## 2 Characteristics Analysis of Stator and Rotor Current
### Rotor Winding Short-Circuit Fault  
A large current flows through the short-circuit ring after the fault occurs in the VPSGM. The short-circuit current can be denoted as \(i_{e}=I_{e}\cos(\omega_{\rm r}t+\varphi_{e})\)[16]. According to the Fourier series decomposition, the magnetomotive force generated by the short-circuit current can be decomposed as [16]:  
\[ \begin{array}{l}F(\theta,t)\!\!=\frac{2I_{e}N_{e}k_{\rm ek}}{\pi P}\frac{ \Sigma}{k}\frac{1}{k}\cos(k\theta)\cos(\omega_{\rm r}t+\varphi_{e})\\ \phantom{F(\theta,t)}=\frac{I_{e}N_{e}k_{\rm ek}}{\pi P}\Sigma\frac{1}{k}[\cos (k\theta+\omega_{\rm r}t+\varphi_{e})+\cos(k\theta\omega_{\rm r}t\varphi_{e})] \end{array}  \tag{9}\]  
where \(N_{e}\) is the turn number of the short-circuit coils, \(k_{\rm ek}\) is the winding factor of the short-circuit coils, \(\theta\) is the space electric angle, \(P\) is the number of pole-pairs, \(k\) is the harmonic order, \(k=1/P\), \(2/P\),... It can be seen from Equation (9) that the magnetomotive force \(F\) is composed of the superposition of two rotating magnetomotive forces with the same rotational speed but opposite turning directions. Among them, the forward rotating magnetomotive force will induce the positive-sequence current component in the rotor winding, while the reverse rotating magnetomotive force will induce the negative-sequence current component; both currents are equal. Therefore, the three-phase currents are no longer symmetrical after the fault, which contains some negative-sequence current components. The rotor three-phase currents after the fault can be recorded as:\[ \begin{array}{l}i_{\rm a.r}(t)=I_{\rm r}\cos(\omega_{\rm r}t+\varphi_{\rm ar})+I_{ \rm r1}\cos(\omega_{\rm r}t+\varphi_{\rm ar1})\ +I_{\rm r1}\cos(\omega_{\rm r}t+\varphi_{\rm ar1})\\ i_{\rm b.r}(t)=I_{\rm r}\cos(\omega_{\rm r}t-\frac{2\pi}{3}+\varphi_{\rm ar})+I_ {\rm r1}\cos(\omega_{\rm r}t-\frac{2\pi}{3}+\varphi_{\rm ar1})+I_{\rm r1}\cos (\omega_{\rm r}t+\frac{2\pi}{3}+\varphi_{\rm ar1})\\ i_{\rm c.r}(t)=I_{\rm r}\cos(\omega_{\rm r}t+\frac{2\pi}{3}+\varphi_{\rm ar})+I _{\rm r1}\cos(\omega_{\rm r}t+\frac{2\pi}{3}+\varphi_{\rm ar1})+I_{\rm r1}\cos (\omega_{\rm r}t-\frac{2\pi}{3}+\varphi_{\rm ar1})\end{array}  \tag{10}\]  
where \(I_{\rm r1}\) is the amplitude of the positive- and negative-sequence current components induced by the positive and reverse rotating magnetomotive force. The amplitude is positively correlated with the severity of the short-circuit fault. \(\varphi_{\rm ar1}\) is the phase angle of the A-phase induced current. The current \(i_{\rm d.r}\) and \(i_{\rm q.r}\) in the dq coordinate system after the fault can be obtained by Clark transformation:  
\[ \begin{array}{l}i_{\rm d.r}(t)=I_{\rm r}\cos(\omega_{\rm r}t+\varphi_{\rm ar })+2I_{\rm r1}\cos(\omega_{\rm r}t+\varphi_{\rm ar})\\ i_{\rm q.r}(t)=I_{\rm r}\sin(\omega_{\rm r}t+\varphi_{\rm ar})\end{array}  \tag{11}\]  
Further, \(f(i_{\rm d.r},i_{\rm q.r})\) after the rotor winding short-circuit fault can be calculated:  
\[ \begin{array}{l}f(i_{\rm d.r},i_{\rm q.r})=i_{\rm d.r}^{2}(t)+i_{\rm q.r}^{2} (t)\\ =I_{\rm r}^{2}+2I_{\rm r1}I_{\rm r1}\cos(\varphi_{\rm ar}-\varphi_{\rm ar1})\ +2I_{\rm r1}\cos(2\omega_{\rm r}t+\varphi_{\rm ar }+\varphi_{\rm ar1})\ +4I_{\rm r1}^{2}\cos^{2}(\omega_{\rm r}t+\varphi_{\rm ar1})\end{array}  \tag{12}\]  
For the stator side, the stator phase current after the rotor winding short circuit contains a small amount of \(1\pm 2\ s\) harmonic components [19]. The stator-side three-phase current after fault can be recorded as:  
\[