# Modeling and Design of a Novel 5-DOF AC-DC Hybrid Magnetic Bearing
## 3 Mathematical Model
### Equivalent Magnetic Circuit  
Figure 3 shows an equivalent bias magnetic circuit. If the rotor is displaced in the positive X, Y, and Z directions, the magnetic permeances of each air gap can be given as  
\[\left\{ \begin{array}{l}G_{IA}=\frac{\mu_{0}S_{r}}{d_{r}-x_{i}},G_{rA}=\frac{ \mu_{0}S_{r}}{d_{r}-x_{r}}\\ G_{IB}=\frac{\mu_{0}S_{r}}{d_{r}+x_{1}+\frac{\sqrt{3}}{2}y_{r}},G_{rB}=\frac{ \mu_{0}S_{r}}{d_{r}+x_{r}-\frac{\sqrt{3}}{2}y_{r}},\\ G_{IC}=\frac{\mu_{0}S_{r}}{d_{r}+x_{1}+\frac{\sqrt{3}}{2}y_{r}},G_{rC}=\frac{ \mu_{0}S_{r}}{d_{r}+x_{2}+\frac{\sqrt{3}}{2}y_{r}}\\ G_{Iz}=\frac{\mu_{0}S_{z}}{d_{z}+z-z\mu\mu},G_{rz}=\frac{\mu_{0}S_{z}}{d_{z}-z \mu\beta}\\ \end{array} \right. \tag{1}\]  
Due to the symmetrical structure of the left and right 2-DOF MBs, the geometric dimensions are the same. The bias fluxes of permanent magnets at each air gap are written as  
\[\left\{ \begin{array}{l}\phi_{IPA}=\frac{G_{Iz}}{G_{Iz}+G_{IA}+G_{IB}+G_{IC}}F_ {m}G_{IA}\\ \phi_{IPB}=\frac{G_{Iz}}{G_{Iz}+G_{IA}+G_{IB}+G_{IC}}F_{m}G_{IB}\\ \phi_{IPC}=\frac{G_{Iz}}{G_{Iz}+G_{IA}+G_{IB}+G_{IC}}F_{m}G_{IC}\\ \end{array} \right. \tag{2}\]  
\[\left\{ \begin{array}{l}\phi_{TPA}=\frac{G_{Iz}}{G_{Iz}+G_{Iz}+G_{IB}+G_{IC}} F_{m}G_{rA}\\ \phi_{TPB}=\frac{G_{Iz}}{G_{Iz}+G_{IA}+G_{IB}+G_{IC}}F_{m}G_{rB}\\ \phi_{rpC}=\frac{G_{Iz}}{G_{Iz}+G_{IA}+G_{IB}+G_{IC}}F_{m}G_{rC}\\ \end{array} \right. \tag{3}\]  
Figure 3: Equivalent bias magnetic circuit: (**a**) left bias fluxes and (**b**) right bias fluxes. \[\left\{ \begin{matrix}\phi_{lpz}=\frac{G_{d1}+G_{d2}+G_{lC}}{G_{l1}+G_{l1}+G_{l1}+G_{ l1}+G_{l1}}F_{m}G_{l2}\\ \phi_{rpz}=\frac{G_{l1}+G_{l1}+G_{l1}+G_{l1}}{G_{r2}+G_{r1}+G_{r1}+G_{r1}+G_{r1}}F _{m}G_{rz}\end{matrix} \right. \tag{4}\]  
Figure 4 shows the equivalent control magnetic circuit. The resultant fluxes at each air gap can be expressed as  
\[\left\{ \begin{matrix}\phi_{lA}=\phi_{lpA}+\phi_{lkA}=\phi_{lpA}+Ni_{lA}G_{lA}\\ \phi_{lB}=\phi_{lpB}+\phi_{lRB}=\phi_{lPB}+Ni_{lB}G_{lB}\\ \phi_{lC}=\phi_{lpC}+\phi_{lRC}=\phi_{lpC}+Ni_{lC}G_{lC}\end{matrix} \right. \tag{5}\]  
\[\left\{ \begin{matrix}\phi_{rA}=\phi_{rpA}+\phi_{rkA}=\phi_{rpA}+N_{r}i_{rA}G_{ rA}\\ \phi_{rB}=\phi_{rpB}+\phi_{rkB}=\phi_{rpB}+N_{r}i_{rB}G_{rB}\\ \phi_{rC}=\phi_{rPC}+\phi_{rRC}=\phi_{rPC}+N_{r}i_{rC}G_{rC}\end{matrix} \right. \tag{6}\]  
\[\left\{ \begin{matrix}\phi_{z1}=\phi_{lxz}+\phi_{lpz}=N_{z}i_{z}G_{lz}+\phi_{lpz }\\ \phi_{z2}=\phi_{rHz}-\phi_{rpz}=N_{z}i_{z}G_{rz}-\phi_{rpz}\end{matrix} \right. \tag{7}\]  
For three-phase AC systems, \(i_{lA}\), \(i_{lb}\), \(i_{lC}\), \(i_{rA}\), \(i_{lB}\), and \(i_{rC}\) meet the condition of \(i_{lA}+i_{lB}+i_{lC}=0\) and \(i_{rA}+i_{lB}+i_{lC}=0\). Therefore, the suspension force in the axial and radial directions can be written as  
\[F_{z}=\frac{\phi_{z1}^{2}-\phi_{z2}^{2}}{2\mu_{0}S_{a}}\cdot\sin\theta \tag{8}\]  
\[\left\{ \begin{matrix}F_{lA}=\frac{\phi_{l2}^{2}}{2\mu_{0}S_{r}}\\ F_{lB}=\frac{\phi_{lB}^{2}}{2\mu_{0}S_{r}}\\ F_{lC}=\frac{\phi_{lC}^{2}}{2\mu_{0}S_{r}}\end{matrix} \right. \tag{9}\]  
\[\left\{ \begin{matrix}F_{rA}=\frac{\phi_{l2}^{2}}{2\mu_{0}S_{r}}\\ F_{rB}=\frac{\phi_{lB}^{2}}{2\mu_{0}S_{r}}\\ F_{rC}=\frac{\phi_{rC}^{2}}{2\mu_{0}S_{r}}\end{matrix} \right. \tag{10}\]  
The levitation force is nonlinear with the rotor offset and is also nonlinear with the current. The suspension force is linearized near the central position. The approximation can meet the engineering requirements. Therefore, the axial and radial suspension force can be further expressed as  
\[F_{z}=k_{fd}z+k_{in}i_{z} \tag{11}\]  
\[\left\{ \begin{matrix}F_{lA}=F_{lp}+k_{lir}i_{lA}\\ F_{lB}=F_{lp}+k_{lir}i_{lB}\\ F_{lC}=F_{lp}+k_{lir}i_{lC}\end{matrix} \right. \tag{12}\]  
Figure 4: Equivalent control magnetic circuit: (**a**) left radial control fluxes, (**b**) right radial control fluxes, and (**c**) axial control fluxes. \[\left\{ \begin{array}{l}F_{rA}=F_{rp}+k_{ri}i_{rA}\\ F_{B}=F_{rp}+k_{ri}i_{rB}\\ F_{rC}=F_{rp}+k_{ri}i_{rC}\end{array} \right. \tag{13}\]  
where  
\[\left\{