Force characteristics and gain determination for a slotless self-bearing motor

In this paper, we use the Maxwell stress tensor method to derive the analytical force and torque expressions for a large-scale slotless permanent-magnet (PM) self-bearing motor actuator that uses common coils to produce both the torque and radial forces, in order to include all the possible interactions among the PMs, currents, and stator/rotor back iron. We solve the radial and angular components of the two-dimensional instantaneous magnetic field distribution in the low-permeability region induced from the rotor PMs and stator windings separately, using the unwrapped geometry, and then superimpose them. Instead of being incorporated into the boundary conditions, the general winding current and PM magnetization distributions are expanded into the Fourier series in the separate source layers, with respect to one motor revolution and one PM pole pair, respectively. For each source, we first solve one homogenous Laplace´s equation in the planar layer without source and one nonhomogenous Laplace´s equation in the planar layer with source simultaneously in Cartesian coordinates for a centered rotor and then add the rotor eccentricity. We compare the analytical solutions for the individual magnetic field distributions, as well as the total force and torque production, to those from the finite-element analysis (FEA), and find excellent agreement between the two. We characterize the open loop current and negative stiffness gains of the SBM from the linearized force-current-displacement relationship, which forms the basis for the linear system modeling and controller design, and validate our results by comparison with the FEA results.