Analytical and numerical modelling of the electrostatic behaviour of highly insulating materials in the time domain

The usual response of an insulator subjected to a voltage step involves time power laws. We present here mathematical tools allowing to calculate this time domain response in open circuit after an initial charge deposit, within the framework of linear systems theory, using linear fractional transfer functions. In the time domain, the inverse Laplace transform of the data taken from the frequency domain leads to Mittag-Leffler functions, generalizing the Debye exponential response to an extended fractional α-order response. The open-circuit boundary conditions are different from the closed-circuit ones. We nevertheless demonstrate that using a transfer function deduced from the Cole-Cole response in closed-circuit, a precise analytical formula of the potential decay after an initial charge deposit may be established, and a numerical computation of this decay may be performed using easily available software. Applying the superposition principle, the voltage return following a brief short circuit may also be deduced. Experimental results are presented and the limits of the superposition principle applied to real materials are discussed.


Introduction
135 years ago, precise electrometer measurements allowed Jacques Curie to establish that the usual response of an insulator subjected to a voltage step involves time power laws, therefore challenging the simple idea of a characteristic time in the charging and discharging processes.This is (or at least, should be) nowadays common knowledge for scientists and engineers operating in the field of electrical engineering.Dielectric spectroscopy data are commonly fitted using Cole-Cole, Havriliak-Negami or Kohlrausch-Williams-Watt phenomenological relaxation models.
It has been established decades ago that a relaxation process following a Cole-Cole or Havriliak-Negami frequency dependence may be described in the time domain by a dielectric response involving Mittag-Leffler functions [1].Numerical software nowadays allows a fast and accurate calculation of these functions.
The literature on surface potential decay is also quite abundant [2].However, to our knowledge, the powerful tool that represents Mittag-Leffler functions, together with the introduction of constant-phase elements in the equivalent circuits, are still unfamiliar to the Electrostatics community.We try here to demonstrate that they provide a convenient way to model the open-circuit relaxation, in a more elegant way than the arrays of classical RC cells, which are often used in this field.

The insulator in the time domain as a linear system
Let us consider a dielectric plate of thickness d on an earthed plane.Assuming an electric field () in the material and null outside, we get a surface charge density (): () being the polarization in the material,  ∞ its permittivity including vacuum permittivity and the polarization phenomena fast enough to follow the electric field variations, the others being included in ().The dielectric considered here as a linear system, which means that the polarization is proportional to the applied field, including however a delayed component.The polarization at a given time depends on the past values of the electric field following a convolution relationship [3]: is the static permittivity (including all the polarization processes) and () a dielectric response function of the material.A complementary dielectric response function  * has to be introduced to deduce the field from the charge density: The Laplace transform of  * may be deduced from the Laplace transform of : Applying a voltage step  0 at t=0, the measurement of the absorption current will provide a direct image of the response function () since the external displacement current density -in the absence of any conductivity -will be: Similarly, applying at t=0 a charge step  0 and leaving the insulator evolve in open circuit leads to a voltage decay proportional to  * (t): A short circuit of the insulator may be performed after a given polarization time.In closed circuit, because of the delayed component of the polarization, a resorption current will flow which is very similar to the absorption current.The analogue of this phenomenon in open circuit is the return voltage   () which occurs when a charged insulator is neutralized after a given relaxation time   , and left in open circuit [4].It may be deduced from the voltage decay   () using the superposition principle:  2).During an absorption current measurement, after an initial peak to charge  ∞ , the current decay is determined by the progressive charge of   , while during a potential decay experiment the system discharges the series association of   and  ∞ .Hence the relaxation time will be smaller for the potential decay than for the absorption current.

Cole-cole relaxation in open circuit
The Laplace transform of the response function (which may be deduced from dielectric spectroscopy) for a dielectric following the Cole-Cole relationship [5] may be written as: In the time domain, the corresponding response function may be written as: Γ() being the Euler gamma function and  , () a generalized Mittag-Leffler function [1].A material following the Cole-Cole dielectric behavior may be modelled by the RC circuit given in figure 1, with   = () −1   ⁄ being a constant phase element (CPE): the ratio between its imaginary and real parts does not depend on frequency [5].The Laplace transform of the potential decay following a charge deposit  0 at t=0 may be computed from ( 4) and (6): Since the Laplace transform of  −1  , [−(/)  ] is 1+    , and using the property [1] that: We obtain: and Figure 2. Computation of the voltage decay given by equation ( 12) and its derivative given by ( 13) for several values of  assuming  initial =  0   ∞ ⁄ = 1 and  final =  0    ⁄ = 0.5 The voltage decay and its derivative may hence be analytically given using simple Mittag-Leffler functions.They depend on three main parameters:  1 , (characteristic time),    ∞ ⁄ (relative polarization amplitude), and  (shape of the potential decay).From the properties of the Mittag-Leffler functions, it may be shown that the time derivative of the potential is decaying as (  1 ⁄ ) 1− for   1 ⁄ ≪ 1 and as Its displays the same shape than the absorption current, replacing  by  1 .A computation of the voltage decay and its time derivative is given on figure 2 for    ∞ ⁄ = 2 and several values of .

An experimental example on an alumina plate
Figure 3 presents the result of a classical experiment: an alumina plate (1mm thick, 96% pure Al2O3) is charged by corona during 1s at ambient temperature and the potential decay is measured with an electrostatic probe during 1000s.This experimental curve is fitted with a simulated decay following equation ( 12) assuming a Cole-Cole behaviour of the insulator with the following parameters:  = 0.45,  1 = 38360s, and    ∞ ⁄ = 1.9.  Figure 4 presents the recording of the voltage return on the same plate after a brief (1s) negative corona charge allowing surface neutralization.The measured curve is accompanied by the result of the simulation using the superposition principle described in equation (7), and equation (12) with the above parameters deduced from the decay.

Conclusions
The parameters coming from the analysis described here (,  1 and    ∞ ⁄ ) provide in our opinion a much better description of the electrostatic behavior of a dielectric than the resistivity value that is usually given.However, we have to underline that the dielectric response on a real material involves several superimposed physical processes which may include relaxation of different dipoles, interfacial polarization, surface conduction, charge trapping and detrapping processes, so that a simple linear RC model, even involving a constant phase element accounting for a time power law response, is not sufficient.It does not account for nonlinearities, and for a description on many time decades, several cells will have to be included in the equivalent network of the insulator.What has been done here could be the description of any of these cells, since, due to disorder and to mutual influence, all these elementary relaxation processes lead to power laws and, mathematically, to Mittag-Leffler functions.

Figure 1 .
Figure 1.Equivalent circuit for the insulator. ∞ includes the instantaneous and fast polarization phenomena while the   −   dipole represents the polarization phenomena occurring during the experiment.Lowercase  and  notations correspond to capacitance and impedance per unit area.On the same insulator, closed and open circuit measurements lead to slightly different dynamics.The equivalent circuit shown figure 1 is deduced from equation (2).During an absorption current measurement, after an initial peak to charge  ∞ , the current decay is determined by the progressive charge of   , while during a potential decay experiment the system discharges the series association of   and  ∞ .Hence the relaxation time will be smaller for the potential decay than for the absorption current.

Figure 3 .
Figure 3. Measurement and computation of the potential decay on Al2O3

Figure 4 .
Figure 4. Measurement and computation of the voltage return on Al2O3