Stability and Passivity of the Super Node Algorithm for EM modelling of ICs Maria Ugryumova...

Stability and Passivity of the Super Node Algorithm

for EM modelling of ICs

Maria UgryumovaSupervisor: Wil SchildersTechnical University Eindhoven,The Netherlands

CASA day, 13 November 2008

Outline

1. Motivation

2. Fasterix and EM simulation

3. System parameters

4. Super Node Algorithm

5. Numerical example

6. Passivity enforcement

7. Conclusions

1

• From the original model to the reduced one

• Realization

Motivation

Solution

EM tools Model Order Reduction

• increasing IC complexity

• smaller feature sizes

• higher frequencies

• multilayer structure

• electromagnetic effects

2

65 nm – 45 nm

1.06 GHz – 3.33 GHz

9 layers

Intel CoreTM2 Processors

84 10 transistors

Fasterix – layout simulation tool for EMmodelling (NXP) properties of ICs

Motivation

• program simulator PSTAR (NXP)

• radiated EM fields

3

Model Order Reduction:

Super Node Algorithm

Motivating example

Time response of the lowpass filter model (300 unknowns). Why is it unstable?

• What is the reason of instability?• How can we avoid the instability?

Key questions?

4

unstable

Outline

1. Motivation






7. Conclusions

5


• Realization

How Fasterix works

6

• Initial data (coordinates, pins, metal, max. frequency, etc.)

• Geometry preprocessor;

[Du Cloux 1993], [Wachters, Schilders 1997]

1. 2.

How Fasterix works

0

0

'

0

'

EdxG

iJ

EJdxGiJ

A

, ,0

)( ,

xnJ

xVx Vfixed

2

1

0

( ) - charge density

( ) - scalar potential

( ) - current density

- irradiation; - permeability

- permittivity; - conductivity

div

L

H

J H

E

7


• Geometry preprocessor; BVP

• Full RLC circuit – inefficient!

3.



• Geometry preprocessor; BVP

• Full RLC circuit – inefficient!

• Super nodes are defined

• Reduced RLC circuit - efficient!

How Fasterix works

8

4. 5.

Super node

algorithm


Outline

1. Motivation





6. Numerical examples

7. Conclusions

9


• Realization

System parameters

1

1

( )n

T io i

i i

RH s B G sC B

s

lims

• Transfer function

– residuals – poles

• Poles are for which C

| ( ) |H s or det( ) 0G C

i.e. poles are eigenvalues of Gx Cx

10

i

( )

( ) ( )

i

To

dC x Gx B u t

dt

y t B x t

• Linear time invariant system

iR

pole , , then Re( ) 0

11

System parameters

• Passive positive real:

H(s) is analytic for Re(s)>0 *( ) ( ) 0 for Re(s)>0H s H s

( )H s

Passive systems

• dissipate power delivered through input and output ports

• synthesizable with positive R,L,C and transformers [Brune ‘31]

Stable •

Outline

1. Motivation






7. Conclusions

12


• Realization

0 0

0 0T

G C

R P L Is

P C V J

1( ) TP R sL P sCJ s V

I - current in the branchesV – voltage in the nodesJ – currents, floating into the sys. through the nodes G – positive real, C – positive definite

• Voltage to current transfer:

Admittance matrix

• Kirchhoff equations

SNA: original (non-reduced) RLC model

• Y(s) is stable and positive real Re( ( , )) 0R L

givenunknown

13

( )Y s

( ) ( ) 0HY s Y s

(( )) n nYJ Vss

SNA: Model Order Reduction

Super Node Algorithm (SNA)

) )( (k kY sJ s V

21 kY YY

1. Elimination of non-super nodes:

2. Two steps of approximations

3. Realization of the circuit

1nY Y

14

Generated by BEM

port

1

port

2port

1

port

2

0 0 (1)

0 0T

R P L Is

P C V J

1. SNA: Elimination of non-super nodes• Kirchhoff equations

• Partitioning

''

' ' '

' 0 '

NN NNN N

N N N N

C CN NV J P P P C

C CN N

R, L, C – positive definite

N – super nodesN’ – all other nodes

15

• •

•

1nY Y

0 0 (1)

0 0T

R P L Is

P C V J

'

'

' ' ' ' '

0 0 0

0 0 0

0 0 0 0

N NT

N NN NN N NT T

N NN N N N

R P P L I

P s C C V J

P C C V


• Partitioning

''

' ' '

' 0 '

NN NNN N

N N N N

C CN NV J P P P C

C CN N

• Substitution into (1)



15

• •

•

1. SNA: Elimination of non-super nodes 1nY Y

0 0 (1)

0 0T

R P L Is

P C V J

'

' ' ' ' '

'

0

0 0N N

NTN N N N N N

xG C Bi

TN N NN NN N

Bo

R P L I Ps V

P C V sC

J P sC x sC V

1( ) ( )T

o iN NNNB s G s BJ C s VsC


• Partitioning

''

' ' '

' 0 '

NN NNN N

N N N N

C CN NV J P P P C

C CN N

• Substitution into (1)

• - Schur complement of• stable: eig(-G,C)<0• positive real



16

1( )Y s

• •

•

1. SNA: Elimination of non-super nodes 1nY Y

1Y nY

G – positive real, C – positive definite

Sketch of the proof: - positive real

1. Stable: Re( ( , )) 0G C

1( )Y s

*1 1*

1 * * *

( ) ( )

2 2 0,

T T

T

Y s Y s P R sL P sC P R sL P sC

P R sL R sL R sL R sL P y R L y

4. Lemma If is positive definite matrix then its Schur

complements are positive definite.

n nA C

5. By Lemma, positive definite positive real1( )Y s

2. - Schur complement of

3. Y(s) – positive definite:

1( )Y s 1( ) TP R sLY P sCs

17

0

(( )) n nYJ Vss

2. SNA: Model Order Reduction


) )( (k kY sJ s V

21 kY YY



3. Realization of the circuit

1nY Y

18

Generated by BEM

port

1

port

2port

1

port

2

' 0 1

0 1

NV V V

I I I

' 0

' 0

' 1

' ' 0 '' 1

0 (3)

0 0 0 0

00 (4)

0 0 0

N N NT

N

NT

N N N N NN

R P IL P Vs

P V

R P ILs

s C V C VP V

2 0 1 ' 0T

N NN NNY P I I sC V sC

• Under the assumption: , - free-space wave number.

• Pairs found from two systems:

2. SNA: 1st approximation:

0 1k h

1 0V ik h

• If then (1,...,1)NV diag

00( ) :ik h

10( ) :ik h

19

0k

1 0I ik h

21Y Y

'

' ' ' ' '

'

0

0 0N N

NTN N N N N N

xG C Bi

TN N NN NN N

Bo

R P L I Ps V

P C V sC

J P sC x sC V

• stable • not positive real • computation of eigenvalues

• Introducing the null space for , we solve:

2. SNA: 2nd approximation: (details)

0( )ULL N CFTPY I Ys s

'NP

0i

20

2 0 1 ' 0T

N NN NNY P I I sC V sC

' 0

' 0

0

0 0 0 0N N N

TN

R P IL P Vs

P V

' 0

0 0

0

TNP I

I I

1

0 ( )T TN NI R sL P V

1

( )n

lC

l lFULL

HsY

sY s

2Y Y

high freq. range

Yc – indefinite!

• •

In the pole-residual form:•

SNA: Comparison of the approximations

• All approximations match well• Capacitances start influence at high frequencies

21

0.5 GHz

Outline

1. Motivation






7. Conclusions

22


• Realization

(( )) n nYJ Vss

SNA: Model Order Reduction (MOR)


) )( (k kY sJ s V

21 kY YY



3.

1nY Y

23

Generated by BEM

kY

• Calculate m<<n eigenvalues

• Choose (m+1) match frequencies

• Solve for

• Circuit elements

, and , 1, 1, , 1, C ij ijy H k m i j N

1

nl

FULL Cl l

HY sY

s

RLC circuit

realization

,

1

( )m

l ijk C ij k

l l k

Hs y y s

s

1 1, , ,, , l l l ij l l ij C ijR H L H C y

i

1

2

3

4 3 5

5

0

min( ) /100

max( ) 10

[ , ]

i

i

s

s

s

s s s

s

24

3. SNA: Realization of the reduced circuit

stablenot positive real

1 1, , ,, , l l l ij l l ij C ijR H L H C y

• N – number of super nodes• m – number of branches between each pair of s.n. l 1, , , 1,m i j N


25

1

ml

FULL Cl l

HY sY

s

RLC circuit

realization

stablenot positive real [Guillemin’68]

• What is the reason of instability in time domain?

Key question?

26


• MNA: dimension of the system ) • Redundancy

2( )O N m

27

1

ml

FULL Cl l

HY sY

s

RLC circuit

realization

stablenot positive real

• N – number of super nodes• m – number of branches between each pair of s.n.

1.0e+006 -0.33075173081148

-0.33075151394768 -0.33075158822141 -0.73063347579307 -0.73063369561798 -0.73063384656739 -0.68099777735205 -0.68099799754699 -0.68099790176943 -0.62258539700220 -0.62258525785232 -0.62258531561929 9.90350498680717 0.00000000000670

1

2

3

4

3.3075e+5

6.8100e+5

7.3063e+5

6.2259e+5

• MNA: finite poles:• Generalized eigenvalues:

• Match frequencies (Fasterix):

Example (Two parallel striplines, 1MHz)

4

1

lRLC C

l l

HY sY

s

sk(1)=0 sk(2)=-526.365 sk(3)=-0.116024e+07 sk(4)=-0.164082e+07sk(5)=-0.232048e+07

0

0 00

T

CG

C BG Ps x u

LP

dim(G,C) 85 x 85

stable notstable

28

RHP

[6 6]lH

How to guarantee that reduced circuit will be described by the same poles?

TheoremSuper node reduced circuit described by Y(s) with n stable poles, in MNA formulation has exactly the same n poles iff all ports: grounded / voltage / current sources [proof in progress]

1

( )n

k

k k

HY s sC

s p

2 2 2 2, kC R H R

11 C,

,

0

1

i

ik

k ij

kk ij

p

pR

H

LH

22 C12C

Two-ports realization

port 1 port 2

29

Outline

1. Motivation






7. Conclusions

30


• Realization

Example (Lowpass filter, 10e9 Hz)

system Dim sys. R L C Mutual.Ind. Time, sec.

original 257 452 452 2763 50950 754.1

reduced(m=4)

98 19012 19012 19110 0 47.9

31

Outline

1. Motivation






7. Conclusions

32


• Realization

' 00 01T

C NN N NNsY s C V P I sC

0( )FULL N CY s P I sY

1' 00 01 1 1( ,... )T

NN NC V P I Vdiag V

Passivity enforcement

pos. definite ( )FULLY spos. real

1

0 , n m

T T i iN N N

i ii i

H HP I P A sB P m n

s s

• not positive real

remedy: Modal approximation [Rommes, 2007]

33

pos. real Not

Not pos. definite

2.)

1.)

Outline

1. Motivation






7. Conclusions

34


• Realization

Conclusions

Achieved

• The reason of instability of SNA models has been found

• Remedy to guarantee stability has been presented

• Passivity enforcement

Main hurdles

• Redundancy of the poles

• For N super nodes, m poles circuit elements

• Positive R,L,C not guaranteed

Future work

• Another approach for simulation of EM effects based on measurement of Y/Z/S parameters

2( )O N m

35

Thank you!

References

Schilders, W.H.A. and ter Maten, E.J.W, Special volume : numerical methods in electromagnetics,

Elsevier, Amsterdam, 2005.

Cloux, R.Du and Maas, G.P.J.F.M and Wachters, A.J.H and Milsom, R.F. and Scott, K.J.,

Fasterix, an environment for PCB simulation, Proc. 11th Int. Conf. on EMC, Zurich, Switzeland,

1993

Rommes J., Methods for eigenvalue problems with applications in model order

reduction, Ph.D. dissertation, Utrecht University, Utrecht, The Netherlands, 2007.

[Online]. Available: http://rommes.googlepages.com/index.html

Guillemin, E.A., Synthesis of Passive Networks, Wiley, New York, 1957

http://rommes.googlepages.com/index.html

Stability and Passivity of the Super Node Algorithm for EM modelling of ICs Maria Ugryumova...

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