Mathematical analysis of conductive and superconductive transmission lines

We present a theoretical study of guided modes in the most commonly used transmission line: the microstrip line. It consists of a thin conductive strip placed on a dielectric substrate, which is located on a perfectly conductive ground plane. Two cases are considered: the case of a perfectly conductive strip; and the case of a superconductive one. In the model used, the dissipative losses are neglected. However, since one is looking for modes which are confined near the strip, the metallic cavity surrounding the line is not taken into account. Consequently, in both cases, the study of guided modes amounts to the spectral analysis of a family of non-compact self-adjoint operators, which is realized thanks to the min-max principle.