chapter 2 interconnect analysis

40
Chapter 2 Interconnect Analysis Prof. Lei He Electrical Engineering Department University of California, Los Angeles URL: eda.ee.ucla.edu Email: [email protected]

Upload: rhiannon-navarro

Post on 31-Dec-2015

37 views

Category:

Documents


0 download

DESCRIPTION

Chapter 2 Interconnect Analysis. Prof. Lei He Electrical Engineering Department University of California, Los Angeles URL: eda.ee.ucla.edu Email: [email protected]. Organization. Chapter 2a First/Second Order Analysis Chapter 2b Moment calculation and AWE - PowerPoint PPT Presentation

TRANSCRIPT

Page 1: Chapter 2 Interconnect Analysis

Chapter 2

Interconnect Analysis

Prof. Lei HeElectrical Engineering DepartmentUniversity of California, Los Angeles

URL: eda.ee.ucla.eduEmail: [email protected]

Page 2: Chapter 2 Interconnect Analysis

Organization

Chapter 2a First/Second Order Analysis

Chapter 2b Moment calculation and AWE

Chapter 2c Projection based model order

reduction

Page 3: Chapter 2 Interconnect Analysis

1

1

1

ˆ

ˆq

q

N

q

x

x

u u

x

x

U

Projection Framework:Change of variables

Note: q << NNote: q << N

reduced statereduced state

original stateoriginal state

Page 4: Chapter 2 Interconnect Analysis

Projection Framework

Original System

Substitute

, TsEx x bu y c x

ˆqx U x

Note: now few variables (q<<N) in the state, but still thousands of Note: now few variables (q<<N) in the state, but still thousands of equations (N)equations (N)

,ˆˆ buxUxsEU qq xUcy qT ˆ

Page 5: Chapter 2 Interconnect Analysis

Projection Framework (cont.)

Reduction of number of equations: test multiplying by VReduction of number of equations: test multiplying by VqqTT

If V and U biorthogonal If V and U biorthogonal

Tq qV U I

,ˆˆ buxUxsEU qq xUcy qT ˆ

,ˆˆ buVxUVxEUsV Tqq

Tqq

Tq xUcy q

T ˆ

,ˆˆˆ ubxxEs xUcy qT ˆ

Page 6: Chapter 2 Interconnect Analysis

E b

ˆTcTqV

E qUE

nxnnxn

qxqqxq

nxqnxq

qxnqxn

Projection Framework (cont.)

T

sEx x bu

y c x

xUcy

buVxxEUsV

qT

Tqq

Tq

ˆ

ˆˆ

Page 7: Chapter 2 Interconnect Analysis

Use Eigenvectors

Use Time Series Data Compute Use the SVD to pick q < k important vectors

Use Frequency Domain Data Compute Use the SVD to pick q < k important vectors

Use Krylov Subspace Vectors?

1 2( ), ( ), , ( )kx t x t x t

1 2( ), ( ), , ( )kx s x s x s

Approaches for picking V and U

Page 8: Chapter 2 Interconnect Analysis

H s

H s

Taylor series expansion:

2 xx b Ab A b

UU

Intuitive view of Krylov subspace choice for change of base projection matrix

2span , , ,x b Eb E b

• change base and use only the first change base and use only the first few vectors of the Taylor series few vectors of the Taylor series expansion: equivalent to match first expansion: equivalent to match first derivatives around expansion pointderivatives around expansion point

0

k k

k

x s E b u

sEx x bu 1x I sE bu

2b Eb E b

Page 9: Chapter 2 Interconnect Analysis

Combine point and moment matching: multipoint moment matching

H s

H s

• Multiple expansion points give larger bandMultiple expansion points give larger band• Moment (derivates) matching gives more Moment (derivates) matching gives more accurate accurate behavior in between expansion pointsbehavior in between expansion points

Page 10: Chapter 2 Interconnect Analysis

Aside on Krylov Subspaces - Definition

2 1, , , , kk E b span b Eb E b E b

The order k Krylov subspace generated The order k Krylov subspace generated from matrix A and vector b is defined asfrom matrix A and vector b is defined as

Page 11: Chapter 2 Interconnect Analysis

ˆ for 0, , 1

l lj j b c

j jl l

H s H sl k k

s s

IfIf

1 1

1 1,..., ,bj

Jq j j jk

span u u I s E E I s E b

andand

1 1,..., ,cj

T TJ Tq j j jk

span v v E I s E I s E c

ThenThen

Projection Framework: Moment Matching Theorem (E. Grimme 97)

Page 12: Chapter 2 Interconnect Analysis

IfIf U and V are such that: U and V are such that:

ThenThen the first q moments (derivatives) of the first q moments (derivatives) of the the reduced system matchreduced system match

1,..., qU V u u

11,..., ( , ) , , , q

q qspan u u E b span b Eb E b

TU U I

Special simple case #1: expansion at s=0,V=U, orthonormal UTU=I

0 s=0

ˆ for k 0, ,

k k

k k

s

H Hq

s s

ˆˆˆ T k T kc E b c E b

Page 13: Chapter 2 Interconnect Analysis

, ,..., can not be computed directlykb Eb E b

b

Eb

2E b3E b

Vectors will line up with dominant eigenspace!Vectors will line up with dominant eigenspace!

Need for Orthonormalization of U

Page 14: Chapter 2 Interconnect Analysis

Need for Orthonormalization of U (cont.)

In "change of base matrix" U transforming to the new reduced state space, we can use ANY columns that span the reduced state space

In particular we can ORTHONORMALIZE the Krylov subspace vectors

1

1

1

q

r

q

r

N

q

x

x

u u

x

x

U

i ju u i j

Page 15: Chapter 2 Interconnect Analysis

1 1

1

1,

1i i

i

i i

u uu

Normalize new vectorNormalize new vector

1 /u b b

For i = 1 to k

1i iu Eu

Generates k+1 vectors!Generates k+1 vectors!

1 1 1T

i i i j j

ji

u u u u u

Orthogonalize new vectorOrthogonalize new vector

For j = 1 to i

Orthonormalization of U:The Arnoldi Algorithm

Page 16: Chapter 2 Interconnect Analysis

ThenThen the first the first 2q2q moments of reduced system match moments of reduced system match

IfIf U and V are such that: U and V are such that:

11,..., ( , ) , , , ( )

>>>>>>>>>>>>>>>>>>>>>>>>>>>>T T T q

q qspan v v E c span c E c E c

TV U I

Special case #2: expansion at s=0, biorthogonal VTU=I

11,..., ( , ) , , , q

q qspan u u E b span b Eb E b

0 s=0

ˆ for 0, , 2

k k

k k

s

H Hk q

s s

ˆˆˆ T k T kc E b c E b

Page 17: Chapter 2 Interconnect Analysis

PVL: Pade Via Lanczos[P. Feldmann, R. W. Freund TCAD95]

PVL is an implementation of the biorthogonal case 2:

11,..., ( , ) , , , ( )

>>>>>>>>>>>>>>>>>>>>>>>>>>>>T T T q

q qspan v v E c span c E c E c

TV U I

11,..., ( , ) , , , q

q qspan u u E b span b Eb E b

Use Lanczos process to biorthonormalize the Use Lanczos process to biorthonormalize the columns of U and V: columns of U and V: gives very good numerical gives very good numerical stabilitystability

Page 18: Chapter 2 Interconnect Analysis

Case #3: Intuitive view of subspace choice for general expansion points

In stead of expanding around only s=0 we can expand around another points

For each expansion point the problem can then be put again in the standard form

1 20 Js s s s s s s

T

sE x x b u

y c x

( )i

T

s s E x x b u

y c x

is s s

1 1( ) ( )i i

T

s I s E E x x I s E b u

y c x

Page 19: Chapter 2 Interconnect Analysis

Case #3: Intuitive view of Krylov subspace choice for general expansion points (cont.)

matches matches first kfirst kjj of of transfer transfer function function around around each each expansion expansion point spoint sjj

1 1( ) ( )i i

T

s I s E E x x I s E b u

y c x

Hence choosing Krylov subspaceHence choosing Krylov subspace

1 1

1 1,..., ,bj

Jq j j jk

span u u I s E I s E b

ss11=0=0

ss11ss22

ss33

Page 20: Chapter 2 Interconnect Analysis

Interconnected Systems

ROM

Can we assure that the simulation of the composite system will be Can we assure that the simulation of the composite system will be well-behaved? At least preclude non-physical behavior of the well-behaved? At least preclude non-physical behavior of the reduced model? reduced model?

In reality, reduced models are only useful when connected In reality, reduced models are only useful when connected together with other models and circuit elements in a composite together with other models and circuit elements in a composite simulationsimulation

Consider a state-space model connected to external circuitry Consider a state-space model connected to external circuitry (possibly with feedback!) (possibly with feedback!)

Page 21: Chapter 2 Interconnect Analysis

Passivity

Passive systems do not generate energy. We cannot extract out more energy than is stored. A passive system does not provide energy that is not in its storage elements.

0)()(Energy

dvit

If the reduced model is not passive it can generate energy from nothingness and the simulation will explode

Page 22: Chapter 2 Interconnect Analysis

Interconnecting Passive Systems

QDC

-++-

QDC

-++-

QDC

-++-

QDC

-++-

The interconnection of stable models is not necessarily stable

BUT the interconnection of passive models is a passive model:

Page 23: Chapter 2 Interconnect Analysis

Sufficient conditions for passivity

Sufficient conditions for passivity:

Cxy

BuAxsEx

xxAAx

xxEEx

BC

H

H

T

allfor ,0)()3

allfor ,0)()2

)1

BAsECsH 1)()(

Note that these are NOT necessary conditions (common misconception)

Page 24: Chapter 2 Interconnect Analysis

Congruence Transformations Preserve Positive Semidefinitness

Def. congruence transformation EUUE Tˆ

same matrix Note: case #1 in the projection framework V=U produces

congruence transformations Property: a congruence transformation preserves the positive

semidefiniteness of the matrix Proof. Just rename Note:

xEUxxU T allfor ,0

Uxy

Page 25: Chapter 2 Interconnect Analysis

PRIMA (for preserving passivity) (Odabasioglu, Celik, Pileggi TCAD98)

A different implementation of case #1: V=U, UTU=I, Arnoldi Krylov Projection Framework:

b

ˆTb

Use Arnoldi: Numerically Use Arnoldi: Numerically very stable stable

xby

buAxsExT

xUby

buUxAUUxEUsU

qT

Tqq

Tqq

Tq

E A

IUU

uuVU

bEAEAbspanbEAuuspan

T

q

qqq

},...,{

})(,...,,{),(},...,{

1

11111

Page 26: Chapter 2 Interconnect Analysis

PRIMA preserves passivity

The main difference between and case #1 and PRIMA:– case #1 applies the projection framework to

– PRIMA applies the projection framework to

PRIMA preserves passivity because– uses Arnoldi so that U=V and the projection becomes

a congruence transformation – E and A produced by electromagnetic analysis are

typically positive semidefinite while may not be.

– input matrix must be equal to output matrix

xByBuAxsEx T

xByBuAxExsA T 11

EA 1

Page 27: Chapter 2 Interconnect Analysis

Compare methods

number of number of moments moments matched by matched by model of order model of order qq

preserving passivitypreserving passivity

case #1 (Arnoldi, case #1 (Arnoldi, V=U, UV=U, UTTU=I on U=I on

sAsA-1-1Ex=x+Bu)Ex=x+Bu)qq nono

PRIMAPRIMA (Arnoldi, (Arnoldi,

V=U, UV=U, UTTU=I on U=I on

sEx=Ax+Bu)sEx=Ax+Bu) qq

yesyes

necessary when necessary when model is used in a model is used in a

time domain time domain simulatorsimulator

case #2 (case #2 (PVLPVL, , Lanczos,V≠U, Lanczos,V≠U, VVTTU=I on sAU=I on sA--

11Ex=x+Bu)Ex=x+Bu)

2q2q

more efficientmore efficient

no no

(good only if model (good only if model is used in frequency is used in frequency

domain)domain)

Page 28: Chapter 2 Interconnect Analysis

Homework 2 (due April 27)

(3) Modify the PRIMA code with single frequency expansion to multiple points expansion. You should use a vector fspan to pass the frequency expansion points. Compare the waveforms of the reduced model between the following two cases:

1. Single point expansion at s=1e4.

2. Four-point expansion at s=1e3, 1e5, 1e7, 1e9.

28

Page 29: Chapter 2 Interconnect Analysis

29

Format of the input matrices for test1 1 19.4595 1.43391e-14 1 2 0.000464141 -2.9702e-15 1 3 -0.000542882 0.0 1 4 0.000152585 -7.5288e-15 1 5 0.000464074 -2.9702e-15 1 6 -0.000542801 0.0 1 68 -19.4595 0.0 2 1 0.0 -2.9702e-15 2 2 3.66672 2.44291e-13 2 3 0.0 -2.3594e-13 2 4 0.0 -5.3806e-15 2 72 -1.425 0.0 2 329 -2.06075 0.0 2 341 -0.091255 0.0 2 343 -0.0897199 0.0 3 1 -2.44188e-06 0.0 3 2 -0.000464141 -2.3594e-13 3 3 40.8898 2.42089e-13 …..

Page 30: Chapter 2 Interconnect Analysis
Page 31: Chapter 2 Interconnect Analysis
Page 32: Chapter 2 Interconnect Analysis
Page 33: Chapter 2 Interconnect Analysis
Page 34: Chapter 2 Interconnect Analysis
Page 35: Chapter 2 Interconnect Analysis
Page 36: Chapter 2 Interconnect Analysis
Page 37: Chapter 2 Interconnect Analysis
Page 38: Chapter 2 Interconnect Analysis

G,C,B,U,L matrices have been generated.Prima begins:Elapsed time is 2.003787 seconds.Prima done!

Calculate original time domain response:Elapsed time is 2.868078 seconds.Original time domain response done!

Calculate reduced time domain response:Elapsed time is 0.553771 seconds.Reduced time domain response done!

Calculate original frequency response:Elapsed time is 1.192908 seconds.Original frequency response done!

Calculate reduced frequency response:Elapsed time is 0.359126 seconds.Reduced frequency response done!

Calculate original impulse response:Elapsed time is 0.052804 seconds.Original impulse response done!

Calculate reduced impulse response:Elapsed time is 0.052701 seconds.Reduced impulse response done!

Page 39: Chapter 2 Interconnect Analysis

100

105

1010

-30

-20

-10

0

10

20

30

40

50

60frequency response, magnitude

Original

Reduced

100

105

1010

-4

-3

-2

-1

0

1

2

3

4frequency response, phase

Original

Reduced

0 0.5 1 1.5 2

x 10-5

-40

-30

-20

-10

0

10

20

30

40time response

Original

Reduced

Page 40: Chapter 2 Interconnect Analysis

100

105

1010

0

5

10

15

20

25

30

35

40frequency response, magnitude

Original

Reduced

100

105

1010

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0frequency response, phase

Original

Reduced