CANAVERO F.

I. MAIO

I.S. STIEVANO

S. GRIVET-TALOCIA

English

CANAVERO, Flavio

GRIVET TALOCIA, STEFANO

MAIO, Ivano Adolfo

STIEVANO, IGOR SIMONE

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEModeling of Interconnect Junctions from measured scattering responses / CANAVERO F.; S. GRIVET-TALOCIA; I.MAIO; I.S. STIEVANO. - STAMPA. - III(2001), pp. 185-188. ((Intervento presentato al convegno 15th EuropeanConference on Circuit Theory and Design (ECCTD ’01) tenutosi a Helsinki (Finland) nel August 28-31, 2001.OriginalModeling of Interconnect Junctions from measured scattering responsesPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1409408 since: 2015-07-14T12:21:18ZModeling of interconnect junctions frommeasured scattering responsesF. G. Canavero,∗ S. Grivet-Talocia∗ I. A. Maio∗ and I. S. Stievano∗Abstract — This paper addresses the developmentof lumped equivalent circuits of interconnect junc-tions from their measured or simulated scatteringresponses. Two methods for the identification oflinear dynamic systems are applied and their per-formances are compared for accuracy and efficiency.1 IntroductionToday’s tendency towards an increasing use ofhigher frequencies both for RF and digital applica-tions results in unwanted parasitic electromagneticeffects on the interconnection structure of digitaland telecommunication systems. Signal distortionsare introduced due to the mutual coupling amongdifferent conductors. Therefore, the modeling ofthese structures has become an unavoidable stepin the analysis and design of high speed electronicsystems.In most cases, a complete electromagnetic simu-lation of the entire structure could be unaffordable(both in terms of computer memory needed andCPU time required). Therefore, it becomes cru-cial to derive equivalent models of subparts of thesystem. Such models must retain, in a given band-width, the major features of the details they repro-duce, and must be simple enough to be handled insubsequent simulations. The possibility to gener-ate circuit equivalents that are SPICE compatibleis usually sought for.This contribution addresses the modeling of lin-ear dynamic multiport elements from measuredor simulated responses, with the aim to generatelumped wideband equivalents of generic junctions.From a formal point of view, the modeling of mul-tiport elements from their responses by means oflumped equivalents is a parametric identificationproblem and many different identification methodscould be tried [1]. In this paper, we assess theperformances of two identification approaches thatseem particularly suited to the application at hand:the Block Complex Frequency Hopping (BCFH)method [2, 3] and the subspace identification meth-ods. The BCFH algorithm allows accurate approx-imations of input data over wide frequency bands∗Politecnico di Torino, Dipartimento di Elettronica,C. Duca Abruzzi 24, I-10129 Torino, Italy. E-mail:canavero@polito.it, Tel: +39-011-5644060, Fax: +39-011-5644099.with simple models having a meaningful physicalfoundation. On the other hand, in the control sys-tem area, subspace methods are well establishedas the most effective approach for the modeling ofmulti-input multi-output linear systems from theirresponses.We test both methods on realistic structures byusing quasi-matched scattering responses. Suchnetwork function are preferred because, for typicalinterconnect structures, they have the simplest be-havior and often allow the factorization of possibleideal delay terms.2 BCFH MethodThe BCFH algorithm is a moment matching or-der reduction method that can be exploited alsofor system identification, because its inputs are themoments of the response data [2, 3]. It works byestimating the true set of poles of the modeled cir-cuit element within the bandwidth of interest (i.e.,in a region of the complex angular frequency planewhere σ < 0 and |ω| < ωMAX). This is achievedby computing the moments of some input scatter-ing functions at several complex expansion points.The set of poles is then used to represent each scat-tering function with a partial fraction expansion,whose residues are computed by least square fittingthe function samples, as in [2].The estimation problem defined by the BCFHmethod is well conditioned and the obtained mod-els have a solid physical foundation, because theirpoles are approximations of the actual poles of themodeled multiport element. This guarantees thestability of estimated models and helps the controlof their passivity. In fact, as far as the accuracy ofthe identified models is good, the passivity condi-tion ∑j|Sij(jω)|2 ≤ 1, ∀i, j, and ω (1)where Sij(ω) is the scattering function describingwave transmission from port i to port j, is guar-anteed within the approximation bandwidth. Ac-tive spurious behavior of estimated models can ariseonly beyond ωMAX and can be compensated a pos-teriori by extending ωMAX to the region where theamplitudes of the scattering functions of the struc-ture become negligible.The applied approach can be extended to dis-tributed structure whose scattering characteristicsallow the factorization of delay terms. For thesecases, mixed models composed of lumped parts andof delay blocks can be used, thereby effectively in-cluding propagation effects without losing the ad-vantages of lumped equivalents.Finally, for the implementation of models asequivalent circuits, we replace wave variables withvoltages and currents in the scattering equationsand synthesize the resulting relations (e.g., see [5]).2.1 BCFH resultsThe BCFH approach has been tested on simpleideal multiport elements, obtaining good results.Even if the poles of quasi-matched scattering re-sponses are far from the imaginary axis, their esti-mations via BCFH turn out to be sufficiently accu-rate to yield good models.In order to check the performances of the ap-proach on a realistic test case we apply it to theintegrated circuit package of Fig. 1. This struc-ture is composed of an ideal conductor referenceplane, a dielectric layer and metallic lands betweenthe chip and the board. It is characterized by thewave variables measured at the ends of each landwith respect to the reference plane. The scatter-ing characteristics of such a multiport element arecomputed in the time domain by means of a threedimensional electromagnetic simulator based on thefinite integration technique. The symmetry proper-ties of the structure are exploited and its complete104×104 scattering matrix is obtained in 3.5 hoursof computation by a Pentium PC @ 450 MHz.Figure 1: 52-pin package modeled by scatteringdata and BCFHThe input data for the modeling process are theinput and output transient wave variables obtainedfrom the time-domain fullwave simulation. Suchresponses are used to compute the moments of theelements of the scattering matrix and to estimatetheir poles via BCFH [4]. Time and frequency nor-malization of responses as well as moment scalingare used to generate well conditioned matrices ofmoments.The search for poles of a scattering matrixelement is computationally inexpensive, since itamounts to compute its first 15 – 20 moments fora few complex expansion points. Also, many polesdetected from different scattering matrix elementsare common and, therefore, only a subset of thescattering matrix elements must be searched. Thecomplete set of estimated poles is then used to ap-proximate every element of the scattering matrixby a partial fraction expansion, whose residues areobtained by least square fitting.In this example, the BCFH search of the elementsSij , i = 1, 2, 3, 4 and j = 1, 2, 3, 4 (ports numberedin counter-clockwise direction as shown in Fig. 1)leads to the same set of three couples of complexconjugate poles. Such poles allow a good modelingof all scattering matrix elements Sij , i = 1, 2, 3, 4j = 1, 2, 3, 4 over the bandwidth of available data,that is 7GHz. As an example, Fig. 2 shows theamplitude of S11(jω) and the phase of S31(jω) ver-sus frequency compared with the corresponding re-sponses of the model defined by the mentioned setof poles. The accuracy of the model can be clearlyappreciated.0 1 2 3 4 5 6 700.10.20.30.40.50.60.7−2.5−2−1.5−1−0.500.511.5frequency    GHzFigure 2: Reference (solid lines) and model (dottedlines) values for |S11| (left scale) and 6 S31 (rightscale, rad)3 Subspace MethodsIn the last years, many contributions on the estima-tion of time invariant, linear dynamic state-spacemodels of multi-input multi-output systems havebeen published [6]. Such estimation approachesare collectively named subspace methods and arerelated to Matrix Pencil methods [7].A discrete-time linear state-space model is de-fined by {z(k + 1) = Az(k) +Bu(k)y(k) = Cz(k) +Du(k)(2)where z ∈ <r is the state vector and matricesA, B,C, D are the unknowns (parameters) of the model.The key step of subspace methods to approx-imate systems with model (2) is the estimationof the column space of the extended observabil-ity matrix from suitable matrices containing theinput/output perturbed data. The estimation isperformed by the span of the column or row spaceof the matrices of input/output data. Then, themodel unknowns, i.e., the A, B, C, D matrices,are directly obtained from the estimation of theextended observability matrix. With the aid ofsubspace methods, the estimation of state-spacemodels for circuit elements with a number of in-put/output of 10 and a number of states of theorder of 100 is practicable and the numerical com-plexity is not critical [6].3.1 Subspace estimation exampleOne of the ideal multiport elements used to test themodeling via subspace methods is the 2-port circuitshown in Fig. 3.   			 	Figure 3: Example 2-port circuit element (R1,3 =150Ω, R2 = 200Ω, C1 = 10µF , C2 = 1µF ).The continuos-time state-space equations forsuch a circuit (i.e., for the element under model-ing) are {ddtz(t) = Acz(t) +Bcu(t)y(t) = Ccz(t) +Dcu(t)(3)where the state vector is z = [va, vb]T , the outputvector collects the outcoming voltage waves y =[b1, b2]T (bj =12(vj − R0ij), j = 1, 2 and R0 =50 Ω), and the input vector collects the incomingvoltage waves u = [a1, a2]T (aj =12(vj + R0ij),j = 1, 2). With such definitions, the matrices of (3)writeAc = −103 5 · 1025 · 103 −104 , Bc = 103 00 104Cc = 0.25 00 0.25 , Dc = 0.5 00 0.5(4)In order to apply subspace methods we need adiscrete-time representation like (2). This is doneby applying standard conversion routines such asthe Matlab function c2d [9] to (3) and (4). In sucha way, with a sampling time T = 30µs, the discrete-time state-space representation (2) for this specificexample turns out to be defined by the followingA, B, C, D matricesA = 0.97144550 0.012761850.12761851 0.74173217B = 0.02956476 0.002020540.00202054 0.25927810C = Cc, D = Dc(5)Such state-space equations are then used to com-pute the response of the element under modelingy(k) (output identification sequence) to suitable in-put identification sequences u(k). Input and outputidentification sequences are processed by the sub-space algorithm to estimate the unknown parame-ters of a state-space model. The algorithm [8] is ap-plied to estimate matricesA andC. This is done onthe basis of the output identification sequence ob-tained by driving the element under modeling withwhite gaussian inputs u(k). Then, in order to getthe best model, matrices B and D are obtained byminimizing the error between the model and thereference responses to a different input identifica-tion sequence (e.g., a multilevel waveform). It canbe shown that, once A and C are estimated, theresponse of a model defined by (2) can be turnedinto a linear combination of the elements of B andD. In such a way, B and D can be obtained bysolving a standard linear least square problem.We checked the ability of the estimation algo-rithm to retrieve the parameters of the original sys-tem (i.e., matrices (5)) when the identification out-put signals are noise free or corrupted with a super-imposed gaussian noise (SNR=26 dB).In the first case (noise free identification), the al-gorithm exactly estimates the matrices of the orig-inal system (5). In the second case (noisy iden-tification), the estimated model approximates theoriginal system very well, as shown by the valida-tion response of Fig. 4.The accuracy of the obtained model can be alsoappreciated by comparing the eigenvalues of the es-timated matrix Aˆ to those of the original matrix A,as shown below.eig(A) = {0.97832914, 0.73484853}eig(Aˆ) = {0.97777425, 0.73098978}(6)20 40 60 80 100 120 140 160 180 20000.020.040.060.080.1 b  1(k)    V k20 40 60 80 100 120 140 160 180 20000.10.20.30.40.5 b  2(k)    V kFigure 4: Discrete-time response y(k) of the origi-nal system and of the estimated model to the inputsequence u(k) = [0, 0]T , k < 0; u(k) = [0, 0.5]T ,k > 0. Solid line: reference response; dotted line:estimated model response.4 ConclusionsWe apply the BCFH and the subspace methodsto the modeling of linear multiport elements fromtheir sampled scattering characteristics. In mosttest cases considered, the accuracy of poles esti-mated by both methods turns out to be sufficientfor a good fitting of the scattering characteristics.From our tests, the poles estimated by BCFHare usually more accurate than those estimated viasubspace methods. The main difficulty of BCFHis the computation of moments of the scatteringcharacteristics and the large amount of time sam-ples needed for such calculation. Such a difficulty,however could be overcome by different approachesto the evaluation of moments. On the other hand,subspace methods sometimes leads to models withspurious poles in the right half plane, whose elimi-nation is an additional problem.Though the present assessment is only prelimi-nary, both methods seem well suited for the gener-ation of models of linear junctions from measuredor simulated scattering data to be used in real sim-ulation problems.AcknowledgementThe Authors are grateful to the Company CST(Darmstadt, Germany) for providing the test caseof Fig. 1. Simulation data were produced by thetool MICROWAVE STUDIO of CST.References[1] L. Ljung, System identification: theory for theuser. Prentice-Hall, 1987.[2] R. Achar, M. Nakhla, and E. Chiprout, “BlockCFH – a model-reduction technique for multi-port distributed interconnect networks,” Proc.of ECCTD’97, pp. 396–401, Sept. 1997.[3] R. Achar, P. K. Gunupudi, M. Nakhla, andE. Chiprout, “Passive interconnect reductionalgorithm for distributed/measured networks,”IEEE Trans. Circ. and Sys.—II: analog anddigital sig. proc., vol. 47, pp. 287–301, Apr.2000.[4] R. Achar and M. Nakhla, “Efficient transientsimulation of embedded subnetworks character-ized by S-parameters in the presence of non-linear elements,” IEEE Trans. MTT, vol. 46,pp. 2356–2363, Dec. 1998.[5] Q. Gu, D. M. Sheen and S. M. Ali, “Analysis oftransients in frequency–dependent interconnec-tions and planar circuits with nonlinear loads,”IEE Proc.-H, vol. 139, pp. 38-44, Feb. 1992.[6] M. Viberg, “Subspace-based methods for theidentification of linear time-invariant systems,”Automatica, Vol. 31, No. 12, pp. 1835–1851,1995.[7] B. D. Rao, “Relationship between matrix penciland state space based harmonic retrieval meth-ods,” IEEE Trans. on Acoustics, Speech andSignal Processing, Vol. 38, No. 1, pp. 177–179,1990.[8] M. Verhaegen, “Identification of the determin-istic part of MIMO state space models given ininnovations form input-output data,” Automat-ica, Vol. 30, No. 1, pp. 61–74, 1994.[9] E.W. Kamen, B.S. Heck, “Fundamentals of sig-nal and systems using Matlab,” Prentice-Hall,1997.

Modeling of Interconnect Junctions from measured scattering responses

https://iris.polito.it/retrieve/handle/11583/1409408/46101/cnf-2001-ecctd-BCFH-4SID.pdf

Modeling of Interconnect Junctions from measured scattering responses

Abstract

Similar works

Full text

Available Versions

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)