## 2 Methods
### Geometric Model of the Air-Gap with Mixed Eccentricity  
The eccentricity caused in the process of motor manufacturing and assembly is an objective factor that cannot be ignored in motor design optimization and performance analysis. Similarly, both static eccentricity and dynamic eccentricity co-exist at the same time during motor operation. Thus, the accurate geometric model of the air-gap with mixed eccentricity is established in this paper. It is assumed that the boundary line of the rotor cross-section is almost an ideal circle, and the shape and position of the stator and rotor do not change in the axial direction. Moreover, the position of the minimum air-gap relative to the stator during static eccentricity remains unchanged [29], and the position of the minimum air-gap relative to the rotor during dynamic eccentricity remains unchanged too [30]. As shown in Figure 2, when there is mixed eccentricity in the motor, the air-gap length \(g(\alpha,t)\) is a function of spatial angle \(\alpha\) and time \(t\). \(R_{s}\) and \(R_{r}\) are motor stator inner diameter and rotor outer diameter, respectively, \(\delta\) is the eccentric angle, \(O_{S}\) is the center of the stator circle, \(O_{R}\) is the center of the rotor circle, and \(O_{R}\) is the center of the motion track of the rotor center. \(d_{SE}=\mid\mid O_{S}\ O_{R}\mid\mid\) is the static eccentricity distance, while \(d_{DE}=\mid\mid O_{S}\ O_{R^{\prime}}\mid\) is the dynamic eccentricity distance. We draw a straight line through point \(O_{R}\) to intersect the circle \(O_{S}\) and the circle \(O_{R}\) at points A and B, respectively. The length of line segment AB is \(g(\alpha,t)\), and the angle between the line segment \(O_{R}A\) and the polar axis is \(\alpha\), namely, the angle of the air-gap space. The angle between the line segment \(O_{R}\ O_{R^{\prime}}\) and the pole axis is \(\omega t\), namely, the rotor rotation angle. According to the trigonometric relationship, Equation (19) can be yielded:  
\[\angle O_{R}O_{R^{\prime}}O_{S}=\left\{ \begin{array}{ll}\pi-\delta+\omega t &\omega t-\delta<2k\pi\\ \pi-\omega t+\delta&2k\pi\leq\omega t-\delta<(2k+1)\pi\\ \omega t-\pi-\delta&(2k+1)\pi\leq\omega t-\delta<2(k+1)\pi\end{array} \right.\quad k =0,1,2\cdots \tag{19}\]  
Further, the length of \(O_{S}O_{R}\) can be yielded as  
\[ \begin{array}{ll}\|O_{S}O_{R}\|=\sqrt{d_{SE}{}^{2}+d_{DE}{}^{2}-2d_{SE}\cdot d _{DE}\cdot\cos{}\angle O_{R}O_{R^{\prime}}O_{S}}\\ =\sqrt{d_{SE}{}^{2}+d_{DE}{}^{2}+2d_{SE}\cdot d_{DE}\cdot\cos{}\angle(\delta- \omega t)}\end{array}  \tag{20}\]  
Therefore, in \(\Delta O_{S}O_{R}O_{R^{\prime}}\)  
\[\angle O_{S}O_{R}O_{R^{\prime}}=\arccos\frac{\|O_{S}O_{R}\|^{2}+d_{DE}{}^{2}-d _{SE}{}^{2}}{2\|O_{S}O_{R}\|\cdot d_{DE}} \tag{21}\]  
and in \(\Delta A\ O_{S}O_{R^{\prime}}\)  
\[\angle AO_{R}O_{S}=\left\{ \begin{array}{ll}\pi-\alpha+\omega t+\angle O_{S} O_{R}O_{R^{\prime}}&\omega t-\delta<2k\pi\\ \pi-\alpha+\omega t-\angle O_{S}O_{R}O_{R^{\prime}}&2k\pi\leq\omega t-\delta< (2k+1)\pi&k=0,1,2\cdots\\ \angle O_{S}O_{R}O_{R^{\prime}}-\alpha+\omega t-\pi&(2k+1\ \pi\leq\omega t-\delta< 2(k+1)\pi\\ \end{array} \right.\quad k=0,1,2\cdots \tag{22}\]  
\[R_{S}{}^{2}=\|O_{S}O_{R}\|^{2}+\|AO_{R}\|^{2}-2\|O_{S}O_{R}\|\cdot\|AO_{R}\| \cdot\cos{}\angle AO_{R}O_{S} \tag{23}\]  
According to Equation (23), it can be obtained that  
\[ \begin{array}{ll}\|AO_{R}\|=\|O_{S}O_{R}\|\cdot\cos{}\angle AO_{R}O_{S}+ \sqrt{\left(\|O_{S}O_{R}\|\cdot\cos{}\angle AO_{R}O_{S}\right)^{2}-\left(\|O_ {S}O_{R}\|^{2}-R_{S}{}^{2}\right)}\\ =\ \|O_{S}O_{R}\|\cdot\cos{}\angle AO_{R}O_{S}+\sqrt{R_{S}{}^{2}-\left(\|O_{S} O_{R}\|\cdot\sin{}\angle AO_{R}O_{S}\right)^{2}}\end{array}