Why RLC Realizations of Certain Impedances Need Many More Energy Storage Elements Than Expected

It is a significant and longstanding puzzle that the resistor, inductor, and capacitor (RLC) networks obtained by the established RLC realization procedures appear highly nonminimal from the perspective of the linear systems theory. Specifically, each of these networks contains significantly more energy storage elements than the McMillan degree of its impedance, and possesses a nonminimal state-space representation whose states correspond to the inductor currents and capacitor voltages. Despite this apparent nonminimality, 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 that realize certain impedances contain more than twice the expected number (McMillan degree) of energy storage elements.