IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. , NO. , 2009 1

Guaranteed Passive Parameterized Model OrderReduction of the Partial Element Equivalent Circuit

(PEEC) MethodFrancesco Ferranti,Member, IEEE, Giulio Antonini, Senior Member, IEEE, Tom Dhaene,Senior Member, IEEE,

and Luc Knockaert,Senior Member, IEEE

Abstract—The decrease of IC feature size and the increaseof operating frequencies require 3-D electromagnetic methods,such as the Partial Element Equivalent Circuit (PEEC) method,for the analysis and design of high-speed circuits. Very largesystems of equations are often produced by 3-D electromagneticmethods. During the circuit synthesis of large scale digital oranalog applications, it is important to predict the response of thesystem under study as a function of design parameters, such asgeometrical and substrate features, in addition to frequency (ortime). Parameterized model order reduction (PMOR) methodsbecome necessary to reduce large systems of equations withrespect to frequency and other design parameters.

We propose an innovative PMOR technique applicable toPEEC analysis, which combines traditional passivity-preservingmodel order reduction methods and positive interpolationschemes. It is able to provide parametric reduced order models,stable and passive by construction over a user defined rangeof design parameter values. Numerical examples validate theproposed approach.

Index Terms—Partial Element Equivalent Circuit method(PEEC), parameterized model order reduction (PMOR), inter-polation, passivity.

I. I NTRODUCTION

Electromagnetic (EM) methods [1]–[3] have become in-creasingly indispensable analysis and design tools for a varietyof complex high-speed systems. Users are usually only inter-ested in a few field (E,H) or circuit (i,v) unknowns at the inputand output ports, in the frequency domain or time domain,while the use of these methods usually results in the computa-tion of a huge number of field or circuit unknowns. Therefore,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]–[8]. The development of a reducedorder model (ROM) of the EM system has become a topicof intense research over the last years. Important applica-tions of EM-based modeling include high-speed packages,

Manuscript received November 2009.Francesco Ferranti, Tom Dhaene and Luc Knockaert are with the De-

partment of Information Technology (INTEC), at Ghent University - IBBT,Sint Pietersnieuwstraat 41, 9000 Ghent, Belgium, email:francesco.ferranti,tom.dhaene, luc.knockaert@ugent.be.

Giulio Antonini is with the UAq EMC Laboratory, Dipartimento di Ingeg-neria Elettrica e dell’Informazione, Universita degli Studi dell’Aquila, Via G.Gronchi 18, 67040, L’Aquila, Italy, e-mail: giulio.antonini@univaq.it.

This work was supported by the Research Foundation Flanders (FWO)and by the Italian Ministry of University (MIUR) under a Program for theDevelopment of Research of National Interest, (PRIN grant n. 20089J4SM9-002).

interconnects, vias, and on-chip passive components [9]–[11].Among EM methods, the Partial Element Equivalent Circuit(PEEC) method has gained an increasing popularity amongelectromagnetic compatibility engineers due to its capabilityto transform the EM system under examination into a passiveRLC equivalent circuit. PEEC uses a circuit interpretation ofthe electric field integral equation (EFIE), thus allowing tohandle complex problems involving EM fields and circuits[2], [12]. The PEEC equivalent circuits are usually connectedwith nonlinear circuit devices such as drivers and receiversin a time domain circuit simulator (e.g. SPICE). 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 tens ofthousands. For this reason, MOR techniques are often usedto reduce the size of a PEEC model [8], [13]. A passivereduced-order interconnect macromodeling algorithm, knownas PRIMA [8], has received great attention due to its capabilityto generate passive models of RLC circuits, which is importantbecause stable, but nonpassive models can produce unstablesystems when connected to other stable, even passive, loads.

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 study asa function of environmental effects, manufacturing variations,and fluctuations in the critical dimensions of transmission linessuch as width and height. Typical design process includesoptimization and design space exploration, and thus requiresrepeated simulations for different design parameter values.It is often not feasible to perform multiple simulations oflarge circuits due to variations in these parameters. Suchdesign 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.

A number of PMOR methods have been developed. Theperturbation technique [14] is one of the early work tocapture small variation around the nominal circuit values.Other PMOR techniques are based on statistical performanceanalysis [15], [16]. Multiparameter moment-matching methodspresented in [17], [18] use a subspace projection approachand guarantee the passivity. However, the structure of suchmethods may present some computational problems, and theresulting reduced models usually suffer from oversize when

2 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. , NO. , 2009

the number of moments to match is high, either because highaccuracy (order) is required or because the number of parame-ters is large. The Compact Order Reduction for parameterizedExtraction (CORE) algorithm [19] is an explicit-and-implicitscheme. It is numerically stable, but unfortunately it does notpreserve the passivity. The Parameterized Interconnect Macro-modeling via a two-directional Arnoldi process (PIMTAP)algorithm presented in [20] is numerically stable, preservesthe passivity of parameterized RLC networks, but, such as allmultiparameter moment-matching based PMOR techniques, itis suitable only to a low-dimensional design space.

This paper proposes a PMOR method applicable to PEECanalysis that provides parametric reduced order models, stableand passive by construction over the design space of interest.It combines traditional passivity-preserving model order re-duction methods and interpolation schemes based on a classof positive interpolation operators [21], to guarantee overallstability and passivity of the parametric reduced order model.A method has been recently proposed in [22], [23] based on anefficient and reliable combination of rational identification andpositive interpolation schemes to build parameterized macro-models, stable and passive by construction over the designspace of interest, starting from multivariate data samples of theinput-output system behavior and not from system equationsas in all PMOR techniques previously discussed. The PMORmethod proposed in this paper starts by computing a setof reduced order PEEC models using the PRIMA algorithmfor different design parameters values. The PEEC modelsare put into a state-space form for which PRIMA preservespassivity. The PEEC models and the corresponding reducedmodels obtained by means of PRIMA describe an admittance(Y) representation. Since at microwave frequencies, theY-representation cannot be accurately measured, the scatteringparameters (S) representation is used to describe the broad-band frequency behavior of microwave systems. Therefore,the reduced models are translated fromY-representation intoS-representation. In the following paper we refer to theseinitial S-reduced order models asroot ROMs. However, itshould be noted that the proposed PMOR technique is notbound to PRIMA, other passivity-preserving MOR methodscan be used, such as the Laguerre-SVD MOR [24] or thepassivity-preserving truncated balanced realization algorithms[25]. Finally, a parametric ROM is built by combining allrootROMsthrough an interpolation scheme that preserves stabilityand passivity properties over the complete design space.

The 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 numerical examples are presented in Section IV,validating the proposed technique.

II. PEECFORMULATION

The PEEC method [2] starts from the integral equationform of Maxwell’s equations. It has received an increasinginterest by electrical engineers since it provides a circuitinterpretation of the EFIE equation, thus allowing to handlecomplex problems involving both circuits and electromagnetic

fields [2], [12], [26]–[30]. In what follows, we describe aquasi-static PEEC formulation [2] that approximates the full-wave PEEC approach [26].

In the standard approach, volumes and surfaces of conduc-tors and dielectrics are respectively discretized into hexahedraand patches that represent elementary regions [30] over whichthe current and charge densities are expanded into a series ofbasis functions. Pulse basis functions are usually adopted asexpansion and weight functions. Such choice of pulse basisfunctions corresponds to assume constant current density andcharge density over the elementary volume (inductive) andsurface (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.

Conductor losses are modeled by their ohmic resistance,while dielectrics requires modeling the excess charge due tothe dielectric polarization. This is done by means of the excesscapacitance which is placed in series to the partial inductanceof each dielectric elementary cell. Magnetic field couplingbetween elementary volume cells is characterized by partialinductances, while electric field coupling is modeled by thecoefficients of potential. An example of PEEC circuit electricalquantities for a conductor elementary cell is illustrated, in theLaplace domain, in Fig. 1 where the current controlled voltagesourcessLp,ijIj and the current controlled current sourcesIcci

model the magnetic and electric field coupling, respectively.Hence, Kirchoff’s laws for conductors can be re-written as

P−1 dv(t)dt

−AT i(t)− ie(t) = 0 (1a)

−Av(t)− Lpdi(t)dt

−Ri(t) = 0 (1b)

where v(t) denotes the node potentials to infinity,i(t) de-notes the currents flowing in volume cells,P and Lp arethe coefficients of potential and partial inductance matrices,respectively,R is a diagonal matrix containing the resistancesof volume cells,A is the incidence matrix,ie(t) representsthe external currents. A selection matrixK is introduced todefine the port voltages by selecting node potentials. The samematrix is used to obtain the external currentsie(t) by thenp

port currentsip(t)

vp(t) = Kv(t) (2a)

ie(t) = −KT ip(t) (2b)

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

d qd(t) [31].As a consequence, the polarization current flowing through thedielectric is modeled in terms of the excess voltage drop aswell. Hence, for dielectric elementary volumes, equations (1)

F. FERRANTI et al.: PARAMETRIC MODEL ORDER REDUCTION 3

become

P−1 dv(t)dt

−AT i(t)− ie(t) = 0 (3a)

−Av(t)− Lpdi(t)dt

− vd(t) = 0 (3b)

i(t) = Cddvd(t)

dt(3c)

1 2

I c 1 I

c 2

I cc 1 I cc 2

3

I c 3

I cc 3

L p 11

+

-

s L p, 12

I 2

I 1

+

L p 22

+

-

s L p, 21

I 1

I 2

R 1

R 2

V p I p I s

1 /P 22

1 /P 33

1 /P 11

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

A. Descriptor representation of PEEC circuits

Let us assume to have discretized the system under studyconsisting of conductors and dielectrics and to have generatedni volume cells where currents flow andnn surface cellswhere charge is located; the resultant number of elementarycells of conductors and dielectrics isnc andnd, respectivelyand that of electrical nodes isnn. Furthermore, let us assumeto be interested in generating an admittance representationY(s) havingnp output currentsip(t) under voltage excitationvp(t). If the MNA approach [32] is used, the global numberof unknowns isnu = ni + nd + nn + np. In a matrix form,the previous equations (1)-(3) read

P−1 0nn,ni 0nn,nd0nn,np

0ni,nn Lp 0ni,nd0ni,np

0nd,nn 0nd,ni Cd 0nd,np

0np,nn 0np,ni 0np,nd0np,np

︸ ︷︷ ︸C

d

dt

v(t)i(t)

vd(t)ip(t)

︸ ︷︷ ︸x(t)

=

−

0nn,nn −AT 0nn,ndKT

A R Φ 0ni,np

0nd,nn −ΦT 0nd,nd0nd,np

−K 0np,ni 0np,nd0np,np

︸ ︷︷ ︸G

·

v(t)i(t)

vd(t)ip(t)

︸ ︷︷ ︸x(t)

+

[0nn+ni+nd,np

−Inp,np

]

︸ ︷︷ ︸B

· [ vp(t)]

︸ ︷︷ ︸u(t)

(4)

whereInp,np is the identity matrix of dimensions equal to thenumber of ports. MatrixΦ is

Φ =

[0nc,nd

Ind,nd

](5)

In a more compact form, the previous equations can berewritten as:

Cdx(t)

dt= −Gx(t) + Bu(t) (6a)

ip(t) = LT x(t) (6b)

where x(t) = [v(t) i(t) vd(t) ip(t)]T and B = L,B ∈ <nu×np . This is an admittancenp-port formulationY(s) = LT (sC + G)−1B, whereby the only sources are thevoltage sources at thenp-port nodes. If we consider N designparametersg = (g(1), ..., g(N)) in addition to frequency, theequations (6a)-(6b) become

C(g)dx(t, g)

dt= −G(g)x(t, g) + B(g)u(t) (7a)

ip(t, g) = L(g)T x(t, g) (7b)

B. Properties of PEEC formulation

When performing transient analysis, stability and passivitymust be guaranteed. It is known that, while a passive systemis also stable, the reverse is not necessarily true [33], which iscrucial when the model is to be utilized in a general-purposeanalysis-oriented nonlinear simulator. Passivity refers to theproperty of systems that cannot generate more energy than theyabsorb through their electrical ports. When a passive system isterminated on any set of arbitrary passive loads, none of themwill cause the system to become unstable [34], [35]. A linearnetwork described by admittance matrixY(s) is passive (orpositive-real) if [36]:

1) Y(s∗) = Y∗(s) for all s, where “∗” is the complexconjugate operator.

2) Y(s) is analytic in<e(s) > 0.3) Y(s) is a positive-real matrix, i.e. :

z∗T(Y(s) + YT (s∗)

)z ≥ 0 ; ∀s : <e(s) > 0 and any

arbitrary vectorz.

Provided that the matricesP−1, Lp, Cd, R are symmetric non-negative definite matrices by construction, it is straightforwardto prove that the matricesC, G satisfy the following properties

C = CT ≥ 0 (8a)

G + GT ≥ 0 (8b)

The properties of the PEEC matricesB = L, C = CT ≥0, G + GT ≥ 0 ensure the passivity of the PEEC admittancemodel Y(s) = LT (sC + G)−1B [37] and allow to exploitthe passivity-preserving capability of PRIMA [8].

III. PARAMETERIZED MODEL ORDER REDUCTION

In this section we describe a parameterized model orderreduction algorithm that is able to include, in addition tofrequency, N design parametersg = (g(1), ..., g(N)) in the re-duced order model, such as the layout features of a circuit (e.g.lengths, widths,...) or the substrate parameters (e.g. thickness,dielectric constant, losses,...). The main objective of the param-eterized MOR method is to accurately approximate the originalscalable system (having a high complexity) with a reduced

4 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. , NO. , 2009

scalable system (having a low complexity) by capturing thebehaviour of the original system with respect to frequency andother design parameters. The proposed algorithm guaranteesstability and passivity of the parametric reduced model overthe entire design space of interest. A flowchart that describesthe different steps of the proposed PMOR method is shown inFig. 2.

for di®erent design parameters values by means offor di®erent design parameters values by means of

a passivity-preserving MOR method, e.g. PRIMAa passivity-preserving MOR method, e.g. PRIMA

the set of equations (8)the set of equations (8)

Yr(s; g(1)k1

; :::; g(N)kN

) ¡! Sr(s; g(1)k1

; :::; g(N)kN

) transformation byYr(s; g(1)k1

; :::; g(N)kN

) ¡! Sr(s; g(1)k1

; :::; g(N)kN

) transformation by

Parametric ROM Sr(s; g) obtained byParametric ROM Sr(s; g) obtained by

Compute Y(s; g(1)k1

; :::; g(N)kN

) models by PEECCompute Y(s; g(1)k1

; :::; g(N)kN

) models by PEEC

for di®erent design parameters valuesfor di®erent design parameters values

MOR step Y(s; g(1)k1

; :::; g(N)kN

) ¡! Yr(s; g(1)k1

; :::; g(N)kN

)MOR step Y(s; g(1)k1

; :::; g(N)kN

) ¡! Yr(s; g(1)k1

; :::; g(N)kN

)

multivariate interpolation of root ROMs Sr(s; g(1)k1

; :::; g(N)kN

)multivariate interpolation of root ROMs Sr(s; g(1)k1

; :::; g(N)kN

)

Sr(s; g(1)k1

; :::; g(N)kN

) called root ROMsSr(s; g(1)k1

; :::; g(N)kN

) called root ROMs

Fig. 2. Flowchart of the proposed PMOR method.

A. Root ROMs

The proposed PMOR technique starts by computing a bunchof stable and passive reduced order models of the PEECadmittance matrixY(s, g) = LT (g)(sC(g) + G(g))−1B(g)using the PRIMA algorithm for a set of points in the designspace, that we call estimation design space grid. The designspaceD(g) is considered as the parameter spaceP(s, g)without frequency. The parameter spaceP(s, g) contains allparameters(s, g). If the parameter space is N-dimensional, thedesign space is (N-1)-dimensional. Two design space gridsare used in the modeling process: an estimation grid and avalidation grid. The first grid is utilized to build therootROMs. The second grid is utilized to assess the capabilityof parametric reduced order models of describing the systemunder study in a set of points of the design space previouslynot used for the construction of theroot ROMs. To clarifythe use of these two design space grids, we show in Fig. 3a possible estimation and validation design space grid in thecase of two design parametersg = (g(1), g(2)). A root ROMis built for each red (x) point in the design space. The set ofroot ROMsis interpolated, as explained in Sections III-C andIII-D, to build a parametric reduced model that is evaluatedand compared with original PEEC models related to the blue(o) design space points. We note that these blue (o) points arenot used for the generation of theroot ROMs.

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. 3. An example of estimation and validation design space grid.

N-dimensional and scattered design space grids can also betreated by the proposed PMOR technique that does not imposeany constraint on the number of design parameters and thedistribution ofg points in the design space. PRIMA algorithmis used to reduce the PEEC admittance representationY(s, g)for each pointg in the estimation design space grid. TheKrylov subspace and a corresponding projection matrixV(g)are built to perform a congruence transformation on theoriginal PEEC matricesC(g), G(g), B(g), L(g)

Cr(g) = V(g)T C(g)V(g)

Gr(g) = V(g)T G(g)V(g)

Br(g) = V(g)T B(g)

Lr(g) = V(g)T L(g) (9)

Since at microwave frequencies, theY-representation cannotbe accurately measured because the required short-circuit testsare difficult to achieve over a broad range of frequencies, theS-representation is used to describe the broadband frequencybehavior of microwave systems. Therefore, the reduced modelsare translated fromY-representation intoS-representationpreserving stability and passivity by the procedure describedin the following Section III-B. In the paper we refer to theseinitial S-reduced order models asroot ROMs. This initial stepallows the separation of frequency from the other parameters,in other words frequency is treated as a special parameter. Thesampling density in the estimation design space grid, whichdecides the number ofroot ROMs, is important to accuratelydescribe the parameterized behavior of an EM system understudy over the entire design space of interest. A techniqueto choose the number of points in the estimation grid, andconsequently the number ofroot ROMs, can be found in [38].

B. Y-S transformation

The definition of S-representation and its relation tothe other system representations depend on the referenceimpedance at each port Z0,i, that in practice is often chosen

F. FERRANTI et al.: PARAMETRIC MODEL ORDER REDUCTION 5

equal to50 Ω. Let Z0 be a real diagonal matrix such thatZ0(i, i) = Z0,i, the S-representation is related to theY-representation by

S(s) =(I− Z1/2

0 Y(s)Z1/20

)(I + Z1/2

0 Y(s)Z1/20

)−1(10)

It is possible to obtain a descriptor formSr(s, g) = [Cr(g), Gr(g), Br(g), Lr(g), Dr(g)]from a descriptor form of the reduced order modelYr(s, g) = [Cr(g), Gr(g), Br(g), Lr(g)] obtained bymeans of PRIMA, using the following equations [39]

Cr(g) = Cr(g)

Gr(g) = Gr(g) + Br(g)Z0Lr(g)T

Br(g) =√

2Br(g)Z1/20

Lr(g) = −Br(g)

Dr(g) = I (11)

We note that the transfer functionY(s) is positive-real if andonly if S(s) is bounded-real [36], i.e.:

1) S(s∗) = S∗(s) for all s, where “∗” is the complexconjugate operator.

2) S(s) is analytic in<e(s) > 0.3) I− ST (s∗)S(s) ≥ 0 ; ∀s : <e(s) > 0.

The bounded-realness property represents the passivity prop-erty for systems described by scattering parameters. Once apassive and stableY-reduced model is obtained by meansof PRIMA, a transformation fromY into S is performedusing the set of equations (11), which results in a reducedS-representation that is still stable and passive. If Krylov-subspace-based MOR algorithms (e.g. PRIMA [8], Laguerre-SVD [24]) are used to build theroot ROMs, no error bound isprovided for them. Hence, the selection of the reduced orderfor eachroot ROM is performed by a bottom-up approach, itis increased as long as the maximum absolute error of eachroot ROM (S-representation) is larger than−60 dB over thefrequency range of interest that is suitably sampled.

C. 2-D PMOR

First, we discuss the representation of a bivariate reducedmodel and afterwards the generalization to more dimensions.Once theroot ROMsare available, the next step is to find abivariate reduced modelSr(s, g) that: 1) is able to accuratelydescribe the system under study in design space points pre-viously not used for the generation of theroot ROMs (thevalidation design space grid is used to test this modelingcapability), 2) is able to preserve stability and passivity overthe entire design space. The bivariate reduced model we adoptcan be written as

Sr(s, g) =K1∑

k=1

Sr(s, gk)`k(g) (12)

whereK1 is the number of theroot ROMs, and the interpola-tion kernels k(g) are scalar functions satisfying the followingconstraints

0 ≤ `k(g) ≤ 1, (13)

`k(gi) = δk,i, (14)K1∑

k=1

`k(g) = 1. (15)

A suitable choice is to select the set`k(g) as in piecewiselinear interpolation

g − gk−1

gk − gk−1, g ∈ [gk−1, gk] , k = 2, ...,K1, (16a)

gk+1 − g

gk+1 − gk, g ∈ [gk, gk+1] , k = 1, ...,K1 − 1, (16b)

0 , otherwise. (16c)

The reduced model in (12) is a linear combination of stableand passive univariate reduced models by means of a class ofpositive interpolation kernels [21]. Stability is automaticallypreserved in (12), since it is a weighted sum of stable rationalmodels ofs. The proof of the passivity-preserving property ofthe proposed PMOR scheme over the entire design space isgiven in Section III-E.

D. (N+1)-D PMOR

The bivariate formulation (12) can easily be generalizedto the multivariate case by using multivariate interpolationmethods. Multivariate interpolation can be realized by meansof tensor product [40] or tessellation [41] methods. Tensorproduct multivariate interpolation methods require that thedata points are distributed on a fully filled, but not neces-sarily equidistant, rectangular grid, while tessellation-basedmultivariate interpolation methods can handle scattered orirregularly distributed data points. For the sake of clarity, weshow in Fig. 4 how a parametric reduced order model is builtin a 2-D design space by means of interpolation ofroot ROMs.

1) Tensor product multivariate interpolation:The paramet-ric reduced model can be written as

Sr(s, g) = (17)

=K1∑

k1=1

· · ·KN∑

kN =1

Sr(s, g(1)k1

, ..., g(N)kN

)`k1(g(1)) · · · `kN (g(N))

where `ki(g(i)), i = 1, ..., N satisfy all constraints (13)-

(15). A suitable choice is to select each set`ki(g(i)) as in

piecewise linear interpolation, which yields to an interpolationscheme in (17) called piecewise multilinear. This method canbe also seen as a recursive implementation of1-D piecewiselinear interpolation. We remark that the interpolation processis local, because the parametric reduced modelSr(s, g) ata specific pointg in the design spaceD(g) only dependson the root ROMs at the vertices of the hypercube thatcontains the pointg. An hypercube inRN has2N vertices,2N

increases exponentially with the number of dimensions, but

6 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. , NO. , 2009

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

g(1)

g(2)

Estimation gridValidation grid

(g(1) , g(2))2

(g(1) , g(2))4(g(1) , g(2))5

Sr(g(1), g(2))5 =

∑4k=1 Sr(s, (g

(1), g(2))k)ℓk(g(1), g(2))

(g(1) , g(2))1

(g(1) , g(2))3

Fig. 4. Example of a parametric reduced model in a 2-D design spaceobtained by means of interpolation ofroot ROMs.

it still remains much smaller than the number of data pointsK1 · K2 · ... · KN in the fully filled design space grid. Thismultivariate interpolation method belongs to the general classof positive interpolation schemes [21].

2) Multivariate simplicial interpolation:Before performingthe multivariate simplicial interpolation process, the designspace is divided into cells using simplices [41], resulting in ageometric structure that connects the points of the estimationdesign space grid. In 2-D this process is called triangulation,while in higher dimensions it is called tessellation. A simplex,or N-simplex, is the N-D analogue of a triangle in 2-D anda tetrahedron in 3-D. Fig. 5 shows a possible triangulationof a 2-D design space starting from a regular and a scatteredestimation grid, respectively.A simplex in N dimensions has N+1 vertices. For eachdata distribution many tessellations can be constructed. Theminimal requirement is that the simplices do not overlap, andthat there are no holes. Kuhn tessellation [42] can be used fora fully filled design space grid, such technique splits everyhypercube inRN into N! simplices. Delaunay tessellation [41]can be used for an irregular design space grid, such techniqueis a space-filling aggregate of simplices and can be performedusing standard algorithms [43]. We indicate a simplex regionof the design space asΩi, i = 1, ..., P and the correspondingN+1 vertices asg Ωi

k , k = 1, ..., N+1. Once the tessellation ofthe design space is accomplished, a tessellation-based linearinterpolation (TLI) is used to build a parametric reduced ordermodel. TLI performs a linear interpolation inside a simplexusing barycentric coordinates [44] as interpolation kernels andit is therefore a local method. If the N-dimensional volumeof the simplex does not vanish, i.e. it is non-degenerate, anypoint enclosed by a simplex can be expressed uniquely as alinear combination of the N+1 simplex vertices. A parametricreduced model can be written as:

Sr(s, g) =N+1∑

k=1

Sr(s, g Ωi

k )`Ωi

k (g) (18)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

g(1)

g(2)

Estimation gridTriangulation

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

g(1)

g(2)

Estimation gridTriangulation

Fig. 5. Triagulation of a 2-D design space starting from a regular (top) anda scattered (bottom) estimation grid.

where Ωi is the simplex the contains the pointg and thebarycentric coordinatesΩi

k (g) satisfy the following properties

0 ≤ `Ωi

k (g) ≤ 1 (19)

`Ωi

k (g Ωii ) = δk,i (20)

N+1∑

k=1

`Ωi

k (g) = 1 (21)

We remark that the interpolation process is local, because theparametric reduced modelSr(s, g) at a specific pointg in thedesign spaceD(g) only depends on the N+1root ROMsatthe vertices of the simplex that contains the pointg. The TLImethod belongs to the general class of positive interpolationschemes [21]. Stability is automatically preserved in (17)-(18), since they are weighted sums of stable rational modelsof s. The proof of the passivity-preserving property of theproposed PMOR schemes over the entire design space is givenin Section III-E. We note that the interpolation kernels wepropose only depend on the design space grid points and theircomputation does not require the solution of a linear systemto impose an interpolation constraint. The proposed PMORtechnique is able to deal with fully filled and scattered design

F. FERRANTI et al.: PARAMETRIC MODEL ORDER REDUCTION 7

space grids. It is general and any interpolation scheme thatleads to a parametric reduced model composed of a weightedsum of root ROMswith weights satisfying (13)-(15) can beused. The error distribution of parametric reduced modelsobtained by the proposed PMOR technique is related to theutilized interpolation scheme and the error of theroot ROMs.

E. Passivity-Preserving Interpolation

In this section we prove that the proposed PMOR methodpreserves passivity over the entire design space. Concerningtheroot ROMs, we have already proven in Section III-B that allthree bounded-realness conditions are satisfied. Condition 1)is preserved in (12) and the proposed multivariate extensions(17)-(18), since they are weighted sums with real nonnegativeweights of systems respecting this first condition. Condition2) is preserved in (12), (17), (18), since they are weightedsums of stable rational reduced models ofs. Condition 3) isequivalent to‖S(s)‖∞ ≤ 1 (H∞ norm) [45], i.e., the largestsingular value ofS(s) does not exceed one in the right-halfs-plane. Using this equivalent condition, in the bivariate casewe can write

‖Sr(s, g)‖∞ ≤K1∑

k=1

‖Sr(s, gk)‖∞ `k(g) ≤K1∑

k=1

`k(g) = 1

(22)

Similar results are obtained for the proposed multivariatecases (17)-(18), so condition 3) is satisfied by constructionusing our PMOR method. We have demonstrated that all threebounded-realness conditions are preserved in the novel PMORalgorithm, using the sufficient conditions (13)-(15) related tothe interpolation kernels.

F. Passivity assessment considerations

The properties of the PEEC matrices, the PRIMA algo-rithm, theY − S transformation procedure and the proposedmultivariate interpolation schemes ensure overall stability andpassivity for the parametric reduced order modelSr(s, g)by construction. Although no passivity check is required forSr(s, g), the authors describe in this section a passivity testfor the sake of completeness. Let us assume thatSr(s, g) isobtained and one wants to carry a passivity test out for aspecific pointg in the design space. If the descriptor matrixCr(g) of Sr(s, g) is singular, the procedure described in [46]is used to convert the descriptor system into a standard state-space model

dx(t)dt

= Ax(t) + Bu(t) (23a)

y(t) = Cx(t) +Du(t) (23b)

otherwise the standard state-space model can be obtained by

A = −Cr(g)−1Gr(g)

B = Cr(g)−1Br(g)

C = Br(g)T

D = Dr(g) (24)

OnceSr(s, g) is transformed into a standard state-space form,its passivity can be verified by computing the eigenvalues ofan associated Hamiltonian matrix [45]

H =

[ A− BR−1DT C −BR−1BT

CTQ−1C −AT + CTDR−1BT

](25)

with R = DTD − I and Q = DDT − I. This passivitytest can only be applied ifDTD − I is not singular. If suchsingularity exists, the modified Hamiltonian-based passivitycheck proposed in [47] should be used.

IV. N UMERICAL RESULTS

A. 2-D example: Microstrip line

In the first example a microstrip line with a dispersiveDriClad dielectricεr = 4.1 and a length = 2 cm has beenmodeled. Its cross section is shown in Fig. 6. The dielectric andconductor thickness values areh = 600 µm andt = 100 µm,respectively. A bivariate reduced order model is built as afunction of frequency and the width of the stripW . Theircorresponding ranges are shown in Table I.

w

t

h

Fig. 6. Cross section of the microstrip.

TABLE IPARAMETERS OF THE MICROSTRIP STRUCTURE.

Parameter Min MaxFrequency (freq) 1 kHz 4 GHzWidth (W) 50 µm 250 µm

The PEEC method is used to compute theC, G, B, Lmatrices in (6a)-(6b) for30 values of the width. The orderof all original PEEC models is equal tonu = 2532. Then,we have built reduced models for12 values of the width bymeans of PRIMA, each with a reduced orderq = 20. A Y−Stransformation has been performed choosing Z0,1 = Z0,2 =50 Ω, which results in a set of12 root ROMs. A bivariatereduced modelSr(s,W ) is obtained by piecewise linearinterpolation of theroot ROMs. The passivity of the parametricreduced model has been checked by the procedure describedin Section III-F on a dense sweep over the design space andthe theoretical claim of overall passivity has been confirmed.Fig. 7 shows the magnitude of the parametric reduced modelof S11(s,W ). Fig. 8 shows the magnitude of the parametricreduced models ofS11(s,W ) and S21(s,W ) for the widthvaluesW = 55, 150, 245 µm. These specific width valueshave not been used in theroot ROMs generation process,nevertheless an excellent agreement between model and datacan be observed. Fig. 9 shows the absolute error distribution

8 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. , NO. , 2009

for S11(s,W ) and S21(s,W ) over a dense reference gridcomposed of200 × 30 (freq,W ) samples. The maximumabsolute error of the bivariate reduced model of theS matrixover the reference grid is bounded by−60.57 dB. As clearlyseen, the parametric reduced model captures very accuratelythe behavior of the system, while guaranteeing stability andpassivity properties over the entire design space.

01

23

4

50

100

150

200

250

0

0.2

0.4

Frequency [GHz]

Width [µm]

|S11

|

Fig. 7. Magnitude of the bivariate reduced model ofS11(s, W ).

B. 3-D example: Multiconductor system with variable sepa-ration

This example reproduces the geometry of a multiconductorsystem composed by six conductors with a length` = 2 cm,a width W = 1 mm and a thicknesst = 0.2 mm. The crosssection is shown in Fig. 10 and depends on the two variablesSx and Sy that represent the horizontal and vertical spacingbetween the conductors. A trivariate reduced order model isbuilt as a function of frequency and the horizontal and verticalspacing. Their corresponding ranges are shown in Table II.

TABLE IIPARAMETERS OF THE MULTICONDUCTOR SYSTEM.

Parameter Min MaxFrequency (freq) 1 kHz 15 GHzHorizontal spacing (Sx) 2 mm 3 mmVertical spacing (Sy) 1 mm 2 mm

The PEEC method is used to compute theC, G, B, Lmatrices in (6a)-(6b) for15 values ofSx and15 values ofSy.The order of all original PEEC models is equal tonu = 702.Then, we have built reduced models for5 values ofSx and8 values ofSy by means of PRIMA, each with a reducedorder q = 54. A Y − S transformation has been performedchoosing Z0,i = 50 Ω, i = 1, ..., 6, which results in aset of 40 root ROMs. The ports of the system are six andare defined between a conductor and the corresponding oneabove. A trivariate reduced modelSr(s, Sx, Sy) is obtained bypiecewise multilinear and multivariate simplicial interpolationof the root ROMs. The passivity of the parametric reduced

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

Frequency [GHz]

|S11

|

DataModel

W = 245 µm

W = 55 µm

0 1 2 3 40.9

0.92

0.94

0.96

0.98

1

Frequency [GHz]

|S21

|

DataModel

W = 55 µm

W = 245 µm

Fig. 8. Magnitude of the bivariate reduced models ofS11(s, W ) andS21(s, W ) (W = 55, 150, 245 µm).

−180 −160 −140 −120 −100 −80 −600

50

100

150

200

250

300

Absolute error [dB]

Num

ber o

f sam

ples

Fig. 9. Histogram : error distributions of the bivariate reduced modelsof S11(s, W ) (light grey) andS21(s, W ) (dark grey) over 6000 validationsamples.

models has been checked by the procedure described inSection III-F on a dense sweep over the design space andthe theoretical claim of overall passivity has been confirmed.Figs. 11-12 show the magnitude of the parametric reducedmodel of the forward crosstalk termS16(s, Sx, Sy) (input

F. FERRANTI et al.: PARAMETRIC MODEL ORDER REDUCTION 9

w

Sx

t

Sy

Sx

Fig. 10. Cross section of the multiconductor system.

port of the first couple of conductors on the left and outputport of the first couple of conductors on the right) obtainedby piecewise multilinear interpolation for the vertical spac-ing valuesSy = 1, 2 mm. Similar results are obtainedusing the multivariate simplicial interpolation scheme. Fig.13 shows the magnitude of the parametric reduced modelof S11(s, Sx, Sy) obtained by multivariate simplicial inter-polation for the horizontal spacing valueSx = 2.5 mmand the vertical spacing valuesSy = 1.07, 1.5, 1.93 mm.These specific spacing values have not been used in therootROMs generation process. Fig. 14 shows the magnitude ofthe parametric reduced model ofS16(s, Sx, Sy) obtained bymultivariate simplicial interpolation for the horizontal spacingvaluesSx = 2.07, 2.5, 2.93 mm and the vertical spacingvalue Sy = 1.5 mm. These specific spacing values have notbeen used in theroot ROMs generation process. Figs. 15-16 show the absolute error distribution forS11(s, Sx, Sy) andS16(s, Sx, Sy) over a reference grid composed of300×15×15(freq, Sx, Sy) samples. The maximum absolute error of thetrivariate reduced model of theS matrix over the reference gridis bounded by−60.5 dB and−61.44 dB, respectively for thepiecewise multilinear and multivariate simplicial interpolationscheme. As in the previous example, the parametric reducedorder model describes the behavior of the system understudy very accurately, while guaranteeing overall stability andpassivity.

0 5 10 15

2

2.5

3

0

0.01

0.02

0.03

0.04

0.05

0.06

Frequency [GHz]Sx [mm]

|S16

|

Fig. 11. Magnitude of the trivariate reduced model ofS16(s, Sx, Sy)(piecewise multilinear interpolation,Sy = 1 mm).

0 5 10 15

2

2.5

3

0

0.01

0.02

0.03

0.04

0.05

0.06

Frequency [GHz]Sx [mm]

|S16

|

Fig. 12. Magnitude of the trivariate reduced model ofS16(s, Sx, Sy)(piecewise multilinear interpolation,Sy = 2 mm).

0 5 10 150

0.2

0.4

0.6

0.8

1

Frequency [GHz]

|S11

|

DataModel

Sy = 1.07 mm

Sy = 1.93 mm

Fig. 13. Magnitude of the trivariate reduced model ofS11(s, Sx, Sy)(multivariate simplicial interpolation,Sx = 2.5 mm, Sy = 1.07, 1.5, 1.93mm).

0 5 10 150

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Frequency [GHz]

|S16

|

DataModel

Sx = 2.07 mm

Sx = 2.93 mm

Fig. 14. Magnitude of the trivariate reduced model ofS16(s, Sx, Sy)(multivariate simplicial interpolation,Sx = 2.07, 2.5, 2.93 mm, Sy = 1.5mm).

10 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. , NO. , 2009

−180 −160 −140 −120 −100 −80 −600

500

1000

1500

2000

2500

3000

Absolute error [dB]

Num

ber o

f sam

ples

Fig. 15. Histogram : error distributions of the trivariate reduced mod-els of S11(s, Sx, Sy) (piecewise multilinear interpolation, light grey) andS11(s, Sx, Sy) (multivariate simplicial interpolation, dark grey) over 67500validation samples.

−180 −160 −140 −120 −100 −80 −600

500

1000

1500

2000

2500

3000

3500

4000

Absolute error [dB]

Num

ber o

f sam

ples

Fig. 16. Histogram : error distributions of the trivariate reduced mod-els of S16(s, Sx, Sy) (piecewise multilinear interpolation, light grey) andS16(s, Sx, Sy) (multivariate simplicial interpolation, dark grey) over 67500validation samples.

V. CONCLUSIONS

We have presented a new parameterized model order re-duction technique applicable to PEEC analysis. The overallstability and passivity of the parametric reduced order modelis guaranteed by an efficient and reliable combination of tra-ditional passivity-preserving MOR methods and interpolationschemes based on a class of positive interpolation operators.Numerical examples have validated the proposed approach onpractical application cases, showing that it is able to buildvery accurate parametric reduced models, while guaranteeingstability and passivity over the entire design space of interest.

REFERENCES

[1] R. F. Harrington,Field Computation by Moment Methods. New York:Macmillan, 1968.

[2] A. E. Ruehli, “Equivalent circuit models for three dimensional multi-conductor systems,”IEEE Trans. Microw. Theory Tech., vol. 22, no. 3,pp. 216–221, Mar. 1974.

[3] J. M. Jin,The Finite Element Method in Electromagnetics, 2nd ed. JohnWiley and Sons, New York, 2002.

[4] L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation fortiming analysis,” IEEE Trans. Comput.-Aided Design Integr. CircuitsSyst., vol. 9, no. 4, pp. 352–366, Apr. 1990.

[5] K. Gallivan, E. Grimme, P. Van Dooren, “Asymptotic waveform evalu-ation via a Lanczos method,”Applied Math., vol. 7, no. 5, pp. 75–80,Sep. 1994.

[6] P. Feldmann and R. W. Freund, “Efficient linear circuit analysis by Padeapproximation via the Lanczos process,”IEEE Trans. Comput.-AidedDesign Integr. Circuits Syst., vol. 14, no. 5, pp. 639–649, May 1995.

[7] K. Gallivan, E. Grimme, P. Van Dooren, “A rational Lanczos algorithmfor model reduction,”Numerical Algorithms, vol. 12, pp. 33–63, 1996.

[8] A. Odabasioglu, M. Celik, and L. T. Pileggi, “PRIMA: passive reduced-order interconnect macromodeling algorithm,”IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 17, no. 8, pp. 645–654, Aug.1998.

[9] A. Dounavis, E. Gad, R. Achar, M. S. Nakhla, “Passive model reductionof multiport distributed interconnects,”IEEE Trans. Microw. TheoryTech., vol. 48, no. 12, pp. 2325–2334, Dec. 2000.

[10] R. Achar, P. K. Gunupudi, M. Nakhla, E. Chiprout, “Passive interconnectreduction algorithm for distributed/measured networks,”IEEE Trans.Circuits Syst. II, vol. 47, no. 4, pp. 287–301, Apr. 2000.

[11] R. Achar, M. Nakhla, “Simulation of high-speed interconnects,”Proc.IEEE, vol. 89, no. 5, pp. 693–728, May 2001.

[12] A. E. Ruehli and A. C. Cangellaris, “Progress in the methodologies forthe electrical modeling of interconnects and electronic packages,”Proc.IEEE, vol. 89, no. 5, pp. 740–771, May 2001.

[13] R. D. Slone, W. T. Smith, Z. Bai, “Using partial element equivalentcircuit full wave analysis and Pade via Lanczos to numerically simulateEMC problems,” in Proc. IEEE Int. Symp. Electromagn. Compat.,Austin, Tx, Aug. 1997, pp. 608–613.

[14] Y. Liu, L. T. Pileggi, and A. J. Strojwas, “Model order-reduction ofRC(L) interconnect including variational analysis,” inProc. IEEE/AMCDes. Autom. Conf., 1999, pp. 201–206.

[15] P. Heydari and M. Pedram, “Model reduction of variable-geometryinterconnects using variational spectrally-weighted balanced truncation,”in Proc. IEEE/ACM Int. Conf. Comput.-Aided Des., Nov. 4–8, 2001, pp.586–591.

[16] J. R. Phillips, “Variational interconnect analysis via PMTBR,” inProc.IEEE/ACM Int. Conf. Comput.-Aided Des., Nov. 7–11, 2004, pp. 872–879.

[17] P. K. Gunupudi, R. Khazaka, M. S. Nakhla, T. Smy, and D. Celo,“Passive parameterized time-domain macromodels for high-speedtransmission-line networks,”IEEE Trans. Microw. Theory Tech., vol. 51,no. 12, pp. 2347–2354, Dec. 2003.

[18] L. Daniel, O. C. Siong, L. S. Chay, K. H. Lee, and J. White, “A mul-tiparameter moment-matching model-reduction approach for generatinggeometrically parameterized interconnect performance models,”IEEETrans. Comput.-Aided Design Integr. Circuits Syst., vol. 23, no. 5, pp.678–693, May 2004.

[19] X. Li, P. Li, and L. T. Pileggi, “Parameterized interconnect orderreduction with explicit-and-implicit multi-parameter moment matchingfor inter/intra-die variations,” inProc. IEEE/ACM Int. Conf. Comput.-Aided Des., Nov. 6–10, 2005, pp. 806–812.

[20] Y.-T. Li, Z. Bai, Y. Su, and X. Zeng, “Model order reduction of param-eterized interconnect networks via a two-directional arnoldi process,”IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 27, no. 9,pp. 1571–1582, Sep. 2008.

[21] G. Allasia, “Simultaneous interpolation and approximation by a class ofmultivariate positive operators,”Numerical Algorithms, vol. 34, no. 2,pp. 147–158, Dec. 2003.

[22] F. Ferranti, L. Knockaert, and T. Dhaene, “Parameterized S-parameterbased macromodeling with guaranteed passivity,”IEEE Microw. WirelessCompon. Lett., vol. 19, no. 10, pp. 608–610, Oct. 2009.

[23] ——, “Passivity-preserving interpolation-based parameterized macro-modeling of scattered S-data,”IEEE Microw. Wireless Compon. Lett.,vol. 20, no. 3, pp. 133–135, Mar. 2010.

[24] L. Knockaert and D. De Zutter, “Laguerre-SVD reduced-order model-ing,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1469–1475,Sep. 2000.

[25] J. R. Phillips, L. Daniel, and L. M. Silveira, “Guaranteed passivebalancing transformations for model order reduction,”IEEE Trans.Comput.-Aided Design Integr. Circuits Syst., vol. 22, no. 8, pp. 1027–1041, Aug. 2003.

[26] P. J. Restle, A. Ruehli, S. G. Walker, G. Papadopoulos, “Full-WavePEEC Time-Domain for the Modeling of On-Chip Interconnects,”IEEETransactions on Computer-Aided Design, vol. 20, no. 7, pp. 877–887,Jul. 2001.

F. FERRANTI et al.: PARAMETRIC MODEL ORDER REDUCTION 11

[27] G. Wollenberg, A. Gorisch, “Analysis of 3-D interconnect structureswith PEEC using SPICE,”IEEE Trans. Electromagn. Compat., vol. 41,no. 2, pp. 412–417, Nov. 1999.

[28] G. Antonini, A. Orlandi, “A wavelet based time domain solution forPEEC circuits,”IEEE Trans. Circuits Syst., vol. 47, no. 11, pp. 1634–1639, Nov. 2000.

[29] J. Cullum, A. Ruehli, T. Zhang, “A method for reduced-order modelingand simulation of large interconnect circuits and its application to PEECmodels with retardation,”IEEE Trans. Circuits Syst. II, vol. 47, no. 4,pp. 261–373, Apr. 2000.

[30] A. E. Ruehli, G. Antonini, J. Esch, J. Ekman, A. Mayo and A. Orlandi,“Non-orthogonal PEEC formulation for time and frequency domain EMand circuit modeling,”IEEE Trans. Electromagn. Compat., vol. 45, no. 2,pp. 167–176, May 2003.

[31] A. E. Ruehli, H. Heeb, “Circuit models for three-dimensional geometriesincluding dielectrics,”IEEE Trans. Microw. Theory Tech., vol. 40, no. 7,pp. 1507–1516, July 1992.

[32] C. Ho, A. Ruehli, P. Brennan, “The modified nodal approach to networkanalysis,”IEEE Trans. Circuits Syst., vol. 22, no. 6, pp. 504–509, Jun.1975.

[33] R. Rohrer, H. Nosrati, “Passivity considerations in stability studies ofnumerical integration algorithms,”IEEE Trans. Circuits Syst., no. 9, pp.857–866, Sep. 1981.

[34] L. Weinberg,Network Analysis and Synthesis. New York: McGraw-HillBook Company, 1962.

[35] E. A. Guillemin,Synthesis of Passive Networks. John Wiley and Sons,New York, 1957.

[36] B. D. Anderson, S. Vongpanitlerd,Network Analysis and Synthesis.Englewood Cliffs, NJ, 1973.

[37] R. W. Freund, “Krylov-subspace methods for reduced-order modelingin circuit simulation,” J. Comput. Appl. Math., vol. 123, no. 1-2, pp.395–421, 2000.

[38] J. De Geest, T. Dhaene, N. Fache and D. De Zutter, “Adaptive CAD-model building algorithm for general planar microwave structures,”IEEETrans. Microw. Theory Tech., vol. 47, no. 9, pp. 1801–1809, Sep. 1999.

[39] C. P. Coelho, J. R. Phillips, and L. M. Silveira, “Passive constrainedrational approximation algorithm using nevanlinna-pick interpolation,”in Proc. Design, Automation and Test in Europe Conference andExhibition, Mar. 4–8, 2002, pp. 923–930.

[40] E. W. Cheney, “Multivariate approximation theory: Selected topics,”in CBMS-NSF Regional Conference Series in Applied Mathematics,vol. 51. Philadelphia, PA: SIAM, 1986.

[41] D. F. Watson, “Computing the n-dimensional delaunay tessellation withapplication to voronoi polytopes,”The Computer Journal, vol. 24, no. 2,pp. 167–172, Febr. 1981.

[42] H. W. Kuhn, “Some combinatorial lemmas in topology,”IBM Journalof Research and Development, vol. 4, no. 5, pp. 518–524, 1960.

[43] C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithmfor convex hulls,”ACM Trans. Math. Softw., vol. 22, no. 4, pp. 469–483,1996.

[44] D. F. Watson,Contouring: A Guide to the Analysis and Display ofSpatial Data. Pergamon Press, 1992.

[45] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan,Linear MatrixInequalities in System and Control Theory. Philadelphia: SIAM, 1994.

[46] M. G. Safonov, R. Y. Chiang, and D. J. N. Limebeer, “Hankel modelreduction without balancing – A descriptor approach,” inProc. 26thIEEE Conference on Decision and Control, vol. 26, Dec. 1987, pp.112–117.

[47] R. N. Shorten, P. Curran, K. Wulff, and E. Zeheb, “A note on spectralconditions for positive realness of transfer function matrices,”IEEETrans. Autom. Control, vol. 53, no. 5, pp. 1258–1261, Jun. 2008.

Francesco Ferranti 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,in 2007. He is currently working towards the Ph.D.degree in the Department of Information Technology(INTEC), Ghent University, Ghent, Belgium. His re-search interests include parametric macromodeling,parameterized model order reduction, EMC numer-

ical modeling, system identification.

Giulio Antonini (M’94, SM’05) received his Laureadegree (summa cum laude) in Electrical Engineeringin 1994 from the Universita degli Studi dell’Aquilaand the Ph.D. degree in Electrical Engineering in1998 from University of Rome ”La Sapienza”. Since1998 he has been with theUAq EMC Laboratory,Department of Electrical Engineering of the Uni-versity of L’Aquila where he is currently AssociateProfessor. His research interests focus on EMCanalysis, numerical modeling and in the field ofsignal integrity for high-speed digital systems. He

has authored or co-authored more than 170 technical papers and 2 bookchapters. Furthermore, he has given keynote lectures and chaired severalspecial sessions at international conferences. He has been the recipient ofthe IEEE Transactions on Electromagnetic Compatibility Best Paper Awardin 1997, the CST University Publication Award in 2004, the IBM SharedUniversity Research Award in 2004, 2005 and 2006, the IET-SMT Best PaperAward in 2008. In 2006 he has received a Technical Achievement Awardfrom the IEEE EMC Society ”for innovative contributions to computationalelectromagnetic on the Partial Element Equivalent Circuit (PEEC) techniquefor EMC applications”. He holds one European Patent.

Tom Dhaene was born in Deinze, Belgium, onJune 25, 1966. He received the Ph.D. degree inelectrotechnical engineering from the University ofGhent, Ghent, Belgium, in 1993. From 1989 to1993, he was Research Assistant at the Universityof Ghent, in the Department of Information Technol-ogy, where his research focused on different aspectsof full-wave electro-magnetic circuit modeling, tran-sient simulation, and time-domain characterizationof high-frequency and high-speed interconnections.In 1993, he joined the EDA company Alphabit (now

part of Agilent). He was one of the key developers of the planar EMsimulator ADS Momentum. Since September 2000, he has been a Professorin the Department of Mathematics and Computer Science at the Universityof Antwerp, Antwerp, Belgium. Since October 2007, he is a Full Professorin the Department of Information Technology (INTEC) at Ghent University,Ghent, Belgium. As author or co-author, he has contributed to more than 150peer-reviewed papers and abstracts in international conference proceedings,journals and books. He is the holder of 3 US patents.

12 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. , NO. , 2009

Luc Knockaert received the M. Sc. Degree in phys-ical engineering, the M. Sc. Degree in telecommu-nications engineering and the Ph. D. Degree in elec-trical engineering from Ghent University, Belgium,in 1974, 1977 and 1987, respectively. From 1979to 1984 and from 1988 to 1995 he was working inNorth-South cooperation and development projectsat the Universities of the Democratic Republic ofthe Congo and Burundi. He is presently affiliatedwith the Interdisciplinary Institute for BroadBandTechnologies (www.ibbt.be) and a professor at the

Dept. of Information Technology, Ghent University (www.intec.ugent.be). Hiscurrent interests are the application of linear algebra and adaptive methods insignal estimation, model order reduction and computational electromagnetics.As author or co-author he has contributed to more than 100 internationaljournal and conference publications. He is a member of MAA, SIAM and asenior member of IEEE.