# Induction Motor Dynamics Regimes: A Comprehensive Study of Mathematical Models and Validation
## 2 Transient Process Modeling in Induction Motor
### Generalized Machine in a Common Coordinate System Rotating at Arbitrary Speed  
The mutual immobility of the stator and rotor magnetic fields can be achieved under the following conditions:  
* The rotor's rotating windings are conceptually treated as stationary (braked); or
* The stator's stationary windings are assumed to rotate at the same speed as the rotor. Additionally, the frequencies of the currents must be adjusted to align with these conditions. To preserve power invariance and operate with actual amplitudes of voltages and currents, an electromotive rotational force is introduced into the voltage and current balance equations. To generalize, we transform the equations of an electric machine into the _x,y_ coordinate system. This system is common to both the stator and rotor and rotates at an arbitrary angular speed (see Figure 4). The stator voltage balance equation, expressed in terms of space vectors \(\dot{U}_{1}\), \(\dot{\Psi}_{1}\), \(\dot{I}_{1}\), is given by:  
\[\dot{U}_{1}=R_{1}\dot{I}_{1}+\frac{d\dot{\Psi}_{1}}{dt} \tag{27}\]  
To account for the rotational difference between the stationary \(1\alpha,1\beta\) coordinate system and the rotating \(x,y\) coordinate system, the equation is multiplied by \(e^{-j(\omega_{x}t+\gamma_{x0})}\). \[\dot{\psi}_{1}(x,y)=\dot{\psi}_{1}e^{-j(\omega_{x}t+\gamma_{x0})}=\psi_{1m}e^{ j((\omega_{1}-\omega_{x})t+\alpha_{0}-\gamma_{x0}-\phi)}; \tag{28}\]  
\[\dot{U}_{1}(x,y)=\dot{U}_{1}\cdot e^{-j(\omega_{x}t+\gamma_{x0})}=U_{1m}\cdot e ^{j(\omega_{1}t+\omega_{0})}\cdot e^{-j(\omega_{x}t+\gamma_{x0})}= \tag{29}\]  
\[=U_{1m}\cdot e^{j((\omega_{1}-\omega_{x})t+\alpha_{0}-\gamma_{x0})};\]  
\[\dot{I}_{1}(x,y)=\dot{I}_{1}e^{-j(\omega_{x}t+\gamma_{x0})}=I_{1m}\cdot e^{j( (\omega_{1}-\omega_{x})t+\alpha_{0}-\gamma_{x0}-\phi)}. \tag{30}\]  
where \(\phi\) represents the displacement angle between the vectors \(\dot{\psi}_{1}\) and \(\dot{I}_{1}\) relative to \(\dot{U}_{1}\) at \(t=0\). The time derivative of the flux linkage is expressed as:  
\[\frac{d\dot{\psi}_{1}(x,y)}{dt}=\frac{d}{dt}\Big{\{}\dot{\psi}_{1}\cdot e^{-j (\omega_{x}t+\gamma_{x0})}\Big{\}}=\left(\frac{d\dot{\psi}_{1}}{dt}\right) \cdot e^{-j(\omega_{x}t+\gamma_{x0})}-j\omega_{x}\dot{\psi}_{1}e^{-j(\omega_{ x}t+\gamma_{x0})}. \tag{31}\]  
Figure 4: Coordinate system and voltages space vectors. Substituting, the stator voltage balance in the x, y coordinate system becomes:  
\[\dot{U}_{1}(x,y)=R_{1}\dot{I}_{1}(x,y)+\frac{d\dot{\psi}_{1}(x,y)}{dt}+j\omega_{x }\dot{\psi}_{1}(x,y). \tag{32}\]  
Separating real and imaginary parts, the voltage balance equations in the \(x\),\(y\) coordinate axes are:  
\[ \begin{array}{c}U_{1x}=\frac{d\dot{y}_{1x}}{dt}-\omega_{x}\psi_{1y}+R_{1y}+R_ {1}i_{1x}\\ \\ U_{1y}=\frac{d\dot{y}_{1y}}{dt}-\omega_{x}\psi_{1x}+R_{1y}+R_{1}i_{1y}\end{array}  \tag{33}\]  
For the rotor windings, the voltage balance equation is:  
\[\dot{U}_{2}=\frac{d\dot{\psi}_{2}}{dt}+R_{2}\dot{I}_{2}\;. \tag{34}\]  
Multiplying by \(e^{-j((\omega-\omega_{x})t+\gamma_{0}-\gamma_{x0})}\) to account for the rotational difference between the \(2\alpha\), \(2\beta\) and \(x\), \(y\) systems, and applying similar transformations, we obtain:  
\[\dot{U}_{2}(x,y)=\frac{d\dot{\psi}_{2}(x,y)}{dt}+j(\omega-\omega_{2})\dot{ \psi}_{2}(x,y)+R_{2}\dot{I}_{2}(x,y). \tag{35}\]  
Breaking this into components yields:  
\[ \begin{array}{c}U_{2x}=\frac{d\dot{y}_{2x}}{dt}-(\omega-\omega_{x})\dot{ \psi}_{2y}+R_{2}\dot{i}_{2x}\\ U_{2y}=\frac{d\dot{y}_{2y}}{dt}-(\omega-\omega_{x})\dot{\psi}_{2x}+R_{2}\dot{ i}_{2y}\end{array} . \tag{36}\]  
By appropriately selecting the angular frequencies \(\omega_{1}\), \(\omega_{2}\), \(\omega\), \(\omega_{x}\), the induction machine can be analyzed in any coordinate system. A key feature of this model is the inclusion of rotational electromotive forces: \(\omega_{x}\psi_{1y}\), \(\omega_{x}\psi_{1x}\), \((\omega-\omega_{x})\psi_{2y}\), \((\omega-\omega_{x})\psi_{2x}\).