Measurement Errors and Uncertainties in the Complex Permittivity Extraction With a Fabry–Perot Open Resonator

Abstract

A Fabry–Perot open resonator (FPOR) has recently become one of the most popular fixtures dedicated to the characterization of dielectric materials in the microwave and millimeter-wave range, mainly due to its quasi-broadband operation combined with large accuracy typical for resonant methods. However, to become a widely recognized measurement standard, an error budget of this approach has to be evaluated, accounting for systematic and random errors in the extraction of the dielectric constant (Dk) and loss tangent. It will be shown in this article that the main sources of the systematic and random Dk errors are the choice of the electromagnetic (EM) model and variation of the thickness of the dielectric sheet, respectively. The first one can be controlled to some extent by the developer of the measurement system, however, thickness variation cannot be easily controlled. Therefore, it has to be accounted for in the form of uncertainty bars. In the case of the loss tangent, systematic and random errors are mainly due to the electric energy filling factor error and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> -factor variation, respectively. Moreover, it is shown in this article that the loss tangent uncertainty (LTU) is the lowest when the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> -factor with the sample is 40%–50% of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> -factor of the empty cavity. In addition to that, it is shown that the noise floor of the measurement is much less important when the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> -factor is estimated with the aid of curve-fitting algorithms instead of the 3-dB-point method. However, complete knowledge on the LTU cannot be acquired from just one measurement as it does not account for the finite repeatability of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> -factor measurement. It is, therefore, proposed in this article how to practically evaluate and suppress this error. Although the main attention has been focused on the FPOR, many conclusions presented in this study are not limited to a particular type of resonant technique.