High- Q H-Shaped Nested Split-Ring Resonator Sensor for Liquid Complex-Permittivity Characterization

We present a single-layer, via-less H-shaped nested split-ring resonator (H-NSRR) for complex-permittivity characterization of small-volume liquids. A square ring with a central H-slot co-locates the electric and magnetic fields in the same gap, improving frequency resolution and sensitivity while preserving a high unloaded Q. At around 5.9 GHz, the prototype achieves an unloaded Q of 346 in air and a normalized frequency sensitivity of approximately 1.19% per unit change in ε′r, using an open PET tray (0.5mL) positioned over the peak-|E| region to avoid microfluidic channels, vias, and multilayer processing. An equivalent-circuit model and an empirical retrieval framework map resonance-frequency shifts and Q-factor variations to the real and imaginary parts of permittivity, with retrieved values in close agreement with full-wave simulations and benchtop measurements. The proposed sensor combines high sensitivity, high unloaded Q, repeatable small-volume handling, and single-layer manufacturability, making it a practical platform for compact, application-ready microwave material characterization.