EE5900 Spring Lecture 5 IC interconnect model order reduction Zhuo Feng

Transcription

1 EE59 Spring Parallel VLSI CAD Algorithms Lecture 5 IC interconnect model order reduction Zhuo Feng 5. Z. Feng MU EE59

2 In theory we can apply moment matching for any order of approximation But in practice it s not so simple: Approximations of stable systems can be unstable Finite precision problems Inherent instability of Pade approximations 5. Z. Feng MU EE59

3 ypical Application : transfer function/impedance and driving pt. impedance for logic gate + interconnect 5.3 Z. Feng MU EE59

4 v o v IN v IN Driving pt. + admittance v v IN o Driving pt. admittance 5.4 Z. Feng MU EE59

5 Moments of the input voltage current represent moments s V IN of the driving point admittance if I s V IN I V V IN s s Y IN s 5.5 Z. Feng MU EE59

6 For an RC tree (all C s to gnd, no loops of R s): Y m Y all capacitors m Y Y Y IN s m s m s m 3 Y 3 s C6 m k C5 m k C m C5 5 k C C6 6m k C m k C m k C3 mk C 4 m k V in m k C C Cm k C m k C C3 3m k C C 4 4m k 5.6 Z. Feng MU EE59

7 Calculate an approximate model via usual approach: Y Y Y m s m s m3 s 3 Y Y Y m m s m3 s q i i s pˆ i kˆ hen multiply by s: Yˆ IN s q s kˆ i s p i ˆ i For many applications we synthesize a ckt to represent this driving pt. admittance 5.7 Z. Feng MU EE59

8 Driving pt. synthesis for RC ckts: ˆ k q kˆ s I q ˆ ˆ p k q q p k ˆ ˆ s V IN s Yˆ q q IN p s k s p s k s s Y ˆ ˆ ˆ ˆ ˆ q s I ˆ ˆ ˆ ˆ ˆ k s p s s I k s p k s I s V IN s V IN s I Z. Z. Feng Feng MU EE59 MU EE

9 Waveform/delay analysis ( nd order) v t v v t Ĥ s v t ransfer fcts are not easily synthesized more later We ll also show that we don t have to synthesize the driving pt. -- can insert num. integ. eqns directly 5.9 Z. Feng MU EE59

10 For lossy interconnects & gates, this flow works well For low-loss RLC ckts it is more difficult to capture dominant poles due to comparable decay rates O O O O O O j vt t 5. Z. Feng MU EE59

11 Attempts have been made at making moment-matching work for such problems (e.g. frequency shifting) But now there are much better ways! What if we use eigenvalue methods to calculate the dominant poles? V at a node of interest s L sa RU s A G C R G B m k Eigenvalues of A are ckt time constants L A k R 5. Z. Feng MU EE59

12 Attempts have been made at making moment-matching work for such problems (e.g. frequency shifting) But now there are much better ways! What if we use eigenvalue methods to calculate the dominant poles? V at a node of interest s L sa RU s A G C R G B m k Eigenvalues of A are ckt time constants L A k R 5. Z. Feng MU EE59

13 A series of papers on model order reduction L. Pillage and R.A. Rohrer, Asymptotic waveform evaluation for timing analysis Computer-Aided Design of Integrated Circuits and Systems, IEEE ransactions on Volume 9, Issue 4, April 99 Page(s): E. J. Grimme, Krylov projection methods for model reduction, Ph.D. dissertation, Univ. of Illinois, Urbana-Champaign, 997. P. Feldmann and R. W. Freund, Reduced-order modeling of large linear subcircuits via a block Lanczos algorithm, in Proc. 3nd IEEE/ACM Design Automation Conf., pp , Jun K. J. Kerns and A.. Yang, Preservation of passivity during RLC network reduction via split congruence transformations, in Proc. 34th IEEE/ACM Design Automation Conf., pp , Jun A. Odabasioglu, M. Celik, and L. Pileggi, PRIMA: Passive reduced-order interconnect macromodeling algorithm, IEEE rans. on Computer-Aided Design of CAS, vol. 8, no. 8, pp , Aug J. Phillips, L. Daniel, L. M. Silveira, Guaranteed passive balancing transformations for model order reduction, Design Automation Conference,. Proceedings. 39th June Page(s):5-57 P. Li and L. Pileggi, NORM: compact model order reduction of weakly nonlinear systems, Design Automation Conference, 3. Proceedings, June 3 Page(s): Z. Feng MU EE59

14 Eigenvalues A i i i n x n i-th n x eigenvector i-th scalar eigenvalue Build an S = hen AS x S x x x n Diagonal matrix of all eigenvalues 5.4 Z. Feng MU EE59

15 If A has a complete set of n eigenvalues (diagonalizable matrix) then the eigenvectors span the entire n-dimensional space --form a basis Example= n=3 z y x We can orthonormalize the eigenvectors so that this basis looks more like x, y, z 5.5 Z. Feng MU EE59

16 Matrix A expressed in terms of this basis is diagonal A i for all directions i i AS S S AS similarity transformation Eigenvalues of A and S AS are identical 5.6 Z. Feng MU EE59

17 If the eigenvalues are distinct, and we orthonormalize the eigenvalues, then S is an orthogonal matrix, and S S S AS S S x x x n x x x n Not true in general! 5.7 Z. Feng MU EE59

18 Congruence transformation B Y AY for some Y A & B are congruent Eigenvalues not necessarily the same (not a similarity transformation) If S is not orthogonal, then the congruence trans- formation does not preserve the eigenvalues S AS if S is not orthogonal! 5.8 Z. Feng MU EE59

19 What if we apply a congruence transformation to G and C directly If AS S GS S CS G C S G CS S CS GS S GS S GS? Eigenvalues are preserved! Amatrix 5.9 Z. Feng MU EE59

20 his congruence transformation is foundation of PRIMA We don t know eigenvalues, but we can approximate them via orthogonalized moments Krylov subspace S R AR A R Instead of an n x n S-matrix, we use an n x q approxi- mation to capture q-most dominant poles 5. Z. Feng MU EE59

21 If S is n x q, then congruence transformations on G and C tend to preserve q most dominant eigenvalues S GS S CS q x n n x n n x q q x q G ~ q q C ~ q q 5. Z. Feng MU EE59

22 PRIMA: use the Krylov subspace (moments) for the congruence transformation instead of eigenvectors erminology: PRIMA paper uses to denote the n x q matrix for congruence transformation instead of S R AR q A q R n 5. Z. Feng MU EE59

23 defines the subspace spanned by the first q moments But we must orthonormalize the Krylov vectors since moments become dominated by first eigenvalue and high frequency information is lost wo approaches to orthonormalization: Arnoldi Process: apply Gram-Schmidt orthogonalization as moments are calculated Lanczos Process: avoids long recurrence through all previous vectors, but at a price of algorithm complexity 5.3 Z. Feng MU EE59

24 PRIMA (with Arnoldi Process). Calculate R and normalize it R G B R first column of (a vector if B is a vector) where. Calculate l AR using G R R C G C Orthonormalize w.r.t. to produce nd column of 5.4 Z. Feng MU EE59

25 Add a term to make the vectors orthogonal: a a a Choose so that is orthogonal to Find y z y, z a such that, a a Inner product: nd column of 5.5 Z. Feng MU EE59

26 & 3. Use to calculate ; then orthonormalize w.r.t. a a a a a a For 3 it s a 3 x 3 problems, and so on 5.6 Z. Feng MU EE59

27 In Matlab it s easier to use QR to orthonormalize the vectors as they re computed Not the most efficient approach, but the simplest: QR of Columns of Q are the vectors that span the Krylov subspace QR of Z. Feng MU EE59

28 Why are Krylov vectors (moments) a good approximation of the dominant eigenvectors? We know that: lim k A k R k x Dominant eigenvalue Dominant eigenvector But since we orthogonalize, we keep focusing on the next most dominant eigen-direction 5.8 Z. Feng MU EE59

29 R is not the first eigenvector direction, but we d expect the first several Krylov vectors to span a subspace similar to that spanned by the first several eigenvectors You ll find that PRIMA works great for high orders of approximation, but is lousy for really low order (e.g. nd ) Probably could try moment shifting to get a more accurate starting vector or even shift to get accurate vector at each order 5.9 Z. Feng MU EE59

30 PRIMA q ~ C C PRIMA terminology: H ~ B H s L G sc B B ~ ~ L ~ L ~ ~ ~ s L G sc B ~ G G scalarq x for one response q x q q x for one input 5.3 Z. Feng MU EE59

31 o solve for poles & residues: ~ ~ s L G sc ~ B H ~ ~ L ~ sa ~ ~ R Eigendecompose ~ ~ ~ A SS A ~ A G R G q x q q x q containing dominant time constants ~ ~ ~ Substitute for G C A ~ ~~ ~ ~ ~ L SS ss S R ~ ~ ~ L S s S ~ R ~ ~ ~ L ~ S s S R C B 5.3 Z. Feng MU EE59

32 s s s s q ~ ~ L S S x q q x ~ ~ G ~ B H ~ s s q k kk s k 5.3 Z. Feng MU EE59

33 RLC example once again for the response at node 5 sl G G 5 sc 5 G 6 G 6 sc C G G G G sc 6 V in sc sc sc sc Z. Feng MU EE59

34 RLC Example L = (Response at Node 5) C = Z. Feng MU EE59

35 G = Need a particular formulation of G & B to preserve passivity: Sec. 6.6, Ref. B = Z. Feng MU EE59

36 A = Columns through Columns 8 through Z. Feng MU EE59

37 R = eigenvalues of A: i i Z. Feng MU EE59

38 normalized R = build up by adding A^i*R terms to it... = Z. Feng MU EE59

39 then orthonormalize using QR: Q = he R part of QR is not needed: r = Z. Feng MU EE59

40 = = Z. Feng MU EE59

41 Q = = and so on Z. Feng MU EE59

42 Using q=6: Gtilda = Ctilda = Btilda = Z. Feng MU EE59

43 Ltilda = Stilda = Columns through i i i i i.636+.i i.54-.4i i.7+.44i i i Columns 5 through Z. Feng MU EE59

44 LAMBDAtilda = i i If we use only 4 moments: LAMBDAtilda = i i 5.44 Z. Feng MU EE59

45 .6.56 s s 4 s 5 Z s poles 3 j Z s.56s 3 s s 5.7s.4s 4.3 Z(s) poles at : 3.74, 5 j, 5 j Drive Z(s) with a current and it oscillates at 5 rads/s Practical example shown in PRIMA paper 5.45 Z. Feng MU EE59

46 Passivity is much more important for multiports -- we can generally find passive models for single ports Example: calculate reduced order Y or Z parameters N-coupled lines example Modeled as N or N ports 5.46 Z. Feng MU EE59

47 One ports and transfer functions are often used for Static iming Analysis Z. Feng MU EE59

48 Multiport models required for interconnect when there is coupling between metal lines Aggressor Aggressor Victim 5.48 Z. Feng MU EE59

49 Analog circuits will often be multiports too M.O.R. works with controlled sources in the RLC circuits For digital or analog, the multiport is often Y or Z parameters v v N linear ckt I I N s Y s Y s V s Y N V s N s YNN s N s Find reduced order Y(s) model 5.49 Z. Feng MU EE59

50 Example: l C c G G 3 4 U i C C s i L i s G U Z. Feng MU EE59

51 Write eqns so that C matrix is non negative Write eqns so that C-matrix is non-negative definite v C C C 3 C C C C v v C C C C C C 4 L i i v L S S i i Z. Z. Feng Feng MU EE59 MU EE

52 v v G G G G G G u u v v G G G u i i S L i S E E N G Z. Z. Feng Feng MU EE59 MU EE E

53 Use L to isolate variables: i i G sc Bu N-port currents are: i L G sc Bu i currents at ports x Y(s) matrix voltages at ports v v v v il is i 3 4 S Z. Feng MU EE59

54 i i u u Y s L G sc B L sa R A G C R G B B is now a matrix Eigenvalues of A are the poles of Y(s) Z. Feng MU EE59

55 A k R Columns of represent the k-th moments when considering each port voltage input individually Y s L sa R R G B matrix Z. Feng MU EE59

56 A k R So we calculate moments in blocks, (size is n x N, where N is the # of ports) i i u u Z. Feng MU EE59

57 G and C are n x n, so the congruence transformation ~ ~ to G, C of q x q size will require to be q x n We match only yq q/n moments (but at all ports simultaneously) R AR A q N R For our examples, N=, so if we want q=6, we match 3 moments for each port Z. Feng MU EE59

58 Apply congruence transformation in the same way as we did for single-input-single-out (SISO) case But it is now MIMO (multi-input input multi-output) ~ C C ~ G G ~ B B Lˆ L Yˆ ~ ~ s Lˆ G sc B ~ Yˆ s B is provably passive as long as and C is nonnegative definite L Z. Feng MU EE59

59 Can stamp reduced order matrices, G ~, C ~ directly into circuit simulation matrix Nonlinear models are in time domain Stamp for all x NL V other NL elements I N ~ u p V p I N L ip ~ ~ ~ d ~ B G C xq dt Note that G ~, C ~ contain negative and off-diagonal terms that behave like controlled sources Z. Feng MU EE59

60 Can also eigendecompose and solve for pole and residues as we did for the SISO case Yˆ s Each term in has the same set of poles A ~ For a -port system, if poles are required, we only match m through m5 For a large N this can make block reduction method overly complex 5.6 Z. Feng MU EE59

61 It is more efficient to use a SIMO model for N-ports, but passivity is not ensured!. Solve for first column of Y(s) i u. Solve for the nd column of Y(s) i u Y(s) no longer has a common set of poles! 5.6 Z. Feng MU EE59

62 Realization of reduced order models in Spice --- State-space representation Y. Liu, Strojwas and Pileggi, ftd: An exact Frequency to ime domain conversion for Reduced Order RLC Interconnect Models, DAC98 S. Kim, Gopal and Pillage, ime domain macromodels for VLSI interconnect Analysis, IEEE CAD 94 Bracken, Raghavan and Rohrer, Interconnect Simulation with AWE, IEEE CAS, Z. Feng MU EE59

63 Passivity is still very important! RLC macromodel Impedance at port is changing with every NR iteration RLC macromodel 5.63 Z. Feng MU EE59

64 If we use successive chords for the nonlinear iteration, the impedance driving the ports can be fixed Newton-Raphson Successive Chord 5.64 Z. Feng MU EE59

65 Now we include the chord resistors as part of the model order reduction process RLC macromodel 5.65 Z. Feng MU EE59