A technique for modeling a superconductor contacted piezoelectric microwave resonator is presented. A transmission line method of modeling the superconductor as a two fluid model is shown to agree with the London penetration depth, and is modeled to take into account the superconductor to normal transition. By including both the normal conductor effects along with the superconductor effects, crystal quality can be factored into this model using an analytical approximation. A two dimensional model for the superconductor, applying experimental results from Y1Ba2Cu3O7-x for circuit elements, is analyzed. The superconducting layer is modeled as an electrical circuit in combination with an acoustical circuit to model its effect on a piezoelectric resonator. The acoustical equivalent of the superconductor including mass loading effects and terminations are taken into account in the Mason model acoustical equivalent circuit.
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Goff, E.D., Weber, R.J. A Modified Transmission Line Approach to The Analysis of Superconducting Microwave Resonators. MRS Online Proceedings Library 360, 323–328 (1994). https://doi.org/10.1557/PROC-360-323