1. Chapter 2 Interconnect 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 Note: q << N reduced state 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 V q T If V and U biorthogonal

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

7. Equation Testing Change of variables Projection Framework

8. Use Eigenvectors Use Time Series Data n Compute n Use the SVD to pick q < k important vectors Use Frequency Domain Data n Compute n Use the SVD to pick q < k important vectors Use Singular Vectors of System Grammians? Use Krylov Subspace Vectors? Approaches for picking V and U

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

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

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

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

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

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

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

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

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

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. Case #3: Intuitive view of Krylov subspace choice for general expansion points (cont.) matches first k j of transfer function around each expansion point s j Hence choosing Krylov subspace s 1 =0 s1s1s1s1 s2s2s2s2 s3s3s3s3

22. Interconnected Systems 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?  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!)

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

24. Interconnecting Passive Systems QD C - + + - QD C - + + - QD C - + + - QD C - + + - 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) (no negative resistors) (real response) 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, U T U=I, Arnoldi Krylov Projection Framework: Use Arnoldi: Numerically stable 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 number of moments matched by model of order q preserving passivity case #1 (Arnoldi, V=U, U T U=I on sA -1 Ex=x+Bu) qno PRIMA (Arnoldi, V=U, U T U=I on sEx=Ax+Bu) qyes necessary when model is used in a time domain simulator case #2 (PVL, Lanczos,V≠U, V T U=I on sA -1 Ex=x+Bu) 2q more efficient no (good only if model is used in frequency domain)