# Effect of Slot Opening Width on the Air-Gap Magnetic Field of a Direct Drive Permanent Magnet Motor
## 2 Analytical Model
\begin{array}{c}\frac{\partial A_{2}}{\partial\theta}\Big{|}_{\theta= \theta_{1}-\frac{\theta}{2}}=0\\ \frac{\partial A_{2}}{\partial\theta}\Big{|}_{\theta=\theta_{1}+\frac{\theta} {2}}=0\\ A_{2}\Big{|}_{r=R_{2}}=A_{1}\Big{|}_{r=R_{2}}\\ A_{2}\Big{|}_{r=R_{3}}=A_{1}\Big{|}_{r=R_{2}}\end{array} \right. \tag{7}\]  
As there is neither current density distribution nor magnetization in the air-gap subdomain, the Laplace equation is  
\[\
abla^{2}A=0 \tag{8}\]  
Thus, the partial differential equation of the vector magnetic position in the air-gap subdomain can be expressed as  
\[\frac{\partial^{2}A_{3}}{\partial r^{2}}+\frac{1}{r}\frac{\partial A_{3}}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}A_{3}}{\partial\theta^{2}}=0\ \ (R_{3}\leq r\leq R_{4},\ 0\leq \theta\leq 2\pi) \tag{9}\]  
where the boundary conditions are  
\[\left\{ \begin{array}{c}\frac{\partial A_{3}}{\partial\theta}\Big{|}_{r=R_{4 }}=\frac{1}{\mu_{r}}\frac{\partial A_{4}}{\partial r}\Big{|}_{r=R_{4}}\\ \frac{\partial A_{3}}{\partial\theta}\Big{|}_{r=R_{3}}=g(\theta)\end{array} \right. \tag{10}\]  
In the rotor permanent magnet subdomains, there is no current distribution, but the magnetization exists. Therefore, the Poisson equation in the rotor permanent magnet subdomains is expressed as follows:  
\[\
abla^{2}A=-\mu_{0}\times M \tag{11}\]  
Subsequently, the partial differential equation of the vector magnetic position in the rotor permanent magnet subdomains can be expressed as  
\[\frac{\partial^{2}A_{4}}{\partial r^{2}}+\frac{1}{r}\frac{\partial A_{4}}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}A_{4}}{\partial\theta^{2}}=- \frac{\mu_{0}}{r}\bigg{(}M_{\theta}-\frac{\partial M}{\partial\theta}\bigg{)} \ (R_{4}\leq r\leq R_{5},\ 0\leq\theta\leq 2\pi) \tag{12}\]  
where the boundary conditions are  
\[\left\{ \begin{array}{c}A_{3}\Big{|}_{r=R_{4}}=A_{4}\Big{|}_{r=R_{4}}\\ \frac{\partial A_{4}}{\partial\theta}\Big{|}_{r=R_{5}}=0\end{array} \right. \tag{13}\]  
In summary, the general formula for solving each DDPMM subdomain is expressed as follows:  
\[\frac{\partial^{2}A}{\partial r^{2}}+\frac{1}{r}\frac{\partial A}{\partial r }+\frac{1}{r^{2}}\frac{\partial^{2}A}{\partial\theta^{2}}=\left\{ \begin{array} []{c}-\mu_{0}J\ \mbox{slot subdomains}\\ 0\ \mbox{air -gap subdomain and slot opening subdomains}\\ -\frac{\mu_{0}}{r}\Big{(}M_{\theta}-\frac{\partial M}{\partial\theta} \Big{)}\ \ \ \mbox{permanent magnet subdomain}\end{array} \right. \tag{14}\]  
According to the winding configuration in Figure 2, the current density in the slot for the double-layer winding can be expressed as  
\[J=J_{00}+\sum_{m=1}^{\infty}J_{im}cos\bigg{[}\tau_{m}\bigg{(}\theta+\frac{ \beta}{2}-\theta_{i}\bigg{)}\bigg{]} \tag{15}\]  
where \(J_{i0}=\frac{I_{0i}+I_{2}}{2},J_{im}=\frac{2(I_{0i}-I_{2i})}{m\pi}\sin\bigg{(} \frac{m\pi}{2}\bigg{)}\), and \(\tau_{m}=\frac{m\pi}{\beta}\). To clarify the general solution of the equations of the different subdomains, two functions are defined as follows:  
\[X_{a}(b,c) =(\frac{b}{c})^{a}-(\frac{c}{b})^{a}, \tag{16}\] \[Y_{a}(b,c) =(\frac{b}{c})^{a}+(\frac{c}{b})^{a}. \tag{17}\]  
Then, according to the boundary and interface conditions, the vector magnetic potential of the air-gap subdomain can be given by the method of separation of variables. \[ \begin{array}{l}A_{3}(r,\theta)=\sum\limits_{n=1}^{\infty}\left[A_{n}^{3} \frac{R_{4}}{n}\frac{Y_{n}(r,R_{3})}{X_{n}(R_{4},R_{3})}+B_{n}^{3}\frac{R_{3}}{ n}\frac{Y_{n}(r,R_{4})}{X_{n}(R_{3},R_{4})}\right]\cdot\cos(n\theta)\\ \quad+\sum\limits_{n=1}^{\infty}\left[C_{n}^{3}\frac{R_{4}}{n}\frac{Y_{n}(r,R _{3})}{X_{n}(R_{4},R_{3})}+D_{n}^{3}\frac{R_{3}}{n}\frac{Y_{n}(r,R_{4})}{X_{n} (R_{3},R_{4})}\right]\cdot\sin(n\theta),\end{array}  \tag{18}\]  
where \(n\) is the harmonic orders and \(A_{n}^{3}\), \(B_{n}^{3}\), \(C_{n}^{3}\), and \(D_{n}^{3}\) are the Fourier coefficients to be determined by the boundary conditions. The problem is solved in 2D polar coordinates.