## 3 Mathematical Modeling of the Six-Phase DSW-12/2-BPMSM
### Basic Model for Six-Phase DSW-12/2-BPMSM
\[L_{ij}=2\pi\mu_{0}lr\left[\langle P_{e}N_{i}N_{j}\rangle-\frac{\langle P_{e}N_{ i}\rangle\langle P_{e}N_{j}\rangle}{\langle P_{e}\rangle}\right] \tag{6}\]  
where \(\mu_{0}\) is the vacuum permeability, \(l\) is the stator core length, \(r\) is the rotor radius, and \(N_{i}\) and \(N_{j}\) are the winding functions of phases \(i\) and \(j\) (\(i\), \(j=a\), \(b\), \(c\), \(d\), \(e\), \(f\)), respectively. The operator \(<\!\!f\!\!>\) denotes that the function f takes the average value over the air gap circumference. Substituting Equation (1) into Equation (6) yields the following:  
\[ \begin{array}{l}L_{ij}=2\pi\mu_{0}lr\left\{\frac{1}{2}P_{e}N_{1}^{2}\cos[ \frac{\pi}{3}(i-j)]\ +\frac{1}{2}P_{e}N_{2}^{2}\cos\times\right.\\ \left.\left[\frac{2\pi}{3}(i-j)\right]+\frac{4}{4}P_{e}^{2}N_{1}N_{2}\cos[ \lambda+\frac{\pi}{3}(i-2j)]\right\}\end{array}  \tag{7}\]  
Based on the multidimensional control characteristics of a multiphase motor [18], a coordinate transformation matrix \(C\) can be established, which can transform the mathematical model of the six-phase BPMSM from the six-phase stationary coordinate system to the synchronous coordinate system, as shown in Equation (8). \[C=\sqrt{\frac{1}{3}}\left[ \begin{array}{cccccc}\cos\theta&\cos(\theta-\frac{ 1}{3}\pi)&\cos(\theta-\frac{2}{3}\pi)&\cos(\theta-\frac{3}{3}\pi)&\cos(\theta -\frac{4}{3}\pi)&\cos(\theta-\frac{5}{3}\pi)\\ \sin\theta&\sin(\theta-\frac{1}{3}\pi)&\sin(\theta-\frac{2}{3}\pi)&\sin( \theta-\frac{3}{3}\pi)&\sin(\theta-\frac{4}{3}\pi)&\sin(\theta-\frac{5}{3}\pi )\\ \cos 2\theta&\cos(2\theta-\frac{3}{3}\pi)&\cos(2\theta-\frac{4}{3}\pi)&\cos(2 \theta-\frac{6}{3}\pi)&\cos(2\theta-\frac{8}{3}\pi)&\cos(2\theta-\frac{10}{3 }\pi)\\ \sin 2\theta&\sin(2\theta-\frac{2}{3}\pi)&\sin(2\theta-\frac{4}{3}\pi)&\sin(2 \theta-\frac{6}{3}\pi)&\sin(2\theta-\frac{8}{3}\pi)&\sin(2\theta-\frac{10}{3 }\pi)\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\end{array} \right] \tag{8}\]  
Setting the \(x\)-\(y\) axis as the synchronous coordinate system of the mechanical rotor, \(\theta\) in Equation (8) represents the included angle in the mechanical space between the rotor magnetic flux linkage and the axis of the A-phase winding (i.e., the rotor position angle) as \(\theta=\omega t\), where \(\omega\) is the rotation speed. The first two rows of the matrix correspond to the \(d\)-\(q\) torque plane in the electrical space; the middle two rows of the matrix correspond to the \(j\)-\(k\) levitation plane in the electrical space, with the electrical angular velocity of the \(j\)-\(k\) plane being twice that of the \(d\)-\(q\) plane; and the last two rows of the matrix correspond to the zero space. Taking the inductance parameter \(L_{ij}\) (\(i\), \(j=a\), \(b\), \(c\), \(d\), \(e\), \(f\)) in Equation (7) as elements, the inductance matrix \(L_{\mathrm{s}}\) of the stator winding of the six-phase BPMSM can be constituted as follows:  
\[L_{\mathrm{S}}=\left[ \begin{array}{cccccc}L_{aa}&M_{ab}&M_{ac}&M_{ad}&M_{ae}& M_{af}\\ M_{ba}&L_{bb}&M_{bc}&M_{bd}&M_{be}&M_{bf}\\ M_{ca}&M_{cb}&L_{cc}&M_{cd}&M_{ce}&M_{cf}\\ M_{da}&M_{db}&M_{dc}&L_{dd}&M_{de}&M_{df}\\ M_{ae}&M_{cb}&M_{cc}&M_{ed}&L_{ce}&M_{ef}\\ M_{fa}&M_{fb}&M_{fc}&M_{fd}&M_{fe}&L_{ff}\end{array} \right] \tag{9}\]By performing coordinate transformation on the inductance matrix \(L_{s}\) using Equation (8), the corresponding inductance matrix in the \(x\)-\(y\) coordinate system can be obtained as follows:  
\[L_{xy}= \begin{bmatrix}L_{1d}\frac{4\xi_{0}^{2}+\vec{\epsilon}^{2}}{4\xi_{0}^{2}} &0&M_{1d2j}&M_{1d2k}\\ 0&L_{1q}\frac{4\xi_{0}^{2}+\vec{\epsilon}^{2}}{4\xi_{0}^{2}}&M_{1q2j}&M_{1q2k}\\ M_{1d2j}&M_{1q2j}&L_{2j}\frac{4\xi_{0}^{2}+\vec{\epsilon}^{2}}{4\xi_{0}^{2}}&0 \\ M_{1d2k}&M_{1q2k}&0&L_{2k}\frac{4\xi_{0}^{2}+\vec{\epsilon}^{2}}{4\xi_{0}^{2}} \end{bmatrix} \]  
\[\approx