## 3 Mathematical Modeling of the Six-Phase DSW-12/2-BPMSM
### Magnetic Levitation Force Model of the Six-Phase DSW-12/2-BPMSM  
Setting \(I_{xy}=[i_{1d},\,i_{1q},\,i_{2j},\,i_{2k}]\), based on the electromechanical energy conversion principle, the magnetic field energy stored in the motor inductance can be expressed as follows:  
\[W_{m}=I_{xy}^{T}L_{xy}I_{xy}/2 \tag{26}\]  
Based on the virtual displacement principle, the radial electromagnetic force acting on the rotor can be expressed as the partial derivative of the electromagnetic energy storage with respect to the radial displacement. The radial electromagnetic force consists of a controllable magnetic levitation force and an uncontrollable unilateral magnetic pull force. In the \(x\)-\(y\) coordinate system, the expressions of the radial electromagnetic force components \(F_{x}\) and \(F_{y}\) are as follows:  
\[F_{x}=M_{12}(i_{2d}i_{f}+i_{1q}i_{2q})+\frac{x}{4g_{0}^{2}}\Big{[}L_{1}(i_{f}^ {2}+i_{1q}^{2})+2L_{2}(i_{2j}^{2}+i_{2k}^{2})\Big{]} \tag{27}\]  
\[F_{y}=M_{12}(i_{2d}i_{f}-i_{1q}i_{2d})+\frac{y}{4g_{0}^{2}}\Big{[}L_{1}(i_{f}^ {2}+i_{1q}^{2})+2L_{2}(i_{2j}^{2}+i_{2k}^{2})\Big{]} \tag{28}\]  
The first parts in Equations (27) and (28) are the controllable magnetic levitation force components \(F_{ix}\) and \(F_{iy}\), which can be written in matrix form as follows:  
\[ \begin{bmatrix}F_{ix}\\ F_{iy}\end{bmatrix} =M_{12} \begin{bmatrix}i_{f}&i_{1q}\\ -i_{1q}&i_{f}\end{bmatrix}\begin{bmatrix}i_{2j}\\ i_{2k}\end{bmatrix}  \tag{29}\]  
For the six-phase DSW-12/2-BPMSM in this work, the torque system and levitation system have two poles and four poles, respectively. Then, in the electrical space, if the \(d\)-\(q\) torque plane and the \(j\)-\(k\) levitation plane jointly correspond to the \(x\)-\(y\) mechanical synchronous coordinate system of the rotor, the rotational speed of the \(j\)-\(k\) levitation plane (or coordinate system) is twice that of the \(d\)-\(q\) torque plane (or coordinate system) [5]. Thus, if the levitation current in the \(j\)-\(k\) plane is transformed to the \(d\)-\(q\) plane, the following coordinate transformation relations should apply:  
\[ \begin{bmatrix}i_{2d}\\ i_{2d}\end{bmatrix} = \begin{bmatrix}\cos(\omega t)&-\sin(\omega t)\\ \sin(\omega t)&\cos(\omega t)\end{bmatrix}\begin{bmatrix}i_{2j}\\ i_{2k}\end{bmatrix}  \tag{30}\]  
From Equations (29) and (30), the following equation can be obtained by coordinate transformation:  
\[ \begin{bmatrix}F_{ix}\\ F_{i\beta}\end{bmatrix} = \begin{bmatrix}\cos(\omega t)&-\sin(\omega t)\\ \sin(\omega t)&\cos(\omega t)\end{bmatrix}\begin{bmatrix}F_{ix}\\ F_{iy}\end{bmatrix} =M_{12} \begin{bmatrix}i_{f}&i_{1q}\\ -i_{1q}&i_{f}\end{bmatrix}\begin{bmatrix}i_{2d}\\ i_{2q}\end{bmatrix}  \tag{31}\]  
In Equations (30) and (31), \(F_{ix}\) and \(F_{i\beta}\) are the controllable magnetic levitation force components in the stationary \(\alpha\)-\(\beta\) coordinate system, and \(i_{2d}\) and \(i_{2q}\) are the \(d\)-\(q\) axis components of the levitation current transformed to the torque plane (or coordinate system). The last two parts in Equations (27) and (28) represent the uncontrollable eccentric magnetic pulling force components, whose expressions are as follows:\[ \begin{cases}F_{ex}=\frac{x}{4g_{0}^{2}}\bigg{[}L_{1}(\hat{t}_{f}^{2}+\hat{t}_{1q}^{ 2})+2L_{2}(\hat{t}_{2j}^{2}+\hat{t}_{2k}^{2})\bigg{]}\\ F_{ey}=\frac{y}{4g_{0}^{2}}\bigg{[}L_{1}(\hat{t}_{f}^{2}+\hat{t}_{1q}^{2})+2L_{ 2}(\hat{t}_{2j}^{2}+\hat{t}_{2k}^{2})\bigg{]}\end{cases}  \tag{32}\]  
where \(F_{ex}\) and \(F_{ey}\) are the eccentric magnetic pull force components in the \(x\)-\(y\) coordinate system. Based on Equation (32), the eccentric magnetic pull force along a certain axial direction is proportional to the displacement in this axis. Thus, according to the structural symmetry of the six-phase DSW-12/2-BPMSM, the expressions of the eccentric magnetic pull force components in the stationary \(\alpha\)-\(\beta\) coordinate system can be written as follows:  
\[