Infinite electrical networks

Abstract

The investigation of infinite electrical networks having no restrictions on their graphs other than countability and connectedness is a recent occurrence. The first rigorous analysis of locally finite purely resistive networks was published in 1971 by Flanders, and the subject has developed apace since then. This article summarizes the current status of the subject and points out some open questions and areas for future research. First of all, various approaches to infinite purely resistive networks are described. They use some ideas from algebntic topology and Hilbert-space theory. These methods no longer work when the network is allowed to have a variety of electrical parameters. Two approaches to the latter situation ate then presented. In one of these, the network is decomposed into an interconnection of a finite number of infinite subnetworks, each of which contains parameters of only one kind. If each subnetwork can be characterized as an operator on Hilbert's coordinate space l 2 , it is possible under certain circumstances to achieve a solution to the network by extending the customary analyses for finite scalar networks to finite operator networks. The second approach, which is inherently very general and is due to Dolezal, treats an incidence matrix a as an operator on a certain Hilbert space and characterizes the solution of the network by means of the null space of a and its orthogonal complement. All these approaches achieve a unique solution by imposing various conditions in addition to Kirchhoff's node and loop laws. The last part of the paper investigates the whole class of all possible solutions when only Kirchhoff's node and loop laws are imposed.