Power law susceptibility function for the analysis of anomalous spectral response

TL;DR

This paper introduces a generalized susceptibility model with power-law and fractional derivatives to better describe anomalous spectral responses in dielectric materials, extending classical models like Debye and Cole-Cole.

Contribution

It develops a new time-domain kinetic equation incorporating fractional derivatives and power-law terms, providing a more flexible model for anomalous dielectric behavior.

Findings

The model captures deviations from classical dielectric responses.

It introduces a constant-phase element with two fractional parameters.

The approach links fractional calculus with spectral response analysis.

Abstract

The extensions of the classical Debye model of susceptibility of dielectric materials to the well-known Cole-Cole, Davidson- Cole, or the Havriliak-Negami models is done by introducing non-integer power parameters to the frequency-domain function. This is very often necessary in order to account for anomalous deviations of the experimental data from the ideal case. The corresponding time-domain descriptions expressed in terms of the relaxation or response functions are in the form of first-order differential equations for the case of Debye model, but involves relatively complex integro-differential operators for the modified ones. In this work, we study the extension of the time-domain kinetic equation describing the Debye polarization function to include two extra degrees of freedom; one to transform the first-order time derivative of the polarization function to the Caputo fractional-order time derivative and another to change the linear term to a power term. From an electrical perspective, it results in a constant-phase element with two fractional parameters.