Parameterized S-Parameter Based Macromodeling With Guaranteed Passivity

Abstract

Abstract-This letter presents a novel parametric macromodeling technique for scattering input-output representations parameterized by design variables such as geometrical layout or substrate features. It provides accurate multivariate macromodels that are stable and passive by construction over the entire design space. Overall stability and passivity of the parametric macromodel are guaranteed by an efficient and reliable combination of rational identification and interpolation schemes based on a class of positive interpolation operators.

Scalable compact models for fast design optimization of complex electromagnetic systems

“…Finally, the set of modified root macromodelsâ 1;k Rðsâ 2;k ; g Xi k Þ; k ¼ 1; …; Q, is interpolated at an input/output level as described in [8,9]. If a generic root macromodel R(s,g Xi k ) has the state-space representation {A,B,C,D}, then a corresponding scaled and frequency shifted versionâ 1;k Rðsâ 2;k ; g Xi k Þ has the state-space representation f e A; e B; e C; e Dg with…”

“…where each ' k i (g (i) ), i ¼ 1,…,M satisfies constraints (7)(8)(9) and is selected as in piecewise linear interpolation. These positive interpolation schemes have been already used in [8,9], where a parametric macromodel is built by interpolating a set of root macromodels treated as input-output systems, while preserving overall stability and passivity.…”

“…Recently, some parametric macromodeling techniques able to guarantee overall stability and passivity of parametric macromodels have been proposed [8][9][10][11]. The techniques described in [8,9] are based on the passive interpolation of a set of stable and passive univariate macromodels, called root macromodels, treated as inputoutput systems. This interpolation process of input-output systems leads to parameterize only the residues.…”

“…This interpolation process of input-output systems leads to parameterize only the residues. A passive interpolation of the state-space matrices of a set of root macromodels is proposed in [10,11], which provides an increased modeling capability with respect to [8,9] due to the parameterization of both poles and residues. Unfortunately, these methods are sensitive to some issues related to the interpolation of state-space matrices [12] and can only deal with rational models of the same order.…”

“…Initially, a set of univariate frequency-dependent macromodels related to different values of the design variables, called root macromodels [8,9], is built by means of the vector fitting (VF) technique [14]. Stability for each root macromodel is enforced by pole-flipping [14], while passivity is checked and enforced by means of standard techniques (see, e.g., [15,16]).…”