# 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 Resistance-Inductance-Capacitor (RLC) Second-Order Response  
In this stage, the DC short-circuit fault occurs instantly, the capacitor \(C\) discharges quickly, and the DC voltage drops quickly from the initial value to zero. The DC fault current correspondingly rises rapidly. During this period, \((R+R_{\rm c})\), \(L\) and \(C\) constitute an Resistance-Inductance-Capacitor (RLC) series second-order circuit. The \(C\) and \(L\) have initial values when the DC short-circuit fault occurs. Due to the presence of the AC inductor \(L_{\rm r}\), the three-phase AC current cannot be mutated, but flows through the six anti-parallel diodes, so there is an external excitation current to the second-order RLC circuit. As shown in Figure 7, this stage can be seen as a full response (including zero-input response and zero-state response) process of the RLC series circuit. Zero input response process: Regardless of the influence of the AC system (without considering the \(i_{\rm s}\)), it is the zero-input response process of the RLC second-order circuit. From Kirchhoff Voltage Laws (KVL):  
\[LC\frac{d^{2}v_{\rm c}}{dt^{2}}+(R~{}+~{}R_{\rm c})C\frac{dv_{\rm c}}{dt}+v_{ \rm c}=0. \tag{6}\]  
According to the relationship of \((R+R_{\rm c})\) and \(2\sqrt{L/C}\), Equation (6) may have different solutions, considering \(R\) and \(R_{\rm c}\) are small, this paper only analyses \((R+R_{\rm c})<2\sqrt{L/C}\). Assume that the short circuit fault occurs at moment \(t_{0}\), the initial condition is \(v_{\rm c}(t_{0})=V_{0}\), \(i_{\rm L}(t_{0})=I_{0}\). The circuit is in the oscillation discharge process and, from Equation (6), the zero-input response components of the DC side voltage and current after fault are as follows:  
\[v_{\rm c}=\frac{V_{0}\omega_{0}}{\omega}e^{-\delta t}\sin(\omega t+\beta)- \frac{I_{0}}{\omega C_{\rm d}}e^{-\delta t}\sin(\omega t) \tag{7}\]  
Figure 6: DC short-circuit fault of the reversible converter. Figure 7: Resistance-Inductance-Capacitor (RLC) second-order circuit response equivalent circuit. \[i_{\rm L1}=C\frac{dv_{\rm c}}{dt}=-\frac{I_{0}\omega_{0}}{\omega}e^{-\delta t} \sin(\omega t-\beta)+\frac{V_{0}}{\omega L}e^{-\delta t}\sin(\omega t) \tag{8}\] \[\omega^{2}=\frac{1}{LC}-\left(\frac{R+R_{\rm c}}{2L}\right)^{2}, \ \delta=\frac{R+R_{\rm c}}{2L},\ \omega_{0}=\sqrt{\delta^{2}+\omega^{2}}=\frac{1}{ \sqrt{LC}},\ \beta=\arctan\frac{\omega}{\delta}.\]  
From Equation (7), when \(C\) discharges to zero, that is, \(v_{\rm c}=0\), the instant time is  
\[t_{1}=t_{0}+\frac{\gamma}{\omega},\ \gamma=\arctan[(V_{0}\omega_{0}C_{\rm d} \sin\beta)/\left(I_{0}-V_{0}\omega_{0}C_{\rm d}\cos\beta\right)]. \tag{9}\]  
Substituting Equation (9) into Equation (8):  
\[\rm i_{L1}(t_{1})=-\frac{I_{0}\omega_{0}}{\omega}e^{-\delta(\frac{\gamma}{ \omega})}\sin(\gamma-\beta)+\frac{V_{0}}{\omega L}e^{-\delta(\frac{\gamma}{ \omega})}\sin\gamma. \tag{10}\]  
Generally, when a metal short-circuit fault occurs in the DC line, \(\left((R+R_{\rm c})/2L\right)^{2}\) is far smaller than \(1/LC\), It can be assumed that \(\omega=\omega_{0}\), that is, the zero-input response component of the DC inductor current can be reduced to  
\[i_{\rm L1}=e^{-\delta t}\Bigg{(}V_{0}\sqrt{\frac{C}{L}}\sin\omega t+I_{0}\cos \omega t\Bigg{)}. \tag{11}\]  
Making \(\theta=\arctan(\frac{I_{0}}{V_{0}}\sqrt{\frac{L}{L}})\); then Equation (11) becomes  
\[i_{\rm L1}=e^{-(\frac{R+R_{\rm c}}{2L})t}\Bigg{[}\sqrt{\frac{C}{L}{V_{0}}^{2}+ {l_{0}}^{2}}\sin(\omega t+\theta)\Bigg{]}. \tag{12}\]  
Zero state response process: As shown in Figure 7, assume that there is no initial value on \(C\) and \(L\). The current supplied to the DC side by the AC system is regarded as a constant value \(i_{\rm s}\).