2011 EMC Directory & Design Guide - Interference Technology

shielding / cables & connectors
Num e ric a l S o l u t i o n of C o mpl e x EMC Probl e m s
Entry Analytical [pF/m] FEM Solver [pF/m] Relative Error [%]
C11 14.9395 14.9718 0.22
C12 -6.0894 -5.5062 9.58
C22 18.8111 18.7565 0.29
C13 -1.6117 -1.4734 8.58
C23 -6.0894 -5.5069 9.56
C33 14.9395 14.9710 0.21
Figure 4. Selection of some of the cable types supported in FEKO.
Figure 5. Parameters for the computation of the per unit length
inductances of widely separated wires above a ground plane.
Table 1. Transmission line per unit length capacitance matrix entries
for three widely spaced conductors above ground: Comparison of
analytical values with the numerical static FEM solution.
the cross sectional dimension information about a specific
line is contained in these parameters.
Under the fundamental transverse electromagnetic field
structure assumption, the per unit length parameters of
inductance, capacitance, and conductance are determined
as a static solution to Laplace’s equation 2 (x,y)=0 in the
two-dimensional cross sectional (x,y) plane of the line.
The determination of the per unit length parameters can
be simple or very difficult depending on whether the cross
sections of the conductors are circular or rectangular, and
whether the conductors are surrounded by a homogeneous
or an inhomogeneous dielectric medium.
In fact, there are very few transmission lines for which
the cross sectional fields can be solved analytically to give
simple formulas for the per unit length parameters. To illustrate,
the self-inductance and mutual inductance terms
of n widely spaced cores above an infinite ground (see Figure
5) can be derived as [9]
Figure 6. Field excitation of a transmission line using only voltage
sources (Agrawal method).
where x denotes the longitudinal direction and parameters
Z and Y are the per unit length impedance and admittance
parameters of the line
Z = jωL + R
Y = jωC + G.
As a generalization to the two-conductor system, a multiconductor
transmission line model is simply a distributed
parameter network for an arbitrary cable cross section (see
Figure 4) where the voltages and currents can vary in magnitude
and in phase over its length.
Per Unit Length Cable Parameters
The per unit length parameters of inductance, capacitance,
resistance and conductance are essential ingredients in the
determination of transmission line voltages and currents
from the solution of the transmission line equations. All of
where r wi
is the wire radius, h i
is the wire height above
ground and s ij
is the center to center spacing between wires.
If however these cores are surrounded by an insulation
medium or the separation is not wide, one has no alternative
but to employ approximate, numerical methods. In
FEKO a 2D static FEM solver to Laplace’s equation is used.
Table 1 compares the analytic to the numerical per unit
length capacitance matrix entries for 3 wires above ground
(r w1
=r w3
=1.0 mm, r w2
= 1.5 mm, h 1
=h 3
=52 mm, h 2
=50 mm,
s 12
=s 23
=15.13 mm, s 13
=30 mm. The 2D FEM solution is based
on minimizing the stored field energy per unit length. The
self-capacitance matrix entries are a direct function of the
total energy in the system, and hence these terms agree
very well with those of the analytic prediction. The mutual
capacitance matrix entries are derived from the FEM solution
(integration over -) and as such use a lower order approximation,
also explaining the larger differences between
the analytic and numerical solutions.
82 interference technology emc Directory & design guide 2011