Modeling the Static Vertical Force of the Core-Type Permanent-Magnet Planar Motor

The vertical electromagnetic force affects the design of the air bearing or magnetic bearing of a synchronous permanent-magnet planar motor (SPMPM), whereas the static vertical electromagnetic force is one of the important components of the vertical electromagnetic force. This paper discusses the problem of modeling the static vertical force of the SPMPM analytically. First, we obtain an analytical model of the scalar potential and the equations for calculating the magnetic field energies (MFE) in the magnet array and in the air gap. Since the MFE in the yokes is small enough to be neglected, we obtain the overall MFE equation by summing both the air-gap MFE equation and the magnet array MFE equation. Second, using the virtual work principle, we establish an analytical model of the static vertical force. Using this analytical model, we discuss the characteristics of the static vertical force. To verify the usability of the vertical force model, we analyze a magnet array by the analytical method proposed in this paper and by the finite-element method (FEM). The results show that the vertical force model proposed in this paper is approximately equal to the result from the FEM. This indicates that the vertical force model can be used for estimating the value of the static vertical force in a core-type SPMPM. Furthermore, the analytical model produces continuous results in shorter calculation time.