## 3 Analysis of Simulation Result
### Streamer Development Process  
This paper analyzes the development process of streamers during surface discharge of C4F7N/CO2 in three stages: Stage 1 is along the upper surface of the insulator; Stage 2 is along the rounded corner transition stage; and stage 3 is along the right surface of the insulator. Figure 3 shows the electric field intensity (V/m) of the three stages in the process of surface discharge. Figure 4 shows the potential distribution (\(V\)) at the initial stage of discharge. Figure 2: Flow chart of the surface discharge. Figure 4: Potential distribution at the initial stage of discharge: (**a**) 0 ns; (**b**) 0.5 ns. Figure 3: Surface discharge process: (**a**) 0.03 ns; (**b**) 0.2 ns; (**c**) 0.5 ns; (**d**) 0.7 ns; (**e**) 1.2 ns; (**f**) 1.7 ns; (**g**) 1.84 ns; (**h**) 2.14 ns; (**i**) 2.34 ns. After applying voltage to the needle electrode, the electric field near the needle electrode is very strong, and a large number of electrons and cations are distributed here. The applied voltage intensifies the movement of electrons, which collides with neutral molecules in the gas mixture, such as C4F7N and CO2, and generates a large number of electrons and cations. Because the charge-mass ratio of cations is much lower than that of electrons, the movement speed of cations is much lower than that of electrons, which can be regarded as stationary within a nanosecond [27]. Under the action of the applied voltage at the needle electrode, electrons continue to move to the needle electrode, while cations almost stay in place. With the accumulation of cations, the space charge effect becomes ever more significant, and the electric field of the streamer head continues to increase. When the electric field at the head of the streamer exceeds the critical reduced field strength of the mixed gas, a new electronic collapse occurs and the streamer continues to develop, as shown in Figure 3a-c. When the streamer is transitioning from the top surface of the insulator to the right surface along the rounded corner, the streamer-branching phenomenon occurs, as shown in Figure 3d-f, which is caused by the uneven distribution of the electric potential. By comparing (a) and (b) in Figure 4, it can be found that at 0.5 ns, the electric potential gradients along the positive x direction and the negative z direction are very large, which means the field strengths in these two directions are significantly larger than those in other directions. When the electric field intensity is higher than the critical reduced electric field intensity of the gas, the streamer develops in the direction of a large electric potential gradient. A similar streamer-branching phenomenon was observed in [27]. After the transition, the streamer continues along the right surface of the insulator, and the branched streamer gradually moves to the cathode. Because the streamer branch has a speed of x direction, this part of the stream gradually moves away from the insulator. By comparing Figure 3g-i, in this process, the streamer branch becomes slower and eventually stops. This is because when the streamer is far away from the insulator, the electric field at the head of the streamer and the background electric field become smaller and smaller, which cannot make the streamer continue to move forward. As the streamer is still approaching the cathode, the electric field intensity at the head of the streamer increases, while the electric field intensity at the tail of the streamer near the needle electrode also increases. This is due to the fact that a large number of electrons outside the streamer enter the streamer channel and flow to the anode, and the space charge effect weakens the electric field inside the channel to a certain extent. When the streamer reaches the cathode, the electric field intensity of the streamer head increases from \(7.84\times 10^{7}\) V/m to \(1.07\times 10^{8}\) V/m. Figure 5 shows the variation of electric field intensity in the streamer head during discharging.