Chapter 2 Interconnect Analysis

1. Chapter 2Interconnect Analysis Prof. Lei He Electrical Engineering Department University of California, Los Angeles URL: eda.ee.ucla.edu Email: lhe@ee.ucla.edu

2. Organization • Chapter 2a First/Second Order Analysis • Chapter 2b Moment calculation and AWE • Chapter 2c Projection based model order reduction

3. Projection Framework:Change of variables reduced state Note: q << N original state

4. Projection Framework • Original System • Substitute Note: now few variables (q<<N) in the state, but still thousands of equations (N)

5. Projection Framework (cont.) Reduction of number of equations: test multiplying by VqT If V and U biorthogonal

6. Projection Framework (cont.) qxn qxq nxn nxq

7. Projection Framework Change of variables Equation Testing

8. Approaches for picking V and U • 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?

9. Intuitive view of Krylov subspace choice for change of base projection matrix Taylor series expansion: • change base and use only the first few vectors of the Taylor series expansion: equivalent to match first derivatives around expansion point U

10. Combine point and moment matching: multipoint moment matching • Multipole expansion points give larger band • Moment (derivates) matching gives more accurate • behavior in between expansion points

11. Compare Pade’ Approximationsand Krylov Subspace Projection Framework • Pade approximations: • moment matching at • single DC point • numerically very • ill-conditioned!!! • Krylov Subspace Projection Framework: • multipoint moment • matching • numerically very • stable!!!

12. Aside on Krylov Subspaces - Definition The order k Krylov subspace generated from matrix A and vector b is defined as

13. Projection Framework: Moment Matching Theorem (E. Grimme 97) If and Then

14. Special simple case #1: expansion at s=0,V=U, orthonormal UTU=I If U and V are such that: Then the first q moments (derivatives) of the reduced system match

15. Need for Orthonormalization of U Vectors will line up with dominant eigenspace!

16. 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

17. For i = 1 to k Generates k+1 vectors! For j = 1 to i Orthogonalize new vector Normalize new vector Orthonormalization of U:The Arnoldi Algorithm

18. Special case #2: expansion at s=0, biorthogonal VTU=I If U and V are such that: Then the first 2q moments of reduced system match

19. PVL: Pade Via Lanczos[P. Feldmann, R. W. Freund TCAD95] • PVL is an implementation of the biorthogonal case 2: Use Lanczos process to biorthonormalize the columns of U and V: gives very good numerical stability

20. 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

21. s2 s1 s1=0 s3 Case #3: Intuitive view of Krylov subspace choice for general expansion points (cont.) Hence choosing Krylov subspace matches first kj of transfer function around each expansion point sj

22. Interconnected Systems • In reality, reduced models are only useful when connected together with other models and circuit elements in a composite simulation • Consider a state-space model connected to external circuitry (possibly with feedback!) ROM • Can we assure that the simulation of the composite system will be well-behaved? At least preclude non-physical behavior of the reduced model?

23. Passivity • Passive systems do not generate energy. We cannot extract out more energy than is stored. A passive system does not provideenergy that is not in its storage elements. • If the reduced model is not passive it can generate energy from nothingness and the simulation will explode

24. - - - - + + + + - - - - + + + + D D D D Q Q Q Q C C C C Interconnecting Passive Systems • The interconnection of stable models is not necessarily stable • BUT the interconnection of passive models is a passive model:

25. Positive Real Functions • A positive real function is a function internally stable with non-negative real part (no unstable poles) (real response) (no negative resistors) Hermittian=conjugate and transposed It means its real part is a positive semidefinite matrix at all frequencies

26. + + - - Positive Realness & Passivity • For systems with immittance (impedance or admittance) matrix representation, positive-realness of the transfer function is equivalent to passivity ROM

27. Necessary conditions for passivity for Poles/Zeros • The positive-real condition on the matrix rational function implies that: • 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.

28. Sufficient conditions for passivity • Sufficient conditions for passivity: Note that these are NOT necessary conditions (common misconception)

29. Congruence Transformations Preserve Positive Semidefinitness • Def. congruence transformation 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:

30. PRIMA (for preserving passivity) (Odabasioglu, Celik, Pileggi TCAD98) A different implementation of case #1: V=U, UTU=I, Arnoldi Krylov Projection Framework: Use Arnoldi: Numerically very stable

31. 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

32. Compare methods