## 2 Electromagnetic Effects of Winding Patterns
### Unbalanced Magnetic Force Analysis  
Depending on the winding combination, it may cause asymmetry or harmonic of the air gap magnetic flux density. This causes an unbalanced magnetic force, and this unbalanced electromagnetic force is a major cause of vibration and noise along with the static and dynamic eccentricity of the rotor. Therefore, to design a high-speed motor with improved performance, the unbalanced electromagnetic force of the winding combination of concentrated and distributed windings was derived and analyzed.  
The electromagnetic force generated in the motor's air gap can be derived through the Maxwell stress tensor, and if this is expressed mathematically in a two-dimensional analysis, it is as follows in Equation (1):  
\[F=\frac{1}{\mu_{0}}\left(B_{n}^{2}-\frac{1}{2}|B|^{2}\right)i_{n}-\frac{1}{\mu _{0}}B_{n}B_{t}i_{t} \tag{1}\]  
Here, \(\mu_{0}\) is the permeability of the vacuum. \(B\) is the magnetic flux density. \(B_{n}\) and \(B_{t}\) are vertical and horizontal magnetic flux densities, respectively. \(i_{n}\) and \(i_{t}\) represent vertical and horizontal directions, respectively. It can be calculated from [PERSON]'s stress method through the force density distribution Equations (2) and (3) in the radial and circumferential directions,  
\[f_{r}=\frac{1}{2\mu_{0}}\left(B_{rg}^{2}-B_{\theta g}^{2}\right) \tag{2}\]  
\[f_{\theta}=\frac{1}{\mu_{0}}\left(B_{rg}B_{\theta g}\right) \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Characteristic** & **Concentrated Winding** & **Distributed Winding** & **Unit** \\ \hline Stack length & 90 & mm \\ \hline Outer diameter of stator/rotor & 96/30 & & mm \\ \hline Turns & 15 & 9 & - \\ \hline Resistance & 0.04 & 0.01 & \(\Omega\) \\ \hline Synchronous Inductance & 0.4 & 0.22 & mH \\ \hline Induced Voltage & 175 & 177 & V\({}_{\text{rms}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: High speed motor design specifications according to winding patterns.  
Figure 2: Stator layouts and magnetic flux line distribution according to armature reaction of a 2 pole/6 slot motor (**a**) concentrated, (**b**) distribute winding model.  
Here, \(B_{Tg}\) and \(B_{gg}\) are the magnetic flux densities in the radial and circumferential directions at the air-gap center, respectively, and \(l_{slk}\) is axial length of the stator core.  
After converting these to a rectangular coordinate system, it is possible to obtain the direction and magnitude of the force received when the rotor rotates by integrating the area, whereas the force density distribution in the radial and circumferential directions exist. This is the same as in Equations (4) and (5) below:  
\[F_{x}=[\frac{rl_{slk}}{2\mu_{o}}]\int_{0}^{2\pi}\Bigl{[}(B_{gg}^{2}-B_{rg}^{2} )\cos\theta+2B_{rg}B_{gg}\sin\theta\Bigr{]}d\theta \tag{4}\]  
\[F_{y}=\frac{rl_{slk}}{2\mu_{o}}\int_{0}^{2\pi}\Bigl{[}(B_{gg}^{2}-B_{rg}^{2}) \sin\theta+2B_{rg}B_{gg}\cos\theta\Bigr{]}d\theta \tag{5}\]  
Through these equations, an analysis of the unbalanced magnetic force (UMF) characteristics of the PM motor and the distribution characteristics of the force density was performed and compared.  
Figure 3a,b shows the magnetic pull force (MPF) analysis results according to the winding method at a rated load, and Figure 3c,d shows the results of UMF analysis according to the winding method at a rated load. Both models have different waveforms, but they are symmetrically distributed. This means that the vibration caused by an unbalanced magnetic force in both models is small.