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.

#### Keywords

- Cole-Cole
- Curie-von Schweidler
- Kohlrausch
- Constant phase element

The Cole Cole model [1] is a generalization of the Debye dielectric relaxation model which fits measurements in many applications including the bioimpedance field, [2, Sec. 9.2.7]. One interpretation is that it represents a distribution of relaxation processes, each described by the Debye model. Since the Debye model has a simple time domain interpretation and both the current and charge responses to a voltage step are exponential, the Cole-Cole responses can therefore be expressed as sums of exponential functions. In practice, however, this result is often too complex to lend itself to interpretation.

In recent years, 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 Mittag-Leffler function, a generalization of the exponential which is named after Gösta Mittag-Leffler (1846–1927). This function is rightly called the “queen function of fractional calculus” [3] showing the close link between non-integer derivatives and the Cole-Cole model. The asymptotes of the Mittag-Leffler function are just as important as the function itself and is what will be emphasized here.

There are two well-established results for non-ideal dielectrics. The first is that for a long time it has been known that the current response to a step voltage for a practical non-ideal capacitor often follows the Curie-von Schweidler power law:

The second is an even older result which is due to Kohlrausch who found that the discharge of a capacitor with glass as a dielectric medium in a Leiden jar follows a stretched exponential. The charge is:

The purpose of this paper is to increase awareness of the time domain properties of the Cole-Cole model by collecting and interpreting some results from in particular [5, 7, 8]. The paper starts with the Debye model in order to define the relevant current and charge responses, called the response function and the relaxation function respectively. It will also be shown that both the Curie-von Schweidler power law and the Kohlrausch-Williams-Watt stretched exponential response are approximations to those of the Cole-Cole model. Finally, it is also shown that just as the Debye model corresponds to an ordinary partial differential equation for the constitutive law between the displacement field and the electric field, the Cole-Cole model corresponds to a similar equation but with non-integer, i.e. fractional derivatives.

The constitutive relation between the displacement field, _{0} is free space permittivity, _{r}

Frequency-dependency can be given either for the susceptibility [5, 9] or for the permittivity [2]. The relationship between the two is:
_{r}_{r}_{r}

This capacitance of such a dielectric material is
^{−1}. The frequency domain response to an input voltage is:
^{−1}, this is:

The current charge relation is :

As a reference and in order to establish terminology, the Debye model will first be analyzed for its current and charge responses. Its permittivity is
_{s}_{∞} < _{s}_{∞}, represents an ideal capacitor which is in parallel with a frequency-varying part.

The current response, (7), is

Likewise the charge response is given by (9) or by integration of the current:
_{∞}. The charge therefore starts with this value and ends up to be proportional to _{s}

The models will be given in terms of a normalized permittivity which for the Debye model is:

In order to characterize subsequent models, the two descriptions of [5] will be used. The first is the response function,

The second function is the relaxation function,

The definitions of

The Debye model can also be expressed as a differential equation between D and E by combining (3) with a rearranged (10):

The permittivity of the Cole-Cole model follows a more general power-law than the Debye model:

The permittivity of the Cole-Cole model in normalized form is:

Section 3.1 of [5] gives the functions for the Cole-Cole model. The response function that characterizes the current response is:
_{α,β} is the Mittag-Leffler function which is a generalization of the exponential function. The two-parameter Mittag-Leffler function is defined by
_{α}(_{α,1}(_{1}(

According to [5], the response function may be approximated:
^{α} ≫ 1 and _{∞} = 0 and then the capacitance is:
^{α} requires a very large argument to be much larger than one for such a small ^{100} for

The exact expression and the approximations are plotted in Fig. 1 for

The Mittag-Leffler function with a negative argument raised to a power can also be approximated [8, 5]:

The Cole-Cole model can also be expressed as a differential equation between D and E. The frequency domain relation building on (19) is:
^{n} transforms to the nth order derivative for integer

There are several alternatives to the Cole-Cole model such as the Cole-Davidson and Havriliak-Negami models. The latter is the more general one:

The familiar frequency domain expression for the Cole-Cole model of order

_{α}(−^{α}_{α}(−^{α}

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