Passive Reduction Algorithm for RLC Interconnect Circuits With Embedded State-space Systems (PRESS

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 9, SEPTEMBER 2004 2215

Passive Reduction Algorithm for RLC InterconnectCircuits With Embedded State-Space

Systems (PRESS)Dharmendra Saraswat, Student Member, IEEE, Ramachandra Achar, Senior Member, IEEE, and

Michel S. Nakhla, Fellow, IEEE

Abstract—With the increasing operating frequencies and func-tionality in modern designs, the resulting size of circuit equationsof high-frequency interconnect and microwave subnetworks arebecoming large. Model-order reduction-based algorithms wererecently suggested to handle the solution complexity of suchcircuits. The major objectives in state-of-the-art model-reductionalgorithms are: 1) achieving accurate and compact models;2) numerically stable and efficient generation of models; and3) preservation of system properties such as passivity. Algorithmssuch as PRIMA generate guaranteed passive reduced-ordermodels for large interconnect circuits described by RLC typeof circuits. However, with the diverse technologies and complexgeometries, it is becoming prevalent to describe some of theembedded linear modules in terms of state-space equations. In thispaper, we show how to extend the scope of PRIMA-type first-levelreduction algorithms for simultaneous reduction of combinedcircuits containing both RLC interconnects and embedded mod-ules described by general passive state-space equations, whilepreserving the passivity of the resulting reduced-order model.Necessary formulation, proof of macromodel passivity, andvalidation examples are given.

Index Terms—Algebraic Ricatti equations, electromagnetic(EM), Hamiltonian matrices, Krylov subspace, measuredsubnetworks, model order reduction, passive macromodels,PRIMA, state-space systems, tabulated data, transmission lines.

I. INTRODUCTION

THE RAPID growth in microwave and very large scale inte-gration (VLSI) technology coupled with the trend toward

complex/miniature devices is placing enormous demands oncomputer-aided design (CAD) tools focused on high-frequencymodules. The design requirements are becoming very stringent,demanding sharper excitations, denser layouts, and lower powerconsumption. Consequently, traditional boundaries betweenthe circuit/EM/mechanical and thermal design considerationsare rapidly vanishing. Managing the modeling and simulationin such a complex environment presents highly demandingchallenges [1]–[5]. In recent years, model-reduction-basedalgorithms have had a tremendous success in addressing these

Manuscript received December 24, 2003; revised June 1, 2004. This workwas supported in part by the Natural Sciences and Engineering ResearchCouncil of Canada, by Micronet, a Canadian Network of Centers of Excellenceon Microelectronics, by the Canadian Microelectronics Corporation, and bythe Gennum Corporation.

The authors are with the Department of Electronics, Carleton University,Ottawa, ON, Canada K1S 5B6 (e-mail: [email protected]).

Digital Object Identifier 10.1109/TMTT.2004.834571

difficulties. These techniques lead to significant computationaladvantages as the size of the reduced model is much smallerthan the original system. In addition, model reduction has cometo be viewed as a method for generating reduced models for allsorts of physical systems [5]–[30].

Krylov-subspace-based model-reduction methods [5]–[11]were proposed to obtain accurate models at reasonable compu-tational cost. Preserving passivity of reduced-order models isimportant because stable, but nonpassive, models may lead tounstable systems when connected to other passive components.Several reduction algorithms (such as PRIMA [9]) that preservepassivity of reduced-order models of large RLC circuits haveappeared.

Recently, due to the diverse nature of high-speed modulesin microwave and integrated-circuit designs, macromodelingbased on state-space equations is gaining importance. Forexample, a given linear subnetwork describing a signal pathcould consist of a large lumped RLC network, distributedtransmission lines, and electromagnetic (EM) and measuredmodules (see Fig. 1). The preferred approach in such casesis to analyze each of these modules separately (using themost appropriate algorithm available for that purpose, e.g.,measured subnetworks are characterized using algorithms suchas discussed in [31]–[40], multiconductor transmission linesare macromodeled using algorithms such as discussed in [6]and [11], etc.), and subsequently represent them by passivestate-space equations. Generally, these embedded state-spacesystems are of relatively low order compared to the RLC partof the circuit. From the transient analysis perspective, it wouldbe of great interest to reduce the large RLC part and multipleembedded state-space systems simultaneously so as to get asingle passive macromodel.

However, PRIMA (referred to as a first-level reduction al-gorithm in literature, which is applicable to large systems) inits current form has practical issues that may prevent its appli-cation to include systems outside the class of RLC circuits. Itemploys congruence transformation and can preserve the pas-sivity of the reduced model of the original system provided itscircuit matrices satisfy the positive semidefinite conditions out-lined in [9]. However, including embedded modules describedby state-space equations may pose difficulties in satisfying theseconditions. If the passive state-space equations are specificallyderived in a form that is suitable for PRIMA (e.g., those re-sulting from macromodeling of EM subnetworks as describedby [13]–[15]), then the passivity of the resulting model can

0018-9480/04$20.00 © 2004 IEEE

2216 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 52, NO. 9, SEPTEMBER 2004

Fig. 1. Linear subnetwork � with diverse high-speed modules (distributed, EM, RLC elements, etc.).

be preserved. However, in the case of general embedded pas-sive state-space systems (e.g., macromodels of measured sub-networks resulting from [31] and [36], from second-level reduc-tion algorithms such as [24], etc.) the positive semidefinitenessof the resulting unified circuit matrices may not be guaranteed,leading to nonpassive reduced model. This is illustrated with anumerical example in Section III.

To address the above difficulty, this paper presents passivereduction algorithm for RLC interconnect circuits withembedded state-space systems (PRESS). The proposed algo-rithm adopts a PRIMA type of formulation for RLC networkswhile providing a means to include multiple passive state-spacedescribed modules. Reduction is performed using a congruenttype of transformations to obtain a single passive macromodelfor the entire linear subnetwork. This paper also presents theproof of passivity of the reduced-order model. Examples areprovided to demonstrate the accuracy and efficiency of theproposed algorithm. It is to be noted that second-level reductionalgorithms (such as positive-real balance realization (PR-TBR)[24], which are applicable to stand-alone and relatively smallerstate-space systems) can still be used to achieve further reduc-tion in the order of the reduced model.

The remainder of this paper is organized as follows.Section II briefly reviews passivity issues and the PRIMAalgorithm. Section III presents the problem formulation perti-nent to this paper. Section IV presents the proposed reductionalgorithm for RLC circuits with embedded state-space systems.Sections V and VI present computational results and conclu-sions, respectively.

II. REVIEW OF PASSIVITY ISSUES AND

THE PRIMA ALGORITHM

A brief background on passivity issues and also passivemodel-order reduction based on PRIMA type of algorithms isprovided here.

Passivity implies that a network cannot generate more en-ergy than it absorbs, and no passive termination of the networkwill cause the system to go unstable. Therefore, when mod-eling linear passive systems, it is desired that the resulting re-duced-order model is also passive. The loss of passivity can bea serious problem because transient simulations of reduced net-works may encounter artificial oscillations when connected tothe rest of the circuitry.

Krylov-subspace-based techniques were suggested in the lit-erature to overcome the ill conditioning with the direct Padé ap-proximation [8] and also to ensure passivity of reduced-order

models [9], [10]. A brief overview of one of these methods,i.e., PRIMA, as relevant to the method proposed in this paper isgiven here. Consider a -port linear subnetwork consisting oflumped RLC components, which can be expressed using time-domain modified nodal analysis (MNA) equations as

(1)

where the following holds.

• is the vector of node voltage waveforms ap-pended by independent voltage source current and linearinductor current. is the node-space of subnetwork .

• is a selector matrix, map-ping port voltages into the node space ( ) of the sub-network , where .and are port voltage and current vectors, respectively.

• , are constant matrices describinglumped memory and memoryless elements of subnetwork

, respectively. and are obtained using the formu-lation in [9], such that

(2)

where , , are matrices containing stamps ofresistors, capacitors, and inductors, respectively. Matrix

corresponds to current variables in a Kir-choff’s current law (KCL) formulation. Provided that theoriginal network is composed of RLC elements, , ,and lead to symmetric nonnegative definite matrices.With this formulation, the resulting is also symmetricnonnegative definite.

Taking the Laplace transform of (1) and solving for port cur-rent variables, the admittance matrix of the -port linear subnet-work can be obtained as

(3)

where is an identity matrix. Next, to reducethe order of the system, the matrix is reduced to a smallupper Hessenberg matrix , such that . This is

SARASWAT et al.: PRESS 2217

accomplished by obtaining an orthonormal basisby spanning the Krylov subspace such that

(4)

(5)

where the operator represents the truncation to the nearestinteger toward zero [9]. For example, assume a three-port char-acterization (i.e., ) and let . In this case, we have

; and (4) can be written as

(6)

Next, the reduced-order admittance matrix is found by mappingthe vector to a smaller vector using theorthogonal matrix and congruence transformation as

(7)

(8)

where

(9)

Therefore, in the Laplace domain, the reduced-order admittancematrix can be written as

(10)

Equation (8) represents the reduced-order model of the system.From (10), it can be seen that the eigenvalues of arethe reciprocal of the poles of the reduced-order system. It is to benoted that, the PRIMA algorithm in its current form is limitedto the RLC type of interconnects so as to satisfy the passivityrequirements of the reduced model.

Section III provides the formulation of the problem for thispaper and discusses the limitations of the PRIMA algorithmwhen applied to RLC circuits with general embedded passivestate-space systems.

III. PROBLEM FORMULATION—EMBEDDED

STATE-SPACE SYSTEMS

Consider an embedded linear module described by passivestate-space equations [33]–[40]. Without loss of generality, con-sider the input to be the port current vector and the outputto be the port voltage vector and the state-space equationscan be written as

(11)

Fig. 2. RLC circuit with an embedded state-space module.

(12)

Let be the total number of state variables, be the numberof ports, and the dimensions of matrices are as follows:

, , , , and .Some examples of such linear subnetworks include the systemsthat result from passive rational approximation of tabulated data[33]–[40], second-level reduced models [24], etc.

Next, the state-space system of (11) and (12) can be includedin time-domain MNA equations of lumped RLC elements (1) as

(13)

where

• is the domain of the lumped part of the network;• with elements where

, with a maximumof one nonzero in each row or column is a selectormatrix that maps the port current vector ofthe embedded module into the node space of thelinear subnetwork . Also, port voltage vectorof the embedded module is related to as

.Combining (11)–(13), we can write the augmented set of equa-tions as

(14)

It is to be noticed that the augmented set of (14) can be easilyextended to handle multiple embedded state-space modules. If aPRIMA like algorithm is used on (14) there is no guarantee thatthe reduced model will be passive. This is because, althoughthe matrices , , and are formulated as per (2), uni-fied matrix does not guarantee passive reduction. We illus-trate this issue through the following example (Fig. 2). Herea two-port linear subnetwork is considered, which consists ofRLC lumped components and an embedded passive two-port


state-space module of order 4 ( , , , ). The unified for-mulation of (14) resulted in a MNA of order 14 and the corre-sponding matrices are given as follows:

Next, the above set of equations are reduced to a system oforder 6 using congruence transforms, as suggested by (8) and(9). However, it is found that the reduced system is nonpassive.Fig. 3 confirms this, which shows the plot of the eigenvalueof real-part-admittance matrix of the combined linear subnet-work for both original and reduced cases. As seen, although theoriginal system has a positive eigenvalue spectrum, the reducedsystem contains some negative eigenvalue spectrum (which in-dicates nonpassivity [41]).

Fig. 3. Eigenvalue spectrum of Real(YYY (j!)) versus frequency.

IV. PROPOSED ALGORITHM: PRESS

The proposed algorithm for a passive reduction of large RLCinterconnect networks with embedded passive state-space sys-tems is presented here. For the purpose of simplicity of presenta-tion, the discussion given below corresponds to the case of RLCcircuits with a single embedded state-space system. The discus-sion can be easily extended to the case of multiple embeddedstate-space systems.

A. Formulation of Unified Network Equations

Consider the case of a single -port embedded state-spacemodule described by (11) and (12) with -states. In order toensure the passivity of the reduced-order model, the followingformulation is used. Pre-multiplying (11) by a real matrix

, we can write

(15)

(16)

Using (15) and (16), the unified network (14) can be rewrittenas

(17)

Let the total number of MNA variables in the above formulationbe such that , , , and . Itshould be noted that the size of the embedded state-space system( ) is generally much smaller than that of the RLC circuit ,i.e., . Typically is of the order of a few hundreds,while is of the order of thousands. Hence, even though thestate matrix of (11) becomes dense after multiplication by ma-trix (15), its impact on the overall computational cost involvedin the solution of the unified system (17) is minimal.


Fig. 4. Eigenvalue spectrum of Real(YYY (j!)) versus frequency.

Matrix is obtained as a solution of the followinglinear matrix inequality (LMI) [42]

(18)

For the given embedded module, if can be obtained suchthat and satisfying (18), then the passivityof the reduced-order model is guaranteed (proof is givenin Section IV-C). Details of computation of is given inSection IV-D.

Taking the Laplace transformation of (17), the admittancematrix of the unified system is given as

(19)

B. Passive Model-Order Reduction

Using the congruence transformation, the original system in(17) can be reduced as

(20)

where and the reduced order matrices are given by

(21)

Here, is an orthonormal matrix spanning the Krylovspace , and is the size of thereduced order system in (21). From (20), the reduced-order ad-mittance matrix can be obtained as

(22)

It has been shown in [9] that the reduced system (22) matchesthe first moments of the original system in (17). A math-ematical proof of the passivity preservation of the proposed re-duced model is given in Section IV-C.

C. Proof of Passivity Preservation

A network with an admittance matrix represented by ispassive iff [41]

where " " is the complex conjugate operator

(C1)

is a positive real (PR) matrix, that is

the product for all

complex values of with

and any arbitrary vector (C2)

For the reduced-order model (22), condition (C1) is satisfiedbecause the reduced matrices , , and are real since the

Fig. 5. Eigenvalue spectrum of Real(YYY (j!)). Original and reduced systemof (14) (Example 1).

Fig. 6. Eigenvalue spectrum of Real(YYY (j!)): Original and proposed[reduced system of (17)] (Example 1).

Fig. 7. Circuit with lumped components and measured subnetwork(Example 2).

transformation matrix is real. Using the formulation of theoriginal system, as in (17), condition (C2) can be proven forthe reduced system as follows. Condition (C2) can be expressedusing (22) as

(23)

Substituting and , (23) can beexpressed as

(24)


Fig. 8. Frequency responses (Example 2).

Substituting for and from (21), (24) can be expressed as

(25)

Noting from (17) that the matrix is symmetric and nonneg-ative definite (since both [obtained by solving (18)] and[formulated as per (2)] are symmetric and nonnegative definite),(25) can be simplified as

(26)Substituting , (26) can be written as

(27)

Since is symmetric and nonnegative definite, it can be con-cluded that

(28)

for any and complex vector . Next, using thedefinition of from (17), can be expressed as

(29)

Since is the conductance matrix corresponding to thelumped circuit elements [formed as per (2)], we have

(30)


Also note that, as per the positive-real lemma [42], there exists amatrix for a passive state-space system, satisfying(18). Using these, it can be inferred from (29) that

(31)Using (27), (28), and (31), it can be easily concluded that thepassivity condition (C2) is satisfied for the reduced-order model(21).

Next, to demonstrate the passivity of the reduced modelnumerically, consider the example described in Section III(Fig. 2). Formulation according to (17) resulted in a set ofunified equations of order 14. A reduced system of order 6was obtained using the congruence transformation (9). Next,the reduced system is verified for passivity by constructingthe corresponding Hamiltonian matrix [42] and checking itseigenvalues. No imaginary eigenvalues were found and, hence,the macromodel is proven to be passive [36]. This is alsonumerically demonstrated by plotting the eigenvalue of thereal part of in Fig. 4 (which now contains positiveeigenvalues). For this example, matrix satisfying (18) isgiven in (32), shown at the bottom of this page.

D. Computation of Matrix

Here, a brief discussion of computation of a real symmetricpositive semidefinite matrix is given. For additional detailsand related computational considerations, readers can refer to[45]. Assuming that the macromodel represented by (11) and(12) is passive, according to the positive-real lemma, there existsa real matrix [42] such that the LMI given in (18)is satisfied. The solution of the inequality (18) is related to thesolution of the associated Riccati equation

(33)Essentially, a solution of (33) also satisfies theLMI (18) [43]. One of the efficient methods to solve (33) isthe Schur method [44], [45]. A summary of the essential stepsinvolved in obtaining the solution is given below. Noting thata solution of (33), satisfying , lies in the stablesubspace of the following Hamiltonian matrix:

(34)can be obtained as follows: compute an orthogonal matrixsuch that

(35)

Fig. 9. Eigenvalues spectrum of Real(YYY (j!)) versus frequency for thereduced system (Example 2).

where ; and

, a quasi-upper-triangular matrix with all the

eigenvalues of lying in the left-half plane. The Schur

vectors comprising span the stable invariant subspace

and the solution of (33) is given by . Thecomputational cost involved in computing is , beingthe order of the state-space system. It is to be noted from (17)that is needed only for the embedded passive state-spacemodules, which are usually of relatively low orders comparedto the RLC part of the circuit.

For cases where the state-space system is in the followingdescriptor form:

(36)

(37)

(32)


Fig. 10. Time-domain responses (Example 2).

and is singular, the methodology based on additive decom-position of the transfer function corresponding to (36) and (37)can be used [46], [24]. Also, for the case where is singular or

, the matrix can be calculated by using the algorithmssuch as in [47].

V. NUMERICAL RESULTS

Here, we present three examples to demonstrate the validityand efficiency of the algorithm presented in this paper.

Example 1 : This example further illustrates the concept ofthe proposed algorithm for passive reduction in the presence ofembedded state-space systems. The same network used in Fig. 2is considered here. The linear measured subnetwork is describedby passive state-space system , , , and of order 26. Here,(14) resulted in an MNA of order 36. When this system is re-duced to a system of order 20 by congruence transformation(9), it is found that the reduced system is nonpassive and is con-firmed by the plot of the eigenvalue of in Fig. 5,which contains negative values [41] (denoted by dashed line).

Next, the proposed unified formulation using (17) is obtainedand is reduced to a order of size 20. Proposed reduced-ordersystem is tested for passivity by constructing the correspondingHamiltonian matrix (34) and checking its eigenvalues. Noimaginary eigenvalues were found and, hence, the macromodelis proven to be passive [36] (also demonstrated numericallyin Fig. 6 by plotting the eigenvalue spectrum ofagainst frequency [41]).

Fig. 11. Circuit for Example 3.

Fig. 12. Eigenvalue spectrum of Real(YYY (j!)). Original and reduced systemof (14) (Example 3).

Fig. 13. Eigenvalue spectrum of Real(YYY (j!)). Original and proposed[reduced system of (17)] (Example 3).

Example 2 : In this example, a two-port linear subnetworkwith an embedded two-port measured module (via), 610resistors, 600 inductors, and 200 capacitors is considered(Fig. 7). The passive state-space macromodel for the measuredmodule was obtained using [37] with an order of 45 45.The overall size of MNA matrices (17) (of the lumped circuitwith the macromodel of measured module) is 1433 1433.Using the proposed passive model-reduction scheme, thereduced macromodel of size 100 100 is obtained. Fig. 8compares the -parameters of the original (non-reduced) andreduced systems and they match accurately up to 6 GHz. Thereduced-order system is tested for passivity by constructingthe corresponding Hamiltonian matrix (34) and checking itseigenvalues. No imaginary eigenvalues were found and, hence,the macromodel is proven to be passive [36] (also demonstratednumerically in Fig. 9 by plotting the eigenvalue spectrum of

against frequency [41] up to 100 GHz).Next, the reduced system is linked to HSPICE and a non-

linear transient analysis is performed for an input pulse havingrise and fall times of 0.25 ns and a pulsewidth of 5 ns. The


Fig. 14. Frequency responses for Example 3.

results at nodes , , and are shown in Fig. 10. Forvalidation purposes, the original system (non-reduced) wasalso subjected to the transient analysis (using HSPICE) withthe similar input and terminations, and the results are comparedin Fig. 10. As seen, both match accurately. The reduced system

took 2.17 s, while the original system took 29.92 s on aSun-Ultra-20 machine.

Example 3 : In this example, a three-port linear subnetworkof Fig. 11 is considered. It consists of 4000 resistors, 2000 in-ductors, 2000 capacitors, and a three-port measured module.


Fig. 15. Transient results of Example 3.

Passive state-space macromodel of order 132 132 was ob-tained for the measured subnetwork using [37].

The overall size of resulting MNA matrices, using theregular PRIMA formulation (14) (including both the lumpedcircuit and passive state-space macromodel) was 4142 4142.This system was reduced to a system of order 192 192 bycongruence transformation (9) and it is found that the reducedsystem is nonpassive and is confirmed by the plot of the firsteigenvalue of in Fig. 12 (denoted by the dashedline). Next, using the proposed unified formulation (17), areduced-order model with order same as the previous case(192 192) is obtained. The time taken by the proposed algo-rithm to create the reduced model was 31.2 s on a Sun-Ultra-20

machine. The resulting macromodel is tested for passivity byconstructing the corresponding Hamiltonian matrix (34) andchecking its eigenvalues. No imaginary eigenvalues were foundand, hence, the proposed macromodel is proven to be passive[36] (also demonstrated numerically in Fig. 13, by plotting thefirst eigenvalue against frequency [41]). Fig. 14compares the -parameters of the original and reduced system(proposed) and they match accurately up to 6.0 GHz.

Next, the reduced system is linked to HSPICE and a nonlineartransient analysis is performed for an input pulse having a riseand fall time of 0.1 ns and a width of 5 ns. The results at node

, , , and are shown in Fig. 15. For validationpurposes, the original system was also subjected to the tran-sient analysis (using HSPICE) with the similar input and termi-nations, and the results are compared in Fig. 15. As seen, bothmatch accurately. The proposed reduced system took 97 s, whilethe original system took 2041 s on a Sun-Ultra-20 machine (ona comparative note, the transient simulation of the concatenatedsystem of individual reduced models of RLC networks and thestate-space model of the measured network required 312 s forproviding comparable responses of Fig. 15).

VI. CONCLUSIONS

In this paper, a new algorithm (i.e., PRESS) has beenpresented to extend the scope of PRIMA-based first-levelmodel-order-reduction algorithms for circuits with both RLCinterconnects and multiple embedded modules described bypassive state-space equations. A new formulation is presentedto guarantee the passivity of the reduced-order model in thepresence of multiple embedded passive state-space modules.The proposed algorithm helps to identify a single passivemacromodel for linear subnetworks with RLC lumped compo-nents and numerous embedded passive devices.

REFERENCES

[1] H. B. Bakoglu, Circuits, Interconnections and Packaging forVLSI. Reading, MA: Addison-Wesley, 1990.

[2] A. Deustsch, “Electrical characteristics of interconnections for high-per-formance systems,” Proc. IEEE, vol. 86, pp. 315–355, Feb. 1998.

[3] M. Nakhla and R. Achar, Multimedia Book Series on SignalIntegrity. Ottawa, ON, Canada: OMNIZ Global Knowledge Corpora-tion, 2002.

[4] C. Paul, Analysis of Multiconductor Transmission Lines. New York:Wiley, 1994.

[5] R. Achar and M. Nakhla, “Simulation of high-speed interconnects,”Proc. IEEE, vol. 89, pp. 693–728, May 2001.

[6] Q. Yu, J. M. L. Wang, and E. S. Kuh, “Passive multipoint moment-matching model order reduction algorithm on multiport distributed in-terconnect networks,” IEEE Trans. Circuits Syst. I, vol. 46, pp. 140–160,Jan. 1999.

[7] W. T. Beyene and J. E. Schutt-Aine, “Krylov subspace based model-order reduction techniques for circuit simulations,” in IEEE MidwestCircuits and Systems Symp., vol. 1, Aug. 1996, pp. 331–334.

[8] P. Feldmann and R. W. Freund, “Efficient linear circuit analysis by Padéapproximation via the Lanczos process,” IEEE Trans. Computer-AidedDesign, vol. 14, pp. 639–649, May 1995.

[9] A. Odabasioglu, M. Celik, and L. T. Pileggi, “PRIMA: Passive reduced-order interconnect macromodeling algorithm,” IEEE Trans. Computer-Aided Design, vol. 17, pp. 645–654, Aug. 1998.


[10] K. J. Kerns, I. L. Wemple, and A. T. Yang, “Preservation of passivityduring RLC network reduction via split congruence transformations,”in Proc. 34th ACM/IEEE Design Automation Conf., Anaheim, CA, June1997, pp. 34–39.

[11] M. Celik and A. C. Cangellaris, “Simulation of multiconductor trans-mission lines using Krylov subspace order-reduction techniques,” IEEETrans. Computer-Aided Design, vol. 16, pp. 485–496, May 1997.

[12] Q. Su, V. Balakrishnan, and C. K. Koh, “A factorization-based frame-work for passivity-preserving model reduction of RLC systems,” inProc. 39th ACM/IEEE Design Automation Conf., New Orleans, LA,June 2002, pp. 40–45.

[13] Y. Zu and A. C. Cangellaris, “A new finite element model for reducedorder electromagnetic modeling,” IEEE Microwave Wireless Comp.Lett., vol. 11, pp. 211–213, May 2001.

[14] A. C. Cangellaris, S. Pasha, J. L. Prince, and M. Celik, “A new discretetransmission line model for passive model order reduction and macro-modeling of high-speed interconnections,” IEEE Trans. Adv. Packaging,vol. 22, pp. 356–364, Aug. 1999.

[15] A. C. Cangellaris, M. Celik, S. Pasha, and L. Zhao, “Electromagneticmodel order reduction for system-level modeling,” IEEE Trans. Mi-crowave Theory Tech., vol. 47, pp. 840–850, June 1999.

[16] J. Cullum, A. Ruehli, and T. Zhang, “A method for reduced-order mod-eling and simulation of large interconnect circuits and its application toPEEC models with retardation,” IEEE Trans. Circuits Systems II, vol.47, pp. 261–273, Apr. 2000.

[17] J. M. Wang, C. C. Chu, Q. Yu, and E. S. Kuh, “On projection based al-gorithms for model-order reduction of interconnects,” IEEE Trans. Cir-cuits Systems I, vol. 49, pp. 1563–1585, Nov. 2002.

[18] L. Knockaert and D. D. Zutter, “Laguerre-SVD reduced-order mod-eling,” IEEE Trans. Microwave Theory Tech., vol. 48, pp. 1469–1475,Sept. 2000.

[19] , “Stable laguerre-SVD reduced-order modeling,” IEEE Trans. Cir-cuits Systems I, vol. 50, pp. 576–579, Apr. 2003.

[20] B. Denecker, F. Olyslager, L. Knockaert, and D. D. Zutter, “Generationof FDTD subcell equations by means of reduced order modeling,” IEEETrans. Antennas Propagat., vol. 51, pp. 1806–1817, Aug. 2003.

[21] A. C. Cangellaris and A. Ruehli, “Model order reduction techniques ap-plied to electromagnetic problems,” in IEEE Electrical Performance ofElectronic Packaging Conf., Oct. 2000, pp. 239–242.

[22] M. Celik and A. C. Cangellaris, “Efficient transient simulation of lossypackaging interconnects using moment-matching techniques,” IEEETrans. Comp., Packag., Manufact. Technol., vol. 19, pp. 64–73, Feb.1996.

[23] J. Roychowdhury, “Reduced order modeling of linear time-varying sys-tems,” in Proc. Int. Computer Aided-Design Conf., Santa Clara, CA,Nov. 1998, pp. 92–96.

[24] J. R. Phillips, L. Daniel, and L. M. Silveira, “Guaranteed passive bal-ancing transformations for model order reduction,” IEEE Trans. Com-puter-Aided Design, vol. 22, pp. 1027–1041, Aug. 2003.

[25] J. E. Bracken, “Passive modeling of linear interconnect networks,” Dept.Elect. Comput. Eng., Carnegie–Mellon Univ., Pittsburgh, PA, 1996.

[26] N. Marques, M. Kamon, J. White, and L. M. Silveira, “Amixed nodal-mesh formulation for efficient extraction and passivereduced-order modeling of 3D interconnects,” in Proc. 35thACM/IEEE Design Automation Conf., San Francisco, CA, June1998, pp. 297–302.

[27] J.-R. Li, F. Wang, and J. White, “Efficient model reduction of inter-con-nect via approximate system grammians,” in Proc. Int. Computer Aided-Design Conf., San Jose, CA, Nov. 1999, pp. 380–383.

[28] P. Rabiei and M. Pedram, “Model order reduction of large circuits usingbalanced truncation,” in Proc. Asia and South Pacific Design AutomationConf., Hong Kong, Jan. 1999, pp. 237–240.

[29] I. M. Jaimoukha and E. M. Kasenally, “Krylov subspsace methods forsolving large Lyapunov equations,” SIAM J. Numer. Anal., vol. 31, pp.227–251, 1994.

[30] P. Rabiei and M. Pedram, “Model reduction of variable-geometry inter-connects using variational spectrally-weighted balanced truncation,” inProc. Int. Computer Aided-Design Conf., San Jose, CA, Nov. 2001, pp.586–591.

[31] C. P. Coelho, J. R. Phillips, and L. M. Silveira, “A convex programmingapproach to positive real rational approximation,” in Proc. IEEE/ACMInt. Computer-Aided Design Conf., November 2001, pp. 245–251.

[32] H. Chen and J. Fang, “Enforcing bounded realness of ‘S’ parameterthrough trace parameterization,” in IEEE 12th EPEP Topical Meeting,Princeton, NJ, Oct. 2003, pp. 291–294.

[33] J. Morsey and A. C. Cangellaris, “Passive realization of interconnectmodels from measured data,” in IEEE 10th EPEP Topical Meeting,Cambridge, MA, Oct. 2001, pp. 47–50.

[34] R. Neumayer, F. Haslinger, A. Stelzer, and R. Weigel, “On the synthesisof equivalent circuit models for multiports characterized by frequency-dependent parameters,” IEEE Trans. Microwave Theory Tech., vol. 50,pp. 2789–2796, Dec. 2002.

[35] W. T. Beyene and J. E. Schutt-Aine, “Efficient transient simulation ofhigh-speed interconnects characterized by sampled data,” IEEE Trans.Comp., Packag., Manufact. Technol., pt. B, vol. 21, pp. 105–114, Feb.1998.

[36] D. Saraswat, R. Achar, and M. Nakhla, “Passive macromodels ofmicrowave subnetworks characterized by measured/simulated data,” inProc. IEEE MTT-S Int. Microwave Symp. Dig., Philadelphia, PA, June2003, pp. 999–1002.

[37] , “A fast algorithm and practical considerations for passive macro-modeling of measured/simulated data,” IEEE Trans. Comp., Packag.,Manufact. Technol., vol. 27, pp. 57–70, Feb. 2004.

[38] M. Elzinga, K. L. Virga, and J. L. Prince, “Improved global rationalapproximation macromodeling algorithm for networks characterized byfrequency-sampled data,” IEEE Trans. Microwave Theory Tech., vol. 48,pp. 1461–1468, Sept. 2000.

[39] W. T. Beyene and J. E. Schutt-Aine, “Accurate frequency-domain mod-eling and efficient simulation of high-speed packaging interconnects,”IEEE Trans. Microwave Theory Tech., pp. 1941–1947, Oct. 1997.

[40] S. Min and M. Swaminathan, “Efficient construction of two port pas-sive macromodels for resonant networks,” in IEEE 10th EPEP TopicalMeeting, Cambridge, MA, Oct. 2001, pp. 230–232.

[41] E. Kuh and R. Rohrer, Theory of Active Linear Networks. San Fran-cisco, CA: Holden-Day, 1967.

[42] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear MatrixInequalities in System and Control Theory. Philadelphia, PA: SIAM,1994, vol. 15, Studies Appl. Math..

[43] S. Bittanti, A. J. Laub, and J. C. Willems, The Riccati Equation. Berlin,Germany: Springer-Verlag, 1991.

[44] A. J. Laub, “A Schur method for solving algebraic Riccati equations,”IEEE Trans. Automat. Contr., vol. AC-24, pp. 913–921, Dec. 1979.

[45] W. F. Arnold and A. J. Laub, “Generalized eigen-problem algorithmsand software for algebraic Riccati equations,” Proc. IEEE, vol. 72, pp.1746–1754, MONTH 1984.

[46] B. Kagstorm and P. Van Dooren, “A generalized state-space approachfor the additive decomposition of a transfer matrix,” J. Numer. LinearAlgebra Applicat., vol. 1, no. 2, pp. 165–181, 1992.

[47] N. Sadegh, J. D. Finney, and B. S. Heck, “An explicit method for com-puting the positive-real lemma matrices,” in Proc. 33rd IEEE Decisionand Control Conf., 1994, pp. 1464–1469.

Dharmendra Saraswat (S’03) received the B.E.degree from the Government Engineering College(GEC), Jabalpur, India, in 1990, the M.A.Sc. degreefrom Carleton University, Ottawa, ON, Canada, in2003, and is currently working toward the Ph.D. de-gree in electrical engineering at Carleton University.

His research interests include modeling and sim-ulation of high-speed interconnect networks, circuitsimulation, and numerical algorithms.

Mr. Saraswat was the recipient of the 2002 BestStudent Paper Award presented at the Electrical Per-

formance of Electronic Packaging Conference, and the Third Best Student PaperAward presented at the 2003 IEEE Microwave Theory and Techniques Society(IEEE MTT-S) International Microwave Symposium (IMS). He was also therecipient of the Natural Sciences and Engineering Research Council (NSERC)Scholarship at the doctoral level.


Ramachandra Achar (S’95–M’00–SM’04) re-ceived the B.Eng. degree in electronics engineeringfrom Bangalore University, Bangalore, India, in1990, the M.Eng. degree in microelectronics fromthe Birla Institute of Technology and Science, Pilani,India, in 1992, and the Ph.D. degree from CarletonUniversity, Ottawa, ON, Canada, in 1998.

He is currently an Assistant Professor with theDepartment of Electronics, Carleton University.He spent Summer 1995 involved with high-speedinterconnect analysis with the T. J. Watson Research

Center, IBM, Yorktown Heights, NY. In 1992, he was a Graduate Traineewith the Central Electronics Engineering Research Institute, Pilani, India, andwas also with Larsen and Toubro Engineers Ltd., Mysore, India, and with theIndian Institute of Science, Bangalore, India, as a Research and DevelopmentEngineer. From 1998 to 2000, he was a Research Engineer with the Com-puter-Aided Engineering (CAE) Group, Carleton University. He is a consultantfor several leading industries focused on high-frequency circuits, systems, andcomputer-aided design (CAD) tools. His research interests include modelingand simulation of high-speed interconnects, model-order reduction, numericalalgorithms, and development of CAD tools for high-frequency circuit analysis.

Dr. Achar serves on Technical Program Committees of several leading IEEEconferences. He was the recipient of several prestigious awards including the2000 Natural Science and Engineering Research Council (NSERC) DoctoralAward, the 1997 Strategic Microelectronics Corporation (SMC) Award, the1996 Canadian Microelectronics Corporation (CMC) Award, and the 1998Best Student Paper Award presented at the Micronet (a Canadian Network ofCentres of Excellence on Microelectronics) Annual Workshop.

Michel S. Nakhla (S’73–M’75–SM’88–F’98)received the M.A.Sc. and Ph.D. degrees in electricalengineering from the University of Waterloo, Wa-terloo, ON, Canada, in 1973 and 1975, respectively.

He is currently Chancellor’s Professor of Elec-trical Engineering with Carleton University, Ottawa,ON, Canada. From 1976 to 1988, he was withBell-Northern Research, Ottawa, ON, Canada, as theSenior Manager of the Computer-Aided EngineeringGroup. In 1988, he joined Carleton University, asa Professor and the Holder of the Computer-Aided

Engineering Senior Industrial Chair established by Bell-Northern Research andthe Natural Sciences and Engineering Research Council (NSERC) of Canada.He is the founder of the High-Speed CAD Research Group, Carleton Univer-sity. He serves as a technical consultant for several industrial organizations andis the principal investigator for several major sponsored research projects. Hisresearch interests include CAD of VLSI and microwave circuits, modeling andsimulation of high-speed interconnects, nonlinear circuits, multidisciplinaryoptimization, thermal and EM emission analysis, microelectromechanicalsystems (MEMS), and neural networks.

Dr. Nakhla was associate editor of the IEEE TRANSACTIONS ON CIRCUITS

AND SYSTEMS—PART I: FUNDAMENTAL THEORY AND APPLICATIONS andguest editor of the IEEE TRANSACTIONS ON COMPONENTS, PACKAGING,AND MANUFACTURING TECHNOLOGY (Advanced Packaging) and the IEEETRANSACTIONS ON CIRCUITS AND SYSTEMS—PART II: ANALOG AND DIGITAL

SIGNAL PROCESSING.

Documents

Passive Reduction Algorithm for RLC Interconnect Circuits With Embedded State-space Systems (PRESS