THE GYRATION OPERATOR IN NETWORK THEORY

Abstract
The gyration operator which generates a matrix from a given matrix is defined. It is shown that the gyration is related to the concept of combinatorial equivalence. The hybrid matrices of network theory are gyration matrices. The ABCD and hybrid matrices are combinatorially equivalent to their impedance and admittance matrices. All combinatorially equivalent matrices have the same degree. The gyration operator also preserves the PR property as well as the rank of the hermitian part. The gyration operators form an involutary Abelian group. The PR property is shown to extend beyond the impedance and admittance matrices of a passive network, and a complete set of PR matrices is given for the description of the network. The relationships between the scattering and gyration operators are detailed. A complete synthesis procedure, based on the gyration operator is given. Any PR immittance matrix can be synthesized. Three worked examples are included. Of special significance is a novel concept of enumerating the degree of a rational matrix function.

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