## 3 Electromagnetic Force and Torque
### Accurate Real-Time Equation of Electromagnetic Force and Torque  
According to the calculation results of straight and corner segments, the electromagnetic force and torque on the concentric winding can be derived as  
\[{}^{c}F_{xj}=C_{z}JB_{z}\sin(\frac{\pi}{\tau_{n}}\left({}^{c}x_{j}+p_{x}\right) )(C_{1}+C_{x}) \tag{36}\]  
\[{}^{c}F_{yj}=-C_{z}JB_{z}\sin(\frac{\pi}{\tau_{n}}\left({}^{c}y_{j}+p_{y} \right))\left(C_{1}+C_{y}\right) \tag{37}\]  
\[{}^{c}F_{zj}=\sqrt{2}C_{z}JB_{xy}\biggl{(}\cos\left(\frac{\pi}{\tau_{n}}\left( {}^{c}x_{j}+p_{x}\right)\right)(C_{1}+C_{x})-\cos\left(\frac{\pi}{\tau_{n}} \left({}^{c}y_{j}+p_{y}\right)\right)(C_{1}+C_{y})\biggr{)} \tag{38}\]  
\[{}^{c}T_{xj}={}^{c}y_{j}{}^{c}F_{zj}-{}^{c}r_{z}{}^{c}F_{yj}+\sqrt{2}C_{z}JB_{ xy}\sin\left(\frac{\pi}{\tau_{n}}\left({}^{c}y_{j}+p_{y}\right)\right)\left(C_{2}+ \frac{\tau_{n}}{\pi}C_{1}+C_{y1}\right) \tag{39}\]  
\[{}^{c}T_{yj}={}^{c}r_{z}{}^{c}F_{xj}-{}^{c}x_{j}{}^{c}F_{zj}+\sqrt{2}C_{z}JB_{ xy}\sin\left(\frac{\pi}{\tau_{n}}\left({}^{c}x_{j}+p_{x}\right)\right)\left(C_{2}+ \frac{\tau_{n}}{\pi}C_{1}+C_{x1}\right) \tag{40}\]  
\[{}^{c}T_{zj}={}^{c}F_{yj}{}^{c}x_{j}-{}^{c}F_{xj}{}^{c}y_{j} \tag{41}\]  
where  
\[C_{1}=\frac{\pi}{\pi}\left(l_{w}\left(\cos\left(\frac{\pi}{\tau_{n}}\left( \frac{l_{w}}{2}+w_{c}\right)\right)-\cos\left(\frac{\pi}{\tau_{n}}\frac{l_{w}}{2} \right)\right)-l_{w}\left(\cos\left(\frac{\pi}{\tau_{n}}\left(\frac{l_{w}}{2}+w _{c}\right)\right)-\cos\left(\frac{\pi}{\tau_{n}}\frac{l_{w}}{2}\right)\right)\right) \tag{42}\]  
Figure 4: Corner segment model. (**a**) Top view of the division of the corner segment; (**b**) Left view of the division of the corner segment. The corner segment is divided into \(n\) fan-shaped sectors and \(m\) sectors in the direction of the axis of \({}^{s}x\). \[C_{2} =\frac{\tau_{n}}{\pi}\left(l_{w}\left(\frac{l_{w}}{2}+w_{c}\right) \sin\left(\frac{\pi}{\tau_{n}}\left(\frac{l_{w}}{2}+w_{c}\right)\right)\ -\left(\frac{l_{w}}{2}\right)\sin\left(\frac{\pi}{\tau_{n}}\frac{l_{w}}{2} \right)\right) \tag{43}\] \[-\frac{\tau_{n}}{\pi}\left(l_{n}\left(\frac{l_{w}}{2}+w_{c}\right) \sin\left(\frac{\pi}{\tau_{n}}\left(\frac{l_{w}}{2}+w_{c}\right)\right)-\left( \frac{l_{w}}{2}\right)\sin\left(\frac{\pi}{\tau_{n}}\frac{l_{w}}{2}\right)\right)\] \[{}^{c}F_{yj}=-C_{z}JB_{z}\sin(\frac{\pi}{\tau_{n}}\left({}^{c}y_{j}+p_{y} \right))\left(C_{1}+C_{y}\right) \tag{37}\]0 (44) \[{}^{c}F_{yj}=-C_{z}JB_{z}\sin(\frac{\pi}{\tau_{n}}\left({}^{c}y_{j}+p_{y} \right))\left(C_{1}+C_{y}\right) \tag{37}\]1 (45) \[{}^{c}F_{yj}=-C_{z}JB_{z}\sin(\frac{\pi}{\tau_{n}}\left({}^{c}y_{j}+p_{y} \right))\left(C_{1}+C_{y}\right) \tag{37}\]2 (46) \[{}^{c}F_{yj}=-C_{z}JB_{z}\sin(\frac{\pi}{\tau_{n}}\left({}^{c}y_{j}+p_{y} \right))\left(C_{1}+C_{y}\right) \tag{37}\]3 \[{}^{c}F_{yj}=-C_{z}JB_{z}\sin(\frac{\pi}{\tau_{n}}\left({}^{c}y_{j}+p_{y} \right))\left(C_{1}+C_{y}\right) \tag{37}\]4 (47) \[{}^{c}F_{yj}=-C_{z}JB_{z}\sin(\frac{\pi}{\tau_{n}}\left({}^{c}y_{j}+p_{y} \right))\left(C_{1}+C_{y}\right) \tag{37}\]5  
The electromagnetic model of the whole winding can be derived when the equations for both the straight and corner segment are added together. Once the permanent magnet array and winding size are determined, the numerical value of \(C_{1}\), \(C_{2}\), \(C_{x}\), \(C_{y}\), \(C_{x1}\), and \(C_{y1}\) can be derived. Once the winding is put on the mover, the position deviation on the \({}^{c}\)_x-\({}^{c}\)_y_ plane can be seen as a constant, therefore, the whole electromagnetic force and torque on the mover is the sum result of that on each winding, taking the position deviation into account.