Stability analysis of electromagnetically levitated spherical bodies in 3D coils

Electromagnetic levitation is a well-known technique for container-less processing of molten metals and their alloys. In experiments, the levitated bodies often exhibit various kinds of instabilities. In the simplest case of static instability, a small displacement of the body from its equilibrium position can cause the failure of levitation. In some other cases the levitated bodies are observed to start to oscillate with growing amplitude or rotate with increasing rate that also can result in failure of levitation. This work presents a theoretical study of oscillatory and rotational instabilities of a solid sphere levitated electromagnetically in 3D coil systems. We have developed a combined numerical and analytical approach to analyze the static and dynamic stability of a sphere depending on the AC frequency and configuration of the magnetic system which is modeled by linear current filaments. First, we calculate numerically the magnetic vector potential in a number of points on the surface of the sphere and use Legendre and fast Fourier transforms to find the expansion of the magnetic field in spherical harmonics about the center of the body. Second, the numerically obtained expansion coefficients are substituted into the analytic solution describing the perturbation of the total electromagnetic force depending on both the displacement of the body from its equilibrium position and its velocity of motion. Thus we find the effective electromagnetic stiffness coefficients which characterize the frequency of small-amplitude oscillations of the body. Each equilibrium position is characterized by three mutually orthogonal principal directions of oscillations and three corresponding principal stiffness coefficients which all have to be positive for the equilibrium state to be statically stable. 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 friction coefficients which are usually small are found by classical perturbation theory approach. For the spin-up instability we propose a new theoretical model applicable for arbitrary field configurations. This model yields three critical AC frequencies for rotations around three mutually orthogonal principal axis found respectively as eigenvalues and eigenvectors of the effective electromagnetic friction coefficient matrix. Our approach may be useful for analysis and design of electromagnetic levitation systems.