## 2 Electromagnetic Effects of Winding Patterns
### Magneto-Motive Force Analysis  
In this chapter, space harmonic magneto-motive force (MMF), one of the main causes of rotor eddy current loss in high-speed PM motors, was comparatively analyzed through analytical methods. The MMF owing to the winding structure and distribution was compared and analyzed using the space harmonic method. The space harmonics method is an electromagnetic field analysis technique that obtains the magnetic vector potential from the Maxwell Equation. Using this analysis method, MMF analysis of a high-speed motor was performed [8]. Based on the Maxwell Equation, the magnetic flux density \(\mathbf{B}\), the magnetic field strength \(\mathbf{H}\), and the magnetization strength \(\mathbf{M}\) are based on the relationship  
Figure 3: MPF analysis results according to (**a**) concentrated winding and (**b**) distributed winding and UMF analysis results according to (**c**) concentrated winding and (**d**) distributed winding. of Equation (6). The magnetic vector potential \(\mathbf{A}\) has a distribution characteristic according to the number of pole pairs \(p\) and the space harmonic coefficient \(n\) in the \(\theta\) direction, and the magnitude has a rate of change in the \(r\) direction. This is summarized as an equation with periodicity, as shown in the following Equation (7),  
\[\
abla\times\mathbf{B}=\
abla\times(\
abla\times\mathbf{A})=\mu_{0}\
abla\times (\mathbf{M}+\mathbf{H}) \tag{6}\]  
\[\mathbf{A}=A_{zn}(r)e^{-jn\theta}i_{z} \tag{7}\]  
Then, the magnetic vector potential Equation (7) and the current modeling of the pole slot are calculated, and the following Equation (8) is derived, where \(\mathbf{J}\) denotes the current density. In Equation (8), the following magnetic vector potential Equation (9) was derived by solving the characteristic equations using the condition of \(\mathbf{J}=0\), when \(r\) is the air-gap center. By applying it to the magnetic vector potential definition, the magnetic flux density equation was derived as shown in Equations (10) and (11):  
\[\
abla^{2}A=-\mu_{0}J=\left(\frac{\partial^{2}}{\partial r^{2}}A(r)+\frac{1}{r }\frac{\partial}{\partial r}A(r)-\frac{(np)^{2}}{r^{2}}A(r)\right)e^{-jnp\theta} \tag{8}\]  
\[A(r)=\sum_{n=odd}^{\infty}\left\{C_{n}^{1}r^{np}+D_{n}^{1}r^{-np}\right\}e^{- jnp\theta} \tag{9}\]  
\[B_{r}(r)=\sum_{n=-\infty}^{\infty}\left\{\frac{-jn\eta}{r}\big{(}C_{n}r^{np}+ D_{n}r^{-np}\big{)}e^{-jnp\theta}\right\} \tag{10}\]  
\[B_{\theta}(r)=\sum_{n=-\infty}^{\infty}\left\{\frac{-np}{r}\big{(}C_{n}r^{np} -D_{n}r^{-np}\big{)}e^{-jnp\theta}\right\} \tag{11}\]  
Finally, the magnetic flux density Equations (10) and (11) were applied to the boundary conditions according to the permeability to derive the final air-gap magnetic flux density graph [9]. Equation (12) below shows the relationship between the magnetic flux density and the magnetic force distribution,  
\[B=\frac{1}{S}\frac{\mu_{0}S}{l_{g}}f_{m}=\frac{\mu_{o}}{l_{g}}f_{m}=P_{r}f_{m} \tag{12}\]  
Here, \(S\) is the area of over which magnetic flux passes. \(f_{m}\) is the MMF, and \(l_{g}\) is the length of the air-gap. At this time, in the case of a rotary machine with a uniform air-gap, the air-gap magnetic flux density distribution is the same as the magnetic force distribution and the waveform according to a certain specific permeability (\(P_{r}\)) [10]. Therefore, based on the derived air-gap magnetic flux density, the final MMF distribution as shown in Figure 4 was derived. At this time, the phase currents of the two winding methods have the same rms value, and both were analyzed based on 1A. As a result, it was confirmed that the MMF waveforms derived through the air-gap magnetic flux density were generally consistent with the ideal MMF waveform considering only the winding pattern. This suggests the validity of Equation (12) applied in this study, and harmonic components were compared for a more detailed analysis. Figure 5 shows the harmonic components of MMF according to the analytical method and FEM used in this study.