Thin-film superconducting rings in the critical state: the mixed boundary value approach

Abstract

In this paper, we describe the critical state of a thin superconducting ring (and of a perfectly conducting ring as a limiting case) as a mixed boundary value problem. The disc is characterized by a three-part boundary division of the positive real axis, so this work is an extension of the procedure used in a previous work of ours for describing superconducting discs and strips, which are characterized by a two-part boundary division of the real axis. Here, we present the mathematical tools to solve this kind of problems—the Erdélyi–Kober operators—in a frame that can be immediately used. Contrary to the two-part problems considered in our previous work, three-part problems do not generally have analytical solutions and the numerical work takes on a significant heaviness. Nevertheless, this work is remunerated by three clear advantages: firstly, all the cases are afforded in the same way, without the necessity of any brilliant invention or ability; secondly, in the case of superconducting rings, the penetration of the magnetic field in the internal/external rims is a result of the method itself and does not have to be imposed, as it is commonly done with other methods presented in the literature; thirdly, the method can be extended to investigate even more complex cases (four-part problems). In this paper, we consider the cases of rings in uniform field and with transport current, with or without flux trapping in the hole and the case without net current, corresponding to a cut ring (washer), as used in some SQUID applications.