Two-Variable Synthesis of Resistively Terminated Cascades of Lossless Transmission Lines, Series Inductors and Shunt Capacitors, and Minimum Gyrator Synthesis of Certain Lumped Lossless N-Ports

摘要

The work consists of two parts. In Part 1, necessary conditions are derived for the driving point impedance function of a resistively terminated cascade of equal lengths of lossless transmission line, series inductors and shunt capacitors. Several approaches are presented which tend to indicate that the conditions are also sufficient. Sufficiency is actually proved for the special case of a resistively terminated cascade of no more than two equal lengths of loosless transmission line and shunt capacitors only. A numerical example is included which illustrates the synthesis procedure. In Part 2, the problem of realizing an arbitrary rational n x n regular paraunitary matrix, S(p), as the scattering matrix of a lossless n-port containing the minimum number of ideal gyrators is considered. The approach used is to try to generate a lossless reciprocal n + k port, N sub a, such that when the last k-ports of N are terminated in k/2 uncoupled ideal gyrators, the resulting n-port realizes S(p). The success of the approach is shown to depend upon the existence of an appropriate factorization of the skew-symmetric part of S(p). It is shown that a minimum gyrator synthesis is always possible for the special class of matrices having a skew-symmetric part which is non-singular for all p = jw, including p = infinity. A non-trivial numerical example of the synthesis technique is presented in an appendix. In addition, techniques for the minimum gyrator synthesis of arbitrary 2 x 2 and degree two n x n regular para-unitary matrices are presented. (Author)