Why RLC realizations of certain impedances need many more energy storage elements than expected

TL;DR

This paper investigates why RLC networks often have more energy storage elements than the minimal number suggested by system theory, proving that for certain impedances, this excess is unavoidable.

Contribution

It introduces new network realizations and proves that existing RLC networks necessarily contain more than twice the minimal energy storage elements for certain impedances.

Findings

RPFG networks contain more than twice the minimal energy storage elements.

New alternative networks are discovered that approach minimal energy storage.

It is proven that excess energy storage elements are unavoidable for certain impedance realizations.

Abstract

It is a significant and longstanding puzzle that the resistor, inductor, capacitor (RLC) networks obtained by the established RLC realization procedures appear highly non-minimal from the perspective of linear systems theory. Specifically, each of these networks contains significantly more energy storage elements than the McMillan degree of its impedance, and possesses a non-minimal state-space representation whose states correspond to the inductor currents and capacitor voltages. Despite this apparent non-minimality, there have been no improved algorithms since the 1950s, with the concurrent discovery by Reza, Pantell, Fialkow and Gerst of a class of networks (the RPFG networks), which are a slight simplification of the Bott-Duffin networks. Each RPFG network contains more than twice as many energy storage elements as the McMillan degree of its impedance, yet it has never been established if all of these energy storage elements are necessary. In this paper, we present some newly discovered alternatives to the RPFG networks. We then prove that the RPFG networks, and these newly discovered networks, contain the least possible number of energy storage elements for realizing certain positive-real functions. In other words, all RLC networks which realize certain impedances contain more than twice the expected number (McMillan degree) of energy storage elements.