Analytical extraction of dielectric loss and reactive circuit parameters using coplanar waveguide resonators

We present an analytical framework for extracting circuit parameters and dielectric loss tangents of a reactive load using superconducting coplanar waveguide resonators. Unlike conventional methods that rely on perturbative approximations or extensive electromagnetic simulations, our approach provides closed-form expressions for reactance and loss tangent and remains valid across a broad range of reactive load conditions. We apply our method to study the dielectric constant and loss tangent of hexagonal boron nitride, and compare the experimental results with circuit simulation. From the statistical analysis across multiple reference resonators and multimode self-calibration measurements, we show/confirm consistent and reproducible measurement of capacitance and loss tangent in close agreement with literature values. This new framework can guide us in designing resonators to maximize energy participation in the load, thereby improving the measurement accuracy.