**5. Discussion**

The results at multiple protrusions have shown that the experimental breakdown E50 values can be interpreted by the predictions of the various physical processes as first electron, streamer inception and leader propagation for a single protrusion. In SF6, breakdown will only occur if all criteria are fulfilled. This is also valid in CO2. However, error bars were much larger for the CO2 experiments, resulting in larger uncertainties of the interpretations. In CO2, streamer crossing and spark transition is a possible mechanism for breakdown at negative polarity, additionally to the leader breakdown, which is dominant at positive polarity. Similarly, this was found in [37]. In CO2, breakdowns at protrusions occur much closer to the critical field, which is also reflected in the predictions. This is likely due to the lower slope of the effective ionization coefficient in CO2, which is only a weakly electron attaching gas and, therefore, needs relatively higher fields for streamer and leader inception compared to the critical field than in SF6 [50]. This leads to lower sensitivity to surface roughness or protrusions in case of CO2 compared to SF6.

Increasing the number of protrusions and comparing the resulting *E*50 breakdown fields with predictions from the enlargement law (1) leads to reasonable agreemen<sup>t</sup> for SF6 at negative polarity. This indicates that the statistical processes for a single protrusion, resulting in the measured breakdown probability distribution, can be scaled according to the enlargement law without reaching other, e.g., physical limitations at the tested number of protrusions. This can be possibly understood by the

very low first electron and streamer inception fields, i.e., these criteria are not limiting the scaling of the breakdown fields and the statistical scatter is probably mainly due to leader transition and propagation. This is different at positive polarity in SF6. Whereas, the reduction of breakdown field with 20 protrusions is still well described by the enlargement law, we see a significant discrepancy using 100 protrusions. This is possibly produced by the availability of a first electron which needs a certain minimum field and can, therefore, not be described by statistical phenomena alone—see [34,38]. Thus, if the field is below a certain threshold using a large number of protrusions cannot compensate for the necessary field for creating the first electron in the critical volume. Note that the first electron depends on the size of the critical volume, which increases with the number of protrusions, but also strongly non-linearly on the applied electric field within the critical volume—see (4). Additionally, in CO2, limits for the reduction of breakdown field with increasing number of protrusions can be seen. At negative polarity, such a limitation is the streamer inception criterion for small protrusions of 250 μm and smaller. For larger protrusions there is a lower limit which could be explained by streamer crossing and spark transition, but possibly also by the leader mechanism. This could not be determined unambiguously so far and would need more detailed optical diagnostic investigations. At positive polarity in CO2, a reasonable agreemen<sup>t</sup> with the enlargement law is seen for 20 protrusions, which is similar to SF6. For 100 protrusions and positive polarity the experimental breakdown fields drop to the predicted leader breakdown field. This drop is underpredicted by the enlargement law, which indicates that the breakdown probability distributions for single protrusions does not include all the relevant phenomena with sufficient precision in this case.

For rough surfaces, we investigated the case of very small protrusions in the range of only a few 10 μm in height. We used here the approach of Pedersen [14] to approximate surface roughness by hemispherical protrusions. Thus, this case has similarities to the protrusion array, only that the number of protrusions is probably much higher than in the case of the protrusion array. A detailed surface structural analysis was not done in the present case but unpublished investigations on a similar surface has shown that we can expect one protrusion of 20 μm height per mm<sup>2</sup> as an order of magnitude. For the small surface of 240 mm<sup>2</sup> we can expect, therefore, about 240 protrusions very roughly. For the larger area this was increased by more than a factor 20 to investigate the enlargement law in a more realistic situation. Thus, for the larger surface we can expect as an order of magnitude about 6450 protrusions of a 20-μm size. For negative polarity and SF6, the area scaling due to this increase is satisfactorily described. This indicates that the probability distributions at negative polarity in SF6 describe well the statistical phenomena. In this case, we do not expect a lack of a first electron and the statistical processes of the breakdown like streamer inception and leader transitions are probably decisive for the breakdown statistical scatter. In CO2 and negative polarity, the reduction of breakdown field with increasing area is less well described by the enlargement law. Here, the statistical scatter of the first electron is also probably important. This is similar at positive polarity, where the breakdown fields in the experiments drop roughly to the streamer inception fields for large surfaces and the enlargement laws underpredict this. Possibly, statistical phenomena of the first electron do not follow a simple enlargement law scaling. This needs further investigations.

The investigation showed that enlargement laws cannot be applied without caution. The physical processes which might limit the scaling of statistical distributions, e.g., first electron or streamer inception, must be taken into account. Additionally, the enlargement law is sensitive to the precise shape of breakdown probability distributions, which might lead to insufficient accuracy if the distributions are not determined with sufficient precision. Finally, underlying physical processes might not follow a simple enlargement law scaling, e.g., due to strongly non-linear effects like for the first electron criterion. This is especially the case if various physical processes are involved.

In the present investigation, the lower limit of the Weibull distribution E0 was set to the streamer inception values, which can be regarded as a minimum requirement for breakdown. As can be seen from the figures, in none of the cases this was reached with multiple protrusions. Only for rough surfaces and large area it was approached, i.e., in this case it influenced significantly the enlargement law. The satisfactory agreemen<sup>t</sup> for this case might indicate that this assumption for *E*0 was justified.

The breakdown images have revealed some further information showing especially many discharges occurring in parallel during breakdown events on the short time scale of only a few microseconds. These discharges could bridge the gap or remain arrested in the gap. This shows that a first electron was not lacking in such cases, in agreemen<sup>t</sup> with the model predictions. Thus, at breakdown electrons were not only available at one protrusion but also at other sites. Interestingly, in CO2 also discharges from the bottom plate could be sometimes observed at negative polarity, similar to a return stroke in lightning. This was not observed in SF6. Possibly, this is linked to a di fferent breakdown mechanism, which might be of streamer to spark transition type in this case.
