# Effect of Slot Opening Width on the Air-Gap Magnetic Field of a Direct Drive Permanent Magnet Motor
## 2 Analytical Model
The adopted assumptions indicate that the magnetic vector potential only has one component along the z-direction and only depends on \(r\) and \(\theta\) coordinates. Thus, the radial and tangential components of magnetic density in the air-gap subdomain can be expressed as follows:  
\[\left\{ \begin{array}{l}B_{r}=\frac{1}{r}\frac{\partial A(r,\theta)}{\partial \theta}\\ B_{\theta}=-\frac{\partial A(r,\theta)}{\partial r}\end{array} \right.. \tag{19}\]  
Substituting (18) into (19), the following equations can be obtained:  
\[\begin{array}{ll}B_{gr}(r,\theta,\delta)=&-\sum\limits_{n=1}^{\infty} \left[A_{n}^{3}\frac{R_{4}}{r}\frac{Y_{n}(r,R_{3})}{X_{n}(R_{4}  
Figure 4 depicts the air-gap flux density waveforms of the DDPMM with open and closed slots. Regions A, B, and C show the air-gap flux density of the three motor units. The waveform in Region A is identical to those in Regions B and C. Positions 1-4 represent the different positions of the slot opening relative to the permanent magnet. The waveform in Region A indicates that the slot opening affects the distribution of the air-gap magnetic field and the effect degree varies at different positions. The slot opening in the stator had symmetry with respect to the position of the permanent magnet. Thus, the centerline of the permanent magnet was considered the criterion in classifying the air-gap magnetic field distribution into four situations, as shown in Figures 4-8. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Inner radius of stator (mm) & 47 & Air-gap length (mm) & 1 \\ Outer radius of stator (mm) & 75 & Permanent magnet thickness (mm) & 4 \\ Inner radius of rotor surface (mm) & 76 & Polar arc coefficient (deg) & 0.75 \\ Outer radius of rotor (mm) & 90 & Slot width (mm) & 7 \\ Radius of the slot bottom in the stator (mm) & 57 & Axial length (mm) & 85 \\ Radius of the slot opening in the stator (mm) & 72 & Winding turns/phase & 80 \\ Inner radius of rotor yoke (mm) & 80 & Number of poles & 24 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the DDPMM. Figure 4: Air-gap flux density waveforms of the DDPMM with open and closed slots. Figure 3: Finite element models of DDPMM: (**a**) DDPMM with open slots; (**b**) DDPMM with closed slots. Figure 5: Magnetic field analysis at position 1: (**a**) position 1; (**b**) air-gap flux density waveforms; (**c**) harmonics of air-gap flux density. Figure 6: Magnetic field analysis at position 2: (**a**) position 2; (**b**) air-gap flux density waveforms; (**c**) harmonics of air-gap flux density. Figure 7: Magnetic field analysis at position 3: (**a**) position 3; (**b**) air-gap flux density waveforms; (**c**) harmonics of air-gap flux density. Figures 4-8 show the analysis of the air-gap magnetic field distributions and their harmonics at different positions. Figures 4-8 indicate that the air-gap flux density waveforms of the DDPMM with open slots differed from those of the DDPMM with closed slots in terms of the amplitude and the distribution of air-gap flux density. In addition, the Fourier analysis in Figures 4-8 shows that the fundamental component and the total harmonic distortions (THDs) of air-gap flux density in the DDPMM with open slots differed from those of the DDPMM with closed slots. Therefore, the slot opening width dose affects the air-gap magnetic field. Furthermore, the comparisons in Figures 4-8 depict the effect degree of slot opening width on the air-gap magnetic field that varies with the different positions. Tables 2 and 3 compare the fundamental amplitude and the THD of the air-gap flux density in the DDPMMs with closed and open slots. The effect proportion in Tables 2 and 3 refers to the ratio of the difference between the value in the DDPMM with a closed slot to that with an open slot. The air-gap flux density was affected by the slot opening width. amplitudes in the air-gap magnetic field changed with the increase in slot opening width.