SLIDE 1 ReductionandRealization Techniques inModelling of Passive ElectronicStructures
Pieter Heres
Scientific Computing Group, EindhovenUniversityofTechnology
SLIDE 2 FunnyexampleofROM:
Animageconsistsofmatrix(or3) Singularvaluedecomposition: Truncateacertainamountofsingularvalues. Example: 139singularvalues
A V U = Σ
T
SLIDE 3
Example:
Thelargest10,20,30and40singularvalues
SLIDE 4 Overview ofmytalk
- Systemformulation
- WhatisROM?
- Krylov subspacemethods
- Orthogonalization
- Realization
SLIDE 5 Electronicstructures
– Interconnects – Analogpartsofchips – Coupledwithdigitalpart
- ManymodelscanberepresentedbyRLC-
networks: – TLM – PEEC – EFIEintegralmodel – FIT – FDTD(spatiallydiscretized)
SLIDE 6 Systemformulation
DAEsystem: Or: Forinstance:voltage-in-current-out
+
− =
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t t t t t t t t dt d
T T
i v L y Bu i v R P P G i v L C
) ( ) ( ) ( ) ( ) ( t t t t t
Tx
L y Bu Gx x C = + − =
SLIDE 7 WhatisROM?
Large(RLC-)circuit replacedby Smallcircuit
(withapproximatelysamebehavior)
) ( ) ( ) ( ) ( ) ( t t t t t
Tx
L y Bu Gx x C = + − =
( ~ ~ ) ( ˆ ) ( ~ ) ( ~ ~ ) ( ~ ~ t t t t t dt d
Tx
L y u B x G x C = + − =
SLIDE 8 Demands
- Behaviorapproximatedwell:
– Forafixedsetofinputs – Uptoamaximumfrequency
- Gainincomputationaltime
- Passivitypreservation!
SLIDE 9 Systemformulation
TransformwithLaplace: Transferfunction: Directrelationbetweeninputandoutput Approximation,infrequencydomain
B C G L H
1
) ( ) (
−
+ = s s
T
) ( ) ( ) ( ) ( ) ( s s s s s s
TX
L Y BU GX CX = + − = ) ( ) ( ) ( ) ( ) ( t t t t t
Tx
L y Bu Gx x C = + − =
SLIDE 10 Krylov-subspacemethods
PRIMA(Odabasioglu,Celik)andPVL(Feldmann, Freund): with:
R A I L B C G L H
1 1
) ) ( ( ) ( ) (
− −
− − = + = s s s s
T T
] ,..., , [ ) , (
1b
A Ab b A b
−
=
n n
C G A
1 0 )
(
−
+ − = s B C G R
1 0 )
(
−
+ = s
SLIDE 11 Krylov-subspacemethods(2)
Definedby: Orthonormal basis:V Projection:
] ,..., , [ ) , (
1b
A Ab b A b
−
=
n
˜
G VT V
SLIDE 12 Krylov-subspacemethods(3)
SVD-Laguerre (Knockaert,DeZutter): Laguerre expansion: Krylov-space:
( )
n n n T
s s s s s
− + − + + = + =
− ∞ = −
α α α α α α B C G C G C G L B C G L H
T 1 1
) ( ) )( ( 2 ) ( ) (
B C G b
1
) (
−
+ = α ) ( ) (
1
C G C G A α α − + =
−
SLIDE 13
Example
PCB Originalsize695by695 Reducedto70by70 Behaviorupto1GHz
SLIDE 14
Example(ACanalysis)
SLIDE 15
Example(transient)
SLIDE 16
Orthogonalization
SLIDE 17 Krylov spaces
Orthogonalizewhilebuildingup: ExtracareformultiplecolumnsofB Choices: – whichcolumnsaregeneratedwhen? – whatisorthogonalizedagainstwhat? EverycolumninBhasitsownKrylov-space.
] ,..., , [ ) , (
1B
A AB B A B
−
=
n
SLIDE 18 Krylov spaces
Whichcolumnsaregeneratedwhen? – ColumnsofBseparately – Binblocks Whatisorthogonalizedagainstwhat? – Afterwards(eg.withSVD) – During
- Againstall
- AgainstcolumnofsameKrylov
space
SLIDE 19
ModifiedGram-Schmidt
Sometimesre-orthogonalization isnecessary. fori=1..j h=viT w; w=w– hvi; end w=w/|w|; Ruleofthumb: re-orthogonalizeifmorethan75%isremoved
SLIDE 20 Krylovspaces
– essentialinformationcanbelost – spuriousinformationcanoccur
- PreservetheshapeoftheHessenbergmatrix
- BlockArnoldi(asinPRIMA)isarightwayandan
efficientway
SLIDE 21
Realization
SLIDE 22 Realization
Projection: Physicalmeaningislost Givenanarbitrarysystem,findacircuit: ACandTransientanalysis
x V x
T
= ~
) ( ~ ~ ) ( ˆ ) ( ~ ) ( ~ ~ ) ( ~ ~ t t t t t dt d
Tx
L y u B x G x C = + − =
SLIDE 23 Needforrealization
Transientanalysiscanbedone,viafrequency domainresult(IFFT).. …orviageneralsolutionandaconvolution integral. Allthesemethodsareexpensiveandspecificfor
Acircuitcanbecoupledwiththerestofcircuit.
SLIDE 24 Realization(2)
Defineacircuitwithq internalnodes Statespacevector:nodevoltages Rows:KCL’s foreverynode =>Circuitwithq nodes:
j j mjx
G i
j j mjx
C q
j j mju
B i
SLIDE 25
Realization(3)
Outputisdefinedassourcestotheterminalsofthe model
SLIDE 26 Results
- Fromlargemodeltosmallmodel
- Abletocombinereducedmodelwithcomponents
andothermodels – ACanalysisandstabletransientanalysis
– non-linearMOR – ParametrizedMOR
SLIDE 27
Mywebsite,emailand… www.ecce.tue.nl/SMURF
P.J.Heres@tue.nl
Thankyouforyourattention!
