Actuator gains for a toothless permanent-magnet self-bearing motor

Permanent-magnet self-bearing motors provide independent bearing and motoring functionality in a single magnetic actuator. Typically, self-bearing motor designs use toothed stators to provide minimum reluctance flux paths that create the magnetic bearing forces necessary to support the rotor. These toothed designs can have significant cogging torque, rendering them ineffective for smooth torque applications such as those found in aerospace. A toothless permanent-magnet self-bearing motor can provide smooth torque production and adequate bearing force for low-gravity environments. Characterization of the open-loop gains for this actuator is necessary for linear controller development. In this paper simple algebraic equations are derived for the motoring and bearing current gains, and an analytical method is presented for computing the negative stiffness. The analytical method solves the Dirichlet boundary value problem (BVP) in the eccentric annulus for the magnetomotive force (MMF) in the air gap subject to harmonic boundary conditions. A conformal transformation to bipolar coordinates is used, yielding a BVP that is solvable by separation of variables. Expressions for the flux density, Maxwell force on the rotor, and the negative stiffness in terms of the MMF are presented. A sample problem is presented that illustrates the flux distribution in the air gap and the operating principals of this actuator type.