Measuring Reactive-Load Impedance with Transmission-Line Resonators Beyond the Perturbative Limit

Abstract

We develop an analytic framework to extract circuit parameters and loss tangent from superconducting transmission-line resonators terminated by reactive loads, extending analysis beyond the perturbative regime. The formulation yields closed-form relations between resonant frequency, participation ratio, and internal quality factor, removing the need for full-wave simulations. We validate the framework through circuit simulations, finite-element modeling, and experimental measurements of van der Waals parallel-plate capacitors, using it to extract the dielectric constant and loss tangent of hexagonal boron nitride. Statistical analysis across multiple reference resonators, together with multimode self-calibration, demonstrates consistent and reproducible extraction of both capacitance and loss tangent in close agreement with literature values. In addition to parameter extraction, the analytic relations provide practical design guidelines for maximizing energy participation ratio in the load and improving the precision of resonator-based material metrology.