The solution of passive electrical networks by means of mathematical trees

Abstract

A new method of solving electrical networks is described which is more rapid than the classical determinant method. The new method is based on the properties of certain sub-networks termed “trees” and “2-trees.”. It is shown that, with suitable algebraic representations, the required sets of trees and 2-trees can be evaluated with the aid of equations in which the terms are networks. Functions known as linkages are defined in terms of sets of trees and 2-trees, and it is shown that these functions obey laws analogous to Kirchhoff's laws for voltages and currents. Hence relations involving voltages and currents can be expressed in terms of linkages. In particular, the transfer admittance of a network can be expressed as the ratio of the set of all trees on the network to an appropriate linkage. Network theorems can be rapidly deduced with the aid of trees and 2-trees. Since the method gives results in terms of admittances, a rule is given whereby admittance functions can be transformed into the corresponding impedance functions.