## 2 Mathematical Model of the Induction Motor
### Generalized Electric Machine Equations  
Generalized electric machine equations are a system of differential equations that include the equilibrium equations for the windings voltages of machine and the equation of motion (torque balance equation on the machine shaft). The rotor windings' parameters are reduced to the stator windings (with the reduction marks omitted). Thus, on axes \(1\alpha\), \(1\beta\) and \(2\alpha\), \(2\beta\)[49]:  
\[U_{1\alpha} =R_{1}i_{1\alpha}+\frac{d}{dt}\psi_{1\alpha}; \tag{1}\] \[U_{1\beta} =R_{1}i_{1\beta}+\frac{d}{dt}\psi_{1\beta};\] (2) \[U_{2\alpha} =R_{2}i_{2\alpha}+\frac{d}{dt}\psi_{2\alpha};\] (3) \[U_{2\beta} =R_{2}i_{2\beta}+\frac{d}{dt}\psi_{2\beta}. \tag{4}\]  
In these equations, the flux linkages of the windings are represented as follows:  
\[\psi_{1\alpha} =L_{1\alpha}i_{1\alpha}+M_{1\alpha 2\alpha}i_{2\alpha}+M_{1 \alpha 2\beta}i_{2\beta} \tag{5}\] \[\psi_{1\beta} =L_{1\beta}i_{1\beta}+M_{1\beta 2\alpha}i_{2\alpha}+M_{1\beta 2 \beta}i_{2\beta}\] (6) \[\psi_{2\alpha} =L_{2\alpha}i_{2\alpha}+M_{2\alpha 1\alpha}i_{1\alpha}+M_{2\alpha 1 \beta}i_{1\beta}\] (7) \[\psi_{2\beta} =L_{2\beta}i_{2\beta}+M_{2\beta 1\alpha}i_{1\alpha}+M_{2\beta 1 \beta}i_{1\beta} \tag{8}\]  
where \(U_{1\alpha},U_{1\beta},U_{2\alpha},U_{2\beta}\)--are the voltages applied to the machine windings;  
\(i_{1\alpha},i_{1\beta},i_{2\alpha},i_{2\beta}\) are the currents in the windings;\(R_{1},R_{2}\) are the resistances in stator and rotor windings, respectively;  
\(L_{1\alpha},L_{1\beta},L_{2\alpha},L_{2\beta}\)--are the self-induction inductances of the stator and rotor windings;  
\(M_{1\alpha 2\alpha},M_{1\alpha 2\beta},M_{1\beta 2\beta},M_{2\alpha 1\alpha},M_{2 \alpha 1\beta},M_{1\beta 2\beta},M_{2\beta 1\beta}\) are the mutual inductances between the stator and rotor windings. Since the air gap of the machine is constant along the stator circumference and the magnetic field is not saturated, the self-inductance coefficients \(L_{1\alpha},L_{1\beta},L_{2\alpha},L_{2\beta}\) are constant. The symmetry of the machine along the \(\alpha\), \(\beta\) axes and the sinusoidal distribution of the winding magnetomotive force allow us to establish the following equalities:  
\(M_{1\alpha 2\kappa}=M_{2\alpha 1\kappa}=M_{1\beta 2\beta}=M_{2\beta 1\beta}=M \cdot\gamma\);  
\(M_{1\alpha 2\beta}=M_{2\beta 1\alpha}=-M\cdot sinsin\ \gamma\) ;  
\(M_{1\beta 2\alpha}=M_{2\alpha 1\beta}=M\cdot sinsin\ \gamma\) ;  
where \(M\)--mutual inductance coefficient when windings axes have a specific angle. Rotor movement equation:  
\[T_{e}=T_{l}+J\frac{d\Omega}{dt}, \tag{9}\]  
where \(T_{e}\)--developed by machine electromagnetic torque;  
\(T_{l}\)--oad torque;  
\(J\frac{d\Omega}{dt}\)--dynamic torque;  
\(J\) is the combined inertia of the rotor and the equipment inertia, reduced to the rotor shaft;  
\(\Omega\)--rotation speed, and when \(2p=2\), where p is the number of pole pairs, \(\Omega=\omega\) (for generalized electric machine). The equations fully define dynamic and static processes in generalized electric machine. Given \(\dot{U}_{1}\) for stator winding and \(\dot{U}_{2}\) for the rotor winding:  
\[\dot{U}_{1}=U_{1m}\cdot e^{j(\omega_{1}t+\omega_{0})}=U_{1m}[cos\ cos\ (\omega_{1}t+\alpha_{0})+j\ sin\ sin\ (\ \omega_{1}t+\alpha_{0})] \tag{10}\]  
\[U_{1\beta} =R_{1}i_{1\beta}+\frac{d}{dt}\psi_{1\beta};\]0  
In these equations:  
\(\omega_{1}\)--angular rotation speed of network;  
\(\omega_{2}\)--angular rotation speed of vector \(\dot{U}_{2}\) comparing to rotor axes;  
\(\alpha_{0}\)--phase angle of vector \(\dot{U}_{1}\) relative to the axis \(1\alpha\) at the time point \(t=0\);  
\(\alpha_{2}\)--phase angle of vector \(\dot{U}_{2}\) relative to the axis \(2\alpha\) at the time point \(t=0\). Obviously, the system of differential Equations (1)-(4) and (9) becomes cumbersome due to periodic coefficients. Additionally, the coefficients of the current in the flux linkages are variable functions, making these equations extremely challenging to solve.