## 3 Electromagnetic Force and Torque
### Force and Torque on Straight Segment  
The goal of this paper is to derive a set of basic equations which predict the forces and torques produced in the actuator. Three simplifications are made to obtain this model:  
* The magnetic flux density distribution of the Halbach permanent magnet array is modeled by a 2-D sine wave;
* The coils are replaced by filament ones;
* The coil and magnet arrays are rigid. The forces and torques are calculated with the [PERSON] force method. According to the [PERSON]'s law of force, the electromagnetic force and torque on the winding can be derived when the magnetic flux density and the current in the winding are known. The force on the mover is opposite to the force on the coils. The Lorentz force and torque equations are  
\[\mathbf{F}=\iiint\limits_{V}\mathbf{J}\times\mathbf{B}\mathrm{d}V \tag{23}\]  
\[\mathbf{T}=\iiint\limits_{V}\mathbf{r}\times(\mathbf{J}\times\mathbf{B})\mathrm{d}V \tag{24}\]  
where \(\mathbf{B}\) is the magnetic flux density of the magnet array, \(\mathbf{J}\) is the volume current density in the coil, \(V\) is the volume of the coil, \(\mathbf{r}\) is the vector from the point about which the torque is calculated. In the maglev planar motor with the concentric winding structure, the magnetic field generated by the Halbach permanent magnet is exponentially decayed in the vertical direction. In the mover local coordinate system, the electromagnetic force on the straight segment of the coil can be derived as  
\[{}^{c}\mathbf{F}_{l}=\iiint\limits_{V}\mathbf{\epsilon}\mathbf{J}\times{}^{c}\mathbf{B}({}^{c }\mathbf{x},\mathbf{p})\mathrm{d}V=C_{z}\iiint\limits^{c}\mathbf{J}\times{}^{c}\mathbf{B}_{xy} ({}^{c}\mathbf{x},\mathbf{p})\mathrm{d}{}^{c}x\mathrm{d}{}^{c}y \tag{25}\]  
where  
\[C_{z}=\int\limits_{c_{l}}^{c_{b}}e^{-\frac{\pi\pi}{\pi\pi}m_{z}}\ \mathrm{d}{}^{m}z=\int\limits_{c_{l}}^{c_{b}}e^{-\frac{\pi\pi}{\pi \pi}(z+p_{z})}\mathrm{d}{}^{c}z \tag{26}\]  
As shown in Figure 2, the concentric winding coil consists of the inner coil and the outer coil. Thus, the force on the whole coil can be expressed as  
\[{}^{c}\mathbf{F}_{l}={}^{c}\mathbf{F}_{lout}+{}^{c}\mathbf{F}_{lin} \tag{27}\]  
The methods to calculate the forces on the inner and outer coils are the same. Taking the outer coil as an example, the force can be calculated as  
\[{}^{c}\mathbf{F}_{lout-j}=C_{z}\begin{pmatrix}{}^{\
icefrac{{\
u_{y}+\frac{\mu_{y }}{\mu}}+\
icefrac{{\
u_{z}}}{{\
u_{z}}}}{{\
u_{z}}}} \begin{bmatrix}J\\ \int\limits_{y_{j}-\frac{\mu_{y}}{\mu}}\end{bmatrix}\begin{bmatrix}J\\ 0\\ 0\end{bmatrix} \times{}^{c}\mathbf{B}_{xy}({}^{c}\mathbf{x},\mathbf{p})\mathrm{d}{}^{c}x \mathrm{d}{}^{c}y+\int\limits_{y_{j}-\frac{\mu_{y}}{\mu}} \begin{bmatrix}J\\ \int\limits_{x_{j}-\frac{\mu_{y}}{\mu}}\end{bmatrix}\begin{bmatrix}0\\ J\\ 0\end{bmatrix} \times{}^{c}\mathbf{B}_{xy}({}^{c}\mathbf{x},\mathbf{p})\mathrm{d}{}^{c}x \mathrm{d}{}^{c}y \tag{28}\]  
where \({}^{c}\mathbf{x}_{j}\) and \({}^{c}\mathbf{y}_{j}\) are the coordinate positions for the \(j\) winding. \[{}^{c}T_{c-j}=C_{z}\lim\limits_{m,n\rightarrow+\infty}\sum\limits_{i=1}^{n}\sum \limits_{k=1}^{n}\pi_{k}m_{k}^{2}\frac{\bar{z}-\frac{1}{44 nm^{2}}}{\bar{z}-\frac {1}{44 nm^{2}}}\left[ \begin{smallmatrix}c_{x_{j}}+{}^{s}x+\frac{i_{p}}{2}\\ \frac{i_{p}}{2}\end{smallmatrix} \right]\] \[\times\left( \begin{array}{c}\sum\limits_{i=1}^{n}\left(\frac{(2k-1 )\pi}{4n}\right)\\ -\cos\left(\frac{(2k-1)\pi}{4n}\right)\\ 0\end{array} \right)\times\left[