# Induction Motor Dynamics Regimes: A Comprehensive Study of Mathematical Models and Validation
## 3 Models of Induction Motors
### The Model of the Induction Motor in the \(d\),\(q\) Coordinate System in Flux Linkages  
If Equations (33) and (36) are rewritten in terms of flux linkage derivatives, the system of equations takes the following form:  
\[\left.\begin{array}{c}\frac{d\psi_{1d}}{dt}=U_{1d}-R_{1}i_{d}+\omega\psi_{1q} \\ \frac{d\psi_{1q}}{dt}=U_{1q}-R_{1}i_{q}-\omega\psi_{1d}\\ \frac{d\psi_{2d}}{dt}=-R_{2}i_{2d}+(\omega_{x}-\omega)\psi_{2d}\\ \frac{d\psi_{2q}}{dt}=-R_{2}i_{2q}-(\omega_{x}-\omega)\psi_{2d}\end{array} \right\}, \tag{45}\]  
The torque balance equation is given by:  
\[T_{M}\frac{d\omega}{dt}=[T_{e}-T_{l}]\, \tag{46}\]  
where the electromagnetic torque is expressed as:  
\[T_{e}=X_{ad}\big{(}i_{2d}i_{1q}-i_{2d}i_{1d}\big{)}.\]  
The stator and rotor currents can be expressed in terms of flux linkages as:  
\[\left.\begin{array}{c}i_{1d}=\left(X_{2}\cdot\psi_{1d}-X_{ad}\cdot\psi_{2d} \right)/\Delta\\ i_{1q}=\left(X_{2}\cdot\psi_{1q}-X_{ad}\cdot\psi_{2q}\right)/\Delta\\ i_{2d}=\left(X_{1}\cdot\psi_{2d}-X_{ad}\cdot\psi_{1d}\right)/\Delta\\ i_{2q}=\left(X_{1}\cdot\psi_{2q}-X_{ad}\cdot\psi_{1q}\right)/\Delta\end{array} \right\}, \tag{47}\]where \(\Delta=X_{1}X_{2}-X_{ad}X_{ad}\).  
Equations (45)-(47) represent the mathematical model of the induction motor (IM) in terms of flux linkages. This IM model, expressed in the \(d\_q\) coordinate system (referred to as model III) enables the analysis of motor operation in autonomous regimes, including scenarios where the motor operates independently or is connected to a power supply network with an infinitely large capacity.  
Induction Motor Model in Flux Linkages with Rotor Windings Electromagnetic Time Constant in \(d\_q\) Coordinate System  
If Equations (33) and (36) are reformulated to include flux linkages with rotor windings' electromagnetic time constant derivatives, the system of equations is as follows:  
\[\left.\begin{array}{l}U_{d}=\frac{d\varphi_{1d}}{dt}-\omega\psi_{1d}+R_{1}i_ {d}\\ U_{q}=\frac{d\varphi_{1d}}{dt}+\omega\psi_{1d}+R_{1}i_{d}\\ 0=-\frac{d\varphi_{2d}}{dt}-\frac{\varphi_{1d}}{T_{R}}-\frac{X_{ad}}{T_{R}}i_ {d}+\varphi_{2d}(\omega_{0 el}-\omega)\\ 0=-\frac{d\varphi_{2d}}{dt}-\frac{\varphi_{1d}}{T_{R}}+\frac{X_{ad}}{T_{R}}i_ {q}-\varphi_{2d}(\omega_{0 el}-\omega)\\ \end{array}\right\}, \tag{48}\]  
where \(T_{e}\) the electromagnetic torque, is calculated as:  
\[T_{e}=X_{ad}(i_{2d}i_{1q}-i_{2q}i_{1d}).\]  
The currents can be expressed in terms of flux linkages:  
\[\left.\begin{array}{l}i_{1d}=(X_{2}\cdot\psi_{1d}-X_{ad}\cdot\psi_{2d})/ \Delta\\ i_{1q}=\left(X_{2}\cdot\psi_{1q}-X_{ad}\cdot\psi_{2q}\right)/\Delta\\ i_{2d}=\left(X_{1}\cdot\psi_{2d}-X_{ad}\cdot\psi_{1d}\right)/\Delta\\ i_{2q}=\left(X_{1}\cdot\psi_{2q}-X_{ad}\cdot\psi_{1d}\right)/\Delta\\ \end{array}\right\}, \tag{49}\]  
where \(\Delta=X_{1}X_{2}-X_{ad}X_{ad}\).  
Model IV incorporates the rotor winding electromagnetic time constant, a critical parameter governing the evolution of flux linkages and currents within the rotor circuit during transient events. This time constant directly influences the motor's response to rapid changes in voltage, load, or other external conditions. For example, during starting or voltage dips, it determines the decay rate of transient components and the stabilization of steady-state performance. By accounting for this parameter, Model IV enables accurate analysis of motor parameter variations during rundown regimes and transient phenomena, such as torque oscillations and current peaks. This detailed modeling is essential for motor protection and optimizing system performance, offering insights beyond those achievable with simpler models.