The Admittance Matrix and Network Solutions

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.