# Research on Ultra-Fast Transient Overvoltage Characteristics of Electric Locomotive
## 4 Analysis and Discussion
### Formation Mechanism of High-Amplitude and High-Steepness Operating Overvoltage
The rapid exchange of electromagnetic energy between the capacitance and inductance components induces complex electromagnetic oscillation. This oscillation will produce a transient overvoltage which is much higher than the normal operating voltage. It is impulsive and strong damping, usually decaying within a few milliseconds to tens of milliseconds [14]. The higher the system-rated voltage, the higher the amplitude of operating overvoltage. For 220 kV and below systems, considering the cost of equipment and the probability of high-voltage accidents, the insulation of electrical equipment is designed according to 3-4 times the rated voltage. When the ultra-fast transient overvoltage with high amplitude and high steepness happens frequently, serious accidents on high-voltage equipment are hard to avoid. Formation Mechanism of Ultra-Fast Transient Overvoltage Caused by Pantograph Lifting and Main Breaker Closing  
The process of pantograph lifting and main breaker closing involves contact connection, which can be classified as a closing operation. Figure 9a shows the equivalent circuit of the pantograph lifting process. The amplitude of operating overvoltage can be calculated by establishing an equivalent circuit [16]. Taking the pantograph lifting process as an example. The closing operation of the switch is used to represent the closing process between the pantograph and the catenary. \(L_{0}\) and \(R_{0}\) represent the equivalent inductance and resistance of the catenary. \(u(t)\) is the catenary voltage. \(L_{m}\) is the equivalent excitation inductance of the high-voltage transformer and \(C_{0}\) is the equivalent ground capacitance of the post-stage circuit. For simplicity, the catenary resistance and transformer inductance are ignored, as shown in Figure 9b. The circuit equation of pantograph lifting is:  
\[L_{0}C_{0}\frac{d^{2}u_{c}}{dt^{2}}+u_{c}=U_{m}\sin(\omega t+\varphi) \tag{2}\]  
Figure 8: Reignition and extinction condition of arc. Figure 9: Pantograph lifting equivalent circuit. (**a**) Equivalent circuit; (**b**) simplified circuit. By solving (2):  
\[u_{c}(t)=\frac{\frac{1}{\omega C_{0}}\cdot U_{m}\sin(\omega t+\varphi)}{\frac{1}{ \omega C_{0}}-\omega L_{0}}+A\cos(\omega_{0}t)+B\sin(\omega_{0}t) \tag{3}\]  
In (3), \(U_{m}\) is the amplitude of the power supply voltage, \(\varphi\) is the initial phase angle, \(A\) and \(B\) are the integral constants. Among them, \(w_{0}=1/(L_{0}C_{0})^{1/2}\) and \(wL_{0}<<1/(wC_{0})\). Suppose that the initial conditions are as follows:  
1. Before the pantograph lifting, the voltage of the residual charge on the catenary is \(U_{0}\). \(u_{c}(0)=U_{0}\) when \(t=0\). 2. The current does not mutate due to inductance in the circuit, when \(t=0\), \(i_{c}(0)=0\). Substituting the initial conditions into (3), the capacitor voltage \(u_{c}(t)\) is derived as: \[u_{c}(t)=U_{m}\sin(\omega t+\varphi)+(U_{0}-U_{m}\sin\varphi)\cos\omega_{0}t- \frac{\omega U_{m}\cos\varphi}{\omega_{0}}\sin\omega_{0}t\] (4)  
If \(w_{0}>>w\), the maximum capacitance voltage \(U_{c\max}\) occurs at \(wt=0\). Based on (4), it is obtained that:  
Figure 6:0  
When the initial value of \(u_{c}(0)\) is \(-U_{m}\), the maximum operating overvoltage of pantograph lifting can reach \(3U_{m}\). Considering the thermal and charge-accumulation effect of the arc, the pantograph-catenary gap is equivalent to the RC circuit shown in Figure 9(a). \(R(t)\) represents the arc resistance while \(C(t)\) represents the arc capacitance. The expressions of \(R(t)\) and \(C(t)\) are deduced as [17]:  
\[\left\{ \begin{array}{c}R(t)=R_{1}e^{-\frac{t}{11}}+R_{2}e^{\frac{t}{2}}\\ C(t)=24.72t^{0.32}\end{array} \right. \tag{6}\]  
In (6), \(R_{1}\) and \(R_{2}\) correspond to the steady-state value and the arcing resistance value in the double exponential arc-resistance model, respectively. \(\tau_{1}\) and \(\tau_{2}\) are the decay time constants of the exponential function, which represent the time of arc combustion and extinguishing during one arc process.