# From Modeling to Failure Prognosis of Permanent Magnet Synchronous Machine
## 2 Finite Element Models of the Permanent Magnet Machine  
Finite element analysis is useful for models having complicated geometry and containing several types of materials with different properties. It consists of dividing the domain subject of analysis into small elements called sub-domains where the differential equation for each domain is built separately. Then, the combination of the whole sub-systems forms the general solution.  
As we stated previously, the dynamic electromagnetic, thermal and vibration aspects of the machine are all very important for accurately modeling it. Hence, a finite element model combining all those sub models is elaborated for a 10 kw surface-mounted permanent magnet used in HEV application.  
An electromagnetic, thermal and vibration model of this machine was developed in paper [17; 18; 19] respectively. The governing equation of the dynamic electromagnetic model are  
* The time-dependent magnetic diffusion equation \[\
abla*\
u\
abla*A=\frac{\sigma\mathrm{Vb}}{1}-\ \sigma\frac{\partial A}{ \partial t}+\ \sigma v*\ \
abla*A\] (1)
* Expression of current in  
* Circuit equation: parallel coil equation \[Vb=Rs\left\{1\right\}^{\prime}\left\{l\right\}+Ls\left\{1\right\}^{\prime}\left\{ \frac{dl}{dt}\right\}+Vc\] (4)
* Mechanical acceleration equation \[m\frac{dv}{dt}+\lambda v=Fem-Fext\] (5)
* Mechanical velocity equation \[v=\frac{dx}{dt}\] (6)  
\(\
u\): reductivity. \(A\): vector potential. \(\sigma\): conductivity. Vb: voltage across rotor's bar (the equivalent of the magnets). \(l\): axial length of the machine. \(v\): linear rotor's speed. \(t\): time. \(l\): current in each conductor. \(Vc\): voltage across series coils (stator's coil). \(db\): a diagonal matrix with entries of \(+1\) or \(-1\), indicating the polarity of each bar in the coil. _Lext_: coils' equivalent inductance. _Rext_: coils' equivalent resistance. _Rs_: parallel conductor's equivalent resistance. _Ls_: parallel conductor's equivalent inductance. _m_: rotor's mass. \(x\): rotor's position. \(\lambda\): damping factor (in a synchronous machine this value is almost zero). _Fem_: electromagnetic force. _Fext_: externally-applied mechanical force (load).  
According to the Demerdash approach, the piece of magnets in the machine can be modeled as two parallel conductors having the same direction of the magnetic flux.  
After discretization and linearization, the global matrix of the system is [18]  
\[\left[\begin{array}{cccc}M11&M12&0&0&M15\\ M12^{\prime}&M22&M23&0&0\\ 0&M23^{\prime}&M33&M34&0\\ 0&0&M34^{\prime}&M44&0\\ M15^{\prime}&0&0&0&M55\end{array}\right]\begin{array}{c}t+\Delta t\\ \Delta Vb\\ \Delta I\\ \Delta Vc\\ \Delta x\end{array}\right]=\left[\begin{array}{c}N1\\ N2\\ N3\\ N4\\ N5\end{array}\right]. \tag{7}\]  
The system is solved using the Newton-Raphson method. For each step of rotation, an update of the system is done using the below transformation coordinates where \(\theta\) is the rotating angle of the rotor.  
\[xnew=xold*cos\theta\ -ydd*sin\theta \tag{8}\]  
\[ynew=yold*cos\theta\ +xold*sin\theta \tag{9}\]  
The solution gives the change in the vector potential, voltages, currents and rotor position. Hence, for example, \(Anew=Aold+\ \Delta A\).  
A sketch of the rotor's sweep at different times during the simulation is presented in Figure 1.  
The purpose of this generalized model is to generate a synthetic data signal of torque temperature and vibration for the machine in its healthy state and in the case of faulty states. Demagnetization, turn-to-turn short circuit and eccentricity are integrated in the FEMs model of the machine. The simulation is done for five revolutions [20].  
To extract these data signals, a fictive sensor, for each measured parameter, is located in the machine. The temperature and vibration sensors are located at the machine's boundary. The torque sensor is assumed to be at the shaft level [21].