Some Properties of Impedance as a Causal Operator

Abstract

As a model for the various problems connected with domains of analyticity and dispersion relations, the concept of impedance (self-impedance) has been studied in general under the physical requirements of linearity, passivity, reproducibility, and causality. After identifying impedance functions with suitably defined positive functions, any impedance function may be decomposed to the sum of a minimum reactive part and two reactive parts, one of which is of unfamiliar nature. In the time domain, the convolution representation of the current in terms of the voltage and vice versa are obtained rigorously. As an example, a reactance obtained from the Cantor function is studied in detail. Its peculiar delta-function response is computed explicitly, and it is also used to construct a counter example to a representation by van Kampen of the S matrix of a Maxwellian field. The general results are also applied to study the Kronig-Kramers dispersion relations for dielectric constants. It is proved that one of the two relations is true under very general conditions, but the other is false in general.