## 3 ITSCF Model and Feature
### The Design of ITSCF Model
\right], \tag{7}\]  
where the fourth item represents the back EMF of the shorted coil, presented using the per-unit concept. The initial voltage \(v_{0}\) is expressed by Equation (8). \[v_{0}=v_{0}\big{[}1\quad 1\quad 1\quad 0\big{]}^{\mathrm{T}}, \tag{8}\]  
By substituting the last term of each equation from Equations (3)-(8) into Equation (2), Equation (9) is obtained. \[0=-\mu r_{s}i_{a}+\Big{(}\mu r_{s}+r_{f}\Big{)}i_{f}-2\omega_{e}\Big{(}-\mu Li _{a}-\mu Mi_{b}-\mu Mi_{c}+\mu^{2}Li_{f}\Big{)}+\Big{(}-\mu Lipi_{a}-\mu Mpi_{ a}-\mu Mpi_{c}+\mu^{2}Lpi_{f}\Big{)}-e_{f}, \tag{9}\]  
The three-phase currents can be obtained from the transformation matrix and are expressed by Equations (9)-(11). \[i_{a}=\sqrt{\frac{2}{3}}\big{[}i_{q}\cos(\theta_{r})+i_{d}\sin(\theta_{r}) \big{]}=\sqrt{\frac{2}{3}}I\sin(\theta_{r}+\delta), \tag{10}\]  
\[i_{b}=\sqrt{\frac{2}{3}}\bigg{[}i_{q}\cos\left(\theta_{r}-\frac{2}{3}\pi \right)+i_{d}\sin\left(\theta_{r}-\frac{2}{3}\pi\right)\bigg{]}=\sqrt{\frac{2 }{3}}I\sin\left(\theta_{r}-\frac{2}{3}\pi+\delta\right), \tag{11}\]  
\[i_{c}=\sqrt{\frac{2}{3}}\bigg{[}i_{q}\cos\left(\theta_{r}+\frac{2}{3}\pi \right)+i_{d}\sin\left(\theta_{r}+\frac{2}{3}\pi\right)\bigg{]}=\sqrt{\frac{2 }{3}}I\sin\left(\theta_{r}+\frac{2}{3}\pi+\delta\right), \tag{12}\]  
Due to the significant difference between \(\mu^{2}\) and the overall value, it is neglected. Subsequently, applying Equations (10)-(12), Equation (9) can be expressed as Equation (13). \[\Big{(}\mu r_{s}+r_{f}\Big{)}i_{f}=-2\omega_{r}\bigg{(}\mu L\sqrt{\frac{2}{3 }}I\sin(\theta+\delta)+\mu M\sqrt{\frac{2}{3}}I\sin\left(\theta-\frac{2\pi}{3 }+\delta\right)+\mu M\sqrt{\frac{2}{3}}I\sin\left(\theta+\frac{2\pi}{3}+ \delta\right)\bigg{)}+e_{f} \tag{13}\]  
After rearranging and simplifying Equation (13), the result is obtained as follows:  
\[i_{f}\cong-\sqrt{6}5\omega_{r}L_{2}I\cos(\theta-\delta)+S\omega_{r}\lambda \cos(\theta), \tag{14}\]  
It can be concluded that the short-circuit circulating current \(i_{f}\) is a cosine function with amplitude proportional to the severity of the fault and the rotor speed ratio, as given by Equation (15) for \(S\). \[S=\frac{\mu}{r_{f}+\mu r_{s}}, \tag{15}\]  
By combining the Clarke and Park transformations, the transformation matrix \(\textbf{T}_{cp}\), as shown in Equation (16), is obtained. This matrix enables the conversion of a three-phase motor ITSC model into a \(d\)-\(q\) axis ITSC model. \[\textbf{T}_{cp}=\frac{2}{3} \begin{bmatrix}-\sin\left(\theta\right)&-\sin \left(\theta-\frac{2}{3}\pi\right)&-\sin\left(\theta+\frac{2}{3}\pi\right)\\ \cos\left(\theta\right)&\cos\left(\theta-\frac{2}{3}\pi\right)&\cos\left( \theta+\frac{2}{3}\pi\right)\end{bmatrix} , \tag{16}\]  
The \(d\)-\(q\) axis model can be obtained as shown in Equation (17). \[\textbf{v}_{qd}=\textbf{T}_{cp}\big{[}v_{a}\quad v_{b}\quad v_{c}\big{]}^{ \mathrm{T}}=\textbf{T}_{cp}\textbf{r}_{s}\textbf{T}_{cp}^{-1}\textbf{i}_{abc}- \textbf{T}_{cp} \begin{bmatrix}\mu r_{s}i_{f}\\ 0\\ 0\end{bmatrix} +\textbf{T}_{cp}L_{s}\frac{\mathrm{d}}{\mathrm{d}t}\left(\textbf{T} _{cp}^{-1}\textbf{i}_{abc}\right)-\textbf{T}_{cp} \begin{bmatrix}\mu L\\ \mu M\\ \mu M\end{bmatrix} \frac{\mathrm{d}}{\mathrm{d}t}i_{f}+\textbf{T}_{cp}\textbf{e}_{ abcf}, \tag{17}\]After expanding the matrix, the result is obtained as follows:  
\[\mathbf{v}_{qdf}= \begin{bmatrix}r_{s}&\frac{3}{2}\omega_{r}L_{d}&A\\ -\frac{3}{2}\omega_{r}L_{q}&r_{S}&B\end{bmatrix}\begin{bmatrix}i_{q}\\ i_{d}\\ i_{f}\end{bmatrix} + \begin{bmatrix}\frac{3}{2}L_{q}&0&-\sqrt{\frac{3}{2}}\mu L_{q} \text{cos}(\theta_{r})\\ 0&\frac{3}{2}L_{d}&-\sqrt{\frac{3}{2}}\mu L_{d}\text{sin}(\theta_{r})\end{bmatrix} \begin{bmatrix}pi_{q}\\ pi_{d}\\ pi_{f}\end{bmatrix}