## 2 Analysis of the Radial Air-Gap Flux Density in a Permanent Magnet Synchronous Motor (PMSM)
### Correction Factor of Relative Permeance  
The air-gap distribution is not uniform when the rotor is eccentric with respect to the stator, and the radial air-gap flux density along the rotor surface is also not uniform. Since the relative permeance is inversely proportional to the air-gap length and the eccentricity can only affect the relative permeance [29], the radial air-gap flux density of a PMSM with eccentricity can be obtained by introducing the correction factor based on the air-gap flux density of a PMSM without eccentricity.  
The position of the stator and the rotor for a cross section in PMSM can be shown as in Figure 1, and the air-gap length can be obtained by calculation. \(C_{0}\) is the geometrical center of the stator, \(C\) is the geometrical center of the rotor under static eccentricity, and \(C_{r}\) is the geometrical center of the rotor during the rotation. The distance between \(C_{0}\) and \(C\) is \(r_{0}\), which is the static eccentricity and \(r_{0}=\sqrt{x_{0}^{2}+y_{0}^{2}}\). The distance between \(C_{r}\) and \(C\) is \(r\), which represents the dynamic eccentricity and \(r=\sqrt{x^{2}+y^{2}}\). \(\theta\) is the position angle of the dynamic eccentricity and \(\cos\theta=x/r\), \(\sin\theta=y/r\)The distance between \(C_{r}\) and \(C_{0}\) is \(e\), which represents the composite eccentricity, and \(\beta\) is the position angle at the instant time. \(R_{r}\) is the radius of the rotor and \(R_{s}\) is the radius of the stator.  
In order to calculate the air-gap length, the geometrical relationship between the stator and the rotor needs to be investigated in detail. As Figure 1 shows, A is a point on the stator surface and B is a point on the rotor surface. The composite eccentricity can be written as follows:  
\[e=\left|\mathbf{C_{0}C_{r}}\right|=\sqrt{\left(x+x_{0}\right)^{2}+\left(y+y_{ 0}\right)^{2}} \tag{6}\]  
The air-gap length can be calculated as:  
\[\delta(\alpha,t)=l_{\text{AB}}=l_{\text{AC}_{r}}-R_{r} \tag{7}\]  
Based on the cosine theorem, the length of \(\text{AC}_{r}\) can be written as:  
\[l_{\text{AC}_{r}}=\sqrt{{R_{s}}^{2}-e^{2}\sin^{2}(\alpha-\beta)}-e\cos(\alpha-\beta) \tag{8}\]  
Considering \(e^{2}\ll R_{s}^{2}\), the length of \(\text{AC}_{r}\) can be approximately expressed as:  
\[l_{\text{AC}_{r}}\approx R_{s}-e\cos(\alpha-\beta) \tag{9}\]  
Then the air-gap length can be obtained as follows:  
\[\delta(\alpha,t)=R_{s}-R_{r}-e\cos(\alpha-\beta)=\delta_{0}-e\cos(\alpha-\beta) \tag{10}\]  
where \(\delta_{0}\) is the average air-gap length when the rotor is not eccentric with respect to the stator.  
Considering the effect of the thickness of the magnet, the effective air-gap length can be expressed as [26]:  
\[\delta^{\prime}=\delta_{0}+h_{m}/\mu_{r} \tag{11}\]  
where \(h_{m}\) is the thickness of the magnet, and \(\mu_{r}\) is the relative recoil permeability of the magnet.  
According to Equations (10) and (11), the air-gap length can be written as follows:  
\[\delta(\alpha,t)=\delta^{\prime}-e\cos(\alpha-\beta) \tag{12}\]  
Figure 1: Air-gap for a cross section in a permanent magnet synchronous motor (PMSM).  
where the position angle \(\beta\) can be written as:  
\[\beta=\left\{\begin{array}{ll}\arccos(\frac{x+x_{0}}{e})&(y+y_{0}\geq 0)\\ 2\pi-\arccos(\frac{x+x_{0}}{e})&(y+y_{0}<0)\end{array}\right. \tag{13}\]  
Since the relative permeance is inversely proportional to the air-gap length, the correction factor of the relative permeance \(\varepsilon\) can be calculated as:  
\[\varepsilon=\frac{\delta^{\prime}}{\delta(\alpha,t)}=\frac{\delta^{\prime}}{ \delta^{\prime}-e\cos(\alpha-\beta)}=\frac{1}{1-e^{\prime}\cos(\alpha-\beta)} \tag{14}\]  
where \(e^{\prime}\) is the relative eccentricity, and \(e^{\prime}=e/\delta^{\prime}\).