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CSE245: Computer-Aided Circuit Simulation and Verification

Learn about explicit and implicit moment matching methods, utilizing congruence transformation, Krylov subspace, PRIMA, SPRIM, and other advanced methods for efficient circuit model reduction. Understand concepts like passivity, stability, and preservation of structure.

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CSE245: Computer-Aided Circuit Simulation and Verification

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  1. CSE245: Computer-Aided Circuit Simulation and Verification Lecture Notes 4 Model Order Reduction (2) Spring 2006 Prof. Chung-Kuan Cheng UCSD CSE 245 Notes – SPRING 2006

  2. Model Order Reduction: Overview • Explicit Moment Matching • AWE, Pade Approximation • Implicit Moment Matching • Krylov Subspace Methods • PRIMA, SPRIM • Gaussian Elimination • TICER, Y-Delta Transformation UCSD CSE 245 Notes – SPRING 2006

  3. Explicit V.S. Implicit Moment Matching • Explicit moment matching methods • Numerically ill-conditioned • Implicit moment matching methods • construct reduced order models through projection, or congruence transformation. • Krylov subspaces vectors instead of moments are used. UCSD CSE 245 Notes – SPRING 2006

  4. Congruence Transformation • Definition: • Property: Congruence transformation preserves semidefiniteness of the matrix UCSD CSE 245 Notes – SPRING 2006

  5. Krylov Subspace • Given an n x q matrix Vq whose column vectors are v1, v2, …, vq. The span of Vq is defined as • Given an n x n matrix A and a n x 1 vector r the Krylov subspace is defined as UCSD CSE 245 Notes – SPRING 2006

  6. PRIMA • Passive Reduced-order Interconnect Macromodeling Algorithm. • Krylov subspace based projection method • Reduced model generated by PRIMA is passive and stable. PRIMA (system of size q, q<<n) (system of size n) where UCSD CSE 245 Notes – SPRING 2006

  7. PRIMA • step 1. Circuit Formulation • step 2. Find the projection matrix Vq • Arnoldi Process to generate Vq UCSD CSE 245 Notes – SPRING 2006

  8. PRIMA: Arnoldi UCSD CSE 245 Notes – SPRING 2006

  9. PRIMA • step 3. Congruence Transformation UCSD CSE 245 Notes – SPRING 2006

  10. PRIMA: Properties • Preserves passivity, and hence stability • Matches moments up to order q (proof in next slide) • Original matrices A and C are structured. • But and do not preserve this structure in general UCSD CSE 245 Notes – SPRING 2006

  11. PRIMA: Moment Matching Proof Used lemma 1 UCSD CSE 245 Notes – SPRING 2006

  12. PRIMA: Lemma Proof UCSD CSE 245 Notes – SPRING 2006

  13. SPRIM • Structure-Preserving Reduced-Order Interconnect Macromodeling • Similar to PRIMA except that the projection matrix Vq is different • Preserves twice as many moments as PRIMA • Preserves structure • Preserves passivity, stability and reciprocity UCSD CSE 245 Notes – SPRING 2006

  14. SPRIM • Recall • Suppose Vq is generated by Arnoldi process as in PRIMA. Partition Vq accordingly • Construct New Projection Matrix UCSD CSE 245 Notes – SPRING 2006

  15. SPRIM • Congruence Transformation • Now structure is preserved • Transfer function for the reduced order model UCSD CSE 245 Notes – SPRING 2006

  16. Traditional Y- Transformation • Conductance in series • Conductance in star-structure n0 n1 n2 n1 n2 n1 n1 n0 n3 n2 n2 n3 UCSD CSE 245 Notes – SPRING 2006

  17. TICER (TIme Constant Equilibration Reduction) • 1) Calculate time constant for each node • 2) Eliminate quick nodes and slow nodes • Quick node: Eliminate if • Slow node: Eliminate if • 3) Insert new R’s/C’s between former neighbors of N • If nodes j and k had been connected to N through gjN and gkN, add a conductance of value gjNgkN/GN between j and k • If nodes j and k had been connected to N through cjN and gkN, add a capacitor of value cjNgkN/GN between j and k UCSD CSE 245 Notes – SPRING 2006

  18. TICER: Issues • Fill-in • The order that nodes are eliminated matters • Minimum Degree Ordering can be implemented to reduce fill-in • May need to limit number of incident resistors to control fill-in • Error control leads to low reduction ratio • Accuracy • Matches 0th moment at every node in the reduced circuit. • Only Correct DC op point guaranteed UCSD CSE 245 Notes – SPRING 2006

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