## 2 Current Vector Analysis of IMs with ITF
### Mathematical Model for ITF
\right]if_{r},\]  
\(\left[v_{a\beta s}\right]\) is the voltage in the stationary coordinate frame; \(\left[i_{a\beta s}\right]\) and \(\left[i_{a\beta r}\right]\) are the current of the stator and rotor in the stationary coordinate frame, respectively; \(\left[\lambda_{a\beta s}\right]\) and \(\left[\lambda_{a\beta r}\right]\) are the flux linkage of the stator and rotor in the stationary coordinate frame, respectively; and \(\omega_{e}\) is the electrical speed. In a steady-state IM, the voltage and current can be expressed as a complex form:  
\[\mathbf{V}_{s}=V_{ps}e^{i\omega_{e}t}+V_{ns}e^{-i\omega_{e}t},\ \ \mathbf{I}_{s}=I_{ps}e^{i\omega_{e}t}+I_{ns}e^{-i\omega_{e}t},\ \ \mathbf{I}_{r}=I_{pr}e^{i\omega_{e}t}+I_{nr}e^{-i\omega_{e}t} \tag{5}\]  
Figure 2: Current vector pattern: (**a**) health condition; (**b**) fault condition. where vs is the phase form of the voltage; \(\mathbf{I}_{s}\) is the phase form of the stator current; \(\mathbf{I}_{r}\) is the phase form of the rotor current; \(V_{ps}\) and \(V_{ns}\) are the positive and negative sequences of the voltage, respectively; \(I_{ps}\) and \(I_{ns}\) are the positive and negative sequences of the stator current, respectively; and \(I_{pr}\) and \(I_{nr}\) are the positive and negative sequences of the rotor current, respectively. By subtracting Equation (5) from Equation (4), the phase equation in terms of the positive and negative sequences can be expressed as follows:  
\[ \begin{array}{l}V_{ps}=\left(R_{s}+j\omega_{e}L_{s}\right)\left(I_{ps}-\frac {2}{3}\mu I_{f}\right)+j\omega_{e}L_{m}I_{pr}\\ V_{ns}=\left(R_{s}+j\omega_{e}L_{s}\right)\left(I_{ns}-\frac{2}{3}\mu I_{f} \right)+j\omega_{e}L_{m}I_{nr}\\ 0=\left(\frac{R_{s}}{s}+j\omega_{e}L_{r}\right)I_{pr}+j\omega_{e}L_{m}\left(I_ {ps}-\frac{2}{3}\mu I_{f}\right)\\ 0=\left(\frac{R_{s}}{2-s}+j\omega_{e}L_{r}\right)I_{nr}+j\omega_{e}L_{m}\left( I_{ns}-\frac{2}{3}\mu I_{f}\right)\\ \mu\left(V_{sp}+V_{sn}\right)=\mu\left(1-\frac{2}{3}\mu\right)\left(R_{s}+j \omega_{e}L_{ls}\right)I_{f}+R_{f}I_{f}\end{array}  \tag{6}\]  
From Equation (6), the stator current can be expressed as follows:  
\[ \begin{array}{l}\begin{bmatrix}I_{ps}\\ I_{ns}\end{bmatrix}=\begin{bmatrix}Y_{pp}&Y_{pn}\\ Y_{np}&Y_{m}\end{bmatrix}\begin{bmatrix}V_{ps}\\ V_{ns}\end{bmatrix}\end{array}  \tag{7}\]  
where  
\[ \begin{array}{l}Y_{pp}=\frac{1}{R_{s}+j\omega_{e}L_{s}-\frac{\omega_{p}L_{s} ^{2}}{R_{s}+j\omega_{e}L_{s}}}+Y_{pn},\\ Y_{nn}=\frac{1}{R_{s}+j\omega_{e}L_{s}-\frac{\omega_{p}L_{s}^{2}}{R_{s}/(2-s)+ j\omega_{e}L_{s}}}+Y_{np},\\ Y_{pn}=Y_{np}=\frac{2}{3}\frac{\mu^{2}}{R_{f}+\mu\left(1-\frac{2}{3}\right) \left(R_{s}+j\omega_{e}L_{ls}\right)},\end{array}  \tag{8}\]  
\(Y_{pp}\) and \(Y_{m}\) are the admittances of the positive and negative sequences, respectively, and \(Y_{pn}\) and \(Y_{np}\) are the admittances between the positive and negative sequences. If the positive-sequence voltage is applied to the IM, the positive and negative currents can be expressed as shown in Equation (7) because the negative-sequence voltage is zero. \[I_{ps}=Y_{pp}V_{ps},\,I_{ns}=Y_{np}V_{ps} \tag{9}\]  
When a fault occurs in Phase A, the admittance between the positive and negative sequences becomes non-zero due to the presence of \(\mu\) in Equation (8). As a result, the negative-sequence current is induced by the applied positive-sequence voltage and is influenced by fault parameters such as the fault resistance and the ratio of shorted turns. The presence of a negative-sequence component under fault conditions changes the current vector pattern differently from that of a healthy condition.