Multiple non-linear regression for microwaves characterization of dielectric materials

Abstract

In this study, we have introduced a new multivariate regression model to solve the inverse problem in microwave characterization of dielectric materials. The probe is an open-ended coaxial line, whereas the dielectric sample is taken at its end. The relative dielectric permittivity (ε = ε'-jε'') is calculated from the measurements of the probe admittance on a broad band frequency (f from 1 MHz to 1.8 GHz). In order to fit the regression coefficients of the Multiple Linear Regression (MLR) model (inverse problem), a data set is generated by solving the direct problem for the probe admittance (Y(f) = G(f) + jB(f)) using Finite Elements Method (FEM). Unfortunately, the MLR model with the original three descriptors (f, G and B) as X bloc and ε' or ε'' as response (Y bloc) has given very bad results. In order to integrate the nonlinearity between inputs and output, more descriptors have been generated mathematically from the original ones (for example: 1/f, B/f², fB, 1/B, G/f, f²G/B, fG²B, f²GB², … etc.) using the extended X bloc method and the Multiple Non-Linear Regression (MNLR) model is created. In order to choose the most relevant or statistically significant descriptors, stepwise selection is adopted. Inversion results of experimental measurements on an ethanol sample have proved that the dielectric permittivity can be measured with excellent accuracy using MNLR as inversion tool.