A Fast Block Structure Preserving Model Order Reduction for Inverse Inductance Circuits

Transcription

1 A Fast Blok Struture Preserving Model Order Redution for Inverse Indutane Ciruits Hao Yu, Yiyu Shi, Lei He EE Dept, University of California Los Angeles, CA 99 David Smart Analog Devies In Wilmington, MA 8 davidsmart@analogom ABSTRACT Most existing RCL iruit redutions stamp inverse indutane L elements by a seond-order nodal analysis (NA) The NA formulation uses nodal voltage variables and desribes indutane by nodal suseptane This leads to a singular matrix stamping in general We introdue a new iruit stamping for RCL iruits using branh vetor potentials The new iruit stamping results in a first-order iruit matrix that is semi-positive definite and non-singular We all this as vetorpotential based nodal analysis (VNA) It enables an aurate and passive redution In addition, to preserve the struture of state matries suh as sparsity and hierarhy, we represent the flat VNA matrix in a bordered-blok diagonal (BBD) form This enables us to build and simulate the maromodel effiiently In experiments performed on several test ases, our method ahieves up to X faster modeling building time, up to X faster simulation time, and as muh as X smaller waveform error ompared to SAPOR, the best existing seond order RCL redution method Categories and Subjet Desriptors: B[Hardware]: Integrated iruits Design aids General Terms: Algorithms, Design Keywords: Indutane and Interonnet Modeling, Model Order Redution INTRODUCTION Indutane is important in analysis of high-performane interonnets The small sale of high-speed signal nets in on-hip integration an be well handled using the existing model order redution [] However, for system on a hip or on a pakage, there exist omponents with high omplexity and strong magneti oupling To ensure signal and power integrity, omplete RLC models are required A detailed D extration using the partial-element equivalent iruit (PEEC) model [] introdues densely oupled indutanes The extrated model is dense and large Sparsifiation and model order redution (MOR) are hene both needed to redue a large sale RLC iruit Compared to the dense partial indutane matrix (L), L matrix is easier to sparsify [, ], L elements are related to the drop of the branh vetor potential [] Existing model order redution tehniques [], however, stamp L in the MNA This paper is partially supported by NSF CAREER award CCR- 9/8 and UC-Miro fund from Analog Devies, Intel and Mindspeed Address omments to lhe@eeulaedu Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for profit or ommerial advantage and that opies bear this notie and the full itation on the first page To opy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speifi permission and/or a fee ICCAD, November 9,, San Jose, CA Copyright ACM // $ (modified nodal analysis) [] matrix with the first-order admittane Beause diretly stamping L in the MNA matrix leads to an asymmetri iruit stamping, a redution using PRIMA does not preserve passivity [] To passively redue the linear iruit ontaining L, ENOR [] stamps the nodal suseptane: Γ = E l L El T (E l is the inident matrix for indutane) It results in a seond-order NA (nodal analysis) matrix, whih an be passively redued SAPOR further improves the auray of orthonormalization by using the seond-order Arnoldi method [8] With detailed disussion in Setion, Γ and G matries in NA are reognized to be singular in general Thus the transfer funtion of NA annot be expanded at d (s = ) As a result, the redued state matries an not be diretly stamped in the iruit matrix together with ative devies for transient simulation To avoid singularity, SuPreme [9] applies the double-inversion to stamp L in MNA However, it introdues additional fatorization osts due to the double inversion In this paper, we introdue the branh vetor potential as a new state variable and obtain a first-order admittane that ontains L elements The new stamping is alled vetor-potential nodal analysis (VNA) The state matries in VNA are semi-positive definite and nonsingular, whih an be passively redued by ongruene transformation with the blok Arnoldi orthonormalization Compared to [9] with double inversions, our method is more effiient In addition, we propose a fast struture-preserving redution to build and simulate the redued maromodel effiiently Different from SPRIM [], a struture-preserving redution that relates the struture of the high-order system to the first-order system, our method aims at preserving the struture of state matries suh as the sparsity and the hierarhy In this paper, we extend a two-level bordered-blok diagonal (BBD) deomposition developed in BSMOR [], whih an onsider ouplings between bloks Note that no indutane was onsidered in [], and its moment mathing was not loalized In this paper, we show that the moments an be mathed loally by using only the diagonal bloks in the BBD state matries The maromodel hene an be effiiently onstruted and simulated Experimental results show that ompared to SAPOR, an existing seond order RCL redution method, our approah is up to X and X faster to build and analyze maromodels, respetively In addition, our method has X smaller waveform error The rest of this paper is organized as follows In Setion, we detail the limitations of stamping L in NA In Setion, we show that iruits ontaining L elements an be written in a first-order form using VNA, whih is non-singular and an be redued with preserved passivity We all it VOR (Vetorpotential based model Order Redution) method In Setion, we further introdue a BBD-struture preserving model order redution (BVOR) for large sized VNA iruits We present experiments in Setion and onlude in Setion Proofs of theorems and more details an be found in a tehnial report [] BACKGROUND Considering branh resistive (G), apaitive (C), indutive (L) elements, the external exitation urrent I(s) as input, and the probing voltage u(s) as output, the KCL and KVL equations in

2 x g v v v l m g i v v v l (a) i x v g s v v v g v v a v b a a a a v v a v () v - (b) s v - Figure : Two oupled wires with (a) indutanes and (b) suseptanes (S = L ) In addition, () is a VNAraph omposed by nodal voltage and branh vetor-potential for the two-wire example, b- branh stands for apaitive branh, and a-branh stands for indutive branh v i i v g+ g - g+ g - i - i - vi i vx -x -x i i x l m m l p p v i i (a) (b) () Figure : Stamping the first-order admittane in MNA matrix, (a), (b) and () represent G, C and B in Eqn () G is non-singular the s-domain are Ei T b =, ET v n = v b, () E = [E g E E l E i ] T is the inident matrix, i b = [i g i i l i i ] T and v b = [v g v v l v i ] T are branh urrent and voltage, respetively, and v n is nodal-voltage The subsripts (g,, l, i) stand for the orresponding omponents for (G, C, L, I) For example, the branh urrents are determined by i g = Gv g, i = scv, v l = sli l, i i = I(s) () By defining nodal admittane matries: E gge T g and C = E CE T, and only reserving the nodal-voltage v n ( R nv, n v is the number of nodal-voltage variables) and indutive branhurrent i l ( R n i, n i is the number of indutive branh-urrent variables) as the state variables, () and () an be represented in the MNA matrix with the first-order admittane and (G + sc)x(s) = BI(s) u(s) = B T x(s) () vn x(s) =, B = il G El E T, C = l Ei, C L () v vg+ g g+ g (a) v vx -x -x x () v v s -s s -s - (b) p p v Figure : Stamping the seond-order admittane in NA matrix, (a), (b) and () represent for G, Γ, C and B in Eqn () G and Γ are both singular for matries Note that S = L in the table when stamping in L Note that the input and output ports are assumed to be idential Fig presents the G, C and B for the iruit example in Fig Note that G and C beome» G E l L E T l, C =» C I when stamping in L The differene between () and () is that the state matrix G in () does not satisfy: G + G T, ie, it is not symmetri and semi-positive definite Therefore, diretly applying the ongruene transformation by projeting () does not preserve passivity [] With G and C matries defined in (), () an be written as (d) () (G + sc)v n + E l i l = E i I(s), si l = L E T l vn () It is in fat the KCL s law Beause in most appliations only nodal-voltage v n is of interest, the indutive branh-urrent i l an be eliminated as an intermediate variable Therefore, () an be further written in the NA matrix with the seond-order admittane (sc + G + Γ/s)x(s) = E i I(s) u(s) = E T i x(s), () x(s) = v n, Γ = E l L E T l Γ is the nodal suseptane similar to G and C In addition, the time-domain response by NA is obtained by (G + C h + Γ h)vn(t + h) = C h vn(t) E li l (t) E i I(t + h) (8) at time instant t with step h Fig shows the stamping of Γ for the example of two wires in Fig In this form, G, C and Γ are all symmetri and positive definite SAPOR [8] an passively redue () by onstruting orthonormalized olumns to span a seond-order Krylov subspae Yet, there exist limitations related to the seond-order NA stamping for L elements It is primarily due to the singularity Here the singularity means: (i) the iruit matrix is indefinite at some frequeny point or time instant; and (ii) the iruit matrix is rank-defiient, ie, there exists linear dependene between olumns or rows Correspondingly, we have the following observations: Observation Nodal admittane is indefinite at d Due to the stamping of Γ, it is obvious that at d (s, or h ), the nodal admittane in () and (8) beomes indefinite (approahing to infinity)

3 Observation A PEEC-based extration results in an interonnet model with π-topology, R and L are serially onneted [] for one piee of interonnet Assuming that there exists d-path for the interonnet network, then stamping R and L separately into G and Γ by NA results in a singular stamping Observation an be understood as follows If there are N R resistors and N L indutors serially onneted, then the number of nodal-voltage variables is N R + N L + In other words, G matrix has only N L stamped resistors but has total N R + N L + nodalvoltage variables Therefore, G is rank-defiient, ie, singular The same reason an explain why Γ is singular These singularities are learly shown in Fig Here we assume that there exists d-path for interonnet network It an be always satisfied when onneting interonnet network with input soures or ative devies (See soure resistors in Fig (a)) But the singularity in NA remains even when onneted externally [] A non-singular iruit matrix is needed by an aurate model order redution and a SPICE-like time-domain simulation Beause Γ is singular, the seond-order admittane K = s C+s G+ Γ beomes indefinite (approahing to infinity) at d (s = ) As a result, the moment generation matrix in SAPOR [8] has large numerial error around d as well, and hene the redued model expanded at d is inaurate (See Fig later on) Therefore, SAPOR needs to hoose a non-zero expansion point s to ensure that A is not singular s often uses a large value due to the magnitude differene between G, C and Γ matries This leads to an inaurate mathing at the low-frequeny region (See Fig later on) More importantly, a redution with nonzero s expansion results in iruit matries eg, ec and eγ that do not have real values As a result, it is inonvenient to stamp them in a iruit matrix together with ative devies for time-domain simulation In ontrast, MNA [] uses indutive-branh urrent i l to desribe indutane (See ()) It ombines the singular G together with E l (desribing branh indutors) The resulting G is nonsingular For example, if we assume there is an indutive-branh urrent i k between voltage node v i and v j, the ombined matrix G below has nonzero entries at node v i and v j, ie, v i v j i k (9) Therefore, the ombined G is non-singular even when there is no ondutane stamped at v i and v j Ensuring the non-singularity of G is ritial for simulation Beause in both model redution and time-domain simulation, G needs to be fatorized aurately Moreover, L is stamped together with C in MNA, and the resulting sc beomes a zero matrix at d (s = ) Hene, the MNA is definite at d, different from NA whih is indefinite (Observation ) In summary, the singularity is only related to how we onstrut the iruit matrix Only MNA an orretly desribe indutanes without a singular stamping However, MNA is not symmetri and positive definite when stamping inverseindutane This will be solved below VNA MATRIX Stamping L by Vetor Potential Variable We start with the differential Maxwell equations in terms of the vetor potential A []: A = µj, A t = E () J is urrent density, E is eletrial field, and µ is permeability onstant For eah filament (a onstant branh urrent), v a vg+ g s -s - g+ g s - -s v a - - v a v x -x -x x a s s p p v a (a) (b) () Figure : Stamp first-order admittane in VNA matrix, (a), (b) and () represent G, C and B in Eqn() G is non-singular vabaaaaa vabaaaaa vg+ v g s -s - - a -ss a s g+ - - g s s ss s b- b- x a - a a - a - a - a - - a - a - - a - a - pp v a b a a a a a (a) (b) () Figure : Stamping the first-oder admittane in a bordered-blok diagonal VNA matrix, (a), (b) and () represent G, C and B in Eqn(9) G is non-singular by defining a branh vetor potential A l as the average volume integral of A within the volume τ: A l = Z Adτ, τ τ () an be rewritten in the following iruit equations: L A l = i l, A l t = E l v n () with E l and v n defined in () The detailed derivation an be found in [] However, [] only presented a SPICE-ompatible equivalent iruit, but did not disuss how to stamp the VPEC in the iruit matrix In this paper, we show that by using branh vetor potential A l as a new state variable to replae the i l, L element an be stamped into a iruit matrix that is non-singular and an be passively redued Note that () leads to (L E T l )vn = sl A l () In addition, aording to KCL law, we have (G + sc)v n + E l (L A l ) = E i I(s) () As a result, by introduing a new state variable vn x =, A l

4 a first-order admittane an be obtained B = Beause Gx(s) + scx(s) = BI(s) u(s) = B T x(s), ()» G (E l L ) (L E T l» ) Ei G + G T = C = C L () G, C + C T C = L, () G + G T and C + C T are symmetri and semi-positive definite in VNA In addition, the G for interonnet is non-singular in VNA Let L kk at branh i k be an element between node v i and v j The produt of E l L is i k v i v j i k v i L kk v j L kk i k i k L kk =, () whih still has nonzero entries at v i and v j Therefore, stamping E l L and its transpose together with G results in a non-singular G similarly as MNA does, and Fig presents G, C and B in VNA for the two-wire example In ontrast, both [] and [8] use the NA form, whih ontains singular G and Γ Reduing L in First-Order Form After stamping, we an perform model order redution similar to PRIMA Solving () results in a transfer funtion H(s) = B T (G + sc) B By expanding of H(s) at s (s = s + σ), it an be easily verified that the result is ontained in the blok Krylov subspae: {A, AR, A q R, } with two moment generation matries: A = (G+s C) C and R = (G+s C) B A projetion matrix Q that spans the qth blok Krylov subspae K(A, R, q) = {A, AR, A q R} Q an be onstruted from the blok Arnoldi method [] Using Q to projet G, C and B matries respetively, we obtain Ĝ = Q T GQ, Ĉ = Q T CQ, ˆB = Q T B (8) The resulting redued transfer funtion is Ĥ(s) = ˆB T (Ĝ+sĈ) ˆB We all this redution as VOR (nodal Vetor-potential based model Order Redution) method The redued model Ĥ(s) has the following properties [] Theorem The first q blok moments expanded at s are idential for Ĥ(s) and H(s) Theorem When the input and output are symmetri, Ĥ(s) projeted by Q is passive However, the redution by VOR is onstruted and projeted in a flat fashion, and hene the aording maromodel is ineffiient to onstrut and simulate To improve effiieny, we further introdue a blok strutured redution below BBD STRUCTURED REDUCTION Bordered-Blok Diagonal Representation The first step in BBD deomposition is to partition the VNA iruit into several bloks by branh-tearing [] Note that branhtearing an onsider ouplings between different bloks A min-ut multilevel algorithm [] is used to partition a VNA graph into user-speified m bloks, eah blok with size n bi and n pi external ports Sine the number of indutive branhes is muh larger than that of apaitive branhes, we define the graph omposed by indutive branhes as VNA graph The resulting m bloks with no oupling in between are at the bottom level of BBD The top-level is one global interonnetion blok that onnets all bloks by orresponding ut matries The interonnetion blok has size n and ontains all oupling branhes between any pair of bloks at the bottom level Preisely, the resulting BBD system equation is Y x = Bu (9) Y X Y X Y = Y m X m (X ) T (X ) T (X m ) T Z and B B B = B m x = [x, x,, x m, i ] T, u = [u, u,, u m, ] T For eah blok Y i, x i is the state variable, the first-order VNA admittane is G ii + sc ii ( R n bi n bi ), B i is the exitationport inidene matrix ( R n bi np i ) for external urrent soures, and X i is the onnetion-port inidene matrix (ut matrix) ( R n b i n ) For the global interonnetion blok Z at the bottom, i is the state variable, (Z g) + s(z ) ( R n n ) is its branh impedane matrix It is diagonal and ontains all oupling branh impedanes desribed by the set of ut matries of X i Beause of the branh-tearing, all external soures are ounted at eah blok by B i, and there are no external soures in Z Note that the off-diagonal blok in the original flat VNA matrix an be reovered by G ij + sc ij = (X i ) T (Z ) (X j ) () As Z is a diagonal matrix, inverting Z is effiient Fig () presents a VNAraph for the two-wire example BBD representation introdues a sparse representation, but the overall system size is inreased by Z Below, we show that the redution and fatorization in BBD form an be performed at the blok level with redued omputational ost and memory requirement BBD-Strutured Redution We first onstrut a blok diagonal projetion matrix, Q = diag[q, Q,, Q ] () Q i R n bi q ( i m) Eah projetion blok Q i is onstruted from eah diagonal blok in (9) independently, K(A ii, R ii, q) Q i () Note that Q is an identity matrix I ( R n n ), beause there are no exitation soures for blok Z and Z ontains only diagonal entries

5 V (db) 9 8 SAPOR (order 8) VOR (order 8) V 8 VOR (order 8) Beause Q T Q = I, ie, eah olumn of Q is still linearly independent and the total olumn-rank of Q (rank=mq) is inreased by a fator of m ompared to the olumn-rank of Q (rank=q) The m-times inreased olumn-rank means m-times more poles an be approximated after projetion Deompose the BBD admittane into the diagonal and offdiagonal parts: Y = (Y g + Y x) + s(y + Y x) Freq (Hz) (a) x Time (ps) (b) the diagonal parts are Y g = diag[g,, G m, (Z g) ], Y = diag[c,, C m, (Z ) ] V (db) SAPOR (order 8) VOR (order 8) V (db) 9 8 SAPOR (order 8) VOR (order 8) and the off-diagonal part is X Y x = Xm X T Xm T Freq (Hz) () Freq (GHz) (d) Figure : Frequeny and time domain waveform omparison of full-mna, SAPOR (order=8) and VOR (order=8) The redued models are expanded lose to d (s = Hz) in (a) and (b)vor and original are visually idential The redued model by SAPOR an not onverge in time-domain simulationin addition, the redued models are expanded at s = GHz in () and (d) VOR is idential to the exat MNA in both ranges,but SAPOR has large error at low-frequeny range We an obtain the order redued state matries by projeting Q: ey g + e Y x = Q T (Y g + Y x)q, e Y + e Y x = Q T (Y + Y x)q, and hene define the transfer funtion e B = Q T B eh(s) = eb T [( ey g + ey x) + s( ey + ey x)] eb () We all this redution as BVOR (BBD based VOR) method In addition, we define the moment generation matries for the original BBD as A = [(Y g + Y x) + s (Y + Y x)] (Y + Y x) R = [(Y g + Y x) + s (Y + Y x)] B, and those for its diagonal bloks as A = (Y g + s Y ) Y, R = (Y g + s Y ) B Aordingly, we have the following properties for redued e H(s) [] Theorem The qth-blok Krylov spae of BBD system is ontained by the qth-blok Krylov spae spanned by the diagonal blok of the BBD system, ie, K(A, R, q) K(A, R, q) Therefore, the first q expanded blok moments at s are idential for eh(s) and H(s) Theorem When the input and output are symmetri, eh(s) projeted by Q is passive Figure : BBD struture-preserving of state matries G and C before and after redution, NZ is the number of non-zero entries Moreover, beause the strutured projetion preservers the BBD struture, it enables a two-level domain-deomposition proposed in [] to solve eah redued blok independently Eah redued blok matrix is first solved individually with LU fatorization and substitution The results from eah redued blok are then further used to solve the oupling blok The final x i of eah redued blok is updated with the result from the oupling urrent i As shown by experiments in Setion, beause building and solving the sparse BBD system are both performed at blok level, the overall runtime is redued although the system size beomes larger than the original MNA However, there is a trade-off of sparsity and partition-time When inreasing the blok number, the resulting BBD form beomes sparser but the partition time beomes larger to onstrut the ut matrix

6 Table : Runtime omparison of SAPOR, VOR and BVOR with similar auray kt size blok Original(MNA) SAPOR(NA) VOR(VNA) BVOR(BBD-VNA) (before:after) # sim(s) build(s) sim(s) build(s) sim(s) build(s) sim(s) bus8x : bu : 9e 9 89 bus8x : 8e 9 e 9 9 busx :8 NA NA NA NA NA 88 lktree : 9e lktree : 8 e 9 lktree : 8e lktree :8 NA NA NA NA NA 8e mesh 8:8 8 8 mesh : e 9 8 mesh : 8e mesh 8: NA NA NA NA NA e 9e NUMERICAL EXPERIMENTS Numerial experiments are presented to demonstrate the auray and effiieny of the proposed V OR and BV OR methods They are ompared with the exat MNA and SAPOR [8], the best existing seond-order NA based redution All methods are implemented in MATLAB and C Experiments are run on a Linux workstation with Intel Pentium IV G CPU and G RAM The RCL iruits are extrated for buses, lok-trees and meshes Model order redution is first performed to obtain the redued state matries that are stamped bak for the frequeny and timedomain simulations The Bakward-Euler method is used to obtain the time-domain transient waveform The first example is an 8-bit bus with segments per bit, similar to the example used by SAPOR [8] We first expand firstorder admittane in VNA and the seond-order admittane in NA around d All redutions have an order of q = 8 The nearend of the first bit is exited with a unit urrent impulse, and the response at far-end is observed Fig (a) and (b) show the frequeny and time domain responses of the exat MNA, redued models by SAPOR and by VOR Beause state matries in NA are lose to singular, SAPOR shows large numerial errors beyond GHz In ontrast, VOR has a similar auray as the exat MNA for up to GHz The redued model by VOR is also as aurate as MNA in time-domain transient simulation However, the one by SAPOR does not onverge In Fig () and (d), we further expand VNA and NA at a non-zero frequeny point s = GHz SAPOR onverges at high frequeny range as VOR does, but it is still inaurate in low frequeny range when ompared to VOR The redued model by BVOR has up to X smaller error than that by SAPOR The seond example is a -bit bus with segments per bit, shielding is inserted for every 8 bits The partitioning results in bloks, eah with 8 resistors, apaitors, and around nodal suseptors The near end of the first bit in eah blok is exited with a unit urrent impulse BVOR is applied to redue eah blok independently, but VOR, SPRIM and SAPOR are applied to the entire iruit, all with order q = Fig (a) and (b) show the non-zero BBD pattern of the G matrix in VNA before and after the redution, and Fig () and (d) show those for the C matrix in VNA Clearly, the redued state matries preserve the BBD struture and are sparse G e and ec have a % and % sparsifiation ratio, respetively Table ompares runtime for different typed RCL iruits with different size (number of state variables), % of nodes in eah iruit have unit impulse urrent soures The runtime here inludes both the maromodel building and simulation time The redution and simulation are preformed in the frequenydomain All redued models have the same size and similar auray For a iruit with elements (bus8x), BVOR redues the orthonormalization ost and hene is X (s versus 9s) faster than VOR to build maromodels BVOR is also X (s versus 9s) faster to build than SAPOR for the same iruit The additional speedup is due to the fat that the firstorder VNA matrix used in VOR is sparser than the NA matrix used in SAPOR Moreover, for the same iruit (bus8x) as the BBD struture is preserved after the redution, the two-level analysis in BVOR further redues the simulation by X and 9X when ompared to SAPOR and VOR, respetively CONCLUSIONS Existing NA based seond-order model redution has singular matrix stamping for RCL iruits This paper presented a vetor-potential based nodal analysis (VNA) to aurately stamp L element and to enable a passive and simpler first-order redution By further representing the flat VNA matrix into a bordered-blok diagonal (BBD) form, moment mathing an be ahieved loally by only using diagonal bloks in the BBD matrix As a result, eah blok an be redued independently and hene orthonormalization ost is minimized Moreover, the redued model is sparse and an be effiiently analyzed in a twolevel fashion In experiments performed on several test ases, our method ahieved up to X faster modeling building time, up to X faster simulation time, and as muh as X smaller waveform error ompared to SAPOR REFERENCES [] A Odabasioglu, M Celik, and L Pileggi, PRIMA: Passive redued-order interonnet maro-modeling algorithm, IEEE Trans on CAD, pp, 998 [] A E Ruehli, Equivalent iruits models for three dimensional multiondutor systems, IEEE Trans on MTT, pp, 9 [] A Devgan, H Ji, and W Dai, How to effiiently apture on-hip indutane effets: introduing a new iruit element K, in Pro ICCAD, [] H Yu and L He, A provably passive and ost effiient model for indutive interonnets, IEEE Trans on CAD, pp 8 9, [] C W Ho, A E Ruehli, and P A Brennan, The modified nodal approah to network analysis, IEEE 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