# Analytical Model to Calculate Radial Forces in Permanent-Magnet Synchronous Machines
## 2 Sources of Magnetic Forces in PMSMs
### Magnetic Forces Due to Eccentricities
\[B_{ge}(t,\theta)=\left[\sum_{n=-\infty}^{n=+\infty}B_{gn}\cdot e^{i\cdot n\cdot p \cdot(\theta+\Omega_{m}t)}\right]\cdot\left[\Delta_{e0}+\sum_{v=-\infty}^{v=+ \infty}\Delta_{ev}\cdot e^{i\cdot p\cdot(\theta+\Omega_{e}t)}\right] \tag{14}\]  
\[ \begin{array}{rl}B_{ge}(t,\theta)&=\sum\limits_{n=-\infty}^{n=+\infty}B_{ gn}\cdot e^{i\cdot n\cdot p\cdot(\theta+\Omega_{m}t)}\cdot\Delta_{e0}+\sum \limits_{n=-\infty}^{n=+\infty}\sum\limits_{v=-\infty}^{v=+\infty}B_{gn}\cdot e ^{i\cdot n\cdot p\cdot(\theta+\Omega_{m}t)}\cdot\Delta_{ev}\cdot e^{i\cdot(v -\theta+\Omega_{e}t)}\\ &=\sum\limits_{n=-\infty}^{n=+\infty}\Delta_{e0}\cdot B_{gn}\cdot e^{i\cdot n \cdot p\cdot(\theta+\Omega_{m}t)}+\sum\limits_{n=-\infty}^{n=+\infty}\sum \limits_{v=-\infty}^{v=+\infty}\Delta_{ev}\cdot B_{gn}\cdot e^{i\cdot(n+\frac {v}{p})\cdot p\cdot\theta+j\cdot n\cdot p\cdot\Omega_{m}t+j\cdot\frac{v}{p} \cdot p\cdot\Omega_{e}t}\end{array}  \tag{15}\]  
From (15), the order of the new magnetic flux density harmonics with both static and dynamic eccentricity are obtained (see Table 8). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Static Eccentricity** & **Dynamic Eccentricity** \\ \hline Spatial distribution & Temporal distribution & Spatial–temporal distribution \\ \(v_{e}\)=\(\pm n\)\(\left|\pm\left(n\pm\frac{v}{p}\right)\right.\) & \(v_{e}\)=\(\pm n\)\(\left|\pm\left(n\pm\frac{v}{p}\right)\right.\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Harmonics of the air-gap flux density in a machine with eccentricity. Figure 2: Types of eccentricity: (**a**) static, (**b**) dynamic, and (**c**) mixed. The red shaft represents the rotation axis. By applying the simplified equation of [PERSON]'s stress tensor law (2) to the expression (15) of the flux density, the magnetic pressure including the eccentricities can be computed. \[ \begin{split} P_{re}(t,\theta)=\frac{1}{2\mu_{0}}&\left( \sum\limits_{n=-\infty}^{n=+\infty}\Delta_{e0}\cdot B_{gn}\cdot e^{j\cdot n \cdot p\cdot(\theta+\Omega_{m}t)}\right.\\ &\left.+\sum\limits_{n=-\infty}^{n=+\infty}\sum\limits_{v=-\infty }^{v=+\infty}\Delta_{ev}\cdot B_{gn}\cdot e^{j\cdot(n+\frac{p}{p})\cdot p\cdot \theta+j\cdot n\cdot p\cdot\Omega_{m}t+j\frac{p}{p}\cdot p\cdot\Omega_{e}t} \right)^{2}\\ \\ P_{re}(t,\theta)=\frac{1}{2\mu_{0}}&\left(\sum \limits_{n=-\infty}^{n=+\infty}\sum\limits_{m=-\infty}^{m=+\infty}\epsilon_{1 }\cdot e^{j\cdot(n+m)\cdot p\cdot\Omega_{m}t}\cdot e^{j\cdot(n+m)\cdot p\cdot \theta}\right.\\ &\left.+\sum\limits_{n=-\infty}^{n=+\infty}\sum\limits_{m=- \infty}^{m=+\infty}\sum\limits_{k=-\infty}^{k=+\infty}\epsilon_{2}\cdot e^{j \cdot(n+m)\cdot p\cdot\Omega_{m}t+j\frac{k}{p}\cdot p\cdot\Omega_{e}t}\cdot e ^{j\cdot(n+m+\frac{k}{p})\cdot p\cdot\theta}\right.\\ &\left.+\sum\limits_{n=-\infty}^{n=+\infty}\sum\limits_{m=- \infty}^{m=+\infty}\sum\limits_{m=-\infty}^{m=+\infty}\epsilon_{3}\cdot e^{j \cdot(n+m)\cdot p\cdot\Omega_{m}t+j\frac{p}{p}\cdot p\cdot\Omega_{e}t}\cdot e ^{j\cdot(n+m+\frac{n}{p})\cdot p\cdot\theta}\right.\\ &\left.+\sum\limits_{n=-\infty}^{n=+\infty}\sum\limits_{v=-\infty }^{v=+\infty}\sum\limits_{m=-\infty}^{m=+\infty}\epsilon_{4}\cdot e^{j\cdot(n +m)\cdot p\cdot\Omega_{m}t+j\cdot(\frac{p}{p}+\frac{k}{p})\cdot p\cdot\Omega_{ e}t}\right.\\ &\left.\cdot e^{j\cdot(n+m+\frac{p}{p}+\frac{k}{p})\cdot p\cdot \theta)}\right.\\ \epsilon_{1}=\Delta_{e0}{}^{2}\cdot B_{gn}\cdot B_{gn}\\ \epsilon_{2}=\Delta_{e0}\cdot\Delta_{ek}\cdot B_{gn}\cdot B_{ gn}\\ \epsilon_{3}=\Delta_{e0}\cdot\Delta_{ev}\cdot B_{gn}\cdot B_{ gn}\\ \epsilon_{4}=\Delta_{ev}\cdot\Delta_{ek}\cdot B_{gn}\cdot B_{ gn}\\ \end{split}  \tag{17}\]  
Analyzing (8), it can be deduced that the new harmonics of the magnetic pressure originating from the eccentricities are the ones shown in Table 9.  
The magnetic forces on a region can be calculated by (37), integrating magnetic pressures along the region. The resulting magnetic force is a temporal waveform.