# 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-Diode (RLD) First-Order Response  
This stage begins with the capacitor DC voltage falling to zero, and continues until the capacitor is recharged. When \(v_{\mathrm{c}}>0\), the capacitor discharges and the current flows through the line impedance. With the capacitor discharge, DC voltage \(v_{\mathrm{c}}\) will decrease. When \(v_{\mathrm{c}}\) drops to zero, that is moment \(t_{1}\), the inductor begins to discharge, and the fault current will flow through the anti-parallel diodes of the three-phase bridge, assuming that the initial condition is \(i_{\mathrm{L}}(t_{1})={I_{0}}^{\prime}\). The equivalent circuit is shown in Figure 8, the \(R_{\mathrm{on}}\) is the total equivalent internal resistance of the six anti-parallel diodes, and \(V_{\mathrm{f}}\) is the their total turn-on voltage drop.  
At this moment, the circuit becomes a first-order circuit and the DC inductor current decreases with the exponential law as follows:  
\[i_{\mathrm{L}}(t)=I_{0}^{\prime}e^{-((R+R_{\mathrm{on}})/L)t}. \tag{16}\]  
This process is the most dangerous time for the diodes, because there is a large inductor discharge current suddenly flowing through the diodes in this period, and the diodes can be damaged instantly because of the large initial current. Therefore, the diode over-current capability should be considered in the selection of switching devices.  
The time that capacitor \(C\) discharges to zero, and the value of the short-circuit current at that time, can be simplified as  
\[t_{1}=t_{0}+\frac{\gamma}{\omega}=t_{0}+\sqrt{LC}\mathrm{arctan}\bigg{(}\frac {V_{0}}{I_{0}}\sqrt{\frac{C}{L}}\bigg{)} \tag{17}\]  
\[i_{\mathrm{L}}(t_{1})=i_{\mathrm{L}1}(t_{1})+i_{\mathrm{L}2}(t_{1})=e^{-\delta t _{1}}\sqrt{\frac{C}{L}{V_{0}}^{2}+{I_{0}}^{2}}+{I_{\mathrm{s}}}-{I_{\mathrm{s }}}e^{-\delta t_{1}}\cos\gamma-\frac{{I_{\mathrm{s}}}(R+R_{\mathrm{c}})\sqrt{ C}}{2\sqrt{L}}e^{-\delta t_{1}}\sin\gamma \tag{18}\]  
where \(\gamma=\mathrm{arctan}\bigg{(}\frac{V_{0}}{I_{0}}\sqrt{\frac{C}{L}}\bigg{)}\), \(\omega=\frac{1}{\sqrt{LC}}\), \(\delta=\frac{R+R_{\mathrm{c}}}{2L}\).  
From Equation (17), the time of capacitor \(C\) discharging to zero, that is, the maximum over-current time of the diode, is related to the value of \(L\) and \(C\), as well as \(V_{0}\) and \(I_{0}\). The relationships among capacitance discharge time, inductance value, and capacitance value are shown in Figure 9 when \(V_{0}=750\) V, and \(I_{0}=1300\) A. It can be seen that the capacitor discharge time will increase with the increase of the total inductance value and capacitance value.  
Figure 8: Equivalent circuit of Resistance-Inductance-Diode (RLD) first-order circuit response.  
It can be seen from Equation (18) that the inductor peak current is related to the value of \(L\) and \(C\), as well as \(V_{0}\), \(I_{0}\) and \(I_{\mathrm{s}}\) at the fault instant. The relationships among the inductor peak current, inductance value, and capacitance value are shown in Figure 10, when \(V_{0}=750\) V, \(I_{0}=1300\) A, and \(I_{\mathrm{s}}=1800\) A. It can be seen that the inductor peak current increases with the increase of the capacitance value, and decreases with the increase of the total inductance, which is in accordance with the analysis conclusions in Section 4.1.  
The discharge time of the DC capacitor and the inductor peak current are analyzed by the RLC second-order discharge circuit. It can be seen from Equation (16) that the peak current begins to discharge as the initial current in the RLD first-order response. The AC system provides current for the DC side through the diode in the actual first-order discharge process, so the actual DC inductor current should be the sum of the inductor discharge current and AC system supply current. As the DC inductor current decays and the AC grid current increases, the capacitor starts charging again and the DC voltage increases at the same time.