The application of polynomial matrix factorization in active network synthesis

Abstract

PurposeTo investigate the general necessary condition for synthesis of square, real rational matrices of complex frequency as admittance matrices of active multiports with resistors, inductors, capacitors and possibly multiport transformers and to prove that this condition is also sufficient for synthesis of stable, square, real rational matrices of complex frequency as admittance matrices of balanced active multiports having only resistors, capacitors and voltage‐amplifiers with sufficiently large amplifications. The main aim of the paper is to provide a new and general method for stable admittance matrices synthesis and to develop strict realization algorithm by active balanced transformerless multiport networks.Design/methodology/approachThe objectives of the paper are achieved by using factorization of regular polynomial matrices in complex frequency with certain degree as products of other regular polynomial matrices with specified degrees. A set of sufficient conditions for such a factorization is presented and derived a pertinent algorithm as the starting point for investigation and solving network synthesis problem and generation of class of equivalent realizations.FindingsTheorem 1 states that sufficient condition for factorization of Pth order, generally regular polynomial matrix P(s) in complex frequency s with degree L, whose determinant has K distinct zeros, in form P(s)=P1(s)·P2(s), where 1≤p2=P20≤L−1 is degree of polynomial matrix P2(s), reads: K>(P−1)·L+p2−1. The coefficient‐matrices of s, s2,… in P1(s) and P2(s) are real or complex depending on whether distinct zeros of det P(s) are real or complex, respectively. Theorem 2 states that: (a) for realization of Pth order matrix of real rational functions in complex frequency s (i.e. RRF matrix) as admittance matrix of active balanced RLC P‐port network with multiport transformers, or without them, P generalized controlled‐sources and P controlling‐ports are necessary, in general; and (b) P balanced voltage‐controlled voltage‐sources (VCVSs) with real and by module greater than unity controlling coefficients (“voltage amplifications”) are sufficient for realization of stable admittance RRF matrix by active, balanced, transformerless, RC P‐port network.Originality/valueThis is a research paper with the following two main contributions (original results). First, a theorem on sufficient conditions for factorization of regular polynomial matrices in complex frequency; and second, a theorem relating to sufficient conditions for synthesis of matrices of real rational functions in complex frequency by active, balanced, transformerless networks. The results may be interesting for network theorists and researchers in the field of electric circuits and systems.