
.
2 2
rr zz Here, k is the modulus of the elliptic integrals of the
first and the second kind K(k) and E(k). The vector
r ra r za
z defines the position of the source point
(B), and the vector r rar zaz
defines the position of
the point of interest (A, D).
The system of integral equations (15) and (16) is solved
by BEM using the same technique mentioned in
capacitance calculation section above, dividing the
interfaces between different magnetic materials into finite
segments and approximating the unknown distribution of
surface magnetization current density with linear
combination of predefined basis functions:
N
Ki( r) Kint in(
r).
(19)
i1
Again, using the simplest adequate approach, the basis
functions are constant on the finite segment (N=1). The
application of (19) to (15) and (16) results in:
N
Ar ( ) K Gr ( , r) dC A( r); rC
K r
0 i i S
0
i1 Ci
HrS ( r) arHzS ( r) azn; r Ci
N
i( ) o i
Ki HT( r, r ) dCi
2 o
i i1 Ci
- 141 - 15th IGTE Symposium 2012
(20)
Here, C0 is boundary at known magnetic vector
potential and Ci is interface between two magnetic
materials. Using the point-matching technique, a linear
equation system for unknown coefficients Ki is derived.
The example of the distribution of surface magnetization
current density on the core of the transformer is shown in
Fig. 7.
As can be seen through inspecting equations (15) and
(16), it is still necessary to determine the magnetic vector
potential contribution of imposed current density AS( r)
and their radial and axial components of magnetic field
strength HrS ( r) and HzS ( r) .
The calculation for magnetic vector potential and
magnetic field strength of circular conductor of
rectangular cross section have been done in [9] and are
presented here for the completeness of proposed method.
Fig. 7. Distribution of surface magnetization current density on
transformer core boundaries
Using Fig. 8. for clarity, the equations are:
T1( R1, R2, r, zZ1) T1( R1, R2, r, Z2 z)
;
z Z1
TA(
R1, R2, r) T1( R1, R2, r, zZ1) AS
T1(
R1, R2, r, Z2 z); Z1 z Z2
T1( R1, R2, r, Z2 z) T1( R1, R2, r, zZ1)
;
z Z2
(21)
T2( R1, R2, r, zZ1) T2( R1, R2, r, Z2 z)
;
z Z1
TB(
R1, R2, r) T2( R1, R2, r, zZ1) H zS
T2(
R1, R2, r, Z2 z); Z1 z Z2
T2( R1, R2, r, Z2 z) T2( R1, R2, r, zZ1)
;
z Z2
(22)
H T ( R , R , r, Z z) T ( R , R , r, zZ ).
rS
3 1 2 2 3 1 2 1
R1 R2
r
Fig. 8. Circular conductor with rectangular cross section
The subfunctions TA, TB, T1, T2 and T3 are:
0Ira
R2 r T1( R1, R2, r, a)
ln
4
R1r 2 2
R2 r a
2 2
R1r a
3
2
0IrR2 sin d 2 X( R , r, a, ) aX( R , r, a,
)
0
z
Z2
Z1
2 2
I

3
2
0 1
sin
IrR d
2
X( R , r, a, ) a X( R , r, a,
)
0
1 1
0Ia
co s X( R2, r, a, ) 2
0
X( R1, r, a, ) d
0Ir rsin arctan
2
a
0
2
R2
X( R2, r, a, ) 2
R
1
X( R1, r, a,
)
2
0Ir
a sincossin
cossind
4 R rcos
X( R , r,
a,
)
0
2 2
2
2
0Ir
a
1
R
1 d X( R2, r, a, ) 4
R
1
X( R
0 1,
r, a,
)
sin cos sin
,
cos ( , , , ) d
R r X R r a
(23)
1 1
I Ia
T2( R1, R2, r, a) R2 R1
2 2
2
ln
R r
R1r 2 2
R2 r a
2 2
R1r a
Iar
2
sin
1 R
0
2 rco s X( R2, r, a, )
R2
X( R2, r, a,
)
d
sin R1
1
d
R
0
1rco s X( R1, r, a, ) X( R1, r, a,
)
2
Ir rsin R2
arctan
2
a
X(
R
0
2,
r, a,
)
2
R
1
sin d, X( R , r, a,
)
(24)
1
2 2 2
X( R, r, a, ) R r a 2Rrcos (25)
2 2
Ir R2 r R2 r a
T3( R1, R2, r, a)
ln
4 2 2
R1 r R1 r a
2
I Ir
co s X( R2, r, a, ) X( R1, r, a, ) d
2
4
0
sincossin R2
1 d
R
0
2 rco s X( R2, r, a, ) X( R2, r, a,
)
0
sincos sin
R1
1d R1rcos X( R1, r, a,
)
X( R1, r, a,
)
I R2 R1 ; r R1
T ( R , R , r) I
R r ; R r R
B
1 2 2 1 2
0; r R2.
(26)
- 142 - 15th IGTE Symposium 2012
0Ir
R2 R1; r R
1
2
3 3
0Ir
r R
1
TA( R1, R2, r) R2 r ; R
2 1 r R2
2
3r
3 3
0I
R2 R
1
; r R2
2
3r
Finally, the inductance calculation can be done using
the equation for the stored magnetic energy:
1 2 1
LI
2 2
JS AdV V
(27)
1
L J ( ) ( ) .
2 S AM r AS r dV
I
V
As can be seen from equation (27), the inductance of ith
conductor can be separated into two parts:
Li LiM LiS,
(28)
where LiM is the contribution of magnetizing currents
and LiS is the contribution of imposed currents
(inductance of conductor in free space).
The self and mutual inductance calculations of circular
conductors with rectangular cross section have been done
in a couple of papers [10]-[13]. Technically, the
equations for L and M are the same with the difference in
the limits of integration. The calculation of the selfinductance
can be observed as the special case of the
mutual-inductance equation.
With the assumption of uniform distribution of current
on conductor's cross section, the total energy stored in the
magnetic field of the conductor is:
2 2
Z2 Z4 R2 R4
0JJ
1 2
W cos
r
2
0zZ1 ZZ3 rR1 RR3 RdRdrdZdzd
r R 2Rrcos zZ 2
.
(29)
Using the expressions:
1
I
W MI1I2; J (30)
2
S
the equation for the mutual inductance of circular
conductor with rectangular cross section is:
0
M
Q,
(31)
( R2 R1)( Z2 Z1)( R4 R3)( Z4 Z3)
where Q represents the above written quintuple integral
in (29).
Using the equivalences:
Z3 Z1; Z4 Z2; R3 R1; R4 R2; I1 I2,
(32)
after analytically solving the four integrals for r, R, z and
Z, according to [13], the final expression for the selfinductance
of circular conductor of rectangular cross
section in free space is:
2
0N
L (
2 2
2, 2, , ) ( 1, 1,
, )
2 1
Q R R H Q R R H
R R H 0
QR ( , R, H, ) QR
( , R, H, ) d
1 2 2 1

4
2
h cos QrRh (, , , )
30sin
2 hcos
bh arctan
sin
2 2 2
2 2
h cosrRsin3h r R cos
2
hsin bh 20
4 2
Rhsincos
hrRcos
arctan
2 2
Rsin bh
4 2
rhsincos
hRrcos
arctan
2 2
rsin bh
2
bh
cos
2 4 4 2 2
3cos
R r Rrcosr R
15
2
2 2
2r R
2
bh
bhcos ln
2
bh h
2
bh h
2
2 2 2 4 4
r R 2cos R r
8
5 2 2
R sin cos ln rRcos
5
b
5 2 2
r sin cos ln Rrcos b
5
3 2 2 2 2
R cos 5h 3R sin
ln rRcos
15
(33)
2
bh
3 2 2 2 2
r cos 5h 3r sin
ln Rrcos
15
2
bh ,
2 2
where N is the number of turns, b r R 2rRcos, and H Z2 Z1.
The integral over in equation (33)
cannot be written in closed form so it has to be solved
numerically, solving the singularities in 0 and
by using the l'Hôpital's rule.
The above presented method for determining the
inductance matrix of the power transformer windings has
been tested for the various conductor positions and
different magnetic permeabilities of the transformer core.
Comparison showed that the difference between the
professional FEM tools (Ansoft Maxwell ®
software
package) is way beyond 1%, which proves the accuracy
and usefulness of the presented method.
V. CONCLUSION
In this paper we present the method for fast and precise
computation of capacitances and inductances of high
power transformer windings with coils of rectangular
cross section based on the boundary element method.
Geometry of the windings is axially symmetric, and the
model may be reduced to two-dimensional axially
symmetric problem. Capacitances are computed from
static electric field solution. Surface charge distribution is
determined by BEM solution of a pair of coupled integral
equations for static electric fields. Inductances are
computed from static magnetic field solution. Surface
magnetization current distribution is determined by BEM
- 143 - 15th IGTE Symposium 2012
solution of a pair of coupled integral equations for static
magnetic fields.
Boundaries and interfaces are divided into line and arc
finite segments, and the unknown distribution of surface
charge or current density is approximated by piecewise
constant functions. System of equations for unknown
coefficients of distribution is obtained by “pointmatching”.
The testing shows that even very coarse discretization
results in satisfactory accuracy of the computation and
therefore proves the applicability of the presented
method.
REFERENCES
[1] E. Bjerkan and H. K. Høidalen, "High frequency FEM-based
power transformer modeling: investigation of internal stresses due
to network-initiated overvoltages", International Conference on
Power Systems Transients (IPST'05), Montreal, Canada, June
2005.
[2] Y. Shibuya, T. Matsumoto and T. Teranishi, "Modelling and
analysis of transformer winding at high frequencies", International
Conference on Power Systems Transients (IPST'05), Montreal,
Canada, June 2005.
[3] K. Pedersen, M. E. Lunow, J. Holboell and M. Henriksen,
"Detailed high frequency models of various winding types in
power transformers", International Conference on Power Systems
Transients (IPST'05), Montreal, Canada, June 2005.
[4] G. Liang, H. Sun, X. Zhang, X. Cui, “Modeling of Transformer
Windings Under Very Fast Transient Overvoltages” IEEE
Transactions on Electromagnetic Compatibility, Vol. 48, No 4,
November 2006.
[5] M. Popov, L. van der Sluis, R. P. P. Smeets and J. L. Roldan,
"Analysis of very fast transients in layer-type transformer
windings", IEEE Transactions on Power Delivery, Vol. 22, No. 1,
pp. 238-247, January 2007.
[6] Ž. Štih, “High Voltage Insulating System Design by Application
of Electrode and Insulator Contour Optimization”, IEEE
Transactions on Electrical Insulation, Vol. EI-21, No.4, August
1986.
[7] P. Zhu, "Field distribution of a uniformly charged circular arc",
Journal of Electrostatics, Vol. 63, pp. 1035-1047, March 2005.
[8] Z. Haznadar, Ž. Štih, "Electromagnetic Fields, Waves and
Numerical Methods", IOS Press, Amsterdam 2000.
[9] J. T. Conway, "Trigonometric integrals for the magnetic field of
the coil of rectangular cross section", IEEE Transactions on
Magnetics, Vol. 42, No. 5, pp. 1538-1548, May 2006.
[10] S. I. Babic and C. Akyel, "New analytic-numerical solutions for
the mutual inductance of two coaxial circular coils with
rectangular cross section in air", IEEE Transactions on Magnetics,
Vol. 42, No. 6, pp. 1661-1669, June 2006.
[11] J. T. Conway, "Inductance calculations for circular coils of
rectangular cross section and parallel axes using Bessel and Struve
functions", IEEE Transactions on Magnetics, Vol. 46, No. 1, pp.
75-81, January 2010.
[12] D. Yu, K. S. Han, "Self-Inductance of Air-Core Circular Coils
with Rectangular Cross Section", IEEE Transactions on
Magnetics, Vol. 23, No. 6, pp. 3916-3921, November 1987.
[13] I. Doležel, "Self-inductance of an air cylindrical coil", Acta
, Vol. 34, No. 4, pp. 443-473, 1989.

- 144 - 15th IGTE Symposium 2012
Numerical and Experimental Investigations of
the Structural Characteristics of Stator Core
Stacks
Mathias Mair ∗ , Bernhard Weilharter †‡ , Siegfried Rainer § , Katrin Ellermann ∗ and Oszkár Bíró †§
∗ Institute for Mechanics, University of Technology Graz, Austria
† Christian Doppler Laboratory for Multiphysical Simulation, Analysis and Design of Electrical
Machines, Austria
‡ Institute for Electric Drives and Machines, University of Technology Graz, Austria
§ Institute for Fundamentals and Theory in Electrical Engineering, University of Technology Graz, Austria
E-mail: mair@tugraz.at
Abstract—The response characteristics of two stator core stacks are investigated by experimental modal analysis.
Furthermore, the modal parameters like the eigenfrequencies and eigenvectors are calculated from a numerical modal
analysis. Afterwards, their frequency response functions are computed and compared with the measured frequency response
functions. In order to achieve a set of material parameters, the computed response characteristics are adjusted to match
the measured response characteristics.
Index Terms—experimental modal analysis, homogeneous material model, numerical modal analysis, stator core stack
I. INTRODUCTION
The development of electrical machines requires an accurate
dynamical analysis in order to reduce side effects
of vibration, e.g. noise and material damage. Since the
electrical machine consists of many complex and heterogeneous
parts, like the stator core stack, the mechanical
modeling of an electrical machine is a complicated task.
Especially for the noise computation of electrical machines,
the structural behavior of the stator core stack is
of interest. It is mainly influenced by forces caused by
the magnetic field in the air gap acting on the stator teeth
[1]–[4]. An analytical method proposed by [5] allows
for the computation of the stator vibration with a twodimensional
ring model. However, the disadvantage of
this method is that it is not possible to consider the
response characteristics along the axial direction. Other
approaches, considering the stator core stack as a thin
cylinder, have been investigated in [6]–[11].
With numerical methods, e.g. the finite element method
(FEM), the three dimensional behavior of the stator can
be determined [12], [13]. The main problem is to set up
an appropriate material model for the numerical analysis,
which considers the heterogeneous composition of the
stator core stack consisting of laminated sheets coated
with resin [14].
Experimental investigations of models consisting of
laminated iron sheets and a comparison with numerical
simulations using three-dimensional homogeneous models
has been presented in [15]. Another investigation of a
stator core stack has been done by [16] in order to acquire
isotropic material parameters. With this approach, it is
possible to calculate modes with pure radial displacement
adequately. A similar investigation has been presented by
[17] with the difference that the used FEM-model has
been computed with a material model of transversally
isotropic elasticity. However, the measurement points
have not been uniformly distributed on the outer ring
surface. As a consequence, the measurement result of the
response characteristics of the stator core stack is limited.
In this paper, the three dimensional structural vibration
behavior of stator core stacks is investigated by using
the finite element method in conjunction with a modal
analysis. The influence of the lamination along the axial
direction will be considered by using a homogeneous material
model with transversally isotropic elasticity. For the
identification of the corresponding material parameters,
two finite element models of stator core stacks have been
set up, one with teeth and one without teeth. A modal
analysis is carried out to determine the eigenfrequencies
and eigenforms (modes) of the finite element models.
Thereafter the frequency response characteristics of the
two structures are computed with a reduced order model
by applying a modal reduction.
Acceleration measurements for which the structures
have been excited with an electrodynamic shaker in a
frequency range of 0 − 6000 Hz have been performed
at 60 points on the stator core stacks. An experimental
modal analysis then provides the measured response
characteristics and eigenfrequencies and eigenforms [18].
Finally, the results of the numerical investigation are
compared with the results from the experimental modal
analysis. The material parameters are adjusted step by
step until the response characteristics of the numerical
model approximate the measured one sufficiently. This
way a set of material parameters for homogeneous material
models for the stator core stacks is obtained, which
describes the structural behavior adequately.

II. EXPERIMENTAL MODAL ANALYSIS (EMA)
An experimental modal analysis allows for the determination
of the response characteristics of the stator core
stacks excited by a shaker. For this, acceleration sensors
are recording the vibration at distinct measurement points
on the stator core stacks. Then, the characteristic response
behavior can be derived, transforming the resulting time
signals into the frequency domain.
A. Test stand for experimental modal analysis
In order to measure the vibration on the stator core
stacks, a test stand as shown in Fig. 1 has been built.
Ropes composed of an elastic material are connected
to a portal frame and suspend the stator core stack,
additionally the table is mounted on air bearings. This
reduces the influence of the adjacent structure to a minimum.
In order to excite the structure, an electromagnetic
shaker is mounted on the table. The connection between
structure and shaker is realized by a push rod and a
force sensor affixed to the test structure with a twocomponent
adhesive. This way, the measurement setup
can be built up and disassembled easily without machine
tools. For the measurement, the shaker is controlled by
a measurement system which also records the signals of
the acceleration sensors.
Fig. 1: Test stand
B. Measurements procedure
The measurement points are located on the outer
surface of the structure at sixty defined positions, see
Fig. 2. The excitation is applied to point no. 1 for all
measurements. At this point, the structure is excited by
the shaker with a so-called periodic chirp signal in a
frequency range from 2 Hz to 6400 Hz. The applied signal
is repeated ten times with the entire frequency range
passed through in each sequence within a short time
period of 2.5 s.
In the course of the measurement procedure, the frequency
response function (FRF) in reference to the excitation
point is determined for each measurement point.
Finally, the arithmetically averaged acceleration and force
signals of these ten measurement runs are used to derive
the FRFs corresponding each of the measurement points.
- 145 - 15th IGTE Symposium 2012
55
43
31
19
7
58
46
34
22
10
49
37
25
13
Fig. 2: Defined measurement points
C. Identification of modal parameters
After all FRFs are measured, the next step is to
identify the modal parameters. This is done by the so
called PolyMAX frequency-domain method [19], which
is a curve fitting technique. In a first step a least-squares
method fits the polynomials
p
p
−1 V0(Ω) =
(1)
z
i=0
i βi z
i=0
i αi
to the measured FRFs. Thereby, Ω denotes the excitation
frequency and V0(Ω) is called the frequency response
matrix. βi are the coefficient numerator matrices, αi are
the coefficient denominator matrices, zi are the complex
basis vectors in the discrete frequency domain and p is
the order of the polynomials.
With the known polynomial functions, the eigenvalues
λi and the modal participation vectors li can be calculated.
To determine the eigenvectors ri as well as the
lower and upper residual matrices LR and UR, a further
least square method, based on the pole residual model
q
ril
V(Ω) =
i=0
T i
λi − jΩ + rilH
i
+
λi − jΩ
LR
+UR , (2)
Ω2 is applied. The dashed symbols mark the conjugate
complex variables. The determined frequency response
matrix V(Ω)
⎡
⎤
f11 f12 ··· f1m
⎢
.
⎢
.
⎥
f21 f22
V(Ω) = ⎢
. ⎥
⎢ .
⎣
. ⎥
(3)
.
.. ⎦
fn1 ··· fnm
is filled with the frequency response functions fkl between
the excitation point l and the measuring point k.
The size of V(Ω) is therewith defined by m excitation
points times n measuring points.
The identified modal parameters and frequency response
functions fkl allow a convenient description of
the measured response characteristics for the later comparison
with numerical results.
D. Measurement results
In order to determine material parameters for a specific
numerical model, the mode-shapes determined with the
1

EMA, corresponding to the estimated eigenvalues, must
be identified. To distinguish the different mode-shapes,
a numbering system is established. The numbering comprises
three numbers and refers to a cylindrical coordinate
system. The first digit describes the number of maxima
of the mode in radial direction along the azimuthal
coordinate axis, see Fig. 3. The second digit represents
the number of zero crossings of the mode in radial
direction along the z-axis. The third digit is a counter to
differentiate modes with the same deformation properties
regarding the first two digits.
z
2
y
Fig. 3: Example for a mode (3,2,0)
1) Results of the stator core stack without teeth:
The sum of all measured as well as the sum of all
approximated frequency response functions fkl, the latter
calculated by the identified modal parameters, are depicted
in Fig. 4, with the different mode patterns indicated
by the introduced numbering system.
magnitude [ m N ]
10 -5
10 -6
10 -7
(2,1,0)
(2,0,0)
(2,2,3)
(2,2,2)
(2,2,1)
(2,2,0)
(3,2,2)
(3,2,1)
(3,2,0)
(3,1,0)
(3,0,0)
(3,1,1)
(3,2,3)
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
(4,0,0)
(4,0,1)
(3,3,0)
(3,4,0)
(3,3,1)
(3,4,1)
(3,3,2)
(3,4,2) (4,2,0)
(3,3,3) (3,4,3) (4,2,1)
(3,3,4)
(4,2,2)
(3,3,5)
(4,4,0)
(4,4,1)
3
Sum of all approximated FRF‘s, using modal parameters
Sum of all measured FRF‘s
frequency [Hz]
x
(5,0,0)
(5,4,0)
(5,4,1)
(5,4,2)
(5,4,3)
Fig. 4: Sum of all frequency response functions of the
stator core stack without teeth
As Fig. 4 shows, with the increase of the excitation
frequency the mean of the magnitude decreases. This
corresponds to mass dominated dynamical behavior. [20,
p.294]
- 146 - 15th IGTE Symposium 2012
The modes (2,0,0), (2,1,0), (3,0,0), (3,1,0), (4,0,0),
(4,0,1) and (5,0,0) have the most distinctive magnitude in
the investigated frequency domain. Some of these modes,
(2,0,0), (3,0,0), (4,0,0) and (5,0,0) could be calculated by
analytical two-dimensional methods, see [7], [5] or [10].
Other modes like (2,2,0) or (3,1,1) with non-uniform
radial displacements along the z-axis can only be treated
by three-dimensional approaches.
Table I summarizes the measurement results for the
stator core stack without teeth and lists modes with their
appropriate eigenfrequencies and damping ratios.
TABLE I: Modes, eigenfrequency and damping ratio of
stator core stack without teeth
mode num. eigenfrequency damping ratio
1/(2, 0, 0) 769,22 Hz 0,411387 %
2/(2, 1, 0) 795,03 Hz 0,959535 %
3/(2, 2, 0) 1282,95 Hz 1,197700 %
4/(2, 2, 1) 1353,39 Hz 1,086200 %
5/(3, 2, 0) 1728,14 Hz 1,442430 %
6/(3, 1, 0) 2092,07 Hz 0,218612 %
7/(3, 0, 0) 2109,96 Hz 0,093046 %
8/(3, 1, 1) 2192,46 Hz 1,122050 %
9/(3, 2, 0) 2278,98 Hz 0,691529 %
10/(3, 3, 0) 2509,73 Hz 0,897709 %
11/(3, 3, 1) 2569,24 Hz 1,057550 %
12/(3, 3, 2) 2631,76 Hz 1,108720 %
13/(4, 0, 0) 3860,04 Hz 0,043218 %
14/(4, 0, 1) 3871,37 Hz 0,050271 %
15/(3, 4, 0) 3947,18 Hz 0,496143 %
16/(3, 4, 1) 4037,01 Hz 1,110130 %
17/(4, 2, 0) 4414,03 Hz 0,606060 %
18/(4, 2, 1) 4482,12 Hz 0,590154 %
19/(4, 4, 0) 4756,58 Hz 0,348687 %
20/(4, 4, 1) 4773,74 Hz 0,360285 %
21/(5, 4, 0) 4940,08 Hz 0,930774 %
22/(5, 0, 0) 5927,25 Hz 0,127039 %
2) Results of the stator core stack with teeth: Similarly
the results given in Fig. 4, the sum of all measured and
the sum of all approximated frequency response function
fkl for the stator core stack with teeth, are plotted in Fig.
5.
magnitude [ m N ]
10 -5
10 -6
10 -7
1e-8
(2,0,1)
(2,0,0)
(2,1,0)
(2,1,1)
(2,2,1)
(2,2,0)
(3,0,0)
(3,1,0)
(0,0,1)
(0,0,2)
(3,2,0)
(4,0,0)
(4,3,2)
(4,3,1)
(4,3,0)
Sum of all approximated FRF‘s, using modal parameters
Sum of all measured FRF‘s
(4,0,1)
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
frequency [Hz]
(4,1,0)
(4,3,7)
(4,3,6)
(4,3,3)
(4,3,4)
(4,3,5)
(5,1,0)
(5,0,0) (6,1,0)
(6,0,0)
(0,1,0)
(5,2,0)
(5,4,0)
(5,2,1)
(5,4,1)
(5,4,2)
(5,4,3)
Fig. 5: Sum of all frequency response functions of the
stator core stack with teeth

Comparing the FRFs depicted in Fig. 4 and Fig. 5,
the number of distinct modes in Fig. 5 is greater. Due
to the higher mass, the corresponding eigenfrequencies
of the stator core stack with teeth are lower than those
of the stator core stack without teeth. Furthermore, the
frequency spacing of distinct modes, e.g. (3,0,0) and
(3,1,0) increases.
It is impossible to identify modes which have higher
eigenfrequencies than 5400 Hz, because the used measurement
grid is too coarse to detect the appropriate
eigenforms. In this frequency range, one would expect
to see modes comprising seven maxima or more with a
radial displacement along the azimuthal axis. This is not
possible to identify with only twelve measurement points
in the azimuthal direction.
Table II lists measurement results for the stator core
stack with teeth.
TABLE II: Modes, eigenfrequency and damping ratio of
stator core stack with teeth
mode num. eigenfrequency damping ratio
1/(2, 0, 0) 661.27 Hz 0.054903 %
2/(2, 0, 1) 667.84 Hz 0.055824 %
3/(2, 1, 0) 720.08 Hz 0.337501 %
4/(2, 1, 1) 727.85 Hz 0.332884 %
5/(2, 2, 0) 1365.85 Hz 1.005810 %
6/(2, 2, 1) 1416.91 Hz 0.939553 %
7/(3, 0, 0) 1767.43 Hz 0.047843 %
8/(3, 1, 0) 1851.83 Hz 0.205125 %
9/(3, 2, 0) 2313.17 Hz 0.640636 %
10/(0, 0, 1) 2372.92 Hz 0.654953 %
11/(0, 0, 2) 2376.10 Hz 0.608697 %
12/(4, 3, 0) 2755.43 Hz 0.382039 %
13/(4, 0, 0) 3107.52 Hz 0.097675 %
14/(4, 0, 1) 3116.28 Hz 0.152128 %
15/(4, 1, 0) 3190.75 Hz 0.180018 %
16/(4, 3, 3) 3288.28 Hz 0.743811 %
17/(0, 1, 0) 3955.81 Hz 0.306280 %
18/(5, 2, 0) 4074.92 Hz 0.188232 %
19/(5, 0, 0) 4423.63 Hz 0.046342 %
20/(5, 1, 0) 4484.19 Hz 0.173786 %
21/(5, 4, 0) 4655.18 Hz 0.661100 %
22/(5, 4, 1) 4745.87 Hz 0.237239 %
23/(6, 0, 0) 5314.37 Hz 0.031471 %
24/(6, 1, 0) 5356.55 Hz 0.039125 %
III. NUMERICAL MODAL ANALYSIS
As a means for the numerical simulation of the dynamical
behavior of the stator core stacks, the finite element
method is used. Therefore, an adequate simulation model
has to be set up.
For the numerical model based on the FEM - model,
some simplifications are made:
• the contacts between the laminations are neglected
• the grain orientation of the cold rolled silicon-ironalloy
is neglected
• a linear and homogeneous material model is assumed
• the FEM model is assumed to be linear
- 147 - 15th IGTE Symposium 2012
Using an adequate FEM-model and performing a numerical
modal analysis, the modal parameters (eigenfrequencies,
eigenforms, and damping coefficients) which
can be directly related to the modal parameters from the
measurements are obtained.
A. Material model
By considering the above limitations, a material model
with transversally isotropic elasticity is implemented.
z
Fig. 6: Coordinate system of the stator core stack
The used transversally isotropic elasticity of the material
model corresponds to the coordinate system, depicted
in Fig. 6. The flexibility matrix S for a material model
with a transversally isotropic elasticity is given by
⎡
⎢
S = ⎢
⎣
1
Ex
νxy
− Ex
1
Ex
νzy
− Ex
νxz
− Ez
νyz
− Ez
1
Ez
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2(1+νxy)
Ex
0
0
0
1
Gxz
0
0
0
1
Gxz
− νyx
Ex
− νzx
Ex
x
y
⎤
⎥ , (4)
⎥
⎦
where E is Young’s modulus, G is the shear modulus and
ν is Poisson’s ratio. The material model and therefore
the flexibility matrix S of the transversally isotropic
elasticity has rotationally symmetrical properties. Due to
the symmetric properties of S, the following conditions
are valid:
νxy = νyx
νxz
Ez
= νzx
Ex
B. Modal description
Based on the finite element method, the equations of
motion of a structural damped system can be formulated
as the linear system of differential equations
(5)
(6)
M ¨ û + D ˙ û + Kû = f . (7)
Thereby, M is the mass matrix, K is the stiffness matrix,
f is the excitation force and D is the proportional

damping matrix defined by the Rayleigh damping model
[12, p.950ff]
D = α M + β K . (8)
as a linear combination of the mass matrix M and the
stiffness matrix K with the scalar coefficients α and β.
Solving the eigenvalue problem of the undamped
system leads to the eigenvalues λi and eigenvectors
ri. The modal matrix R consisting of mass-normalized
eigenvectors defines the transformation
û = Rz (9)
of the displacement in the global state space û to the
displacement in the modal state space z.
Using (9) and multiplying (7) with the transformed
modal matrix R from the left, the proportional damped
equations of motion are transformed into a noninteracting
system of the dimension q
˜M ¨z + ˜ D ˙z + ˜ Kz = ˜ f . (10)
Since, the eigenvectors are mass-normalized, the transformation
of the mass matrix M into the modal state
space leads to the identity matrix I
˜M = R T MR= I . (11)
The transformed stiffness matrix becomes
˜K = R T KR= Λ = diag ω 2 i , (12)
where ωi is the undamped angular eigenfrequency of the
i−th mode. The damping matrix yields a diagonal matrix
expressed as
˜D = R T DR= diag [2ζiωi] , (13)
Thereby ζi denotes the modal damping ratio of the i−th
mode [20, p.63ff].
This approach yields simultaneously a modal reduction
of the equation system. This reduction has the advantage
that, without it, the computing effort increases rapidly.
The disadvantage is that the disregarded modes create
an error in the response characteristics. However, the
error resulting from the material model with transversally
isotropic elasticity is expected to be much higher than this
error and thus the latter is neglected.
Assuming a harmonic excitation, the excitation force
can be expressed as
f = ˇ f e jΩt
(14)
in the frequency domain. Here, ˇ f is the amplitude
of the excitation force and Ω describes the excitation
frequency. This leads to a harmonic ansatz for the modal
displacement:
z = ˇz e jΩt
(15)
where ˇz denotes the amplitude of the displacement in the
modal state space. Hence, (10) becomes
−Ω 2 I ˇz + jΩ ˜ Dˇz + ˜ Kˇz = ˇ f . (16)
- 148 - 15th IGTE Symposium 2012
Finally, the backward transformation in the global state
space leads to
ǔ = R
−Ω 2 I + jΩ ˜ D + ˜ −1 K
R T ˇ f = ˜V ˇ f . (17)
Therewith, the numerically estimated frequency response
matrix ˜V is
˜V (Ω) = R −Ω 2 I + jΩ ˜ D + ˜ −1 K R T
(18)
Using (11), (13) and (12), the entries of ˜V (Ω) can be
expressed by the frequency response functions ˜ fkl
˜fkl(Ω) =
q
i=1
ir ∗ k ir ∗ l
−Ω 2 + ω 2 i + jΩ 2ζi ωi
, (19)
which can be related directly to the corresponding frequency
response functions fkl(Ω) estimated by the measurement.
Thereby ir∗ k denotes the k-th entry of the i-th
mass-normalized eigenvector ri.
IV. INFLUENCE OF MATERIAL PARAMETERS ON
TRANSMISSION BEHAVIOR
In order to estimate the influence of each material
parameter, simulations as explained in section III-B are
carried out for the stator model without teeth. The
material parameters, except for the density, are varied
in a distinct range and their influence on the structural
behavior is investigated.
The density is determined by measurements. With a
mass of 149.8kg and a volume from 0.019905 m 3 , a
density of 7525 kg/m 3 results. This value is used for all
calculations of this study.
The material parameters of interest for the influence
on the dynamical behavior are the Young’s moduli Ex
and Ez, the shear module Gxz and the Poisson ratios
νxy and νxz. The initial set of material parameters is
shown in Table III. Based on this set, each parameter is
TABLE III: Initial dataset of material parameters for a
variation of each parameter
material parameters values
Ex
190 · 109 N/m 2
Ez
25 · 109 N/m 2
Gxz
10 · 109 N/m 2
νxy
0.3
νxz
0.3
ϱ 7525 kg/m 3
varied to a lower and a higher value. The influence of
each parameter on the frequency response functions is
shown below. The results of these investigations are the
basis on which the adjustment of the simulated response
characteristics on the measured response characteristics
is done.

A. Variation of the Poisson ration ν
In order to analyse the influence of the Poisson ratios
on the response characteristics, the values are varied from
0.2 to 0.4 . The Poisson ratios νxy and νxz are set equal
for the calculations. Fig. 7 shows the sum of all frequency
response functions for the varied Poisson ratios.
magnitude [ m N ]
-4
10 v = 0.2
v = 0.3
v = 0.4
10 -5
10 -6
10 -7
500 1000 1500 2000 2500 3000
frequency [Hz]
Fig. 7: Sum of all frequency response functions of the
stator core stack without teeth for varied Poisson ratio
This comparison evidences that the influence of the
Poisson ratios on the response characteristics is insignificantly
small. Therefore, for νxy and νxz a value of 0.3
is chosen.
B. Variation of the elastic modulus Ex
In Fig. 8, the sum of the calculated frequency response
functions for the variation Ex is depicted. It can be ob-
magnitude [ m N ]
10 -4
10 -5
10 -6
10 -7
1
2
4
5
11 11a
500 1000 1500
frequency [Hz]
2000 2500 3000
7
8
9 2
E x = 170· 10 N/m
9 2
E x = 190· 10 N/m
9 2
E x = 210· 10 N/m
Fig. 8: Sum of all frequency response functions of
the stator core stack without teeth for varied Young’s
modulus Ex
served that some eigenfrequencies, for example for mode
4, 5 or 11, are not influenced by the Young’s modulus Ex.
Other modes, such as 7, 8, or 11a, are heavily affected by
it. A small variation of the Young’s modulus Ex yields
a large shift of distinct eigenfrequencies. If the value of
Ex decreases, the eigenfrequencies corresponding to the
modes 7, 8, or 11a are declining and vice versa. Also
- 149 - 15th IGTE Symposium 2012
the eigenfrequencies of the corresponding modes 1 and
2 depend on the Young’s modulus Ex but not as much
as the previous ones.
C. Variation of the elastic modulus Ez
The variation of the material parameter Ez, depicted
in Fig. 9, leads to a different dynamical behavior than
the variation of Ex. The eigenfrequencies corresponding
to the modes 1, 2 and 7 are not influenced by a variation
magnitude [ m N ]
10 -4
10 -5
10 -6
10 -7
2
1
3
4
500 1000 1500
frequency [Hz]
2000 2500 3000
7
9 2
E z = 20· 10 N/m
9 2
E z = 25· 10 N/m
9 2
E = 30· 10 N/m
Fig. 9: Sum of all frequency response functions of
the stator core stack without teeth for varied Young’s
modulus Ez
of the Young’s modulus Ez. When increasing the value
of Ez, all other eigenfrequencies are shifted upwards
in the considered frequency range and when decreasing
Ez, these eigenfrequencies are shifted downwards. It is
interesting to note, that the eigenfrequencies which are
independent of the Young’s modulus Ez (mode 1, 2, 7),
depend on the Young’s modulus Ex.
D. Variation of the shear modulus Gxz
Finally, theresults for the variation of the shear modulus
Gxz is shown in Fig. 10. It can be seen that the
magnitude [ m N ]
10 -4
10 -5
10 -6
10 -7
1
2
500 1000 1500
frequency [Hz]
2000 2500 3000
6
9
z
10
9 2
G xz = 8· 10 N/m
9 2
G xz = 10· 10 N/m
9 2
G = 12· 10 N/m
Fig. 10: Sum of all frequency response functions of the
stator core stack without teeth by varied shear modulus
Gxz
5
xz
11
12

eigenfrequencies corresponding to the modes 1 and 2 are
independent of the variation of Gxz. Other modes, like
5, 6, 9, 11 or 12, are heavily affected by the variation
of the shear modulus. These eigenfrequencies are shifted
downwards by decreasing and upwards by increasing the
value of Gxz.
Summing up, the influence of the material properties
on the eigenfrequencies is strongly influenced by the
corresponding eigenmode occurring at these frequencies.
Depending on the characteristics (radial, azimuthal or axial
bending) of the mode, Ex, Ez and Gxz have different
influences. This background is important to know for the
latter adjustment of the response characteristics.
V. ADJUSTMENT OF MATERIAL PARAMETERS
The dynamical behavior of the numerical model
strongly depends on the chosen material parameters.
An iterative process optimizes the eigenfrequencies and
eigenvectors based on the known influence of the material
parameters as discussed in section IV. Material
parameters can be found by an adequate adjustment of
the measured and simulated response characteristics.
A. Stator core stack without teeth
First, a dataset of material parameters is chosen which
describes the behavior of isotropic elasticity. Thereafter,
the material parameters of the transversally isotropic
elasticity are identified for the stator core stack without
teeth.
1) Comparison of measured and calculated frequency
response functions by using the isotropic dataset: The
results from the numerical simulation using isotropic
material parameters (Table IV) are compared with the
results from the experimental investigation, see Fig. 11.
It can be seen that there is no analogy between the simu-
TABLE IV: Initial dataset of material parameters of
isotropic elasticity
material parameters values
Ex
210 · 109 N/m 2
Ez
210 · 109 N/m 2
Gxz
Ex
2(1+ν)
ν 0.3
ϱ 7525 kg/m 3
lated and measured response characteristics. Only the first
eigenfrequency of the measured and simulated FRFs are
similar. Furthermore, in the investigated frequency range
less eigenfrequencies arise for the simulation model. This
comparison shows that a material model with isotropic
elasticity is clearly unsuitable.
2) Adjustment of the measured and simulated frequency
response function by using transversal isotropic
elasticity: In a next step, a material model with transversally
isotropic elasticity is used and the needed material
parameters are adjusted. Table V lists these material
parameters used as an initial dataset for the simulation.
- 150 - 15th IGTE Symposium 2012
magnitude [ m N ]
10 -5
10 -6
10 -7
10 -8
Sum of simulated FRF‘s in radial direction
Sum of measured FRF‘s in radial direction
500 1000 1500 2000 2500 3000
frequency [Hz]
Fig. 11: Comparison of the sum of all measured and
calculated FRFs with isotropic material model of the
stator core stack without teeth
Thereby, the Young’s modulus Ez and the shear modulus
Gxz are chosen considerably lower with 40·10 9 N/m 2 and
15 · 10 9 N/m 2 . This shifts the eigenfrequencies downward
in the considered frequency range as explained in section
IV.
Looking at Fig. 12, it can be seen that in the regarded
frequency range the measured and simulated eigenfrequencies
of the modes (2,0,0) and (3,0,0) are close
together.
TABLE V: Dataset of material parameters of transversal
isotropic elasticity
magnitude [ m N ]
10 -5
10 -6
10 -7
material parameters values
Ex
210 · 109 N/m 2
Ez
40 · 109 N/m 2
Gxz
15 · 109 N/m 2
νxy
0.3
νxz
0.3
ϱ 7525 kg/m 3
mode (2,0,0)
Sum of simulated FRF‘s in radial direction
Sum of measured FRF‘s in radial direction
mode (3,0,0)
10
500 1000 1500 2000 2500 3000
-8
frequency [Hz]
Fig. 12: Comparison of the sum of all measured and calculated
FRFs with transversally isotropic material model
of the stator core stack without teeth
Now the modes are adjusted by decreasing the elastic

modulus Ex. The value is optimized manually step-bystep
until an adequate match is attained. For the Young’s
modulus Ex, a value could be found which aligns the
measured and simulated modes (2,0,0) and (3,0,0). The
next step in the optimization is to adjust the mode
(3,1,0). Therefore the shear modulus Gxz is reduced.
Finally, with the Young’s modulus Ez, the other modes
between (2,0,0) and (3,0,0) can be influenced. A stepwise
reduction of this value yields an adequate correlation of
the other modes.
The resulting material parameters of this optimization
of the stator core stack without teeth are listed in Table
VI.
TABLE VI: Resulting dataset of material parameters
for the stator core stack without teeth of transversally
isotropic elasticity
material parameters values
Ex
191, 8 · 109 N/m 2
Ez
24, 7 · 109 N/m 2
Gxz
11 · 109 N/m 2
νxy
0, 3
νxz
0, 3
ϱ 7525 kg/m 3
Fig. 13 depicts the response characteristics of the
simulation results, using optimized material parameters
and the measurement results for the stator core stack
without teeth. A good approximation for the simulated
magnitude [ m N ]
10 -5
10 -6
10 -7
mode (2,0,0)
Mode mode (2,0,0) (2,1,0)
Mode mode (2,2,0)
mode (2,2,1)
mode (2,2,3)
mode (3,1,0)
mode (3,2,1)
Sum of simulated FRF‘s in radial direction
Sum of measured FRF‘s in radial direction
mode (3,0,0)
mode (3,2,3)
mode (3,3,0)
mode (3,3,4)
10
500 1000 1500 2000 2500 3000
-8
frequency [Hz]
Fig. 13: Comparison of the sum of all measured and
calculated FRFs of the stator core stack without teeth,
by using a material model with transversally isotropic
elasticity and optimized parameters
response characteristics can be observed. Hence, the
identified material parameters for a linear and homogeneous
material model can represent a similar dynamical
behavior as the real structure of the stator core without
teeth in a frequency range from 0Hzto 3000 Hz.
The coincident eigenfrequencies and their corresponding
modes are listed in Table VII.
- 151 - 15th IGTE Symposium 2012
TABLE VII: Coincident measured and simulated eigenfrequencies
and modes of the stator core stack without
teeth, resulting from adjustment
modes measured eigenfreq. simulated eigenfreq.
(2, 0, 0) 769.22 Hz 749.11 Hz
(2, 1, 0) 795.03 Hz 757.16 Hz
(2, 2, 0) 1280.95 Hz 1278.38 Hz
(2, 2, 1) 1353.39 Hz 1362.88 Hz
(2, 2, 3) 1606.08 Hz 1607.62 Hz
(3, 2, 1) 1881.54 Hz 1869.87 Hz
(3, 1, 0) 2092.07 Hz 2095.87 Hz
(3, 0, 0) 2109.96 Hz 2097.67 Hz
(3, 2, 3) 2278.98 Hz 2313.00 Hz
(3, 3, 0) 2509.72 Hz 2512.20 Hz
(3, 3, 4) 2826.32 Hz 2830.84 Hz
B. Stator core stack with teeth
As an initial dataset for the stator core stack with teeth,
the resulting material parameters of the stator core stack
without teeth have been chosen.
For the investigation of the stator core stack with
teeth the density has to be determined. With a weight of
196.4kgand a volume of 2.61335·10−2 m3 the density of
the stator core stack with teeth results in 7515.3 kg/m 3 .
This is 0.14 % less than the density of the stator core
stack without teeth. Hence, all simulations for the stator
core stack with teeth use the newly found density.
1) Comparison of measured and calculated frequency
response functions by using the initial dataset: The identified
material parameters are validated by a comparison
of the measured and the calculated response characteristics
of the stator core stack with teeth, see Fig. 14. The
comparison shows that the match of the measured data
with the simulation results using the material parameters
in Table VI for the stator core stack without teeth is not
satisfactory. Only the modes (2,0,0) and (3,0,0) have a
smaller deviation than the other modes. Therefore, the
material parameters of the stator core stack with teeth
are determined again.
magnitude [ m N ]
10 -5
mode (2,0,0)
10 -6
10 -7
10 -8
mode (2,1,0)
mode (3,0,0) mode (3,1,0)
mode (2,2,x)
Sum of simulated FRF‘s in radial direction
Sum of measured FRF‘s in radial direction
mode (3,2,x)
mode (3,3,x)
500 1000 1500
frequency [Hz]
2000 2500 3000
Fig. 14: Validation of the identified material parameters
by comparing results of a the simulation and measurement
of the stator core stack with teeth

2) Adjustment of measured and simulated frequency
response function by using transversal isotropic elasticity:
The procedure of the adjustment of the material
parameters is the same as for the stator core stack without
teeth. For the initial dataset, the material parameters
identified in section V-A are used. The optimization
yields a set of material parameters listed in Table VIII.
TABLE VIII: Dataset of material parameters of transversally
isotropic elasticity with optimized Ez
material parameters values
Ex
199.8 · 109 N/m 2
Ez
20.1 · 109 N/m 2
Gxz
9.9 · 109 N/m 2
νxy
0.3
νxz
0.3
ϱ 7515.3 kg/m 3
Fig. 15 shows that eigenfrequencies exist in the simulation
which do not occur in the measurement. Furthermore
the linear and homogeneous model does not adjust
the modes (2,1,0), (3,1,0) and (4,1,0) to the measured
eigenfrequencies of the equivalent simulated modes. The
measured distance in the frequency range of about 100 Hz
between the modes (2,0,0) and (2,1,0) or (3,0,0) and
(3,1,0) etc. could not be represented.
magnitude [ m N ]
10 -5
10 -6
10 -7
10 -8
mode (2,0,0)
mode (2,1,0)
mode (3,0,0)
mode (2,3,0)
mode (2,2,x)
mode (2,2,0)
mode (3,1,0)
Sum of simulated FRF‘s in radial direction
Sum of measured FRF‘s in radial direction
mode (3,2,x)
mode (3,3,0)
mode (4,3,2)
mode (4,3,1)
mode (4,0,0)
500 1000 1500 2000
frequency [Hz]
2500 3000
Fig. 15: Comparison of measured and simulated frequency
response functions of the stator core stack with
teeth in a frequency domain from 500 Hz to 3300 Hz
Table IX lists modes and their corresponding measured
and simulated eigenfrequencies which could be approximately
adjusted in a frequency range from 500 Hz to
3300 Hz.
VI. SUMMARY AND CONCLUSION
For the investigation of electrical machines, the dynamical
behavior of the stator core is of high interest. The
mechanical characterization is difficult since the structure
of the stator core is inhomogeneous. In this paper, an
approach has been presented which yields a linear and
homogeneous description of a stator core stack.
For the investigation of the dynamical behavior, two
stator core stacks, one without stator teeth and the other
- 152 - 15th IGTE Symposium 2012
TABLE IX: Coincident measured and simulated eigenfrequencies
and modes of the stator core stack with teeth,
resulting from adjustment
modes measured eigenfreq. simulated eigenfreq.
(2, 0, 0) 661.27 Hz 662.79 Hz
(2, 1, 0) 720.08 Hz 611.87 Hz
(2, 2, 0) 1365.85 Hz 1358.87 Hz
(3, 0, 0) 1767.43 Hz 1777.50 Hz
(3, 1, 0) 1854.83 Hz 1782.64 Hz
(4, 3, 1) 2819.37 Hz 2955.25 Hz
(4, 3, 2) 2962.95 Hz 2955.25 Hz
(4, 0, 0) 3107.52 Hz 3134.49 Hz
with stator teeth, have been chosen. The experimental
modal analysis have been carried out on both stator core
stacks. The measurement results have been used for the
adjustment of the simulation data.
The numerical modal analysis has been applied in
conjunction with the finite element method. For that,
adequate models had to be chosen. The inhomogeneous
structure has been represented by a linear and homogeneous
FEM model and the lamination of the stator cores
has been considered by a transversally isotropic material
model.
A study of the influence of the transversally isotropic
material parameters has been carried out. Thereby, each
material parameter has been varied and the resulting
FRFs have been compared. It could be identified,
which material parameter influences which mode. This
knowledge is the basis of the adjustment of the simulated
response characteristics of the stator cores.
Before adjusting the response characteristics, a comparison
of the simulated results using a material model
of isotropic elasticity with the measurement results has
been carried out for the stator core stack without teeth.
It could be shown that a material model with isotropic
elasticity is not appropriate.
The stepwise adjustment of the simulated FRFs to
the measured have shown a good match of the response
characteristic of the stator core stack without teeth up to
a frequency of 3kHz. However, some measured modes
could not be identified with the used linear and homogeneous
numerical model. The approximated material
parameters have then been validated by a comparison of
the measured and simulated dynamical behavior of the
stator core stack with teeth. This validation shows that
the response characteristic of the measured and simulated
results did not coincide except for some simulated
eigenfrequencies.
In order to improve the numerical model of the stator
core stack with teeth, a further optimization of the material
parameters has been carried out. The match of the
measured and simulated response characteristics is not
as good as for the stator core stack without teeth. In the
investigated frequency range of the measured data some
eigenfrequencies and eigenmodes could not be identified
in the simulation.
However, a working method has been introduced,

which describes the three dimensional dynamical behaviour
of a stator core stack by using a linear and
homogeneous numerical model.
REFERENCES
[1] B. Weilharter, O. Biro, H. Lang, G. Ofner, and S. Rainer, “Validation
of a comprehensive analytic noise computation method for
induction machines,” Industrial Electronics, IEEE Transactions
on, vol. 59, no. 5, pp. 2248 –2257, may 2012.
[2] J. Li, X. Song, D. Choi, and Y. Cho, “Research on reduction
of vibration and acoustic noise in switched reluctance motors,”
in Advanced Electromechanical Motion Systems Electric Drives
Joint Symposium, 2009. ELECTROMOTION 2009. 8th International
Symposium on, july 2009, pp. 1 –6.
[3] S. Rainer, O. Biro, B. Weilharter, and A. Stermecki, “Weak coupling
between electromagnetic and structural models for electrical
machines,” Magnetics, IEEE Transactions on, vol. 46, no. 8, pp.
2807 –2810, aug. 2010.
[4] B. Weilharter, O. Bi? andro? and, S. Rainer, and A. Stermecki,
“Computation of rotating force waves in skewed induction machines
using multi-slice models,” Magnetics, IEEE Transactions
on, vol. 47, no. 5, pp. 1046 –1049, may 2011.
[5] H. Jordan, Geräuscharme Elektromotoren. Essen: Verlag W.
Girardet, 1951.
[6] R. N. Arnold and G. B. Warburton, “Flexural vibrations of the
walls of thin cylindrical shells having freely supported ends,” Proceedings
of the Royal Society of London. Series A, Mathematical
and Physical Sciences, vol. 197, no. 1049, pp. 238–256, Jun. 1949.
[7] H. Frohne, “Über die primären bestimmungsgrößen er
lautstärke bei asynchronmotoren,” Ph.D. dissertation, Technische
Hochschule Hannover, 1959.
[8] E. C. Naumann and J. L. Sewall, “An experimental and analytical
vibration study of thin cylindrical shells with and without longitudinal
stiffeners,” NASA Langley Research Center, Tech. Rep.
NASA-TN-D-4705, Sep 1968.
[9] Y. Stavsky and R. Loewy, “On vibrations of heterogenous orthotropic
cylindrical shells,” Journal of Sound and Vibration,
vol.18,no.3,pp.i–i,1971.
[10] P. L. Tímár, A. Fazecas, J. Kiss, A. Miklós, and S. J. Yang,
Noise and Vibration of Electrical Machines. New York: Elsevier
Science Publishing Company, Inc., 1989.
[11] C. Wang and J. C. S. Lai, “Prediction of natural frequencies of
finite length circular cylindrical shells,” Applied Acoustics, vol. 59,
no. 4, pp. 385 – 400, 2000.
[12] K.-J. Bathe, Finite-Elemente-Methoden, 2nd ed. Berlin Heidelberg:
Springer, 2002, aus dem Englichen übersetzt von Peter
Zimmermann.
[13] R. Gasch and K. Knothe, Strukturdynamik: Band 1: Diskrete
Systeme. Springer, 1987.
[14] W. Yixuan and W. Ying, “Research on dynamic characteristics
of stator core of large turbo-generator,” in Power and Energy
Engineering Conference (APPEEC), 2010 Asia-Pacific, march
2010, pp. 1 –4.
[15] H. Wang and K. Williams, “Effects of laminations on the vibrational
behaviour of electrical machine stators,” Journal of Sound
and Vibration, vol. 202, no. 5, pp. 703–715, May 1997.
[16] M. van der Giet and K. Hameyer, “Identification of homogenized
equivalent materials for the modal analysis of composite structures
in electrical machines,” in the 9th International Conference on
Vibrations in Rotating Machinery, VIRM 2008, Exceter, UK, 2008,
pp. 437–448.
[17] J. Roivainen, “Unit-wave response-based modeling of electromechanical
noise and vibration of electromechanical machines,”
Ph.D. dissertation, Helsinki University of Technology, Helsinki,
2009.
[18] S. Verma and A. Balan, “Experimental investigations on the
stators of electrical machines in relation to vibration and noise
problems,” Electric Power Applications, IEE Proceedings -, vol.
145, no. 5, pp. 455 –461, sep. 1998.
[19] B. Peeters, H. Van der Auweraer, P. Guillaume, and J. Leuridan,
“The polymax frequency-domain method: a new standard for
modal parameter estimation?” Shock and Vibration, vol. 11, no. 3,
pp. 395–409, Jan. 2004.
- 153 - 15th IGTE Symposium 2012
[20] D. J. Ewins, Modal Testing: Theory, Practice and Applications,
2nd ed. Letchworth, Hertfordshire, UK: Research Studies Press,
2000.

- 154 - 15th IGTE Symposium 2012
Proper Location of the Regulating Coil in Transformers
from Short-Circuit Forces Point of View
*, O. Sonmez, * B. Duzgun, * G. Komurgoz
* Istanbul Technical University Electrical and Electronics Faculty, 80626 Istanbul, Turkey
Abstract—A transformer has complicated network of internal forces acting on and stressing the conductors, support and
insulation structures. These forces are fundamental to the interaction of current-carrying conductors within magnetic fields
involving an alternating-current source. Location of the regulating coil in transformer determines electrodynamic forces
effect on the operational behavior of the transformer. This paper presents design principles of the regulating coil in
transformers and shows the electrodynamics forces and their deformation results by using finite element method.
Index Terms—Electrodynamic Forces, Deformation Analysis, FEA, Regulating Coil.
I. INTRODUCTION
The transformer is a very critical and costly important
component in power generation and transmission systems
as regarding reliable and performance. The capacity of
transformers is increasing with the rapid development. As
the voltage level is higher, the time needed to design a
transformer is of great importance. One of the important
problems in the design of transformers is radial and axial
forces, being proportional to the square of the short
circuit current. By the interaction of leakage field and
the short circuit current, which makes the windings be
published or pulled, the huge short circuit force is
generated in the windings (large power). The leakage
flux not only causes the additional losses and forces, but
also creates heating to the internal components. Short
circuit current is 8 to 10 times the rated current in larger
transformers and 20 to 25 times in smaller units. Forces
arising during short-circuit may be as high as ten
thousand to million N. By the effect of so large forces
and thermal expansion of wires, the insulation of
transformer windings can be distorted, even collapsed,
short circuit error occurs or damage to the clamping
structures. Furthermore, the location of the tapings has
the predominant effect on the axial forces since it
controls the residual ampere turn. Failure of transformers
due to short circuits is major concern for power utilities
and manufactures. These hazards can be avoided by
proper design of windings structure against thermal and
mechanical strains to prevent permanent deformations
and movement of windings if forces can be calculated
correctly.
In the past, many technical papers have been published
which give equations for calculation the electromagnetic
forces acting on the windings in transformers [1-9].
Electromagnetic force computations methods have been
proposed in the literature mainly based on static and
transient formulations [10]. Classical methods can be
used to compute the short circuit forces in windings [11].
In these methods, it is used simplified configurations
with some assumptions. Furthermore these methods are
simple, fast and easy, but not accurate and not suitable
for predicting the performance of special types of
Sönmez Transformer Company, 41410 Kocaeli, Turkey
e-mail: komurgoz@itu.edu.tr
transformers, especially the axial length of windings is
not equal [12]. It is, however, obvious that by using
modern computerized methods, sophisticated methods, it
is possible to calculate forces acting on the elements of
winding, the effect of any arrangements of parts and
asymmetries. If magnetic field is calculated accurately, it
is possible to define electromagnetic forces in the
detailed transformer model by using numerical methods,
Finite Element Methods (FEM), Finite Difference
Methods (FDM) and Boundary elements (BEM) etc. In
recent years, a significant development of FEM software
has enabled the force calculation to be accomplished
easily in where the winding and tapping arrangement is
complex.
This paper concentrates on the use of FEM to models.
This method provides a comprehensive view of the
overall transformer mechanic and electromagnetic
behavior under normal and disturbance conditions. The
effect of tap winding configurations is also analyzed. The
results obtained from FEM of transformers using
MAXWELL® and ANSYS® are validated by the
mathematical models.
II. FORCES ACTING ON THE TRANSFORMER
When the electromagnetic force becomes greater than the
strength of the windings, the windings will fail. The types
of failure,Electromagnetic forces, acting on transformer
can be classified as “radial forces” which develop in the x
direction and “axial forces” develop in the y direction.
For the calculation of these forces, both analytical and
numerical methods are presented such as residual
ampere-turn method, Robin’s solution, Smythe’s
solution, calculation using Fourier series, two
dimensional method of images, FEM, image method with
discrete conductors etc. [13].
Axial forces creates slipping or breakdown of windings as
a whole standing-up of part of windings, tilting and
deformation of coils. Radial forces creats buckling
phenomena of inner windings, excessive elongation of
outer windings.
A. Axial Forces
One of the elementary and simplest methods, residual

ampere-turn method, gives closer approximations and
reliable results for the calculation of axial forces.
Concentric windings are separated into two groups and
each group has balanced ampere-turns. The radial
ampere-turns produce radial flux which causes axial
force in the windings as it seen in Figure 1. This
assumption allows calculation of the axial forces.
Figure 1: Axial and radial forces in concentric axially nonsymmetrical
windings [13].
The algebraic sum of the ampere-turns of low voltage
and high voltage windings at any point and at end of the
windings gives the radial ampere-turns at that point in the
winding. A curve is plotted for every points called
residual or unbalanced ampere-turn diagram which the
method derives its name [12]. It is clear that windings
without axial displacement and windings have the same
length have no residual ampere-turns or forces between
windings. However, there are some internal compressive
forces and forces on the end coils, although there is no
axial thrust between windings.
Figure 2 gives the methodology for the determining
distribution of radial ampere-turns. ‘a’ is the length
tapped out at the end of the outer windings. Summation
of I and II shown in Figure 2(b) are both balanced
ampere-turn groups. If these groups are superimposed,
they produce the given ampere-turn arrangement. The
triangle as shown in Figure 2(c) presents the diagram of
the radial-ampere turns. This diagram plotted against
distance along the winding. a(NImax) is the maximum
value, where (NImax) represents the ampere-turns of either
the low voltage or high voltage winding.
Figure 2: Determination of residual ampere-turns [12].
- 155 - 15th IGTE Symposium 2012
Tapings location on the winding has a great effect on the
axial forces since it controls the residual ampere-turn
diagram.
B. Radial Forces
The radial forces develop due to interaction of coil
currents with the axial component of its own magnetic
flux. In a transformer with concentric windings, radial
forces considered insignificant because, the radial
strength of the winding is high. Most problems occur
because of axial forces and axial movement results more
damage to the winding and insulation than radial
movements.
The inner coil is subjected a pressure tends to collapse
to the core. At the same time, the outer coil is under a
pressure to extend the diameter of the coil which
produces a stress as shown in Figure 1. Preferable choice
in a transformer is circular coils, because they are the
strongest shape to withstand the radial pressure
mechanically [14].
III. CALCULATION OF ELECTROMAGNETIC FORCES
A. Short-Circuit Current
Short-circuit currents on the windings have a
significant effect on calculation of electromagnetic
forces. Generally, the short-circuit current is calculated
for different situations by considering [15];
Tapping arrangement
Fault position
Short-circuit power combination (network and
transformer)
Short-circuit type (e.g. three phase symmetrical)
To see the effects of the short-circuit current on power
transformers, the simplest fault scenario, three phase
short-circuit scenario is investigated. Symmetrical shortcircuit
current can be calculated according IEC 60076-5
as [16];
I
U
Z Z
3 t s
9 And the amplitude
is;
Imax of the first peak of the current
I Ik 2 10
max
The factor k is the initial offset of the current and
2 stands for the peak to r.m.s. value of sinusoidal wave.
This k 2 factor depends on the X/R ratio and the
values of k are shown in standards IEC 60076-5 [16].
This current is based on the following expression for
the peak factor;
R/ X
2
k 2 1
e
sin
2 11

Y1 [kA]
25.00
12.50
0.00
-12.50
Curve Inf o
InputCurrent(Winding_LV_A)
Setup1 : Transient
InputCurrent(Winding_LV_B)
Setup1 : Transient
InputCurrent(Winding_LV_C)
Setup1 : Transient
Name X Y
Phase C_sc 131.5000 21.6972
Phase A_sc 118.5000 21.6972
Phase B_sc 105.0000 21.7270
Input Current LV Model2D_coils ANSOFT
Phase B_sc
Phase A_sc
Phase C_sc
-25.00
75.00 87.50 100.00
Time [ms]
112.50 125.00 135.00
Figure 3: Input currents of low voltage windings.
The given short-circuit has two components as steady
state and exponentially unidirectional component. In
Figure 3, applied steady-state and short-circuit currents
on the windings of the power transformer in Maxwell
software is shown. The exponentially unidirectional
component is ignored to make calculations simpler.
B. Electromagnetic Forces
Transient analysis allows calculating electromagnetic
forces for every time step by calculating the leakage flux
and full field in winding region. Fully coupled dynamic
physics solution is;
A
AJs V Hc vA t
The differential equation and the boundary conditions
of transient axial symmetric electromagnetic field can be
expressed in the cylindrical coordinate as;
- 156 - 15th IGTE Symposium 2012
12 rA rA rA
: v' Z Z
v' r r
Js
'
t
13
S1: rA rA0
14 rA
S2: v' Ht
n
15 For 2D analysis, the radial and axial components of the
magnetic flux density can be expressed as;
A
Br
z
B
0
16 1 rA
Bz
r r
17 When the magnetic flux density is decomposed into its
radial and axial components;
F J ˆ B rˆB zˆ d F rˆF zˆ
18
r z r z
In brief, the force on the power transformer is
expressed by the Lorentz force as
dF idlB And the radial force of unit length
F B I dl
x y
max
The axial force of unit length
F B I dl
y x
max
19 20 21 IV. RESULTS &DISCUSSIONS
A. Model
Electrical machines require an accurate mathematical
model for system simulation and performance evaluation.
Detailed knowledge of the flux distribution of a
transformer plays a very important role in a safe
estimation of the forces of the transformer. Complex
computer programs are required to obtain a reasonable
representation of the field in different parts of the
windings. Using the above models for determinate forces,
a numerical application (FEM) has been implemented for
a 25 MVA power transformer. 3-D model of the general
structure is shown in Figure 4. To reduce computing time
and avoid excessive use of ram, the insulating materials
and supporting structure are neglected, besides analyses
were done in 2-D structure.
Figure 4: 3-D model of analyzed power transformer.
The characteristics of the studied transformer are
presented in Table I and geometry details of the analyzed
transformer are shown in Figure 5.
TABLE I
TRANSFORMER DATA
Rated Power 25 [MVA]
Rated Frequency 50 [Hz]
Rated Voltages 120 / 11 [kV]
Rated Currents 120 / 1310 [A]
Turns Ratio 1000 / 159
Connection Yd11
Tap setting ± 15 %
Transformer short circuit voltage (%) 9
Figure 5: Geometry details of analyzed transformer.

Figure 6: 2-D model of analyzed transformer tapped at upper side.
B. Electromagnetic Results
Transformers require an accurate mathematical model
for system simulation and performance evaluation. In this
study, magnetic analysis of the designed machines has
been investigated using Maxwell 2D program and total
deformations have been investigated using ANSYS®
program (Figure 6). The simulations were completed
using the following steps;
1) Geometric model creation,
2) The appointment of the materials that make up
the structure of the machine,
3) Boundary conditions and mesh process,
4) The appointment of currents in windings,
5) Analyze,
6) Examination of the results.
In Figure 7 and 8 leakage flux distributions are shown
for +15% tapping position and -15% tapping position. As
the leakage flux increases, electromagnetic forces are
occurring rapidly.
Figure 7: Leakage flux distribution at +15% tapping position of HV
windings.
Figure 8: Leakage flux distribution at -15% tapping position of HV
windings.
- 157 - 15th IGTE Symposium 2012
The graphs of distribution of radial magnetic flux
density along the transformer window are shown in
Figure 9 and 10 for +15% tapping, -15% tapping at upper
part of HV windings, respectively.
Mag_B [tesla]
1.50
1.25
1.00
0.75
0.50
0.25
Axial Flux Density Distribution Model2D_coils ANSOFT
0.00
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75
Distance [meter]
Curve Inf o
Mag_B
Setup1 : Transient
Time='115000000ns'
Figure 9: Axial flux distribution at +15% tapping position of HV
windings.
Figure 9 shows axial flux distribution with respect to
height of the winding for at +15% tapping position of HV
windings. To determine the axial forces, it is necessary to
find the radial flux produced by the radial ampere-turns.
As seen from figure, axial flux density is approximately
constant along the winding due to symmetrical windings
(with fully balanced ampere-turns)
Mag_B [tesla]
3.50
3.00
2.50
2.00
1.50
1.00
0.50
Axial Flux Density Distribution Model2D_coils ANSOFT
0.00
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75
Distance [meter]
Figure 10: Axial flux distribution at -15% tapping position of HV
windings.
Curve Inf o
Mag_B
Setup1 : Transient
Time='115000000ns'
If there is an asymmetry in the winding heights due to the
tap position or for some other reasons such as failure,
flux distribution changes as shown in Figure 10. Flux
density distribution makes maximum in one place along
the height of the winding.
The electromagnetic forces in the winding of the
power transformer are calculated with the leakage flux
and transient currents. The radial and axial forces of each
conductor coil in the HV windings are given in Figure
11-14. Figure 11 and 12 shows radial and axial forces at
+15% tapping position of HV windings.
Radial Forces
x 105
3.5
3
2.5
2
1.5
1
0.5
0
Radial Forces on the HV Coils for +15% tapping at 118.5 ms
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Coil Numbers
Figure 11: Radial Forces at +15% tapping position of HV windings.

Axial Forces
x 104
8
6
4
2
0
-2
-4
-6
-8
Axial Forces on the HV Coils for +15% tapping at 118.5 ms
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Coil Numbers
Figure 12: Axial Forces at +15% tapping position of HV windings.
Axial and radial forces in windings when the windings
are axially non-symmetrical are calculated as given in
Figure 13 and 14.
Due to the symmetry of winding and regular
distribution of flux, forces values are smaller than
asymetrical winding arrangement. If there is an
asymmetry in the winding heights due to the tap position
(or for some other reasons), the ampere-turn unbalance
increases and gives rise to forces, and result of this,
tending to break the winding.
Radial Forces
x 105
6
5
4
3
2
1
0
Radial Forces on the HV Coils for -15% tapping at 118.5 ms
1 2 3 4 5 6 7 8 9 10 11 12
Coil Numbers
Figure 13: Radial Forces at %-+15 tapping position of HV windings.
Axial Forces
x 105
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
Axial Forces on the HV Coils for -15% tapping at 118.5 ms
1 2 3 4 5 6 7 8 9 10 11 12
Coil Numbers
Figure 14: Axial Forces at -15% tapping position of HV windings.
The total body force density and total deformation are
determined by using ANSYS program and shown in
Figure 15-18. Figure 15 and 17 shows the effect of forces
on winding at +15% tapping position of HV windings.
- 158 - 15th IGTE Symposium 2012
Figure 15: Total body force density at -15% tapping position of HV
windings (115 ms)
Figure 16: Total body force density at -15% tapping position of HV
windings (115 ms)
Deformations in windings when the windings are
axially non-symmetrical are obtained as given in Figure
16 and18. Total deformation depends on the tap position
and at +15% tapping position of HV windings, they are
bigger than which at -15% tapping position of HV
windings. The location of forces shifts to the upper side
of the winding.
Figure 17: Total deformations at +15% tapping position of HV windings
(115 ms).
V. CONCLUSION
In this paper, leakage magnetic field and electrodynamic
force of the 25 MVA power transformers were analyzed
under short circuit conditon of the low voltage windings
of the transformer by using ANSYS® and MAXWELL®

Figure 18 Total deformations at -15% tapping position of HV windings
(115 ms).
based on the FEM.Two different conditions when the
power transformer is under mximum tap are analyzed.
The location of the regulating coil is changed.
Afterwards, deformation result is showed by using
calculated force values. Undesirable stresses values can
be prevented on the transformers by making appropriate
coil arrangements. The insertation of tap sections in the
windings, which produces asymetries between LV and
HV windings, tends to cause an inrease of radial and
axial forces annd then damages in transformers. The
method of calculation offeres a reference to the design of
transformer.
LIST OF PRINCIPLES SYMBOLS
a fractional difference in winding heights
A magnetic vector potential
Br, B , Bz components of the flux density of (in Tesla)
dl
F
unit length of wire
force
h axial height of the winding
Hc coercive magnetic field strength of the PM
Ht tangential component of magnetic intensity
Imax maximum current
Js current source density
J - directional short-circuit current density
ˆr , ˆ and ˆz unit vectors in cylindrical coordinate
S1 parallel boundary condition
S2 vertical boundary condition
U rated voltage
v velocity
V electric scalar potential
Zs short-circuit impedance of the system
Zt short-circuit impedance of the transformer
phase angle
conductivity
studied domain
v ' reluctivity
' conductance
REFERENCES
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transformer under the short circuit situation," Electrical Machines
- 159 - 15th IGTE Symposium 2012
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17-20 October 2008.
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4423-4426, October 2008.
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[6] K. Kurita, T. Kuriyama, K. Hiraishi, and et all. "Mechanical
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Conditions," IEEE Transactions on Power Apparatus and Systems,
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[8] R. E. Ayres, G. O. Usry, M R Patel, and et all. "Dynamic
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- 160 - 15th IGTE Symposium 2012
Robust Design of IPM motors using
Co-Evolutionary Algorithms
*Min Li, † André S. Ruela, † Frederico G. Guimarães, † Jaime A. Ramírez and *David A. Lowther
*McGill University, 3480 University, H3A 2K6 Montreal, Canada
† Federal University of Minas Gerais, Belo Horizonte, MG 31270-010, Brazil
E-mail: *david.lowther@mcgill.ca, † jramirez@ufmg.br
Abstract—A robust design formulation is developed considering the minimization of the torque ripples of an interior
permanent magnet (IPM) machine in the presence of uncertainties in the values of the design variables. This optimization
problem is first solved using the worst vertex prediction and a deterministic search. In addition, a competitive co-evolutionary
algorithm is applied to the minimax optimization problem to find a robust solution, in which one population evolves values for
the design variables and the other one evolves values in the uncertainty set. Through a worst case analysis, the result from the
co-evolutionary algorithm is proven to have a more robust performance than that of the non-robust optimization if
manufacturing tolerance is taken into account. The computation time of the co-evolutionary algorithm may be largely
reduced through the use of parallel computing environments.
Index Terms—Evolutionary algorithms, IPM motor, Minimax optimization, Robust design.
I. INTRODUCTION
In recent years, interior permanent magnet (IPM)
motors have become popular for many applications that
require variable speed and torque. As an alternative to the
traditional induction motor, IPM motors have the
advantages of higher efficiencies and lower noise. The
design of an IPM motor is a complicated task that
involves the consideration of many different aspects,
such as the size and the weight of the machine, the
desired output torque, the cost of the permanent magnet,
etc. In this paper, we focus on the robust design of IPM
machines for which the objective is the reduction of
vibrations and noise of the device caused by errors in
manufacturing, in order to improve the quality and to
extend the life of product.
The idea of robust design was introduced to electrical
machine design over two decades ago. Robustness is
often defined in terms of the performance of the device
being less sensitive to manufacturing errors and
variations of the operation conditions. Dr. Taguchi, with
his statistical based methods, is considered as one of the
pioneers of engineering robust design as he developed
the foundations of robust design to meet the challenges of
producing high-quality products. A Taguchi-based
optimization method has been applied to the design of
brushless DC motors in [1], where the signal-to-noise
ratio was used to estimate the robustness of the product.
A robust shape optimization was applied by Yoon to the
design of electromagnetic devices in [2], where the mean
and the standard deviation of the performance were
treated as multi-objectives for the design problem. This
paper also employed a sensitivity based approach to
compute the approximation of the standard deviation and
took feasibility robustness into account. Another useful
formulation of robust design is to apply the worst case
analysis and to optimize the worst performance of the
objective function in the presence of uncertainties [3] [4].
The robustness measure is integrated into the
optimization process by using a robust target function
defined on the uncertainty set of the design variables; and
the vertices of the uncertainty set were used to predict the
worst value of the objective function. Several different
robust design formulations were reviewed and discussed
in [5] and the authors proposed that the standard
deviation can be approximated using the difference
between the worst performance and the nominal
performance and the computation cost could be largely
reduced. In the recent development of sensitivity based
robust design optimization, the authors defined a gradient
index (GI) using the sensitivity of the performance
function with respect to some critical uncertainty
variables [6]. This simple and efficient algorithm was
illustrated with an example of MEMS devices where
robustness is crucial for high yield rate but information
on uncertainties is hard to obtain. This gradient index
based robust design method was also tested with the
TEAM workshop problem 22 in [7]. The worst case
analysis and the robust target function have also been
applied to topological design problems [8], where a
robust topological gradient (TG) was used to evaluate the
robustness for a certain topological design.
In addition to deterministic optimization approaches,
the worst case analysis based robust design problems (i.e.
minimax optimization) can also be solved using genetic
algorithms and evolutionary algorithms [9] [10]. In
particular, co-evolutionary algorithms have been used to
solve constrained optimization problems formulated as
the minimax optimization problem [11, 12]. In this case,
one population evolves solutions for the problem while
the second one evolves the terms for the Lagrange
penalty function. Co-evolutionary methods have been
applied to robust design as well [13, 14]. In [13] a coevolutionary
algorithm is used to design a robust
nonlinear control under uncertainties. In [14], the authors
have also reviewed a few different formulations of
competitive co-evolutionary genetic algorithms.
In this paper, a robust design problem is defined for
minimizing the torque ripples of an IPM motor while
considering the uncertainties in the design variables. Two
different approaches based on the worst case scenario are
being considered. The first one employs a deterministic
search and uses the computed sensitivity information to

predict the worst performance of the design. The second
algorithm is based on a competitive co-evolutionary
strategy. It introduces a competition between two
populations, one evolves values for the design variables
and the other one evolves values for the uncertainty
variables. Details of the two approaches are presented in
the second section of the paper and the results from the
robust design are tested against a non-robust design in
section III. In the last section, the performance and the
limitations of the two robust design approaches are
discussed.
II. ROBUST TOPOLOGY OPTIMIZATION
A. Robust Design formulation
A practical way to treat the robust design problem is to
use the worst case scenario. A robust objective function
can be defined as:
min max f ( ) , (1)
x U
( x)
where f(x) is the nonrobust objective function and U(x) is
an uncertainty set containing all the possible variations of
the design variable x,
n
U ( x)
{ R : ( 1
i
) xi
( 1
i
) xi}
. (2)
Then the worst performance of the objective function f
can be approximated using the value of f evaluated at one
of the vertices of U, i.e. the worst vertex.
max
U
( x)
f ( ) f ( x
pred
Nevertheless, in a constrained optimization problem, if
a nominal optimal solution x* is located close to the
boundary of the feasible region, which may happen in
some cases, some perturbed solutions, due to the
variations of the design variables, will no longer be
feasible. In such a situation, the robust solution must be
placed away from the boundary of the feasible region to
make sure that the entire uncertainty set of x* stays in the
feasible region. To ensure feasibility robustness, a robust
constraint function is defined as:
max
U
( x )
)
- 161 - 15th IGTE Symposium 2012
(3)
g ( ) 0 , (4)
where gi(x) are the original constraints for the problem.
Finally, a robust design formulation using a robust
objective function and a robust constraint function is
given as:
min
x
max
U
( x )
L
i
f ( )
max g i ( ) 0 , (5)
s.
t.
U
( x )
X x X
U
B. Topological gradient
Applications of topological shape optimization to the
design of electromagnetic devices are relatively new [15]
[16]. Unlike the classical shape optimization for which
only the size and boundary of the design object is
allowed to vary, in a topological shape design process,
the topology of the domain can change as well, for
instance, by drilling an air hole in the domain or filling
this hole with a different material than the rest of the
domain.
The topological gradient (TG) is defined as the
derivative of an objective function with respect to an
infinitely small hole Q as:
obj
( \ Q(
x,
r))
obj
TG ( x)
lim
. (6)
r
0 ( )
where is an arbitrary objective function, x is the center
of the hole Q, r is the radius of Q, \Q(x,r) is the new
topology after the small hole is present and () is the
volume change of the domain , which is the volume of
Q but with a negative sign. Thus a positive value of
TG(x) means a negative change of the values of the
objective function after the small hole Q is created.
Hence the topological gradient can provide information
on whether a topology change (creating a small hole in
the system) will result in a decrease of the objective
function.
Now we can define a robust objective function based
on the worst performance of a non-robust function J due
to the perturbation of the design variable x as:
f
w
J ( ) , (7)
max
U
( x )
and U(x) is the uncertainty set similar to (2), while in a
topological design using TG, the design variables are the
three dimensional coordinates, x, of the center of the
potential topology change. Hence n = 3 and the vector
= {1 2 3} represents the largest variation to the
nominal value of x of the three dimensional coordinates.
This robust objective function fw can be easily estimated
using the worst vertex of the rectangular uncertainty set
U.
If a topology change is taking place in the design
domain , the scalar objective function J can be
approximated near the point using a first order local
expansion, as:
2
J ( ) (
\ Q(
, r))
(
)
TG(
)
( r)
o(
r ) .(8)
Since () and (r) are both constants with respect to
(note that (r) is the volume of Q with a negative sign),
J() has the largest value where TG() is the smallest.
Therefore, the worst performance of J due to the
perturbation of the design variables x is determined by
the point in the uncertainty set, where TG() has the
smallest value. Hence we can obtain a robust topological
gradient as:

TG
R
( x)
min TG ( ) . (9)
U
( x )
Figure 1 is used to illustrate the robustness of a
topology. There exist two areas for a potential
topological change in the design domain . However, the
first area, which has the highest TG value, is close to a
large area which has the lowest TG value, i.e. the TG
value drops drastically in the neighborhood of the first
area. Therefore the second area, which has the second
largest TG values, is superior to the first one for a
topological change in terms of the topological robustness.
This is, in fact, equivalent to the robust topological
design using second-order sensitivity analysis.
Figure 1. Robustness of topology
C. Worst vertex prediction using sensitivity
In robust topology optimization, first, we use the
robust TG to determine the topology change in the
problem domain. After we “drilled” a hole in the system,
the boundary of the hole is parameterized and is
optimized using a shape optimizer. Thus a new
uncertainty set is defined for the new design variables,
which are the coordinates of the controlling points on the
boundary of the hole. However, the robust objective
function remains the same through the entire design
process.
In order to find the worst vertex, we can use the
information of the gradient computed at the point x. The
following figure gives an example of the worst vertex
prediction in R 2 , where the opposite direction of the
gradient of the objective function points to the worst
vertex of the uncertainty set.
- 162 - 15th IGTE Symposium 2012
Figure 2. Worst vertex prediction using gradient
D. Algorithm
Finally, an algorithm for robust topology optimization
based on topological shape optimization is described as
follows:
1. Set the iteration number k =0.
2. Calculate the robust topological gradient TGR at the
center of each element.
3. Define the new domain k where the topology
changes take place by removing the material in the
elements where TGR is greater than zero.
4. Apply standard shape optimization method with a
robust objective function to determine the shape of the
boundary.
5. Check convergence and exit if the optimality
condition is satisfied.
6. Set k=k+1 and go to 2.
Note that uncertainties related to both topology and
shape are being handled throughout the entire design
process.
III. COMPETITIVE CO-EVOLUTIONARY ALGORITHMS
Co-evolutionary algorithms are suitable for solving
minimax optimization problems. The robust design
formulation based on the worst case analysis defined in
(1) can be generalized as
min max f ( x,
u)
, (10)
xX uU
where f(·,·) is an objective or fitness function, x is a
vector of the design variables and u is a vector of the
uncertainty variables. Equation (1) is a special case of
(10) where = x + u. This formulation presents a
competitive relationship between the two players, where
the leader selects a value in X and the follower chooses a
value in U in correspondence.
In a worst case analysis based robust design, the
system seeks for the best design under its worst case
scenario. Thus it is possible to decompose this design
process into two tasks, to find the design with the best
performance and to find the worst performance of a

design subject to small variations of the design
parameters. Similarly, in a competitive co-evolutionary
algorithm, one population competes with the other
leading to an “arms race”. The first population (denoted
as population A in the rest of the paper) represents
candidate solutions in the design space, which are
evolving to minimize the objective; while the second
population (population B) represents disturbances in U,
an uncertainty set applied to the design variables in order
to maximize the objective function. In other words, the
former population provides a solution and the second
population tries to attack the solution in the worst case
scenario. Through the evolutions of the two populations,
a more robust solution, i.e. with the best possible worst
case performance, can be found after several generations.
Therefore, this competitive model will guide the
evolution towards robust solutions.
A. Fitness computation
In co-evolutionary algorithms, the two populations, A
and B, evolve independently, but the fitness evaluations
of the populations are related to each other. The fitness of
an individual in one population is evaluated against
values of the individuals from the other population. For
instance, the fitness of an individual in population A is
defined as,
F( x)
f ( x,
u*)
, (11)
where u* is the current best solution from population B.
The goal of evolution of population A is to minimize the
fitness function F(x).
The fitness of an individual in population B is assigned
against the value of the current best individual x* in
population A. Therefore the fitness function for
population B is defined as,
G( u)
f ( x*,
u)
. (12)
The goal of evolution of population B is to maximize the
fitness function G(u).
B. Alternating co-evolutionary GA
One typical co-evolutionary approach that can be
applied to the continuous minimax problem is called the
alternating co-evolutionary GA (ACGA). Figure 3 shows
a diagram of the ACGA. The two populations (A and B)
are initialized randomly. The finesses of the individuals
in one population are evaluated against the other
population using the functions defined in (11) and (12).
For instance, after the initialization of population A (i.e. a
set of random values is assigned to the design variables x
between the lower bound and the upper bound), the
algorithm fixes the values of the uncertainty variables u
and evolves population A for several generations to
minimize the fitness function F. Then the algorithm
switches to the evolution of population B, while the
values of the design variables archived for the best fitness
are kept and the values of the uncertainty variables are
- 163 - 15th IGTE Symposium 2012
updated towards the maximization of the fitness function
G. This alternating process repeats until the stopping
criterion is met, e.g. the maximum number of
generations.
Figure 3 Alternating Co-Evolutionary GA
In the algorithm implemented in this paper, both
populations have a total of = 100 individuals each. At
the beginning of the execution, these individuals are
randomly generated, respecting the bounds. The
candidate solutions are represented by a one-dimensional
array of real values. An individual has three genes
representing the values of the design variables or the
uncertainty variables. Individuals are selected for
reproduction by means of a binary tournament where two
individuals are randomly selected and their fitness values
are compared, and that individual with the best fitness is
selected for reproduction.
The crossover operator used in the algorithm is a
combination of an extrapolation method with a one-point
crossover method [17]. Each pair of the selected
individuals undergoes crossover with a recombination
rate r = 1.0, and produces two offspring. The operator
performs a blend crossover of the gene at the crossing
point, with a random factor within the interval [0, 1].
After crossover, a mutation operator is applied to the
offspring, with a mutation rate m = 0.2. The mutation
operator is very simple. If a gene is under mutation, the
algorithm randomly generates a new real value within the
bounds. The genetic algorithm implemented is
generational, i.e. all offspring replace their parents in the
next generation.
All the offspring are then evaluated and the best
individual is stored and is used as a population
representative and passed as argument for the opponent’s
evaluation, as described in equations (11) and (12).

The algorithm runs for a maximum of 100 generations
and returns the best stored pair (x, u).
C. Parallel co-evolutionary GA
The parallel co-evolutionary GA (PCGA), shown in
figure 4, is very similar to the ACGA, except that the two
competitive populations evolve simultaneously. As a
parallel model, this can be implemented easily for a
parallel computing environment and the computational
time will be reduced to half in theory.
Figure 4 Parallel Co-Evolutionary GA
Several other methods of co-evolutionary algorithms
using different schemes of the fitness assignment can be
seen in [18]–[20]. In this paper, the alternating coevolutionary
GA is used to solve the robust design
problem.
IV. RESULTS
Figure 5 A simulation model of an IPM motor.
- 164 - 15th IGTE Symposium 2012
A. Numerical model
Figure 5 shows a quarter of a 3-phase 4-pole IPM
machine. The quarter of the rotor core has one slot in the
center and the rest of the core is made of steel. A
permanent magnet bar made of NdFeB magnet is inserted
in the center of the slot. The goal of the design is to find
the optimal shape of the permanent magnet bar and the
flux barriers which minimize the torque ripples of this
motor, while maintaining an adequate average torque.
The objective function can be defined, without
considering the manufacturing uncertainties, as,
Ti
Tavg
minmax
F ( )
x u
i Tavg
. (13)
s.
t.
minT
0.
4Nm
u
avg
The design variables chosen for the optimization are the
length of the permanent magnet, L, the width of the
permanent magnet, h and the distance from the
permanent magnet to the surface of the rotor, d.
This numerical model is solved using a 2-D nonlinear
finite element solver (MagNet [21]). At each iteration of
the optimization, torques are evaluated at different
positions of the rotor. The rotor mesh is regenerated after
a new geometry of the rotor is archived during the
optimization process.
B. Results from RTO
The robust topological optimization method is applied
to a rotor core filled only with iron [22]. The topological
gradient is evaluated in the design region in order to find
potential topological changes which reduce the value of
the cost function. The permanent magnet and air
materials are created in the region according to the TG
values, as shown in figure 6.
Figure 6 Topology of the rotor generated by RTO
This shows a rough topology with one permanent
magnet block and two air flux barriers of the design,
which serves as the starting point of the shape
optimization process. The system then optimizes the
shape of the boundaries between different materials in
order to achieve more accurate values of the geometries.
The value of the design variables of the robust optimal is:
H = 1.616 mm, L = 19.879 mm and d = 12.243 mm.

C. Results from ACGA
It is not practical to combine the topology optimization
with the alternating co-evolutionary GA due to the huge
computational cost. Thus the ACGA is applied to the
model shown in figure 5 to find the robust optimal values
of the design variables. The manufacturing tolerances of
the design variables are considered as the uncertainties of
the problem. The optimal results, from the robust
formulation, are given in table I.
TABLE I
VALUES OF DESIGN VARIABLES OF THE NOMINAL AND ROBUST OPTIMA
Design
variables
Unit Nominal
optimal
Robust
Optimal
(by RTO)
Robust
Optimal
(by CGGA)
H mm 1.588 1.616 1.453
L mm 18.33 19.879 19.345
D mm 13.426 12.243 12.695
Nominal
Performance
Nm 0.2358 0.2583 0.2547
Worst
Performance
Nm 0.2709 0.2791 0.3039
Feasibility
robustness
No Yes Yes
Table 1 shows the values of the non-robust optimal and
the robust optimal. The worst cases of the performances
are evaluated. The uncertainty is set to be 5% of the
design variables.
V. CONCLUSION
This paper discusses robust design issues and
formulations for IPM design problems. Two different
methods have been applied, and they can both find robust
solutions for the problem.
The robust topology optimization method employs a
deterministic search based on the topological gradient
and the shape sensitivity. The algorithm requires two
FEM solutions per evaluation of the robust objective
function (one FEM solution for the nominal cost function
value and sensitivity calculation, and one for the worst
performance). In the deterministic search, the maximum
number of objective function evaluation is set to 200 and
the total time of execution is around a few hours. Thus
the method is very efficient and fast to converge.
However, the robust objective function defined in [1] is
not necessarily partially differentiable and this may pose
some difficulties to the optimization. Also, convexity of
the objective function is not guaranteed, thus the worst
performance point may be found inside the uncertainty
set instead of on the corner. The worst performance
prediction is only an approximation.
On the other hand, the co-evolutionary GA does not
rely on the sensitivity information. The algorithm
maintains a population of the uncertainty variables and
seeks for the exact worst performance point in the
uncertainty set. The algorithm maintains two populations
with 100 individuals for each population. The maximum
number of generations of evolution is set to 100. The coevolutionary
GA requires a total number of 20000 FEM
- 165 - 15th IGTE Symposium 2012
solutions and the total execution time for the algorithm is
around 5 days. Although a huge computation time is
required for the co-evolutionary GA, this algorithm is
parallelizable, thus the time may be reduced by choosing
an appropriate scheme of parallel computing. Also,
depending on the nature of the optimization problems,
some modifications can be applied to the GA to reduce
the number of the function evaluations.
[1]
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- 166 - 15th IGTE Symposium 2012

IGTE Symposium, TU Graz 2012
- 167 - 15th IGTE Symposium 2012
Free-form Optimization for Magnetic Design
Z. Andjelić 1 , S. Sadović 2
1 ABB Corporate Research, Baden, Switzerland;
2 Sadovic Consulting, Paris, France
E-mail: zoran.andjelic@ch.abb.com
Abstract— The paper presents an approach for free-form optimization of the magnetic problems. The approach is based on
the novel simple sensitivity analysis and does not require the calculation of the adjoint problem. The solution engine in the
background is IEM. The developed approach is illustrated on some typical benchmark problems.
Index Terms— Free-form optimization, IEM, Sensitivity analysis
Also, in free-form optimization the meshing of the
I. INTRODUCTION
analysed objects using mesh generator is performed
When speaking about free-form optimization of industrial
only in the first iteration. For all further iterations the
problems we distinguish two different approaches: direct
mesh is updated directly in the Analysis module.
and indirect approach. In direct approach we try to As we use the non-gradient approach, the calculation
minimize the maximal field quantities laying directly on time is much faster than with the gradient approach,
the interface between different media by changing the requiring the costly calculation of the gradients.
form of those interfaces in the normal direction. Typical As mentioned above we distinguish between the direct
applications are optimization of the structural problems and indirect approaches for free-form optimization. One
[1], or dielectric design of electrical apparatus [2], [3], of the additional key differences between direct and
[4]. In indirect approach we are searching for the indirect approach is that for the optimization problems
prescribed distribution of the objective function in the following the direct approach it is not necessary to
space of interest by changing the shape of the structures calculate any sensitivity function [2], [3]. In the current
outside of such space of interest. In this paper we discuss contribution we shall focus us on the indirect approach
in more details the second approach, illustrated by some illustrated by some applications in magnetic design. It is
typical benchmark problems.
important to note that the proposed approach is
independent of the application class and can be used for
optimization of not only magnetic but also dielectric,
acoustic or similar class of problems. It also has a generic
character and can be used having FEM or other numerical
method as the numerical engine in the background. In the
present contribution we use IEM (Integral Equation
Method) for the solution of the magnetostatic field
problems.
II. FREE-FORM OPTIMIZATION
For automatic shape optimization we follow a nonparametric,
non-gradient approach, which in
combination with IEM (Integral Equation Method)
enables fast and robust optimization of the real-world 3D
problems [5]. The main benefits of such an approach
comparing to the standard parametric, gradient-based
approaches are:
The applied procedure usually leads to the global
optimum contrary to the parametric optimization
approach where the optimum can be searched only
within the “parametric space” defined by the design
parameters (radii, distances, etc.).
Due to the fact that we don’t need as input any design
parameter it is not necessary to “communicate” with
the CAD system during the optimization iterative
procedure. As shown in Figure 1 the iterative
framework in free-form optimization requires
communication only between Analysis and
Optimization module, whereby by parametric
optimization in each iteration a new set of parameters
has to be generated in CAD tool, meshed in mesh
generator and then passed to Analysis module for
further processing.
Figure 1: Free-form vs. parametric optimization framework
III. IEM FORMULATION
The analysis of the non-linear problems in
magnetostatic by IEM is performed using the improved
procedure described initially in [6], and more detailed
elaborated recently in [7]. The magnetic field in any space
point can be found as:
J M
H H H
J
where H is a field component produced by the excitation
M
current in free space and H is a field produced by the
magnetic charges. The first field component can be easily
calculated by Bio-Savarot law. For the calculation of the
second one we use the formula:
M 1 1
H J 1dSJ N
2dVN
(2)
4 J K 1 dS
4
N 2 2dV
N (2)
4 4
K
S
J
VN
where J and N are the fictitious surface and volume
magnetic charges, and 1 K and K 2 are the kernels of the
3
type r / r . The surface charges are obtained by solving
second Fredholm integral equation:
(1)

IGTE Symposium, TU Graz 2012
1 1
2 (3)
(3)
2
1
I
J
J GdS 1 I 2 I I N NGdV 2 2d
N
I 2 2 2
s V N
JGdS 1 2H
n
Here 1 G and 2
12
1 2
G are the kernels of the type
3
rn/ r and
where the 1 and 2 are the relative
permeabilities of the surrounding media and magnetic
materials. When solving the linear problems the last term
on the right-hand side of equation (3) is equal to zero.
Here is important to stress some of the main features
of IEM when solving the non-linear magnetostatic
problem. In spite of the fact that it is necessary to mesh
the volume of the non-linear magnetic parts, the number
of unknowns for the non-linear problem is same as the
number of unknowns for the linear one. This is due to the
fact that the non-linear contribution - second term on the
right-hand side of (3) - appears just as the correction term
and is calculated throughout the iteration procedure from
the previous iteration. Also, as the material parameter
appears only in the diagonal term, the matrix
calculation is performed only during the first iteration. In
other iterations only the diagonal term is changed together
with the RHS term taking into account the contributions
due to volume charges.
IV. INDIRECT APPROACH
Contrary to the direct approach where it is not necessary
to calculate any sensitivity functions, in indirect approach
this function has to be established. To establish such
function we use the analogy to the sensitivity analysis
typically used in the signal-processing (SP) problems. In
SP the objective of controller design is to keep the error
between the controlled output and the external input as
small as possible. In signal processing the sensitivity
function S(s) is typically calculated as:
Es ()
Ss () ; Es () Rs () Ys
() (4)
Rs () ds ()
where E(s) is feedback error, R(s) and d(s) are the
external input and disturbance.
To calculate the sensitivity for our optimization tasks we
use the analogy to the above SP scheme. Here we take as
example the quantities from the magnetic problem,
Figure 2.
Figure 2: Sensitivity calculation scheme for optimization
tasks
In magnetic problems the magnetic field in the space
point of interest can be calculated using BEM [5] as:
- 168 - 15th IGTE Symposium 2012
1
H(
j) ( i) K( i, j) d
(5)
4
Sensitivity of changing the field H in the space of interest
with the changes of the geometry of the magnetized body
can then be obtained as:
H H
S
H H
G C
G C
max
In the above case the external input H G is a given i.e.
prescribed (desired) field distribution in the space of
interest, H C is a calculated field in the same space. The
C
disturbance H is calculated as:
max
C C
max max
(6)
H ( i) max[ H ( i, j), j1, N ]; (7)
The displacement vector D of the moving of the mesh
nodes can then be calculated as:
D Sn More information on the calculation of the sensitivity
function for free-form optimization tasks can be found in
[8].
For illustration the above procedure has been used to
optimize the Die mold problem, Example 1 and Field
homogenization problem, Example 2.
V. EXAMPLE 1: DIE MOLD OPTIMIZATION
This is a TEAM benchmark problem No. 25 used up to
now for the benchmarking of the codes dealing with 2D
parametric optimization [9], Figure 3.
Figure 3: TEAM benchmark problem No. 25
Here we use the same model as a 3D problem adding the
extrusion in y-direction of 200 mm, Figure 4. The
objective is to obtain the homogeneous radial field
distribution within the cavity shown in Figure 3. One of
the die molds is keept fix (inner cylinder) and the other
one is in our approach subjected to the free-optimization
process in order to get the radial field distribution in the
j
8

IGTE Symposium, TU Graz 2012
cavity. 3D model is shown in Figure 4 and the detailed
2D view in Figure 5.
Figure 4: 3D model of the Team problem No. 25
Figure 5: Details of the Team problem No. 25
Applying module for free-form optimization governed by
the above given approach for sensitivity calculation we
have after 24 iterations obtained the optimal form of the
magnetic poles, Figure 6 (in red).
Figure 6: Outer magnetic mold before and after optimization
Such new form of magnetic poles has provided a desired
radial field distribution in the cavity. Figure 7 shows the
form of the magnetic poles befor otpimization, after 10 th
iteration and as the optimal form after 24 iterations. The
field vectors illustrate the changes of the field during the
optimization process. Only at the end of the die molds
some deviations are observed caused by the end-region
field disturbances.
- 169 - 15th IGTE Symposium 2012
Figure 7: Magnetic field homogenization during the
optimization process.
VI. EXAMPLE 2: AIR GAP FIELD HOMOGENIZATION
In this example the objective function is to achieve the
homogeny field distribution over the prescribed space of
interest lying in the air gap between the magnetic poles,
Figure 8.
Figure 8: Model of the magnetic structure
The core is made of the material with 1500
, and is
excited by the current-carrying coil with I=12240A.
Before doing any optimization the magnetic field
distribution over the space of interest is shown in Figure 9
and Figure 11, a.). The field over the space of interest
varies from 35737 A/m to 57555 A/m. As the
optimization objective we define here the desired value of
the homogeneous field over the space of interest
(50x50mm) as H d =50000 A/m. After applying the
optimization modus governed by the above sensitivity
calculation, we have obtained after 37 iterations the
optimal form of the magnetic pole shoes that deliver the
desired field distribution within the error less than 10%,
Figure 10.

IGTE Symposium, TU Graz 2012
Figure 9: Magnetic field distribution over the space of
interest before any optimization
Figure 10: Magnetic field distribution over the space of
interest after 37 iterations. The magnetic poles have changed
the form in order to provide prescribed homogeneous field of
50000A/m.
Figure 11 shows in more details the field distribution over
the space of interest before (a) and after optimization (b).
Figure 11: Detailed view on the field distribution over the
space of interest before (a) and after (b) optimization
The field variation for optimal design (with threshold
error of 10%) is between Hmin = 46489A/m and
Hmax=55027 A/m.
VII. CONCLUSION
The paper elaborates the procedure for free-form
optimization of magnetic problems. The procedure is
- 170 - 15th IGTE Symposium 2012
based on the novel approach for the simple sensitivity
calculation. The proposed approach does not require
calculation of the adjoint problem and has a generic
character with respect to both the classes of the
application (magnetic, dielectric, acoustics...) and the
numerical methods used within the simulation engine
(BEM, FEM).
REFERENCES
[1] R. Meske: “Non-parametric gradient-less shape optimization in
solid mechanics”, Shaker Verlag,2007, ISBN 978-3-8322-6373-7
[2] Z. Andjelic, S. Sadovic: “Reduction of breakdown appearance by
automatic geometry optimization”, IEEE Conf. on El. Insulation
and Dielectric Phenomena, Vancouver BC, Canada, 2007
[3] Z. Andjelic, D. Pusch, T. Schoenemann, S. Sadovic: “Multi-load
optimization in electrical engineering design, Part 1: Simulation,
EngOpt 2008- Int. Conf. on Engineering Optimization, Rio de
Janeiro, Brazil, 01-05. June 2008
[4] Z. Andjelic, S. Sadovic, Jean-Claude Mauroux: “Preventing
breakdown by direct optimization approach”, IEEE Int. Power
Modulator and High Voltage Conf, San Diego, CA-June 3-7,
2012
[5] Z. Andjelic at al: “BEM-based simulations in engineering
design”, In Boundary Element Analysis, Mathematical Aspects
and Applications, Springer Verlag 2007, ISBN: 3-540-47465-X
[6] B. Krstajic, Z. Andjelic, S. Milojkovic, S. Babic, S. Salon:
“Nonlinear 3D magnetostatic field calculation by the integral
equation method with surface and volume magnetic charges”,
IEEE Tran. on Mag., vol.28, No.2, March 199
[7] Z. Andjelic, G. Of, O. Steinbach, P. Urthaler: “Fast BEM for
industrial applications in magnetostatic”, in Lecture Nodes in
Applied and Computational Mechanics, Springer-Verlag, Vol. 63,
2012
[8] Z. Andjelic: “Simple sensitivity approach for optimization tasks in
electrical engineering”, OIPE Workshop, Gent, Belgium, 2012
[9] N. Takahashi, M. Natsumeda, M. Otoshi and K. Muramatsu:
“Examination of optimal design method using die press model
(problem 25)”, COMPEL 17 5/6, 1982

- 171 - 15th IGTE Symposium 2012
Optimization for ECT treatment planning
1 P. Di Barba, 3 L.G. Campana, 2 F. Dughiero, 3 C.R. Rossi, 2 E. Sieni
1 Department of Industrial and Information Engineering, Pavia University, via Ferrata 1, 27100 Pavia (Italy)
2 Department of Industrial Engineering, Padova University, via Gradenigo 6/A, 35131 Padova (Italy)
3 Melanoma and Sarcoma Unit, Istituto Oncologico Veneto (IOV),Via Gattamelata 64, 35128 Padova (Italy)
E-mail: paolo.dibarba@unipv.it,{fabrizio.dughiero, carlor.rossi, elisabetta.sieni}@unipd.it, luca.campana@ioveneto.it
Abstract—Treatment planning of Electrochemotherapy (ECT) is designed by means of a genetic multi-objective optimization
method: the needle position maximizing the electric field in the treated volume is searched for. NSGA algorithm is coupled
with penalty function technique in order to identify the constrained Pareto front to select the best compromise solutions and
discard the unfeasible ones.
Index Terms—Electrochemotherapy, conduction field, Finite Element, Pareto front, NSGA.
I. INTRODUCTION
ECT uses pulses of electric field in order to improve the
delivery of chemotherapeutic drugs into cancer cells [1]-
[2]. A suitable electric field intensity is able to induce cell
membrane permeabilization that improves the
chemotherapy drug delivery. However, a high electric
field intensity in healthy tissues, and in some critical
regions like e.g. large vessels, is to be prevented. A
conduction electric field is applied to tumor tissues by
means of needle electrodes suitably positioned in the
target volume. In order to improve the therapy success,
the positioning of electrodes is considered an
optimization problem. The research group in Ljubljana
University has proposed some solutions to optimal
electrode positioning in deep-seated tumor like in [3-6].
In this paper a multiobjective optimization method, based
on a modified NSGA-II algorithm, that includes
constraints and penalty functions in order to prevent
unfeasible solutions, is proposed for the optimal
positioning of needles in the tumor mass [7-10]. The
optimization problem is solved using a 2D model of
steady conduction field.
II. CLINICAL ECT
ECT is a medical therapy based on cell electroporation
for patients with cutaneous and subcutaneous tumor
nodules on the basis of the synergistic association of
locally applied brief electrical currents (reversible
electroporation) and low permeant anticancer agents [11-
13]. Electroporation is a local electric treatment that uses
a physical behavior of cells when a pulsed electric field is
applied in order to open some pores on the cell
membrane. Those opening can be used as channels as a
delivery system to enhance the penetration of drugs,
genes, or molecular probes into cancer cells. This is an
applied electrical fields with suitable intensity that
increase cell membrane permeability [14-17]. Figure 1
shows the most important phase of chemotherapy drug
administration using ECT technique: In the phase I of the
treatment the clinician injects the drug (e.g. bleomicine),
then during phase II he applies the electric pulse, and
finally the drug penetrate the membrane cells.
Since its development at the Institute Gustave Roussy,
this technique has been quickly tested in the clinical
setting and recently is entered in the clinical practice [11-
12], [18-21]. At Melanoma and Sarcoma unit of the
“Istituto Oncologico Veneto” (IOV) in Padova, Italy,
clinical application of ECT using standard electrodes [23]
has shown yet satisfying results [22]. Standard electrodes
are a set of 7 needles with a length between 10 and 30
mm on a rigid support, [23]. The ECT equipment
manufacturer produces also a long needles machine that
can be used to treat with ECT some deep-seated tumors
like sarcoma [23-25]. In this case the clinician implants
single 20 cm length electrodes on the tumor mass
accordingly to medical image of the tumor and clinical
practice. So, in the case of flexible long-needle
equipment, it is of interest to improve the therapy success
studying the electric field produced by some
configurations of electrodes implanted on the tumor mass
using optimization algorithms.
ECT electrode
E
Skin surface
Figure 1: Description of the ECT application.
III. DIRECT PROBLEM: ELECTRIC FIELD ANALYSIS
In general, the case study models three regions: the
tumor, T, with an average radius of 3 cm, the
surrounding healthy tissue, H, and a region close to the
treated region that might be a critical one, C. Each
region is attributed the relevant electric conductivity [3].
The needle electrodes are represented as a set of nine
points. In particular, the fixed main electrode is located in
the center of the lesion whereas the other eight electrodes,
the ones that can be moved, are around the central one.
The ECT process forces a sequence of voltages in the
range 1 to 3 kV for each electrode pair. The imposed
voltages represent the boundary conditions of the field
problem. Then, the electric field is computed by means of
the finite-element method (FEM) solving a steady
conduction problem for each electrode pair: specifically,
16 field analyses on the same grid are needed to compute
the electric field for each needle configurations [26]. The

solved equation is:
V
0
(1)
imposing Neumann condition on electric scalar potential
on the domain boundary:
V
n
0
And finally the electric potential has been fixed to a
constant value, U, in two of the ne electrodes in the
following way
V U
i
0 V U
(
i,
j)
i j i 1,...
n
j
e
U
i
Then, given an electrode configuration, solving the direct
problem implies to repeat the field analysis, i.e. solving
(1), for all possible (i,j) pairs of electrodes.
H
C
T
Electrode
U i
Main electrode
Figure 1: Geometry of the 2D conduction field.
Given all the ne field analyses considering the mesh
nodes of each problem region the highest value of the
electric field is searched for each node of the problem
domain and recorded in sets named Emax(i), one for each
of examined region.
IV. INVERSE PROBLEM: OPTIMAL ELECTRODE
POSITIONING
The therapy efficacy depends on the electric field
intensity applied to the cells. In some practical cases the
proximity to a prescribed therapeutic value of the
temperature is searched for [27-28], whereas in our case
the overcoming of a given electric field threshold is to be
controlled. The ideal configuration of needle electrodes is
the one that maximizes the sub-volume of the tumor
region covered with an electric field intensity over the
electropermeabilization threshold [3], ERE, and
simultaneously minimizes the volume of healthy tissues or
critical organs that have an electric field higher than ERE
[29-30]. Accordingly, the following objective functions,
to be minimized, have been defined:
N E ( E E
f1(
E)
100
1
N E,
tot
RE
)
that represents the complementary sub-volume of the
tumor region for which the electric field is under ERE,
evaluated as the number of nodes, NE, in which the
U j
(2)
(3)
(4)
- 172 - 15th IGTE Symposium 2012
electric field intensity is higher than ERE. NE,tot is the total
number of nodes in which the electric field is evaluated in
the tumor region. The design criterion considered
evaluates the nodes of the healthy tissue region or the
region of a critical organ (e.g. large vessel), in which the
electric field exceeds a prescribed threshold ETH:
g(
E,
E
TH
)
N ( E E
N
E
TH
100
(5)
E,
tot
)
Starting from (5) three objective functions have been
generated. Namely:
(a) g(
E,
E )
f (6)
2 IRE
in which the threshold of electric field is fixed to the
irreversible electroporation value, EIRE = 10 5 V/m, and is
computed in the tumor region T;
(b) g E,
E )
f (7)
3 ( ETH 1
in which the threshold of electric field is fixed to the
reversible electroporation, ETH1 = 410 4 V/m, computed
on the healthy tissue H. In this case it is desirable that
the electric field does not exceed the threshold ETH1 in
order to preserve healthy tissue; and finally:
(c) g E,
E )
f (8)
4 ( ETH 2
In this case the electric field cannot exceed the ETH2 = 10 3
V/m in the critical region C to prevent the damage of
critical organs. Generally this threshold is chosen lower
than electroporation threshold in order to ensure an
electric field lower the ERE.
All the objective functions (4) and (6)-(8) are computed
using the Emax(i) set of values in the corresponding
region of the computation domain.
A1
A2 A2
Figure 2: f1 and f3 optimization goal.
Accordingly, a sequence of bi-objective optimization
problems have been considered and solved: find the
Pareto front minimizing the couple of functions (f1, fk)
with k=2,3 and 4 subject to the solution of the direct
problem (1) and a set of geometrical constraints on the
electrode position. For instance the minimum distance
between two electrodes must be greater than 10 mm.
Constraints have been incorporated in the objectives
functions by means of a penalty term as in [7]. For
instance Figure 2 shows the f1 and f3 optimization goal: f1
tends to maximize the area A1, whereas f3 tends to
A1

minimize the area A2. Moreover, Figure 3 shows the
penalty constraint effect: if the non-penalty algorithm is
used, two electrodes can be at a distance lower than the
prescribed minimum (10 mm) like the one marked with a
circle in Figure 3 (a). In contrast, if the penalty algorithm
is used, too near electrodes configuration are discarded
and a possible configuration is like the one in Figure 3
(b).
(a) (b)
Figure 3: Penalty algorithm effect.
V. RESULTS
Results of some optimized configurations are here
presented.
A. Case 1: penalty vs non-penalty
The optimization problem considers the electric field in
the tumor region T that must exceed the electroporation
threshold ERE (f1) and the electric field in the healthy
tissue region, T, that must be lower than the
electroporation threshold ETH1=ERE (f3).
In this case, results obtained using penalty algorithm are
compared with results obtained using non-penalty
algorithm. Figure 4 reports the two Pareto fronts obtained
starting from the same initial population and using the
two algorithm: the Pareto front is reshaped.
Figure 4: Pareto Front for the case 1 using penalty and
non-penalty algorithm.
200103 E [V/m]
15010 3
10010 3
5010 3
0,00
Not feasible
Figure 5: Optimized electrodes configurations: Electric
field in the examined region using (a) penaltyand (b) nonpenalty
algorithm.
- 173 - 15th IGTE Symposium 2012
In Figure 5 the highest value of the electric field obtained
at each domain point applying the whole sequence of
electrodes discharges during an ECT treatment is reported
for the tumor region, Emax(T), and the healthy tissue,
Emax(H). The corresponding electrodes position is also
indicated by dots.
A. Case 2: preventing irreversible electroporation
The optimization problem considers the electric field in
the tumor region T that must exceed the electroporation
threshold ERE (f1) and must be lower than the irreversible
threshold, EIRE (f2).
Figure 6 reports the Pareto front obtained starting from an
initial population and using the penalty algorithm.
Figure 6: Pareto Front for the case 2 using penalty
algorithm.
In Figure 7 the highest value of the electric field obtained
at each domain point applying the whole sequence of
electrodes discharges during an ECT treatment is reported
for the tumor region, Emax(T). The corresponding
electrodes position is also indicated by dots.
200103 E [V/m]
15010 3
10010 3
5010 3
0,00
Figure 7: Optimized electrodes configurations: Electric
field in the examined region using penalty algorithm.
In this case electrodes are positioned in the healthy tissue
because the irreversible electroporation is avoided in
order to prevent cells necrosis.
A. Case 3: preserving critical organ (blood vessel)
The optimization problem considers the electric field in
the tumor region T that must exceed the electroporation
threshold ERE (f1) and the electric field in the critical
region, C, that must be lower than the threshold ETH1
(f4).
Figure 8 reports the Pareto front obtained starting from an
initial population and using the penalty algorithm. In
Figure 9 the highest value of the electric field obtained at
each domain point applying the whole sequence of
electrodes discharges during an ECT treatment is reported

for the tumor region, Emax(T) and the critical region,
Emax(C). The corresponding electrodes position is also
marked by black and green dots.
Figure 8: Pareto Front for the case 3 using penalty
algorithm.
200103 E [V/m]
15010 3
10010 3
5010 3
Figure 9: Optimized electrodes configurations: Electric
field in the examined region using penalty algorithm.
In this case the electrode are far from the critical region,
whereas in Figure 5 are close and even inside the critical
region. Then different objective functions allow to
identify different electrodes configurations depending on
the problem constraints.
VI. CONCLUSIONS
The NSGA-II algorithm modified including constraints
and penalty function has been applied to design ECT
electrodes positioning. Various objective function pairs
have been implemented in order to compare results
obtained considering different problem targets.
VII. ACKNOWLEDGES
This project has been developed in the frame of a Post-
Doc granted by the Padova University, Italy.
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sparing treatment of bleeding melanoma recurrence by
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Chiarion-Sileni, A. Vecchiato, L. Corti, C. Rossi, D. Nitti,
“Bleomycin-Based Electrochemotherapy: Clinical Outcome from
a Single Institution’s Experience with 52 Patients”, Annals of
Surgical Oncology, 16(1), 191–199, 2009.
[23] Cliniporator, Igea: http://www.igea.it (last visited October 2012).
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“Robustness of treatment planning for electrochemotherapy of
deep-seated tumors”, J. Membrane Biol., 236(1), 147–153, 2010.
[25] I. Edhemovic, E. M. Gadzijev, E. Brecelj, D. Miklavcic, B. Kos,
A. Zupanic, B. Mali, T. Jarm, D. Pavliha, M. Marcan, G.
Gasljevic, V. Gorjup, M. Music, T. P. Vavpotic, M. Cemazar, M.
Snoj, G. Sersa, “Electrochemotherapy: a new technological
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“Adaptive Ablation Treatment Based on Impedance Imaging”,
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dimensional optimal current patterns for radiofrequency ablation
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- 175 - 15th IGTE Symposium 2012
Investigation of the Electroporation Effect
in a Single Cell
Jaime A. Ramirez ∗ , William P.D. Figueiredo ∗ , Joao Francisco C. Vale ∗ , Isabela D. Metzker ∗ , Rafael G. Santos ∗ ,
Matheus S. de Mattos ∗ Elizabeth R.S. Camargos ∗ , and David A. Lowther †
∗ Federal University of Minas Gerais, Belo Horizonte, Brazil
† McGill University, Montreal, Canada
E-mail: jramirez@ufmg.br
Abstract—This paper investigates the electroporation phenomenon in a single cell exposed to ultra short (μs) and high voltage (kV/m)
electric pulses. The problem is addressed by two complementary approaches. First, numerical simulations based on an asymptotic
approximation derived from the Smoluchowski theory are used to calculate the pore generation, growth and size evolution at the
membrane of a spherical cell model, immersed in a suspension medium and consisting of cytoplasm and membrane. The numerical
calculations are solved using the finite difference method. Second, an in vitro experiment with LLC-MK2 cells is carried out in which
electroporation was monitored with molecules of propidium iodide. This part also comprehended the design and manufacturing of a
portable electric pulse generator capable of providing rectangular pulses with amplitude of 1,000V and duration in the range of 1-μs
to 100-μs. The pulse generator is composed of three modules: a high voltage dc source, a control module, and an energy storage and
high voltage switching. The numerical simulations considered a 5-μm radius cell submitted to a 500kV/m rectangular electric pulse
for 1-μs. The results indicate the formation of ∼3,500 pores at the cell membrane, most of them, ∼950, located at poles of the cell
aligned to the applied electric pulse, with radii sizes varying from 0.5-nm to 13-nm. The in vitro experiment considered expositon
of LLC-MK2 cells to pulses of 200V, 500V, and 700V, and 1-μs. Images from fluorescence microscopy exhibit the LLC-MK2 cells
with intense red, a strong evidence of the electroporation.
Index Terms—Electroporation, electric fields, finite difference method.
I. INTRODUCTION
Electroporation is the process of applying pulsed electric
fields to biological cells to induce the formation of transient
“pores” in the cell membrane. Depending on the magnitude
and duration of the electric pulse, the membrane may recover
to its original state (the pores reseal) -areversible process;
otherwise, the cell dies - an irreversible process. This phenomenon
was first reported by [1] and is well discussed in the
contributions [2]- [4].
Earlier studies have focused on relatively low external fields,
i.e. less than a kilovolt per centimeter, applied over time periods
ranging from several tens of microseconds to milliseconds.
Recently, the use of high electric fields (∼ 100kV/cm), or
higher, with pulse durations in the nanosecond range has been
employed and opened a new area of research in bioelectrics
[5]. A controlled electroporation process can, therefore, be
used to deliver substances to the cell cytoplasm in a wide range
of applications, including gene therapy, drug delivery, nonthermal
inactivation of micro-organisms and cancer treatment,
see for instance [4].
From the practical point of view, controlling the electroporation
process involves two complementary challenges. First,
a comprehensive simulation analysis is required. The time
dependent electric field, induced at the cell membrane by
the external pulse, need to be obtained. It is this field that
provides the dynamic driving force for the physical process.
In addition, the dynamical evolution of the pores at the
cell membrane under the influence of this field need to be
adequately treated. Second, a detailed experimental laboratory
test is necessary to confirm the simulation. This involves the
building of an electric pulse generator in which the pulse
width and electric field strength are controlled. Moreover,
appropriate microscopy techniques are required to confirm the
pore formation at the cell membrane.
In terms of numerical simulations, most works that consider
the dynamics aspects of the electroporation phenomenon are
based on the Smoluchowski theory. Krassowska et al. [6]- [9]
employ an asymptotic approximation of the Smoluchowski
theory in a single cell model to determine the formation of
pores in a spherical cell submitted to electric pulses of a
few kV/m in the millisecond range. Schoenbach et al. [10]-
[13], [5], use the full equations of Smoluchowski theory to
establish the nucleation of pores in spherical cell models
exposed to high intensity (thousands of kV/m) and ultra short
(nano seconds) electric pulses. The approaches used by both
groups yield acceptable results that are used in a wide range
of applications, which in the first case are confirmed indirectly
and in the second case are verified experimentally; however
there is the need to address the electroporation phenomenon
for electric pulses in the micro second range.
This work investigates the electroporation phenomenon in
a single cell when submitted to electric pulses of magnitude
in the order of 1kV/mm and duration of 1-μs. The
phenomenon is addressed by two complementary approaches,
numerical simulations and an in vitro experiment. The Material
and Methods is divided into three subsection. First, the
mathematical modeling of electroporation describes how the
numerical simulations can be used to assess the dynamics
of the pore formation process, i.e. pore generation, growth
and size-evolution at the cell membrane. This is based on

an asymptotic approximation based on the Smoluchowski
theory and is solved using the finite difference method. The
formulation is capable of providing the voltage induced across
the cell membrane and important features for the practical
application of electroporation, i.e. the number of pores and the
distribution of pore radii as a functions of time and position
on the cell membrane. A detailed description on how the
numerical calculations are made is also given. Second, the
electric pulse generator subsection discusses the theory used
to design and build the generator. The first module is a high
voltage d.c. source, the second is a control module and the
third is responsible for the energy storage and high voltage
switching. The generator is capable of providing retangular
pulses with amplitude of 1,000V and duration in the range
of 1μs to 100μs, with resting intervals of 10μs between
the pulses. Third, the cell culture subsection describes the
procedures used to prepare the LLC-MK2 cells for the in
vitro experiment with molecules of propidium iodide. Finally,
the Results presents the numerical analysis in a spherical cell
of 5-μm radius, exposed to an electric pulse of 500-kV/m,
duration of 1-μs; and the in vitro exposition of LLC-MK2 cells
to electric pulses of of 200-kV/m, 500-kV/m, and 700-kV/m,
duration of 1-μs, and fluorescence microscopy analysis.
II. MATERIAL AND METHODS
A. Mathematical Modeling of Electroporation
For the mathematical description of the electroporation at
the cell membrane, let us consider the model consisting of
a cell in a suspension medium within the parallel plates of
a cuvette, as indicated in Fig.1. For convenience, the cell
is considered spherical and composed only by a membrane
and cytoplasm, which are characterized by a conductivity and
permittivity (σm, εm) and (σc, εc), respectively. The outer
region, or suspension medium, is also characterized by its
conductiviy and permittivity (σo, εo).
σ , ε
Fig. 1. Cell model - not to scale
σ , ε
σ, ε
- 176 - 15th IGTE Symposium 2012
The dynamics of the electroporation process, i.e. behavior
of pore generation, growth and size-evolution at the cell membrane,
can be calculated using the continuum Smoluchowski
theory [6], [11], with the following governing equation for the
pore density distribution function n (r, t),
∂n ∂
+ D −
∂t ∂r
n∂E
1 ∂n
− = S(r); (1)
∂r kT ∂r
where S(r) is the source, or pore formation term; D is a pore
diffusion constant; r is the pore radius; T is the temperature;
kB is the Boltzmann constant; and E(r) is the energy. This
expression can be simplified by an asymptotic approximation
based on [6]- [9]. This is discussed next.
1) The formation of pores: It is assumed that the pores
are hydrophilic and, thus, able to conduct current, and created
with an initial radius r∗ at a rate determined by,
dn
= αe(Vm/Vep)2 1 −
dt n
(2)
neqVm
where n(t, θ) is the pore density and neq is the equilibrium
pore density for a given transmembrane voltage Vm,
neq(Vm) =n0e q(Vm/Vep)2
2) The evolution of pore radii: The rate of change of the
pore radii is given by
drj
dt = U(rj,Vm,σeff ),j =1, 2, ..., K (4)
where U is the advection velocity given by
U(r, Vm,σeff )= D
kT
V 2 mFmax
1+rh/(r + rt) +4β
r∗
r
4 1
r
+ D
kT [−2πγ +2πσeff r]
The last term represents the effective tension of the membrane,
σeff , which is a function of Ap, the combined area of all pores
existing on the cell,
σeff (Ap) =2σ ′ − 2σ′ − σ0
(6)
1 − Ap/A
where σ0 is the tension of the membrane without pores, σ ′ is
the energy per area of the hydrocarbon-water interface, Ap =
k j=1 πr2 j
(3)
(5)
, and A is the surface area of the cell. In this work
it is assumed that changes of cell shape, area, and volume,
can be ignored for microsecond pulses.
3) The voltage in the cell membrane: The voltage in the
cell membrane Vm can be calculated as the difference between
Vi and Ve, i.e. the difference between the potential at the
interfaces of the cell membrane, as indicated in the next Fig.2.
Vm(t, θ) =Vi(t, R2,θ) − Ve(t, R1,θ) (7)
The potential Vi at the inner interface (cytoplasm) and Ve
at the outer interface (outer region) of the cell membrane can
be obtained by two systems defined by Laplace’s equations,
∇ 2 Vi =0 and ∇ 2 Ve =0 (8)

σ , ε
σ , ε
σ , ε
Fig. 2. Cell membrane interface - not to scale
The applied electric pulse is imposed as a boundary condition
on the outer region Ve,
Ve(t, r, θ) =−Ercos(θ) as r →∞ (9)
where r is the distance from the center of the cell and θ
is the polar angle measured with respect to the direction of
the applied pulse E. In terms of numerical calculation, it is
sufficient to set r ≥ 3R1, i.e. the outer region at least three
times greater than the cell radius. This will be discussed in
detail in the numerical calculation subsection.
The other boundary conditions can be defined for t

TABLE I
PARAMETERS FOR THE CELL NUMERICAL MODEL [9]
Parameter Value
σc(Sm −1 ) 3.0 × 10 −1
σo(Sm −1 ) 3.0 × 10 −1
σm(Sm −1 ) 3.0 × 10 −7
εo(AsV −1 ) 6.4 × 10 −10
εc(AsV −1 ) 6.4 × 10 −10
εm(AsV −1 ) 4.4 × 10 −11
R1(m) 5.0000 × 10 −6
R2(m) 4.9995 × 10 −6
CM (Fm −1 ) 8.8 × 10 −3
n0(m −2 ) 1.5 × 10 9
α(m −2 s −1 ) 1 × 10 9
Vep(V ) 0.258
r∗(m) 0.51 × 10 −9
rm(m) 0.8 × 10 −9
rh(m) 0.97 × 10 −9
rt(m) 0.31 × 10 −9
T (K) 310
q 2.4606
D(m 2 s −1 ) 5 × 10 −14
γ(Jm −1 ) 1.8 × 10 −11
β(J) 1.4 × 10 −19
Fmax(NV −2 ) 0.7 × 10 −9
σ ′ (Jm −2 ) 2 × 10 −2
σ0(Jm −2 ) 1 × 10 −6
Vrest(V ) −0.08
It is capable of providing rectangular pulses with amplitude
of 1,000V and duration in the range of 1μs to100μs with
resting intervals of 10μs between the pulses. The modules are
discussed in detail next.
Fig. 4. Pulse generator modules.
1) The high voltage d.c. source: The high voltage source
module is based on [17] and uses a six capacitor doubler
stage, as indicated in Fig.5. It is capable of converting a.c.
voltage into d.c. voltage up to 2kV. It uses MUR1560 diodes
and 330μF capacitors. A 1:1 transformer is used to provide
insulation between the electrical grid and the module; in
addition, a variable autotransformer is used to control the a.c.
input and the d.c. output.
2) The control module: The control circuit uses a
PIC18F4550 microcontroller that operates at 12 Mips. The
control driver is based in a push-pull amplifier and composed
of a pair of Mosfets models ZVN2106a and ZVP2106a, both
- 178 - 15th IGTE Symposium 2012
Fig. 5. High voltage dc source.
capable of working with voltages up to 60V and currents
up to 500mA, as indicated in Fig.6. The control driver is
responsible for adjusting the control signal produced by the
microcontroller to the ratings required to fast charge the
capacitances of the switching circuit, which can be achieved
by increasing the output voltage and the maximum current.
Fig. 6. Driver circuit.
3) High voltage storage and switching: The high voltage
switching is based on a IGBT (IRGPS60B120KD) that operates
with voltages up to 1,2kV and current pulses up to 240A.
It has a 45ns rise time, 58ns fall time and 400ns turn off
delay that provides pulses with amplitude and width in the
range required in this work. The IGBT is connected in series
to a 73μF capacitor and the load, as indicated in Fig.7. The
capacitor charges through a 1kΩ resistor until it reaches the
same voltage of the high voltage source. When the IGBT is
activated, it inverts the voltage on the capacitor and a negative
voltage, that equals the one that charged the capacitor, appears
on the load.
Fig. 7. Switching circuit.

4) The pulse generator: The pulse generator built is shown
in Fig.8. For safety reasons, the high voltage source and the
high voltage storage and switching circuit are inside an acrylic
box; the control module is outside the box.
Fig. 8. Pulse generator.
C. In vitro experiment
1) Cell culture procedures: LLC-MK2 (monkey kidney
epithelial cell line) were maintained in DMEM (Dulbecco’s
modified Eagle’s medium - Invitrogen) with 10% fetal bovine
serum (FBS, Invitrogen), 1% penicillin-streptomycin (Invitrogen)
and 1% glutamine (Sigma-Aldrich) at 37C under 5%
CO2 atmosphere. Cell culture was kept at an optimal density
through weekly passages. Briefly, LLC-MK2 cells were seeded
until 70-90% confluent cell monolayer. To perform subculture,
the cell culture medium was removed and the cells were
rinsed twice with PBS -/-. Trypsin was used to remove
adherent cells (0,5ml/25cm2 surface area). The cells were then
resuspended in 4,5ml of fresh serum-containing medium for
trypsin inactivation. Cell viability was assessed directly by
Trypan Blue staining. Cultures were split 1:10 and placed in
a new flask with DMEM 10%.
2) Exposure to electroporation: LLC-MK2 cells were collected
from culture media and suspended in HBS (Hepes
Buffered Saline) at a concentration of 2, 6 × 106 cells/ml
in rectangular electroporation cuvettes with 1-mm electrode
separation. All the manipulations were done in sterile condition
in a vertical laminar flow cabinet Veco. Electroporation
was monitored with molecules of propidium iodide, 1,5-nm
× 2-nm, (25 μg/ml, Sigma-Aldrich), a fluorochrome that is
excluded from cells with intact membrane.
3) Fluorescence microscopy: Aliquots of control and
pulsed cells were placed into glass slides and observed in
Axioplan 2 Zeiss fluorescence microscope (UV emission 630
nm).
III. RESULTS AND DISCUSSION
A. Numerical simulations
The simulations considered a 5-μm radius cell submitted to
a 500kV/m rectangular electric pulse for 1-μs. The results for
the total number of pores, number of pores at the polarized
poles and the maximum radii of the pores are indicated at
- 179 - 15th IGTE Symposium 2012
Figs.9-11. It can be seen from Figs.9-11 that the pore nucleation
starts at approximately 0.8μs, when Vm is approximately
1.25V, an reaches ∼3,500 pores at the cell membrane, most of
them, ∼950, located at poles of the cell aligned to the applied
electric pulse (θ =0 ◦ and θ = 180 ◦ ), with radii sizes varying
from 0.5-nm to 13-nm. After the initial stage, the number of
pores increases until around 1-μs, when the pulse ends. In the
final stage, the radii of the pores decrease very fast but the
number of pores stays stable for a longer period. This is in
agreement with the literature, large pores tend to decay faster
but the complete resealing of all pores take a longer period.
Fig. 9. Total number of pores.
Fig. 10. Number of pores in the 1st sector (θ =0 ◦ ) and in the 180th sector
(θ = 180 ◦ ).
B. Experiments
A series of in vitro experiments were carried out with the
LLC-MK2 cells following the methodology described in the
previous section. After exposure to 1-μs pulse, the LLC-MK2
cells exhibited intense red fluorescence mostly in the cell
nuclei where propidium iodide binds to double-stranded DNA,
as illustrated in Fig.13. This is a strong evidence that pores
with radii of at least 2-nm (the size of the propidium iodide
molecule) were created at the cell membrane. The numerical

Fig. 11. Maximum radii evolution.
calculation predicts the creation of most pores at poles of
the cell (θ=180 and θ=0) with radii sizes varying from 0.5nm
to 13-nm; the fluoresce microscopy can only partially
confirm this. Further tests with other microscopy techniques
are required to confirm the region in the cell membrane where
the pores are created, their sizes and how long they last.
Fig. 12. Pulse of 500V × 1μs measured at the cuvette.
Fig. 13. Propidium iodide influx into LLC-MK2 cell exposed to pulses of
1-μs, 200V (a), 500V (b) and 700V (c).
IV. CONCLUSIONS
This paper considered the study of the electroporation
phenomenon in a single cell, using numerical simulations and
- 180 - 15th IGTE Symposium 2012
an in vitro experiment. The numerical simulations considered
a5-μm radius cell submitted to a 500kV/m rectangular electric
pulse for 1-μs, which was addressed using an asymptotic
approximation based on the Smoluchowski theory and the
finite difference method. The results indicate the formation
of ∼3,500 pores at the cell membrane, most of them, ∼950,
located at poles of the cell aligned to the applied electric pulse,
with radii sizes varying from 0.5-nm to 13-nm. The in vitro
experiment considered expositon of LLC-MK2 cells to electric
pulses of 200kV/m, 500kV/m, and 700kV/m, and 1-μs. Images
from fluorescence microscopy confirm the electroporation at
the LLC-MK2 cells. The methodology employed is adequate
to investigate the electroporation phenomenon in simple cells
exposed to electric pulses of kV/m and in the μs range.
V. ACKNOWLEDGMENT
This work was supported by CNPq, Brazil, under
Grants 306910/2006-3, 482185/2010-4, 507810/2010-4,
504978/2010-1; by FAPEMIG, Brazil, under Grants Pronex:
TEC 01075/09 and TEC-PPM-489/10; by CAPES, Brazil and
DFAIT, Canada.
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[9] W. Krassowska and P.D. Filev, “Modeling electroporation in a single cell,”
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Blackmore, E.S. Buescher, S.J. Beebe, “Ultrashort electrical pulses open a
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[17] E. Kuffel, W.S. Zaengl, J. Kuffel. High Voltage Engineering Fundamentals.
Second Edition, Butterworth Heinemann, Oxford, UK, 2000.

- 181 - 15th IGTE Symposium 2012
Anisotropic Model for the Numerical
Computation of Magnetostriction in
Grain-Oriented Electrical Steel Sheets
M. Kaltenbacher∗ ,A.Volk † , and M. Ertl ‡
∗Institute of Mechanics and Mechatronics, Vienna University of Technology, Austria
† Department of Sensor Technology, University of Erlangen-Nuremberg, Germany
‡ Siemens Energy Sector, Nuremberg, Germany
E-mail: manfred.kaltenbacher@tuwien.ac.at
Abstract—We present a recently developed physical model for magnetostriction in transformer cores and its efficient
numerical computation by applying the Finite Element (FE) method. Thereby, we fully take the anisotropic behavior of the
material into account, both in the computation of the nonlinear electromagnetic field as well as the induced magnetostrictive
strains. Numerical computations demonstrate the importance of modeling the anisotropy of grain oriented electrical steel
sheets as used in electric transformers. Both the magnetic field along the joint regions, and furthermore the mechanical
vibrations especially in thickness direction differ strongly as compared to computations with an isotropic material model.
Index Terms—magnetostriction, finite element method, anisotropic material behavior, nonlinearity
I. INTRODUCTION
Magnetostrictive materials are widely used for actuator
and sensor applications. However, often the magnetostrictive
behavior of these alloys is an undesirable effect, as
e.g. in electric machines and transformers, where it is one
of the main sources for noise generation. Unfortunately,
these materials exhibit nonlinear behavior for the magnetic
properties as well as the mechanical characteristics
leading to the well-known magnetic hysteresis loop and
the magnetostrictive hysteresis loop (so-called butterfly
curve), respectively (see, e.g., [1], [2], [3]). A quite
important aspect – especially for grain-oriented electrical
steel as used in transformers – is the anisotropic material
behavior both concerning the magnetic properties as well
as the induced mechanical strains [4].
The modeling of magnetostrictive effects is a topic of
intensive research. Among the huge amount of publications
one can find three main approaches. The first one,
which is widely used, is based on introducing a magnetostrictive
strain tensor, where the entries depend on the
magnetic induction (see, e.g., [5], [1]). Thereby, these
additional strains result in mechanical forces modeled as
a right hand side term in the partial differential equation
(PDE) for mechanics. In a second approach, a free
energy as a tensor function depending on the mechanical
strain and magnetic induction is used (see, e.g., [6], [7]).
Thereby, a fully coupled constitutive relation between
mechanical and magnetic quantities is achieved. The last
approach is based on a thermodynamic consistent model,
where the mechanical strain and magnetic induction is
decomposed in a reversible and an irreversible part [2],
[3]. Furthermore, the full constitutive model is based on a
free energy function. Whereas in [2] the irreversible part
is modeled by a switching criterion using inner variables,
[3] uses hysteresis operators. Common to all models
is the current restriction to isotropic and / or uniaxial
behavior.
Our goal is the precise investigation of the magnetic
field and resulting mechanical vibrations caused by magnetostriction
along the joint regions of electric transformers.
Therefore, we cannot apply any homogenization
technique and fully resolve each individual steel sheet.
This is clearly not possible for a whole transformer
core, and so we restrict our investigation to some few
steel sheets. To reduce the complexity, we choose an
ansatz, in which we neglect the reaction of the mechanical
stresses and strains on the magnetic properties and
therefore decouple the computation of the magnetic and
mechanical field. By help of an Epstein frame and a SST
(Single Sheet Tester), we measure the magnetic as well
as the mechanical hysteresis curves of the grain-oriented
electrical steel sheets with different orientations (w.r.t the
rolling direction). From these curves we then extract for
each orientation the corresponding commutation curve,
so that the hysteretic behavior is simplified to a nonlinear
one. This approach is then applied to a stack of six
electrical steel sheets with a 90 o joint region, excited
by two current loaded coils. We compare this anisotropic
model to an isotopic one, where the nonlinear magnetic
and mechanical material parameter are just used from the
rolling direction.
The rest of the paper is organized as follows. In Sec.
II we describe our physical model and its in-cooperation
into the magnetic and mechanical PDE as well as their Finite
Element (FE) formulation. The measurement setups,
which provide us the nonlinear curves, are discussed in
Sec. III. In Sec. IV the numerical results are presented,
demonstrating the importance of taking anisotropy for
grain-oriented electrical steel sheets as used in transformers
into account. Finally, Sec. V summarizes our
achievements.

II. PHYSICAL MODELING AND FE DISCRETIZATION
Magnetostrictive materials are characterized by the
magnetic hysteresis between the magnetic induction B
and magnetic field intensity H as well as the mechanical
hysteresis between the mechanical strain S and magnetic
induction B (see Fig. 1).
Fig. 1. Magnetic and mechanical hysteresis (butterfly curve).
According to a thermodynamically consistent model,
we decompose the physical quantities magnetic induction
and mechanical strain into a reversible and an irreversible
part1 (indicated by the superscripts r and i, respectively)
S = S r + S i , B = B r + B i . (1)
To allow for the history of the driving magnetic field
intensity, the irreversible magnetic induction Bi is set to
be equal to the magnetization M, which is modeled, e.g.,
by a Preisach hysteresis operator [3]
B i = M = H[H] eM . (2)
The irreversible strain can be, e.g., expressed by the
following polynomial ansatz [3]
i
S = 3
2 (β1 ·H[H]+β2 · (H[H]) 2 + ···
+βn · (H[H]) n
) eM ⊗ e t M − 1
3 [I]
, (3)
while the parameters β1, ···,βn need to be fitted to
measurement data and [I] denotes the identity tensor.
Now, magnetostriction is a property of ferromagnetic
materials and can be described as a coupling between the
mechanical and the magnetic field. This relation is described
by the well-known magnetostrictive constitutive
equations modeling the linear coupling of the magnetic
and the mechanical deformation [2]
σ = [c H ]S r − [e] t H (4)
B r = [e]S r +[μ S ]H . (5)
In (4), (5) σ denotes the Cauchy stress tensor in Voigt notation,
[cH ] the tensor of mechanical moduli (at constant
magnetic field intensity), [e] the piezomagnetic coupling
tensor and [μS ] the tensor of magnetic permeability (at
constant mechanical strain).
By using these constitutive relations, we have presented
in [3] a formulation based on the magnetic
1 With [S] we denote the tensor of mechanical strain and with S
the algebraic vector containing the three normal and three shear strains
according to Voigt notation.
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scalar potential, and in [8] have even extended it for
the magnetic vector potential to also take eddy current
effects into account. However, both models are currently
restricted to scalar hysteresis operators, and do not take
into account the anisotropic behavior, which is a crucial
point for grain-oriented electrical steel used in transformer
cores [4]. Furthermore, both models are quite
expensive concerning computational time. Therefore, we
have developed a dedicated physical model for grainoriented
electrical steel. In doing so, we first assume that
the entries of the piezomagnetic coupling tensor [e] are
small, and we are allowed to neglect this coupling in (4),
(5). Next the anisotropic and nonlinear magnetic behavior
of the steel sheets is modeled by its vector relation
between the magnetic induction B and field intensity H
B = B (H) =Bϕ(H)eB ; eB = B
. (6)
B
Here, we compute the unit vector eB and evaluate the
magnetic commutation curve Bϕ for which the orientation
fits best with eB. Therefore, the defining partial
differential equation (PDE) for the magnetic field reads
as
γ ∂A
∂t −∇×ν(Bϕ)∇×A = Ji (7)
with A the magnetic vector potential, Ji the impressed
current density, ν the magnetic reluctivity depending on
Bϕ (see (6)) and γ the electric conductivity.
The PDE for mechanics is given by
ρ ∂2u ∂t2 −Btσ =0 (8)
with u the mechanical displacement, ρ the density, and
B = ∇s the differential operator. As in (3), we assume the
conservation of volume for the irreversible strain. However,
we now model instead of the hysteretic behavior
a nonlinear, anisotropic behavior, and denote it by [Sm ]
(magnetostrictive induced strain tensor), which computes
as follows
[S m ]= 3
2
eB × e t B − 1
3 I
S m ϕ (B) . (9)
Here, we compute the direction of B and evaluate the
magnetostrictive commutation curve S m ϕ (B) for which
the orientation fits best with eB. Now, we can express
the reversible mechanical strain S r by the difference of
the total strain S = Bu and the irreversible (magnetostrictive)
strain S i = S m via
S r = Bu − S m . (10)
This relation in combination with (4) by neglecting [e]
results for (8) into
ρ ∂2 u
∂t 2 −Bt [c H ]Bu = −B t [c H ]S m . (11)
The Finite Element (FE) formulation of (7) and (11)
is straight forward. For (7) we use edge finite elements
and solve the arising algebraic system of equations by
an efficient Newton scheme utilizing a two level solver

[9]. For (11) we apply nodal finite elements (for details,
see e.g., [10]).
Summarizing, the developed magnetostrictive model
has the following features:
• Decoupling of magnetic and mechanical PDEs; so
both PDEs can be solved separately with optimal
conditions.
• Anisotropy and eddy currents are taken into account
• No hysteresis considered; instead it uses commutation
curves computed from measured hysteresis
curves.
• Change of magnetic properties due to the mechanical
field within a working point is neglected
(working point can be determined by pre-stressing
of measured samples).
III. MEASUREMENT SETUPS
First of all, to obtain reliable measurement data for the
magnetic behavior, we have constructed an Epstein frame
according to IEC 60404-2 (see Fig. 2). The 25 cm Epstein
Steel sheets (overlap at the corners)
Coil to compensate flux in air
Excitation and
measurement
coils
Fig. 2. 25 cm Epstein frame for measuring the magnetic properties
of grain-oriented electrical steel sheets.
apparatus consists of 4 coils with primary windings,
secondary windings, a compensation coil and the material
sample as core. The sheets are stratified in stripes. The
measurement setup represents in this way a transducer,
whose characteristics are specified. The primary outer
windings are used to magnetize the material and the
secondary inner windings are needed for magnetic flux
density determination over the induced voltage. We have
performed measurements for steel sheets, which have
been cut out at different angles according to the rolling
direction. Thereby, for each stack of steel sheets, we
have measured the outer and all inner hysteresis loops, as
demonstrated in Fig. 3. Out of all the hysteresis loops,
we compute for each angle a commutation curve (see
Fig. 4), which we then use in our numerical computation
for the magnetic field.
To measure the mechanical hysteresis of the electrical
steel sheets a second measurement setup was constructed
on the basis of a Single Sheet Tester (SST) as displayed
in Fig. 5. This extended setup also captures
- 183 - 15th IGTE Symposium 2012
B (T)
H (kA/m)
Angle 0 o
Angle 15 o
Angle 30 o
Angle 45 o
Angle 60 o
Angle 75 o
Angle 90 o
Fig. 3. Magnetic hysteresis curves for grain-oriented electrical steel
sheets being cut out at different angles according to the rolling direction
(0 o corresponds to the rolling direction).
B ( (T) )
H (kA/m)
Angle 0 o
Angle 15 o
Angle 30 o
Angle 45 o
Angle 60 o
Angle 75 o
Angle 90 o
Fig. 4. Nonlinear BH curves for different angles (0 o corresponds to
the rolling direction).
the magnetic induction as well as the magnetic field
intensity. However we did not use the SST to determine
magnetic hysteresis since it has to be calibrated to a
certified setup to obtain reliable measurement results. To
capture the mechanical hysteresis the SST was extended
by a lifting mechanism to unload the sample sheet to
ensure its stress-less vibration. The mechanical vibration
due to magnetic excitation of the SST is measured by
Ferrite core
Excitation n and
Steel sheet measurement ment coil
(material under test)
Laser-
vibrometer
Fig. 5. Single sheet tester as used to obtain the mechanical hysteresis
(principle and manufactured setup).

a laser vibrometer, which compared to strain gauges
provides high accuracy without electromagnetic crosssensitivity
and is contact-free. The measurement of the
mechanical strain as a function of the magnetic field
results in the magnetostrictive hysteresis loop (so-called
butterfly curve). Additionally the extended SST permits
pre-stressing of the steel sheets in order to capture the
reaction on the magnetic properties which corresponds to
a working point that is used in the simulation. To consider
anisotropy, again a series of measurement is performed
with different electrical steel sheets which have been cut
out with varying cutting angles with respect to the grain
orientation of the steel. As for the magnetic hysteresis, we
also convert the mechanical hysteresis in a single commutation
curve, which leads to angle-dependent nonlinear
magnetostriction curves, as displayed in Fig. 6.
S (μm/m)
Angle 0 o
Angle 15 o
Angle 30 o
Angle 45 o
Angle 60 o
Angle 75 o
Angle 90 o
B (T)
Fig. 6. Nonlinear SB curves for different angles (0 o corresponds to
the rolling direction).
IV. NUMERICAL RESULTS
For the numerical investigation, we choose a setup of
six stacked electrical steel sheets with a 90 degree joint
and an excitation coil along each yoke as displayed in
Fig. 7. We model just a quarter symmetry by applying
Steel sheets
Excitation coils
Zoomed and scaled
in thickness direction
Fig. 7. Computational model: quarter symmetry is considered.
appropriate boundary conditions at the symmetry planes.
Our main goal is to study the difference between an
isotropic and anisotropic magnetostrictive computation.
- 184 - 15th IGTE Symposium 2012
Thereby, we choose for the isotropic computation the
measured material curves along the rolling direction
(angle of zero degree), whereas for the anisotropic computation
we use all measured material curves (see Fig. 4
and 6).
In a first step, we compute the magnetic field and
compare the flux lines at the joints. Figure 8 displays the
flux lines for the isotropic and Fig. 9 for the anisotropic
case at the time step of maximal magnetic induction
(about 1.7 T). We display the flux lines just for the two
Fig. 8. Magnetic flux lines for the two upper steel sheets in case
of isotropic computation. For better visualization we have scaled the
thickness direction by a factor of ten.
last layers and zoom into the joint region. Comparing
the results, one can clearly see the difference. For the
Fig. 9. Magnetic flux lines for the two upper steel sheets in case
of anisotropic computation. For better visualization we have scaled the
thickness direction by a factor of ten.
isotropic case, the amplitude of the magnetic induction
immediately drops to a low one at the beginning of the
joint due to the increased effective cross section when
turning the flux direction in the rectangular joint region.
Since the magnetic material properties are homogeneous
and independent of direction, the change of the magnetic
flux direction itself is continuously across the joint region.
The transition of the magnetic flux between the
two vertical stacked steel sheets is mainly limited when
entering and leaving the joint region. Accordingly the
magnetic flux density reaches its full value just at the
end of the joint when entering the opposite yoke.
In the anisotropic case, the guiding effect of the
preferred magnetic direction in the grain orientation of
the electrical sheet keeps the amplitude and direction of
the magnetic flux for some distance in the joint region.

In the area of the central diagonal of the joint region (at
45 o ), the reduced magnetic permeability perpendicular
to the grain orientation forces the magnetic flux to a
vertical transition into neighbouring steel sheets. This
x-Displacement (nm)
y-Displacement (nm)
10
0
-25
20
0
-50
0 20 40 60
Time (ms)
0 20 40 60
Time (ms)
z-Displacement (nm)
2.5
0
-2.5
Evaluation point
0 20 40 60
Time (ms)
Isotropic case
Anisotropic case
Fig. 10. Mechanical displacement at an observation point along the
yoke.
behavior of the magnetic field has a strong impact on the
mechanical vibrations. In a second step we use the computed
magnetic induction and calculate the mechanical
deformation according to the additional magnetostrictive
strain. In Figs. 10 and 11 we display all three components
of the mechanical displacement over time at
two different observation points. In general, we observe
that the displacement in plane direction (x− and y−
displacement) show almost no difference. However, the
displacement in thickness direction (z− displacement) is
quite different both concerning amplitude and frequency
content. Especially at the joint region the amplitude of the
mechanical vibration is a factor of about 1000 larger in
the anisotropic case as in the isotropic case. Furthermore,
we can state that the computation for the isotropic
material model exhibits mainly the 100 Hz component
(current excitation is at 50 Hz). In the anisotropic case
higher harmonics are predominant. The related frequency
spectrum in vibration and noise is typical what can be
measured at real transformers.
V. CONCLUSION AND OUTLOOK
We have presented a magnetostrictive constitutive
model which fully takes the anisotropy of grain-oriented
electrical steel sheets as used in electrical transformers
into account. The model itself is simplified in this
sense that the magnetic as well as mechanical hysteretic
behavior is reduced to a nonlinear one by computing
commutation curves out of the corresponding hysteresis
measurements. Furthermore, we neglect the impact of
the mechanical field on the magnetic properties within a
working point, which can be determined by pre-stressing
the measured sample sheets. However, the model needs
measurements provided by an Epstein frame and a SST
- 185 - 15th IGTE Symposium 2012
x-Displacement (nm)
y- Displacement (nm)
0
-45
0
0 20 40 60
Time (ms)
-45
0 20 40 60
Time (ms)
z - Displacement (nm)
8
0
-6
Evaluation point
Scaled by 1000
0 20 40 60
Time (ms)
Isotropic case
Anisotropic case
Fig. 11. Mechanical displacement at an observation point at the joint
region.
(Single Sheet Tester). The computations show strong differences
both in the magnetic field as well as mechanical
vibrations when comparing this anisotropic model to an
isotropic one, which just uses measured curves in rolling
direction of the steel sheets.
Currently we are working on an experimental validation
setup, where we can study different joint techniques,
especially step-lap joints.
REFERENCES
[1] L. Vandevelde, J. A. Melkebeek. Modeling of Magnetoelastic
Material. IEEE Trans. on Magnetics, 38(2), 2002.
[2] K.Linnemann, S. Klinkel, W. Wagner. A constitutive model for
magnetostrictive and piezoelectric materials. International Journal
of Solids and Structures 46, 2009.
[3] M. Kaltenbacher, M. Meiler, M. Ertl. Physical modeling and
numerical computation of magnetostriction. Compel, 28(4), 2009.
[4] B. Weiser, H. Pfützner, J. Anger. Relevance of Magnetostriction
and Forces for the Generation of Audible Noise of Transformer
Cores IEEE Trans. on Magnetics, 36(5), 2000.
[5] K. Delaere, W. Heylen, K. Hameyer, R. Belmans. Local Magnetostriction
Forces for Finite Element Analysis. IEEE Trans. on
Magnetics, 36(5), 2000.
[6] A. Dorfmann and R. W. Ogden. Magneto-elastic modeling of
elastomers. Eur. J. Mechanics and Solids, 22, 2003.
[7] K. Fonteyn, A. Belahcen, R. Kouhia, P. Rasilo, A. Arkkio. FEM
for Directly Coupled Magneto-Mechanical Phenomena in Electrical
Machines. IEEE Trans. on Magnetics, 46(8), 2010.
[8] A. Volk, M. Kaltenbacher, A. Hauck, M. Ertl, R. Lerch. Finite Element
Scheme based on Magnetic Vector Potential and Mechanical
Displacement for Modeling Magnetostriction. Proceedings of the
8th International Conference on Computation in Electromagnetics
CEM, 2011.
[9] A. Hauck, M. Ertl, J. Schöberl, M. Kaltenbacher. Accurate Simulation
of Transformer Step-Lap Joints using Anisotropic Higher
Order FEM. 15 th IGTE Symposium, Graz, Austria, 2012.
[10] M. Kaltenbacher. Numerical Simulation of Mechatronic Sensors
and Actuators. Springer, 2nd edition, 2007.

- 186 - 15th IGTE Symposium 2012
Analytic Approximation Solution for the
Schwarz-Christoffel Parameter Problem
Norbert Eidenberger∗ and Bernhard G. Zagar∗ ∗Institute for Measurement Technology, Altenberger Strasse 69, A-4040 Linz, Austria
E-mail: norbert.eidenberger@jku.at
Abstract—We present a novel analytic approximation method for the Schwarz-Christoffel parameter problem based on
linearization. The modeling requirements for successful linearization are discussed. The linearization introduces a mapping
error which can be virtually eliminated by applying an optimization method. Thus, the proposed method yields conformal
mapping functions which can provide solutions for inverse problems and are suited for sensitivity analyses.
Index Terms—Schwarz-Christoffel parameter problem, Schwarz-Christoffel transform, conformal mapping, potential
problem.
I. INTRODUCTION
Conformal mapping methods provide useful tools for
the analysis of many physical phenomena. In particular,
these methods can be utilized to solve two dimensional
potential problems which appear e. g. in electromagnetics,
fluid dynamics, or heat transfer [1], [2]. The
general idea consists in transforming a potential problem
bounded by a complicated geometry to a simpler one,
for which the solution can be computed more easily.
The transformation is performed by a conformal mapping
function. Subsequently, this function also transforms the
solution of the simpler problem to the complicated one
which provides the solution of the original problem.
Some recent examples for the application of conformal
mapping methods are [3], [4], and [5].
For the purpose of conformal mapping the coordinates
of two dimensional problems are interpreted as the real
and imaginary parts of complex numbers. Thus, the
mapping functions represent complex valued functions
in complex variables [6]. Different methods are available
for the construction of suitable mapping functions [2].
One often utilized method, the Schwarz-Christoffel transform
(SCT), constructs mapping functions for polygonal
geometries. Many relevant technical problems involve
polygon shaped boundaries therefore the SCT plays an
important role in many applications.
In order to employ the SCT, its problem dependant
parameters need to be computed. It is not possible to
compute the parameters for polygons with more than
three corners analytically, although a unique solution for
the SCT parameters exists. This constitutes the so-called
SCT parameter problem [7]. Nowadays numerical methods
are routinely employed to solve the SCT parameter
problem [8]. However, numerical methods yield solutions
which are disconnected from the original problem geometry.
This prevents further analysis with respect to the
geometric parameters of the original problem.
In this paper we propose a solution method for the SCT
parameter problem based on a series expansion of the
SCT base function (1). The method yields an approximate
analytic solution for the SCT parameters containing the
geometry parameters. Due to the approximation error
the resulting mapping function produces mapping errors.
We show that the mapping errors can be eliminated by
prewarping the geometric parameters appropriately. This
is achieved through an optimization method which minimizes
the mapping error. The advantage of the proposed
method over the standard numerical solution consists in
the presence of the geometry parameters in the mapping
function which permits further analysis of the problem,
e. g. sensitivity analyses or solving inverse problems.
This paper consists of three main parts. The first part
gives a short introduction to the SCT together with
the corresponding parameter problem. The second part
describes the approximation method for the solution of
the SCT parameter problem. The third part presents the
minimization method which eliminates the mapping error.
Finally, the conclusion sums up the properties of the
proposed method and highlights its potential advantages
and applications.
II. THE SCHWARZ-CHRISTOFFEL TRANSFORM
The SCT represents a widely utilized method for
constructing conformal mapping functions. The SCT base
equation,
n
z = f(w) =A (w − wi) αi π −1
dw + B, (1)
i=1
maps the upper half of the image (w-) plane to the
inside of a polygon in the object (z-) plane [9] which
is illustrated in Fig. 1. It contains several unknown
parameters. Parameter A represents a scaling and rotation
factor, parameter B represents a translation, and the
parameters wi represent the corner coordinates in the
w-plane with the known parameters αi representing the
corresponding interior angles.
The unknown parameters A, B and wi are computed
by comparing the polygon corners coordinates in both
planes via the relation z = f(w). The resulting number of
equations equals the number of unknowns, which means
that a unique solution exists. However, for polygons with
more than three corners the integral in (1) yields special

- 187 - 15th IGTE Symposium 2012
Fig. 1. Example setup for the Schwarz-Christoffel transform showing the geometric parameters of a rectangle in the z-plane and its image in the
w-plane.
functions for which no inverse functions exist. This
prohibits the analytic computation of the SCT parameters
even though it is known that a unique solution exists. This
constitutes the SC parameter problem [7].
Nowadays, the parameter problem is usually solved
numerically. A thorough discussion of numerical solution
methods for the SC parameter problem is presented in
[8]. The authors of [8] also provide a Matlab toolbox
[10] which permits an easy application of the SCT.
For quadrilaterals such as rectangles the SCT produces
mapping functions which consist of elliptic functions.
In these cases the elliptic modulus can be utilized to
efficiently compute the parameters numerically [7]. An
application example for this approach is presented in
[11].
III. ANALYTIC APPROXIMATION OF THE PARAMETER
PROBLEM
The disadvantage of purely numerical solutions of the
SC parameter problem consists in the missing relation to
the original geometry. This prevents subsequent analyses
of problems with respect to their geometry. Preserving
this relation requires an analytic approach which is
developed below.
The proposed method consists of several steps. The
first step consists in constructing the SCT for the problem
at hand. The evaluation of the integral in (1) then yields
a mapping function for which the SC parameters need to
be computed. In order to compute them analytically, the
mapping function is linearized. Then the SC parameter
problem is solved for the linearized mapping function.
The results are inserted back into the original nonlinear
mapping function which then contains geometry parameters
of the original problem.
Unfortunately, the procedure is not quite that straight
forward. Several problems which may occur during linearization
need to be addressed, before the method can
be applied successfully.
A. Method Development
An analysis of (1) shows that the SCT lends itself to
Taylor series expansion. Equation (1) can be rewritten as
z = A g(w)dw + B (2)
where
g(w) =
n
i=1
(w − wi) α i
π −1
which represents the transformation core of the SCT
defining the general polygon shape. Equation (2) indicates,
that beginning with the first order term, the Taylor
series consists only of the transformation core g(w) and
its derivatives. Because g(w) represents a product, this
means that the series expansion consists mainly of simple
functions.
Expanding (2) as a Taylor series yields
(3)
z = f(w0)+Ag(w0)(w − w0)+O(w 2 ) (4)
where O(w2 ) represents the higher order terms of the
series. In order to be able to solve the parameter problem,
the SC parameters must not appear within non-invertible
functions. The non-invertible special functions contained
in f(w) disappear in the first and higher order terms
of the series. However, there remains a special function
within the zeroth order term
f(w0) =A g(w)dw
+ B. (5)
w=w0
The special function vanishes if the result of the integral
at w0 equals zero. Parameter B represents a translation
of the mapped polygon in the z-plane, thus it can be set
to B =0without loss of generality. This corresponds
to mapping the origins of the w- and z-plane onto each
other, so the Taylor series expansion is centered at w0 =
0 and f(w0) =0. If a translation of the mapping result is
truly required, this can be achieved by a simple additional
mapping function.
Truncating (4) after the first order term and incorporating
the above considerations yields
z = Ag(w0)(w − w0) (6)
which represents a mapping function linearized with
respect to the image coordinates w. Note, that (6) usually
will not be linear with respect the geometric parameters.
The condition developed in the previous step requires
that the origins of the planes are mapped onto each
other. This can only be guaranteed if the coordinates of
a point are known in both planes. This is the case for the
polygon corners which are present in the transformation
core g(w).

- 188 - 15th IGTE Symposium 2012
Fig. 2. Example for a symmetric polygon which features a point besides the corners for which its coordinates are known in both planes.
The transformation core consists of a product of terms
which contain the coordinates of the polygon corners
in the w-plane. However, for a corner at w0 = 0 the
transformation core evaluates either to zero or infinity,
depending on the angle αi in the exponent corresponding
to the corner.
g(0) = 0 if α>π (7)
g(0) = ∞ if α

Fig. 3. Illustration of the mapping error introduced into the mapping
function by the linearization.
In this case the mapping error can be defined as the
distances between the ideal and the actual mapped corner
positions in the z-plane
i=1
Δz = f
w,
d − z, (13)
where the components of w represent the coordinates
wi of the polygon corners in the w-plane, and z and
Δz consist of the corresponding ideal polygon corner
positions and mapping errors in the z-plane.
Equation (13) is not suited as an objective function
for minimizing the mapping error because the sign of
the mapping error may change. In order to avoid this,
the sum of the square of the real and imaginary part of
the mapping errors is utilized as the objective function
n
min Re(Δzi) 2 +Im(Δzi) 2
, (14)
where n represents the number of polygon corners.
Equation (14) can be reformulated as
min Δz T Δz ∗ , (15)
where Δz ∗ represents the vector containing the complex
conjugate mapping error. The objective function in (15)
forms a convex function with a global minimum at 0
which is known to exist. In addition, (15) consists of a
sum of squares. For this type of functions exist specialized
optimization algorithms [12], [13], which ensures
that a solution can be computed easily.
V. CONCLUSION
In this paper we have presented a method for computing
an analytic approximation solution of the SC
parameter problem. The proposed method takes advantage
of the structure of the SCT base equation and
shows that a linearization of the problem is possible
under certain conditions. These conditions are defined
and their consequences regarding the modeling process
are discussed.
The resulting conformal mapping function contains
approximation errors. We presented a formulation of the
corresponding mapping error which permits its minimization
to arbitrarily small dimensions by varying the
geometric parameters of the original polygon. Thus, the
proposed method yields a conformal mapping function
which produces the desired map.
- 189 - 15th IGTE Symposium 2012
The advantage of the proposed analytic over the conventional
numeric approximation consists in the presence
of the geometry parameters in the mapping function.
Together with the minimization procedure this permits
• the solution of inverse problems e.g. in capacitive
sensing applications,
• the sensitivity analysis of sensor setups with respect
to their geometry [14].
ACKNOWLEDGMENT
The authors gratefully acknowledge the partial financial
support for the work presented in this paper by the
Austrian Research Promotion Agency and the Austrian
COMET program supporting the Austrian Center of
Competence in Mechatronics (ACCM).
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Wiley & Sons, 1974, vol. 1.
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[8] T. A. Driscoll and L. N. Trefethen, Schwarz–Christoffel Mapping,
P. Ciarlet, A. Iserles, R. Kohn, and M. Wright, Eds. Cambridge
University Press, 2002.
[9] V. I. Smirnov, Lehrbuch der höheren Mathematik, 14th ed. Harri
Deutsch, 1995.
[10] T. Driscoll. (2012, Sep.) Schwarz-Christoffel toolbox
for MATLAB. University of Delaware, Department
of Mathematical Sciences. [Online]. Available:
http://www.math.udel.edu/ driscoll/software/SC/
[11] R. Igreja and C. Dias, “Extension to the analytical model of the
interdigital electrodes capacitance for a multi-layered structure,”
Sensors and Actuators A: Physical, vol. 172, no. 2, pp. 392–399,
2011.
[12] R. Fletcher, Practical methods of optimization, 2nd ed. Wiley,
2000.
[13] P. E. Gill, W. Murray, and M. H. Wright, Practical optimization.
Academic Press, 1981.
[14] N. Eidenberger and B. G. Zagar, “Sensitivity of capacitance
sensors for quality control in blade production,” in ICST 2011,
Dec. 2011.

- 190 - 15th IGTE Symposium 2012
Additional Eddy Current Losses in Induction
Machines Due to Interlaminar Short Circuits
P. Handgruber∗ , A. Stermecki∗ ,O.Bíró∗ , and G. Ofner †
∗Institute for Fundamentals and Theory in Electrical Engineering (IGTE),
Christian Doppler Laboratory for Multiphysical Simulation, Analysis and Design of Electrical Machines,
Inffeldgasse 18/I, A-8010 Graz, Austria
† ELIN Motoren GmbH, Elin-Motoren-Straße 1, A-8160 Preding/Weiz, Austria
E-mail: paul.handgruber@tugraz.at
Abstract—A novel three-dimensional eddy current model to account for the additional losses caused by interlaminar short
circuits is presented and applied to the loss estimation of an induction machine. The method is based on a single sheet model
with appropriate boundary conditions on the interlaminar contact surface avoiding cumbersome full three-dimensional
models with multiple short circuited laminations. The results of the single sheet model without short circuits are compared
to measured no-load iron loss data. The interlaminar model is validated by means of full models comprising several
interconnected sheets and used for the quantification of extra losses caused by conductive joints and shearing burrs. It has
been found that particularly burrs occurring on the tooth edges lead to a significant loss increase.
Index Terms—AC-machines, eddy currents, electric machines, electromagnetic modeling, finite element methods, magnetic
losses
I. INTRODUCTION
Active iron parts in electrical machines are commonly
built of thin steel laminations. In an ideally assembled
core, the individual laminates are isolated from each
other in order to minimize the effects of eddy currents.
During the cutting process, mechanical deformations lead
to microscopic shearing burrs on the cutting edges. This
edge burrs can break down the insulation resulting in
conductive connections between the stacked sheets. If
the burr-induced short circuits cover several laminations,
high currents begin to circulate leading to a significant
loss increase and hence to local overheatings [1].
The core stacks are frequently held together by conductive
joints, such as bolts, welds or clamping bars.
In most cases, these fixations as well as the shaft and
parts of the frame are mounted uninsulated on the core,
short-circuiting a large number of laminations. Further
interlaminar contacts are induced by small insulation
faults on the lamination surfaces inside the core middle.
However, the probability of such inner short circuits is
very low and stochastic [2]. Therefore, their effects are
not considered in this work.
Up to now, the complex problem of interlaminar short
circuits has been chiefly treated by statistical and/or empirical
methods accompanied by vast measurement series
[3], [4]. Attempts to quantify the resulting additional
losses are mostly based on analytical approaches. For
instance, in [5], [6] and [7] artificial burrs have been applied
in a controlled manner to a distribution transformer.
Comparisons of the performed measurements to an analytical
eddy current model showed only poor correlation
due to the simplifications made. In [8], [9] and [10] the
losses have been evaluated using a resistance network
analogy. The circuit parameters have been derived from
small-scale models based on simple geometries.
In order to enable studies on more complicated structures
like an electrical machine, a novel method based
on a three-dimensional (3-D) finite element analysis
is proposed in this work. The method is capable to
compute the true paths of the eddy currents and values
of the ensuing losses. A full 3-D model with multiple
joined laminations is avoided by introducing appropriate
boundary conditions on the interlaminar contact surface
of a single sheet model.
This paper is organized as follows: In section II a
novel 3-D eddy current model considering the effects of
interlaminar short circuits is introduced. In section III
the method presented is applied to the loss estimation
of a megawatt rated slip-ring induction machine. First,
the single sheet model without short circuits is compared
to no-load iron loss measurements. The validation of
the interlaminar contact model is performed using full
models with several short circuited laminations. Finally,
the effects of conductive joints and shearing burrs are
subjected to an in-depth analysis.
II. 3-D EDDY CURRENT MODEL
The proposed method can be subdivided into two steps:
first, a transient two-dimensional (2-D) field analysis is
carried out for the whole machine. In a second step, a
transient nonlinear 3-D eddy current problem is solved
separately for an individual stator and rotor sheet.
The 3-D model is excited by time-dependent boundary
conditions obtained from the 2-D field analysis. This
allows separate treatment of the stator and rotor sheets
and avoids cumbersome and computationally expensive

transient 3-D finite element simulations including rotor
motion [11]. Nevertheless, all loss relevant effects, like
high-order field harmonics and the rotor movement are
included in the analysis.
Only a single sheet is considered in the 3-D model,
the effects of interlaminar short circuits are taken into
account by boundary conditions on the contact surface.
This approach is valid insofar as, for multiple interconnected
laminates, the electromagnetic quantities in the
sheets within the core become periodic.
A nodal based A,V -A formulation is employed for the
analysis of the 2-D problem. The 2-D iron regions are assumed
to be non-conductive. The massive rotor windings
are modeled as eddy current domains but not the stator
conductors. The starting transients of the machine are
bypassed by using the initial steady state solution from
a time harmonic analysis. The 3-D eddy current problem
is solved solely in the conductive laminations using the
A,V formulation based on isoparametric hexahedral edge
elements with quadratic shape functions [12]. Introducing
the magnetic vector potential A, in the whole problem
domain Ω and the electric scalar potential V , in the eddy
current region Ωc as
B = ∇×A,
E = − ∂ ∂
A −
∂t ∂t ∇V
⎫
⎬
⎭ in Ωc, (1)
B = ∇×A in Ω − Ωc
(2)
where B is the magnetic flux density, E is the electric
field intensity and t is time, the Maxwell equations under
quasistatic approximation can be written as
∇×ν∇×A + σ ∂ ∂
A + σ ∇V = 0,
∂t ∂t
∇· σ ∂ ∂
A + σ
∂t ∂t ∇V
⎫
⎪⎬
in Ωc, (3)
=0 ⎪⎭
∇×ν∇×A = J0 in Ω − Ωc. (4)
Herein ν denotes the reluctivity and σ the conductivity of
the sheet material, J0 represents the given current density
of the stranded coils in the 2-D model.
Fig. 1 shows the specifications of the boundary conditions
for the 3-D model of a stator sheet. The boundary
conditions are prescribed anew for every time step. On
the boundaries along the lamination thickness, the model
is excited by the normal component of B derived form
the 2-D field analysis. The prescription of the normal
component of B is equivalent to specifying the tangential
component of A. This component is obtained from the
2-D field vector potential A2-D and is assumed to be
constant along the thickness. The boundaries along the
sheet thickness are hereinafter called axial boundaries.
Since flux in the axial direction is neglected, the tangential
component of A is set to zero on the boundaries
parallel to the laminations.
Setting the electric scalar potential free on the boundaries
of Ωc denoted by Γi ensures the satisfaction of the
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Fig. 1. Specification of the boundary conditions for a 3-D stator sheet
model. The sheet thickness is increased for better visibility. Symmetry
boundary conditions on the bottom surface allow to consider only half
of the thickness.
natural boundary condition of vanishing normal component
of the current density J here. At the interlaminar
contact surface Γc, the current flow is assumed to be
normal to the face, resulting in a constant and unknown
electric scalar potential. Furthermore, it has to be guaranteed
that the exchanged current through the contact spots
between the sheets is zero by introducing the surface
integral relation
Γc
σ
∂ ∂
A +
∂t ∂t ∇V
· n dΓ = 0 (5)
as an additional equation into the finite element equation
system [13]. The symbol n stands for the outer normal
vector.
For interlaminar contacts occurring on the sheet boundaries
like shearing burrs, the effects of the high circulating
eddy currents on the given magnetic boundary field
cannot be neglected. In such cases, the magnetic field
density is not prescribed directly on the axial sheet
boundaries, but on an additional outer finite element layer
surrounding the lamination. This layer is modeled as nonconductive
extending the 3-D problem to an A,V -A one.
Thus, the required independence of the prescribed field
from the eddy currents is ensured again.
III. APPLICATION AND RESULTS
The method presented has been applied to the loss
estimation of a megawatt rated, 50 Hz, 690 V, deltaconnected
three-phase, four pole slip-ring induction machine
fed by sinusoidal voltage. In the case of a healthy
machine without short circuits present in the core, the
computed total iron losses are compared to no-load
measurements. After validating the proposed interlaminar
contact model against full models with many short
circuited sheets, the additional losses due to conductive
joints and shearing burrs will be quantified.

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(a) (b)
Fig. 2. Eddy current loss density distribution in a stator (a) and rotor (b) sheet at rated no-load current and a specific time instant.
A. No-load iron losses
According to the statistical loss theory, the total iron
losses are composed of eddy current, hysteresis and
excess losses [14]. The computed 3-D eddy current
loss distribution for a healthy stator and rotor sheet
is presented in Fig. 2. The losses are quite uniformly
distributed in the stator sheet and mainly attributable to
the 50 Hz fundamental frequency component. The rotor
losses are concentrated in the vicinity of the air-gap
and primarily evoked by the first stator slot harmonic at
1800 Hz. In order to cover all relevant harmonic effects,
the time step size Δt was fixed to Δt = T/500 for the stator
and Δt = T/1000 for the rotor sheet simulation; T is the
time period of the fundamental frequency. The total eddy
current losses are obtained by integrating the product of
E and J over the sheet volume, averaged over time and
multiplied by the number of laminations. The proportion
of the losses in the rotor core is remarkable and nearly
as high as in the stator (see Fig. 3(a)).
The measured and computed no-load iron losses as
a function of the supply current are compared in Fig.
3(b). During the measurements, the rotor windings were
(a) (b)
Fig. 3. Simulated no-load eddy current losses (a) and separated total
iron losses (b) as a function of the supply current as well as measured
losses.
kept open in order to avoid additional rotor currents
and hence further losses. The test machine was driven
by a second machine at synchronous speed. Thereby,
the friction losses are covered by the driving machine.
The measured losses on the stator terminals of the test
machine correspond to the iron losses and stator winding
losses. For the computation, the hysteresis and excess
losses are obtained by a method developed in [15] based
on the evaluation of the shape of dynamic magnetization
curves. The hysteresis losses are calculated by a static
vector Preisach model [16], [17], the excess losses using
the statistical loss theory [14]. The good agreement
between the measurement and simulation results confirms
that the methods used are able to cope with the complex
electromagnetic phenomena arising in an induction machine,
i. e. rotating flux and high-order field harmonics.
The stator eddy current losses for no-load can even be
evaluated correctly using time harmonic analyses, since
they are almost exclusively caused by the fundamental
field component. The methodology of the proposed twostep
approach can be adopted in a straightforward way
to time harmonic problems. For rated no-load current
(I0=288 A), the losses obtained from a transient analysis
are 901.1 W, the time harmonic analysis yields 903.7 W.
When load is getting applied, the field harmonics increase
strongly and transient analyses are required [18]. In
order to shorten the computation time, the following
investigations on the effects of interlaminar short circuits
have been performed for the stator sheet at rated noload
current using time harmonic calculations. However,
all relevant factors influencing the behavior underlying
an interlaminar short circuit are still incorporated in the
computation. The use of time harmonic analyses constitutes
no restriction of the introduced approach which is
easily expandable for transient simulations.
B. Validation of the interlaminar model
The proposed single sheet model combined with
boundary conditions considering the interlaminar interaction
has been validated using two different examples:
a simple conductive ring and a stator sheet sector of the
induction machine investigated.
1) Conductive ring: As shown in Fig. 4(a), the 2-D
model of the ring is excited by a conductor arranged
symmetrically in the bore separated from the core by
an air-gap. The 3-D model (see Fig. 4(b)) consists of
ten and a half stacked lamination quarters short circuited
by two conductive joints through the entire core stack.
The joint material is the same as those of the laminates.
The isolation between the sheets is modeled as a nonconductive
A-region with a relative permeability of one
and a thickness of one hundredth of those of the laminations.
On the curved boundaries, the 3-D problem is

(a) (b)
Fig. 4. Validation geometry of the conductive ring: 2-D (a) and 3-D
(b) model.
excited by boundary conditions derived form the 2-D
field solution. Periodic boundary conditions have been
applied on the left and right boundaries. On the top
surface, the normal component of B and that of J are set
to zero; on the bottom surface, the problem is mirrored
using symmetric boundary conditions. The 2-D problem
is treated with the time harmonic A formulation, the
3-D one with the A,V -A formulation. Linear material
properties are assumed for the sake of simplicity.
Figs. 5(a) and 5(b) show the current and flux density
distributions computed by the full model. The electromagnetic
quantities in the center sheets repeat periodically
suggesting the applicability of the reduced
approach. Consequently, the currents circulating through
the contact spots affect the original 2-D field distribution
(see Figs. 5(c) and 5(d)). The undermost half lamination
serves as a validation reference for the reduced model
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with a single sheet. Fig. 6 compares the obtained field
solutions. The good agreement is also verified in terms
of losses wich are 10.93 mW for the full model and
11.37 mW for the reduced one.
2) Stator sheet sector: Since 3-D simulations with
multiple short circuited stator laminations are hardly
feasible, only a small sheet sector is considered in the
full model. Fig. 7(a) shows the used 2-D model of
the previously examined induction machine, Fig. 7(b)
the 3-D model comprising hundred and a half laminations
interconnected by shearing burrs over two teeth
throughout the stack. A continuous burr width of 100 μm
(a) (b)
Fig. 7. Validation geometry of the stator sheet sector: 2-D (a) and
3-D (b) model.
has been selected requiring a rather fine mesh near the
burred region. Results for different burr widths will be
discussed in section III-C2. The burr material properties
are the same as that of the sheets. The isolation thickness
is specified as one fiftieth of the lamination thickness
(a) (b)
(c) (d)
Fig. 5. Current density (a) and flux density (b) at a specific time instant for the full model as well as the corresponding vector plots (c,d) for the
framed sectors.

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(a) (b)
(c) (d)
Fig. 6. Current density and flux density distribution computed for the bottom lamination of the full model (a,b) and the reduced one (c,d).
(d=0.5 mm). The 2-D field solution is prescribed on an
outer non-conductive layer enclosing the teeth and on the
stator back. The normal component of both B and J is
set to zero on the top surface and on the boundaries intersecting
the yoke. Again, the problem region is mirrored
at the bottom surface and solved in the frequency domain
using the A,V -A formulation. Linear media are used for
validation purposes, nonlinearity is considered in the next
sections by means of well established methods.
The eddy current distribution for rated no-load current
calculated by the full model is shown in Fig. 8. In
the burr region, high currents are closing over the teeth
(see also Fig. 12). The induced currents increase with
Fig. 8. Current density distribution at rated no-load current and a
specific time instant for the full model.
the number of short circuited laminations and approach
asymptotically a final value. Accordingly, the reduced
model presents a worst case scenario for a suitably
large number of interconnected sheets. The required sheet
number depends on various factors, such as size and
position of the short circuits, sheet dimensions or material
properties. The undermost lamination of the full model
and the reduced method again give similar current density
distributions as shown in Fig. 9. The resulting losses
for the full model are 392.0 mW, the reduced approach
predicts 363.9 mW.
C. Effects of interlaminar short circuits
Two examples of interlaminar short circuits will be
addressed in more detail. The first one involves conductive
joints represented by clamping bars. Second, the
impact of shearing burrs will be discussed by means of
parametric studies. All of the following simulations have
been carried out for rated no-load current in the frequency
domain using the proposed interlaminar model. Nonlinearity
is taken into account by an effective reluctivity
approach [19].
1) Conductive joints: The iron stacks of mid-power
range machines are often axially fixed by clamping bars
installed after pressing the core. These clamping plates
are attached uninsulated on the core back leading to
a conductive connection among the sheets. Different
numbers of bars installed along the stator back circumference
have been investigated using the method presented.
Fig. 10 demonstrates the computed current and flux
density distribution when four clamping bars per pole
pitch are considered in the simulation. On the one hand,
the induced currents in axial direction cause additional
losses. On the other hand, they oppose the original field
distribution increasing the magnetic flux density and
hence the losses near the clamping areas. A detailed view
of the electromagnetic quantities near a contact area is
shown in Fig. 11. The eddy current losses in the stator
core increase by 3.93 % for one installed connection bar
per pole. For four bars they rise by 13.65 %.

- 195 - 15th IGTE Symposium 2012
(a) (b)
Fig. 9. Current density distribution for the bottom lamination of the full (a) and reduced (b) model.
(a) (b)
Fig. 10. Maximal flux density (a) and loss density (b) distribution for four clamping bars per pole installed.
(a) (b)
(c) (d)
Fig. 11. Current density (a) and flux density (b) in a contact area at a specific time instant as well as the corresponding vector plots in the bar
section (c,d).
2) Shearing burrs: The quantification of the extra
losses caused by edge burrs is carried out performing
simulations for different numbers of teeth afflicted by
burrs and various burr widths. Contrary to the former
validation example, a complete stator sheet quarter has
been considered in the computations. Fig. 12 shows the
eddy current paths in the burr layer for different numbers
of teeth burred. The high magnetizing flux in the tooth
area induces interlaminar currents enclosing the teeth and
resulting in a steep rise of the losses. The trends in Fig.
13 indicate a cubic dependence of the additional losses on
the number of burred teeth. The application of burrs on
the stator back led to no significant loss increase owing
to the small flux density near the outermost rim.

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(a) (b)
(c) (d)
Fig. 12. Current density vector plots for one (a), two (b) three (c) and five (d) burred teeth at a specific time instant and a burr width of 100 μm.
The burr region only is considered in the plot.
Eddy Current Losses per Sheet Pcl in W
10
9
8
7
6
5
4
3
2
1
bburr =10μm
bburr =20μm
bburr =40μm
bburr =60μm
bburr =80μm
bburr = 100 μm
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Number of Teeth Burred
Fig. 13. Losses as a function of the number of burred teeth for different
burr widths.
The width of the burr layer has been varied in a
practically relevant range from 10 to 100 μm. Even
broader interlaminar contacts can be present in laser cut
sheets due to the heat induced insulation burn-off. As
shown in Fig. 14, the losses rise linearly with the burr
width. The loss behavior for different no-load supply
currents can be seen in Fig. 15. The losses increase
quadratically for lower values, whereas at higher currents,
saturation effects occur limiting the maximal attainable
flux densities and thus losses.
IV. DISCUSSION AND CONCLUSIONS
The method presented enables to compute the true 3-
D eddy current distribution underlying an interlaminar
Eddy Current Losses per Sheet Pcl in W
10
9
8
7
6
5
4
3
2
1
1 tooth burred
2 teeth burred
3 teeth burred
4 teeth burred
5 teeth burred
0
10 20 30 40 50 60 70 80 90 100
Burr Width bburr in μm
Fig. 14. Losses as a function of the burr width for different numbers
of burred teeth.
short circuit allowing a quantitative assessment of the
arising additional losses. In order to avoid full models
comprising multiple short circuited laminations, only a
single sheet is considered. The interlaminar interaction is
taken into account by boundary conditions on the contact
surface using a generalized A,V -A formulation.
First, the method has been applied to the no-load
iron loss estimation of a slip-ring induction machine.
Therefore, simulations for a healthy machine with no
short circuits present in the core have been carried out.
The transient 3-D eddy current problem has been excited
by boundary conditions derived from a classical 2-D field
analysis and solved separately for a stator and rotor sheet.
The hysteresis and excess losses required for comparisons

Eddy Current Losses per Sheet Pcl in W
2
1.5
1
0.5
bburr =10μm
bburr =20μm
bburr =40μm
bburr =60μm
bburr =80μm
bburr = 100 μm
0
50 100 150 200 250 300 350 400
No-Load Current I0 in A
Fig. 15. Losses as a function of the supply current for two burred
teeth and different burr widths.
to no-load iron loss measurements have been evaluated
by a static Preisach vector model and the statistical
loss theory, respectively. Good agreement was obtained
between measured and simulated results.
The effects of interlaminar short circuits have been
studied for the stator sheet at rated no-load using timeharmonic
computations, since it was found that the occurring
eddy current losses are almost completely caused
by the fundamental field component. Comparisons of the
interlaminar contact model against full models with many
laminations joined confirmed the validity of the reduced
method as long as the electromagnetic quantities in the
short-circuited sheets stay periodic. For a low number of
interconnected laminations, the full model yields lower
losses than the reduced one. Consequently, the reduced
model constitutes a worst-case approximation.
Applications of the proposed approach revealed a
significant loss increase for conductive paths introduced
by shearing burrs on the tooth edges. It should be noted
that the burrs studied will not be present to such an extent
in a healthy machine, but the trends and findings will
still apply, indicating the strong necessity to minimize
bur-induced short circuits as far as practicable.
The magnetic and electric properties have been assumed
to be unaffected by the manufacturing process. In
[20], [21] and [22] it was shown that especially the magnetic
permeability near the edges can vary considerably
due to the mechanical stress applied during punching.
The incorporation of these effects as well as combinations
of the developed method with measurement-based
statistics have to be carried out in future work.
V. ACKNOWLEDGMENT
This work has been supported by the Christian
Doppler Research Association (CDG) and by the ELIN
Motoren GmbH.
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- 198 - 15th IGTE Symposium 2012
Evaluating the influence of manufacturing
tolerances in permanent magnet synchronous
machines
I. Coenen, T. Herold, C. Piantsop Mbo’o, and K. Hameyer
Institute of Electrical Machines, RWTH Aachen University, Schinkelstrasse 4, D-52056 Aachen, Germany
E-mail: isabel.coenen@iem.rwth-aachen.de
Abstract—Manufacturing tolerances can result in an unwanted behavior of electrical machines. Undesired parasitic effects
such as torque ripples may be increased. A quality control of machines subsequent to manufacturing is therefore required
in order to test whether the machines comply with its specifications. This is useful to ensure a high reliability of the
manufactured machines. This paper describes the consideration of rotor tolerances due to non-ideal manufacturing processes.
The idea is to estimate the influence of the manufacturing tolerances for realization of a reliable quality control. To study
various fault scenarios numerical field simulations are employed which are parameterized by measurements.
Index Terms—electrical machines, Finite Element Analysis (FEA), manufacturing tolerances, quality control.
I. INTRODUCTION
The reliability of electrical drives [1] is an important
aspect to ensure a high availability. In industrial applications,
permanent magnet excited synchronous machines
(PMSM) are widely employed as they offer advantages
in efficiency and power density. However, especially the
rotor of PMSMs is susceptible to tolerances caused during
mass production. Variations from the ideal machine
influence its operational behavior [2]. Therefore, it is
important to verify the machine’s quality prior to its
installation.
Reliable and widely used diagnostic methods are vibration
and current monitoring [3]. In this study, electrical
quantities are focused because this offers the advantage
that no additional sensors need to be installed [4].
A. Proposed monitoring setup
The most often proposed end-of-line test is back-EMF
monitoring [5], [6]. Fig. 1 shows a possible setup for its
realization. Here, the motor under test is driven under
open-circuit conditions. For attenuation of the drive’s
influence, a flywheel is employed between drive and
motor under test.
Fig. 1. Back-EMF monitoring setup.
This approach presents a non invasive monitoring
method being benefical for diagnosis. However, such a
setup is very expensive. It is cost-expensive because a
drive is required and a certain device is needed to damp
possible influences of the drive. Above all, it is timeexpensive
due to the fact that the motor under test is
mechanically coupled to the drive. This is not an efficient
solution when a large number of machines needs to be
tested during mass production.
In this study, an additional approach is investigated
where the current is being monitored. The corresponding
setup is shown in Fig. 2. Here, a start-up of the motor
up to a certain speed is performed in such a way that the
current is measured at various speeds. The benefit of this
method is its time- and cost-saving setup. When compared
to the back-EMF setup, less hardware is needed.
No mechanical coupling to a drive is required, simply
the motor is connected to the inverter.
However, for evaluating the current, it needs to be considered
that the current is a controlled quantity. Impacts
caused by the control system or the inverter supply might
lead to misinterpretation of the results.
Fig. 2. Current monitoring setup.
In the following, the back-EMF and current characteristic
of a PMSM is determined. In order to study various
fault scenarios, numerical field calculation is employed
considering tolerance affected rotor components. The aim
is to evaluate the influence of such tolerances to be able to
determine distinguishing characteristics. This information

can be helpful to develop an appropriate end-of-line test
and to reveal which of the proposed setups is most
qualified.
II. MOTOR UNDER STUDY
The machine studied within this work is a three-phase
permanent magnet synchronous machine with tooth-coil
winding system. It presents six stator slots and four pole
pairs p. The eight magnets of the rotor are arranged in a
spoke configuration.
III. INFLUENCE OF ROTOR TOLERANCES
During the manufacturing process material dependant
failures, geometrical or shape deviations may occur. Such
tolerances influence the machine’s behavior. For instance,
increased torque ripples are caused [7].
The considered tolerances within this paper concern
the magnet’s material and its dimensions. The magnetization
faults are illustrated Fig. 3. Possible deviations
affect the magnitude of the remanence flux density BR
and the angle β of the magnetization direction. Further
Fig. 3. Magnetization faults.
BR
examples of rotor tolerances, not considered within this
study, would be a displacement of the magnet and rotor
eccentricity.
A. Theoretical analysis
Within this study, the influence of rotor tolerances onto
electrical signals is focused. According to [5], for nonideal
rotor components new harmonic orders nrf are
expected to appear in the back-EMF spectrum which are
a function of the pole pair number p:
nrf =1± k
with k ∈ N. (1)
p
In the following, this relation shall be approved and
specialized for the certain machine investigated.
The back-EMF Vi is the induced voltage at no load
condition (open circuit). For a coil with w numbers of
turns Vi is defined as follows:
Vi = −w dφ d
= −w
dt dt (
Bd A). (2)
Applied to a machine’s winding, it means that the
back-EMF in one coil of the winding is determined by the
air gap flux density B. Therefore the back-EMF presents
the same harmonic orders which appear in the spectrum
of the flux density. The latter will be considered for an
order analysis.
β
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The magnetic flux density at the air gap of the machine
is a rotating wave which is a function of relative position
at the air gap α and time t [8]. It is given as the product
of the magnetomotive force Θ (MMF) and the air gap
permeance Λ:
B(α, t) =Θ(α, t) · Λ(α, t). (3)
The functions of permeance, magnetomotive force and
flux density can generally be represented by a series of
space and time harmonics [9]:
Λ(α, t) =
Λyl,zl · cos(yl · α − zl · t), (4)
yl,zl
Θ(α, t) =
yt,zt
B(α, t) =
yb,zb
Θyt,zt · cos(yt · α − zt · t), (5)
Byb,zb · cos(yb · α − zb · t). (6)
At this, ω1 is the supplying angular frequency. For
reasons of illustration the phase angle is neglected.
For derivation of the new harmonic orders caused by
non-ideal rotor components only the rotor fundamental
component of the MMF is considered, meaning yt = p
and zt = ω1. Furthermore, a constant air gap width is
considered, meaning yl =0and zl =0. For the faultless
case, this implies:
Bp(α, t) =Bp,ω1 · cos(p · α − ω1 · t). (7)
The above mentioned rotor tolerances lead to an
asymmetrical distribution of the air gap field. With the
described approach, this means a modulation of the
MMF caused by the rotor magnets. New space and time
harmonics k with k ∈ N appear resulting in the following
expression for the flux density considering non-ideal rotor
components:
Brf(α, t) =
Bk · cos((p ± k) · α − (ω1 ± kωm) · t).
k
Here, ωm is the rotational speed with ωm = ω1
p . Hence,
Brf can be expressed as follows:
Brf(α, t) =
Bk · cos((p ± k) · α − (1 ± k
) · ω1t).
p
k
This expression indicates the new harmonic orders appearing
in the spectrum of the flux density and equally
in the back-EMF spectrum due to deviations at the machine’s
rotor as predicted in expression (1). However, it
is only valid considering one single coil [10]. Derivating
the harmonics for one phase, the coil configuration needs
to be considered. One phase of the investigated machine
contains two coils which are displaced by 180 ◦ .
According to (9) the back-EMF of the first coil in
one phase assuming faulty rotor components can be
(8)
(9)

determined as follows:
Virf1(α, t) =
Vik · cos((p ± k) · 0 ◦ − (1 ± k
) · ω1t).
p
k
(10)
Similarly, the back-EMF of the second coil in the same
phase is:
Virf2(α, t) =
Vik · cos((p ± k) · 180 ◦ − (1 ± k (11)
) · ω1t).
p
k
The resulting back-EMF Virfph for one phase can be
determined by the superposition of the two coils:
Virfph = Virf1 + Virf2 =
Vikcos((1 ±
k
k
p )ω1t) · [1 + cos((p ± k) · 180 ◦ )].
(12)
For odd numbers (p ± k), (12) is equal to zero. With
p =4it is obvious that only even numbers of k appear
in the back-EMF spectrum.
Finally, (1) can be specialized for the analyzed machine,
indicating the new harmonic orders appearing in
the spectrum of back-EMF in case of non-ideal rotor
components:
n ′ rf =1± 2k′
p with k′ ∈ N. (13)
For the faultless case where the air gap field is symmetrical,
the appearing orders are determined by the winding
arrangement [5]. Considering a three phase winding,
these harmonic orders are:
n =6m ± 1 with m ∈ N. (14)
The mentioned new harmonic orders caused by faults
appear in addition to (14).
For the current, the harmonic orders can be derived
analogously. Ampere’s law reveals the general relation
between electrical current I and magnetic flux density
B:
μ0 · I = Bds. (15)
In practice, the current may additionally be affected by
the control system and by the inverter supply. These
impacts need to be considered in order to avoid wrong
interpretation of the measured signals.
IV. METHODOLOGY
To determine the influence of the rotor tolerances
onto the back-EMF and current characteristic, numerical
field simulations are used. Reliable analysis requires a
sufficiently large number of experiments which means
less effort performing with simulation instead of measurements.
In addition, the interpretation of measurement
results is difficult within this context, as for a certain
prototype the real existing deviations are unknown. The
intentional construction of tolerances is very difficult to
realize.
S
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Back−EMF [p.u.]
A. Finite Element Analysis
To calculate the back-EMF, a two-dimensional timestepping
Finite Element Analysis (FEA) is applied. Noload
operation at a speed of 3000 rpm is assumed and the
voltage is calculated by use of the time derivative of the
magnetic flux, as in equation (2).
For analysis, a discrete Fourier transform (DFT) is performed
which yields the spectrum of back-EMF as shown
in Fig. 4. For the ideal faultless case with symmetrical
air gap field, harmonic orders appear according to (14).
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 2 4 6 8
Harmonic order
Back−EMF [p.u.]
0.05
0.04
0.03
0.02
0.01
0
0 2 4 6 8
Harmonic order
Fig. 4. Back-EMF spectrum assuming faultless case.
1) Parameterization by statistical measurements: For
parameterization of the FEA model, a statistical verification
is performed. The back-EMF is measured for ten
prototypes of the machine. The resulting first order is
evaluated in form of a histogram shown in Fig. 5. Based
on this results, the magnets material properties (BR)
within the model are adjusted in order that simulated
value of first order and mean value of measured first
order agree.
Absolute Frequency
5
4
3
2
1
0
0.99 1 1.01
First order back−EMF [p.u.]
Fig. 5. Measured Back-EMF histogram.
B. Extended d-q model
A d-q model is a common way to describe the PMSM’s
dynamical behavior considering the control system. Here,
the application of such a model is studied to calculate
the current. Since the common d-q model only takes the
fundamental wave into account, it is not able to consider
non-ideal behavior such as local defects studied within
this work. Hence, an extended d-q model [11] is applied

to calculate the current. Here, FEA is used to extract
additional elements to extend the common d-q equations.
In the following, a start-up of the machine from zero to
3000 rpm is simulated and the stator current is analyzed
by use of a short-time Fourier transform (STFT). This
yields the spectrum including the frequency distribution
over time of the non-stationary current signal. Fig. 6
shows the result for the faultless case. The value of
current is represented by a color range, where light colors
mean a high and dark colors a low value. It can be seen
that the harmonic orders are the same as for the back-
EMF.
Fig. 6. Current spectrum assuming faultless case.
Here, sine-wave excitation is assumed. With the presented
model it is also possible to simulate inverter
operation. However, modeling the inverter leads to a
computationally expensive model. Fig. 7 shows the result
for the faultless case assuming inverter supply. Due to
the high intensity of computation it is illustrated with
lower resolution. Besides the main harmonic orders some
new orders appear. However, these do not interfere with
the orders which are expected to appear correspending to
(13) due to tolerances. Therefore, inverter supply is not
considered within this study because of the computational
costs.
Fig. 7. Current spectrum assuming faultless case and inverter supply.
V. SIMULATION RESULTS
To develop a reliable end-of-line check, the most
common and important fault modes should be captured.
In the following, different approaches are applied to
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Back−EMF [p.u.]
simulate various fault scenarios. The choice of the corresponding
approach depends on the particular fault, the
prior knowledge of the fault and the available data.
A. Worst-case analysis
For PMSMs cogging torque is an undesired effect as it
leads to rotational oscillations of the drive train. Cogging
torque is strongly influence by deviations caused by
the manufacturing process [2],[7]. However, measuring
cogging torque is very time- and cost-expensive [12] and
therefore no appropiate method for an end-of-line control.
In the following it shall be studied how back-EMF and
current are influenced at a faulty machine presenting a
high value of cogging torque due to magnetization faults.
Considering a deviation in the magnitude of the magnets’
remanence flux density BR, the amount of variation
of cogging torque is depending on which and how
many permanent magnets are affected. In [13] Designof-Experiments
is applied to find out the worst-case
configuration of magnetization faults concerning cogging
torque. Applied to the studied machine, the configuration
shown in Fig. 8 presents the highest value of peak-to-peak
cogging torque. Thereby, the filled magnets represent the
ones that are defective. Considering a deviation at BR
of -10%, the value of cogging torque is about seventimes
higher compared to the reference value of the ideal
machine.
Fig. 8. Worst-case configuration of magnetization faults.
Fig. 9 and Fig. 10 show the results for back-EMF and
current in case of the described worst-case magnetization
fault.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Faulty case
Faultless case
0
0 2 4 6 8
Harmonic order
Back−EMF [p.u.]
0.05
0.04
0.03
0.02
0.01
Faulty case
Faultless case
0
0 2 4 6 8
Harmonic order
Fig. 9. Back-EMF spectrum assuming worst-case magnetization fault.
The spectra show new harmonic orders, especially
n ′ rf =0.5 and n′ rf =2.5 are apparent according to (1).
Compared to the faultless case the first harmonic order
is reduced.

Back−EMF [p.u.]
Fig. 10. Current spectrum assuming worst-case magnetization fault.
B. Sample cases
For the studied machine the height of the magnet can
vary between 97% and 100% of its desired value and
the width can vary by ± 1.5%. The dimensions of some
sample magnets have been measured which are used
as input data for this approach. None of the measured
dimensions exceed the allowed tolerance range. Five
cases are created where every magnet is subjected to
the given tolerances. Fig. 11 and Fig. 12 exemplarily
show the back-EMF and current spectrum of one faulty
case. The first harmonic order is reduced compared to
the faultless case and the new ordinal numbers appear
corresponding to (13).
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Faulty case
Faultless case
0
0 2 4 6 8
Harmonic order
Back−EMF [p.u.]
0.05
0.04
0.03
0.02
0.01
Faulty case
Faultless case
0
0 2 4 6 8
Harmonic order
Fig. 11. Back-EMF spectrum assuming faulty magnet dimensions.
Fig. 12. Current spectrum assuming faulty magnet dimensions.
When compared to the results from the worst-case
magnetization fault studied in V-A, one can see that the
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influence of deviations at the magnets’ dimensions within
the allowed tolerance range is not significant. The spectra
show the same specific charateristics but less amounts.
For all five studied cases the simulated spectra do not
differ considerably from the faultless case.
C. Stochastic analysis
In [14] the influence of varying qualities of the permanent
magnet has been investigated applying a stochastic
analysis. This is applied here to compare the influences
of deviations in the magnetization magnitude and magnetization
direction. Overall, 60 failure configurations are
studied. For 20 cases the remanence flux density BR
is assumed to be Gaussian distributed with a standard
deviation of 3σ equal to 10% of the nominal value.
For 20 other cases the magnetization direction is also
assumed to be normally distributed with a standard
deviation of 5 ◦ . The other 20 cases present both kind
of deviations. Applying FEA, cogging torque and back-
EMF are calculated and evaluated employing histograms.
The distribution of the first harmonic order of the back-
EMF is shown in Fig. 13. It shows the range in which
the back-EMF is influenced because of the different
magnetization deviations.
Absolute Frequency
Absolute Frequency
Magnetization magnitude
7
6
5
4
3
2
1
0
0.98 1 1.02 1.04
First order back−EMF [p.u.]
7
6
5
4
3
2
1
Magnitude and direction
0
0.98 1 1.02 1.04
First order back−EMF [p.u.]
Absolute Frequency
20
15
10
5
Magnetization direction
0
0.98 1 1.02 1.04
First order back−EMF [p.u.]
Fig. 13. Back-EMF histogram considering magnetization faults.
It can be seen that the influence of the deviations
concerning the magnetization direction is very small. The
magnitude fault is prevailing. This is to be expected for
the studied machine as it presents interior magnets.
The same analysis is performed for the peak-to-peak
cogging torque, which is presented in Fig. 14. Again
it can be concluded that the influence of magnetization
direction is small when compared to the deviation in
magnitude as the corresponding distribution shows.

Absolute Frequency
Absolute Frequency
8
6
4
2
Magnetization magnitude
0
0 2 4 6
Peak−to−peak cogging torque [p.u.]
8
6
4
2
Magnitude and direction
Absolute Frequency
0
0 2 4 6
Peak−to−peak cogging torque [p.u.]
20
15
10
5
Magnetization direction
0
0 2 4 6
Peak−to−peak cogging torque [p.u.]
Fig. 14. Cogging torque histogram considering magnetization faults.
Generally, magnetization faults influence cogging
torque and back-EMF characteristic simultaneously [15].
For both quantities new harmonic orders arise depending
on the caused asymmetry in the air gap field. Based
on the presented studies, it becomes apparant that the
influence of magnetization tolerances on cogging torque
is more significant as on back-EMF. This means, that
on the one hand the machine’s behavior is strongly
influenced in general. But on the other hand, as a cogging
torque test is excluded for an end-of-line concept, it
implies high functional requirement of the measurement
devices to detect faults by use of back-EMF analysis.
This shows the importance of an influence analysis such
as presented in this study. It is required to study the
impact of tolerances to be able to separate it towards
measurement inaccuracy.
VI. CONCLUSION
In this work, the influence of non-ideal manufactured
rotor components of a PMSM on its back-EMF and
current characteristic is studied. It has been shown that
electrical quantities are applicable to realize tolerance
diagnosis. Especially the stator current approves to be
a promising approach due to its time- and cost-saving
setup. The new harmonic orders caused by rotor faults
are derived within a theoretical analysis and confirmed
by the simulation results.
An end-of-line check could be realized in such a
way that all machines presenting a certain level in these
specific characteristics are rejected. With the presented
methods, the range of these distinguishing characteristics
can be evaluated to detect the corresponding levels for
rejection. At this, measurement accuracy should be taken
into account.
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Comparing the different rotor tolerances, all present
the same characteristics but with different amounts depending
on the fault’s intensity and arrangement. As a
feedback for manufacturing, a differentiation of various
tolerances would be gainful but can not be achieved with
the presented analysis. However, the focus of the endof-line
check is to verify the machines’ quality which
can be realized by the suggested approach. The results
of this study evince to be valueable for application of an
accurate quality control for PMSMs finally improving its
reliability.
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- 204 - 15th IGTE Symposium 2012
Hai Van Jorks, Erion Gjonaj and Thomas Weiland
TU Darmstadt, Institute of Computational Electromagnetics, Schloßgartenstraße 8, 64289 Darmstadt, Germany
Abstract— High frequency eddy currents are investigated and the Common Mode Input Impedance of a PWM controlled
induction motor is calculated from finite element simulations. In order to determine machine parameters accurately, two
modelling approaches are compared. The first is a two-dimensional simulation approach where iron core lamination effects
are included by means of an equivalent material approximation. The second approach consists in fully three-dimensional
analysis taking into account explicitly the eddy currents induced in the laminations. It is shown that homogenised equivalent
material models may lead to large errors in the calculation of machine inductances, especially at high frequencies. However,
the Common Mode Input Impedance, which is the final parameter of interest, seems to be less affected by the lamination
modelling.
Index Terms—eddy currents, finite element, lamination, PWM
I. INTRODUCTION
In modern drive systems fast switching inverters are
the source of high frequency common mode voltages at
the motor terminals. Due to stray capacitances between
windings and grounded iron parts of the machine, a
common mode current is excited and for its part may
cause circulating bearing currents which may damage the
bearing [1]. The phenomena can be described by
equivalent circuit representation which employs the
frequency dependent Common Mode Input Impedance
being the ratio of common mode voltage and current
Common Mode Input Impedance can be computed
using
Figure 1: Lumped parameter model of a two conductor system.
Parameters can be gathered in the impedance and capacitance matrix
Firstly, the stray capacitances are extracted from
electrostatic and winding impedances from
magnetoquasistatic simulations. Ohmic conductivity of
winding insulation is negligible.
Secondly, the winding scheme is taken into account to
match the corresponding voltages and currents at the
front and rear end of the machine. At this point, endwinding
inductances can be included in the model, but
require distinct modelling approaches and are, therefore,
neglected in the present analysis.
Considering the laminated middle part of the motor, it
is common to use two-dimensional (2D) models of the
motor cross-section to compute the self and coupling
impedances. In earlier literature it is proposed to apply
the finite element (FE) method within a single stator slot
while the magnetic field was assumed to be zero outside
the slot perimeter [2]. However, as shown in a recent
investigation [3], strong inductive coupling at high
frequencies (several to ) may occur even
between distant slots. The effect is caused by the core
lamination, which, despite the small skin depth (7),
promotes the spreading of high frequency magnetic
fluxes over the iron sheets’ surfaces. In [4] the lamination
was approximated by an additional impedance.
Moreover, it is possible to include the lamination already
in the FE model. Therefore, we consider the entire motor
cross section in the magnetoquasistatic simulations.
In the case of 2D analysis, modelling the laminated
core in a plane requires the application of
homogenization techniques. We investigated the accuracy
of the widely used formulation (2) for a broad frequency
range . The reference solution was
obtained from fully three-dimensional (3D) simulations.
Since resolving the small skin depth in motor laminations
in a 3D mesh is computationally very costly, general
purpose simulation software cannot be utilized. On that
account, we developed a specialised 3D FE simulation
tool, which takes advantage of the periodicity of the
lamination stack, but otherwise does not use any
approximation on motor geometry or on the material
properties of the laminated core.
II. 2D LAMINATION MODELLING
A well-known homogenization model for laminated
cores [5] utilizes a frequency dependent equivalent
permeability for the iron core given by,
where 0r is the permeability of iron, 2b the thickness of
the plate and the skin depth at a given frequency. The
magnetic field problem for the homogenized core is
reduced to a planar problem. While this approach allows
for efficient 2D FE modelling, accuracy at higher
frequencies may not be sufficient. Figure 2 shows the Bfield
plot of the motor model obtained from simulations

with “FEMM” [6], a 2D open source software which
employs the approach (2). In order to obtain self and
mutual impedances of the multi-conductor system, only
one conductor was excited by a current. The spreading of
the flux over the lamination as well as across the periodic
boundary of the computational model can be observed.
The impedance matrix was extracted and will be
compared to the 3D reference in Section IV.B.
Figure 2: 2D model of cross-sectional motor geometry (60° section)
with magnitude of magnetic flux density at 1 MHz
III. FULLY 3D FE ANALYSIS
A. 3D FE formulation
Maxwell’s equations in frequency domain expressed
by a magnetic vector potential yield
where is the permeability, the conductivity, is a
voltage gradient used for excitation. Two types of
boundary conditions on are applied. Firstly,
where is the unit normal vector and is the
triangular prims are chosen, because they allow for an
efficient discretisation of the very thin iron sheet (Fig. 3).
Applying Galerkin’s method to (3) the matrix equation
is generated, where the complex matrix combines the
discrete operators corresponding to the left hand side of
(3), vector holds the degrees of freedom (DOFs) of
the vector potentials and vector the exciting currents.
In the case of voltage excitation of individual conductors
the relation
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can be employed [7], where
, is the
angular frequency and is the coupling matrix for the
vector of the wire voltages . The
important case where a single conductor n is exited can
be obtained by setting in (6) and for all
other conductors (see also Section IV.A).
B. Reduction of the problem size
A high frequency 3D FE model of the complete motor
geometry is still far beyond today’s computing capacities.
But even if we consider just a slice of the motor
with the thickness of half a lamination sheet ,
an appropriate 3D discretisation will lead to several
million mesh cells. In order to model eddy currents at
frequencies up to the discretisation has to
resolve the small skin depth in the high-permeability iron
In the analysis, the following parameters were used:
,
,
where is is the electrical conductivity and the
permeability of iron, respectively.
Parallelization of our 3D FEM code is a key feature,
nevertheless, further reduction of the problem size is
particularly important. In the case of common mode
excitation of a 3-phase 4-pole induction machine, the
field pattern in the motor cross section is periodic with
respect to a 60° rotation around the longitudinal axis of
the motor. This reduces the computational domain to a
60° section while periodic boundary conditions are
applied to the cut planes (Fig. 3).
Figure 3: Simulation mesh and magnitude of the magnetic flux density
at 1 MHz for the 3D motor model. For better visibility, the model is
scaled by a factor 100 in the transversal direction.
IV. IMPEDANCE MATRIX CALCULATION
A. Extraction procedure
The standard procedure to extract the impedances of
the cross-sectional conductors from FE analysis, is to
excite the -th conductor with a current of and set all
the other conductors to . After running the simulation,
the induced voltages in all conductors have to be
computed from the magnetic vector potential solution.

This procedure has to be repeated for all
conductors. A section of the analyzed
induction motor holds 120 stator and 8 rotor conductors.
If a general purpose FEM software is used, this leads to a
large computational overhead, which makes the method
inconvenient. However, the extraction procedure can be
condensed into a single simulation cycle. In this way,
common steps like loading of the mesh, setup of the curlcurl
matrix and its LU decomposition have to be
performed only once (see Fig. 4). Referring to the 3D
simulation of the motor cross section, a speedup factor of
could be obtained. As was shown in [7],
impedance matrix can be computed from
Still, in order to avoid the explicit inversion of the sparse
matrix , the equation system (5) has to be solved
times, with being the number of conductors. A detailed
overview of the implemented algorithm is depicted in
Fig. 4.
Figure 4: Flowchart of computational algorithm
B. Simulation results
The same motor geometry is used in the 2D (Fig. 2)
and the 3D (Fig. 3) case. The simulation time of a 3D
model with DOFs on a cluster with 60 nodes
was for a sweep of 7 frequency points. Figure 5
shows the magnitude of self-impedance of one stator
conductor extracted from 2D and 3D simulations. The 2D
solution which employs the lamination model (equivalent
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permeability) shows major discrepancies from the 3D
reference case. For additional validation of the simulation
models, the same geometry was tested without laminated
materials, i.e. the iron core forms a massive block. Thus,
2D analysis does not require the lamination formulae and
therefore should give the same results as 3D analysis. The
corresponding impedance is shown in Fig. 6. 2D and 3D
impedances can be found in very good agreement.
Z / Ω
10 0
10 -7
9%
10 1
Figure 5: Laminated iron with . Comparison of conductor
impedance extracted from field simulations and relative error between
2D and 3D results.
Z / Ω
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -3
10 -4
10 -5
10 -6
10 0
10 -7
0.00%
53%
0.02%
10 1
Figure 6: Massive iron with . Comparison of conductor
impedance extracted from field simulations and relative error between
2D and 3D results.
V. COMMON MODE INPUT IMPEDANCE
A. Assembling of transmission line model
The state of a multi-conductor line in the frequency
domain is described by telegrapher’s equations. Taking
into account the lumped parameter approximation, three
matrix equations are obtained
Z 1,1 laminated μ r,Fe = 1000
94%
10 2
10 2
97%
10 3
f / Hz
10 3
f / Hz
where , , and are the voltage and
current vectors at the front (z=0) and rear end (z=l) of the
95%
10 4
Z 1,1 massive iron μ Fe = 1000
0.13%
1.82%
6.43%
10 4
88%
10 5
2D
3D
9.58%
10 5
2D
3D
80%
6.04%
10 6
10 6

motor, respectively, and and
are the lumped element matrices in the pi-equivalent
circuit (Fig. 1). The vector holds the currents through
the impedances and can be eliminated by inserting one
equation into the others. In the next step, the winding
scheme is taken into account which further reduces the
degrees of freedom in the system (9a-c). The proceeding
can be found in [3]. It is important to set the current at the
star point of the machine to zero, according to the
common mode measuring setup, where the star point is
not grounded. When solving the final equation system, a
given voltage at the motor terminals will yield a certain
input current. The ratio of the two quantities is the
Common Mode Input Impedance . B. 240 kW induction motor
The 2D and 3D plots (Fig. 7) show very good
agreement, despite the deviations in the impedance
matrices (Fig. 5). This is understandable since the stray
capacitances are dominant in the common mode circuit of
the machine. However, impedances are important to
predict resonance points. The frequency of the first
resonance in the 2D curve is shifted by 6 kHz while its
magnitude differs by 4%.
Z / Ω
10 3
10 2
10 1
10 3
first resonance
| Z com | 2D vs. 3D
10 4
f / Hz
Figure 7: Comparison of the Common Mode Impedance between 2D
and 3D results.
VI. CONCLUSION
We developed a specialized 3D FE simulation code
which is able to efficiently extract the impedance matrix
of a multi-conductor setup, e.g. a motor cross section.
Since the laminated iron is modelled with its actual
geometry and material properties, 3D results are more
accurate than those from 2D simulations with
homogenized material approach. Finally, capacitance
matrix, impedance matrix and the winding scheme are
combined to obtain the frequency-dependent Common
Mode Input Impedance of the machine. It can be
observed that the solution is dominated by the
10 5
2D
3D
10 6
- 207 - 15th IGTE Symposium 2012
capacitances and, therefore, less sensitive to inaccuracies
in the impedance matrix. In future work the nonlinear
properties of the iron will be considered in the
simulations and the end regions of the motor will be
included in the Transmission Line Model.
Acknowledgement This work is founded by the
Deutsche Forschungsgemeinschaft (DFG) under the
collaborative research group grant FOR 575.
[1]
REFERENCES
S. Chen, T.A. Lipo, D. Fitzgerald, “Source of induction motor
bearing currents caused by PWM inverters”, IEEE Trans. on
Energy Conv., Vol. 11, Iss. 1, 1996, pp. 25–32.
[2] I. Boldea, S. A. Nasar, “The induction machine handbook”, CRC
Press, 2002.
[3] H. Jorks, E. Gjonaj, T. Weiland, O. Magdun, “Three-dimensional
simulations of an induction motor including eddy current effects in
core laminations”, IET Science, Measurement & Technology, Vol.
6, Iss. 5, Sep. 2012, pp. 344 – 349.
[4] P. Maeki-Ontto, J. Luomi, “Induction motor model for the analysis
of capacitive and induced shaft voltages”, Proc. of IEMDC '05,
May 2005, pp. 1653-1660.
[5] R.L. Stoll, “The analysis of eddy currents”, Oxford University
Press, 1974.
[6] FEMM by David Meeker, version 4.2, available at
[7]
www.femm.info.
A. Bossavit, "Two dual formulations of the 3-D eddy-current
problem", COMPEL, Vol. 4, Iss. 2, 1985, pp. 103 – 116.
[8] H. De Gersem, O. Henze, T. Weiland, A. Binder, “Simulation of
wave propagation effects in machine windings”, COMPEL, Vol.
29, Iss. 1, 2010, pp. 23 – 38.

- 208 - 15th IGTE Symposium 2012
Computation of end-winding inductances of rotating
electrical machinery through three-dimensional
magnetostatic integral FEM formulation
F. Calvano 1 , G. Dal Mut 2 , F. Ferraioli 2 , A. Formisano 3 , F. Marignetti 4 ,
R. Martone 3 , G. Rubinacci 1 , A. Tamburrino 4 and S. Ventre 4
1 Dip. di Ingegneria Elettrica, Università di Napoli Federico II, Via Claudio 25, I-80124, Naples, Italy
2 Ansaldo Energia, Via N. Lorenzi 8, I-16152, Genova, Italy
3 Dip. di Ing. Industriale e dell’Informazione, Seconda Università di Napoli, Via Roma 29, I-81031, Aversa (CE), Italy
4 Dip. di Ing. Elettrica e dell’Informazione, Univ. di Cassino e del Lazio Merid., Via Di Biasio 43, I-03043, Cassino,
Italy
Abstract—An effective numerical technique to calculate end-winding inductances of rotating electrical machinery is presented.
The algorithm is based on a 3D integral formulation; it allows to take into account non linearities, relative speed between
stator and rotor and is well suited for the treatment of regions with large air volumes. Numerical implementation concerning
the analysis of a large synchronous generator highlights the advantages of the proposed method. The aim of the paper is to
assess, by means of an accurate 3D model, the correction factor to be applied to the inductances computed through 2D models
to take into account the effects due to end windings.
Index Terms— End windings, Integral FEM approach, Inductances, Flux density numerical computation, Synchronous
generators.
I. INTRODUCTION
Although the computation of the inductances of
rotating electrical machinery is a key issue both in the
design stage and in performance assessment, its accurate
calculation by Finite Elements approaches is still an open
problem.
The inductances can be split into two contributions:
main and leakage inductances.
The end-winding effect affects both the contributions:
main and, especially, leakage inductances. End-winding
inductances are at the base of both the steady-state
operation and the dynamical behavior of electrical
machinery [1-3]. The main inductances are related to the
flux linkages while the leakage inductances can be
divided into slot inductances, tooth tip inductances,
skewing inductances and zigzag leakage inductances. In
large machines, end windings contribute significantly to
the values of both main and more significantly leakage
inductances [4].
The contributions to the inductances due to the active
length of the conductor can easily be computed either
numerically, by standard 2D FEM analyses, or
analytically by means of infinite length models. On the
contrary, the end winding contribution can only be
computed from the actual 3D magnetic field distribution.
Most analytical approaches to the magnetic field
computation [5, 6], including the most recent ones [7], are
based on the solution of the Biot-Savart law through
equivalent current sheet representations, using
axisymmetric hypothesis [8] or the theory of images [9].
Both three dimensional and two-dimensional
techniques based on FEM can be used as an alternative,
but they also rely on rough approximations to reduce the
number of nodes [10,11]. End regions fields and fluxes
can be very complex to be computed, especially for large
power machines, where end regions may occupy up to
one third of the total machine length.
The aim of the paper is to propose a method based on
the use of an accurate 3D FEM simulation to improve the
accuracy of the standard 2D model achieved via a
commercial software.
The 3D FEM technique here proposed takes advantage
from an integral formulation implemented in an noncommercial
code. Such an approach has been previously
applied to compute end winding forces [12,13].
In this paper the analysis of a large synchronous
generator is considered as an example. Field simulations
in different working conditions are used to assess the
influence of end effects on flux linkages.
The main advantages of the proposed approach are
manifold: (1) a reduced number of elements is required to
model the end regions; because the integral formulations
do not require the discretization of the air region but,
rather, of the material regions only (conducting and/or
magnetic materials); (2) neighboring elements do not
need to share nodes allowing for more freedom in
meshing complex geometries; (3) the formulation
provides an inherent capability to include air-spaced
moving parts, as no interface mesh is needed [13].
This approach is therefore particularly effective to
model generator end regions because it can also take into

account rotor motion and magnetic nonlinearities.
The paper is organized as follows: Section II presents
the basis of the integral formulation and its numerical
implementation. Flux expressions coming from integral
formulation are also discussed. Then, the 3D correction
with respect to 2D fluxes and inductances is introduced in
Section III.
Such formulation is used in Section IV to look for the
3D flux density in the end regions of a large synchronous
generator. The 3D correction to the 2D quantities are then
performed according to the proposed formulation.
Section IV contributes in particular to extend the
knowledge of rotating electrical machinery by providing a
powerful numerical tool to compute lumped parameters in
the analytical model of synchronous generators with a
precision superior to that achieved by classical 2D finite
element models. As matter of fact, with the proposed
model, the contribution to terminal quantities such as the
reactances coming from the end winding region can be
accurately taken into account. Terminal quantities are
finally compared with experimental measurements.
II. INTEGRAL FORMULATION AND ITS NUMERICAL
IMPLEMENTATION
It is well known that for a synchronous machine
operating at steady state for the computation of the
inductances it is sufficient to refer to a nonlinear
magnetostatic model [16, 17]. The currents in the stator
and in the rotor coils are supposed to be assigned for any
position of the rotor in order to focus the attention on the
magnetostatic problem formulation. However for assigned
voltage, active and reactive power field currents and
inductances can be computed by solving a sequence of
non linear mangetostatic problems [17]. In any case it is
possible to neglect the effects of the eddy currents in the
massive conductive parts of the device.
The numerical model is based on an integral
formulation of the nonlinear magnetostatic problem in
terms of the unknown magnetization M. The solution is
obtained by means of a Picard-Banach iteration whose
convergence can be theoretically proved when the
magnetic constitutive equation is uniformly monotonic
and Lipschitzian [14, 15].
In particular, by using the Biot-Savart law, the
magnetic induction can be expressed in terms of its
sources as:
( ) =
( ) =
ˆ S ( ) +
0 ( )
μ0
(
( r−r') ) 3
Br Mr B r
μ Mr − ∇⋅ 'Mr' 4π Vf
r−r' dV '+
μ
( r−r') + ( ') ⋅ ˆ ( ') dS ', for∈V
3
f
4 Mr nr
r
r−r' 0
π ∂Vf
- 209 - 15th IGTE Symposium 2012
(3.1)
where BS is the magnetic induction produced in the free
space by the stator and rotor currents, Vf is the region
filled by the magnetic materials, ∂Vf is its boundary and
ˆn is the outward unit normal on ∂Vf.
The nonlinear constitutive relationship in Vf can be
expressed by introducing the local operator as
M(r)=[B(r)] in Vf
(3.2)
Therefore M is the solution of the following nonlinear
problem:
M(r)=[M] in Vf
(3.3)
As shown in [14], the operator is a contraction if is
uniformly monotonic and Lipschitzian. Therefore, the
solution exists, is unique and can be found by the fixed
point iteration.
From the numerical point of view, the magnetization
can be expressed in terms of piece-wise constant vector
shape functions in each elementary volume arising the
after discretization of Vf, such as
M
() r = M jP
j () r in V f
j
(3.4)
where the Pj's are discontinuous shape functions obtained
multiplying the scalar pulse functions pk's (pk = 1 in the kth
element and it is zero elsewhere) by the (three) unit
vectors along the coordinate axes.
The numerical model is obtained by applying the
Galerkin method to (3.3), rewritten as -1 [M]=[M] in
Vf:
-1
Pi [ M] Pi [
M]
⋅ dV = ⋅ dV, ∀i
Vf Vf
The fixed point iteration is therefore rewritten as:
V f
k+
1
k
[ M ] dV Pi
⋅ [ M ]
-1
Pi ⋅
V f
V f
P ⋅ P dV
i
i
=
V f
P ⋅ P dV
i
i
dV
, ∀i
(3.5)
(3.6)
where the subscript k indicates the approximation of M
and B at the k-th iteration. Note that being the term
the volume of the i-th element, the r.h.s of
P ⋅ P dV
V f
i j
(3.6) is the average of the magnetic induction B k =[M k ]
in the i-th elementary volume at the iteration k. Being the
magnetization piece-wise constant, eq. (3.6) can be solved
for M k+1 in each element, by applying the constitutive
relation to the average magnetic induction in the same
element.
Therefore, after discretization, (3.6) gives rise to the
following fixed point iteration [14, 15]:
( )
k −1
k
B = D EM + W
(3.7)

where:
( )
k+ 1
k
M = G B
(3.8)
( ) ⋅ ( ) ( ) ⋅ ( )
μ ˆ ˆ ' '
0 nr Pi r nr Pj r
Eij = μ0Dij−
dSdS'
4π
r−r' ∂Vi ∂Vj
Dij
= Pi⋅PjdV (3.9)
Vf
Wi
= Pi⋅BSdV Vf
Vi is the volume of the i-th element, M k the column
vector of the coefficients in (3.4) at the k-th iteration, B k
the column vector made by the average magnetic
induction in the elements and G is the global relationship
corresponding to after the discretization process.
The flux Φn linked with the n-th circuit of volume τ and
produced by the currents flowing in a set of coils is
defined as:
Φ n = Ar () ⋅Jn()
r dτ
τ n
(3.10)
where A is the magnetic vector potential associated to all
the sources and Jn is the current density associated to the
unit current impressed in the n-th circuit.
This definition is consistent with the definition of the
magnetic energy in the linear case and with the definition
of the flux linked with a circuit of infinitely small crosssection.
As a matter of fact, in this case it results:
Jn() r
Φ n = Ar () ⋅ dτ
=
I
τ n
n
In
1
= () ˆ
Ar ⋅ tSd
n =
S
n
n I
γ
n
= Ar () ⋅tˆd
γ n
(3.11)
being γn the closed curve defining the axis of the circuit,
ˆt the unit tangent vector and In the unit current flowing
in the n-th filamentary circuit.
The magnetic vector potential appearing in (3.11) can
be calculated from both free and (magnetic) polarization
currents after (3.3) is solved with suitable boundary
conditions by applying the Biot-Savart law for the
magnetic vector potential:
( ) = ˆ ( )
Ar r
μ
( ', t)
× ( − ')
Mr r r
0 AS+ dV', for
r∈V
3
f
4π V r−r' f
(3.12)
In (3.12) A has been written as the sum of the
contribution of the free and magnetizing currents. All the
procedure is quite time consuming when high number of
unknowns are treated; however a very effective
computational tool has been recently proposed [18] based
on a suitable use of high performance computing
- 210 - 15th IGTE Symposium 2012
architecture.
III. THE 3D CORRECTION TO THE 2D SOLUTION
In the case here treated the 2D solution provides an
accurate solution in a large part of the domain of interest.
As a consequence, the 3D analysis can be limited just to
the region where it is really required.
The classical theory [19], of the electrical machinery
suggests to split fluxes and inductances in a 2D and 3D
contributions, each corresponding to one of the two
geometrical parts of the system geometry.
Unfortunately such a separation is rather questionable
and ambiguous. Then here a quite different approach is
suggested: the actual 3D magnetic flux linked with a close
line, is split in (a) the 2D part Φn (2D) (evaluated by 2D
flux per the unit length, in the axial symmetrical region,
multiplied for the length) and (b) the complement ΔΦn (3D)
defined as the 3D correction requested for the case at
hand. Similar consideration can be applied to other
parameter including the main or flux leakage coefficients.
IV. NUMERICAL EXAMPLE
Among the rotating electrical machinery, large turbogenerators
are endowed with quite long end windings
which contribute poorly to produce linkage flux but,
unfortunately, to produce significant leakage flux [20].
Such a contribution is generally evaluated by simplifying
considerably the complex geometry and, in addition, by
neglecting the non linearity’s [7].
The integral approach presented in the Section II is a
powerful tool able provide a deeper analysis of the
machinery and to overcome both limitations. This kind of
information is particularly relevant in the design process
of the turbine-generator where the flux leakage has a
preeminent significance.
In the following a numerical example is presented in
order to assess the consistency of the 3D corrections to be
considered for different operating conditions.
The turbine generator simulated has a rated apparent
power in the range 300-350 MVA depending on the room
temperature affecting the cooling system. It has two poles
and the nominal frequency is 50 Hz.
Some details of the finite element mesh of the iron
regions denoted (Vf ) as well as of the field and armature
coils, are reported in Fig.1.
The 3D FEM model is characterized by a number of
45593 nodes (24201 in the iron region Vf including stator
and rotor iron as well as the enclosure) and 17774
elements (12478 in Vf ). It is worth noticing that the a non
conformal mesh of the iron regions Vf has been adopted in
order to exploit some additional geometrical degrees of
freedom in the sub-regions where a particular refinement
of the mesh is necessary.

Armature coil
Field coil
Stator
iron
Rotor
iron
Enclosure
Figure 1: Section of the Finite element Mesh used in the
computations.
Boundary conditions. In principle the complete
geometry of the machine should be treated. However just
a part has been considered while the effect of the
remaining part has been forced by suitable boundary
conditions in a cutting plane in the region where the 3D
solution actually meets the 2D approximation.
Material characterization. The magnetic iron properties
has been represented in the (3.2) form by substituting the
BH curve in B=μ0(H+M).
In the following the no load operation mode is
considered: the rotor is assumed to rotate at the nominal
angular speed, a DC current is applied to the field coil
and an open circuit is imposed to the armature terminals.
Notice that, for a synchronous machine such an
operative condition can be examined by means of just a
single magneto-static problem. As a matter of fact, such
solution provides as many samples of the time evolution
of the terminal voltage as the number of the stator slot if
the stator winding is a double layer one [16].
Of course the non-linear iron effects affects the main
flux and, consequently, the output voltages. Therefore the
magnetic analysis has been repeated for several rotor
currents, including, 100, 500 and 1000A, respectively.
In particular the field current 500 A corresponds to the
rated voltage at the generator terminals. In the following,
the 500 A case is discussed in details while the other two
currents are considered just to evaluate the saturation
effect on ΔΦn (3D) .
The three-dimensional distribution of the magnetic
vector potential on the field and armature coils are
sketched in figs. 2, 3 and the flux density in figs. 4, 5,
assuming the boundary condition imposed on the
symmetry plane and an excitation current of 500 A.
Flux waveforms as well as their amplitude spectrum is
then calculated according to (3.11); the results are
reported in fig. 6. In order to look for higher harmonics, a
spectrum analysis of the principal flux has been
performed (fig. 7).
- 211 - 15th IGTE Symposium 2012
Figure. 2: amplitude of the magnetic vector potential
[Tm] in the armature coil.
Figure. 3: amplitude of the magnetic vector potential
[Tm] in the field coil.
Figure. 4: amplitude of the magnetic induction [T] in the
armature coil.
Figure. 5: amplitude of the magnetic induction [T] in the
field coil.

As mentioned before, the 2D field coincides with the
3D field in the neighborhood of the symmetry plane. The
analysis of the vector potential and flux density
components of the 3D model can be compared to the 2D
solution, to evaluate the validity of the 2D approximation.
The 2D solution matches the 3D one in a large part of the
active length, with a good approximation (in the order of
90%).
The comparison of magnetic flux, inductance
coefficient per unit length of both solutions provides the
desired correction factors. The flux waveforms as well as
the amplitude spectrum from the 2D solution are reported
in figs. 6, 7 and the results are compared with those from
3D solution.
Figure. 6: Flux linkage [Wb] waveforms of a single a
stator coil
From the comparison of the main fluxes it follows the
2D solution underestimates the flux as well as the no load
voltage of ΔΦn (3D) =2%.
Unfortunately, due to its complexity, both the accuracy
and the resolution of the 3D solution could be
unsatisfactory for a number of applications. Of course, the
2D analysis is able to provide more accurate and robust
solution in the plane. Therefore, in order to assess the
quality of the 2D solution given by the 3D analysis, a 2D
analysis of linkage fluxes has been carried out by using
the commercial software package Ansys (Release 13) and
the voltage computed from the principal flux has been
compared with both the 3D evaluations and the
experimental measurements.
The finite element mesh used in this case is shown in
fig. 8 and consists of 125549 nodes and 5754 second
order triangular elements.
The no load voltage ha been computed and compared
to the 3D calculations. The actual end-winding effect,
neglected by the 2D solution, is highlighted in fig. 9
where the air gap radial magnetic induction as a function
of the angle θ and of the axial position Z is plotted. In
addition, to further assess the analysis the 2D no load
voltage has been also compared with a set of experimental
measurements.
The discrepancy is rather vanishing (below 1%). Of
- 212 - 15th IGTE Symposium 2012
course such a result comes by an equilibrium of two
conflicting effects: the first is lack of the end winding
contribution and the second the error introduced by
neglecting the 3D effects in the 2D evaluations.
Figure. 7: Flux linkage [Wb] amplitude spectrum.
Fig. 8: 2D finite element mesh used in the 2D
calculations.
Figure. 9: 3D rendering of the radial magnetic flux at the
air gap.

Similar results can be achieved with different currents
(discrepancy in the order 2-3% of the actual flux linkage
with 100 A or 1000 A) showing that, in no load operation
the effect of iron non linearity is quite limited.
The same procedure is applied to evaluate the leakage
flux and its contribution coming from 3D effects. The
results show that the discrepancy is much more relevant.
In the order of 10, 15, 20 % for an exciting current of
100, 500 and 1000 A, respectively.
V. CONCLUSION
The computation of end winding inductances of large
turbo generators requires proper mathematical tools to be
performed. This paper proposes an integral FEM
formulation to compute the 3D vector potential and flux
density distribution in the end regions. The approach does
not requires the meshing of the free space than allowing a
significant reduction of computer burden.
A large synchronous generator with power in the range
300-350 MVA has been analyzed as a case study. Both
axial and radial components of the flux density generated
by stator coils have been computed.
The flux linkages for all coils and the no load voltages
have been computed.
In order to assess the influence of the end regions,
different stack lengths have been simulated for the same
end windings length. The results achieved include the
definition of 3D correction to be applied to 2D
simulation and, in addition, the variation of the correction
as a function of the exciting currents has been evaluated.
The comparison between terminal quantities coming from
widely overspread 2D models and experimental
measurements have been reviewed by considering 3D
effects evaluated by using the proposed model.
REFERENCES
[1] B. Hosninger, “Theory of end-winding leakage reactance”, Power
Apparatus and Systems, Part III, vol. 78, pp. 417-426, Aug. 1959.
[2] W. M. Arshad, H. Lendenmann, Y. Liu, J.-O. Lamell and H.
Persson, “Finding end winding inductances of MVA machines”
Proc. Fortieth IAS Meeting, vol. 4, pp. 2309-2314, 2005.
[3] M.F. Hsieh, Y.C. Hsu, D.G. Dorrell, and K.H. Hu, "Investigation
on end winding inductance in motor stator windings", IEEE
Trans. on Magn., vol. 43, pp. 2513–2515, June 2007.
[4] J. Pyrhönen, T. Jokinen and V. Hrabovcová, Design of Rotating
Electrical Machines, John Wiley and Sons, Ltd, 2008.
[5] J. A. Tegopoulos, "End component of armature leakage reactance
of turbine generators", IEEE Trans. on PAS, vol. 83, pp. 632-637,
June 1964.
[6] J. A. Tegopoulos, “Current sheets equivalent to the end-winding
currents of turbine generator stator and rotor,” AIEE Trans. Pt. III,
vol. PAS-81, pp. 695–700, February 1963.
[7] V.S Lazarns, A.G Kladas, A.G Mamalis, and J.A. Tegopoulos,
"Analysis of end zone magnetic field in generators and shield
optimization for force reduction on end windings", IEEE Trans.
on Mag., vol. 45, pp.1470–1473, March 2009.
- 213 - 15th IGTE Symposium 2012
[8] D. J. Scott, S. J. Salon, and G. L. Kusik, “Electromagnetic forces
on the armature end windings of large turbine generators I—
Steady state conditions”, IEEE Trans. PAS., vol. PAS-100, pp.
4597–4603, Nov. 1981.
[9] Q. Li and F. Wang, “Application of image method to calculate 3-
D magnetic field and parameters of SC alternator”, IEEE Trans.
on Mag., vol. 25, pp. 1850–1853, Feb. 1989.
[10] D. Ban, D. Zarko, and I. Mandic, “Turbo-generator end winding
leakage inductance calculation using a 3D analytical approach
based on the solution of Neumann integrals”, IEEE Trans. on En.
Conv., vol. 20, pp. 98–105, March 2005.
[11] A.T. Brahimi, A. Foggia, and G. Meunier, "End winding
reactance computation results using a 3D finite element program"
IEEE Trans. on Mag., vol. 29, pp. 1411-1414, March 1993.
[12] R. Albanese, F. Calvano, G. Dal Mut, F. Ferraioli, A. Formisano,
F. Marignetti, R. Martone, G. Rubinacci, A. Tamburrino and S.
Ventre, "Coupled three dimensional numerical calculation of
forces and stresses on the end windings of large turbo generators
via Integral Formulation", IEEE Trans. on Mag., vol. 48, pp. 875
- 878, Feb. 2012.
[13] F. Calvano, G. Dal Mut, F. Ferraioli, A. Formisano, F. Marignetti,
R. Martone, G. Rubinacci, A. Tamburrino and S. Ventre, “A
novel technique based on integral formulation to treat the motion
in the analysis of electric machinery”, International Journal of
Applied Mathematics and Mechanics, in press.
[14] R. Albanese, F. I. Hantila, and G. Rubinacci, “A nonlinear eddy
current integral formulation in terms of a two-components current
density vector potential”, IEEE Trans. Mag. 32, pp. 784-787,
March 1996.
[15] R. Albanese, and G. Rubinacci, “Finite elements methods for the
solution of 3D eddy current problems”, Advances in Imaging and
Electron Physics, vol. 102, pp. 1-86, 1998.
[16] N. Bianchi, Electrical machine analysis using finite elements,
Taylor and Francys, pp. 141-162, 2005.
[17] M.V.K. Chari, S.H. Minnich, S.C. Tandon, Z.J. Csendes, J.
Berkery, “Load characteristics of synchronous generator by the
finite element method”, IEEE Trans. on PAS, vol. 100, pp.1-13,
January 1981.
[18] R. Albanese, F. Calvano, G. Dal Mut, F. Ferraioli, A. Formisano,
F. Marignetti, R. Martone, G. Rubinacci, A. Tamburrino and S.
Ventre, “Electromechanical analysis of end windings in turbo
generators”. COMPEL, vol. 30, pp. 1885-1898, 2011.
[19] J. Pyrhonen, T. Jokinen, V. Hrabokova, Design of Rotating
Electrical Machines, John Wiley and sons, Ltd, 2008, pp.246-249
[20] M.V. Deshpande, Design and Testing of Electrical Machines,
Phi learning Pvt. Ltd., 2010.

- 214 - 15th IGTE Symposium 2012
Magnetomechanical Coupled FE Simulations of
Rotating Electrical Machines
*A. Belahcen, *K. Fonteyn, † R. Kouhia, *P. Rasilo, and *A. Arkkio
*Aalto University, Dept. of Electrical Engineering, POBox 13000, FIN-00076 Aalto, Finland
† Tampere University of Technology, Dept. of Mechanics and Design, P.O BOX 589, 33101 Tampere, Finland
E-mail: anouar.belahcen@aalto.fi
Abstract— Regardless of the relatively large amount of published models of magnetostriction, only few of them have been
applied to describe this phenomenon in electrical steel and even less have been incorporated in the FE simulation of electrical
machines. In this paper we review the models of magnetostriction and magnetomechanical coupling in electrical steel and their
incorporation into the FE analysis of rotating electrical machines. We also discuss the advantages and disadvantages of the
different models and present an energy-based coupled magnetomechanical set of constitutive equations that describe both the
magnetostriction and the magnetic nonlinearity and its dependency on stresses in the electrical steel. We further present the
implementation of these equations into in-house 2D FE software for the simulations of electrical machines. The simulations
carried out show that the energy based model describes well the vibrations of electrical machines due to magnetostriction and
reluctance forces. A discussion on how the model should be improved to account for hysteresis is also presented.
Index Terms—coupled models, electrical machines, finite elements, magnetostriction.
current density and the geometry of the windings are
known. However, some issues related to the skin-effect
and the eddy-currents make this computation rather
complex in some special cases as explained by Islam et al.
[1]. In iron, and due to its magnetic domain structure and
its finite electric conductivity, the flow of a time-varying
flux produces hysteresis and eddy-current losses (in some
approaches also excess losses). The computation of these
so-called iron losses is still one of the most active
research fields in the simulation of electrical machines.
Finally, the motion of the rotating parts of the machine
and the friction in the bearings of the machine as well as
the one between the moving parts and the air produces
mechanical losses that can be computed through complex
CFD models in case of high speed machines or
approximated by semi-empirical equations.
The knowledge of the above loss components is
valuable information for the designers of electrical
machines as they allow them to optimize the structure of
the machine with regards to the cooling and the
mechanical strength as well as the use of magnetic and
other materials.
Besides the structural and cooling optimization of the
machine, the designers are bind by environmental aspects
such as the level of acoustic noise and the estimation of
the lifecycle of the machine. The acoustic noise is
produced by the vibrations of the structure of the machine
under the effect of different forces and other excitations
and by the airflow in different channels.
I. INTRODUCTION
Owing to the high demand on energy efficient and
environmental friendly apparatuses, the designers of
electrical machines, among others, are more and more
interested in accurate computational methods for the
analysis of their design. The energy conversion in
electrical machines occurs within three different but
tightly coupled subsystems, i.e., the electrical system, the
magnetic system and the mechanical system as shown in
Fig. 1. The electric system consists of the windings of the
machine that are connected to the supply in the case of a
motor or to the load in case of generator operation. The
electric supply/load is nowadays typically a voltage
source frequency converter, the current of which is
controlled according to the operation point of the
machine. Such current depends on the load of the
machine and consists of a torque producing component as
well as a component necessary for the magnetization of
the iron core and another compensating for the core
losses. The magnetic system consists of the iron core of
the machine as well as the airgap and other construction
parts in which a magnetic flux is produced by the coils’
currents according to the Ampere’s law of induction.
These fluxes and their interaction with the magnetic
materials produce forces and torque that are transferred to
the mechanical system consisting of the rotor, the shaft
and the bearings of the machine as well as a possible
cooling fan mounted on the shaft of the machine.
The electric, magnetic and mechanical systems,
although represented as separate subsystems, are tightly
coupled to each other and their operation quantities
cannot be solved separately. Indeed the current drawn by
the machine, the torque it produces and the magnetic flux
density in the airgap and the iron core are usually solved
simultaneously especially if the machine is voltage fed.
The operation of the above subsystems is known to be
dissipative as there are energy losses related to each
subsystem. The Joule losses resulting from the currents
flowing in the windings are usually easy to compute if the
Electrical power
Lorentz forces
Electrical
System (windings)
Electrical
Losses
Magnetic forces and
Magnetostriction
Coupling field
(iron and air)
Air flow and Friction
Vibrations Wearing and Noise
Magnetic
Losses
Mechanical system
(bearings and fan)
Mechanical
Losses
Mechanical power
Fig. 1. Illustration of the energy conversion process with the
related electric, magnetic and mechanical subsystems and
related losses, forces and vibrations and their origins.

The interaction between the flux and the currents
produces Lorentz forces acting on the windings, while the
flow of the magnetic flux in the iron core and the airgap
gives rise to magnetic forces and strains in the core.
The vibrations produced by the interaction between the
magnetic forces, the deformations and the structure result
in acoustic noise and mechanical wearing of different
parts of the machine such as the winding insulation and
the bearings. The knowledge of these parasitic effects at
the design stage will help in reducing the vibrations and
noise of the machine as well as estimating the lifecycle of
the machine and optimizing the structure for longer life
too. An illustration of the different losses and vibration
phenomena occurring at different subsystems of the
energy conversion is given in Fig. 1.
The strains and stresses in the iron core of an electrical
machine are produced by different sources and have
strong degrading effect on the quality of the iron, thus
reducing the efficiency of the energy conversion process
and increasing the amount of iron needed for a given
power of the machine. The effect of the mechanical stress
on the power losses in electrical steel have been known
for quite long time. Already in the 70’s of the last century
Moses [2], [3], among others, showed through magnetic
measurements that the mechanical stress affects the
losses, the magnetization, and the magnetostriction of
electrical steel. By magnetostriction it was meant the
relative change in the length of a specimen of magnetic
material when it is subjected to a magnetic field.
For a better understanding let us first clarify what is
magnetostriction. In 1842 W. P. Joule discovered what is
today called Joule magnetostriction, which is a volume
conserving deformation of magnetic material caused by
its magnetization. Such a deformation results in an
elongation of the material in the direction of
magnetization and a shrink in the orthogonal directions
for positive magnetostrictive materials and vice versa for
negative magnetostrictive materials. Later on, it was
observed that at high values of the magnetization the
deformation is no more volume conserving and this
phenomena was called volume magnetostriction. Both
types of magnetostriction are called forced
magnetostriction in a sense that they are caused by the
magnetization that forces the magnetic domain walls to
move and the domains to rotate thus producing the
mechanical deformation. On the other hand, when the
magnetic material is cooled down from a high
temperature, it undergoes a strong isotropic change in its
dimensions as it goes though the Curie temperature. Such
a change in volume was explained by the formation of
magnetic domains and the orientation of elementary
magnetic moments within the domains. Several other
experimental works have been conducted on magnetic
materials and separate magnetomechanical phenomena
obtained separate names according to their respective
discoverers. The skew magnetostriction ,e.g., resulting
from a helical magnetization and producing a bending of
some electrically conducting magnetic material has been
called Wiedemann-effect and the inverse effect of
- 215 - 15th IGTE Symposium 2012
magnetostriction which results in a change of the
magnetization properties of magnetic materials under the
action of applied mechanical stress was called Vilari
effect. Also the apparent change in the Young modulus of
magnetic material, which is due to the intrinsic
rearrangement of the magnetic domains and thus the
intrinsic elongation following this rearrangement, has
been called Delta-E effect. A comprehensive description
of these phenomena can be found, e.g., from [4]. The
intrinsic forms of magnetostriction are nowadays referred
to as spontaneous magnetostriction. They are of great
interest for metallurgist but are not of importance for the
simulation of electrical machines under operation as they
do not occur anymore at this stage. Fig. 2 shows an
illustration of the difference between spontaneous and
forced magnetostriction.
Cooling through
Curie temperature
H
Positive Joule
magnetostriction
H=0
Applying external
magnetic field
Spontaneous
Negative Joule
magnetostriction
H
Forced magnetostriction
magnetostriction
Fig. 2. Illustration of spontaneous and forced magnetostriction.
The mechanical stress or strain acting on the magnetic
material can originate from different phenomena some of
them produce static stresses and other dynamic stresses.
The shrink fitting of the stator into the frame of the
machine produces static compressive stresses of the order
of 10 MPa as shown by Fujisaki et al. [5]. These stresses
can be evaluated by means of numerical simulations or
through analytical approximations, but due to the
manufacturing tolerances they may have excessive local
values. On the other hand the so called reluctance forces
occurring between the stator and the rotor of the machine
and which are time and space dependent, produce
dynamic tensile and compressive stresses at different
locations of the machine. These stresses are at the origin
of the so called magnetic noise, i.e., the acoustic noise
due to magnetically excited vibrations. The level of these
stresses depends on the magnetic flux density in the air
gape of the machine and can be of the order of 200 MPa.
Further, the rotating and alternating magnetization of the
iron core produces dynamic magnetostrictive strains the
level of which depends on the state of stress in the core.
At last but not least, the punching of the magnetic
material in the manufacturing process produces residual
stresses and plastic strains at some regions of the
magnetic material. The level of these stresses and strains
depend the manufacturing process and the quality of
punching tools.
All these stresses and strains will affect both the
magnetization characteristics and the energy losses of the
magnetic material as well as the vibrations of the core and
thus the acoustic noise and the wearing of the materials
and parts of the machine.
The modeling of magnetostriction started In the 50’s of
the last century as related to the so called giant
magnetostrictive materials and their applications in
ultrasound generators and receptors. The effect of

magnetostriction on the noise of power transformer was
also investigated through experimental studies starting in
the 70’s by Moses [6] and later by Weiser and Pfutzner
[7] but the modeling of such phenomena in electrical
machines started only at the beginning of the last two
decades as it was noted that the prediction of the losses
and vibrations of the machine requires adequate models
able to account for the magnetomechanical coupling as
well as the other electromagnetic couplings.
In this paper we will concentrate on the modeling of
magnetostriction and the related magnetomechaical
effects and their incorporation into the simulation of
electrical machines as well as the effect of these
phenomena on the computation of iron losses. The
coupled electromagnetic simulation methodology for
electrical machines, which made it possible to develop the
magnetomechanical models is now quite established as
reported by Arkkio [8] and Salon et al. [9] among others
and will not be handled here.
In section II we will explore the force-based models of
magnetostriction and in section III the stress-strain based
models. In section IV we will introduce the energy based
model and in section V we will discuss the incorporation
of the different models into 2D finite element simulation
of electrical machines. In Section VI we will discuss the
impact of these models on the computation of iron losses
and sketch the necessity for future developments in view
of accurate simulations and computation of losses and
vibrations.
II. FORCE-BASED MODELS OF MAGNETOSTRICTION
The main idea behind the force-based models of
magnetostriction is that the magnetostrictive deformation
of a sample of magnetic material under homogeneous
magnetization can be produced by a distributed set of
forces acting on the boundaries of the sample as
explained by Delaere et al. [10] and sketched in Fig. 3 for
an arbitrary element.
Such a representation of the magnetostriction emanates
from earlier work of Besbes et al. [11], where the
magnetostrictive forces were directly derived from the
principle of virtual work and by accounting for the
variation of the permeability with the mechanical stress. It
should be mentioned that in [11], the magnetostriction has
not been well described as sever assumptions such as
linear magnetization and its linear dependency on the
stress have been made. The local application of the
principle of virtual work for the computation of magnetic
forces itself was developed by Bossavite [12] after its
introduction at a global level by Coulomb [13].
The development of such force-based models and their
coupling with the electromagnetic simulation of electrical
machines was also reported by Mohamed et al. [14],
Vandevelde et al. [15] and Belahcen [16]. Vandevelde
and Belahcen used a stress approach to compute the
magnetostrictive forces, whereas Delayer used a strain
approach. In all these works, the other magnetic forces
were computed either according to the principle of virtual
- 216 - 15th IGTE Symposium 2012
work and introduced into the FE simulation as
generalized nodal forces or through the Maxwell stress
tensor. The two methodologies have been earlier
demonstrated by Kameari [17] to be equivalent. Lately,
many authors applied the concept of magnetostrictive
forces in the FE simulation of electrical machines.
The equation for the computation of the generalized
nodal magnetic forces is given bellow
B
T 1
F A A d dSˆ
J e
Sˆ
e
H
B J
(1)
e U U
0
where B and H are the magnetic flux distribution and
field strength respectively. J is the Jacobian matrix for the
transformation from the reference finite element to actual
one and ˆ Se stands for the reference element. U is the
vector of nodal displacement. The integration with respect
to the magnetic flux density in (1) as well as the
differentiation with respect to the displacements is carried
out analytically. For this purpose and for FE computation,
a cubic spline representation of the HB-curve of the iron
sheets is used. The different approaches for the
computation of nodal magnetostrictive forces can be
found in [10], [14], [15], and [16]. Fig. 4 shows the
magnetic and magnetostrictive force computed for the
stator core of a synchronous machine [18]. Similar forces
for the induction machines have been reported in [19].
Fig. 3. Sketch of the computation of magnetostrictive force from
magnetostrictive deformation after Delaere [10].
Fig. 4. Generalized magnetic forces (left) and equivalent
magnetostrictive forces computed in the stator core of a
synchronous machine. The forces have been normalized.
The structural deformation can be computed either in a
coupled or uncoupled methodology. In the coupled
approach the mechanical and magnetic problems are
solved simultaneously and the forces are updated at
iteration level. In the uncoupled approach the magnetic
problem is first solved and the forces computed as postprocessing
quantities from the magnetic problem then
introduced in the mechanical problem as loads. The
results of the mechanical problem are the nodal
displacements from which the deformation as well as the
strains and stresses can be computed.
Although the concept of magnetostrictive forces
describes quit well the deformation of the magnetic

material it has a major drawback that consist of resulting
in erroneous stress state in the material. Indeed, the
magnetostrictive forces result into tensile stress if the
boundaries of the element are free to move (refer to Fig.
3). However, the actual state of stress in this case should
be a zero stress in the element. If the boundaries of the
element are fixed, the magnetostrictive forces will result
into a zero stress, while the actual stress is a compressive
one, the magnitude of which depends on the material
properties and the level of magnetization. This erroneous
behavior due to equivalent magnetostriction forces is
illustrated in Fig. 5, where the magnetostrictive
elongation is solved correctly but the stress is wrong.
From the above discussion it is clear that the concept of
magnetostrictive force is not able to describe the stress
dependency of the magnetic properties of the material and
thus of the magnetostriction itself when it has to be
computed under different boundary conditions dictated by
the geometry and the topology of the electrical machine.
III. STRAIN-BASED MODELS OF MAGNETOSTRICTION
In the strain-based models of magnetostriction there is
no need for the calculation of equivalent forces. The
magnetostrictive strains are incorporated in the structural
analysis in a similar way to the thermal dilatation of
metals. Here also, the magnetostriction can be modeled in
a decoupled or coupled approach. In the decoupled
approach [20] the magnetostrictive strains are computed
from per element magnetic flux densities and the
measured single valued flux densty-elongation
relationship. These strains are then incorporated in a
structural finite element model of the electrical machine
to produce the deformation. If the effect of stress is to be
accounted for in the computation of the magnetostriction,
a coupled model should be used. Such a model emanates
from the coupled magnetomechanical constitutive
equations of the material [21] as explained in the next
Section.
Equivalent forces:
Equivalent forces:
= 0 ; = 0 = ms ; = -ms
Iron
Actual behavior:
Actual behavior:
= 0 ; = ms = ms ; = 0
Fig. 5. Illustration of the magnetostriction and the state of
stress in the sample under the effect of magnetostrictive forces.
Left clamped sample and right free sample. In both case the
elongation is correct but the stress is wrong.
IV. THE ENERGY-BASED CONSTITUTIVE EQUATIONS
The energy-based, coupled constitutive equations of
the electrical steel are derived from an appropriate
representation of the Helmholtz free energy [21], which
itself is based on previous empirical observations made
from the measurement of magnetostriction under different
stresses and flux densities [22]. A summary of the results
of these measurements all together with the model
prediction are shown in Fig. 6. The Helmholtz free energy
in a sample is written as:
Iron
- 217 - 15th IGTE Symposium 2012
1
I
2
2
1
4 1
g ( I ) I
i 1 4 1 1
2
2 4 5 5 6 6
2
i0 i 1
B
2 2
ref
2GI
I I I
where the invariants I .. I are:
1 6
1 2 1 3
I tr( )
, I tr( ) , I tr( )
1 2
3
2
3
(3)
I
4 B B, 2
I BB, I BB 5
6
(4)
is the first Lamé parameter,G the shear modulus of the
material, the mass density and
3 3 1
g exp( I ) 0 0 1 0 5
4 4 3
(5)
3( i 1) 4( i 1)
g exp
I
; i i
1
4
3
i 1..4(6)
; i 0..6 are model parameters and B is a reference
i ref
flux density. The magnetomecjanically coupled
constitutive equations are derived from (2) as
6
6
Ii
Ii
and M
i1, i3I i
i1, i3IB i
(7)
where and are the stress and strain tensors and M the
magnetization. An extensive derivation of the model and
its equations is given in [23]. The model results in an
explicitly coupled formulation for the stress tensor and
the magnetic field strength vector in terms of the magnetic
flux density and the mechanical strain tensor as
( B,
) I + (
I ) 2G
1 1 4
1 1
( )
2 ( )
0
B B BB
B B BB 4
2
B B( BB)
5
2
2
B B B B
6 2
2 ( ) +( )
B B B B B B
(8)
1
5
2
HB ( , ) B2 B B B
0 4 5
2 2
(9)
Magnetostriction (m/m)
(2)
In (8) and (9) 1 and are used to shorten the notation:
4
x 10-6
5
4
3
2
1
0
-1
1 g
I and
4
i i1
1 4
2
i0
I1
3.9 MPa
0.0 MPa
-1.7 MPa
-6.1 MPa
-2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Flux density (T)
1
g
(10)
4
i i
I
4 4
2i0i1 Fig. 6. Measured magnetostrictive strains at different flux
densities and applied mechanical stresses (left) and comparison
with model prediction (right). Positive stresses are tensile and
negative ones are compressive.
V. INCORPORATION INTO FE SIMULATIONS
The starting point for the implementation of the
magnetomechanically coupled equation in FE analysis of
electrical machines is an in-house 2D software package.
The field equations have been previously coupled with

the electrical circuit equations of the machine winding,
which makes it possible to feed the model from a voltage
source and also resolve the induced voltages and currents
in other parts of the machine, e.g., cage winding or solid
rotor, through time stepping analysis [8].
The coupled constitutive equations (8) and (9) were
first linearized to the first order and then the weak form of
Galerkin method was applied to (8) and the principle of
virtual work applied to (9). This resulted in:
H ( w)
d H
( w)
B d
B
(11)
( w) H d wH ds 0
0 0
T ˆ
d
T
ˆ
d
ˆ d uˆ ( f f ) d
B
T
0
B
T
T
uˆf ds 0
surf
mech inert
- 218 - 15th IGTE Symposium 2012
(12)
where w is a test or weight function and the quantities
with hat are virtual ones.u is the mechanical displacement
vector and f , f , f are respectively mechanical,
mech inert surf
inertia body forces and surface forces. In the
implementation the shape functions of the finite element
approximation are used as weight function.
Equations (11) and (12) are then spatially descitized
using standard finite element procedure and inserted in
the in-house code, thus replacing the nonlinear model of
iron cores. The insertion of these equations in the code
does not affected the electromagnetic coupling as this
latter one takes place in the windings and conducting
regions only whereas the magnetomechanical coupling
takes place in non conduction iron. Special attention
however has to be given to the region formed by the
airgap or more properly the interface between any noniron
region and the iron core. This is because the Maxwell
stress tensor makes sense only if it is computed from both
regions at any interface. Thus when assembling the
system matrix, the contribution to the nodal values of the
Maxwell stress are computed from both iron and non-iron
element with common interface with the iron core.
In the case of force based model of magnetostriction,
the approach is quite similar to the one presented above,
except that the equivalent magnetostrictive forces as well
as the other magnetic forces computed with (1) have been
inserted in the model as external mechanical forces. Such
implementation was already reported in [19].
The implemented formulation and software were first
tested on a simple model consisting of an iron disc
excited through Direchlet boundary condition on its outer
edge. The magnetic vector potential on this edge was set
to time dependent values as to create a rotating field
uniformly distributed on the surface of the test sample.
The boundaries of the sample were free to move and only
the center of the disc was fixed in both x- and y-direction.
Fig. 7. Shows the original and deformed mesh used in the
model when the flux density was 1.5 T either along the xaxis
or at an angle of 45 deg. to it. The results show that
thanks to the tensor representation and formulation the
effect of the shear stress and strain are correctly
computed. The model was also applied to a induction
machine-like device without the airgap in view of
minimizing all the other magnetic forces. The results from
this verification are reported in [23] and show good
agreement between the measured and computed
displacements. The model was applied to the computation
of the deformation and vibrations of two induction
machines, the parameters of which are given in Table I.
The extensive results from the simulations of the two
machines as well as a comprehensive analysis of these
results are reported in [24]. Here we present a comparison
between the computed displacements of nodes on the
tooth of the machines when the magnetostriction only is
accounted for and when the so called reluctance forces
(Maxwell stress) are also accounted for. Although, the
vibrations depend on the machine construction and could
not be generalized, this result gives the reader an estimate
of the effect of magnetostriction on the vibrations of
rotating electrical machines. Fig. 8 shows this comparison
for both machines. Due to the differences in the number
of pole pairs the vibration behaviors are different too.
y-coordinate (m)
0.15
0.1
0.05
0
-0.05
-0.1
-0.1 -0.05 0 0.05 0.1 0.15
x-coordinate (m)
y-coordinate (m)
0.15
0.1
0.05
0
-0.05
-0.1
-0.1 -0.05 0 0.05 0.1 0.15
x-coordinate (m)
Fig. 7. Test model, original and deformed mesh computed with
the developed method with a flux density of 1.5 T along the xaxis
(left) and at 45 deg (right).
Table I. Parameters of the simulated induction machines
Parameter Machine I Machine II
Rated voltage 380 V 380 V
Slip 2 % 3.2 %
Rated current 60 A 27 A
Rated power 30 kW 15 kW
Number of pole pairs 1 2
Outer diameter of the stator core 323 mm 235 mm
Inner diameter of the stator core
190.2 mm 145 mm
Number of stator slots
36
36
Outer diameter of the rotor core 188.37 mm 144.1 mm
Number of rotor slots 28 34
Fig. 8.Comparison between computed displacements of a
node on the stator tooth in machine 1 (left) and machine 2
(right). Subscripts r and stand for the radial and tangential
directions. 1 is the case with only magnetostriction and 2 when
both magnetostriction and reluctance forces are considered.

VI. FUTURE TRENDS AND DEVELOPMENTS
The developed coupled magnetomechanical model
although bidirectional in magnetic and mechanics is
single valued and thus does not take dissipation into
account. On the other hand, existing hysteresis models
either for magnetic [25] or mechanics [26] are based on
the Preisach approach, which is mathematically rigorous
but does not give clear insight into the energetic balance
between the two subsystems. The only energy-based
hysteresis model that describes both magnetism and
mechanics is the one developed by Jiles [27] but its
application in electric steel and further to the simulation
of electrical machines has not been reported yet. This
might be due to the sharp saturation of the magnetization
curves of electrical steel, which cause convergence
problems but also to the fact that the original model traits
the compressive and tensile stresses in a symmetric way,
which is not adequate for the electric steel where the
compressive stresses have much pronounced effect on the
magnetic and magnetostrictive properties of the material.
The dynamic behavior of magnetostriction also needs to
be addressed [28] as well as the effect of anisotropy [20].
These shortcuts in the modeling and simulation of
electrical machines still need to be addressed in view of
better estimation of the vibrations of electrical machines
and iron losses, which are known to depend on the state
of stress in the material. The presented model already
estimates the effect of magnetostriction on the state of
stress but other causes of stress have to be added too.
The models to be developed and used need both
characterization of the material under different flux
densities and mechanical stress and verification
procedures to assess the validity of the models. The work
presented in [29] is a good start for the characterization
work. We have also developed characterization
methodologies and analyzed their accuracy [30]; this
work is still continuing. The verification work needs still
some development.
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[5] K. Fujisaki, R. Hirayama, T. Kawachi, S. Satou, C. Kaidou, M.
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[8] A. Arkkio, Analysis of induction motors based on the numerical
solution of the magnetic field and circuit equations, Doctoral
dissertation, 1987, Helsinki University of Technology, Espoo, Finland
- 219 - 15th IGTE Symposium 2012
[9] S. J. Salon, R. Palma, C. C. Hwang, “Dynamic modeling of an
induction motor connected to an adjustable speed drive,” IEEE Trans.
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[11] M. Besbes, Z. Ren, A. Razek, “Finite element analysis of magnetomechanical
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1266, Mar 1992.
[13] J. L. Coulomb, “A methodology for the determination of global
electromechanical quantities from a finite element analysis and its
application to the evaluation of magnetic forces, torques and
stiffness,” IEEE Trans. Magn., vol. 19. 6, pp. 2514-19, 1983.
[14] O.A. Mohammed, T. Calvert, R. McConnell, “Coupled
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2001.
[15] L. Vandevelde, J.A.A. Melkebeek, “Magnetic forces and
magnetostriction in electrical machines and transformer cores,” IEEE
Trans. Magn., vol. 39, no. 3, pp. 1618- 1621, May 2003.
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magnetomechanical coupling and magnetostriction,” IEEE Trans.
Magn., vol. 42, no. 4, pp. 971-974, Apr. 2006.
[17] A. Kameari, “Local calculation of forces in 3D FEM with edge
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[18] A. Belahcen, “Magnetoelastic coupling in rotating electrical
machines,” IEEE Trans. Magn., vol. 41, no. 5, pp. 1624-1627, May
2005.
[19] A. Belahcen, Magnetoelasticity, magnetic forces and
magnetostriction in electrical machines. Doctoral dissertation, 2004,
Helsinki University of Technology, Espoo, Finland.
[20] S. Somkun, A. J. Moses, P. I. Anderson, P. Klimczyk,
“Magnetostriction anisotropy and rotational magnetostriction of a
nonoriented electrical steel,” IEEE Trans. Magn., vol. 46, no. 2, pp.
302-305, Feb. 2010.
[21] A. Belahcen, K. Fonteyn, A. Hannukainen and R. Kouhia, “On
numerical modeling of coupled magnetoelastic problem”, Nordic
Seminar on Computational Mechanics NSCM-21, pp. 203-206, Oct.
16-17.2008, Trondheim, Norway.
[22] A. Belahcen and M. El Amri, “Measurement of stress-dependent
magnetisation and magnetostriction of electrical steel sheets,”
International Conference on Electrical Machines ICEM, Sep. 5-
8.2004, Cracow, Poland.
[23] K. A. Fonteyn, Energy-based magneto-mechanical model for
electrical steel sheets, Doctoral dissertation, 2010, Aalto University,
Finland
[24] K. A. Fonteyn, A. Belahcen, P. Rasilo, R. Kouhia, A. Arkkio,
“Contribution of Maxwell stress in air on the deformations of
induction machines,” Journal of Electrical Engineering & Technology,
vol. 7, no. 3, pp. 336-341, 2012.
[25] E. Dlala, A. Belahcen, K. Fonteyn, M. Belkasim, “Improving loss
properties of the mayergoyz vector hysteresis model,” IEEE Trans.
Magn., vol. 46, no. 3, pp. 918-924, March 2010.
[26] A. Bergqvist, On magnetic hysteresis modelling, Doctoral
dissertation, 1994, Royal Institute of Technology, Stockholm,
Sweden
[27] D. C. Jiles, D. L. Atherton, “Theory of ferromagnetic hysteresis
(invited),” Journal of Applied Physics, vol. 55, no. 6, pp. 2115-2120,
Mar 1984.
[28] P. Rasilo, A. Belahcen, “Iron losses, magnetoelasticity and
magnetostriction in ferromagnetic steel laminations,” IEEE
Conference on Electromagnetic Field Computation CEFC, 11-
14.11.2012, Oita, Japan.
[29] Y. Kai, Y. Tsuchida, T. Todaka, M. Enokizono, “Influence of stress
on vector magnetic property under rotating magnetic flux conditions,”
IEEE Trans. Magn., vol. 48, no. 4, pp. 1421-1424, April 2012.
[30] A. Belahcen, P. Rasilo, K. Fonteyn, R. Kouhia and A. Arkkio,
“Modeling the stress effect on the measurement of magnetostriction in
electrical sheets under rotational magnetization,” IEEE Conference on
Electromagnetic Field Computation CEFC, 11-14.11.2012, Oita,
Japan.

- 220 - 15th IGTE Symposium 2012
Magnetic Saturation Effect on Modeling Squirrel-cage
Induction Motors with Stator Inter-turn Fault
*Jawad Faiz, † Mansour Ojaghi and † Mahdi Sabouri
*Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer Engineering,
College of Engineering, University of Tehran, Tehran, Iran, E-mail: jfaiz@ut.ac.ir
† Department of Electrical Engineering, University of Zanjan, Zanjan, Iran
Abstract— Coupled circuits model (CCM) of squirrel-cage induction motors is the most detailed and complete analytical
model for analyzing the performance of the faulty induction motors. This paper extends the CCM to a saturable model
including variable degrees of the saturation effects using an appropriate air gap function and novel techniques for locating
the angular position of the air gap flux density and estimating the saturation factor. Comparing simulated and experimental
magnetization characteristics shows the accuracy of the new saturable model. Using saturable and non-saturable models,
various simulations are carried out on faulty induction motors, and then, by comparing the results, the impacts of the
saturation on the performance of the faulty motor are presented.
Index Terms—Induction motors, Inter-turn fault, Magnetic saturation, Coupled circuits model.
I. INTRODUCTION
Implementing a proper condition monitoring system is
essential to prevent squirrel-cage induction motors
(SCIMs) from catastrophic failure. The stator inter-turn
fault is a relatively frequent fault and if not diagnosed on
time, causes major breakdown of the SCIM. SCIM
performance under the inter-turn fault can be analyzed
using magnetically coupled circuits model (CCM) [1],
[2]. Such analysis helps to realize the faulty SCIMs
performance, to extract proper indexes for the various
faults and to develop effective fault diagnosis and
condition monitoring techniques for SCIMs. To do so,
CCM must be as exact as possible; however, ignoring the
magnetic saturation may keep it away from the required
exactness [3].
For economical utilization of the magnetic material,
electrical machines operating regions have to be extended
above the knee of the magnetization characteristic, which
forces the machine into the saturation region. Many
attempts have been so far made to include saturation
effects in SCIM models including CCM [4]–[9]. An
extension to CCM of healthy SCIM, which includes
variable degrees of saturation, has been reported in [9].
The proposed saturable CCM (SCCM) needs to track the
air gap rotating flux density, and this has been done by a
rather simple technique in [9]. However, distortion of the
air gap flux density distribution due to the fault causes
the application of that technique erroneous. Thus, the
existing SCCM is not viable to analyze the faulty SCIMs.
In this paper, the rotor meshes are used as the air gap
flux samplers. The flux-linkages of the rotor meshes,
calculated at each step, are used to estimate the air gap
flux density distribution. Then, Fourier series analysis is
used to determine the space harmonics of the air gap flux
density, including fundamental harmonic amplitude (B1)
and its phase angle (1). B1 is used to determine
saturation factor (Ksat) properly and 1 is utilized to track
the air gap flux density. Therefore, a modified version of
the SCCM is obtained, whose accuracy is proved by
comparing the magnetization characteristics determined
through the simulation and experiments. Then, using
both the modified SCCM and the normal CCM, the
performance of SCIM with some inter-turn faults are
simulated. By comparing the results, impacts of magnetic
saturation on the faulty SCIMs performance and their
fault indexes will be clear. Comparisons are also made
with experimental results, which confirm the accuracy of
the proposed model.
II. SCCM OF SCIM WITH INTER-TURN FAULT
Any loop on the rotor of SCIM consisting of any two
adjacent rotor bars is considered as a circuit mesh. The
shorted turns in the stator are also considered as an
independent circuit (phase 'd'). Applying KVL to the
rotor meshes, stator phases and stator shorted turns and
adding the related torque and mechanical equations,
CCM dynamic equations are attained. These equations
with complete details are presented in [2]. Self/mutual
inductances of the various circuits and their derivatives
versus the rotor position are the most important
parameters of CCM equations. These inductances are
calculated using the following equation [10]:
2
1
Lxy or
l
g n N d
0 x y
(1)
where x and y can be any phase of the stator (a, b, c or d)
or any mesh of the rotor (1 to R), μ0 is the air magnetic
permeability, r is the air gap mean radius, l is the stack
length, g -1 is the inverse air gap function, nx is the x
phase (mesh) turn function and is the angle in the stator
stationary reference. In the case of the uniform air gap
machine, Ny is the winding function of the y circuit, but
in the case of the non-uniform air gap machine, it is the
modified winding function of the y circuit.
Saturation of the magnetic material causes its
reluctance to be increased against the machine's flux.
Similar increase of the reluctance can be achieved by a
proportional increase in the air gap length along the main
flux path [4]. Anywhere within the core material,
reluctance increase caused by the saturation depends on
the exact value of the flux density, but independent of the
flux direction. Thus, it is expected that the fictitious air
gap length (gf) fluctuates a complete cycle every half

cycle of the air gap flux density distribution around the
air gap. Assuming a sinusoidal form for this fluctuation,
the following satisfies the mentioned requirements [9]:
g f g[
1
cos( 2P(
f ))] (2)
where f is the angular position of a zero crossing point
of the air gap flux density in the stator reference, P is the
pole pairs, g' is the mean value of gf and is the peak
value of its fluctuation. g' and are determined as
follows:
3K
sat
g
ge
(3)
K 2
2( Ksat
1)
(4)
3Ksat
where ge is the effective air gap length of the unsaturated
machine, which is related to the mechanical air gap
length (g0) by the Carter's coefficient (ge=kcg0). Ksat is the
saturation factor which is defined as the ratio of the
fundamental components of the air gap voltage for the
saturated and unsaturated conditions [4].
Replacing the inverse of (2) into (1), using modied
winding function theory, assuming the turn functions to
have only step variation in the center of the slots and
determining the indefinite integrals, exact analytic
equations are obtained for the various inductances. Then,
differentiating the equations versus the rotor position
(r), exact analytic equations also are obtained for the
derivatives of the inductances. The equations are
functions of Ksat and f, thus by using them; variable
degree of saturation effect enters to the model of the
SCIM. The equations are included in the appendix.
For the proposed SCIM (see Section V), Figure 1
shows the turn functions of the phase 'a' winding before
and after inter-turn fault. The winding has 4 concentric
coils each with 90 turns in healthy condition. In the faulty
case 14 turns from the outer coils are short-circuited.
Figure 1 also shows the turn function of the shorted turns
(phase 'd'). As an example, Figure 2 shows the variations
of the self inductance of the phase 'd' (Ldd) by the
variations of Ksat and f, which has been calculated using
the proposed analytical equations. As seen, by increasing
Ksat and Ldd more variations occur around its decreasing
mean value. The increase of g' and by increasing Ksat
are respectively the reasons for the decreasing mean
value and increasing uctuations amplitude of the
inductance. A complete rotation of f causes two
complete cycles of variation for Ldd, which is due to the
two poles of the SCIM.
sat
- 221 - 15th IGTE Symposium 2012
Figure 1: Turn function of phase 'a' winding: a) before and b) after
inter-turn fault with 14 shorted turns in the outer coil and c) turn
function of phase 'd' after the fault occurrence
Figure 2: Phase 'd' self inductance (Ldd) variation versus Ksat and f
III. DETERMINING K SAT AND F IN FAULTY SCIM DURING
SIMULATION STEPS
Generally, the air gap flux density distribution is
disturbed in the faulty SCIMs and this leads to an error in
the use of the technique for determining f [9]. In
addition, Ksat was determined using the air gap voltage,
which depends on the rotation speed of the air gap flux as
well as its amplitude, while the saturation degree depends
only on the flux density amplitude. This bring difficulty
when applying the model in variable-speed drives
systems, as the air gap voltage before saturation is no
more a constant, but varies by the reference speed
variation.
When simulating SCIM using the CCM or SCCM,
flux-linkages of all the rotor meshes are evaluated within
any simulation step. Since any rotor mesh consists of
only one turn, these flux-linkages are the total fluxes
passing through the meshes. Considering the short air
gap length and ignoring the small flux leakages, the air
gap flux density next to any rotor mesh i can be estimated
as:
Bai i
A
(5)
where i is the flux-linkage of mesh i and A is the area
above the mesh in the air gap:
A 2rl
R
(6)
where R is the rotor bars number. Therefore, the air gap
flux density distribution is estimated within any
simulation step. Then, using Fourier series analysis, the
space harmonics of the air gap flux density is determined
as follows:
1 2
R 1
i1
Bsn
B(
) sin( nP
) d
Bai
sin( )
0
nP
d
i
(7)
B
cn
1
np
R
i1
B [cos( nP
) cos( nP
) ]
ai
i1
R
1
Bai[sin(
nPi
1)
sin( nPi
)]
np
i1
where Bsn and Bcn are the sine and cosine components of
the nth space harmonic of the estimated air gap flux
density respectively, is the angle in the rotor reference
and i the angle of center of the rotor bar i. Then, the
phase angle of the space harmonics (in electrical
i1
R
1 2
1
i1
B(
)
cos( nP)
d
Bai
0
i
i 1
i
cos( nP)
d
(8)

adians) and their amplitudes can be calculated as follows
respectively:
tan ( )
1 Bsn
n
(9)
B
cn
2
Bsn
2
Bn Bcn
(10)
Having the phase angle of the fundamental harmonic
(1), f could be estimated by:
1
f r
(11)
P 2P
Also using the amplitude of the fundamental harmonic,
Ksat obtained by:
1 0 B B Ksat (12)
where B0 is related to the flux density of knee (Bkp) of the
core material within the teeth.
These modifications in f and Ksat estimations make
the SCCM applicable to the simulation of mains-/driveconnected
SCIMs under the inter-turn fault. However,
knee point is not a distinct point on the magnetization
characteristic and there is not precise analytic method to
determine the Carter's coefficient (kc). Therefore, Genetic
Algorithm (GA) is used for the optimal estimation of the
required B0 and kc in the next section.
IV. ESTIMATION OF B0 AND K C
GA is a heuristic searching method for the optimal
solution based on mechanics of natural selection and
natural genetics. It evolves into new generations of
individuals by using knowledge from the previous
generations and generally includes three fundamental
genetic operations of reproduction, crossover and
mutation. The searching process is independent of the
form of the objective function, and will not be trapped in
the rapid descending direction introduced by the local
optimum solutions. The solution of a complex problem
can be started with weak initial estimations and then be
corrected in evolutionary process of fitness. Figure 3
shows a flowchart of the applied GA. More details about
GA can be found in [11].
To use GA for estimating B0 and kc in the proposed
SCIM, a proper objective (fitness) function must be
defined. Such fitness function may be achieved by
closely fitting the magnetization characteristic (i.e. the
no-load voltage versus no-load current curve) of the
SCIM obtained from SCCM to that obtained by
experiments. To do so, the no-load stator RMS line
currents are measured in the laboratory with n different
stator line voltages up to
- 222 - 15th IGTE Symposium 2012
Reproduction
No
Figure 3: Flowchart of the applied Genetic Algorithm
Figure 4: Convergence rate of the algorithm
the nominal voltage (I e ai, I e bi, I e ci, for i=1,2,…,n). Then,
for any distinct values of B0 and kc, corresponding line
currents with the same stator voltages are obtained by
simulation (I s ai, I s bi, I s ci, for i=1,2,…,n). Now, the fitness
function is defined as follows:
n
i1
e
ai
s 2
ai
e
bi
Start
Select variables
(Solution space)
Construct initial
population randomly
Calculate the fitness for
each population
Apply crossover
Apply mutation
Ending
condition
reached?
Select the best population
End
Yes
s 2
bi
Fit.
(( I I ) ( I I ) ( I I ) ) (13)
e
ci
s 2
ci
The lower the fitness, the better will be the estimation of
B0 and kc. With a population size of 15, GA converges to
the required solution after about 130 iterations. Figure 4
shows the convergence rate of the algorithm. The
optimum values for B0 and kc are 0.5007 and 1.2058
respectively. Figure 5 compares the
magnetization characteristics obtained from the
experiment and SCCM with optimal B0 and kc. Good
agreement between the simulated and experimental
results is evident.
V. EXPERIMENTAL TEST RIG
A test rig consisting of a 750 W, 380 V, 50 Hz, 2-pole,
Y-connected SCIM was set up in the laboratory. Three-

Figure 5: Magnetization Characteristic obtained from simulation ()
and experiments (---).
Figure 6: Photograph of the test rig.
phase windings of the stator of the SCIM removed and
replaced by similar windings with various taps taken out
from different turns of the phase ‘a’ winding. Inter-turn
fault with variable number of shorted turns is produced in
the SCIM by connecting any two of the taps. The motor
is mechanically coupled with a magnetic powder brake to
produce adjustable mechanical load. A digital scopemeter
is used for sampling the line currents of the SCIM
[12]. Two independent current or voltage signals can be
sampled and recorded with 5000 samples per second.
Figure 6 shows a photograph of the test rig.
VI. SIMULATION AND ANALYSIS
The proposed SCIM is simulated using the
developed SCCM under various loads, supply and fault
conditions and corresponding tests are performed on the
real SCIM in the laboratory. The results are presented,
compared and analyzed in this section. Figure 7 shows
the variation of f with time in an interval between - to
obtained during a simulation by the method introduced in
the Section III. Constant slop of the variation of f is due
to the constant speed of the air gap rotating magnetic
field (the synchronous speed).
Figure 8 shows the normalized spectra of the stator
line current in the faulty SCIM with 21 shorted turns
under no load. As seen, all the even/odd harmonics are
present in the experimental and SCCM result but not in
the CCM result. The amplitude of the 3 rd harmonic of the
stator line current (150 Hz) might be considered as an
index for the inter-turn fault [13], [14]. As seen, this
amplitude in the SCCM result is very closer to the
experimental result than that in the CCM result.
The stator negative-sequence current component at the
fundamental frequency is one of the old indexes
introduced to diagnose the inter-turn fault [15], [16]. In
the healthy symmetrical SCIM with the balanced threephase
supply, the negative-sequence current is zero.
- 223 - 15th IGTE Symposium 2012
Figure 7: Simulated time variations of f
Figure 8: Normalized spectra of stator line current under no load with
21 shorted turns obtained by: a) experiment, b) SCCM and c) CCM
However, the inter-turn fault quickly increases this
current component. To determine the negative sequence
current at the required frequency, the related line current
phasors of the stator are obtained first by using the
sampled currents and the Fourier algorithm which is
conventional in the field of the digital protection [17].
Then, using the line current phasors, the negative
sequence current phasor is determined [15]. Knowing the
amplitude, phase angle and frequency of the negative
sequence current, its waveform can also be sketched. For
the proposed SCIM with 21 shorted turns under full load,
the negative-sequence currents obtained through the
simulation and experiments have been shown in Figure 9.
As seen, the saturation effect, introduced by the SCCM,
increases the amplitude of the current in order to
approach the experimental results
Simulations and experiments on the proposed SCIM
with 14 and 21 shorted turns were repeated under
various load levels. Table I compares the attained
amplitudes of the stator negative sequence current at the
fundamental frequency. As seen, the negative sequence
current increases by increasing the fault degree, while the
load level change has negligible impact on the current.
Also, the SCCM results follow the experimental results
more
closely than the CCM results.
However, any negative sequence component in the
TABLE. I

Current
(mA)
- 224 - 15th IGTE Symposium 2012
AMPLITUDE OF STATOR NEGATIVE SEQUENCE CURRENT UNDER VARIOUS LOAD LEVELS
Faulty SCIM with 14 short-circuited turns Faulty SCIM with 21 short-circuited turns
No
load
20%
rated
load
40%
rated
load
60%
rated
load
80%
rated
load
Full
load
No
load
20%
rated
load
40%
rated
load
60%
rated
load
80%
rated
load
Experimental 470 478 481 482 489 494 698 690 690 712 726 730
SCCM 467 490 480 456 462 487 693 700 668 701 702 702
CCM 413 457 438 411 422 430 619 623 609 647 648 650
Figure 9: Stator negative sequence current obtained by a) experiment, b)
SCCM and c) CCM in faulty SCIM with 21 shorted turns under fullload.
mains voltage, which is permissible to some small extent
in real mains, produces negative sequence current in the
stator of the healthy SCIM. Simulation result indicates
that with 2% negative sequence in the stator voltage of
the healthy SCIM, the negative sequence current changes
by 0.54% from no-load to full-load, by 1.86% from nonsaturable
(CCM) to saturable motor (SCCM) and by
6.65% from healthy to single-turn shorted condition (the
weakest fault), while changing the negative sequence
voltage from 2% to 5% changes the negative sequence
current by 148.8% in the healthy SCIM. Therefore, the
negative sequence current as an inter-turn fault index is
highly sensitive to the voltage imbalance level, which is
not pleasing as shown in Figure 10.
The negative sequence apparent impedance of the
SCIM is also used as an index to diagnose the stator
inter-turn faults [18], [19]. This impedance is the ratio of
the stator negative sequence voltage phasor to its
negative sequence current phasor. Figure 11 shows the
similar results with Figure 10 for this index which is
obtained using simulation. As seen, this index has very
smaller sensitivity to the voltage imbalance and the
magnetic saturation is the most effective lateral factor
affecting this index.
VII. CONCLUSION
The flux-linkages of the rotor meshes, calculated in
every simulation step of the CCM for the SCIM was used
to estimate the air gap flux density distribution. Then,
space harmonic components of the air gap flux density
were determined using Fourier series analysis. The phase
angle of the space fundamental harmonic was utilized to
locate the air gap flux density during simulation of the
faulty SCIMs. Also, the amplitude of this fundamental
harmonic is applicable to evaluate the saturation factor
more reasonably. Therefore, a saturable CCM was
Full
load
Figure 10: Sensitivity of stator negative sequence current to the weakest
fault (single-turn), saturation, voltage imbalance and load level,
evaluated by simulations.
Figure 11: Sensitivity of the negative sequence apparent impedance of
the SCIM to weakest fault (single-turn), saturation, voltage imbalance
and load level, evaluated by simulations.
developed which is capable to analyze faulty SCIMs.
Comparing the simulation results with the corresponding
experimental results indicates that the saturable model is
more precise than the non-saturable model. Further study
showed that the magnetic saturation affects the inter-turn
fault indices more than the load level and the stator
voltage imbalance.
APPENDIX
The indefinite integral of the inverse air gap function
(gf -1 ) is determined first as follows:
f ( ,
, K
f
sat
1
) g f ( ,
f , K sat ) d
1
cos( 2P(
))
1
cos
f
2
2 P g 1
1 cos( 2P(
f ))
(A1)
Then, the analytical equation for the inductances of the
`stator windings is obtained as:
L
x y
m
o
r l
nx
( ti )[ n y ( ti ) f s ( f , K sat )]
(A2)
i1
[ f (
i1
, , K
f
sat
) f ( , , K
where x and y accounts for the stator phases, m is the
number of the stator slots, i is the angle of center of the
stator slot i, ti is the angle of center of the stator tooth
after the stator slot i, and fs is:
2
g
1
m
fs ( f , Ksat
) ny
( ti)
[ f ( i1,
f , Ksat
) f ( i,
f , K
2
i1
i
f
sat
)]
sat
)]
(A3)

Equation (A2) is independent of r and its derivative
versus r is zero for all x and y. For the rotor meshes the
inductance equation is:
Luv o r l [ C fr
( f , K sat )] [ f ( x1,
f , K sat ) f ( x,
f , K sat )] (A4)
now u and v accounts for the rotor meshes 1 to R, f is f
in the rotor reference (f = f - r), u is the angle of center
of the rotor bar number u in the rotor reference, C=1 for
u = v and C=0 otherwise and fr is:
2
g
1
fr ( f , K sat ) [ f ( y1,
f , K sat ) f ( y , f , K
2
sat
)]
- 225 - 15th IGTE Symposium 2012
(A5)
Equations (A4)-(A5) depend on r because of f.
Considering the relationship between these two variables
yields Luv/r= -Luv/f and thus:
Luv
or
l[
C fr
( f , K
r
where:
f
r ( f , K
o
rl
f
sat
f
( u1,
f , Ksat)
f
( u,
f , Ksat)
)] [
]
sat
)]
[ f ( , , K ) f ( , , K
u1
f
f
sat
u
f
f
sat
)]
(A6)
f r ( f , Ksat
) g
f
2
1
2
f
( y 1,
f , Ksat
) f
( y , f , Ksat
)
[
]
f
f
(A7)
f ( i , f , Ksat
)
1
; i x1,
x , y1,
y
f g[
1
cos( 2P(
i
f ))]
(A8)
For the mutual inductances between the rotor meshes and
stator phases the following equation is obtained:
k2
1
i1
i
Lmn
o
r l
[ nn
( ) f s ( f , K sat )]
ik
2
1
(A9)
[ f ( , , K ) f ( , , K )]
i
f
sat
i1
where x and y account for the rotor meshes and stator
phases respectively, k1-1 and k2+1 are the angles of the
two bars of mesh x in the stator reference, k1 to k2 are the
stator slots between k1-1 and k2+1 and e.g. k1 is the angle
of the stator slot k1. Now k1-1 and k2+1 are responsible to
the dependency of inductances on r. The derivatives of
the two mentioned parameters versus r are equal to 1,
while the derivatives of the other parameters in (A9)
versus r are zero. Using these facts and some
differentiation rules leads to:
Lx
y
r
o r l
k1
k11
[ n y ( ) f s ( f , K sat )]
2
g[
1
cos( 2P(
k11
f ))]
o
r l
k 21
k 2
[ n y ( ) f s ( f , K sat )]
2
g[
1
cos( 2P(
))]
k 21
REFERENCES
f
f
sat
(A10)
[1] X. Luo, Y. Liao, H. A. Toliyat, A. El-Antably and T. A. Lipo,
“Multiple coupled circuit modeling of induction machines,”
IEEE Trans. Ind. Applications, vol. 31, pp. 311 - 318,
March/April 1995.
[2] A. Raie, and V. Rashtchi, "Using a genetic algorithm for
detection and magnitude determination of turn faults in an
induction motor", Springer-Verlag, vol. 84, pp. 275–279, August
2002.
[3] S. Nandi, “Detection of stator faults in induction machines using
residual saturation harmonics,” IEEE Trans. Ind. Applications,
vol. 42, no. 5, pp. 1201 - 1208, 2006.
[4] J. C. Moreira and T.A. Lipo, “Modeling of saturated ac machines
including air gap flux harmonic components,” IEEE Trans. Ind.
Applications, vol. 28, pp. 343 - 349, March/April 1992.
[5] D. Bispo, L. M. Neto, J. T. Resende and D. A. Andrade, “A new
strategy for induction machine modeling taking into account the
magnetic saturation ,” IEEE Trans. Ind. Applications, vol. 37, no.
6, pp. 1710 - 1719, Nov./Dec. 2001.
[6] T. Tuovinen, M. Hinkkanen, and J. Luomi, “Modeling of
saturation due to main and leakage fux interaction in induction
machines,” IEEE Trans. Ind. Applications, vol. 46, no. 3, pp.
937 - 945, 2010.
[7] Tu Xiaoping, L.-A. Dessaint, R. Champagne, and K. Al-Haddad,
“Transient modeling of squirrel-cage induction machine
considering air-gap flux saturation harmonics,” IEEE Trans. Ind.
Electronics, vol. 55, no. 7, pp. 2798 - 2809, 2008.
[8] S. Nandi, “A detailed model of induction machines with
saturation extendable for fault analysis,” IEEE Trans. Ind.
Applications, vol. 40, pp. 1302 - 1309, September/October 2004.
[9] M. Ojaghi and J. Faiz, “Extension to multiple coupled circuit
modeling of induction machines to include variable degrees of
saturation effects,” IEEE Trans. Magn., vol. 44, no. 11, pp.
4053-4056, Nov. 2008.
[10] J. Faiz, and I. Tabatabaei, "Extension of winding function theory
for nonuniform air gap in electric machinery," IEEE Trans. Magn.,
vol. 38, pp. 3654-3657, November 2002.
[11] Z. Michalewicz, Genetic Algorithms & Data Structures, Evalution
Programs, Springer-Verlag, 1992.
[12] Fluke 196c/199C Scope-Meter User's Manual, Fluke
Corporation, Oct. 2001, Netherlands.
[13] G. Joksimovic, J. Penman, “The detection of interturn short
circuits in the stator windings of operating motors,” IEEE Trans
Ind. Electronics, vol. 47, no.5, pp.1078–1084, Oct. 2000.
[14] J.H. Jung, J.J. Lee, and B.H. Kwon, “Online diagnosis of
induction motors using MCSA,” IEEE Trans. Ind. Electronics,
vol. 53, no. 6, pp. 1842–1852, Dec. 2006.
[15] A.Bellini, F.Filippetti, C.Tassoni, G.A.Capolino, “Advances in
Diagnostic Techniques for Induction Machines,” IEEE Trans.
Industrial Electronics, vol.55, no12, pp. 4109-4126, Dec. 2008.
[16] Wu Qing, and S. Nandi, “Fast single-turn sensitive stator interturn
fault detection of induction machines based on positive- and
negative-sequence third harmonic components of line currents,”
IEEE Trans. Ind. Applications, vol. 46, pp. 974 - 983, 2010.
[17] A.T.Johns and S.K. Salman, Digital Protection for Power
Systems. IEE Power series 15, London, UK 1995.
[18] J.L. Kohler, J. Sottile, and F.C. Trutt, “Condition monitoring of
stator windings in induction motors: I. Experimental
investigation on effective negative-sequence impedance
detector,” IEEE Trans. Ind. Application, vol. 38, pp. 1447–1453,
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[19] L. Sang Bin, R.M. Tallam, and T.G. Habetler, “A robust, on-line
turn-fault detection technique for induction machines based on
monitoring the sequence component impedance matrix,” IEEE
Trans. Power Electronics, vol. 18, pp. 865–872, 2003.

- 226 - 15th IGTE Symposium 2012
Accurate Magnetostatic Simulation of Step-Lap
Joints in Transformer Cores Using Anisotropic
Higher Order FEM
A. Hauck∗ , M. Ertl † ,J.Schöberl ‡ and M. Kaltenbacher §
∗ SIMetris GmbH, Erlangen, Germany † SIEMENS Transformers, Nuremberg, Germany
‡ Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria
§ Institute of Mechanics and Mechatronics, Vienna University of Technology, Austria
E-mail: andreas.hauck@simetris.de
Abstract—We present a simulation scheme for the accurate simulation of thin magnetic structures, specifically the nonlinear
magnetic flux distribution in a core step-lap joint with interest in the local saturation near the air gaps. Due to the high
aspect ratio of the model, we utilize hierarchical higher order finite elements, where the polynomial degree is spatially
adapted to resolve the flux distribution within the steel sheets. The deterioration of convergence in the iterative conjugate
gradient (CG) solver is handled by an anisotropic Schwarz-type block preconditioner, grouping the unknowns depending
on the aspect ratio of the elements. The resulting Newton scheme can optionally be accelerated by a 2-step solution strategy,
where a start value is computed on a coarse subspace of lowest order in analogy to a full multigrid scheme.
Index Terms—Step-Lap Joints, Higher Order Finite Elements, Block Preconditioner, Nonlinear Solver
I. INTRODUCTION
Precise knowledge about the accurate flux distribution
in transformer cores is both important for reducing the
magnetic losses, as well as for localizing sources for
forces (magnetostriction, interlaminar forces). In recent
years significant reduction of both effects was achieved
using the multi-step-lap technique (see Fig. 1 and 2),
where the overlap region of transformer sheets is shifted
in several steps [1]. In order to optimize the layout
further, a detailed simulation of the fluxes, including the
nonlinear B-H curve of the core has to be performed.
Accurate simulation of such a problem poses some diffi-
Fig. 1. Sketch of transformer core
with step-lap corners.
A
A
View A – A
Fig. 2. Magnetic flux concentration
(45 ◦ -view).
culties, as the thickness of the steel sheets is typically in
the range of 200 - 300 μm, while the length can be a few
meters and the number of vertically stacked sheets add up
to some thousand layers. In addition, the flux variation is
very high in the vicinity of the air gaps in the corner, but
rather smooth in some distance away from it, making it
difficult to resolve accurately. Furthermore, the nonlinear
permeability of the grain-oriented electrical steel sheets
has to be taken into account and modeled correctly.
Within this paper we solve the nonlinear magnetostatic
problem using finite elements of higher order, together
with an iterative preconditioned conjugate gradient (CG)
method. By exploiting the special structure of the Finite
Element (FE) basis [2], we can build an effective
preconditioner for handling elements with high aspect
ratios, while at the same time the spatial accuracy can be
adapted to the discretization of the model. The nonlinear
Newton scheme can be further accelerated by calculating
a good initial start value on a coarse sub-space. Finally,
the applicability of the method is demonstrated for the
aforementioned step-lap core model.
II. NONLINEAR MAGNETOSTATIC FORMULATION
The nonlinear magnetostatic problem can be written in
terms of the magnetic vector potential A as
∇×ν(|∇ × A|)(∇×A) =J + ∇×νB0
B = ∇×A , (1)
where ν(|∇ × A|) is the nonlinear reluctivity (e.g. of
steel), J the impressed current density (e.g. of a coil)
and B0 an additional prescribed flux density. As the
impressed current density J and the curl of the prescribed
flux density B0 are assumed to be divergence free, a
unique solution can either be guaranteed by enforcing
the Coulomb gauge
∇·A =0 (2)
explicitly (leading to a mixed formulation) or by adding
a small regularization term αA to (1), where α ≈ 10 −6 ν
[2]. We will modify this strategy in Section III.
For physical reasons, the vector potential A is only
continuous in the tangential part, which requires the use
of H(curl)-conforming vectorial elements, which will be
introduced in Sec. III.

The nonlinear problem is solved using a Newton
formulation
F ′ (Ak)[ΔA] =−F(Ak) (3)
Ak+1 = Ak + ηΔA . (4)
Improvement of convergence is achieved by computing
a line search parameter η ∈ ]0, 1], which is determined
by minimizing the residual of (3). The magnetic B-H
commutation curve is extracted from measured hysteresis
curves, reducing the problem to a simplified nonlinear
one, which is given in terms of measured (Bi,Hi)-value
pairs. By applying a C 1 -spline approximation, a smooth
monotone approximation is calculated, from which the
reluctivity ν and its derivative ν ′ (entering the Fréchet
derivative F ′ ) can be derived (see Fig. 3). For details we
refer to [3].
Fig. 3. Approximation of measured B-H-data by C 1 -splines.
III. HIGHER ORDER FINITE ELEMENTS
The two key requirements for our choice of higher
order H(curl)-conforming FE shape functions are the
ability to choose the polynomial degree p independently
in each local direction (ξ,η,ζ), as well as the availability
of efficient iterative solution techniques (i.e. an efficient
preconditioner). The first requirement leads to the use
of hierarchical shape functions, where we we utilize the
hierarchical shape functions of [2], which can be written
as
N(T )=
N 0 E ⊕
N ∇ E ⊕
NF ⊕
E
E
F
N ∇ F ⊕ NI ⊕ N ∇ I
The shape functions N are composed of unknowns
defined on edges, faces and in the interior (subscripts
E, F and I), see Fig. 4 for a hexahedral element.
The lowest order Nédélec functions N0 E , which have a
pζ pη pξ E 5
E 12
F 2
E 4
E 8
E 1
ζ
η
ξ
F 1
F 6
E 9
E 11
E 3
E 6
E 10
E 2
F4 E7 Fig. 4. Degrees of freedom for the hexahedral element.
- 227 - 15th IGTE Symposium 2012
(5)
constant tangential component along one edge (p = 0),
are explicitly included. In addition, higher order gradient
components on edges N∇ E , faces N∇F and in the interior
N∇ I are represented separately. This key feature - also
known as the local exact sequence property - is equivalent
to fulfilling the so-called De-Rham complex
R id
−→ H 1 (Ω) grad
−→ H(curl,Ω)
curl
−→ H(div,Ω) div
0
−→ L2(Ω) −→ {0} (6)
already on the finite element level, i.e. the gradients,
forming the null-space of the curl-operator, can be completely
omitted for each type of unknowns (edge, face,
interior) separately if only the flux density B is of
interest. In [2], this is denoted as the reduced basis and
can be used to gauge the problem in the following way:
• For the lowest order Nédélec functions N0 E , we add
the regularization term α as described in Sec. I.
• For the higher order terms, we simply skip the
gradient functions N∇ E , N∇F and N∇I .
Another unique advantage of this basis according to [9]
is that shape functions of arbitrary order are available
for all types of elements in 2-D and 3-D, utilizing any
kind of hierarchical 1-D shape functions, e.g. Legendre
or Gegenbauer.
A. Anisotropic Adapted Polynomial Degree
In general the magnetic flux density B is defined as
⎛
B = ⎝ Bx
⎞
⎛
⎞
∂Az ∂Ay
∂y − ∂z
By ⎠ ⎜ ∂Ax ⎟
= ∇×A = ⎝
⎠ . (7)
Bz
∂z
∂Ay
∂x
∂Az
− ∂x
∂Ax
− ∂y
In case of thin structures, the in-plane components
(Bx,By) are dominant (see Fig. 5). Additionally, the
variation of the in-plane components of A in z-direction
( ∂Ax ∂Ay
, ) in (7) is already resolved accurately by the
∂z
∂z
FE-discretization in thickness direction of the single sheet
, i.e.
layers. Thus the dominant terms left are ∂Az
∂y
A (z)
z
z
y
x
z
B (x)
y
A (x)
z
A (x)
z
x
and ∂Az
∂x
Fig. 5. Flux / potential distribution in face on (x, z)-plane.
the magnetic vector potential A should be approximated
quite accurately in the in-plane direction.
Therefore, it is advantageous to reflect his behavior in
the anisotropic polynomial degree as
pη,pξ >pζ , (8)
assuming that the global z and the local ζ direction
coincide. The increase of the polynomial degree only
affects the face NF and inner NI degrees of freedoms,

as we skip higher order gradient functions N ∇ E and N∇ F
due to gauging. This leads to practical order templates
like paniso =(2, 2, 1) or paniso =(3, 3, 1). Although it
seems that the lowest order anisotropic template should
be paniso =(1, 1, 0), this does not lead to more accurate
results, as only faces in (x, y)-direction get additional
unknowns, which do not contribute to an improved
resolution of Az.
As the permeability in air μ0 is typically several
orders of magnitude smaller compared to the one in the
ferromagnetic core, the flux is mostly concentrated in
the core. This allows us the choose a small isotropic
polynomial degree of pair = 0 for the approximation.
The last step is especially effective, if structured grids
are utilized or if the air domain is significantly large.
B. Single Step Iterative Solution Scheme
The resulting system of equations after FE discretization
can be written as
with
⎛
⎝
KN0N0 KN0F KN0I
KF N0 KFF KFI
KIN0 KIF KII
K(A)A = f , (9)
⎞ ⎛
⎠ ⎝
AN0
AF
AI
⎞ ⎛
⎠ = ⎝
fN0
fF
fI
- 228 - 15th IGTE Symposium 2012
⎞
⎠ .
(10)
The interior unknowns AI can be eliminated by static
condensation as
AI = K −1
II (fI − KIN0 AN0 − KIFAF ) , (11)
where K −1
II can be inverted on the element level. Substituting
this result back in (10) results in the reduced
system
ˆKN0N0 ˆKN0F
ˆfN0
AN0 = , (12)
ˆKF
ˆKFF AF ˆfF
N0
with the modified matrices and RHS vectors as
−1
= KN0N0 − KN0I(KII )KIN0
ˆKN0F = KN0F − KN0I(K −1
II )KIF
ˆKN0N0
ˆKF N0 = KF N0 − KIF(K −1
II )KN0I
(13)
(14)
(15)
ˆKFF = KFF − KIF(K −1
II )KFI (16)
ˆfN0 = fN0 − KN0I(K −1
II )fI (17)
ˆfF = fF − KFI(K −1
II )fI . (18)
The two main effects of the static condensation are:
• The number of unknowns is reduced significantly,
as with increasing polynomial degree p only face
and interior unknowns are added.
• The condition number κ of the reduced system (12)
is much smaller compared to the one of the full
system (10), causing less iterations of the CG solver.
In order to solve the reduced system (12), we apply a
Preconditioned Conjugate Gradient (PCG) method. It was
shown in [2], that a α-robust preconditioner C−1 can be
simply defined by a block Jacobian preconditioner (i.e.
an additive Schwarz method, ASM), defined by
C =
ˆKN0N0
0
0
Kˆ B
FF
, (19)
where the single blocks are formed as follows:
• ˆ KN0N0 : The lowest order Nédélec functions can be
either solved by a sparse direct solver [10] or by a
suitable iterative method, respecting the Helmholtzdecomposition
of the magnetic vector potential (see
e.g. [11]).
: For every face, all unknowns are grouped in
• ˆ KB FF
one block (superscript B). The application of the
preconditioner basically is just the inversion of the
single face blocks K −1
FF .
IV. ANISOTROPIC BLOCK PRECONDITIONER
If the preconditioner (19) is applied for structures with
a very high aspect ratio (AR), the convergence of the
iterative solver deteriorates. This can be explained by an
increase in the condition number κ = λmax/λmin, asthe
entries in the stiffness matrix K scale with 1/h, where h
is the mesh size, leading to strongly coupled entries for
nearly parallel edges / faces and thus to nearly singular
systems with high condition numbers [5].
The idea proposed in [5] is based on a singularity
decomposition technique, where new unknown variables
are introduced and assigned to groups of parallel edges
with small distance. However, this method introduces
new matrix entries, as all edges in one group couple via
the auxiliary variable. In addition, the method is only
applicable to 1st order elements, as only edge degrees of
freedom are considered.
An alternative approach is taken in [4], where a plane
smoother for nodal and edge components of the A − φformulation,
respecting the Helmholtz-decomposition, is
applied within a geometric multigrid (MG) solver. Again,
explicit knowledge of the anisotropic direction is needed
a priori. In our approach, the idea of [4] is extended to
η
ζ
ξ
F3
F2
F1
F8
F7
F6
F5
F4
thin direction(s)
Fig. 6. Thin structure with 2 distinct face groups {F1, F2, F3} and
{F4, ..., F10}, where η is the long direction.
the p-version of the FEM. As the lowest order edge contributions
KN0N0 are already solved with a direct solver
and the inner degrees of freedom KII get eliminated by
static condensation, only the face contributions ˆ KFF are
affected by the anisotropy. Thus we can modify the initial
face blocks ˆ KB FF of the preconditioner matrix C (19) by
grouping all unknowns of the faces perpendicular to the
F10
F9

thin direction in one diagonal block ˆ K Bai
FF , if the aspect
ratio of the element exceeds a user-defined threshold
ARth. We can even generalize the idea, allowing for two
anisotropic / thin directions within one element, e.g. in
Fig. 6 the faces F4 to F10 couple strongly, as the size in
both, ξ- and ζ-direction, is small compared to the extend
in η-direction. This is especially useful in meshes with
tensor-product structure.
The modified preconditioner matrix is then defined as
ˆKN0N0 0
C =
0 Kˆ Bai . (20)
FF
The procedure for computing ˆ K Bai
FF without explicit
knowledge of the thin direction(s) is sketched in Algorithm
1. It collects strongly coupled (thin) faces in a graph
and determines the blocks of ˆ K Bai
FF by calculating the set
of connected components of it. As the only information
needed for the algorithm is the size of the elements in
each direction, the procedure can be applied to general
3-D elements (tetrahedron, wedge, pyramid).
Algorithm 1: Definition of anisotropic face blocks.
Input: elements e of mesh T
Output: groups of thin faces Gi,i=1,...,nG
Data: graph of connected anisotr. faces G=(V,E)
foreach e ∈T do
if ARmax(e) ≥ ARth then
compute size of element w.r.t. local
directions (hξ,hη,hζ)
hmax = max(hξ,hη,hζ)
foreach d ∈{ξ,η,ζ} do
if hd/hmax ≥ ARth then
get faces F1, F2 perpendicular to
d-direction
insert (F1,F2) in G
nG = # of non-connected components of G
for i =1to nG do
Gi = connected faces in i-th component of G
V. NONLINEAR TWO-STEP STRATEGY
In order to accelerate the resulting Newton scheme,
we utilize a 2-step strategy, motivated by full multigrid
methods (see [6] and [8]). The idea is to compute a
good start approximation for the Newton scheme on a
coarse sub-space T H , which is formed in the p-version
by the lowest order functions KN0N0. The fine space T h
is spanned by the complete set of polynomials [7]. Thus
the stiffness matrix K in (10) can be splitted into KH
and Kh as follows
KH = K00
Kh = K =
= KN0N0 (21)
. (22)
K00 K01
K10 K11
Due to the hierarchical basis functions, the interpolation
operator Ih H is trivially defined as
Ah =[I, 0] T AH = I h HAH . (23)
- 229 - 15th IGTE Symposium 2012
15 cm
y
z
x
30 cm
0.96 mm
1 layer
air gaps
(exploded view)
Fig. 7. Model of step-lap core without air domain (scale factor 30 in
thickness direction).
This allows us to perform a 2-step solution strategy as
follows:
1) Solve a few Newton steps on the small coarse
system KN0N0 (21), using a direct solver.
2) Interpolate the coarse space solution AH to the fine
space Ah according to (23).
3) Proceed with the full system Kh (22) using the
solution strategy of Section III-B.
VI. APPLICATION: STEP-LAP CORE MODEL
The applicability of the method is demonstrated for a
typical 45◦-multi-step-lap joint region of a transformer
core (see Fig. 7) with 4 layers of steel sheet, each
0.24 mm in thickness, with a step-lap of 2 air gaps
(width: 1 mm) in each sheet. The model is discretized
by 2568 hexahedral elements and 3172 nodes. The used
nonlinear B-H curve is the one depicted in Fig. 3. As
excitation, we apply a prescribed flux density B0 =
0.1 − 2.5 Tiny-direction.
Fig. 8. Concentration of magnetic flux lines in corner (45 ◦ -view) for
uniform p =0(top) and p =3(bottom) with B0 =1.0 T (scale
factor 30 in thickness direction).
A. Initial Results
Initially, we choose an isotropic polynomial degree
p =0,...,3 and compare the spatial resolution of the
magnetic flux density near the air gaps. Here, the Newton
algorithm takes between 3 and 9 iterations.
The results of the simulation are visualized on a very
fine postprocessing mesh. In Fig. 8 it is clearly visible,
B 0

Fig. 9. Flux distribution for uniform p =0(top) and p =3(bottom)
with B0 =1.0 T (scale factor 30 in thickness direction).
that the continuation of the fluxlines across the air gaps
is poorly approximated for p =0and that the curvature
of the streamlines is unphysical between the air gaps. In
contrast, the simulation using p =3resolves accurately
the flux concentration above and below the air gaps
(depicted in red). The same observation holds true for
the absolute value of the magnetic flux density in Fig.
9, where the flux concentration between the air gaps is
smeared over a large area for p =0.
From Fig. 10 we deduce, that the iterative method is
by a factor 2 to 10 slower compared to the direct method
for all excitation values B0. In contrast, the memory
consumption is only about 50% compared to the direct
one for higher polynomial degrees p, as seen in Table I
(SC denotes the use of static condensation).
Fig. 10. Simulation time for direct and iterative solution approach
without anisotropic block preconditioner.
TABLE I
MEMORY REQUIREMENT AND DOFS FOR DIFFERENT POLYNOMIAL
DEGREES (SC: STATIC CONDENSATION)
Polynomial Degree piso
0 1 2 3
# Total DOFs 7824 43956 141840 332292
# Inner DOFs - 12840 71904 208008
Memory Usage (GB)
Direct Solver 0.24 0.39 1.12 3.61
Direct Solver (SC) 0.24 0.37 0.92 2.32
Iterative 1-step (SC) 0.24 0.28 0.53 1.61
- 230 - 15th IGTE Symposium 2012
B. Use of Anisotropic Block Preconditioner
The poor runtime performance of the iterative 1-step
scheme in Fig. 10 can be explained by the extremely
high aspect ratios up to 1:1000 in Fig. 11: All elements
within the steel sheets have aspect ratios higher than
1:400, leading to over 3000 CG iterations on average.
Fig. 11. Aspect ratio of step-lap setup (not shown for elements in air).
If we utilize the anisotropic preconditioner
(20) for varying aspect ratio thresholds
ARth = {1000, 500, 100, 50, 10} the iteration numbers
and time for solving the linear equation system drops
significantly (see Fig. 12). The results are compared for
the iterative 1-step solver with B0 = 1.0 T. From an
Fig. 12. Reduction of CG iterations (left) and solution time (right)
for varying aspect ratio threshold ARth.
initial CG iteration count of about 3000 (ARth = 1000)
we achieve an average reduction to 100-150 iterations
for ARth = 10, corresponding to a factor of 20-30,
depending on the polynomial degree.
The effect on the solution time is similar, where
a reduction by a factor of 9 (p=3) to 25 (p=1) can
be achieved, making it comparable in runtime to the
direct solver. The increase in memory for storing larger
diagonal blocks is very moderate, being in the range of
5-15% compared to the non-blocked version.
The rate of reduction in iterations is not heavily
depending of the polynomial degree, making the preconditioner
a p-robust method for practical applications. For
all the following results, we apply the preconditioner with
a default threshold of ARth =10.

C. Application of Two-Step Approach
By applying the 2-step strategy of Section V, an
additional decrease in runtime can be observed (see Fig.
13). We start here with 2 Newton iterations on the coarse
Fig. 13. Runtime comparison of standard 1-step iterative solution
approach and 2-step approach.
space T H with p =0and use it as start value for the
fine space T h . On average, this saves 1 to 2 Newton
iterations on the fine space, resulting in a reduction of
runtime between 4% and 48%, which is in a similar range
as reported in [8]. However, the effect diminishes with
higher flux values for all polynomial degrees.
D. Anisotropic Polynomial Degree
Finally, we utilize the strategy as explained in Section
III-A by reducing the polynomial degree anisotropically
in thickness direction pζ

- 232 - 15th IGTE Symposium 2012
Parameter Identification of a Finite Element
Based Model of Wound Rotor Induction Machines
*Martin Mohr, *Oszkár Bíró, *Andrej Stermecki and † Franz Diwoky
* Christian Doppler Laboratory for Multiphysical Simulation, Analysis and Design of Electrical Machines at the
Institute for Fundamentals and Theory in Electrical Engineering, Inffeldgasse 18, A-8010 Graz, Austria
† AVL List GmbH, Hans-List-Platz 1, A-8020 Graz, Austria
E-mail: martin.mohr@TUGraz.at
Abstract—This paper presents two efficient algorithms for the parameter identification in a finite element based circuit model
of a wound rotor induction machine. This approach uses magneto-static finite element method simulations for building lookup
tables and employs quint-cubic splines for the interpolation. A reduction of the finite element simulation cost is achieved by
decreasing the number of nonlinear iterations using a special simulation order keeping the magneto-motive force in the
machine for several sampling points constant. Furthermore, the quint-cubic spline parameter calculation method has been
revised using a dimensional recursive evaluation approach allowing a fast parameter calculation with lower memory demand
and offering possibility of parallelization.
Index Terms— Finite element methods, motor drives, numerical models.
I. INTRODUCTION
Finite element (FE) based models, in particular
physical phase variable (PPV) models, use look-up tables
(LUT) generated by magneto-static finite element method
(FEM) simulations. During the following transient
simulations, only the evaluation of these LUTs by an
appropriate interpolation method is needed. Several
applications of this approach to electrical machines have
already been published [1]-[8].
For permanent magnet machines, this approach is
straightforward [1]-[5]. Typically three independent state
variables are sufficient for prescribing the state of the
machine. Therefore, no more than a few hundred or
thousand FEM simulations are needed and a three
variable interpolation method is sufficient.
In contrast, wound rotor induction machines (WRIM)
necessitate a higher number of state variables. This is a
consequence of the second coil system on the rotor. The
number of simulations Ntot increases exponentially with
the number of state variables:
Ntot Ni
(1)
states
where Ni is the number of sampling points for the i th state
variable. Furthermore, a higher dimensional interpolation
method is necessary. Nevertheless, several
implementations of FE-based WRIM models are found in
the literature [6]-[8].
This work refers to the PPV-model of a WRIM with
stranded coils introduced in [8] and can be perceived as
an addendum to it. Since the parameter identification has
not been treated in [8], this paper presents the practical
application of this model.
The drawbacks of the PPV-WRIM model are the very
high number of needed FEM-simulations and the
demanding quint-cubic spline parameter calculation. Both
are caused by the number of state variables of the model
being five as shown in the next section.
II. INTRODUCTION TO THE PPV-WRIM MODEL
This model uses the electrical rotor position Rot and
S S
two transformed currents for each of the stator i, i and
R R
the rotor i , i as state variables. This reduction can be
done under the assumption that the rotor and stator
systems are in Y-connection with isolated star point.
The current state variables used are the phase currents
S S S
R R R
of the stator iA, iB, i C and rotor iA, iB, i C transformed
into the rotor related reference system:
S S S R R R
iA iB iC 0, iA iB iC 0,
S 1 0 S
i
cosRot sin
Rot 2 i
1 2 A
S , S
i
sinRot cosRot 3 i
(2)
B
3 3
R 1 0 R
i
2 i
1 2 A
R R .
i
3 i
B
3 3
The rotor position is derived from the mechanical
model and is an input variable for the electrical machine
model.
The used LUTs of this model approach contain the
S S S
phase flux linkages of the stator A, B, C
and of the
R R R
rotor A, B , C
for the electrical system of equations as
well as the machine torque T.
All these quantities are parameterized by the five state
variables:
S S R R
LUT f Rot , i, i, i , i
.
(3)
In contrast to [6], the total machine torque T is also
included in the LUT.
The electrical system of equations uses the line to line
quantities defined as
S S S S S S S S S S S S
vAB: vBvA, vBC : vCvB, iAB : iBiA, iBC : iCiB R R R R R R R R R R R R
vAB: vBvA, vBC : vCvB, iAB : iBiA, iBC: iC iB
(4)
S S S
with the phase voltages of the stator v , v , v and of the
A B C

R R R
rotor vA, vB, v C . This system can be rewritten as
S S S S S
AB AB AB AB
Rot
AB
dt
Rot iS iS iR i
R
S
di
S
S S
S S S S S
v
AB RCUi AB
BC BC BC BC
BC
S
dt
S S
v
BC RCUi
BC Rot iS iS iR i
S
R
di
R R R R
v
AB RCUi
R R R R
AB
(5)
AB AB AB AB AB
R R R
dt
v BC RCUi R
BC
Rot
iS iS iR i
R
di R R R R R
BC BC BC BC BC dt R
di
Rot i S i S i R i
R
dt
S
with the stator phase resistance R CU and the rotor phase
R
resistance R CU .
For the evaluation of the function values as well as the
partial derivatives, a quint-cubic spline interpolation has
been suggested in [8]. Furthermore, it has been shown
that this model is equivalent to a FEM model in all but
interpolation errors. However, the parameter
identification process has not been discussed there.
In this work, an improved simulation workflow for the
needed FEM simulations is proposed and presented.
Furthermore, a revised quint-cubic spline parameter
calculation method is introduced in detail.
III. IMPROVED FEM SIMULATION WORKFLOW
The motivation for an improved simulation workflow is
the circumstance that the number of magneto-static FEM
simulations needed to achieve an accurate interpolation is
higher than in usual applications. However, there are only
slight changes in the input data of these simulations.
Utilizing this circumstance, a reduction of the FEM
simulation time can be achieved as shown below.
A. Finite Element WRIM Model for Parameter
Identification
The FEM model of the WRIM under investigation is
shown in Fig. 1. This three phase machine has three
magnetic pole pairs. The rotor and stator coil systems are
in Y-connection with isolated star points. Furthermore, all
coils are modeled as stranded coils, skin effect is not
considered. Due to the symmetry of the machine, only a
third of the geometry is modeled, decreasing the number
of finite elements and thus the simulation time. Periodic
boundary conditions are used at the cutting planes. Rotor
and stator are coupled with constraint equations, thus no
conforming mesh is needed, and the rotation of the rotor
is taken into account [9].
The number of finite elements is 16128 and the
number of DOFs is 48 025. A single magneto-static FEM
simulation with ANSYS 12.1 needs approximately 21
seconds on a computer with Intel Core2 Quad CPU
(Q9400) and 8GB RAM. During this simulation, in
average 20 nonlinear iterations are needed. Simulation
time and number of iterations depend on the operating
point.
For a test example of nine sampling points for each
state current and 15 different rotor positions, 98 415
- 233 - 15th IGTE Symposium 2012
FEM simulations have to be carried out. The overall
calculation time is approximately 574 hours. In order to
reduce the computation time, the approach described
below and called method of constant magneto-motive
force (MMF) is proposed.
Figure 1: FEM model of wound rotor induction machine with rotor and
stator coil system.
B. Method of Constant Magneto-Motive Force
A reduction of the FEM simulation time can be
achieved in general by reducing
o the number of finite elements,
o the number of simulations or
o the number of nonlinear iterations.
A reduction of the number of elements decreases the
model quality and a reduction of the total number of
simulations decreases the quality of the interpolation.
However, a reduction of the number of nonlinear
iterations reduces the simulation time without decreasing
the model quality. Nevertheless, a direct manipulation of
this quantity is not possible because it depends on the
nonlinear behavior of the material used in the model.
However, the material properties of a prior simulation can
be used as initial guess for the new simulation.
This can be done for several magneto-static simulations
although they are independent from each other under the
assumption that the saturation state between these
simulations is approximately the same. This can be
accomplished if the magneto-motive force between
these simulations does not change significantly.
If the same reference system is used for the rotor and
stator current transformation, then can be easily
calculated component-by-component using
S R
Θα i i
α α
= =
NS S + NR
R
, (6)
Θβ iβ iβ
with the number of windings on the stator side NS and on
the rotor side NR. So, a constant MMF can be reached by

keeping the MMF components and constant as
shown in Fig.2.
R
S
S
R
Figure 2: Several parameter setups (different stator and rotor currents)
with constant total magneto-motive force.
This leads to a simple algorithm using a special
simulation order for the different parameter setups.
S
i
S
i
Rot
R
i
R
i
Figure 3: a) Blockdiagram of algorithm with five independent loops for
each state variable. b) Blockdiagram of constant MMF algorithm
Figure 3a shows the straightforward approach without
any optimization. All loops are independent, the loop
order is arbitrarily and massive parallelization is possible.
Figure 3b shows the improved constant MMF
algorithm. Instead of five independent loops for each state
variable, this algorithm consists of two coupled loops for
each of the orthogonal - and -components and one
independent loop for the rotor position. The outermost
loops define the MMF-vector component-by-component,
the innermost loops change the rotor and stator current
ratio for the relevant components. By this coupling, the
step sizes of the state variables are also coupled and
cannot be chosen freely. Nevertheless, a high number of
simulations with constant MMF can be carried out.
The variation of the rotor position is done between
these loops. This is suggested because the saturation state
between adjacent rotor positions does not change
significantly. However, this loop can be easily made the
outermost one, allowing a parallelization for each rotor
position.
Rot
S R
Ni Ni S R
S R
Ni Ni S R
- 234 - 15th IGTE Symposium 2012
A direct comparison between the two algorithms in
Fig. 3 is shown in Fig. 4, highlighting the benefits of this
simple algorithm.
Figure 4: Comparison of the number of nonlinear iterations for 100
magneto-static FEM simulations with and without constant MMF.
The number of nonlinear iterations has been reduced
approximately by a factor of four, the simulation time by
a factor of three. This lower improvement for the
simulation time can be explained by computational costs
during the simulation setup and initialization. For the test
example, the total simulation time has decreased to 190h
from 574h, without any parallelization.
IV. QUINT CUBIC SPLINE PARAMETER CALCULATION
As shown in [8], a quint cubic spline interpolation
method is well suited to be applied in the PPV-WRIM
model, because it allows a continuous interpolation of the
function value as well as the first partial derivatives of the
function defined by the LUT. Nevertheless, the number of
parameters per segment is 1024 and this results in a high
memory demand for the LUTs holding these spline
parameters.
For the test example, the memory demand per LUT is
approximately 480MB. Thus, approximately 3.3GB are
needed in total for the PPV-WRIM model. However, this
is not a problem for state of the art computers and
moreover it is not necessary to load all data into the
RAM. But the calculation of the parameters is very
demanding, since their number is as high as 62.9 millions
(Table III).
Nevertheless, under the assumption of a regular and
orthogonal grid and with the continuity conditions for
quint-cubic splines utilized, a fragmentation of this
system of equations is possible, allowing a fast
calculation of the spline parameters with less memory
demand and a parallelization capability. In this section,
this method is presented and explained in detail.
A. Quint-cubic spline interpolation
A quint-cubic spline interpolation is a piecewise third
order polynomial interpolation in five dimensions as
shown in (8). C 1 -continuity (continuity of the function
value and its first order partial derivatives) at the segment
boundaries can be forced by the right choice of continuity
conditions. Within the segments all derivatives are
continues due to the properties of polynomials.
Each segment in this sense is a five dimensional (5D)

hyper-cuboid with 32 corners and 10 adjacent segments
sharing a four dimensional (4D) hyper-cuboid. The
number of such segments NSeg depends on the number of
sampling points of each coordinate variable x1, x2, x3, x4,
x5 and can be calculated as
NSeg Ni1. (7)
ix , x , x , x , x
1 2 3 4 5
The quint-cubic spline is defined as
x : x1, x2, x3, x4, x5,
3 3 3 3 3
i j k l m
f xaijklmx 1x2x3x 4x5 ,
(8)
i0 j0 k0 l0 m0
with the spline parameters aijklm of the segment fulfilling
L U L U L U
x1 x1 x1 , x2 x2 x2 , x3 x3 x3
,
(9)
L U L U
x4 x4 x4 and x5 x5 x5
,
L L L L L
with the lower segment boundaries x1 , x2, x3, x4, x 5 , the
U U U U U
upper segment boundaries x1 , x2 , x3 , x4 , x 5 and the
local coordinates x1, x2, x3, x4, x5
defined by
L L L
x1 x1x1 , x2 x2x2, x3
x3x3, (10)
L L
x4 x4x4 and x5
x5x5. The first order partial derivative with respect to x1 can
be interpolated with
3 3 3 3 3
f i1j k l m
fx1 ia ijklmx1
xxxx 2 3 4 5 (11)
x1 i1 j0 k0 l0 m0
and in a similar manner those with respect to the other
variables.
Linear extrapolation or a periodic behavior at the
domain boundaries can be easily realized using
appropriate additional constraints [11].
B. Continuity conditions for quint-cubic splines
The choice of the continuity conditions for the spline
parameter determination is important for the continuity at
the segment boundaries. C 1 -continuity even at the
segment boundaries can be achieved for tri-cubic splines
if the function value f, the first order partial derivatives
fx1, fx2, fx3, and all higher order pure mixed derivatives
fx1x2, fx1x3, fx2x3 and fx1x2x3 are continuous at the corners of
each cuboid-seqment. This prerequisite is proven in [10]
for tri-cubic splines.
However, this proof can also be used for higher
dimensional splines. This leads to the conclusion that for
C 1 -continuity, the same prerequisites are necessary. In the
case of quint-cubic splines, these are the function value,
the five first order partial derivatives and all 26 higher
order pure mixed derivatives. These are in sum 32
equations per corner and in total 1024 constraints for
1024 parameters per segment. It can easily be proven that
these equations are linear independent.
C. Continuity of f, fx1, fx2, fx3, fx4 and fx5 on the segment
faces
For the interpolation of the partial derivatives is
necessary to show their continuity also on the segment
faces. Therefore, let us assume a regular and orthogonal
grid for the sampling variables and two adjacent segments
- 235 - 15th IGTE Symposium 2012
S 1 and S 2 that share a face with x1=const. Without loss of
U
generality, the face defined by x 1 is used for S 1 and for
S 2 L
the face defined by x 1 is used. On both faces, the
quint-cubic splines become the quad-cubic splines f S1 and
f S2 below:
3 3
S1U L
i
j k l m
ijklm 1 1 2 3 4 5
jklm , , , 0 i0
S1 quad-cubic spline parameter bjklm
S2
3
jklm , , , 0
j k l m
0jklm 2 3 4 5 with:
S2
jklm 0jklm
3 3
S1U L
i1
j k l m
x1 ijklm 1 1 xxxx 2 3 4 5
jklm , , , 0
i1
S 1
quad-cubic spline
parameter b
jklm
S2 x1
3
jklm , , , 0
j k l m
1jklm 2 3 4 5
S2
with:
jklm 1jklm
f a x x x x x x
f a x x x x b a
f i a x x
f a x x x x b a
(12)
For continuity of f, fx2, fx3, fx4 and fx5 at this face, it is
sufficient that the quad-cubic splines f S1 and f S2 are equal.
S1
Therefore the all spline parameters b jklm of f S1 must be
S 2
equal to the corresponding parameter b jklm of f S2 .
This equality can easily be shown, since both of these
quad-cubic splines fulfill the same 16 constraints
f, fx2, fx3, fx4, fx5, fx2x3,..., f x2x3x4x5 in all 16 involved
nodes and these 256 equations for 256 unknowns are
linearly independent. Therefore, the solution is unique
and the two splines are the same.
For continuity of fx1 at this face, the two additional
S1
S 2
quad-cubic splines f x1
and f x1
representing the partial
derivative with respect to x1 on this face, must be equal.
This can be proven in a similar manner as for the other
constraints fx1, fx1x2, fx1x3, fx1x4, fx1x5,..., f x1x2x3x4x5 per
node.
Furthermore, the same proof can be used for the other
faces. Thus, C 1 -continuity for the quint-cubic spline
interpolation has been shown for a regular and orthogonal
grid.
D. Segmented parameter calculation
The proof of continuity shows that f, fx1, fx2, fx3, fx4, fx5,
fx1x2, fx1x3,…, fx2x3x4x5 and fx1x2x3x4x5 must be continuous in
each node. If all of those 32 conditions per node can be
determined in advance, it is not necessary to assemble a
single system of equations for all segments. Instead, each
segment could be solved independently. This will lead to
NSeg systems of equations with 1024 unknowns each and
furthermore allows parallel evaluation.
Under the assumption of a regular and orthogonal grid
for the sampling data, only one parameter changes along
each edge of the segments. Therefore, the quint-cubic
spline becomes a normal cubic spline
i
f x aˆ x 1
2, x3, x4, x i x
5const
i0
3 3 3 3
j k l m
i
ijklm 2 3 4 5
j0 k0 l0 m0
with aˆ a xxxx,
3
(13)

as shown for an edge with constant values for x2, x3, x4
and x5. The resulting cubic splines along the straight lines
with constant values for x2, x3, x4 and x5 could be
evaluated independent of each other. This can also be
done for all other straight lines where only one parameter
changes. This leads to a number of NCSP simple cubic
splines, where
N N with M : x
, x , x , x , x .
(14)
CSP j
iM jM/ i
1 2 3 4 5
With these cubic splines, all first order partial
derivatives in each node can be evaluated as for x1 below:
3
i 1
f x i aˆx
. (15)
x1 1 i 1
i1
These derivatives can be used in a similar manner for
determining all higher order mixed derivatives.
This approach needs a high number of cubic spline
determinations, but all these systems of equations have
only three unknowns per segment and thus in total
3Nx1 1
unknowns for the cubic splines along the x1direction,
for example. Furthermore, the system matrices
for all splines in one direction are equal. This leads to a
single system of equations with many right hand sides that
can be solved very effectively.
E. Dimensional recursive approach
Obviously, not all 1024 parameters per segment are
unknown. For example the constant parameter a00000 of
each segment is equal to the function value in the segment
L L L L L
reference node x1 , x2, x3, x4, x 5 where all local
coordinates are zero
L L L L L
a00000 f x1, x2, x3, x4, x5
(16)
and all derivatives in the reference node can be directly
identified as parameters of the quint-cubic spline
L L L L L
a10000 fx1x1, x2, x3, x4, x5,
L L L L L
a01000 fx2x1, x2, x3, x4, x5,
(17)
Nevertheless, most of the parameters per segment must
still be calculated. However, the construction of the quintcubic
spline by using a reference node can be easily
combined with the continuity conditions. This leads to a
dimensional recursive approach allowing a well
structured determination of all parameters.
There are five faces of the segment including the
reference node. For each of these faces, one coordinate is
constant zero, resulting in a quad-cubic spline, as shown
for x1=0:
3 3 3 3
j k l m
f 0, x2x3x4x5a 0 jklm x2x3x4x5 . (18)
j0 k0 l0 m0
All parameters of this quad-cubic spline are also
parameters of the quint-cubic spline as already indicated
by the parameter indices in (18). Furthermore, the first
order partial derivative with respect to x1 for the face
x1=0 leads to a quad-cubic spline
3 3 3 3
j k l m
fx10, x 2x3x4x5a1jklmx 2x3x4x5 . (19)
j0 k0 l0 m0
- 236 - 15th IGTE Symposium 2012
All of these parameters can be also identified as quintcubic
spline parameters. Doing this also for the other four
coordinates, leads to ten quad-cubic splines. Each of them
is defined by the continuity conditions of the
corresponding nodes. Finally, 992 quint-cubic parameters
per segment can be determined by this method, only the
32 parameters aijklm with i, j, k, l, m 2,3 have to be
solved for separately. This is due to the segment diagonal
U U U U U
node x1 , x2 , x3 , x4 , x 5 ,
since this node is not one of
the determined quad-cubic splines.
For a quad-cubic spline, this approach can be used in
the same manner, leading to eight tri-cubic splines and an
additional system of equations of 16 unknowns for the
diagonal node per segment. The further dimensional
recursion is shown in Fig. 5.
Figure 5: Dimensional recursive approach for quint-cubic spline
parameter determination.
It is pointed out that the function calls shown in Fig. 5
are multi-function calls. This means that each call has to
be multiplied with the number of sampling points for the
corresponding dimension to get the real number of single
function calls. As an example, the number of single quadcubic
spline function calls NQ4 can be calculated as
NQ4 2Nx1Nx2 Nx3 Nx4 Nx5,
(20)
i.e. two multi-function calls per dimension.
Furthermore, there is also some redundancy within this
recursion. It is not necessary to call 80 multi-function
calls for tri-cubic splines as Fig. 5 hypothesizes. Some of
them are equal for different quad-cubic splines. This can
easily be shown by the fact that each quad-cubic spline is
defined by 256 parameters and each of them is also a
parameter of the quint-cubic spline. However, the ten
quad-cubic splines define only 992 quint-cubic
parameters.
TABLE I
RECURSIVE MULTI-FUNCTION CALLS
QuintQuadTriBi- Cubic
cubiccubiccubiccubic 1 10 40 80 80
Table 1 shows the actual number of multi-function

calls during this recursive approach. Table 2 shows an
overview for different sampling rate setups and Table 3
shows the corresponding calculation time and memory
demand for an implementation in Fortran95. These
calculations were carried out without parallelization on a
computer with Intel Core2 Duo CPU (E8400) and 8GB
RAM.
TABLE II
SETUP OVERVIEW FOR VARIOUS SAMPLING RATES
Setup Nx1 Nx2 Nx3 Nx4 Nx5 NSample NSeg
1 6 6 6 6 6 7776 3125
2 8 8 8 8 8 32768 16807
3 11 11 11 11 11 161051 100000
4 12 12 12 12 12 248832 161051
5 16 8 8 8 8 65536 36015
6 16 9 9 9 9 104976 61440
TABLE III
SIMULATION TIME AND MEMORY DEMAND
Setup Number of Calculation Memory
Parameters time demand
in Mio [s] [MByte]
1 3.20 3,04 60,8
2 17.21 15,61 131,3
3 102.40 83,80 781,3
4 1649.16 134,53 1258
5 36.88 33,76 281,4
6 62.91 54,39 480,0
These setups illustrate the scaling of this approach. A
comparison shows that the calculation time increases
slightly slower than the number of segments does.
The last setup corresponds to the test example of
section III-A. For this example, Nx1=16, although there
are 15 different rotor positions. This can be explained by
the periodicity of the rotor position, i.e. the first sampling
point is identical with the last one.
V. CONCLUSION
The method of constant magneto-motive force
presented in this paper reduces the number of nonlinear
iterations and thus the over-all simulation time for the
FEM simulations. This is achieved by just changing the
order of parameter setups and thus makes this method
simply usable in every commercial FEM simulation tool.
The revised quint-cubic spline parameter calculation
method introduced in this work allows a very fast and
memory saving pre-calculation of the needed spline
- 237 - 15th IGTE Symposium 2012
parameters. This was achieved by a dimensional recursive
approach that leads to a segmentation of the whole system
of equations. The parameters can also be evaluated in
parallel.
These two improvements make the PPV-WRIM model
approach introduced in [8] applicable to simulation tasks
during the design stage of electrical drive chains.
VI. ACKNOWLEDGEMENT
This work has been supported by the Christian Doppler
Research Association (CDG) and by the industrial partner
AVL List GmbH.
[1]
REFERENCES
Mohammed, O.A.; Liu, S.; Liu, Z.; , "Physical modeling of
electric machines for motor drive system simulation," Power
Systems Conference and Exposition, 2004. IEEE PES , vol., no.,
pp. 781-786 vol.2, 10-13 Oct. 2004
[2] Liu, Z.; Mohammed, O.A.; Liu, S.; , "An improved physics-based
phase variable model of PM synchronous machines obtained
through field computation," Computation in Electromagnetics,
2008. CEM 2008. 2008 IET 7th International Conference on ,
vol., no., pp.166-167, 7-10 April 2008
[3] Kallio, S.; Karttunen, J.; Andriollo, M.; Peltoniemi, P.;
Silventoinen, P.; , "Finite element based phase-variable model in
the analysis of double-star permanent magnet synchronous
machines," Power Electronics, Electrical Drives, Automation and
Motion (SPEEDAM), 2012 International Symposium on , vol.,
no., pp.1462-1467, 20-22 June 2012 doi:
[4]
10.1109/SPEEDAM.2012.6264434
Mohammed, O.A.; Liu, S.; Liu, Z.; Khan, A.A.; , "Improved
physics-based permanent magnet synchronous machine model
obtained from field computation," Electric Machines and Drives
Conference, 2009. IEMDC '09. IEEE International , vol., no.,
pp.1088-1093, 3-6 May 2009, doi:
[5]
10.1109/IEMDC.2009.5075339
Mohr, M.; Bíró, O.; Stermecki, A.; Diwoky, F.; ,”An Improved
Physical Phase Variable Model for Permanent Magnet Machines”,
submitted and accepted at ICEM, Marseille, France, 2012
[6] Sarikhani, A.; Mohammed, O.A.; , "Development of transient FEphysics-based
model of induction for real time integrated drive
simulations," Electric Machines & Drives Conference (IEMDC),
2011 IEEE International , vol., no., pp.687-692, 15-18 May 2011
[7] Sarikhani, A.; Mohammed, O. A.; , "Non-linear FE-based
modeling of induction machine for integrataed drives,"
[8]
Computation in Electromagnetics (CEM 2011), IET 8th
International Conference on , vol., no., pp.1-2, 11-14 April 2011
Mohr, M.; Bíró, O.; Stermecki, A.; Diwoky, F.; ,” An Improved
Physical Phase Variable Model for Wound Rotor Induction
Machines”, submitted and accepted at CEFC, Oita, Japan, 2012
[9] ANSYS® Academic Research, Release 12.1, Help System, “Low-
Frequency Electromagnetic Analysis Guide”, ANSYS, Inc.
[10] Lekien, F.; Marsden, J.; , “Tricubic interpolation in three
dimensions,” Internat. J. Numer. Methods Engrg., 63 3 (2005),
pp.455-471
[11] Boor, C.D.; , “A practical guide to splines,” New York: Springer-
Verlag, 1978, p39f., ISBN: 9780387903569

- 238 - 15th IGTE Symposium 2012
Post Insulator Optimization Based on Dynamic
Population Size
Peter Kitak, Arnel Glotic, Igor Ticar
University of Maribor, Faculty of Electrical Engineering and Computer Science, Smetanova 17, SI-2000 Maribor,
Slovenia
E-mail: peter.kitak@uni-mb.si
Abstract—This paper suggests the use of dynamic population size throughout the optimization process which is applied on the
numerical model of a medium voltage post insulator. The main objective of the dynamic population is reducing population
size, to achieve faster convergence. Change of population size can be done in any iteration by proposed method. The multiobjective
optimization process is based on the PSO algorithm, which is suitably modified in order to operate with the principle
of the optimal Pareto front.
Index Terms—Dynamic population size, Insulation elements, Multi-objective optimization, particle swarm optimization.
reductions in the population size are presented in fifth
section. The sixth section is a conclusion with
fundamental summarizing thoughts.
I. INTRODUCTION
Principal function of medium voltage insulator is
electrical insulation of the conductive parts from the
earthed parts of the device, and the mechanical fixing of
equipment and conductors which have different electrical
potential. This element often contains built-in capacitive
voltage divider and thus is capable of performing voltage
indication function.
Particle swarm optimization method [1] is a very
efficient algorithm and it is applied onto many
engineering problems. In comparison to the original
version, many modifications have been made to the
algorithm, which has gained many improvements [2], [3].
Also, the PSO algorithm is extended into a multiobjective
particle swarm MOPSO [4], [5], on the basis of a nondominant
solution sorting (Pareto concept) [6]. Although,
the PSO algorithm does not contain genetic operators in
its fundamentals, it has been proven that the introduction
of the mutation is very useful [7]. The presented method
enables the enhancement of the solution space. The more
recent modifications of the algorithm introduce the
dynamic population size throughout the optimization
process [8]. A variable [9], [10] or fixed [11] change of
the population size, during the evolution, is possible. The
method that includes variable changes of population size
enables the change in any iteration. The method with
fixed change of population is determined with a
predefined step of iteration.
Reduction of the population is desired, when the lasting
time of the optimization needs to be shortened. However,
the efficiency and robustness of the modified algorithm
must not change.
The remainder of this paper is organized as follows.
Section two describes numerical model of medium
voltage post insulator and also a multi-objective
optimization model. Section three presents a classical
Particle Swarm Optimization (PSO) algorithm and also an
updated version of such algorithm with the Cauchy
mutation operator. Section four describes a dynamic
population PSO algorithm, where two procedures for
population reduction are presented. Results of the
classical optimization PSO algorithm and for all dynamic
II. NUMERICAL MODEL AND OPTIMUM
SIGNIFICATION
Post insulator is used as a voltage indicator in medium
voltage switchgear. Post insulators are installed in a
switchgear device or in any other input element, where
voltage is present. Fig. 1 shows parametrically written 3D
numerical model of a post insulator. The metal fitting of
the insulator (upper connection) for fastening to the
conductive part of the upper side, is elongated with a
special electrode of the divider, which has the same
potential as the conductive part. The metal fitting for
insulator fastening to the earthed part (lower connection)
is situated at the bottom of the insulator. An electrically
separated cylindrical metal mesh is mounted around this
metal connection, which is the other divider’s electrode.
Modeling of the mentioned switchgear element
demands design regarding the exact value of capacitance,
which is calculated from electric energy. Modeling also
requires the lowest possible magnitude of electric field
inside insulation material, because increased values of
electric field lessen the life-time of these switchgear
elements. Optimization provides electric field strength
reduction on electrodes’ edges and on other critical points
on its crossing between insulation and air. Switchgear
stable performance requires that switchgear elements
must be constructed to endure the highest voltages, such
as lightning strike (125 kV).
Requirements described above, which have an opposite
tendency, are presented with two objective functions (fC,
fE). Every objective function presents an individual
electric characteristic with different quantities, therefore
objective functions must be written relatively. Both
objective functions are written with bell shaped fuzzy sets
with the maximal value of 1. Function fE is presented in
the Fig. 2a, whereas function fC is in Fig. 2b. The
numerical calculation is based on FEM analysis, with
solver of the EleFAnT program package [12].

upper electrode
epoxy insulation
cylindrical metal mesh
lower electrode
Figure 1: Post insulator illustrative model with
optimization parameters.
The entire model is parametrically written and
described with eight parameters (Fig. 1). It is necessary to
perform FEM calculation for each evaluation of the
objective functions.
f E
a)
f c
1,2
1
0,8
0,6
0,4
0,2
0
1 2 3 4 5
1,2
1
0,8
0,6
0,4
0,2
p7
p5
p4
p2
E (MV/m)
p8
p6
p3
p1
0
54 59 64 69 74 79 84 89 94
b)
C (μF)
Figure 2: Determination of objective function: a) fE, b) fC.
PSO algorithm [1] has been used as a multi-objective
optimization algorithm. Among other methods, the
weighted sum method is used to solve the multiobjective
optimization problem. Equation (1) describes how the
functions fE and fC are merged into a unified objective
function f
f wEfE wCfC (1)
where wE and wC are the weights of the individual
quantities.
The weighted sum method requires special attention
when selecting the objective functions weights, which
enable transformation to optimization with a composed
single objective function. This method also requires a
detailed knowledge of the applicative problem. Knowing
the presented problem, according to (1), the authors have
selected the following weights: wE = 0.6 and wC = 0.4.
- 239 - 15th IGTE Symposium 2012
III. PSO ALGORITHM AND CAUCHY MUTATION
The PSO algorithm is placed in a population-based
stochastic search technique, that imitates social behavior
of the birds while they fly and does not contain genetic
operators. Instead of the genetic operators, the population
members are exposed to the cooperation between each
other, at the same time, they compete with each other
throughout generations.
Each and every particle adjusts its flying ability to the
leading particle - the best individual. Each particle of the
population, which represents a possible solution to the
problem, is treated as a point in the D-dimensional space.
The i th particle is presented as xi ( xi1, xi2,..., xiD)
. The
best former position (position, which gives the best result
in the previous iteration) of the each and every particle is
stored and presented as pi ( pi1, pi2,..., piD)
. The velocity
of the i th particle is presented as vi ( vi1, vi2,...., viD)
.
The velocity changes vi and the new position xi of the
i th particle changes in accordance with the (2) and (3):
vi( t1) wvi() t c1rand() ( pi() t xi()) t
(2)
c2Rand() ( pg( t) xi(
t))
x( t1) x( t) v( t
1)
(3)
i i i
where t indicates the iteration, c1 and c2 are positive
constants, rand() and Rand() are random functions of the
dimension [0,1]. Index g represents the position of the
best particle among other particles from the optimization
process. Equation (2) is used for calculation of the new
particle velocity on the basis of the previous particle
velocity and the distance between its instantaneous
distance and distance of the leading particle. Equation (3)
represents a flight of the particle towards a new position.
When the new population is entirely formed, the
algorithm is being carried out until the interruption
criterion is reached. Two approaches are used in this
paper: a classical approach with the static population
(number of the population members is always the same)
and a dynamic approach, where the population changes
the number of members throughout the optimization
process. The quality of each particle is evaluated on the
basis of the defined objective function.
With the intention to prevent too fast convergence and
consequentially to trap into a local minimum, the classical
PSO algorithm has been upgraded with the Cauchy
mutation [13]. Cauchy mutation operator that is used in
the PSO is determined with a weighted vector.
1 NP
W v , (4)
i ji
NP j1
where vji is velocity of vector j th particle in the
population. The best particle is mutated from (2)
according to the following equation:

p () i p () i W N( X , X ) , (5)
'
g g i
min max
where N is Cauchy distribution function over the
interval (Xmin, Xmax).
IV. DYNAMIC POPULATION SIZE IN THE PSO
ALGORITHM
In this paper the optimization procedure considers two
approaches for the dynamic population size, employed in
the multiobjective optimization problem. The first
approach is based on the gradual reduction of the
population size by half (subsection 4A). In the second
approach, a dynamic reduction of population size is
proposed, which is described in subsection 4B.
A. Gradual reduction of the population size by half
The original idea is presented by Brest [11] and
proposes gradual reduction of the population size by half
in each block of a predefined iteration number. This
means that the reduction is not applied throughout all
iterations. Fig. 3 shows the example where the population
reduction has been carried out four times and the
coefficient that defines the reduction is pmax = 4. In each
reduction step, the population is reduced by a half in
comparison to its former size.
p=1
p=2
p=3 NP/4
p=4 NP/8
NP/2
NP
Figure 3: Schematic presentation of the population
reduction.
The stopping criterion for the optimization process is a
predefined number of function evaluations maxnfeval. In
relation to the population size reduction, there are two
possibilities to determine the size of iteration blocks iterp.
First possibility is an equal number of function
evaluations throughout each single iteration block.
Therefore the number of iterations is defined as:
maxnfeval
iterp
pmax NP
p
The second possibility offers a constant number of
iterations iterp, therefore the number of function
evaluations for each reduction block is:
(6)
nfeval NPp iterp
(7)
For easier explanation, the Table I shows values for
number of objective function evaluations (NP times iterp).
These values are valid for individual population size
blocks with a constant number of iterations iterp = 10 and
the number of population reduction pmax = 4.
- 240 - 15th IGTE Symposium 2012
TABLE I
RUN DATA, MAXNFEVAL=1200, PMAX=4
p 1 2 3 4
NP 56 28 14 7
iterp 10 10 10 10
NP x iterp 560 280 140 70
Selection procedure is based on the idea from the
selection mechanism in DE optimization algorithm [11].
Individual from the first half of the population xi(t) and
the suitable individual from the second half xNP/2+i(t) are
compared based on the corresponding objective function
values. Afterwards, the individual with a better objective
function value takes the position i and therefore becomes
the member of the new and reduced population. After the
last step of selection is performed, the new and reduced
population is obtained
xNP/2
i( t) if f( xNP/2i( t)) f( xi( t))
xi( t1)
xi(
t)
other
B. Dynamic reduction of population size in individual
iteration
Alteration of population size through an optimization
process is realized based on objective function
evaluations, respectively according to (9):
avr i best
NP() t NPmax
f( xmax
)
(8)
f ( x ) f( x )
, (9)
where the favr(xi) is average objective function value of
the observed population. f(xbest) and f(xmax) are the
objective function values of the best and worst particle
ever found up to the observed iteration. NPmax is initial
(max) population size.
As the optimization algorithm approaches to optimal
solution, the value of the objective function alters.
Generally, the population’s average value of the objective
function value is getting smaller along with iteration
number. However, this does not hold true for all
iterations, because the particles move also through the
non-promising areas of the search space. Because of this,
the population size is, according to (9), generally
reducing its size; however there are also iterations, where
the population size has been extended. Each population
extension in individual iterations appear usually when the
average objective function value is increased. The
missing particles are obtained from the set of particles
that have been discarded on account of population
reduction in previous iterations. This improves the
algorithms reliability.
V. RESULTS
Optimization processes with the PSO algorithm are
performed under the following settings: c1=0.5, c2=1.2,
w=0.8. Maximal number of iterations for all calculations
is maxiter = 40.

The optimization results are shown in Table II, where
first two examples are showing results obtained by
standard PSO algorithm and different size of population,
third one showing results of standard PSO algorithm
upgraded with Cauchy mutation and last two are showing
results obtained by using dynamical population size in
standard PSO algorithm. This paper presented two
different concepts of dynamical population size, gradual
reduction and dynamic reduction proposed in section IV.
TABLE II
OPTIMIZATION RESULTS OBTAINED WITH PSO ALGORITHM
description min f NP maxnfeval
standard PSO 0.345 30 1200
standard PSO 0.328 56 2240
standard PSO + mutation 0. 326 56 2240
dyn. populated PSO
0. 326 56/28/14/7 1050
(gradual reduction)
dyn. populated PSO
(proposed reduction)
0. 326
max 56
min 10
829
Optimization process convergences for all mentioned
examples in Table II are shown in Fig. 4.
Objective function value
0,6
0,55
0,5
0,45
0,4
0,35
standard PSO (NP=56)
standard PSO (NP=30)
standard PSO + mutation
dyn. populated PSO (gradual reduction)
dyn. populated PSO (proposed reduction)
0,3
0 10 20
Iteration
30 40
Figure 4: Objective function values of PSO algorithm
during the optimization process
Algorithm with using small population size has not
reached global solution, because of stuck in local
optimum. Global solution can be reached by increasing
population size which leads increased number of function
evaluations and longer computation time. By using
proposed dynamical population size algorithm achieved
global minimum with decreased number of function
evaluation and computation time.
Size of population
60
50
40
30
20
10
Standard procedure with a static population size
Dynamically populated PSO
(proposed reduction)
Dynamically populated PSO
(gradual reduction)
0
0 10 20
Iteration
30 40
Figure 5: Changing of population size during the
optimization process.
Changing population size along the optimization process
is shown on Fig. 5 – for the gradual reduction, proposed
reduction and fixed population size (static size).
- 241 - 15th IGTE Symposium 2012
VI. CONCLUSION
Results show comparison of the optimization process for
different population size reduction methods. Reduction of
the population is desirable, when computation time
should be decreased and although efficiency and
robustness of algorithm should not be changed.
The important impact of proposed PSO algorithm with
using dynamical population size can be seen in decreased
number of function evaluation.
In each iteration is tendency to decrease the population
size. However, the population number can be also
increased by adding the new members, which refreshes
the population. Therefore, the algorithm’s ability to
search the minima is increased. It is important, already at
the beginning, to select the appropriate, respectively
enough large population. Therefore, the global search of
environment is enabled. Smaller population size is
sufficient just for local search solutions.
[1]
REFERENCES
J. Kennedy and R. C. Eberhart, “Particle swarm optimization,”
Proc. IEEE International Conference on Neural Networks, Vol.
IV, Piscataway, NJ, pp. 1942-1948, 1995.
[2] S.L. Ho, Y. Shiyou, Ni Guangzheng and H.C. Wong, “A particle
swarm optimization method with enhanced global search ability
for design optimizations of electromagnetic devices,” IEEE Trans.
on Magn., vol. 42, no.4, pp. 1107-1110, 2006.
[3] G. Toscano Pulido and C.A. Coello Coello, “Using clustering
techniques to improve the performance of a particle swarm
optimizer,” Proceedings of Genetic and Evolutionary
[4]
Computation Conference, Seattle, WA, pp. 225-237, 2004.
L. dos Santos Coelho, H.V.H. Ayala and P. Alotto, “A
Multiobjective Gaussian Particle Swarm Approach Applied to
Electromagnetic Optimization,” IEEE Trans. on Magn., vol. 46,
no.8, pp. 3289-3292, 2010.
[5] L. dos Santos Coelho, L.Z. Barbosa and L. Lebensztajn,
“Multiobjective Particle Swarm Approach for the Design of a
Brushless DC Wheel Motor,” IEEE Trans. on Magn., vol. 46,
no.8, pp. 2994-2997, 2010.
[6] U. Baumgartner, C. Magele and W. Renhart, “Pareto optimality
and particle swarm optimization,” IEEE Trans. on Magn., vol. 40,
no.2, pp. 1172-1175, 2004.
[7] L. Jize, S. Ping and L. Kejie, “A Modified Particle Swarm
Optimization with Adaptive Selection Operator and Mutation
Operator,” International Conference on Computer Science and
Software Engineering CSSE 2008, Vol. 1, Wuhan, China, pp.
1199-1202, 2008.
[8] W. F. Leong and G. G. Yen "PSO-based multiobjective
optimization with dynamic population size and adaptive local
archives", IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38,
no. 5, pp.1270 - 1293 , 2008.
[9] M. Greeff and AP. Engelbrecht, “Dynamic multi-objective
optimisation using PSO,” Studies in Computational Intelligence,
(Series Ed: Kacprzyk, Janusz), pp. 105-123, 2010.
[10] G. G. Yen and W. F. Leong "Dynamic multiple swarms in
multiobjective particle swarm optimization", IEEE Trans. Syst.,
Man, Cybern. A, Syst. Humans, vol. 39, p.890 - 911, 2009.
[11] J. Brest and M. Sepesy Maucec, “Population Size Reduction for
the Differential Evolution Algorithm,” Applied Intelligence, 29(3)
pp. 228–247, 2008.
[12] Program tools ELEFANT. Graz, Austria: Inst. Fundam. Theory
Elect. Eng., Univ. Technol. Graz, 2000.
[13] H. Wang, Y. Liu, S. Zeng, H. Li, and C. Li, "Opposition-based
Particle Swarm Algorithm with Cauchy Mutation", Proc. of the
2007 IEEE Congress on Evolutionary Compulation, 2007, pp.
4750-4756.

- 242 - 15th IGTE Symposium 2012
Simulation of the Absorbing Clamp Method for
Optimizing the Shielding of Power Cables
Szabolcs Gyimóthy∗ ,József Pávó∗ ,Péter Kis † , Tomoaki Toratani ‡ , Ryuichi Katsumi ‡ and Gábor Varga∗ ∗Budapest University of Technology and Economics, Egry J. u. 18, H-1111 Budapest, Hungary
† Furukawa Electric Institute of Technology, Vasgolyó u. 2-4, H-1158 Budapest, Hungary
‡ Furukawa Electric R&D Center for Automotive Systems & Devices, Hiratsuka, Japan
E-mail: gyimothy@evt.bme.hu
Abstract—An efficient numerical simulation tool based on FEM is proposed, by which the EMC shielding effect
characteristics of power cables can be predicted in the 30–1000 MHz frequency range, as if it would be measured by the
Absorbing Clamp Method. The proposed simulation method is based on decomposition: a 2D axisymmetric RF FE model
is used for describing the whole measurement setup, while a 3D quasistatic FE model is used for the symmetry cell of the
shielding layer in order to capture the effect of its fine geometric details. The two models are coupled via the concept of
the equivalent shielding layer obtained by homogenization. Comparison with real measurements show that the shielding
characteristics can be reliably predicted this way, with some deviation in the low end of the frequency range though. This
simulation tool can be applied in the design and optimization of braided cable shields to be used in automotive industry.
Index Terms—EMC testing, cable shielding, homogenization, automotive industry
I. INTRODUCTION
The effective reduction of emitted radio frequency (RF)
disturbances in electric vehicles –generated mainly by
power semiconductors having high slew rates– becomes
more and more important nowadays [1]. Partly for
this reason, the network configuration of cars is being
changed from unshielded single core multi wire harnesses
into coaxial conductor layouts. Consequently, optimization
of the shielding of such cables is a current topic.
The absorbing clamp method (ACM) is a well known
technique for measuring electromagnetic interference
(EMI) generated by electric cables in the range 30–
1000 MHz [2]. Nowadays it is commonly used in the
automotive industry for testing electromagnetic compatibility
(EMC) of braided shields and connectors of
wire harnesses [3]. The measurement set-up is shown in
Fig. 1. The central element of the device is the “clamp”
consisting of a set of split lossy ferrite rings (see Fig. 2)
and a sensing loop [4]. The shielding effect (SE) to be
measured is –in essence– the ratio of the output signal
of the unshielded cable to that of the shielded.
This method became de facto an industry standard in
many areas. For instance –in addition to RF compliance
testing of cables– it is already used for the quantitative
Fig. 1. ACM device for measuring the efficiency of cable shielding.
evaluation and comparison of the performance of shields
through their measured SE characteristics. Therefore,
although there are other practical parameters by which
one may characterize and optimize a shielding, the design
cycle would be much more efficient if the SE curve of
a shield prototype (as taken by ACM) could be directly
predicted by numerical simulation.
ACM is simple and relatively cheap, but it was not
developed –in the late 60’s– with numerical modeling
in mind. Although there are some theoretical studies on
its operation [5], they are far from being applicable for
quantitative prediction. Actually, the simulation of this
measurement is quite challenging from the numerical
point of view: a) the set-up consists of several components,
among others ferrite; b) a wide frequency range is
studied; c) the arrangement is “large” but has some very
fine details.
Several types of numerical models have been worked
out to simulate this measurement, based on e.g. coupled
transmission line system (TLS), finite element method
(FEM) and method of moments (MoM), with little success
though [6][7]. Surprisingly, none of them was able
to catch even the main tendency of the characteristics.
Anyone might conclude from the physical picture that
the shielding effect gets better at higher frequencies,
and actually the above mentioned computing models
just confirmed this behavior. However, the real ACM
Fig. 2. Absorbing clamp of type R&S R○ MDS-21 [4].

measurement of a typical braided cable shield usually
results in a decaying SE characteristics as frequency
increases (c.f. Fig. 13). Whether this behavior is intrinsic
to the shielding, or rather just a “side-effect” of the
measurement method, was a question.
II. SUMMARY OF THE MODELING APPROACH
It was realized that detailed 3D modeling of the measurements
is not doable because of the enormous computational
resources needed for correct analysis of the
complicated arrangement.
Our idea is to use decomposition and homogenization.
Different models are used for the braided shield details
and the overall set-up, respectively. The ACM measurement
at higher frequencies tend to show little dependence
on the larger environment (e.g. support, ground, walls,
etc.). This suggests the use of a simplified axisymmetric
2D finite element (FE) model of the arrangement, which
can be analyzed efficiently.
On the other hand, realistic cable shields do not
exhibit such symmetry. To overcome this difficulty, we
introduced the concept of equivalent homogeneous (bulk)
shielding layer that may have frequency dependent complex
conductivity, such that its shielding characteristics
approximates that of the original braided shield. Also a
method was developed by which the equivalent conductivity
parameter can be identified. Although this latter
requires a 3D FE model of the shield, the domain of
computation extends to only a single symmetry cell of the
geometry. As a result, this 3D analysis is manageable in a
relatively moderate computing environment, too. Putting
the equivalent shield with its properly selected parameters
into the 2D model of the experimental set-up provides the
simulated output of the measurement.
III. THE 2D AXISYMMETRIC RF FE MODEL
By omitting the support and the surroundings of the
measurement device and taking the two reflector plates as
disc-shaped, we get an axially symmetric arrangement, of
which a 2D longitudinal section is enough to be considered.
We used the RF Module of Comsol Multiphysics R○
in the application mode “Electromagnetic Waves / TM
Waves / Harmonic propagation” and with model space
dimension “Axial symmetry (2D)” [8].
The geometry of the 2D model can be seen in Fig. 3.
Cable conductor and reflector plates are both modeled
as perfect electric conductor (PEC) boundary conditions.
The terminal impedance is modeled by means of a
50 Ω coaxial cable section with non-reflecting boundary
condition at its end. The arrangement is considered
open in the radial direction, which is modeled by a
cylindrical perfectly matching layer (PML). Finally, the
excitation is realized as a 50 Ω coaxial cable, fed by an
incident coaxial-mode TEM wave, which is prescribed as
“port” boundary condition and characterized by the input
voltage Uin.
- 243 - 15th IGTE Symposium 2012
Fig. 3. The 2D axisymmetric FE model built in Comsol RF.
The output signal is the induced voltage Uout of an
assumed loop encircling the first ferrite ring (the one
which is closest to the feed). The transfer function is
defined as the ratio of output voltage to input, and
normally its gain k versus the frequency f is used,
k(f) = 20 log10 |Uout/Uin| [dB]. (1)
The shielding effect (SE) of a given shield is defined here
as the ratio of the output voltage measured for the bare
cable core (i.e. for the cable with its shielding removed)
to that one measured for the shielded cable. Provided the
input voltage is kept fixed, the SE characteristics (given
in dB, as function of the frequency) can be expressed as
follows,
SE(f) =kun(f) − ksh(f), (2)
where “un” and “sh” stand for the unshielded and
shielded configurations, respectively.
Figure 4 is a snapshot about the circumferential component
of the magnetic field, Hϕ, taken in the unshielded
case. It can be observed how two waves –one along the
line with a longer wavelength and one along the series
of ferrite rings with shorter– are coupled with each other.
The damping effect of the ferrite is also observable on
the plot. Notably, this FE model performs well even in
the reconstruction of the stray field of bulk aluminum
shielding layers, for which SE can be as high as 130 dB.
A. Parameter Dependency of the 2D Model
We have carefully investigated the sensitivity of the
computed results on several model parameters –including
some of the numerical implementation– in order to filter
out all possible sources of inconsistency between the
model and the real measurement. First comes a list of
those parameters which have little or no effect: value
of the terminal impedance; the implemenation of open
boundary (e.g. PML, absorbing boundary condition, etc.);
fluctuation of the input voltage.

Fig. 4. Snapsot of the circumferential magnetic field.
The latter needs some explanation. Since the device is
not matched with the signal generator source, the voltage
along the feeding cable becomes location dependent due
to reflections, moreover this dependence is function of the
frequency. In the FE model we consider a chunk of the
feeding cable of fixed length, which implicitly defines the
quantity Uin for the model. However, this may not be the
same as its real (measured) counterpart. Fortunately, the
computed SE curves do not show significant dependence
on the “definition” of Uin, as simulations testified.
Two factors that really affect the results are the permeability
characteristics of the ferrite rings, as well as the
assumption of axial symmetry. These are detailed in the
next two subsections.
B. Permeability of the Ferrite Rings
The permeability of the ferrite material plays a dominant
role in forming the SE characteristics predicted by
the model, hence its accurate description is critical. Since
permeability data for the given absorbing clamp were
not available, we made measurements. The obtained frequency
dependence of the complex relative permeability
(real and imaginary parts) can be seen on the graph of
Fig. 5. Note that the data above 100 MHz are extrapolated
values.
C. Limitations of Assuming Axial Symmetry
Considering the field plot in Fig. 4 one can conceive
that the ferrite clamp acts a waveguide and –at higher
- 244 - 15th IGTE Symposium 2012
Relative permeability
800
700
600
500
400
300
200
100
0
10 7
10 8
Frequency (Hz)
real part
negative imaginary part
Fig. 5. Complex relative permeability of the ferrite material.
frequencies– can give rise to higher order modes. Although
with the use of coaxial feeding the axisymmetric
mode(s) are deemed predominant, non-axisymmetric
modes can appear as well wherever symmetry is violated
in the cross section of the device.
For studying this behavior we carried out the mode
analysis for the cross section of the device using FEM.
In order to break axial symmetry, a large grounded
conducting plate –being parallel with the axis of the cable
under test– was added to the arrangement. In addition
to this, the full 3D FE model of the arrangement was
analyzed. Note that an unshielded cable was examined
in both cases for the sake of simplicity.
Figure 6 compares the transfer functions obtained from
the full 3D and the 2D axisymmetric model, respectively.
Although several possible propagating modes exist towards
1GHz, there is only a little deviation between the
curves, mostly limited to the lower frequencies. From
this we concluded that the effect of higher order nonaxisymmetric
modes is negligible, and that the axisymmetric
model is acceptable within this frequency range.
Fig. 6. Comparison of the transfer functions computed by the 2D and
3D FE models, respectively, for the case of an unshielded cable.
10 9

IV. HOMOGENIZATION AND EQUIVALENT SHIELDING
The goal is to replace the shield having complex geometry
with a homogeneous cylindrical shielding layer,
which is called hereinafter the “equivalent shield”. The
task is to determine the parameters of the latter so that
its shielding effect be the same as if it was measured
on the real cable shielding. Of course, this equivalence
is allowed to satisfy approximately, and only on (and
outside of) an observation surface, i.e. at and beyond a
certain distance from the shield (Fig. 7).
3D Observation surface 1D
Real texture
Wire
Shield
Isolation
Fig. 7. Illustration of the concept of homogenization.
Equivalent
homogeneous material
Figure 8 demonstrates how the roughness of the stray
field distribution is smoothing as we increase the distance
taken from the shield. (the radial component of the
electric field, Er, is plotted along a line parallel with
the cable, at 1GHz). This smoothing behavior justifies
the use of the homogeneous shield as replacement in the
2D model. Note that the shield tested here is not braided
but leaky (c.f. Fig. 10), hence for braided shields one can
expect stronger homogenization effect.
There are several parameters that can be varied in order
to find an equivalence, like for instance the inner and
outer radii of the layer, or the specific electric conductivity
and magnetic permeability of the material. In order to
simplify and regularize the inverse problem we decided
to keep the geometry fixed, and choose non-magnetic
material. Hence only the complex valued conductivity,
σ, of the layer remained to be determined. The proposed
values for the inner and outer radius of the equivalent
shield are r1 =10mmand r2 =11mm, respectively.
The observation surface is at the radius robs =23.5mm
which coincides with the inner radius of the ferrite rings.
A. Evaluation of the Scalar Conductivity Parameter
For those shielding configurations which show certain
symmetry in the ϕ direction, like the one on the left
hand side in Fig. 9, the azimuthal components of the
shielding currents are symmetric too. As a consequence,
the axial component of the magnetic fields caused by
those currents are compensated, and thereby vanish.
This allows us to investigate only the “axial electric –
azimuthal magnetic” field mode, and hence to describe
the conductivity σ by a complex scalar value.
If the current in the wire is given and the parameter σ is
known, the electric and magnetic fields can be determined
from Maxwell’s equations. Since the problem with homogeneous
shielding is essentially one dimensional (see
- 245 - 15th IGTE Symposium 2012
Fig. 8. Smoothness of the field at various distances from the shield.
Fig. 7, right), the solution can be given analytically. We
can describe the electric and magnetic fields, each, by
one single component,
E(r, ϕ, z, t) =Re Ez(r)e −jωt ez (3)
H(r, ϕ, z, t) =Re {Hϕ(r)e −jωt eϕ (4)
where time harmonic fields of angular frequency ω are
assumed, and j denotes the imaginary unit. Using the
quasi static approximation in the conducting region,
one can easily derive the following Bessel’s differential
equation from Maxwell’s equations:
d2Ez 1 dEz
+
dr2 r dr − jωμ0σEz =0, r1

• Hϕ is specified at r0 (this is equivalent to prescribing
the total current of the wire),
• both Ez and Hϕ are continuous at r1 and r2,
• the asymptotic behavior of the fields for r →∞is
known.
We omit further details of the solution as they can
be found in several textbooks on electromagnetism [9].
The closed formula expressing the magnetic field was
implemented as a Matlab R○ function, where the input is
the complex conductivity σ, and the output is the complex
amplitude (phasor) of Hϕ at r = robs.
We use the following procedure for the evaluation of
the equivalent conductivity of the homogeneous shielding
layer (let us denote it with σeq in the following):
1) We create the 3D FE model of the real shielding
together with an exciting wire centered in its axis.
We compute the magnetic field at some selected
frequencies between 30 − 1000 MHz.
2) We take samples of the azimuthal magnetic field on
the observation surface, and calculate its average.
This is what we call the “observed magnetic field”
and denote with Hϕ,obs.
3) In an optimization loop we attempt to find σeq
for which Hϕ (a function of σ) and Hϕ,obs match
the best (this is carried out for each frequency
separately):
σeq =argmin|Hϕ(σ)
− Hϕ,obs| (8)
σ
Some remarks on the procedure. Notably, in step 1) it
is enough to take one symmetry cell of the arrangement
as Fig. 10 demonstrates. We used the AC/DC module
of Comsol Multiphysics R○ for computing the fields with
quasi static approximation [10]. In step 2) we also verify
whether the z component magnetic field, Hz itself, as
well as the fluctuation of Hϕ on the observation surface
are really negligible. Finally, in step 3) we used the
Matlab R○ function fminsearch for the purpose.
B. Evaluation of the Conductivity Tensor
For shielding configurations like the spiral-structure in
Fig. 9 on the right, the above described method is not
suitable because the magnetic field is no longer pure azimuthal.
In this case we can still use the anisotropic conductivity
tensor in the equivalent homogeneous shielding
layer to imitate a similar phenomenon. The conductivity
tensor is assumed to have the following form:
⎡
σ = ⎣ σrr
⎤
0 0
⎦ (9)
0 σϕϕ σϕz
0 σzϕ σzz
That is, the r − ϕ and r − z cross-effects are supposed to
have negligible contribution to the observable magnetic
field. Moreover, σϕz = σzϕ is expected.
To demonstrate the treatment of anisotropy we derive
the equations for the field components in the conductive
region. First, the governing Maxwell’s equations are
- 246 - 15th IGTE Symposium 2012
Fig. 10. Symmetry cell of the shielded cable used for determining
the σ parameter of the equivalent homogeneous shielding layer. This
structure was inspired by leaky coaxial (LCX) cables.
written in the quasi static approximation, in the time
harmonic regime:
∇×H = σE, ∇×E = −jωμ0H (10)
Taking into account the obvious symmetries of the configuration
with homogeneous shielding layer, i.e. ∂/∂z =
0 and ∂/∂ϕ =0, the resolution of (10) written for the
three cylindrical components is the following:
r : 0 = σrrEr, 0=−jωμ0Hr (11)
⎧
⎪⎨ −
ϕ :
⎪⎩
∂Hz
∂r = σϕϕEϕ + σϕzEz
− ∂Ez
(12)
= −jωμ0Hϕ
⎧
∂r
⎪⎨
1 ∂
z :
r ∂r
⎪⎩
(rHϕ) =σzϕEϕ + σzzEz
1 ∂
r ∂r (rEϕ)
(13)
=−jωμ0Hz
It can be seen that the radial components vanish, and
also that σrr does not play a role. However, there are no
longer separable Hϕ − Ez and Hz − Eϕ modes, as in
the isotropic case. The solution of the equations (11-13)
is not easy, but as the domain is one dimensional, fast
solution method may be established. The algorithm given
in section IV-A should slightly be modified here:
1) The same as in the isotropic case, but we have
to carry out the FE analysis for two different
excitations: one in which Hϕ is prescribed at the
wire surface (r = r0), and one in which Hz is
prescribed there (whatever would be the physical
meaning of the latter). These two solutions are
marked with ( ′ ) and ( ′′ ) in the following.
2) We take samples of the axial and azimuthal
magnetic field on the observation surface from
both FEM solutions, and calculate their average.
This way we obtain the observed magnetic fields,
, H′′
H ′ ϕ,obs
ϕ,obs , H′ z,obs
and H′′
z,obs respectively.

3) We attempt to optimize the components of σ as
above. The generalization of the scalar case is
straightforward:
σeq =argmin
σ
H ′
ϕ − H ′
ϕ,obs + H ′′
ϕ − H ′′
+ H ′ z − H ′
z,obs + H ′′
z − H ′′
V. TEST RESULTS
ϕ,obs
z,obs
+
(14)
For testing the method we chose a simple shield structure
(Fig. 11) inspired by leaky coaxial (LCX) cables. The
shield is is made of aluminum; the inner radius of the
tube is 10 mm; the wall thickness is 1mm. The shield
has circular holes of 5mmradius in a regular distribution;
there are 4 holes along the circumference.
Fig. 11. Leaky aluminum shield used for testing the method.
Since the geometry is symmetric with respect to the
azimuthal (ϕ) direction, we are allowed to use the equivalent
scalar conductivity (c.f. section IV-A). By solving
(8) we obtained the σeq curves presented in Fig. 12.
Note that these curves are not the only suitable ones,
because the solution of (8) is not unique. However, we
experienced that quite different σeq curves resulted in
the same SE characteristics at the end, so they can be
considered equally good in this respect.
Figure 13 shows the SE curve computed by building
the σeq characteristics of Fig. 12 into the 2D axisymmetric
FE model. For comparison, the figure also shows the
curve obtained by real measurement. Obviously, the SE
Fig. 12. The computed equivalent complex conductivity, σeq.
- 247 - 15th IGTE Symposium 2012
Shielding Effect (dB)
60
55
50
45
40
35
30
25
20
15
10 7
10 8
Frequency (Hz)
measured
simulated
Fig. 13. Comparison of measured and simulated shielding effects.
characteristics has been reliably predicted by the model,
with some deviation in the low end of the frequency
range.
VI. CONCLUSIONS
A numerical simulation method has been elaborated that
can be used to predict the shielding effect of shields with
various patterns. Using this tool the designer can predict
the usability of a given shield construction. This simulation
method has been thoroughly verified by theoretical
considerations, numerical experiments, and also by some
experimental data. In the authors’ opinion, the error of
prediction at low frequency might be due to either the
inexact knowledge of the ferrite permeability characteristics
or the insufficiency of the 2D axisymmetric model.
REFERENCES
[1] M. Reuter, S. Tenbohlen, W. Köhler, and A. Ludwig, “Impedance
analysis of automotive high voltage networks for EMC measurements,”
in 10th Int. Symposium on Electromagnetic Compatibility
(EMC Europe), York (UK), 26-30 Sept 2011.
[2] A. Tsaliovich, Electromagnetic Shielding Handbook for Wired and
Wireless EMC Applications, ser. Kluwer international series in
engineering and computer science. Kluwer Academic, 1999. [Online].
Available: http://books.google.hu/books?id=4vl0S6fZo-IC
[3] S. Miyazaki, S. Kihira, and T. Nozaki, “New shielding construction
of high-voltage wiring harnesses for Toyota Prius – winning
of Toyota Superior Award for cost reduction,” Sumitomo Electric
Industries Technical Review, no. 61, pp. 21–23, Jan 2006.
[4] Rohde & Schwarz R○ MDS-21 Absorbing Clamp – Data
sheet. [Online]. Available: http://www2.rohde-schwarz.com/file/
MDS-21 EZ-24 dat en.pdf
[5] D. Williams and S. Jones, “Time domain characterization and
modelling of the absorbing clamp. a device for measuring radiated
radio frequency power,” in Eighth International Conference on
Electromagnetic Compatibility, 21-24 Sept 1992, pp. 149–159.
[6] L. Fejérvári, “Simulation of wire harness radiation,” Furukawa
Electric Institute of Technology, Budapest, Tech. Rep., March
2009.
[7] P. Kis, “Simulation of wire harness radiation,” Furukawa Electric
Institute of Technology, Budapest, Tech. Rep., March 2010.
[8] Comsol Multiphysics RF Module User’s Guide, COMSOL AB,
November 2008.
[9] K. Simonyi, Foundations of Electrical Engineering: Fields, Networks,
Waves. London: Pergamon, 1963.
[10] Comsol Multiphysics AC/DC Module User’s Guide, COMSOL
AB, November 2008.
10 9

- 248 - 15th IGTE Symposium 2012
A Neural Network Approach to Sizing an Electrical
Machine
Steven Bielby, David A. Lowther
Electrical and Computer Engineering Department, McGill University, 3480 University Street, Montreal, Quebec,
Canada. H3A 2A7
E-mail: david.lowther@mcgill.ca
Abstract—The first stage in the design of an electrical machine, or an electromagnetic device, is usually referred to as
“sizing”. It produces an approximate description (or design) of the desired device in terms of its major physical dimensions.
This is traditionally performed using simple magnetic circuits or electric equivalent circuits. This paper proposes an
approach based on a data base of device performance data and a neural network to estimate the values of the parameters
necessary to provide a complete initial description of the device.
Index Terms—Design, electrical machines, sizing, neural networks.
Not only is the speed of the process an issue in that the
I. INTRODUCTION
design can take too much time, many of the parameters
The design of any artifact is a process of searching an that are needed for a complete field solution of the device
appropriate space at increasing levels of complexity. This have little or no effect on the main performance
space is usually referred to as the design space. It is, parameters. For example, the magnitude of the torque in
fundamentally, a high dimensional space relating the an electrical machine is dependent on the variation of the
performance of a device to the values of a set of stored energy in the airgap of the device as the rotor
parameters that describe the structure and physical changes position. The energy, in turn, depends on the
properties of the device. The performance, of course, fields in the airgap and, to a first approximation, these
might not be a single variable but several. For example, may be related to the current densities and the size of the
the cost, size and weight as well as output or functional airgap. This information can be expressed through a basic
capabilities might all be performance variables. The magnetic circuit.
physical parameters could be the actual dimensions Consequently, the process usually employed for
describing the physical structure; the material properties; relatively conventional structures relies on the knowledge
or the external environment including the excitation of the designer and simple equivalent circuits for the
sources, the mechanical loads, etc. Thus the design space initial steps. The design starts at an extremely coarse high
can be extremely complex and the design process for a level and works downwards. The equivalent circuit
device needs to be able to balance output objectives and model of an electrical machine provides a way of relating
input specifications. The difficulty in design is that the electrical, mechanical and, possibly, thermal performance
challenge posed by the specification is, in general, an of a device to the values of a relatively small number of
inverse problem, i.e. the designer is asked to produce a components and thus is easy to work with as well as
physical structure which will meet objectives which are providing fast initial iterations. However, it is not always
the input to the design. For example, for an electrical easy to relate the equivalent circuit values to the physical
machine, the design input might well be the torque structures themselves and the knowledge base of the
required from the device over a particular speed range. In designer helps to bridge this gap at the early stages of the
addition, there might be electrical constraints imposed by design process. Fig. 1 illustrates the hierarchical process
the form of the power supply.
involved where the initial search for a design solution
While it might be possible to achieve the design of an takes place at a high level with a limited set of parameters
electrical machine from first principles, i.e. starting from but a large space to explore. As the design space is
the equations of physics and knowledge of the properties narrowed down, the number of parameters is expanded to
of various materials, this involves working at the provide a more detailed examination of the local space.
maximum detail of the device. To use this approach, the At each level of the process, the analysis performed
problem requires the solution of the field equations and becomes more complex and, consequently, more
these require that all the physical parameters are expensive. If design is considered to be an optimization
identified and given values. This generates a huge design process, then the phases of exploration and exploitation
space to work in. The approach has been demonstrated re-occur at each level. At some point, the “virtual”
on simple models but is computationally extremely (computer based) design is terminated because an
expensive [1]. Searching such a space without an initial increase in the level of detail will have no effect on the
idea of where a solution might be is extremely slow on
existing computational platforms. In addition, such an
required accuracy of the performance parameters .
approach has difficulty including economic,
manufacturing, and other constraints on the design.

Figure 1. Hierarchical Design Process
The starting point at the highest level can, clearly,
affect the convergence and time taken to complete the
process. In an industrial organization, determining the
starting point is performed through one or both of two
processes. The first involves a database of previous
designs [2], while the second uses simple magnetic
circuit models (mentioned earlier) which vary in their
effectiveness. However, if little or no design experience
exists and a magnetic circuit is ineffective, for example in
the case of systems with non-linearity and eddy currents,
these two approaches may fail. The approach suggested
in this paper is to use a neural network to map the desired
machine performance onto a set of parameter values for
the device.
II. THE CONCEPT OF SIZING
Following from the above discussion, the first stage in
the design process of an electrical machine is to estimate
the approximate size of the device needed to meet a set of
specifications. Traditionally, this is based on a few,
relatively basic, rules. For example, the torque that can be
delivered by an electrical machine is given in Equation 1:
2
T 0.
5D
L.(
B.
J )
(1)
Where D is the outer diameter of the rotor, J is the
effective stator current sheet in amps per meter of
circumference, L is the length of the rotor, and B is the
flux density crossing the air gap.
How these quantities are created is the job of the rotor
and stator structures. In a simple machine, the flux
density in the airgap can be estimated from a basic
magnetic circuit where the only component that provides
a magnetic “resistance” to the magnetic flux is the airgap.
The benefit of this approach is that it can provide
approximate values for many of the key parameters in the
machine design and it, in effect, locates the most likely
area in the design search space for a solution. The levels
of accuracy needed here are relatively low – the goal is to
find a “ball park” estimate for the parameters so a
solution within 10% or 20% of the real answer is good
enough to begin the design process. The issue is speed –
these results can be obtained very quickly and the
exploration of a possible design space can be completed
at a much reduced cost.
However, given a need for only an approximate
- 249 - 15th IGTE Symposium 2012
solution but to deliver it extremely quickly, there are
several alternate paradigms which could achieve this on a
relatively unsophisticated computing device. In modern
terms, there are two approaches which can be used. The
first is to generate a surrogate model [3]. This is, in
effect, the approach taken by the equivalent circuit
approach, i.e. the real model is replaced by a simplified
structure which performs in much the same way. The
effectiveness and accuracy of the surrogate can be
controlled relatively easily. The relationship between the
output performance and the input parameters is often
referred to as the “Response Surface” [4] and the
accuracy or fidelity of the response surface depends on
the surrogate model chosen. An alternate approach is to
generate a series of points on the response surface and
then to develop a curve fitting or interpolation system to
estimate other points on the surface. In this case, the
response surface can be modeled in a way that estimating
a new point and determining the values of the input
parameters for this point can be achieved very quickly.
This is a variant of the approach suggested in [5].
However, the development of the surrogate can be
computationally expensive since full field solutions may
be necessary. The gain, of course, over the simple
magnetic circuit approach is that the synthesized initial
design is likely to be much more detailed and to take into
account more of the real behavior of the device than the
simple assumption of perfect magnetic materials and only
an air gap.
The approach being proposed in this paper is a
combination of the two “conventional” systems. The
surrogate model is based on a neural network and an
existing database of solutions is used to train the
network, thus providing an approximation to the response
surface. Such a system can deliver an initial estimate of a
design with minimal computational effort and has the
added advantage of improving its capabilities after each
design as the new design can be added to the training
database.
III. NEURAL NETWORKS
A neural network is an interconnected system of basic
processing elements [6]. Each element performs a simple
computation based on its inputs and produces an output.
Each neuron “sees” a different weighted set of inputs and
the outputs are combined to generate the output of the
network. Fig. 2 illustrates the process.

Figure 2. Basic Neural Network Architecture
The architecture shown in Fig.2. is often referred to as a
“feed-forward network” in that the input data is fed in
one direction through the network and the network
operates synchronously.
The operation of the network is controlled through the
values of the weights on the neuron inputs and the
combination of the neuron outputs. If a vector of input
values, representing a point in a multi-dimensional space,
is presented at the inputs, the network will respond with
an appropriate output. In a sense, the network operates
like an active read-only memory. The output from the
memory system being determined not by addressing a
particular location in the memory but based on the
content of a memory location. Thus, neural networks are
often referred to as “auto-associative” or “contentaddressable”
memories. The feed-forward network is
only one of several possible neural interconnections and
is interesting in this application because it is able to
interpolate between examples it has been shown.
The process of defining the operation of a particular
network is known as “training”. In this phase, the
network is provided with a set of input vectors and the
output corresponding to each vector. The weights seen by
each neuron and their final combination into the output
are adjusted to minimize the error between the required
output and the one that is actually generated. This can
often be a difficult process depending on the form of the
neurons themselves.
The neurons can be based around several different
functions. Traditionally, neurons have used a paradigm
based on a summing amplifier. Each neuron provides the
weighted sum of its inputs which is then processed by a
thresholding function. The outputs of all the neurons are
then summed at the output. Each neuron operates over
the entire input space. In this case, a neuron is said to
“fire”, i.e. produce an output, when the weighted sum of
the inputs exceeds some threshold value. Thus the
evaluation of the weights requires the satisfaction of a set
of inequalities and the solution is non-unique. In
addition, in order to model sophisticated functions,
several layers of neurons may be needed and this can lead
to difficulties in the training operation.
- 250 - 15th IGTE Symposium 2012
For the work described in this paper, neurons based
on radial basis functions are used. In this case, the
output function of the network is described by:
Where ci represents the center of the area of interest of a
single neuron and x is the position of the current input
point in the parameter space being considered. Wi
represents the trained weight of neuron i.
The function, , is given by:
2
y
2
(3)
(
y) e
Where controls the domain of influence of the neuron.
The training process can thus determine the values of
W and for each neuron. Each neuron thus has a local
effect. The determination of the weights for a network
based on these functions can be expressed as an
optimization problem and the approach results in a
network that is easier to train.
Once trained, the network can reproduce the examples
that it was shown. However, there is also an emergent
property in that it is able to “generalize”, i.e. it can
generate outputs for input vectors it has not “seen”
before. The process of building a neural network can be
considered similar to fitting a surface in a multidimensional
space to a set of data points. In fact, there is
some commonality here with methods used in meshless
systems to evaluate field solutions [7]. The network can
function extremely quickly since the individual neuron
operations are computationally simple and it acts as a
look-up table for the unknown surface.
How well the network can match the input data and
corresponding outputs and how good the generalization
capabilities are depends on the network design. The
number of neurons can be considered to be similar to the
number of basis functions used to represent the surface.
If too few are used, the network will have a problem
training to the presented data with sufficient accuracy; if
too many are used, the network will have difficulty
generalizing and may generated large errors between the
known data points (a sort of high frequency oscillation
between the points). For this reason, the training set is
generally split into two pieces: the first is used to train
the network; the second, which has not been seen by the
network during training, is used to test the generalization
capabilities. This can then lead to a higher level process
where the network architecture, i.e. the number of
neurons used, is modified during the training process to
try to improve the generalization performance.
IV. THE PROPOSED SIZING PROCESS
From the above, the process of sizing an electromagnetic
device, in particular, an electrical machine, could be
implemented using a neural network. This is based on the
fact that the process of sizing is usually fairly limited, e.g.
(2)

for a specific torque requirement and architecture of
machine, determine the key diameter values and the air
gap size. If several designs of a specific class of machine
already exist, then a neural network can be trained on this
data and the generalization capability will allow it to
estimate the “size” of the new device. As stated above,
the goal of sizing is not to produce a perfect solution to
the design problem, rather it is to get within a reasonable
range in the design space of a possible solution. Thus the
system does not have to be highly accurate; an error of 10
or 20 percent in the performance of the proposed design
is probably acceptable since a conventional optimization
system can take the design from that point to completion.
The process of developing a neural network based
sizing system is shown in Fig. 3.
Figure 3. Sizing Network Development Process.
Since the goal of the sizing process is to develop an
approximate synthesized prototype, the database shown
in Fig.3 is used primarily to identify the major features
and parameter values. Hence, in fact, a database of
existing designs is (a) probably too limited and will not
cover the design space particularly well and (b) is
unlikely to be structured to provide the information
needed for sizing. Instead, a more controlled database can
be constructed by using existing analysis programs.
Using this approach, the database can be developed to
provide effective coverage of the design space. In
addition, certain parameters of the device, e.g. the
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number of poles, the maximum frequency, etc., can be
fixed and thus the network can be trained on a subset of
the machine design space. This lowers the dimensionality
of the space and hence, simplifies the network and
reduces the training time. It also simulates most existing
sizing processes where certain key parameters are set in
the specifications. In the event that these are not set, a
higher level network can be developed to first make the
choice of these key parameters before moving into the
sizing process.
V. A SIMPLE SIZING TEST
Given the issues facing machines designers due to the
costs of permanent magnets, a possible design scenario
for demonstrating the effectiveness of the neural network
approach is the replacement of a permanent magnet rotor
with that for an induction machine while keeping the
stator design constant. Thus the goal is to design a rotor
structure that can produce a specific torque-speed
performance. Note that, since the native torque-speed
curve for an induction machine is very different to that
for a permanent magnet design, the substitution is only
possible with the additional use of power electronics and
an external control system.
Conventional sizing approaches, which work well for
permanent magnet machines, are not very effective in
dealing with induction machine sizing and somewhat
more sophisticated models based around equivalent
circuits are needed. Thus the induction machine is an
ideal candidate for the process being described in this
paper.
The proposed system was tested on two different rotor
architectures. The first was a drag-cup servo rotor where
the rotor architecture is a conducting (copper) cylinder
around a permeable (iron) core. The design parameters
here are simple: just the thickness and radius of the
conducting cylinder. The second design involved a
squirrel-cage rotor which increased both the
electromagnetic complexity of the problem and the
number of design parameters.
A. The Drag-Cup Rotor
Fig.4 shows the basic design of the drag-cup rotor
being considered.
Figure 4. A Drag-Cup Rotor for an Induction Machine.

The typical torque-speed curve for this device is
shown in Fig.5.
Figure 5. Typical Torque-Speed Curve for a Drag-Cup
Rotor.
TABLE II Drag-Cup Rotor from Neural Net (Torque in
Nm)
Test#
TABLE I Drag-Cup Rotor Simulations
Test# Inner Outer Starting Maximum
Radius
(mm)
Desired
Start
Torque
Radius
(mm)
Desired
Max
Torque
Start
Torque
Torque
(Nm)
Max
Torque
Torque
(Nm)
1 25 27 6.12 17.93
2 25 28 6.74 25.56
3 25 29 6.82 32.84
4 25 30 6.65 39.25
Averag
eError
1 6.12 17.93 6.52 17.87 3.40%
2 6.74 25.56 7.02 24.85 3.44%
3 6.82 32.84 6.76 31.4 2.57%
4 6.65 39.25 6.58 36.31 4.23%
Table I shows a typical set of parameters for the drag-cup
rotor. A large range of values over each parameter was
used to generate the training and testing sets for the
neural network and the torque results computed using a
finite element code (MagNet [7]). The network was
constructed and trained using the MatLab Neural
Network toolbox. Once trained, the network predictions
were tested on a set of 50 samples. Each sample was also
evaluated using the finite element analysis and the results
were compared. The average error over the whole set was
4%. Table II shows some typical results.
Following on from these results, the complexity was
increased by considering a squirrel-cage rotor, i.e. a
structure consisting of a set of conducting bars in slots on
the rotor.
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B. The Squirrel-Cage Rotor.
A range of squirrel-cage rotor designs were
constructed to work with a 4 pole, 3 phase stator, shown
in Fig. 6. The variables in the rotor were the number of
bars, the size of the bars and the diameter of the rotor.
Fig. 7 shows the architecture of the squirrel-cage. A
number of combinations of these parameters were
produced and the torque-speed curves generated, again
using the MagNet software. Results were generated for a
range of values of each parameter resulting in 144
models in the database Table III shows the parameters
and the ranges used. The number of conduction bars was
set to an integer corresponding to the most commonly
used values for a 4 pole system. Each rotor geometry was
simulated for a range of frequencies from 0 to 60 Hz and
the starting and peak torques recorded, as well as the
torque-speed curve.
Table III Ranges of Parameters for Squirrel-
Cage Rotor
Parameter Minimum Maximum
Radiusof
Conduction
Bars(mm) 0.5 2
Radiusof
Rotor(mm) 28 36
Thenumberofconductionbarswassettoone
of15,20,30,35
Figure 6. The 4 Pole, 3 Phase Stator Design used with
the Sizing System.
The network was developed following the process
described in Fig. 3 and, once trained, was used to size a
rotor for a particular specification. The neural network
sizing estimates were then compared with an analysis of
the designed rotor in MagNet and an average error of
around 9% was generated over all the samples for a
network with 20 neurons. Thus it is reasonable to state

that the proposed system provided a “sizing” estimate for
the rotor design which was within the tolerance expected
at this point in the design process.
As a last test, the network architecture was varied, i.e.
to determine the effect of the number of neurons on the
error in prediction. The resulting errors are shown in Fig.
8 as a function of the number of neurons. The data in Fig
8 show the lack of approximation capability of the
network for low numbers of neurons and the inability to
generalize for high numbers. The ideal number for this
problem appeared to be around 20 neurons in the
network. It is not clear what caused the slight increase in
error for a 15 neuron network and this bears further
investigation.
Figure 7. Basic Conductor Layout for a Squirrel Cage
Rotor.
Figure 8. Error between the Finite Element and Neural
Network Solutions against the Number of Neurons in the
Network for Starting and Maximum Torques
VI. CONCLUSIONS
The paper has described an approach to developing an
initial prototype of an electromagnetic device based on a
limited number of specifications. This is conventionally
known as “sizing”. The use of a neural network together
with a pre-computed database of examples, developed
from a finite element analysis of a range of devices
covering the design space, has been shown to be effective
in developing an initial solution. The process of training
the network is similar to developing the response surface
- 253 - 15th IGTE Symposium 2012
for the particular machine examples. The neural network
acts as a form of surrogate but it is capable of providing a
solution to the inverse problem unlike the more
conventional usage of these techniques where the goal is
to develop an effective forward model. The accuracy of
the neural network is within the range of existing sizing
approaches and can probably be improved with a better
training database.
REFERENCES
[1] Dyck, D.N., Lowther, D.A., “Automated Design of Magnetic
Devices by Optimizing Material Distribution,” IEEE Transactions
on Magnetics, Vol.32, 3, 1996, pp. 1188-1193.
[2] Ouyang, J., Lowther, D.A., “A Hybrid Design Model for
Electromagnetic Devices,” IEEE Transactions on Magnetics, Vol.,
45, 3, 2009, pp. 1442-1445.
[3] Hawe, G,I,. Sykulski, J.K., “The Consideration of Surrogate
Model Accuracy in Single-Objective Electromagnetic Design
Optimization,” Proceedings of the 6 th International Conference on
Computational Electromagnetics, 2006, pp.1-2.
[4] Wang, L., Lowther, D.A., ”Reducing the Design Space of
Standard Electromagnetic Devices using Bayesian Response
Surfaces,” IEEE Transactions on Magnetics, Vol. 46, 2010, pp.
2884-2887.
[5] Hawe, G., Sykulski, J., “Considerations of Accuracy and
Uncertainty with Kriging Surrogate Models in Single-Objective
Electromagnetic Optimisation,” IET Proceedings on Science,
Education and Technology, Vol. 1, 2007, pp.37-47.
[6] Aleksander, I., Morton, H., “An Introduction to Neural
Computing,” London, UK, International Thomson Computer
Press, 1991.
[7] Benbouza, N, Louai, F.Z., Nait-Said, N. “Application of Mexhless
Petrov Galerkin (MLPG) Method in Electromagnetics using
Radial Basis Functions,” Proceedings of the 4 th IET Conference on
Power Electronics, Machines and Drives, 2008, pp. 650-655.
[8] MagNet Users Manual, Infolytica Corporation, 2012.

- 254 - 15th IGTE Symposium 2012
Exploring and Exploiting Parallelism in the Finite
Element Method on Multi-core Processors: an
Overview
Hussein Moghnieh and David A. Lowther
Department of Electrical and Computer Engineering, McGill University Montreal, Quebec, H3A 2A7, Canada
E-mail: hussein.moghnieh@mail.mcgill.ca
Abstract—Exploring parallelism requires identifying parts of a method or a kernel that can run concurrently. Exploiting
parallelism involves utilizing techniques aimed at devising an efficient parallel implementation on a given processor.
Different stages of the Finite Element Method have been found to require different approaches to explore and exploit their
parallelism. While data locality is essential to gain performance, many approaches to parallelism have been found to not
exhibit data locality by nature.
Index Terms—Finite Element Method, incomplete Cholesky preconditioner, matrix assembly, mesh generation, multi-core
processor, sparse matrix-vector multiplication.
structure (i.e. maximum number of non-zeros per row and
I. INTRODUCTION
the average number of non-zeros per row) has been
examined. The resulting matrices are shown in TABLE I.
The matrix naming convention used is an indicator of the
problem, element mesh size and the type of finite element
formulation applied. For instance, BDC-1-0.07, indicates
that the matrix is generated from the BDC problem, and a
first order (i.e. 1) nodal formulation has been applied on a
mesh where the maximum size of any triangular element
is 0.07mm, while BDC-0-1 denotes a matrix that was
assembled by applying an edge element formulation on a
mesh where the maximum size of any triangular element
is 1mm.
Further, an initial 3D mesh of a transformer (ET)
model has been refined multiple times and a first-order
nodal finite element formulation has been applied on each
of the refined meshes. The resulting matrices are denoted
by ET and are shown in TABLE I.
The introduction of the multi-core processor by IBM
(i.e. the POWER4) in 2001, and later by Intel and AMD,
has rekindled the interest in using parallel computing to
accelerate computations in an electromagnetic (EM) field
simulation software running on a desktop computer.
Since then, a considerable amount of research effort has
been invested in investigating the methods and kernels
executed in field simulation software; these include mesh
generation, matrix assembly, sparse matrix-vector
multiplication (SMVM) and iterative solver
preconditioning techniques such as incomplete LU
factorization (ILU). Despite having achieved a degree of
performance gain, several shortcomings have reduced the
effectiveness of those techniques in achieving the
ultimate performance goal of a field analysis software,
which is the reduction of the overall time to design a
device. These impediments include the problem size and
structure as well as the architecture of the multi-core
processor.
This paper intends to illustrate the degree of
parallelism which might be expected in each of the design
and analysis stages of a process based around the finite
element method (FEM), in addition to discussing several
issues and bottlenecks that arise while exploiting
parallelism on a multi-core processor. In particular, it is
intended to examine the gains due to parallelism on
realistic electromagnetic design examples, i.e. a 2D
brushless DC motor model and a 3D transformer model.
II. METHODOLOGY
An initial 2D mesh of a brushless DC (BDC) motor
model has been refined multiple times, by setting an
upper limit on the area of the elements in each refinement
step, in order to create a range of typical meshes and
mesh sizes. Subsequently, first order and second order
nodal formulations, in addition to an edge formulation
have been applied on each mesh and a matrix has been
assembled in each case. The effect of applying different
formulations and mesh sizes on the matrix size (i.e.
degrees of freedom and number of non-zeros) and matrix
TABLE I
MATRICES PROPERTIES
Matrix DOF NNZ Ave.
(Max)
nnz/row
CSR size
(MB)
BDC-1-0.5 38,084 259,188 7 (12) 3.2
BDC-1-0.07 1,194,044 8,334,798 7 (22) 100
BDC-1-0.04 3,152,216 22,000,128 7 (33) 264
BDC-2-3 48,031 407,733 9 (37) 5
BDC-2-0.07 4,787,651 40,664,669 9 (43) 484
BDC-2-0.04 12,660,592 107,560,044 9 (60) 1,280
BDC-0-1 55,772 278,168 5 (5) 3.4
BDC-0-0.07 3,492,389 17,931,763 5 (5) 219
BDC-0-0.04 9,492,389 47,437,511 5 (5) 579
ET-1-0.08 38,234 549,047 15 (36) 6.4
ET-1-0.04 409,531 5,999,230 15 (31) 70
ET-1-0.01 1,975,427 28,927,159 15 (39) 339
Subsequently, the parallel performance and bottlenecks
encountered in an efficient implementation of important
FEM kernels, particularly matrix assembly, sparse
matrix-vector multiplication, and preconditioning
techniques based on incomplete LU factorizations, are
investigated.

III. PARALLEL MATRIX ASSEMBLY
The process of matrix assembly is not considered to be
time consuming. It is an process, since it consists of
iterating once over all mesh elements. For each element,
two operations are performed. The first is to approximate
the solution of the field within each element which would
result in a dense matrix structure for each mesh
element e where u depends upon the formulation and the
number of unknowns in an element. The second operation
is to map each entry of the dense matrix to a global
matrix A. The latter step constitutes a significant portion
of the total assembly cost mainly because the global
matrix A is sparse. Inserting and updating entries in a
sparse matrix, even when its structure is a priori known,
is not trivial, such is the case when using compressed
sparse row (CSR).
In the case of matrix assembly in FEM, the maximum
number of non-zeros in any row can be roughly estimated
since it depends on the FEM formulation used. In such a
case, a more suitable choice of a sparse storage than the
CSR is to use the ELLPACK sparse storage scheme [1].
The ELLPACK sparse format stores a sparse matrix into two dense data structures
(ELL_values and ELL_column_ind) as shown in Figure 1.
ELL_values stores the values of non-zeros in each row in
a condensed form and pads the remaining spaces with
zeros. ELL_column_ind stores the column index of each
corresponding non-zero in the ELL_values and “-1” for
the padded non-zeros. The size W corresponds to the
maximum number of non-zeros per row. When the
number of non-zeros per row is less than W, zeros are
padded to fill the remaining locations.
Mutex
objects
Values per
row counter
00 01
2 00 01 0 0 0 1 1 1
11 14
2 11 14 0 0 1 4 1 1
20 22 25
3 20 22 25
0
0 2 5 1
32 33 35
3 32 33 35
0
2 3 5 1
44 45
2 44 45 0 0 4 5 1 1
50 51 52 55
4 50 51 52 55 0 1 2 5
nxn sparse matrix nxw values nxw
column indices
Figure 1: Synchronized ELLPACK sparse storage.
ELLPACK sparse format
ELL_values ELL_colum_ind
The performance of parallel matrix assembly using
atomic operations on multi-core processors has been
investigated. Mutual exclusion (mutex) objects from the
POSIX threads (Pthread) library were used to
synchronize access of multiple threads to a shared
resource, which in our case, is the matrix A. For this
purpose, an array of mutex objects was created where
each object corresponds to a row in the global matrix as
shown in Figure 1. Typically, in order for a thread to add
or modify entries on a row of the global matrix, it must
acquire a lock on the mutex object corresponding to that
row. After the thread finishes its modifications, it releases
the lock to make it available for other threads. For
example, a thread that is assembling an element of 3
unknowns (1, 2, 3) must aggregate the total of
entries in the global matrix. Each 3 of these entries is
added onto the same row of the global matrix; hence, a
- 255 - 15th IGTE Symposium 2012
total of 3 locks are required on 3 different mutex objects.
This is illustrated in Algorithm 1 (line 6).
Algorithm 1: Parallel matrix assembly using atomic operations.
Figure 2 shows the runtimes in seconds of the parallel
assembly of 3 matrices using a first-order nodal finite
element formulation on a quad-core Intel i7 processor.
The sequential runtimes are small (a few seconds) despite
the fact that these matrices are considered to be those of
realistic average size problems. The runtimes were
reduced by more than 50% relative to 1-thread execution
when the number of threads was 4. Notice the difference
in runtimes between sequential execution (no
synchronization) shown in horizontal lines and runtimes
of 1 thread. This difference highlights the cost of calling
the Pthread Application Programming Interface (API)
times. The overhead of calling a Pthread API
although it appears to be large in here, is not the main
concern in multi-threaded applications. Instead it is the
wait time that could incur when a thread is waiting for a
mutex object to be released by another thread. In matrix
assembly, this occurs when threads are simultaneously
processing mesh elements that share vertices and edges.
In the case of FEM, the possibility of threads waiting to
acquire a lock is small since the number of shared
vertices or edges is low; it is related to the average
number of non-zeros per row.
Figure 2: Parallel matrix assembly timings in seconds on an
Intel quad-core i7-860 processor.

A. Parallel Matrix Assembly Synchronization and
Cache Data Locality
The time it has taken to complete the matrix assembly
process in the previous experiments was very small (only
a few seconds), hence, it was not possible to accurately
measure the total time spent on synchronization (i.e.
calling the Pthread API and waiting to acquire a mutex
lock). Instead, Intel’s VTune Amplifier [2] was used to
count the number of execution cycles spent on
synchronization. In the case of matrix assembly using 1
thread, this number constituted around 9% of the total
cycles spent on matrix assembly (see Figure 3). This
number reflects only the time to call the Pthread API,
since there was no time or cycles wasted waiting to
acquire a lock (no other threads were competing to
acquire a lock). When using 4 threads, more cycles were
halted during synchronization, and in this case the
percentage of time wasted increased to 24% (see Figure
4).
91%
Matrix assembly execution cycles
Pthreads Lock / Unlock execution cycles
Figure 3: Execution cycles of matrix assembly using 1 thread.
76%
Figure 4: Execution cycles of matrix assembly using 4 threads.
IV. SPARSE MATRIX-VECTOR MULTIPLICATION
It is well established that matrix-vector multiplication
( ) exhibits a low floating-point operations
(FLOP) count to memory access ratio, regardless of
whether A is dense or sparse [3, 4]. This low ratio of
FLOP/BYTE makes SMVM a memory bandwidth
limited problem requiring the use of optimization
techniques which efficiently use the memory hierarchy
system (main memory, caches and registers).
The experiments conducted and presented in this
section aim at analyzing both the effectiveness and the
limitation of the commonly used SMVM optimization
techniques when applied on the matrix set described in
TABLE I.
Instead of using the ELLPACK storage described
above which could incur a large number of padded zeros
in matrices arising from a high order finite element
formulation, a variation of this storage, known as the
Hybrid (HYB) storage, is used instead. In this storage
scheme, some of the non-zeros are stored in a coordinate
list format (COO) so as to minimize the number of
padded zeros in the ELLPACK storage as illustrated in
9%
Matrix assembly execution cycles
Pthreads Lock / Unlock execution cycles
24%
- 256 - 15th IGTE Symposium 2012
Figure 5.
00 01
11 14
20 22 25
40
41
32 33 35
44 45
50 51 52 55
ELLPACK
sparse format
ELL_values ELL_colum_ind
00 01
11 14
0
0
0
1
1
4
1
1
Coordinate (COO) list
sparse format
20 22 25 0 2 5
COO_values 45 55
32 33 35 2 3 5 COO_row_ind 4 5
40 41 44 0 1 4 COO_col_ind 5 5
50 51 52 0 1 2
Fillin
Nonzeros stored in
COO
Figure 5: Hybrid (HYB) storage scheme. Some non-zeros are
stored in a coordinate list format (COO) in order to reduce the
total number of padded zeros.
In order to evaluate the magnitude of the impact of
accessing X on SMVM performance, the multiplication
by X[column] was replaced by X[i] (Algorithm 2, line 7).
Although this multiplication yielded an incorrect result,
the aim was to show an upper bound on performance gain
in cache blocking (i.e. no cache misses on X).
Algorithm 2: Modified SMVM to eliminate the effect of cache
misses on .
Figure 6: BDC-1: SMVM performance when using cache
blocking on .
Eliminating the cache misses of has increased the
performance of SMVM significantly (as anticipated)
when the matrix was unstructured (i.e. BDC-1) as shown
in Figure 6 (Natural ordering). To further validate the
results, the set of matrices in TABLE I (BDC-1) were
ordered to reduced their bandwidth using the Reverse

Cuthill-McKee (RCM) technique [5]. When the matrices
were ordered using RCM, the performance of SMVM
using cache blocking was close to the performance of
SMVM without cache blocking (Figure 6), since cache
misses were reduced due to the ordered access pattern on
.
A. Loop Setup Overhead
One of the factors that has been argued to be contributing
to reducing the performance of SMVM is the low number
of non-zeros per row[6]. For each row of the matrix A,
the inner loop of the SMVM code, whether using the
CSR storage (as shown in line 5 of Algorithm 2) or using
the HYB storage, iterates over the row's non-zeros and
multiplies them by the corresponding entries in . When
only a few non-zeros are present, the inner loop setup
overhead time would dominate the calculation time and
would not be able to be amortized over the short
calculation time of a few non-zeros. Since the set of FEM
matrices used in this work falls within this category (i.e.
low per row) a test examining the degradation of
the SMVM performance due to the inner loop setup
overhead has been carried out by replacing the inner loop
of SMVM with a set of instructions which explicitly
multiply each element of by its corresponding element
in ; this technique is often referred to as “loop
unrolling”. “Loop unrolling” has been made possible by
the use of the ELLPACK (or Hybrid) sparse format since
the number of non-zeros per row is fixed, hence the
number of times an inner loop executes its inner
instruction is fixed. In such a case, the inner loop can be
eliminated and the instruction within the inner loop can
be replaced by explicitly writing the set of instructions
that would have been executed by the inner loop.
Algorithm 3 illustrates a sparse matrix-vector
multiplication using the ELLPACK storage. Assuming
that the width of the ELLPACK storage is 7, the inner
loop which multiplies the non-zeros of a row by the
corresponding locations in is replaced by seven
instructions. The effect of this technique on the
performance of SMVM when applied on BDC-1 matrix
test set is shown in Figure 7. It can be seen that while loop
unrolling did increase SMVM performance, it was not as
significant as the performance gain obtained from
eliminating cache misses on .
Algorithm 3: SMVM loop unrolling using NVIDIA's Hybrid
sparse storage.
- 257 - 15th IGTE Symposium 2012
Figure 7: BDC-1: Loop unrolling and cache blocking (singleprecision
floating-point operations).
B. SMVM memory bandwidth
Figure 8 shows the memory bandwidth when executing
SMVM using different optimization techniques. The
sustainable memory bandwidth obtained from executing
the STREAM benchmark [7] on an Intel i7-860 processor
is also shown on the same figure. The widely used
STREAM benchmark serves as an indicator of the
realistic performance of the memory subsystem of a
particular processer. In this benchmark, a set of kernels is
applied on a dense data structure chosen to be larger than
the available cache of a particular processor.
Figure 8: BDC-1: SMVM sustainable memory bandwidth
(MB/s) on Intel i7-860 processor.
In general, a naïve implementation of SMVM (i.e. no
optimization) would work well below 50% of the
STREAM benchmark sustained memory bandwidth,
while an optimized SMVM (with cache blocking)
attained 70% of the STREAM benchmarks. The
implication of these results highlights the effect of using
sparse storage, which introduces additional memory
fetches due to indirect addressing which also prevents
efficient memory pre-fetching by the processor. A similar
observation has been found when running the same
experiments on an older generation of quad-core

processor; AMD’s dual-socket, dual-core Opteron 2214
processor.
C. Parallel SMVM
This section compares the sequential and parallel
performance of SMVM kernels when applied to matrices
obtained from the set described in TABLE I and a
miscellaneous matrix test set obtained from “the
University of Florida Sparse Matrix Collection” [8]
shown in TABLE II. The latter set has been widely used in
the past few years by researchers to evaluate the
performance of SMVM algorithms. The results of our
evaluation are shown in Figure 9.
TABLE II
MISCELLANEOUS MATRIX TEST SET
Matrix DOF NNZ Ave. (Max)
nnz/row
CSR
size
(MB)
Protein 36,417 4,344,765 120 (204) 50
Sphere 83,334 6,010,480 73 (81) 69
Cant. 62,451 4,007,383 65 (78) 46
Tunnel 217,918 11,524,432 53 (180) 133
CFD 46,835 2,374,001 50 (145) 27
Ship. 140,874 7,813,404 26 (68) 42
Econ. 206,500 1,273,389 7 (74) 16
Epidem. 525,825 2,100,225 4 (4) 26
Circuit 170,998 958,936 6 (353) 12
The following observations were concluded from the
results shown in Figure 9:
The sequential performance of SMVM kernels when
the size of a matrix fits in the available processor
cache is significantly higher than when the matrix
does not fit in the cache (e.g. BDC-1-0.5 and ET-0-
0.5).
Figure 9: Parallel SMVM using HYBRID storage (doubleprecision
floating-point operations)
- 258 - 15th IGTE Symposium 2012
Matrices that have a high percentage number of nonzeros
per row attained higher GFLOPS than matrices
with short row lengths. This is not due to the overhead
caused by the inner-loop of SMVM (as demonstrated
in section III.A), but to the ratio of the DOF and
NNZ. In general, matrices arising in FEM have high
ratios of DOF over NNZ, which explains the low
performance relative to other matrices. This explains
also why matrices obtained from 3D first-order finite
element analysis attained higher GFLOPS than
matrices obtained from 2D analysis.
The performance of parallel SMVM is affected by the
distribution of non-zeros in a row. Matrices arising
from FEM have a balanced distribution of the number
of non-zeros per row, leading to better thread
utilization and subsequently to higher GFLOPS.
V. PRECONDITIONING TECHNIQUES:INCOMPLETE
CHOLESKY AND INCOMPLETE CHOLESKY WITH FILL-INS.
There are two techniques to solve a system of linear
equations where is the coefficient matrix and
is the right hand side vector. The first is to use direct
solver methods and the second is to use iterative methods.
The direct solver methods rely on decomposing the
coefficient matrix, , into upper and lower triangular
matrices and, where . This is a robust
method. However, it is not useful for large systems, since
the triangular matrices L and U lose their sparsity, as zero
entries in the coefficient matrix turn into non-zero
entries in and . Those new entries are referred to as
fill-ins.
A less robust technique is based on iterative
approaches, such as the conjugate gradient method (CG).
This method requires a large number of iterations over
the system of linear equations to reach the solution. The
number of iterations depends upon the condition number
of the matrix in . This condition number can be
reduced (i.e. leading to less CG iterations) if a
preconditioner that is based on the incomplete
factorization of is applied to the CG method[9].
Incomplete factorization derives its name from the
direct method discussed above. It uses the same
elimination algorithm to decompose the matrix into an
and , which are an approximation of and,
obtained by dropping some fill-in entries. One of the
dropping strategies during ILU factorization is to drop all
fill-ins so that the sparsity of and matches that of the
original matrix A. This dropping rule gives rise to an
ILU(0) or IC(0) (incomplete Cholesky in the case of
symmetric matrices) preconditioner [10], where the zero
denotes that no fill-ins are allowed. Incomplete Cholesky
with no fill-ins has been the preconditioner of choice on a
desktop computer mainly due to its ability to reduce the
number of iterations of a PCG while being inexpensive to
produce and to compute on a desktop computer. The
structures of the factors and are a priori known,
making it easy to pre-allocate the storage requirement,
without the need for symbolic factorization. An efficient
implementation would be to duplicate the lower part of A
and then perform an in-place factorization by going in an

ordered manner over the entries of each row. A very
efficient implementation is found in the SparseLib++
library [11]. TABLE III shows the execution times of
creating an IC(0) preconditioner.
Matrix Degrees of
freedom
TABLE III
INCOMPLETE CHOLESKY PERFORMANCE
Upper
triangle
NNZ
CSR size
(MB)
IC(0) time
(sec.)
BDC-1-0.5 38,084 147,636 1.9 0.0275
BDC-1-0.1 632,883 2,521,428 31.3 0.4987
BDC-1-0.04 3,152,216 12,576,171 155 2.594
ET-0.08 38,324 293,643 3.5 0.1359
ET-0.04 409,531 3,204,372 38.2 1.554
ET-0.01R 2,666,039 21,100,983 252 10.3244
In order to improve the convergence rate of PCG
beyond that provided by using the IC(0) preconditioner,
much research has focused on extending the idea of the
incomplete Cholesky preconditioner by allowing fill-ins
to occur. There are two heuristics used to control the
amount of fill-in. The first is based on a drop tolerance
criterion, known as the Incomplete LU Threshold (ILUT)
through which entries are dropped if their values are
below a preset threshold. The second is based on the level
of fill-in known as ILU, where symbolic factorization,
using graph theory, is carried out to identify the locations
of the fill-ins and their level in the graph. The fill-in
entries that exceed a given level are dropped. Matrix
elements are assigned a level 0, hence IC(0) discards all
fill-ins and the resulting factorized matrix has the same
sparsity pattern as the original matrix. One way to
calculate the level of a fill-in is to use the sum rule as
shown in (1). This rule gives rise to a symbolic
factorization algorithm described by Hysom [12] that is
amenable to parallelization. The sparsity of each row in
the final preconditioner can be evaluated independently
from the other rows. Figure 10 demonstrates the
scalability of this algorithm. Despite that, the runtime of
the symbolic factorization is considered to be a
bottleneck in our case mainly due to the large number of
fill-ins that incurred in the final ILU preconditioners
(where =1, 2 or 3) as shown in TABLE IV.
level(i, j) min {level(i, k) level(k, j) 1} (1)
1hmin{i, j}
- 259 - 15th IGTE Symposium 2012
Figure 10: Execution times of parallel symbolic factorization of
BDC-1-0.1 where . The results demonstrate that the multithreaded
implementation of Hysom’s algorithm is highly
scalable.
Matrix IC(0)
TABLE IV
FILL-INS
ILU(1) ILU(2) ILU(3)
BDC-1-0.5 148,636 225,244 321,101 429,566
(51%) (116%) (189%)
BDC-1-0.1 2,521,428 3,848,266 5,544,448 7,380,993
(52%) (120%) (193%)
ET-0.08 293,643 700,914 1,368,930 2,555,737
(139%) (366%) (770%)
ET-0.04 3,204,372 7,927,979 15,963,746 30,758,154
(147%) (398%) (860%)
VI. PRECONDITIONER BACKWARD-FORWARD
SUBSTITUTION
The next step is to investigate the degree of parallelism
(i.e. the number of operations that can be executed
simultaneously) that can be attained when solving a
preconditioner (by backward and forward substitution)
within a PCG iteration obtained from the matrix BDC-1-
0.5 (i.e. a 2D problem) and the matrix ET-0.08 (i.e. a 3D
problem). A histogram will be used to depict the
maximum degree of parallelism and the number of steps
required to solve each of the preconditioners. The x-axis
shows the number of steps required to solve a matrix, and
the y-axis of the histogram shows the number of rows
that can be solved simultaneously at a given step. Figure
11 and Figure 12 show the rows dependency histograms of
ILU(1) and ILU(3) of the matrix BDC-1-0.5 respectively.
The maximum degree of parallelism of ILU(1) was 1,151
and the number of steps required to solve the
preconditioner was 196. On the other hand, the ILU(3)
preconditioner of the same problem had a maximum
degree of parallelism equal to 453 and 399 steps were
required to solve it. The more fill-ins that existed in a
preconditioner, the less parallelism could be exploited.
Solving a preconditioner obtained from a 3D problem
is less amenable to parallelism than that obtained from a
2D problem. For instance, an ILU(1) preconditioner
obtained from ET-0.08 (3D electric transformer problem)
can be solved in 12,559 steps where the maximum
number of rows that could be solved simultaneously is
only 16 (see TABLE V) and an ILU(3) preconditioner of
the same problem can be solved in 24,125 steps where the

maximum attainable degree of parallelism is only 16 (see
TABLE VI). A 2D problem that has the same number of
degrees of freedom as ET-0.08 (i.e. BDC-1-0.5) was
more amenable to parallelism.
1200
1000
800
600
400
200
0
0 20 40 60 80 100 120 140 160 180 200
Figure 11: Degree of parallelism (y-axis) attained when solving
ILU(1) of BDC-1-0.5. The x-axis represents the number of
sequential steps to finish the solve stage.
500
450
400
350
300
250
200
150
100
50
0
0 50 100 150 200 250 300 350 400
Figure 12: Degree of parallelism (y-axis) when solving ILU(3)
of BDC-1-0.5. The x-axis represents the number of sequential
steps to finish the solve stage.
TABLE V
ILU(1) OF ET-0.08
Preconditioner NNZ Max. Solving
level and ordering
degree of
parallelism
steps
ILU(1)-Natrual 700,914 69 12,559
ILU(1)-AMD 673,511 207 569
ILU(1)-RCM 585,646 36 3,491
TABLE VI
ILU(3) OF ET-0.08
Preconditioner NNZ Max. Solving
level and ordering
degree of
parallelism
steps
ILU(3)-Natrual 2,555,737 16 24,125
ILU(3)-AMD 1,937,345 119 1,690
ILU(3)-RCM 1,984,093 12 10,548
Since the preconditioner obtained from the 3D
transformer problem exhibited a low degree of
parallelism, approximate minimum degree (AMD) [13]
and Reverse Cuthill–McKee (RCM) orderings were first
applied on the ET-0.08 matrix before generating ILU(1)
and ILU(3) preconditioners. Although, it has been
established in the literature that orderings to reduce fillins
or increase parallelism (RCM and AMD) degrade the
quality of the preconditioner and lead to more PCG
- 260 - 15th IGTE Symposium 2012
iterations than when using Natural ordering [14], [15], the
aim of this experiment was to only focus on the
implication of ordering in terms of reducing the overall
solver time.
The number of non-zeros in the upper triangle
preconditioner, the maximum degree of parallelism and
the solving steps of both preconditioners ILU(1) and
ILU(3) using different orderings are summarized in
TABLE V and TABLE VI respectively. AMD ordering
resulted in a relatively higher parallelizable
preconditioner solver than Natural and RCM orderings.
On the other hand, RCM exhibited a similar degree of
parallelism to that of the Natural ordering but required
less solve steps. The reason being that RCM reduces the
bandwidth of the matrix and balances the distribution of
non-zeros between rows. This implies that there is a
balance in the degree of parallelism among steps, which
will translate into balanced threads utilization.
VII. CONCLUSION
A. Matrix Structure
Given the dependency of the sparse storage upon the
problem structure, it is important to devise matrix test
sets that are relevant to the problem domain (i.e. low
frequency electromagnetic analysis using the Finite
Element Method). One of the advantages of matrices
generated in FEM is the absence of a large discrepancy in
the number of non-zeros between rows. This enables the
use of a sparse storage technique such as ELLPACK that
pre-allocate memory to store the coefficient matrix.
B. 2D vs 3D Analysis
Matrices generated from 2D finite element analysis are
less dense that those generated from 3D problems.
SMVM attained more GFLOPS in the case of a denser
matrix (i.e. first-order finite element formulation of a 3D
problem). However, ILU preconditioner generated in the
case of a 3D problem was less amenable to parallelism
than a preconditioner of a 2D problem.
C. Matrix Ordering
Overall, although parallelism can be explored and
exploited in most of the examined FEM kernels, the main
bottleneck remains the solver part of the FEM process.
There are many sub-kernels that are executed within a
large number of loops. This places a stringent
requirement on sparse data structures as there is no gain if
these structures change in between sub-kernels. Further,
in each sub-kernel within the solver, there is a large
number of simple operations to be executed. The number
of operations is related to the degrees of freedom of the
matrix and the operations’ complexity is related to the
number of non-zeros per row. These simple operations
are memory bandwidth limited requiring that each
operation be optimized in terms of memory access.
Hence, single thread optimization remains the most
essential part of the solver’s optimization.
Reverse Cuthill-McKee (RCM) ordering has been
found to be beneficial assuming that it will not degrade
the performance of PCG. It balances threads utilization

when solving a preconditioner and also in enhances cache
performance in SMVM.
REFERENCES
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User's Guide," CNA-150, Center for Numerical
Analysis,University of Texas,Austin,Texas
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[2] Intel, "Intel VTune Amplifier XE 2011," 2011 ed: Intel
Corporation, 2011.
[3] S. Toledo, "Improving the memory-system performance of sparsematrix
vector multiplication," IBM Journal of Research and
Development, vol. 41, pp. 711-725, 1997.
[4] R. Vuduc, "Automatic performance tuning of sparse matrix
kernels," PhD thesis, University of California,Berkley, 2003.
[5] E. Cuthill and J. McKee, "Reducing the bandwidth of sparse
symmetric matrices," presented at the Proceedings of the 1969 24th
national conference, 1969.
[6] S. Williams, L. Oliker, R. Vuduc, J. Shalf, K. Yelick, and J.
Demmel, "Optimization of sparse matrix-vector multiplication on
emerging multicore platforms," Parallel Computing, vol. 35, pp.
178-194, 2009.
[7] J. D. McCalpin, "STREAM: Sustainable memory bandwidth in
high performance computers," STREAM: Sustainable Memory
Bandwidth in High Performance Computers, 1991.
[8] T. A. Davis and Y. Hu, "The University of Florida sparse matrix
collection," ACM Transactions on Mathematical Software (TOMS),
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[9] H. A. Van der Vorst, "Preconditioning by incomplete
decompositions," Ph.D. Thesis, University of Utrecht, 1982.
[10] J. Meijerink and V. DER VORST, "An iterative solution method
for linear systems of which the coefficient matrix is a symmetric
M-matrix," Mathematics of Computation, vol. 31, pp. 148-162,
1977.
[11] R. Pozo and K. Remington, "Sparselib++ v. 1. 5 sparse matrix class
library. reference guide," NASA, 1996.
[12] D. Hysom and A. Pothen, "Level-based incomplete LU
factorization: Graph model and algorithms," Preprint UCRL-JC-
150789, US Department of Energy, p. 17, 2002.
[13] P. R. Amestoy, T. A. Davis, and I. S. Duff, "An approximate
minimum degree ordering algorithm," SIAM Journal on Matrix
Analysis and Applications, vol. 17, pp. 886-905, 1996.
[14] I. S. Duff and G. A. Meurant, "The effect of ordering on
preconditioned conjugate gradients," BIT Numerical Mathematics,
vol. 29, pp. 635-657, 1989.
[15] S. Doi and T. Washio, "Ordering strategies and related techniques
to overcome the trade-off between parallelism and convergence in
incomplete factorizations," Parallel Computing, vol. 25, pp. 1995-
2014, 1999.
- 261 - 15th IGTE Symposium 2012

- 262 - 15th IGTE Symposium 2012
Diagnosis of real cracks from the three spatial
components of the eddy current testing signals
M. Rebican∗ , L. Janousek † ,M.Smetana † , T. Strapacova † ,A.Duca∗and G. Preda∗ ∗ University Politehnica of Bucharest, Spl. Independentei 313, Bucharest 060042, Romania
† Faculty of Electrical Engineering, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakia
E-mail: mihai.rebican@upb.ro
Abstract—This paper presents a novel approach for diagnosis of real cracks from two-dimensional eddy current testing
signals by means of a stochastic method, such as tabu search. A new testing probe driving uniformly distributed eddy
currents is employed for the inspection. Three spatial components of the perturbation field due to partially conductive
cracks are sensed as the response signals in order to enhance information level of the inspection. The signals are simulated
by a fast forward FEM-BEM solver using a database. Two crack models are proposed for the inversion: a crack with cuboid
shape and a crack with more complex shape. In the both cases, the cracks have uniform conductivity. The length, depth,
width and conductivity of the crack are unknown in the inversion process. Numerical results of the 3D reconstruction of
partially conductive cracks from simulated 2D signals with added noise are presented and discussed.
Index Terms—partially conductive cracks, diagnosis, eddy current testing, tabu search.
I. INTRODUCTION
Eddy current testing (ECT) is one of the most common
electromagnetic methods employed in non-destructive
evaluation of conductive materials. The principle of
ECT underlies in the interaction of induced eddy currents
with a structure of an examined conductive body
based on the electromagnetic induction phenomena. The
method is widely applied in various fields accounting for
measurements of material thickness, proximity measurements,
corrosion evaluation, sorting of materials based on
the electromagnetic properties. However, the most wide
spread area of its application in present is the detection
and possible diagnosis of discontinuities.
Real cracks, such as stress corrosion cracks (SCC),
usually appear in steam generator (SG) tubes of pressurized
water reactor (PWR) of nuclear power plants.
Recently, quite satisfactory results are reported by several
groups for automated evaluation of artificial slits [1] and
even for several parallel notches [2] using eddy current
testing (ECT). However, evaluation of real cracks from
ECT signals remains still very difficult.
In the case of artificial EDM notches, the width is
usually considered fixed in the inversion process of ECT
signals. However, for cracks with non-zero conductivity
the width affects the signal and it has to be considered
unknown during reconstruction [3]. It means that the
additional variable should be taken into account for
evaluation of a detected SCC what considerably increases
ill-posedness of the inverse problem [4]. Thus, many unsatisfactory
results are reported when the automated procedures
originally developed for non-conductive cracks
are employed in the evaluation of SCCs. It is stated
that one of the possible reasons is lack of sufficient
information [1].
Several studies of the authors focused on enhancing
information level of eddy current testing signals and on
decreasing uncertainty in evaluation [5], [6]. Promising
results create new challenges concerning development of
automatic procedures for diagnosis of real cracks.
In a previous work [2], the authors developed an
algorithm for reconstruction of multiple artificial slits
from ECT signals by means of a stochastic optimization
methods, such as tabu search. The reconstruction of multiple
cracks was a 3D one. Therefore, the scheme is also
appropriate for reconstruction of a partially conductive
crack, when the width is not considered fixed.
The paper proposes a novel approach for the threedimensional
reconstruction of partially conductive cracks
from simulated two-dimensional ECT signals, consisting
of all the three spatial components of the perturbation
field. Two crack models are proposed for the inversion.
The first one has a cuboid shape and the other reflects
a more complex geometry. Both the crack models consider
uniform distribution of the partial conductivity. The
length, depth, width and conductivity of the cracks are
unknown in the inversion process of the signals. The
validity of the approach is proved using perturbed ECT
signals by added noise in the inversion process.
II. EDDY CURRENT TESTING PROBLEM DEFINITION
A plate specimen having the electromagnetic parameters
of a stainless steel SUS316L is inspected in this
study. The specimen has a thickness of t =10mm, a
conductivity of σ =1.35 MS/m and a relative permeability
of μr =1.
Figure 1 shows the configuration of the plate (region
Ω0) with a single surface breaking crack (shadow zone)
located inside the region Ω1. The crack region Ω1 (22 ×
2 × 10 mm3 ) is uniformly divided into a grid composed
from nx × ny × nz (11×5×10) cells defining a possible
crack geometry. The dimensions of each cell are 2.0 mm
in length, 0.4 mm in width, and 1.0 mm in depth.
A new eddy-current probe proposed by the authors
is employed for the near-side inspection of the plate
[7]. It consists of two circular exciting coils positioned
apart from each other and oriented normally regarding

2
1
1
nz
Crack
n y
Ω 1
Probe
1 2 nx
y
x
Scanning
Fig. 1. Configuration of the plate specimen with a crack.
the plate surface. The circular coils are connected in
series but magnetically opposite to induce uniformly
distributed eddy currents in the plate. The exciting coils
are supplied from a harmonic source with a frequency
of 5kHz and the current density 1A/mm 2 . A detection
system of the probe is composed of three small circular
coils oriented along three axes perpendicularly to each
other [5]. The detection system is located in the center
between the exciting coils to gain high sensitivity as
the direct coupling between the exciting coils and the
detectors is minimal at this position.
Figure 2 shows the configuration of the new probe.
Dimensions of the detecting coils are as follows: an inner
diameter of 1.2mm, an outer diameter of 3.2mm and a
winding height of 0.8mm.
Two-dimensional scanning, so called C-scan, is performed
over the cracked surface with a lift-off of 1mm.
The real and imaginary parts of the induced voltages in
all three detecting coils corresponding to three spatial
components of the perturbation electromagnetic field are
sensed and recorded during the inspection.
III. PARTIALLY CONDUCTIVE CRACK MODELS
Partially conductive cracks with a uniform conductivity
smaller than the conductivity of the base material are
considered in this paper. Two crack models are proposed
for the partially conductive cracks.
In the first model shown in Figure 3, the crack has a
cuboid shape. The crack parameter vector c consists of
z
23
x
plate
35
14
Fig. 2. ECT probe configuration.
exciting
coils
10
detectors
Ω 0
- 263 - 15th IGTE Symposium 2012
1
1
2
n
z
2
ny
1 2 n
x
Fig. 3. Crack region division - cuboid shape of the crack (model 1).
6integers,c =[ix1,ix2,iy1,iy2,iz,s], whereix1 and
ix2 are the indices of the first and last cells of the crack
along the length of crack, iy1 and iy2 are the indices of
the first and last cells of the crack along the width of
crack, iz is the number of cells of the crack along the
depth of crack, and σc = s%σ (σc - the conductivity of
crack, σ - the conductivity of base material).
In Figure 3, for a uniform grid with 13×5×10 cells, the
parameter vector is c =[6, 13, 1, 3, 4, 20]. Thus, 8×3×4
cells form the crack, and the crack conductivity is σc =
20%σ.
The second crack model shown in Figure 4 adopts
a more complex shape. The crack depth is considered
as variable along the crack length. The crack parameter
vector c consists of nx +3 integers, c =
[iz1,iz2,...,iznx,iy1,iy2,s], whereizk, k = 1,nx is
the number of cells of the crack along the depth of crack,
iy1 and iy2 are the indices of the first and last cells of
the crack along the width of crack, and σc = s%σ (σc
- the conductivity of crack, σ - the conductivity of base
material).
In Figure 4, for a uniform grid with 13 × 5 × 10
cells, the parameter vector contains 16 integers, as c =
[0, 0, 0, 0, 0, 8, 4, 1, 2, 5, 3, 6, 4, 1, 3, 30]. Thus, (8+4+1+
2+5+3+6+4)× 3 cells form the crack, and the crack
conductivity is σc = 30%σ.
In the both models, the cracks have the same orienta-
1
1
2
n
z
2
ny
1 2 n
x
Fig. 4. Crack region division - complex shape of the crack (model 2).

tion. The width of crack can have the values: 0.4, 0.8,
1.2, 1.6, 2 mm.
IV. DIAGNOSIS OF PARTIALLY CONDUCTIVE CRACKS
The fast-forward FEM-BEM analysis solver using
database [8], [9] is adopted here for the ECT signals simulation.
Actually, a version of the algorithm of database
upgraded by the authors in previous works [2], [10],
for the computation of the ECT signals due to multiple
cracks is used in this paper. The database is designed for
a three-dimensional defect region, and not as usually for a
two-dimensional one where the crack width is considered
fixed. Thus, the ECT response signals can be simulated
also for partially conductive cracks with variable width
using the same database generated in advance.
The authors have already developed an algorithm for
the reconstruction of multiple cracks from ECT signals
by means of a stochastic optimization method, such as
tabu search [2]. The reconstruction of multiple cracks
validated by experimental data was a 3D one. Therefore,
the scheme is also appropriate for the reconstruction
of a partially conductive crack, when the width is not
considered constant. It is well known that the width
significantly affects the signal for cracks of non-zero
conductivity [3].
Tabu search is employed for the three-dimensional
diagnosis of a partially conductive crack [2]. The error
function ε to be minimized is defined as:
ε(c) =
j=X,Y,Z
N
i=1
|ΔVij(c) − ΔV m
ij |2
, (1)
N
i=1
|ΔV m
ij |2
where c is the crack parameter vector of the crack,
ΔVij(c) and ΔV m
ij are the simulated (reconstructed) and
true (measured) induced pick-up voltages of the coils
(ECT signal) for each spatial component (X, Y and Z
according to the coordinate system shown in Figures 1
and 2) at the i-th scanning point respectively, and N is
the number of scanning points.
Figures 5-7 show the simulated ECT signal for each
spatial component (X, Y, Z) caused by a partially conductive
crack with a cuboid shape, which has the parameters:
lc =6mm, wc =0.8mm, dc =4mm, σc =5%of σ.
In this paper, the simulated signals are affected by
added noise before the inversion process in order to prove
the validity and robustness of the proposed approach. The
perturbed signal is computed as:
(ΔV m
i )ns =ΔV m
i (1 ± ns%), (2)
where ΔV m
i and (ΔV m
i )ns are the initial and perturbed
true signals at the i-th scanning point respectively, ns is
a random value of an imposed maximum level, NOISE.
Figure 8 shows the perturbed ECT signal for Z component
when noise of maximum level 40% is added to
the simulated signal shown in Figure 7.
- 264 - 15th IGTE Symposium 2012
Absolute voltage [mV]
0.8
0.6
0.4
0.2
0
1
1.8
1.6
1.4
1.2
15
10
5
-20 -15 0
-10 -5 0
-5 y [mm]
5
x [mm] 10 -10
15 20-15
Fig. 5. X component of the simulated ECT signal.
Absolute voltage [mV]
0.8
0.6
0.4
0.2
0
1
1.8
1.6
1.4
1.2
15
10
5
-20 -15 0
-10 -5 0
-5 y [mm]
5
x [mm] 10 -10
15 20-15
Fig. 6. Y component of the simulated ECT signal.
V. NUMERICAL RESULTS AND DISCUSSION
The numerical simulations of the cracks reconstruction
are performed using an ordinary PC: Intel Core 2 Quad
2.4GHz, 3GB RAM.
In Table I the numerical results of the reconstruction
are presented, when a partially conductive crack is
modeled as a cuboid shape (crack model 1, Figure 3).
The column denoted ”Real” gives the true dimensions
(lc × wc × dc) and partial conductivity (σc in % of σ)
of the crack. The results of the diagnosis are provided
in the column labelled as ”Reconstructed” for various
maximum levels of the noise added to simulated ECT
signals (2). The error function ε(c) (1) of the diagnosis
are reported, too.
The time required for reconstruction of one crack is
approximately 90-120 minutes. When there is no noise

Absolute voltage [mV]
0.8
0.6
0.4
0.2
0
1
1.8
1.6
1.4
1.2
15
10
5
-20 -15 0
-10 -5 0
-5 y [mm]
5
x [mm] 10 -10
15 20-15
Fig. 7. Z component of the simulated ECT signal.
Absolute voltage [mV]
0.8
0.6
0.4
0.2
0
1
1.8
1.6
1.4
1.2
15
10
5
-20 -15 0
-10 -5 0
-5 y [mm]
5
x [mm] 10 -10
15 20-15
Fig. 8. Z component of the perturbed ECT signal by added noise of
maximum level 40%.
added to signal (shown in Figures 5-7), the reconstructed
crack is the same with the real crack (the error function
ε =0). In the case of diagnosis from signals perturbed
by noise (maximum level NOISE =10, 20, 30, 40%),
TABLE I
RESULTS OF THE PARTIALLY CONDUCTIVE CRACK
RECONSTRUCTION FOR THE CRACK MODEL 1
Crack Real Reconstructed
NOISE (%) 0 10 20 30 40 40
lc [mm] 6.0 6.0 6.0 6.0 6.0 6.0 6.0
wc [mm] 0.8 0.8 0.8 0.8 0.8 0.8 1.2
dc [mm] 4.0 4.0 4.0 4.0 4.0 4.0 4.0
σc[%] 5 5 5 5 4 4 9
ε · 10 −3 - 0 11 21 32 42 45
- 265 - 15th IGTE Symposium 2012
the crack is exactly reconstructed even if the maximum
level of noise is high (30, 40%). Starting from another
initial population of tabu search, and for a perturbed
signal by added noise of highest level, NOISE =40%
(the Z component is shown in Figure 8), the result is
slightly different (the last column in Table I): the length
and depth are equal to the true values; the width and
partial conductivity are estimated not exactly but with
good precision. However, the last two parameters are not
very important from structural integrity point of view.
The results clearly show that the crack parameters are
estimated quite precisely from the noisy ECT signals
using the proposed approach.
Figures 9 and 10 show the results of three-dimensional
diagnosis of the partially conductive crack described in
Table I (column ”Real”) from 2D ECT signals without
and with added noise of maximum level 20%, respectively,
when the complex crack model (Figure 4) is
employed for the inversion. The inversion procedure
takes around 5-7 hours.
In the case of the reconstruction from signal without
noise, the crack is precisely localized and also its length
width
depth
real reconstructed
σ c=5%σ
σ c=3%σ,
ε=0.004
Fig. 9. Reconstruction of a conductive crack from signal without noise.
width
depth
real reconstructed
σ c=5%σ
σ c=4%σ,
ε=0.027
Fig. 10. Reconstruction of a conductive crack from perturbed signal
by added noise of maximum level 20%.

width
depth
real reconstructed
σ c=8%σ σ c=6%σ,
ε=0.024
Fig. 11. Reconstruction of an elliptical conductive crack.
and width are exactly estimated. The depth profile does
not perfectly copy the true one. However, the maximum
depth is accurately assessed. But, for the reconstruction
from signal with added noise of maximum level of 20%,
the width is smaller with a minimum value of 0.4 mm
than real width.
A crack with elliptical profile is also reconstructed
from signal without noise. The crack opening has a value
of wc =0.4 mm, its surface length is lc =14mm, the
maximum depth is dc = 4mm and the crack partial
conductivity is adjusted to σc =8%of the base material
conductivity σ. The reconstruction result is shown
in Figure 11. The crack width and its surface length
are accurately assessed. The estimated crack position is
minimally shifted (0.4mm) in the crack width direction
comparing the true position. The maximum depth is
slightly overestimated of 1mm. When the signal caused
by the elliptical conductive crack is perturbed by added
noise of maximum level of 20%, and then is used in
reconstruction, the maximum depth is overestimated of
2mm, but the crack width and its surface length are
precisely estimated, too.
The presented results proved effectiveness of the proposed
novel approach of three-dimensional diagnosis of
partially conductive cracks, even if cracks with complex
shape and signals with added noise are considered. ECT
response signals gained during C-scan together with
acquiring all three spatial components of the perturbation
electromagnetic field significantly improve the preciseness
of inversion process using tabu search stochastic
method.
VI. CONCLUSION
A novel approach for three-dimensional diagnosis of
partially conductive cracks has been proposed in the
paper. A special eddy current probe driving uniformly
distributed eddy currents was used for the inspection of
a plate specimen. A detection system of the probe was
designed in such a way that all three spatial components
of the perturbation electromagnetic field were acquired.
- 266 - 15th IGTE Symposium 2012
The tabu search stochastic method was employed for the
reconstruction of partially conductive cracks profile from
eddy current response signals gained during the C-scan
of the probe. The signals were perturbed by added noise.
Two crack models were proposed: a crack with cuboid
shape and the other one with more complex shape.
The length, depth, width and conductivity of the crack
were considered unknown in the inversion process. The
conductivity of the crack was uniform.
The presented results proved that the proposed approach
allows quite precisely reconstructing threedimensional
profile of a crack together with its partial
conductivity from signals with added noise.
Further work of the authors will concern more realistic
shapes of cracks and validation with measured data from
natural cracks (SCC).
ACKNOWLEDGEMENTS
This work has been co-funded by the Sectoral Operational
Programme Human Resources Development
2007-2013 of the Romanian Ministry of Labour, Family
and Social Protection through the Financial Agreement
POSDRU/89/1.5/S/62557.
This work was supported by the Slovak Research and
Development Agency under the contracts No. APVV-
0349-10 and APVV-0194-07, and by grants of the Slovak
Grant Agency VEGA, projects No. 1/0765/11, 1/0927/11.
REFERENCES
[1] N. Yusa, “Development of computational inversion techniques to
size cracks from eddy current signals,” Nondestructive testing and
evaluation, vol. 24, pp. 39–52, 2009.
[2] M. Rebican, Z. Chen, N. Yusa, L. Janousek, and K. Miya, “Shape
reconstruction of multiple cracks from ECT signals by means of
a stochastic method,” IEEE Transactions on Magnetics, vol. 42,
pp. 1079–1082, 2006.
[3] Z. Chen, M. Rebican, N. Yusa, and K. Miya, “Fast simulation of
ECT signal due to a conductive crack of arbitrary width,” IEEE
Transactions on Magnetics, vol. 42, pp. 683–686, 2006.
[4] N. Yusa, H. Huang, and K. Miya, “Numerical evaluation of the illposedness
of eddy current problems to size real cracks,” NDT&E
International, vol. 40, pp. 185–191, 2007.
[5] L. Janousek, M. Smetana, and K. Capova, “Enhancing information
level in eddy-current non-destructive inspection,” International
Journal of Applied Electromagnetics and Mechanics, vol. 33, pp.
1149–1155, 2010.
[6] L. Janousek, M. Smetana, and M. Alman, “Decreasing uncertainty
in size estimation of stress corrosion cracking from eddy-current
signals,” Studies in Applied Electromagnetics and Mechanics,
vol. 35, pp. 53–60, 2011.
[7] L. Janousek, M. Smetana, and M. Alman, “Decline in ambiguity
of partially conductive cracks depth evaluation from eddy current
testing signals,” International Journal of Applied Electromagnetics
and Mechanics, vol. 34, 2012 (in press).
[8] Z. Chen, K. Miya, and M. Kurokawa, “Rapid prediction of eddy
current testing signals using A−φ method and database,” NDT&E
International, vol. 32, pp. 29–36, 1999.
[9] Z. Chen, K. Aoto, and K. Miya, “Reconstruction of cracks with
physical closure from signals of eddy current testing,” IEEE
Transactions on Magnetics, vol. 36, pp. 1018–1022, 2000.
[10] M. Rebican, N. Yusa, Z. Chen, K. Miya, T. Uchimoto, and
T. Takagi, “Reconstruction of multiple cracks in an ECT roundrobin
test,” International Journal of Applied Electromagnetics and
Mechanics, vol. 19, no. 1-4, pp. 399–404, 2004.

- 267 - 15th IGTE Symposium 2012
An Adaptive Galaxy-Based Search Approach for
Electromagnetic Optimization Problems
* Θ Leandro dos Santos Coelho, Θ Teodoro Cardoso Bora and † Piergiorgio Alotto
* Industrial and Systems Eng. Graduate Program, Pontifical Catholic University of Parana, Curitiba, PR, Brazil
Θ Department of Electrical Engineering (PPGEE), Federal University of Parana (UFPR), Curitiba, PR, Brazil
† Dip. Ingegneria Industriale, Università di Padova, Italy, E-mail: piergiorgio.alotto@dii.unipd.it
Abstract—Optimization metaheuristics have become very popular methods for electromagnetic device design. The Galaxybased
search algorithm (GBSA) is a recently proposed algorithm, inspired by the movement of the arms of spiral galaxies in
outer space. In this work, a standard and an adaptive version of GBSA (AGBSA) based on historic knowledge are applied to
an analytical testcase and to Loney’s solenoid benchmark problem, showing the suitability of this technique for
electromagnetic optimization. Furthermore, both algorithmic variants are compared with other well-known stochastic
optimizers.
Index Terms— Electromagnetic optimization, Galaxy-based search algorithm, Loney’s solenoid.
I. INTRODUCTION
Optimization algorithms which include stochastic
components are nowadays commonly classified as
metaheuristics and many of them, e.g. Particle Swarm
Optimization (PSO), Genetic Algorithms (GA) and
Evolution Strategies (ES), Differential Evolution (DE),
just to name a few well-known ones, are known to be
powerful techniques for the solution of optimization
problems related to the design of electromagnetic
devices. Such methods have been studied extensively in
the last decades with growing interest in recent years (see
e.g. [1]-[4]).
A recently introduced metaheuristic which has not yet
received much attention in the electromagnetic
optimization community and which is starting to show
interesting performances in other application areas is the
the Galaxy-based search algorithm (GBSA) [6],[7].
GBSA is a nature-inspired optimization method which
mimics the movement of the arms of spiral galaxies in
outer space.
The objective of this paper is to review the basic
algorithmic features of the relatively uncommon GBSA
optimizer and to present a modified and improved
adaptive GBSA (AGBSA) variant. Both algorithms are
then tested on Loney’s solenoid benchmark problem [5],
which features a rough objective function surface typical
of many electromagnetic problems in which the direct
problem is solved by numerical methods.
The rest of this paper is organized as follows. Section
II provides a detailed description of the GBSA algorithm,
while section III is devoted to the application of GBSA to
a multiminima analytical test problem. In Section IV, we
describe Loney’s solenoid benchmark problem and
presents the optimization results for the GBSA and
AGBSA algorithmic variants and comparisons with other
metaheuristics, finally the paper concludes with a brief
discussion in Section V.
II. FUNDAMENTALS OF THE GBSA ALGORITHM
GBSA searches the input space using a spiral chaotic
movement approximating the behavior of one arm of a
spiral galaxy. This movement is driven by a chaotic
process using a logistic map [7]. The main steps of
GBSA are given in Fig. 1, where S represents the current
solution. The algorithm consists of two main
S ← GenerateInitialSolution
S ← LocalSearch (S)
While (termination condition is not met) do
Flag ← False
SpiralChaoticMove (S, Flag)
If Flag then
S ← LocalSearch (S)
Endif Endif
End while while
Fig. 1. Pseudo code of classical GBSA.
componentes which are repeated in sequence:
SpiralChaoticMove, shown in Fig. 2, and LocalSearch,
shown in Fig. 3. The SpiralChaoticMove has the role of
searching around the current solution denoted by S. When
the SpiralChaoticMove procedure finds an improved
solution, it updates S with the improved solution, and the
variable Flag is set to true. When Flag is true, the
LocalSearch component of GBSA is activated in order to
locally search around the current optimal solution.
The SpiralChaoticMove component is iterated for
MaxRep times. However, whenever it finds a solution
better than the current optimal solution,
SpiralChaoticMove is terminated and the control of the
algorithm is transferred to the main procedure of GBSA.
The SpiralChaoticMove component searches the space
around the current best solution using a spiral movement
enhanced by a chaotic variable generated by the logistic
map:
(1)
Where, λ = 4 and x 0 = 0.19 (a sample output for the
case of two degrees of freedom is given in Fig. 4). It
should be mentioned that the first 5000 iterations of the
logistic map are discarded in order not to include in the
generating sequence the transient motion leading to the
chaotic attractor.
The LocalSearch component of GBSA may either
find a locally optimal solution or it will exceed the
maximum number of iterations kMax without
improvement.
The SpiralChaoticMove component of GBSA is the

input:
S, the current best solution (Si is the ith component
i=1:N)
output:
SNext is the first found solution better than S.
Flag if set to true indicates that a better solution has
been found.
parameters:
Each θi is initialised by (–1 + 2 NextChaos()).
Δθ =0.01.
r =0.001.
Δr is set by NextChaos() at each procedure call.
MaxRep is the maximum number of local iterations in
SpiralChaoticMove. (e.g. 100)
θ = –π
While rep < MaxRep
For i = 1 to N
SNexti ← Si + NextChaos() r cos(θi)
End
If (f(SNext) ≥ f(S)) then
Flag ← true
Return
Endif
For i = 1 to N
SNexti ← Si - NextChaos() r cos(θi)
End
If (f(SNext) ≥ f(S)) then
Flag ← true
Return
Endif
r ← r + Δr
For i = 1 to N
θi ←θi +Δθ
End
For i = 1 to N
If(θi > π) then
θ ← –π
Endif
End
rep←rep +1
Endwhile
mechanism which is used for exploring the search space
Fig. 2. Pseudo-code of the SpiralChaoticMove component of
GBSA
in order to find the promising area which may include the
optimal solution. In contrast, the LocalSearch is the
GBSA component which is used to explore the promising
area to find within this area the minimum of the objective
function. In summary, exploration is conducted by
SpiralChaoticMove while exploitation is carried out by
LocalSearch.
Both exploration and exploitation mechanisms are
necessary for the success of any metaheuristic: without
the exploitation mechanism, the metaheuristic may not be
able to obtain accurate solutions whereas without the
exploration mechanism, the metaheuristic may get easily
trapped into a local optimum.
The advantage of using the LocalSeach component
with respect to some other exploitation mechanisms, such
as mutation typical of Genetic Algorithms, is that the
- 268 - 15th IGTE Symposium 2012
input:
S, the current best solution (Si is the ith component
i=1:N)
output:
SNext is a solution better than S
parameters:
ΔS is the step size
α is a dynamic parameter .
KMax is the maximum number of local iterations in
LocalSearch. (e.g. 100).
For i = 1 to N
a←1
k←0
while k < kMax
SLi ←Si –α·ΔS·NextChaos()
SUi ←Si +α·ΔS·NextChaos()
If f(SL) < f(S) and f(SU) < f(S) then
Goto Endrepeat
Endif
If f(SU) > f(S) then
Si ← SUi
SLi ← Sui
α ← α + 0.01 × NextChaos()
k←k+1
ElseIf f(SL) > f(S) then
Si ← SLi
SUi ← SLi α ← α + 0.01 × NextChaos()
k←k+1
Else
α ← α + 0.05 × NextChaos()
k←k+1
Endif
Endwhile
SLi ← Si
SRi ← Si Endrepeat
SNext ← S
proposed local search never allows the algorithm lose the
current best solution, thus increasing the greadyness of
the algorithm.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Fig. 3. Pseudo-code of the LoccalSearch component of GBSA
0.2
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fig. 4. Sample output of the logistic map for the two dimensional
case

Moreover, also the SpiralChaoticMove never loses the
best solution found so far. In addition, it uses chaotic
movements in order to reduce the chance of getting
trapped into local optima (although this may still happen
as will be seen in the paragraph devoted to the analytical
test case). Due to the chaotic process, GBSA does not
return to the same solution and thus diversity of the found
solutions is kept high. Keeping diversity high is
obviously especially important in dealing with
multimodal problems.
The tuning of the step size ΔS in the LocalSearch
procedure is a very delicate task in the classical GBSA
and the convergence properties of the method strongly
depend on its specific value, which is also problemdependent.
The proposed AGBSA efficiently tunes the
step size using history knowledge of mean distances for
the best solution in the previous iteration. The use of
history knowledge is typical of cultural algorithms [9].
III. ANALYTICAL BENCHMARK
The analytical benchmark refers to minimization
of the so-called six-hump camel back function
/3+
. The function has features which are typical of
many real problems, i.e. a bowl-shaped large-scale
behavior, shown in Fig. 5a, which incorporates a
relatively flat plateau with a rather rough small-scale
behavior with several local minima, shown in Fig. 5b.
The function has two global minima, i.e. at [-
0.089842, 0.712656] and [0.089842, -0.712656] with
value f=-1.031628453 and an additional four local
minima.
f(x1,x2)
f(x1,x2)
−50
2
6
5
4
3
2
1
0
−1
2
200
150
100
50
0
1
1
0
x2
0
x2
−1
−1
−2
−2
−2
−4
Fig. 5: a) large-scale and b) small-scale behavior of the six-hump camel
back function
−1
−2
x1
x1
0
0
2
1
4
2
160
140
120
100
80
60
40
20
0
5
4
3
2
1
0
- 269 - 15th IGTE Symposium 2012
Fig 6. shows the position of the optimal
solutions obtained over 30 runs of the algorithm with
maximum number of objective function evaluation set to
100 (Fig. 6a) and 500 (Fig. 6b).
The picture clearly shows that even with a rather
small number of function evaluations, the areas of the
global minima are usually correctly identified by the
algorithm, while a larger number of function evaluations
allows a good precision. The ability of the algorithm to
escape local minima is also clearly shown, since none of
the runs terminated in one of the four local minima at a
higher number of function evaluations, while some
trapping in local minima is to be seen with a lower
number of function evaluations.
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5 −1 −0.5 0 0.5 1 1.5
Fig. 6: Optimal solution over 30 runs : a) max. 500 function evaluations b)
max. 3000 function evaluations
IV. LONEY’S SOLENOID DESIGN
The objective in Loney’s solenoid benchmark problem
[5] is to produce a uniform magnetic flux density B within
a given interval on the axis of a main solenoid (Fig. 7). The
problem is described by two degrees of freedom (the
separation s and the length l of the correcting coils) both
constrained by box bounds

Fig. 7. Axial cross-section of Loney’s solenoid (upper half-plane).
Three different basins of attraction can be recognized in
the domain of the objective function F with values of F >
4·10 -8 (high level region: HL), 3·10 -8 < F < 4·10 -8 (low
level region: LL), and F < 3·10 -8 (very low level region -
global minimum region: VL). The very low level region is
a small ellipsoidally shaped area within the thin low level
valley. In both VL and LL areas, small changes in one of
the parameters result in changes in objective function
values of several orders of magnitude, as shown in Fig. 8.
In the numerical tests, a stopping criterion of 3000
objective function evaluations in each run was used. Tables
I and II show the results over 30 runs. Table II also shows a
comparison with other metaheuristics [10], [11].
It can be noticed that the proposed improvement allows
GBSA to become almost as good as some other wellknown
stochastic optimizers, especially as far as the best
solution is concerned, while improvements in the standard
deviation and mean value are still required.
Fig. 8. Objective function landscape and detail of the VL area
Optimization
Method
TABLE I
SIMULATION RESULTS OF F IN 30 RUNS
F(s, l)·10 -8
Maximum Mean Minimum StandardD
(Worst)
(Best) eviation
GBSA 1510.646 95.960 2.121 282.727
AGBSA 1510.631 88.313 2.068 283.834
SOMA [10] 3.8761 3.2671 2.0595 0.5078
Tribes [11] 3.9526 3.4870 2.0574 0.5079
- 270 - 15th IGTE Symposium 2012
TABLE II
BEST SOLUTIONS FOR LONEY’S SOLENOID IN 30 RUNS
Optimization separation length F(s, l)·10
Method s (cm) l (cm)
-8
GBSA 11.8212 1.6519 2.121
AGBSA 1.6072 1.5145 2.068
V. CONCLUSION
This paper proposes an improved AGBSA based on
historic knowledge. Results on Loney’s solenoid design
problem, a benchmark featuring many of the
characteristics of typical electromagnetic design problems
show promising results. Since the main remaining
weakness of the algorithm, compared with other
metaheuristics, is a rather large standard deviation of
optimal solutions, future research will be targeted at
introducing additional improvements in order to decrease
the spread of solutions.
ACKNOWLEDGMENTS
This work was supported by the National Council of
Scientific and Technologic Development of Brazil —
CNPq — under Grant 476235/2011-1/PQ.
REFERENCES
[1] N. Al-Aaawar, T. M. Hijazi, and A. A. Arkadan, “Particle swarm
optimization of coupled electromechanical systems,” IEEE
Transactions on Magnetics, vol. 47, no. 5, 2011, pp. 1314-1317.
[2] G. Crevecoeur, P. Sergeant, L. Dupré, and R. Van de Walle, “A
two-level genetic algorithm for electromagnetic optimization,”
IEEE Transactions on Magnetics, vol. 46, no. 7, 2010, pp. 2585-
2595.
[3] K. Watanabe, F. Campelo, Y. Iijima, K. Kawano, T. Matsuo, T.
Mifune, and H. Igarashi, “Optimization of inductors using
evolutionary algorithms and its experimental validation,” IEEE
Transactions on Magnetics, vol. 46, no. 8, 2010, pp. 3393-3396.
[4] P. Alotto. A hybrid multiobjective differential evolution method
for electromagnetic device optimization. COMPEL, Vol. 30, No.
6, 2011, pp.1815 – 1828.
[5] P. Di Barba and A. Savini, “Global optimization of Loney’s
solenoid by means of a deterministic approach,” Int. J. of Applied
Electromagnetics and Mechanics, vol. 6, no. 4, pp. 247-254, 1995.
[6] H. Shah-Hosseini, “Principal components analysis by the galaxybased
search algorithm: a novel metaheuristic for continuous
optimization,” Int. J. of Comp. Sci. and Eng., vol. 6, no. 1-2, pp.
132-140, 2011.
[7] H. Shah-Hosseini, “Otsu’s criterion-based multilevel thresholding
by a nature-inspired metaheuristic called galaxy-based search
algorithm,” Proc. of 3rd World Congr. on Nature and Biologically
Inspired Computing, Salamanca, Spain, pp. 383-388, 2011.
[8] G. Ciuprina, D. Ioan and I. Munteanu, “Use of intelligent-particle
swarm optimization in electromagnetics,” IEEE Transactions on
Magnetics, vol. 38, no. 2, pp. 1037-1040, 2002.
[9] R. L. Becerra and C. A. C. Coello, “Cultural differential evolution
for constrained optimization,” Comp. Methods in Appl. Mechanics
and Engineering, vol. 195, no. 33-36, pp. 4303-4322, 2006.
[10] L. S. Coelho and P. Alotto, “Electromagnetic optimization using a
cultural self-organizing migrating algorithm approach based on
normative knowledge,” IEEE Trans. on Magnetics, vol. 45, no. 3,
pp. 1446-1449, 2009.
[11] L. S. Coelho and P. Alotto, “Tribes optimization algorithm applied
to the Loney’s solenoid,” IEEE Trans. on Magnetics, vol. 45, no.
5, pp. 1526- 1529, 2009.

Abstract—Finite element modeling of a magnetic circuit used in
automotive technologies is presented. A 3D magnetic analysis
was performed in order to calculate the field distribution on the
surface of giant magnetoresistance (GMR) sensors. Model
results were compared with experiments, which showed good
agreement. The validated model was further used to optimize
the magnetic circuit design and to improve the working
performance sensors.
Index Terms— Sensors, Finite element method, Magnetic
circuits, Magnetic fields, Giant Magnetoresistance..
I. INTRODUCTION
Magnetic sensors play an important role in automotive
applications. They are reliable, cost effective with high
performance and provide contactless measurements. They are
majorly employed for applications such as measuring pedal
position, engine transmission control, rotational speed of the
wheels, and for anti-lock braking system (ABS) [1].
A new type of magnetic sensor based on the Giant
Magneto-resistance phenomenon (GMR) was developed by
Infineon [2]. They offer key benefits such as high sensitivity,
linear operation over the sensing range, good temperature
stability over a wide range and low field detection
capabilities. Therefore, they are capable of being more precise
on measuring the position or operating at large distances
from the gear wheel in applications. Another benefit of using
GMR elements is the low resistance noise. Presently, GMR
sensors can be used in small fields such as 10 nT at 1 Hz and
up to 10 8 nT. They can operate under temperatures between -
55°C up to 150°C. Unfortunately due to GMR’s high
sensitivity and their low field detection capability GMR
elements can easily drive on saturation, if the detected
magnetic field reached a crucial strength value. Therefore it
is very important to ensure that GMR sensors always stay in
their linear range.
This can be achieved using an experimental procedure to
measure the field distribution of the magnetic circuit.
Unfortunately the experimental method is time consuming
and cost expensive. In order to overcome these problems, this
paper presents a model development of GMR sensors based
on finite element method which can predict the field strength
on the surface of GMR elements with high accuracy.
- 271 - 15th IGTE Symposium 2012
Implementation of a 3D magnetic circuit model
for automotive applications
Ioannis Anastasiadis 1, 3 , Andreas Buchinger 1 , Tobias Werth 2 ,Lukas Bellwald 1 and Kurt Preis 3
1 KAI Kompetenzzentrum Automobil- und Industrieelektronik GmbH, Europastrasse 8, Villach, 9524 Austria
2 Infineon Technologies Austria AG, Siemensstrasse 2, 9500 Villach, Austria
3 Institute for Fundamentals and Theory in Electrical Engineering, Kopernikusgasse 24/3, A-8010 Graz, Austria
II. GMR SENSOR CONCEPT
Magnetoresistance is the change in resistance of a
ferromagnetic material caused by an external magnetic field.
The measure of magnetoresistance is usually given by the
ratio ΔR/R, where R is the resistance for zero magnetic field
and ΔR is the change in resistance when magnetic field
changes by an amount ΔH. Usually ΔR/R value is small and
hence the change in DC voltage remains low. In applications
by using a Wheatstone bridge configuration to place the
magnetoresistance elements, it is possible to minimize the DC
offset.
In some cases of ferromagnetic multilayer’s stack (Fe/Cr)n,
it was reported that at low temperatures their resistance can
change up to 50 % [3]. Due to this major change in
resistance, this phenomenal behavior was named as Giant
Magneto-Resistance (GMR). GMR elements consist of a
sequence of ferromagnetic and antiferromagnetic layers,
which drastically change their resistance under an external
magnetic field. The simplest GMR technology structure is the
spin valve consisting of three layers, of which two
ferromagnetic layers are separated by an antiferromagnetic
layer [4]. One layer has a fixed magnetization direction called
pinned layer-hard layer and the other layer is free to rotate
with external fields and magnetization direction, termed as
free layer-soft layer. For industrial use, this pinned layer can
be created in two ways. The first, using the current flow
which provides heating to the layer and the second method is
to use laser pulses for heating the selected layer. During
cooling, the pinned magnetization is formed. In sensor
technology, the pinned layer has its magnetization direction
perpendicular to the free axis of the free layer. This setup
gives a linear response of the change in GMR resistance when
an external magnetic field is applied.
GMR is a quantomechanic phenomenon created due to the
orientation of conducting electrons while they pass through
the GMR stack. If the spin orientation of the electrons is
parallel to the magnetic orientation of the layer, they move
freely and the resistance remains low. If the spin orientation
is antiparallel to the orientation of the layer, resistance
increases due to collisions with the atoms of the layers. For
application purposes, the trigger/external field should have a
magnitude bigger than the saturation field of the free layer

and smaller than the standoff field of the pinned layer. If this
is not the case, then the magnetization direction of the layers
will be affected, which will change the overall characteristics
of the magnetic sensor. The equation that describes the
change in the resistance R of the GMR element is related to
the angle θ between the magnetization directions of the free
and pinned layer. In the simplest form of the GMR elements
the change in resistance is proportional to the cosine of angle
θ between the magnetization layers [5]:
R − R||
ΔR
ΔR
− ( 1−
m1
⋅ m2
) = [ 1−
cosθ
] (1)
R||
2 R||
2R||
Where R is the resistance of the stack, R|| is the resistance
of the stack in the parallel state, ΔR is the difference between
the resistance of the stack in parallel and antiparallel state,
m1 and m2 are the unit magnetization vectors.
The magnetic-sensor consists of 4 GMR elements situated
at the two edges of the sensor chip. Typical size of the GMR
stacks is approximately 1 μm in length, with 1 mm depth and
a thickness of few nanometers. They are connected to each
other with a Whitestone bridge configuration to measure the
speed signal. In addition, an extra GMR element is placed at
the center of the IC to calculate the directional movement.
The GMR element configuration is shown in figure 1.
Fig. 1: GMR element configuration
By using the above bridge configuration, it is possible to
compensate the DC-offset signal coming from the magnetic
sources. The output signal is given by the following equation:
R4
R2
Vsign = Vleft
−Vright
= VDD
−VDD
R + R R + R
3 4
2 1
≈ Bxleft − Bxright
(2)
Calculating the magnetic field will provide an indication of
the sensor output signal. The above equation is valid, since
the change in resistance of the GMR elements has a linear
response with the change in magnetic field. This assumption
is correct for fields around zero, but for larger applied fields
the overall characteristics of GMR elements will change as
they will saturate.
≈
- 272 - 15th IGTE Symposium 2012
III. MAGNETIC CIRCUITS
Typical magnetic circuits used in automotive technologies
consist of a gear wheel, sensor and a magnet. This magnet,
termed as back-bias magnet is the source for the circuit. The
sensor and the back-bias magnet are fixed, while the wheel is
subjected to rotation. The ferromagnetic gear wheel acts as an
accumulator of the magnetic field –passive target- and the
fluxes bend according to the position of the gear wheel, either
if the static part of the circuit faces a tooth or not. This
difference of the field distribution is sensed by the GMR
elements and is transformed to an electrical signal as the
output of the magnetic sensor. A schematic of this typical
circuit is shown in Figure 2.
magnetic sensor
Fig.2. Basic magnetic circuit application
back-bias magnet
Previously, investigations and optimization process for
back-bias magnets and gear wheels geometries was carried
out for GMR magnetic sensors applications. [6, 7]. This paper
presents the investigation and model development of the
circuit as shown in Figure 3. The gear wheel consists of 44
teeth with a circular pitch of 8°.18. The gear wheel is 10 mm
long in y-axis dimension. The back-bias magnet structure
consists of two magnets formed together with magnetization
directions on the xz plane tilted at an angle of 20° in the z
axis as shown in figure 3. The dimensions of the magnet are
10 x 10 x 4 mm. The magnet is a ferrite with a remanence of
287 mT.
y
z
x
3 mm
20° 20°
3mm
airgap
4 mm
3mm
Fig.3: Magnetic circuit under investigation
The magnetic sensor is placed between the gear wheel and
the magnet. The distance from the end of the sensor to the top
of a tooth is the airgap distance of the circuit. When the gear
wheel is rotated, change in magnetic field distribution on the

GMR element surface takes place. By the rotation of the
wheel with respect to the MS location and for a distance of
one pitch, the field distribution along x-axis has a sinusoidal
form. It is of interest to calculate this field distribution and to
compare with experimental results.
IV. MODEL CREATION
The field distribution on the xy plane along the surface of
the GMR elements was investigated. Additionally, it is
necessary to check also the field strength in the normal
direction (z-direction) in order to determine the sensor circuit
response due to change in the airgap distance. For the above,
it is necessary to investigate the magnetic circuit’s field
distribution in three dimensions (3D). Because of the
complexity of the problem, it is not possible to derive an
analytical 3D solution. Therefore, finite element method was
used to for this purpose [8]. Within this method the geometry
of the problem is discretized in smaller regions, where the
field distribution is calculated by means of approximated
polynomial shape functions. The approach of the problem is
bottom-up. Model was first created in two dimensions and
then extracted in the third direction. 3D scalar magnetic
element was used for this model. Figure 4 shows the field
distribution for an airgap of 2 mm.
Fig.4: field distribution for an airgap of 2 mm
For the model creation, back-bias magnet and the gear
wheel were created surrounded by air. Since the GMR
elements do not interfere in the field distribution but only
used to measure the magnetic field they are ignored in the
model. Precautions have to be made for the magnet
surrounding free space. The dimensions should be taken
around 5 times the respective dimensions of the magnet for
convergence reasons. Larger the model size will increase the
computational time and smaller may lead to distorted
calculations of the field due to calculation errors. All
simulations reported in the paper were carried out using a
commercial FEM tool [9].
A. MS attached to back-bias magnet
Initially, investigations for the case where magnetic sensor
is attached to the back-bias magnet were performed. The air
gap was 2 mm. For a rotation of 1 pitch, the field distribution
- 273 - 15th IGTE Symposium 2012
was calculated on the surface of the GMR elements. The Bx
field distribution on the surface of the left GMR and center
GMR element measured at a point in the center of their
surface and for a rotation of 1 pitch is shown in figure 5.
Bx(mT)
Bx(mT)
0
-2
-4
-6
-8
-10
-12
-14
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
Bx field on the left GMR
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.18
angle(°)
Fig. 5a:Bx field distribution on the left GMR
Bx field on the center GMR
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.18
angle(°)
Fig. 5b:Bx field distribution on the center GMR
As it can be seen from Figure 5a, the Bx field on the left
GMR element is approximately -10 mT which is big enough
to drive GMR elements on saturation. Therefore MS should
not be attached to the back-bias magnet but kept a distance
between MS and magnet to prevent driving GMR elements on
saturation.
B. MS placed in a distance from the magnet
In this case, a distance of 2 mm is kept between the
magnet and the magnetic sensor as shown in figure 6.
20°
20°
Fig.6: The new circuit geometry under investigation
By setting the airgap distance also to 2 mm and considering
that the package of the sensor has in normal direction 1 mm
length, the total distance between the magnet back face and
gear wheel teeth was 5 mm.
y
z
x

Hence magnetization of the free layer of GMR stack follows
the magnetization direction of the external plane field
distribution on the surface of the element. We calculated the
plane field Bx and By distribution along the surface of GMR
stripes. GMR stripes have a length of approximately 1 mm in
y-direction. For investigations of the plane field at the length
of GMR stripe along y-direction, the Bx and By distributions
were calculated at points which were equidistant spaced. The
bottom point at the surface of GMR stripe is denoted to be the
0 point and the next point is spaced by 0.05 mm until the last
point of calculations on the top point of the stripe. The Bx and
By filed distributions for those points and for two GMR
elements, one which is located on the left half-bridge and
another on the center of the sensor (GMR5) can be seen on
figures 7 and 8.
Bx(mT)
By(mT)
Bx(mT)
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
1
1.5
2
2.5
3
3.5
angle(°)
4
4.5
5
5.5
6
6.5
7
7.5
8
8.1818
Bx @ 0mm
Bx @ 0.05mm
Bx@ 0.1mm
Bx@ 0.15mm
Bx@ 0.2mm
Bx@ 0.25mm
Bx@ 0.3mm
Bx@0.35mm
Bx@0.4mm
Bx@0.45mm
Bx@0.5mm
Bx@ 0.55mm
Bx@ 0.6mm
Bx@ 0.65mm
Bx@ 0.7mm
Bx@ 0.75mm
Bx@ 0.8mm
Bx@ 0.85mm
Bx@ 0.9mm
Bx@ 0.95mm
Bx@ 1mm
Fig. 7a: Bx field distribution on a GMR on the left half-bridge
0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.182
angle(°)
By@ 0mm
By@ 0.05mm
By@ 0.1mm
By@ 0.15mm
By@ 0.2mm
By@ 0.25mm
By@ 0.3mm
By@ 0.35mm
By@ 0.4mm
By@ 0.45mm
By@ 0.5mm
By@ 0.55mm
By@ 0.6mm
By@ 0.65mm
By@ 0.7mm
By@ 0.75mm
By@ 0.8mm
By@ 0.85mm
By@ 0.9mm
By@ 0.95mm
By@ 1mm
Fig. 7b: By field distribution on a GMR on the left half-bridge
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.18
angle(°)
Bx@ 0mm
Bx@ 0.05mm
Bx@ 0.1mm
Bx@ 0.15mm
Bx@ 0.2mm
Bx@ 0.25mm
Bx@ 0.3mm
Bx@ 0.35mm
Bx@ 0.4mm
Bx@ 0.45mm
Bx@ 0.5mm
Bx@ 0.55mm
Bx@ 0.6mm
Bx@ 0.65mm
Bx@ 0.7mm
Bx@ 0.75mm
Bx@ 0.8mm
Bx@ 0.85mm
Bx@ 0.9mm
Bx@ 0.95mm
Bx@ 1mm
Fig. 8a: Bx field distribution on a center GMR element
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By(mT)
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
angle(°)
8
8.1818
By@ 0mm
By@ 0.05mm
By@ 0.1mm
By@ 0.15mm
By@ 0.2mm
By@ 0.25mm
By@ 0.3mm
By@ 0.35mm
By@ 0.4mm
By@ 0.45mm
By@ 0.5mm
By@ 0.55mm
By@ 0.6mm
By@ 0.65mm
By@ 0.7mm
By@ 0.75mm
By@ 0.8mm
By@ 0.85mm
By@ 0.9mm
By@ 0.95mm
By@ 1mm
Fig. 8b: By field distribution on a center GMR element
The Bx fields along the GMR stripes have always the same
response for a rotation of 1 pitch. On the other hand the By
field is shifted each time we move from the bottom side of the
stripe towards the upper side. The By fields are
homogeneously distributed along the stripes and have the
same values on the surfaces of all GMR stripes. Due to
symmetry reasons, a GMR element situated on the right halfbridge
should also have along their surface similar By and Bx
distribution.
C. Experimental results
Experiments were performed for the configuration shown in
figure 6. To compare with simulations, the gear wheel speed
was set to 1.5 rpm. Such a small speed was chosen, because
the finite element model was built for every corresponding
magnet positions over the rotated angle. As the gear wheel
rotates, the magnetic sensor provides speed and the
directional signal. These measured signals are compared with
simulation results. Drawback of the experimental procedure is
that there is no possibility to directly derive the Bx and By
distribution along the GMR stripe, but only the plane field
distribution on the surface of GMR elements while gear wheel
is rotated towards magnetic sensor. This experimental field
distribution along the GMR surface is compared with the field
distributions derived from the model.
Hence Bx and By distributions are changed along the GMR
stripes we have to calculated the average field that the
elements are sensed and that field we have to compare it with
the experimental results. Substituting the signal from the left
and right Whetstone bridge, we measure the speed signal
while the center GMR element shows the directional signal.
The results are shown in figure 9. The signal is calculated on
field distribution along the surface of the stripes.
Fig. 9a: comparison of directional signal

Fig. 9b: comparison of speed signal
Again, results are shown for a rotation of 1 pitch.
Comparisons for the speed signal between experimental
and simulation results reveal a small deviation of
approximately 3%. On the other hand, the comparison for the
directional signal which comes from the measurements of the
middle GMR element shows a bigger deviation with a mean
value of 9%. Differences between simulation and
experimental results are due to inaccuracies in the simulation
model, such as how dense the model is. Another important
issue is that in reality the geometry of gear wheel has
deviations from the theoretical geometry due to construction
reasons, for example each pitch distance has not exactly the
same dimensions of 6mm or there may be a small deviation
on the height of all the teeth of the gear wheel. Those
problems can bring an error on the calculated signal.
V. CONCLUSION
A 3D model describing the rotation of a GMR sensor
around a gear wheel was developed and verified with
experiments. The model is based on finite element analysis
and is used to calculate the variations of the field distribution,
when the gear wheel is rotated around the stator of the
magnetic circuit, MS and back-bias magnet. Calculation of
the field distribution was performed along the GMR element
surface. In parallel, experiments were performed for the same
configuration to support model development.
Comparison shows a small deviation between the compared
values given an indication of the valid of the model. Such a
model can give us a fast and accurate estimation on the
magnetic circuit’s functionality showing also the maximum
airgap performance of this circuit.
ACKNOWLEDGEMENTS
This work was jointly funded by the Federal Ministry of
Economics and Labour of the Republic of Austria (contract
98.362/0112-C1/10/2005) and the Carinthian Economic
Promotion Fund (KWF) (contract 98.362/0112-C1/10/2005).
REFERENCES
[1] C.P.O. Treutler, “Magnetic sensors for automotive applications,” Elsevier,
Sensors and Actuators A, vol. 91,2001, pp. 2-6
[2] Dirk Hammersdchmidt, Ernst Katzmaier, et. all, “Giant magneto resistorssensor
technology & automotive applications”, SAE 2005 World Congress
& Exhibition, SAE, Detroit, 01-01-2005, pp. 1-16.
- 275 - 15th IGTE Symposium 2012
[3] M.N.Baibich, M.Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P.
Etienne, G. Creuzei, A. Frederick and J. Chazelas, “Giant
Magnetoresistance of (001) Fe/(001) Cr Magnetic Superlattices,” Phys.
Rev. Lett., vol. 61, num. 21, Nov. 1988, pp. 2472-2475.
[4] Robert L. White, “Giant Magnetoresistance: A Primer” IEEE Trans. on
Magn., vol. 28, Sept. 1992, pp. 2482-2487.
[5] S.E. Russek, R.D: McMichael, M.J: Donahue and S. Kaka, “High Speed
Switching and Rotational Dynamics in Small Magnetic Thin Film
Devices,” Springer, Spin Dynamics in Confined Magnetic Structures 2,
vol.87, 2003, pp. 93-156
[6] I. Anastasiadis, T. Werth, K. Preis, “Evaluation and optimization of backbias
magnets for automotive applications using Finite Element Methods”,
IEEE Transactions on Magnetics, vol. 45 (March 2009) no.3, pp. 1332-
1335.
[7] I. Anastasiadis, T. Werth, K. Preis, “Investigation and optimization of
magnetic sensor gear wheels for automotive applications”, 14 th IGTE
Symposium on Numerical Field Calculation in Electrical Engineering,
19-22 Sept. 2010, submitted.
[8] A. Bonderson,T. Rylander, P. Ingelström, Computational
Electromagnetics (Book style), Springer Inc. 2005.
[9] Tutorial, Electromagnetic Field Analysis Guide, ANSYS Release 11.0,
ANSYS Inc Book International Inc., Canonsburg, PA, 2006

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Mixed Order Edge-based Finite Element Method
by Means of Nonconforming Mesh Connection
Yoshifumi Okamoto and Shuji Sato
*Department of Electrical and Electronic Systems Engineering, Utsunomiya University
7-1-2 Yoto, Utsunomiya, Tochigi 321-8585, Japan
E-mail: okamotoy@cc.utsunomiya-u.ac.jp
Abstract—The first order element is widely used as discretized order of edge-based finite element method. The second order
discretization has a tendency of escape from the practical magnetic field analysis for the reason of many nonzero entries in
global matrix and convergence deterioration of a linear solver. Therefore, the performance of higher order elements should
be successfully utilized by the restriction of analyzed region. This paper presents the mixed order finite element method with
reasonable computational costs by using nonconforming mesh connection. The higher order discretization is applied to the
main region with higher accuracy, and is connected with the outer space discretized by first order element using
nonconforming connection. This paper shows the detailed characteristics of nonconforming mixed order edge-based finite
element analysis.
Index Terms— Higher order element, higher order interpolation, mixed order edge-based finite element analysis,
nonconforming mesh connection.
linear combination [5] – [8]. We applied the linear
combination to nonconforming connection to retain the
symmetry of the global matrix. Furthermore, we newly
propose the finite element analysis using the
nonconforming connection between 2nd order elements,
and comparisons have been made with conventional
conforming analysis and mixed-order analysis.
I. INTRODUCTION
The edge-based finite element method is widely
recognized as a powerful and practical numerical method
in the design environment for the evaluation of temporal
changes of electromagnetic field and various
characteristics of electrical machine. Furthermore, the
mesh generator has been advanced in order to product the
finite element mesh for the complicated target. From
these technical background, 1st order discretization is
widely adopted as an edge-based finite element method.
On the other hand, the 2nd order discretization has the
tendency of escape from the practical magnetic field
analysis owing to the many nonzero entries in global
matrix and convergence deterioration of a linear solver
for the algebraic equation. However, 2nd order element
has better convergence characteristics to the true value of
physical quantity such as the magnetic energy than the
1st order element when the element size is shortened.
Then, it is assumed that 2nd order element should be
adopted at the region where the accuracy is required and
1st order element is adopted at other region. This
combinatorial technique might be capable of improving
the convergence characteristics of linear solvers and
shortening the elapsed time with lower computational
cost than conventional 2nd order discretization. The
reference [1] describes the mixed order analysis based on
the hierarchical elements. Furthermore, the mixed order
analysis in the discontinuous Galerkin method is applied
to the eddy current problem [2]. These references show
the effectiveness of mixed order analysis. However, the
combinatorial technique between higher order elements is
not reported and the degree of nonconforming mesh
connection between 2nd and 1st order is not verified.
This paper shows the detailed effectiveness of mixed
order edge-based finite element analysis which is realized
using nonconforming mesh connection. The
nonconforming mesh connection is mainly classified into
two methods, in which one is the method based on
Lagrange multiplier [3], [4], and the other is based on the
II. MIXED ORDER FINITE ELEMENT METHOD BY MEANS
OF NONCONFORMING MESH CONNECTION
A. Weak form for edge-based finite element method
The weak form Gi for Maxwell equation in
magnetostatic field is given as follows:
G
i
V
(
N ) (
A)
dV N J 0dV
i
Vm
Vc
( N )
B dV 0,
where Ni is the edge-based shape function, is the
reluctivity, A is magnetic vector potential, J0 is the
current density vector, Br is the remanence, respectively.
The domain for volume integral V, Vc, and Vm denote the
whole region, the region for magnetizing winding, and
the permanent magnet, respectively. When the magnetic
nonlinearity is taken into account, Newton-Raphson (NR)
method supported by the line-search based on functional
(0, 1.0) [9] is adopted as the nonlinear analysis method.
The ICCG method with shifted parameter [10] is used as
a linear solver for the algebraic equation derived from
edge-based finite element method.
B. Nonconforming mesh connection
This subsection describes the nonconforming mesh
connection using the linear combination, which is
formulated as follows:
i
i
r
(1)
A N A dl
,
(2)
b
ab
a
k
k
k
where Aab is the vector potential of nonconforming edge
ab as shown in Figure 1 (a). In following formulation,
suppose that the global coordinate at the node k is (xk, yk),

the global coordinate of node a and b is (xa, ya), (xb, yb)
and the local coordinate of node a and b is (a, a), (b,
b). The element shape on master side is assumed as a
square or rectangle. Hence, the global coordinate (x, y)
on master side element is as follows:
x1
x2
x1
x2
x ,
(3)
2 2
y1
y4
y1
y4
y ,
(4)
2 2
where and are local coordinate on the master side
element. , is given as follows:
b
a
i ,
(5)
x x
b
a
b
a
j ,
(6)
yb ya
where i and j are the unit vectors in x- and y-direction.
The linear combination using 1st and 2nd order mesh on
the master side is mentioned below.
Coefficients for the 1st order nonconforming mesh
connection
Adopting the 1st order element as the discretization of
master side, (2) becomes the expression:
4
A ab I ke Ake
,
(7)
k 1
where Ike is a coefficient which is evaluated by a line
integral of edge-based shape function Nke along the edge
ab as follows:
b
1
dl
I ke N d ( ) d
,
a
ke l N
1
ke tab
d
(8)
where tab is the unit vector in the direction ab, dl/d is the
Jacobian,and is an additional parameter to perform the
analytical line integrals, respectively. Using , the local
coordinates and are transformed into
b
a a b
,
2 2
(9)
b
a a
b
.
2 2
(10)
Subsequently, tab and dl/d for the analytical evaluation
of (8) are as follows:
1
tab { ( xb
xa
) i ( yb
ya
) j}
,
l
(11)
2e
N 4e
ab
a
l ab
N 1e
b
1e
N 3e
3e N2e 4e
2e
N 7e
8e
N 8e
a
N 2e
N 9e
l ab
5e
N 10e
N 1e
b
1e
N 5e
7e
N 6e
3e N4e 6e N3e 4e
(a) (b)
Figure 1. Interpolation to the slave edge ab from master
side element. (a) 1st order master side element and (b)
Serendipity 2nd order master side element.
- 283 - 15th IGTE Symposium 2012
2
dl
d
x
x
lab
,
2
(12)
where lab is the length of edge ab. Then, the 1st order
edge-based shape functions on master side surface are as
follows:
1
( 1
ke
)
N 4
ke
1
( 1
ke
)
4
( 1 k 2)
,
( 3 k 4)
(13)
where ke and ke are the local coordinates on the edge k
according to TABLE I. Substituting (11), (12), and (13),
are substituted for (8), all components of Ike are
analytically evaluated as follows:
I
ke
1
( b
a ){ 2
ke ( a
b )}
8
1
( b
a ){ 2 ke ( a b
)}
8
2
( 1
( 3
TABLE I
LOCAL COORDINATES FOR 1ST ORDER
EDGE-BASED SHAPE FUNCTION ON MASTER SIDE
k 1 2 3 4
ke 0 0 1.0 -1.0
ke 1.0 -1.0 0 0
k 2)
. (14)
k 4)
Coefficients for the 2nd order nonconforming mesh
connection
Even in the case of the linear combination adopting the
2nd order elements as the master side mesh, the
derivation of the coefficients for linear combination is in
the same way as 1st order nonconforming connection.
Firstly, 2nd order edge-based shape functions of
Serendipity type [10] on the master side surface shown in
Figure 1 (b) as follows:
1
( 1
ke
) ( 4
ke
ke
)
( 1 k 4)
4
1
( 1
ke
) ( 4
ke
ke
)
( 5 k 8)
4
N ke
, (15)
1 2
( 1
)
( k 9)
2
1 2
( 1
)
( k 10)
2
where ke and ke are the local coordinates on the edge k
according to TABLE II. Then, adopting the 2nd order
element as master side discretization, (2) becomes next
expression:
10
ab ke
k 1
A I A .
(16)
ke
When the shape of master side element is a square or
rectangle, the order of coordinates x and y can be defined
TABLE II
LOCAL COORDINATES FOR 2ND ORDER
EDGE-BASED SHAPE FUNCTION ON MASTER SIDE
k 1 2 3 4 5 6 7 8 9 10
ke 0.5 -0.5 0.5 -0.5 1.0 1.0 -1.0 -1.0 0 0
ke 1.0 1.0 -1.0 -1.0 0.5 -0.5 0.5 -0.5 0 0

as 1st order. Therefore, , , tab, dl/d are all same
as (5), (6), (11), and (12). Substituting these equations
and (15) for (8), all components of Ike are analytically
obtained as follows:
b
a
[ ke
( b
a ){ 4
ke ( b
a )
48
ke ( b
a )} 3{
2
ke ( a
b )}
{ 4
ke ( a
b
)
ke ( a
b )}]
( 1 k 4)
b
a
[ ke ( b
a ){ 4
ke ( b
a )
48
ke ( b
a )} 3{
2 ke ( a b
)}
I ke
. (17)
{ 4
ke ( a
b
) ke ( a b
)}]
( 5 k 8)
b
a
2
{ 12 3(
a
b ) ( b
a )}
24
( k 9)
b
a
2
{ 12 3(
a b
) ( b
a )}
24
( k 10)
In the case of nonconforming connection between 1st
order elements, 1st order connection (14) is adopted.
Similarly, in the case between 2nd order elements, 2nd
order connection (17) is adopted. When the
nonconforming connection between same order elements
is performed, the coarser side is adopted as the master
side of the linear combination and the finer side is
adopted as slave side.
On the other hand, 1st order connection (14) is adopted
as the connection order for mixed order analysis, in
which 1st and 2nd order elements are connected. The
reasons are follows: If the 2nd order connection (17) is
applied to the interface for mixed order connection, the
coefficients I9e and I10e to the edges on 1st order edge 1e-
4e become zero. Therefore, the diagonal component
related to the connection interface may be zero, and the
difficulty for solving the algebraic equation causes.
III. ANALYSIS MODEL
Figure 2 shows a square coil model in order to verify
the performance of various nonconforming connections.
The current density is determined by the electric scalar
potential to be I = 1000 AT in the current input surface.
The meshes for the region including a square coil and the
outer space are nonconforrmally connected. The
nonconforming connection is performed at the three
surfaces composed of 1st surface (x = 120, y: [0, 120], z:
[0, 120]), 2nd surface (x: [0, 120], y = 120, z: [0, 120]),
and 3rd surface (x: [0, 120], y: [0, 120], z = 120). The
range of whole region is set to x: [0, 300], y: [0, 300], and
z: [0, 300], and the all element shapes are the cube to
remove the error caused by the element distortion.
Figure 3 shows an open type MRI model [12], in which
the main object is to compute the uniform magnetic flux
distribution in the imaging region with high accuracy.
The nonconforming connection is performed at the three
- 284 - 15th IGTE Symposium 2012
surfaces composed of 1st surface (x = 400, y: [0, 400], z:
[0, 400]), 2nd surface (x: [0, 400], y = 400, z: [0, 400]),
and 3rd surface (x: [0, 400], y: [0, 400], z = 400). The
magnetic nonlinearity of SS400 is considered in the yoke
and pole piece. The remanence of two facing magnets is
set to 1.2 T, and the nonlinear magnetostatic analysis is
performed by NR method.
IV. NUMERICAL RESULTS
A. Verification using square coil
Figure 4 shows the two examples of finite element
meshes. The element coefficient matrix is computed by
Gaussian quadrature 3×3×3 points. The linear equation is
stopped when the condition ||rk||2/||b||2 < cg is satisfied,
where ||rk||2 and ||b||2 are 2-norm of the residual at the k-th
iteration and right side vector in the algebraic equations
and cg is set to 10 -6 . is the typical element size of
standard model, and h is the element size of target model.
Therefore, (h/) 2 of standard model becomes 1.0 as
shown in Figure 4 (a). On the other hand, h in the
nonconforming case (hexa-1st + hexa-1st, hexa-2nd +
hexa-2nd, and hexa-2nd + hexa-1st) is defined as the
element size of the inner mesh including magnetizing
winding.
Figure 5 shows the convergence characteristics of
magnetic energy. All characteristics have an asymptotic
behavior as h shortens. The W values in nonconforming
case (hexa-1st + hexa-1st and hexa-2nd + hexa-2nd) at
(h/) 2 = 1.0 is equivalent to those in conforming case.
These nonconforming characteristics are slightly
detached from conforming characteristics in the range
(h/) 2 < 1.0.
coil(I = 1000 AT)
currentoutput
y
z
Figure 2: Square coil model.
y
400 130 350 400
imagingregion
polepiece(SS400)
z
unit:[mm]
directionofcurrent
currentinput
40
x
Figure 3: Open type MRI model.
unit:[mm]
magnet
(Br = 1.2 T)
yoke(SS400)
x

Comparing the nonconforming characteristic (hexa-2nd +
hexa-2nd) with (hexa-2nd + hexa-1st), the (hexa-2nd +
hexa-2nd) characteristic is superior to the result of (hexa-
2nd + hexa-1st) from the viewpoint of asymptote to the
behavior of conforming hexa-2nd.
TABLE III shows the effect of the size ratio on the
computational accuracy. shows the ratio of the element
size in outer space mesh for the element size in inner
region, for example, becomes 1.5 in Figure 4 (b). In
nonconforming case, the mesh for outer space is
subdivided on the condition that the mesh for inner
region is fixed. The number of elements for 2nd order is
set to one eighth of 1st order element. When gets
larger, the relative error of W tends to be worse owing to
being coarse size of outer space element. The results of
nonconforming connection have the tendency, in which
the accuracy of the nonconforming connection using 2nd
order element is superior to 1st order on the whole. The
outerregion
y
currentoutput
innerregion
nonconf.
boundary
y
outerregion
currentoutput
innerregion
z
(a)
z
x
currentinput
x
currentinput
(b)
Figure 4: Finite element meshes of a square coil model.
(a) conforming (h/) 2 = 1.0 and (b) nonconforming (h/) 2
= 0.444.
- 285 - 15th IGTE Symposium 2012
elapsed time using (nonconf. 2nd + 1st, = 2.0) is the
shortest among the nonconforming results, in which the
relative error of W is less than 0.1 %. Whereas the
accuracy of W using (nonconf. 2nd + 2nd, = 2.0) is the
best among the above mentioned nonconforming types,
the elapsed time approximately quintuples against the
case of (nonconf. 2nd + 1st, = 2.0). Hence, it is shown
that the enough accuracy is provided by the mesh type
(nonconf. 2nd + 1st, = 2.0) from the viewpoint of
practical analysis.
Figure 6 shows the z-component of magnetic flux
density on z-axis. The all characteristics coincide with the
standard characteristic, and the relative error between
(nonconf. hexa-2nd + hexa-1st) and (conf. hexa-2nd) is
less than 0.05 %. The accuracy of magnetic flux in local
area as well as that of magnetic energy is retained even in
W [J]
nonconf.hexa2nd+hexa2nd
nonconf.hexa2nd+hexa1st
0.0512
0.0508
0.0504
stand.
conf.hexa2nd
conf.hexa1st
0.0500
0.0 0 0.2 0.4 0.6 0.8 1.0
(h /) 2
nonconf.
hexa1st+hexa1st
Figure 5: Convergence characteristics of magnetic
energy.
nonconf.boundary
0.012 12.0
inner outer
B z [mT]
0.010 10.0
0.008 8.0
0.006 6.0
0.004 4.0
0.002 2.0
0.00 0 0.04 40 0.08 80 0.12 120
z [mm]
TABLE III
ANALYZED RESULTS OF SQUARE COIL MODEL
nonconf.hexa2nd+hexa2nd
nonconf.hexa1st+hexa1st
conf.hexa2nd conf.hexa1st
7.72
B z [mT]
7.70
stand.
nonconf.
hexa2nd+hexa1st
7.68
59.8 60.0 60.2
z [mm]
Figure 6: Distributions of Bz on z-axis of square coil
model.
mesh type
inner
NoE
outer total
size ratio DoF nonzero
time for
global matrix [s]
ICCG
ite.
time for
ICCG [s]
W [mJ]
relative error (%) of W
vs. conf. 2nd (stand.)
conf. 2nd (stand.) 37,044 1,120,581 1,157,625 1.0 13,759,410 576,830,675 224.2 *
466 1042.3 *
51.0553 0
conf. 1st
1,672,704 1,728,000 1.0 5,126,520 86,040,332 67.5 147 67.1 50.9912 0.125
nonconf. 1st + 1st
55,296
209,088
26,136
264,384
81,432
2.0
4.0
775,368
236,424
12,873,180
3,901,596
10.7
3.5
96
59
6.8
1.4
50.9840
50.9548
0.140
0.197
3,267 58,788 8.0 170,289 2,819,055 2.6 49 0.9 50.8334 0.435
conf. 2nd
209,088 216,000 1.0 2,548,920 105,755,060 51.4 263 142.4 51.0505 0.009
nonconf. 2nd + 2nd
26,136
3,267
33,048
10,179
2.0
4.0
383,256
115,971
15,585,392
4,574,483
8.2
2.7
141
98
12.5
3.6
51.0505
51.0179
0.009
0.073
6,912 1,672,704 1,679,615 0.5 5,037,828 86,450,712 67.7 214 96.7 51.0477 0.015
nonconf. 2nd + 1st
209,088
26,136
216,000
33,048
1.0
2.0
693,576
154,632
13,378,746
4,398,882
10.2
2.9
114
84
8.4
2.3
51.0408
51.0117
0.028
0.085
3,267 10,179 4.0 88,497 3,312,651 2.0 84 2.4 50.8903 0.323
CPU: Intel Core i7-2620M 2.7 GHz & 16 GB
CPU * : Intel Core i7-3930K 4.2 GHz with over-clocked & 32 GB

outer region
imaging region
(inner region)
y
outer region
imaging region
(inner region)
y
outer region
hexa-1st
imaging region
(inner region)
hexa-1st
y
outer region
hexa-1st
imaging region
(inner region)
hexa-2nd
y
z
r
(a)
z
r
(b)
z
r
(c)
z
r
(d)
Figure 7: Finite element meshes of open type MRI
model. (a) conforming (hexa-1st), (b) conforming (hexa-
2nd isoparametric), (c) nonconforming (hexa-1st + hexa-
1st), and (d) nonconforming (hexa-2nd + hexa-1st).
B z [T]
-0.20
nonconf.boundary
inner outer
-0.22
-0.24
-0.26
-0.28
-0.30
0.0 0.2 0.4 0.6 0.8
r [m]
x
x
x
x
- 286 - 15th IGTE Symposium 2012
the nonconforming connection.
B. Application of mixed order finite element analysis to
open type MRI model
This subsection shows the effectiveness of mixed order
finite element analysis with nonconforming connection in
the open type MRI model shown in Figure 3. The
convergence criterion cg of ICCG method is set to 10 -3 ,
and when the maximum correction of magnetic flux
density is to be 10 -3 T, NR iteration is stopped.
Figure 7 shows the finite element meshes for open type
MRI model. Figure 7 (a), (b), (c), and (d) show the mesh
for conforming hexa-1st, conforming hexa-2nd
isoparametric, nonconforming (hexa-1st + hexa-1st), and
nonconforming (hexa-2nd + hexa-2nd), respectively. The
nonconforming mesh connection is performed at the
interface between imaging region and other region in
order to reduce the number of elements with keeping the
accuracy of magnetic flux density in imaging region.
Number of elements in conforming hexa-2nd (b) is one
eighth size in conforming hexa-1st (a). The mesh for
imaging region of (c) is exactly the same as that of (a),
and the mesh for outer region of (c) and (d) is completely
the same as that of (b).
Figure 8 shows the distributions of z-direction
magnetic flux density Bz on 45° direction r-axis which is
located on the surface (x, y) = (0, 0) shown in Figure 7.
There are some noise spikes in the characteristics of
(conf. hexa-1st), (nonconf. hexa-1st + hexa-1st), and
(nonconf. hexa-2nd + hexa-1st). The generation of noise
is likely to be caused by the element distortion of 1st
order hexahedral elements. The distributions of 1st order
discretization have the concave and convex owing to the
interpolation of inner flux using edge-shape function.
The mixed order characteristic of (nonconf. hexa-2nd +
hexa-1st) seems to be combined two properties, in which
the property of 2nd order is confirmed in the inner region
(imaged region) and 1st order property appeared in the
outer region.
TABLE IV shows the analysis results for open type
MRI model. Even in MRI model, the DoF of (conf. 2nd)
is a half of (conf. 1st); nevertheless, the elapsed time of
(conf. 2nd) is longer than that of (conf. 1st). There is a
possibility that the condition number of the global matrix
B z [T]
-0.26
-0.27
nonconf.hexa1st+hexa1st
nonconf.hexa2nd+hexa1st
conf.hexa2nd
stand.
conf.hexa1st
-0.28
0.50 0.62
r [m]
Figure 8: Distributions of Bz on r-axis in open type MRI model.

derived from 2nd order get worse than that of 1st order.
On the other hand, the elapsed time of (nonconf. 2nd +
1st) is the almost same as that of (nonconf. 1st + 1st).
Furthermore, the iteration number for NR method in
mixed order analysis is quite same as other mesh type.
V. CONCLUSION
We proposed a mixed order finite element method
using nonconforming mesh connection technique. The
obtained results are summarized as follows:
1. We propose the nonconforming mesh connection
between 2nd order elements supported by the linear
combination, in which the coefficient for 2nd order
connection can be derived from the line integral.
2. The performances of the nonconforming connection
for 2nd order elements and mixed order elements are
verified using a square coil model. The accuracy of the
nonconforming connection including the 2nd order
discretization is superior to that of the meshes
discretized by only 1st order element.
3. The accuracy of mixed order analysis has the good
agreement with that of conforming 2nd order
discretized mesh and the mesh with 2nd order
nonconforming connection.
4. Mixed order analysis has superiority than the
conventional conformal mesh from a point of view of
the elapsed time.
Finally, we will investigate the effectiveness of mixed
order edge-based finite element analysis including the
distorted finite elements as a future works.
ACKNOWLEDGMENT
The authors would like to thank Mr. Y. Tominaga for
helpful support. This work was supported by Japan
Society for Promotion of Science (JSPS) Grant-in-Aid
for Young Scientists (B) (Grant Number: 23760252).
REFERENCES
[1] M. Hano, T. Miyamura, and M. Hotta, “Fast and high-accuracy
finite-element electromagnetic analysis by mixed-order vector
elements,” The Papers of Joint Technical Meeting on Static
Apparatus, SA-02-14, RM-02-14, pp. 13-18, Jan. 2002. (in
Japanese)
[2] P. Houston, I. Perugia, and D. Schötzau, “Nonconforming mixed
finite-element approximations to time-harmonic eddy current
problems,” IEEE Trans. Magn., Vol. 40, No. 2, pp. 1268-1273, Feb.
2004.
[3] D. Rodger, H. C. Lai, and P. J. Leonard, “Coupled elements for
problems involving movement,” IEEE Trans. Magn., Vol. 26, No.
2, pp. 548-550, Feb. 1990.
- 287 - 15th IGTE Symposium 2012
TABLE IV
ANALYSIS RESULTS OF OPEN TYPE MRI MODEL
mesh type
inner region
NoE
outer space total
DoF nonzero NR ite.
time for
global matrix [s]
total
ICCG ite.
time for
ICCG [s]
conf. 2nd (stand.)
conf. 1st 8,000
556,528 564,528
6,680,172
1,662,234
278,902,794
27,727,823
7
1035.0
171.7
2,254
1,714
2268.6
262.6
nonconf. 1st + 1st
223,699 3,663,713 7 23.3 617 14.2
conf. 2nd
nonconf. 2nd + 1st
1,000
69,566 77,566 823,308
212,099
33,753,786
3,709,345
7
7
127.5
23.3
315
624
526.3
14.9
CPU: Intel Core i7-2620M 2.7 GHz & 16 GB
[4] E. Lange, F. Henrotte, and K. Hameyer, “A variational
formulation for nonconforming sliding interfaces in finite element
analysis of electric machines,” IEEE Trans. Magn., Vol. 46, No. 8,
pp. 2755-2758, Aug. 2010.
[5] C. Golovanov, J.-L. Coulomb, Y. Marechal, and G. Meunier, “3D
mesh connection techniques applied to movement simulation,”
IEEE Trans. Magn., Vol. 28, No. 2, pp. 3359-3362, Feb. 1992.
[6] H. Kometani, S. Sakabe, and A. Kameari, “3-D analysis of
induction motor with skewed slots using regular coupling mesh,”
IEEE Trans. Magn., Vol. 36, No. 4, pp. 1769-1773, Apr. 2000.
[7] K. Muramatsu, Y. Yokoyama, N. Takahashi, A. Nafalski, and Ö.
Göl, “Effect of continuity of potential on accuracy in magnetic field
analysis using nonconforming mesh,” IEEE Trans. Magn., Vol. 36,
No. 4, pp. 1578-1582, Apr. 2000.
[8] Y. Okamoto, R. Himeno, K. Ushida, A. Ahagon, and K. Fujiwara,
“Dielectric heating analysis method with accurate rotational motion
of stirrer fan using nonconforming mesh connection,” IEEE Trans.
Magn., Vol. 44, No. 6, pp. 806-809, Jun. 2008.
[9] Y. Okamoto, K. Fujiwara, and R. Himeno, “Exact minimization of
energy functional for NR method with line-search technique,” IEEE
Trans. Magn., Vol. 45, No. 3, pp. 1288-1291, Mar. 2009.
[10] K. Fujiwara, T. Nakata, and H. Fusayasu, “Acceleration of
convergence characteristic of the ICCG method,” IEEE Trans.
Magn., Vol. 29, No. 2, pp. 1958-1961, Mar. 1993.
[11] A. Kameari, “Calculation of transient 3D eddy current using edgeelements,”
IEEE Trans. Magn., Vol. 26, No. 2, pp. 466-469, Mar.
1990.
[12] C. Lee and K. Miyata, “Large-scale magnetic field analysis on
MRI with hysteresis,” The papers of Joint Technical Meeting on
Static Apparatus and Rotating Machinery, SA-06-20, RM-06-20,
pp. 25-30, Jan. 2005. (in Japanese)

- 288 - 15th IGTE Symposium 2012
Topology Optimization Using Parallel Search
Strategy for Magnetic Devices
1 Takumi Nagano, 1 Shogo Yasukawa, 1 Shinji Wakao, and 2 Yoshihumi Okamoto
1 Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555, Japan
2 Utsunomiya University, 7-1-2, Yoto, Utsunomiya, Tochigi 321-8585, Japan
E-mail: wakao@waseda.jp
Abstract— In this paper, we propose a topology optimization method using parallel search strategy for magnetic devices.
Here, we use the gradient method to minimize an object function from the viewpoint of convergence speed, i.e., steepest
descent method. By applying parallel computing, we will carry out calculations simultaneously for some patterns of initial
variables. With these calculation results, new patterns of initial variables are efficiently created for restarting new search
processes, which results in the better topologies than previous ones. Compared with the conventional method, the proposed
search method enables us to decrease the whole CPU time with keeping the optimization quality.
Index terms— density method, parallel computing, topology optimization.
I. INTRODUCTION
Structural optimization is categorized into three types
of problems, i.e., size optimization, shape optimization,
and topology optimization. Topology optimization is a
useful method especially in terms of weight saving. And
there is possibility of discovering a new topology
unthinkable by conventional methods. However, the
computational load of topology optimization will be
heavier than that of other optimization, because of the
large search space. In this paper, we applied “density
method with gradient method” to make the optimization
process more efficient. In density method, topology of the
target is expressed with the density value of elements
which consist of the design domain [1]. And gradient
method has good convergence speed as the search
method. However, the result of minimization with
gradient method will be mostly local minimum
depending on initial variables. Therefore this paper
proposes an efficient topology optimization method based
on parallel computing for magnetic devices. In the
proposed method, we can efficiently escape local
minimums by simultaneous searches from various initial
variables and creating new initial variables with the
results of parallel optimization.
II. PROPOSED METHOD
A. Density method based on sensitivity analysis
In density method, topology of the target is expressed
with the density value of elements. In this paper, the
target is magnetic circuit. The magnetic permeability i
of ith element can be formulated as
2
{ 1
1
(1)
},
i 0 r i
where i is the density value of ith element, and r is
relative permeability of the material. In this paper, r is
1000, and the property of magnetic system is regarded as
linear one. In density method, the number of elements in
the design domain should be large to express the detail of
topologies. When density method is combined with
gradient method, the first derivative of object function
with respect to density value (sensitivity) of each element
in the design domain needs to be calculated. As the
efficient sensitivity analysis, “adjoint variable method” is
applied [2]. The algebraic equation of FEM can be
formulated as
HA G,
(2)
where H is whole coefficient matrix, A is unknown
magnetic vector potential, and G is right side vector.
(3) is obtained by differential calculus of (2) with respect
to density vector .
A
G
H
H = A.
(3)
ρ
ρ
ρ
(4) is obtained by multiplying H -1 to (3).
A 1
G
H
~
= H
A
,
(4)
i
i
i
where A ~ is the solution of (2).
Sensitivity of the object function W is obtained as
T
dW W W
A
= . (5)
d
A
i
i
In this paper, W is defined with the value of magnetic
flux density vector. Therefore, the 1 st term of (5) is
invisible. By substituting (4) into (5), (6) is obtained.
T
W W
1
G
H
~ T G
H
~
H
A
λ
A
.
(6)
i
A
i
i
i
i
As the property of FEM matrix, H is symmetric. Taking
account of this point, (7) is obtained by transforming (6).
Hλ.
A
W
(7)
The is called “adjoint variable”. is obtained by
solving (7). Finally, the whole sensitivity vector is
obtained by substituting into (6).
B. Basic concept of proposed method
Here, we carried out parallel computing based on Open
MP for topology optimization by using a computer with 8
cores. The basic concept of proposed method is shown in
figure 1.
i

Figure 1: Basic concept of proposed method.
The proposed method is based on gradient method, where
the solution depends on initial values. Therefore, we
regard the data set of initial topology, optimized
topology, and object function value as the properties of
one “individual”.
First, we prepare 8 individuals i.e., the number of
cores, by creating random initial topologies and
computing their optimized topologies. Next, a new initial
topology is created with property information of 2
selected individuals. The process of creating a new
topology with 2 individuals’ information is named
“intercross”. We prepare new 8 initial topologies for the
next generation, and obtain the data sets by optimizing
new initial topologies. The better solution will be
obtained by repeating the above cycle.
In the next section, we explain how to create initial
topologies in the next generation with 2 individuals.
C. How to select 2 individuals for intercross
Three manners of selection of 2 individuals for
intercross are proposed.
a) We combined 2 individuals with superior object
function value to give properties of superior individuals
to the next generation. In this paper, the combinations of
individuals in the top 4 from the viewpoint of object
function superiority are selected for intercross. We create
3 initial topologies in the next generations based on this
selection.
b) We combined 2 individuals with different optimized
topologies. The subject of the selection is to make
diversity for intercross. The difference between 2
topologies is evaluated in the following rule.
As shown in figure 2, design domain is separated into
some areas. Next, we define a reference value in the i th
area with the density values as
area
i
Ni
2
j
j (8)
.
N
Ni is the number of elements in the i th area. We define a
vector C as in (9).
i
C area , area ,..., area ). (9)
( 1 2
8
- 289 - 15th IGTE Symposium 2012
Figure 2: Separation of design domain.
The vector C expresses the character of the optimized
topology. The characteristic difference of 2 topologies A
and B, is evaluated as (10).
2
B
Diff C C . (10)
AB
The combination of 2 individuals with larger value of
(10) is selected for intercross. We create 3 initial
topologies in the next generations based on this selection.
c) We combined 2 individuals chosen randomly. The
subject of the selection is also to make diversity of
intercross. We create 2 initial topologies in the next
generations based on this selection.
8 initial topologies in the next generation are created
based on the selections a)-c).
D. How to make new initial topologies
Here, 2 individuals selected for intercross are named as
IA and IB, and new individual is named as IC. The initial
topology of IC is created in the following three kinds of
manners, which are adopted randomly.
a) The density values of IC’s initial topology are created
as the weighted mean of those of initial topologies of IA
and IB.
b) The density values of IC’s initial topology are created
as the weighted mean of those of optimized topologies of
IA and IB.
To inherit the properties of superior individuals to the
next generation, the weight coefficients of IA and IB are
formulated as
,
Ci
Ai
W
W
3
A
3
3
A WB
,
Bi
A
W
W
3
B
3
3
A WB
(11)
,
where, WA and WB are object function values of IA and IB,
and i stands for the element number.
c) The density values of IC’s initial topology are created
as the multiplication of initial topology of IA and
optimized topology of IB.
. (12)
Ci
Ai
Bi

The distribution of density value i of 0 in optimized
topology will be strongly inherited to the next generation.
The probabilities of occurrence of a),b), and c) are 3/7,
3/7, 1/7 respectively.
III. NUMERICAL EXAMPLE
A. C-shaped iron core model
To demonstrate the validity of the proposed method, we
carried out the optimization of the model shown in figure
3.
Figure 3: Magnetic force model. (unit : mm)
The main subject of this optimization problem is to
maximize the electromagnetic force generated in the
magnetic bar lying in the right side of design domain. The
density values of elements in the design domain are
design variables of this optimization [3]. To evaluate the
electromagnetic force, we adopt the following equation as
the object function of this optimization,
1
2
By
W , (13)
where, By is the y component of magnetic flux density
vector generated in the target element. This model is
discretized by triangular elements. The numbers of
elements, in the design domain and whole domain, are
1,575 and 5,576 respectively.
Next, the detail of the proposed method is explained.
Initial topologies of the 1 st generation are created as
random values i which range from 0 to 1. The initial
magnetic permeability distribution is determined as to
(1). The best object function value before the n th
generation is Wbest, and the best object function value in
the n+1 th generation is Wn+1. If Wn+1 Wbest , we update
the value of Wbest as Wn+1. If the update doesn’t occur over
10 generations, the calculation is terminated.
- 290 - 15th IGTE Symposium 2012
Figure 4: Flow chart of proposed method.
Now, for the comparison with the proposed method, the
optimization without intercross is also applied to this
model. In the method, the initial topologies in whole
generation are created as randomly as that of the 1 st
generation. This method is named “random method”.
2 pattern of initial topologies are prepared as those of in
the 1 st generation, which are named as ITA and ITB. An
example of initial topology in 1 st generation is shown in
figure 5.
Figure 5: Density distribution example of initial topology.

Wbest
best object funciton value
68
67.5
67
66.5
66
65.5
65
64.5
64
63.5
Initial
pattern
IT A
IT B
random method (ITA) proposed method
random method (ITB) proposed method (ITB) 0 10 20
generations
30 40
Figure 6: Convergence characteristic of each method.
TABLE I
THE OPTIMIZATION RESULTS OF BOTH METHODS.
method
total
generations
Wbest
(IT A)
(a) The obtained topology with random method
from initial pattern ITA.
CPU
time(sec.)
Random 20 65.0926 764.56
Proposed 34 64.0327 347.91
Random 25 64.2958 954.66
Proposed 17 63.9976 229.98
- 291 - 15th IGTE Symposium 2012
Figure 7: Comparison of CPU time.
We carry out the optimizations with these topologies by
the proposed method and the random method. The
convergence characteristics of both methods are shown in
figure 6. In random method, the improvement of object
function value is obviously inefficient, and the
computational result is in the local minimum. By
contrast, the proposed method enables us to improve the
object function value more efficiently, and to achieve
better solution than that of the random method. The
object function values Wbest of both methods are shown in
table 1.
(b) The obtained topology with proposed method
from initial pattern ITA.
(c) The obtained topology with random method
(d) The obtained topology with proposed method
from initial pattern ITB. Figure 8: Optimization result of each method.
from initial pattern ITB.
frequency
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
proposed method
random method
2 6 10 14 18 22 26 30 34 38 42 46
calculation time[CPU sec.]

The frequency distribution of calculation time for each
individual is shown in Figure 7. Figure 7 indicates that
the calculation time for each individual is sufficiently
reduced in the proposed method because the starting
points for optimization, i.e., the initial topologies, are
effectively created near stationary points. We can
estimate the total CPU time by multiplying the
calculation time for individuals in one generation by the
required number of generations. As the results, the CPU
time of the proposed method is much less than that of the
random method in spite of their total generations required
for convergence as shown in table I.
The obtained topologies by both methods are shown in
Figure 8. Black elements correspond to the magnetic
material with i = 1, and white elements to air elements
with i = 0. Gray elements have an intermediate property
between air and magnetic material with 0 < i < 1 which
are named as “gray scale”. The topologies obtained by
the random method strongly depend on their initial
variables and contain many gray scales. On the contrary,
we can obtain the similar topologies without gray scales
by using the proposed method.
B. Magnetic shielding model
Proposed method is applied to the optimization of the
shield model shown in Figure 9.
y
x
Figure 9: shield model. (unit : mm)
The main subject of the optimization problem is to
minimize the magnetic flux entering into target domain
generated by 2 surrounding coils. The density values of
elements in the design domain are design variables of the
optimization. The object function is defined as (14).
W
2
Bx
t arg et
domain
2
By
.
(14)
- 292 - 15th IGTE Symposium 2012
This model is discretized by triangular elements. The
number of elements, in the design domain and the whole
domain, are 3,800 and 12,106 respectively.
The optimization results of the random and the proposed
methods are shown in Figures 10-13.
Wbest
best object funtion value
ferquency
0.0012
0.001
0.0008
0.0006
0.0004
0.0002
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
random method
proposed method
0 20 40 60 80
generations
Figure 10: Convergence characteristic of object function.
0
Wbest = 4.0610 -4
0
24
48
72
96
120
144
168
192
216
240
264
288
312
336
360
384
408
432
calculation time (sec.)
Wbest = 6.9010 -5
random method
proposed method
Figure 11: Comparison of CPU time.
Fig 12: Result of optimization. (random method)

Figure 13: Result of optimization. (proposed method)
The computational results demonstrate the effectiveness
of the proposed method compared with the random
method as is the case with the magnetic force model.
We can successfully obtain the multilayer structure as
an effective topology of the shield as shown in Figure 13.
IV. CONCLUSION
In this paper, a topology optimization using parallel
search strategy for magnetic devices is proposed to
efficiently obtain more global solutions. In the proposed
method, we effectively introduce the novel concept of
intercross into the density method with sensitivity
analysis, which results in the CPU time reduction with
keeping the optimization quality.
We will investigate various algorithms of intercross,
and apply the proposed method to more practical
nonlinear problems as future works.
REFERENCES
[1] H. P. Mlejnek and R. Schirrmacher, "An engineer's approach to
optimal material distribution and shape finding," Comput.
Methods Appl. Mech. Eng., Vol. 106, pp. 1-26 (1993).
[2] S.Gitosusastro, J.L.Coulomb and J.C. Sabonnadiere, "Performance
derivative calculations and optimization process," IEEE Trans.
Magn, Vol.25, No.4, pp. 2834-2839(1989)
[3] Yoshihumi Okamoto, and Norio Takahashi, "Investigation of
Topology Optimization of Magnetic Circuit by Using Density
Method", IEEJ Trans. IA, Vol.124, No.12, pp. 1228-1235(2004).
[4] Jin-kyu Byun, Il-han Park, and Song-yop Hahn, "Topology
optimization of electrostatic actuator using design sensitivity,"
IEEE Trans. Magn. Vol.38, No. 2, pp. 1053-1056 (2002).
[5] Jin-kyu Byun and Song-yop Hahn, "Application of topology
optimization to electromagnetic system," International journal of
applied electromagnetics and mechanics, Vol. 13, No. 1-4, pp. 25-
33 (2002).
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Interaction Magnetic Force Calculation of Axial
Passive Magnetic Bearing Using Magnetization
Charges and Discretization Technique
*Saša S. Ilić, *Ana N.Vučković and *Slavoljub Aleksić
*University of Niš, Faculty of Electronic Engineering of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia
E-mail: ana.vuckovic@elfak.ni.ac.rs
Abstract— The paper presents calculation of the force between two ring permanent magnets whose magnetization is axial.
Such configuration corresponds to a passive magnetic bearing. The simple and fast analytical approach is used for this
calculation based on magnetization charges and discretization technique. The results for interaction magnetic force obtained
using proposed approach are compared with finite element method using FEMM 4.2 software.
Index Terms— Permanent magnet, interaction magnetic force, magnetization charges, discretization technique, Finite
Element Method (FEM).
charges for uniform magnetization do not exist. nˆ is the
unit vector normal to surface.
I. INTRODUCTION
Permanent magnets are used nowadays in many
applications, and the general need for dimensioning and
optimizing leads to the development of calculation
methods. Permanent magnets are commonly used in
many electrical devices and their own quality depends on
the magnet material, magnetization and dimensions. Two
major kinds of applications can be identified: the ones
which use block magnets and the ones which use
cylindrical magnets. Block permanent magnets are easy
to manufacture and to magnetize, and it’s easier to
calculate magnetic field they create [1-3]. Indeed, most
engineering applications need several ring permanent
magnets and the determination of the magnetic force
between them is thus required.
Magnetic bearings are contactless suspension devices
with various rotating and translational applications [4].
Depending on the ring permanent magnet magnetization
direction, the devices work as axial or radial bearings and
thus control the position along an axis or the centering of
an axis. Knowledge of the interaction magnetic force is
required to control devices reliably.
There are numerous techniques for analyzing permanent
magnet devices and different approaches for determining
interaction forces between them [5-8]. Many authors
are proposing simplified and robust formulations of the
interaction forces created by permanent magnets. The
authors generally use the Ampere's current model [9],[10]
or the Columbian approach [11],[12]. Several application
examples were previously presented in [13],[14] where
levitation forces for magnetic bearings were calculated.
II. THEORETICAL BACKGROUND
Axial passive magnetic bearing [5] that is considered in
the paper is presented in the Fig.1.
Since the boundary condition for surface magnetization
charges density has to be satisfied for both magnets,
m ˆ
1 n M1
and m ˆ
2 n M2
, (1)
it is obvious that fictitious surface magnetization charges
[2] exist only on the bottom and the top bases of each
permanent magnet, because volume magnetization
Figure 1: Axial passive magnetic bearing.
The simplest procedure for interaction magnetic force
determination is to discretize each base of both
permanent magnets into system of circular loops. The
interaction force between two magnetized circular loops
will be calculated first. That will be performed by
calculating the magnetic field and magnetic flux density
generated by the arbitrary magnetized circular loop of the
upper magnet first and then the force that acts on the
arbitrary loop of the lower magnet (Figure 2). Magnetic
field of the upper loop will be determined by calculating
the magnetic scalar potential. Using results for interaction
magnetic force between two circular loops, magnetic
force of the axial magnetic bearing can be obtained by
summing the contribution of both magnet bases of lower
and upper permanent magnets by using uniform
discretization technique.
The goal of this approach is to determine the interaction
magnetic force between two circular loops uniformly
loaded with magnetization charges Qm1 and Q m2
.
Dimensions and positions of the loops are presented in
the Figure 2. For determining the interaction force
between two circular loops, magnetic scalar potential,
magnetic field and magnetic flux density generated by
the upper loop will be calculated. Elementary magnetic
scalar potential generated by the elementary point
magnetization charge, dQ m1,is

Figure2: Two circular loops.
dQm1
1
d m . (2)
4
R
Qm1
Qm1
Since dQ
m1
Qm
1 dl
a d '
d '
,
2a
2
elementary magnetic scalar potential has the following
form
Qm1
1
d m
d '
, (3)
2
8
R
and the resulting magnetic scalar potential generated by
the upper circular loop at an arbitrary point P( r,
,
z)
is
2
Qm1
1
m
d '
,
2
8
2 2
2
0 r r0
zz0 2r0r
cos'
(4)
Considering the existing symmetry, in 0 plane,
magnetic scalar potential has the following form
Qm1
1
m ( r,
z)
d '.
2
4
2 2
2
0 r r0
zz0 2rr0
cos '
(5)
Substituting θ' 2
in Eq. (5), magnetic scalar
potential is obtained as:
2
Qm1
m ( r,
z)
2
2
0
1
2
2
2
( r r0
) 4rr0
sin zz0 d
. (6)
After some simple operations the magnetic scalar
potential can be given in the form:
where
m
( r,
z)
Qm1
2
2
2
K
, k
2
2
( r r0
)
2 z z
1
K , k K
,
2
2 2
1
k sin
0
0
d
- 301 - 15th IGTE Symposium 2012
(7)
is complete elliptic integral of the first kind with modulus
2 4rr0
k .
2
2
( r r0
) ( z z0
)
External magnetic field (magnetic field generated by the
upper loop) at an arbitrary point can be determined as
ext
H ( r, z)
grad
m
( r,
z)
Hr
( r,
z)
rˆ
H z ( r,
z)
zˆ
,
(8)
External magnetic flux density is
ext
ext
ext
ext
B ( r, z)
H ( r,
z)
, (9)
ext
r
ext
r
ext
z
with components
ext
Br
2
r
2r
and
0
B ( , z)
B ( r,
z)
rˆ
B ( r,
z)
zˆ
. (10)
( r,
z)
ext
ext H
( r,
z)
Br ( r,
z)
0
, (11)
r
2 2
2
rr0zz0 E,
k
2
2
2
2
( r r ) zz ( r r ) zz 0
( r r
K
, k
2
0
)
B z
0
2
Q
m1
2
2
0
zz 0
2
,
0
ext
0
2
(12)
ext H
( r,
z)
( r,
z)
0
, (13)
z
ext Qm1
Bz ( r,
z)
0
2
2
zz0E, k
2
2 2
2
2
( r r ) z z ( r r ) z z
0
0
2
0
0
(14)
2 2
where E , k E 1
k sin d ,
2
0
is complete elliptic integral of the second kind with
modulus
2 4rr0
k
2
2
( r r0
) ( z z0
)
.
The interaction magnetic force on elementary
magnetization charge of lower circular loop
Q
m2
Qm2
dQm
2 Qm2
dl
bd
d
is
2b
2
ext
d m2
m m
F dQ
B ( r , z ) . (15)
Finally, interaction magnetic force components can be
expressed as:

Qm1Qm
2
Fr ( r,
z)
0
2
2
2
rm
m 0 m 0 m 0
K
, k0
2
2rm
2
( rm
r0
) m
2 2
2
rmr0zmz0 E,
k0
2
2
2
2
( r r ) zz ( r r ) zz Qm1Qm
2
Fz ( r,
z)
0
2
2
zmz0E, k0
2
m
zz 0
2
0
2
0
2 2
2
2
( rm
r0
) zm
z0
( rm
r0
) zm
z0
(16)
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(17)
Q
( , ) m1Q
F m 2
z r z 0
F ( r0,
rm,
z0,
zm)
2 z
, (18)
p
2
with elliptic integrals modulus
2 4r0rm
k0
2
2
( rm
r0
) ( zi
zm
)
.
The axial component of the force (17) presents
interaction force between two magnetized circular loops.
The simplest procedure for levitation magnetic force
determination is to discretize each bases of permanent
magnets into system of circular loops, where N 1 is the
number of discretized segments of each bases of upper
permanent magnet and N 2 is the number of discretized
segments of each bases of lower permanent magnet.
Figure 3: Discretizing model.
By taking into account the ring geometry of permanent
magnets (Figure 3), the radius of each discretized
segment of both bases of upper magnet is
r n
2n
1
a ( b a),
n 1,
2,
,
N1
2N1
, (19)
and magnetization loop charges of upper permanent
magnet bases are
b a
Qm n M12rn
, n 1,
2,...,
N1
N1
. (20)
For lower magnet bases the radius of each discretized
segments is
r i
2i
1
C ( d c),
i 1,
2,
,
N2
2N
2
. (21)
Magnetization loop charges of lower permanent magnet
bases are
d c
Qm i M 2 2ri , i 1,
2,...,
N2
N2
. (22)
Using results for interaction magnetic force between
two circular loops, Eqs. (17), the levitation magnetic
force between two ring permanent magnets can be
obtained. It can be achieved by summing the contribution
of both magnet bases of lower and upper permanent
magnets by using uniform discretization technique,
1 2
20
M1M
2
Fz
( b a)(
d c)
rnri
N N
F
F
zp
zp
1
2
( r , r , h,
0)
F
n
( r , r , h L , 0)
F
n
i
i
1
Lh zp
n1
i1
( r , r , h,
L )
n
zp
E
, k3
2
i
N
2
N
( rn
, ri
, h L1,
L2
)
; (23)
2
0M
1M
2
Fz
( b a)(
d c)
N1N
2
N1
N2
hE
, k1
2
rn
ri
2 2
2 2
n1
i1
( ri
rn
) h ( ri
rn
) h
L2hE, k2
2
2
2
2
2
( ri
rn
) L2h ( ri
rn
) L2h 2
2
2
( r r ) Lh ( r r ) Lh i
1
LLh
E
, k4
2
2
2
2
( r r ) LLh ( r r ) LLh i
n
n
where
2 4rnri
k1
,
2 2
( ri
rn
) h
2 2 4rnri
k2
,
2
ri
rn
L2
h
2 2 4rnri
k3
,
2
ri
rn
L1
h
2 2
4rnri
k4
.
2
ri
rn
L2
L1
h
2
1
2
1
1
i
n
i
n
1
2
2
1
2
(24)
III. NUMERICAL RESULTS
We are working under presumption that the both ring
permanent magnets are made of the same material and
magnetized uniformly along their axis of symmetry, but
in opposite direction, M1 M 2 M .

Distribution of magnetic flux density obtained using
FEMM 4.2 software [15] is presented in Fig. 6. The
values of the geometrical parameters used in the
numerical computation are: 2 1,
L a , 2 2 L b
c L2
3,
d / L2
4,
L1
/ L2
0.
5,
2 1.
5,
L h
2 1mm
L and kA/m 900 M .
Convergence of the normalized interaction force,
nor Fz
Fz
obtained using presented approach is
2 2
0M
L2
,
given in Table I for magnetic bearing dimensions:
a L2
1, b L2
2,
2 3,
L c , 4 2 L d , 5 . 0 / L1
L2
h / L2
0.
1 .
Figure 4: Distribution of magnetic flux density for magnetic bearing
obtained using FEMM 4.2 software.
TABLE I
CONVERGENCE OF LEVITATION MAGNETIC FORCE FORCE VERSUS
NUMBER OF SEGMENTS.
N tot
nor
F z
nor
F z (FEM)
10 -0.0732327
20 -0.0741579
30 -0.0743319
50
100
-0.0744213
-0.0744591
-0.07491167
200 -0.0744685
300 -0.0744703
500 -0.0744712
In order to save the calculation time, the number of
segments is limited on N tot N1
N2
200 because it
is not necessary to take a greater number of segments to
obtain a desired accuracy.
Compared results for normalized interaction magnetic
force of two identical ring permanent magnets, obtained
using presented analytical approach and finite element
method (FEM) versus 2 L h , for parameters: , 1 2 L a
2 2,
L b c L2
3, d L2
4 and 5 . 0 / L 1 L2
are
given in the Table II.
Comparative results for normalized interaction
magnetic force of axial passive magnetic bearing versus
ratios 2 L a and 2 L b , obtained using presented approach
and finite element method (FEM), for parameters:
c L2
3, d L2
4,
L1
/ L2
0.
5 and h / L2
1.
5 are
shown in the Table III.
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TABLE II
COMPARED RESULTS FOR INTERACTION MAGNETIC VERSUS h L2
h / L2
nor
F z
nor
F z (FEM)
0 -0.120071 -0.120515
0.1 -0.074469 -0.074911
0.2 -0.025238 -0.025659
0.3 0.025238 0.024873
0.4 0.074469 0.074099
0.5 0.120071 0.119765
0.6 0.159974 0.159691
0.7 0.192597 0.192327
0.8 0.216978 0.216771
0.9 0.232821 0.232627
1.0 0.240465 0.240293
1.1 0.240766 0.240654
1.2 0.234930 0.234829
1.3 0.224326 0.224240
1.4 0.210322 0.210279
1.5 0.194163 0.194138
TABLE III
COMPARED RESULTS FOR INTERACTION MAGNETIC FORCE VERSUS
a L AND
2 b L2
a / L2
b/
L2
nor
F z
nor
F z (FEM)
1.0 2.0 0.194163 0.194138
1.5 2.5 0.320992 0.321208
2.0 3.0 0.209864 0.210493
2.5 3.5 -0.614301 -0.613222
3.0 4.0 -1.341280 -1.339868
3.5 4.5 -0.714116 -0.712579
4.0 5.0 0.272002 0.273619
4.5 5.5 0.491236 0.492676
5.0 6.0 0.352195 0.353756
IV. CONCLUSION
Determination of the interaction forces of axial passive
magnetic bearing is presented. It is preformed using
magnetization charges and discretization technique.
Presumption was that both magnets are made of the
same material and magnetized uniformly along the
magnet axis of symmetry, with the same intensity, but in
opposite directions. The derived algorithm is easily
implemented in any standard computer environment and
it enables rapid parametric studies of the interaction
force. The results of the presented approach are
successfully confirmed using FEMM 4.2 software. Table
I shows that it is not necessary to take a great number of
segments (not more then 200) to obtain a desired
accuracy so the computational time can be saved.
Interaction forces calculation using presented approach
for mentioned parameters and N tot 200 is performed
with Intel Core 2 Duo CPU at 2.4GHz and 4GB RAM
memory and it took less than two seconds of run time.
Interaction forces are also determined on the same
computer using FEMM 4.2 software and the computation
time was 14 minutes for about 1.8million finite elements.
Therefore, the advantage of presented analytical approach
is its accuracy, simplicity and time efficiency.

V. ACKNOWLEDGEMENT
The work presented here was partly supported by the
Serbian Ministry of Education and Science in the frame
of the project TR 33008.
REFERENCES
[1] J. S Agashe and D. P Arnold, “A study of scaling and geometry
effects on the forces between cuboidal and cylindrical magnets
using analytical force solutions”, J. Phys. D: Appl. Phys. 41
105001, pp.1-9, 2008.
[2] A. N. Vučković, S. R. Aleksić, S. S. Ilić.: “Calculation of the
Attraction and Levitation Forces Using Magnetization Charges”,
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PES 2011, Proceedings of full papers (CDROM), pp. 33-55, 25-
29 September, Niš, Serbia, 2011.
[3] G. Akoun, J. P. Yonnet.: “3d Analytical Calculation of the Forces
Exerted between two Cuboidal Magnets”, IEEE Transactions on
Magnetics, Vol. 20, No. 5, pp. 1962-1964, September 1984.
[4] S. I. Babic, C. Akyel.: “Magnetic Force Calculation between Thin
Coaxial Circular Coils in Air”, IEEE Transactions on Magnetics,
Vol. 44, No. 4, pp. 445-452, April 2008.
[5] V. Lemarquand, G. Lemarquand.: “Passive Permanent Magnet
Bearings for Rotating Shaft: Analytical Calculation”, Magnetic
Bearings, Theory and Applications, Sciyo Published book, pp. 85-
116, October 2010.
[6] R. Ravaud, G. Lemarquand, V. Lemarquand.: “Force and
Stiffness of Passive Magnetic Bearings Using Permanent Magnets.
Part 1: Axial Magnetization”, IEEE Transactions on Magnetics,
Vol. 45, No. 7, pp. 2996-3002, July 2009.
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“Cylindrical Magnets and Coils: Fields, Forces and Inductances”,
IEEE Transactions on Magnetics, Vol. 46, No. 9, pp. 3585-3590,
September 2010.
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[8] M. Greconici, Z. Ž. Cvetković, A. N. Mladenović, S. R. Aleksić,
D. Vesa.: “Analytical-numerical Approach for Levitation Force
Calculation of a Cylindrical Bearing with Permanent Magnets
Used in an Electric Meter” Proceedings of full papers OPTIM
2010, pp. 197-201, 20-21 May, Brasov, Romania, 2010.
[9] Furlani, E. P., S. Reznik, & A. Kroll. 1995. A three-dimensional
field solution for radially polarized cylinders. IEEE Trans. Magn.,
vol. 31, no.1, pp. 844–851.
[10] M. Braneshi, O. Zavalani and A. Pijetri.: “The Use of Calculating
Function for the Evaluation of Axial Force between Two Coaxial
Disk Coils”, 3 rd International PhD Seminar Computational
Electromagnetics and Technical Application, pp. 21-30, 28
August - 1 September, Banja Luka, Bosnia and Hertzegovina,
2006.
[11] Rakotoarison, H. L., J.-P. Yonnet, & B. Delinchant.2007. Using
Coulombian Approach for Modeling Scalar Potential and
Magnetic Field of a Permanent Magnet With Radial Polarization.
IEEE Transactions on Magnetics, Vol. 43, No. 4, pp. 1261-1264.
[12] R. Ravaud, G. Lemarquand, V. Lemarquand.: “Force and
Stiffness of Passive Magnetic Bearings Using Permanent Magnets.
Part 2: Radial Magnetization”, IEEE Transactions on Magnetics,
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[13] Ana N. Vučković, Saša S. Ilić & Slavoljub R. Aleksić: Interaction
Magnetic Force Calculation of Ring Permanent Magnets Using
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[15] Meeker, D. n.d. Software package FEMM 4.2. Available on-line
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2007).

- 305 - 15th IGTE Symposium 2012
Magnet deviation measurements and
their consideration in
electromagnetic field simulation
Peter Offermann ∗ , Isabel Coenen ∗ , David Franck ∗ and Kay Hameyer ∗
∗ Institute of Electrical Machines
RWTH Aachen University
Schinkelstrasse 4
D-52062 Aachen, Germany
E-mail: Peter.Offermann@IEM.rwth-aachen.de
Abstract—Due to their manufacturing process arc segment magnets for the use in permanent-magnet synchronous machines
(PMSM) may show deviations from their intended ideal magnetization. Using magnets with unfavourable error constellations
in one rotor of a PMSM will result in a spatial unsymmetric air gap field, causing undesired parasitic effects as e.g. torque
pulsations. Most manufacturer information only contain the mean values of the magnetization as well as certain guaranteed
error bounds, not stating if (and how) the magnetization will vary spatial over a set of magnets. In order to allow an
accurate consideration of these deviations in the machine simulation, the emitted radial field of a set of magnets has been
measured and compared to their assumed magnetisation using finite element method (FEM). As a result, the measured
deviations can be quantified and the influence of magnet deviations can be estimated using e.g. stochastic collocation
methods in combination with the FEM.
Index Terms—finite element method, magnetization errors, measurements, stochastics variations
I. INTRODUCTION
The simulation of an electrical machine employing
the finite element method (FEM) requires the exact
knowledge of the machine’s geometry, its excitations and
its material properties. For machines which are manufactured
in mass production, the material or geometry of
one specific instance of the designed machine may vary
from its specified targets [1], leading in the worst case
to a non-fulfilment of the rated machine’s data.
For geometry variations a typical cause is the abrasion
of the punching tools. Varying material properties
may be caused e.g. by a stochastic jitter in the orientation
of the punched stator lamination sheets, which
can be tainted with anisotropy. Causes for variations
in excitations can either arise from the converter or –
in case of a permanent-magnet synchronous machines
(PMSM) – from magnet deviations [2] with respect to
their intended ideal magnetization [3]. Using magnets
with unfavourable error constellations in one rotor of a
PMSM will result in a spatial unsymmetric air gap field,
causing undesired parasitic effects as torque pulsation [4],
[5].
Most manufacturer information only contain the mean
values of the magnetization as well as certain guaranteed
error bounds, not stating if (and how) the magnetization
will vary spatial over a set of magnets. The goal of
this publication hence is to improve the simulation of
electrical machines by reducing the described epistemic
uncertainty of magnet variations. Therefore, a magnet
test-bench has been created, in order to measure the
emitted radial field of a set of magnets. From this, the
modality and probability distribution of the occurring
variations have been deduced.
The comparison of the magnets’ FEM-simulations
with their measurements may allow the calculation of
improved simulation parameters for complete machine
simulations. For the measured magnets, which were
diametrally magnetized, three error-types have been identified:
A general variation of the flux-density’s strength
of up to 11.6%, a maximal local, angle deviation at the
magnet’s outer borders of 8 ◦ and local errors of up to
9.1%.
II. MAGNETIZATION MEASUREMENT TEST-BENCH
In order to obtain reliable data about possible magnetisation
errors, a test bench for the evaluation of surface
magnets has been built. In the following the sensor
selection (sec. II-A) and the test-bench construction (sec.
II-B) are described.
A. Sensor selection
Typical methods to measure the magnetic flux-density
are Hall-sensors and Helmholtz-coils. In this paper, a
Hall-sensor as depicted in fig. 1 has been selected, due
to the following reasoning:
For best results, both methods require that the measured
magnetic field is oriented perpendicular to the
measuring coil respectively Hall-sensor. This can be
easier accomplished for larger sensors than for very small

devices. Hall-sensors can be miniaturized due to the fact
that an interaction with a given current is measured.
Therefore the concomitant reduction of the Hall-constant
CH, being a consequence of a reduction in material
volume, can be compensated to certain extents with an
increase in the measurement current (fig. 1). This allows
to measure field components nearly pointwise.
d
ϕ1
I
B
ϕ2
Fig. 1. Hall-sensor and its distinctive input sizes.
Helmholtz-coil configurations – in contrast to Hallsensors
– always measure the the overall magnetic fluxdensity.
Due to this integration over the magnet’s surface
flux-density, however, a pointwise selective resolution of
the magnetic field is no longer possible. Global angle
offsets in the magnetization can be detected with both
measurement methods by either using multiple sensors
respectively coils or by turning the magnet under test.
For this purpose, coils are preferable, because their
orientation is better adjustable and an integration over
all local values for a single angle value is implemented
intrinsic in the coil. Local angle errors however cannot
be detected using such a setup. Lastly, coil measurements
are less noise sensitive because the integration already
smoothes some measurement noise.
The decisive factor for Hall-sensors was the interest
in local magnet variations, since most publications until
now focus only on global magnet variations [6], [7] in
electrical machines. Furthermore, this selection allows
the analysis of possible locational misalignments of the
magnets and will enable a later use of the measured
variations in conformal mapping Ansatz functions [8],
[9].
B. Test-bench construction
For the construction of the magnet test bench, Hallsensors
of the type HE-244 [10] were selected. Table II-B
summarizes the main features of the selected sensor:
TABLE I
PROPERTIES OF THE USED HALL SENSOR.
value unit
supply current up to 10 mA
sensitivity 90 to 190 V / (A · T)
linearity
hall voltage typical ≤ 0.2 %
Three sensors for the measurement of the magnetic
field components Bx, By and Bz are located on an index
- 306 - 15th IGTE Symposium 2012
arm with predefined 90 degree edges, in order to achieve
a good positioning. The sensors are positioned directly
on adjacent edges to measure the field at approximately
one point as depicted in fig. 2.
y x
z
Fig. 2. Positions and labelling of the used Hall-sensors on the
measurement anchor.
The index arm itself is mounted on a gibbet, which
is constructed in such a way, that it allows a position
adjustment in all three dimensions. Below the index arm
the magnets under test can be mounted upon a cylindric
shaft which rotates around its symmetry-axis (fig. 3, 4).
z
encoder
y
x
step motor
magnet mounting
rotation axis
hall sensor
magnet under test
Fig. 3. Schematic scetch of the created test bench for magnet
measurements.
This allows the use of a connected stepper-motor to
measure the field along a circular line over the magnet’s
surface. To avoid field distortion by flux guidance all
relevant test bench components have been constructed
from aluminium. Data acquisition and the stepper-motor
control are implemented using a dSpace-system in combination
with a PC.
III. RESULTS
In this study 52 magnets with diametral magnetization
and a field strength of Br = 1.04T were analysed,
consisting of two equally sized groups with either northor
south-pole on the outer magnet circumference. For
each magnet, the Hall-voltage of the radial outwards
pointing flux-density was measured 1.5mm above the
magnet’s surface. The magnet’s dimensions are given in
fig. 5.
A. Simulations
In the simulations, the magnet (as depicted in fig. 5)
is surrounded by an air layer which measures ten times

Fig. 4. Photograph of the constructed magnet test bench.
Br =1.04T
3mm
Fig. 5. Dimensions of the measured magnets.
15mm
the magnet’s height in every direction [11]. The applied
solver implements the magnetic vector-potential formulation.
All boundaries were set as Neumann conditions. The
radial flux-density was sampled along a circumference of
1.5mm above the magnet.
B. Measurements
1) Repetition measurements:
Repetitive measurements were executed to determine the
test-bench’s measurement reproducibility. The average
error between two arbitrary measurements of the same
magnet is below 0.5% and mainly caused by very small
positioning errors of the magnet in the tangential direction
of the measurement shaft. Fig. 6 depicts five
repetitive measurements of magnet #7.
2) Post-processing of measurements:
For data acquisition, every magnet is inserted, measured,
and removed from the test-bench five times (fig. 6).
Afterwards, the repetitive data of each magnet data are
scanned for obvious misplacement errors. If they exist,
the worst deviating measurement is removed. Thereafter,
- 307 - 15th IGTE Symposium 2012
V(Brad)[V ]
2
0
−2
−4
−6
200 220 240 260 280 300 320 340
angle [ ◦ −8
]
Fig. 6. Five repetitive measurements of magnet #7, showing the testbench’s
reproduction quality.
the repetitive measurements are aligned to have their
outer minima centred at around fixed value. Ultimately,
the remaining, centred flux-density values of the magnet
are averaged. Fig. 7 shows – for the purpose of demonstration
exaggerated – examples of the described process.
raw measurements
delete errors
x-align measurements
average
Fig. 7. Post-processing of measured flux-density curves.
3) Variation measurements:
Figure 8 presents the results of the variation measure-

ments for all magnets which have their north pole located
on the outer side. Two obvious variations can be directly
identified:
• Strength variations in the overall remanence fluxdensity
per magnet,
• Strong deformations from the expected curve shape
in terms of local variations.
V(Brad)[V ]
8
6
4
2
0
−2
200 220 240 260 280 300 320 340
angle [ ◦ ]
Fig. 8. Measured radial flux-density 1.5mm above each magnet’s
centre in the magnet group ’north-up’.
Fig. 9 shows accordingly the likelihood of occurrence
for the radial outwards pointing flux-density over the
magnet angle for the opposite magnet group. Due to the
envelope shape of the resulting curve, the strong influence
of the variations is even more obvious.
Fig. 9. Probability of measured magnetisation strength, probabilities
ranging from low (dark) to high (light).
C. Comparison of measurements and simulations
In order to quantify the strength of the occurring
deviations in terms of changes in excitation (in contrast
to changes in the resulting flux-density), the excitation
of each magnet had to be reconstructed from the given
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measurements. To solve this inverse problem [12], a
straightforward approach was to compare the measured
radial flux-density component of each magnet to a set
of simulations. In these simulations, the magnet’s remanence
flux-density Br was varied as parameter ξ1,
applying the simulation conditions presented in section
III-A. However, the resulting shapes did not agree to
the measured curves. The employed magnetisation model
was therefore extended to include a second deviation
parameter ξ2, allowing an angle spread in magnetisation
as given in fig. 10 and yealding the excitation given in
eq. 1:
⎛
B(Δα, ξ1,ξ2) =Br(ξ1) · ⎝ cos(αmid
⎞
+Δα(ξ2))
sin(αmid +Δα(ξ2)) ⎠ (1)
0
Δα
Fig. 10. Determined second deviation parameter ξ2 (grey) from the
ideal, unidirectional magnetisation.
Applying both variation types, the magnet excitation
parameters could be reconstructed sufficiently in most
cases using the least-square minimization from eq. 2 for
parameter determination:
min
ξ1,ξ2
310 ◦
α=230 ◦
[Brad,sim(α, ξ1,ξ2) − Brad,mes(α)] 2
(2)
Fig. 11 shows the comparison of the measured radial
flux-density (dashed) in comparison to the best fitting
simulated curve (solid). The divergence of both curves at
V(Brad)[V ]
8
6
4
2
0
−2
200 220 240 260 280 300 320
angle [ ◦ −4
]
Fig. 11. Measured (dashed) radial outwards pointing flux-density in
comparison to its best fitting siumlation for magnet #1.

the outer side of both graphs can safely be neglected here,
because they are caused by effects of the 2D-simulation
and are considered as not relevant, as this area is not
above, but beside the magnet.
Figure 12 finally shows the comparison of measured
and simulated radial outwards pointing flux-density for a
magnet having a local magnetisation error. As the graph
clearly shows, this behaviour cannot be reproduced by the
applied model yet. The three identified error-types finally
have been identified to: flux-density’s strength variations
of up to 11.6%, a maximal local, angle deviation at the
magnet’s outer borders of 8 ◦ and local errors of up to
9.1%
V(Brad)[V ]
8
6
4
2
0
−2
200 220 240 260 280 300 320
angle [ ◦ −4
]
Fig. 12. Measured (dashed) radial outwards pointing flux-density in
comparison to its best fitting siumlation for magnet #13. Local errors
cannot be reproduced yet.
IV. CONCLUSIONS
The presented methodology allows an accurate determination
of remanence flux-density variations above the
surface of a set of magnets or rotors. A comparison of
the measured curves with the magnet’s simulated and
intended remanence flux-density reveals, in which way
the used FE-magnet-models have to be adopted to be
used in stochastic considerations of parameter variations
in electrical machines. Necessary implementations are a
scalable magnetization strength and an over the magnet
changing deviation angle. Optional, local errors can be
considered as well. The resulting magnet parameters
finally can be used for uncertainty propagation applying
appropriate tools as stochastic collocation [13] or polynomial
chaos approaches [14] to propagate the magnet
deviations onto output sizes of interest.
V. ACKNOWLEDGEMENT
The results presented in this paper have been developed
in the research project Propagation of uncertainties
across electromagnetic models granted by the Deutsche
Forschungsgemeinschaft (DFG).
- 309 - 15th IGTE Symposium 2012
REFERENCES
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of tolerances in robust magnets design,” IEEE Transactions on
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[3] K.-C. Kim, S.-B. Lim, D.-H. Koo, and J. Lee, “The shape
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[4] D. Torregrossa, A. Khoobroo, and B. Fahimi, “Prediction of
acoustic noise and torque pulsation in pm synchronous machines
with static eccentricity and partial demagnetization using field
reconstruction method,” IEEE Transactions on Industrial Electronics,
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[5] G. Heins, T. Brown, and M. Thiele, “Statistical analysis of the
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permanent magnet motors,” IEEE Transactions on Magnetics,
vol. 47, no. 8, pp. 2142 –2148, aug. 2011.
[6] F. Jurisch, “Production process based deviations in the orientation
of anisotropic permanent magnets and their effects onto the operation
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107), no. 1, pp. 255–261, 2007.
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evaluation of non-ideal manufacturing process on the cogging
torque of a permanent magnet excited synchronous machine,”
COMPEL, vol. 30, no. 3, pp. 876–884, 2011.
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in conformal mapping modeling of a permanent magnet
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[9] D. Zarko, D. Ban, and T. Lipo, “Analytical calculation of magnetic
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November 2011.
[11] P. Offermann and K. Hameyer, “Non-Linear stochastic variations
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Phenomena in Nonlinear Circuits,EPNC 2012. Pula,
Croatia: PTETIS Publishers, June 2012, pp. 21–22.
[12] A. Mohamed Abouelyazied Abdallh, “An inverse problem based
methodology with uncertainty analysis for the identification of
magnetic material characteristics of electromagnetic devices,”
Ph.D. dissertation, Ghent University, 2012.
[13] E. Rosseel, H. De Gersem, and S. Vandewalle, “Nonlinear
stochastic Galerkin and collocation methods: application to a
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[14] B. Sudret, “Uncertainty propagation and sensitivity analysis
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BLAISE PASCAL - Clermont II, Ecole Doctorale Sciences pour
l’Ingenieur, 2007.

- 310 - 15th IGTE Symposium 2012
Potential of Spheroids in a Homogeneous
Magnetic Field in Cartesian Coordinates
Markus Kraiger∗ and Bernhard Schnizer †
∗Institute for Radiopharmacy - PET Center, Helmholtz-Zentrum Dresden - Rossendorf e.V., Bautzner Landstr. 400,
D-01328 Dresden - Schönfeld/Schullwitz, Germany. Email: m.kraiger@hzdr.de
† Institute for Theoretical Physics - Computational Physics, Technische Universität Graz, Petersg. 16, A-8010 Graz,
Austria
E-mail: schnizer@itp.tu-graz.ac.at
Abstract—The potential and the field of a prolate or an oblate magnetic spheroid in a static homogeneous field are computed
and expressed in Cartesian coordinates. The directions of both the primary magnetic field and of the symmetry axis are
completely arbitrary. These expressions are used to investigate trabecular structures built from spheroids having different
symmetry axes and positions for Magnetic Resonance (MR-) Osteodensitometry.
Index Terms—Prolate or oblate spheroid in homogeneous field, building flexible models for magnetic resonance imaging or
spectroscopy.
I. INTRODUCTION
In gerneral, the potential of a magnetic spheroid in a given
external magnetic field is derived in spheroidal coordinates,
whose symmetry axis is the z-axis. Models of biological tissues,
as e.g. trabecular bones, are arrays of such spheroids with symmetry
axes having various directions. Having such applications
in mind, we derived potential and field expressions for prolate
and oblate spheroids in a homogeneous field. These expressions
depend on Cartesian coordinates for arbitrary directions of both
the field and the symmetry axes.
II. METHOD OF SOLUTION
A spheroid (permeability μi = μ0(1 + χi); semi-axes
a, a, c) is in a medium (permeability μe = μ0(1 + χe))
and a static homogeneous field H0 = (H0x,H0y,H0z) =
H0(sin β cos α, sin β sin α, cos β) of arbitrary direction. At
first the problem of a prolate spheroid is solved in prolate
spheroidal coordinates ([1], Fig.1.06)
x + iy = ep sinh η sin θe iψ
(1)
z = ep cosh η cos θ
or in the corresponding oblate spheroidal coordinates ([1],
Fig.1.07)
x + iy = eo cosh η sin θe iψ
(2)
z = eo sinh η cos θ.
for an oblate spheroid as shown e.g. in [2] to [4]. The particular
solutions of the potential equation are obtained by separation
giving Legendre functions and polynomials of cosh η, i sinh η
respectively multiplied by Legendre polynomials of cos θ and
by trigonometric functions of ψ. A solution of this problem
is found by the usual method, namely by expanding the
potential in the interior and in the exterior of the spheroid
w.r.t. the particular solutions fulfilling the appropriate boundary
conditions: i) the total potential must be finite at η =0; ii)
the total potential must agree with that of the primary field
(5) at η = ∞. The expansion coefficients are determined
by the continuity conditions that the total potential must be
continuous Φ0 +Φ σ e =Φ0 +Φ σ i and the corresponding normal
component of the magnetic induction must be continuous at the
interface of the two media ((7) with n = ez). The solutions
contain only Legendre funtions and polynomials of order 1
since the inhomogeneity (5) is of that order. Thereafter the
Legendre functions and polynomials may be replaced with
elementary functions of η and θ. These may be in turn expressed
by functions of Cartesian coordinates by use of (1),
(2) respectively and by cosh η = up(r, ez)/ √ 2, sinh η =
uo(r, ez)/ √ 2, eq.(24) respectively. The expansion coefficients
L σ 0 ,L σ 1 ,M σ 0 ,M σ 1 obtained from matching the two pieces of
the potential at the interface are first expressed in Legendre
functions and polynomials of argument ηp,ηo respectively:
ηp = Arcoth(cp/ap) (3)
ηo = Artanh(co/ao). (4)
The coefficients are also reexpressed in elementary functions
of these geometrical parameters and by the magnetic susceptibilities
χe,χi to give eqs.(8) to (11), (13) to (16) respectively.
In the last step the potential in both domains is transformed to
an arbitrary direction n of the spheroidal symmetry axis. All
vectors in the potential are decomposed into vectors parallel to
or perpendicular to the z-axis. Finally all vectors ez occuring
in these expressions are replaced by n.
This description is rather concise; full details may be found
in the papers [3] and [4] and in the notebooks at the website
quoted. But the next paragraph gives a complete listing of all
formulas needed for the applications.
III. RESULTS
The primary field is homogeneous with the potential
Φ0(x, y, z) = − (H0x x + H0y y + H0z z). (5)
A. The potentials of the reaction fields
The presence of a spheroid induces a reaction field with
potential (r =(xβ)) :
Φ σ k(x, y, z) =
3X
α,β=1
H0αt σ,k
αβ xβ = H0 · T σ,k · r (6)
with σ = p (= prolate) or = o (= oblate) and k = e (= external)
or i (= internal) to the ellipsoid
Eσ := r2 − (n · r) 2
a 2 σ
+ (n · r)2
c 2 σ
=1. (7)

For p a prolate spheroid, ap < cp, the excentricity is ep =
c2 p − a2 p; for an oblate one, co

the additional contribution, originating from the local field
inhomogeneities, to the effective transversal relaxation rate R ∗ 2.
Further, R ′ 2 ≈ γΔB with ΔB representing the field variation
and γ the gyromagnetic ratio.
B. Theory: Computersimulation
The aim of the current simulation is to investigate effects on
the induced line broadening of the resonance spectra evoked
through micro cracks as examples of trabecular rarefaction.
Thus, the evaluation of the magnetic field distribution was
performed utilizing a two-compartment model, consisting of
marrow and bone. In oder to mimic the known trabecular micro
structure within a vertebra [13] prolate ellipsoids were arranged
appropriately within a three-dimensional unit cell.
The precession frequency of spins in a homogeneous magnetic
field is determined through the magnetic induction B.
Hence, in a first step the reaction fields induced by the susceptibility
difference between the ellipsoids (trabeculae) and the
background (bone marrow) were computed [14].
Introducing a sample with a different susceptibility, in the
current experiment trabecular bone (χ2) is surrounded by bone
marrow (χ1), the resulting magnetic induction Bz can be
generally written as:
Bz = μ (H0z + Mz (r)) = μ0(1 + χ)(H0z + Mz (r)) , (31)
with Mz characterising the induced reaction field. Herin the
units are given in the MKS-system, and susceptibility units are
per unit volume.
Since the transversal magnetization decay of mineralized
bone is several magnitudes faster comparing to bone marrow,
the received resonance signal in MR-Osteodensitometry is governed
by the magnetization arising within the marrow. Thus Mz
corresponds to the computed reaction fields ΔHr1,z caused by
the difference in magnetic property between bone and marrow.
The resulting magnetic field distribution within the unit cell
was determined as the sum of the individual contributions Hzi
originating from all ellipsoids n:
nX
ΔHr1,z (r) = Hzi (r) . (32)
i=1
Interactions between the trabeculae have been neglected. This
assumption is valid, since interactions between such structures
include susceptibility effects of the second order, which will
give rise to field contributions of the order of H0 (Δχ) 2 ,or
≈ H0 · 10 −12 .
In a simple MR experiment, excitation followed by an
acquisition period, the signal of the free induction decay (FID)
can be written as:
S(t) =const
Z
VOI
with ω(r) =γBz(r) it follows:
Z
S(t) =const
VOI
d 3 r e −iω(r)t e −T2/t ; (33)
d 3 r e −iγBz(r)t e −T2/t . (34)
Using again expression (31) the following expression in
ΔHr1,z can be found:
Z
S(t) =const
VOI
d 3 r e −iγtμ0(1+χ)(H0z+ΔHr1,z(r)) e −T2/t .
(35)
This integral must be extended over the entire unit cell enclosing
the ellipsoids.
In order to compare the simulation results with MR images
the magnitude of S(t) must be found. Except for the dissipative
relaxation phenomenon e −T2/t
the expressions in (35) are
- 312 - 15th IGTE Symposium 2012
purely oscillatory in H0z. Hence, for the analysis of the signal
course the essential decay can be expressed as:
Z
|S(t)| = const d 3 r e −iγtμ0(1+χ)ΔHr1,z(r)
. (36)
VOI
ΔHr1,z(r) can be computed according to (32) as the sum
over all the reactions fields of the individual ellipsoids, where
μ0(1+χ) describes the magnetic permeability at the location r.
1) Algorithm: Utilizing the expression developed for the
reaction field (28) the simulation was implemented in Mathematica
(Wolfram Research, Inc.). The program computed the
field distribution of ΔHr1,z(r) in the sense of a histogram and
generated the MR signal curve according to (36).
As input parameters the spacing of the trabeculae in x-,
y- and z-direction, the dimensions of the ellipsoids and the
position of the symmetry axis with respect to the z-axis
of the coordinate system had to be defined. Further, the
susceptibilities of the bones and the background as well as the
orientation of the applied homogenous main magnetic field had
to be set. The results of the simulations were the histograms
of the magnetic field distribution and the signal curve, which
was further utilized within a fitting-procedure yielding the
relaxation constant R ′ 2.
2) Data fitting: Utilizing the simulated signal curves a
exponential signal model was applied in order to approximate
the relaxation time T ′ 2 [15]. The computed signal intensities (36)
at the echo times ranging from 0 to 50 ms, 5 ms increment, were
used to generate a single T ′ 2 value by means of a non linear
least-squares-approximation to a two parameter fit function:
S(t) =Ae −t/T ′ 2 . (37)
C. Model of vertebra
The three-dimensional unit cell was composed out of thirty
prolate ellipsoids, fifteen aligned along the x- and z-direction
each, mimicing the initial intact trabeculae. The interruptions
were simulated in the way, that each trabecula was replaced by
two ellipsoids, which were displaced along the x/z-axis by 50
μm forming a crack. The configuration of the three-dimensional
vertebra model and the applied parameter setting are given in
Fig.1.
Fig. 1. Depiction of the 3.75 × 3.75 × 3.75 mm 3 unit cell; the
x/z aligned sets are built up of three planes displaced by 750 μm.
The trabeculae in each plane were modelled with a trabecular spacing
and width of 500 μm and 120 μm respectively. The trabecular micro
fractures were simulated by replacing each of the intact trabeculae with
two opposed shifted versions.

D. Results
The resulting reaction fields Hr1 pre- and post bone rarefaction
are depicted in Fig.2. Note, that the field distribution is
directly affected by the shape of the micro cracks, whereby the
resulting field inhomogeneities in the vicinity of the spiky edges
lead to the observed major field broadening. Prior rarefaction,
the inital field distribution ranged approximately around ±1
A/m, afterwards field values from almost ±2 A/m were found
within the three-dimensional vertebra model. The effect of the
interrupted bone mesh on the MR signal decay and the resulting
estimated relaxation time T ′ 2 is presented in Fig.3. The modelled
cracks gave rise to a change of the initial T ′ 2 of 26.1 ms to
approximately 14.4 ms.
Fig. 2. Resulting field distribution of the reaction field Hr1,z within
the applied three-dimensional vertebra model. The trabecular cracks
causing a broadening of the distribution, resulting in a more Lorentzian
like line shape. A main magnetic field H0 =2.38732 · 10 6 A/m with
α =30 ◦ and β parallel z-axes, and values of χ1 = −0.62·4·π ·10 −6
and χ2 = −0.9 · 4 · π · 10 −6 were applied.
V. CONCLUSION
The advantage of this new approach is that it is very easy
to build and investigate structures built from spheroids with
different axes and positions. There is no need of complicated
coordinate transformations.
The analytical solutions of the Laplacian potential problem
of spheroids in Cartesian coordinates were successfully applied.
Fig. 3. Resulting resonance signal decays affected by the reaction field
Hr1,z of the vertebra model in the two situations. As a consequence
of the increasing inhomogeneous reaction field a rapid signal decay in
case of micro cracks is visible (green curve). The signals are normalized
to the values at the first echo time TE, markers are indicating the
computed signal values at TE.
- 313 - 15th IGTE Symposium 2012
A three-dimensional magnetostatic problem in the area of MR-
Osteodensitometry, susceptibility effects in the vicinity of micro
cracks, was analysed. Within vertebrae affected by pathologies
such as osteoporosis horizontally arranged structures get typically
interrupted at first. The novel expressions make it possible
to study the bone rarefaction along such pathologies, whereby
either cracks of the horizontal, the vertical or arbitrary structures
are accessible for modelling.
In the present work just one application of the analytical
expressions, the modelling of bone disorders in the area of
MR-Osteodensitometry, was given. For example in the field
of functional MRI the devoloped toolbox eases the analysis of
the BOLD (blood oxygenation level-dependent) contrast, where
induced reaction fields in the surrounding of vascular networks
are of great interest [16]. A fast and precise computation of the
magnetic distortion is essential for improving the precision of
the temperature determination in techniques using the proton
resonance frequency (PRF) shift method [17], [18]. Temperature
mapping in the vicinity of the needle electrode is a crucial
determinant of MRI guided interventional radiofrequency ablations
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magnetic susceptibility. [20].
In summary, the authors believe that the novel formulation
of solutions depending solely on the Cartesian coordinates will
facilitate the modelling of countless magnetostatic problems.
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- 314 - 15th IGTE Symposium 2012

- 315 - 15th IGTE Symposium 2012
Application of Signal Processing Tools for Fault
Diagnosis in Induction Motors-A Review
*Jawad Faiz, *Amir Masoud Takbash, *Bashir Mahdi Ebrahimi and †Subhasis Nandi
*Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer Engineering,
College of Engineering, University of Tehran, Tehran, Iran
†Department of Electrical and Computer Engineering, University of Victoria, Victoria, BCV8W 3P6, Canada
E-mail: jfaiz@ut.ac.ir
Abstract—Use of efficient signal processing tools (SPTs) to extract proper indices for fault detection in induction motors (IMs)
is the essential part of any fault recognition procedure. In this paper, all utilized SPTs employed in fault identification of IMs
are analyzed in details. Then, their competency and their drawbacks for extracting indices in transient and steady-state modes
are criticized from different aspects. The considerable experimental results are used to certificate demonstrated discussion.
Different kinds of faults including eccentricity, broken bars and bearing faults as major internal faults in IMs, are
investigated.
Index Terms—Fault detection, Fast Fourier Transform, Hilbert, Wavelet Transform.
I. INTRODUCTION
Ever increasing application of induction motors (IMs)
and importance of its uninterrupted operation in
production lines make it necessary to diagnose internal
faults in IMs quickly and precisely. The internal faults in
IMs consist of electrical and mechanical faults. Electrical
faults occur in the stator and rotor. Electrical faults of
squirrel-cage rotor of IM consist of bars and end-rings
breakage which are about 10% of the internal faults of
squirrel-cage IM [1]. The reasons for these faults are as
follows:
1. Thermal stresses due to over-load and asymmetrical
dissipation of heat which may change the hot spot.
2. Magnetic stresses arising from electromagnetic
forces.
3. Mechanical stresses due to mechanical fatigue of
different parts, bearing damage, etc.
4. Stresses due to assembling process and centrifugal
forces arising from shaft torque.
Some impacts of broken bars on IM are: Increasing
core losses and total losses in faulty machine [2]-[4] and
asymmetrical vector diagram of rotor current [2]. Broken
bars and end rings faults have been studied more than
other internal faults of IM. The faulty motor is studied
using experimental or modeling and simulation methods.
Following analysis of faulty motor by test or modeling, a
proper signal must be selected. The signals used in the
fault diagnosis process, consist of mechanical and
electrical signals. There are three following reasons that
make the stator current an appropriate signal for fault
diagnosis:1) unique effect of motor internal fault on this
signal. 2) there is no need to have sensor for monitoring
the signal. 3) this method is economical. After testing or
modeling motor and selecting a proper signal, it must be
processed and effect of the proposed fault upon the signal
is determined. The signal processing methods are based
on the mathematical transformations. The well-known
transformations are Fourier, Wavelet, Hilbert and
multiple signal classification (Music). These processors
are widely used in the fault diagnosis; however, recently
intelligent methods such as Genetic, Fuzzy and Neural
Network algorithms have been applied to make fault
diagnosis methods more efficient [5]-[7]. Thus
considering fault type, load conditions and the proposed
processor characteristics, a particular processor will be
suitable for each case. To choose an appropriate
processor, different faults and operating conditions such
as load are considered.
II. FAST FOURIER TRANSFORM
Fourier transform expresses signal as a sum of
sinusoidal functions. This transform expresses a timedomain
signal to frequency-domain signal. This transform
determines the frequency components arising from the
fault. In application of Fourier transform to a signal, the
signal must have two basic features: stability and
alternating. A fast Fourier transform (FFT) is a faster
version of the discrete Fourier transform
(DFT). Application of these processors includes sampling
and applying Fourier transform. Sampling has a series of
rules and laws as described in [7].
A. Rotor Bars and End Ring Breakage Fault Diagnosis
For broken bars and end ring fault diagnosis in IM,
FFT base processor is often used and frequency spectrum
of torque, speed, instantaneous power, body vibration and
stator current signals are obtained. Torque signal has been
employed as reference for fault diagnosis in [3].
Harmonics 2sfs are produced in the torque frequency
spectrum which used for fault detection [8]. Some
references use the speed signal for fault diagnosis and
here harmonics 2sfs of frequency spectrum are again
proposed. Figure 1 exhibits the frequency spectrum of
speed signal and its variations due to the broken rotor
bars [3]. Torque and speed signals depend on the external
factors such as load and this makes hard to diagnose the
fault. Also the procedure for acquiring these signals is
important, because using sensors and other devices affect
the accuracy of the operation. Another signal that is
considered for fault diagnosis is the case vibration signal
The reason for this vibration is air gap radial

Figure 1: Frequency spectrum of motor speed for
different numbers of rotor broken bars [4].
electromagnetic forces. Broken bars lead to the odd
harmonics in the frequency spectrum of vibration signal.
Although signal with twice supply frequency has been
used for fault diagnosis, this signal is not suitable because
it also appears in the healthy motor vibration frequency
spectrum. The above-mentioned signal depends on the
load and its detection requires a sensor [9]. On the other
hand, Fourier transforms application to the transient
signals such as speed and vibration does not yield
accurate results. Pendulum oscillations and increment of
[10], [11
proposed in [11] with some simplifying assumptions. As
seen, broken bars generate 2sfs
[10]. Instantaneous power signal can be also utilized for
broken bars and end rings fault diagnosis. Advantages of
using instantaneous power spectrum are listed in [12].
Output voltage harmonics of motor following the power
supply interruption can be used to diagnose the fault. The
main idea of this method is eliminating the harmonics
generated by the voltage supply [13]. However, this
signal is a transient signal and FFT application on this
signal leads to inaccurate fault detection. Intelligent
algorithms can be used to diagnose the fault through
current signal envelop [14]. The drawback of this method
is that the harmonics of the envelop signal depends on the
severity of the rotor bar fault as well as their locations
[13]. Current signal has been considered as the most
appropriate signal for internal fault diagnosis. Some
references [15]-[17] use time signal of the line current for
fault diagnosis but this fault is often detected through line
current harmonics [1]-[3]. The most important harmonics
used for fault diagnosis are (1±2s)fs. The amplitudes of
these harmonics are larger than that of other harmonics
and their diagnosis is easier. Amplitude of harmonic (1-
2s)fs depends on the rotor broken bars fault and its
intensity and harmonic (1+2s)fs is mostly depends on the
speed variations [18]. It is noted that harmonic (1-2s)fs
may be disappeared when broken bars has 90 degrees
increased by the broken bars fault. The reason for such
amplitude rising is asymmetry of the rotor due to the fault
and consequently generating a negative rotating field
[22]. Table I shows these harmonics before and after the
- 316 - 15th IGTE Symposium 2012
TABLE I
AMPLITUDES OF CURRENT SIDEBANDS FOR MOTOR WITH DIFFERENT ROTOR
BROKEN BARS [4]
NBB fs+2fr fs-2fr
0 -58 -57
1 -54 -55
2 -53 -48
3 -48 -42
4 -46 -40
Figure 2: Frequency spectrum of stator current for broken
end-ring [23].
fault versus number of broken bars (NBB) [3]. However,
raising the fault degree produces lower changes in the
amplitudes of the sidebands. The reason is increasing the
number of parallel paths of currents and saturation due to
asymmetry of the currents passing the bars. Influence of
bars inner current in the broken bars fault has been
proposed in [23] and its effects consisting of harmonics
amplitude reduction has been mathematically proved. In
[22], broken end-ring has been considered. Figure 2
presents stator current frequency spectrum for such a
case. Starting current signal may be used for fault
diagnosis [3] where broken bars generate harmonics
(3±4k)fr in the current spectrum (Figure 3). Frequency
spectrum of starting transient current signal is determined
STFT in which the problem of processor with the
transient signal is solved. However, dimensions of the
window are fixed and therefore it has not good frequency
and time resolution at the same time [24]. Sometimes
current is indirectly used, for instance Park transform of
stator current has been used for fault diagnosis [25]. Of
course, this method has some drawbacks such as no-clear
fault effect and susceptible to noise, so it seems that
application of this method beside other techniques such as
intelligent methods is useful. However, in this case a set
of full data is necessary. Park transform of stator current
leads to iD+jiQ Modulus and harmonics arising from the
broken bars fault in line current are as 2sfs and 4sfs. The
advantage of these harmonics is that these are far from the
fundamental harmonic so its detection is simple.
B. Impacts of Load Variation
Side-band components vary with the load torque
fluctuations [26], [27]. Figure 4 shows the impact of the
load upon the high and low side-bands of the stator
current spatial vector [21]. Load fluctuation decreases the
amplitude of low-band and increases the amplitude of
high-band.
C. Impact of Drive
In the presence of drive and closed-loop circuits the
situation differs. In PWM-driven motor odd harmonics as

Figure 3: Frequency spectrum of starting current: (a)
healthy motor, (b) motor with 4 rotor broken bars [3].
Figure 4: Impact of load upon high and low side-bands of
stator current spatial vector [21].
well as third harmonic are injected to the motor. These
harmonics are fb=(m±2nks)fs (m=supply odd harmonicorders,
n=odd harmonics due to rotor induced currents
and k=integer number) and odd-order harmonics currents
are induced in the rotor, that subsequently produces oddorder
rotor flux in the air gap. Therefore, a new frequency
pattern is introduced in faulty motors under PWM supply
[28]. In the closed-loop drive, mutual effects of electrical
and mechanical oscillations amplify each other and
amplitude of the above-mentioned frequency spectrum
increases. Figure 5 shows rotor asymmetry signature in
inverter-fed motor line current spectrum for healthy motor
and motor with broken bars [29].
D. Impact of Broken Rotor Bars Location
Rotor bar location and its impact upon the fault
diagnosis have been investigated and reported in [30] and
effect of the broken bars location on the waveform and
frequency spectrum of stator current and side-bands
components (Figure 6) have been given. Influence of the
broken bars location on the amplitude of the torque
harmonics has been pointed out in [28]. Amplitude of
torque harmonic is increased by more concentration of the
broken bars [31].
III. WAVELET TRANSFORM
Wavelet transform is a method that transforms the
signal to time and frequency spectrum. This transform is
based on transforming a signal to different kinds of scaled
and shifted of mother wavelet function [13]. Wavelet
transform enables to show some characteristics of the
- 317 - 15th IGTE Symposium 2012
Figure 5: Rotor asymmetry signature in inverter-fed
motor line current spectrum (a) around fundamental, (b)
around fifth and seventh harmonics [29].
signal such as non-continuity of high-order derivatives of
the function and sharp point of maximum of the function
that cannot be shown by other transforms, because they
eliminate these characteristics during transform [13].
Considering the above-mentioned points, wavelet
transform gives a detailed and fully localized view of the
function. Having frequency components caused by the
internal fault of the motor, this transform can concentrate
on particular regions and this can enhance the precision,
while Fourier series provides a general view over a period
of signal [12].
A. Rotor Bars and End Ring Breakage Fault Diagnosis
Various wavelet transforms have been so far used for
fault diagnosis. Most of these methods are based on the
sidebands components of frequency spectrum of the
current signal. In [28], energy of a bandwidth is used to
diagnose the fault in which the load impact is also taken
into account. Since discrete wavelet transform (DWT) has
a better clarity over the low frequencies, the use of the
current spatial vector which has harmonics with lower
frequencies will yield more precise results [32]. In [33], a
method based on CWT has been used to diagnose the
fault in different drives. However, there is no physical
interpretation for fault diagnosis using the Figurers. In
[34], power spectral density (PSD) values of details signal
in any level of transform is fault diagnosis criterion.
Figure 7 shows the pattern of current signal wavelet
transform of healthy and rotor broken bars motor [34]. In
[35], the reason for application of DWT in the papers has
been noted. There are not suitable physical description for
results, complicated trend and algorithm of other wavelet
methods and ambiguous results. In [35], fault has been
diagnosed using envelope of the starting current signal
and procedure has been introduced for extracting the
envelope signal considering the impact of the broken bars
on the settling time and amplitude of the envelope of the
starting current. Also determination of wavelet main
function is important in fault diagnosis. Harmonics due to
torque ripples and unbalanced voltage generate harmonics
similar to that of the broken bar and this reduces the

- 318 - 15th IGTE Symposium 2012
Figure 6: Impact of bars location on amplitude of sidebands; (a) three broken bars in one pole and one broken bar in
adjacent pole, (b) Two broken bars in one pole and two broken bars in adjacent pole, (c) One bar under each pole [30].
a b
Figure 7: Pattern of current signal wavelet transform, (a)
healthy, (b) rotor broken bar motor [34].
accuracy of the fault diagnosis process. However, this can
be solved by application of DWT transform [36]. The
analytical wavelet transform (AWT) is one of the wavelet
transforms which has been used to diagnose the rotor
broken bars fault. Advantage of this wavelet transform is
keeping the characteristics of time domain, amplitude and
phase as well as frequency. Amplitude is related to the
proposed signal envelope and the phase is related to the
time characteristics of the signal. In [37], AWT has been
used to diagnose the rotor broken bars fault by the help of
starting signal of the IM under low level loads.
B. Impacts of Load Variation
Impact of load fluctuations on wavelet coefficients of
the stator current spectrum of a motor under broken bars
fault has been studied in [38]. Table II summarizes the
variations of D4 coefficient and values of a function (that)
defined in the reference. The un-decimated discrete
wavelet transform (UDWT) is a type of DWT in which
shift- invariant has been included. This leads to a good
time precision over high frequency harmonics, and good
time and frequency precision over low frequency
harmonics. In addition to DWT and CWT, there is
another wavelet called wavelet packet decomposition
(WPD), which yields more precise results but it is time
consuming method [37], [39]. Sidebands move to higherorder
nodes WPD transform due to load fluctuations [39].
One important point in the application of this transform is
the use of a proper node for fault diagnosis. For high
loads the low-order nodes and for low loads high-order
nodes are investigated [39]. In [40], the impact of the
drive in broken bar diagnosis using wavelet transform has
been proposed. Although fault diagnosis procedure and
load impact have been considered in the above-mentioned
reference, the location of the broken bars has not been
taken into account.
TABLE II
VARIATIONS OF D4 COEFFICIENT OF WAVELET TRANSFORM OF CURRENT
SIGNAL OF MOTOR UNDER BROKEN GAR FAULT AGAINST LOAD [38].
% of rated
load
Mean Current
(A)
Mean distortion in D4 Index2
0 9.54 0.0923 0.97%
33 8.92 0.3220 3.61%
71 8.81 0.4044 4.59%
100 8.74 0.5469 6.25%
133 8.57 0.5674 6.62%
IV. HILBERT PROCESSOR
Due to the drawbacks of the above-mentioned
processors some new processors such as Hilbert processor
have been introduced. Hilbert transform similar with
Fourier transform is orthogonal in respect of its main
transform. In addition one function and its Hilbert
transform has identical energy. One type of Hilbert
transform is the Hilbert- Huang transform (HHT). In
HHT the energy distribution in time-frequency domain is
obtained by estimation of the local energy of signal in
different times and frequencies. On contrary to other
time- frequency transforms which depend on the size of
window and sampling frequency in Fourier transform and
mother wavelet in wavelet transform, HHT is independent
on aforementioned parameters, that is an advantage of this
transform. No-load and light load cases in rotor broken
bars and end-rings have been emphasized in [41].
A. Rotor Bars and End Ring Breakage Fault Diagnosis
In no-load IM there is no harmonic arising from the
load, but harmonics are very close to the fundamental
frequency. Here a Hilbert vector is defined for signal and
using this vector instead of the proposed signal has some
advantages:
1. Requirement of phase current.
2. Generation of harmonic components due to fault and
deletion of non-applicable harmonics.
Figure 8: Hilbert modulus: (a) healthy motor, (b) motor
with two broken bars [42].
3. Elimination of frequency scattering.
4. Nonexistence of fundamental frequency that lets to

use linear scale on the vertical axes instead of
logarithm scale and this clarifies the graphs.
5. No need to sample with twice of Nayquest
frequency.
Considering the proposed low frequencies the sampling
speed is reduced to 0.1 of the normal case values and this
is useful in practice. In [42], a fault diagnosis method
based on fundamental harmonic deletion and
determination of Hilbert Modulus has been introduced.
Figure 8 shows Hilbert modulus for a healthy motor
and a motor with broken bars. By increasing the fault
degree, this modulus becomes larger due to the
harmonics. In the following part a dimensionless
numerical criterion with a low dependency on the load is
introduced.
V. MUSIC PROCESSOR
Recently some methods with high frequency precision
such as Music and Root-Music have been proposed [43].
These methods are used where keeping a particular
frequency is necessary. So in these methods, precise
information on frequency components are required [19].
This mathematical transform is similar with other
transforms and it converts a signal to sum of several
signals with identical feature. Music transform consists of
transform of a signal and expressing it based on K
complex sinusoidal pair and a signal e(n).
C. Rotor Bars and End Ring Breakage Fault Diagnosis
Combination of FFT and Music methods and a
method of fundamental harmonic deletion have been used
for fault diagnosis in [44]; because lonely application of
Music leads to error. This method provides clearer results
compared to FFT method. In Figure 9 the results of two
methods have been compared [44]. In [45], frequency
spectrum of output voltage after disconnecting the input
supply obtained by FFT and Music methods and they
compared. It has been shown that a series of particular
harmonics in the frequency spectrum is excited due to the
broken bars and variations of these variations are seen.
Reliability and low impact of noise are the advantages of
this method compared to FFT method [45]. Music- based
methods similar with Hilbert transform are new methods
which have precise results and computation is quicker
than Wavelet method. However, it needs improved
algorithms for deletion of the fundamental harmonic
which complicates these methods. Since these methods
are new, many fault diagnosis indexes have not been yet
modeled by these methods.
VI. CONCLUSIONS
Different methods and processors used for diagnosis
of rotor bars and end ring breakage fault in IMs were
investigated briefly. At this end, four types of processors
and their advantages and drawbacks were studied. It is
clear that a single method and a common processor
cannot be specified for fault detection in all conditions.
Fourier processor as a most applied processor for broken
- 319 - 15th IGTE Symposium 2012
Figure 9: Comparison of FFT and Music-based methods
for motor with 1 broken bar: (a) FFT, (b) Music [44].
bars fault has weak and strong points. Its most important
weakness is in processing of transient signals. To
overcome this problem, application of wavelet processor
was suggested which provide more detailed time and
frequency view of the signal. Following wavelet packet
with simultaneous high precision of time and frequency is
commonly used. These processors often are used for
broken bars fault but there are no appropriate researches
about number of broken bars and their location. Other
drawbacks of this method are that it is time consuming
and complicated. In recent years, Hilbert-based methods
with high frequency precision methods such as Music
have been proposed. A common point that must be taken
into account in an appropriate fault diagnosis method in
industry beside on-line case; is the method must quick
and at the same time have good accuracy.
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- 320 - 15th IGTE Symposium 2012
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Sept. 2008.0

†
- 321 - 15th IGTE Symposium 2012
Experimental Calibration of Numerical Model
of Thermoelastic Actuator
*L. Voracek, *V. Kotlan and *B. Ulrych
*University of West Bohemia, Faculty of Electrical Engineering, Univerzitní 26, 30614, Pilsen, Czech Republic
Abstract—A numerical model of the thermoelastic actuator for accurate settings of position is compared with the data
obtained by measurements on an experimental prototype. Some disagreements between the results became the reason for a
calibration of the model carried out using an appropriate iterative process.
Index Terms—Actuator, FEM, Measurement, Thermoelasticity.
I. INTRODUCTION
Various industrial technologies work with extremely
small and accurate shifts on the order of 10 –3 to 10 –6 m.
One way of reaching that small shifts and exact positions
is using a thermoelastic actuator. Its theoretical
backgrounds are described in previous papers written by
our group (i.e., [1], [2]). Recently, we also built a
prototype of the device and started experimental verifying
the calculated results. Certain discrepancies between the
computations and measurements lead to necessity of
calibrating the material parameters and their temperature
dependences.
II. FORMULATION OF THE PROBLEM
The arrangement of the device is depicted in Fig. 1.
The dilatation element 2 made of a suitable electrical
conductive metal is inserted into a coil 3 fixed in frame 4.
The coil is supplied by harmonic current. The whole
system is placed in an insulating shell 1. The device is
clamped by its bottom part 5 in the basement 6 that is
supposed to be perfectly stiff. The time-variable magnetic
field generated by the field coil 3 induces in the dilatation
element 2 eddy currents. These eddy currents produce
heat and consequent geometrical changes (mainly in its
longitudinal direction z ) of the dilatation element.
r
6 5 4 3 2 1
Figure 1. The basic arrangement of the device
1 –shell, 2 – dilatation element, 3 – field coil, 4 – fixing frame, 5 –front,
6 – stiff wall
III. MATHEMATICAL MODEL
Mathematical modelling of the device represents a
triply coupled problem. Its mathematical model consists
of three partial differential equations describing the
distribution of magnetic field, temperature field and field
of thermoelastic displacements.
The device does not contain any ferromagnetic part. If
the field coil 3 (Fig. 1) carries harmonic current of
z
density J , the magnetic field in the system may be
ext
described by the Helmholtz partial differential equation
for the phasor A of the magnetic vector potential A in
the form [3]
curlcurlA j A J
. (1)
Here, symbol denotes the magnetic permeability, is
the electric conductivity, stands for the angular
frequency, and J is the phasor of external harmonic
ext
current density in the field coil. The conditions along the
axis of the device and artificial boundary placed at a
sufficient distance from the system are of the Dirichlet
type ( A 0 ).
Heat power in the system is generated by currents in
the field coil and induced currents in the dilatation
element. The distribution of the temperature in the system
can be described by equation [4]
T
div grad T cp pJ
,
t
where stands for the thermal conductivity, is the
specific mass, and c p denotes the specific heat at a
constant pressure. Finally, the symbol p stands for the
J
time average internal sources of heat represented by the
volumetric Joule losses. These are given by the formula
2
ext
(2)
J
p J , J J + J , J = j
A , (3)
ext ind ind
where symbol J denotes the current density in the
ind
electrically conductive parts of the device.
The boundary conditions should take into account
both convection and radiation. But since between the
dilatation element 2 and field coil 3 there is a ceramic
tube characterized by a very poor thermal conductivity
and all parts are placed in a Teflon insulating shell
(characterized also by a poor thermal conductivity),
radiation can be – with a practically negligible error –
disregarded.
The last considered field is the field of thermoelastic
displacements. The distribution of displacements in the
dilatation element 2 follows from the solution of the
Lamé equation [5]

grad div
3 2 T grad T
u u
f 0 ,
where 0, 0 are coefficients associated with
material parameters by the relations
E E
1 1 2 2 1
, .
Here E is the modulus of elasticity and denotes
the Poisson coefficient of the contraction. Finally, symbol
u ur , u, u z represents the vector of the displacement,
is the coefficient of the linear thermal dilatability of
T
the material, and f stands for the vector of the internal
volumetric forces. These consist (at least in the dilatation
element 2) of the gravitational and Lorentz volumetric
forces. But in comparison with the thermoelastic strains
and stresses they are very small and may be neglected.
The boundary conditions depend on the particular
arrangement. In the solved case the displacements of the
dilatation element 2 at the place of clamping are assumed
to be equal to zero.
IV. NUMERICAL SOLUTION
The numerical solution was performed by a
combination of professional codes COMSOL
Multiphysics and Matlab that were supplemented with a
lot of own procedures and scripts. Special attention was
paid to the convergence of the results (dependence of the
distribution of physical fields on the density of
discretization meshes and in the case of electromagnetic
field also on the position of the artificial boundary). Some
part of solution was verified with results obtained from
code Agros2D. This is a new program from the category
of FEM based software developed by group at our
department. This program is very strong for the solution
of various physical fields but nowadays it is still in a test
version and some specific parts are not finished yet.
Therefore it cannot be used for the nonlinear systems.
V. ILLUSTRATIVE EXAMPLE AND RESULTS
An illustrative example concerns a prototype of the
thermoelastic actuator built in our laboratory. Its
arrangement and dimensions are depicted in Fig. 2. The
field coil is wound from a copper wire with diameter
D w 0,63 mm and has approximately 2400 turns. This
coil is supplied by a harmonic current of frequencies
519 Hz, 1005 Hz and 2210 Hz, respectively, whose
RMS values were 0.5 A and 1 A. The dilatation element
is made of brass UNS C26000, whose nonlinear
temperature-dependent characteristics of most important
physical parameters are depicted in next Figs. 3–5. Other
parts exhibit mainly the function of electrical and thermal
insulation. All remaining parameters for computation are
listed in Tab. 1. The entire device is fixed in a firm steel
construction for its correct functioning.
- 322 - 15th IGTE Symposium 2012
(4)
(5)
Figure 2: Arrangement of the considered actuator (dimensions in mm):
1–nylon shell, 2–brass core, 3–copper coil, 4–Teflon fixing part of the
coil, 5–nylon stand, 6–metallic stand, 7–ceramic tube, 8–nylon cap
Figure 3: Temperature dependence of thermal expansion coefficient for
brass UNS C26000 (data from [6]).
Figure 4: Temperature dependence of the thermal conductivity for brass
UNS C26000 (data from [6]).
Figure 5: Temperature dependence of the electrical conductivity for
brass UNS C26000 (data from [6]).

Table 1: Physical properties of particular materials for the initial step of
simulation.
brass Teflon ceramic nylon copper
rel. permeability [-] 1 1 1 1 1
rel. permittivity [-] 1 1 1 1 1
el. conductivity [S/m] 1.5e7 0 0 0 5.7e7
therm. cond. [W/(mK)] 115 0.24 1.6 0.26 395
density [kg/m 3 ] 8440 2220 2500 1150 8930
spec. heat cap [J/(kgK)] 375 1050 1090 1100 313
Young. modulus [Pa] 9.79e10
Poisson ratio [-] 0.301
therm. expans. coeff. [1/K] 18.7e–6
A. Measurements
The physical model of the thermoelastic actuator (see
Fig. 6) was designed by our group at the University of
West Bohemia. This is the first prototype that can help us
to validate the results of several projects of devices based
on the principle of thermoelasticity.
Figure 6: The thermoelastic actuator – manufactured prototype.
We also designed and manufactured a measurement
stand intended for fixing of the device, which
significantly contributes to the accuracy of
measurements. The rigidity of the measurement device is
of extreme importance for measuring such small shifts.
The measuring stand is supposed to be connected with a
dynamometer in the future with the aim of making use
the device for another purpose: the actuator can also act
as a source of large forces produced by small shifts of the
dilatation element.
Figure 7: Arrangement of the measurement.
The measuring circuit (see Fig. 7) consists of a
measuring rack, thermoelastic actuator, function
generator, amplifier, capacitance decade, auxiliary
resistor and oscilloscope. The measurements of the
thermoelastic actuator are carried out in several regimes
- 323 - 15th IGTE Symposium 2012
characterized by the above frequencies and RMS values
of the field current. A sinusoidal signal delivered from
the frequency generator is amplified by the amplifier to
the desired value of the current at a given frequency.
A small resistor connected to the thermoelastic
actuator in series is used for measuring the current
through the circuit. We oscilloscopically measured the
voltage at the above resistor and the value of the current
was determined using the Ohm law from the measured
voltage and the known resistance. Using of an ammeter is
inappropriate due to its internal resistance. This would
increase the total resistive load and the input signal could
not be amplified sufficiently by the used amplifier. The
thermoelastic actuator can be considered an RL circuit
and we obtain the maximum current through it by getting
it into resonance. This can be achieved by adding a serial
capacitor. For given values of R and L and prescribed
frequency it is very easy to calculate its capacitance
C using the well-known Thomson relation
2 f
1
LC
.
(6)
For compensation we used rolled capacitors whose
capacitance was determined with respect to the given
frequency. But we had to respect the available types of
the capacitors, which did not allow reaching exactly the
desired values of capacitances from (6). And this explains
the above values of frequencies that differ from
“reasonable” values of 500 , 1000 , and 2000 Hz.
The displacement of the top front of the dilatation
element was measured using digital indicator MarCator
1088 (accuracy 0.001mm) in the time interval from 0 to
300 seconds with increments equal to 30 seconds. In the
same intervals we measured the temperature inside the
brass core. The measurements of the internal temperature
were performed on the coil. The corresponding values
were plotted into graphs for a comparison with the results
from the mathematical model.
The measurements performed on an experimental
prototype were used for calibration. The measured results
are shown in Figs. 8 and 9. Figure 10 shows the
dependence between the temperature and displacement in
the brass core 2.
Figure 8: Time dependence of the temperature for the specified
parameters (current RMS value 1 A and frequencies 519, 1005 and 2210
Hz) of the field current).

B. Numerical simulation
For numerical solution we used the previously
mentioned programs and algorithms based on the FE
method. At the beginning we created a geometrical model
according to the technical drawing, which was used for
manufacturing of the prototype. In the first step of
simulation we used the material properties according to
the Tab. 1, whose were found in the base material
datasheets.
Figure 9: Time dependence of the displacement for the specified
parameters (current RMS value 1 A and frequencies 519, 1005 and 2210
Hz) of the field current.
Figure 10: Dependence of the displacement on the temperature for the
specified parameters (current RMS value 1 A and frequencies 519, 1005
and 2210 Hz) of the field current.
The time dependence of the temperature and
displacement (Figs. 11 and 12) show the large
discrepancy between the measurement and simulation.
Figure 11: Time dependence of displacement for the first solution –
current RMS value 1 A and selected frequency 1005 Hz.
- 324 - 15th IGTE Symposium 2012
These discrepancies were obviously brought about by
the temperature-dependent material properties used in the
numerical simulation (that differed from the real
properties of those used for building the physical model)
and also differences between the real geometry of the
prototype and geometry used for the numerical model.
Therefore, we checked all dimensions of the model and
prototype. The next step was usage of some iteration
processes to found new material properties to get the
better agreement of the results. Especially we focused on
the brass, because we did not know its exact designation
and chemical composition of this material. All materials
used in the prototype have the relative permeability near
to one and are not ferromagnetic, therefore all used
nonlinear characteristics are just the temperature
dependencies (see Figs. 3, 4 and 5). After several
corrections we found the satisfying values of material
properties and comparing them with available database of
materials [6] we found that the used brass should be UNS
C26000, whose characteristics are in the mentioned
figures and were used for the final solution.
Figure 12: Time dependence of temperature for the first solution –
current RMS value 1 A and selected frequency 1005 Hz.
The next two figures show an acceptable agreement
between the measurement and simulation, for the
frequency f 1005 Hz and RMS value of current
I 1A.
in
Figure 13: Time dependence of displacement for the final solution –
current RMS value 1 A and selected frequency 1005 Hz.

Figure 14: Time dependency of temperature for the final solution –
current RMS value 1 A and selected frequency 1005 Hz.
Figures 15 and 16 depict the distribution of the
temperature and displacement in the dilatation element
for time t 300 s , frequency f 1005 Hz and current
I 1A.
in
Figures 17 and 18 show the final comparison of the
measured data with the results of simulation for all three
frequencies ( 519 Hz , 1005 Hz and 2210 Hz ).
From the results is visible a small discrepancy,
especially for the highest frequency. This can be brought
about by the incorrect temperature-dependent
characteristic of the thermal conductivity. Therefore, it is
necessary to perform next steps to find the exact model.
Figure 15: Graphical presentation of obtained results for total
displacement of the dilatation element in time t = 300 s, for frequency
f = 1005 Hz and current Iin = 1 A.
- 325 - 15th IGTE Symposium 2012
Figure 16: Graphical presentation of obtained results for temperature in
the model of the thermoelastic actuator in time t = 300 s, for frequency
f = 1005 Hz and current Iin = 1 A.
Figure 17: Comparison of the final solution results of the time
dependency of displacement with the measured data – for current RMS
value Iin = 1 A.
Figure 18: Comparison of the final solution results of the time
dependency of temperature with the measured data – for current RMS
value Iin = 1 A.

VI. CONCLUSION
Thermoelasticity may prove to be a mighty tool in
some applications where setting of accurate position is
needed. The process of reaching the required dilatation is
slow, but reliable.
Nevertheless, accuracy of the results strongly depends
on correctness of the input data as is shown in this paper.
Further research will be, therefore, aimed at possibilities
of their improvement. At this time we are preparing the
measurements of material properties of the available
materials and we want to make the brass spectroscopy to
find the correct chemical composition to improve the
model. And, of course, we need to improve the material
properties of other materials in the model that are not so
important like the brass, but they can affect the results as
well.
Another important aim is to accelerate the heating
process. New possibilities are investigated in this
direction, based on using variable amplitude of the field
current, which can be realized, for example, by pulsewidth
modulation.
VII. ACKNOWLEDGMENT
This work was supported by the University of West
Bohemia grant system (project No. SGS-2012-039) and
by the Grant Agency of the Czech Republic (project
102/11/0498).
REFERENCES
[1] I. Dolezel, B. Ulrych and V. Kotlan, “Combined Actuator for
Accurate Setting of Position Based on Thermoelasticity Produced
by Induction Heating”, in IEEE Transaction on Industry
Applications, Vol. 47, No. 5, 2011, ISSN 0093-9994, p. 2250–
2256.
[2] Doležel, I., Kotlan, V., Ulrych, B. Electromagnetic-thermoelastic
actuator for accurate wide-range setting of position. Przeglad
Elektrotechniczny, 2011, Vol. 87, No. 5, p. 22-27. ISSN: 0033-
2097
[3] Kuczmann, M.: Iványi, A.: The Finite Element Method in
Magnetics. Akademiai Kiado, Budapest, 2008.
[4] Holman, J.P.: Heat Transfer. McGrawHill, NY, 2002.
[5] Boley, B., Wiener, J.: Theory of Thermal Stresses. NY, 1960.
[6] MPDB Database of materials: www.jahm.com.
- 326 - 15th IGTE Symposium 2012

- 327 - 15th IGTE Symposium 2012
Scattering Calculations of Passive UHF-RFID
Transponders
*Thomas Bauernfeind, *Gergely Koczka, *Kurt Preis and *Oszkár Bíró
*Institute for Fundamentals and Theory in Electrical Engineering, Inffeldgasse 18, 8010 Graz, Austria
E-mail: t.bauernfeind@TUGraz.at
Abstract—Beside the energy extraction capability from impinging electromagnetic waves provided by the interrogator, the
signal strength of the field scattered by the transponder is also a quality criterion of UHF-RFID transponder tags. In general,
the transponder antenna is conjugate complex matched to the nominal transponder IC impedance to achieve maximum
energy extraction. The reverse link from the transponder to the reader is commonly not taken into account. Due to the
modulation technique and a voltage limiter circuit at the analog IC frontend, the input impedance of the IC is strongly
nonlinear. In the present paper a method is proposed which is able to analyze the influence of this nonlinearity on the
scattered signal by separating the scattered field of the transponder to a reference scattering problem and a pure radiation
problem.
Index Terms— Antenna Impedance, Radar Cross Section, UHF-RFID.
I. INTRODUCTION
In passive backscattering applications like UHF-RFID
(ultra high frequency-radio frequency identification) the
communication between the interrogator unit (reader) and
the transponder is established by means of modulating the
radar cross section (RCS) of the transponder. In general,
for UHF-RFID applications, this is done by switching the
analog input impedance between two states in phase with
the data stream to be transmitted, e.g. the EPC (electronic
product code) value [1], [2]. Unfortunately, the analog IC
input impedance is not constant, indeed it has a strong
nonlinear behavior versus applied power e.g. as shown in
Figure 1. This behavior is mainly caused by a voltage
limiter at the transponder IC’s frontend and on the power
consumption of the IC which is determined by the actual
mode of operation of the transponder [3]. Hence, the
transponder input impedance is a function of the induced
antenna voltage [4]. To capture this nonlinear behavior,
an iterative full-wave simulation of the whole channel
including the reader antenna, the air volume and the tag
antenna as proposed in [5] should be applied, taking into
account the feedback of the tag on the reader. The IC
behavior is commonly taken into account by means of
circuital co-simulation [6]. Due to the huge problem
domain, this technique is unpractical to carry out
optimization investigations. A possibility to reduce the
computational costs is to describe the scattered field of
the nonlinearly terminated tag antenna in terms of a
reference scattering field and a pure radiated field from
the excited transponder antenna [7]. Since, in general, the
effect of the tag on the reader field is small, the feedback
on the reader is neglected in a first approximation hence,
the scattered field can be calculated by superimposing
those fields applying the finite element method.
II. GENERALIZED SCATTERING MATRIX
In Figure 2a) a typical UHF-RFID application
consisting of a transponder tag and a reader antenna is
shown. Following the approach described in [7], the
situation at the RFID-transponder tag can be modeled as
shown in Figure 2b) where the field situation is described
in terms of spherical wave modes. A mathematical
description of the simplified transponder model is given
by the generalized scattering matrix [7], [8]:
Resistance in Ohm
100
90
80
70
60
50
40
30
20
10
0
b S S S a
d S c
d S S c
00 01 0 N
1 S 10 ... c 1 . (1)
N N0NN N
Real{Z_IC} un-modulated
Real{Z_IC} modulated
Imag{Z_IC} un-modulated
Imag{Z_IC} modulated
-10 -8 -6 -4 -2 0 2 4 6 8 10
Pa in dBm
Figure 1: Nonlinear transponder IC impedance versus applied power
(NXP Ucode G2X).
In (1), a relationship between the complex applied (a)
and reflected (b) waves at the transmission line
connecting the load impedance to the antenna and the
incoming (cn) and outgoing (dn) spherical wave mode
series is given.
a) b)
Figure 2: a) Typical RFID application. b) Simplified transponder
model.
0
-50
-100
-150
-200
-250
Reactance in Ohm

In case of an antenna driven by external waves
(c1,… cN), the mode on the feed transmission line reflects
at the load impedance defined by the load reflection
coefficient given by
Z L Z 0
Z ZL Z 0 Z 0 . (2)
0
So the total field can be written as:
N
b 1 S 00 S0m c m
m 1 m
N
S0 1 0 , (3)
t
dn S nn0 0 b
N
Snmc m
m 1
c
1 m
N
S
1 nm S . (4)
Combining (3) and (4), the total outgoing field dn t can be
calculated as:
t
dn S N
n
0
S
0
m c m
1 S00 00 m 1 N
S nm nmc c m
m
1
m
0
S S0
. (5)
In (5), S00 is the antenna port reflection coefficient given
to be
Z Ant A t Z 0
S00
Z Ant Z0
0 Z 0 . (6)
0
The coefficients S0m are the transmission coefficients
from the antenna to the transmission line, Sn0 the
transmission coefficients from the transmission line to the
antenna and Snm are the mode reflection coefficients
directly connecting the incoming and outgoing wave
modes. The first term in (5) can be thought of as only
load impedance dependent and the second term as
structure dependent, respectively.
If one is interested in the scattered field only, one has
to subtract the field in absence of the antenna (dn = cn)
from (5) yielding
N N
s S N
N
n
0
dn S 0
m c cm m c cn n S Snm
nm c cm
m
1
S00 00 m 1 m 1
m
0
. (7)
S S0
Since a short circuit condition can easily be achieved, it is
reasonable to use the short circuited case as reference.
With = -1, one can calculate the scattered field in the
short circuit case too:
s S N N
sc n
0
dn S S00m 0
m c cm m c cn n S Snm
nm c m
1 S 00 m 1 m
1
m
0
. (8)
S S0
Using (7) and (8) it is now possible to rewrite (7) in terms
of the reference scattered field from the short circuit
condition as:
1 0
N
s s 1 S
sc
n 0
dn d dn n S0mcm 1
S00 1 S00 m 1
sc s
N
S0cm 1 m 0 S0 00 00 m
S 00 1 S0
1
. (9)
In (9) it is assumed that the incoming spherical wave
mode series cm is known. Since the finite element method
should be applied, the description in terms of the wave
modes is not practical. Introducing the antenna short
circuit current Isc and describing the radiated field in
terms of an antenna driving current IAnt as presented in
[7], one can eliminate the incoming wave mode series cm
from (9) yielding:
- 328 - 15th IGTE Symposium 2012
0 1 1
s s I
sc rad sc
Z 1 S
rad 0 1
00
dn d sc sc 0 1 S
sc rad
00 000
n d n
n
1...
N
. (10)
2 I IAnt Z ZAnt 1
S S00
000
Equation (10) is the key expression enabling the
description of a scattered field in terms of a reference
scattered field and a pure radiated one. Finally, using (2)
and (6), (10) can be written as:
s s ssc
rad IscZ d L
n d sc
n d
n
I IAnt ZAnt Z L Z . (11)
L
The electric field E is the summation of the spherical
wave mode series, so E is given by [7], [11]:
IZ
E E E 0 L
Scattered E EShort Short E EAntenna
Antenna
. (12)
IAnt ZAnt Z L
Since, especially for UHF-RFID applications in
general, the transponder antenna is conjugate complex
matched to the nominal transponder IC impedance it is
advantageous not to use the short circuit case as reference
but rather the conjugate complex matched case. In [7], [8]
it is described how to eliminate the short circuit case from
(12) to get the final relationship:
* *
EScattered ( ZL) E Scattered ( Z Ant ) E EAntenn
Antenna
. (13)
* *
I I
m I
I
m
Ant
In (13), Im * is the current at the terminal of the antenna
for a conjugate complex matched transponder antenna in
case of a pure scattering problem and * is the conjugate
matched reflection coefficient:
*
* Z ZAnt AAnt t Z L
Z ZAnt ZL
L Z . (14)
L
Finally, this means that the field scattered by an antenna
terminated with a certain load impedance ZL can be
calculated by a superposition of a reference scattering
field and a scaled radiated field of the antenna driven
with a current IAnt.
III. NUMERICAL INVESTIGATIONS
The basic electromagnetic field problem shown in
Figure 3 is analyzed with a finite element based in-house
code. Introducing the magnetic vector potential A and the
modified scalar potential V the electric field intensity E
and the magnetic field intensity H in the time harmonic
case can be written as:
E j A j V
, (15)
1
H A = A. (16)
Using n1 edge basis functions Ni for the magnetic vector
potential A and n2 nodal basis functions Ni for the
modified electric scalar potential Vh as proposed in [9],
the Galerkin equations to be solved become:
N i A h hd j c c N i i A
h
h d
j l
j c N
i
gradV h hd N
i A
h hd
d 0
Z IC w
SIBC
( i 1,2,..., 12 1,2,..., , , , n )
N c i
c i (17)
1

j c cgradN
gra adNi
A h d
j c gradN gra adNi
gradV h d 0 (18)
( i 1, 12 , 2,..., 22,...,
, , n2).
In (17) and (18), the approximations of the potential
functions are given by:
n 1
Ah a i N
i
i 1
N
i 1 i
, (19)
n
2
Vh VVN i
N i
i
1 i
. (20)
The needed truncation of the problem domain has been
realized by applying perfectly matched layers (PMLs) as
proposed in [10]. For the first basic scattering
investigations it was refrained from modeling the reader
antenna structure e.g. as shown in Figure 2a). Instead, the
actual scattering problem was excited by means of a
Hertz-dipole as proposed in [4] since the Hertz-dipole can
be modeled by a filament current with a given length. The
main advantage of this excitation technique beside the
reduction in the degree of freedom is, that the radiated
power of a Hertz-dipole can be calculated analytically
[11], which offers the possibility of validating the quality
of the results gathered within the post processing. On the
other hand, the feedback of the transponder tag on the
reader is neglected in this case. Since the influence of the
transponder on the reader for UHF-RFID applications is
small in general, it is assumed that this effect is negligible
in a first approximation [4].
The excitation of the pure radiation problem is done by
impressing a voltage U0 at the feed gap of the dipole
structure by prescribing a constant vector potential for the
length y of the feed gap:
Ee y y j j A y y U 0 . (21)
As it can be seen from (17), the IC impedance is
modeled with a surface impedance boundary condition
(SIBC) as proposed in [12]:
E Ett1 t 1 dds
s
U E
l
t 1 l l
Z l
t 1
l
IC Z
SIBC , (22)
I H t 2 d ds s K
w
t1
w w
t
c
where l is the length of the impedance geometry and w is
the width. The surface impedance ZSIBC in (22) is given
by the relationship of the tangential component of the
electric field intensity Et1 and the tangential component
Ht2 of the magnetic field intensity, assuming constant
tangential components at the surface impedance.
Due to the choice of the basis functions, the resulting
system of equations (17) and (18) becomes singular
which is not a drawback applying an iterative solver
method. Unfortunately, the resulting system of equations
is ill conditioned as described in [13], [14]. Hence, a
direct solver method [14] has to be applied to avoid
impractically long simulation durations. Due to the
singularity of the system of equations, a tree-gauging [15]
is needed to be able to apply the direct solver method.
- 329 - 15th IGTE Symposium 2012
IV. BASIC EXAMPLE
The proposed method is tested on a very basic example
shown in Figure 3. Applying a electric boundary
condition
E n 0 (23)
(which is a Dirichlet boundary condition for A) in the x-zplane
and a magnetic boundary condition
H n 0 (24)
(which is a Neumann boundary condition for A) in the yz-plane,
only a quarter of the problem has to be modeled.
The half wavelength dipole at a frequency of f = 1 GHz
with a length of 0.5 l Ant 7.5 cm and a width of
0.5 w Ant 2.5 mm has a thickness of d = 1 mm and is
placed at a distance of 25 cm to the Hertz-dipole
excitation which is 0.8 times the wavelength . Hence,
far field conditions can be assumed. In Figure 3b) a detail
of the feed gap with a width of wgap = 200 μm is shown.
The SIBC is connected via perfect electric conductors
(PEC) to the excitation and the antenna structure.
a) b)
Figure 3: a) Typical RFID application. b) Simplified transponder
model.
The input impedance of the dipole antenna structure in
the present model is calculated to be
ZAnt = 100.15 + j 44.05 . Hence, the input impedance of
the fictitious IC must be ZIC = 100.15 – j 44.05 to
fulfill the conjugate complex matching. With the given IC
impedance, the needed reference field can be calculated
by subtracting the field of the Hertz-dipole excitation in
absence of the dipole structure from the total scattering
problem as proposed in [4]. The results are shown in
Figure 4a) to c).
a) b) c)
Figure 4: a) Scattering problem. b) Hertz-Dipole excitation.
c) Scattered field from the dipole structure.
Next the scattered field in case of the modulated IC
impedance should be calculated with the proposed
method. For the modulated IC impedance, it is assumed

that a resistance of Rmod = 150 is parallel to the IC
impedance which is a typical value for UHF-RFID
transponder ICs. So the modulated IC impedance is given
to be ZICmod = 62.76 – j 15.6
In Figure 5, the result obtained by the proposed method
is compared with the result gathered by the method of [4].
As it can be seen, a good qualitative agreement between
the two methods is obtained. The current distribution
along the dipole structure has also been investigated. This
is done by comparing the magnetic field intensity in the
vicinity of the antenna structure along the antenna for
different phase angles in the excitation. The results are
shown in Figure 6. In Figure 7, the relative error between
the two methods is shown. As it can be seen, the good
agreement between the two methods is also quantitative.
The difference can be explained by uncertainties in the
determination of the antenna input impedance ZAnt, since
the scaling factor in (13) is directly related to this
number.
a) b)
Figure 5: a) Scattered field calculated with the method from [4].
b) Scattered field calculated with the proposed method.
|H| in A/m
1,6
1,4
1,2
1,0
0,8
0,6
0,4
0,2
0,0
0 0,02 0,04 0,06 0,08
Distance in m
Figure 6: Current distribution along the dipole antenna in terms of the
magnetic field intensity.
8
6
4
2
-4
-6
-8
-10
Magnetic field intensity along the dipole antenna
Scattering 0°
Proposed method 0°
Scattering 45°
Proposed method 45°
Scattering 90°
Proposed method 90°
Scattering 180
Proposed method 180
Relative error of the magnetic field intensity in %
0
0
-2
0,02 0,04 0,06 0,08
Relative error 0°
Relative error 45°
Relative error 90°
Relative error 180°
Distance in m
Figure 7: Relative error of the current distribution along the antenna.
- 330 - 15th IGTE Symposium 2012
V. CONCLUSION
It has been shown on a very basic example that the
scattering from passive objects like UHF-RFID
transponder tags can be described in terms of a reference
scattering problem and a pure radiation problem. Hence,
the proposed method offers the possibility of a certain
reduction in the computational effort, since if multiple IC
impedances have to be taken into account the whole
channel including the reader antenna has to be simulated
for the reference case only. All other IC states can be
modeled as pure radiation problems without having to
model the reader structure.
REFERENCES
[1] K. V. S. Rao, P. V. Nikitin and S. F. Sander, “Antenna design for
UHF RFID tags: a review and a practical application,” IEEE
Trans. on Ant. and Prop., vol. 53, no. 12, pp. 3870-3876, 2005.
[2] V. Chawla and D. S. Ha, “An overview of passive RFID,” IEEE
Communications Magazine, vol. 45, no. 9, pp. 11-17, Sept. 2007.
[3] A. Moretto, E. Colin, C. Ripoll and S. A. Chakra, “Shunt behavior
in RFID UHF tag according to ISO standard and manufacturer
requirements,” Proceedings of the IEEE, vol. 98, no. 9, pp. 1550-
1554, 2010.
[4] T. Bauernfeind, K. Preis, G. Koczka, S. Maier and O. Biro,
“Influence of the Non-Linear UHF-RFID IC Impedance on the
Backscatter Abilities of a T-Match Tag Antenna Design,” IEEE
Trans. on Magn., vol. 48, no. 2, pp. 755-758, 2012.
[5] R. Wang and J. Jin, “A Flexible Time-Stepping Scheme for
Hybrid Field-Circuit Simulation Based on the Extended Time-
Domain Finite Element Method,” IEEE Trans. on Advanced
Packaging, vol. 33, no. 4, pp. 769-776, 2010.
[6] G. Manzi and U. Mühlmann, “Passive UHF RFID sensor /
transponder antenna optimization for backscatter operation by
electromagnetic-circuital co-simulation,” Proceedings of the 11 th
International Conference on Telecommunications, pp. 17-22,
2011.
[7] R. C. Hansen, “Relationships Between Antennas as Scatterers and
as Radiators,” Proceedings of the IEEE, vol. 77, no. 5, pp. 659-
662, 1989.
[8] R.G. Green, “Scattering from conjugate-matched antennas,” IEEE
Trans. on Ant. and Prop., vol. 14, no. 1, pp. 17-21, 1966.
[9] O. Biro, “Edge element formulations of eddy current problems,”
Comput. Methods Appl. Mech. Eng., vol. 169, pp. 391-405, 1999.
[10] I. Bardi, R. Remski, D. Perry and Z. Cendes “Plane wave
scattering from frequency-selective surfaces by the finite-element
method,” IEEE Trans. on Magn., vol. 38, no. 2, pp. 641-644,
2002.
[11] C. Balanis, Antenna Theory: Analysis and Design, Hoboken: John
Wiley & Sons, 2005.
[12] K. Hollaus, O. Biro, K. Preis and C. Stockreiter, “Edge finite
elements coupled with a circuit for wave problems,” International
Conference on Electromagnetics in Advanced Applications, pp.
956-959, Torino, 2007.
[13] G. Koczka, T. Bauernfeind, K. Preis and O. Biro, “Schur
Complement Method Using Domain Decomposition for Solving
Wave Propagation Problems,” The 10th International Workshop
on Finite Elements for Microwave Engineering, FEM2010, pp. 53,
Meredith, 2010.
[14] G. Koczka, T. Bauernfeind, K. Preis and O. Biro, “An Iterative
Domain Decomposition Method for Solving Wave Propagation
Problems,” The 11th International Workshop on Finite Elements
for Microwave Engineering, FEM2012, pp. 66, Estes Park, 2012.
[15] R. Albanese and G. Rubinacci, “Solution of three dimensional
eddy current problems by integral and differential methods,” IEEE
Trans. on Magn., vol. 24, pp. 98-101, 1988.

- 331 - 15th IGTE Symposium 2012
Simulation of a High Speed Reluctance Machine
Including Hysteresis and Eddy Current Losses
B. Schweighofer∗ , H. Wegleiter∗ , M. Recheis∗ , and P. Fulmek †
∗Graz University of Technology, Institute of Electrical Measurement and Measurement Signal Processing
Graz, Austria
† Vienna University of Technology, Institute of Sensor and Actuator Systems, Vienna, Austria
E-mail: bernhard.schweighofer@TUGraz.at
Abstract—Flywheel energy storage systems in automotive applications require a compact design, which typically uses the
rotor of the electrical machine as storage mass. In order to minimise friction losses the rotor is running in vacuum. The
heat generated in the rotor can only be transferred by thermal radiation and thermal conduction through the bearings,
eventually. Therefore, a precise estimation of the expected rotor losses is needed to design an efficient thermal management
of the whole machine. In this paper a switched reluctance machine is analysed by finite–element simulations with focus on
hysteresis losses and eddy current losses.
Index Terms—Flywheel, Reluctance Motor, Hysteresis, EM Model
I. INTRODUCTION
Flywheel energy storage systems are an important
area of research in the automotive industry, to satisfy
the demands for low–emission or zero–emission by
hybridisation and electrification of vehicles, trucks and
buses. Typical systems have to be designed for high
power and medium energy content, e.g. 120 kW and
1.5 kWh. Advantages of flywheel systems are the high
energy density in comparison to double–layer capacitors,
the higher power density in comparison to batteries, the
State–Of–Charge is always known, almost no degradation
of performance with age, etc. Additionally several constraints,
such as size, weight, and costs, have to be fulfilled.
Typically a compact design, in which the rotor of
the electrical machine acts as the flywheel storage mass
is chosen. Permanent magnet (PM) electrical machines
seem to be the optimum choice for flywheel systems
due to their high efficiency, high power density, and
lower rotor losses in comparison to induction machines.
Especially for automotive applications, however, several
drawbacks have to be considered:
– high power rare earth (RE) magnets (samariumcobalt
or neodymium-iron-boron) are very expensive
– even moderate temperatures may dramatically degrade
the RE–magnets performance by demagnetisation
or even destroy them
– high mechanical stress due to centrifugal forces on
the rotor in flywheel machines necessitate additional
supporting structures to protect the RE–magnets
– the combination of a RE–magnet–rotor with an
iron stator leads to significant zero–torque losses,
limiting the storage capabilities of the flywheel.
Reluctance machines (RM) generally show a higher mechanical
robustness. They can be produced from high–
strength electrical steels to easily withstand the centrifugal
forces, the high torques, and the pulse accelerations
during operation in the vehicle. In comparison to PM–
machines the rotational speed can be increased, leading to
higher storage densities. Additionally, no electromagnetic
losses have to be expected for the free running flywheel,
which means no zero–torque losses. A comparison
between synchronous (synRM) and switched reluctance
machine (SRM) shows an advantage for the SRM due
to its simple coil design and the higher power density.
For the design of the SRM the rotor losses need special
attention. As the rotor is operated in vacuum to avoid
frictional losses, heat dissipation can only happen by
thermal radiation and, eventually, by thermal conduction
through conventional mechanical bearings. The stator is
cooled by conventional water–cooling. For an accurate
modeling of the heat flow as well as to predict the
efficiency of the machine the knowledge of the losses
inside the motor, especially in the rotor is needed.
This paper deals with the analysis of the rotor losses in
a switched reluctance machine (SRM). We have chosen
an external rotor 12/8 SRM with a power of 120 kW and
a rated speed of 25000 rpm.
II. METHODOLOGY
Losses in ferromagnetic materials are divided into the
static, rate–independent hysteresis loss and dynamic rate–
dependent electromagnetic losses (eddy–current losses,
excess losses) [1], [2], [3]. The rate–independent iron–
losses are determined by the materials’ hysteresis loop.
If the course of the local vectorial flux–density and
magnetic field is known, the integral over the respective
vectorial hysteresis loops gives the hysteresis loss in the
material. Existing models for ferromagnetic hysteresis
(e.g. Jiles–Atherton, Preisach, Energetic Model) describe
scalar BH–loops, only.
The calculation of rate–dependent losses is usually
based on the time variation of the flux density obtained
from static finite element (FE) simulations. With increasing
frequencies, however, the skin–depth decreases leading
to an increasing deviation of the real flux density from
the static simulation results. For the rate dependent losses

several formulations for sinusoidal flux densities have
been proposed [1], [4], suitable only for linear materials,
in the frequency domain. The correspondence between
simulation results and experiments is not satisfying if the
material is used at high flux densities, at saturation, due
to the assumption of linear material behaviour. During the
operation of a high performance electrical machine the
magnetic fields and the flux densities in the core material
are neither unidirectional nor sinusoidal.
Loss calculations taking into account the non–linear
materials’ properties, the eddy current distribution and
the skin effect require an exorbitantly high computational
effort for 3D dynamic FE simulations.
We have prepared a 2D FE–model of the machine to
calculate the distribution of the static flux density for
arbitrary rotor angles and excitation patterns. An external
source current is used to establish the magnetic field. The
model is prepared to analyse the influence of different
current pulse patterns, and to develop efficient control
strategies. The non–linear ferromagnetic materials’ properties
have been modelled by the scalar Energetic Model
(EM) [5], which has been parameterised by Epstein frame
experiments and from manufacturers data–sheet. The EM
is used to derive the single valued BH–commutation
curve used to characterise the material in FEM, and
to calculate the static hysteresis loss for arbitrary flux
density waveforms [6], [7]. The variation of the flux
density with time, obtained by the FE–simulation, is
used to estimate eddy current losses by approximating
equations [8]. Finally, an expression for the losses in
the machine can be found by integrating these local loss
expressions over the whole volume.
III. RELUCTANCE MACHINE FE–MODEL
Fig. 1. Geometry of the reluctance machine RM. Three sets of coils
produce a rotating magnetic quadrupole field. With the 12/8 ratio the
principle step angle is 15 Degrees, π/12. The shown position of the
external rotor is defined as α =0 ◦ , rotation direction chosen is clock–
wise. Two points are chosen to evaluate the magnetic field, flux density
and hysteresis loss: Point 1 at the center of a rotor tooth, Point 2 at the
surrounding yoke area.
We have chosen a 12/8 SRM (double 6/4 machine [9])
with an external rotor. The geometrical dimensions of
- 332 - 15th IGTE Symposium 2012
the SRM are described in Table I. With the chosen ratio
of air–gap–diameter to length of the machine ≈ 12:16,
the 3rd dimension has been omitted leading to a more
simple 2D FEM model of the SRM Figure 1. Three sets
of coils (ABC) build a rotating quadrupole field in the
machine. With 12 stator teeth and 8 rotor teeth the step
angle is 15 ◦ . The central steel shaft of the stator is used
for cooling.
Both, stator and rotor, are built up from steel sheet.
The stator is made from a standard electrical steel. For
the rotor material we have chosen the Vacodur50S high
strenght cobalt–steel from Vacuumschmelze.
The rated speed of the SRM is 25000 rpm, the stator
coils switching frequency results to 10 kHz.
length of motor
external rotor
160.0 mm
No. of teeth 8
material Vacodur 50S
outer diameter 180.0 mm
inner diameter 121.0 mm
tooth depth 14.5 mm
gap width
stator
28.0 mm
No. of teeth 12
material Armco electrical steel
outer diameter 120.0 mm
inner diameter 60.0 mm
gap depth 17.4 mm
tooth width 14.0 mm
shaft
material construction steel
outer diameter 60.0 mm
TABLE I
GEOMETRIC DIMENSIONS OF SRM.
Fig. 2. FEM: Gmsh/GetDP 2D–model for rotor position α =15 ◦ .
17516 vertices and 37224 elements in a free triagonal mesh.
For the FEM simulation we use two different software
packages: the commercial Comsol–Multiphysics [10] and
the general open–source packages Gmsh/GetDP [11].
Figure 2 shows the final mesh of the model with 37224
triangle elements. In both FEM–packages we defined the
magnetic materials’ property as BH–table, calculated by
the Energetic Model.

IV. MATERIALS MODEL
The static magnetic materials’ properties are described
by the Energetic Model (EM) of ferromagnetic hysteresis
[5]. The EM describes the non–linear, hysteretic behaviour
of magnetic polarisation and magnetic field in
the material based on the concept of minimising the
total energy in a statistical description of the magnetic
domain structure. Consequently, many physical factors
influencing the magnetisation process can be included:
e.g. magnetocrystalline anisotropy, internal demagnetisation,
anisotropy of magnetostriction, etc. The EM can
simulate major and minor hysteresis loops, it also simulates
the effect of slowly evolving closed minor loops, in
contrast to other models (e.g. Preisach model, wipeout–
property). The simplified scalar formulation of the EM
[5], [6] is perfectly prepared for integration into FEM.
The parameters of the EM are found by evaluating several
important points of a measured BH–loop (e.g. saturation,
coercivity, remanence, initial susceptibility).
The rotor material has to be chosen with respect
to soft–magnetic properties (low coercivity, high flux
density, low losses), and mechanical properties, as well.
A material with optimum properties for high speed
rotors is the soft–magnetic cobalt–iron alloy Vacodur50
manufactured by Vacuumschmelze [12]. Due to the Co–
content it exhibits a very high saturation flux density of
2.35 T. At a magnetic field strenght of 800 A/m more
than 2.0 T are reached. The coercivity is in the range
of 100–200 A/m. Figure 3 shows the BH–loop from the
Vacodur50S datasheet and the corresponding results of
EM–simulations.
Fig. 3. Hysteresis loops of Vacodur50S. Red lines: BH–loop from
datasheet (Vacuumschmelze), blue line: BH–loop (virgin curve and
major hysteresis loop) from EM–simulation.
Vacodur50S Armco
Js 2.40 2.10
q 40.12 15.26
k 314.95 16.31
Ne 2.09e-5 3.15e-6
g 11.30 23.41
h 3.40 3.1e-3
TABLE II
EM–PARAMETERS FOR VACODUR50S AND ARMCO
- 333 - 15th IGTE Symposium 2012
For our simulations of the stator material we used the
EM parameterised for soft–magnetic Armco electrical
steel sheets (GO, FeSi)[13]. This grain oriented FeSi–
steel exhibits a coercivity as low as 7 A/m, the technical
saturation is limited to 2.0 T. Epstein measurements
provided the data necessary to parameterise the EM.
Table II shows the parameters used for the EM simulations
of rotor and stator material. The single valued
BH–function, required for our FEM simulations, has been
found by EM calculations of the commutation curves
Fig. 4 and 5. Both FEM packages (Comsol, GetDP)
use tables of the simulated BH–commutation curve to
interpolate the required BH–point.
Fig. 4. Hysteresis loops for Armco electrical steel. EM–simulations
for several complete symmetrical loops build the commutation curve.
Fig. 5. Vacodur50S: simulated symmetrical hysteresis loops build the
commutation curve.
V. FEM RESULTS
The above described model setup (geometry and materials)
was used for magnetostatic FEM simulations with
current excitation in Comsol and GetDP. Both packages
gave almost exactly identical results. A series of static
calculations has been done to simulate the rotating machine
for various coil currents. Typical results of both
FEM simulations are depicted in the following figures.
The excitation current was applied to Coil–set A (see
Fig. 1), producing a quadrupole magnetic field. The
angular rotor position has been changed in 1◦ steps, and
the corresponding flux densities in 2 points of the rotor

and the flux in the coil A stator tooth have been evaluated
exemplarily to estimate the rotor losses.
Figure 6 shows the corresponding relative permeability
values, i.e. the quotient B/(μ0H), for a moderate coil
current density of 1 A/mm 2 . Different μr–scales are
used for stator and rotor. As the respective range of flux
densities Fig. 7 is below 300 mT, the permeability is still
rising with flux density (cmp. Fig. 4 and 5), accordingly
the maximum permeability is at the loci of maximum
flux density.
Figure 8 shows Comsol results for a coil current
density of 5 A/mm 2 . The flux density at the partly overlapping
stator and rotor teeth approaches the materials
saturation.
Fig. 6. GetDP simulation: relative magnetic permeabilities μr at coil
A current density s =1A/mm 2 (I = 112 A), rotor angle α =10 ◦ .
Different scales are used for rotor (Vacodur50) and stator (Armco).
Fig. 7. GetDP simulation: flux density B at coil A current density
s =1A/mm 2 (I = 112 A), rotor angle α =10 ◦ .
The evaluation of the flux in the stator teeth is shown
in Fig. 9. The line integral of B over a plane stator tooth
cross–section, multiplied by the length of the machine,
gives the total flux in Wb for a quarter of the machine.
Perfect alignment of stator and rotor teeth at a rotor angle
of 22.5 ◦ gives the steepest flux versus field curve. As the
rotor teeth are moved towards the gap between the stator
teeth, the demagnetising effect of the increasing length of
the effective air–gap is clearly visible. When the stator
tooth faces a rotor gap exactly at 0 ◦ the flux vs. field
characteristic becomes a rather flat, almost straight line.
- 334 - 15th IGTE Symposium 2012
Fig. 8. Comsol simulation: flux density B at coil A current density
s =5A/mm 2 (I = 560 A), rotor angle α =10 ◦ .
Fig. 9. Flux in stator tooth versus coil current for varying rotor angle.
Maximum flux for α =22.5 ◦ , stator tooth exactly aligned with rotor
tooth, minimum demagnetisation. Minimum flux for α =0 ◦ , stator
tooth exactly between rotor teeth, maximum demagnetisation, almost
linear dependence.
VI. LOSSES
The conventional three–term iron loss model [1], [8]
contains expressions for hysteresis loss, eddy current loss,
and excess loss. All these loss–contributions depend on
the flux density itself and its time derivative. The local
flux densities are usually estimated by finite element analyses,
completely neglecting hysteresis and eddy currents,
sometimes even the single valued non–linear BH curve.
In a first step FEM calculations approximately determine
the local flux density, from the flux density the iron losses
are determined.
A. Hysteresis loss
In our work we use the EM hysteresis model described
above, to calculate the hysteresis loss for any arbitrary
course of flux densities in the material. Series of simulations
for constant coil current and varying rotor angle
are used to identify the course of the flux density at two
chosen distinct points of the rotor (see Fig. 1). Figure 10
shows the evolution of the vectorial components of the
flux density at point 1 (rotor tooth) versus rotor angle
when coil A is switched on. The symmetry with a change
in sign at 90 ◦ and the periodicity of 180 ◦ are evident.

Both components of the B–vector, radial and tangential,
and the total of the flux density are shown.
Under normal operation each set of coils (ABC) is
active during a rotation angle of 15 ◦ . Figure 11 shows
the radial component of the flux density at point 1 (rotor
tooth) when all coils A–B–C are excited sequentially,
leading to a periodicity of 60 ◦ for a fixed point on the
rotor. The total hysteresis loss at point 1 for a complete
revolution of 360 ◦ is Wh =6·555.5 J/m 3 = 3332.8 J/m 3
(Fig. 12),
Fig. 10. Flux density at rotor point 1 (tooth) versus rotor angle, coil A
activated with 5 A/mm 2 . The thick lines indicate the normal switched–
on range for coils A.
Fig. 11. Radial component of the flux density at rotor point 1 (tooth)
versus rotor angle. Coils A–B–C are excitated sequentially with 5
A/mm 2 .
The same procedure, described above for a rotor tooth,
is applied to point 2 on the yoke part of the rotor (see
Fig. 1). Figure 13 shows the evolution of the vectorial
components of the flux density at point 1 (rotor tooth)
versus rotor angle when coil A is switched on. In the rotor
yoke area there exists only a tangential component of the
flux density, the radial component completely vanishes.
Figure 14 shows the tangential component of the flux
density at point 2 (rotor yoke) when all coils A–B–C
are excited sequentially, leading to a periodicity of 60 ◦
for a fixed point on the rotor. The total hysteresis loss
at point 2 for a complete revolution of 360 ◦ is Wh =
6 · 672.1 J/m 3 = 4032.6 J/m 3 (Fig. 15). The two extra
minor loops lead to a significant increase of the hysteresis
loss in the rotor’s yoke area.
- 335 - 15th IGTE Symposium 2012
Fig. 12. Simulated BH–loop for a complete period of the magnetisation
process point 1, covering 60 ◦ rotational angle.
Fig. 13. Flux density at rotor point 2 (yoke) versus rotor angle, coil A
activated with 5 A/mm 2 . The thick lines indicate the normal switched–
on range for coils A.
Figure 16 shows the hysteresis losses at two points of
the rotor for a complete rotor revolution in dependence of
coil current. Although the amplitude of the flux density
is significantly larger in the rotor tooth than in the rotor
yoke, the losses in the yoke are dominant due to the
existence of a pair of extra minor loops. As the yoke
reaches local saturation, however, the flux density becomes
distributed more uniformly, and the amplitude of
the minor loops decreases, accompanied by a decreasing
hysteresis loss.
Fig. 14. Tangential component of the flux density at rotor point 2
(yoke) versus rotor angle. Coils A–B–C are excitated sequentially with
5 A/mm 2 .

Fig. 15. Simulated BH–loop for a complete period of the magnetisation
process at point 2, covering 60 ◦ rotational angle.
Fig. 16. Hysteresis loss for one complete rotor revolution versus coil
current. The losses are calculated for two points in the rotor (see Fig. 1).
B. Eddy current loss
Eddy current losses depend on the rate of change of
flux density, the electrical conductivity of the material,
and the geometry. As long as the induced eddy currents
are small enough to allow the flux density to completely
penetrate the material, eddy current losses can be locally
calculated straight forward. This criterion would
allow a maximum frequency for sinusoidal excitation
below 1 kHz (Vacodur50: h = 0.35 mm, μr ≈ 4000,
σ =2.83 · 106 S/m). A modified expression [8] can be
used to determine eddy current loss and excess loss:
W ′ e ∼ = κσh2 1
·
12 T ·
T 2 dB
dvdt
0 dt
The time derivative dB/dt is defined by our FEM
results, σ is the electrical conductivity of the rotor material,
h is the thickness of the rotor steel sheets, κ is the
modified coefficient for excess loss, T is the time period.
Figure 17 shows the resulting eddy current losses for
a complete revolution at 12000 rpm. The modified loss
coefficient was chosen as κ =1. For higher frequencies
the skin–effect has to be taken into account additionally.
VII. CONCLUSIONS
A 2D finite element model of a switched reluctance
machine is presented, using the EM–hysteresis model to
- 336 - 15th IGTE Symposium 2012
Fig. 17. Eddy current loss per revolution at 12000 rpm versus coil
current. The losses are calculated for two points in the rotor (see Fig. 1).
derive the non–linear, hysteretic BH–function. The EM
is parameterised from BH–loops from experiment (e.g.
Epstein measurement) or data–sheet information. The
FEM calculations use only the single–valued commutation
curve, derived from EM simulations. Hysteresis loss
is calculated based on the local flux densities resulting
from 2D–FEM, by numerical integration of the respective
EM BH–loops. Under the necessary assumption of a
negligible influence of the skin depth, we can estimate
eddy current and excess losses in the rotor, as well.
VIII. ACKNOWLEDGMENT
Part of this research has been supported by the Austrian
FFG, project No. 824164.
REFERENCES
[1] G. Bertotti. Hysteresis in magnetism. Academic Press, 2008.
[2] I. D. Mayergoyz. Mathematical Models of Hysteresis. New York,
Springer, 1991.
[3] D. C. Jiles, D. L. Atherton. J. Magn. Magn. Mater. vol. 61, pp.
48–60, 1986.
[4] D. Lin et.al. IEEE Trans. Magn. 40 (2), pp. 1318–1321, 2004.
[5] H. Hauser. J. Appl. Phys. 96 (5), pp. 2753–2767, 2004.
[6] P. Fulmek, P. Haumer, H. Wegleiter, B. Schweighofer. COMPEL
29 (6), pp. 1504–1513, 2010.
[7] P. Fulmek, N. Mehboob, P. Haumer, M. Kriegisch, R. Grössinger.
SMM19, Book of Abstracts, B1–16, 2009.
[8] K. Yamazaki, N. Fukushima. IEEE Trans. Magn. 46 (8), 3121 –
3124, 2010.
[9] T. J. E. Miller. Switched Reluctance Motors and their Control.
Magna Physics Publishing and Oxford Science Publications
(1993)
[10] Comsol Multiphysics, http://www.comsol.com.
[11] C. Geuzaine, J.–F. Remacle, Gmsh, http://www.geuz.org/gmsh.
P. Dular, C. Geuzaine, GetDP, http://www.geuz.org/getdp.
[12] Vacodur50, Soft magnetic Cobalt Iron http://www.
vacuumschmelze.com/
[13] http://www.aksteel.eu/en/1-products/3-electrical-sheet/

- 337 - 15th IGTE Symposium 2012
An Iterative Domain Decomposition Method for
Solving Wave Propagation Problems
*Gergely Koczka, *Thomas Bauernfeind, *Kurt Preis and *Oszkár Bíró
*Institute for Fundamentals and Theory in Electrical Engineering, Inffeldgasse 18, A-8010 Graz, Austria
E-mail: gergely.koczka@TUGraz.at
Abstract—Solving wave propagation problems with FEM results in a huge number of unknowns due to the large air volume to
be modeled. These equation systems are very ill-conditioned, because of the big material differences, element-size changes and
due to the fact that the system matrices are indefinite. Common iterative methods (CG, GMRES) exhibit bad convergence due
to these conditions. The memory requirement is the weakness of the direct methods. The aim of this paper is to present a
method, which has smaller memory requirement than the direct methods, and converging faster than iterative methods.
Index Terms—no more than 4 in alphabetical order.
I. INTRODUCTION
It is always an open question how to solve huge
indefinite, ill-conditioned equation systems efficiently.
Common iterative methods (CG, GMRES) exhibit bad
convergence due to these conditions [1]. Assembling the
system matrix of wave propagation problems in frequency
domain with finite element method (FEM) results in this
kind of equation systems because of the huge material
differences and element-size changes.
Applying direct solver methods to overcome the
problem of the ill-conditioned system of equations results
in high memory requirements [2]. The aim of this paper is
to present a method with smaller memory requirement
than the direct methods and better convergence quality
than common iterative methods.
The memory requirement of the classical LU
decomposition applied to sparse matrices can be reduced
with fill-in reduction algorithms: the minimum degree
algorithm [3] or the nested dissection algorithm [4].
These algorithms are implemented in the Intel® Math
Kernel Library (PARDISO: sparse linear equation system
solver routine). However, the memory requirement of the
direct method is higher than first order: between
2
Onlog n and
1.3
1.4
O n , typically On ,
O n .For
huge systems a method is required which is capable of
decreasing it.
Fig. 1. The geometrical decomposition of the domain
II. DOMAIN DECOMPOSITION
With geometrical domain decomposition it is possible
to decrease the memory requirement of the direct method.
The problem domain should be subdivided into n
open disjunctive sub-domains, namely
n
,
i1
i
(1)
i, j 1,2,.., n : i j . (2)
i j
The interfaces between the domains are
ij : i j ,
n
(3)
: .
(4)
i, j1
With these notations, the equation system can be
written in block-form:
AII AIxI bI ,
AIA
x
b
where the sub-matrix AII corresponds to the domains,
ij
(5)
A to the interfaces between the domains and I A and
AI to the connections of the sub-domains and the
interface.
The Schur-complement equation system of (5) is
obtained as
A A A A x b A A b .
1 1
I II I I
II I
S
c
(6)
To build the Schur-complement matrix S ,
it is
necessary to have regular subsystems corresponding to
the domains (the matrix AII has to be regular).
The memory requirement of the method can be
estimated as follows:
Let us assume that the memory requirement of the

direct method is One
, where ne is the number of
equations and 1 2.
If all the sub-domains have the
same number of unknowns, then
1
ne
n nen ,
n
1
i.e. the overall memory requirement will be decreased by
a multiplier which depends on the number of the subdomains.
Assembling the Schur complement matrix takes
a long time. However, for solving the reduced equation
system, not the full Schur complement matrix is
necessary. Applying the biconjugate gradient method
(BiCG) to solve the Schur complement equation system
will result in an efficient iterative solver (DD-BiCG) for
solving wave propagation problems.
III. ANALYSIS OF THE METHOD
A. Stability
Since the original equation system is symmetric, the
matrices corresponding to the sub-domains and the
interfaces are also symmetric. To improve the
conditioning of (6), the following form is used:
T T
I L A A A L L x L b L
A A b
where L
T A LL 1 1 1 1 1
I II I I
II I
1T L SL y 1
L c
is the Cholesky decomposition of A
. Using (8) with BiCG without
preconditioning is theoretically equivalent to the form (6)
using A as a preconditioner. The practical examples
show that the symmetric form (8) is more stable and
therefore converging faster.
B. Condition number
The convergence speed of the BiCG depends on the
condition number of the equation system.
Since the A,v formulation is used to solve the
Maxwell’s equations, the resulting linear equation system
is singular. To formulate a regular system a tree has to be
eliminated in the discretized domain. Due to the fact that
the singular system is better conditioned than the regular
one, it is better to work with the singular system.
If and only if the original matrix is singular, the Schurcomplement
matrix is also singular, if the sub-domain
matrix AII is regular.
1
AII :
1
A AIAII AIv0AII AIvI 0 1
v A I A
v
0
.(9)
I AII AIv
So to let the Schur complement matrix be singular, it is
enough to not eliminate edges on the interface.
A huge advantage of this method is that the matrix
,
- 338 - 15th IGTE Symposium 2012
(7)
(8)
corresponding to the sub-domains is block-diagonal, and
its blocks can be inverted in parallel. The speed of one
iteration-step of the DD-BiCG can be increased by
choosing the sub-domains with about the same number of
unknowns and using a multi-core computer.
IV. NUMERICAL RESULT
The efficiency of this method is shown on the example
of a dipole. The dipole has a length of 140 mm, a width of
1 mm and thickness of 20 μm. There is a 160 μm air gap
in the middle (see Fig. 3.). An air volume of 250 mm
radius is modeled around the antenna (see Fig. 2.). The
air volume is truncated by a first order absorbing
boundary condition (ABC).
Fig. 2. The structure of the dipole antenna and the truncation of the air
volume. 1/8 model.
The voltage is prescribed in the air gap (1 V, 1.5 GHz).
Modeling an eighth of the problem, using A,v formulation
(A is the magnetic vector potential, v is the modified
electric scalar potential), and second order hexahedral
finite elements (20 nodes, 36 edges) the resulting problem
has 1.986.152 edges and 669.398 nodes.
Fig. 3. The structure of the dipole antenna (yellow) near the air gap and
the prescribed electric field in the gap (blue). 1/8 model.
The efficiency of the DD-BiCG method compared with
the incomplete Cholesky preconditioned Biconjugate
gradient method (IC-BiCG) is shown in Fig. 6. The
convergence criterion was the global relative residual to
become smaller than 10 -7 .
To demonstrate the efficiency of the method, two
different decompositions were tested. In the first case the
domain has been subdivided into 5 sub-domains (see Fig.
4.), in the second case 8 sub-domains. (Fig. 5.).

Fig. 4. The problem with five sub-domains.
In the first case the first domain is the antenna, the second
is a small air volume around the antenna, and the huge air
volume is subdivided into three parts.
Fig. 5. The problem with eight sub-domains.
In the second case, the antenna and a small air volume
again build the first two sub-domains, but the air volume
is subdivided into six parts.
Method
name
TABLE I
Comparison of the methods
Memory
Requirement
Iterations Run
time (h)
ICCG 9,0 GB 68.750 77,86
DDCG
5 Domains
DDCG
8 Domains
41,8 GB 975 4,41
31,8 GB 1.036 4,66
LU 81,0 GB - -
To solve the problem an “Intel(R) Xeon(R) CPU 2x
X5570@2.93GHz 8 cores 64 GB RAM” computer was
used.
- 339 - 15th IGTE Symposium 2012
Fig. 6. The best residuum of the methods during the iterations.
Blue line: DD-BiCG (5 domains) global residuum;
Red line: DD-BiCG residuum on the interface;
Blue dotted line: DD-BiCG (8 domains);
Red dotted line: DD-BiCG (8 domain) residuum on the interface;
Black line: IC-BiCG residuum in the whole domain.
V. CONCLUSION
Applying the domain decomposition method for
solving huge indefinite equation system iteratively results
in an efficient method with reduced memory requirement
compared to direct methods, and accelerates the iteration
by decreasing the number of iterations. It enables an
efficient parallelization technique in implementating of
the algorithm.
The choice of the sub-domains is very important to
increase the efficiency of the method. The sub-domains
should have about the same number of unknowns.
Increasing the number of domains decreases the memory
requirement but results in a higher condition number.
REFERENCES
[1] O. Nevanlinna, Convergence of iterations for linear equations.
Birkhauser Verlag AG, Basel, 1993, pp. viii+177
[2] G. Koczka , T. Bauernfeind, K. Preis and O. Bíró, "Schur
complement method using domain decomposition for solving
wave propagation problems," presented at The 10th International
Workshop on Finite Elements for Microwave Engineering,
Meredith, New Hampshire United States, Oct. 12-13, 2010.
[3] J.W.H. Liu. Modification of the Minimum-Degree algorithm by
multiple elimination. ACM Transactions on Mathematical
Software, 11(2):141-153, 1985.
[4] G. Karypis and V. Kumar. A Fast and High Quality Multilevel
Scheme for Partitioning Irregular Graphs. SIAM Journal on
Scientific Computing, 20(1):359-392, 1998.

- 340 - 15th IGTE Symposium 2012
On Effectiveness of Model Order Reduction
for Computational Electromagnetism
*Yuki Sato, *Hajime Igarashi
*Graduate School of Information Science and Technology, Hokkaido University
Kita 14, Nishi 9, Kita-ku, Sapporo, 060-0814
E-mail: yukisato@em-si.eng.hokudai.ac.jp
Abstract— This paper presents the model reduction method based on the method of snapshots for time-domain finite element
analysis of quasi-static electromagnetic fields. In this method, the snapshots of transient electromagnetic fields for relatively
short periods are stored to build the variance-covariance matrix, from whose eigenvalues the basis functions for reduced
analysis are constructed. In this paper, the effect of various parameters in the present method such as the number of
snapshots, snapshot intervals on the results of the reduced field computations is discussed.
Index Terms—Model order reduction, finite element method, method of snapshots, eddy current problem.
applying it to three dimensional eddy current problems.
Moreover, we discuss a possible method to determine the
adequate values of the parameters for this method.
I. INTRODUCTION
In recent years, finite element method (FEM) has
widely been applied to transient analysis of quasi-static
and high-frequency electromagnetic fields. However,
since FE equations must be solved at each time step, it
has significant computational burden. Therefore, a lot of
efforts have been made to reduce the computational times
for analysis of transient electromagnetic fields.
One of the most promising methods to shorten the
computational time would be the time-period explicit
error correction (TP-EEC) method [1], [2]. It has been
shown that TP-EEC method applied to non-linear eddy
current problems reduces the computational time for
transient analysis and gives correct steady state solutions
[1], [2]. However, TP-EEC method cannot be applied to
analysis of non-time-periodic problems or high-frequency
problems.
On the other hand, there is yet another method, called
the model order reduction, which can reduce the
computational time for transient analysis [3], [4]. In this
method, the snapshots of transient solution are stored for
initial short period. Then, using these snapshotted
solutions, a variance-covariance matrix is constructed and
the eigenvectors of this matrix is computed. The reduced
FE matrix is then constructed using the transform matrix
whose column space is spanned by the dominant
eigenvectors. There are a few merits in this method; it can
accurately analyze the transient solutions and can be
applied to non-time-periodic problems and highfrequency
problems. However, in order to realize accurate
analysis, it is important to determine adequate values of
the parameters in this method such as snapshot interval
and period, and number of the basis vectors that are
chosen from the eigenvectors of the variance-covariance
matrix. However, the dependence of the accuracy on
these parameters has not been clarified. Moreover, though
the effectiveness of the model reduction for twodimensional
eddy current problems has been discussed
[5], that for three dimensional problems has not been
shown yet.
In our study, the dependence of the accuracy in the
model reduction method on its parameters is evaluated by
II. REDUCTION TECHNIQUE
A. Time-Domain Finite Element Method
The A-φ (A-V) method is used for FE analysis of
quasi-static electromagnetic analysis. The governing
equations derived from Maxwell’s equations is expressed
as
A
rot rotA
grad J , (1)
t
t
A
div grad 0 ,
(2)
t
t
where ν is magnetic resistivity, is conductivity and J is
forced current density. The vector potential A and scholar
potential φ are discretized as follows
e
A a N ,
(3)
j
n
j
j
, (4)
j
j j N
where e and n is the number of edges and nodes, and Nj
and Nj are vector and scholar interpolation functions
respectively. The weighted residual method with Galerkin
method applied to (1) and (2) results in the FE equation
given by
K 0a
d N
S a
b
,
t
0 0
d
S M
0
(5)
t
where
K rotN
rotN
dV
,
(6)
ij
V
i
j
Nij
N i N jdV
,
(7)
V
Sij
N i grad N jdV
,
(8)
V
M grad N grad N dV
, (9)
ij
V
i
j

i
N i JdV.
V
(10)
Moreover, time derivative is approximated by the finite
difference and the unknown variables and right hand
vector are interpolated as
k
x x
k1
( 1
) x ,
(11)
k
b b
k1
( 1
) b ,
(12)
where x = [a φ] t , 0 ≤ θ ≤ 1 and k represents time steps.
Equation (5) now becomes
1 N
t
t
S
S K
M
0
0
k
0
x
1 N
t
t
S
S K
( 1
)
M
0
k
k 1
0
k 1
b ( 1
) b
,
0
x
0
(13)
where Δt is the time step interval. The transient solutions
can be obtained by solving (13) at each time step.
B. Model Reduction
As mentioned above, solution of (13) at each time step
is computationally expensive if the number of unknowns
is large. To reduce the computational time, the reduced
equation is obtained from (13) using the model reduction
method. To do so, after obtaining snapshots for the initial
periods by solving (13), the variance-covariance matrix
Cm
t
m XX C (14)
is constructed where
1
2
s
X [
x μ x μ x μ]
, (15)
x i , i=1,2,..,s are snapshotted solution vectors, s is the
number of snapshots (m>>s) and μ is the mean vector of
these solutions. Note that the matrix Cm is a dense matrix
whose size is the same as that of the FE matrix.
Therefore, numerical solution to the eigenvalue problem
for (14) is computationally prohibitive. In order to
alleviate this problem, we consider the smaller matrix of
s×s defined by
t
Cs
X X
(16)
instead of Cm. The eigenvalues of Cm and Cs are identical
except m-s zero eigenvalues of Cm as shown below. The
singular value decomposition of matrix XR m×s is given
by
X
s
i1
t
t
u v UV
,
(17)
i
i
i
where UR m×s , VR s×s and
diag[ 1 2 s ] , (18)
σ1 ≥σ2 ≥ ... ≥σs ≥ 0 and σi is singular value of X. The
matrices U and V satisfy
t
U I ,
(19)
U s
t
V V Is
,
(20)
where Is denotes the s×s unit matrix. Then, the matrices
Cm and Cs can be decomposed as follows:
C
2 t
U
U ,
(21)
m
- 341 - 15th IGTE Symposium 2012
2 t
Cs V
V . (22)
From eqs. (21) and (22), we find that the eigenvalues of
Cm and Cs are essentially identical. Moreover, since
X vi iu
(23)
i
holds, the eigenvectors of Cm can be easily obtained by
solving the eigenvalue problem for Cs.
Then, the dominant r eigenvectors are chosen to
construct the matrix defined by
W [ 1 2
r ] . w w w
(24)
It is assumed that the original unknown variable x k R m
can be expressed by the linear combination of the reduced
variables y k R r in the form
W .
k
k
x y
(25)
Using the transform (25), the original FE equation (13),
which is simply expressed by Ax=b, can be reduced to
t n t n
W AWy
W b .
(26)
Since the size of the coefficient matrix W t AW is r×r
(m>>s>r), eq. (26) can be solved much faster than (13).
III. ANALYSIS OF BULK CONDUCTOR MODEL
The bulk conductor model shown in Fig. 1 is analyzed
by using the present method. The FE model has 125000
nodes, 117649 elements, 367500 edges and 369036
unknown variables. The conductivity and relative
permeability in the magnetic material are 0.510 7 S/m
and 1000. The driving frequency is set to 50 Hz and time
step is ∆t=10 -4 sec. Under these conditions, as the time
constant τ is estimated to be about 0.01 sec and the period
T is 0.02 sec, the relation T>τ holds.
A. Dependence on snapshot interval and period
We change the snapshot intervals and the period during
which the snapshots are taken to clarify the dependence of
the solution on them. In this study, the solution to eq. (13)
is snapshotted from 0 to T with snapshot intervals 4∆t,
2∆t and ∆t. The snapshot period is T, T/2 and T/4.
Moreover, the number of the basis functions is set as r=40
and 50.
The time variation in the magnetic flux density |B| at
the center of magnetic material is shown in Fig. 2. In this
problem, we set the number of basis function as r=40. In
Fig. 2(a), we find that there are no significant differences
between the original solution and the solutions obtained
by the conventional method and model reduction method
with different snapshot intervals. Also, Fig. 2(b) plots the
time changes in |B| for initial period, 0

35
20
15
y(mm)
magnetic
material
J(A/m 2 )
TABLE I
DEPENDENCE OF COMPUTATIONAL TIME AND ERROR ON SNAPSHOT
INTERVAL AND PERIOD
(A) SNAPSHOT INTERVAL
Snapshot intervals Δt 2Δt 4Δt
computational time (%) 43.2 37.3 29.6
error e(%) 0.07 0.19 0.65
Snapshot period
(B) SNAPSHOT PERIOD
T T/2 T/4
computational time (%) 42.5 30.0 24.7
error e(%) 0.02 0.32 0.60
TABLE II
DEPENDENCE OF COMPUTATIONAL TIME AND ERROR ON NUMBER OF
BASIS FUNCTION
Number of basis function 20 30 40
computational time (%) 40.2 41.8 43.2
error (%) 2.38 0.29 0.07
TABLE III
DEPENDENCE OF COMPUTATIONAL TIME AND ERROR ON NUMBER OF
BASIS FUNCTION LONGER TIME CONSTANT
Number of basis function 40 60 80
computational time (%) 46.6 56.0 65.5
error (%) 0.39 0.15 0.07
conductor, shown in Fig. 4, obtained by the conventional
method and present method in which the snapshot period
is set to T/4. The discrepancy can also be found in the
initial responses shown in Fig. 3(b). This suggests that the
long range errors can be predicted from the initial
responses. That is, by performing the analysis using the
present method for initial short period changing the
snapshot period, we could know the appropriate value for
it.
The error between the original solution and that
obtained by the present method is defined by
e
H i
i
i
red
H
H
i
i
z(mm)
x(mm)
15 20 35
15
coil
Figure1 : Bulk conductor model
100(%)
- 342 - 15th IGTE Symposium 2012
(27)
where Hi red is the magnetic field obtained by the present
method and Hi is computed from the original solution.
Table I and II summarize the error e evaluated at t=0.08
sec where the solutions sufficiently converge to steady
state and corresponding computational time. We can see
that the errors e become small as the snapshot interval
decreases or the snapshot period increases. However, the
35
5
x
magnetic
material
20
coil
35
x(mm)
Magnetic Density (T)
Magnetic Density (T)
2.00E-04
1.50E-04
1.00E-04
5.00E-05
0.00E+00
Original solution without reduction
T/4
T/2
T
-5.00E-05
0 0.02
Time (s)
0.04 0.06
(a) |B| during 0

(a) Original solution.
(b) Solution obtained by present method.
Figure 4 : Eddy current distribution.
computational time simultaneously increases. This means
that we must determine these parameters considering both
effects.
B. Dependence on number of basis functions
In this section, we discuss the dependence of the
solutions obtained by the present method on r, the number
of the basis functions. The snapshot period and interval
are set to T and Δt, respectively. The analysis results are
shown in Fig. 5. It can be found both in Fig. 5 (a) and (b)
that the solutions obtained by the present results approach
the original solution as r increases.
The computational time and error e depending on r are
summarized in Table II. We can see that e decreases as r
increases while there are little dependence of the
computational time on r.
C. Bulk conductor with longer time constant
To test the validity of the present method for systems
with longer time constants, we increase the conductivity
- 343 - 15th IGTE Symposium 2012
Magnetic Density (T)
Magnetic Density (T)
2.00E-04
1.50E-04
1.00E-04
5.00E-05
0.00E+00
-5.00E-05
0 0.01 0.02 0.03 0.04 0.05
(a) |B| during 0

(a) Original solution
(b) Primal basis vector w1
(c) Second basis vector w2
(d) Fifth basis vector w5
Figure 7 : Distribution of original solution and basis vector.
- 344 - 15th IGTE Symposium 2012
35
20
15
Magnetic Density (T)
2.50E-04
2.00E-04
1.50E-04
1.00E-04
5.00E-05
Original solution without reduction
Δt
0.00E+00
-5.00E-05
2Δt
4Δt
0 0.01 0.02
Time (s)
0.03 0.04 0.05
(a) |B| during 0

sec. The snapshot period is set to T/2, T/4 and T/8. The
snapshot interval and the number of basis vectors are set
to Δt and 40, respectively.
The time change in |B| at the center of the model is
shown in Fig. 9, where (a) and (b) show the relatively
long-range and initial responses, respectively. Due to the
structure of the stacked iron core, the time constant of this
system is much smaller than that of the bulk iron shown in
Fig. 1. We can see that in Fig. 9 that the solutions are in
good agreement with the original solution except during
the initial short period.
V. CONCLUSION
In this paper, the three dimensional time-domain FE
analysis using the model order reduction based on the
method of snapshots has been presented. Effectiveness of
this present method is shown for bulk conductor and
stacked iron model. It has been found that the snapshot
period and number of basis functions have great influence
on the transient solutions obtained by the present method.
It has been suggested that these parameters could be
appropriately determined by performing time marching
for initial some steps for the different parameter values.
In future, we plan to apply the present method to nonlinear
eddy current problems and high-frequency
problems.
REFERENCES
[1] Y. Takahashi, T. Tokumasu, M. Fujita, S. Wakao, T. Iwashita,
and M.Kanazawa, “Improvement of convergence characteristic in
nonlinear transient eddy-current analyses using the error
correction of time integration based on the time-periodic FEM and
the EEC method,” (in Japanese) IEEJ Trans. PE, vol. 129, no. 6,
2009.
[2] H. Igarashi, Y. Watanabe and Y. Ito, ”Why Error Correction
Methods Realize Fast Computations,” IEEE Trans. Magn., vol.
48, no. 2, pp.415-418, 2012.
[3] Krysl, P., S. Lall, and J. Marsden, “Dimensional Model Reduction
in Non-linear Finite Element Dynamics of Solids and Structures,”
International Journal for Numerical Methods in Engineering,
vol. 51, pp479-504, 2001.
[4] G. Kerschen J. Golinval, AF. Vakakis, LA. Bergman, The
method of proper orthogonal decomposition for dynamical
characterization and order reduction of mechanical systems: an
overview, Nonlinear Dynamics, vol. 41, pp. 147 169, 2005.
[5] S. Rutenkroger, B. Deken, S. Pekarek, Reduction of Model
Dimension in Nonlinear Finite Element Approximations of
Electromagnetic, Computers in Power Electronics, 2004,
Proceedings. IEEE Workshop on, pp. 20-27, Aug., 2004.
[6] P. Holmes, JL. Lumley, G. Berkooz, Turbulence; Coherent
Structures; Dynamical Systems and Symmetry. Cambridge
University Press: Cambridge, 1996.
- 345 - 15th IGTE Symposium 2012

- 346 - 15th IGTE Symposium 2012
Calculation of eddy-current probe signal for a
3D defect using global series expansion
Sándor Bilicz, József Pávó and Szabolcs Gyimóthy
Budapest University of Technology and Economics
Department of Broadband Infocommunications and Electromagnetic Theory
Goldmann Gy. tér 3.,1111 Budapest, Hungary
E-mail: bilicz@evt.bme.hu
Abstract—A novel eddy-current modeling technique of volumetric defects embedded in conducting plates is presented in
the paper. This problem is of great interest in electromagnetic nondestructive evaluation (ENDE) and has already been
exhaustively studied. The defect is modeled by a volumetric current dipole density which satisfies an integral equation.
The latter is solved by the classical method of moments. This have been usually based on the volume discretisation of
the defect. Contrarily, –as a new contribution– we propose the use of globally defined, continuous basis functions for the
expansion of the current dipole density. This global expansion lets us expect for an improvement of the numerical stability
and the performance of the simulation. The proposed method is tested against both measured and synthetic data obtained
by a different defect model.
Index Terms—eddy-current modeling, integral equation, global expansion, moment method
I. INTRODUCTION
Eddy-current nondestructive testing (ECT) is a widely
used technique to reveal and characterize in-material
flaws (inclusions, voids, cracks, etc.) within conducting
specimens. The principle of ECT is based on the local
changes in the specimen’s electromagnetic (EM) parameters
due to the flaw. These changes result in an EM field
different from the field in the flawless case. Either the
field directly, or a deduced quantity (e.g., impedance of
a probe coil) is measured during a nondestructive test and
the acquired data are used for the flaw reconstruction.
The inverse problem of nondestructive testing can be
ill-posed. This means that any of the existence, unicity
and stability of the solution is not necessarrily provided.
Beyond these theoretical challenges, the flaw characterization
can be numerically demanding as well: the
inversion algorithms are often iterative, i.e., several flaws
are to be sequentially simulated in an optimisation loop.
Consequently, a key element of the inversion is a fast and
reliable numerical simulation of flaws.
Classical attempts of flaw modeling are the integral
approaches. They can cope with the difficulties arisen by
the relatively small size of flaws compared to the excited
region (yielding discretisation issues). The classical work
[1] presents a flaw simulation where the flawed volume is
discretised by a regular grid. The yielded volume integral
eqution is resolved by the Method of Moments (MoM)
[2], assuming a piecewise constant approximation of the
EM field over each cell of the grid. By now, this method
has been implemented in commercial softwares, e.g., [3],
and has been successfully applied in inversion algorithms
as well [4]. The volume integral method has recently
been revisited in [5], where the EM field is expanded
by means of locally defined splines. This provides the
smoothness of the field, which is violated in the previous
approach. Variational formalisms have also been tried
with success: in [6], a Finite Element Method (FEM)
scheme is presented for the separated computation of the
field in the flawless specimen and the “reaction field”
risen by the presence of the flaw. Coupled methods
have been introduced, e.g., in [7]: a FEM code for the
computation of the flawless field is coupled with a surface
integral scheme of the ideally thin crack model.
In [8], an ideally thin crack is considered which is
modeled by a surface integral equation, again resolved
by MoM, using a piecewise constant approximation.
Though some of the works above were carried out
decades ago, several pitfalls are still present in eddycurrent
flaw modeling. Today’s challenges are mainly
related to the increasing needs of flaw inversion in the
sense of accuracy and speed. Beyond being small, flaws
can have bad aspect ratio as well, making the volumetric
models fail. It is also not straightforward how to choose
between the volumetric and the ideally thin crack models
for an arbitrary defect. The optimisation-based inversion
schemes can badly perform if the sensitivity data are
inaccurate. Another important issue is the convergence of
the simulation with respect to the discretisation applied.
In case of a grid-discretisation, this can only be controled
at the price of computational load.
These challenges inspired the improvement of the
MoM-based discretisation techniques of the integral
equation models. The above cited formalisms are resolved
by using locally defined basis functions for the
expansion of the EM field. A new approach has been
presented in [9], where the basis functions were globally
defined (i.e., all over the surface of the ideal crack)
harmonic functions. The use of such global expansion

provided considerable advantages over local expansions.
In this paper, we present the use of global expansion
functions for volumetric flaw modeling. In a certain
sense, this is an extension of the method formalized for
the ideally thin flaws in [9].
II. THE VOLUME INTEGRAL METHOD
Let us consider a non-magnetic, conducting specimen
(to be tested against material flaws) with a homogeneous
conductivity σ0. A time-harmonic source (typically, a
coil) near the specimen induces eddy-currents within the
conductive medium. In the presence of a flaw embedded
in the volume region V , the otherwise constant conductiviy
of the specimen will locally change: σ = σ(r),
r ∈ V , so the EM field will change, too. The EM field
can be decomposed into a so-called incident term and a
defect term :
E(r) =E i (r)+E d (r), (1)
where only Ei (r) would exist in the flawless (σ(r) ≡ σ0)
specimen, whereas Ed (r) rises due to the flaw. The latter
is imagined as the field corresponding to a fictious source
distribution which takes place in the flawless specimen
and has exactly the same effect as imposed by the flaw
[1]. Formally, let the secondary source be a current dipole
density P =(σ(r) − σ0)E, r ∈ V .ThenEd (r) can be
expressed as
E d
(r) =−jωμ0 G(r|r ′ )P(r ′ )dV ′ , (2)
V
where G(r|r ′ ) is the electric-electric dyadic Green’s
function transforming the current density excitation at
the point r ′ to the generated electric field at the point
r. ω is the angular frequency of the source and μ0 is
the vacuum permeability. By substituting (2) into (1),
using the definition of P, we get a Fredholm-type integral
equation of the second kind for the unknown current
dipole density:
1
P(r)+jωμ0
σ(r) − σ0
G(r|r
V
′ )P(r ′ )dV ′ =
= E i (r).
Once the integral equation is solved, P(r) can be used
to derive quantites that can be measured during the
nondestructive test. In the illustrative cases that we will
present in this paper, a probe coil is used for both the
excitation of the field and the acquisition of the measured
data via its complex impedance variation ΔZ. As a
consequence of the reciprocity principle [10], ΔZ can
be computed as
ΔZ = − 1
I2
E i (r) · P(r)dV, (4)
V
with I being the amplitude of the probe coil’s current.
This decomposition (1) let E i (r) and G(r|r ′ ) be
separately computed, which provides the well-known
advantages from the viewpoint of numerical evaluation.
Moreover, the formula (4) can also be easily evaluated.
- 347 - 15th IGTE Symposium 2012
(3)
Let us also highlight that the volume integral method
bears the potential pitfall of properly computing the
Green’s function. Due to its singularity, a numerically
stable expression of G(r|r ′ ) often requires special efforts,
as it will be shown in Subsection III-C.
III. SOLUTION OF THE INTEGRAL EQUATION
A. The studied configuration
We restrict our studies to a special, but practically
important configuration, outlined in Fig. 1. The specimen
is assumed to be a homogeneous conducting plate with
a finite thickness. The dimensions of the plate in the x
and y directions are assumed to be infinite. The flaw
is of cuboid shape and it has four edges perpendicular
to the plate surface. The flaw edges are A, B and D,
respectively, and the volume V is defined as
|x| ≤ A B
, |y| ≤ and |z − C| ≤
2 2
D
, (5)
2
where C is the center of the crack along z. The conductivity
within the flaw volume V is known, σ(r), typically,
σ(r) =0.
r 1
r2
Coil
x
c
Coil
Plate
z=C
y
c
A
y
z
B
z=0 l
A
z=−d
D
Plate
Flaw
TOP VIEW
d
h
SIDE VIEW
Figure 1. The studied configuration. An air-cored pancake-type coil
scans above the infinite plate near the flaw. For generality, a burried
flaw is sketched, however, we deal with ID and OD flaws.
The probe coil is actually an air-cored pancake-type
probe, driven by a time-harmonic current. During the
nondestructive test, the coil scans above the damaged
zone and its impedance variation is measured at given
coil positions.
B. Global approximation of the current dipole density
Let us approximate the solution P(r) of the integral
equation (3) by means of a finite series. Let the basis
x
x

functions of this expansion be products of three factors,
each depending only on one Cartesian coordinate:
P(r) =
M
N
Q
m=−M n=−N q=−Q
Pmnqf m x (x)f n y (y)f q z (z),
(6)
The key idea in this paper is the choice of the basis
functions: in contrary with the classical schemes, herein
each basis function is globally defined, i.e., all over the
flaw volume V . Let us note that the special restrictions
for the shape of the flaw are needed here. We propose
the use of the following harmonic factors in the basis
functions:
f m
1
x (x) =
A exp
2πj mx
,
A
f n
1
y (y) =
B exp
2πj ny
,
B
f q
1
z (z) =
D exp
(7)
q(z − C)
2πj .
D
In fact, this leads to a three-dimensional complex Fourierseries;
the integers m, n and q are the harmonic orders.
The basis functions form an orthonormal set with respect
to the scalar product:
g(r) ,h(r) := g(r)h ⋆ (r)dV. (8)
Let us notice that our choice for the basis functions
provides a smooth approximation of P, instead of the
piecewise constant approximation discussed in [1]. In
Section IV, the advantages provided by the global expansion
are discussed along with numerical examples.
C. Discretisation by the Method of Moments; computation
of the matrix elements
The R =(2M +1)(2N+1)(2Q+1) unknown vectorial
coefficients Pmnq in the series (6) are determined by
means of the Method of Moments. The testing functions
are the same as the basis functions (Galerkin-method) and
a linear system of R vectorial equations is obtained. For a
handy formalization, let the basis functions be ordered so
as each triplet of harmonic orders (m, n, q) has a unique
index k (1 ≥ k ≥ R) and denote the kth basis function
as
wk(x, y, z) =f m x (x)f n y (y)f q z (z). (9)
The elements in the system matrix of the linear equations
are in the form
a λκ
lk =(eλ · eκ) wl(r) ,wk(r)/(σ − σ0) +
jωμ0 wl(r) , eλ · G(r|r
V
′ )(eκwk(r ′ ))dV ′ ,
(10)
where l and k are the indices of the test and basis
functions (l, k =1, 2,...,R), eλ and eκ are the unitvectors
(λ, κ = x, y, z) and , stands for the scalar
product.
V
- 348 - 15th IGTE Symposium 2012
The evaluation of the integral with respect to r ′ dV ′
needs special numerical treatment due to the singularity
of the Green’s function. However, in the case of the
considered planar geometry and of the proposed basis
functions, one can cope with the singular kernel by
means of the spectral method, presented in detail in
[11]. In brief, by using the 2-dimensional spatial Fouriertransform
in the xy plane, the spectrum of the Green’s
function can be represented as a sum of planar waves
traveling along z. Due to the product-separation form (7)
of the bais functions, the integral with respect to r ′ dV ′
in (10) splits up to a factor depending only on z and z ′
and to an other factor which is represented by its 2D
Fourier-transform. The integral with respect to z ′ can
be analytically evaluated in the spectral domain, and the
remaining factor (to be inverse transformed) is no longer
singular.
As a useful consequence of the Galerkin-method,
certain elements of the yielded system matrix must be
the same. In (10), the volume integral with respect to
rdV (due to the scalar product) and to r ′ dV ′ can be
commuted. According to the reciprocity theorem, we
have
a λκ
lk ≡ a κλ
k ′ l ′, (11)
for all k and l if wk(r) ≡ wk ′(r)⋆ and wl(r) ≡ wl ′(r)⋆
hold, respectively. This equivalence can be applied to
check the numerical computations and/or to reduce the
computational load.
Finally, let us notice that the presence of the probe coil
is neglected in the expression of the Green’s function.
This is usual and does not cause considerable error.
D. Computation of the incident field
The Ei (r) incident field can be analytically computed
in the studied case. The pancake-type coil generates an
axisymmetric field which depends only on z and r =
(x − xc) 2 +(y − yc) 2 . This field can be expressed in
the form of an integral of first-order Bessel-functions, as
detailed in the classical work [12].
More complicated probes (e.g., including ferrit core or
having rectangular-shaped turns) can also be considered.
However, in such cases, Ei (r) is obviously more difficult
to compute (e.g., by a Finite Element Method).
Once the incident field is obtained within the flaw
volume V , the excitation vector of the linear system of
equations yielded by the MoM can be assembled from
the entries
b λ l = wl(r) , eλ · E i (r) , (12)
where l =1, 2,...,Ris the index of testing function and
eλ is the unit vector (λ = x, y, z). As a consequence of
the axial symmetry, bz l ≡ 0 holds for all l.
E. Implementation issues
The algorithms are coded in Matlab R○ . The spectral
domain expression of the Green’s function is inverse
transformed by a 2-dimensional Fast Fourer Transform

(FFT2) routine. The width of the FFT2’s spatial window
in the xy plane is estimated from the skin depth within
the conductive medium, whereas the spectral window is
assigned with respect to the harmonic orders m and n,
respectively.
The integrals involved by the scalar products in (10)
and (12) are evaluated numerically, based on a regular
discretisation of the flaw volume. The number of samples
along each axis is set with respect to the harmonic orders
m, n and q of the basis and testing functions, respectively.
IV. TEST CASES AND COMPARISONS
In this section, the proposed method is illustrated
and its main advantages are highlighted via numerical
examples.
A. Definition of the configurations
The illustrative test cases are presented in Fig. 1. The
air-filled rectangular flaw has constant zero conductivity.
Though a buried flaw is outlined in the sketch, we present
cases for ID-type (“inner defect”, opening to the top
surface of the plate: C = −D/2) and OD-type (“outer
defect”, C = −d + D/2) flaws only.
Experimental data of the variation of the coil’s
impedance (ΔZ) are available on the xc = 0 line in
function of yc, at discrete coil positions. The cases #1
and #2 are JSAEM Benchmarks [13], whereas case #3 is
an also frequently cited TEAM Benchmark no. 15 [14].
The parameters of each case are given in Tab. I.
Table I
NUMERICAL DESCRIPTION OF THE TEST CASES.(NOTATION IS
ACCORDING TO FIG.1.)
#1 #2 #3
Specimen
d (mm) 1.25 1.25 12.22
σ0 (MS/m) 1 1 30.6
Flaw
A (mm) 0.21 0.21 0.28
B (mm) 10 10 12.6
D (mm) 0.75 0.5 5
C (mm) −d + D/2 −D/2 −D/2
Probe coil
r1 (mm) 0.6 0.6 6.15
r2 (mm) 1.6 1.6 12.4
h (mm) 0.8 0.8 6.15
l (mm) 0.5 1.0 0.88
f (kHz) 150 300 0.9
Turns 140 140 3790
B. Convergence of the series
One of the main advantages of the global expansion
method is the easy access to the convergence with respect
to the maximal harmonic orders of M, N and Q in the
series of P. By adding further terms of higher orders,
the previously computed elements of the system matrix
remain unchanged. Consequently, convergence studies
can be performed at a much lower computational cost
than in the case of local basis functions (the latter needs
a new grid whenever a finer discretisation is set).
- 349 - 15th IGTE Symposium 2012
We have computed ΔZ in the test case #1 using different
MNQ maximal orders. The discrepancy between
two impedance signals is expressed by the norm
ΔZ := (1/K)ΣK k=1 |ΔZ(yc,k)| 2 , (13)
where the number of coil positions is actually K =11
and yc,k =(k−1) mm. In Fig. 2, the normalised discrepancy
between the first impedance signal (MNQ = 121)
and some others obtained by higher order approximations
is shown. A fast convergence is experienced: e.g., there
is ca. 5% discrepancy between the impedance signals
computed with MNQ = 121 and with the higher orders
MNQ = 163. Let us note that the variation of the current
dipole density in the x-direction is smooth enough to be
modeled by a first order Fourier series, i.e., the choice
M =1seems to be appropriate.
||ΔZ − ΔZ 121 || / ||ΔZ 121 ||
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
121
131
151161
141
122
132
152162
142
123
133
153163
143
Figure 2. Normalised discrepancy between impedance signals obtained
by different maximal harmonic orders MNQ (marked above each bar)
in test problem #1.
The fast convergence is also reasoned by the behavior
of the coefficients Pmnq. Again in the test case #1, we
examined the coefficients of the x-directed current dipole
density P x (r) (note that P x is much more dominant than
P y and P z in this case) for a centered coil location (xc =
yc =0). Some coefficients of the largest magnitude are
plotted in Fig. 3. A fast decrease of the magnitudes is
experienced as the harmonic orders increase.
C. Comparison to exparimental data and to the ideally
thin crack model
The volumetric flaw model using global expansion
is a sort of extension of the model proposed in [9].
Therein, ideally thin cracks are considered and modeled
by a surface layer of current dipole density. This can be
imagined as if the A edge length of the crack (Fig. 1)
would collapse to zero whereas the x component of
the total electric field E vanishes on the crack surface.
For the surface current dipole density, certain boundary
conditions must hold, thus, the basis functions in the

x
Normalized |P |
mnq
1.2
1
0.8
0.6
0.4
0.2
0
0 1 0
0 −1 0
1 1 0
−1 1 0
1 −1 0
−1 −1 0
0 −2 0
0 2 0
1 −2 0
−1 −2 0
1 2 0
−1 2 0
0 3 0
0 −3 0
−1 1 1
−1 −1 1
1 1 1
1 −1 1
−1 1 −1
−1 −1 −1
1 1 −1
1 −1 −1
1 1 2
−1 1 2
1 −1 2
−1 −1 2
1 1 3
1 −1 3
−1 1 3
−1 −1 3
1 1 −2
−1 1 −2
1 −1 −2
−1 −1 −2
Figure 3. Normalized magnitudes of the coefficients of the x-directed
current dipole density P x (r) in test problem #1. The 34 highest magnitudes
are plotted (totally there are (2M + 1)(2N + 1)(2Q + 1) = 273
coefficients, as M =1, N =6and Q =3is chosen); the triplets
mnq (any of the indices can be negative as it can be seen) are marked
above each bar.
series expansion –e.g., for ID cracks– are products of
the following sine and cosine functions:
g n
2
y (y) =
B sin
y + B/2
nπ ,
B
g q
2
z (z) =
D cos
(2q − 1)π z
(14)
,
2D
with the integers n and q, as “harmonic orders”. Herein
we do not deal with the surface model in detail, but
only present some results for comparison, provided by
the authors of [9].
In Figs. 4, 5 and 6, comparisons of impedances (i)
computed by our volumetric model, (ii) by the surface
model and (iii) measured data (provided with the benchmarks)
are presented. The comparisons let us conclude
the followings:
• The volumetric model can appropriately reconstruct
the measured data at very low maximal harmonic
orders. Let us note again that for instance, when
MNQ = 122, we have only 75 (vectorial) unknowns
in the series (6).
• Though there is no straightforward connection between
the harmonic orders of the volumetric and
surface models, in the presented cases, the volumetric
model provides better results in the sense of |ΔZ|
than the surface model with more-or-less the same
harmonic orders. In Figs. 4 and 5, the surface model
appears to be unstable at NQ =44. (However, the
surface model also has good convergence properties
but at considerably higher N and Q which is not
presented herein.)
• The volumetric model tends to slightly overestimate
|ΔZ| and underestimate the phase arg{ΔZ}.
Whereas the phase error is acceptable in Figs. 4 and
5, it becomes considerable in Fig. 6.
- 350 - 15th IGTE Symposium 2012
|ΔZ| (mΩ)
arg{ΔZ} (rad)
80
60
40
20
0
1.5
1
0.5
0
JSAEM OD−60 Benchmark 150 kHz
measured
vol 011
vol 122
surf 44
surf 66
surf 88
0 2 4 6 8 10
Coil position y (mm)
c
Figure 4. Impedance variation in the test case #1. Legend: “vol”
and “surf” refer to the volumetric and the surface model. The maximal
harmonic orders MNQ and NQ used in the simulations are also given.
|ΔZ| (mΩ)
arg{ΔZ} (rad)
150
100
50
0
3
2
1
JSAEM ID−40 Benchmark 300 kHz
measured
vol 011
vol 122
surf 44
surf 66
surf 88
0
0 2 4 6 8 10
Coil position y (mm)
c
Figure 5. Impedance variation in the test case #2. Legend notations
are explained in Fig. 4.
The computation times are quite short. The assemblation
of the system matrix took, e.g., 285 s for the OD60
flaw and 202 s for the ID40 flaw when a basis function
set with maximal orders M =1, N =4, Q =8was
used for both. Though the number of samples within the
flaws for the numerical integration is the same in both
cases, the OD60 flaw computation needs a wider spatial
window for the FFT2 as the lower frequency yields a
higher skin-depth.
V. CONCLUSION AND PERSPECTIVES
The classical integral equation models of volumetric
flaws have been used for decades in the simulation
of ECT. Though these schemes have many advantages,
several bottlenecks are still present. In this paper, we
proposed a new discretisation technique for the numerical
solution of the volume integral equation. Instead of the
locally defined, pulse basis functions, we use globally
defined, harmonic basis functions for the expansion of
the unknown current dipole density distribution. Thanks

|ΔZ| (Ω)
arg{ΔZ} (rad)
20
15
10
5
0
2
1.5
1
0.5
TEAM Benchmark
measured
vol 011
vol 122
surf 22
surf 44
surf 66
0 5 10 15 20
Coil position y (mm)
c
Figure 6. Impedance variation in the test case #3. Legend notations
are explained in Fig. 4.
to this choice, an improvement in the accuracy and
performance of the simulation has been experienced in
the test cases. The results obtained so far are promising,
the research certainly needs to be continued, with special
emphasis on the followings:
• More parametric studies are needed for a various
range of flaw sizes and frequencies to confirm that
the new scheme outperforms the existing ones, and
at the same time, to point out its limitations. We
have not considered yet, for instance, through-plate
flaws.
• The method could easily be extended to the case
of flaws embedded in thick plates (modeled as halfspace).
The extension must be possible to the case
of layered medium as well, which might be of more
practical interest.
• As the aspect ratio of the flaw (e.g., width length)
•
gets worse, the volumetric model is expected to become
less accurate and the ideally thin crack model
should be applied instead. However, the relation between
the two models has not been exactly revealed
yet. One expects the results of the volumetric model
to converge to the results of the surface model as the
width of the flaw collapses. This should be studied
both in theoretical and in numerical senses as well.
The inverse problem is often formalised as an
optimisation task of minimizing the discrepancy
between the measured and simulated data. The
gradient-based schemes require the sensitivity data
with respect to the parameters of the flaw. This
sensitivity is accessible, e.g., via the the adjoint
problem [15]. However, the numerical stability of
the gradient computation strongly depends on the
precision of the EM field calculation near the boundaries
of the flaw. The proposed global expansion
of P could improve the precision in these cruical
regions.
- 351 - 15th IGTE Symposium 2012
The authors think that the contribution of this paper
can be of industrial interest as well, if the further numerical
studies remain convincing about its performance.
VI. ACKNOWLEDGEMENTS
This research is supported by the Hungarian Science
Research Fund (OTKA grant no. K105996).
REFERENCES
[1] J. R. Bowler, S. A. Jenkins, L. D. Sabbagh, and H. A. Sabbagh,
“Eddy-current probe impedance due to a volumetric flaw,” Journal
of Applied Physics, vol. 70, no. 3, pp. 1107 –1114, 1991.
[2] R. F. Harrington, Field computation by moment methods.
Macmillan, 1968.
[3] CIVA. “CIVA: State of the art simulation platform for NDE”.
[Online]. Available: http://www-civa.cea.fr
[4] S. Bilicz, E. Vazquez, M. Lambert, S. Gyimóthy, and J. Pávó,
“Characterization of a 3D defect using the expected improvement
algorithm,” COMPEL: The International Journal for Computation
and Mathematics in Electrical and Electronic Engineering,
vol. 28, no. 4, pp. 851–864, 2009.
[5] C. Reboud, D. Prémel, D. Lesselier, and B. Bisiaux, “New discretisation
scheme based on splines for volume integral method:
Application to eddy current testing of tubes,” COMPEL: The
International Journal for Computation and Mathematics in Electrical
and Electronic Engineering, vol. 27, no. 1, pp. 288–297,
2008.
[6] Z. Badics, Y. Matsumoto, K. Aoki, F. Nakayasu, M. Uesaka, and
K. Miya, “Accurate probe-response calculation in eddy current
NDE by finite element method,” Journal of Nondestructive
Evaluation, vol. 14, pp. 181–192, 1995. [Online]. Available:
http://dx.doi.org/10.1007/BF00730888
[7] Y. Le Bihan, J. Pavo, M. Bensetti, and C. Marchand, “Computational
environment for the fast calculation of ect probe signal by
field decomposition,” Magnetics, IEEE Transactions on, vol. 42,
no. 4, pp. 1411 –1414, 2006.
[8] J. R. Bowler, “Eddy-current interaction with an ideal crack. I. The
forward problem,” Journal of Applied Physics, vol. 75, no. 12, pp.
8128–8137, 1994.
[9] J. Pávó and D. Lesselier, “Calculation of eddy current testing
probe signal with global approximation,” IEEE Transactions on
Magnetics, vol. 42, no. 4, pp. 1419–1422, 2006.
[10] R. F. Harrington, Time-harmonic electromagnetic fields.
McGraw-Hill, 1961.
[11] J. Pávó and K. Miya, “Reconstruction of crack shape by optimization
using eddy current field measurement,” IEEE Transactions on
Magnetics, vol. 30, no. 5, pp. 3407–3410, 1994.
[12] C. V. Dodd and W. E. Deeds, “Analytical solutions to eddy-current
probe-coil problems,” Journal of Applied Physics, vol. 39, no. 6,
pp. 2829–2838, 1968.
[13] T. Takagi, M. Uesaka, and K. Miya, “Electromagnetic NDE
research activities in JSAEM,” in Electromagnetic Nondestructive
Evaluation, ser. Studies in Applied Electromagnetics and Mechanics,
T. Takagi, J. R. Bowler, and Y. Yoshida, Eds. IOS Press,
1997, vol. 1, pp. 9–16.
[14] T.E.A.M. Benchmark Problems. Accessed on 7.08.2012. [Online].
Available: http://www.compumag.org/jsite/team.html
[15] S. J. Norton and J. R. Bowler, “Theory of eddy current inversion,”
Journal of Applied Physics, vol. 73, no. 2, pp. 501–512, 1993.

- 352 - 15th IGTE Symposium 2012
Computation of the Motion of Conducting Bodies
Using the Eddy-Current Integral Equation
*Mihai Maricaru, † Ioan R. Ciric, *Horia Gavrila, *George-Marian Vasilescu and *Florea I. Hantila
*Department of Electrical Engineering, Politehnica University of Bucharest, Spl. Independentei 313,
Bucharest, 060042, Romania, E-mail: mihai.maricaru@upb.ro
† Department of Electrical and Computer Engineering, The University of Manitoba, Winnipeg, MB R3T 5V6, Canada
Abstract—The analysis of the motion of a system of solid conductors in the presence of magnetic fields is performed by
solving the classical mechanics equation of motion under the action of magnetic forces. Application of the eddy-current
integral equation and the usage of the local coordinates attached to the bodies in motion allow the determination of
electromagnetic field without being necessary to reconstruct the discretization grid at each new position of the conducting
bodies. Only the submatrices associated with the coupling between the bodies in relative motion are modified in the global
system matrix. A time-domain method of solution is first presented for the electromagnetic field problem, coupled with the
equation of motion, which can be efficiently applied at high frequencies when the time steps are small. The eddy-current
integral equation for the derivative of current density contains a term that takes into account the relative motion of the
bodies. Since the electromagnetic quantities vary much more rapidly than the mechanical quantities, a second method is also
proposed in this paper, where the eddy-current integral equation is solved in the frequency domain by assuming that the
bodies are motionless, but by adding supplementary terms due to the actual motion of the bodies. Thus, only the average
force over a period of time is now computed. This method is extremely efficient especially at higher frequencies when the
time steps are very small.
Index Terms—eddy-current integral equation, electrodynamics of moving conductors, levitation.
I. INTRODUCTION
The equation of translational motion of a solid
conducting body of mass m under the action of the
magnetic force F is
2
d r dr
m F(
r,
, )
G
(1)
2
dt dt
where r is the position vector of a point of the body, for
instance of its center of gravity, is a vector
representing the imposed current distribution and G is the
gravitational force acting on the body. Equation (1) is
discretized in time and F is determined at each time step
by solving an electromagnetic field problem in the region
with moving bodies. The application of the Finite
Element Method requires a tremendous amount of
computation since it is necessary to reconstruct the
discretization mesh at each time step. Moreover, the
modifications of the discretization mesh are, usually,
accompanied by undesired cumulative errors in the
successive solutions of the electromagnetic field. A
substantial improvement can be achieved when adopting
hybrid Finite Element – Boundary Element Methods [1].
Using the “laboratory” frame of references complicates
the field problem solution due to the presence of the
motional electric field intensity v0 B , where v 0 is the
body velocity in this frame of references and B the
magnetic induction. This disadvantage is eliminated
when employing local frames of reference, attached to
the bodies in motion [1], [2]. This also allows the usage
of the simpler eddy-current integral equation for the
bodies at rest, as it has been done in the case the
velocities of the bodies are known [2]. However, in many
situations the velocities of the bodies are not known, as,
for instance, in the case of the electromagnetic levitation,
their determination constituting one of the objectives of
the present work.
In the case of the electromagnetic levitation, to ensure
the stability of the solution it is necessary to choose a
sufficiently small time step. Since for same accuracy of
the results the time period has to be divided practically in
the same number of intervals (for example, at 50 Hz in
200 intervals [1]), at higher frequencies the time step
decreases. Unfortunately, as the time step decreases, the
successively computed solutions tend to be very close to
each other and the errors in the solution differences
increase considerably, the computation procedure
becoming inefficient.
In the present paper, a new procedure is described for
the time-domain solution of the eddy-current integral
equation applicable to small time steps. As well, a
technique is proposed for accelerating the determination
of the trajectory of the moving bodies, based on the
frequency-domain solution of the eddy-current integral
equation.
II. TIME-DOMAIN SOLUTION OF THE EDDY-CURRENT
INTEGRAL EQUATION
For two-dimensional field problems, the time-domain
eddy-current integral equation for motionless conductors
is
d
1
J ( r, t)
J ( r',
t)
ln dS'
dt R
d
1
Ji
(r',
t)
ln dS'
(2)
dt
R
i
where r and r ' are the position vectors of the observation
point and of the source point, respectively, is the

egion containing the solid conductors, i is the region
where the imposed current density J i is confined,
0
R |
r r'|
, , 0 being the permeability of free
2
space.
In the three-dimensional case, the eddy-current integral
equation has the form
d J(
r',
t)
J(
r,
t)
dV '
2 dt R
d Ji
( r',
t)
dV ' grad
2 dt R
i
where is the electric scalar potential.
To simplify the formulation, we consider here a
two-dimensional structure with a single solid conductor.
Using a frame of reference attached to the conducting
body in motion, the time discretization of (3) leads to
t
1
( J J0
) ( J1
J0
) ln dS '
2
R
1
- 353 - 15th IGTE Symposium 2012
(3)
1
1
J0t Ji
ln dS'
ln '
1 Ji
dS (4)
0
R
R
i1
where the subscript “0” indicates the time t and the
subscript “1” the time t t
. Dividing (4) by t yields
i0
J
t
J
1
ln dS'
J
t
1 2 t
1 R
2
2
Ji
1
1
ln ' ( ) 1 dS
ln '
1 n v
Ji
dl (5)
t
R
2 R
i
1
2
2
i
1 t
where the subscript “ ” refers to the time t
2
2
, i
is the boundary of i , v is the velocity of the i in the
frame of references attached to , and n is the outward
unit vector normal to i
. The last term in (5) is due to
the relative motion of and i . Solution of (5) gives
the current distribution J 1 at the time step t t
in
terms of that at the time step t in the form
1
2
J
J1 t
J
1
0
t
2
. (6)
The magnetic force is evaluated by applying Ampère’s
force formula, i.e.,
r ri
F
J
( r,
t)
Ji
( ri
, t)
dSi
dS (7)
2
| r r |
i
i
0
with r and r i being the position vectors of the points of
and i , respectively.
The spatial discretization grid in the two-dimensional
case is constructed by dividing the region into
polygonal surface elements m , with the induced current
density considered to be constant through each m . The
region i is divided into surface elements i , with the
k
imposed current density being constant through each
i . Integrating (5) over each
k
m yields the following
J
matrix equation for the vector :
t
t
J
J
A B AJ
i
0 Bi
C i
2
t
t
1
2
1
2
1
2
J1 2
where A is a diagonal matrix with entries Am mSm
,
m being the resistivity of the material for m and S m
its area, and B is a symmetric matrix with its entries
corresponding to the elements m of having the form
B
1
ln dS
dS
S mS
m,
k
k m = k
R
mk
4
1 2
m k
(8)
1
( n m nk
) R ln dlkdlm
. (9)
R
The entries of the matrix B i are defined as in (9), but
with the elements k of being replaced with the
elements i belonging to
k
i , while the entries of the
matrix C are
C
m,
i
k
1
2
m ik
1
( n m R)(
ni
v)
ln dl
k
i dl
k m . (10)
R
All integrals in (9) and (10) are evaluated by analytic
expressions, the entries of the matrix B being calculated
only once, but those of the matrices B i and C are to be
calculated for each new position of i .
Taking into account the small dimensions of the
elements m , a rapid numerical computation of the force
in (7) is performed using the approximation
where the vector
Jm
F Sm Pi
(11)
k
m k
P i is expressed in the form
k
1
P i n
k
i ln dl
k
i (12)
k
| rm
ri
|
Jik
ik
k

with r m being the position vector of the center of the
element m of and r i the position vector of the
k
point of integration on i
. When the ratio of the linear
k
dimensions of i to the distance between its center and
k
the center of m is sufficiently small, P i can be
k
calculated by subdividing i in a number of elements
k
p in terms of this ratio and by using the summation
k
rm
rpk
P i J
k ik
S p (13)
2 k
p | rm
rp
|
k
where r p is the position vector of the center of p
k
k and
S p the area of the element p
k
k . The same technique is
applied for a rapid numerical calculation of the entries in
the matrices B i and C in (8), making also use of the
relation
1 R
( n i v)
ln dl'
v
k dS'
. (14)
R
2
R
i
i
k
In the case the imposed currents are periodic, the initial
distribution of the induced current can be obtained by
performing a Fourier expansion and by employing the
phasor form of the eddy-current integral equation (see
Section IV).
III. SOLUTION OF EQUATION OF MOTION
Equation (1) is solved iteratively. We choose an
appropiate time step t and assume that the magnetic
force F has a linear variation during t . At the time t the
body has a position defined by the vector r 0 and a
magnetic force F 0 is exerted upon it. The iterative
process is started by imposing the value F1 F0
at the
time t t
and the position vector r 1 results from
solving (1). The electromagnetic field problem is then
solved for the new r 1 and a new value of the force F 1 is
determined for the time t t
. This operation is repeated
until the difference between two successive values of the
magnetic force for the time t t
is sufficiently small
and, then, we proceed to the next time step.
IV. FREQUENCY-DOMAIN SOLUTION OF THE
EDDY-CURRENT INTEGRAL EQUATION
Since the region i is moving with the velocity v in
the frame of reference attached to , (2) is written in the
form
J
( r',
t)
1 Ji
( r',
t)
1
J ( r, t)
ln dS'
ln dS'
t R
t
R
i
k
1
( n v)
Ji ( r',
t)
ln dl'
. (15)
R
i
k
- 354 - 15th IGTE Symposium 2012
If the imposed currents are sinusoidal, the phasor
representation of (15) is
1
1
J ( ) j J ( ')
ln dS'
j J i ( ')
ln dS'
R
R
r
r r
1
( n v)
J i ( r')
ln dl'
(16)
R
i
where 2
f
, f being the frequency, and
re im
J J jJ
is the phasor form of the current density,
with j 1
. The two terms on the right side of (16)
show the contribution to the induced current density due
to the time variation of the imposed currents and that due
to the relative motion. The same technique as in the case
of the time-domain analysis is used for the space
discretization of (16). One obtains the following
algebraic system with complex coefficients:
re im im re
AJ BJ Bi
Ji
CJi
im re re im
AJ BJ Bi
Ji
CJi
. (17)
The average magnetic force over a period is evaluated
using the relation
* r ri
Fav
ReJ
( r)
J i ( ri
) dSi
dS
2 (18)
| r ri
|
i
where the asterisk indicates the complex conjugate.
For a multiple-conductor systems, one uses local
frames of reference attached to each of the conductors.
For the conductor q, occupying the region ,
q 1,
2,
,
(16) is written in the form
i
q
1
1
J ( r) j
J ( r')
ln dS'
j
J ( r')
ln dS'
R
R
pq
q
( q)
1
1
( n v p ) J ( r')
ln dl'
j
J i ( r'
) ln dS'
R
R
pq
p
i
( q)
1
( n
vi
) J i ( r')
ln dl'
(19)
R
i
(q)
(q)
where v p and v i are, respectively, the velocities of
the conductor p and of i with respect to the conductor
q. Equation (1) is always solved separately for each body.
V. SOLUTION ACCELERATION FOR THE EQUATION OF
MOTION
The computation of the motion of conducting bodies
can be spectacularly accelerated by using the average
value of the force over a period, evaluated using the
p

Figure 1: Discretization of the levitated plate.
y (m)
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13
t (s)
Figure 2: Evolution in time of the coordinate y of the plate for
f = 2,000 Hz.
y (m)
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0.0 0.1 0.2 0.3 0.4 0.5
t (s)
Figure 3: Detail regarding the motion at the beginning for f = 2,000 Hz.
phasor representation of current density. Using the
algorithm described in Section III, the time step is chosen
to be a multiple of the period and is adjusted according to
the force value, such that when the force decreases the
time step is increased and when the force increases it is
reduced.
VI. ILLUSTRATIVE EXAMPLE
A copper plate of width 80 mm, thickness 4 mm (see
8
Fig. 1), resistivity 210
m
and of mass density
3 3
8. 9
10 kg / m is levitated using two coils of 200 turns
each, of 10 mm 10 mm in cross section and a distance
between the axes of their sides of 70 mm and 30 mm,
respectively. The current direction is the same in the
outer and inner coils, the current intensity in each turn
being i I 2 sin 2ft
, with I = 10 A and f = 2,000 Hz.
- 355 - 15th IGTE Symposium 2012
y
y (m)
0.050
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0 1 2 3 4 5 6 7
t (s)
Figure 4: Evolution in time of the coordinate y of the plate for
f = 2,000 Hz, with a direct current of 10 A added in the outer coil.
y (m)
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
0 1 2 3 4 5 6 7 8 9 10 11 12 13
t (s)
Figure 5: Evolution in time of the coordinate y of the plate for
f = 200 Hz.
Initially, the conducting plate is located at 10 mm
above the coils. It is assumed that the plate only moves in
the vertical direction, but the procedures described in the
paper are also applicable when more degrees of freedom
are considered. The plate cross section is discretized in
180 rectangular elements, as indicated in Fig. 1.
For the time-domain method presented in this paper,
the period was divided in 48 intervals and the motion of
the plate was observed during 26,000 periods, i.e. for
1,248,000 time steps. The result is presented in Fig. 2,
with the detailed motion at the beginning shown in Fig. 3.
The computation took about 6 hours employing a 2.128
GHz Intel processor notebook. A great reduction in the
amount of computation is obtained by approximating the
conducting region with a thin strip of thickness equal to
the field depth of penetration [3]. The oscillations of the
plate can be attenuated by adding a dc component in the
current coils or by using a permanent magnet. If a direct
current of 10 A is added in the outer coil, the plate
motion becomes as it is shown in Fig. 4.
For a frequency of 200 Hz, the motion of the plate is
shown in Fig. 5, the attenuation of the mechanical
oscillations being much stronger than for a frequency of
2,000 Hz.
It should be remarked that the same results in Figs. 2
and 3 were obtained by using the proposed
frequency-domain procedure (see Sections IV and V).
Only 4,393 variable time steps, of magnitude between a
period and 50 periods, were necessary for determining
the motion of the plate between t = 0 and t = 13 s. The

equired computation time was only 124 s, i.e., about 170
times less than for the time-domain solution.
VII. CONCLUSION AND REMARKS
Two efficient methods are presented for computing the
motion of the solid conductors under the action of
electromagnetic forces. Practically, for the same accuracy
of the results, a tremendous reduction in the amount of
computation is achieved when using a frequency-domain
procedure.
The proposed methods can be extended to nonlinear
media. For the time-domain procedure one can utilize the
polarization method [4], which allows the formulation of
the eddy-current integral equation [2]. For the
frequency-domain procedure, one can adopt the method
proposed in [5], [6]. Since in some problems, for instance
in electromagnetic levitation problems, the air regions are
relatively large with respect to the conducting or/and
ferromagnetic regions, the weight of the fundamental
harmonic in the harmonic spectrum is significant and,
thus, for the convergence acceleration in the polarization
method one can efficiently employ the technique
proposed in [7].
The methods presented can be applied to
three-dimensional structures as well, by adopting the
eddy-current integral equation proposed in [8] and
extended in [2] to nonlinear media and to moving bodies.
In this case, the spatial discretization of (16) is done by
decomposing the induced current density using functions
of the form W
, where W are edge elements. When
edge elements of the first order are used, then the current
density is constant inside the tetrahedral volume elements
and the relations presented in this paper remain valid.
Now, the integrals similar to those in (9) and (10) can
only partially be evaluated analytically.
Finally, it should be remarked that the procedures in
- 356 - 15th IGTE Symposium 2012
[2], [5], [6] and [7] allow the extension to nonlinear
media of the proposed frequency-domain technique
which, as illustrated in this paper, could yield a
spectacular reduction in the amount of computation
needed.
ACKNOWLEDGMENT
This work was supported in part by the Romanian
Ministry of Labour, Family and Social Protection through
the Financial Agreement POSDRU/89/1.5/S/62557 and
by a grant of the Romanian National Authority of
Scientific Research, CNDI-UEFISCDI, project number
PN-II-PT-PCCA-2011-3.2-0373.
REFERENCES
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formulation for 3D eddy current problems with moving bodies
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Trans. Magn., vol. 34, no. 5, pp. 3068-3073, Sep. 1998.
[2] R. Albanese, F. Hantila, G. Preda, and G. Rubinacci, “Integral
formulation for 3-D eddy current computation in ferromagnetic
moving bodies,” Rev. Roum. Sci. Techn., Electrotechn. et Energ.,
vol. 41, no. 4, pp. 421-429, 1996.
[3] I. R. Ciric, F. I. Hantila, and M. Maricaru, “Field analysis for thin
shields in the presence of ferromagnetic bodies,” IEEE Trans.
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[4] F. Hantila, “A method of solving magnetic field in nonlinear
media,” Rev. Roum. Sci. Techn., Electrotechn. et Energ., vol. 20,
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[5] I. R. Ciric and F. I. Hantila, “An efficient harmonic method for
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[6] I. R. Ciric, F. I. Hantila, M. Maricaru, and S. Marinescu, “Efficient
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[7] I. R. Ciric, F. I. Hantila, and M. Maricaru, “Convergence
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- 357 - 15th IGTE Symposium 2012
Adaptive Inductance Computation on GPUs
A.G. Chiariello, A. Formisano and R. Martone
Dipartimento di Ingegneria Industriale e dell’Informazione
Seconda Università di Napoli, Via Roma 29, Aversa (CE), Italy
E-mail: Alessandro.Formisano@Unina2.it
Abstract—Inductances computation involving highly complex geometries and linear materials can be tackled by discretizing
coils into simpler elements, whose magnetic behavior is analytically expressible but, to achieve reliable results, very high
numbers of elements may be required. In such cases, advantages can be taken from GPU capabilities of dealing efficiently
with simple computational tasks. In the paper, a code able to compute self and mutual inductances of any 3D coils, taking
advantage of GPU capabilities, is presented.
Index Terms— GPU, High Performance Computing, Inductance Computation
I. INTRODUCTION
The computation of self and mutual inductances for
complex 3D shaped coils is a demanding task, since no
general formulas exist. Numerical computations, in order
to achieve reliable results, require large discretization
efforts and, in general, multiple runs. As a consequence,
in a number of practical cases the computer burden could
be very high.
Accuracy for generally shaped coils and computational
promptness represent conflicting objectives, especially in
optimal design [1, 2], and various computational
paradigms have been proposed to achieve reasonable
trade-offs. One of the most promising approaches is the
adoption of High-Performance Computing (HPC)
architectures; among HPC approaches, an effective
solution is the use of Graphic Processing Units (GPU),
easily available even on desktop class hardware.
On the other hand, in order to exploit at their best these
peculiar architectures, a revising of simulation codes is
often necessary, and new solutions, well suited for CPUbased
computational environments, must be adopted.
If assuming that no magnetic materials are present (i.e.
the relative permeability rel is equal to 1 everywhere),
following well established approximation formulas [3],
self and mutual inductances can be computed using
“segmented” approximations. The basic idea is to
decompose coils into simpler elements, for which self or
mutual inductances can be easily computed, eventually
using closed form expressions. Superposition is then used
to get the final value of coils self or mutual inductance.
In this paper a method able to compute self and mutual
inductances in air for generally 3D shaped massive coils,
based on coil decomposition into filamentary elements,
called current sticks, is presented, and its implementation
on HPC environments, based on GPUs, is briefly
discussed. The method is able to adapt the discretization
level to the required accuracy, and was implemented in
the INDIANA code (INDuctance Iterative and Adaptive
Numerical Assessment).
The basic objective of INDIANA is the computation of
self and mutual inductances of coils wound with multiple
series-connected turns of conductors. In mutual
inductance computation, INDIANA decomposes both
coils into a suited number of current sticks, and computes
the line integral along each stick of the “target” coil of the
vector potential A generated by each stick of the “source”
coil, with unit current. Then, the procedure, taking
advantage of the concept of “partial inductance” [4] and
of the linearity assumption, sums all contributions to get
the final, overall required value. Self inductances are
computed by using the same coil for both source and
target, but extracting the singular case of “self”
computations for each element, which are treated using
the expression for self inductance of a current stick [3].
This scheme suits quite well for GPU-based
computations, as will be further discussed in Sect. III.
In the following, a short overview of the relevant
points in the method will be presented (Sect. II), some
comments on the GPU implementations are reported
(Sect. III), and a few examples are discussed to help
assessing the method capabilities (Sect. IV). Finally, in
two final annexes, a brief description of the GPUs
architecture and programming paradigms is given.
II. MATHEMATICAL FORMULATION
The decomposition of massive coils into elementary
components is performed in two steps, taking into
account the structure of the winding, the distance between
the coils, and the local curvature of each coil.
As a first step, each coil is decomposed in as many
filamentary conductors as required by accuracy needs.
The typical figure adopted is as many as the actual
conductors in the Winding Pack (WP) of each coil.
However the figure can be increased according to the
adopted technology (e.g. in superconducting cable in
conduit conductor technology, the decomposition can be
extended down to petals level) to meet the accuracy
needs.
As a second step, each conductor is described using an
interpolating line, typically a spline, defined by a limited
number of parameters, such as the coordinates of a
suitable number of control points, constraining the shape
of the conductor to the required geometrical accuracy. In
addition the continuous curve is reduced to a collection of
sticks, defined by a number of suitable break points (see
Fig. 1 for a schematic view).
The number of sticks is selected on the basis of the
accuracy required by magnetic field computation, and can
vary depending on the local curvature of the interpolating
curve and on the distance from field points. INDIANA
performs a first guess for the distribution of break points
along conductors of the source coil on the basis of
average curvature radius and on minimum distance

Actual
Coil
Current
Sticks
Approximation
First coil: source Second coil: target
Figure 1: Segmentation of coils centerline into “sticks”: source coil
(left) and target coils (right).
between the midpoint of the section of the source coil
being considered and the target coil.
If the required accuracy is not fulfilled, the inductance
computation is assessed by increasing the number of
break points, and performing a new inductance
computation. The process iterates until the convergence,
in Cauchy sense within a prescribed accuracy, is
achieved.
The mutual inductance Mjk between the k-th stick on
the source coil and j-th stick on the target coil can be
computed either using the classical formulas from [3], or
by line integrating (numerically) the vector potential
Ak(x) generated by stick k on stick j:
(1)
ˆ
M jk Akx j tˆ dl j
j
where j is straight line along the j-th stick, xj is a generic
point along j, and ˆt is the stick unit vector. The
expression for Ak is given in [5]:
1
ˆ
cba A 0 a ln
(2)
4
cba where c=j+1-xi, b=j-xi, and a=j+1-j and the coordinates
of the stick tips.
c
xj
Figure 2: Basic elements form computation of vector potential using
(2b).
INDIANA implements a slightly modified version of
(2), to treat the singularity when computing A on points
on the source stick axis [6]. The number of Gauss points
for numerical integration of (2) is chosen, according to
the requested accuracy, on the basis of the distance
between centers of source and target points, using a linear
relationship based on the length of the target stick and the
distance between mid points of source and target sticks.
If source and target sticks are coincident, the self
inductance Mkk of the k-th stick can be computed using
the expression for a thin beam [3], providing the value in
Henry if the stick length Lk is given in meters:
4 2Lk
Mkk 210 Lkln
1
(3)
r
Lk
where r is the geometric mean distance and is the
a
Ak b
A
- 358 - 15th IGTE Symposium 2012
arithmetic mean distance on the corresponding k-th beam
cross section. Values of r and for different cross
sections are given in [3], while INDIANA adopts the
expression for circular cross section and long beams,
where r is the radius of the cross section and /Lk is
negligible.
III. GPU IMPLEMENTATION
In this section attention will be focused on the porting
of INDIANA code on the peculiar GPUs hardware. In
Appendix A, to the benefit of non experts, a short
introduction to GPU’s architecture is given [7-11], while
in Appendix B some hints strictly related to the
peculiarity of the GPU’s hardware are provided for the
interested programmers [7, 8].
The typical computational GPU architecture includes a
classical CPU section where the GPUs are grafted. In
order to profitably use the parallel nature of the GPU, any
code implementing numerical computations needs to be
split into sequential parts, which are performed on the
main CPU (or CPUs), and into the numerically intensive
parts, which can be more effectively performed on the
GPUs. In this way the best exploitation of GPUs and
CPUs execution capabilities can be achieved.
In the INDIANA code the basic computational task,
i.e. the evaluation of (2), the magnetic vector potential A
generated by a single stick in a single field point, can
benefit of the GPUs architecture. As a matter of fact this
task, which must be repeated a very high number of
times, can be simply assigned to a thread; then, suitable
grouping of treads onto computational blocks, can be
organized to exploit at best the data structure and the
available resources.
In order to achieve the peak performance [9], the
computational kernel needs to use GPU registers; in
addition, it needs to treat a large number of independent
instructions to exploit the scheduler capability of the
graphic card (further details can be found in Appendix
B). For these reasons the code was structured following
the flowchart:
1) Load the field point associated
to the considered thread. (Global
memory access)
2) Load the start and end points of
the sticks (Global memory access)
3) Compute the contribution to the
field of each stick in the bundle
using (2)
4) Accumulate all the contributions;
in a register
Last
stick?
Yes
5) Store the computed vector field
in the global memory
Figure 3: Flowchart showing the INDIANA kernel computation on
GPU architecture.
No

The final integration step (1) is executed in the CPU
side, since it is well suited to CPU characteristics, and its
impact on the computational burden is quite marginal.
INDIANA was implemented using the MATLAB©
parallel computational toolbox, and the core of the code
has been parallelized on the GPU using CUDA©, an
extension of the C language for Nvidia© GPU
programming. A few details are given in Appendix B.
IV. EXAMPLES OF APPLICATION
In this section, two groups of examples are presented.
In the first group, in order to assess accuracy, results from
INDIANA are compared to standard results for simple
geometries where analytical formulas are available. In the
second group, results from INDIANA are compared to
computations from 3D finite elements to assess speed-up
of computations for generally shaped coils.
Computational times are all referred to an intel i7–based
PC, with 8Gb ram, running Matlab© Ver.7 for the CPU
computations, and CUDA© Ver. 4.2 for GPU
computations.
a. Accuracy Assessment
Three filamentary single-turn coils have been
considered in this group. The first one is a circular
coil, while the other two are elliptical coils. Analytical
expressions and reference figures were taken from
[12]. Geometrical details are reported in Table I,
while a comparison of results is reported in Table II,
for increasing accuracy (expressed in Parts Per
Million -p.p.m.- of the reference value), and
consequently for increasing number of sticks in
INDIANA calculations.
z
C3
Figure 4: Coils used for Accuracy Assessment
TABLE I
COILS USED FOR ACCURACY ASSESSMENT
Coil # Centre position [m] Radii (a,b) [m]
C1 (0.0, 0.0, 0.0) 1 (circular)
C2 (0.0, 0.0, 0.5) (1/3, 2/3)
C3 (0.6, 2.0, 0.1) (1/3, 2/3)
Required
Figure
x
b
TABLE II
INDUCTANCES FOR INCREASING ACCURACY
Required
Accuracy
[p.p.m.]
C2
a
C1
Number of
sticks
Computational
times [s]
Reference
Value [H]
MC1-C2 1.0 10 3 1.13 0.30871178
MC1-C2 0.1 10 4 36.9 0.30871178
MC1-C3 1.0 10 3 1.17 0.03963496
MC1-C3 0.1 10 4 35.8 0.03963496
y
- 359 - 15th IGTE Symposium 2012
b. Speed Assessment
For this second analysis, the mutual inductance of two
coaxial circular coils has been considered. This
benchmark case is treated either using INDIANA with
300 sticks and the 3D FEM package COMSOL
multiphysics 4.2a (27524 2 nd order tetrahedral elems.,
neglecting any symmetry for the sake of generality). The
two coils are described in Table III, while results are
reported in Table IV.
TABLE III
COILS USED FOR SPEED ASSESSMENT
Coil # Centre position [m] Radius [m]
C1 (0.0, 0.0, 0.0) 0.20
C3 (0.0, 0.0, 0.1) 0.25
Method
TABLE IV
INDUCTANCES FOR SPEED ASSESSMENT
Mutual
Inductance [H]
Computational time
[s]
INDIANA 0.2487 4
3D FEM package 0.2486 490
Reference Value [2] 0.2488 ---
As a second speed test, the computation of mutual
inductance between a massive solenoidal source coil
(radius 1.7 m, length 2.0 m, thickness 0.7 m, 14 layers, 38
turns per layer) and a filamentary coil (Rin=4.77 m,
Rout=5.83 m, Zlow=5.02 m, Zup=5.11 m, =17.3°) used to
measure flux across a test surface was considered (See
Fig. 5). This test case is relevant for flux measurements in
magnetic confinement fusion devices [13, 14]. The
accuracy requirements on the flux measurement are rather
severe, in order to achieve that accuracy a high number of
sticks can be needed, in these cases the GPU speed
enhancement can be very useful to complete the
simulation in a reasonable amount of time. All methods
proved able to give the correct result of 2.5665048e-3 H.
The massive source coil has been represented by as
many conductors as actually present in its WP (that is,
14×38), while the discretization level along each
conductor has been varied to improve accuracy.
Comparison of computational times for various
discretization levels either for GPU computations, and for
purely CPU computations for the sake of comparison, are
reported in Table V.
Source Coil
Partial Flux
Measurement
Loop
Figure 5: Sketch of source coil for flux generation in Tokamak devices
and Partial Flux measurement loop

TABLE V
SPEED UP FOR MUTUAL INDUCTANCE
Computational Times [s] 1000 sticks 5000 sticks
GPU tGPU 5.31 19.4
CPU tCPU 35.8 125
Speed up (tCPU/ tGPU) 6.74 6.44
V. CONCLUSIONS
A numerical code able to compute mutual inductance
between couples of any massive coils has been presented.
The code is called INDIANA, and is able to adaptively
modify its computational parameters to achieve a tradeoff
between accuracy and computational speed.
INDIANA code benefit of a significant acceleration
(up to 7x) thanks to the GPU parallelization. This
performance allows to easily meet high accuracy request
in mutual inductance calculations for complex 3D shaped
coils.
INDIANA performance has been assessed either in
terms of accuracy and speed with respect to simple
geometries presented in literature or complex shapes,
compared to FEM computations.
Future activity will be addressed the MPI
parallelization over a computer cluster where each node
is equipped with GPU, in order to analyze more complex
structures.
ACKNOWLEDGEMENTS
Authors wish to thank Mr. M. Nicolazzo from
CREATE and Mr. M. Fatica from Nvidia for fruitful
discussions, and valuable hints and suggestions.
This work was partly supported by Seconda Università
di Napoli under PRIST grant “Generazione distribuita di
energia da fonti tradizionali e rinnovabili: aspetti
ingegneristici e giuridici-economici-ambientali”, partly
by NVIDIA Corporation and partly by
ENEA/EURATOM CREATE association.
REFERENCES
[1] M. Cioffi, A. Formisano, R. Martone, “Increasing design
robustness in evolutionary optimisation“, COMPEL, vol. 23,
pp.187-196, 2004.
[2] M. Cioffi, A. Formisano, R. Martone, G. Steiner, D. Watzenig, ”A
fast method for statistical robust optimization”, IEEE Transactions
on Magnetics , Vol. 42, pp. 1099-1102, 2006.
[3] F. Grover, Inductance Calculation, New York: D. Van Nostrand,
1946.
[4] C. R. Paul, Introduction to Electromagnetic Compatibility,
Hoboken (NJ): J. Wiley & Sons, 2006.
[5] H. A. Haus, J. R. Melcher, Electromagnetic Fields and Energy,
Englewood Cliffs, NJ: Prentice Hall, 1989.
[6] J. Hanson, S. Hirshman, “Compact expressions for the Biot–
Savart fields of a filamentary segment”, Phys. of Plasmas, vol. 9,
pp.4410-4412, Oct. 2002.
[7] D. Kirk, W. Hwu, Programming Massively Parallel Processors: A
Hands-on Approach, Elsevier, 2010.
[8] M. Garland et al., “Parallel computing experiences with CUDA”,
IEEE Micro, vol. 28, 2008 pp. 13–27.
[9] V. Volkov. “Better performance at lower occupancy”, Proceedins
of NVIDIA GPU Technology Conference 2010, San Jose, USA,
pp. 20-23, Sept. 2010.
[10] R. Farber, CUDA Application Design and Development, Morgan
Kaufmann, 2011.
[11] F. Calvano, G. Rubinacci, A. Tamburrino, G. Vasilescu, S.
Ventre, “Parallel MGS-QR sparsification for fast eddy current
- 360 - 15th IGTE Symposium 2012
NDT simulation” Studies in Applied Electromagnetics and
Mechanics, vol. 36, pp. 29-36, 2012.
[12] J. T. Conway, “Exact Solutions for the Mutual Inductance of
Circular Coils and Elliptic Coils”, IEEE Trans. on Magn., vol 48,
pp. 81-94, 2012.
[13] A. J. Donné et al., “Progress in ITER Physics basis, Chapetr 7:
Diagnostics”, Nucl. Fusion, vol. 47, pp. S337-S384, (2007).
[14] A. Formisano, J. Knaster J., R. Martone et al., “ITER nonaxisymmetric
error fields induced by its magnet system”, Fusion
Engineering and Design, vol. 86, pp. 1053-1056, 2011.
APPENDIX A: GPU ARCHITECTURES
Hardware used in computer Central Processing Unit
(CPU) seems to be reaching the physical limits beyond
which increase of clocking frequency or of integration
scale is very hard with present technology. As a possible
alternative, CPU manufactures are moving to multiplecores
CPU’s, but the number of the cores is usually
limited to a few tens. On the other hand, realistic
treatment of real world applications gives rise to
computationally demanding numerical models. In order
to speed up the computations, different parallelization
paradigms can be considered, a few examples being
reported in [11].
Recently, the Graphic Processing Units (GPUs),
present on virtually all graphic cards of computers, have
been proposed as data-parallel coprocessors, used to
solve compute-intensive science and engineering
problems, since it was observed that the mathematical
processing in high resolution images are very similar to
the computations usually required in numerical models of
physical phenomena.
A modern GPU can have up to 1024 processor cores or
Streaming Processors (SPs) grouped in Streaming
Multiprocessors (SMs), each containing eight processor
cores.
In order to reduce the dimension of chip area dedicated
to the control unit, each core in an SM use a parallel
computation paradigm called Single Instruction, Multiple
Data (SIMD), where concurrent processor execute the
same code (called Kernel) on different data.
The tasks submitted at each core are called threads; the
threads are grouped in thread blocks (fig. 6b); the
maximum dimension of a block is presently 512 threads;
hence, a code need to launch a lot of thread blocks. For
these reason the thread blocks are grouped into a grid of
thread blocks (fig. 6c). The threads in a block can be
indexed using a 3D identifier, a block in a grid can be
indexed using a 2D identifier.
A block of threads is assigned at a SM; each SM can
use 8,192 registers; the registers are the faster memory
inside a GPU but they are dynamically partitioned among
the threads inside the blocks; each thread can only access
its own registers (fig. 6a). All threads inside a block can
cooperate with the others sharing memory using an onchip
low latency memory called shared memory (48 kB);
the shared memory bandwidth is about 6x lower respect
the register bandwidth.
The GPU has a memory, called Global memory, where
the CPU can upload the input data and download the
result of the computation. The global memory is available
at all the SMs; it is the largest memory inside a GPU, up
to 6 GB in modern solutions.

APPENDIX B: GPU PROGRAMMING STRATEGY
In order to obtain the best results, a programmer needs
to take into account the peculiar hardware characteristic
of the GPUs, in each step of the program [7, 10].
The access to Global Memory are time consuming, in
order to increase the access rate each time a location is
accessed, many consecutive locations are accessed by the
hardware.
a) thread
b) thread blocks
…
c) grid of thread blocks
…
Figure 6: GPU memory model
A typical GPU program will follow the steps showed
in Fig.7. In order to obtain the peak performance the
programmer need to reorganize, in the host side (Fig.7
step a), the input data in such a way that adjacent threads
operate on adjacent data in global memory (Coalescing
Access). In this case, the hardware combines, or
coalesces, all of these accesses into a unique access to
consecutive locations. An example related to geometric
data is showed in figure 8. In order to allow coalescing
accesses, duplication of data can be needed.
Figure 7: Flowchart of a typical GPU code
Register
… …
Shared
memory
a) Upload part of the input data from the
CPU memory to the GPU global memory
b) Use the thread and block index to select
the data from the GPU global memory
c) Compute the task and store the results
in GPU global memory
d) Download the result from the GPU global
memory to the CPU memory
Global memory
- 361 - 15th IGTE Symposium 2012
a)
b)
X1 Y1 Z1 X2 Y2 Z2 …. XN YN ZN
Thread 1 (th1) th2 thN
X1 X2 …XN Y1 Y2 … YN Z1 Z2 ZN
th1 th2 thN
Figure 8: a) Uncoalescing access pattern, b) Coalescing access pattern
As already discussed, a thread can access three kinds
of memory: global memory, shared memory and the
registers. The registers are the fastest memory; hence, in
order to achieve the best performance, the programmer
has to use as many registers as possible.
Of course, the actual speed up can be limited by the
amount of on chip memory resources (as registers and
shared memory) and by the shared memory bandwidth.
The GPU has a sophisticated scheduler very effective
in minimizing the performance loss due to access to
global memory. If a sufficient number of threads is
available, the scheduler can concurrently run the threads
on the multiprocessor, masking the memory accesses.
This approach is called in literature Thread Level
Parallelism (TLP) approach [7-9].
The larger is the number of threads, the fewer are the
registers available per each thread (the registers are 8192
and are partitioned among the threads inside a block).
Unfortunately the actual availability of a limited number
of registers per thread can be a bottleneck for the entire
code.
In order to increase the speed up, the scheduler is
provided by the capability to analyze the instruction flow
and evaluate how reliable could be to execute two
instructions at the same time. If the answer is positive, the
hardware localizes possible free units and increases the
parallelism and executes more than one instruction during
the same clock cycle. Then, a small number of threads
per block usually are recommended (64 threads per block
can be a good compromise) and, in addition, the thread
have to be designed to present as many independent
instructions as possible: in such a way the Instruction
Level Parallelism (ILP) [7-9] increases the performance.

- 362 - 15th IGTE Symposium 2012
The Reduced Basis Method Applied to
Transport Equations of a Lithium-Ion Battery
Stefan Volkwein∗ ∗ †
, Andrea Wesche
∗Universität Konstanz, Fachbereich Mathematik und Statistik , Universitätsstraße 10, D-78457 Konstanz,
E-mail: stefan.volkwein@uni.konstanz.de
† Adam Opel AG, Bahnhofsplatz, D-65423 Rüsselsheim, E-mail: Andrea.Wesche@de.opel.com
Abstract—In this paper we consider a coupled system of nonlinear parametrized partial differential equations (P 2 DEs),
which models the concentrations and the potentials in lithium-ion batteries. The goal is to develop an efficient reduced
basis approach for the fast and robust numerical solution of the P 2 DE system. Numerical examples illustrate the efficiency
of the proposed approach.
Index Terms—finite volume method, greedy algorithm, lithium-ion battery, reduced basis method
I. INTRODUCTION
The modelling of lithium-ion batteries has received an
increasing amount of attention in the recent past. Several
companies worldwide are developing such batteries
for consumer electronic applications, in particular, for
electric-vehicle applications. To achieve the performance
and lifetime demands in this area, exact mathematical
models of the battery are required. Moreover, the multiple
evaluation of the battery model for different parameter
settings involves a large amount of time and experimental
effort. Here, the derivation of reliable mathematical
models and their efficient numerical realization are very
important issues in order to reduce both time and cost in
the improvement of the performance of batteries.
In the present work we consider a mathematical model
for lithium-ion batteries which describes the transport
processes by a partial differential equation system. This
model is developed in the paper by Popov et al. [17]. The
physical and chemical details can be found in [13] and
[14]. The equation system models a physico-chemical
micro-heterogeneous battery model.
We discretize this by the finite volume method and the
backward Euler method. The reduced basis methodology
for by finite volumes discretized systems can be found
in [10]. The discretized model is reduced by the reduced
basis method [16]. Our numerical tests will illustrate the
efficiency of this approach.
A popular battery model is the one developed by Newman
[4], [15], which was implemented and tested [5]. Let
us also refer to the work [6], where a different battery
model is derived. For an equation system which describes
a physico-chemical macro-homogeneous battery model
the well posedness is shown by Wu et al. [19].
II. BATTERY MODEL
PDE Model
Let Ω ⊂ R be an open interval, which is divided in three
disjunct open sub-intervals Ωc, Ωe, Ωa ⊂ R, see Figure
2.1. For tend > 0 we define Q ∶= Ω ×(0,tend) and let
c, φ ∶ Q → R and α, β, λ, κ ∶ R2 → R, the notations
positive electrode
electrolyte
negative electrode
Ωc Ωe Ωa
BUTLER-VOLMER-equation
Fig. 2.1. Structure of the considered battery domain
can be found in Table 1.2 in the appendix. The transport
processes in a battery, i.e. transport of mass and charge,
are described by the equations [17]
∂c
−∇⋅(α (c, φ)∇c + β (c, φ)∇φ) =0
∂t
(2.1a)
−∇ ⋅ (λ (c, φ)∇c + κ (c, φ)∇φ) =0 (2.1b)
in Ωc ×(0,tend), Ωe ×(0,tend) and Ωa ×(0,tend), where
“∇” denotes the gradient and “∇⋅” the divergence.
The positive electrode is Ωc (cathode for discharge),
the electrolyte Ωe and the anode Ωa (anode for discharge).
Boundary/Interface Conditions
The boundary conditions are
∂c
∂ν = 0, φ = 0 on (∂Ω ∩ ∂Ωc)×(0,tend) (2.2a)
∂c
= 0,
∂ν
∂φ I
=−
∂ν σa
on (∂Ω ∩ ∂Ωa)×(0,tend) (2.2b)
where ν is the outer unit normal vector, I ∶ R + → R
(time-dependent current) and σa ∈ R + /{0} (electric conductivity
multiplied with the cross section). The initial
condition is given by
c(x, t0 = 0) =c0 (x) ,x∈ Ω (2.3)

The interface conditions are given by
−(α (c, φ)∇c + β (c, φ)∇φ)
⎧⎪ I(cec,cc,φec,φc) ∣ in (∂Ωc ∩ ∂Ωe)
(0,tend)
= ⎨
⎪⎩
−I (cea,ca,φea,φa) ∣ in (∂Ωe ∩ ∂Ωa)
(0,tend)
−(λ (c, φ)∇c + κ (c, φ)∇φ)
⎧⎪ J(cec,cc,φec,φc) ∣ in (∂Ωc ∩ ∂Ωe)
(0,tend)
= ⎨
⎪⎩
−J (cea,ca,φea,φa) ∣ in (∂Ωe ∩ ∂Ωa)
(0,tend)
(2.4a)
(2.4b)
where cea is the concentration in the electrolyte at the
negative electrode interface:
cea (t) =lim
h→0 c (x∣ Ωe∩Ωa − h, t)
and h > 0 is small enough, i.e. x∣ − h ∈ Ωe.
Ωe∩Ωa
Analogously
cec (t) =lim
h→0 c (x∣ Ωe∩Ωc + h, t)
cc (t) =lim
h→0 c (x∣ Ωe∩Ωc − h, t)
ca (t) =lim
h→0 c (x∣ Ωe∩Ωa + h, t)
for sufficient small h > 0. Thevariablesφc, φa, φec, φea
are defined in the same way. We write cs for the concentration
in the solid part, i.e. in the negative and positive
electrode, and ce for the concentration in the electrolyte,
φs and φe are analogously denoted. The scalar functions
I∶R 4 → R and J∶R 4 → R are defined by
I(ce,cs,φe,φs) = J(ce,cs,φe,φs)
√ √
F
√
ce cs
J(ce,cs,φe,φs) =k
1 − cs
c 0 e
c 0 s
cs,max
⋅ 2sinh( F
(φs − φe − U0 (cs)))
2RT
where F = 96486 A⋅s
is the Faraday constant, R =
mol
A⋅V ⋅s
8.314 is the gas constant and T > 0 [K] is the
K⋅mol
temperature. The function U0 ∶ R → R is the over
potential and depends on the concentration c in the
electrodes. The coefficient functions are defined as
α (c, φ) ∶=De (c, φ)+ RT
F 2
(t+ (c, φ)) 2 κ (c, φ)
c
t+ (c, φ)
β (c, φ) ∶=κ (c, φ)
F
λ (c, φ) ∶= RT
F
t+ (c, φ) κ (c, φ)
c
[ cm2
s ]
[ mol
V ⋅ cm ⋅ s ]
[ A ⋅ cm2
mol ]
where the transference number t+ is zero in the electrodes
and larger than zero in the electrolyte and κ
is the ionic/electric conductivity; κ, t+, De ∶ R 2 → R.
To measure the battery parameters experimentally it is
assumed, that they are polynomials in c and φ.
The homogeneous Neumann boundary conditions for
the concentration (2.2) mean that no flux of lithium(-ions)
- 363 - 15th IGTE Symposium 2012
can pass through. The inhomogeneous Neumann boundary
condition for the potential is Ohm’s law, the homogeneous
Dirichlet boundary condition have no physical
meaning. It ensures the uniqueness of the solution if one
exists.
The interface conditions describe the exchange of the
lithium-ions at the interfaces which are modeled by the
Butler-Volmer-equation [1].
For physical reasons we assume that
c (x, t) ≥0 ∀x ∈ Ω, t∈(0,tend)
We remark that the coefficient functions α and κ are
larger than zero for physical reasons: the diffusivity De
and the conductivity κ are larger than zero. Because of
the definition of the transference number t+ the coefficient
functions β and λ are equal or larger than zero.
Discretization of the Problem
We discretize the partial differential equation system
(2.1a)-(2.1b) with the appropriate boundary (2.2) and
interface conditions (2.4) by the cell centered finite
volume method. We divide therefore Ωc in Nc ∈ N, Ωe in
Ne ∈ N and Ωa in Na ∈ N, ND = Nc+Ne+Na, equidistant
control volumes of the width Δx. We use the method of
lines and solve the equation system for every time step.
The time step size is Δt. The integrals over the spatial
we approximate by the middle point rule, the integrals
over the time by the backward Euler method, for details
cf. [17]. These discretized equations are implemented in
MATLAB 7.10.0 (R2010a).
III. REDUCED BASIS METHOD
Initial Point
We consider a parametrized PDE which we want to
solve for many parameter sets, e.g. for parametric studies.
The better the numerical model approximates the physical
phenomenon, the more expensive the computation
gets. So in some cases e. g. a parameter analysis needs
too much effort, because a single computation is too
expensive. Therefore one has to develop a reduced model
to get cheap solutions.
The reduced basis method is based on the discretized
model: the idea is to compute a “few” times an expensive
solution to different parameter sets which are in the range
of interest. With the knowledge of these so called “true”
solutions basis vectors are computed. The approach is
that the reduced solutions in the parameter set of interest
are linear combinations of these basis vectors.
An assumption is that the error between the “exact”
analytical solution and the “true” numerical solution is
small in contrast to the error between the “true” and the
“reduced” solution.
A big advantage of the present method is that you
determine the error between the true and reduced solution
during “developing” your reduced model (→ Greedy
algorithm). A further property is that the method has
two phases: the offline computation in which the reduced

model is set which fulfills the given error tolerance and
in which the needed true solutions are computed and
the online phase in which the reduced solution(s) are
computed. The offline part is expensive and the online
phase is cheap.
Approach
In the following we have to resolve how to choose the
true solution and how to estimate the error between the
reduced and true solution. The (POD-)Greedy algorithm
ensures both issues.
We now describe how to apply the reduced basis
method on our battery model. The transport equations of
the battery (2.1a)-(2.1b) depend on many parameters: on
geometrical parameters (e.g. the width of the electrode)
on state parameters (e.g. temperature) and on battery
parameters (e.g. the diffusion coefficient). We note these
parameters with μ ∈Dand assume that all these different
parameters are polynomials in c and φ, some are of
course constant and so polynomials of the degree zero.
We write u N ∈X N for a piecewise linear functions,
its coefficient are denoted with u N ∈ R N . The discretized
problem is now the following: Find a u N ∈X N so that
F N (u N ; μ) =0
⇔⟨F N (u N ; μ) ,v N ⟩W =0 ∀v N ∈X N
(3.1)
where the mass matrix W is in the present case given by
W = Δx ⋅ 1 ∈ R N×N .
We approximate the finite volume space by a Ndimensional
space X N which is spanned by “snapshots”.
A snapshot is a true solution to a specific parameter set
μ ∈D and time node t ∈(0,tend). We assume now, that
we have a (orthonormalized) basis Ξ =(ξ1,...,ξN )∈
R N×N of this space and that the reduced solution can be
written as
N
u N (μ) = ∑ θ uN
j (μ) ξj (3.2)
j=1
If we replace u N in equation (3.1) by u N of equation
(3.2) and choose v N = ξi ∈ R N we get:
Ξ T ⋅ W ⋅ F N (θ uN
(μ)⋅Ξ; μ) =∶F N (θ uN
(μ))
=0 (3.3)
The start vector for the coefficient vector for Newton’s
method one can get by
u start
coeff (μ) =ΞT ⋅ W ⋅ u N (⋅,t0 = 0; μ)
To find a basis we use for our time dependent problem
the POD-Greedy algorithm, cf. algorithm 1 and [10]. It
consists of two loops: the outer loop is the “standard”
Greedy, cf. for instance [16], which finds the new parameter
set to which the error between the reduced and
- 364 - 15th IGTE Symposium 2012
true solution is the largest. For this we need a problem
specific error estimator; an error estimator for a linear
by finite volumes discretized problem can be found in
[10]. The inner loop reduces the trajectory u N (⋅,tn; μ ∗ )
in time with the POD (proper orthogonal decomposition)
algorithm. This algorithm returns for each snapshot matrix
u N (μ) ℓ ∈ N eigen-/basis vectors. One can state the
number of basis vectors or the projection error of the
POD method. For details to the POD algorithm cf. for
instance [11], [18].
Usually we take an error estimator to estimate the error
between the true and reduced solution to each parameter
of the discretized parameter set. Until now we have no
error estimator for the present problem, so we compare
the true solution to a parameter set of the training set,
to the reduced solution computed by the so far reduced
basis vectors. The function u (i)
RB (μj) is the reduced
solution constructed by i basis functions evaluated at the
parameter set μj ∈Dtrain ⊂D.
Algorithm 1 POD-Greedy algorithm, c.f [10]
Require: ● Limit the parameter range, discretize the
parameter set Dtrain ={μ1,...,μNP }
● Ξtrain = {u N (μ1) ,...,u N (μNP )}, uN (μi) ∈
R Nx×Nt , ∀i ∈{1,...,NP }
● Choose a tolerance for the Greedy: TOLGreedy
● Choose the exactness for the POD basis per ∈
[0, 1], it is just a measurement for the projection
error (or directly choose the number of POD
basis elements in each “Greedy step” ℓ).
Ensure:
1: Initializing:
● Choose μ (1) ∈Dtrain → ˜ ξ (1) ∈ Ξtrain
ξ (1) = POD( ˜ ξ (1) ) ∈ R Nx×ℓ (1), with the POD
tolerance per
● Set: i = 2, ɛ = 1
2: while i ≤ NP and ɛGreedy > TOL do
3: [μ (i) ,ɛ]=max j∈{1,...,NP } ∣u (i−1)
RB (μj)−u N (μj)∣
4: ˜ ξ (i) = POD(u N (μ (i) )) NT
n=0 ∈ RNx×ℓ (i), where
u N (μ (i) ) ∈ Ξtrain
5: (ξ (1) ,...,ξ (i) )
= Gram-Schmidt (ξ (1) ,...,ξ (i−1) , ˜ ξ (i) )
6: end while
An essential property of the reduced basis method is
that you can decompose the computation into an offline
and online phase.
Offline: After determination of the set of parameter
sets and the accuracy of the reduced solutions,
we start the Greedy algorithm to compute the
basis vectors, the true solutions to the chosen
parameters respectively. The offline phase is
computationally expensive and so the reduced
basis method is only worth if you want to solve
the equation system many times.
Online: In the present case we have to compute the

coefficient vector for the basis to the different
parameter sets. We get it by the damped Newton’s
method. This phase is cheap.
The big advantage of the reduced solution is that
you know how good your reduced solution approximates
the true solution. A big disadvantage is, that if you
change your parameter ranges you usually have to do
the expensive offline computation again.
RBM applied on the battery model
In the following we explain how to apply the reduced
basis method on our discretized problem (3.1).
We denote by F N 1 (C, Φ) ∶R N × R N → R N equation
(2.1a) with boundary and interface conditions discretized
by the finite volume method, analogously F N 2 (C, Φ) ∶
R N × R N → R N stands for the finite volume discretized
equation (2.1b) with the boundary and interface conditions.
We choose a parameter training set for C and Φ 1 :
Ξ C
train = (c N (μ1) ,...,c N (μNP ))
Ξ Φ
train = (φ N (μ1) ,...,φ N (μNP ))
Let us assume that we have a basis matrix for C
) ∈ RND×NBc and for Φ ΨΦ =
ΨC = (ψc 1,...,ψc NBC (ψ φ
1 ,...,ψφ )∈R NBφ ND×NBφ then the reduced models,
reduced functions respectively, are given by
(Ψ c ) T ⋅ W ⋅(F N
1 (Ψ c Ccoeff , Ψ φ Φcoeff )) ! = 0 (3.4a)
(Ψ φ ) T
⋅ W ⋅(F N
2 (Ψ c Ccoeff , Ψ φ Φcoeff )) ! = 0 (3.4b)
We should add that the snapshots for Ψ c and Ψ φ are
taken at the same parameter sets (outer POD-Greedy),
but the number of the POD basis elements could differ
to achieve the same accuracy (inner POD-Greedy).
A further issue is how to choose the next parameter
in the (POD-)Greedy algorithm. The error estimation we
have to do for two functions c and φ. So we usually
get two different parameter values μc and μφ, where
the L 2 -errors of the concentration and of the potential,
respectively, attain their maximum values. Then, we
choose the parameter μ ∈{μc,μφ} corresponding to the
greater L 2 -error of both.
IV. NUMERICAL EXPERIMENTS
In this section we apply the reduced basis method to
the discretized equations describing the transport processes
in a lithium-ion battery (3.1). The step size in
spatial is Δx = 1μm, the time step size Δt = 5s and we
compute Nt = 10 time steps. The Newton tolerance to
compute the true solution is set 10−6 relatively and 10−9 absolutely. The discretization error of the finite volume
solution is ɛFVM =O( 1
100 ).
1 One can also choose different training set, e.g. cf. [7]
- 365 - 15th IGTE Symposium 2012
pos. electrode electrolyte neg. electrode
De, [ cm2
s ] 1.0 ⋅ 10−9 7.5 ⋅ 10−7 3.9 ⋅ 10−10 κ, [ A
]
V ⋅cm
c
0.038 0.002 1.0
0 , [ mol
cm3 ]
cmax, [
0.020574 0.001 0.002639
mol
cm3 ]
U0, [V ]
t+, [−]
k, [
0.02286
0.001
0 0.2
0.02639
0
0
A
cm2 ] 1.3716 ⋅ 10−4 5.2780 ⋅ 10−7 N⋅, [−]
A⋅, [cm
10 30 10
2 ] (50 ⋅ 10−4 ) 2
(50 ⋅ 10−4 ) 2
TABLE 4.1
BATTERY PARAMETERS, [17]
The Newton tolerance for the reduced solution is 10 −5 .
So the L 2 -error between the finite volume and reduced
solution is at best less than
ɛ L 2 = 10 −5 ⋅ Nx ⋅ Nt = 0.005 =∶ ɛGreedy
The tolerance of the POD method is set 99%, cf. for the
POD method for instance [18].
Our “standard” battery parameter set is listed in table
4.1, notations can be found in Table 1.2 in the appendix.
We charge the battery with 1.5913 ⋅ 10 −8 A which corresponds
to 1C-rate and set the temperature T = 300K.
Test 1
In this subsection we variate the open circuit voltage
in the positive electrode: μ = Uc ∈[0.001, 4.501]. We
discretize this parameter set with the equidistant step
width ΔUc = 0.1V .Sowehavea46-dimensional training
set for the parameter. All the other parameters are fixed,
cf. Table 4.1.
In Figure 4.1 and 4.2 the finite volumes solutions
chosen by the first two iterations of algorithm 1 are
presented. The associated parameters are Uc = 3.001V
and Uc = 4.501V . The concentration seems to be less sensitive
to the circuit voltage than the electrical potential.
The L2-error for the concentration becomes worse after
the second Greedy step, but already after the first Greedy
step the error is smaller than the L2-error tolerance
ɛL2 and stays smaller; the L2-error for the electrical
potential gets denotative smaller with the information of
a second true solution, cf. Figure 4.3 and 4.4. The same
observation can be done for the L∞-error which is not
presented here. The basis functions are shown in Figure
4.5: they have the same “structure” as the finite volume
solutions for one fixed time step and there is just one
basis function for each Greedy step.
The speed up of the reduced solution in comparison
to the true solution is 17.54.
Test 2
In this section we variate a few parameters:
μ ={Dec,Dee,Dea,t+,kc,ka}
The subscript c denotes the parameter in the positive
electrode, e in the electrolyte and a in the

c [mol/cm 3 ]
0.025
0.02
0.015
0.01
0.005
0
20 40 60
x [μ m]
0
t [s]
50
U [V]
0
−1
−2
−3
−4
0 20 40 60
x [μ m]
Fig. 4.1. Test 1: Finite volume solution fort the concentration (left)
and the potential (right) for Uc = 3.001V
c [mol/cm 3 ]
0.025
0.02
0.015
0.01
0.005
0
20 40 60
x [μ m]
0
t [s]
50
U [V]
0
−2
−4
−6
0 20 40 60
x [μ m]
Fig. 4.2. Test 1: Finite volume solution fort the concentration (left)
and the potential (right) for Uc = 4.501V
x 10−5
6.5
6
5.5
5
4.5
1 2 3 4
U [V]
0c
0.08
0.06
0.04
0.02
0
0
0
t [s]
t [s]
1 2 3 4
U [V]
0c
Fig. 4.3. Test 1: L 2 -error after the first Greedy step for the
concentration (left) and the potential (right).
x 10−5
7.019
7.019
7.019
7.019
7.019
7.019
1 2 3 4
U [V]
0c
x 10−3
3.6949
3.6949
3.6949
3.6949
3.6949
1 2 3 4
U [V]
0c
Fig. 4.4. Test 1: L 2 -error after the second Greedy step for the
concentration (left) and the potential (right).
40
20
0
−20
−40
−60
Basis functions for the concentration
−80
0 10 20 30 40 50
Spatial
Greedy step 1
Greedy step 2
30
20
10
0
−10
Basis functions for the potential
−20
0 10 20 30 40 50
Spatial
50
50
Greedy step 1
Greedy step 2
Fig. 4.5. Test 1: Basis functions for the first Greedy step for the
concentration (left) and the potential (right).
negative electrode. We choose the following parameter
set range: for the diffusion coefficients Dec ∈
[1.0 ⋅ 10 −9 , 1.1 ⋅ 10 −9 ], Dee ∈ [7.5 ⋅ 10 −7 , 7.6 ⋅ 10 −7 ],
- 366 - 15th IGTE Symposium 2012
Dea ∈ [3.9 ⋅ 10 −10 , 4.0 ⋅ 10 −10 ], the transference number
t+ ∈ [0.2, 0.3] and the reaction rates kc,ka ∈
[0.02, 0.022]. We discretize the parameter set in the
following way: for the diffusion coefficients we choose
the boundary values, for the other parameters we also
take the boundary values and a value in between. With
this discretization we get a 216-dimensional trainings
set. All the other parameters are fixed like in table 4.1
noted, but in contrast to the previous subsection the POD
tolerance is set 1 − 1 ⋅ 10 −8 %.
The graphical results are listed in the Figures 4.6 and
4.7: In Figure 4.6 the finite volume solutions for the first
parameter set can be seen. The first parameter set is the
one denoted in table 4.1. The L 2 -error of the reduced
solutions in comparison to the finite volume solution
is for the concentration smaller than 10 −5 and for the
electrical potential 10 −4 to all 216 parameter sets. The
L ∞ -error is smaller than 10 −3 for the concentration as
well as for the potential after one Greedy step. The basis
functions are plotted in Figure 4.7: there are four basis
functions for the concentration and three for the potential.
c [mol/cm 3 ]
0.025
0.02
0.015
0.01
0.005
0
20 40 60
x [μ m]
0
t [s]
50
−1
−2
−3
−4
x 10
0
−3
−5
0 20 40 60
x [μ m]
Fig. 4.6. Test 2: Finite volume solution fort the concentration (left)
and the potential (right) for first parameter set
80
60
40
20
0
−20
−40
Basis functions for the concentration
−60
0 10 20 30 40 50
Spatial
U [V]
30
20
10
0
−10
−20
−30
0
t [s]
Basis functions for the potential
−40
0 10 20 30 40 50
Spatial
Fig. 4.7. Test 2: Basis functions after the first Greedy step for the
concentration (left) and the potential (right).
The L 2 -error between the reduced solution and the true
solution is after one Greedy step smaller than the L 2 -error
tolerance ɛ L 2 to all 216 parameter set. That means that
just the information of one computational expensive true
solution is needed to compute the reduced solutions to
all 216 parameter sets with an acceptable error.
The speed up of the reduced solution in comparison
to the true solution is 12.37.
V. DISCUSSION
In the present document we do the same approach like
in [10].
The reduced basis approach works for the transport
equation in a lithium-ion battery: in the above numerical
50

Notation
c, [ mol
cm3 ] concentration of the lithium/lithium-ions
φ, [V ] electrical potential
α, [ cm2
] coefficient function
s
β, [ mol
] coefficient function
V ⋅cm⋅s
λ, [ A⋅cm2
] coefficient function
mol
I, [A] current
De, [ cm2
]
s
κ, [
diffusion coefficient
A
]
V ⋅cm
c
electric/ionic conductivity
0 , [ mol
cm3 ] start concentration of lithium in the
electrodes/electrolyte
cmax, [ mol
cm3 ]
U0, [V ]
t+, [−]
k, [
maximum of lithium the electrode can store
open circuit potential
transference number
A
cm2 ]
N⋅, [−]
A⋅, [cm
reaction rates
number of control volumes
2 ] cross section
TABLE 1.2
NOTATIONS OF THE BATTERY PARAMETERS
tests there are at most two Greedy steps needed to reach
the L 2 -error tolerance ɛ L 2. If the knowledge of all finite
volume solutions to the training set is needed, the method
would not be sufficient. In the above numerical tests
we see that the limiting factor is the potential: for the
concentration one Greedy step is sufficient but for the
potential we need in some cases an additional Greedy
step.
The application of the reduced basis method to this
problem is not completed yet: We have to develop an
a posteriori error estimator so that we do not have to
compute all true solutions to the discretized parameter
set. Further the computational time of the reduced
solutions in comparison to the finite volume solutions
for the presented numerical tests are fast but not so
fast as it could be. In every Newton step we have to
evaluate the nonlinearities completely. Also we have no
affine parameter dependence. If you have a linear(ized)
affine parameter dependent problem you can separate the
parameter dependence from the bilinear form and from
the linear form. To get an affine parameter dependent
problem as well as a linearized problem we have to
apply the (discrete) empirical interpolation method, cf.
for instance [2], [3].
VI. ACKNOWLEDGMENTS
The authors gratefully acknowledge support by the
Adam Opel AG. Besides Competence Center The Virtual
Vehicle (Graz) supported the lecture by Mr. Volkwein
within the scope of the IGTE Symposium.
APPENDIX
NOTATION
In Table 1.2 one can find some notations for the battery
parameters.
- 367 - 15th IGTE Symposium 2012
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- 368 - 15th IGTE Symposium 2012
Surrogate Parameter Optimization based on
Space Mapping for Lithium-Ion Cell Models
Matthias K. Scharrer∗ , Bettina Suhr∗ , and Daniel Watzenig∗† ∗Kompetenzzentrum – Das Virtuelle Fahrzeug Forschungsgesellschaft mbH (ViF),
Inffeldgasse 21/A/I, A-8010 Graz, Austria
† Institute of Electrical Measurement and Measurement Signal Processing,
Kopernikusgasse 24/4, A-8010 Graz, Austria
E-mail: matthias.scharrer@v2c2.at
Abstract—Optimizing batteries of electric cars is a complex and time consuming task. In order to reduce the number of
prototypes, development costs and time, reliable numerical models are highly required. But optimizing models reflecting the
fundamental electrochemical processes is typically computationally expensive. In this paper we present a surrogate model
optimization approach based on space mapping to reduce computation time. This technique is applied to the parameter
estimation problem of an electrochemical cell model by linking a coarse linearized model to the accurate model. We
present results of two synthetical fitting problems solved directly and by our surrogate optimization method to validate the
approach. As a remarkable result 15% reduction of computation time for the one dimensional case and 25% for the two
dimensional case are obtained. We discuss a simple measure that doubles the achieved reduction to 48% for the latter.
The method can easily be adopted to speed up other gradient–based optimization problems. Since the used electrochemical
model shows strong non–linear behaviour, the achieved speed up indicates even better performance in the case of relaxed
conditions.
Index Terms—multi physics, space mapping, surrogate optimization
I. INTRODUCTION
In terms of pollutant emissions during vehicle operation,
battery–powered and hybrid vehicles are clearly
more environmentally friendly than those purely based
on combustion engines. In order to reduce the number of
expensive prototypes the fast and reliable simulations of
the electrical and chemical behavior of cells are becoming
increasingly important. Also a more efficient operation of
a battery can be achieved, when simulation models are
used to estimate the battery’s internal states – e.g. state–
of–charge (SOC), state–of–function and state–of–health –
from measurement data.
Many internal variables and material properties are
difficult to access or not measurable. Several approaches
exist to get insight into the internal dynamic processes in
a lithium–ion cell. The field was pioneered by Newman
and co–workers [1], [2]. Overviews can be found in [3]
and [4]. The authors focus on modeling the cell in terms
of transport equations for lithium ions, chemical interaction
and electronic field computations in active particles
of anode and cathode. These are coupled by modeling
electrode kinetics occuring on the particle surfaces of
such electrodes.
Recently, much effort is put into estimating parameters
for cell models to gain better knowledge about effects
occuring during life time. In this work we focus on non–
invasive methods only, i.e. methods that estimate parameters
by matching predicted cell model voltages for a given
current profile to experimental measurements without
the need to destructively open the cell. For example,
Santhanagopalan and White [5] devised an online SOC–
estimation method by applying an extended Kalman
filter to a simplified ordinary differential equation model.
In [6], [7] a gradient based method to parameter estimation
is introduced – based on Levenberg–Marquardt
optimization. Here Santhanagopalan et al. investigate
both single and multi–particle systems and successfully
identify five parameters for constant current charge and
discharge cycles. Schmidt et al. estimate parameters
of a single–particle model in a combined approach by
performing Fisher–information based parameter analysis
and applying a pattern search algorithm consecutively [8].
In contrast to the usage of single–particle models (SPM)
and deterministic estimation algorithms above, Forman et
al. [9] focus on fitting the Doyle–Fuller–Newman (DFN)
model [1] to battery cycling data by application of genetic
algorithm.
The above literature provides an overview of a range
of different methods and models to estimate the intrinsic
material properties and unknown states. In contrast to
using a SPM, we try to find a mechanism that allows
us to use the higher detailed DFN model, as suggested
in [9]. But, as opposed to the latter, we try to further
improve the speed of parameter estimation up to online
application if possible.
In this paper the application of the so called space
mapping method – first mentioned in [10] – to the
parameter estimation of the cell model is investigated.
Thus, it is possible to apply a fast gradient based Gauss–
Newton optimization method to the complex DFN model
and use a simplified model as a surrogate to both, speed
up direct model evaluation and gradient computation at
once.

The remainder of this paper is structured as follows:
Section II defines the simulation framework of the cell
model and briefly summarizes the solution procedure. A
mechanistic model describing the electrochemistry of a
lithium–ion cell motivated by the DFN model [1] has
been implemented as a system of coupled non–linear
partial differential equations (PDEs) in one dimension
[11]. In Section III the optimization problem and solution
algorithm to estimate the parameters is defined.
Section IV presents a framework how to replace many of
the time consuming evaluations of the complex forward
model by a surrogate model – a linearization of the
complex model in this case – and how a link between
the two models is established. In Section V we present
and discuss the results. Finally, Section VI summarizes
and concludes the paper.
II. FORMULATION OF THE PROBLEM
In order to mathematically describe the internal dynamic
processes in a lithium–ion cell, a mechanistic
electrochemical model has been realized as a system
of coupled non–linear partial differential equations in
one dimension [11]. A lithium–ion cell with two porous
intercalation electrodes (cathode in Ωc and anode in
Ωa) and an electronically isolating separator in Ωs in
between is considered. For homogenization purpose each
electrode is assumed to consist of two phases. We assume
spherical particles in both cathode (in Λc) and anode
(in Λa), which line up continously in x direction. The
liquid phase modeled in each electrode is electrolyte.
In the separator Ωs we only consider electrolyte, as the
solid phase in the separator does not participate in the
reactions. In Figure 1 a schematic view of the modeled
domain is given.
Ri
, a
a
a
Ro,
a
a
r r
c
s
a,
s c,
s
c
Ro, c
Fig. 1. Problem Domain: The spatial domains are defined as Ω=
Ωa ∪ Ωs ∪ Ωc ⊂ R, Ω ′ =Ωa ∪ Ωc, Λa =Ωa × [0,Ra] ⊂ R 2 ,
Λc =Ωc × [0,Rc] ⊂ R 2 , Λ=Λa ∪ Λc and Ra,Rc ∈ R.
The implemented model is similar to the widely
used DFN approach [1], extended to include additional
aspects, e.g. from [2] and [12]. The full model will
be described in [13]. It is a coupled system of non–
linear partial differential equations in one dimension.
The variables are potentials and concentrations for the
electrolyte (ϕℓ,cℓ), for the cathode (ϕsc,csc) and for
the anode (ϕsa,csa). The one–dimensional cell model
considered is defined by the system (1).
c
Ri , c
x
- 369 - 15th IGTE Symposium 2012
TABLE I
LIST OF SYMBOLS
Ai
inner surface (m 2 m −3 )
Dℓ
diffusivity in electrolyte (m 2 s −1 )
Ds diffusivity in solid (m 2 s −1 )
Cdl double layer capacity (Fm −1 )
F Faraday’s constant (= 96485C mol −1 )
R universal gas constant (= 8.31447 Jmol −1 K −1 )
T temperature (K)
UOCV(cs) equilibrium potential function (V )
cℓ
Li + –concentration in electrolyte (mol m −3 )
cℓ,0 initial Li + –concentration in electrolyte (mol m −3 )
cs
Li + –concentration in active particles (mol m −3 )
cs,0 initial Li + –concentration in particles (mol m −3 )
i(t) cell current density (Am −2 )
j ∗
BV Butler–Volmer current density (Am −2 k
)
exchange current density and reaction rate (mol m −2 s −1 )
t +
ℓ
z
transference number of cations (1)
number of transferred electrons per unit (Li + : z =1)(1)
αA,αK anodic/cathodic charge transfer coefficients (1)
εℓ
electrolyte volume fraction (1)
κ (cℓ) ionic conductivity function (Sm −1 )
μℓ
migration coefficient
ϕs
electrochemical potential of the active material (V )
ϕℓ
electrochemical potential of the electrolyte (V )
σs
electronic conductivity (Sm −1 )
−∇ · (σs∇ϕs) =−Aij ∗
BV in Ω ′
−∇ · κℓ(cℓ)∇ϕℓ + RT
+ 1
κℓ(cℓ)t ℓ ∇cℓ = Aij
zF cℓ
∗
BV in Ω
∂ (ɛℓcℓ)
∂t
−∇·
Dℓ ∇cℓ + zF
RT μℓcℓ∇ϕℓ
= Ai
zF j∗ BV in Ω
∂cs 1
−
∂t r2
∂
Dsr
∂r
2
∂cs
=0 in Λ
∂r
j ∗
BV =
⎧
αAzF(ϕs−ϕℓ −UOCV(cs))
zFk exp
RT
+
⎪⎨
−(1−αK)zF(ϕs−ϕℓ −UOCV(cs))
−zFk exp
RT
+
∂(ϕs−ϕℓ) ⎪⎩
+Cdl ∂t
in Ω ′
(1)
0 else
where the system variables are defined as ·(t, x) at time
t ∈ [0,T], T ∈ R and at space point x and (x, r),
respectively. A comprehensive overview of symbols is
given in Table I.
Homogenous Neumann conditions are applied at the
boundaries except for the outer boundaries of potentials
and concentrations in solid phase:
ϕs =0 on Γa × [0,T]
−σs∇ϕs = −i (t) on Γc
∂cs 1
−Ds =
∂r zF j∗
BV on ΓRo,a ∪ ΓRo,c
The concentrations are restricted by the following
initial conditions:
cℓ = cℓ,0
cs = cs,0
in Ω
in Λ
The potentials are consistently initialized at rest by
the condition j (x, 0) = 0. The solution of this system of
four non–linearly coupled partial differential equations
is done by application of the Finite Element Method
with linear test functions for spatial discretization and
Backwards Euler Method for time integration. The non–
linearity is solved by a damped Newton method – see
[11] for details.
(2)
(3)

III. PARAMETER ESTIMATION
The system described in the previous section contains
many parameters which cannot be measured directly. To
formulate the parameter estimation problem in a general
way, we merge the parameter set of interest into the
parameter vector μ ∈ Pad ⊂ R m , where Pad is defined
as the admissible parameter set. The basis optimization
problem is introduced as
μ ∗ =argminH (f (μ)) , (4)
μ∈Pad where an optimal set of parameters μ ∗ ∈ Pad is sought,
which minimizes a merit function H of a model response
f (μ) depending on the parameters μ.
Since we focus on parameter estimation based on cell
voltages, we set H to compute the difference with respect
to a predescribed function ˆy. We rewrite (4) to
μ ∗ =argminwi (y(ti; μ) − ˆy(ti))
μ∈Pad 2
2
where we want to minimize the difference between measured
cell voltages, ˆy, and computed voltages, f (μ) =
y (·; μ) =ϕs| Γc − ϕs| Γa , at predefined times, ti. Variations
in time step sizes are taken into account by the
weights wi.
Classical optimization using this objective function
yields unacceptable response times, since not only the
solution of the system defined in Section II has to be
computed, but additionally the derivative of the objective
function with respect to every parameter in our set of
interest μ has to be estimated. Since this might be
intractable for non–linear PDE constraint problems, we
revert to numerical gradient estimation by finite differences.
As execution time of a single simulation on current
hardware varies between seconds and hours – depending
on the prescribed input profile – direct evaluation of (5)
is not acceptable due to its enormous computation times.
IV. PROPOSED FRAMEWORK
To speed up parameter estimation, we introduce space
mapping – first mentioned in [10] – to the cell model optimization.
The idea behind is best described as follows:
We have a very complex and accurate – fine – model
that describes a process on basis of a couple of parameters.
We search for an optimal set of parameters with
respect to some cost function by repeatingly evaluating
our model and computing model responses for intermediate
sets of parameters. Since a single evaluation of the
model is expensive, we replace the responses by results of
a much cheaper and less accurate – coarse – model (also
known as surrogate model) describing the same process
by using a similar parameter set. Thus we only get a
vague idea of where the optimal parameters are with
respect to the fine model. Finally, we link the results
by evaluating the fine model and establish a mapping
between the individual parameter spaces of both, the fine
and the coarse model. Since this results in fewer calls of
the fine model, the optimization time can be reduced.
- 370 - 15th IGTE Symposium 2012
(5)
In our case this means to substitute evaluations of the
fine model u = F(μ), where u is the tuple representing
the solutions to (1) and F(·) is the solution operator,
by evaluations of a coarse model v = C(λ), where λ ∈
Lad ⊂ R m is the parameter set of interest of the coarse
model, v is the tuple representing the solutions of the
coarse model and C(·) is the solution operator.
A mapping function p : Pad → Lad enables us
to establish a link between the two models such that
the response of the coarse model c(p(μ)) is a good
approximation for f(μ). Of course, the coarse model
response c (·) has to be defined the same way as the
fine model response f (·). Since directly evaluating the
mapping function p is not possible, we introduce a new
optimization problem:
λ =argminc( ˜λ∈L ad
˜
λ) − f(μ) 2
2
A problem with this approach is the computation of the
Jacobian of the space mapping. Bandler suggested a time
consuming way in [10]. This unnecessary big effort can
be circumvented by applying Broyden’s formula [14], as
discussed in [15]. The latter will be used in this paper.
Using the coarse model response c (·), we reformulate
the optimization problem in (4) as follows:
˜μ ∗ =argminH (c (p (μ))) , (7)
μ∈Pad where ˜μ ∗ is a coarse approximation of μ ∗ . By iteratively
updating p(·), the solution of the surrogate problem ˜μ ∗
is supposed to converge towards the real solution μ ∗ .
The algorithm applied to indirectly solve the optimization
problem defined in (5) by means of the space mapping
is stated in Algorithm 1.
Following the idea of [16], we obtain the simplified
coarse model by linearizing the right hand side of the
original system by approximation on the basis of Taylor
series expansion:
j (v; λ) ≈ ˆj (û; ˆμ)+ ∇uˆj T
(û; ˆμ) (v − û)+ ∇μˆj T (û; ˆμ) (λ − ˆμ),
(8)
where û denotes the state of the original system for a
reference parameter set ˆμ, ˆj (û;ˆμ) denotes the function
j∗ BV in a working point – throughout the rest of the paper
we write ˆj for a function ˆj (û;ˆμ). The parameters of the
linearized model are denoted by v and λ, respectively.
The non–linear factors on the left hand side, i.e. the
ionic conductivity κℓ and direct occurrences of the Li + –
concentrations in solution cℓ, are fixed to their initial
values. In (11) the coarse model is stated as used. In
addition, the boundary condition of the concentrations in
solid phase cs changes to:
∂cs 1
−Ds = ∇u
∂r zF
ˆj
T
v + ∇μˆj T λ + ˆ
Jc on ΓRo,a ∪ ΓRo,c
(9)
where the constant parts of the linearization (8) are given
as:
ˆJc = ˆj − ∇uˆj T
û − ∇μˆj T ˆμ (10)
(6)

- 371 - 15th IGTE Symposium 2012
∂ˆj
−∇ · (σs∇ϕs) +Ai ϕs +
∂ϕs
∂ˆj
ϕℓ +
∂ϕℓ
∂ˆj
cℓ +
∂cℓ
∂ˆj
cs = −Ai ∇μ
∂cs
ˆj
T λ + ˆ
Jc in Ω ′
−∇ · κℓ(ĉℓ)∇ϕℓ + RT
+ 1
∂ˆj
κℓ(ĉℓ)t ℓ ∇cℓ − Ai ϕs +
zF ĉℓ
∂ϕs
∂ˆj
ϕℓ +
∂ϕℓ
∂ˆj
cℓ +
∂cℓ
∂ˆj
cs = Ai ∇μ
∂cs
ˆj
T λ + ˆ
Jc in Ω
∂ (ɛℓcℓ)
∂t
−∇·
Dℓ ∇cℓ + zF
RT μℓĉℓ∇ϕℓ
+ Ai
∂ˆj
ϕs +
zF ∂ϕs
∂ˆj
ϕℓ +
∂ϕℓ
∂ˆj
cℓ +
∂cℓ
∂ˆj
cs = −
∂cs
Ai
∇μ
zF
ˆj
T λ + ˆ
Jc in Ω
∂cs 1
−
∂t r2
∂
Dsr
∂r
2
∂cs
=0 in Λ
∂r
Algorithm 1 Space mapping surrogate optimization
Input: Initial μ0 ∈ Pad; set i =0,λ0 = μ0, and B0 = I.
1: Evaluate f (μ0) and H (f (μ0))
2: repeat
3: Define mapping function
pi (μ) ← Bi (μ − μi) +λi
4: Compute new candidate parameter
˜μ ∗
i+1 ← arg min H (c (pi (μ)))
μ∈Pad 5: Evaluate f ˜μ ∗ ∗
i+1 and H f ˜μ i+1
6: if H f ˜μ ∗
i+1

implementation of Algorithm 1 reached its final residual
of 4.110 −7 after 4 hours and 4 iterations which results in
a reduction of runtime of 15%. This speed up is induced
by the number of evaluations of the fine model being
reduced from 20 to 5. But the additional 54 evaluations of
the coarse model in typically 189.6±2.4s during the two
optimization processes undermines this large reduction.
The differences in the curves resulting from inital and
final parameters can be mainly related to the strong
impact of the non–linearity of the equilibrium potential
function U OCV(cs). Different inital Li + –concentrations
cs,0 lead to a different working window of the curve,
so that the shape changes. Additionally, by changing the
initial amount of Li + , the available amount to move
inside the system is changed. This intrinsic value can
be estimated by the volume integral of the initial Li + –
concentrations in the electrodes
Ω εscs,0 dΩ. The limits
of Li + –concentrations of each electrode – commonly
referred to as full and empty, respectively – will result in
the cell capacity. By modifying the cell capacity and the
effective load, which can differ from the preset 0.2h−1 ,
the lower cut–off voltage is reached at different times.
The model response and the reference data where padded
with their final value, to match the length of one another.
Because of the sharp voltage drop near the empty cell,
the sensitivity of the objective function is very high.
This simplifies finding the optimum and interpreting the
results.
The second task was to simulate a 100s short charge
pulse of 0.5h−1 load to find the exchange current density
and reaction rate μ = k = {ka,kc} in both anode
Ωa and cathode Ωc starting from 50% SOC. As shown
in [6] and confirmed by our own investigations, the
exchange current density and reaction rate are showing
high impact on the results. The reference value μ∗ in this
case was set to 10−7 , 10−7 mol m−2s−1 , optimization
was initialized at μ0 = 10−8 , 10−6 mol m−2s−1 –
see Figure 3 for resulting voltage curves. In this case,
stopping criteria were applied tighter as before, because
of the smaller number of points in time of the problem:
• absolute value of the function value
H (f (μi))

TABLE II
COARSE MODEL SPEED UP COMPARISON
T model runtime model Total Speed
evaluations runtime up
1 2.95 ± 0.07s 549 (7) 1469s 1.3
2 2.39 ± 0.05s 474 (7) 1241s 1.6
5 2.04 ± 0.05s 474 (7) 1071s 1.8
10 1.92 ± 0.04s 486 (7) 1038s 1.9
20 1.87 ± 0.04s 549 (8) 1145s 1.7
non–linear 11.8 ± 0.19s 165 1947s 1.0
Speed up achieved by space mapping surrogate optimization of
kinetic rate parameters k for different reassembly periods T
compared to the non–linear case (last line). Model evaluations in
parentheses (·) are non–linear model evaluations.
TABLE III
PROGRESS OF RESIDUALS DURING OPTIMIZATION
H (f(μk))
i T =1 T =2 T =5 T =10 T =20
0 9.566E-02
1 5.159E-01 5.161E-01 5.159E-01 5.126E-01 4.261E-01
2 5.276E-02 5.280E-02 5.287E-02 4.127E-02 4.179E-02
3 2.414E-03 2.426E-03 2.439E-03 3.793E-03 3.802E-03
4 5.399E-06 5.484E-06 5.648E-06 9.178E-06 1.049E-05
5 3.229E-11 3.726E-11 4.589E-11 6.074E-11 3.258E-11
6 7.749E-13 7.840E-13 7.796E-13 9.783E-13 1.035E-12
7 — — — — 9.031E-13
Residuals achieved by space mapping surrogate optimization of
kinetic rate parameters k at iterations i for different reassembly
periods T .
Additionally, an optimum exists for some reassembly
period T , which appears in the decrease of the speed
up factor at some point. This is related to the additional
iteration necessary to reach the target residual threshold.
Yet, the performance of each iteration of the optimization
procedure shows similar progression, as can be seen in
Table III for different reassembly periods which strengthens
the before mentioned assumption of adaptivity of the
algorithm.
VI. CONCLUSION
This paper shows the application of the space mapping
approach to speed up estimation of the parameters of a
DFN motivated battery model. This is done by substituting
model evaluations by the response of a fast surrogate
model. Afterwards, the obtained parameters are mapped
into the original model’s parameter space by an iteratively
refined mapping function.
We have validated the algorithm by applying it to
two synthetic fitting problems, where parameters of a
simulation are recovered starting from a different point
in parameter space. The one dimensional quasi–stationary
case results in a reduction of 15%, whereas in the two
dimensional case shows 25% as compared to the direct
optimization computation times. Further simplification of
the coarse model increases the latter to 48%.
There is another advantage that evolves out of the
usage of a linear coarse model, which is the possibility
to state the adjoint system, which can be used to estimate
the cost functions’s exact gradient after only one
- 373 - 15th IGTE Symposium 2012
additional evaluation instead of one per parameter using
finite differences to approximate the gradient. This way
of optimization seems to be prospective, for example, for
estimating parameters of a real system or optimization
of the battery itself, with much less effort and increased
efficiency than by using direct methods.
VII. ACKNOWLEDGMENT
The authors gratefully acknowledge
financial support from Climate- and
Energy Fund “Klima- und Energiefonds”
as part of the program New Energy 2020
“NEUE ENERGIEN 2020” of the Federal
Province of Styria/Austria for the project in which the
above presented research results were achieved.
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- 374 - 15th IGTE Symposium 2012
Large Scale Energy Storage with Redox Flow Batteries
Piergiorgio Alotto, Massimo Guarnieri, Federico Moro and Andrea Stella
Dipartimento di Ingegneria Industriale, Università di Padova, Via Gradenigo 6/a, 35131 Padova, Italy
E-mail: name.surname@unipd.it
Abstract— The expected expansion of renewable energy sources is calling for large and efficient energy storage systems.
Electrochemical ones are considered the solution of choice in most cases, since they present unique features of localization
flexibility, efficiency and scalability. Among them, Redox Flow Batteries (RFBs) exhibit remarkably high potential for several
reasons, including power/energy independent sizing, high efficiency, room temperature operation and extremely long life.
The most developed RFBs are the all-vanadium based ones (VRB), but other research programs are underway in many
countries. They aim at substantial improvements which can lead to more compact systems, capable of taking the technology
to a real breakthrough in stationary grid-connected applications, but which can also prove suitable for powering electric
vehicles. This paper gives an overview of the RFB technology state-of-the-art, highlights its pros and cons, and indicates
current research challenges.
Index Terms— Electrochemical storage, redox flow batteries, vanadium flow batteries.
I. INTRODUCTION
Presently, renewable sources except hydroelectric,
particularly solar and wind, provide roughly 4% of the
worldwide electricity production, but they are expected
to grow substantially in the near future (up to 26% by
2030 [1]).
In contrast with conventional electrical power plants,
wind, solar, and other primary renewable sources are
intermittent, because the generated electrical power
depends on the time of the day and on the climatic
availability of resources. The integration into the grid of
such primary energy sources, each with different
peculiarities, requires their careful design and control.
Furthermore, traditional grids have not been designed for
such operational conditions so that they are not always
able to work satisfactorily when many renewable-source
generators are connected. In fact, recent studies suggest
that grids can become unstable if such sources provide
more than 20% of the whole generated power without
adequate energy storage.
Thus, future power grids provided with relevant
amounts of renewable sources call for adequate energy
storage systems capable of storing production surplus
during some periods and of contributing to face higher
demand during others, while at the same time
contributing to stabilizing the grid operation. Applied in
this way, energy storage systems will allow to
substantially undersize the primary power plants with
respect to peak demand, since relevant quantity of power
will be provided by the storage systems.
Two main different application scenarios can be
identified: i) “peak shaving” and “sag compensation”
refer to charge/discharge cycles on short timescales (secmin)
and are needed for grid stabilization; ii) “load
leveling” concerns charge/discharge cycles on longer
timescales (hour) and allow to improve load factor of the
grid.
Several recent surveys show that electrochemical
storage systems will be the solution of choice for
complementing intermittent photovoltaic and wind
generation facilities with long-time-scale energy storage.
In fact, such storage technologies feature site versatility,
modularity, scalability, ease of operation, and no moving
parts [2]. Worldwide important funding programs have
been established for the scientific and technological
Fig. 1: Discharge times vs. power for mainstream energy
storage technologies
development of innovative electrochemical storage
systems.
Among them, Redox Flow Batteries (RFBs) are
particularly promising They have emerged in recent
years as a very promising solution for stationary
applications, in combinations with renewable sources,
for applications such as peak shaving, sag compensation,
and load leveling [3,4,5]. The reason for their potential
success depends on the fact that, with respect to
competing technologies, they cover a very wide range of
discharge times (energy) and powers, as shown in Fig. 1.
RFBs exploit reduction and oxidation (redox) reactions
of ion metals (i.e. electrochemical species) solved in
aqueous or nonaqueous liquids. The storage of these
solutions is performed in external tanks, potentially of
very high capacity, and they are circulated into the RFB
battery when power exchange is required. The main
appealing features of RFBs are: scalability and
flexibility, independent sizing of power and energy, high
round-trip efficiency, high depth of discharge (DOD),
long durability, fast response times, reduced
environmental impact, and absence of expansive noblemetal
based catalyzers.
In the rest of this paper, the most important features of
RFBs will be presented together with an overview of the
current state-of-the art of commercial systems.
Furthermore, current research and development issues of
RFB systems will be highlighted.

II. RFB BASICS AND FEATURES
A. RFB basics
RFB cells operate on the basis of electrochemical
reduction and oxidation reactions of two liquid
electrolytes containing ionized metals [6]. One electrode
performs the reduction half-reaction of one electrolyte
that releases one electron and one ion while the other
electrode performs an oxidation half-reaction that
recombines them into the other electrolyte (Fig. 2). Ions
can then migrate from one electrode to the other (from
anode to cathode) through a membrane which can not be
crossed by electrons, which are instead forced to pass
through the external circuit thus exchanging electric
energy. Typical RFB cells must operate at room
temperature in order to keep the solutions in the liquid
phase. This condition implies that the ion-conducting
membrane between the two electrodes is a polymeric
one. Both half-cells are connected to external storage
tanks where the solutions are stored and circulated by
means of two suitable pumps. In order to design such a
storage system based on a RFB, expertise in
electrochemistry, chemistry, chemical engineering,
electrical engineering, power electronics, and control
engineering are required, thus calling for highly
multidisciplinary research and development teams.
B. RFB features
RFBs can be considered as a particular type of Fuel
Cell (FC), since they can generate electrical power as
long as they are fed with fuel (in this case the electrolyte
solutions), and indeed the cell structure is very similar to
the one of Polymer Electrolyte Membrane Fuel Cells
(PEMFCs). Another similarity between RFBs and FCs is
that energy is stored in components, the tanks, which are
physically separated from the cells themselves, were
power conversion takes place.
Independent sizing of power and energy in typical
RFB systems is therefore possible and this feature allows
for virtually unlimited capacity simply by using larger
storage tanks, without altering the battery itself or
Fig. 2: Schematic of a RFB energy storage system: RFB
stack and electrolyte tanks are separated
- 375 - 15th IGTE Symposium 2012
Fig. 3: Schematic of a typical RFB cell structure with
MEA (membrane-electrode assembly) and flow-by
solution distribution in bipolar plates with parallel-channel
layout (gaskets are not shown)
power conversion devices. Compared to other
electrochemical systems that incorporate energy and
power in a single device, RFBs are usually more
advantageous when generation for 4-6 hours or more at
maximum power is required. Furthermore, RFBs can
also be fully charged or discharged and left in such
conditions for long periods with no negative effects.
RFB cells consist of a sandwiched structure composed
of electrodes and interposed proton conducting
membranes that are very similar to the Membrane
Electrode Assembly (MEA) typical of PEMFCs (Fig. 3).
The electrolyte solutions reach the electroactive sites
within the electrodes by flowing through a porous
structure consisting of materials such as carbon felt. In
contrast with FC storage systems, which require a
specific device, i.e. the electrolyzer, for converting
electrical energy into hydrogen and oxygen, RFBs can
perform both conversions, from electrical to chemical
and from chemical to electrical, in a single device.
A second advantage of RFBs with respect to FCs is that
their fuels are not hazardous gases such as hydrogen and
oxygen, but much less dangerous electrolyte solutions,
which makes handling and storage much simpler and
cheaper. As shown in Fig. 2, only two tanks and two
pumps are required for these functions.
Moreover, RFBs operate by changing the metal ion
valence, but the ions themselves are not consumed. This
feature allows extremely long cyclic service with very
low maintenance.
The optimal electrolyte temperatures are in a narrow
range, roughly between 15°C and 35°C, and outside this
range unwanted side effects such as solution
precipitation may occur. On the other hand, the
temperature of the battery can be controlled rather easily
by appropriately regulating the electrolyte flow. The
control RFBs is also relatively easy: in fact, the cell
voltage allows to monitor easily the state of charge
(SOC) of the battery and, at the same time, very deep
discharges are possible because no damage occurs to the
morphology of the cell with such operations.
Furthermore, self-discharge is prevented by the
separation of the two electrolytes in two different

circuits. The very fast reaction kinetics provides very
fast response times and high overloading is tolerable on
short time scales.
On the other hand, at present even the most advanced
RFBs have relatively low power and energy densities
compared to other competing electrochemical storage
technologies. Consequently, RFBs tend to have large
active areas and ion conducting membranes and
therefore their overall size is usually bulky, making them
unsuitable for mobile applications. Also, the large active
areas cause high transverse gradients of the solutions
that feed the electrochemically active sites, particularly
when operating at high power and with high flows. This
causes the current density to be far from uniform on the
active areas, with average values quite lower than the
maximum ones.
III. RFB TECHNOLOGIES
A. Fe-Cr systems
The first commercial RFBs were of the Fe-Cr type,
featuring open circuit voltages of about 1 V for the
single cell. Test systems in the range of 10-60 kW were
produced in the late 1980s by several Japanese
companies including: Mitsui Engineering and
Shipbuilding Co. Ltd, Kansai Electric Power Co. Inc,
and Sumitomo Electric Industries Ltd (SEI). Beside
relatively low energy density, the main drawbacks of
such systems included: slow reaction of the Cr ion,
membrane aging and cell degradation due to the mixing
of the two ions. Due to these problems, Fe-Cr cells are
inferior to VRBs, so that they were abandoned after the
emerging of the latter.
B. VRB systems
VRBs (Vanadium Redox Batteries), also called allvanadium
RFBs, are currently the most successful RFB
technology, the only one that has reached substantial
quite commercial maturity. Such systems use only
vanadium, dissolved in aqueous sulfuric acid (~5 M). A
positive feature with respect to other RFBs is that, since
they use the same metal on each electrode, the electrodes
and membrane are not cross-contaminated, preventing
capacity decrease and providing longer lifetimes.
Exploiting the ability of vanadium to exist in solution
in four different oxidation states, vanadium II-III
(bivalent-trivalent) is used at one electrode while
vanadium IV-V (tetravalent-pentavalent) is used at the
other one. The electrochemical half-reactions are:
positiveelectrode
VO 2+ charge
+ H2O↽⇀ + + −
VO2 + 2H + e
discharge
negativeelectrode
V 3+ + e − charge
↽ ⇀V 2+
discharge
- 376 - 15th IGTE Symposium 2012
(1)
Fig. 4: Polarization curve of a RFB
During charge, tetravalent vanadium ions VO2+ are
oxidized to pentavalent vanadium ions VO2 + at the
positive electrode, while trivalent ions V3+ are reduced
to bivalent ions V2+ at the negative electrode. The
hydrogen ions 2H + created at the positive electrode flow
to the negative one through the membrane thus
maintaining the electrical neutrality of the electrolytes.
The theoretical open circuit voltage (OCV) of VRB cell
is Eo =1.26 V at 25°C, but, in fact, real cells exhibit
Eo =1.4 V in operating conditions. The cell voltage v in
load operation differs from Eo due to the activation
overpotentials η of the electrodes which are modeled by
the Butler-Volmer equation:
c
j = j r (0,t)
o exp
cr *
αF
RT η
⎛ ⎞
⎝
⎜
⎠
⎟ − c ⎡
p(0,t) ⎛ (1− α )F ⎞ ⎤
⎢
exp η
cp * ⎝
⎜
RT ⎠
⎟ ⎥
⎣⎢
⎦⎥
where j is the current density at the electrode, jo is the
exchange current density, ci are the species
concentrations at the electrochemical activity sites of the
reagents and products indicated in (1) (i = r regents,
i = p products), α is the transfer coefficient (with a value
around 0.5), F is the Faraday constant, R is the gas
constant, and T is the absolute temperature. The ci/ci
fractions express the dynamic reduction of the
concentrations normalized to steady state equilibrium
values.
According to (2) v =Eo – η is higher than Eo in the
charge phase, i.e. where the current density is j 0 when electric power
is released. jo is a parameter which depends on the
reactions and on the physical-chemical structure of the
electrodes, and is crucial for the cell operation, since the
higher jo the lower η for a given j. In fact, the activation
overpotentials are the major culprits for the cell’s
internal losses at lower current densities (with ci/ci≅1,
Fig. 4). Thus, increasing jo by means of appropriate
electrode designs allows to get performance
improvements and higher round trip efficiency. jo can be
increased with higher concentrations, lower activation
(2)

Fig. 5: Schematic of a RFB stack with side fluid feedings –
series of about 100 cells with active areas as large as
0.4x0.4 m 2 are usual
barriers (i.e. higher activity provided by efficient
catalysts), and larger effective active areas, achievable
by means of highly porous electrodes (e.g.
nanostructured materials).
At medium current densities internal losses mainly
depend on the ion conducting membrane that separates
the electrodes (Fig. 3). The material of choice is a
perfluorosulfonic acid polymer that allows, if properly
hydrated, the transport of ions by binding cations to its
sulfonic acid sites. It is a rather expensive material
patented by DuPont and commercially available under
the name Nafion. Electrically the membrane behaves as
a linear resistor, if temperature and hydration are kept
constant.
At higher current densities the losses are dominated by
transport phenomena in the electrode diffusion layers,
which dramatically reduce the concentrations (ci/ci

system so far, intended for smoothing power output
fluctuations at the Subaru Wind Villa Power Plant which
is rated at 30.6 MW, is a 4 MW / 6 MWh installation
built by Sumitomo Electric Industries (SEI), Japan, for J-
Power in 2005. The system consists of 4 banks, each of
24 stacks and rated at 1 MW (which can be overloaded
up to a maximum of 1.5 MW). Individual stacks consist
of 108 cells, with a rated power of 45 kW each. During
over 3 years of operation, the system completed more
than 270,000 complete cycles, thus demonstrating its
reliability.
The abovementioned SEI is one of the largest
manufacturers of VRB systems for the smoothing and
leveling of the fluctuating power generated by wind
farms. Most of such systems have been built by SEI and
later by VRB Power Inc., based in Vancouver, CA,
which acquired SEI patents around 2005. In 2009, all
vanadium redox battery assets of VRB Power Inc. where
acquired by Prudent Energy, controlled by investors
from China and the U.S.A., in a plan of business
expansion in China and abroad. Further important efforts
in the development of commercial RFB technologies in
China are those of the Chengde Wanlitong Industrial
Group. The reason of this interest lies in Chinese plans
to expand the exploitation of intermittent renewable
energy sources, especially wind. In fact, wind power
production in the country is expected to rise from about
20 GW in 2010 to 100 GW in 2015, and almost $50
billion per year are expected to be invested in power grid
improvements in the next decade to handle this
increasing amount of energy production from
intermittent sources.
Significant developments are also taking place in other
Asian countries, e.g. Cellennium Company Ltd. of
Thailand produces VRB systems under license, while
Samsung Electronics Co. Ltd. in South Korea is engaged
in the development of RFBs with nonaqueous
electrolytes.
Further interesting developments are taking place in
Australia, where V-Fuel Pty Ltd is pursuing innovative
V-Br technology in cooperation with the University of
New South Wales (UNSW). Other Australian companies
working on RFBs, are ZBB Energy Corp. and Redflow
Ltd., both involved in the development and installation
of Zn/Br2 batteries.
In the U.S., the Department of Energy (DoE) launched
an RFB development program which identified Ashlawn
Energy, LLC for the design of a 1 MW / 8 MWh VRB
test plant while Primus Power Corp. was funded to
develop a 25 MW / 75 MWh system based on Zn/Cl2
RFBs. Premium Power Corp. is also developing Zn/Br2
batteries.
In Europe, Renewable Energy Dynamics (RED-T),
Ireland, Cellstrom GmbH, Austria, and RE-Fuel
Technology Ltd., UK, are some of the most active
companies developing and producing VRB systems.
High-energy density innovative RFBs are also being
investigated in Germany, where the Fraunhofer-
Gesellschaft is researching nonaqueous electrolytes, and
in the UK where Plurion Ltd is working on Zn-Ce
systems.
Overall, since the market for smart grid technologies is
expected to grow significantly worldwide in the near
- 378 - 15th IGTE Symposium 2012
future, the market for VRB systems, which is already
starting to flourish, is also expected to expand
vigorously.
V. RESEARCH ISSUES
In spite of the previously described initial commercial
successes, RFB technology has yet to witness a complete
technical and commercial breakthrough and substantial
R&D programs are still required in order to fully unleash
its industrial potential. The next generation of systems,
expected within the next 5 years, will be even more
economically competitive and will be able to provide the
capital and lifecycle cost reductions that are essential for
widespread commercial success.
The basis for more compact and efficient systems,
exhibiting higher power and energy densities will be
provided by non-aqueous electrolytic solutions and by
improved electrode activity. Improved electrolytes will
also allow to expand the operation temperature range.
For example, the nonaqueous 2MW/20MWh RFB
system under development at the Fraunhofer Institute
will consists of 8 blocks of 7 stacks, with 100-cell
stacks, and will have an output of 2 kV, 1 kA, while
being fed from 2x300 m3 tanks. Further improvements
will come also from nanostructured electrodes, currently
under development, with increased effective surface area
and hence improved exchange current density.
In next generation systems, the currently common and
expensive Nafion ion conducting membrane will be
substituted with alternative ones having significantly
reduced cost and, at the same time, lower ohmic losses
in the cells. Incidentally, further material cost reduction
will also be provided by higher power densities, through
more compact designs.
Apart from the above mentioned developments which
involve mainly materials science and basic chemistry,
important engineering efforts are being aimed at system
scale-up and at the structural and operational
optimization of flow geometries, state-of-charge
monitoring and supervisor systems. Numerical modeling
and simulation are instrumental in improving the current
systems which are far from optimal in many respects.
Multi-scale, multidimensional, multi-physic, both
steady-state and dynamic, models can accurately
simulate the behavior of the whole system and its
components and thus speedup the development of more
efficient components and systems. Many modeling
problems encountered in RFB systems are similar to
those posed by direct alcohol fuel cells which also
consist of the same basic building blocks (MEA-based
cells, bipolar plates and stacks) and are also fed with
liquid solutions instead of gases, so that some of the
numerical tools developed in that context [10] may be
adapted to the simulation of RFB systems. Sophisticated
modeling tools are also aimed at designing advanced
bipolar plates with either flow-by or flow-through
diffusion of the electrolytic solutions, were the aim is to
minimize transverse gradients and, at the same time, to
reduce longitudinal conductance for lowering the shunt
currents. Advanced computational techniques are needed
to deal with the very challenging numerical problems
arising from cell elements which exhibit multi-physic

material behavior and high aspect ratio geometries
[11,12].
In the area of controls engineering, advanced control
systems will provide automatic electrolyte rebalancing
and capacity correction and will possibly allow the
remote operation of large RFB systems. Optimized
electrolyte flow-rates will also minimize pumping
energy requirements, which are one of the main factors
affecting the overall efficiency (together with shunt
currents and internal cell losses). Such control systems
will eventually cope with the conflicting requirements
arising from the strong dependence of the cell voltage
vs. current polarization curve on the solution flow-rates.
As far as the electrical interface of RFB systems is
concerned, modeling, simulation and optimization are
aimed at designing supervisor and control sub-systems
with proper feed-back loops and reduced response times
which are required to assure improved performance for
peak shaving, sag compensation and load leveling in the
smart-grid context. Flexible solutions for interfacing
both the DC renewable energy sources and AC grid and
load can be obtained with DC/DC converters coupled to
inverters. Non-linear control techniques of the inverter
can allow RFB systems to provide active as well as
reactive power to the smart-grid connected loads. The
success in designing such power management subsystems,
including both the DC/DC converter and the
inverter, strongly depends on the accuracy in modeling
the various components and the whole system.
Further research is also needed for optimizing the
technological solutions from the economical (operating
earning and savings arising from the RFBs operation)
and environmental (primary energy savings, carbon
dioxide emission reductions) point of view. The results
of these analyses will allow assessing the viability of
RFB technologies within the context of modern energy
hubs.
All the above described scientific challenges raised by
RFBs require strongly interdisciplinary development
programs and collaborative efforts among researchers
with different and complementary expertise. If such
efforts are successful the next generation of RFB
systems will be low cost, highly efficiency and durable,
and thus be suitable for large-scale industrial
exploitation, overcoming the limitations of more
conventional systems.
Finally, more compact and more flexible RFB systems,
such as the ones mentioned above, may one day become
suitable for powering some classes of electric vehicles.
VI. CONCLUSIONS
Redox flow batteries (RFBs) are already a promising
energy storage technology and first generation systems,
based on all-vanadium solutions, have already been
successfully demonstrated in test installations
worldwide, and their commercial exploitation is
undergoing. The next generation of systems, with
increased power and energy densities, are currently
under development, but further progresses in
electrochemical materials and system engineering are
expected to produce the final technical and commercial
breakthrough. RFB systems are expected to become a
- 379 - 15th IGTE Symposium 2012
key technology for stationary smart-grid-oriented
applications supporting the load leveling and peak
shaving of intermittent renewable energy sources. Future
high-density systems may also become suitable for some
automotive applications.
REFERENCES
[1] European Commission, “Proposal for a COUNCIL
DECISION
establishing the Specific Programme Implementing Horizon
2020 - The Framework Programme for Research and
Innovation (2014-2020), COM(2011) 811 final, 2011/0402
(CNS).
[2] B. Dunn, H. Kamath, J.Tarascon, “Electrical Energy
Storage for the Grid: A Battery of Choices”, Science, 334, pp.
928-935, 2011.
[3] Z. Weber, M. M. Mench, J. P. Meyers, P. N. Ross, J. T.
Gostick, Q. Liu, “Redox flow batteries: a review”, J. Appl.
Electrochem. 41, pp. 1137-1164, 2011.
[4] T. Shigematsu, “Redox Flow Batteries for Energy Storage”,
SEI Technical Review, 73, pp. 4-13, 2011.
[5] C. Ponce de León, A. Frías-Ferrer, J. González-García,
D.A. Szánto, F. C. Walsh, “Redox flow cells for energy
conversions”, J. Power Sources, 160, pp. 716-732, 2006.
[6] C. Menictas, M. Skyllas-Kazacos, “Performance of
vanadium-oxigen redox fuel cell”, J. Appl. Electrochem., 41,
pp. 1223-1232, 2011.
[7] M. Skyllas-Kazacos, G. Kazacos, G. Poon, H. Verseema,
“Recent advances with UNSW vanadium-based redox flow
batteries”, Int. J. Energ. Res., 34, pp. 182-189, 2010.
[8] Kaneko H, Negishi A, Nozaki K, Sato K, Nakajima M
(1992) Redox battery. US Patent 5318865.
[9] C. Menictas, M. Skyllas-Kazacos, “Vanadium-oxygen
redox fuel cell”, Final report. SERDF Grant, NSW Department
of Energy, 1997.
[10] V. Di Noto, M. Guarnieri, F. Moro: “A Dynamic Circuit
Model of a Small Direct Methanol Fuel Cell for Portable
Electronic Devices”, IEEE Tran.s Ind. Electronics, Vol. 57, N.
6, pp. 1865-1873, 2010.
[11] P. Alotto, M. Guarnieri, F. Moro, A. Stella: “A Proper
Generalized Decomposition Approach for Fuel Cell Polymeric
Membrane Modelling”, IEEE Trans. Mag., Vol. 47 No. 5, pp.
1462-1465, 2011.
[12] P. Alotto, M. Guarnieri, F. Moro, A. Stella: “Multi-physic
3D dynamic modelling of polymer membranes with a proper
generalized decomposition model reduction approach”,
Electrochimica Acta, pp. 250-256, 2011.
[1]

- 380 - 15th IGTE Symposium 2012
Model Order Reduction via Proper Orthogonal
Decomposition for a Lithium-Ion Cell
B. Suhr∗ , J. Rubeˇsa∗ ∗Kompetenzzentrum - Das Virtuelle Fahrzeug Forschunggesellschaft mbH (VIF), Graz, Austria
E-mail: bettina.suhr@v2c2.at
Abstract—The simulation of lithium-ion batteries is a challenging research topic. Since there are many electrochemical
processes involved in dis-/charging, models which aim to include these processes are in general complex and therefore slow.
For many tasks, e.g. in optimization, a repeated solution of a model is necessary. In this paper a speed up in simulations,
with acceptable error in results, is obtained by combining proper orthogonal decomposition with empirical interpolation
method. We report a speed up factor between 10 and 15.
Index Terms—electrochemical model, empirical interpolation method, model reduction, proper orthogonal decomposition
I. INTRODUCTION
The accurate and fast simulation of lithium-ion batteries
is of a growing interest in the automotive industry.
As fossil fuels are limited, more and more research is
conducted on electric, especially on hybrid cars. Here, the
quality and the speed of the battery simulation is a crucial
point. Often battery models are simplified strongly, in
a physical meaning, to obey the need for speed of on
board usage or optimization purposes. In contrary, here a
speed up in simulation will be gained by using model
reduction via proper orthogonal decomposition (POD)
combined with a fast evaluation of nonlinearities, the
empirical interpolation method (EIM).
Cai and White in [4] applied POD method to a battery
model, but the mayor nonlinearity of the system was
assumed to be constant. Starting from the full model and
very fine discretization in space, a speed up factor of 4
was obtained. In their work a comparison between full
and reduced for only constant discharge simulations were
done.
We follow the work introduced by Lass and Volkwein
in [7] where POD and EIM were applied to the battery
model of Wu-Xu-Zou [10]. We use this procedure and
apply it on more general but more complex battery model
of Cifrain [5]. In our work, as in the work of Lass,
no simplifications of the battery model, as mentioned
previously, are necessary and a speed up factor of 15
was obtained for constant discharge simulations.
The paper is organized in the following manner: In
Section II the nonlinear parabolic dynamical system that
describes considered battery model is formulated. Section
III is devoted to the reduced order model (ROM) utilizing
proper orthogonal decomposition (POD) method.
We describe the method of POD in general and its
application to the battery system. Moreover, the empirical
interpolation is introduced. In Section IV numerical
results are presented. Finally, in Section V conclusions
are drawn and an outlook on future work is given.
II. BATTERY MODEL
The battery cell consists of two electrodes, an anode
and a cathode, and a separator between them. Each
electrode consists of particles and an electrolyte, while
in the separator we consider only the electrolyte.
The mathematical model described in [5] and used
here is an electrochemical model similar to the well
known model of Newman [6]. It is a coupled dynamical
system of four nonlinear partial differential equations.
The system variables are potentials and concentrations
for the electrolyte, φl,cl, for the cathode, φsc,csc, and
for the anode φsa,csa. All state variables describing the
potential and the liquid concentration variable are one
dimensional system variables. Those four variables are
time t ∈ [0,T], T ∈ R, and space x ∈ Ω dependent,
where Ω ⊂ R. Two remaining variables, the variables
for the solid concentration are two dimensional variable,
i.e., cs := cs(x, r, t) ∈ Λ × (0,T) where Λ ⊂ R 2 . Such
model is also referred to as the pseudo-two-dimensional
model; see Figure 1. Considered battery model is given
in the following way:
∂ (φs − φl)
CDSAi
−∇·(σs∇φs) =−AiθjBV in Ω
∂t
′
∂ (φs − φl)
−CDSAi
−∇ ·
RT
zF
∂ (ɛlcl)
∂t
∂t
κl(cl)t +
l
−∇ · Dl ∇cl + zF
∂cs
∂t
= 1
r 2
−∇·(κL(cl)∇φl)+
(1a)
1
∇cl = AiθjBV in Ω (1b)
cl
∂ (φs − φl)
− CDSAi
+
∂t
RT μlcl∇φl
= Aiθ
F jBV in Ω (1c)
∂
Dsr
∂r
2 fRK(cs) ∂cs
in Λ (1d)
∂r
strongly coupled with
⎧
⎪⎨
⎪⎩
αAzF(φs−φ
zFkθ exp
l−UOCV (cs))
+
RT
−(1−αK )zF(φs−φ
−zFkθ exp
l−UOCV (cs))
RT
in Ω ′
0 in Ωs
jBV =

where the spatial sub-domains are introduced
as Ω = Ωc∪ Ωs ∪ Ωa, Ω ′ Λa
= Ωc∪ Ωa,
=Ωa × [0,Ra] ⊂ R2 , Λc =Ωc × [0,Rc] ⊂ R2 ,
Λ=Λa∪ Λc and Ra,Rc ∈ R. See Section VI for a
list of symbols. The system is initialized with constant
values which correspond to an equilibrium state, i.e.,
jBV =0. The boundary conditions are all homogeneous
Neumann conditions at the external boundaries and
continuous flux conditions at internal boundaries
(between electrodes and separator). Exceptions are
the solid potential where either a current is specified
−σS ∂φS
∂x = − Γa
I(t)
or a voltage φS
= U(t). Also
Acell Γa
the solid concentration has a non zero boundary condition
DSfRK(cS) ∂cS
∂r = − j∗ BV 1 ∂(φS−φL)
F − F
CDS ∂t .
r=Rp
The nonlinear initial value problem (1) is discretized
in space with the Finite Element (FE) method, in time
using the implicit Euler method and linearized with the
damped Newton method. The details on numerics, initial
values and boundary conditions one can find in [8]. After
obtaining the numerical solution of the model, the need
for model reduction and speed up has occurred.
Fig. 1. Pseudo-two-dimensional model.
III. MODEL REDUCTION
The reduction of linear dynamical systems is a classical
research topic and there exist several well known
methods for this task, e.g. balanced truncation, Krylov
methods, reduced basis and POD. Detailed information
one can found in the standard literature or in [1], [3]
and [9]. A relatively new area in research is the model
reduction of nonlinear dynamical systems of equations.
A. The POD method.
POD is a commonly used model order reduction technique,
when repeated simulations are to be conducted.
Relevant information is extracted from snapshots generated
with the full model and saved into the POD basis.
Using the POD basis a ROM will be build and for all
further simulations the ROM is used.
Computation of the POD basis: The solutions yj ∈
Rm , j =1,...,n, of the full system we refer to as
snapshots and we define the following:
Y := [y 1 , ..., y n ] ∈ R m×n .
- 381 - 15th IGTE Symposium 2012
The goal of the POD is to find l ≤ d = dim span (Y ) ≤ n
orthonormal vectors {Ψi} l i=1 in Rm that minimize the
cost function and does the best approximation of Y , i.e.,
J(Ψ1,...,Ψl) =
n
αjyj −
j=1
l
〈yj, Ψi〉XΨi 2 , (2)
i=1
where x = √ x T x is the Euclidean norm, αj ≥ 0 are
the weights and 〈x, y〉 L 2 is the inner product. Further,
from the Lagrange functional:
L(Ψ1,...,Ψl,λ11,...,λll) =J(Ψ1,...,Ψl)+
l
+ λij(〈Ψi, Ψj〉X − δij) ,
i,j=1
with the Kronecker symbol δij, we obtain two necessary
optimality conditions:
n
1) αjyj〈yj, Ψi〉X = λiiΨi ,λij =0for i = j
j=1
2) 〈Ψi, Ψj〉X = δij ,i,j=1,...,l.
The second condition is satisfied under the assumption
that the vectors {Ψi} l i=1 are orthonormal and the first
condition is equivalent with
YBY T MΨi = λiΨi , i =1,...,l, (3)
where M ∈ Rm×m is the mass matrix from the FE
simulation, λi = λii, B := diag(α1,...,αn), α1 = δt1
2 ,
αj = δtj+δtj−1
2 ,j=2,...,n− 1 and αn = δtn
2 . There
exist more possibilities to solve (3) and we will consider
the following approach.
a) K-Ansatz (Covariance matrix): By setting ¯ Y :=
M 1
2 YB 1
2 we can define the symmetric matrix K, i.e.,
K := ¯ Y T ¯ Y = B 1
2 Y T MYB 1
2 ∈ R n×n .
After the application of the eigenvalue decomposition on
K we obtain the following relation:
Kvj = λjvj , j =1,...,n.
The n eigenvalues of the matrix K we denote by λj
and the eigenvectors by vj. We sort the eigenvalues in
decreasing order, i.e., λ1 ≥ λ2 ≥ ... ≥ λl ≥ ... ≥ λn,
and we cut the first l eigenvalues such that following
holds:
n
λj ≤ tolerance ,
j=l+1
since the error of the cost function (2) can be calculated
by J(Ψ1,...,Ψl) = n j=l+1 λj. Hence, the choice of
l strictly depends on the eigenvalues λj, j =1,...,n
and how fast same decay. In Figure 2, the computed
eigenvalues are shown which are scaled by trace of matrix
K.
The POD basis consists of l functions:
Ψi(x) = 1
n
√ α
λi
1
2
j (¯vi)jy j (x) ∈ R m , i =1,...,l.
j=1

Fig. 2. Example plot of decaying eigenvalues.
Denoting the FE basis functions as ϕ, the snapshots
yj have the standard Galerkin representation: yj
=
m
k=1 yj
kϕ(x). This directly yields:
m
Ψi(x) = ψ i kϕ(x), ψ i k = 1
n m
√ α
λi
1
2 j
j (¯vi)jy k .
k=1
j=1 k=1
For later use we define (Ψi)k := ψ i k , Ψi ∈ R m ,i =
1,...,l, and the matrix ˆ Ψ:=[Ψ1, ..., Ψl] ∈ R m×l .
B. Application of POD to the battery model.
The POD basis for the battery model is calculated
for each variable separately. The snapshots of the full
FE solutions are split up in the data for each of the
six variables φl,cl,φsc,φsa,csc,csa and the six matrices
of snapshots, Yi ∈ RNi×n , i =1,...,6, are obtained.
Here n is the number of time steps and Ni is the
number of degrees of freedom in the FE Ansatz for the
corresponding variable, i.e., m = 6 i=1 Ni. With these
prerequisites the computation of the POD basis for each
of the six variables can be carried out, as described above.
The differences in the dimension of the domains of the
six variables, cause no further changes.
Before we describe how the reduced order model can
be build we need to give a quick overview on how the
full FE system is solved and to introduce some notations.
As the FE method is well known we will not give details
about how one can transform the battery system (1) in
its weak form and apply the Galerkin method on it.
The system, which is discretized in space, is still
continuous in time and has the following form:
M ˙u + K(u)u = f(u) , (4)
where u := [φl,cl,φsc,φsa,csc,csa] T , M is the mass
matrix, K(u) is the stiffness matrix and f(u) is the right
hand side of the system. Please note that we solve all
six equations simultaneously, therefore all matrices are
block matrices. For time discretization the implicit Euler
method is applied on (4) and we use
1
M + K(uk+1) uk+1 − f(uk+1) =
Δt 1
Δt Muk , (5)
- 382 - 15th IGTE Symposium 2012
in every time step k = 1, 2, ..., n. As this system is
fully discretized but still nonlinear, we apply now the
Newton method in order to linearize the system (5). After
introducing the following notation:
1
A(uk+1) := M + K(uk+1)
Δt
we can rewrite equation (5) and define the operator G as
and F := 1
Δt Muk ,
G(uk+1) :=A(uk+1)uk+1 − f(uk+1) − F =0. (6)
Applying the Newton method we end up calculating the
Newton step
Δu = −[JG(u i k+1)] −1 G(u i k+1) , (7)
in every time step k until the predefined stopping criterion
is met for the sequence {ui k }∞i=0 , ui+1
k+1 = ui k+1 +Δu.
The derivative JG is calculated using the derivatives of
K and f such that
JG(uk+1) = 1
Δt M + JK(uk+1) − Jf(uk+1). (8)
1) Reduced order model for the battery: Next step
is to build the reduced order model using the POD
basis. We point out that it is necessary to formulate
non homogeneous Dirichlet boundary conditions with a
penalty method.
To obtain the reduced order model the Galerkin method
is used with the POD basis functions, instead of the FE
basis functions as before. We denote our calculated POD
basis for each of the six variables with:
ˆΨi ∈ R Ni×li 6
, i =1,...,6 , lPOD = li.
i=1
In the case of one single equation the mass matrix M of
the full system and the mass matrix ˆ M of the reduced
system are connected via the POD basis ˆ Ψ as follows:
ˆM = ˆ Ψ T M ˆ Ψ , M ∈ R m×m , Mˆ l×l
∈ R . (9)
This can be adapted to a system of equations using the
block structure of the mass matrix. The mass matrix
M ∈ R m×m in (4) is a block matrix with the following
structure: Mij ∈ R Ni×Nj , i,j =1,...,6. The corresponding
matrix ˆ M ∈ R lPOD×lPOD , in the reduced order
model, can be obtained as follows:
ˆMij = ˆ Ψ T i Mij ˆ Ψj , i,j =1,...,6 ,
where ˆ Mij ∈ R li×lj for i, j =1,...,6. For the right
hand side in (5) it holds analogously:
ˆFi = ˆ Ψ T i Fi ∈ R li , i,j =1,...,6 ,
where Fi ∈ R Ni , F =[F1, ..., F6] T ∈ R m and ˆ F =
[ ˆ F1, ..., ˆ F6] T ∈ R lPOD . To improve readability we use
from now on L(M) := ˆ M and L(F ):= ˆ F .
There are many hidden nonlinearities in equations (7)
which have to be evaluated for each time and Newton
step in the full FE dimension m. This can be accessed
via the relation (uk+1)i = ˆ Ψi (ûk+1)i.

The corresponding equations to (6) and (8), for the
reduced order model, are
L(G(ûk+1)) :=L(A( ˆ Ψûk+1))ûk+1 −L(f( ˆ Ψûk+1))
−L(F ) (10a)
L(JG(ûk+1)) := 1
Δt L(M)+L(JK( ˆ Ψûk+1))
−L(Jf( ˆ Ψûk+1)) (10b)
then the Newton step and the iterations are defined by
Δû = −[L(JG(ûk+1))] −1 L(G(ûk+1)) , (11a)
ûk+1 = Luk +Δû. (11b)
C. Application of EIM
The evaluation of nonlinearities in (10) is very slow.
EIM can be used to reduce number of necessary evaluations
of these nonlinearities, see [2] for details on
the method. Let g be a given nonlinear function, which
can be evaluated point wise. Needed are snapshots of
this function with fast decaying eigenvalues. For the
basis construction a greedy algorithm for the biggest
residual is used, which stops at a given tolerance. The
returned basis, Υ=[υ1,...,υlEIM ], has lEIM entries
and also the indices for the evaluation of the nonlinearity
p =[p1,...,plEIM ] are given. The nonlinear function g
is approximated as follows:
g(y) =Υc(y) =
lEIM
i=1
υici(y). (12)
To compute the coefficients c the above equality can be
transformed to:
P T Υc(y) =P T g(y) ⇔ c(y) =(P T Υ) −1 P T g(y) (13)
with the matrix P =[ep1,...,epl ] where epi ∈ Rm
EIM
is the pith unit vector.
Since the singular values of the nonlinearity on the
right hand side of the system decay slowly, the use of
EIM is not useful here. Luckily, the eigenvalues on the
left hand side of the system decay fast and EIM can be
used. This substitutes the matrix assembly.
In equation (10) the nonlinearities are part of the
stiffness matrix K and its derivative JK. As both cases
are handled analogously we explain the case for the
stiffness matrix only. The nonlinearities in the different
blocks of the stiffness matrix are dealt separately and
therefore we drop one pair of indices and denote the
entries of an arbitrary block of the stiffness matrix as
Kij = g(y)∇ϕi∇ϕjdx, i, j =1,...,m, (14)
Ω
where g(y) is the nonlinear function to be approximated.
When we now insert equation (12) in (14) and combine
this with the reduced order stiffness matrix ˆ K = ˆ Ψ T K ˆ Ψ
- 383 - 15th IGTE Symposium 2012
we obtain:
ˆK = ˆ Ψ T
=
lEIM
lEIM
k=1
k=1
ck(y) υk∇ϕi∇ϕjdx
Ω
ck(y) ˆ Ψ T
υk∇ϕi∇ϕjdx ˆΨ . (15)
Ω
ij
The underlined expressions can be precomputed, these
are lEIM matrices of dimension lPOD×lPOD. Instead of
the assembly of the stiffness matrix the coefficients c have
to be calculated using (13), this includes lEIM function
evaluations. These coefficients are then used in the above
formula where a linear combination of the precomputed
matrices is calculated. When lEIM is small a massive
speed up can be gained by this method.
IV. RESULTS
In general, our goal is to replace a slow FE simulation
with a fast ROM simulation by allowing a small relative
error between those two simulations. In this section we
will compare the FE solutions and the ROM solutions
for accuracy and speed archived by three different approaches.
We will start with one very simple example.
A. Simple simulation of battery.
We consider a simulation where the battery is charged,
discharged and charged again between 3.8V and 2.5V (all
three times with 0.2C); see the solid blue line in Figure 3.
To solve the full model the following spatial discretization,
given in degrees of freedom (DOFs), for the single
variables is used:
φl φsc φsa cl csc csa total
241 101 101 241 5151 5151 10986
The simulated time is 61586s and 1451 time steps are
used due to the adaptive time stepping algorithm. The
solutions of the full FE system is plotted in Figure 4.
These solutions were taken as snapshots for the POD
basis computation for each of the six variables. When we
cut the scaled eigenvalues, at the tolerance of 10−8 , the
number of POD basis functions for the single variables
is as follows:
Fig. 3. Simple simulation of battery.
ij
ˆΨ

Fig. 4. 3d plots of full FE solution for first example.
φl φsc φsa cl csc csa total
5 3 3 3 14 12 42
The total dimension of the ROM is therefore 42. Also
EIM bases for the different nonlinearities are computed.
The ROM is then solved using these POD and EIM bases.
For a comparison between the FE and the ROM solution
we consider the L2-error. For voltage and current the
absolute error is 6.6 · 10−06 and 1.0 · 10−04 , respectively.
A difference between the voltage of the FE solution and
the ROM solution is shown with asterisk line in Figure 3.
Also we consider the relative error of each variable which
is defined for φl as:
eφl :=
||φFE l − φROM
l || L2 (Ω)
||φFE l || L2 (Ω)
L 2 (0,T )
, (16)
and for all other variables analogously. The resulting
relative errors for each variable are:
φl φsc φsa
2.2 · 10 −6
1.3 · 10 −6
cl csc csa
1.7 · 10 −7
4.2 · 10 −6
2.6 · 10 −5
5.5 · 10 −6
All of these relative errors are acceptably small. Now
that we checked the accuracy of the ROM solution we
are interested in the speed of the computation. The full
FE solution took 2828s in CPU time while the ROM
solution lasted 189s, which means that a speed up of
factor 15 was achieved.
B. More general POD and EIM bases.
We need POD and EIM bases which are more general
and applicable for a broader range of use cases, concerning
the battery behavior. This is achieved this by building
one basis out of a set of similar simulations.
In battery simulation a natural choice for this set of
simulations is the discharge of the battery at different
C-rates. To be more precise: we discharge the battery
from 3.8V until 2.5V at the C-rates 0.1, 0.2, 0.5, 1,
2, 3, 4 and 5C. From these set of snapshots we build
the POD and EIM bases by using again the tolerance of
10−8 . The ROM is of a dimension of 85, compared to
the full dimension of 10986 DOFs. To make sure that
- 384 - 15th IGTE Symposium 2012
Fig. 5. Voltage of simulations at different C-rates.
Fig. 6. Voltage of simulations not used for POD basis calculation.
the dynamics of the different C-rates are captured well,
we repeat all discharge simulations used for building
the basis with the ROM. In Figure 5 we plotted the
calculated voltage of the full FE solution and the ROM
solution using the new POD and EIM bases. We detected
a extremely small L 2 relative error and we conclude that
the results are in good agreement for all used C-rates.
Next we want to find out whether C-rates not used
for the basis computation, are also simulated well by the
ROM. For this purpose we simulate the C-rates: 0.15,
0.3 and 3.5 with the full FE system and the ROM. The
results are in good accordance as can be seen in Figure 6.
We conclude that all C-rates between 0.1 and 5 can be
simulated well with the ROM using the improved, more
general, basis.
C. POD basis switching.
With the single POD basis we can simulate well
different discharges in the 0.1 - 5 C-rate range and now
we want to use the ROM for a simulation that involves
charge and discharge processes. We have the set of POD
and EIM bases for a discharge. Equivalently using the
same C-rates, we calculate the POD and the EIM basis for
the charge processes. Using the same tolerance, 10−8 ,we

Fig. 7. Voltage of simulations where bases for ROM computation
were switched.
obtain 110 POD basis functions. In the ROM simulation
we can then switch between these two sets of bases.
This is demonstrated through the simulation of
charge/discharge processes were charge and discharge are
simulated at C-rates not included in the bases computation.
The voltage plot of the full FE solution and the
ROM solution can be seen in Figure 7.
The voltage curves are in good accordance, only at the
switching points there occurs an error. Both the discharge
and the charge POD basis span a subspace in the space of
all possible solutions. When switching between discharge
to the charge basis (or vice versa) the ROM solution is
projected from one subspace to the other and an error
occurs.
Below the relative errors for each variable are given,
see (16) for the definition of the error norm.
φl φsc φsa
1.4 · 10 −5
3.4 · 10 −6
cl csc csa
4.1 · 10 −6
1.3 · 10 −5
3.8 · 10 −5
3.0 · 10 −5
Naturally the error in this simulation is bigger than in
the first example, nevertheless the results are acceptable.
This method seems to be very suitable to allow a fast
simulation of different processes, as pulses, rest steps,
cycling etc. To find appropriate POD and EIM bases for
these processes, which also give a small projection error,
remains our next task.
V. CONCLUSION
For speeding up the numerical simulation of the
lithium-ion battery model, a reduced order model was
build utilizing proper orthogonal decomposition method
and empirical interpolation method.
Speed up factors of 10-15 (depending on the considered
case) with acceptable error was obtained. The
developed method of switching between POD and EIM
bases for different purposes, shows good results for
charging/discharging processes at different C-rates. This
- 385 - 15th IGTE Symposium 2012
method is very promising to allow a fast simulation of
different processes, e.g. pulses, rest steps, cycling etc.
Our future research will aim for the construction of
appropriate POD bases for these processes. Also the
question whether the projection error can be minimized
will be considered.
Acknowledgment:
The authors gratefully acknowledge financial support
from “Zukunftsfonds des Landes Steiermark” of the Federal
Province of Styria/Austria for the project in which
the above presented research results were achieved.
VI. LIST OF SYMBOLS
cs
concentration of Li + in active material
cl
concentration of Li + in electrolyte
Φs
electrochemical potential of active material
Φl
electrochemical potential of electrolyte
Ai
inner active surface
αA,αK anodic/cathodic charge transfer coefficients
CDS double layer capacity
Dl
solution diffusivity
Ds
solid diffusivity
εl
electrolyte volume fraction
jBV
Butler-Volmer current density
F Faraday’s constant (= 96485Cmol −1 )
k exchange current density and reaction rate
κl(cl) ionic conductivity function
μl
migration coefficient
σs
electronic conductivity
R universal gas constant (= 8,31447 Jmol −1 K −1 )
T temperature
t time
t +
transference number
UOCV (cs) equilibrium potential function
USEI ohmic loss of potential due to solid electrolyte interface at
ΩA
z number of transfered electrons (for Li + : z =1)
REFERENCES
[1] A.C. Antoulas, “Approximation of Large-Scale Dynamical Systems.”
in Siam, 2005.
[2] M. Barrault, Y. Maday, N. Nguyen, A. Patera, “An empirical
interpolation method: Application to efficient reduced basis
discretization of partial differential equations.” Comptes Rendus
Mathematique, 339: 667-672, 2004.
[3] P. Benner, V. Mehrmann, D.C. Sorensen et. al., “Dimension Reductin
of Large-Scale Systems.” Springer, 2003.
[4] L. Cai and R.E. White, “An Efficient Electrochemical–Thermal
Model for a Lithium-Ion Cell by Using the Proper Orthogonal
Decomposition Method.” in J. Electrochem. Soc., vol. 157, pp.
A1188-A1195, 2010
[5] M. Cifrain et. al., “Elektrochemisches Zellmodell.” publication in
preparation, 2012.
[6] M. Doyle, T.F. Fuller, J. Newman, “Modeling of Galvanostatic
Charge and Discharge of the Lithium/Polymer/Insertion Cell.” in
J. Electrochem. Soc., vol. 140 (7), pp. 1526–1533, 1993.
[7] O. Lass and S. Volkwein, “POD Galerkin schemes for nonlinear
elliptic-parabolic systems.” submitted for publication in 2011
[8] F. Pichler, “Anwendung der Finite-Elemente Methode auf ein
Litium-Ionen Batterie Modell.” Master Thesis, University of Graz,
2011.
[9] S. Volkwein, “Proper orthogonal decomposition (POD) for nonlinear
systems.” PhD program in Mathematics for Technology
Catania, 2007.
[10] J. Wu, J. Xu, H. Zou, “On the well-posedness of a mathematical
model for lithium-ion battery systems.” Methods and Applications
of Analysis, 13:275-298, 2006.

- 386 - 15th IGTE Symposium 2012
Automatic domain detection for a meshfree postprocessing
in boundary element methods
André Buchau, Matthias Jüttner, and Wolfgang M. Rucker
Universität Stuttgart, Institut für Theorie der Elektrotechnik, Pfaffenwaldring 47, 70569 Stuttgart, Germany
E-mail: andre.buchau@ite.uni-stuttgart.de
Abstract—Modern advanced visualization techniques for three-dimensional electromagnetic fields evaluate field values in
some points in space, which are determined only during the computation of visual objects like streamlines. Furthermore, a
meshfree post-processing in boundary element methods along with a bidirectional coupling of numerical field computations
with a visualization tool are advisable to reduce significantly computational costs and the total amount of stored data. However,
a completely automatic domain detection method is then required. Domain data like material values are not explicitly
defined due to the lack of a volume mesh but are necessary for a correct computation of field values in arbitrary points. Here,
a robust and fast octree-based method is presented to determine the domain data of an evaluation point efficiently even for
large and complex field problems. The computational costs of position detection of a single evaluation point are kept small by
filtering of relevant boundary elements. Furthermore, position data of other evaluation points is used if possible.
Index Terms—boundary element methods, domain detection methods, meshfree post-processing, octree-based schemes
I. INTRODUCTION
A very important step in numerical field computations
is an extensive post-processing including a vivid visualization
of the obtained results. Today, visualization tools
like virtual and augmented reality along with modern
visualization techniques enable even experienced engineers
a deep insight into the physical properties of the
studied problem [1, 2]. However, an expressive visualization
of three-dimensional fields is still a challenge. One
possibility is to use volume rendering in the case of scalar
data [3]. An interesting approach, which has been presented
for two-dimensional magnetic fields, is to compute
visual objects that represent the topology of the vector
field [4]. Further techniques, which are more commonly
used, are to filter three-dimensional data first and to visualize
three-dimensional fields in slices or to compute
streamlines or isosurfaces. Data exchange between the
numerical field computation tool and the visualization
tool is normally done with the help of volume meshes of
all considered domains. The field values are precomputed
in all nodes of this mesh and transferred to the
visualization tool that performs the post-processing independently
of the numerical field computation tool.
The boundary element method (BEM) is a very attractive
method for the solution of three-dimensional electromagnetic
field problems, which consist of multiple,
piece-wise homogeneous, linear media. Then, a modeling
and discretization of domain surfaces suffices. Hence, the
discretized model is much smaller than in volume-based
methods like the finite element method (FEM). Furthermore,
well-established compression techniques for the
linear system of equations exist to enable a fast and efficient
solution of large and complex field problems [5].
However, an additional volume mesh is normally created
for the post-processing [6]. Domain data like material
values are assigned to the auxiliary mesh and are available
for the computation of field values in the mesh nodes.
The application of the fast multipole method (FMM)
enables the computation of field values in a huge number
of points at acceptable computational costs [7], but the
amount of data, which must be stored and transferred to
the visualization tool, is relatively large and a bottleneck.
A better approach, which fits much more the basic
concept of a BEM, is a meshfree post-processing [8].
There, field values are only computed at points, which are
necessary for visualization. Hence, the number of evaluation
points is dramatically reduced in comparison to the
classical approach, which uses an auxiliary volume mesh.
Furthermore, the meshfree approach is much more flexible.
Post-processing domains and visualization techniques
are defined completely after the solution of the
problem. The creation of an expensive volume mesh is
unnecessary and the BEM is integrated into the visualization
tool. However, automatic domain detection is required
to make a meshfree post-processing applicable for
problems, which consist of multiple domains.
Here, a novel method is presented that completely automatically
detects the position of an arbitrary evaluation
point directly from the given boundary element mesh.
The octree-based method is fast and efficient to enable
extensive post-processing of large and complex field
problems. A very small number of boundary elements are
extracted from the total model to perform the position
detection efficiently with a method that is similar to the
well-known ray tracing method [9]. Furthermore, domain
data of other evaluation points is used if possible.
The paper is structured as follows. First, the problem
of position detection directly from boundary elements is
formulated and the concept of the presented method is
shown. Then, a flexible octree-based scheme is introduced
to enable a grouping of all boundary elements
regarding their position in three-dimensional space. It is
applied to determine the position of an evaluation point
with the help of other evaluation points of the same domain
or to filter candidate boundary elements for the
actual position detection, which is described afterwards.
There, a method similar to ray tracing is presented to find
a boundary element that is the closest boundary element
to the given evaluation point with the help of a small list
of candidates. Two numerical examples have been studied
to demonstrate robustness and efficiency of the presented
automatic domain detection method. Finally, significance
of the method to BEM and an outlook to future
work are given in the conclusions.

II. NUMERICAL METHOD
In general, automatic domain detection is required for
each evaluation point in a meshfree BEM postprocessing.
Since both the number of evaluation points
and the number of boundary elements are often very large
in practical three-dimensional problems, a fast algorithm,
which exploits properties of BEM, is advisable.
The concept of the presented approach is presented in
the first sub-section. There, a technical formulation of the
problem is given along with a comparison to ray tracing
methods. The second sub-section is about the novel fast
octree-based scheme. It is the key to a successful application
of meshfree post-processing in large BEM problems.
Finally, an efficient and general method of ray tests based
on gradient search is given in the third sub-section. Note,
the presented approach has been developed to achieve
two goals, an efficient and robust automatic domain detection
method and a flexible octree for fast and efficient
post-processing computations using the fast multipole
method (FMM).
A. Concept of automatic position detection
The aim of the presented automatic domain detection
method is to determine the domain of an arbitrary
evaluation point, which is defined by its global Cartesian
coordinates, e. g. during the computation of a streamline,
=
. (1)
The complete BEM problem is discretized by in total
boundary elements. The shape of boundary elements
and the order of their shape functions can be arbitrarily
chosen for the presented domain detection method. The
direction of the normal vector of a boundary element
is known. Furthermore, the domain , which lies in
direction of , and the domain , which lies in
direction of , are stored for each boundary element.
The concept of the presented approach is to construct a
single ray, which starts at the given evaluation point
= + , (2)
where is the direction of the ray and 0 is a parameter.
Then, the domain is determined with the
help of the first boundary element, which is intersected
by the ray (2). An example configuration is given in
Fig. 1. The red point is the given evaluation point, the
blue line is the ray, and the green and yellow lines are
two boundary elements. Here, the domain of the evaluation
point is = of the green boundary element.
Fig. 1: Example of domain detection with the help of a ray
The concept of domain detection is similar to the wellknown
ray tracing method [9]. The main difference is that
the direction of the ray is unknown. In ray tracing, the
direction corresponds to the view direction. Furthermore,
a single ray suffices for successful domain detection. In
ray tracing of computer graphics, a ray for each pixel of
the screen is constructed. Hence, an adapted octree-based
algorithm for BEM is presented in the next sub-section.
- 387 - 15th IGTE Symposium 2012
B. Fast octree-based algorithm
A fast algorithm is necessary to enable an application
of the automatic domain detection method to large BEM
problems. The costs of a standard implementation of the
concept, which has been presented in the previous subsection,
are proportional to . Hence, an efficient filtering
of relevant boundary elements is required to reduce
computation time. Furthermore, the computational costs
are proportional to the number of evaluation points , which are given by the visualization tool. However, an
evaluation point is often close to a previous evaluation
point, for instance points on a streamline. An approach to
reduce the computational costs is to use domain data of
previous evaluation points if possible.
Trees are a common method to group and select
boundary elements in three-dimensional space. Here, an
octree has been chosen, since the FMM is used to accelerate
BEM computations and the FMM is based on an
octree, too [10]. An implementation of the octree, which
uses modern software techniques, enables the application
of the same code foundations for both domain detection
and post-processing computations. Then, code is more
reliable and the complete method works more robust.
The first step is to initialize the octree using all boundary elements. The so-called root cube at octree
level 0 is the smallest cube, which encloses all boundary
elements. Its edges are parallel to the axes of the global
Cartesian coordinate system. Then, the root cube is subdivided
into eight equal sized cubes and the boundary
elements are assigned to these so-called child cubes. Each
child cube is again subdivided into child cubes. The subdivision
is continued while the boundary elements of a
cube can be assigned to its child cubes or the total number
of octree levels is smaller than a given limit.
Boundary elements are assigned to a cube, if their centroid
lies inside the cube. Of course, the position of
boundary elements and their real dimensions must be
considered. The bounding box of a cube is determined
including all its boundary elements. This bounding box
must completely lie inside a test cube with the same center
as the considered cube and an edge length of

The domain detection starts with the addition of the
evaluation point to the octree. If is outside the
root cube of the octree, the spatial domain of the octree is
enlarged by creating a new root cube. Of course, is
increased in that case. The evaluation point is assigned to
a cube in the same way as the boundary elements during
initialization. Here, the subdivision of a cube with an
evaluation point is aborted, if no boundary elements are
assigned to the cube of the evaluation point. The number
of evaluation points in a cube is not taken into account.
Hence, the cubes of evaluation points are chosen as large
as possible.
An example is given in Fig. 2. The black lines with
black points represent some boundary elements. The red
point is an evaluation point. The thick black lines are the
octree cubes, which have been created according to the
above-described rules. One boundary element is assigned
to one cube and the boundary elements stick slightly out
of the cubes. In contrast, the cube of the evaluation point
is relatively large.
Fig. 2: Example of an octree for position detection
A strategy is to reduce the number of actual domain
detections. Hence, a goal is to determine the domain not
only for the given evaluation point but also for the cube
of the evaluation point if possible. Consequently, an initial
cube is searched that fulfills the following conditions.
The octree level of a cube of an evaluation point is
maximal the finest octree level of cubes of boundary
elements. While the octree level criterion is satisfied, the
cube of the evaluation point is refined until no boundary
elements are assigned to that cube or no boundary elements
stick into this cube. To avoid expensive tests based
on the real position of boundary elements, first a cube is
searched that has no neighbors with boundary elements.
If the cube of the evaluation point already has no neighbors
with boundary elements, its parent cube is tested for
the above-described rules.
The black cube of the evaluation point in Fig. 2 has
neighbor cubes with boundary elements. Hence, the black
cube is refined and the blue cube is obtained. Since the
blue cube has also neighbor cubes with boundary elements,
it is refined as well and the green cube is the initial
cube for domain detection.
If no elements are lying inside the cube , the domain
is determined for including all its child cubes. The
domain data of the cube is then used for all evaluation
points, which are assigned to the cube at a later moment.
Neighbor cubes of the cube are at several octree levels
due to the adaptive octree rules of octree initialization.
Furthermore, octree structure is changing during the post-
- 388 - 15th IGTE Symposium 2012
processing. Hence, it is not possible to determine cube
neighbors in advance. Neighbor cubes, or cubes in general,
are searched by the position of the cube center. Here,
the centers of possible neighbor cubes are computed and
the cubes are searched starting from the root cube by
simple and fast comparison of Cartesian coordinates.
If the domain data of a neighbor cube of cube has
been already determined, it can be used for the cube ,
too. Otherwise, a cube with boundary elements of the
second neighbors of cube has to be chosen for domain
detection. Since no elements are assigned to the cube
and its neighbor cubes, one second neighbor cube, which
adjoins a neighbor cube of , suffices for domain detection.
At least one second direct neighbor cube with
boundary elements exists, because the cube is chosen
as large as possible as described above.
After a cube has been chosen, the neighbor cube
of , which lies between and , is determined.
First, all boundary elements, which stick into , are
searched. Note, most neighbors of are without
boundary elements and only neighbors, which adjoin
, have to be considered. In practice, the number of
relevant cubes is small. Furthermore, the boundary elements,
which are assigned to and which stick into
, are determined. In total, a list with relevant
boundary elements is obtained. To improve robustness of
the domain detection method, a new point for domain
detection is defined inside and close to .
Some examples of typical situations of domain detection
are depicted in Fig. 3.
Fig. 3: Some typical situations of domain detection
First, the domain of the red evaluation point is determined.
Since no boundary elements are assigned to the
red cube, the domain of the red cube is evaluated. The
black cube with the red lines is chosen as cube . The
orange evaluation point inside is used for the domain
detection of the read evaluation point. Next, the
domain of the blue evaluation point is detected. Since the
blue evaluation point lies within a cube with boundary
elements, these boundary elements are used for domain
detection. Finally, the domain of the green evaluation
point is determined with the help of domain data of its red
neighbor cube.
As already mentioned, the list of relevant
boundary elements for domain detection includes all
boundary elements, which are assigned to a relevant cube
and all boundary elements of neighbor cubes, which stick
into the relevant cube. A fast method to test whether a
boundary element sticks into a cube is to test an intersection
of the bounding box of the cube and of the bounding

ox of the boundary element. Two examples of boundary
elements inside a cube are given in Fig. 4. Although no
boundary elements are assigned to the blue cube, the blue
boundary element of its neighbor cube must be taken into
account, since it sticks into the blue cube. In the case of
the red cube, one red boundary element, which is assigned
to the red cube, and one red boundary element of
its neighbor cube have to be considered.
Fig. 4: Examples of elements inside a cube
In total, the presented octree-based method is fast,
since is approximately independent of , at least
in the case of large problems. If possible, domain data for
cubes is determined. The domain detection with the help
of the filtered boundary elements is described in
the following sub-section.
C. Domain detection from boundary elements
Starting position of a domain detection from boundary
elements is a very small list of boundary elements,
which is determined using the octree-based method of the
previous sub-section, and the given evaluation point (1).
Furthermore, domain detection from boundary elements
is only necessary, if domain data of octree cubes cannot
be used. Hence, general applicability and extension possibilities
of the following method are more important than
pure efficiency considerations.
The initial step is to sort the boundary elements
by their distance to the evaluation point
= , 0

III. NUMERICAL EXAMPLES
The presented automatic domain detection method for
BEM has been tested on two numerical examples. The
first example is a capacitor, which can be simply discretized
with different sizes of boundary elements. Hence, it
is well suited for fundamental tests of the automatic domain
detection method. The second example is an inductor
of a micro-electro-mechanical system. It represents a
typical configuration in an application of BEM with often
changing domains in a slice. There, domain data of evaluation
points cannot be easily defined and the power of
the presented automatic domain detection method is
clearly demonstrated.
The domain detection method has been implemented in
C# using the .NET framework 4.0 [11]. C# is a managed
language and it supports very well necessary data handling
of the domain detection method. Furthermore, the
interface of C# to native C++ enables the use of existing
high-performance code of numerical methods, for instance
the used BEM and FMM implementation. Bounding
boxes including intersection tests or vector operations
are standard methods of the Windows Presentation
Framework (WPF), which is part of the .NET framework.
Furthermore, the Windows Communication Foundation
(WCF) of the .NET framework enables an interactive
data exchange between different processes based on extensible
markup language (XML) over hypertext
transport protocol (http). Here, WCF is applied to couple
the process of the visualization tool HLRS COVISE,
which is developed at the High Performance Computing
Center at the University of Stuttgart, with the implementation
of the domain detection method. HLRS COVISE
visualizes three-dimensional data with the help of virtual
and augmented reality techniques [1].
The surfaces of both numerical examples have been
discretized with second order, quadrilateral boundary
elements. The Galerkin method has been applied to indirect
and direct BEM formulations. The matrix of the
linear system of equations has been compressed with the
help of the fast multipole method [12]. The matrix has
been assembled in parallel using the OpenMP standard.
The system of linear equations has been solved in parallel,
too. An implementation of the BEM in combination
with the FMM in C++ has been executed on a workstation
with two six-core Intel Xeon E5649 2.53 GHz
processors.
Although the presented domain detection method supports
all kinds of boundary elements, the second order,
quadrilateral boundary elements have been converted into
linear, triangular boundary elements for the postprocessing.
The reason is that in computer graphics only
linear elements, often only linear triangles, are well supported.
Rendering and graphics processing on linear triangles
is much faster than on other types of elements and
linear triangles are supported by modern graphics processors.
The implementation of the domain detection method
has been executed on an Intel Core 2 Duo T9900
3.06 GHz laptop processor using a single core. Graphical
objects have been rendered on a NVIDIA Quadro FX
770M graphic card.
- 390 - 15th IGTE Symposium 2012
A. Capacitor
The electric field of a capacitor has been studied as
first example. The capacitor consists of two quadratic
electrodes and a homogeneous, linear, isotropic dielectric
between the electrodes. The potential of the electrodes
has been set to 0.5 V and -0.5 V respectively. The relative
permittivity of the dielectric is 10.
The capacitor has been discretized with 9600 second
order, quadrilateral elements (Fig. 5). An indirect BEM
formulation is applied. The Dirichlet boundary conditions
are the potential at the two electrodes. The Neumann
boundary condition is the continuity of the electric flux
density at the surface between the dielectric and the surrounding
free space domain. The corresponding linear
system of equations with in total 29442 unknowns has
been solved iteratively using generalized minimal residual
method (GMRES) along with a Jacobi preconditioner
within 91 iteration steps in approximately 3 minutes.
Fig. 5: Discretized BEM model of a capacitor
The original second order boundary elements have
been converted into 19200 first order, triangular elements
for the post-processing including the domain detection.
The boundary elements are grouped by an octree, which
consists of 9 octree levels and 2 elements assigned to a
cube in average. The maximum number of elements of a
cube is 6. The domain data in a slice in 40000 evaluation
points has been determined in 23 s (Fig. 6). The red color
represents the domain inside the dielectric and the green
color represents the air domain. Furthermore, the two
electrodes are depicted. The color at the electrodes displays
the surface charge density, which equals the solution
of the linear system of equations. The surface of the
dielectric has been omitted for graphical reasons.
The presented octree-based scheme reduces the number
of boundary elements, which must be considered for
correct domain detection, from 19200 to maximal 39. The
domain data of 93 % of the given evaluation points could
be obtained from position data of the octree cubes without
expensive ray hit tests.

Fig. 6: Detected domains in a slice through the capacitor
B. Inductor in micro-electro-mechanical systems
The electric current inside an inductor of a microelectro-mechanical
system (MEMS) has been studied as
second example. The inductor has been discretized using
9168 second order, quadrilateral elements (Fig. 7). The
potential at the ports of the inductor has been set as Dirichlet
boundary condition of a direct BEM formulation.
The linear system of equations with in total 27594 unknowns
has been solved within 166 iteration steps of
GMRES in approximately 4 minutes.
Fig. 7: Discretized BEM model of a inductor in MEMS
The boundary elements have been converted into
18336 first order, triangular elements for the postprocessing.
The domain in 40000 evaluation points,
which are lying in a slice through the inductor (Fig. 8),
has been detected in 106 s. The yellow color represents
the domain inside the conductor of the inductor and the
blue color represents the surrounding free space domain.
Although the slice is often intersected by boundaries of
the inductor, the number of actual domain detections is
reduced by 45 % by using domain data of the octree cubes.
is maximal 26.
Fig. 8: Detected domains in a slice through the inductor
- 391 - 15th IGTE Symposium 2012
IV. CONCLUSION
A fast and efficient automatic domain detection method
for a meshfree post-processing in three-dimensional
boundary element methods has been presented. Relevant
boundary elements are filtered with the help of an adaptive
octree-based method. Hence, the number of boundary
elements, which have to be considered for domain detection,
is extremely reduced and approximately independent
of the total number of boundary elements for large problems.
The application of the octree, bounding boxes, and
optimized standard libraries results in very low computational
costs. As a result, the shown domain detection
method enables an efficient post-processing in a large
number of evaluation points even for large and complex
BEM problems. Furthermore, the complete method including
its implementation is very flexible and supports
all types of elements. Hence, it is not only restricted to
pure BEM applications, but volume elements of a volume
integral equation can be used, too. Finally, the numerical
examples show that domain data is detected in arbitrary
chosen evaluation points reliably and efficiently.
The shown method is a very important step towards
flexible and powerful post-processing in boundary element
methods. A direct coupling of visualization tools
with a boundary element method is enabled. Postprocessing
objects as streamlines or isosurfaces can be
computed totally meshfree even in the case of multiple
domains. The octree, which is used here, is the basis of
fast and efficient field computations using the fast multipole
method, too.
[1]
REFERENCES
U. Lang and U. Wössner, “Virtual and augmented reality developments
for engineering applications”, Proceedings of
[2]
ECCOMAS 2004, Jyväskylä, July 24-28, pp. 24-8., 2004
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reality in teaching of electrodynamics”, COMPEL, vol.
28, no. 4, pp. 948-963, 2009
[3] D. Weiskopf, „GPU-Based Interactive Visualization Techniques“,
Springer, 2006
[4] S. Bachthaler, F. Sadlo, R. Weeber, S. Kantorovich, Ch. Holm,
and D. Weiskopf, “Magnetic Flux Topology of 2D Point Dipoles”,
Eurographics Conference on Visualization (EuroVis) 2012, vol.
31, no. 3, 2012
[5] A. Buchau, W. M. Rucker, O. Rain, V. Rischmüller, S. Kurz, S.
Rjasanow, “Comparison Between Different Approaches for Fast
and Efficient 3D BEM Computations”, IEEE Transactions on
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[6] W. Hafla, A. Weinläder, A. Bardakcioglu, A. Buchau, and W. M.
Rucker, “Efficient Post-Processing with the Integral Equation
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[7] A. Buchau, W. Rieger, and W. M. Rucker, “Fast Field Computations
with the Fast Multipole Method”, COMPEL, vol. 20, no. 2,
pp. 547-561, 2001
[8] A. Buchau and W. M. Rucker, “Meshfree Visualization of Field
Lines in 3D”, 14 th IGTE Symposium, pp. 172-177, Graz, 2010
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for Ray Tracing”, IEEE Computer Graphics and Applications,
vol. 7, no. 5, pp. 14-20, 1987
[10] A. Buchau, Ch. J. Huber, W. Rieger, W. M. Rucker, ”Fast BEM
Computations with the Adaptive Multilevel Fast Multipole Method”,
IEEE Transactions on Magnetics, vol. 36, no. 4, pp. 680-684,
2000
[11] “.NET Framework Developer Center”, Microsoft Corporation
[12] A. Buchau, W. Hafla, F. Groh, and W. M. Rucker, ”Parallelized
Computation of Compressed BEM Matrices on Multiprocessor
Computer Clusters”, COMPEL, vol. 24, no. 2, pp. 468-479, 2005

- 392 - 15th IGTE Symposium 2012
Efficient modeling of coil filament losses in 2D
L. Lehti∗ ,J.Keränen †∗ ,S.Suuriniemi∗ , T. Tarhasaari∗ , and L. Kettunen∗ ∗Tampere University of Technology - Electromagnetics, P.O. Box 692, FI-33101 Tampere, Finland
† VTT Technical Research Centre of Finland, P.O. Box 1300, FI-33101 Tampere, Finland
E-mail: leena.lehti@tut.fi
Abstract—Practical estimates for losses in coil filaments of a FEM model are sought for. A low-dimensional function space
is introduced on the filament-air interface and then suitably extended into the filament to significantly reduce the number
of unknowns per filament. Careful choice of extensions enables good loss estimate accuracy. The result is a system matrix
assembly block that can be used verbatim for all filaments, further reducing the cost. Both net current and voltage per
length of the filament are readily available in the problem formulation.
Index Terms—coil modeling, FEM, winding loss estimate
I. INTRODUCTION
Improving the efficiency of electrical machines is an
important aspect of machine design. Modeling methods
are required to be as fast and accurate as possible to
help the designer optimize the machines. One important
aspect of machine design are Ohmic coil loss estimates.
They enable the designer to choose the placement of the
conductors such that the losses are minimized.
Solving for the conductor losses is not a straightforward
task. The losses depend on the conductivity and
the current density. The current density is affected by
numerous factors, which cannot be separately solved for.
The different elements include the feeding current, fields
generated by neighboring conductors, placement of the
permeable materials, and—depending on the frequency—
skin effect.
Different methods have been developed to overcome
these difficulties. One method is to replace the conductivity
of the material by other parameters which
transform the eddy-current losses to hysteresis losses
[1]. In this approach the parameters for resistance and
inductance are sought for. A separate mesh for magnetic
and electric problems are introduced in [2]. The results
are acceptable, but the method has computationally slow
segments. In [3] a cell-model is used to reduce the
computational effort and to extract the resistance of
windings. A homogenization technique in [4] derives
parameters to characterize skin and proximity effects in
windings. Surface impedance methods have also been
used [5], where it is assumed that the magnetic flux does
not penetrate into the conducting material. Therefore the
conducting material can be approximated on the surface
only and the interior of the material can be ignored. This
can be used for conductors with low curvature and small
skin depth.
We aim at a good trade-off between moderate calculation
time and accuracy of loss estimates while maintaining
an explicit connection to the exterior circuit.
In addition, the conductors can be placed freely, i.e. a
periodical spacing is not required, as in [3] and [4].
Previously [6], magnetic flux was not allowed to enter the
filaments and a separable problem was achieved, i.e. the
field problem was divided into the filament interiors and
the exterior. The exterior and interior had only limited
interaction through net current and constant stream function
values on the boundary. To improve the accuracy,
it is necessary to admit some magnetic flux into the
filaments while solving for the exterior problem. Ignoring
the magnetic energy and losses inside the filaments on
the exterior problem enables subsequent solving for the
interior problem, but it leads to a very small reluctance
inside the filaments. When the filaments are close to each
other, this is detrimental.
Consequently, the filament interiors have to be coupled
with the exterior problem. To save computational effort,
the function space on the filament interface is significantly
limited. The basis representing the result inside
the filaments is spanned by solutions of eddy current
problems that use the functions from the interface as
boundary conditions. The use of these solutions improves
the loss estimates significantly compared to [6] and the
same solutions can be used for all filaments with similar
cross-section. The resulting method is called an interface
technique.
II. EXTERIOR FORMULATION
Here, we concentrate on two-dimensional cases, since
they are widely used in industry. Solving for 2D problems
is simple, and they provide enough accuracy for many industrial
applications. Since we look for a time-harmonic
solution, the materials used are assumed to be linear.
However, the method could be extended to nonlinear
time-domain problems with convolution techniques if the
material of the filaments stays linear.
A few terms are represented in Figure 1 for convenience.
Exterior problem refers to Ωe and interior
problem to Ωin = Ωj. The geometry contains K
conducting filaments, Ωj, and the relative permeability of
the core is 1000. The whole domain is Ω= Ωj ∪ Ωe
and on the boundary of Ω the stream function is set to

zero. This geometry, with K =25and f =50Hz, is
also used as an example in the computations. The radius,
r, for each filament is 0.01m to produce a challenging
modeling problem (skin depth/radius ≈ 1).
symmetry axis
∂Ωj
Ω=Ωin ∪ Ωe
Ωj
Ωe
b · n =0
Fig. 1. An example geometry for a transformer with an E-shaped core.
There are 25 conductors in the coil and their net current is set to one.
The core material’s relative permeability is 1000 and f =50Hz. Ω is
the whole domain that consists of Ωin = Ωj and Ωe.
In our previous work [6], the floating potential approach
with a constant but unknown stream function on
each filament boundary was used. This prevented the flux
from entering the filament and the eddy currents were
similar in all filaments. Now, we replace the constant
potential with a low-dimensional subspace of functions,
L, that enables the magnetic flux to enter the filament.
The function space comprises of a constant function and
trigonometric functions. Let ˆ L be the space of Whitney
nodal basis function interpolations of L. We construct the
following formulation for the exterior problem
div 1
μ grad a =0 in Ωe, (1)
a =0 on ∂Ω, (2)
∂Ωj
a =
M−1
i=0
cijfi ,fi ∈ ˆ L on ∂Ωj ∀ j, (3)
h · dl = Ij on ∂Ωj ∀ j. (4)
Here μ is the permeability, a the stream function,
M the number of functions on the interface, cij are
complex scalar coefficients, h the magnetic field, and Ij
the net current in j th filament. There can be multiple
trigonometric functions, i.e., (3) can be written
a = c0 +
N
(c2n−1 sin nα + c2n cos nα), (5)
n=1
- 393 - 15th IGTE Symposium 2012
where N is the number of trigonometric functions and
α an angle parameter. 1 For N =0this reduces to floating
potential. On the interface, other than trigonometric
functions could be used.
The solution for (1)–(4) is sought for in the form
a =
diλi +
cij ˆ fi, (6)
i
where each λi is a Whitney nodal basis function associated
to the interior nodes of Ωe, and ˆ fi are approximated
by Whitney interpolants on the boundary and the extension
of these interpolants into the domain Ωe is done
canonically.
In practice, the function space L on the interface is
reduced to ˆ L by using a projection matrix Q. InQ we
have a row for each basis function of ˆ L for each filament
and columns for all nodes on the boundary. The i th row
of Q consists of the values of fi in the boundary nodes
of one filament. In a standard system, the system matrix
is formed from blocks
i,j
AΩΩ AΩΓ
AΓΩ AΓΓ
, (7)
where Ω refers to the exterior of the filaments and Γ
refers to the filament boundary. We use Q as follows
ÃΩΩ = AΩΩ
ÃΩΓ = AΩΓQ T
ÃΓΩ = QAΓΩ
ÃΓΓ = QAΓΓQ T
(8)
(9)
(10)
(11)
to build the new system matrix à in the same format as
(7). Note that the block with most nodes, i.e. ÃΩΩ, is not
transformed. The dimensions of the projection matrix are
(KM) × (boundary nodes).
Remark 1. If the energy stored and dissipated in the
filaments is neglected the exterior problem can be independently
solved for. However, if we use equations (1)–
(4) without including the effects of the filament interiors’,
the results are not satisfactory. As an example, we have
the E-magnet from Figure 1. The model lacks the reluctance
and eddy current effects from the filament interiors,
and thus the filaments offer no reluctance to the flux. This
effect gets more pronounced when the filaments are close
to each other. In a tightly wound coil the stored energy
inside the filaments is considerable and, in addition, eddy
current losses inside the filaments have an effect on
the exterior magnetic field. In Figure 2, flux lines are
shown for an exterior solution with a constant, sine, and
cosine functions. In Figure 3 a reference result with A-
V-formulation and a finely discretized mesh in the whole
domain is shown for comparison. The A-V-formulation
provides an accurate solution, but the computational cost
is high.
1 Here sin and cos are to be understood as the Whitney nodal basis
function interpolations of the trigonometric functions.

Fig. 2. The solution to E-magnet problem of Fig.1 with a constant,
sine and cosine on the boundary. Since the interior energy is ignored,
the filaments offer a zero reluctance path for the flux and the flux gets
attracted into the filaments.
Fig. 3. The E-magnet problem solved in Ω with an A-V-formulation as
a reference result. The solution is very accurate, but the computational
cost is high.
III. INTERIOR FORMULATION
A. Expansion of the Basis Functions to the Interior
Because of the stored and dissipated energy inside
the filaments and its effect on the exterior, the interiors
cannot be separated from the exterior completely. Thus,
we need to have a model for the interior part and solve
for it simultaneously with the exterior. We take the lowdimensional
function space ˆ L on the filament interface
as the starting point.
For the interior problem we have
div 1
Vz
grad a = jωσ(a +
μ jω ) in Ωj,
(12)
h · dl = Ij on ∂Ωj ∀ j, (13)
∂Ωj
aj = ae on ∂Ωj ∀ j, (14)
where Vz the filamentwise constant potential gradient and
ae is the value of the exterior problem solution on the
- 394 - 15th IGTE Symposium 2012
filament boundary. 2 We want to represent the solution to
this eddy-current problem inside the filament by using
only the unknowns cij related to the functions of ˆ L plus
one unknown associated to the net current condition (13).
Note that computational effort is saved in the interiorexterior
coupling, since we restrict ae to be spanned by
the functions of ˆ L.
Once we have extensions of ˆ L into the filament, we
can assemble a FEM assembly block and corresponding
excitation for the problem (12)–(14). The single block
can then be efficiently used for all filaments of identical
cross section. It is important to notice that even though
we have to solve for one boundary value problem (BVP)
in the filament per basis element of ˆ L, we only have
to solve them once for filaments with the same crosssection.
Usually the number of BVPs is much smaller
than the number of similar filaments.
To produce accurate loss estimates, we choose the
extensions to be the solution to (12)–(14) with the basis
of ˆ L as boundary conditions. Hence, we decompose the
problem into M +1 separately solvable problems and
the solution to the eddy-current problem (12)–(14) is a
linear combination of these solutions. 3 These solutions
are then used to extend the basis { ˆ fj} (of (6)) inside
the filaments and to add K extra unknowns for the net
currents to the exterior problem. Note that we are not
restricted to a specific geometry, because these solutions
can be obtained by any means.
The first M problems to solve are
div 1
μ grad ai − jωσai =0 (15)
with boundary condition ai = fi on ∂Ωj, wherefi∈ˆ L.
The remaining one problem is used to account for the
filamentwise constant potential Vz with the following
div 1
μ grad aξ − jωσaξ = jωσ1 (16)
with aξ =0on ∂Ωj and Vz
jω is spanned by the constant 1.
As the term (aξ +1) equals 1 on the interface, we can use
this term to impose the net current of the filament with a
circulation of h, wherehis the magnetic field. Note that
the restriction of a0 into Ωj qualifies as (aξ +1), so that
it involves no extra cost. The solution within the filament
for the electric field (divided by jω) is expressed by the
space of linear combinations
a + Vz
jω = cξ(aξ +1)+
M
ciai, (17)
i=0
where a is the magnetic vector potential inside the
filament and cξ = Vz
jω .
2 At first glance, it might seem like the problem is overdetermined,
because (14) states a Dirichlet condition on all of the boundary.
However, we have as many extra conditions (13) as we have constants
Vz in (12), namely K.
3 This requires the material to be linear inside the filaments.

B. Assembly Block in Entire Problem
Let us see how the system matrix of the exterior
problem is modified when these extended basis functions
(ai’s and aξ’s) are used. We form an assembly block that
is added to the system matrix from the problem
div 1
μ
Vz
grad a = jωσ(a + ) in Ω, (18)
jω
because we combine the solution from the interior to the
exterior problem. The variational formulation for this is
Ω
w(div 1
μ
Vz
grad a − jωσ(a + )) dΩ =0, (19)
jω
where w is a test function. After integration by parts,
(19) becomes
grad w 1
grad a dΩ =
μ
Ω
∂Ω
w 1
grad a · n dl −
μ
Ω
wjωσ(a + Vz
) dΩ. (20)
jω
Because of homogenous Dirichlet condition for a everywhere
on ∂Ω, the boundary integrals are zero for w = ai
Ω
1
grad ai
μ grad a + jωσai(a + Vz
) dΩ =0. (21)
jω
With weight w = aξ +1, the net current condition is
imposed for each filament. Now, aξ +1 is one at the
boundary of its support Ωj, and (20) becomes
Ωj
∂Ωj
grad (aξ +1) 1
grad a+
μ
jωσ(aξ +1)(a + Vz
) dΩ =
jω
1 1
grad a · ndl = h · dl = Ij, (22)
μ
∂Ωj
where Ij is the imposed net current.
After substituting (17) into (21) and (22), we can
reduce most of the assembly block elements of (21) and
(22) to an integral on the boundary of the filament. For
solutions ai and aj of (15), consider an integral equation
Ωj
ai( div 1
μ grad aj − jωσaj) dΩ =0, (23)
which holds because aj is a solution of (15). After
integration by parts we get
∂Ωj
Ωj
1
ai
μ grad aj · n dl =
grad ai
1
μ grad aj + jωaiσaj dΩ, (24)
which occurs as a basic building block in (21) and (22).
- 395 - 15th IGTE Symposium 2012
∂Ωj Ωj
∂Ωj
Fig. 4. Extension of the basis function inside the filament with the
constant boundary condition in the E-magnet example, real part in solid
line and imaginary part in dashed line.
∂Ωj
Ωj
∂Ωj
Fig. 5. Extension of the basis function inside the filament with the
sine boundary condition in the E-magnet example, real part in solid
line and imaginary part in dashed line.
C. Example: Circular conductors
Circular conductors of radius a admit exact solutions
for (15) in closed form for a constant, sines and cosines:
a0 = J0( 1
−jωσμr)
J0( √ (25)
−jωσμa)
a2n−1 = Jn( 1
−jωσμr)
Jn( √ cos nφ (26)
−jωσμa)
a2n = Jn( 1
−jωσμr)
Jn( √ sin nφ (27)
−jωσμa)
where Jn are Bessel functions of first kind with order
n. In Figure 4, we see a cross-sectional view of the
extension of the constant function (a0) into the filament
and in Figure 5 the extension of the sine function (a2).
In Figure 6, flux lines are shown for the interface
technique with one sine and cosine with the extended
basis functions. The comparison to A-V-formulated case
(Fig. 3) shows the flux to be very similar. The figures
cannot be identical, since the function space of the interface
technique is severely restricted from the interface
function space of A-V-formulation and also the basis
functions inside the filaments differ.

Fig. 6. The flux lines obtained with the interface technique for the
E-magnet are shown. Here we have taken into account the effect of
the interior of the filaments to the exterior solution and have a good
correlation with the reference solution.
Fig. 7. Filament numbers used in Table I for the E-magnet.
IV. LOSS ESTIMATES
By using the interior solution (17), we compute in Ωj
the time average of the losses
P = 1
Re{e · j
2
∗ } da, (28)
where e = −jω(a + Vz/jω) and j = σe.
Losses for selected filaments (see Figure 7) of the
example in Fig. 1 are shown in Table I. Losses are
computed for five different filaments with the interface
technique and A-V-formulation throughout for comparison.
For the interface technique, the assembly block was
produced with (25)–(27). The loss estimates were also
computed analytically. The reference results for the A-
V-formulation were obtained with GetDP [8] to verify
our loss estimated from MATLAB R○ . The greatest error
is in filament 2, where the error is -4.0%.
In Table II some computationally relevant figures for
the reference solution and the interface technique are
shown. The mesh outside the filaments is of an equal
density in both methods. The system of equations was
solved using the backslash-operator in MATLAB R○ and
- 396 - 15th IGTE Symposium 2012
Filament A-V [mW/m] Interface
technique [mW/m]
Error %
1 1.012 1.009 0.2
2 0.7185 0.7470 -4.0
3 0.2634 0.2703 -2.6
4 0.04252 0.04200 1.2
5 0.03497 0.03480 0.5
TABLE I
LOSSES IN NUMBERED FILAMENTS WITH UNIT CURRENT AND
f =50HZ. A-VRESULTS FROM GETDP AND INTERFACE
TECHNIQUE FROM MATLAB R○ . δ/r =0.92, WHERE
δ = 2/(ωσμ) AND r THE RADIUS OF THE FILAMENT.
A-V Interface technique Difference (%)
Nodes 153 796 55 206 -64.1
DoFs 153 435 49 645 -67.6
Time [s] 3.928 0.5230 -86.7
nnz
No. of DoFs
1 280 535 367 240 -71.3
in filaments 98 525 100
TABLE II
-99.9
SOME PERFORMANCE INDICATORS FOR COMPUTATIONS.NNZ IS
THE NUMBER OF NONZERO ELEMENTS.
the number of nonzero elements (nnz) in the system
matrix is much lower than in the A-V-formulation. Most
of the saved elements are in the conducting regions and
this saves computation time. Additionally, for the A-Vformulation,
the number of nodes required to maintain
accuracy inside the filaments has to increase significantly
with increasing frequency.
V. CONCLUSION
An approach to model coil filament losses was proposed.
We expanded the function space on the filament
boundaries from the floating potential approach with
trigonometric functions. We observed that the filament
interiors need to be considered as well due to the
significant effect of the magnetic energy stored and
dissipated inside them. When interface basis functions
were extended into filaments with solutions of magnetoquasi-static
problems, and these were used as FEM basis
functions, the loss estimates are at the most 4% away
from A-V-formulated estimates. The computation time is
significantly reduced in a small problem consisting of 25
filaments.
ACKNOWLEDGEMENTS
The authors thank Professor Stefan Kurz for discussion
and comments.
REFERENCES
[1] O. Moreau, L. Popiel and J. Pages, ”Proximity Losses Computation
with a 2D Complex Permeability Modelling,” IEEE Trans. Magn.,
vol. 34, pp. 3616-3619, 1998.
[2] H. de Gersem and K. Hameyer, ”A Multiconductor Model for
Finite-Element Eddy-Current Simulation,” IEEE Trans. Magn., vol.
38, pp. 533-536, 2002.
[3] A. Podoltsev, I. Kucheryavaya and B. Lebedev, ”Analysis of
effective resistance and eddy-current losses in multiturn winding
of high-frequency magnetic components,” IEEE Trans. Magn., vol.
36, pp. 539-548, 2003.

[4] J. Gyselinck, R. Sabariego and P. Dular, ”Time-Domain homogenization
of windings in 2-D finite element models,” IEEE Trans.
Magn., vol. 43, pp. 1297-1300, 2007.
[5] T. Le-Duc, G. Meunier, O. Chadebec and J.-M. Guichon, ”A
new integral formulation for eddy current computation in thin
conductive shells,” IEEE Trans. Magn., vol. 48, pp. 427-430, 2012.
[6] L. Lehti, J. Keränen, S. Suuriniemi, and L. Kettunen, ”Subsystem
separation by flux linkage in coil filament modelling,” ACOMEN
2011, Liège.
[7] P. Dular, W. Legros, H. De Gersem, and K. Hameyer, ”Floating
potentials in various electromagnetic problems using the finite
element method,” Proc. of the 4th int. workshop on electric and
magnetic fields, 1998, Marseille.
[8] P. Dular and C. Geuzaine, GetDP: a General Environment for the
Treatment of Discrete Problems, available: http://geuz.org/getdp/
- 397 - 15th IGTE Symposium 2012

- 398 - 15th IGTE Symposium 2012
Optimization of Energy Storage Usage
Arnel Glotic 1 , Peter Kitak 1 , Igor Ticar 1 , Adnan Glotic 2
1 University of Maribor, Faculty of electrical engineering and computer science, Smetanova 17, SI-2000 Maribor,
Slovenia
2 Holding Slovenske elektrarne Group, Koprska ulica 92, SI-1000 Ljubljana, Slovenia
Abstract — Energy storage is a physical storage for energy, like Batteries, Flywheels, Compressed Air Storages, Pumped
Storages, etc. This paper presents the use of the optimization algorithm in order to achieve the optimal usage of Energy Storage.
Reservoirs of cascade Hydro Power Plants have been used as model of Energy Storage, and these are known as complex
optimization problems. Optimization algorithm used in this paper was the adapted differential evolution algorithm.
Index Terms — Differential evolution, energy storage, optimization, hydro power plants.
I. INTRODUCTION
Energy storage [1] is a physical storage for energy and
can be found in different types. Authors’ research has
been focused to cascade hydro power plants (HPP), where
each individual plant has its own reservoir and energy
storage, respectively.
Various combinations of reservoirs’ charging and
discharging produces different amount of electricity. In
order to achieve optimal production, several methods can
be implemented [2], such as Lagrangian relaxion and
Benders decomposition-based methods, Mixed-integer
programming, Dynamic programming, Evolutionary
Computing Methods, Artificial intelligence methods and
Interior-point methods.
Differential Evolution (DE) Algorithm [3] is an
efficient and robust global optimization algorithm and
therefore it has been selected in this paper as an
appropriate optimization technique.
Short-term optimization using DE with self-adaptive
parameter settings authors in [4] has been used on four
cascades HPP, where the best objective value has reached
after 2000 generations. The modified DE presented in [5]
includes penalty factor during the objective function
evaluation, which preserves the satisfied final reservoirs
levels of four cascades HPP. In [6] authors combined
advantages of the two modified DE algorithms, where the
grouping and shuffling operation is carried out over the
population periodically.
Optimization of reservoirs scheduling HPP is known as
a complex problem, where large number of HPP in
cascade, means much larger number of reservoirs
scheduling combinations and convergence time,
respectively. The main goal of this paper was to modify
DE in order to be capable of reaching the global optimal
solution with fast convergence. This means the adequate
distribution of individual HPP electrical energy
production by scheduling reservoirs in order to satisfy the
demand for 24 hours. Besides satisfying the demand, the
decreased usage of water quantity per electrical energy
unit (m 3 /MWh) has to be also achieved. Also, the
optimization results must be feasible in range of couple
minutes.
Mathematical model of cascade hydro power plants is
E-mail: arnel.glotic@uni-mb.si
described in section II, standard and modified differential
evolution algorithm in section III, results in section IV,
and conclusion in section V.
II. MATHEMATICAL MODEL OF CASCADE HYDRO POWER
PLANTS
The mathematical model describes cascade HPP on
Drava River in Slovenia, owned by Dravske elektrarne
Maribor (DEM). DEM is a subsidiary company of
Holding Slovenske elektrarne (HSE), which is the biggest
producer and trader with electricity in Slovenia. DEM
provides approximately 25.5% of produced energy in
Slovenia, with maximum output of 587 MW.
The mathematical model consists of eight cascades,
t
where i-th HPP has natural inflow Qi ,NI in the observed
hour t of the day. The first HPP in decade structure has
t
the inflow Qi,I of Drava River coming from Austria. The
source of Drava River lies in Italy, near Austrian-Italian
border.
The total inflow for the first HPP in the observed hour t
is,
t t t
Qi,TI Qi,I Qi,NI
, (1)
i 1, t 1,2,...24
t
where Qi ,TI is the sum of inflows. The total inflow for the
following seven HPP is expressed as
t t t
Qi,TI Q( i1) Qi,NI
i 2,3,...6, t 1,2,...24 , (2)
t
Q is the outflow of the upper HPP, expressed as
where ( i 1)
t t t
i i,T i,O
Q Q Q
i 1,2...8, t 1,2,...24 , (3)
which represents the sum of the flow through the turbine
and the overflow in the observed hour t. The last two
HPP’s, HPP 7 and HPP 8, are canal based type HPP’s
where flows merge with the riverbed at the end of the
canal. Both of these HPP’s have the required biological
minimum flow Q i,B
, which must be provided to the
riverbed.

t
Q1,TI
V
t
HPP1
t
Q1,O
HPP 1
dam
t 1
H V
t
Q1,T
t
Q1
V2,min
Total inflow for the last two HPP in chain is expressed as
t t t
Qi,TI Q( i1) Qi,NI Qi,B
(4)
i 7,8, t 1,2,...24 .
Inflow water can be used for charging reservoir up to the
maximal reservoir height V i,max
or used in combination
with flow gained from discharging reservoirs. However it
must be considered that in the observed hour t the
reservoirs values
t
V i must be between minimal or
maximal allowed value of the individual reservoir. All the
reservoirs also have the prescribed maximal discharging
value.
The hydro generator output power is expressed as
t
i i,1
t
i
2
i,2
t
i
2
i,3
t
i
t
i i,4
t
i
ci,5 t
Qi ci,6
P c V c Q c V Q c V
,(5)
where c represents the hydropower generation
t
coefficient. In cases where the inflow Q i,TI
is larger than
the maximal allowed flow through the turbines of the i-th
HPP and the reservoir level t
V i reaches the maximal
t
value allowed, then the overflow Q i,O
is unavoidable and
it can be expressed as
t t t t
Qi,O Qi,TI Qi
Pi,max ,
i 1,2,...8, t 1,2,...24
(6)
t t
where i i,max
Q P is the flow throughout the turbines,
which provides the maximal output power. Power
generation consider also the head effect,
t t t
Hi HVi Hi,O
i 1,2...8, t 1,2,...24
,
(7)
t
where H i is the difference between the inlet and outlet
t
H V i
t
is the level of reservoir at volume V i and
head,
t
i,O
H is the level of the outlet. Both levels are expressed
with the polynomial of the sixth degree:
- 399 - 15th IGTE Symposium 2012
V2,max
t
t
V Q
2
2,TI
Figure 1: Layout of two hydropower plants
t
Q2,O
HPP 2
dam
t 2
H V
t
Q2,T
3
t
2
t
1
t
6 5 4
t t t t
i,O i,1 i i,2 i i,3 i
H k Q k Q k Q
k Q k Q k Q k
i,4 i i,5 i i,6 i i,7
t
Q2
, (8)
where ki are the coefficients of the polynomial obtained
by experimental measurements of each reservoirs and
provided by DEM personnel.
III. OPTIMIZATION ALGORITHM
Differential evolution (DE) algorithm has been used as
effective global optimizer and was proposed by R. Storn
and K. Price [3]. The main steps of DE algorithm are
initialization, mutation, crossover, evaluation and
selection. The initialization step is defined as a randomly
chosen population. Each individual xi of the initial
population is composed of j variables:
x jG , x j,upp rand(0,1) ( x j,upp xj,low
)
(9)
j 1,2..., D
xiG
, x1, x2,... xD
(10)
i 1,... NP
where UPP x j,upp
and LOW j,low
x are upper and lower
bounds defined for each variable x j , G denotes
generation, NP number of population, D number of
parameters or problem dimension and i the number of the
population member and individual, respectively. The
population size depends on number of the problem
variables D and parameters of the objective function,
respectively.
For the proposed mathematical model the optimization
algorithm has upper and lower bounds defined as minimal
and maximal value of the individual reservoirs.
Therefore, after the initialization, the population is
composed of NP D-dimensional vectors:
1 24 1 24
xiG
, Vi,1 ,... Vi,1 ,... Vi,8 ,... V
i,8
(11)
i 1,2... NP
where V is the volume of individual HPP reservoir in
time t. At the initialization step of DE the volumes are
randomly chosen for each individual HPP and for each
individual hour in 24 hour period. Therefore the

dimension of the problem D is 192 and the population
size is five times larger. Therefore the population size is
960.
The mutation stem if followed after the initialization
step. For each target individual and sometimes referred to
as vector x iG , , the mutant vector is created according to
the selected strategy. The applied strategy in this paper is
formulated as
viG , xiG , Fxbest, GxiG , Fxr 1, Gxr2, G
,
i 1,2..., NP
(12)
where xr and x
1 r are randomly chosen individuals from
2
interval [1,NP], x best,G represents the best individual of
the generation G and F is the weight.
The following step is crossover, where for the each
mutant vector a new trial vector u iG , is produced via
“binary” crossover:
vi, j, G if rand(0,1) CR or j jrand
ui,
j, G
xi, j, G if rand(0,1) CR or j jrand
i 1,2..., NP, j 1,2,...,
D
(13)
where CR is crossover constant selected by the user. The
j rand is a randomly chosen integer from interval [1,…D],
which ensures that the trial vector obtains at least one of
the parameters from the mutant vector.
In the last step, known as selection, DE evaluates trial and
target vector, commonly referred to as parent vector:
iG , if f iG , f ,
u u xiG
xiG
, 1
xiG , if f uiG , f x (14)
iG ,
i 1,2..., NP
where the lower objective function value occupies the
position in next generation (G+1). This comparison is
made for each of NP individuals and the new population
in generation G+1 is selected and steps of DE algorithm
start once again in the following order; mutation,
crossover, evaluation and selection. The algorithm repeats
all steps until one of the stopping criterions is reached.
DE control parameters F, CR and strategy are selected
by the user and have an important influence on the
convergence time, global or local search and manner of
creating new mutants. Use of the standard DE for solving
the presented optimization problem may not always lead
towards the global solution, regardless of the effort given
in order to choose the adequate control parameters. The
algorithm can be easily trapped into local optimum and
also the convergence time can be drastically increased. In
order to overcome these problems the modified algorithm
uses self-adaptive F and CR. For the initial generation
both of the control parameter are selected by the user and
vary along with the iteration number according to (15)
and (16):
FRif f( xbest, G1) f( xbest,
G)
FiG
, 1
FiG , Otherwise.
(15)
i 1,2..., NP
If the algorithm finds a better solution in generation G
- 400 - 15th IGTE Symposium 2012
compared to generation G - 1, then a randomly selected
FR is employed in generation G + 1.
CRR if FiG , 1
FiG , and rand(0,1)
CRiG
, 1
CRiG , Otherwise.
(16)
i 1,2..., NP
A random CRR in generation G+1 is also provided if a
FR is previously employed and at the same time a
randomly selected value from interval [0, 1] is lower than
0.1 . The described modification loads toward the
global solution and improves the convergence time. A
further improvement in convergence time can be achieved
by parallel computation.
The presented optimization problem is a multiobjective
problem [7], where three different objectives
are merged into a single one by using the weighted sum
method [8]. The first goal of the optimization process is
the satisfied demand for 24 hours by scheduling
reservoirs of cascade HPP. The satisfied demand should
be followed by the decreased usage of water quantity per
3
electrical energy unit ( m MWh) which represents the
second objective. The third objective represents the
decreased and eliminated overflow, respectively. The
objective function for each individual objective is
expressed as:
2
24 8
t t
1 demand i,opt
t1 i1
1
f W W
24
8 24
t Qi,T
i1 t1
2
8 24
t Wi,opt
i1 t1
8 24
t
3 i,O
i1 t1
f
f Q
(17)
t
Demand energy Wdemand and optimal production energy
t
i,opt
W is formulated as a product of power P and time t
W PtWh (18)
The unified objective function f is defined as
f f1w1 f2w2 f3w3 (19)
where each individual objective is normalized and
weights are set according to the selected priority of the
individual objective. The values selected for a given
problem were 0.6, 0.15 and 0.25, respectively.
IV. RESULTS
The proposed modified DE algorithm has been used in
order to achieve globally optimal production of the
cascade of the HPP and to satisfy the demand,
respectively. The test data used was a real 24 hours
demand plan from SCADA. It has been shown in Table I
and it is valid for the observed day in the past and
practically realized by scheduling reservoirs.

Time
- 401 - 15th IGTE Symposium 2012
Table I: The satisfied demand by scheduling reservoirs for the dispatcher, standard and modified DE
Demand scheduling the
reservoirs by dispatcher
(real data from SCADA )
Energy
( MWh )
Satisfied demand by scheduling
reservoirs - standard DE
Satisfied demand by scheduling
reservoirs - modified DE
Water discharge Energy Water discharge Energy Water discharge
( /h ) ( MWh ) ( / h ) ( MWh ) ( / h )
1 19.8 828000 0 0 19.0 738446
2 6.8 349200 6.0 284205 7.0 218980
3 0 0 0 0 0 0
4 0 0 0 0 0 0
5 0 0 0 0 0 0
6 0 0 3.0 154185 0 0
7 89.2 2080800 39.0 1372293 89.0 2106432
8 368.6 8355600 368.1 9151060 368.6 8696690
9 417.5 10018800 416.4 10288591 417.5 10553592
10 315.5 7592400 314.7 6764925 315.5 8525579
11 244.2 5968800 244.0 5853691 244.3 5827843
12 313.3 7408800 312.7 7426771 313.3 7453704
13 322.4 7740000 321.9 7392661 322.4 7548595
14 332.7 8089200 332.4 8707231 332.6 7625784
15 320.1 7776000 319.6 7223638 320.2 8137026
16 306.7 7315200 306.4 6806393 306.7 6557976
17 403.3 9518400 403.2 9281914 403.3 9743537
18 397.2 9378000 396.7 9344417 397.1 9186977
19 391.9 9381600 391.4 9250750 391.9 9260050
20 306.5 7452000 306.6 6604082 306.5 7079875
21 290.1 7185600 289.6 6610318 290.0 7440151
22 272.3 6494400 272.2 5624925 272.3 5849220
23 265.2 6091200 265.1 6813626 265.2 5808105
24 249.0 5763600 248.6 5207242 248.9 5301717
Total 5632.2 134787600 5557.4 130162918 5631.3 133660279
The scheduling has been made by the dispatch personnel
of the DEM Company. This data has been used as a
reference followed by optimization algorithm – the DE
and the modified DE – with the objective to satisfy the
given demand by determining the optimal production of
individual HPP during the 24 hour period. According to
the results from Table I, the modified DE compared to
manual dispatch saved approximately 1.12 million
3
m of
water, which equals to approximately 50 MWh less of
potential energy used.
Authors [8] showed DE’s control parameters impact on
convergence and global optimization performance. By
using smaller F values, a local optimum can be reached
faster, while a global one can be reached by choosing
larger values. Selection of larger CR values can reduce a
convergence time. In order to improve algorithm’s
performance on a given optimization problem, a search
for suitable DE control parameters was not successful,
although the best result have been obtained by using the
following parameters F = 0.5, CR = 0.8 and strategy = 3.
However, the standard DE was not able to achieve the
global optimum and to minimize the difference between
the demand and the production down to zero,
respectively.
Authors [9] have shown the benefits of self-adaptive
parameters control. This was a solid research direction in
this paper and in order to achieve the global optimum, the
modified DE with self-adjusting F and CR has been
proposed. The modified DE (Fig. 2) stopped the
evolution process after 200 generations with the
convergence time of 280 seconds, while the standard DE
(Fig. 3) stopped after 1000 generations and 1050 seconds.
The stopping criterion in this case was 500 generations
without of any change in objective function value.
The final result at the end of the optimization process by
using the modified DE algorithm is an optimal 24 h
production of each individual HPP. Such a production
completely satisfies the demand. The optimal 24 hours
production of individual HPP is shown in Fig.4 and the
corresponding reservoir volumes during the 24 hours
period are shown in Fig. 5.

Objectives
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 20 40 60 80 100 120 140 160 180 200
Generation
Figure 2: Convergence of the unified and three individual
objective functions values by using the modified DE
Electrical Enegy Production [MWh]
450
400
350
300
250
200
150
100
50
HPP1
HPP2
HPP3
HPP4
HPP5
HPP6
HPP7
HPP8
Demand
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
Figure 4: Optimal production of individual HPP proposed
by the modified DE
Vmax
Volume [m 3 ]
HPP 1
HPP 2
HPP 3
HPP 4
HPP 5
HPP 6
HPP 7
HPP 8
Vmin
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time [h]
Figure 5: Charging and discharging of reservoirs during
the optimal production of individual HPP
V. CONCLUSION
The modified DE algorithm in this paper was capable
of solving complex optimization problem. As shown on
the presented optimization problem, the algorithm was
f
f
1
f 2
f 3
- 402 - 15th IGTE Symposium 2012
Objectives
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 100 200 300 400 500 600 700 800 900 1000
Generation
Figure 3: Convergence of the unified and three individual
objective functions values by using the standard DE
able to satisfy the demand along with the fast
convergence speed. The optimization problem was
observed in period of 24 hours and includes 8 HPP’s.
Therefore it has 192 variables that need to be identified.
Despite this fact, the algorithm ensures the exploration of
large solution space in time span of several minutes and
provides the global solution. The dispatch personnel is
unable to explore such a large solution space and the
production of individual HPP is determined on the basis
of previous experiences and therefore, non-optimally. By
using the modified DE algorithm, the dispatch personnel
obtain the guide which improves the overall production
efficiency. The algorithm can also be used to indicate
whether a given demand plan is feasible.
REFERENCES
[1] S. Vazquez, S. M. Lukic, E. Galvan, L. G. Franquelo and J. M.
Carrasco “Energy Storage System for Transport and Grid
Applications”, IEEE Transactions on industrial electronics, Vol.
57, pp. 3881-3895, December 2010.
[2] I. A. Farhat, M.E. El-Hawary, “Optimization methods applied for
solving the short-term hydrothermal coordination problem,”
Electric Power System Research, pp. 1308-1320, 2009.
[3] R. Storn, K. Price, “Differential Evolution – A simple and
efficient adaptive scheme for global optimization over continuous
spaces,” Journal of Global Optimization, pp. 341-359, 1997.
[4] X. Yuan, Y. Zhang, L. Wang, Y. Yuan, “An enhanced differential
evolution algorithm for daily optimal hydro generation
[5]
scheduling,” Computers and Mathematics with Applications, pp.
2458-2468, 2008.
L. Lakshminarasimman, S. Subramanian, “Short-term scheduling
of hydrothermal power system cascaded reservoirs by using
modified differential evolution,” IEEE Proc.-Gener. Transm.
Distrib., Vol. 153, pp. 693-700, 2006.
[6] Y. Li, J. Zuo, “Optimal Scheduling of Cascade Hydropower
System Using Grouping Differential Evolution Algorithm,”
International Conference on Computer Science and Electronic
Engineering, pp. 625-629, 2012.
[7] J. Grobler, A.P. Engelbrecht, V.S.S. Yadavalli, “Multi-objective
DE and PSO Strategies for Production Scheduling,” IEEE
Congres on Evolutionary Computation, pp. 1154-1161, 2008.
[8] R. Gamperle, S.D. Muller, P. Koumoutsakos, “A parameter Study
for Differential Evolution,” Conf. on Adances in Intelligent
System, Fuzzy Systems, pp. 293-298,2002.
[9] J. Brest, V. Zumer, M.S. Maucec, “Self-Adaptive Differential
Evolution Algorithm in Constrained Real-Parameter
Optimization,” IEEE Conges on Evolutionary Computation, pp.
215-222, 2006.
f
f
1
f 2
f 3

- 403 - 15th IGTE Symposium 2012
Adaptive Surrogate Approach for Bayesian
Inference in Inverse Problems
M. Neumayer∗ ,H.R.B.Orlande ‡ ,M.J.Colaço ‡ , D. Watzenig∗ ,G.Steiner∗ , B. Brandstätter † ,andG.S.
Dulikravich §
∗Institute of Electrical Measurement and Measurement Signal Processing, Graz University of Technology, Graz,
Austria, † Elin Motoren GmbH, Elinmotorenstrasse 1, A-8160 Preding/Weiz, Austria, ‡ Department of Mechanical
Engineering, Federal University of Rio de Janeiro, UFRJ Rio de Janeiro, RJ, Brazil, § Department of Mechanical
and Materials Engineering, Florida International University Miami, Florida, U.S.A.
E-mail: neumayer@TUGraz.at
Abstract—Bayesian inference forms a flexible and versatile solution strategy for inverse problems. Its advantage lies in the
straight forward formulation of the solution process, the ability to incorporate any existing knowledge, as well as in the
output of the method itself, which provides statistical knowledge about unknown parameters. The costs of the mentioned
benefits are often largely increased numerical efforts due to the use of sampling methods. This especially holds if the
underlying physical problem requires the solution of a partial differential equation. In this paper we present a simple, yet
versatile and effective strategy to accelerate Bayesian inference using an adaptive surrogate approach.
Index Terms—surrogate technique, adaptive, Bayesian inference, MCMC
I. INTRODUCTION
Inverse problems and parameter estimation problems
belong to the class of indirect measurement problems
where one tries to estimate a parameter vector x ∈ R N
from observations ˜ d ∈ R M [1]. They arise in many
disciplines of engineering and science. The term inverse
problem is most often associated with imaging techniques
like electrical capacitance/impedance tomography (ECT
and EIT), or computed tomography, but their mathematical
common is their inherent ill-posed nature. Parameter
estimation has not that massive association with imaging
like inverse problems, but is in many ways of even larger
importance in engineering. Such an example is given by
the determination of material parameters from ”simple”
a simple measurements setup.
Formally the physical measurement process P can be
denoted by P : x ↦→ ˜ d. Hereby ˜ d is the corrupted version
of the otherwise noise free measurements d. Formost
practical examples an additive noise model of form ˜ d =
d + v is valid, where v ∈ R M follows a certain noise
distribution described by a probability density function
(pdf). The modern model based approach to estimate x
from ˜ d maintains a model F : x ↦→ y (y ∈ R M )whichis
referred to as forward map. For most real world problems
this is a computer model solving the underlying partial
differential equations (PDEs) for P in a numerical way.
In this paper we will assume that P = F holds.
Classical deterministic inversion methods then manipulate
the vector x in order to minimize some useful norm
of the residual vector e = y − ˜ d. In addition ill-posed
problems require a regularization term for the numerical
stabilization of such an optimization problem. The single
result of the approach is referred to as point estimate,
which we will denote by xMAP (maximum a posteriori)
as the value of x which provides the smallest misfit.
A contrastable approach to solve inverse problems is
provided by the framework of Bayesian inference [2].
Rather than providing a single result, Bayesian inference
approaches provide the summary distribution π(x| ˜ d) (the
posterior distribution, MAP). Out of this any statistics,
like mean, variance, correlations, MAP estimates, etc.
about x can be computed. The cost for this gain in
information are the increased computational costs, as
the approach requires numerous evaluations of F .This
especially holds for the case that Markov chain Monte
Carlo (MCMC) methods are applied. Thus, the practicability
for the application of Bayesian methods is limited
if the evaluation of F requires computational expensive
operations like the numerical solution of PDEs.
In this paper we will present a simple and versatile
strategy to speed up Bayesian inference for inverse problems
and parameter estimation problems. The speed up
is provided by the use of an approximation or surrogate
model [3]. The paper is structured as follows. In section
II we will introduce the theory about Bayesian inversion
and the exploration of the posterior distribution by the
Metropolis Hastings (MH) algorithm. In section III we
explain an acceleration approach for the MH which is
based on the use of approximations. Finally we will
present a numerical example where we estimate thermal
material parameters from a heated slab.
II. BAYESIAN INFERENCE AND MARKOV CHAIN
MONTE CARLO
In this section we will present the framework of
Bayesian inversion for the solution of inverse problems.
Having measurements ˜ d from a measurement process P
and a model F to simulate P , the solution process is

marked by the use of Bayes law [1]
π(x| ˜ d)= π(˜ d|x)π(x)
π( ˜ ∝ π(
d)
˜ d|x)π(x). (1)
The law connects the so called likelihood function
π( ˜ d|x) and the prior π(x) to formulate the posterior
distribution π(x| ˜ d). π( ˜ d) is termed the evidence and
has the role of a normalization constant to ensure the
property
RN π(x| ˜ d)dx =1of a pdf. Hence, it can be
skipped leading to the right hand formula in equation (1).
The likelihood function π( ˜ d|x) provides the probability
measure for x causing the data ˜ d given the model and
statistical knowledge about the measurement noise. For
an additive noise model ˜ d = d + v, the likelihood is
given by π( ˜ d|x) =πv(y − ˜ d), whereyisthe output
of the forward map. For many practical problems zero
mean white Gaussian noise, i.e. v ∝N(0, Σv), where
Σv is the covariance matrix, can be assumed. Then the
likelihood function becomes
π( ˜
d|x) ∝ exp − 1
y −
2
˜ T
−1
d Σ y − ˜
d
, (2)
where Σ is set to Σv.
The prior π(x) provides a probability measure about x
being the solution. While the design of the likelihood has
to follow strict mathematical rules due to its definition
the prior provides a very flexible way to incorporate
expert knowledge about x. I.e. if we know that the ith
component of x has a lower and an upper bound the
corresponding prior is given by the uniform distribution
xi ∝U(xi,min,xi,max). For the case that a mean value
about the j-th component is known a Gaussian distribution
xj ∝N(μxj ,σxj ) can be used to express the prior
where σxj controls the deviation.
The posterior π(x| ˜ d) expresses the probability for x
being the solution given the data ˜ d, the model and the
prior. Rather than a single result, the posterior covers
all possible solutions. For a post analysis of π(x| ˜ d) one
could look at a specific realization of x and evaluate its
probability. However, as this procedure is not of big use
some meaningful point measures have become popular.
One of them is the maximum a posteriori (MAP) estimate
xMAP =argmaxx π( ˜ d|x), which is the mode of the
posterior. The other one is the conditional mean (CM)
estimate
xCM = xπ( ˜ d|x)dx, (3)
R N
which summarizes the complete distribution. It can be
easily seen, that the MAP estimate can be found by
solving an optimization problem by either maximizing
the posterior, or minimizing its logarithm. This corresponds
to classical regularized approaches except, that
the likelihood introduces statistical knowledge about the
noise. This fact results in generally higher modeling
efforts when using Bayesian methods. The CM estimate
requires the evaluation of a high dimensional integral.
An analytic solution of the integral is often not possible,
- 404 - 15th IGTE Symposium 2012
as the integral is of high dimension and also because
of the complicated interaction of the forward map. Also
standard numerical schemes like the well known Gauss
quadrature cannot be applied for such integrals, due to
the lack of knowledge about the support. The numerical
tool to solve such integrals is known as Monte Carlo
integration. Hereby a set of samples x (N) from the posterior
is generated, where the frequency of the samples
follows the target distribution. Then the CM integral can
be approximated by
xCM =
R N
xπ( ˜ d|x)dx ≈
N
i=1
x (N)
i . (4)
In the same way any other integral (also about functions
of x) can be solved. The generation of samples from
a distribution belongs to the discipline of computational
Bayesian inference and will be discussed in the following
subsection.
One important aspect about the Bayesian framework
which was not stated so far is the possibility to treat
nuisance parameters ν in the same way as the state vector
x. I.e.iftheforwardmapF is in fact a function Fν(x)
it is possible to do inference about both, x and ν in
the same natural way. This can be used if parameters
of a measurement system are unknown or provide an
uncertain factor.
A. The Metropolis Hastings (MH) Algorithm
Algorithms for practical computational Bayesian inference
are typically sampling algorithms [4]. They can be
seen as random number generators which compute independent
samples from an arbitrary target distribution. For
inverse problems the target distribution is the posterior.
On a discrete state space the frequency of certain samples
corresponds to the probability of the sample, enabling the
powerful tool of Monte Carlo integration. As can be seen
by equation (2), the evaluation of π( ˜ d|x) requires one
evaluation of the forward map F . This already indicates
the fact, that sampling methods result in generally higher
computational cost. For computational inference a class
of algorithms termed MCMC methods were developed,
as they rely on an underlying Markov chain X. TheMH
algorithm [5] is one prominent example out of this class
of methods. The algorithm works as the following:
1) Pick the current state x = Xn from the Markov
chain.
2) With proposal density q(x, x ′ ) generate a new
state x ′ .
3) Compute α = min 1, π(x′ | ˜ d)q(x ′ ,x)
π(x| ˜ d)q(x,x ′
.
)
4) With probability α accept x ′ and set Xn+1 = x ′ ,
otherwise reject x ′ and set Xn+1 = x.
Starting from the current state x of the Markov chain
(line 1) X the MH algorithm generates a proposal
candidate x ′ (line 2) using the proposal kernel q(x, x ′ ).
Then the acceptance ration α is evaluated in line 3 for the
proposal x ′ , which requires one evaluation of the forward

map. If the proposal is accepted it becomes the new state
of the Markov Chain, otherwise it gets rejected. The
rejection of proposal candidates is critical with respect
to the computational efficiency of the MH algorithm, as
a high rejection rate, leads to a large number of forward
map evaluations without generating a new state. This is
strongly affected by the proposal kernel q(x, x ′ ) which
drives the exploration of the posterior distribution.
III. ACCELERATION OF THE MH USING SURROGATES
The strategy we use to speed up the classical MH
algorithm is based on the use of an approximation or
surrogate F ∗ [6]. An approximation F ∗ has a considerable
lower runtime with respect to F but at the cost
of an approximation error e = y − y∗ . Subsequently
we introduce the likelihood function π∗ ( ˜ d|x) to indicate
the use of F ∗ . Then the delayed acceptance Metropolis
Hastings (DAMH) algorithm [7] is given by
1) Pick the current state x = Xn from the Markov
chain.
2) With proposal density q(x, x ′ ) generate a new
state x ′ .
3) Compute α = min 1, π∗ (x ′ | ˜ d)q(x ′ ,x)
π∗ (x| ˜ d)q(x,x ′
.
)
4) With probability α accept x ′ to be a proposal for
the standard MH algorithm. Otherwise set x ′ = x
and return to 2.
5) Compute β = min 1, π(x′ | ˜ d)q(x ′ ,x)
π(x| ˜ d)q(x,x ′
.
)
6) With probability β accept x ′ and Xn+1 = x ′ ,
otherwise reject x ′ and set Xn+1 = x.
As can be seen, the DAMH algorithm consists of two
nested MH algorithms (in the original MH algorithm
step 3 and 4 do not exist). The DAMH tries to gain
its advantage from a pre-evaluation of the proposal candidates
x ′ on the distribution π∗ (x ′ | ˜ d). An evaluation
of π(x ′ | ˜ d) in the inner MH is only performed if the
proposal is accepted in the outer MH. In this sense the
outer MH algorithm of the DAMH can be seen as a filter
for bad proposals or as an improved proposal generator.
It is obvious that the gain in performance gain strongly
depends on the difference between π∗ (x ′ | ˜ d) and π(x ′ | ˜ d).
An interesting point about the DAMH is the availability
of the deterministic approximation error e = y − y∗ in line 5. This knowledge can be used to improve the
algorithm by two points:
• Learn about the approximation error to adapt the
likelihood π∗ (x ′ | ˜ d).
• Adapt the approximation F ∗ to improve the quality.
The approach to incorporate knowledge about the approximation
error is referred to as enhanced error model
(EEM) [1]. Hereby the deterministic approximation error
e is treated as a random variable. Mostly a Gaussian
distribution about e is assumed, describing the error as
e ∝N(μe, Σe). Then the likelihood function π∗ (x ′ | ˜ d)
becomes π∗ ( ˜ d|x) ∝
exp − 1
y
2
∗ + μe − ˜ T
−1
d Σ y ∗ + μe − ˜
d
,
(5)
- 405 - 15th IGTE Symposium 2012
where Σ is the sum of Σv and Σe. In its original idea the
distribution N (μe, Σe) is computed using samples over
the prior π(x). However, with the availability of current
value of en in the DAMH an adaptive approximation
error model can be built by [8]
μe,n = 1
(n − 1)μe,n−1 + en , (6)
n
Ce,n = Ce,n−1 + ene T n , (7)
1
Σe,n = (n − 1)Ce,n − nμ
n − 1
e,nμ T
e,n . (8)
Due to this the likelihood π∗ ( ˜ d|x) adapts to the posterior
during the runtime, which means that no sampling of
π(x) is necessary to built N (μe, Σe) in the priming of
the solution process for the data ˜ d.
The second point addresses the possibility to use
the knowledge about e to improve the quality of the
approximation F ∗ during the runtime. A considerable
simple update is possible if F ∗ is of form y ∗ = Pxa.
Hereby xa denotes the augmented state vector, which
holds x in an adequate form, i.e. arbitrary functions
of the components of x or additional variables like the
simulation time for transient problems.
Thus, the approximation can be turned nonlinear with
respect to x but it is linear with respect to the elements of
P . This is important, as for this class of approximations
a number of update algorithms exist. In this work we use
the least mean squares (LMS) algorithm given by [9]
P n+1 = P n + γenx T a,n , (9)
where γ is a step width parameter known as adaptation
coefficient. The simpleness of the LMS update provides
almost no computational costs and helps to improve
the quality of the the approximation F ∗ for the
posterior distribution. Again the initial matrix P can
be determined by samples over the prior distribution
π(x). For this the overdetermined equation system
X aP T = Y has to be assembled and solved, where the
matrix X a holds the augmented state vectors from the
samples, and Y contains the exact solutions evaluated
by F . There is also the possibility to run the standard
MH for some time to learn about P and then switch
to the DAMH. The choice of the adaptation parameter
γ affects the learning speed of the LMS algorithm. For
stability reasons γ has an upper limit which depends on
the problem and can only be derived under restrictive
conditions. However, as an MCMC algorithm provides a
enormous number of evaluations it is less critical to set
μ to a small value, as even this provides an improvement
(although slower) to the approximation P and the LMS
algorithm operates in a stable state.
To use both, the update of the approximation y ∗ =
Pxa and the adaptive error model, the approximation
should be reevaluated for the current state vector x.
This requires a second evaluation of F ∗ , but this is
computational cheap due to the design of F ∗ .

IV. A NUMERICAL EXAMPLE
To demonstrate our approach on a numerical example
we consider an indirect measurement problem where we
want to estimate thermophysical properties of a slab
from a transient heat transfer experiment. We consider
a slab of length L which we model by means of a
1D simulation in the domain Ω : 0 ≤ x ≤ L. The
slab is initially at the uniform temperature ϑ0. Onthe
left side (x = 0) a uniform heat flux J is applied
by an electric heater. On the right side at x = L the
temperature ϑ(L, t) is measured over time. The heat on
this side is exchanged by convection with the surrounding
media at the temperature ϑ0. This exchange depends on
a heat transfer coefficient α in Wm −2 K −1 .Thereareno
heat sources within the medium and the thermophysical
properties are supposed constant in the first assumption.
The mathematical formulation for this heat conduction
problem is given by:
1 dϑ
k dt = ∂2ϑ ∂x2 −λ
in 0 0 (11)
∂ϑ
∂x + αϑ = αϑ0 at x = L, fort>0 (12)
ϑ = ϑ0 for t =0,in0

σ e
- 407 - 15th IGTE Symposium 2012
TABLE I
SUMMARY OF THE RESULTS FOR THE LINEAR CASE.
Nr. Experiment σv
K
μλ
W
mK
σλ
W
mK
μα
W
m
σα μk1
σk1
Tsim,r
2K W
m2K Ω
m2 Ω
m2 true 0.12 11 4.5 × 10
%
−3
1 F 0.1 0.12 6.6 × 10−3 11.4 0.68 4.6 × 10−3 2.3 × 10−4 100
2 F ∗ 1 0.1 0.13 6.3 × 10−3 10.7 0.59 4.3 × 10−3 1.7 × 10−4 3 F
75
∗ 2 0.1 0.12 8.4 × 10−3 10.2 0.23 4.3 × 10−3 6.0 × 10−4 4 F
12
∗ 3 0.1 0.12 4.3 × 10−3 12.1 0.29 4.6 × 10−3 2.3 × 10−5 5 F 0.5 0.13 7.0 × 10
12
−3 12.7 1.72 4.7 × 10−3 4.2 × 10−4 100
6 F ∗ 1 0.5 0.13 5.9 × 10−3 9.8 1.42 4.2 × 10−3 3.5 × 10−4 7 F
110
∗ 2 0.5 0.13 3.1 × 10−3 10.5 1.10 4.4 × 10−3 2.9 × 10−4 8 F
44
∗ 3 0.5 0.13 3.3 × 10−3 10.5 1.05 4.4 × 10−3 2.6 × 10−4 44
TABLE II
SUMMARY OF THE RESULTS FOR THE NONLINEAR CASE.
Nr. Experiment σv
K
μλ
W
mK
σλ
W
mK
μα
W
m
σα μk1
σk1
μk2
σk2
Tsim,rel
2K W
m2K Ω
m2 Ω
m2 W
mK2 W
mK2 true 0.12 11 4.5 × 10
%
−3 0.02
1 F 0.1 0.123 5.5 × 10−3 12.0 0.31 4.8 × 10−3 1.1 × 10−4 0.013 4.6 × 10−3 100
2 F 0.05 0.127 3.9 × 10−3 11.3 0.19 4.6 × 10−3 7.1 × 10−5 0.015 3 × 10−3 100
3 F 0.5 0.130 9.8 × 10−3 13.4 1.42 5.2 × 10−3 4.3 × 10−4 0.018 11 × 10−3 100
4 F ∗ 1 0.1 0.129 4.4 × 10−3 10.7 0.27 4.4 × 10−3 1.1 × 10−4 0.016 5.6 × 10−3 5 F
97.6
∗ 1 0.5 0.128 5.8 × 10−3 11.9 1.24 4.8 × 10−3 3.7 × 10−4 0.012 9.4 × 10−3 6 F
140
∗ 3 0.1 0.122 3.1 × 10−3 11.8 0.22 4.7 × 10−3 8.1 × 10−5 0.019 4.6 × 10−3 7 F
20.9
∗ 3 0.5 0.128 5 × 10−3 10.8 1.29 4.6 × 10−3 4.2 × 10−4 0.017 9.4 × 10−3 72.5
0.12
0.1
0.08
0.06
0.04
0.02
0
20
15
10
dt (s)
5
10
8
6
# FE
(a) Standard deviation of e over
the discretization for F ∗ 1 .
4
2
18000
16000
14000
12000
10000
Fig. 1. Approximation error e for F ∗ 1 and F ∗ 2
8000
6000
4000
2000
0
−1.5 −1 −0.5 0 0.5 1 1.5
e
(b) Distribution (pdf) of e for F ∗ 2 .
over the prior.
how to analyze the results we refer to [4]. For
the simulation we used the following parameters:
L = 0.1 m, ρ = 1040 kgm −3 , c = 1350 Jkg −1 K −1 .
The ambient temperature ϑ0 was set to ϑ0 = 20 ◦ C,
the electrical current I was set to I = 100 A. We
assumed that ϑ(L, t) is measured every 20 seconds for
3000 s. The priors for the state vector are given by a
Gaussian distribution with μλ = 0.13 Wm −1 K −1 and
σλ =0.01 Wm −1 K −1 for λ and uniform distributions
with the boundaries 1Wm −2 K −1 ≤ α ≤ 15 Wm −2 K −1 ,
0.001 Ωm −2 ≤ k1 ≤ 0.01 Ωm −2 , and
0Wm −1 K −2 ≤ k2 ≤ 0.04 Wm −1 K −2 (nonlinear case)
for the remaining variables. The proposal generation is
done by randomly selecting a component of the state
vector x. Forλ the proposal is generated from the prior
about λ. Forα and k1 an additive Gaussian distributed
random variable with a standard deviation being 4% of
the range given by the prior is added to the current state.
In the nonlinear case we only use the approximations
F ∗ 1 and F ∗ 3 .
Figure 2 depicts the output of the Markov chain for
TABLE III
BEHAVIOR OF THE CHAINS FOR THE LINEAR CASE.
Nr. Experiment σv Acα Ac β|α Acβ τIACT
K % % %
1 F 0.1 16.4 X X 392
2 F ∗ 1 0.1 18.0 75.8 13.7 1620
3 F ∗ 2 0.1 10.3 65.8 6.8 420
4 F ∗ 3 0.1 10.2 68.0 6.9 228
5 F 0.5 56.3 X X 263
6 F ∗ 1 0.5 54.4 82.9 45.1 110
7 F ∗ 2 0.5 38.8 78.6 30.5 44
8 F ∗ 3 0.5 38.2 78.6 29.8 37
TABLE IV
BEHAVIOR OF THE CHAINS FOR THE NONLINEAR CASE.
Nr. Experiment σv Acα Ac β|α Acβ τIACT
K % % %
1 F 0.1 30.5 X X 57
2 F 0.05 17.2 X X 63
3 F 0.5 67.9 X X 132
4 F ∗ 1 0.1 33.4 79.8 26.6 38
5 F ∗ 1 0.5 64.6 87.1 56.3 72
6 F ∗ 3 0.1 18.8 70.9 13.4 509
7 F ∗ 3 0.5 56.0 86.1 48.3 121
λ. From the histogram in figure 2(b) we can see the
distribution. Table I and II summarize the results for the
linear and the nonlinear case including the true values for
the state vector x for different standard deviations σv of
the additive measurement noise. As it can be observed, all
estimates meet the true values with reasonable accuracy.
This especially holds for the linear case. For the nonlinear
case some deviations occur, but that can be linked to
the complexity of the problem. An interesting effect can
be seen in the standard deviations of table I. Increased
noise levels have a stronger effect on σk1 with respect to
the other standard deviations. In the linear case a speed
improvement of up to a factor of 10 for the linear, and 5

λ (Wm −1 K −1 )
0.16
0.15
0.14
0.13
0.12
0.11
0 1 2 3 4 5 6
x 10 4
0.1
# MCMC
(a) MCMC output for λ.
Fig. 2. MCMC output and analysis for λ.
1000
900
800
700
600
500
400
300
200
100
0
0.1 0.11 0.12 0.13
λ (Wm
0.14 0.15 0.16
−1 K −1 )
(b) Histogram plot for λ.
for the nonlinear case could be achieved. The speed up
also depends on the noise level, as the noise has direct
influence on π(x| ˜ d) and thus affects the proposal kernel
is important. Table III and IV provide statistics about the
behavior of the algorithms. For the MH, the ratio Acα
states the percentage of accepted proposal candidates. For
the DAMH it states the ratio of accepted proposals in
the first step and Acβ states the overall acceptance in
the second step. The value Ac β|α states the acceptance
in of a proposal in the second step, given an acceptance
in the first step. Hence, this number provides a quality
measure for the approximation F ∗ . The almost same
level of Ac β|α in line 3 and 4, and line 7 and 8 in
table III indicates, that the approximation already has
a high quality, and that no further improvement could
be achieved by the adaption. This corresponds to the
observations of figure 1. The tables III and IV also
explain the larger computation times for some DAMH
variants with respect to the MH. In this case a high value
of Acα results in a large number of evaluations of F .
Thus, the sum of all evaluations of F and F ∗ increases
the pure evaluation of F only. This also indicates, that
the proposal kernel is yet not optimal for sampling from
the posterior. The value τIACT is referred to as integrated
auto correlation time (IACT). It was computed with the
methods explained in [10] and provide a measure about
the statistical efficiency of an MCMC algorithm by the
distance between independent samples in the Markov
chain. As can be seen, the DAMH variants have a lower
IACT τIACT and thus are statistically more efficient.
In section IV we stated the linear dependence of the
parameters due to the equations (11) and (12) and that
this circumstance can be observed in the results. Figure
3 depicts a scatter plot for α and k1. The correlation
factor can be computed with 0.98. From this we can
conclude, that the measurement system is inappropriate
and a second spatial distributed measurement would be
required. This is a direct conclusion from the Bayesian
analysis - an optimization based solution is not able to
provide such inside information.
VI. CONCLUSION
In this work a general approach for accelerating
Bayesian inference for indirect measurement problems
is presented. The approach features the use of simple
approximations by incorporating error knowledge and
- 408 - 15th IGTE Symposium 2012
k 1 (Ωm −2 )
4.9
4.8
4.7
4.6
4.5
4.4
4.3
4.2
4.1
x 10−3
5
4
9.5 10 10.5 11 11.5 12 12.5 13
α (Wm −2 K −1 )
Fig. 3. The scatter plot of α and k1 indicates indicates a strong
correlation between the variables. This indicates, that more spatial
distributed measurements are required.
can even be used to update approximation models during
the runtime. The presented framework can be easily
applied to different problem types, e.g. electrical capacitance
tomography, to perform Bayesian inference for the
solution of indirect measurement problems.
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4B3NPMC-3/2/94bd1b60aba9b7a9ea69ac39d7372fc5

A
Aleksić, Slavoljub, 73, 300
Alotto, Piergiorgio, 267, 374
Anastasiadis, Ioannis, 271
Andjelic, Zoran, 167
Arkkio, Antero, 214
B
Balabozov, Iosko, 59
Bardi, Istvan, 1
Bauernfeind, Thomas, 327, 337
Bavastro, Davide, 101
Belahcen, Anouar, 214
Bellwald, Lukas, 271
Benabou, Abdelkader, 95
Besser, Bruno, 7
Bielby, Steven, 248
Bilicz, Sandor, 346
Bíró, Oszkár, 31, 41, 144, 190, 232, 327,
337
Brandstätter, Bernhard, 67, 403
Brochet, Pascal, 78
Buchau, André, 89, 386
Buchinger, Andreas, 271
Burgard, Stefan, 13
C
Calvano, Flavio, 208
Campana, Luca Giovanni, 171
Canova, Aldo, 101
Cardoso Bora, Teodoro, 267
Chiariello, Andrea Gaetano, 357
Ciric, Ioan R., 352
Clénet, Stéphane, 95
Coenen, Isabel, 198, 305
Colaco, Marcello J., 403
Cvetkovic, Nenad, 294
D
Dal Mut, Giorgio, 208
Dessoude, Maxime, 78
Di Barba, Paolo, 171
Diwoky, Franz, 232
dos Santos Coelho, Leandro, 267
Duca, Anton, 262
Düzgün, Bilal, 154
Dughiero, Fabrizio, 171
Dulikravich, George S., 403
Dyczij-Edlinger, Romanus, 13, 19
E
Ebrahimi, Bashir Mahdi, 125, 131, 315
Eidenberger, Norbert, 186
Elistratova, Vera, 78
Ellermann, Katrin, 144
- 409 - 15th IGTE Symposium 2012
Author Index
Ertl, Michael, 181, 226
F
Faiz, Jawad, 125, 131, 220, 315
Farle, Ortwin, 13, 19
Farnleitner, Ernst, 31, 41
Ferraioli, Fabrizio, 208
Figueiredo, William, 175
Fonteyn, Katarzyna, 214
Formisano, Alessandro, 108, 208, 357
Fornieles, Jesús, 7
Fujita, Yoshihisa, 53
Fujiwara, Koji, 113
Fulmek, Paul, 331
G
Gavrila, Horia, 352
Gergely, Koczka, 337
Ghorbanian, Vahid, 125
Giaccone, Luca, 101
Gigov, Georgi, 63
Gjonaj, Erion, 204
Glotic, Adnan, 398
Glotic, Arnel, 238, 398
Göhner, Peter, 89
Guarnieri, Massimo, 374
Gueorgiev, Vultchan, 59
Guimaraes, Frederico, 160
Gyimóthy, Szabolcs, 242, 346
H
Hameyer, Kay, 198, 305
Handgruber, Paul, 190
Hantila, Florea I., 352
Hauck, Andreas, 226
Hecquet, Michel, 78
Herold, Thomas, 198
Hinov, Krastio, 59
I
Iatcheva, Ilona, 63
Igarashi, Hajime, 276, 340
Ikuno, Soichiro, 47, 53
Ilić, Saša, 73, 300
Iovine, Renato, 25
Itoh, Taku, 47, 53
J
Janousek, Ladislav, 262
Jorks, Hai Van, 204
Jüttner, Matthias, 89, 386
K
Kaimori, Hiroyuki, 84
Kaltenbacher, Manfred, 181, 226
Kamitani, Atsushi, 47, 53

Karastoyanov, Dimitar, 59
Kastner, Gebhard, 31, 41
Katsumi, Ryuichi, 242
Keränen, Janne, 392
Kettunen, Lauri, 392
Kiss, Péter, 242
Kitak, Peter, 238, 398
Klomberg, Stephan, 41
Koczka, Gergely, 327
Kömürgöz, Güven, 154
Kotlan, Vaclav, 321
Kouhia, Reijo, 214
Kraiger, Markus, 310
Krstic, Dejan, 294
Kunov, Georgi, 63
L
La Spada, Luigi, 25
Lambert, Nancy, 1
Lehti, Leena, 392
Li, Min, 160
Lichtenegger, Herbert I. M., 7
Lowther, David, 160, 175, 248, 254
M
Magele, Christian, 37
Mair, Mathias, 144
Manca, Michele, 101
Maricaru, Mihai, 352
Marignetti, Fabrizio, 208
Martone, Raffaele, 108, 208, 357
Metzker, Isabela, 175
Miyagi, Daisuke, 84
Moghnieh, Hussein, 254
Mohr, Martin, 232
Moro, Federico, 374
N
Nagano, Takumi, 288
Nakata, Susumu, 53
Nandi, Subhasis, 315
Neumayer, Markus, 67, 403
O
Offermann, Peter, 305
Ofner, Georg, 190
Ojaghi, Mansour, 220
Okamoto, Yoshifumi, 113, 282, 288
Orlande, Helcio R.B., 403
P
Pávó, József, 242, 346
Perić, Mirjana, 73
Petersson, Rickard, 1
Piantsop Mboo, Christelle, 198
Portí, Jorge, 7
Preda, Gabriel, 262
Preis, Kurt, 271, 327, 337
- 410 - 15th IGTE Symposium 2012
R
Raicevic, Nebojsa, 73
Rainer, Siegfried, 144
Ramarotafika, Rindra, 95
Ramirez, Jaime, 160, 175
Rasilo, Paavo, 214
Rauscher, Michael, 89
Rebican, Mihai, 262
Recheis, Manes, 331
Renhart, Werner, 37
Rossi, Carlo Riccardo, 171
Rubesa, Jelena, 380
Rubinacci, Guglielmo, 208
Rucker, Wolfgang M., 89, 386
Ruela, Andre, 160
S
Sabouri, Mahdi, 220
Sadovic, Salih, 167
Salinas, Alfonso, 7
Santos, Rafael, 175
Sato, Shuji, 113, 282
Sato, Yuki, 340
Scharrer, Matthias, 368
Schnizer, Bernhard, 310
Schöberl, Joachim, 226
Schrittwieser, Maximilian, 31
Schweighofer, Bernhard, 331
Shimoyama, Kouske, 84
Sieni, Elisabetta, 171
Silva, Elizabeth, 175
Silvestro, John, 1
Simioli, Marco, 101
Smetana, Milan, 262
Sommer, Alexander, 19
Sonmez, Oluş, 154
Stancheva, Rumena, 63
Steiner, Gerald, 67, 403
Stella, Andrea, 374
Stermecki, Andrej, 190, 232
Štih, Željko, 137
Stojanovic, Miodrag, 294
Strapacova, Tatiana, 262
Suhr, Bettina, 368, 380
Suuriniemi, Saku, 392
Szabo, Zsolt, 119
T
Takahashi, Norio, 84
Takbash, Amir Masoud, 131, 315
Tamburrino, Antonello, 208
Tarhasaari, Timo, 392
Ticar, Igor, 238, 398
Toledo-Redondo, Sergio, 7
Toratani, Tomoaki, 242
Trkulja, Bojan, 137
Tsuburaya, Tomonori, 113
Tuerk, Christian, 37

U
Ulrych, Bohus, 321
V
Vale, Joao Francisco, 175
Varga, Gábor, 242
Vasilescu, George-Marian, 352
Vegni, Lucio, 25
Ventre, Salvatore, 208
Vizireanu, Darius, 78
Volk, Adrian, 181
Volkwein, Stefan, 362
Voracek, Lukas, 321
Vuckovic, Ana, 300
Vuckovic, Dragan, 294
W
Wakao, Shinji, 288
Watanabe, Yuta, 276
Watzenig, Daniel, 67, 368, 403
Wegleiter, Hannes, 331
Weiland, Thomas, 204
Weilharter, Bernhard, 144
Werth, Tobias, 271
Wesche, Andrea, 362
Y
Yasukawa, Shogo, 288
Yatchev, Ivan, 59
Z
Zagar, Bernhard G., 186
Zhao, Kezhong, 1
Župan, Tomislav, 137
- 411 - 15th IGTE Symposium 2012