Post on 08-Feb-2021
www.cst.com | CST UGM 2010 | April 10 | 1
Optimization of Components using
Sensitivity and Yield Information
Franz Hirtenfelder, CST AG
franz.hirtenfelder@cst.com
www.cst.com | CST UGM 2010 | April 10 | 2
Abstract
Optimization is a typical component of any design engineer’s workflow. With the introduction of
new global (genetic, particle swarm) and local (Nelder-Mead) optimizers, CST addressed this issue
with release 2009 of CST STUDIO SUITE™. With version 2010 CST goes one step further: CST MWS
2010 features a sensitivity analysis algorithm which is capable of evaluating the s-parameter
dependencies on various model parameters after a single 3D electromagnetic simulation run.
This means that all further evaluations for different model parameter sets and optimization runs
based on sensitivity data can be derived without restarting the full-wave simulation.
The efficiency of this new sensitivity analysis approach now makes Monte-Carlo based yield analysis
feasible even for complex multi-parametrical three-dimensional structures. This is due to the s-
parameter results being available at virtually no additional effort or computational cost.
www.cst.com | CST UGM 2010 | April 10 | 3
Optimization: a typical task in a workflow CST Studio Suite 2009
Global (Genetic, Swarm)
Local (Nelder-Mead, Quasi-newton)
CST Studio Suite 2010 Sensitivity
Yield
How can sensitivity data be used in an optimization?
Introduction
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Introduction to Sensitivity
Matrix to solve: QEK
[K]: symmetric, complex, contains geometry, material, frequency
xi, yi
Example: Linear Shape functions for a 2D element in xy
jkijkkjijkkjkjijk
kji
kji
kji
xxcyybxyyxa
ccc
bbb
aaa
yxN ;;;.,,12
1
xj, yj
xk, yk
k
j
i
kji
z
z
z
NNNz ],,[
zi
[E]: unkowns z
[Q]: Sources
kjinmdxdyx
N
x
N
x
N
x
Nk nmy
nm
xy
xnm ,,,;)(,
Example: electrostatic
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),(),(),(1
),(0
pEpKpEj
pS TT
Introduction to Sensitivity
S-Parameters:
[K] …left hand side, E (Fields at ports, p… any parameter
Ep
KE
p
Sj T0
Sensitivity of S-parameter vs. parameter change:
Direct analytical derivation of
K-matrix elements via e.g. [N]
3D Fieldsolution
Same 3D Fieldsolution
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Numerical calculation of gradients is expensive and unstable
Here: Sensitivity of S-parameter vs. parameter change
no additional 3D solution required (only another S-Parameter computation)
Very efficient computation of sensitivities
Result: S-parameter ranges for tolerant parameters
Currently available for FD-Tet solver
Introduction to Sensitivity
Ep
KE
p
Sj T0
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pp
SSS
p
MWSDnmnm .)3(
What is it good for?
The sensitivity helps estimate „new“ S-parameters due to the (small) change of the
parameter, at no extra cost
Suppose the parameter p changes by a quantity ∆p :
exact computation of the Sensitivity
(Approximated by 1st order Taylor expansion)
Introduction to Sensitivity
The various sensitivities are used in an optimizer to solve for ∆p as variables to
best fit the S-parameter goals.
∆p … face constraints
pp
SxSpxS
p
.)()(
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For every product, there are:
Technical specifications
Fabrication tolerances
The fabrication tolerances will lead to
some products not fulfilling the
specifications
Yield:
What is the Yield Analysis
Total
Passedyield
#
#
Gaussian
Uniform
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How is yield calculated typically?
Parameters vary according to a known probability curve
Repeat
Change the value of all parameters
Simulate
Check if specification (in our case for S-params.) is met
Until the number of simulations is statistically relevant
This is a large number of EM simulations - typicaly hundreds or thousands!!!
Knowing the sensitivity, there is no need to perform 3D simulations, at least if the parameters vary in a small range.
The efficiency of this new sensitivity analysis approach makes Monte-Carlo based yield analysis feasible even for complex multi-parametrical three-dimensional structures
Typical Approach vs. CST Approach
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Example 1: 2-Post Bandpass Filter
Rel Bandwidth:0.9 %
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Definition of Face Constraints Ivariation of resonator‘s lengths
+
-
+
-
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Definition of Face Constraints IIvariation of input coupling lengths
+-
+ -
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Sensitivity of facedistances
|S| linear > S1,1 arg(S) > S1,1
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Optimization using Sensitivity Data
A dummy model is defined using two
parameters P1 and P2,
ResultsTemplates are defined to import
sensitivity and S11 data from the 3D model.
compute S11 according to the equation
psensSSp
pMWSD ._3_1111
Dummy
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PostProcessing Templates
psensSSp
pMWSD ._3_1111
Optimization using Sensitivity Data
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Perform e.g. Parameter Sweeps
Shift of the input coupling
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Perform e.g. Parameter Sweeps
Shift the resonator‘s lengths+ -
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Perform Optimizations : 0D Results
New desired band: 570-575 MHz
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Perform Optimizations : Simplex
Inital parameters
Goal
0
---- Initial
-----optimized
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Check Results of the modified 3D Model
+p2
+p1
Found parameters p1
and p2 are used to
modify the new 3D
model geometry.
Frequency band has
been shifted ok.
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Second Iteration
The sensitivity data is fed back again into the optimizer model to run a 2nd loop:
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Results of the 2nd opt. loop
Simplex optimizer
3D
Opt.results
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3rd Iteration
Centering f-band
Opt.results
3D ---- Initial
-----optimized
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S-Parameter Results: Comparison
Several iteration steps are required since the linear sensitivity
approach is applied to a nonlinear system (3D-filtermodel)
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Yield-Specifications: < -25dBIn the range .569 to .574, S11 is under -25dB, the 3-sigma Lines are partially above and
below -25dB
For this spec, the yield will tell us what percentage of the devices are within this limit of
< -25 dB for the given frequency band.
Yield Analysis
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Another yield specification: S11 < -26dB
Yield Result: only 0.71%
Yield Analysis
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Example 2: 7-Post Filter *)
Center frequency: 7.55 GHz
Bandwidth: 100 MHz
Rel Bandwidth: 1.3 %
Insertion Loss: -0.01 dB
Return Loss: – 26 dB
Best results obtained by Group delay Tuning
*) By courtesy of MESL Microwave Limited, Edinburgh
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Method of InvChirpZ
Comparison of the InvChirpZ-resonse between
groupdelayed tuned filter and Theory
Resonator3
and 4
Note the high sensitivity
at resonator 3 and 4
(parameter ph3 and ph4
respectively)
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Final results by individually tuned posts
Method of InvChirpZ: final results
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Opimization using Sensitivity Data I
Results of the lin. optimization
Note high magnitude!
---- Initial
-----optimized Note small dimensional changes! (µm)
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Modifying the geometry according to the found
Delta-Ps of the lin. Opt. system:
Result of the new 3D simulation
Opimization using Sensitivity Data I
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Perform a 2nd optimisation loop
Result of new δP
Opimization using Sensitivity Data II
---- Initial
-----optimized
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Opimization using Sensitivity Data II
Result of the new 3D simulation
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Summary
CST Studio Suite 2010 offers
Sensitivity and Yield Analyses
S-Parameter Sensitivity is computed exactely without any meshvariation,
broadband, at no additional costs
It has been demonstrated that sensitivity data is useful to speed up optimization loops
Optimization performed on the smaller linear matrix system and runs
much faster
10 design parameter
final design
Only 16 solver runs (FD TET),
i.e. almost a factor 100 faster
compared to the current method!
initial design
CST Studio Suite 2011: New Optimization Strategy (based on sensitivity information)