# Effect of Slot Opening Width on the Air-Gap Magnetic Field of a Direct Drive Permanent Magnet Motor
## 1 Introduction  
Direct drive permanent magnet motors (DDPMMs) are becoming a popular key technology that can be applied in home appliances, industrial tools, and electric vehicles, given their high efficiency, high power density, and low maintenance costs. The slotting effect on the armature reaction field is important because it influences torque ripple [1], radial force distribution [2], inductance [3, 4], and copper loss [5, 6]. Therefore, the effect of slot opening width on the air-gap magnetic field of the direct drive permanent magnet motor (DDPMSM) should be investigated.  
In recent years, scholars at home and abroad have conducted numerous studies on the analytical calculation of air-gap magnetic field. In reference [7, 8], an air-gap magnetic field that considered the slotting effect was calculated by the product of the slotless air-gap magnetic field and the air-gap relative permeability function, where the relative permeability function was obtained by the conformal transformation method. However, this method cannot consider the influence of slotting on the tangential component of the air-gap flux density and has a certain error. The air-gap relative permeability method was also adopted in reference [9], in which the slot effect was considered. However, this does not study the distribution of the air-gap magnetic field when the motor operates on load. According to the boundary condition of each subdomain and the interface conditions between subdomains,reference [10] took the vector magnetic position as the variable to calculate the distribution of the air-gap magnetic field by establishing the partition subdomain model. Nevertheless, this method ignores the slot effect. In reference [11], a novel analytical model of the air-gap magnetic field was established, which considered influences such as armature reaction and slot effect. Reference [12; 13] used a combination of different analytical methods to calculate the air-gap magnetic field. The difference method was used to determine the scalar magnetic position and the vector magnetic position of each stator slot point. The calculation speed of this method is fast, but its calculation accuracy is limited by the number of iterations. Compared with these methods, the finite element method (FEM) can provide accurate air-gap field distribution regardless of the geometry and the saturation in ferromagnetic material [14]. However, the finite element analysis (FEA) requires longer computing time and a sophisticated computational environment, thereby making it computationally expensive [15].  
To analyze the air-gap magnetic field accurately and rapidly, this study combined the advantages of the analytical method and FEM. An exact analytical model was established to determine the structural parameters that affect air-gap magnetic field distribution, which refers to the magnetic field distribution generated by the permanent magnets at the air gap between the stator and the rotor. Then, the FEM was adopted to calculate the air-gap magnetic field. The analytical general solution of the air-gap magnetic field shows that the slot opening greatly affects the air-gap magnetic field distribution when the amount of the permanent magnet is constant. The FEM results verified that the slot opening width greatly affects the air-gap magnetic field.  
The remainder of the paper is organized as follows. Section 2 introduces the analytical method by deducing the expression of air-gap magnetic density. Section 3 compares the air-gap magnetic fields of the DDPMM with closed and open slots at different positions. Several finite element motor models composed of different numbers of unit motors are established in Section 4, based on different slot opening widths to investigate the effect of slot opening width on the air-gap magnetic field. Section 5 presents a brief summary of this paper.