## 2 Methods
### Analytical Model of the Normal Air-Gap Magnetic Field
#### 2.1.2 Model of the Air-Gap Magnetic Field of Permanent Magnet under No-Load  
According to the assumptions, when the stator winding is an open circuit, the vector magnetic potential of each subdomain satisfies the Laplace Equation [1]:  
\[\frac{\partial^{2}\varphi}{\partial r^{2}}+\frac{1}{r}\frac{\partial\varphi}{ \partial r}+\frac{1}{r^{2}}\frac{\partial^{2}\varphi}{\partial\alpha^{2}}=0 \tag{7}\]  
The general solution of vector magnetic potential in air-gap is  
\[\mathbf{A}_{gp}=\sum_{k=1}^{\infty}\left[\mathbf{A}_{gpk}(\frac{r}{R_{s}})^{k} +\mathbf{B}_{gpk}(\frac{r}{R_{s}})^{-k}\right]\cos(k\alpha)+\sum_{k=1}^{\infty }\left[\mathbf{C}_{gpk}(\frac{r}{R_{s}})^{k}+\mathbf{D}_{gpk}(\frac{r}{R_{s}} )^{-k}\right]\sin(k\alpha) \tag{8}\]  
where \(\mathbf{A}_{gp}\) is the vector magnetic potential of the permanent magnet with no-load, and \(\mathbf{A}_{gpk}\), \(\mathbf{B}_{gpk}\), \(\mathbf{C}_{gpk}\) and \(\mathbf{D}_{gpk}\) are the Fourier series undetermined coefficients of the air-gap magnetic field of the permanent magnet, \(R_{S}\) is the inner radius of the stator, \(k\) is the harmonic order of the air-gap magnetic field, \(\alpha\) is the circumferential angle of the air-gap space, and \(r\) is the length from the center of the radial rotor to the inner boundary of the stator. Then, the radial component and tangential component of the flux density are described by Equation (9) and Equation (10), respectively  
\[B_{r}=\frac{1}{r}\frac{\partial\mathbf{A}_{gp}}{\partial\alpha} \tag{9}\]  
\[B_{t}=-\frac{\partial\mathbf{A}_{gp}}{\partial r} \tag{10}\]  
by substituting Equation (8) into Equations (9) and (10)  
\[B_{rpg}=-\sum_{k=1}^{\infty}\frac{k}{r}\bigg{[}\mathbf{A}_{gpk}(\frac{r}{R_{s} })^{k}+\mathbf{B}_{gpk}(\frac{r}{R_{s}})^{-k}\bigg{]}\cos(k\alpha)+\sum_{k=1}^ {\infty}\frac{k}{r}\bigg{[}\mathbf{C}_{gpk}(\frac{r}{R_{s}})^{k}+\mathbf{D}_{ gpk}(\frac{r}{R_{s}})^{-k}\bigg{]}\sin(k\alpha) \tag{11}\]  
\[B_{tpg}=-\sum_{k=1}^{\infty}\frac{k}{r}\bigg{[}\mathbf{A}_{gpk}(\frac{r}{R_{s} })^{k}-\mathbf{B}_{gpk}(\frac{r}{R_{s}})^{-k}\bigg{]}\cos(k\alpha)-\sum_{k=1}^ {\infty}\frac{k}{r}\bigg{[}\mathbf{C}_{gpk}(\frac{r}{R_{s}})^{k}-\mathbf{D}_{ gpk}(\frac{r}{R_{s}})^{-k}\bigg{]}\sin(k\alpha) \tag{12}\]  
where \(B_{rpg}\) and \(B_{tpg}\) are the radial and tangential components of air-gap flux density of permanent magnet with no-load, respectively, and \(k\) is the harmonic frequency of air-gap flux density. The Fourier undetermined coefficients can be solved by the interface conditions of each subdomain of the IPMSM under no-load conditions. The interface conditions of each subdomain satisfy the conditions that: the vector magnetic potential and tangential flux density are both continuous on the interface between the stator slot and slot opening; the vector magnetic potential and tangential flux density are also continuous on the interface between the slot opening and air-gap; the scalar magnetic potential and air-gap magnetic flux are continuous on the interface between rotor and air-gap. According to the above interface conditions, the constraint Equations of the undetermined coefficients can be derived as  
\[\left[ \begin{array}{cccccccc}0&0&0&\mathbf{K}_{14}&\mathbf{K}_{15}&\mathbf{ K}_{16}&\mathbf{K}_{17}&\mathbf{K}_{18}\\ 0&\mathbf{K}_{22}&\mathbf{K}_{23}&0&0&\mathbf{K}_{26}&\mathbf{K}_{27}&0\\ 0&\mathbf{K}_{32}&\mathbf{K}_{33}&\mathbf{K}_{34}&\mathbf{K}_{35}&\mathbf{K}_{36 }&\mathbf{K}_{37}&0\\ 0&0&0&\mathbf{K}_{44}&\mathbf{K}_{45}&0&0&\mathbf{K}_{48}\\ 0&0&0&0&0&\mathbf{K}_{56}&\mathbf{K}_{57}&\mathbf{K}_{58}\\ \mathbf{K}_{61}&\mathbf{K}_{62}&\mathbf{K}_{63}&0&0&0&0&0\\ 0&\mathbf{K}_{72}&\mathbf{K}_{73}&\mathbf{K}_{74}&\mathbf{K}_{75}&0&0&0\\ \mathbf{K}_{81}&\mathbf{K}_{82}&\mathbf{K}_{83}&0&0&0&0&0\\ \end{array} \right]\left[ \begin{array}{c}\Omega_{r}\\ \mathbf{A}_{gpk}\\ \mathbf{B}_{gpk}\\ \mathbf{C}_{gpk}\\ \mathbf{D}_{gpk}\\ \mathbf{D}_{gpk}\\ \mathbf{A}_{sopi}\\ \mathbf{B}_{sopi}\\ \mathbf{A}_{spi}\end{array} \right]=\left[ \begin{array}{c}\mathbf{Y}\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}