# Application of a High-Power Reversible Converter in a Hybrid Traction Power Supply System
## 4 Analysis of the DC Short-Circuit Characteristics of the Reversible Converter
### Stage of the Uncontrolled Rectification  
When a short-circuit fault occurs, the reversible converter cannot be immediately separated from the AC and DC network by breakers. As a result, the anti-parallel diodes will form a three-phase bridge rectifier circuit, and the AC system will continue to deliver power to the DC side until the breakers open, as shown in Figure 11.  
Figure 10: The relationships among inductor peak current, the equivalent inductance \(L\), and capacitance \(C\) when \(V_{0}=750\) V, \(I_{0}=1300\) A, and \(I_{\mathrm{s}}=1800\) A.  
Figure 9: The relationships among capacitor discharge time, the equivalent inductance \(L\), and capacitance \(C\) when \(V_{0}=750\) V, \(I_{0}=1300\) A.  
The system is equivalent to a three-phase uncontrolled rectifier working in the DC-side short-circuit state. The short-circuit current can be calculated by the transient analysis method. Assume that the phase current before short-circuit occurs is as follows:  
\[i_{\mathbf{a}|0|}=I_{\mathrm{m}|0|}\sin\Bigl{(}\omega t+\alpha-\varphi_{|0|} \Bigr{)} \tag{19}\]  
where \(I_{\mathrm{m}|0|}\) is the current amplitude, \(\omega\) is the synchronous angular frequency, \(\alpha\) is the phase angle, and \(\varphi_{|0|}\) is the impedance angle.  
Three-phase short-circuit current can be obtained:  
\[\begin{split} i_{\mathbf{a}}&=I_{\mathrm{m}}\sin( \omega t+\alpha-\varphi)+\Bigl{[}I_{\mathrm{m}|0|}\sin(\alpha-\varphi_{|0|})-I _{\mathrm{m}}\sin(\alpha-\varphi)\Bigr{]}e^{-t/\tau}\\ i_{\mathbf{b}}&=I_{\mathrm{m}}\sin(\omega t+\alpha -120^{\circ}-\varphi)+\Bigl{[}I_{\mathrm{m}|0|}\sin(\alpha-120^{\circ}- \varphi_{|0|})-I_{\mathrm{m}}\sin(\alpha-120^{\circ}-\varphi)\Bigr{]}e^{-t/ \tau}\\ i_{c}&=I_{\mathrm{m}}\sin(\omega t+\alpha+120^{\circ}- \varphi)+\Bigl{[}I_{\mathrm{m}|0|}\sin(\alpha+120^{\circ}-\varphi_{|0|})-I_{ \mathrm{m}}\sin(\alpha+120^{\circ}-\varphi)\Bigr{]}e^{-t/\tau}\end{split} \tag{20}\]  
where \(I_{\mathrm{m}}\) is the short-circuit current period component amplitude, \(\varphi\) is the impedance angle of the circuit loop, \(L_{\mathrm{r}}\) is the AC inductance, and \(\tau\) is the time constant of circuit loop.  
\[\varphi=\arctan[\omega(L_{\mathrm{r}}+L)/R],\;\tau=(L_{\mathrm{r}}+L)/R,\;I_ {m}\approx\frac{\sqrt{2}V_{\mathrm{n}}}{R_{\mathrm{x}}+\sqrt{3}\omega L_{ \mathrm{r}}} \tag{21}\]  
where \(V_{\mathrm{n}}\) is the root-mean-square voltage of the transformer secondary.  
From Equation (21), when the AC inductance \(L_{\mathrm{r}}\) is constant, the short-circuit current period component amplitude \(I_{\mathrm{m}}\) is related to the line equivalent resistance \(R\). The farther the location of the short-circuit point is from the reversible converter, the smaller the short-circuit current period component \(I_{\mathrm{m}}\) will be.  
Three-phase uncontrolled rectifier transfers the short-circuit current, whose value is greater than zero, to the DC side. Thus, the short-circuit current on the DC side is as follows:  
\[i_{\mathrm{s}}=i_{1}+i_{2}+i_{3}=i_{\mathbf{a},(>0)}+i_{\mathbf{b},(>0)}+i_{ \mathbf{c},(>0)}. \tag{22}\]