Stability analysis of an electromagnetically levitated sphere

We present a combined numerical and analytical approach to analyze the static and dynamic stability of an electromagnetically levitated spherical body depending on the ac frequency and the configuration of a three-dimensional (3-D) coil made of thin winding which is modelled by linear current filaments. First, we calculate numerically the magnetic vector potential in grid points on the surface of the sphere and then use Legendre and fast Fourier transforms to find the expansion of the magnetic field in terms of spherical harmonics. Second, we employ a previously developed gauge transformation to solve analytically the 3-D electromagnetic problem in terms of the numerically obtained expansion coefficients. Using this solution, we obtain the electromagnetic reaction force due to both a small displacement of the sphere from its equilibrium position and its velocity of motion which are defined by symmetric stiffness and damping matrices, respectively. Eigenvalues and eigenvectors of the stiffness matrix yield three principal stiffness coefficients, which all have to be positive for the equilibrium state to be statically stable, and three mutually orthogonal directions of principal oscillations. Dynamic instabilities are characterized by critical ac frequencies which, when exceeded, may result either in a spin-up or oscillations with increasing amplitude. The effective electromagnetic damping coefficients are found by using a classical eigenvalue perturbation theory. A new theoretical approach based on the vector field transformation by a small rotation in combination with a parametric frequency derivative is introduced to find the electromagnetic reaction torque due to a slow rotation of the sphere in a 3-D ac magnetic field.