Time Domain Characterization of the Cole-Cole Dielectric Model.

Sverre Holm

doi:10.2478/joeb-2020-0015

Abstract

The Cole-Cole model for a dielectric is a generalization of the Debye relaxation model. The most familiar form is in the frequency domain and this manifests itself in a frequency dependent impedance. Dielectrics may also be characterized in the time domain by means of the current and charge responses to a voltage step, called response and relaxation functions respectively. For the Debye model they are both exponentials while in the Cole-Cole model they are expressed by a generalization of the exponential, the Mittag-Leffler function. Its asymptotes are just as interesting and correspond to the Curie-von Schweidler current response which is known from real-life capacitors and the Kohlrausch stretched exponential charge response.

Highlights

The Cole Cole model [1] is a generalization of the Debye dielectric relaxation model which ts measurements in many applications including the bioimpedance eld, [2, Sec. 9.2.7]
Dielectrics may be characterized in the time domain by means of the current and charge responses to a voltage step, called response and relaxation functions respectively
The familiar frequency domain expression for the Cole-Cole model of order can be expressed in the time domain

Summary

Introduction

The Cole Cole model [1] is a generalization of the Debye dielectric relaxation model which ts measurements in many applications including the bioimpedance eld, [2, Sec. 9.2.7]. Since the Debye model has a simple time domain interpretation and both the current and charge responses to a voltage step are exponential, the ColeCole responses can be expressed as sums of exponential functions This result is often too complex to lend itself to interpretation. There has been a development in understanding of the responses of the Cole-Cole model found in a direct way These results depend on the MittagLeer function, a generalization of the exponential which is named after Gösta Mittag-Leer (1846-1927). Second and "r more important, it is which is directly reected in the properties of the macroscopic capacitance of the medium, and since impedance or capacitance is what can be measured, it makes sense to specify properties in "r terms of This is how models are justied in the bioimpedance eld as [2, Sec. 3.1.3] says: Polarization P cannot itself be measured.

The frequency domain response to an input voltage is

Conclusion