A Fractional Complex Permittivity Model of Media with Dielectric Relaxation

Armando Ciancio,Bruno Flora

doi:10.3390/fractalfract1010004

Abstract

In this work, we propose a fractional complex permittivity model of dielectric media with memory. Debye’s generalized equation, expressed in terms of the phenomenological coefficients, is replaced with the corresponding differential equation by applying Caputo’s fractional derivative. We observe how fractional order depends on the frequency band of excitation energy in accordance with the 2nd Principle of Thermodynamics. The model obtained is validated with respect to the measurements made on the biological tissues and in particular on the human aorta.

Highlights

The frequency domain response function of a media dielectric, well-known how complex permittivity, e(iω ), one obtains from spectral measurement of electrical displacement field d(iω )respect to applied electric field e(iω ): d e = (1) e √with ω = 2π f, i = −1, and f being frequency.The polarization does not follow instantaneous changes of the applied electric field, so the dielectric material is in a state of non-equilibrium
Dielectric relaxation is a process through which dielectric media reach the state of equilibrium, with one or more time constants in relation to corresponding polarization phenomena
The fractional model of the complex permittivity (36) is determined uniquely from the possible values of the parameters C0, F0, C1, F1 that satisfaction (36) with C0 > 0, C1 > 0; this is in accordance with the fact that entropy variation is positive, reference [11], for the 2nd principle of the thermodynamics

Summary

Introduction

The frequency domain response function of a media dielectric, well-known how complex permittivity, e(iω ), one obtains from spectral measurement of electrical displacement field d(iω ). Respect to applied electric field e(iω ): d (iω ) e (iω ) = (1) e (iω ) √. The polarization does not follow instantaneous changes of the applied electric field, so the dielectric material is in a state of non-equilibrium. Dielectric relaxation is a process through which dielectric media reach the state of equilibrium, with one or more time constants in relation to corresponding polarization phenomena. Debye [2] has proposed the following complex permittivity to take into account dielectric relaxation corresponding to a linear differential equation of the first order, with constant time τ: e (iω ) = e∞ + es − e∞

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Conclusion