## 2 Transient Characteristics of DC Forced Interruption
### di/dt in Current-Commutation Stage  
\(di/dt\) and \(dv/dt\) are important parameters in the current-commutation stage. We calculate \(di/dt\) as the average current-change rate over the time from the start of forced commutation to the current zero condition, and \(dv/dt\) as the average voltage-change rate over the time from the current zero condition to the first peak of the recovery voltage [26].  
In the current-commutation stage, when the arc resistance is ignored, the commutating current \(I_{\mathrm{C}}\) can be approximated as per the following equation:  
\[I_{\mathrm{C}}=i_{cmax}\sin(2\pi f\cdot t) \tag{1}\]  
where \(f=1/2\pi\sqrt{\mathrm{LC}}\); the reverse peak current is shown in Equation (2):  
\[i_{cmax}=U_{0}/\sqrt{L/C} \tag{2}\]  
The reverse peak-current coefficient \(k\) is set by the equation:  
\[k=i_{\mathrm{cmax}}/i_{0} \tag{3}\]  
\(i_{0}\) is the main circuit current. \(U_{0}\) is the precharged voltage of the commutating capacitor in the forced-commutation branch, which should satisfy:  
\[U_{0}=k\cdot i_{0}\cdot\sqrt{L/C} \tag{4}\]  
As the arc current decreases approximately linearly in the current-commutation stage in Figure 2, \(di/dt\) is calculated using the following equation [27]:  
\[\frac{di}{dt}/_{\mathrm{max}}=\frac{U_{0}}{L} \tag{5}\]  
where \(L\) is the inductance of the commutating inductor in the forced-commutation branch. As can be seen from Equation (5), when the forced-commutation branch parameters are certain, \(di/dt\) can be reduced through the appropriate reduction of the precharged voltage. Similarly, when \(U_{0}\) and \(C\) are fixed, the increase of \(L\) is beneficial to the reduction of \(di/dt\). In fact, the above two methods to  
Figure 2: Typical waveform of DC forced interruption obtained from the simulation with a main circuit current of 3.5 kA and arcing time of 1 ms (\(U_{\mathrm{A}}\) is the contact voltage and \(U_{\mathrm{C}}\) is the voltage of the capacitor in the forced-commutation branch).  
reduce \(di/dt\) lead to the reduction of the reverse peak current in Equation (2). It can be concluded that the magnitude of \(di/dt\) is directly related to the reverse peak-current coefficient, when the main circuit current \(l_{0}\) is fixed. The selection of the charging margin directly affects the value of \(di/dt\). When the circuit parameters are determined, the precharged voltage \(U_{0}\) is determined by the value of \(k\), which affects the transient characteristics in the interruption process. Substituting Equation (4) into Equation (5), we obtain Equation (6), as follows:  
\[\frac{di}{dt}/_{\max}=\frac{k\cdot i_{0}}{\sqrt{L\cdot C}}=2\pi f\cdot k\cdot i _{0} \tag{6}\]  
If the decay of the oscillation current in the current-commutation stage is ignored, \(di/dt\) can be determined from the oscillation frequency and the coefficient \(k\) under the same breaking current level. Under the same oscillation frequency, the selection of the reverse peak-current coefficient \(k\) directly determines the value of \(di/dt\). In order to ensure reliable commutation, the value of \(k\) is generally chosen to be between 1.3 and 2. In practical applications, in order to calculate the \(di/dt\) in the current-commutation stage accurately, the following formula is used:  
\[\frac{di}{dt}/_{\max}=k\cdot i_{0}\cdot\omega\cdot e^{\delta\cdot\beta/\omega }\cdot\sin\beta \tag{7}\]  
Here, \(\omega=\frac{1}{LC}-(\frac{R}{2L})^{2}\), \(\delta=\frac{R}{2L}\), \(\beta=\arctan\frac{\omega}{\delta}\), and \(R\) is the equivalent stray resistance in the forced-commutation branch, which is 22 m\(\Omega\) in our research.  
Table 1 shows the comparisons of the \(di/dt\) obtained through the methods of Formulation (7), simulation, and experiments. The results calculated using Equation (7) are in good agreement with the simulation and experimental results.