Scattering Matrix Synthesis Via Reactance Extraction

This paper considers the problem of multiport passive network synthesis for a rational bounded real scattering matrix S(p) using state-space ideas. The technique is to extract reactances from a network synthesizing the prescribed S(p) to yield a resistive coupling network that may contain transformers, resistors, and if there is no reciprocity constraint, gyrators. Here a minimum number of reactive elements are always sufficient to give a synthesis, reciprocal and nonreciprocal. The synthesis method relies on an algebraic criterion for a class of rational matrices termed “discrete bounded real” that solves the passive synthesis problem. Starting with an arbitrary coordinate basis, a state-space description is found for a discrete bounded real matrix derived from the given bounded real S(p). A coordinate transformation of the state-space is found from the solution of an algebraic equation, which yields almost immediately a passive synthesis for S(p). When a synthesis for a symmetric bounded real S(p) is required to use no gyrators, a further state-space basis transformation is found that preserves passivity and simultaneously inserts reciprocity. The contributions of this paper, to scattering matrix synthesis, are twofold: first, to solve the nonreciprocal synthesis problem including the mimimal reactive element equivalence problem via calculations that need not rely on application of any classical ideas, and second, to present for the first time a reciprocal synthesis of a passive symmetric scattering matrix with a minimum number of reactances using state-space ideas.