# Analytical Model to Calculate Radial Forces in Permanent-Magnet Synchronous Machines
## 2 Sources of Magnetic Forces in PMSMs
### Magnetic Pressures on the Rotor
#### 2.2.2 Oscillating Magnetic Pressure Created by the Stator Winding  
The principal component of the magnetic flux created by the winding of the stator propagates along the air gap in synchronism with the main component of the magnetic flux produced by the magnets. That means that it propagates in the same direction and with the same mechanical speed as the rotor, \(\Omega_{m}\). Meanwhile, the rest of the spatial components propagate asynchronously with the rotor at \(\Omega_{n}\). \[\Omega_{m}=\frac{w_{s}}{p} \tag{7}\]  
\[\Omega_{n}=\frac{w_{s}}{s\cdot n\cdot t_{p}} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{ \begin{tabular}{c} **Flux Density** \\ **Harmonics** \\ \end{tabular}  }} & \multicolumn{2}{c}{ \begin{tabular}{c} **Magnetic** \\ **Pressure** \\ **Harmonics** \\ \end{tabular}  } & \multicolumn{2}{c}{\ \begin{tabular}{c} **Flux Density** \\ **Harmonics** \\ \end{tabular}  } & \multicolumn{2}{c}{ \begin{tabular}{c} **Magnetic** \\ **Pressure** \\ **Harmonics** \\ \end{tabular}  } \\ \hline \(m\) & \(l\) & \(\mu\) = \(\pm\)(m \(\pm\) 1) & \(m\) & \(l\) & \(\mu\) = \(\pm\)(m \(\pm\) 1) \\ \hline \multirow{6}{*}{\(\pm\)Qs/\(p\)} & \(\pm\)Q/s/\(p\) & 0 \(\mid\) \(\pm\)2Q/s/\(p\) & \(\pm\)Q/s/\(p\) & \(\pm\)2Q/s/\(p\) \(\mid\) \(\pm\)4Q/s/\(p\) \\  & \(\pm\)2Q/s/\(p\) & \(\pm\)Q/s/\(p\) \(\mid\) \(\pm\)3Q/s/\(p\) & \(\pm\)2Q/s/\(p\) & \(\pm\)Q/s/\(p\) \(\mid\) \(\pm\)5Q/s \\ \cline{1-1}  & \(\pm\)3Q/s/\(p\) & \(\pm\)2Q/s/\(p\) \(\mid\) \(\pm\)4Q/s/\(p\) & \(\pm\)3Q/s/\(p\) & 0 \(\mid\) \(\pm\)3Q/s/\(p\) & 0 \(\mid\) \(\pm\)6Q/s/\(p\) \\ \cline{1-1}  & \(\pm\)4Q/s/\(p\) & \(\pm\)3Q/s/\(p\) \(\mid\) \(\pm\)5Q/s/\(p\) & \(\pm\)4Q/s/\(p\) & \(\pm\)4Q/s/\(p\) & \(\pm\)4Q/s/\(p\) \(\mid\) \(\pm\)7Q/s/\(p\) \\ \cline{1-1}  & \(\pm\)5Q/s/\(p\) & \(\pm\)4Q/s/\(p\) \(\mid\) \(\pm\)6Q/s/\(p\) & \(\pm\)5Q/s/\(p\) & \(\pm\)2Q/s/\(p\) \(\mid\) \(\pm\)8Q/s/\(p\) \\ \hline \multirow{6}{*}{\(\pm\)2Q/s/\(p\)} & \(\pm\)Q/s/\(p\) & \(\pm\)Q/s/\(p\) & \(\pm\)Q/s/\(p\) & \(\pm\)3Q/s/\(p\) \(\mid\) \(\pm\)5Q/s/\(p\) \\ \cline{1-1}  & \(\pm\)2Q/s/\(p\) & 0 \(\mid\) \(\pm\)4Q/s/\(p\) & \(\pm\)2Q/s/\(p\) & \(\pm\)2Q/s/\(p\) \(\mid\) \(\pm\)6Q/s/\(p\) \\ \cline{1-1}  & \(\pm\)3Q/s/\(p\) & \(\pm\)Q/s/\(p\) \(\mid\) \(\pm\)5Q/s/\(p\) & \(\pm\)4Q/s/\(p\) & \(\pm\)3Q/s/\(p\) \(\mid\) \(\pm\)7Q/s/\(p\) \\ \cline{1-1}  & \(\pm\)4Q/s/\(p\) & \(\pm\)2Q/s/\(p\) \(\mid\) \(\pm\)6Q/s/\(p\) & \(\pm\)4 Qwhere \(w_{s}\) is the electrical speed of the rotor, \(t_{p}\) is the periodicity, which is defined as the greatest common divisor of \(Q_{s}\) and \(p\), and \(s=\pm 1\) defines the direction of the propagation. The pulsation of the \(n\)-order harmonic of the stator flux density in the rotor reference frame, \(\Omega_{rn}\), is the difference between the speed of the rotor, \(\Omega_{m}\), and the propagation velocity of this component, \(\Omega_{n}\). \[\Omega_{rn}=\Omega_{n}-\Omega_{m}=\left(\frac{p}{s\cdot n\cdot t_{p}}-1\right) \Omega_{m} \tag{9}\]  
Consequently, the frequency of this component referenced to the rotor, \(f_{rn}\), is as follows:  
\[f_{rn}=\frac{n\cdot t_{p}\cdot\Omega_{m}}{2\pi}=\left(s\frac{p}{t_{p}}-n\right) \frac{t_{p}}{p}f_{s} \tag{10}\]  
where \(f_{s}\) is the supplying frequency. Consequently, the order of the harmonics of the magnetic flux density can be computed as follows:  
\[n_{r}=\left(s\frac{p}{t_{p}}-n\right)\frac{t_{p}}{p} \tag{11}\]  
From (11), it can be deduced that the harmonic orders of the stator winding flux density referenced to the rotor depend on the \(p/t_{p}\) relationship. Therefore, it could be stated that through the modification of this ratio, undesired or critical frequencies could be eliminated. Furthermore, as demonstrated in Section 3.2, the magnitudes of undesired harmonics could also be minimized or even eliminated by adjusting the winding factor, which depends on the chosen winding configuration.