Stability analysis for axially-symmetric magnetic field levitation of a superconducting sphere

For a magnetically levitated superconducting sphere, the stability analysis in both axial and radial directions is analyzed theoretically using a direct boundary-value problem approach. The external magnetic field is produced by a system of current-carrying coils with axial symmetry. We also assume complete exclusion of magnetic fields from the interior of the superconductor. Use of the Pl1 associated Legendre functions as a basis set allows for a simple series solution for the magnetic stiffness coefficients as tridiagonal quadratic forms in terms of the expansion coefficients. Analysis shows that stability for this problem is not guaranteed in general. Stability is guaranteed though, if coil windings are designed in such away that all but one of the expansion coefficients are zero. Further cases with two and three non-zero coefficients are also solved. In particular the important 1, 3 and 2, 4 cases are treated in detail, and stability maps have been produced. Finally, the case where levitation is attempted by one pair of discrete current coils has been solved and stability mapping has been thoroughly explored.