On the internal signature and minimal electric network realizations of reciprocal behaviors

TL;DR

This paper extends classical results by establishing new lower bounds on the number of capacitors and inductors in minimal electric network realizations of reciprocal behaviors, including uncontrollable systems.

Contribution

It introduces lower bounds on the number of even and odd parity states in minimal realizations of reciprocal systems that are not necessarily controllable.

Findings

Established lower bounds on capacitor states

Established lower bounds on inductor states

Extended classical results to uncontrollable systems

Abstract

In a recent paper, it was shown that (i) any reciprocal system with a proper transfer function possesses a signature-symmetric realization in which each state has either even or odd parity; and (ii) any reciprocal and passive behavior can be realized as the driving-point behavior of an electric network comprising resistors, inductors, capacitors and transformers. These results extended classical results to include uncontrollable systems. In this paper, we establish new lower bounds on the number of states with even parity (capacitors) and odd parity (inductors) for reciprocal systems that need not be controllable.