## 3 Magnetic Circuit Design and Theoretical Modeling
### Calculation of Bearing Capacity of Permanent Magnet Thrust Bearing  
In order to calculate the stiffness characteristics of the permanent magnet bearing, the first step is to determine the bearing capacity index under the action of axial displacement. In the operation of a permanent magnet bearing, an axial load is applied when a small axial displacement, denoted as \(e\), occurs. At this point, the attractive and repulsive forces in the air gap between the permanent magnets hinder the axial movement of the rotor. The force that resists this displacement is referred to as the bearing capacity of the permanent magnet bearing. For permanent magnet thrust bearings, the permanent magnet material is typically NdFeB (neodymium iron boron), which exhibits high magnetic strength. The air gap load line and demagnetization line for the permanent magnet material are depicted in Figure 9. The working point of the permanent magnet, denoted as \(P\), represents the point on the air gap load line at which the bearing operates under the applied axial displacement. The relationship between these forces governs the bearing's load-carrying capacity and stiffness characteristics. Understanding the interaction between the axial displacement and the magnetic forces in the air gap is crucial for calculating the bearing capacity and stiffness of the permanent magnet thrust bearing. This enables engineers to optimize the design and performance of the bearing in various applications. Figure 9: Air gap loading lines and demagnetization lines of permanent magnetic materials. According to the principle of the magnetic circuit of a permanent magnet [27], the magnetic flux density at the working point of a permanent magnet can be expressed as follows:  
\[B_{m}=\frac{B_{r}H_{c}d_{1}\Lambda}{B_{r}S+H_{c}d_{1}\Lambda} \tag{8}\]  
where \(B_{r}\) is the remanent magnetic induction of the permanent magnet, \(H_{c}\) is the coercive force of the permanent magnet, \(d_{1}\) is the thickness of the permanent magnet in the radial direction, and \(S\) is the cross-sectional area of the magnetic circuit of the magnetic ring. According to the flux continuity principle [28], the radial flux of the \(k\)-layer gap of the permanent magnet thrust bearing is  
\[\Phi_{gk}=\frac{2\pi R_{k}nlB_{r}H_{c}d_{1}\Lambda}{2\pi R_{k}nlB_{r}+H_{c}d_{ 1}\Lambda} \tag{9}\]  
where \(R_{k}\) is the equivalent radius of the \(k\)th layer permanent magnet ring. The gap magnetic energy of the \(k\)th layer permanent magnet ring can be obtained by a flux and permeability calculation:  
\[W_{g}=\Phi_{g}^{2}/2\Lambda \tag{10}\]  
Based on the principle of the virtual work of the permanent magnet bearing in the working process [29], the axial magnetic force generated by the permanent magnet ring can be expressed as follows:  
\[F_{z}=\frac{\partial W_{g}}{\partial e}=-\frac{\Phi_{S}^{2}}{2\Lambda_{S}^{2} }\times\frac{\partial\Lambda_{g}}{\partial e} \tag{11}\]  
Based on the above analysis, the axial magnetic force generated by the suction of the \(k\)th ring gap is  
\[F_{gk}=\frac{\left(\frac{2\pi RnlB_{r}H_{c}d_{1}\Lambda_{a}}{2\pi RnlB_{r}+H_{ c}d_{1}\Lambda_{a}}\right)^{2}e^{2}}{2\mu_{0}\pi\left[d+kd_{1}+(1+k/2)h_{g} \right]L_{n}h_{g}} \tag{12}\]  
The axial magnetic force generated by the repulsive of the kth ring gap is  
\[F_{hk}=\frac{\left(\frac{2\pi RnlB_{r}H_{c}d_{1}\Lambda_{a}}{2\pi RnlB_{r}+H_{ c}d_{1}\Lambda_{a}}\right)^{2}e^{2}}{2\mu_{0}\pi\left[d+kd_{1}+(1+k/2)h_{g} \right]L_{n-1}h_{g}} \tag{13}\]  
The combined force in the axial direction of the permanent magnet thrust bearing is  
\[