Power law susceptibility function for the analysis of anomalous spectral response

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.