A Note on the Graphical Analysis of Alternating-Current Networks

A simple proof is given of the following observed property of alternating-current networks: In a network consisting of any number of linear bilateral impedances, connected in any manner whatsoever, and any number of generators of the same frequency, the locus of any voltage or current in the system will be a circle as any one self-impedance is varied so that its locus is a straight line in the complex plane. The proof given is based upon the application of the fundamental theorems of network analysis to a general alternating-current network and upon the identification of a circular form of the linear fractional transformation of function theory. A simple analysis of alternating-current networks is frequently possible by virtue of this property. Reference is made to examples of circuit analysis based upon this property. It is further shown that as a result of the observed property the impedance looking into any two terminals of a general passive network of linear bilateral elements has a circular locus as any impedance in the system is varied so that its locus is a straight line.