## 3 Analytical Model Calculation and Validation  
### Analytical Calculation  
#### 3.1.1 Establishment of Equation  
The space region in Figure 4b is divided into three parts [9], as shown in Figure 5. Region I (\(\Omega_{\rm I}\)) is the nonconductive region between the PM and the conductive plate with a coordinate range \(0<y<d\), region II (\(\Omega_{\rm II}\)) is the conductive region with a coordinate range \(-h_{2}<y<0\), and region III (\(\Omega_{\rm III}\)) is the nonconductive region below the conductive plate with a coordinate range \(y<-h_{2}\). The assumptions of this analytic model are listed below:  
1. The length \(l_{2}\) and width \(w_{2}\) of the conductive plate are large enough to be regarded as infinite, and the thickness \(h_{2}\) is finite. 2. The conductive plate is continuous with constant conductivity and is nonmagnetic. 3. The PM can move translationally along the \(x\), \(y\) and \(z\)-axes. 4. The moving speed of the PM is much less than the speed of light, which makes the frequency of the magnetic field low enough, and the system approximates a quasi-static model. Using the SOVP to calculate the model. The SOVP is expressed as \(W\) and is defined as [10]:  
\[B=\
abla\times A=\
abla\times\
abla\times W \tag{1}\]  
where \(B\) is the magnetic flux density, and \(A\) is the magnetic vector potential. \(W\) can be split into two components of transverse electric (TE) and transverse magnetic (TM) in the \(y\)-direction as follows [11]:  
\[W=\overrightarrow{y}W_{\rm a}+\overrightarrow{y}\times\
abla W_{\rm b} \tag{2}\]  
Figure 4: Structure of the suspension system. (**a**) Three-dimensional structure of the suspension system; (**b**) Two-dimensional structure of the suspension system. Figure 5: Space region division. where \(\overrightarrow{y}\) is the unit vector along the \(y\)-direction, and \(W_{\text{a}}\) and \(W_{\text{b}}\) are the TE and TM potential, respectively. Substituting Equation (2) into Equation (1) gives:  
\[ \begin{array}{l}A_{\text{x}}=-\frac{\partial W_{\text{a}}}{\partial x}-\frac{ \partial W_{\text{b}}}{\partial x\partial y}\\ A_{\text{y}}=\frac{\partial^{2}W_{\text{b}}}{\partial x}+\frac{\partial^{2}W_{ \text{b}}}{\partial z^{2}}\\ A_{\text{z}}=\frac{\partial W_{\text{a}}}{\partial x}-\frac{\partial^{2}W_{ \text{b}}}{\partial z\partial y}\end{array}  \tag{3}\]  
It is noticed that the \(A_{\text{y}}\) in Equation (3) is a function of \(W_{\text{b}}\) only because the unit vector of Equation (2) is chosen along the \(y\)-direction. Since it is assumed that the conductive plate is infinite, the eddy current generated in the conductive plate is always parallel to the upper surface of the conductive plate, and the \(A_{\text{y}}\) component is zero. Therefore, \(W_{\text{b}}\) is always zero. There is only a TE component in the SOVP; then Equation (2) can be written as:  
\[W=\overrightarrow{y}W_{\text{a}} \tag{4}\]  
In the conductive region, the governing equation for the conductive plate in terms of the magnetic vector potential satisfies Equation (5):  
\[\
abla^{2}A=\mu_{0}\sigma\bigg{(}-v_{\text{x}}\frac{\partial A}{\partial x}-v _{\text{y}}\frac{\partial A}{\partial y}-v_{\text{z}}\frac{\partial A}{ \partial z}\bigg{)} \tag{5}\]  
where \(\mu_{0}\) is the magnetic permeability of the vacuum, \(\sigma\) is the conductivity of the conductive plate and \(v_{\text{x}}\), \(v_{\text{y}}\), \(v_{\text{z}}\) are the velocity of PM in the \(x\), \(y\) and \(z\)-directions, respectively.