## 3 Mathematical Modeling of the Six-Phase DSW-12/2-BPMSM  
### Basic Model for Six-Phase DSW-12/2-BPMSM  
One of the conditions for realizing rotor levitation is that the pole-pair numbers of the torque magnetic field and the levitation magnetic field must differ by one. However, in a typical multiphase AC motor with symmetrical windings, there are generally only odd harmonics but no even harmonics; that is to say, there is no even magnetic field in a common multiphase motor [12]. Therefore, the six-phase DSW-12/2-BPMSM in this work adopts a modified winding structure. Specifically, the common full-pole distribution winding is replaced by a half-pole distribution winding for each phase, so that each phase winding is asymmetrical regarding the air gap circumference. Next, the magnetomotive force of the improved winding is analyzed, which depends on the winding structure of the six-phase DSW-12/2-BPMSM and the excitation current. Here, the winding structure is expressed by a winding function, which refers to the spatial distribution function of the magnetomotive force generated by the unit excitation current along the air gap circumference. For the convenience of analysis, the winding function is expressed using Fourier series decomposition. The axis of the A-phase winding is set to the \(0^{\circ}\) position. Then, the winding function of each phase can be obtained as follows:  
Figure 3: Magnetic levitation force generation principal diagram. \[\left\{ \begin{array}{l}N_{a}=N_{1}\cos(\theta)+N_{2}\cos(2\theta)+\cdots\\ N_{b}=N_{1}\cos(\theta-\frac{\pi}{3})+N_{2}\cos(2\big{(}\theta-\frac{\pi}{3} \big{)})+\cdots\\ N_{c}=N_{1}\cos(\theta-\frac{\pi\pi}{3})+N_{2}\cos\big{(}2\big{(}\theta-\frac{ 2\pi}{3}\big{)}\big{)}+\cdots\\ \vdots\\ N_{f}=N_{1}\cos(\theta-\frac{5\pi}{3})+N_{2}\cos\big{(}2\big{(}\theta-\frac{5 \pi}{3}\big{)}\big{)}+\cdots\end{array} \right. \tag{1}\]  
where \(N_{1}\) and \(N_{2}\) are the first and second space harmonic magnitudes of the winding distribution function, respectively, which can be obtained using Fourier decomposition for each phase winding function, as follows:  
\[N_{1}=\frac{4}{\pi}\frac{N}{2}\sin(\frac{1}{2}y_{1}\alpha_{1}),\ N_{2}=\frac{2 }{\pi}\frac{N}{2}\sin(y_{1}\alpha_{1}) \tag{2}\]  
where \(N\) is the number of series turns per phase winding and \(\alpha_{1}\) is the slot pitch angle. Figure 4 shows the schematic diagram of the rotor eccentricity, where \(\theta\) is the spatial position angle along the air gap circumference, \(g_{0}\) is the average value of the equivalent air gap length, \(\lambda\) is the eccentric angle of the rotor, \(d\) is the eccentric distance, and \(\alpha\) and \(\beta\) are the radial displacement components along the horizontal and vertical directions, respectively. From Figure 4, the air gap length at any angle \(\theta\) position can be expressed as follows:  
\[g(\theta,\lambda)=g_{0}-d\cos(\theta-\lambda) \tag{3}\]  
Then, the inverse air gap function is as follows:  
\[P_{e}(\theta,\lambda)=\frac{1}{g(\theta,\lambda)}=\frac{1}{g_{0}-d\cos(\theta -\lambda)} \tag{4}\]  
In order to improve the calculation accuracy of the established analytical model, this work simultaneously considers the influence of the thickness and relative magnetic permeability parameters of the permanent magnet on the average equivalent air gap length \(g_{0}\). For the surface-mounted permanent magnet motor in this work, the calculation expression of \(g_{0}\) is as follows:  
\[g_{0}=g+\frac{L_{m}}{\mu_{r}} \tag{5}\]  
Figure 4: Schematic diagram of rotor eccentricity. where \(g\) is the actual air gap length, and \(L_{m}\) and \(\mu_{r}\) are the thickness and relative permeability of the permanent magnet, respectively. During the operation of a six-phase DSW-12/2-BPMSM, the rotor eccentricity will result in uneven distribution of the air gap length. Then, the inductance can be calculated by the improved winding method [17]. The calculation model for the inductance of the six-phase BPMSM with a surface-mounted permanent magnet rotor is established as Equation (6).