Synthesis of Finite Passive n-Ports with Prescribed Positive Real Matrices of Several Variables

Abstract

Positive real functionsandmatrices of several variables arose in the problem of synthesizing a passive network composed of lumped elements with variable parameters. The importance of these functions and matrices has recently been emphasized by the considerable attention concerning their application to the problem of synthesizing passive networks composed of noncommensurable transmission lines and lumped elements. The problem of synthesizing positive real functions and matrices of several variables has been discussed by several authors. However, the problem has not been solved generally, except for the two-variable lossless case and the case where a two-variable positive real function is prescribed as a bilinear function with respect to one of the two variables. In this paper, a general solution to the above synthesis problem is presented. It is shown that an arbitrarily prescribed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \times n</tex> positive real matrix, symmetric or nonsymmetric, of several variables is realizable as the impedance or admittance matrix of a finite passive multivariable <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port. It is further shown that, if the matrix is symmetric, then it is realizable as a bilateral passive <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port. Related problems and discussions are also given.