Voltage differences as functions of a matrix eigensystem

Abstract

We firstly consider an electrical network that consists of multiple subnetworks. We refer to some basic definitions which relate net current flows and voltages, the edges and nodes respectively of each subnetwork, and define matrix G, as the direct sum of the admittance matrices that correspond to each subnetwork. Then, we prove that any voltage difference at two arbitrary nodes can be explicitly written as a function only of the eigensystem of G, namely its eigenvalues and their corresponding eigenvectors. Next, we consider a DC/AC circuit network and by using duality properties of a matrix pencil related to the network, we obtain expressions for voltage differences as functions of the pencil’s eigenvalues and their corresponding eigenvectors. To validate our findings, we present illustrative numerical examples.