Online proceedings - EDA Publishing Association

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Check applicability
Analytic / FE- modelling
Check sensitivity
Check orthogonality
Development of test structures
Characterization
Fine grid of measurement points
Selection of frequency modes for
identification
Parametet identification & validation
Adaption of FE model
Wafer-Test
Fig. 2: Phases of the parameter identification
The measurement time of a one point measurement is 2
seconds. The measurement respectively software system is
not yet optimized, the lower measurement time limit given
by physics is about 200 milliseconds.
Precondition for the identification is on one hand the
measurement unit which delivers a FRF, and the simulation
unit with a parameter matrix as result on the other hand. The
automatic identification is done by a tool implemented in
C++ with respect to a fast data processing. The
identification tool can be structured into three submodules.
The frequency values has to be extracted from the measured
FRF which is done in one submodule, and the parameter
matrix is approximated by usually polynomials in another
submodule due to a fast and efficient data handling. Based
on an user defined accuracy (default value 0.1%) the degree
of the polynomial is selected by the program.
Finally the optimization respectively identification is
realized by the nonlinear least square method.
Measurement
system
Frequency
response
Peak detection
Identification tool
Optimization
FE-Simulation
Polynomial
approximation
11-13
May 2011, Aix-en-Provence, France
first step a conventional algorithm searches for local maxima
considering the estimated signal-to-noise ratio (SNR). At the
peaks found, starting values for a nonlinear least square fit to
the Lorentzian function
Parameter
matrix
i
2
, ih
LfL
, ip 2 2
,
)(
+−
, i hi
)(
f
= (1)
fff
with the peak amplitude L p,i , the peak frequency f p,i and the
half-width f h,i. of the ith peak are estimated. The iterative
fitting procedure based on Levenberg-Marquardt algorithm
eliminates wrongly preselected peaks and delivers the peak
parameter including the quality factor.
A. FE Modeling and Simulation
The FE model which delivers the parameter matrices is
implemented in Ansys. The ratio thickness to lateral
dimension of the membrane leads to a modeling by twodimensional
shell elements. The default mesh of the
membrane perforated by several thousand holes will be
irregular. To prevent such an inefficient irregular mesh
substructures are generated. Square areas with a centered
hole permit a regular meshing.
Fig. 4: FE modell with prestructured membrane elements
A prestressed modal analysis as well as a prestressed
harmonic analysis is performed. The multitude of small
structures causes a large number of finite elements
respectively nodes. With regard to the measurement time the
membrane symmetry is used by the calculation of a quarter
model. Symmetric boundary conditions are applied to the
static analysis. The modal analysis is executed with three
load steps with different symmetry conditions at the x and y
axes (symmetric/symmetric, asymmetric/symm. and
asym./asym.) to deliver all modal frequencies .
For the modeling of the squeeze film damping the
corresponding element types of Ansys are used. The macro
RMFLVEC.MAC which extracts the damping parameters
from the modal frequencies is adapted to the quarter model
with the multiple loadsteps.
Sensor parameter
Fig. 3: Structure of the parameter identification
From the measured FRF, the peak frequency values are
extracted automatically by a two level algorithm. Within a
a) f 11 b) f 12
Fig. 5: Simulated modal frequencies
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