On Passivity Characterization of Symmetric Rational Macromodels

This paper provides a theoretical framework for passivity characterization of symmetric rational macromodels. These may be obtained for linear and time-invariant reciprocal structures if structural symmetry is preserved during the rational fitting stage of the macromodel generation. Standard Hamiltonian-based methods can be used to characterize any passivity violation of such macromodels. Recent developments have suggested, however, that the same results may be obtained at a reduced computational cost, using so-called ¿half-size¿ passivity test matrices. In this paper, we generalize such results by providing a complete theoretical framework. In addition to imaginary Hamiltonian eigenvalues, we present a complete characterization of associated eigenvectors, allowing for precise localization of passivity violations. Since half-size matrices are also used for computing the eigenvectors, the overall computational cost is reduced up to a factor of eight. Several numerical examples validate and confirm the theoretical developments.