The Admittance Matrix and Network Solutions

TL;DR

This paper reviews classical electrical network theory focusing on the properties of the admittance matrix and their implications for both DC and AC networks, including circuit theorems, topological relations, and impedance characteristics.

Contribution

It synthesizes fundamental properties of the admittance matrix and connects them to network topologies and classical circuit theorems, extending understanding from DC to AC networks.

Findings

Determinant of admittance matrix relates to network topology

No universal impedance metric exists for AC networks

Semi-orientability can be used in AC power flow analysis

Abstract

This overview presents a collection of results from classical electrical network theory concerning properties of the network admittance matrix, and the relationship between electrical characteristics of the network and various mathematical properties of the admittance matrix. The derivation of standard circuit theorems taught to electrical engineers from properties of the admittance matrix and its cofactors is presented, and the determinant of the matrix is related to topological properties of the network. Later sections review key properties of DC and AC networks including monotonicity and orientability of DC networks, the impedance matrix for DC networks, and the extensions and limitations of these properties when translated to AC networks. It is shown that in general there is no impedance metric for AC networks extending that for DC networks but in many cases a form of semi-orientability can be exploited when analysing the AC power flow. The overview concludes with the derivation of standard properties of positive-real functions and impedance functions associated with AC networks, and of Thevenin and Norton equivalent sources for subnetworks.