General Topological Formulas for Linear Network Functions

Abstract

This paper presents a development of topological formulas for the vertex admittance functions of a network which includes components that are mutually coupled. The results are proper generalizations of those which have been presented previously for networks without mutually coupled components. With each network, which excludes generators but which includes mutually coupled coils, linear vacuum tubes, and transistors, is associated a linear graphG. Each edge (element) ofGis either a single edge or a pair edge which belongs to an edge pair ofG. For each graphGthere is a companion graph\underline{G}such thatGand\underline{G}constitute a graph pair. The vertex driving point and transfer admittance functions ofGare topologically related toGby complete tree sets and complete two-tree sets. A complete tree (two-tree) set is a set of edges ofGthe corresponding subgraphs of which are trees (two-trees) of bothGand\underline{G}. Corresponding to each such set is a weight and an admittance. The admittance is the product of the admittance weights of the edges which belong to the set. The weight determines the sign of the admittance and depends upon the pair edge members of the set, their orientations and their topological arrangement in the corresponding subgraphs ofGand\underline{G}. The vertex driving point admittance function associated with the vertex pairp_i , p_vofGisy_{i,\sigma} = \frac{V(Y)}{W_{I,V-I,V}.V(Y)denotes the sum, over all possible complete tree sets ofG, of the product of the complete tree admittance and the associated weight.W_{I,V-I,V}denotes a corresponding sum of products for the complete two-tree admittance and associated weight. Similar expressions are given for the vertex transfer admittance functions ofG.