# The Cluster Computation-Based Hybrid FEM-Analytical Model of Induction Motor for Fault Diagnostics
## 4 Inductances Calculations  
In the case of electrical machines, most of the faults such as eccentricity, broken bars, and stator inter-turn short circuits make the machines unsymmetrical. Due to this fact, the complexity reduction techniques such as exploitation of symmetry by considering symmetric and non-symmetric boundary conditions and model order reduction becomes more tedious. The only best optimal and reliable way for fault diagnostics is to simulate the entire machine at various rotor positions. Since the solution at a distinct rotor position is independent of the solution at subsequent rotor positions, the total \"n\" rotor steps can be divided into various segments. The magneto-static problems of different rotor position sectors such as, (\(0\rightarrow\theta_{1}\)), (\(\theta_{2}\rightarrow\theta_{3}\)),..., (\(\theta_{n-1}\rightarrow\theta_{n}\)), (\(\theta_{n+1}\to 2\pi\)) can be divided among the workers for parallel processing.  
Figure 3 shows the required steps to calculate the inductances at different rotor positions. The computer cluster consists of four computers making a local area network (LAN). Each computer is Intel(R) Core(TM) i7-7500 CPU @ 3.41 GHz with 8 GB RAM and four cores. The finite element method (FEM) based model is constructed using open-access software FEMM 4.0. For making the model and collecting the results, FEMM is interfaced with MATLAB. After making the machine geometry and winding configuration on FEMM, MATLAB works as a job scheduler. It divides the total number of rotor steps among worker computers and their cores and receives the end results.  
For better accuracy, a considerable number of mesh elements (250,160) with (125,177) nodes are solved having a precision of \(1\times 10^{-8}\) at each rotor step. The FEMM 4.0 achieves this precision through a conjugate gradient solver with the help of multiple successive approximation iterations. Since the inductance profiles change with the change in the air gap, the selection of an appropriate rotor step size is very important. The changing air gap is the function of stator and rotor slot openings [37], which becomes very prominent when the openings on both sides align with each other and lead to the abruptly changing inductance derivatives. As the rotor and stator, slots are different in number, their least count multiple (LCM) can be a minimum choice regarding the number of rotor steps. By doing so, some phase inductances can be considered as shifted copies of the other phase inductances. In this case, the number of total rotor steps or solution samples will be divisible by the total number of rotor bars and the number of samples corresponding to 120 and 240 degrees for stator phases. For example,  
Figure 3: The division of rotor steps among various computers and their cores for parallel computation and the procedure of inductances calculation.  
if the number of rotor steps is (\(40\times 48=1920\)), the \(L_{bb}\) will be equal to (\(120\times 1920/360\)) samples shifted copy of \(L_{aa}\). The same is true in the case of rotor bar inductances, only self and mutual inductances of a single loop need to be calculated, the rest of them are shifted copies. However, special attention is needed if the fault changes all the inductances symmetrically or not.