# Analytical Model to Calculate Radial Forces in Permanent-Magnet Synchronous Machines
## 2 Sources of Magnetic Forces in PMSMs
### Magnetic Forces Due to Eccentricities  
It is well known that three types of eccentricities can appear in electrical machines: static, dynamic, or mixed. It is known as static eccentricity when the rotating shaft of the rotor is the center of the rotor, but it is not aligned with the center of the stator (Figure 2a). Dynamic eccentricity occurs when the rotating shaft of the rotor is not aligned with its own center, but it is aligned with the center of the stator (Figure 2b). Finally, mixed eccentricity appears when the previous two cases are mixed. In this case, the rotating shaft of the rotor is aligned with neither the center of the rotor nor the center of the stator (Figure 2c). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Flux Density**} & \multicolumn{2}{c}{**Magnetic Pressure**} & \multicolumn{2}{c}{**Flux Density**} & \multicolumn{2}{c}{**Magnetic Pressure**} \\ \multicolumn{2}{c}{**Harmonics**} & \multicolumn{2}{c}{**Harmonics**} & \multicolumn{2}{c}{**Harmonics**} \\ \hline \(nr\) & \(kr\) & \(\mu=\pm(nr+kr)\) & \(nr\) & \(kr\) & \(\mu=\pm(nr+kr)\) \\ \hline \multirow{4}{*}{0} & 0 & 0 & & 0 & \(\pm 12\) \\  & \(-6\) & \(\pm 6\) & \(-12\) & \(-6\) & \(\pm 18\) \\  & \(-12\) & \(\pm 12\) & & \(-12\) & \(\pm 24\) \\  & \(-18\) & \(\pm 18\) & & \(-18\) & \(\pm 30\) \\ \hline \multirow{4}{*}{\(-6\)} & 0 & \(\pm 6\) & & 0 & \(\pm 18\) \\  & \(-6\) & \(\pm 12\) & \(-18\) & \(-6\) & \(\pm 24\) \\ \cline{1-1}  & \(-12\) & \(\pm 18\) & & \(-12\) & \(\pm 30\) \\ \cline{1-1}  & \(-18\) & \(\pm 24\) & & \(-28\) & \(\pm 46\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Relationship between the harmonics of the flux density and the harmonics of the magnetic pressure created by the armature of the Qs48p8 machine on the rotor surface. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Flux Density**} & \multicolumn{2}{c}{**Magnetic Pressure**} & \multicolumn{2}{c}{**Flux Density**} & \multicolumn{2}{c}{**Magnetic Pressure**} \\ \multicolumn{2}{c}{**Harmonics**} & \multicolumn{2}{c}{**Harmonics**} & \multicolumn{2}{c}{**Harmonics**} & \multicolumn{2}{c}{**Harmonics**} \\ \hline \(nr\) & \(kr\) & \(\mu=\pm(nr+kr)\) & \(nr\) & \(kr\) & \(\mu=\pm(nr+kr)\) \\ \hline \multirow{4}{*}{0} & 0 & 0 & & 0 & \(\pm 2.4\) \\  & \(-1.2\) & \(\pm 1.2\) & \(-2.4\) & \(-1.2\) & \(\pm 3.6\) \\  & \(-2.4\) & \(\pm 2.4\) & \(-2.4\) & \(\pm 4.8\) \\  & \(-3.6\) & \(\pm 3.6\) & & \(-3.6\) & \(\pm 6\) \\ \hline \multirow{4}{*}{\(-1.2\)} & 0 & \(\pm 1.2\) & & 0 & \(\pm 3.6\) \\  & \(-1.2\) & \(\pm 2.4\) & \(-3.6\) & \(-1.2\) & \(\pm 4.8\) \\ \cline{1-1}  & \(-2.4\) & \(\pm 3.6\) & & \(-2.4\) & \(\pm 6\) \\ \cline{1-1}  & \(-3.6\) & \(\pm 4.8\) & & \(-3.6\) & \(\pm 7.2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Relationship between the harmonics of the flux density and the harmonics of the magnetic pressure created by the armature of the Qs36p15 machine on the rotor surface. The effect of the eccentricities can be taken into account by multiplying the magnetic flux density of the air gap in a machine without eccentricity by (12). This distribution function, (13), defines the variation of the air-gap reluctance due to the eccentricity. \[B_{ge}(t,\theta)=B_{g}(t,\theta)\cdot\Delta_{e}(t,\theta) \tag{12}\]  
\[\Delta_{e}(t,\theta)=\Delta_{e0}+\sum_{v=-\infty}^{v=+\infty}\Delta_{ev}\cdot e ^{i\cdot p\cdot(\theta+\Omega_{e}t)} \tag{13}\]  
where \(\Omega_{e}\) represents the velocity of change of the air-gap length. In the case of dynamic eccentricity, this velocity is the same as the mechanical rotation velocity of the rotor, \(\Omega_{e}=\Omega_{m}\), and when there is a static eccentricity this velocity is zero. When there is not any eccentricity the sum term is zero, so the air gap is constant along the whole perimeter, \(\Delta_{e}=\Delta_{e0}\). Nevertheless, when there is an eccentricity, the complete expression has to be considered. Substituting (13) into (12), (14) is obtained and can be developed to give (15).