While measuring the effective permittivity of dispersive material it may be of interest to distinguish between conductivity losses (caused by free electrons) and dielectric losses (caused by bounded electrons) which both are included in the imaginary part. This usually turns out to be a non-trivial task unless suitable dispersion models for the dielectric and/or the conductivity properties of the material are assumed. In this paper we present a more general method based on the Kramers-Kronig transformations to separate the conductivity from the effective complex permittivity of a dispersive material. The Kramers-Kronig transforms (or KK-transforms) are unique integral relations between the real and the imaginary part of a complex quantity describing a causal system. The proposed method and the corresponding algorithm are tested by first supposing some fictitious values of the complex permittivity satisfying the KK-transforms. Then, different values of a conductivity are added leading to a change of the imaginary part of the effective permittivity while the real part remains the same. The effective permittivity (including a conductivity part) does generally not satisfy the KK-transforms. This fact will be employed to retrieve the conductivity from that effective complex permittivity. Finally the method is applied to measured values found in the literature to retrieve the conductivity from the effective permittivity of composite material.

In electrical engineering the dispersion phenomenon is well-known as a
frequency-dependent variation of the phase velocity of electromagnetic waves
due to specific material properties. Dispersion occurs in any medium which is
not vacuum though its effect can often be neglected within a properly limited
frequency band. Dispersion is closely related to the causality principle
meaning that a cause can never happen after the corresponding effect.

In practice, the main difficulty in numerically solving the KK-transforms is
the indefiniteness of the integrals whereas the available data are given on a
limited frequency range only. Several algorithms were proposed to overcome
this problem by either calculating the data sets below the lowest available
frequency to force the system to satisfy the KK-transforms

We propose a systematic method that exploits the KK-transforms to distinguish between dielectric and conductive losses, i.e., to retrieve the conductivity from a given effective permittivity. Usually the imaginary part of the measurement of the complex permittivity includes both, dielectric losses and conductivity losses. However, the KK-transforms between the real and imaginary parts of the effective permittivity are satisfied only if the imaginary part does not contain any conductivity. The proposed method exploits that fact while two different strategies are possible: We numerically perform the KK-transform on the given (i.e. measured) real part to find find the corresponding imaginary part which is solely due to dielectric losses. The difference between that KK-transformed imaginary part and the given (measured) imaginary part must be due to the conductivity losses. A second approach is to numerically perform a KK-transform on the measured imaginary part. If it does not fit the measured real part, we conclude that there is some conductivity which can be found by solving a corresponding integral equation. To check the algorithm used in this paper, we will start from some fictitious values of the complex permittivity and of the conductivity. Afterwards, the method will be applied to real-world data (measured data found in the literature) of the effective permittivity of composite material.

The Ampere-Maxwell equation associates the magnetic field intensity

With the definition

Starting from Eq. (

Starting from Eq. (

Basically the KK transforms can be applied if the real or the imaginary part
of the complex relative permittivity is known over the entire angular
frequency range

Imaginary part

We start with a check of the self-consistency of the proposed method. The red
solid curve in Fig.

We observe that the data obtained for the imaginary part of the relative permittivity obtained after two KK-transforms relatively well fits the initial data. Even the three-times KK-transformed real part shows a reasonable agreement apart from a deviation at the end of the observation interval.

Now we add some conductivity to the previously defined fictitious material.We distinguish two cases:

frequency-independent conductivity

frequency-dependent conductivity

Imaginary part of the effective complex permittivity for different
values of the conductivity

Conductivity retrieval for a frequency-independent

Imaginary part

The situation is shown in Fig.

Now we use Eq. (

In Fig.

Conductivity retrieved by the proposed method from the measured
values of the effective permittivity of different CNT composite materials.
The measured values were found in

Finally, we applied the proposed method to investigate some measured values
of the effective permittivity which is found in the literature

Figure

We introduced a systematic method to retrieve the electric conductivity from given values of the effective permittivity using Kramers-Kronig relations. The method has been applied to fictitious values of the complex permittivity and also to real-life data for a carbon nano-tubes composite material used for absorbers. The most crucial task in this method is the numerical evaluation of the Kramers-Kronig integrals. Therefore, future work will focus on the further investigation of related strategies including to find an a-priori estimate of the corresponding error.

The data are available from the authors upon request.

The authors declare that they have no conflict of interest.

The responsibility for the content of this publication is with the authors.

This article is part of the special issue “Kleinheubacher Berichte 2017”. It is a result of the Kleinheubacher Tagung 2017, Miltenberg, Germany, 25–27 September 2017.

This work was supported by the Deutscher Akademischer Austauschdienst (DAAD) and the Egyptian Ministry of Higher Education and Scientific Research within the GERLS framework. Edited by: Thomas Eibert Reviewed by: Volkert Hansen and one anonymous referee