# From Modeling to Failure Prognosis of Permanent Magnet Synchronous Machine
## 2 Finite Element Models of the Permanent Magnet Machine
### Outcomes of Thermal FEM Model  
The thermal FEM model is capable of computing the temperature in the whole machine. The importance of monitoring the boundary temperature for fault detection and localization is clarified in paper [18].  
The governing equation of heat transfer for any application is [25]  
\[\rho\ Cp\ tz\frac{\partial T}{\partial t}-k\ tz\ \
abla^{2}T=Qd+Qc+Qr. \tag{13}\]  
\(\rho\): material density, kg/m\({}^{3}\). _Cp_: the specific heat, J/kg.K. _tz_: the thickness model. _k_: thermal conductivity, W/m\(\cdot\)K. _t_: time, sec. _T_: temperature at a particular x and y location, \(k\). _Qd_: heat source, watt/m\({}^{2}\).  
The amount of heat transferred from a surface, per unit area, due to convection is expressed as  
\[Qc=hc\star(T-Ta). \tag{14}\]  
_Ta_: ambient temperature. _hc_: convection coefficient, W/m\({}^{2}\cdot\)K.  
The amount of heat transferred, per unit area, due to radiation is expressed as  
\[Qr=\epsilon\ \sigma\left(T^{4}-Ta^{4}\right). \tag{15}\]  
\(\epsilon\): emissivity of the face. \(\sigma\): Stefan-Boltzmann constant, W/m\({}^{2}\)K  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Machine’s State** & **Average Torque (N\(\cdot\)m)** \\ \hline Healthy & 133.49 \\ Crack 1 mm in one magnet & 153.28 \\
10\% eccentricity fault & 148.68 \\ Turn-to-turn short circuit fault & 95.05 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Average torque for the different states of the machine.  
In this study, a thermal steady state analysis is executed for each machine revolution, where the input is considered a pure three phase sine wave voltage. The machine has 12 poles and a nominal speed of 668 rpm [26].  
The general heat equation is a parabolic partial differential equation. The aim of the thermal analysis is to know the distribution of temperature throughout the machine in general, at the magnet level in specific and at the machine's boundary.  
Figure 8 shows the temperature detected by this sensor for the case of healthy machine and faulty machine with short circuit in the first slot. A high difference between the sensed temperatures in the two cases can be seen. This is expected since the short circuit increases the value of current in the faulty phase; this increases  
For the state of 10% eccentricity fault in the machine, four cases are considered: rotor shifted to the right, rotor shifted to the left, rotor shifted upward and rotor shifted downward. The temperature collected by the sensor, in this case, is illustrated in Figure 11.  
It should be mentioned that temperature is not a good indicator for eccentricity and demagnetization faults since the value of the temperature is almost the same for those types of faults.  
After collecting the temperature data for the different machine states, statistical and spectral features were extracted. Statistical features were not a good indicator for fault detection using temperature data signal; hence, we will take advantage of the spectral characteristics [27].  
Figure 12 illustrates the Fourier transformation of the collected temperature data for all the investigated states of the machine. Figure 13 is a zoomed view of Figure 12. As we can see, a sharp peak is detected at the 10 th harmonic of the spectrum where the frequency is equal (10*fundamental) and where ffundamental is the fundamental frequency. A zoomed view on those sharp peaks in Figure 13 shows distinction between the different machine states. We mention that the amplitude of this Fourier transformation is not normalized.  
Figure 11: Sensor temperature for machine with 10% eccentricity faults.  
Figure 10: Sensor temperature for 1 mm crack in one magnet at a time.