Fault tolerance of magnetic bearings with material path reluctances and fringing factors

An equivalent magnetic circuit of an eight-pole heteropolar magnetic bearing with path reluctances is developed with nondimensional forms of flux, flux density, and magnetic force equations. The results show that fluxes and magnetic forces are considerably reduced for the magnetic circuit with relatively large path reluctances. A Lagrange multiplier optimization method is used to determine current distribution matrices for the magnetic bearing with large path reluctances. A cost function is defined in a manner that represents load capacity in a specific direction. Optimizing this cost function yields distribution matrices calculated for certain combinations of five poles failed out of eight poles. Position stiffnesses and voltage stiffnesses are calculated for the fault-tolerant magnetic bearings. Fault-tolerant control of a horizontal rigid rotor supported on multiple-coil failed magnetic bearings including large path reluctances is simulated to investigate the effect of path reluctances on imbalance response.