## 3 Magnetic Circuit Design and Theoretical Modeling
### Calculation of Magnetic Permeability  
Based on the equivalent magnetic circuit model of the permanent magnet thrust bearing, the total magnetic permeability of the bearing mainly consists of the permanent magnet ring permeability and the air gap permeability, in which the permanent magnet ring permeability can be expressed according to the magnetic permeability equation [24]:  
\[\Lambda=2\mu\pi L/\ln(R_{2}/R_{1}) \tag{1}\]  
where \(R_{1}\) and \(R_{2}\) are the radii of the outer and inner rings of the permanent magnet ring, respectively, \(L\) is the axial length of the permanent magnet ring, and \(\mu\) is the permeability, the value of which can be calculated from the remanent magnetization and coercivity of the permanent magnet [25].  
For a permanent magnet bearing with an \(N\)-layer nested structure, the permeability of the \(k\)th layer of the permanent magnet ring is  
\[\Lambda_{rk}=\frac{2\mu\pi L_{n}}{\ln\left\{\frac{d+kd_{1}+(k-1)h_{g}}{d+(k-1)( d_{1}+h_{g})}\right\}} \tag{2}\]  
where \(L_{n}\) is the length of the permanent magnet ring in the axial direction, \(d\) is the radius of the center axis of the permanent magnet bearing, \(d_{1}\) is the thickness of the permanent magnet in the radial direction, and \(h_{g}\) is the air gap of the permanent magnet bearing.  
For the air gap permeability of the permanent magnet bearing, it is also necessary to consider the change of the air gap area under the action of the bearing for axial displacement. When the axial displacement \(e\) is generated, according to the principle of virtual displacement [26], the magnetic line length of the air gap between the two permanent magnet rings of the permanent magnet bearing has \(h_{g}\) change:  
\[l=\sqrt{h_{g}^{2}+e^{2}} \tag{3}\]  
The change of the length of the magnetic force line leads to the equivalent cross section \(S\) of the magnetic flux expressed as follows:  
\[S=\frac{2\pi(R+h_{g}/2)L_{n}h_{g}}{\sqrt{h_{g}^{2}+e^{2}}} \tag{4}\]  
According to the magnetic permeability formula and considering the difference of suction and repulsion between the air gaps, the \(k\)th layer suction air gap permeability  
Figure 8: Equivalent magnetic circuit model of \(N\)-layer nested \(N\)-pole crossover permanent magnet thrust bearing.  
and the \(k\)th layer repulsion air gap permeability of the permanent magnet bearing \(\Lambda_{hk}\) can be expressed as follows:  
\[\Lambda_{gk}=\frac{2\mu_{0}\pi\big{[}d+kd_{1}+(1+k/2)h_{g}\big{]}L_{n}h_{g}}{h_{ g}^{2}+e^{2}} \tag{5}\]  
\[\Lambda_{hk}=\frac{2\mu_{0}\pi\big{[}d+kd_{1}+(1+k/2)h_{g}\big{]}L_{n-1}h_{g}}{ h_{g}^{2}+e^{2}} \tag{6}\]  
Based on the above analysis, the total permeability of the permanent magnet bearing can be calculated from the superposition of the \(N\)-layer nested structure of the permanent magnet permeability and the \(N\)-1 layer air gap permeability:  
\[\begin{array}{ll}\frac{1}{\Lambda_{a}}&=\sum\limits_{k=1}^{N} \frac{1}{\Lambda_{rk}}+\sum\limits_{k=1}^{N-1}\frac{1}{\Lambda_{gk}+\Lambda_{ hk}}\\ &=\sum\limits_{k=1}^{N}\frac{\ln\bigg{\{}\frac{d+kd_{1}+(k-1)h_{g}}{d+(k-1)( d_{1}+h_{g})}\bigg{\}}}{2\mu\pi L_{n}}\\ &+\sum\limits_{k=1}^{N-1}\frac{h_{g}^{2}+e^{2}}{2\mu_{0}\pi\bigg{[}d+kd_{1}+ \bigg{(}\frac{2+k}{2}\bigg{)}h_{g}\bigg{]}(L_{n}+L_{n-1})h_{g}}\end{array} \tag{7}\]