## 3 Mathematical Modeling of the Six-Phase DSW-12/2-BPMSM
### Basic Model for Six-Phase DSW-12/2-BPMSM
\begin{bmatrix}L_{1d}&0&M_{1d2j}&M_{1d2k}\\ 0&L_{1q}&M_{1d2j}&M_{1q2k}\\ M_{1d2j}&M_{1q2j}&L_{2j}&0\\ M_{1d2k}&M_{1q2k}&0&L_{2k}\end{bmatrix}  \tag{10}\]  
where \(L_{1d}\) and \(L_{1q}\) are the \(d\)-\(q\) axis inductance components in the torque plane; \(L_{2j}\) and \(L_{2k}\) are the \(j\)-\(k\) axis inductance components in the suspension plane; and \(M_{1d2j}\), \(M_{1d2k}\), \(M_{1q2j}\), and \(M_{1q2k}\) represent the mutual inductance between the torque windings and suspension windings. The expression of the stator mutual inductance matrix is as follows:  
\[ \begin{bmatrix}M_{1d2j}&M_{1d2k}\\ M_{1q2j}&M_{1q2k}\end{bmatrix} =M_{12}d \begin{bmatrix}\cos(\omega t-\lambda)&- \sin(\omega t-\lambda)\\ \sin(\omega t-\lambda)&\cos(\omega t-\lambda)\end{bmatrix} =M_{12} \begin{bmatrix} \alpha&\beta\\ -\beta&\alpha\end{bmatrix}\begin{bmatrix}\cos(\omega t)&-\sin(\omega t)\\ \sin(\omega t)&\cos(\omega t)\end{bmatrix}  \tag{11}\]  
The transformation relationship between the synchronous \(x\)-\(y\) coordinate system of the rotor and the stationary \(\alpha\)-\(\beta\) coordinate system is as follows:  
\[ \begin{bmatrix}x\\ y\end{bmatrix} = \begin{bmatrix}\cos(\omega t)&\sin(\omega t)\\ -\sin(\omega t)&\cos(\omega t)\end{bmatrix}\begin{bmatrix}\alpha\\ \beta\end{bmatrix}  \tag{12}\]  
Combining Equations (9)-(12), the inductance matrix in the \(x\)-\(y\) coordinate system is obtained as follows:  
\[L_{xy}= \begin{bmatrix}L_{1}&0&M_{12}x&M_{12}y\\ 0&L_{1}&-M_{12}y&M_{12}x\\ M_{12}x&-M_{12}y&L_{2}&0\\ M_{12}y&M_{12}x&0&L_{2}\end{bmatrix}  \tag{13}\]  
where the self-inductance coefficients \(L_{1}\) and \(L_{2}\) of the equivalent two-phase torque winding and suspension winding, as well as the mutual inductance coefficient \(M_{12}\) between them, can be expressed as follows:  
\[\left\{ \begin{array}{l}L_{1}=L_{1d}=L_{1q}=3\pi\mu_{0}lrN_{1}^{2}/g_{0}\\ L_{2}=L_{2d}=L_{2q}=3\pi\mu_{0}lrN_{2}^{2}/g_{0}\\ M_{12}=3\pi\mu_{0}lrN_{1}N_{2}/2g_{0}^{2}\end{array} \right. \tag{14}\]  
where \(N_{1}\) and \(N_{2}\) are the turn numbers of the equivalent two-phase torque winding and suspension winding, respectively. In the stationary coordinate system, the voltage equation and torque equation are as follows:  
\[U_{S}=R_{S}I_{S}+\frac{d\Psi_{S}}{dt} \tag{15}\]  
\[T_{e}=\frac{P_{e}}{\Omega}=\frac{EI_{S}}{\Omega} \tag{16}\]where \(U_{S}\) is the phase winding voltage, \(R_{S}\) is the phase winding resistance, \(I_{S}\) is the phase winding current, \(\Psi_{S}\) is the phase winding magnetic chain, \(P_{e}\) is the electromagnetic power, \(\Omega\) is the mechanical angular speed of the rotor, and \(E\) is the counter electromotive force, whose value is equal to the change rate of flux linkage, as follows:  
\[E=d\Psi_{S}/dt \tag{17}\]  
The flux linkage \(\Psi_{S}\) of each phase winding consists of the flux linkage generated by the winding itself, the coupled mutual inductance flux linkage generated by other phase windings, and the flux linkage generated by the permanent magnet. The expression of \(\Psi_{S}\) is as follows:  
\[\Psi_{S}=L_{S}I_{S}+\Psi_{f}(\theta) \tag{18}\]  
where \(\Psi_{f}\) is the equivalent flux linkage generated by the permanent magnet in the phase winding. The mathematical model of the six-phase DSW-12/2-BPMSM in the synchronous coordinate system can be obtained using coordinate transformation. The voltage equation, torque equation, and flux linkage equation are as follows:  
\[\left[ \begin{array}{c}u_{1d}\\ u_{1q}\\ u_{2j}\\ u_{2k}\end{array} \right]=R_{S}\left[ \begin{array}{c}i_{1d}\\ i_{1q}\\ i_{2j}\\ i_{2k}\end{array} \right]+L_{xy}\frac{d}{dt}\left[ \begin{array}{c}i_{1d}+i_{f }\\ i_{1q}\\ i_{2j}\\ i_{2k}\end{array} \right] \tag{19}\] \[+\omega\left[ \begin{array}{cccc}-1&0&0&0\\ 0&1&0&0\\ 0&0&-2&0\\ 0&0&0&2\end{array} \right]L_{xy}\left[ \begin{array}{c}i_{1q}\\ i_{1d}+i_{f}\\ i_{2j}\\ i_{2k}\end{array} \right]\]  
\[T_{e}=\left(i_{1q}\Psi_{1d}-i_{1d}\Psi_{1q}\right)+2\left(i_{2k}\Psi_{2j}-i_{2 j}\Psi_{2k}\right) \tag{20}\]  
\[\left\{