Time Domain Characterization of the Cole-Cole Dielectric Model

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.


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]. 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-Leer function, a generalization of the exponential which is named after Gösta Mittag-Leer . 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-Leer 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 rst 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: I@tA G t H ; (1) where H is an order which is dened after (19). In [4] such responses are measured for many practical capacitors and the law is attributed to Curie in 1889 and von Schweidler in 1907. This is especially relevant for nonideal dielectrics.
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: Q@tA G exp @t=£A ; (2) where £ is a time constant and is given after (19).

Denitions
The constitutive relation between the displacement eld, D, and the electric eld, E is: D a " H " r E a " H E C E a " H E C P; (3) where P is the polarization charge density, " H is free space permittivity, " r is relative permittivity, and is susceptibility.
In the time domain, D@tA, represents a charge density.
Frequency-dependency can be given either for the susceptibility [5,9] or for the permittivity [2]. The relationship between the two is: a " H @" r IA: (4) There are two reasons why we consider the permittivity here. First, in the bioimpedance eld " r ) I so there is little dierence in practice and D % P . Second and more important, it is " r which is directly reected in the properties of the macroscopic capacitance of the This capacitance of such a dielectric material is C a A d " H " r ; (5) where A and d are the area and the plate distance of the capacitor. The capacitance is complex for the models considered here and its impedance is Z a @j!CA I . The frequency domain response to an input voltage is: When the input voltage is a step, U@!A a @j!A I , this is: In the time domain, the current step response is found as the inverse Fourier transform of the relative permittivity.
The current charge relation is : where J@tA is the charge density. Charge is therefore found by an integration of the result for the current plus a constant. Integration is equivalent to division by j! in the frequency domain and therefore the charge response to a step in voltage is related to Q step @!A a I step @!A j!A a " H d " r @!A j! : (9) Debye model As a reference and in order to establish terminology, the Debye model will rst be analyzed for its current and charge responses. Its permittivity is " r @!A a " I C " s " I I C j! ; (10) where " s is the static value and " I < " s is the value at innity frequency, is a characteristic time constant for the medium. The rst term, represented by the constant " I , represents an ideal capacitor which is in parallel with a frequency-varying part.

Time-domain characterization
The current response, (7), is I step @tA a " H A d p I f" r @!Ag a " H A d " I @tA C " s " I e t= : (11) The current has an initial impulse due to the charging of an ideal capacitor followed by a current that dies out with a time constant .
Likewise the charge response is given by (9) or by integration of the current: Q step @tA a I A t H I step @uAdu: (12) This gives Q step @tA a " H d " I @" s " I Ae t= C K a " H d " s @" s " I Ae t= ; t ! H; (13) where K is a constant which is such that the initial value for the charge is proportional to " I . The charge therefore starts with this value and ends up to be proportional to " s for large time. This is in agreement with the example Characterization of general models The models will be given in terms of a normalized permittivity which for the Debye model is: " D @!A a " r @!A " I " s " I a I I C j! : (14) In order to characterize subsequent models, the two descriptions of [5] will be used. The rst is the response function, @tA, which characterizes the current response.
It is given as the inverse Fourier transform of the normalized relative permittivity, "@!A. In the Debye example this is @tA a p I f"@!Ag a p I I I C j! ; a I e t= ; (15) which can be recognized to be the main time-varying part of (11).
The second function is the relaxation function, ©@tA, which characterizes the charge. It is given by ©@tA a p I I "@!A j! a e t= ; t ! H; (16) which is seen in (13)  functions and that the Laplace transform is non-negative [5]. In this way the physical realizability of the considered system is guaranteed [10].

Constitutive law
The Debye model can also be expressed as a dierential equation between D and E by combining (3)

Cole-Cole model
The permittivity of the Cole-Cole model follows a more general power-law than the Debye model: " r @!A a " I C " s " I I C @j!A I H a " I C " s " I I C @j!A : (19) In electromagnetics, the model is sometimes presented with an exponent of I H , so that H a H corresponds to the Debye model. It may also be expressed with an exponent a I H where H < I and here that convention will be followed in order to conform to [5]. The function i ; is the Mittag-Leer function which is a generalization of the exponential function. The twoparameter Mittag-Leer function is dened by i ; @tA a I naH t n @n C A ; H < I; (23) where @xA is the gamma function, a generalization of the factorial for non-integer arguments and where @nA a @n IA3 for integer arguments. Setting a I gives the standard Mittag-Leer function E @tA a E ;I @tA of (22). Another special case is E I @tA which is the exponential function. In this article, it is in particular the asymptotes of the responses which are important.

Approximation of the response function
According to [5], the response function may be approxi- The small time approximation corresponds to the Curievon Schweidler law of (1) as mentioned in [12]. An example may also be found in [4] where the current in capacitors followed the Curie-von Schweidler law over days.
On rst sight, this seems inconsistent with the small time approximation above, but in fact it ts well. The argument is that the capacitors were modeled by a constant phase element.
The Cole-Cole model approaches such an element for @!A ) I and " I a H and then the capacitance is: An example in [4] is a polypropylene dielectric with H of (19) between 0.999 and 1, i.e. between 0 and 0.001, where a H is an ideal capacitor. The factor @!A requires a very large argument to be much larger than one for such a small , e.g. ! needs to be IH IHH for a H:HI in order for the factor to reach a value of say ten. Therefore, even for large frequencies, has to be very much larger than some days. The point is that even when capacitors followed the Curie-von Schweidler law for as long as several days, this indicates that the small argument approximation of (24) was valid.
The exact expression and the approximations are plotted in Fig. 1 for a H:U using numerical code from [13,14]. The two approximations t very well for small time and large time respectively.  Approximation of the relaxation function The Mittag-Leer function with a negative argument raised to a power can also be approximated [8,5]: © CC @tA $ exp @t=A @CIA ; t ( @t=A @I A ; The small time approximation is the stretched exponential or Kohlrausch-Williams-Watt function of (2). For large values of t the Mittag-Leer function approaches a power law [8].The relaxation function along with both its approximations are plotted in Fig. 2 "@!A a " I C " s " I @I C @j!A A ; H < I; H < < I; (29) where a I gives the Cole-Cole model and a I gives the Cole-Davidson model. All three models yield an ideal capacitor in parallel with the constant phase element of (25) in the limit of a large . These models are also analyzed in [5] and the main thing to note is that the Curie-von Schweidler law and the Kohlrausch-Williams-Watt function t the asymptotes of these models just as well as they t the Cole-Cole model [12]. The link between the constant phase element, these early empirical results, and the Cole Cole model is therefore not unique.

Conclusion
The familiar frequency domain expression for the Cole-Cole model of order can also be expressed in the time domain. The response function, which is related to the current response to a voltage step excitation, is expressed with a two-parameters Mittag-Leer function.
Its asymptote for small time is a power-law function