1. QD C - + + - QD C - + + - QD C - + + - QD C - + + - Chapter 2 Interconnect Analysis Model Order Reduction Prof. Lei He Electrical Engineering Department University of California, Los Angeles URL: eda.ee.ucla.edu Email: lhe@ee.ucla.edu

2. 2 Math Representation of RLC Circuits Any RLC circuit can be represented by a first order differential equation G x(t) + C = B u(t) (G+ sC)x(s) = Bu(s) (Laplace(s) domain) u is an mx1 vector for the inputs of the circuit (e.g. current sources) x is an Nx1 vector for the response of the circuit (e.g. node voltage) G and C are NxN sparse matrices corresponding to the R, L, C element values and their connections B is an Nxm matrix indicating the locations of the current sources dx(t) dt

3. 3 The Curse of Complexity Number of nodes in an RLC Circuit: N  Need to play with N x N matrices  Ω(N 2 ) Floating Point Operations (FLOP) Number of possible input patterns m  Ω(N 2 m) FLOP E.g. One block in Intel Pentium μP [ICCAD’04]  N=349,706, m=36,129  4,418,370,274,646,244 FLOP  4,418,370 seconds (1 FLOP/ns)  51 days …

4. 4 Can we save our computers? The circuit may need to be repeatedly solved for many different inputs u(t) => Model Order Reduction x(t) can be obtained in two ways:  x(t) = h(t) conv u(t), x(s) = H(s)*u(s) => reduce h(t)  Directly solve differential equation => reduce G, C sizes transfer function

5. 5 Outline  Part I:  Moment calculation  AWE  PRIMA  Part II:  1 st and 2 nd order delay model  Delay of one stage

6. 6 Transfer Function of RLC Circuits The system equation in s domain (G+sC)x(s)=Bu(s) For simplicity, we will consider a single-input-single-output (SISO) circuit, then u is a scalar for the input current source x is an Nx1 vector for the node voltage of the circuit B is an Nx1 vector indicating the location of the current source, e.g. B=(0 0 1 0 0 … 0) T indicates the current source at node 3 x(s) can be solved x(s) = (G+sC )-1 Bu(s) The output voltage at one node can be expressed as y(s)=L T (G+sC) -1 Bu(s) L is an Nx1 vector selecting the output node location, e.g. L=(0 0 0 1 … 0) T selects the voltage at node 4 as output Transfer function H(s) = L T (G+sC) -1 B

7. 7 Moments of H(s) Moments of H(s) are coefficients of the Taylor’s Expansion of H(s) about s=0

8. 8 Some Calculations H(s) = L T (G + sC) -1 B = L T [G(I + sG -1 C)] -1 B = L T (I + sG -1 C) -1 G -1 B (AB) -1 = B -1 A -1 = L T (I - sG -1 C + s 2 (G -1 C) 2 - …)G -1 B (1+x) -1 =1-x+x 2 -… = L T G -1 B - s L T (G -1 C)(G -1 B) + s 2 L T (G -1 C) 2 (G -1 B) - … 0th order moment m (0) : L T G -1 B 1st order moment m (1) : -L T (G -1 C)(G -1 B) 2nd order moment m (2) : L T (G -1 C) 2 (G -1 B) …… kth order moment m (k) : (-1) k L T (G -1 C) k (G -1 B)

9. 9 Expansion at Arbitrary Frequency m (k) = (-1) k L T (G -1 C) k (G -1 B) implies that G is invertible. If not, we can do expansion at some frequency different from s=0 Let s = s 0 + σ, where s 0 is an arbitrary, but fixed expansion point such that G+s 0 C is non-singular, then H(s) = L T (G + sC) -1 B => H(σ) = L T (G + s 0 C + σ C) -1 B = L T [I-σ(G+ s 0 C) -1 C] -1 (G+s 0 C) -1 B Denote A= -(G+ s 0 C) -1 C, R= (G+s 0 C) -1 B, then H(σ) = L T (I-σ A) -1 R

10. 10 Taylor Expansion and Moments Expansion of H(σ) about  = 0  Expansion of H(s) about s=s 0 Recursive moment computation: H(σ) = L T (I-σ A) -1 R

11. 11 Taylor Expansion and Moments (Cont’d) Expansion of H(s) around Recursive moment computation:

12. 12 Interpretation of Moment Computation Compute: Convert: InductorVoltage source CapacitorCurrent source When s 0 = 0, equivalent to DC analysis: – setting shorting inductors (0V) and opening capacitors (0A) –compute currents through inductors and voltages across capacitors as moments System equation: (G+sC)x(s)=Bu(s)

13. 13 Interpretation of Moment Computation (Cont’d) Compute: When s 0 = 0, equivalent to DC analysis: –setting voltage sources of inductor L= Lm L, current sources of capacitor C = Cm C –external excitations = 0 –compute currents through inductors and voltages across capacitors as moments Convert: InductorVoltage source CapacitorCurrent source System equation: (G+sC)x(s)=Bu(s)

14. 14 Interpretation of Moment Computation (Cont’d) Compute: When s 0 = 0, equivalent to DC analysis: –setting moments as currents through inductors and voltages across capacitors –external excitations = 0 –compute voltage sources of inductors and current sources of capacitors Convert: InductorVoltage source CapacitorCurrent source System equation: (G+sC)x(s)=Bu(s)

15. 15 Moment Computation by DC Analysis  DC analysis: modified nodal analysis (used in original AWE ) sparse-tableau ……  Time complexity to compute moments up to the p- th order:p  time complexity of DC analysis  Perform DC analysis to compute the (i+1)-th order moments voltage across C j => the (i+1)-th order moment of C j current across L j => the (i+1)-th order moment of L j

16. 16 Advantage and Disadvantage of Moment Computation by DC Analysis  Computation of u k corresponds to vector iteration with matrix A ( ) Converges to an eigenvector corresponding to the eigenvalue of A with largest absolute value  Recursive computation of vectors u k is efficient since the matrix (G+s 0 C) is LU-factored exactly once

17. 17 Numerical Problems for Matrix Power Assume λ 1, λ 2, … λ N are the eigenvalues of matrix A, with λ 1 the largest in absolute value, then

18. 18 Outline  Moment calculation  AWE  PRIMA

19. 19 Pade Approximation H(s) can be modeled by Pade approximation of type (p/q): where p < q << N Or modeled by q-th Pade approximation (q << N): Formulate 2q constraints by matching 2q moments to compute k i & p i

20. 20 General Moment Matching Technique Basic idea: match the moments m -(2q-r), …, m -1, m 0, m 1, …, m r-1 (i) initial condition matches, i.e. (ii) When r = 2q-1: Final value theorem

21. 21 Compute Residues & Poles match first 2q-1 moments EQ1

22. 22 Basic Steps for Moment Matching Step 1: Compute 2q moments m -1, m 0, m 1, …, m (2q-2) of H(s) Step 2: Solve 2q non-linear equations of EQ1 to get Step 3: Get approximate waveform Step 4: Increase q and repeat 1-4, if necessary, for better accuracy

23. 23 Moment Matching by AWE [Pillage-Rohrer, TCAD’90] Recall the transfer function obtained from a linear circuit When matrix A is diagonalizable

24. 24 q-th Pade Approximation Pade approximation of type (p/q): q-th Pade approximation (q << N): Equivalent to finding a reduced-order matrix A R such that eigenvalues l j of A R are reciprocals of the approximating poles p j for the original system

25. 25 Asymptotic Waveform Evaluation Recall EQ1: Let

26. 26 Asymptotic Waveform Evaluation (Cont’d) Rewrite EQ1: Let where Solving for k: Need to compute all the poles first

27. 27 Structure of Matrix A R Matrix: Therefore, A R could be a matrix of the above structure has characteristic equation: Note that: Characteristic equation becomes the denominator of H q (s):

28. 28 Solving for Matrix A R Consider multiplications of A R on m l produces

29. 29 Solving for Matrix A R (Cont’d) After q multiplications of A R on m l produces Equating m’ with m:

30. 30 Summary of AWE Step 1: Compute 2q moments, choice of q depends on accuracy requirement; in practice, q  5 is frequently used Step 2: Solve a system of linear equations by Gaussian elimination to get a j Step 3: Solve the characteristic equation of A R to determine the approximate poles p j Step 4: Solve for residues k j

31. 31 Numerical Limitations of AWE  Due to recursive computation of moments Converges to an eigenvector corresponding to an eigenvalue of matrix A with largest absolute value Moment matrix used in AWE becomes rapidly ill- conditioned Increasing number of poles does not improve accuracy Unable to estimate the accuracy of the approximating model  Remedial techniques are sometimes heuristic, hard to apply automatically, and may be computationally expensive

32. 32 Outline  Moment calculation  AWE  PRIMA

33. 33 General Idea Any RLC circuit can be represented by a first order differential equation G x(t) + C = B u(t) (G+ sC)x(s) = Bu(s) (Laplace(s) domain) Can we reduce the equation size? Reduce the number of variables (column # of G and C) Reduce the number of equations (row # of G and C) dx(t) dt

34. 34 Reduce the Number of Variables Note: q << N reduced original Guess x can be represented by linear combination of some vectors u 1, …, u q

35. 35 Illustration dx(t) dt G x(t) + C = B u(t) N N N 1 1 m 1 m q m + = B u(t) N 1 q 1 1 m q q

36. 36 Reduce the Number of Equations + = B u(t) N 1 q 1 1 m Left multiply N q q q q + = u(t) q 1q 1 1 m q q q m m

37. 37 nxn qxq nxq qxn Projection Framework G

38. 38 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 Krylov Subspace Vectors? Approaches for picking V q and U q

39. 39 Aside on Krylov Subspaces - Definition The order k Krylov subspace generated from a matrix E and a vector b is defined as

40. 40 Taylor series expansion: A=-G -1 C, R=G -1 B Intuitive view of Krylov subspace choice for U q 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

41. Combine point and moment matching: multipoint moment matching Multiple expansion points give larger band Multiple 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

42. 42 Vectors will line up with dominant eigenspace! Need for Orthonormalization of U q cannot be computed directly

43. 43 Need for Orthonormalization of U q (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

44. 44 Normalize new vector For i = 1 to q-1 Generates k+1 vectors! Orthogonalize new vector: Remove the projection on other normalized vectors For j = 1 to i Orthonormalization of U q :The Arnoldi Algorithm end

45. 45 We know how to select U q now… but how about V q ?

46. 46 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!)

47. 47 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

48. 48 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:

49. 49 Sufficient conditions for passivity Sufficient conditions for passivity: Note that these are NOT necessary conditions (common misconception)

50. 50 Congruence Transformations Preserve Positive Semidefinitness Def. congruence transformation same matrix  Property: a congruence transformation preserves the positive semidefiniteness of the matrix  Proof. Just rename

51. 51 PRIMA (for preserving passivity) (Odabasioglu, Celik, Pileggi TCAD98) Select V q =U q with Arnoldi Krylov Projection Framework: Use Arnoldi: Numerically stable Use Arnoldi: Numerically very stable

52. 52 PRIMA preserves the moments of the transfer function up to the q-th order, i.e., Moment Matching Theorem

53. 53 Summary: Conventional Design Flow Funct. Spec Logic Synth. Gate-level Net. RTL Layout Floorplanning Place & Route Front-end Back-end Behav. Simul. Gate-Lev. Sim. Stat. Wire Model Parasitic Extrac.

54. 54 Summary: Parasitic Extraction Parasitic Extraction thousands of wires e.g. critical path e.g. gnd/vdd grid tens of circuit elements for gate level spice simulation identify some ports produce equivalent circuit that models response of wires at those ports

55. 55 Summary: Model Order Reduction Electromagnetic Analysis Model Order Reduction tens of elements million of elements thin volume filaments with constant current small surface panels with constant charge

56. 56 Summary We have presented how to calculate moments of RLC circuits We have discussed about AWE and PRIMA  Both are based on the moment matching  AWE has numerical problems and can only match 3-4 moments  PRIMA is inherently stable and can match high order moments  PRIMA can preserve passivity

57. 57 References L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Trans. Computer-Aided Design, vol. 9, pp. 352–366, Apr. 1990. Altan Odabasioglu, Mustafa Celik, and Lawrence T. Pileggi, “PRIMA: Passive Reduced-Order Interconnect Macromodeling Algorithm”, IEEE Trans. Computer-Aided Design, Vol. 17, pp. 645-653, Aug. 1998.