# Analytical Model to Calculate Radial Forces in Permanent-Magnet Synchronous Machines
## 2 Sources of Magnetic Forces in PMSMs
### Magnetic Pressures on the Stator  
The flux density on the periphery of the stator is formed by a vector field that can be decomposed into radial and tangential components. These components generate a magnetic radial pressure that can be computed using the law of [PERSON]'s stress tensor.  
\[P_{r}(t,\theta)=\frac{1}{2\mu_{0}}\Big{[}(B_{sr}^{m}(t,\theta))^{2}-(B_{st}^{ m}(t,\theta))^{2}\Big{]} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Machine** & **Slot Number** & **Pole Pair Number** & **Nominal Power** & **Nominal Speed** \\ \hline Q48p8 & 48 & 8 & 4.5 kW & 145 rpm \\ Qs36p15 & 36 & 15 & 4.5 kW & 170 rpm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Main parameters of the considered machines.  
where \(\mu_{0}\) is the permeability of the vacuum, and \(B_{sr}^{m}(t,\theta)\) and \(B_{st}^{m}(t,\theta)\) are the radial and tangential components of the flux density around the periphery of the tooth of the stator.  
Due to the high permeability of the material of the stator, the radial component of the flux density is much bigger than the tangential component, so it can be disregarded to obtain the next simplified equation.  
\[P_{r}(t,\theta)\ \approx\frac{1}{2\mu_{0}}\Big{[}(B_{sr}^{m}(t,\theta))^{2}\Big{]} \tag{2}\]  
The radial component of the magnetic flux density on the periphery of the stator can be computed by Fourier spatial series as shown in (3).  
\[B_{sr}^{m}(t,\theta)=\sum_{n=-\infty}^{+\infty}B_{sn}^{m}\text{cos}(pn(\varphi +\Omega_{m}t-\theta)) \tag{3}\]  
where \(B_{sn}^{m}\) are the coefficients of the Fourier series, \(p\) is the number of pole pairs, \(n\) is the order of the harmonic, \(\varphi\) is the initial position of the magnet, \(\Omega_{m}\) is the mechanical speed, \(t\) is the time, and \(\theta\) is the position around the path.  
Substituting (3) into (2), the following expression can be deduced.  
\[P_{r}(t,\theta)=\frac{1}{2\mu_{0}}\Bigg{[}\sum_{n=-\infty}^{+\infty}B_{sn}^{m }\text{cos}(pn(\Omega_{m}t-\theta))\cdot\sum_{k=-\infty}^{+\infty}B_{sk}^{m} \text{cos}(pk(\Omega_{m}t-\theta))\Bigg{]} \tag{4}\]  
Using the trigonometric property \(cos(\alpha)\text{cos}(\beta)=\frac{1}{2}[\text{cos}(\alpha+\beta)+\text{cos}( \alpha-\beta)]\), the following equation for the magnetic pressure is obtained.  
\[P_{r}(t,\theta)=\sum_{n=1}^{+\infty}\sum_{k=1}^{+\infty}\frac{B_{sn}^{m}B_{sk }^{m}}{4\mu_{0}}\cdot\text{cos}(\pm(n\pm k)p(\Omega_{m}t+\theta)) \tag{5}\]  
From (5), Table 2 is obtained, representing the relationship between the harmonics of the flux density and the harmonics of the magnetic pressure.  
It is important to note that all the harmonics of the magnetic flux density are odd. Therefore, all the harmonics of the magnetic pressure are even for any combination of pole pairs and stator slots. In the same way, all the harmonics of the flux density are odd. Table 2 also provides useful information for designers regarding the flux density harmonics that could be minimized or eliminated in order to modify the spectrum of the magnetic force. It is well known that the harmonics of flux density can be modulated by acting upon design parameters such as the magnet pitch or the winding distribution.  
\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 \(n\) & \(k\) & \(\mu=\pm(n\pm k)\) & \(n\) & \(k\) & \(\mu=\pm(n\pm k)\) \\ \hline \multirow{5}{*}{\(\pm 1\)} & \(\pm 1\) & 0 \(\mid\pm 2\) & & \(\pm 1\) & \(\pm 4\mid\pm 6\) \\  & \(\pm 3\) & \(\pm 2\mid\pm 4\) & & \(\pm 3\) & \(\pm 2\mid\pm 8\) \\  & \(\pm 5\) & \(\pm 4\mid\pm 6\) & \(\pm 5\) & \(\pm 5\) & \(0\mid\pm 10\) \\  & \(\pm 7\) &