Modeling linear and nonlinear inductive effects in superconducting planar circuits

Superconducting planar circuits possess properties that can be very different from their normal-metal counterparts, which makes modeling efforts more complicated for superconducting components. In this paper we present closed-form expressions for the resistance and internal inductance per unit length of superconducting planar transmission lines in terms of the relevant material parameters. We also include a simple calculation of the nonlinear response of superconducting devices based on a current-dependent superconducting penetration depth. Taken together, these two descriptions can be used to predict the linear and nonlinear response of superconducting planar devices in a manner that is conducive to use in circuit design. The results of these calculations compare well with 76 K measurements of coplanar waveguide transmission lines and resonators fabricated from thin films of the high T_c superconductor YBa₂Cu ₃O_7-δ