Magnetization and AC loss in a superconductor with an elliptical cross-section and arbitrary aspect ratio

An analytical approximation is developed for the magnetization of an infinitely long superconductor with an elliptical transverse cross-section. The superconductor is modeled in the critical state with a critical current density that is not dependent on the magnetic field. The aspect ratio of the ellipse is varied from one (=circle) to infinitely large. The magnetic field is applied perpendicular or parallel to the broadest face. The analytical expression is compared with a more detailed model that utilizes a numerically optimized contour for the boundary of the saturated zone. The two methods are compared and the maximum error is estimated at 2% for the optimized contour approach and 5% for the analytical approximation. The analytical model is compared with a magnetization loss measurement on a high-Tc superconducting tape with an aspect ratio of nearly 20. A good agreement is obtained for a magnetic field pointing perpendicular as well as parallel to the broadest face of the tape. An interesting result for the magnetic behavior determined for the ellipse is that it contradicts with the behavior that is predicted for an infinitely thin strip in perpendicular field. The difference is attributed to the two specific assumptions made in the thin strip model: the constant critical current density distribution across the tape and the magnetic-field profile that does not exclude unsaturated currents in the shielded zone.