Chapter 2

Interconnect Analysis

Prof. Lei HeElectrical Engineering DepartmentUniversity of California, Los Angeles

URL: eda.ee.ucla.eduEmail: lhe@ee.ucla.edu

Organization

Chapter 2a First/Second Order Analysis

Chapter 2b Moment calculation and AWE

Chapter 2c Projection based model order

reduction

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

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

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

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

spaceqV

Equation TestingEquation Testing

ˆ ˆEx

ˆ ˆT Tq q qEx EV U x xV E

Ex

ˆqx xU

x

spaceqU

Change of variablesChange of variables

xE

ˆ Tq qE V EU

Projection Framework

xcyubxxEsxcybuxsEx TT ˆˆ,ˆˆˆ,

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 Singular Vectors of System Grammians?

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

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

Combine point and moment matching: multipoint moment matching

H s

H s

• Multipole expansion points give larger bandMultipole 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

Compare Pade’ Approximationsand Krylov Subspace Projection

Framework

Krylov Subspace Krylov Subspace Projection Projection Framework:Framework:• multipoint moment multipoint moment matchingmatching• numerically very numerically very stable!!!stable!!!

H s

H s

H s

H s

Pade Pade approximations:approximations:• moment matching moment matching at at single DC pointsingle DC point• numerically very numerically very ill-conditioned!!!ill-conditioned!!!

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

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)

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

, ,..., 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

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

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

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

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

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

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

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!)

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

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:

Positive Real Functions

A positive real function is a function internally stable with non-negative real part

0Refor 0,)()(

0Refor )()(

0Refor analytic is )(

(s)ss

(s)ss

(s)s

HHH

HH

H

)()()( svsHsi

(no unstable poles)

(no negative resistors)

(real response)

Hermittian=conjugate and transposed

It means its real part is a positive semidefinite matrix at all frequencies

Positive Realness & Passivity

For systems with immittance (impedance or admittance) matrix representation, positive-realness of the transfer function is equivalent to passivity

ROMout1v

out2v

in1I

in2I

)(

)()(

)(

)(out2

out1

in2

in1

sI

sIsZ

sv

sv

Necessary conditions for passivity for Poles/Zeros

The positive-real condition on the matrix rational function implies that:

)(sH

– If H(s) is positive-real also its inverse is positive real– If H(s) is positive-real it has no poles in the RHP, and hence also

no zeros there.

Occasional misconception : “if the system function has no poles and no zeros in the RHP the system is passive”.

It is necessary that a positive-real function have no poles or zeros in the RHP, but not sufficient.

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)

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

xUxEExU HT allfor ,0)(

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

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

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)