# Effect of Slot Opening Width on the Air-Gap Magnetic Field of a Direct Drive Permanent Magnet Motor
## 2 Analytical Model  
Analytical modeling is based on the following assumptions:  
1. The stator and rotor iron cores are infinitely permeable. 2. End effects are neglected. 3. Permanent magnets have constant permeability and a linear demagnetization characteristic. 4. The rotor has no eccentricity and all the air gaps are the same. 5. Stator slots and slot openings are simplified into regular sector structures, as shown in Figure 1. In addition, the current density of the double-layer winding in the slot is uniformly distributed, as shown in Figure 2.  
Figure 1: Configuration of the the direct drive permanent magnet motor (DDPMSM). Figure 1 shows the geometric representation of the model used in studying the slot opening on the air-gap magnetic field. The geometrical parameters include the radius of the slot bottom \(R_{1}\), the radius of the slot opening \(R_{2}\), the outer radius of the stator \(R_{3}\), the inner radius of the rotor surface \(R_{4}\), and the inner radius of the rotor yoke \(R_{5}\), where \(R_{1}<R_{2}<R_{3}<R_{4}<R_{5}\). \(\theta=0\) is the initial position of the two-dimensional (2D) polar coordinates. \(\beta\) is the slot width, \(\delta\) is the slot opening width, and N denotes the number of slots. Then, the center position of the \(i\)th slot-opening can be expressed as follows:  
\[\theta_{i}=\frac{2\pi}{N}(i-1) \tag{1}\]  
To facilitate the calculation, the solution domain of the field problem of the DDPMM was divided into four types of subdomains, namely, stator slot subdomains, the stator slot-opening subdomains, the air-gap subdomain, and the rotor permanent magnet subdomains. Then, the boundary condition of each subdomain and the interface conditions between subdomains were applied. These boundary and interface conditions can be used to create the relationships between the coefficients and the sources, such as permanent magnets and currents. After determining the Fourier coefficients of the expressions in each subdomain, the general expression of the field distribution in each subdomain was obtained. Since there was current distribution in the stator slot subdomains without magnetization, the Poisson equation on the stator slot subdomains is expressed as follows:  
\[\
abla^{2}A=-\mu_{0}J \tag{2}\]  
Based on the current density \(J\) in the stator slot subdomains, the partial differential equation of the vector magnetic position can be expressed as  
\[\frac{\partial^{2}A_{1}}{\partial r^{2}}+\frac{1}{r}\frac{\partial A_{1}}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}A_{1}}{\partial\theta^{2}}=-\mu_ {0}J\ \ \left(R_{1}\leq r\leq R_{2},\ \theta_{i}-\frac{\beta}{2}\leq\theta\leq\theta_{i}+\frac{\beta}{2}\right) \tag{3}\]  
where the boundary conditions are  
\[\left\{ \begin{array}{l}\frac{\partial A_{1}}{\partial\theta}\left|{}_{ \theta=\theta_{i}-\frac{\theta}{2}}=0\\ \frac{\partial A_{1}}{\partial\theta}\left|{}_{\theta=\theta_{i}+\frac{\theta }{2}}=0\\ \frac{\partial A_{1}}{\partial\theta}\right|{}_{r=R_{1}}=0\\ \frac{\partial A_{1}}{\partial\theta}\left|{}_{r=R_{2}}=0\end{array} \right.\right. \tag{4}\]  
In the stator slot-opening subdomains, there was no current distribution and magnetization. Therefore, the Laplace equation is expressed as follows:  
\[\
abla^{2}A=0 \tag{5}\]  
Then, the partial differential equation of the vector magnetic position in the stator slots-opening subdomains can be expressed as  
\[\frac{\partial^{2}A_{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial A_{2}}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}A_{2}}{\partial\theta^{2}}=0\ \ \left(R_{2}\leq r\leq R_{3},\ \theta_{i}-\frac{\delta}{2}\leq\theta\leq\theta_{i}+\frac{\delta}{2}\right) \tag{6}\]  
Figure 2: Current density distribution of double-layer winding in the slot. where the boundary conditions are  
\[\left\{