  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.
