## 3 Results
### Field Inhomogeneity  
High-voltage electrodes with varying radii were utilized to investigate the impact of the field inhomogeneity degree (represented by inhomogeneity coefficients \(f\)) on the stability. Figure 7 displays the mathematical expectation of breakdown voltages and the normalized breakdown voltage dispersion for the corona-stabilized switch using rod electrodes in group R at different PRFs. Figure 6: Typical time integral image of the breakdown channel during repetitive breakdowns (PRF is 100 Hz, SF\({}_{6}\)–N\({}_{2}\), 0.1 MPa, gap spacing of 5.0 mm) recorded by the COMS optical device. The exposure time was 1000 ms. The breakdown voltage decreases as the repetition frequency increases. This is due to a shorter recovery interval and lower degree of insulation recovery caused by high frequency. At high frequencies, the breakdown voltage decreases as the degree of field inhomogeneity decreases, while at low frequencies, the breakdown voltage increases with a decrease in the degree of field inhomogeneity. When the insulation recovery time is 1 s (corresponding to the case of 1 Hz), it is generally considered that the insulation strength of the gap has almost fully recovered [14; 22; 23]. This indicates that improving the homogeneity of the electric field can improve the single breakdown voltage of the switch, but when the recovery time is short, increasing the uniformity of the field will reduce the working voltage, indicating that the insulation recovery performance of the switch in this situation deteriorates. For a switch operating at high frequencies, it is appropriate to use an electrode gap with a higher field inhomogeneity to achieve a higher operating voltage. Switches with a highly inhomogeneous electric field exhibit minor differences in breakdown voltage between 1 Hz and 2000 Hz, while slightly inhomogeneous switches show significant discrepancies in breakdown voltage at low- and high-repetition frequencies. This may be because a switch with a more inhomogeneous field exhibits a faster rate of insulation recovery. After a preceding breakdown of 500 microseconds, the switch had recovered most of its insulation capacity. It is speculated that the strong corona stabilization effect dominates the insulation recovery when the gas density recovery degree is low, which accelerates insulation recovery [7; 24; 25]. The use of electrodes with lower inhomogeneity of the electric field will result in a deterioration in the insulation recovery performance of the corona-stabilized switch, and its insulation strength mainly recovers between 0.5 ms and 1.0 s, with a significantly slower recovery rate. The difference between the two voltage curves in Figure 7a essentially characterizes the recovery in the gap insulation strength within the time interval corresponding to different frequencies. The non-uniform variation in breakdown voltage curves reflects the impact of the high-voltage electrode radius (field inhomogeneity) on the insulation recovery characteristics. The electric field becomes more homogeneous as the radius of the rod electrode increases, resulting in a gradual increase in the difference of the voltage curve at high- and low-repetition frequencies. Based on experience, a typical highly inhomogeneous field switch (\(r=0.5\) mm) can restore approximately 95% of the insulation strength within about 2.0 ms [13]. This indicates that for switches with a more homogeneous electric field distribution, most of their insulation recovers within a long interval of 5 ms to 1 s (corresponding to 2 kHz and 1 Hz respectively), resulting in a slow recovery rate. If the difference between the voltage curves is very small, it indicates that the insulation has barely recovered within the corresponding time interval [14; 22; 25]. There may be two  
Figure 7: The influence of rod radii on the repetitive breakdown voltage and its dispersion of the switch (SF\({}_{6}\), 0.1 MPa, gap spacing is 5.0 mm): (**a**) breakdown voltage; (**b**) voltage dispersion.