## 3 ITF Diagnosis Based on Current Vector Pattern
### FEA Results  
As discussed in Section 2, the current vector exhibits an elliptical pattern under ITF conditions in an IM, primarily due to the presence of negative-sequence currents. These negative-sequence currents are influenced by fault parameters such as the fault resistance, the number of shorted turns, and the specific faulted phase. To investigate the current vector pattern under various fault conditions, co-simulations are conducted. Figure 4 illustrates the current vector patterns for each faulted phase. Under healthy conditions, the current vector forms a circle. However, in the presence of an ITF, the pattern transitions into an ellipse due to the unbalanced impedance introduced by the fault resistance. The main axis of the ellipse varies depending on the faulted phase. Specifically, when Phase A is faulted, the ellipse's main axis appears between \(0^{\circ}\) and \(60^{\circ}\); if Phase B is faulted, it appears between \(-60^{\circ}\) and \(0^{\circ}\); and if Phase C is faulted, it appears between \(-120^{\circ}\) and \(-60^{\circ}\). This main axis of the ellipse corresponds to a \(30^{\circ}\) angular displacement between the ellipse main axis and the magnetic motive force center of the faulted phase, as depicted in Figure 4. In the case of a Y-connected winding instead of a \(\Delta\)-connected one, the current phase lags by \(30^{\circ}\), which shifts the main axis. In this case, the main axis of the ellipse for Phase A lies between \(-30^{\circ}\) and \(30^{\circ}\); for Phase B, it lies between \(30^{\circ}\) and \(90^{\circ}\); and for Phase C, it lies between \(-90^{\circ}\) and \(-30^{\circ}\). A summary of the main axis angle for each faulted phase is provided in Table 2. The magnitude of the negative-sequence current is influenced by the fault parameters, such as the number of shorted turns and the fault resistance, as illustrated in Figure 5a,b. As the number of shorted turns increases or the fault resistance decreases, the elliptical distortion becomes more pronounced, indicating a higher degree of current imbalance. Furthermore, the positive- and negative-sequence currents are influenced by the speed and slip, as shown in Equation (8). The current vector pattern is analyzed for speed and slip, as illustrated in Figure 5c,d. Although the magnitude of the current vector \begin{table}
\begin{tabular}{c c c} \hline \hline
**Content** & **Value** & **Unit** \\ \hline Rated power & 12 & kW \\ Rated speed & 2010 & rpm \\ Maximum speed & 7250 & rpm \\ Rated frequency & 70 & Hz \\ DC voltage & 52 & V \\ Rated voltage & 20 & V\({}_{\mathrm{rms}}\) \\ Rated current & 248 & A\({}_{\mathrm{rms}}\) \\ DC link voltage & 48 & V \\ Switching frequency & 10 & kHz \\ Modulation method & SVPWM & - \\ Stator outer/inner diameter & 200/125 & mm \\ Rotor outer/inner diameter & 124/36 & mm \\ Airgap length & 0.5 & mm \\ Stack length & 120 & mm \\ Number of poles/slots & 4/36 & - \\ Total series turns & 13 & \\ Number of parallel branches & 4 & - \\ Core material & 35 PN230 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specification of IM for Twizy. depends on the speed and slip, the main axis of the ellipse appears between 0\({}^{\circ}\) and 60\({}^{\circ}\). These trends are consistently observed across all simulation scenarios, confirming that the proposed method not only detects the occurrence of faults but also provides insights into their severity. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Winding Connection** & **Fault Phase** & **The Angle of the Maximum Current Vector** \\ \hline \hline  & A & 0–60\({}^{\circ}\) \& 180–240\({}^{\circ}\) \\ \(\Delta\) & B & 300–360\({}^{\circ}\) \& 120–180\({}^{\circ}\) \\  & C & 240–300\({}^{\circ}\) \& 60–120\({}^{\circ}\) \\  & A & 0–30\({}^{\circ}\) \& 150–210\({}^{\circ}\) \& 330–360\({}^{\circ}\) \\ Y & B & 270–300\({}^{\circ}\) \& 90–150\({}^{\circ}\) \\  & C & 30–90\({}^{\circ}\) \& 210–270\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The angle of the maximum current vector for the fault phase and winding connection.