# The Cluster Computation-Based Hybrid FEM-Analytical Model of Induction Motor for Fault Diagnostics
## 5 The Simulation Results  
At each rotor position, every individual stator phase is energized with a unity DC current, and relevant inductances are calculated by integrating the magnetic vector potential over the coil area as shown by the following equations. \[L_{self}=\frac{\int_{Si}A.J\ da}{i^{2}} \tag{15}\]  
\[L_{mutual}=\frac{n_{2}}{i_{1}a_{2}}\biggl{(}\int_{+si}A_{1}da_{1}-\int_{-si}A_ {1}da_{2}\biggr{)} \tag{16}\]  
where \(A\) is the vector potential, \(J\) is the current density, \(n\) is the number of turns per phase, \(i\) is the phase current, the subscript \"\(si\)\" is for surface integral. The first bracket term in Equation (16) is the integration of the vector potential of coils with positive current or the coils pointing out of the page. The second term corresponds to the coils with negative current or pointing into the page, \(a_{2}\) is the cross-sectional area of the coil, which is approximately equal to the slot area multiplied with the filling factor. The motor's magnetic flux distribution after the energizing phase \"\(a\)\" with 1 A DC current is shown in Figure 4. The highlighted slots contain phase \"\(a\)\" winding whose surface integral of magnetic potential is equal to self-inductance \(Laa\) at the specific rotor position as in Equation (15). Similarly, all stator-stator self and mutual and stator-rotor mutual inductances can be calculated. The rotor-rotor self and mutual inductances are calculated by energizing a single rotor loop with unity DC current as shown in Figure 5. Only one rotor loop needs to be energized and solved for its inductances, and the rest of them possess the same solution with a phase shift equal to the angle difference between them. Figure 4: The flux distribution with energized phase “\(a\)” with I\({}_{\text{DC}}\) = 1 A, and the selection of slots for vector potential integral for calculation of self-inductance \(L_{aa}\). Since the solution is for the 2D model, the effect of end windings is compensated by using additional end winding leakage inductance and resistance using the following analytical formulas. \[L_{ew}=\ \frac{Q_{s}}{m}q\bigg{(}\frac{Z_{q}}{a}\bigg{)}^{2}\mu_{o}l_{w}\lambda_{w} \tag{17}\]  
where \(Q\)s is the number of stator slots, \(Zq\) is the number of conductors per slot, \(a\) is the number of winding parallel paths per phase, \(m\) is the total number of stator phases, and \(q\) is the number of slots in a pole captured by an individual phase. \(lw\) is the average length of the end winding, \(\lambda w\) is the permeance factor which is 0.20 for the motor under investigation. The same formula can be used to calculate the leakage inductance of the rotor end rings. For building the resistance matrix, various stator- and rotor-related resistances are calculated using resistivity formula where the skinning and proximity effects are neglected because of the DC supply current;  
\[R=\ \frac{\rho l}{A} \tag{18}\]  
where \(\rho\) is the resistivity, A is the cross-sectional area, and l is the length of the conductor. The effective slot area (ESA) is equal to the total area of slot multiplied with the filling factor, which is 0.60 for the machine under investigation. The conductor cross-sectional area can be calculated by dividing the ESA with the total number of conductors in the slot, which are 17 in this case. The resistance of the end windings is included by increasing the length of the per phase conductor corresponding to the length of the end winding. Since the stator and rotor windings are energized with unity DC current, the effects such as proximity, skinning, material saturation, and eddy currents are neglected as the focus is towards the simulation time reduction. However, they can be included in the online section analytically. Figure 6 shows various inductances as a function of the rotor position. All self and mutual inductances are the functions of the air gap, which changes with the stepping rotor. This phenomenon is evident in the inductance profiles, which are calculated with a rotor step size of 0.1875 degrees.