The topological foundations of electrical networks — complete models

The aim of this paper is to identify the topological entities underlying the construction of a complete (not reduced) network model. Consider an active network as a pair of physically different structures, i.e. a ‘dead network’ and an ‘excitation’. First, we define the general structure of the excitation which can be applied to a dead network; then the graphs Ga and G of the active and dead networks, respectively, are introduced. It is shown that, if a complete network model is to be constructed, the topological relations among the voltages and those among the currents of the network elements must be written choosing maximal independent sets of loops and cut‐sets of Ga. It is successively pointed out that the graph G can be obtained from Ga by means of a reduction operation, and the topological entities of G corresponding to the loops and to the cut‐sets of Ga are singled out. It has been found that a complete model of an electric network implies the use of four topological entities defined on the set of G‐edges, i.e. the closed path, the open path, the cut‐set and the pseudo‐cut, together with two topological entities defined on a maximal set of independent vertex couples of G, i.e. the ‘junction pair’ extreme of an open path and the ‘family of independent node couples’ split by a cut‐set. Furthermore the different reference frames and the relative topological matrix transformations which allow a complete model of the network to be built up are singled out and discussed.