IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 1, NO. 3, MARCH 2011 399

Physics-Based Passivity-Preserving ParameterizedModel Order Reduction for PEEC Circuit Analysis

Francesco Ferranti, Member, IEEE, Giulio Antonini, Senior Member, IEEE, Tom Dhaene, Senior Member, IEEE,Luc Knockaert, Senior Member, IEEE, and Albert E. Ruehli, Life Fellow, IEEE

Abstract— The decrease of integrated circuit feature size andthe increase of operating frequencies require 3-D electromagneticmethods, such as the partial element equivalent circuit (PEEC)method, for the analysis and design of high-speed circuits.Very large systems of equations are often produced by 3-Delectromagnetic methods, and model order reduction (MOR)methods have proven to be very effective in combating such highcomplexity. During the circuit synthesis of large-scale digital oranalog applications, it is important to predict the response of thecircuit under study as a function of design parameters such asgeometrical and substrate features. Traditional MOR techniquesperform order reduction only with respect to frequency, andtherefore the computation of a new electromagnetic model andthe corresponding reduced model are needed each time a designparameter is modified, reducing the CPU efficiency. Parameter-ized model order reduction (PMOR) methods become necessaryto reduce large systems of equations with respect to frequencyand other design parameters of the circuit, such as geometricallayout or substrate characteristics. We propose a novel PMORtechnique applicable to PEEC analysis which is based on a para-meterization process of matrices generated by the PEEC methodand the projection subspace generated by a passivity-preservingMOR method. The proposed PMOR technique guarantees overallstability and passivity of parameterized reduced order modelsover a user-defined range of design parameter values. Pertinentnumerical examples validate the proposed PMOR approach.

Index Terms— Interpolation, parameterized model order re-duction, partial element equivalent circuit method, passivity.

I. INTRODUCTION

ELECTROMAGNETIC (EM) methods [1]–[3] havebecome increasingly indispensable analysis and design

tools for a variety of complex high-speed systems. The useof these methods usually results in very large systems of

Manuscript received March 18, 2010; revised November 3, 2010; acceptedDecember 10, 2010. Date of publication February 28, 2011; date of currentversion April 8, 2011. This work was supported in part by the ResearchFoundation Flanders and by the Italian Ministry of University under aProgram for the Development of Research of National Interest under Grant2006095890. This work was recommended for publication by Associate EditorJ. Tan upon evaluation of the reviewers comments.

F. Ferranti, T. Dhaene, and L. Knockaert are with the Department of In-formation Technology, Interdisciplinary Institute for BroadBand Technology,Ghent University, Ghent 9000, Belgium (e-mail: francesco.ferranti@ugent.be;tom.dhaene@ugent.be; luc.knockaertg@ugent.be).

G. Antonini is with the UAq EMC Laboratory, Dipartimento di IngegneriaElettrica e dell’Informazione, Universita degli Studi dell’Aquila, L’Aquila67100, Italy (e-mail: giulio.antonini@univaq.it).

A. E. Ruehli is with the IBM T. J. Watson Research Center, YorktownHeights, NY 10598 USA. He is also with the Missouri University of Scienceand Technology, Rolla, MO 65409 USA (e-mail: albert.ruehli@gmail.com).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCPMT.2010.2101912

equations which are prohibitively expensive to solve. Hence,model order reduction (MOR) techniques are crucial to reducethe complexity of EM models and the computational cost ofthe simulations while retaining the important physical featuresof the original system [4]–[7]. The development of a reducedorder model (ROM) of EM systems has become a topicof intense research over the last years, with applications tovias, high-speed packages, interconnects, and on-chip passivecomponents [8]–[11]. The partial element equivalent circuit(PEEC) method has achieved increasing popularity amongEM compatibility engineers since it is able to transform theEM system under examination into a passive RLC equivalentcircuit. PEEC uses a circuit interpretation of the electricfield integral equation (EFIE) [12], thus allowing handling ofcomplex problems involving EM fields and circuits [2], [13],[14]. Nonlinear circuit devices such as drivers and receiversare usually connected with PEEC equivalent circuits using atime domain circuit simulator (e.g., SPICE [15]). However,inclusion of the PEEC model directly into a circuit simulatormay be computationally intractable for complex structures,because the number of circuit elements can be in the tensof thousands. In this case, a first solution consists in the useof fast multipole methods [16], [17]. The drawback of thesetechniques relies on the fact that they are dependent on theGreen’s function of the problem. Another option is representedby MOR techniques, which are adopted to reduce the size ofthe PEEC model [7], [18], [19].

Traditional MOR techniques perform model reduction onlywith respect to frequency. However, during the circuit syn-thesis of large-scale digital or analog applications, it is alsoimportant to predict the response of the circuit under studyas a function of design parameters such as geometrical andsubstrate features. A typical design process includes optimiza-tion and design space exploration, and thus requires repeatedsimulations for different design parameter values. Such de-sign activities call for parameterized model order reduction(PMOR) methods that can reduce large systems of equationswith respect to frequency and other design parameters of thecircuit, such as geometrical layout or substrate characteristics.

Over the years, a number of PMOR methods have beendeveloped. In order to model and analyze interconnect be-havior with process variations, various techniques have beenproposed for variational interconnect order reduction [20],[21]. These approaches apply projection operator and generatereduced-order interconnect models. In addition, the projec-tion subspace and/or the reduced-order system matrices are

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400 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 1, NO. 3, MARCH 2011

approximated as low-order polynomials of process parameterssuch that the process variation effects can be incorporatedinto the interconnect model. These process parameters, forexample, can be the width and thickness of the interconnectmetal wires. The authors in [22] propose to approximatethe system transfer function by low-order polynomials ofprocess parameters instead of the projection subspace and/orthe reduced order system matrices. The algorithm describedin [22] computes the projection subspace and generates para-meterized ROMs such that the multiparameter moments arematched. However, the structure of such methods may presentsome computational problems, and the resulting parameterizedROMs usually suffer from oversize when the number ofmoments to match is high, either because high accuracy (order)is required or because the number of parameters is large.The compact order reduction for parameterized extractionalgorithm [23] applies a two-step explicit and implicit schemefor multiparameter moment matching. It is numerically stablebut, unfortunately, does not preserve passivity. The para-meterized interconnect macromodeling via a two-directionalArnoldi process algorithm presented in [24] is numericallystable and preserves the passivity of parameterized RLCnetworks, but as in the case of all-multiparameter moment-matching-based PMOR techniques it is suitable only to alow-dimensional design space.

This paper proposes a PMOR method applicable to PEECanalysis, which is based on a parameterization process ofmatrices generated by the PEEC method and the projectionsubspace generated by a passivity-preserving MOR method.The Laguerre-SVD MOR method [19] is used in this paper.Overall stability and passivity of parameterized ROMs areguaranteed by construction over the design space of interest.PEEC models and parameterized ROMs describe an admit-tance (Y) representation. However, it should be noted that theproposed PMOR technique is not bound to the Laguerre-SVDmethod, other passivity-preserving MOR techniques based ona projection subspace approach can be used, such as thePRIMA method [7].

This paper is organized as follows. Section II describesthe modified nodal analysis (MNA) equations of the PEECmethod. Section III describes the proposed PMOR method.Finally, some pertinent numerical examples validate theproposed technique in Section IV.

II. PEEC FORMULATION

The PEEC method [2] stems from the integral equation formof Maxwell’s equations.

The main difference of the PEEC method with otherintegral-Equation-based techniques such as the method ofmoments [1] resides in the fact that it provides a circuitinterpretation of the EFIE [12] in terms of partial elements,namely, resistances, partial inductances, and coefficients ofpotential. Thus, the resulting equivalent circuit can be studiedby means of SPICE-like circuit solvers [15] in both time andfrequency domains.

Over the years, several improvements of the PEEC methodhave been Performed, thus allowing handling of complex

problems involving both circuits and EM fields [2], [13], [14],[25]–[28].

In the standard approach, volumes and surfaces arediscretized into elementary regions, hexahedra, and patchesrespectively [27] over which the current and charge densitiesare expanded into a series of basis functions. Pulse basisfunctions are usually adopted as expansion and weight func-tions. Such choice of pulse basis functions corresponds toassuming constant current density and charge density over theelementary volume (inductive) and surface (capacitive) cells,respectively.

Following the standard Galerkin’s testing procedure, topo-logical elements, namely nodes and branches, are generatedand electrical lumped elements are identified modeling boththe magnetic and electric field coupling.

Conductors are modeled by their ohmic resistance, whiledielectrics requires modeling the excess charge due to thedielectric polarization [29]. Magnetic and electric field cou-pling are modeled by partial inductances and coefficients ofpotential, respectively.

The magnetic field coupling between two inductive volumecells α and β is described by the partial inductance

Lpαβ = μ

4π

1

aαaβ

∫uα

∫uβ

1

Rαβduαduβ (1)

where Rαβ is the distance between any two points in volumesuα and uβ with aα and aβ their cross sections. The electricfield coupling between two capacitive surface cells δ and γ ismodeled by the coefficient of potential

Pδγ = 1

4πε

1

Sδ Sγ

∫Sδ

∫Sγ

1

Rδγd Sδd Sγ (2)

where Rδγ is the distance between any two points on surfacesδ and γ , while Sδ and Sγ denote the area of their respectivesurfaces.

Generalized Kirchoff’s laws for conductors can be rewrittenas

P−1 dv(t)

dt− AT i(t) + ie(t) = 0 (3a)

−Av(t) − Lpdi(t)

dt− Ri(t) = 0 (3b)

where A is the connectivity matrix, v(t) denotes the nodepotentials to infinity, and i(t) and ie(t) represent the currentsflowing in volume cells and the external currents, respectively.

When dielectrics are considered, the resistance voltage dropRi(t) is substituted by the excess capacitance voltage drop,which is related to the excess charge, by vd(t) = C−1

d qd(t)[29]. Hence, for dielectric elementary cells, (3) become

P−1 dv(t)

dt− AT i(t) + ie(t) = 0 (4a)

−Av(t) − Lpdi(t)

dt− vd(t) = 0 (4b)

i(t) = Cddvd(t)

dt. (4c)

A selection matrix K is introduced to define the portvoltages by selecting node potentials. The same matrix isused to obtain the external currents ie(t) by the currents

FERRANTI et al.: PARAMETERIZED MODEL ORDER REDUCTION 401

is(t), which are of opposite sign with respect to the n p portcurrents ip(t)

vp(t) = Kv(t) (5a)

ie(t) = KT is(t). (5b)

An example of PEEC circuit electrical quantities for aconductor elementary cell is illustrated, in the Laplace domain,in Fig. 1, where the current-controlled voltage sources sL p,i j I j

and the current-controlled current sources Icci model themagnetic and electric field coupling, respectively.

A. Descriptor Representation of PEEC Circuits

Let us assume that the system under analysis consistsof conductors and dielectrics. Let the current and chargedensity be defined in volumes and surface of conductors anddielectrics, respectively. The Galerkin’s approach is applied toconvert the continuous EM problem described by the EFIE toa discrete problem in terms of electrical circuit quantities, e.g.,currents i(t) and node potentials v(t). Let us denote with nn

the number of nodes and ni the number of branches wherecurrents flow. Among the latter, we denote with nc and nd thenumber of branches of conductors and dielectrics, respectively.Furthermore, let us assume that we are interested in generatingan admittance representation having n p output currents ip(t)under voltage excitation vp(t). Since dielectrics require theexcess capacitance to model the polarization charge [30],additional nd unknowns are needed in addition to currents.Hence, if the MNA approach [31] is used, the global numberof unknowns is nu = ni + nd + nn + n p . In a matrix form,(3)–(5) read

⎡⎢⎢⎣

Inn ,nn 0nn,ni 0nn ,nd 0nn,n p

0ni ,nn Lp 0ni ,nd 0ni ,n p

0nd ,nn 0nd ,ni Cd 0nd ,n p

0n p,nn 0n p,ni 0n p,nd 0n p,n p

⎤⎥⎥⎦

︸ ︷︷ ︸C

d

dt

⎡⎢⎢⎣

v(t)i(t)

vd(t)is(t)

⎤⎥⎥⎦

︸ ︷︷ ︸x(t)

= −

⎡⎢⎢⎣

0nn,nn −PAT 0nn,nd PKT

A R � 0ni ,n p

0nd ,nn −�T 0nd ,nd 0nd ,n p

−K 0n p,ni 0n p,nd 0n p,n p

⎤⎥⎥⎦

︸ ︷︷ ︸G

·

⎡⎢⎢⎣

v(t)i(t)

vd(t)is(t)

⎤⎥⎥⎦

︸ ︷︷ ︸x(t)

+[

0nn+ni +nd ,n p

−In p,n p

]

︸ ︷︷ ︸B

· [ vp(t)]

︸ ︷︷ ︸u(t)

(6)

where In p,n p is the identity matrix of dimensions equal to thenumber of ports. Matrix � is

� =[

0nc,nd

Ind ,nd

]. (7)

Then, potentials v(t) are expressed in terms of charges as

v(t) = Pq(t). (8)

1 2

Ic1

Ic2

Icc1

Icc2

3

Ic3

Icc3

Lp11

sLp,12

I2

sLp,21

I1I

1

+

Lp22

+ −+ −

I2

R1

R2

VpI

pI

e

1/P22

1/P33

1/P11

Fig. 1. Illustration of PEEC circuit electrical quantities for a conductorelementary cell.

Hence, (6) can be recast as⎡⎢⎢⎣

P 0nn,ni 0nn,nd 0nn,n p

0ni ,nn Lp 0ni ,nd 0ni ,n p

0nd ,nn 0nd ,ni Cd 0nd ,n p

0n p,nn 0n p,ni 0n p,nd 0n p,n p

⎤⎥⎥⎦

︸ ︷︷ ︸C

d

dt

⎡⎢⎢⎣

q(t)i(t)

vd(t)is(t)

⎤⎥⎥⎦

︸ ︷︷ ︸x(t)

= −

⎡⎢⎢⎣

0nn,nn −PAT 0nn ,nd PKT

AP R � 0ni ,n p

0nd ,nn −�T 0nd ,nd 0nd ,n p

−KP 0n p,ni 0n p,nd 0n p,n p

⎤⎥⎥⎦

︸ ︷︷ ︸G

·

⎡⎢⎢⎣

q(t)i(t)

vd(t)is(t)

⎤⎥⎥⎦

︸ ︷︷ ︸x(t)

+[

0nn+ni +nd ,n p

−In p,n p

]

︸ ︷︷ ︸B

· [ vp(t).]

︸ ︷︷ ︸u(t)

. (9)

In a more compact form, (9) can be rewritten as

Cdx(t)

dt= −Gx(t) + Bu(t) (10a)

ip(t) = LT x(t) (10b)

where x(t) = [q(t) i(t) vd(t) is(t)

]T. Since this is an n p-

port formulation, whereby the only sources are the voltagesources at the n p-port nodes, B = L where B ∈ �nu×n p .

B. Scaling

The system of (9) is typically ill conditioned becausecharges are usually much smaller than currents and voltages.Correspondingly, the entries of the matrix P are larger thanother elements in matrices C and G by several orders ofmagnitude. The ill-conditioning of (9) prevents MOR methodsto be efficiently applied. In order to mitigate such a problem,scaling can be adopted. The units of the electrical quantitiesare changed consistently as shown in Table I.

C. Properties of PEEC Formulation

In order to apply the proposed PMOR technique, it isimportant to specify the properties of the matrices involvedin the PEEC formulation (9).

Both matrices describing electric and magnetic field cou-pling, P and Lp , respectively, are full symmetric matrices.

402 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 1, NO. 3, MARCH 2011

TABLE I

SCALED UNITS

Voltage V

Current mA

Charge pC

P pF−1

Cd pF

R k

L p μH

f GHz

s ns

In the case of orthogonal geometries, mutual partial induc-tances corresponding to orthogonal currents are equal to zero.Even in this case, rows and columns can be recast so thatthe partial inductance matrix Lp is a block-diagonal matrix.Since each block is symmetric and positive definite, the overallmatrix Lp is symmetric and positive definite as well. Thecoefficient of potential matrix P is also symmetric and positivedefinite [32].

When pulse basis functions are used, as is the case withthe standard PEEC formulation [2], resistance and excesscapacitance matrices, R and Cd , respectively, are diagonaland symmetric positive semidefinite and definite. The matrixR is diagonal, with positive diagonal elements correspondingto conductor elementary cells, while the diagonal elementscorresponding to dielectric elementary cells are equal to zero.The matrix Cd is diagonal with all the diagonal elementspositive.

Assuming the previous matrix properties, it is easy to provethat the matrices C, G satisfy the following properties:

C = CT ≥ 0 (11a)

G + GT ≥ 0. (11b)

The properties of the PEEC matrices B = L, C = CT ≥0, and G + GT ≥ 0 ensure the passivity of the PEEC admit-tance model Y(s) = LT (sC+G)−1B [33] and allow exploitingthe passivity-preserving capability of the Laguerre-SVD MORalgorithm [19]. When performing transient analysis, stabilityand passivity must be guaranteed. It is known that, while apassive system is also stable, the reverse is not necessarilytrue [34], which is crucial when the reduced model is tobe utilized in a general-purpose analysis-oriented nonlinearsimulator (e.g., SPICE). Passivity refers to the property ofsystems that cannot generate more energy than they absorbthrough their electrical ports. When the system is terminatedon any arbitrary passive loads, none of them will cause thesystem to become unstable [35], [36].

III. PMOR

In this section, we describe a PMOR algorithm that isable to include, in addition to frequency, N design parametersg = (g(1), . . . , g(N)) in a parameterized ROM, such as thelayout features of a circuit (e.g., lengths, widths, etc.) or thesubstrate parameters (e.g., thickness, dielectric permittivity,losses, etc.). The main objective of the PMOR method is to

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

g(1)

g(2)

Estimation gridValidation grid

Fig. 2. Example of estimation and validation design space grid.

accurately approximate the original scalable system (havinga high complexity) with a reduced scalable system (havinga low complexity) by capturing the behavior of the originalsystem with respect to frequency and other design parameters.The design space D(g) is considered as the parameter spaceP(s, g) without frequency. The parameter space P(s, g) con-tains all parameters (s, g). If the parameter space is (N + 1)dimensional, the design space is N dimensional. The proposedalgorithm guarantees stability and passivity of a parameterizedROM over the entire design space of interest. Two data gridsare used in the modeling process: an estimation grid and avalidation grid. The first grid is utilized to build parameterizedROMs, while the second grid, denser than the previous one,is used to assess the capability of parameterized ROMs ofdescribing the system under study in points of the design spacepreviously not used for its construction. To clarify the use ofthese two design space grids, we show in Fig. 2 a possibleestimation and validation design space grid in the case of twodesign parameters g = (g(1), g(2)).

A. PMOR Algorithm

Considering the influence of the design parameters g =(g(1), . . . , g(N)), the MNA formulation (10a)–(10b) becomes

C(g)dx(t, g)

dt= −G(g)x(t, g) + Bu(t) (12a)

ip(t, g) = LT x(t, g). (12b)

We assume that a topologically fixed discretization meshis used and it is independent of the specific design parame-ters values. It preserves the size of the system matrices aswell as the numbering of the mesh nodes and mesh edges.The mesh is only locally stretched or shrunk when shapeparameters are modified. In general, the global coordinatesof the nodes as well as the length and orientation of theedges of the topologically fixed mesh change when shapeparameters change, however, these changes neither introducenew state variables nor eliminate existing state variables.The matrices B, LT are uniquely determined by the circuittopology and therefore remain constant, while the matrices Cand G are defined as functions of the design parameters. At a

FERRANTI et al.: PARAMETERIZED MODEL ORDER REDUCTION 403

deeper level, in the MNA equations (12a)–(12b), the previousassumptions lead to P(g), Lp(g), Cd (g), and R(g), while theother internal PEEC matrices A,�, and K are constant. Theproposed PMOR method starts from computing the multivari-ate models P(g), Lp(g), Cd (g), and R(g), guaranteeing somematrix properties, as explained in Section III-B.

When the multivariate models P(g), Lp(g), Cd (g),and R(g) are computed, instead of assembling a PEEC modeland performing a MOR step for each point of interest g =(g(1)

k1, . . . , g(N)

kN) in the design space, the Laguerre-SVD MOR

method [19] is applied to each PEEC model related to theestimation design space grid:

1) choose a value for α (positive scaling parameter ofLaguerre basis functions) and the reduced order q;

2) solve (G + αC)Q0 = B;3) for k = 1, . . . , q−1 solve (G+αC)Qk = (G−αC)Qk−1;4) Kr = [Q0, . . . , Qq−1];

and a corresponding set of Krylov matrices Kr is computed.Then, this set of Krylov matrices is interpolated and modeledas Kr (g). The sampling density in the estimation design spacegrid is important to accurately describe the parameterizedbehavior of an EM system under study over the entire designspace of interest. A technique to choose the number ofpoints in the estimation grid can be found in [37]. Once themultivariate models P(g), Lp(g), Cd (g), R(g), and Kr (g) arebuilt, a PEEC model Y(s, g) = LT (sC(g)+ G(g))−1B can beassembled and a projection matrix U(g) can be computed bymeans of the singular value decomposition [38] of Kr (g) [19]for any point g = (g(1)

k1, . . . , g(N)

kN)

U(g)(g)V(g)T = SVD[Kr (g)]. (13)

Finally, a congruence transformation is applied onC(g), G(g), L, and B using U(g) [19]

Cr (g) = U(g)T C(g)U(g) (14a)

Gr (g) = U(g)T G(g)U(g) (14b)

Br (g) = U(g)T B (14c)

Lr (g) = U(g)T L (14d)

to obtain the parameterized reduced model. A flowchart thatdescribes the different steps of the proposed PMOR methodis shown in Fig. 3. Concerning the reduced order, whichrepresents the column dimension of Kr (g) and U(g), it ischosen by a bottom-up approach, it is increased as long asa certain root mean square (RMS)-error threshold is satisfiedin the validation design space grid.

B. Multivariate Interpolation of the Internal PEEC Matrices

Starting from multivariate data samples {gk, P(gk),

Lp(gk), Cd (gk), R(gk), and Kr (gk)}Ktotk=1, the multivariate

models P(g), Lp(g), Cd (g), R(g), and Kr(g) are built. Whilethe interpolation process of the set of Krylov matrices isperformed without any constraint, the multivariate models ofthe internal PEEC matrices preserve the positive definitenessof P(g), Lp(g), and Cd (g) and the positive semidefinitenessof R(g). Consequently, the properties (11a)–(11b) of an ad-mittance PEEC model and its related passivity are satisfied for

P(g), Lp(g), C

d(g), R(g), g � (g(1), ..., g(N))

Compute multivariate models of the internal PEEC matrices

MOR step Y(s, g) →Yr(s, g) by means of

a congruence transformation

Compute multivariate model of the Krylov matrix

Kr(g) (Laguerre-SVD MOR method)

Fig. 3. Flowchart of the proposed PMOR method.

any point g = (g(1)k1

, . . . , g(N)kN

) over the design space. Sincethe matrices R(g) and Cd (g) are diagonal, only the diagonalelements are interpolated by means of a positivity-preservinginterpolation scheme. Multivariate interpolation schemes thatbelong to a class of positive interpolation operators [39] can beused, e.g., Shepard’s method [40], multilinear, and simplicialmethods [41]. Such interpolation schemes have interpolationkernel functions that only depend on the design space gridpoints. In the case of multilinear interpolation, each interpo-lated matrix T(g(1), . . . , g(N)), being in turn Cd(g), R(g), canbe written as

T(g(1), . . . , g(N))

=K1∑

k1=1

· · ·KN∑

kN =1

T(g(1)

k1,...,g(N)

kN

)�k1(g(1)) · · · �kN (g(N)) (15)

where T(g(1)

k1,...,g(N)

kN

) are in turn Cd,

(g(1)

k1,...,g(N)

kN

),R(g(1)

k1,...,g(N)

kN

),

and therefore the discrete set of Cd , R matrices related tothe estimation design space grid. Each interpolation kernel�ki (g(i)), i = 1, . . . , N is selected as in piecewise linearinterpolation

g(i) − g(i)ki −1

g(i)ki

− g(i)ki −1

, g(i) ∈[g(i)

ki −1, g(i)ki

], ki = 2, . . . , Ki , (16a)

g(i)ki +1 − g(i)

g(i)ki +1 − g(i)

ki

, g(i) ∈[g(i)

ki, g(i)

ki +1

],

ki = 1, . . . , Ki − 1, (16b)

0, otherwise (16c)

hence, the interpolation kernels �ki (g(i)), i = 1, . . . , N areindependent of the matrices used in the interpolation processand depend only on the design space grid points. Otherinterpolation schemes have kernel functions that depend on thematrices used in the interpolation process, e.g., multivariatecubic spline interpolation [42]. It is a useful technique tointerpolate multivariate data points because of its stable andsmooth characteristics and because it performs elementwiseinterpolation. Unfortunately, although ordinary spline schemes

404 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 1, NO. 3, MARCH 2011

are generally well behaved, they do not prevent overshoot andundesired oscillations at intermediate points, which can violatesuch inherited data features as positivity. Some modified splineinterpolation schemes that are able to preserve positivity of thedata samples in the univariate case are described in [43]–[45].Another simpler and straightforward approach to preservepositivity using the ordinary splines is proposed in this paperand it is composed of three steps: 1) an analytical mappingof the data samples is performed; 2) the new data samplesare interpolated using ordinary splines without any constraint;3) the interpolated data samples are transformed back by theinverse mapping. The mapping function has to be able toensure the positivity of the interpolated data samples after theinverse mapping. For any positive diagonal matrix entry f (g)of the matrices Cd(g), R(g) under modeling, the followingmapping function is used:

M(g) = log

(f (g)

min( f (g))

), g ∈ {gk}Ktot

k=1 . (17)

Once the transformed data samples are modeled by usingmultivariate splines, the inverse mapping function

Minv (g) = min( f (g))exp(M(g)), g ∈ {gk}Ktot,interpk=1 (18)

is used for the back transformation. It is straightforward toverify that the following procedure ensures the positivity of thefinal interpolated values. While a diagonal matrix is positivedefinite if and only if all the diagonal elements are posi-tive, a nondiagonal matrix requires more general conditions.The matrix P(g) is full, symmetric, and positive definite,while the matrix Lp(g) is symmetric and positive definite,and in general a certain degree of sparsity can be presentdue to orthogonal elementary cells. It is easy to show thatmultivariate interpolation schemes that belong to a class ofpositive interpolation operators [39]–[41] are able to preservethe Positive-definiteness property when they are applied topositive definite matrices. When the interpolation of positive-definite nondiagonal matrices is performed elementwise byschemes with kernel functions that depend on the matricesused in the interpolation process (e.g., multivariate cubicspline interpolation), the following procedure can be used toguarantee the positive-definiteness property. Let us denote

S(R) = {Q ∈ M(R), QT = Q} (19)

the space of all R × R real symmetric matrices with M(R)the space of R × R real matrices and

P(R) = {Q ∈ S(R), Q > 0} (20)

the space of all R × R real symmetric positive-definite ma-trices. It is well known that the matrix exponential is aone-to-one map from S(R) to P(R). In other words, thematrix exponential of any real symmetric matrix is a realsymmetric positive-definite matrix, and the inverse of thematrix exponential (i.e., principal matrix logarithm) of anyreal symmetric positive-definite matrix is a real symmetricmatrix [46], [47]. Exploiting such property of the exponentialmap, the matrices P(g) and Lp(g) that are symmetric andpositive definite are mapped from P(R) to S(R) using the

principal matrix logarithm operator, and then only the loweror upper triangular part is interpolated elementwise using theordinary splines. Finally, the matrices are mapped back bythe matrix exponential operator, which results in symmetricpositive-definite matrices and therefore the original propertiesof the matrices P(g) and Lp(g) are preserved. We proposea multivariate interpolation process that is able to preservethe positive definiteness of P(g), Lp(g), and Cd (g) and thepositive semidefiniteness of R(g). Consequently, the proper-ties (11a)–(11b) of the admittance PEEC model Y(s, g) =LT (sC(g) + G(g))−1B and its related passivity are satisfiedfor any point g = (g(1)

k1, . . . , g(N)

kN) over the design space.

The overall computational complexity of the presented PMORalgorithm can be divided into: 1) complexity of computing themultivariate models of P(g), Lp(g), Cd(g), R(g), and Kr (g)by interpolation; 2) complexity of the SVD operation on Kr (g)to obtain the projection matrix U(g); 3) complexity of thecongruence transformation by means of U(g). Which step isthe most computationally expensive cannot be established inadvance, since the computational complexity of the interpo-lation process depends on the chosen interpolation scheme.Concerning the SVD operation, it can be replaced by cheapermodified Gram–Schmidt and Householder QR operations [19],[38], which are computationally cheaper.

C. Passivity Assessment Considerations

The properties of the PEEC matrices, the multivariate in-terpolation approach, and the Laguerre-SVD MOR algorithmensure overall stability and passivity for a parameterized ROMYr (s, g) by construction. Although no passivity check isrequired for Yr (s, g), we describe in this section a passivitytest for the sake of completeness. Let us assume that Yr (s, g)is obtained and one wants to perform a passivity test for aspecific point g in the design space. If the descriptor matrixCr of Yr (s, g) is singular, the procedure described in [48] isused to convert the descriptor system into a standard state-space model

d x(t)

dt= Ax(t) + Bu(t) (21a)

y(t) = Cx(t) + Du(t). (21b)

Otherwise, the standard state-space model can beobtained by

A = −Cr−1Gr

B = Cr−1Br

C = BrT

D = Dr. (22)

Once Yr (s, g) is transformed into a standard state-spaceform, its passivity can be verified by computing the eigenval-ues of an associated Hamiltonian matrix [49]

H =[ A − BR−1C BR−1BT

−CTR−1C −AT + CTR−1BT

](23)

with R = D + DT . The system Yr (s, g) is passive ifH has no purely imaginary eigenvalues. This passivity testcan be applied only if D + DT is not singular. If such

FERRANTI et al.: PARAMETERIZED MODEL ORDER REDUCTION 405

w S

t

h

w

Fig. 4. Cross section of the coupled microstrips.

0 1 2 3 41

2

3

410−10

10−5

100

105

Frequency [GHz]

Spacing [mm]

|Y11

| [S]

Fig. 5. Magnitude of the bivariate ROM of Y11(s, S).

singularity exists, the modified Hamiltonian-based passivitycheck proposed in [50] should be used.

IV. NUMERICAL RESULTS

This section presents two numerical examples that validatethe proposed PMOR method. Let us define the weighted RMSerror as

Err(g)

=

√√√√√∑(Nport )2

i=1

∑Ksk=1

∣∣∣wYi (sk, g)(

Yr,i (sk, g) − Yi (sk, g))∣∣∣2

(Nport )2 Ks

(24)

with

wYi (s, g) = |(Yi (s, g))−1| (25)

where Nport is the number of system ports and Ks is thenumber of frequency samples. The worst case RMS error overthe validation grid is chosen to assess the accuracy and thequality of parameterized ROMs

gmax = argmaxg

Err(g), g ∈ validation grid (26)

Errmax = Err(gmax) (27)

and it is used in the numerical examples. The proposed PMORalgorithm was implemented in MATLAB R2009A [51] and allexperiments were carried out on Windows platform on IntelCore2 Extreme CPU Q9300 2.53-GHz machines with 8 GBRAM.

DataModel

S = 3.875 mm

S = 1.125 mm

DataModel

S = 1.125 mm

S = 3.875 mm

0 1 2 3 4

Frequency [GHz]

10−6

10−4

10−2

100

102

|Y11|[S

]

0 1 2 3 4

Frequency [GHz]

10−6

10−4

10−2

100

102

|Y11|[S

]

Fig. 6. Magnitude of the bivariate ROMs of Y11(s, S) and Y12(s, S) (S ={1.125, 2.375, and 3.875} mm).

TABLE II

PARAMETERS OF THE COUPLED MICROSTRIPS

Parameter Min. Max.

Frequency ( f req) 1 kHz 4 GHz

Spacing (S) 1 mm 4 mm

A. Two Coupled Microstrips with Variable Spacing

Two coupled microstrips (length L = 2 cm) have beenmodeled in this example. The cross section is shown in Fig. 4.The conductors have width W = 500 μm and thickness t =50 μm, and the dielectric is 800 μm thick. A bivariate ROM isbuilt as a function of the spacing S between the microstrips inaddition to frequency. Their corresponding ranges are shownin Table II.

The PEEC method is used to compute the C, G, B, and Lmatrices in (10a)–(10b) for 25 values of the spacing. The orderof all original PEEC models is equal to nu = 2640. Themultivariate models P(g), Lp(g), Cd(g), R(g), and Kr (g) arecomputed by spline interpolation using only nine spacingvalues and with a CPU time equal to 9.6 s. Then, the bivariateROM Yr (s, S) is obtained with a reduced order q = 38. Fig. 5shows the magnitude of the parameterized ROM of Y11(s, S),while Fig. 6 compares the magnitude of Y11(s, S), Y12(s, S)

406 IEEE TRANSACTIONS ON COMPONENTS, PACKAGING AND MANUFACTURING TECHNOLOGY, VOL. 1, NO. 3, MARCH 2011

1 1.5 2 2.5 3 3.5 410−6

10−5

10−4

10−3

Spacing [mm]

min

(|rea

l(ei

g(H

))|)

Fig. 7. Minimum absolute value of the real part of the Hamiltonian matrixeigenvalues.

TABLE III

PARAMETERS OF THE SPIRAL INDUCTOR

Parameter Min. Max.

Frequency ( f req) 10 kHz 30 GHzHorizontal length (Lx ) 0.46 mm 0.93 mmVertical length (L y ) 0.46 mm 0.93 mm

and their parameterized ROMs for the spacing values S ={1.125, 2.375, and 3.875} mm. These specific spacing valueshave not been used during the construction of the multivariatemodels P(g), Lp(g), Cd(g), R(g), and Kr (g), nevertheless,an excellent agreement between model and data can be ob-served. The worst case RMS error defined in (27) is equalto 1.8 × 10−2 and it occurs for gmax = S = 3.875 mm.Fig. 7 shows the minimum absolute value of the real part ofthe Hamiltonian matrix eigenvalues over a dense sweep of thedesign space. Since there are no purely imaginary eigenvalues,the parameterized ROM is passive over the design space ofinterest. As clearly seen, the parameterized ROM capturesthe behavior of the system very accurately while guaranteeingstability and passivity over the entire design space.

B. Spiral Inductor with Variable Horizontal and VerticalLength

An integrated spiral inductor has been modeled in thisexample. The structure is shown in Fig. 8. The Conductor’swidth is equal to 46 μm. A trivariate ROM is built as afunction of the horizontal Lx and vertical L y length of thespiral inductor in addition to frequency. Their correspondingranges are shown in Table III.

The PEEC method is used to compute the C, G, B, and Lmatrices in (10a)–(10b) for 11 values of Lx and 11 values ofL y . The order of all original PEEC models is nu = 801. Themultivariate models P(g), Lp(g), Cd (g), R(g), and Kr (g) arecomputed by spline interpolation using only six values ofLx and six values of L y and with a CPU time equal to43.7 s. Then, the trivariate ROM Yr (s, Lx , L y) is obtainedwith a reduced order q = 91. Figs. 9 and 10 show the

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x [mm]

y [m

m]

Ly

Lx

Fig. 8. Structure of the spiral inductor.

010

2030

0.4

0.6

0.8

1

10−5

100

Frequency [GHz]

Lx [mm]

|Y11|[S

]

Fig. 9. Magnitude of the trivariate ROM of Y11(s, Lx , L y) (L y = 0.46 mm).

010

2030

0.4

0.6

0.8

1

10−5

100

Frequency [GHz]

Lx [mm]

|Y11|[S

]

Fig. 10. Magnitude of the trivariate ROM of Y11(s, Lx , L y) (L y =0.93 mm).

magnitude of the parameterized ROM of Y11(s, Lx , L y) forthe vertical length values L y = {0.46, 0.93} mm, whileFig. 11 compares the magnitude of Y11(s, Lx , L y) and itsparameterized ROM for the horizontal and vertical lengthvalues Lx = 0.63 mm, Ly = {0.50, 0.63, and 0.76} mm.These specific horizontal and vertical length values have notbeen used during the construction of the multivariate models

FERRANTI et al.: PARAMETERIZED MODEL ORDER REDUCTION 407

0 5 10 15 20 25 3010−6

10−4

10−2

100

102

Frequency [GHz]

DataModel

Ly = 0.50 mm

Ly = 0.76 mm

|Y11|[S

]

Fig. 11. Magnitude of the trivariate ROM of Y11(s, Lx , L y) (Lx = 0.63 mm,Ly = {0.50, 0.63, and 0.76} mm).

0.4

0.6

0.8

1 0.4 0.6 0.8 1

0.05

0.1

0.15

0.2

0.25

Ly [mm]

Lx [mm]

min

(|rea

l(ei

g(H

))|)

Fig. 12. Minimum absolute value of the real part of the Hamiltonian matrixeigenvalues.

P(g), Lp(g), Cd(g), R(g), and Kr (g), nevertheless, an excel-lent agreement between model and data can be observed. Theworst case RMS error defined in (27) is equal to 5×10−2 andit occurs for gmax = {Lx , L y} = {0.86, 0.76} mm. Fig. 12shows the minimum absolute value of the real part of theHamiltonian matrix eigenvalues over a dense sweep of thedesign space. Since there are no purely imaginary eigenvalues,the parameterized ROM is passive over the design space ofinterest. As in the previous example, the parameterized ROMis able to accurately describe the parameterized behavior ofthe system while preserving overall stability and passivity.

V. CONCLUSION

We have presented a new PMOR technique applicable toPEEC analysis which is based on a parameterization processof matrices generated by the PEEC method and the projec-tion subspace generated by the Laguerre-SVD MOR method.Overall stability and passivity of parameterized ROMs areguaranteed by construction over the design space of inter-est. Numerical examples have validated the proposed PMORapproach on practical application cases, showing that it is ableto build very accurate parameterized ROMs of highly dynamic

EM systems while guaranteeing stability and passivity over theentire design space of interest.

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Francesco Ferranti (M’10) received the B.S. degree(summa cum laude) in electronic engineering fromthe Universita degli Studi di Palermo, Palermo,Italy, in 2005, and the M.S. degree (summa cumlaude and honors) in electronic engineering from theUniversita degli Studi dell’Aquila, L’Aquila, Italy, in2007. He is currently pursuing the Ph.D. degree inthe Department of Information Technology, GhentUniversity, Ghent, Belgium.

His current research interests include paramet-ric macromodeling, parameterized model order

reduction, electromagnetic compatibility numerical modeling, and systemidentification.

Giulio Antonini (M’94–SM’05) received the Laureadegree (summa cum laude) in electrical engineer-ing from the Universita degli Studi dell’Aquila,L’Aquila, Italy, in 1994, and the Ph.D. degree inelectrical engineering from the University of Rome“La Sapienza,” Rome, Italy, in 1998.

He has been with the UAq EMC Laboratory,Department of Electrical Engineering, University ofL’Aquila, since 1998, where he is currently an Asso-ciate Professor. He has authored or co-authored morethan 170 technical papers and two book chapters.

Furthermore, he has given keynote lectures and chaired several special sessionsat international conferences. He holds one European Patent. His currentresearch interests include electromagnetic compatibility (EMC) analysis, nu-merical modeling, and signal integrity for high-speed digital systems.

Prof. Antonini has been the recipient of the IEEE TRANSACTIONS ON

ELECTROMAGNETIC COMPATIBILITY Best Paper Award in 1997, the Com-puter Simulation Technology University Publication Award in 2004, the IBMShared University Research Award in 2004, 2005 and 2006, and the Institutionof Engineering and Technology, Science – Measurement & Technology BestPaper Award in 2008. He has received the Technical Achievement Award fromthe IEEE EMC Society in 2006 for “innovative contributions to computationalelectromagnetics on the partial element equivalent circuit technique for EMCapplications.”

Tom Dhaene (SM’05) was born in Deinze, Belgium,on June 25, 1966. He received the Ph.D. degree inelectrotechnical engineering from the University ofGhent, Ghent, Belgium, in 1993.

He was a Research Assistant in the Departmentof Information Technology, University of Ghent,from 1989 to 1993, where his research focused ondifferent aspects of full-wave electromagnetic circuitmodeling, transient simulation, and time domaincharacterization of high-frequency and high-speedinterconnections. In 1993, he joined the Electronic

Design Automation company Alphabit (now part of Agilent), San Jose, CA.He was one of the key developers of the planar electromagnetic simulatoradvanced design system “Momentum.” In 2000, joined the Departmentof Mathematics and Computer Science, University of Antwerp, Antwerp,Belgium, as a Professor. Since October 2007, he has been a Full Professorin the Department of Information Technology, Ghent University. As authoror co-author, he has contributed more than 150 peer-reviewed papers andabstracts in international conference proceedings, journals, and books. He isthe holder of three U.S. patents.

FERRANTI et al.: PARAMETERIZED MODEL ORDER REDUCTION 409

Luc Knockaert (SM’00) received the M.Sc. degreein physical engineering, another M.Sc. degree intelecommunications engineering, and the Ph.D. de-gree in electrical engineering from Ghent University,Ghent, Belgium, in 1974, 1977, and 1987, respec-tively.

He was working in the North-South Cooperationand in development projects at the Universities ofthe Democratic Republic of the Congo and Burundifrom 1979 to 1984 and from 1988 to 1995. Heis presently associated with the Interdisciplinary

Institute for BroadBand Technologies and is a Professor in the Department ofInformation Technology, Ghent University. As author or co-author, he haspublished more than 100 papers in international journals and conferenceproceedings. His current research interests include application of linearalgebra and adaptive methods in signal estimation, model order reduction,and computational electromagnetics.

Prof. Knockaert is a member of Mathematical Association of America andthe Society for Industrial and Applied Mathematics.

Albert E. Ruehli (LF’03) received the Ph.D. de-gree in electrical engineering from the Universityof Vermont, Burlington, in 1972, and an HonoraryDoctorate from the Lulea University, Lulea, Sweden,in 2007.

He has been a member of various projects withIBM including mathematical analysis, semiconduc-tor circuits and devices modeling, and as Man-ager of a Very Large Scale Integration Design andComputer-Aided Design Group. Since 1972, he hasbeen at IBM’s T. J. Watson Research Center in

Yorktown Heights, NY, where he was a Research Staff Member in theElectromagnetic Analysis Group. He is now an Emeritus of IBM Research andan Adjunct Professor in the Electromagnetic Compatibility Society, MissouriUniversity of Science and Technology, Rolla. He is the editor of two books,Circuit Analysis, Simulation and Design (New York: North Holland, 1986,1987), and he is an author or co-author of over 190 technical papers.

Dr. Ruehli has served the IEEE in numerous capacities. In 1984 and 1985,he was the Technical and General Chairman, respectively, of the InternationalConference of Computer Design. He has been a member of the IEEE Ad-ministrative Committee for the Circuit and System Society and an AssociateEditor for the IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN. Hehas given talks at universities including keynote addresses and tutorials atconferences, and has organized many sessions. He is a recipient of the IBMResearch Division or IBM Outstanding Contribution Awards in 1975, 1978,1982, 1995, and 2000. He received the Guillemin-Cauer Prize Award in 1982for his work on waveform relaxation, and in 1999 he received a Golden JubileeMedal, both from the IEEE Casualty Actuarial Society. In 2001, he receiveda Certificate of Achievement from the IEEE Electromagnetic CompatibilitySociety for Inductance Concepts and the Partial Element Equivalent Circuitmethod. He received the Richard R. Stoddart Award in 2005, and in 2007 hereceived the Honorary Life Member Award from the IEEE ElectromagneticCompatibility Society for outstanding technical performance. He is a memberof Society for Industrial and Applied Mathematics.