## 3 Mathematical Models  
Only the air gap reluctances and the magnet reluctances are considered, and other nonlinear factors are ignored. According to the magnetic circuit and the suspension mechanism, the equivalent magnetic circuits are shown in Figure 2. \(\phi_{CR1}\)-\(\phi_{CR6}\) represent control fluxes of inner air gaps, while \(\phi_{PR1}\)-\(\phi_{PR6}\) are bias fluxes of inner air gaps. \(\phi_{PRA}\)-\(\phi_{PRC}\) represents bias fluxes of outer air gaps. In Figure 2, it is assumed that the magnetomotive force (MMF) of the outer circle is 0. \(F_{m}\) and \(R_{\mathrm{m}}\) are the MMF and reluctance of each magnet, respectively. \(F_{1}\)-\(F_{4}\) are the MMF of four nodes, as shown in Figure 2b.  
Figure 1: Structure and magnetic circuit of novel AC 6-pole heteropolar HMB, 1—external stator core, 2—control windings, 3—magnets, 4—internal stator core, 5—inner air gap, 6—shaft, 7—rotor core, 8—outer air gap, 9—stator magnetic poles, 10—bias fluxes, 11—control fluxes. (**a**) Three-dimensional structure diagram; (**b**) Magnetic circuit diagram.  
Figure 2: The equivalent magnetic circuit, (**a**) The equivalent control magnetic circuit, (**b**) The equivalent bias magnetic circuit.  
Based on the magnetic circuit Kirchhoff's laws, the magnetic circuit equations are derived by node magnetomotive force method.  
\[\begin{bmatrix}Q_{1}&-\frac{1}{R_{1}}&-\frac{1}{R_{3}}&-\frac{1}{R_{5}}\\ -\frac{1}{R_{1}}&Q_{2}&0&0\\ -\frac{1}{R_{3}}&0&Q_{3}&0\\ -\frac{1}{R_{5}}&0&0&Q_{4}\end{bmatrix}\times\begin{bmatrix}F_{1}\\ F_{2}\\ F_{3}\\ F_{4}\end{bmatrix}=\begin{bmatrix}0\\ \frac{2F_{m}}{R_{m}}\\ \frac{2F_{m}}{R_{m}}\\ \end{bmatrix} \tag{1}\]  
\[\begin{cases}Q_{1}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}+\frac{1}{R_ {4}}+\frac{1}{R_{5}}+\frac{1}{R_{6}}\\ Q_{2}=\frac{1}{R_{m}}+\frac{1}{R_{1}}+\frac{1}{R_{4}}\\ Q_{3}=\frac{2}{R_{m}}+\frac{1}{R_{3}}+\frac{1}{R_{6}}\\ Q_{4}=\frac{2}{R_{m}}+\frac{1}{R_{5}}+\frac{1}{R_{C}}\end{cases} \tag{2}\]  
Reluctances of the inner and outer air gap can be expressed as  
\[R_{i}=R_{j}=\frac{g_{i}}{\mu_{0}S_{i}},j=1,2,3,4,5,6 \tag{3}\]  
\[R_{o}=R_{A}=R_{B}=R_{\text{C}}=\frac{g_{o}}{\mu_{0}S_{o}} \tag{4}\]  
where \(\mu_{0}\) is the permeability of vacuum, \(S_{i}\) is the internal magnetic pole area, \(g_{i}\) is the average inner air-gap length, \(S_{o}\) is the outer magnetic pole area, and \(g_{o}\) is the outer air-gap length.  
The bias fluxes of inner air gaps can be expressed as  
\[\begin{cases}\phi_{PR1}=\frac{F_{1}-F_{1}}{R_{1}}\\ \phi_{PR2}=\frac{F_{1}}{R_{2}}\\ \phi_{PR3}=\frac{F_{3}-F_{1}}{F_{1}}\\ \phi_{PR4}=\frac{F_{4}}{R_{4}}\\ \phi_{PR5}=\frac{F_{4}-F_{1}}{R_{1}}\\ \phi_{PR6}  
According to the equivalent magnetic circuit, then the maximum suspension forces in radial 2-DOF can be deduced as  
\[\left\{\begin{array}{l}F_{+x\text{max}}=F_{-x\text{max}}=\frac{3B_{\text{s}}^{2} S_{\text{s}}}{2\mu_{0}}\\ F_{+y\text{max}}=F_{-y\text{max}}=\frac{\sqrt{3B_{\text{s}}^{2}S_{\text{s}}}}{2 \mu_{0}}\end{array}\right. \tag{8}\]  
where \(B_{\text{s}}\) is the saturated flux density. \(B_{0}\) is bias flux density. \(F_{+x\text{max}}\), \(F_{-x\text{max}}\), \(F_{+y\text{max}}\), \(F_{-y\text{max}}\) are the maximum suspension force in +X, \(-X_{+}+Y\), and \(-Y\) directions.  
Form Equation (8), it is concluded that the maximum suspension force in the positive X and negative X directions are equal, the suspension force in the negative Y and positive Y directions is equal, and the maximum suspension force in the X direction is greater than that in the Y direction.