Application of Schwarz-Christoffel Mapping to Permanent-Magnet Linear Motor Analysis

Several well-known analytical techniques exist for the force profile analysis of permanent-magnet linear synchronous motors. These techniques, however, make significant simplifications in order to obtain the magnetic field distribution in the air gap. From the field distribution, the force profile can be found. These widely used techniques provide a reasonable approximation for force profile analysis, but fail to give really accurate results in the sense of the exact shape of the force profile caused by effects that due to simplification are not fully included. To obtain the exact shape for the force profile in these cases, the computationally expensive finite-element method (FEM) is often applied. In this paper, an elegant semianalytical approach is presented to acquire the force profile. First, the magnetic field distribution in the air gap is determined by means of Schwarz-Christoffel (SC) mapping. The SC mapping allows a slotted structure of the machine to be mapped to a geometrically simpler domain for which analytic solutions are available. Subsequently, the field solution in the slotted structure can be determined by applying the mapping function to the field distribution in the simplified domain. From the resulting field distribution, the force profile is calculated by means of the Maxwell stress tensor. The results are compared with those from the commonly used equivalent magnetic circuit modeling and 2-D FEM software to demonstrate the accuracy which can be reached by application of the SC method.