## 2 Analysis of the Radial Air-Gap Flux Density in a Permanent Magnet Synchronous Motor (PMSM)
### Radial Air-Gap Flux Density with Eccentricity  
By introducing the correction factor \(\varepsilon\), the radial air-gap flux density of PMSM with eccentricity can be obtained according to Equations (5) and (14). \[B_{r}(\alpha,t)=[\Lambda_{0}\sum_{n=1,3,5...}^{\infty}B_{\rm mr}\cos(np\alpha -n\omega_{e}t)+\Lambda_{0}\sum_{v=1}^{\infty}B_{\rm ar}\cos(pv\alpha\mp\omega_ {e}t)]\times\varepsilon \tag{15}\]  
In order to verify the analytical model of the radial air-gap flux density, the results are compared with the finite element model. Ansoft Maxwell is the leading software for the simulation of electromagnetic fields and it is based on the method of finite elements. The RMxprt module in Ansoft Maxwell includes many kinds of motors and it is very convenient for the calculation of electromagnetic fields. The model of PMSM in this paper was established by the RMxprt module and the main parameters are as shown in Table 1. Figure 2 shows the models for different pole-pair numbers in the finite element method (FEM). Based on the assumption that the static eccentricity is 1 mm and the dynamic eccentricity is ignored, the radial air-gap flux density of PMSM with different pole-pair numbers is calculated by the analytical model and the finite element model, and the results are shown in Figure 3. Figure 3a shows the distribution of the radial air-gap flux density on the rotor surface at a given time, and it is nonuniform due to the eccentricity. It can be seen that the analytical results are basically consistent with the finite element results. There are also some slight differences between the FEM and analytical waveforms at some space positions and the reason for them is the effect of stator slotting in the calculation of FEM. The field distribution in the air-gap will be affected due to the stator slotting [26], however, these differences do not have a great impact on the dynamic model. For convenience of calculation, the effect of stator slotting is ignored in the analytical method. Figure 3b shows the distribution of the radial air-gap flux density at a given point on the rotor surface in the time history. In the time history, the distribution varies periodically with time and the time for the rotor rotates with one pole-pair as one period. Since the rotating frequency is 25 Hz, the period of the field distribution is 0.04 s (\(p=1\)), 0.02 s (\(p=2\)) and 0.01 s (\(p=4\)), respectively. It can also be seen that the analytical results are basically consistent with the finite element results and there are also some slight differences between them due to the effects of stator slotting. Therefore, the analytical model of the radial air-gap flux density is verified and can be used in the following study. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Outer radius of the rotor core (\(R_{r}\)) & 80 mm \\ Inner radius of the stator core (\(R_{z}\)) & 925 mm \\ Pole-arc/pole-pitch ratio (\(\alpha_{p}\)) & 0.85 \\ Slot-opening (\(b\)) & 4 mm \\ Vacuum permeability (\(\mu_{0}\)) & \(4\pi\times 10^{-7}\) H/m \\ Current amplitude (\(l\)) & 10 A \\ Slot number (\(Z\)) & 48 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of PMSM. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Magnet thickness (\(\Omega_{\text{m}}\)) & 10 mm \\ Magnet remanence (\(B_{r}\)) & 1.25 T \\ Relative recoil permeability (\(\mu_{r}\)) & 1.05 \\ Rotor rotational speed (\(n\)) & 1500 \(\mathrm{\SIUnitSymbolMicro}\)/min \\ Length of the stator core & 165 mm \\ Outer radius of the stator core & 135 mm \\ Steel type of the stator core & DW315.50 \\ Slot type of the stator core & No. 2 \\ Number of winding layers & 1 \\ Winding type & Whole-Colded \\ Number of parallel branches of stator winding & 1 \\ Conductors per slot & 50 \\ Inner radius of the rotor core & 35 mm \\ Steel type of the rotor core & DW315.50 \\ Length of the rotor core & 165 mm \\ Stacking factor of the rotor core & 0.95 \\ \hline \end{tabular}
\end{table}