## 2 Theoretical Background  
### The Modeling of Induction Motor  
Electrical machines' modeling is the first milestone in machine design, control, analysis and diagnostics. The famous modeling techniques are divided into two major categories: analytic and numerical. The commonly used analytic methods are two-axis theory [25], winding function method (WFM) [26; 27; 28], reluctance network [29] and convolution theorem-based models [30]. The finite element method (FEM)-based models are typical examples of numerical modeling of electrical machines. Although the numerical models have the least number of approximations, the complexity and computational time is the biggest drawback [31]. It becomes worst in the field of fault diagnostics, where the motor's symmetry cannot be exploited to reduce the simulation time. For the fault diagnostic in the transient regime, the V-shaped pattern made by the sideband frequencies is significant. This pattern generates because of the varying slip until the steady state region. This nonstationary signal moves in a bandwidth from 0 Hz to 50 Hz. Thus, the higher-order slotting harmonics become less significant for the analysis of the fault-based, nonstationary signal. The winding function-based model with much less complexity than the corresponding FEM-based models is essential to differentiate among several harmonics. Hence, the air gap is considered constant by neglecting the slot openings with the magnetic material's infinite permeability. The stator winding is approximated as a pure sinusoid, while the rotor turn function represents its actual cage structure. Various self and mutual inductances can be calculated using the winding function formula as in Equation (1). The analytical formulas for the calculation of different inductances given in Equations (2)-(6) are the solution of (1) for the winding functions, as described in [32]:  
\[L_{ij}(\theta)=\ \mu_{\sigma}rI\ \int_{0}^{p\pi}g^{-1}(\varphi,\theta)N_{i}( \varphi,\theta)N_{j}(\varphi,\theta)d\theta \tag{1}\]  
\[L_{m}=\frac{\mu_{\sigma}rI}{g}\ \Big{(}\ \frac{p}{2}\Big{)}\bigg{(}\ \frac{N_{s}}{p}\ \Big{)}^{2}\pi \tag{2}\]  
\[L_{AB}=L_{BC}=L_{CA}=-\frac{L_{m}}{2} \tag{3}\]  
\[L_{rkk}=\bigg{(}\frac{\mu_{\sigma}rI}{g}\bigg{)}\alpha_{r}\ \bigg{[}1-\frac{ \alpha_{r}}{p\pi}\bigg{]}\Big{(}\frac{p}{2}\Big{)} \tag{4}\]  
\[L_{rki}=\bigg{(}\frac{\mu_{\sigma}rI}{g}\bigg{)}\bigg{[}-\frac{\alpha_{r}^{2} }{p\pi}\bigg{]}\Big{(}\frac{p}{2}\Big{)} \tag{5}\]  
\[L_{sri}(\theta_{r})=\bigg{(}\frac{\mu_{\sigma}rlN_{s}}{g}\bigg{)}\ \sin(\frac{\alpha_{r}}{2})\ cos(\theta_{r}+(i-1)\alpha_{r}+\frac{\alpha_{r}}{2}) \tag{6}\]  
where \(\mu_{\sigma}\) is the permeability of the free space; \(r\) is the machine's radius, which is taken as the center of the air gap; \(l\) is the machine's effective length; \(g\) is the air gap width; \(\alpha_{r}\) is the angular displacement between two consecutive rotor bars; \(p\) is the number of the poles; \(N_{s}\) is the number of effective turns per phase in the stator; and \(\theta\) and \(\varphi\) are the rotor and stator angular positions from a reference point. In Equation (6), the inductance is multiplied with a cosine function of \(\theta r\) to emulate the rotating rotor, where integer \(i\) changes the angle of the respective rotor bar from the stator reference point. \(Lm\) is the magnetization inductance, \(L_{AB}\), \(L_{BC}\) and \(L_{CA}\) are stator mutual inductances, \(L_{rkk}\) and \(L_{rki}\) are rotor self and mutual inductances and \(L_{sri}\) is the mutual inductance between the stator and rotor. The performance parameters, such as torque, speed, currents and voltages, can be calculated using Equations (7)-(9):  
\[\left[ \begin{array}{c}\text{Vs}\\ 0\end{array}