## 2 Mathematical Model of the Induction Motor
### Generalized Machine in Common Coordinates Axes Rotating at Arbitrary Speed  
To simplify the analysis of an electric machine, we aim to make the magnetic fields of the stator and rotor appear stationary relative to each other. This can be achieved in two ways:  
* Braking the rotor's rotating windings: this means that, from a mathematical perspective, we adjust the reference frame so that the rotor's windings appear stationary, even though they are physically rotating;Rotating the stator's windings at the same speed as the rotor: in this case, we imagine that the stator windings are rotating at the same speed as the rotor, effectively making the stator and rotor fields appear stationary relative to each other. In both scenarios, we must adjust the frequency of the currents in the windings to reflect this new reference frame, where the fields are stationary. To account for this, an electromotive rotational force is introduced into the voltage and current equations. This adjustment ensures that the real electrical behavior of the machine is accurately captured in the equations. Next, we convert the system of equations into the \(x,y\) coordinate system, which rotates at an arbitrary speed. This common coordinate system simplifies the analysis and makes it easier to model the dynamics of the machine (see Figure 1). The stator-winding voltage balance equation expressed using space vectors \(\dot{U}_{1}\); \(\dot{\psi}_{1}\); \(l_{1}\) is  
\[\dot{U}_{1}=R_{1}\dot{I}_{1}+\frac{d\dot{\psi}_{1}}{dt}.\]  
Thus, equation should be multiplied by \(e^{-j(\omega_{x}t+\gamma_{x0})}\). This multiplier accounts for the rotation frequency difference between the stationary \(1\alpha,1\beta\) coordinate system and the rotating \(x,y\) coordinate system. \[\dot{\psi}_{1}(x,y)=\dot{\psi}_{1}e^{-j(\omega_{x}t+\gamma_{x0})}=\psi_{1m}e^{ j((\omega_{1}-\omega_{x})t+\omega_{0}-\gamma_{x0}-\phi)}; \tag{12}\]  
\[\dot{U}_{1}(x,y)=\dot{U}_{1}\cdot e^{-j(\omega_{x}t+\gamma_{x0})}=U_{1m}\cdot e ^{j(\omega_{1}t+\omega_{0})}\cdot e^{-j(\omega_{x}t+\gamma_{x0})}= \tag{13}\]  
\[=U_{1m}\cdot e^{j((\omega_{1}-\omega_{x})t+\omega_{0}-\gamma_{x0})};\]  
\[\dot{I}_{1}(x,y)=\dot{I}_{1}e^{-j(\omega_{x}t+\gamma_{x0})}=I_{1m}\cdot e^{j( (\omega_{1}-\omega_{x})t+\omega_{0}-\gamma_{x0}-\phi)}. \tag{14}\]  
where \(\phi\) is the angle of displacement between the vectors, \(\dot{\psi}_{1},\dot{I}_{1}\) relative to \(\dot{U}_{1}\) at the time point \(t=0\). The derivative:  
\[ \begin{array}{rl}\frac{d\dot{\psi}_{1}(x,y)}{dt}=\frac{d}{dt}&\left\{\dot{ \psi}_{1}\cdot e^{-j(\omega_{x}t+\gamma_{x0})}\right\}\\ &=\left(\frac{d\dot{\psi}_{1}}{dt}\right)\cdot e^{-j(\omega_{x}t+\gamma_{x0}) }-j\omega_{x}\dot{\psi}_{1}e^{-j(\omega_{x}t+\gamma_{x0})}.\end{array}  \tag{15}\]  
From this, we obtain the following:  
\[\dot{U}_{1}(x,y)=R_{1}\dot{I}_{1}(x,y)+\frac{d\dot{\psi}_{1}(x,y)}{dt}+j\omega _{x}\dot{\psi}_{1}(x,y). \tag{16}\]  
Figure 1: Coordinate system and voltages space vectors. The real and imaginary parts of this equation yield the voltage balance equations in the \(x,y\) axes:  
\[U_{1x}=\frac{d\psi_{1x}}{dt}-\omega_{x}\psi_{1y}+R_{1y}+R_{1}i_{1x}\ \ U_{1y}= \frac{d\psi_{1y}}{dt}-\omega_{x}\psi_{1x}+R_{1y}+R_{1}i_{1y} \tag{17}\]  
Similarly, the voltage balance equation for the rotor windings is the following:  
\[\dot{U}_{2}=\frac{d\dot{\psi}_{2}}{dt}+R_{2}\dot{I}_{2}\.\]  
Multiplying by \(e^{-j((\omega-\omega_{x})t+\gamma_{0}-\gamma_{x0})}\) accounts for the relative difference in rotation speed between the \(2\alpha,2\beta\) and \(x,y\) coordinate systems. Applying analogous transformations as for the stator equations, we obtain the following:  
\[\dot{U}_{2}(x,y)=\frac{d\dot{\psi}_{2}(x,y)}{dt}+j(\omega-\omega_{2})\dot{\psi }_{2}(x,y)+R_{2}\dot{I}_{2}(x,y). \tag{18}\]  
Transforming the generalized electric machine equations into a coordinate system that is common for both the stator and rotor and rotates with the free rotation frequency, we obtain the voltage balance equations in the \(x,y\) axes.