The Implicit McMillan degree and network descriptions

Abstract

Abstract The paper addresses the problem of evaluating the Implicit McMillan degree$\delta{m}$ of $W^{-1}(s)$, where $W^{-1}(s)$ denotes the transfer function of a passive $RLC$ electrical network.1 The Implicit McMillan degree$\delta{m}$ specifies the minimum number of dynamic elements needed to completely characterize the passive $RLC$ network, i.e. an electrical network that contains only passive elements (capacitors, inductors and resistors) and associates it with the rank properties of the passive element matrices. A fact that in the circuit literature is intuitively accepted but not rigorously proved is that this degree must be equal to the minimum number of independent dynamical elements in the network Livada (2017, Implicit network descriptions of RLC networks and the problem of re engineering. Ph.D. Thesis, City, University of London) and Leventides et al. (2014, McMillan degree of impedance, admittance functions of RLC networks. In 21st International Symposium on Mathematical Theory of Networks and Systems. The Netherlands: Groningen). In this paper, we investigate this finding, showing that the maximum possible Implicit McMillan degree$\delta{m}$ of such networks is given by $rankL+rankC$ and that this value is reached when certain necessary and sufficient conditions are satisfied.1