Minimal Synthesis of Two-Variable Reactance Matrices

Two-variable reactance functions and matrices, originally introduced to represent the characteristics of lumped passive networks with variable elements, 1,2 have become more important because of their application to the synthesis of lumped-distributed networks. Ansell first showed the two-variable reactance property of networks composed of lossless transmission lines and lumped reactances. 3 The two-variable theory has also been applied to the synthesis of networks consisting of lumped resistors capacitors and uniformly distributed RC lines, 4,5 which are of importance in microelectronic structures. 0, 7 Besides the various applications, the two-variable reactance theory is of theoretical interest in itself since it can be shown t h a t passive R L C synthesis is a special case of two-variable reactance synthesis. 2 Koga 8 demonstrated that every nXn two-variable reactance matrix W(p, s) can be realized as the impedance seen at the first n ports of a lossless {n+qr) -port network in the p-plane terminated at its last qr ports with unit inductors in the s-plane; q is the rank of W{p, s), and r is the highest degree of s in the least common denominator of * This work is based on Chapter III of the author's dissertation, "Synthesis of Lumped-Distributed RC Networks" submitted in partial fulfilment of the requirements for the Ph.D. degree at Stanford University, May 1967. 163