On a concept of genericity for RLC networks

Abstract

A recent definition of genericity for resistor-inductor-capacitor (RLC) networks is that the realisability set of the network has dimension one more than the number of elements in the network. We prove that such networks are minimal in the sense that it is not possible to realise a set of dimension n with fewer than n − 1 elements. We provide an easily testable necessary and sufficient condition for genericity in terms of the derivative of the mapping from element values to impedance parameters, which is illustrated by several examples. We show that the number of resistors in a generic RLC network cannot exceed k + 1 where k is the order of the impedance. With an example, we show that an impedance function of lower order than the number of reactive elements in the network need not imply that the network is non-generic. We prove that a network with a non-generic subnetwork is itself non-generic. Finally we show that any positive-real impedance can be realised by a generic network. In particular we show that sub-networks that are used in the important Bott-Duffin synthesis method are in fact generic.

Minimal Series–Parallel Network Realizations of Bicubic Impedances

“…, q n are all polynomial functions in the network's element values (inductances, capacitances, etc.) that can be obtained from Kirchhoff's tree formula [30]. Following [30], we call the set of impedances realized by a given network class N the realizability set of N , which can be characterised by the vector of coefficients (p 0 , .…”

“…For any given N ∈ N , the dimension 1 of the realizability set of N is at most one greater than the number of elements in N [30, Lemma 2], and N is called generic if every single network in N contains (strictly) fewer elements than the dimension of the realizability set of N [30, Definition 1]. It follows from [30,Lemma 2] that almost all networks from a given generic network class are minimal in the sense that their impedance cannot be realized by a network containing strictly fewer elements. 2 Definition 1 (Generating/ minimal generating sets): Let Z m,n be the set of impedances realized by series-parallel networks containing at most m capacitors and n inductors; and let Z M = ∪ m,n|m+n=M Z m,n be the set of impedances realized by series-parallel networks containing at most M energy storage elements.…”

“…Note from[30, Corollary 2] that the network classes N 10 -N 14 are not generic (and neither are the network classes N i…”