A Closed-Form Matrix Solution for High-Order Wave Reflection in an Open-Ended Coaxial Line for Rapid Dielectric Spectroscopy

Permittivity spectroscopy using open-ended coaxial probes for material characterization has applications in various fields, including biomedical engineering. The frequency-dependent permittivity of a material is extracted from the measured reflection coefficient through a coaxial probe. Current models that relate the reflection coefficient to the dielectric properties of the material struggle to balance accuracy and computational efficiency, limiting their utility in near real-time applications. This article introduces a novel matrix-based closed-form solution of the reflection coefficient of an open-ended coaxial probe. The approach combines full-wave analysis with a Taylor series expansion, leading to a straightforward matrix calculation. By reformulating the forward problem to decouple the material properties from the geometric properties of the probe, the required numerical integral only needs to be calculated once for each probe geometry. This significantly reduces computational time while providing similar or greater accuracy than existing methods. The model has been validated experimentally using two coaxial probes and four reference liquids, achieving an average error of 3.15%. Further validation through 9600 simulations in Ansys HFSS demonstrated an average error of 2.9%. When applied to inverse problems for estimating material permittivity, the model exhibited an average error of 4.35% while being 376 times faster than existing state-of-the-art models, with similar or enhanced accuracy. These advancements facilitate real-time, full-wave permittivity spectroscopy, offering substantial benefits for medical diagnostics and monitoring.