# From Modeling to Failure Prognosis of Permanent Magnet Synchronous Machine
## 2 Finite Element Models of the Permanent Magnet Machine
### Outcome of Electromagnetic FEM  
The electromagnetic model of the investigated permanent magnet machine is presented in [15; 17; 22]. It computes the flux density in the whole machine. The air gap flux density will be used to calculate the torque [19; 23]. Torque [24]:  
\[T=\frac{\Pi}{2}*D_{\rm r}^{2}*B_{gav}*Q_{rms}*l_{a}, \tag{10}\]  
where electric loading  
\[Q_{rms}=\frac{J*A_{s}*K_{f}*N_{s}}{\Pi*D_{\rm r}}. \tag{11}\]  
The average air gap flux density can be expressed as  
\[Bgav=\frac{2}{tp}\int_{0}^{tp/2}Bg(x)dx. \tag{12}\]  
In Figure 2, the flux lines in the healthy machine at four different moments is illustrated. The magnitude of these flux lines is represented by arrows (zoomed out) representing the magnitude and sense of flux density at the centroid of each finite element. _Bgav_: average air gap flux density over one pole for an instant of time. Dr: rotor's diameter. _la_: the axial length of the machine. _J_: average current density. _As_: slot's area. _Kf_: slot's filing factor. _Ns_: number of slots in the stator. _Bg(x)_: the flux density at equidistant points in the air gap. _tp_: the tooth pitch. In our model, the air gap region is considered to be moving as the rotor rotates. The air gap flux density function of time for the healthy case, case of crack in a magnet, case of turn-to-turn short circuit in one stator's coil and case of eccentricity fault is illustrated in Figures 3-6 respectively. To draw the air gap flux density in the time domain, an arbitrary point in the air gap is considered. This point has almost the same vertices for all the studied cases of the machine. Figure 2: Flux lines in the machine at four different moments (healthy machine). Figure 4: Air gap flux density in time domain (machine with 1 mm crack). Figure 5: Air gap flux density in time domain (machine with 10% eccentricity fault). Figure 3: Air gap flux density in time domain (healthy machine). The analytical described equation to calculate the torque is integrated in the time stepping model of the machine, which generates the torque function of time. The simulation is done for the healthy case and different types of fault previously mentioned. Figure 7 illustrates the torque for the healthy case, case of 10% eccentricity fault, turn-to-turn short circuit in one slot and 1 mm crack in one piece of magnet. Contemplating Figure 7, we can see the impact of the faults on the torque which shows that the torque is a good indicator for fault detection. The transient part of the machine's operation is clearly illustrated in the graph; this is the main advantageous of the time stepping dynamic model. Comparing the graphs in Figure 7, several notes can be recorded. The blue graph is the torque in the healthy case. The red graph is the machine torque in the case of demagnetization represented by a 1 mm crack in one piece of magnet. This graph is close to that of the healthy case; however, this fault has impacted the torque with ripples clearly visualized on the graph. The average torque in this case of demagnetization is greater than the healthy case (Table 2) due to the numerical calculation where the average is calculated over the first tooth pitch that contains the peak. This increased the value of air gap flux density and, subsequently, the torque. Figure 6: Air gap flux density in time domain (machine with turn-to-turn short circuit fault). Figure 7: Torque versus time for the different states of the machine. The yellow graph is the torque of the machine when a turn-to-turn short circuit occurred in one slot of phase A. When a short circuit occurs, temperature rises in the whole machine in general and in the magnet in particular. This rise in temperature weakens the magnet, its flux density decreases; the cause is the high dependence of the magnet's electromagnetic performance on the operating temperature. Accordingly, the torque decreases, being directly proportional to the air gap flux density. This torque will continue decreasing as the number of short circuited turns increases.