Advances in Materials Science and Engineering

Volume 2019, Article ID 3646979, 7 pages

https://doi.org/10.1155/2019/3646979

## Electromagnetic Characteristics Measurement Setup at Variable Temperatures Using a Coaxial Cell

^{1}Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France^{2}DGA, Balard, Paris, France

Correspondence should be addressed to Thibaut Letertre; rf.lenserf@ertretel.tuabiht

Received 17 January 2019; Revised 25 March 2019; Accepted 8 April 2019; Published 8 May 2019

Academic Editor: Matjaz Valant

Copyright © 2019 Thibaut Letertre et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This article presents an easy way to measure frequency dependence of the complex permittivity and complex permeability for any kind of material from 10 MHz to 6 GHz, with temperature variation between the ambient temperature and 85°C. This work is based on a well-known transmission/reflection technique using a coaxial line equipped with a thermoregulation system to manage temperature variations in the sample confinement area. The paper underlines some effects which have to be taken into account with temperature variation. APU10 and cyclohexanol are presented as examples of solid and liquid reference materials.

#### 1. Introduction

Measuring the electromagnetic properties of materials is widely studied nowadays. In the last twenty years, many useful measurement methods in microwave range [1] have been developed for specific applications. Waveguide [2], as the transmission/reflection method, resonators [3, 4], or free space [5] are the most known techniques. These devices are designed to operate at room temperature over a specific frequency range. In many cases, it becomes important to determine the dependence of dielectric quantities as a function of temperature, up to temperatures that can be very high (several hundred degrees in the case of aircraft materials located near aircraft engines). In this context, a new setup, using a coaxial line [6] equipped with a wideband thermoregulated PID system to manage temperature variations in the sample area is proposed. In the literature, a few setups have been proposed to study the temperature behaviour of materials, but mainly using the resonator [4, 7, 8] or free space methods to reach high temperature [9]. Nonetheless, in these cases, it is not possible to measure on a wideband of frequency. A few studies of waveguided setups in temperature have been done [10], but these techniques are often complex and bulky. That is why we decided to propose an easy-to-use solution to measure the permittivity and permeability of any kind of material based on a coaxial line [2], with temperature variation between ambient and 85°C. This paper is divided into two parts; the first part is focused on the frequency measurement method, followed by the temperature variation method. The second part presents the results obtained on a few different materials. The first reference material is a solid magnetic sample, a silicon-metal composite. As a second reference material, cyclohexanol, which is a dielectric alcohol well known in the literature [11], is used.

#### 2. Measurement Method

In this section, the different processes to determine the frequency and temperature dependence of the electromagnetic characteristics of materials are detailed. We start with a brief definition of the classical mathematical equations and of an uncertainty estimation for the frequency dependence of the complex permittivity and the complex permeability. Afterwards, we propose a process to accurately manage the temperature in the sample to obtain the permittivity and permeability behaviours.

##### 2.1. Frequency Measurement

###### 2.1.1. Mathematical Equations

The permittivity and permeability are defined as complex values:where and correspond to the relative permittivity and relative permeability. On the other hand, and correspond to the vacuum permittivity and vacuum permeability, respectively.

The setup is composed by a vector network analyser (Anritsu MS2038 C) and a tapered coaxial line. The coaxial line is divided into three parts.

The two tapered parts were designed to convert the 7 mm diameter line of the coaxial connector, into a 13 mm line, to significantly increase the sample volume. Furthermore, assuming the conical part as lines, it is possible to use a de-embedding method [2] to obtain the *S*-parameters of the sample.

The relation between the *S*-parameters and the transmission parameter and the reflection parameter of a sample arewhere and can be expressed as

With . Next, the refraction index and the line impedance are calculated:

can be expressed as

Finally, with equation (1), permittivity and permeability are obtained:

This method is useful to directly obtain the permittivity and permeability without any assumption on the material properties. Hence, the study of any kind of material (liquid, solid, powder, etc.) is possible.

###### 2.1.2. Uncertainty Estimation

In transmission/reflection method, we expect different sources of errors:(i)Errors in measuring *S*-parameter(ii)Gaps between sample and sample holder(iii)Uncertainties in the length of the sample(iv)Uncertainties in the plane reference position

In our case, *S*-parameter measurement uncertainties have been obtained in the datasheet manual of the vector network analyser (Anritsu MS2038C). Those errors only depend on the calibration. Samples have been made carefully in order to fit perfectly to the sample holder. In this way, we do not care about gaps in this uncertainties model. While the *S*-parameter errors can be compared as offset error in our measurement, the length’s uncertainties and the reference position uncertainties are difficult to determine. The reference positions are presented in this paper with the de-embedding method. This method transforms the *S*-parameters of the cell into the *S*-parameters of the sample because the sample is locked up in the middle of the cell. We define 2 errors for 2 different lengths ( and ) of the conical parts (Figure 1) and . Furthermore, to obtain the permittivity and the permeability of the sample, its length is needed. Because of this, we have to consider the error of the length of the sample . In order to estimate the uncertainties of the *S*-parameter of the sample considering the error of the *S*-parameter, , , and , we derive the equations of the de-embedding method [2]. We assume that the total error of the de-embedded *S*-parameter can be written aswhere , , , and correspond to the uncertainties of the magnitude and the phase of the *S*-parameter of the cell. , , , and correspond to the uncertainties of the magnitude and the phase of the *S*-parameter of the sample. Finally, we estimate the error of the permittivity and the permeability as a function of the error of the de-embedded *S*-parameter and the error of the sample’s length [12]:where *α* = 11 or 21 corresponding to the reflection and the transmission parameter. Among all these error sources, we studied especially the error of the sample’s length which is measured manually with a caliper. Considering the mechanical nature (rigidity) of the material to characterize, the accuracy of the thickness determination using a caliper can then be obtained between 0.5 and 0.1 mm. We compared the two of them to show the impact on the general error (Figure 2).