Reference-Plane-Invariant and Unique Permittivity Determination of Thin Samples Positioned on a Substrate Using One-Port Waveguide Measurements

A microwave extraction method is proposed for relative complex permittivity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon _{\text {sam}} = \varepsilon _{\text {sam}}^{\prime } - j \varepsilon _{\text {sam}}^{\prime \prime }$</tex-math> </inline-formula> determination of thin low- or medium-loss solid samples using reference-plane-invariant (RPI) expressions. Different from other reflection-only methods (one-layer structure), it considers a model based on a thin sample positioned on a substrate material (two-layer structure) and relies on reflection-only matched and short-circuited scattering (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i>)-parameter measurements of direct and reversed configurations of this two-layer configuration. The method has the capability of evaluating one solution for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon _{\text {sam}}$</tex-math> </inline-formula> by using two derived objective functions while additionally providing an approximate and sufficiently accurate solution for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon _{\text {sam}}^{\prime }$</tex-math> </inline-formula>. After its validation followed by an uncertainty analysis, the method is tested for different scenarios, and its accuracy is compared with other methods in the literature using X-band (8.2–12.4 GHz) waveguide <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i>-parameter measurements.