On Connections between the Cauchy Index, the Sylvester Matrix, Continued Fraction Expansions, and Circuit Synthesis

Abstract

A fundamental result in circuit synthesis states that the McMillan degree of a passive circuit’s impedance is less than or equal to the number of reactive elements in the circuit. More recently, Hughes and Smith connected the individual numbers of inductors and capacitors in a circuit to a generalisation of the Cauchy index for the circuit’s impedance, which was named the extended Cauchy index. There is a close connection between the Cauchy index of a real-rational function and many classical algebraic results relating to pairs of polynomial functions. Using this connection, it is possible to derive algebraic constraints on circuit impedance functions relating to the precise numbers of inductors and capacitors in that circuit. In this paper, we first present these algebraic constraints. We will then show a relationship between the extended Cauchy index and properties of continued fraction expansions of real-rational functions, which we use to provide insight into circuit synthesis procedures.