ABSTRACT: We propose a new parametric macromodeling technique for complex electro-magnetic (EM) systems described by scattering parameters, which are parameterized bymultiple design variables such as layout or substrate feature. The proposed technique isbased on an efficient and reliable combination of rational identification, a procedure to findscaling and frequency shifting system coefficients, and positive interpolation schemes. Para-metric macromodels can be used for efficient and accurate design space explorationand optimization. A design optimization example for a complex EM system is used tovalidate the proposed parametric macromodeling technique in a practical design processflow. V C 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE 00:000–000, 2011. Keywords: parametric macromodeling; rational approximation; interpolation; design optimiza-tion; complex systems I. INTRODUCTION Robust computer-aided design flows are essential for theminimization of signal integrity and electromagnetic (EM)compatibility issues in the development of modern digitaland mixed-signal systems [1–5]. Chip packages, data bus-ses, connectors, and cables require to be properlydesigned, since from the early design stage, they can seri-ously limit transmission data rates or increase the bit errorrate beyond acceptable limits. In order to avoid theseproblems, design space exploration, design optimization,and sensitivity analysis are usually performed by multiplefrequency-domain simulations for different design parame-ter values (e.g., layout features), trying to meet govern-ment regulations and customer requirements. In this per-spective, parametric (scalable) macromodels are wellsuited to efficiently and accurately perform these designactivities, while using multiple EM simulations is oftenhigh computationally expensive due to the high computa-tional cost per simulation. Parametric macromodels aremultivariate models that capture the complex behavior ofEM systems, which is typically characterized by the fre-quency (or time) and several design parameters, such aslayout or substrate features.Over the last years, several different parametric macro-modeling techniques have been developed. In [6, 7], bothpoles and residues are parameterized, and it results inaccurately modeling dynamic multivariate data. Overallstability and passivity of parametric macromodels are notguaranteed. Recently, some parametric macromodelingtechniques able to guarantee overall stability and passivityof parametric macromodels have been proposed [8–11].The techniques described in [8, 9] are based on the pas-sive interpolation of a set of stable and passive univariatemacromodels, called root macromodels, treated as input–output systems. This interpolation process of input–outputsystems leads to parameterize only the residues. A passiveinterpolation of the state-space matrices of a set of rootmacromodels is proposed in [10, 11], which provides anincreased modeling capability with respect to [8, 9] due tothe parameterization of both poles and residues. Unfortu-nately, these methods are sensitive to some issues relatedto the interpolation of state-space matrices [12] and can