Highly Sensitive Microwave Permittivity Measurements of Coal–Water Mixtures Using a Trisymmetric Stepped-Impedance Resonator and Deep-Learning-Assisted Inversion

This article presents a high-performance, compact microwave sensor based on a trisymmetric stepped-impedance resonator (SIR) for the nondestructive, contactless, and real-time characterization of complex permittivity in coal-water mixtures. The sensor, with a mere 12 × 16 mm footprint, generates a sharp resonance at 3.2 GHz with a high quality factor (Q-factor) of 549 and a maximum sensitivity of 4.33% per-unit change in real part of the dielectric constant (ε′_r). A single-hidden-layer back-propagation neural network (BP-NN) is seamlessly integrated to directly map S-parameters to permittivity, achieving measurement errors below 4.52% for both ε′_r and loss tangent (tan δ) using only 220 training samples. Experimental validation on coal powders (0%-48% water content) confirms the system’s capability for industrial coal processing, and the method is readily extendable to liquids, powders, and composite materials.