# Induction Motor Dynamics Regimes: A Comprehensive Study of Mathematical Models and Validation
## 2 Transient Process Modeling in Induction Motor
### Coordinate Conversion
Therefore, the currents \(i^{\prime}_{1A}=i_{1A}-i_{0}\), \(i^{\prime}_{1B}=i_{1B}-i_{0}\), \(i^{\prime}_{1C}=i_{1C}-i_{0}\) can be represented as the projections of \(\dot{I}\) onto the 1A, 1B, 1C ensuring that \(i^{\prime}_{1A}+i^{\prime}_{1B}+i^{\prime}_{1C}=0\), as shown in Figure 2.  
The relationship between the space vector components and the phase currents can be expressed as follows:  
\[i^{\prime}_{1A}+i^{\prime}_{1B}+i^{\prime}_{1C}=0 \tag{4}\]  
Figure 2: Current space vector. \[i^{\prime}_{1B}=Icos\bigg{(}\gamma+\frac{\pi}{2}-\theta_{0 el}+\frac{2\pi}{3}\bigg{)} =-Isin\bigg{(}\gamma-\theta_{0 el}+\frac{2\pi}{3}\bigg{)}, \tag{5}\]  
\[i^{\prime}_{1C}=Icos\bigg{(}\gamma+\frac{\pi}{2}-\theta_{0 el}-\frac{2\pi}{3} \bigg{)} =-Isin\bigg{(}\gamma-\theta_{0 el}-\frac{2\pi}{3}\bigg{)} \tag{6}\]  
In the longitudinal axis \(d\), and transverse axis \(q\), the currents are given as:  
\[i_{d}=Icos\Big{(}\gamma+\frac{\pi}{2}\Big{)}=-Isin\gamma, \tag{7}\]  
\[i_{d}=Icos\Big{(}\gamma+\frac{\pi}{2}\Big{)}=-Isin\gamma. \tag{8}\]  
Substituting Equations (4)-(6) in Equations (1)-(3) and using (7) and (8), the phase current can be expressed as:  
\[i_{1B} =i^{\prime}_{1B}+i_{0}\]0  
\[i_{1B} =i^{\prime}_{1B}+i_{0}\]1  
\[i_{1B} =i^{\prime}_{1B}+i_{0}\]2  
Equations (9)-(11) can be written in matrix form as:  
\[ \begin{bmatrix}i_{1A}\\ i_{1B}\\ i_{1C}\end{bmatrix} = \begin{bmatrix}cos\theta_{0 el}&-sin\theta_{0 el}&1\\ cos\big{(}\theta_{0 el}-\frac{2\pi}{3}\big{)}&-sin\big{(}\theta_{0 el}-\frac{2 \pi}{3}\big{)}&1\\ cos\big{(}\theta_{0 el}+\frac{2\pi}{3}\big{)}&-sin\big{(}\theta_{0 el}+\frac{2 \pi}{3}\big{)}&1\end{bmatrix} . \tag{12}\]  
Here, the transformation matrix is:  
\[[A_{1}^{-1}]= \begin{bmatrix}cos\theta_{0 el}&-sin\theta_{0 el}&1\\ cos\big{(}\theta_{0 el}-\frac{2\pi}{3}\big{)}&-sin\big{(}\theta_{0 el}-\frac{2 \pi}{3}\big{)}&1\\ cos\big{(}\theta_{0 el}+\frac{2\pi}{3}\big{)}&-sin\big{(}\theta_{0 el}+\frac{2 \pi}{3}\big{)}&1\end{bmatrix}  \tag{13}\]  
The Park transformation matrix \(A_{1}\), is given as:  
\[[A_{1}]=\frac{2}{3} \begin{bmatrix}cos\theta_{0 el}&-cos\big{(}\theta_{0 el}- \frac{2\pi}{3}\big{)}&cos\big{(}\theta_{0 el}+\frac{2\pi}{3}\big{)}\\ -sin\theta_{0 el}&-sin\big{(}\theta_{0 el}-\frac{2\pi}{3}\big{)}&-sin\big{(} \theta_{0 el}+\frac{2\pi}{3}\big{)}\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}\end{bmatrix}  \tag{14}\]  
Using this matrix, phase currents \(i_{1A},i_{1B},i_{1C}\) can be transformed into \(d\), \(q\), \(0\) coordinates:  
\[i_{d}=\frac{2}{3}\bigg{[}i_{1A}cos\theta_{0 el}+i_{1B}cos\bigg{(}\theta_{0 el}- \frac{2\pi}{3}\bigg{)}+i_{1C}cos\bigg{(}\theta_{0 el}+\frac{2\pi}{3}\bigg{)} \bigg{]}, \tag{15}\]  
\[i_{q}=\frac{2}{3}\bigg{[}-i_{1A}sin\theta_{0 el}-i_{1B}sin\bigg{(}\theta_{0 el}- \frac{2\pi}{3}\bigg{)}-i_{1C}cos\bigg{(}\theta_{0 el}+\frac{2\pi}{3}\bigg{)} \bigg{]} \tag{16}\]  
\[i_{0}=\frac{1}{3}(i_{1A}+i_{1B}+i_{1C}). \tag{17}\]  
Formula (17) corresponds to \(i_{0}\) expression, which was received in the beginning of chapter. Equations (15)-(17) we can obtain directly, projecting phase currents \(i_{1A},i_{1B},i_{1C}\) on the coordinate axes \(d\). Then, received expressions should be multiplied to two out of three. The matrices \(A_{1}^{-1}\) and \(A_{1}\) establish the relationship between the actual stator winding currents and the transformed variables in the \(d\), \(q\), and \(0\) axes. Similarly, analogous expressions can be derived for voltages:  
\[ \begin{bmatrix}u_{d}\\ u_{q}\\ u_{0}\end{bmatrix} =[A_{1}]\cdot \begin{bmatrix}u_{1A}\\ u_{1B}\\ u_{1C}\end{bmatrix}  \tag{18}\]  
The same transformation applies to the electromotive forces (EMFs) and flux linkages in the stator windings:  
\[ \begin{bmatrix}\psi_{d}\\ \psi_{q}\\ \psi_{0}\end{bmatrix} =[A_{1}]\cdot \begin{bmatrix}\psi_{1A}\\ \psi_{1B}\\ \psi_{1C}\end{bmatrix} . \tag{19}\]