## 3 Magnetic Field Analysis and Calculation of the Eddy Current Braking Force  
### Magnetic Field Produced by the Halbach Magnet Array  
Figure 2 shows the analytical model of the PMLSM for the magnetic field analysis. The sources of the permanent magnet array are expressed in the form of a Fourier series, so that higher order harmonics can be taken into consideration. As the doubled-sided linear motor has a symmetric structure along the \(y\)-axis, only one side is selected as the analysis region. To simplify, some assumptions are made, as follows [9; 10; 11; 12]:  
1. All regions are extended infinitely in the \(x\)-direction
2. Permanent magnets are periodically distributed along the \(x\)-axis  
Figure 1: The novel ironless linear synchronous motor with cooling system. The permeability of the yoke is infinite and the permeability of the permanent magnet material is equal to the permeability of free space
4. The effects of the slots inside the cooling jacket are ignored If the transverse edge effects are temporarily ignored, i.e., the eddy currents are considered to flow only in the \(z\)-direction, the actual 3-D problem is then reduced to a 2-D one [13; 14; 15]. The analytical model is divided into three regions as shown in Figure 2. In region 1, the magnetic vector potential satisfies the Laplace equation, in region 2 and 3, the magnetic vector potential satisfies the Poisson equation. \[\left\{ \begin{array}{l}\frac{\partial^{2}A_{\rm{z1}}}{\partial x^{2}}+\frac {\partial^{2}A_{\rm{z1}}}{\partial y^{2}}=0\\ \frac{\partial^{2}A_{\rm{z2}}}{\partial x^{2}}+\frac{\partial^{2}A_{\rm{z2}}}{ \partial y^{2}}=-\mu_{0}\frac{\partial M_{\rm{x}}}{\partial x}\\ \frac{\partial^{2}A_{\rm{z2}}}{\partial x^{2}}+\frac{\partial^{2}A_{\rm{z2}}}{ \partial y^{2}}=\mu_{0}\frac{\partial M_{\rm{x}}}{\partial x}\end{array} \right. \tag{1}\]  
where \(\stackrel{{\rightharpoonup}}{{J}}_{\rm{M}}=\
abla\times\stackrel{{ \rightharpoonup}}{{M}}\), \(A_{z}\) is the \(z\)-axis component of the vector magnetic potential, \(J_{M}\) is equivalent magnetizing current density. \[\left\{ \begin{array}{l}\stackrel{{\rightharpoonup}}{{M}}=M_{ \rm{x}}\stackrel{{\rightharpoonup}}{{i}}+M_{\rm{y}}\stackrel{{ \rightharpoonup}}{{j}}\\ M_{\rm{x}}=\sum\limits_{n=1}^{\infty}\frac{4B_{\rm{r}}}{(2n-1)\pi\mu_{0}} \cos\frac{(2n-1)\pi\alpha_{\rm{p}}}{2}\sin\frac{(2n-1)\pi\alpha_{\rm{x}}}{ \tau}\\ M_{\rm{y}}=\sum\limits_{n=1}^{\infty}\frac{4B_{\rm{r}}}{(2n-1)\pi\mu_{0}}\sin \frac{(2n-1)\pi\alpha_{\rm{p}}}{2}\cos\frac{(2n-1)\pi\alpha_{\rm{r}}}{\tau} \end{array} \right. \tag{2}\]  
Equation (1) satisfies the boundary conditions (3). \[\left\{ \begin{array}{l}B_{\rm{y1}}\Big{|}_{y=\frac{1}{2}S}=B_{\rm{y2}}\Big{|} _{y=\frac{1}{2}S}\\ H_{\rm{y2}}\Big{|}_{y=\frac{1}{2}S}-H_{\rm{y1}}\Big{|}_{y=\frac{1}{2}S}=M_{\rm {x}}\\ B_{\rm{y1}}\Big{|}_{y=-\frac{1}{2}S}=B_{\rm{y2}}\Big{|}_{y=-\frac{1}{2}S}\\ H_{\rm{y2}}\Big{|}_{y=-\frac{1}{2}S}-H_{\rm{y1}}\Big{|}_{y=-\frac{1}{2}S}=-M_{ \rm{x}}\\ H_{\rm{y2}}\Big{|}_{y=\frac{1}{2}S+h_{\rm{m}}}=M_{\rm{x}}\\ H_{\rm{y2}}\Big{|}_{y=-\frac{1}{2}S-h_{\rm{m}}}=-M_{\rm{x}}\end{array} \right. \tag{3}\]  
According to the boundary condition (3), Variable Separation Method is used to solve the differential equations in the regions. The magnetic fields induced by permanent magnets can be derived as follows. The correctness of the analytical calculation of the Halbach array field distribution has been verified in our previous work [15]. \[B_{\rm{y1}}=B_{\rm{ym}}\cosh\Bigl{(}\frac{\pi}{\tau}y\Bigr{)}\cos\Bigl{(} \frac{\pi}{\tau}x\Bigr{)} \tag{4}\]  
Figure 2: The analytical model of the permanent magnet linear synchronous motors (PMLSM). \[