664 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 31, NO. 4,NOVEMBER2008 Modal Vector Fitting: A Tool For Generating Rational Models of High Accuracy With Arbitrary Terminal Conditions Bjørn Gustavsen, Senior Member, IEEE, and Christoph Heitz Abstract—This paper introduces a new approach for rational macromodeling of multiport devices that ensures high accuracy with arbitrary terminal conditions. This is achieved by refor- mulating the vector fitting (VF) technique to focus on eigenpairs rather than matrix elements. By choosing the least squares (LS) weighting equal to the inverse of the eigenvalue magnitude, the modal components are fitted with a relative accuracy criterion. The resulting modal vector fitting (MVF) method is shown to give a major improvement in accuracy for cases with a high ratio between the largest and smallest eigenvalue, although it is computationally more costly than VF. It is also shown how to utilize the impedance characteristics of the adjacent network in the fitting process. The application of MVF is demonstrated for a two-conductor stripline, a coaxial cable, and a transformer measurement. We also show a simplified procedure which achieves similar results as MVF if the admittance matrix can be diagonalized by a constant transformation matrix. The extracted model is finally subjected to passivity enforcement by the modal perturbation method, which makes use of a similar LS formulation as MVF for the constrained optimization problem. Index Terms—Interconnect, macromodel, passivity, pole-residue model, rational model, state-space model, vector fitting. I. INTRODUCTION W IDEBAND modeling of devices and systems from tab- ulated data is becoming of major importance for the design and verification of high-speed electronic systems. The modeling is usually based on “fitting” a model to a set of pa- rameters that characterize the model behavior, such as admit- tance (Y), impedance (Z), or scattering (S) parameters in the frequency domain or the time domain. The fitting process can be based on a ratio of polynomials [1], [2] or orthogonal poly- nomial functions [3]. Recently, the pole relocating vector fitting technique [4] has become widely applied, and several enhance- ments have been proposed [5]–[7]. The modeling consists of fitting the model parameters to some given data, minimizing a specific error criterion based on Manuscript received October 02, 2007; revised October 01, 2008. First pub- lished August 19, 2008; current version published November 28, 2008. This work was supported in part by Deutsch, in part by FMC Technologies, in part by Framo, in part by Norsk Hydro, in part by Petrobras, in part by Siemens, in part by Statoil, in part by Total, and in part by Vetco Gray. The work of B. Gustavsen was supported by the Norwegian Research Council (PETROMAKS Programme). The work of C. Heitz was supported by the Swiss Commission of Technology and Innovation CTI. This work was recommended for publication by Associate Editor F. Canavero upon evaluation of the reviewers comments. B. Gustavsen is with the SINTEF Energy Research, N-7465 Trondheim, Norway (e-mail: bjorn.gustavsen@sintef.no). C. Heitz is with the Zurich University of Applied Sciences, CH-8401 Win- terthur, Switzerland (e-mail: christoph.heitz@zhaw.ch). Digital Object Identifier 10.1109/TADVP.2008.927810 a comparison of the model with the given data. In most cases, quadratic error measures are used, leading to a least-squares (LS) fitting. The modeling is complete when the parameters have been fitted to a given accuracy level. Depending on how the error criterion is constructed, some properties of the physical system are approximated with higher accuracy than others. For instance, when fitting the model to match the elements of a given admittance matrix , the re- sulting model will yield a good accuracy of the -matrix, but not necessarily a good accuracy of the -matrix. As we will show below, using this model for calculating currents for given applied voltages will yield accurate results, whereas using it for calculating voltages for given applied currents can generate poor results. Generally speaking, fitting to admittance or impedance parameters yields models that are optimized to a specific ter- minal condition. Such models may behave unsatisfactory with other terminal conditions. As will be shown in the paper, large error magnifications can take place, depending on the eigen- value structure. In this paper, we introduce a more general way of character- izing accuracy by requiring that the model behaves accurately with arbitrary terminal conditions. This is achieved by focusing on the relative accuracy of eigenvalues (modes) rather than ma- trix elements. This concept is merged with the vector fitting (VF) technique, leading to modal vector fitting (MVF) [8]. The paper is organized as follows. In Section II, we intro- duce a new type of model error characterization which focuses on the relative accuracy of modal contributions. This concept is merged with the VF methodology in Section IV, leading to MVF. In Section V, it is shown how to take into account the external network since it may render the small eigenvalues of little importance. In Section VI, we introduce a simplified mod- eling approach which can achieve similar accuracy properties as MVF by usage of a constant transformation matrix. In Sec- tion VII, we demonstrate the application of MVF to a stripline transmission line, demonstrating the ability of retaining the ac- curacy of small eigenvalues. We also show how to retain this ac- curacy in the subsequent passivity enforcement step by usage of fast modal perturbation (FMP). The shortcomings of assuming a constant transformation matrix are demonstrated in Section VIII for the modeling of a coaxial cable, and the limitations of direct high-order fitting are shown in Section IX. II. ACCURACY CONSIDERATIONS As an example we consider a multiport system which is de- scribed by its admittance parameters. The admittance matrix 1521-3323/$25.00 © 2008 IEEE Authorized licensed use limited to: Sintef. Downloaded on December 3, 2008 at 05:20 from IEEE Xplore. Restrictions apply.