3688 IEEE TRANSACTIONS ON MICROWAVE THEORY AND …pose analysis-oriented nonlinear simulator....

3688 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 58, NO. 12, DECEMBER 2010

Passivity-Preserving Parametric Macromodelingfor Highly Dynamic Tabulated Data

Based on Lur’e EquationsFrancesco Ferranti, Member, IEEE, Luc Knockaert, Senior Member, IEEE, Tom Dhaene, Senior Member, IEEE,

and Giulio Antonini, Senior Member, IEEE

Abstract—We present a new method for the construction ofparametric macromodels for admittance and impedance input-output representations starting from multivariate data samplesthat depend on frequency and additional design variables suchas geometric and material parameters. Poles and residues areparameterized indirectly, while stability and passivity of the para-metric macromodel are guaranteed over a user defined range ofdesign parameter values. Pertinent numerical examples validatethe proposed parametric macromodeling approach.

Index Terms—Interpolation, Lur’e equations, parametricmacromodeling, passivity, rational approximation.

I. INTRODUCTION

D URING the circuit synthesis of large scale digital oranalog applications, it is important to predict the re-

sponse of the circuit under study as a function of environmentaleffects, thermal effects, manufacturing variations, and fluc-tuations in the critical dimensions of transmission lines suchas width and height. A typical design process includes opti-mization and design space exploration, and therefore requiresmultiple simulations for different design parameter values. Itis often not feasible to perform multiple simulations of largecircuits due to variations of these parameters. Such designactivities call for the development of robust parametric macro-modeling techniques. Parametric macromodels approximatethe complex electromagnetic (EM) behavior of EM systems atthe input and output ports in the frequency (or time) domain.This behavior is typically characterized by frequency (or time)and several design parameters that describe physical propertiesof the structure, such as metallizations or substrate parameters.A parametric macromodel is obtained through an identificationprocess starting from a set of multivariate data samples.

The development of parametric macromodels of EM systemshas become a topic of intense research over the last years. Some

Manuscript received February 18, 2010; accepted September 01, 2010. Dateof publication October 28, 2010; date of current version December 10, 2010.This work was supported by the Research Foundation Flanders (FWO).

F. Ferranti, L. Knockaert, and T. Dhaene are with the Department ofInformation Technology (INTEC), at Ghent University—IBBT, 9000 Ghent,Belgium (e-mail: [email protected]; [email protected];[email protected]).

G. Antonini is with the UAq EMC Laboratory, Dipartimento di IngegneriaElettrica e dell’Informazione, Università degli Studi dell’Aquila, 67100,L’Aquila, Italy (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TMTT.2010.2083910

parametric macromodeling techniques parameterize both polesand residues [1]–[3] and are able to accurately model highlydynamic multivariate data samples without requiring a highsample density. Unfortunately, such techniques are not ableto guarantee overall stability and passivity of the parametricmacromodel. In other formulations, poles are kept fixed andonly residues are parameterized [4]–[6]; the modeling poweris reduced with respect to the previous techniques, but thepreservation of stability and passivity of the parametric macro-model is feasible. In [4] the stability problem is addressed bycomputing a parametric macromodel with barycentric inter-polation of univariate stable macromodels. Overall stabilityof the macromodel is guaranteed by construction and a pas-sivity enforcement scheme is proposed by perturbation of thebarycentric weights. However the passivity violations must bereasonably small and the convergence of the passivity enforce-ment procedure is not guaranteed. The techniques presented in[5], [6] are able to build parametric macromodels that are stableand passive over the entire design space, for admittance ,impedance and scattering representations. They com-bine rational identification and interpolation schemes based ona class of positive interpolation operators [7], [8], to guaranteeoverall stability and passivity of the parametric macromodel.

This paper presents a novel parametric macromodeling tech-nique that parameterizes poles and residues to own a high mod-eling power and is able to guarantee stability and passivity overthe entire design space, for admittance and impedancerepresentations. It turns out that the choice of a direct param-eterization of poles and residues is not appropriate, due to thepossible highly non-smooth behavior of the poles with respectto the design parameters. It is well-known that model poles arevery sensitive to even small variations of the parameters. Bifur-cation effects may occur, resulting in quite irregular variationsof each pole in the design space [3]. The direct parameterizationof poles and residues is avoided in our new technique, where de-scriptor state-space matrices are parameterized and hence alsopoles and residues indirectly, while stability and passivity areguaranteed over the design space of interest. The proposed tech-nique starts by computing multiple univariate frequency do-main macromodels using the Vector Fitting (VF) technique [9]for different combinations of the design variables, as in [5]. Inthis paper we refer to these initial univariate macromodels asroot macromodels. A simple pole-flipping scheme is used toenforce stability [9] for each root macromodel, while passivityis checked and enforced by means of standard techniques (see

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FERRANTI et al.: PASSIVITY-PRESERVING PARAMETRIC MACROMODELING 3689

e.g., [10]–[12]). The model formulation presented in [3] is thenused to project the set of root macromodels on a set of commonfrequency-dependent basis functions. Afterwards, a state-spacerealization is obtained for each stable and passive root macro-model and the solution of associated Lur’e equations [13] is usedto convert each root macromodel from the standard state-spaceform to a descriptor form with specific matrix properties. Thefinal step of the algorithm is focused on gluing together the com-puted descriptor form of the root macromodels by a multivariateinterpolation scheme to obtain a parametric macromodel. Themultivariate interpolation scheme preserves stability and pas-sivity over the entire design space. The proposed technique isvalidated by some pertinent numerical examples.

II. PARAMETRIC MACROMODELING

This section treats the generation of a multivariate represen-tation which accurately models a large set of mul-tivariate data samples , while preservingstability and passivity over the entire design space. These datasamples depend on the complex frequency , and sev-eral design variables , such as the layout featuresof a circuit (e.g., ) or the substrate parame-ters (e.g., ). The ra-tional dependence on frequency of the parametric macromodel

is required to synthesize it as an equivalent lumped cir-cuit and use it in system-level simulations and optimizationsusing standard circuit solvers (e.g., SPICE). Since a rationalformulation (poles/residues, zeros/poles, or numerator/denom-inator polynomials) suffers from severe ill-conditioning whenthe dependence of external parameters is included [2], a para-metric macromodel in the form

(1)

is adopted. The basis function with the predefined max-imal order are used to describe the frequency dependenceof the system under study, while parameters variations are in-duced by their expansion coefficients , . The fre-quency-dependent basis functions are chosen in the par-tial fraction form [9]

(2a)

(2b)

since these basis functions are numerically reliable. Two datagrids are used in the modeling process: an estimation grid and avalidation grid. The first grid is utilized to build the root macro-models that are used to provide the parametric macromodel. Thesecond grid, more dense than the previous one, is utilized to as-sess the modeling capability of the parametric macromodel, itscapability of describing the system under study in points of thedesign space previously not used for the construction of the rootmacromodels.

A. Root Macromodels

Starting from a set of data samplesa frequency-dependent rational model is built for all estimationgrid points in the design space by means of VF. A pole-flippingscheme is used to enforce stability [9] and passivity enforcementcan be accomplished using one of the robust standard techniques[10]–[12]. The result of this initial procedure is a set of rationalunivariate macromodels, stable and passive, that we call rootmacromodels. These are the starting points to build a parametricmacromodel. Each root macromodel related to a generic point

in the design space is converted fromthe rational pole-residue form

(3)

obtained by means of VF, into the barycentric form [3], [14]

(4)

The basis poles affect the numerical conditioning ofthe basis functions and of the conversion from the VFform (3) to the barycentric form (4). The proposed set of rationalbases (2a)–(2b) is well conditioned if the basis poles are linearlydistributed over the available bandwidth [9]. Consequently, weadopt this rule to define the basis poles . A state-spacerealization is performed for each root macromodel

(5)

after the conversion from the VF form (3) to the barycentricform (4).

B. Descriptor Form by Lur’e Equations

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 [15], which iscrucial when the macromodel is to be utilized in a general-pur-pose analysis-oriented nonlinear simulator. Passivity refers tothe property of systems that cannot generate more energy thanthey absorb through their electrical ports. When the system isterminated on any arbitrary passive loads, none of them willcause the system to become unstable [16], [17]. A linear net-work described by admittance matrix is (strictly) passive(or (strictly) positive real) if [13]:

1) for all , where “ ” is the complex conju-gate.

2) is analytic in .3) is a (strictly) positive real matrix, i.e.,:

; : and anyarbitrary vector .

Similar results are valid for a linear network described byimpedance matrix . Once a state-space realization for eachstable and passive root macromodel is performed, a conversionfrom the standard state-space form in (5) to a descriptor form is


performed using the solution of Lur’e equations [13] for posi-tive real systems. Let be a minimal realizationof a positive real system. Then, the system is positive real ifand only if there exist matrices , and symmetric positivedefinite which satisfy the Lur’e equations [13]

(6a)

(6b)

(6c)

Several proofs of this theorem, or related results, appeared in theliterature. In particular, the sufficiency part is obtained by ele-mentary algebraic manipulations, while the necessity is more in-volved. See [13] for several different proofs. However, the min-imality assumption is not really crucial in the previous theorem.Minimality is crucial if we want the matrix to be positivedefinite; if minimality is not assumed, in general will be pos-itive semidefinite [18]. Assuming that each root macromodel isstrictly positive real and the corresponding term(strict asymptotic passivity), the associated Lur’e equations are

(7a)

(7b)

(7c)

Equation (7a) is a Lyapunov equation for the matrix . If thematrix is positive (semi)definite, then so is the matrix ;therefore, the matrix is positive definite for strictly positivereal systems, and hence invertible. Once a matrix is ob-tained by solving (7a)–(7c) for each root macromodel, it is pos-sible to convert it from the form (5) to the descriptor state-spaceform

(8)

with the properties , with. Assuming that each root macromodel

is strictly positive real and the corresponding term ,the associated Lur’e equations are

(9a)

(9b)

Once a matrix is obtained by solving (9a)–(9b) for eachroot macromodel, it is possible to convert it from the form (5)to the descriptor state-space form

(10)

with the properties , ,with that guarantee strict

passivity [19]. The assumption is not restrictive, it isof particular interest in IC applications and is usually satisfiedin full-wave methods as the Method of Moments [20] and thePartial Element Equivalent Circuit method [21], [22].

C. Solution of Lur’e Equations

In this section, we discuss some algorithms to solve Lur’eequations and find the matrix . Solution of Lur’e equationsand solution of algebraic Riccati equations (ARE’s) are closelyrelated. An overview of basic numerically robust computationalprocedures is given in [23]. The Lur’e (7a)–(7c) can be equiva-lently formulated as the linear matrix equality (LMI)

(11)

which is a convex formulation. Hence, the direct enforcement ofthese constraints admits an unique optimal solution, which canbe achieved in a finite number of iterations within any prescribedtolerance [24]. The LMI (11) can be solved via convex optimiza-tion using one of the several available solvers. In this work, weemploy SDPT3 [25] as the optimization engine, in combinationwith the CVX interface [26] for the MATLAB environment. Inthe case that , the LMI (11) is feasible if and onlyif there exists a real matrix satisfying the ARE

(12)

It is clear that Lur’e equations and the LMI approach are moregeneral than Riccati equations. If , the method presentedin [27] can be used to solve Lur’e equations based on a mod-ified ARE solution, otherwise the Lur’e equations can still besolved as an LMI problem. The key point in [27] is convertingthe state-space realization with into another equivalentform where the new is different from zero. Another approachto find when is based on solving the following ARE

(13)

This is obtained by the following manipulations on (9a)–(9b):postmultiplying (9a) by , we obtain

(14)

and defining , we obtain the modified Lur’e equations

(15a)

(15b)

(15c)

from which the ARE (13) is obtained. This result is of courseonly valid when , which is the case [27]when the system is strictly positive real and .

D. Passivity Preserving Interpolation

The next step of the parametric macromodeling algorithm isfocused on gluing together the computed descriptor form of theroot macromodels by a multivariate interpolation scheme thatacts on the descriptor matrices and preserves stability and pas-sivity over the entire design space. We treat the cases

and differently.Assuming that , any interpolation scheme

based on a class of positive interpolation operators [7], [8] can


be used to interpolate the state-space matrices , , ,in (8). In the bivariate case each interpolated

state-space matrix can be written as

(16)

where each interpolation kernel is a scalar function satis-fying the following constraints

(17)

(18)

A suitable choice is to select each set as in piecewise linearinterpolation

(19a)

(19b)

(19c)

The bivariate formulation can easily be generalized to the multi-variate case by using multivariate interpolation methods that be-long to the general class of positive interpolation operators: e.g.,the piecewise multilinear method [28], the multivariate simpli-cial method [29], the well-known Shepard’s method [30]. Wenote that the interpolation kernel functions of these methodsonly depend on the data grid points and their computation doesnot require the solution of a linear system to impose an interpo-lation constraint. In the case of piecewise multilinear interpola-tion, each interpolated state-space matrix can be written as

(20)

where each , respects both constraints(17) and (18). A suitable choice is to select each set asin piecewise linear interpolation. It is straightforward to provethat, if each state-space matrix in (8) is interpolated by a pos-itive interpolation scheme with interpolation kernel functionsthat only depend on the data grid points, it is equivalent to inter-polate with the same interpolation scheme the LMI (11)

Since any nonnegative linear combination of positive (negative)definite real matrices is a positive (negative) definite real ma-trix, stability and passivity are preserved over the entire designspace. These interpolation schemes based on a class of posi-tive interpolation operators have been already used in [5], [6]to build a parametric macromodel starting from a set of rootmacromodels, while preserving overall stability and passivity.The interpolation process is applied to a set of root macromodelsin [5], [6], while it is applied to the internal state-space matricesof the root macromodels in the proposed new parametric macro-

modeling technique. Concerning the matrix , the interpo-lation method that is able to preserve the positive definitenessand described in what follows can be used.

Assuming that , the multivariate interpolationscheme has to preserve the positive definiteness of and

, and hence the stability and passivity of theparametric macromodel over the entire design space. Thematrix can be freely interpolated. Let us denote

(21)

the space of all real symmetric matrices with thespace of real matrices and

(22)

the space of all real symmetric positive-definite ma-trices. The matrix exponential is a one-to-one map fromto . In other words, the matrix exponential of any real sym-metric matrix is a real symmetric positive-definite matrix, andthe inverse of the matrix exponential (i.e., principal matrix log-arithm) of any real symmetric positive-definite matrix is a realsymmetric matrix [31], [32]. Exploiting this property of the ex-ponential map, the set of matrices is mapped fromto using the principal logarithm operator, and only thelower or upper triangular part is interpolated. Any multivariateinterpolation scheme can be used, e.g., the piecewise multilinearmethod [28], the multivariate simplicial method [29], Shepard’smethod [30], the multivariate cubic splines method [33]. Finally,the interpolated matrix is mapped back by the matrix exponen-tial operator resulting in a symmetric positive definite matrix

. Concerning the matrix , it has to satisfy the condi-tion over the entire design space. As iswell-known, any square matrix can be written as the sum of asymmetric and a skew-symmetric matrix, , where

and . The symmetric partof has to be positive definite, therefore the same interpola-tion process performed to obtain is applied to the set of ma-

trices . Regarding the skew-symmetric parts

, only the strictly lower (or strictly upper) tri-angular parts are interpolated.

Any interpolation scheme can be used in the interpolationprocess of the state-space matrices in (10) when , there-fore this particular case shows a bigger flexibility in the con-struction of a parametric macromodel than the other case where

and for this reason it is investigated in the numer-ical examples.

III. NUMERICAL EXAMPLES

This section presents three pertinent numerical examples thatvalidate the proposed parametric macromodeling approach. Letus define the weighted RMS-error as

(23)


Fig. 1. Cross section of the microstrip.

TABLE IPARAMETERS OF THE MICROSTRIP STRUCTURE

with

(24)

where is the number of system ports and is equal tothe number of frequency samples. The worst case RMS-errorover the validation grid is chosen to assess the accuracy and thequality of the parametric macromodels

(25)

(26)

and it is used in the numerical examples.

A. One Microstrip With Variable Length

In this example a microstrip has been modeled. The cross sec-tion is shown in Fig. 1. The conductor has width m,the substrate has height m and relative permittivity

. A bivariate macromodel is built as a function ofthe length of the line in addition to frequency. Their corre-sponding ranges are shown in Table I.

The admittance matrix has been computed over avalidation grid of 150 30 samples . We have builtroot macromodels for 12 values of the length by means of VF,each with a order equal to 8. The passivity of each model hasbeen verified by checking the eigenvalues of the Hamiltonianmatrix [34] and enforced if needed. Then, the numerical manip-ulations described in Section II are used to obtain a descriptorstate-space form for each root macromodel. Finally, a bivariatemacromodel is obtained by spline interpolation of the descriptormatrices. Fig. 2 shows the magnitude of the parametric macro-model of . The worst case RMS-error defined in (26)is equal to and it occurs for cm.Figs. 3–4 compare and its macromodel for the lengthvalue corresponding to . As clearly seen, a very goodagreement is obtained between the original data and the pro-posed passivity preserving macromodeling technique. Similarresults are obtained for . The parametric macromodelcaptures the behavior of the system very accurately, preservingstability and passivity over the entire design space.

B. Two Coupled Microstrips With Variable Spacing

A three-conductor transmission line (length cm) withfrequency-dependent per-unit-length parameters has been mod-eled. It consists of two coplanar microstrips over a ground plane.

Fig. 2. Magnitude of the parametric macromodel of� �� .

Fig. 3. Magnitude of the parametric macromodel of � �� cm�.

Fig. 4. Phase of the parametric macromodel of� �� cm�.

The cross section is shown in Fig. 5. The conductors have widthm, the dielectric is m thick and characterized

by a dispersive and lossy permittivity which has been modeledby the wideband Debye model [35]. The parameters of the usedwideband Debye model are: 1) relative permittivityat the frequency , 2) loss tangentat the frequency . A bivariate macromodel is


Fig. 5. Cross section of the two coupled microstrips.

TABLE IIPARAMETERS OF THE TWO COUPLED MICROSTRIPS STRUCTURE

Fig. 6. Magnitude of the parametric macromodel of� �� .

built as a function of the spacing between the microstrips inaddition to frequency. Their corresponding ranges are shown inTable II.

The frequency-dependent per-unit-length parameters havebeen evaluated using a commercial tool [36] over a validationgrid of 150 20 samples, for frequency and spacing respec-tively. Then, the admittance matrix has been computedusing transmission line theory [37]. We have built root macro-models for 7 values of the spacing by means of VF, each with anorder equal to 22. The passivity of each model has been verifiedby checking the eigenvalues of the Hamiltonian matrix [34]and enforced if needed. Then, the numerical manipulations de-scribed in Section II are used to obtain a descriptor state-spaceform for each root macromodel. Finally, a bivariate macromodelis obtained by linear interpolation of the descriptor matrices.Fig. 6 shows the magnitude of the parametric macromodel of

. The worst case RMS-error defined in (26) is equalto and it occurs for m. Figs. 7–10compare , and their macromodels forthe spacing value corresponding to . As in the previousexample, the parametric macromodel describes very accuratelythe behavior of the system, guaranteeing stability and passivityover the entire design space.

C. One Microstrip With Variable Width and Height

In this example a microstrip (length cm) has beenmodeled. The cross section is shown in Fig. 1. The substrate

Fig. 7. Magnitude of the parametric macromodel of� �� m�.

Fig. 8. Phase of the parametric macromodel of� �� m�.

Fig. 9. Magnitude of the parametric macromodel of� �� m�.

has relative permittivity . A trivariate macromodel isbuilt as a function of the width of the strip and the heightof the substrate in addition to frequency. Their correspondingranges are shown in Table III.

The admittance matrix has been computed over avalidation grid of 200 20 20 samples . We havebuilt root macromodels for 7 values of the width and 6 values of


Fig. 10. Phase of the parametric macromodel of� �� m�.

TABLE IIIPARAMETERS OF THE MICROSTRIP STRUCTURE

Fig. 11. Magnitude of the parametric macromodel of � �� m�.

the height by means of VF, each with a order equal to 14. Thepassivity of each model has been verified by checking the eigen-values of the Hamiltonian matrix [34] and enforced if needed.Then, the numerical manipulations described in Section II areused to obtain a descriptor state-space form for each root macro-model. Finally, a trivariate macromodel is obtained by spline in-terpolation of the descriptor matrices. Figs. 11 and 12 show themagnitude of the parametric macromodel of for

m and m, respectively. The worst caseRMS-error defined in (26) is equal to and it occurs for

m m . Figs. 13 and 14 com-pare and its macromodel for the width and heightsubstrate value corresponding to . As clearly seen, a verygood agreement is obtained between the original data and theproposed passivity preserving macromodeling technique. Sim-ilar results are obtained for . The parametric macro-model captures very accurately the behavior of the system, pre-

Fig. 12. Magnitude of the parametric macromodel of � �� m�.

Fig. 13. Magnitude of the parametric macromodel of � �� m�, � � �� m�.

Fig. 14. Phase of the parametric macromodel of� �� m,� � �� m).

serving stability and passivity properties over the entire designspace.

IV. CONCLUSION

We have presented a new parametric macromodeling tech-nique for the generation of parametric macromodels of admit-


tance and impedance representations. The indirect parameteri-zation of poles and residues provides a high power modeling,while overall stability and passivity are guaranteed by applyinga special kind of interpolation on some descriptor state-spacematrices related to Lur’e equations. Pertinent numerical exam-ples have validated the proposed approach on practical applica-tion cases, showing that it is able to build very accurate para-metric macromodels, while guaranteeing stability and passivityover the complete design space.

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Francesco Ferranti (M’10) received the B.S. degree(summa cum laude) in electronic engineering fromthe Università degli Studi di Palermo, Palermo, Italy,in 2005 and the M.S. degree (summa cum laude andhonors) in electronic engineering from the Univer-sità degli Studi dell’Aquila, L’Aquila, Italy, in 2007.He is currently working towards the Ph.D. degree inthe Department of Information Technology (INTEC),Ghent University, Ghent, Belgium.

His research interests include parametric macro-modeling, parameterized model order reduction,

EMC numerical modeling, system identification.

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

From 1979 to 1984 and from 1988 to 1995 he wasworking in North-South cooperation and develop-ment projects at the Universities of the DemocraticRepublic of the Congo and Burundi. He is presentlyaffiliated with the Interdisciplinary Institute forBroadBand Technologies and a Professor at the

Department of Information Technology, Ghent University, Belgium. Hiscurrent interests are the application of linear algebra and adaptive methods insignal estimation, model order reduction and computational electromagnetics.As author or coauthor he has contributed to more than 100 international journaland conference publications.

He is a member of MAA and SIAM.


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.

From 1989 to 1993, he was Research Assistantat the University of Ghent, in the Department ofInformation Technology, where his research focusedon different aspects of full-wave electro-mag-netic circuit modeling, transient simulation, andtime-domain characterization of high-frequencyand high-speed interconnections. In 1993, he joined

the EDA company Alphabit (now part of Agilent). He was one of the keydevelopers of the planar EM simulator ADS Momentum. Since September2000, he has been a Professor in the Department of Mathematics and ComputerScience at the University of Antwerp, Antwerp, Belgium. Since October 2007,he is a Full Professor in the Department of Information Technology (INTEC) atGhent University, Ghent, Belgium. As author or co-author, he has contributedto more than 150 peer-reviewed papers and abstracts in international conferenceproceedings, journals and books. He is the holder of 3 US patents.

Giulio Antonini (M’94–SM’05) received the Laureadegree (summa cum laude) in electrical engineeringin 1994 from the Università degli Studi dell’Aquila,L’Aquila, Italy, and the Ph.D. degree in ElectricalEngineering in 1998 from University of Rome “LaSapienza,” Rome, Italy.

Since 1998 he has been with the UAq EMCLaboratory, Department of Electrical Engineeringof the University of L’Aquila where he is currentlyAssociate Professor. His research interests focus onEMC analysis, numerical modeling and in the field

of signal integrity for high-speed digital systems. He has authored or co-au-thored more than 170 technical papers and 2 book chapters. Furthermore, hehas given keynote lectures and chaired several special sessions at internationalconferences. He holds one European Patent.

Dr. Antonini was the recipient of the IEEE Transactions on ElectromagneticCompatibility Best Paper Award in 1997, the CST University PublicationAward in 2004, the IBM Shared University Research Award in 2004, 2005and 2006, the IET-SMT Best Paper Award in 2008. In 2006 he has received aTechnical Achievement Award from the IEEE EMC Society “for innovativecontributions to computational electromagnetic on the Partial Element Equiva-lent Circuit (PEEC) technique for EMC applications.”

3688 IEEE TRANSACTIONS ON MICROWAVE THEORY AND …pose analysis-oriented nonlinear simulator....

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