A Derivative Approach to Permittivity Extraction from Free-Space S-Parameters Measurements

The contactless characterization of dielectric materials at microwave frequencies in terms of permittivity is a long-standing need in materials science and electromagnetic engineering, and free-space techniques are attractive because they require no sample preparation and operate over broadbands. Conventional Nicolson-Ross-Weir inversion and its iterative variants either suffer from the half-wavelength phase-ambiguity problem or require a reliable initial guess to converge; this work introduces a method that avoids both limitations. The permittivity is retrieved from the frequency derivative of the transmission coefficient T, extracted from the measured S-parameters. As the formulation involves only real-valued quantities, it is intrinsically immune to phase ambiguity and needs no initial estimate. Measurements used two patch antennas separated by 15 cm and a Keysight P9375A Vector Network Analyzer, with the derivative computed through a local polynomial fit. After confirming the negligible-loss regime, the real permittivity was extracted for PMMA, Polymethyl methacrylate (PMMA, ε_r ≈ 3.31), dry sand (ε_r ≈ 3.59), and mineral oil (ε_r ≈ 1.85), all of which are consistent with the literature data.