# Induction Motor Dynamics Regimes: A Comprehensive Study of Mathematical Models and Validation
## 2 Transient Process Modeling in Induction Motor
### Generalized Electric Machine  
The theory of electric machines traditionally examines different types of electromechanical transformations separately. This approach stems from historical developments in the field, where scientists in electromechanics established distinct theories for various machine types [70; 71]. While these theories emphasize the unique features of each machine, they also highlight significant similarities in design principles and calculation methods [72]. One key similarity is the presence of a sinusoidal magnetic field in the motor air gap. To develop a generalized theory of electric machines, we consider an idealized two-pole, two-phase symmetrical electric machine. This idealized machine features mutually perpendicular windings on both the rotor and stator, as illustrated in Figure 3[67]. A two-pole machine was selected because the processes in two-pole and multi-pole machines are analogous. In a two-pole machine, electrical radians align with geometric radians, simplifying the comparison of the rotor position relative to the stator phases, especially during transient processes. In a two-phase machine, spatially displacing the windings by \(\pi/2\) generates a circular magnetic field. This configuration combined with a uniform air gap eliminates mutual inductive coupling between windings and simplifies the machine equations. For this reason, three-phase machines are frequently converted into two-phase machines for analytical purposes [73; 74]. The generalized electric machine assumes a uniform air gap, with no slots on the stator or rotor. They are modeled as current layers with a sinusoidal magnetomotive force distribution. Feeding the windings with alternating current produces a sinusoidal magnetic field in the air gap, given the uniform magnetic circuit and the absence of saturation. This  
Figure 3: Generalized electric machine scheme. generalized model represents a pair of intermoving windings and serves as a mathematical framework to analyze the processes in various electric machines. By reducing real machines to this model, operating and transient processes can be studied more effectively [75]. The mathematical model of the generalized electric machine consists of a system of differential equations. These include equilibrium equations for the winding voltages and a torque balance equation describing the machine's motion. For simplicity, rotor winding parameters are reduced to the stator side, with the reduction factors omitted for clarity. In the 1\(\alpha\), 1\(\beta\) and 2\(\alpha\), 2\(\beta\) coordinate systems, the following equations govern the system [22]:  
\[U_{1\alpha}=R_{1}i_{1\alpha}+\frac{d}{dt}\psi_{1\alpha}, \tag{20}\]  
\[U_{1\beta}=R_{1}i_{1\beta}+\frac{d}{dt}\psi_{1\beta}, \tag{21}\]  
\[U_{2\alpha}=R_{2}i_{2\alpha}+\frac{d}{dt}\psi_{2\alpha}, \tag{22}\]  
\[U_{2\beta}=R_{2}i_{2\beta}+\frac{d}{dt}\psi_{2\beta}. \tag{23}\]  
Here:  
\(U_{1\alpha},U_{1\beta},U_{2\alpha},U_{2\beta}\) are the applied voltages in the stator and rotor windings;  
\(i_{1\alpha},i_{1\beta},i_{2\alpha},i_{2\beta}\) represent the winding currents;  
\(R_{1}\), \(R_{2}\) are the stator and rotor winding resistances, respectively. The rotor's motion is described by the equation:  
\[T_{e}=T_{l}+J\frac{d\Omega}{dt}, \tag{24}\]  
where  
\(T_{e}\) is the electromagnetic torque generated by the machine;  
\(T_{l}\) is the applied load torque;  
\(J\frac{d\Omega}{dt}\) is the dynamic torque;  
\(J\) represents the combined inertia of the rotor and any connected equipment (reduced to the rotor shaft);  
\(\Omega\) is the angular velocity. For a generalized electric machine, when 2\(p=2\), \(\Omega=\omega\). These equations comprehensively describe both dynamic and static behaviors of a generalized electric machine.