3.2.2. Streamer Inception

Streamer inception fields were calculated using the semi-empirical approach in [38]

$$K = \int\_0^{l\_G} \overline{\alpha} \cdot dz \tag{5}$$

With the ionization integral parameter *K* = 10.5 and 13 for SF6 [45] and CO2 [37], respectively. The effective ionization coefficient α was taken from [38] for SF6 and [37] for CO2. The integration is done along the axis of symmetry starting from the protrusion tip at *z* = 0 where the electric field is above the critical field, up to the distance *z* = *lcr*, where the field has dropped to the critical field.

### 3.2.3. Streamer Crossing and Spark Transition

Streamer crossing followed by spark transition is only relevant for CO2, since in SF6 this occurs at the critical field, which is much higher than leader propagation fields [38]. In [37], it was shown that streamer crossing for long duration voltage application times of several 100 μs is approximately sufficient for breakdown in the pressure range investigated here. In case of short-duration voltage application time, a higher field than that necessary for streamer crossing is needed to allow for the spark transition within the short available time of 10 μs, since the heating processes in the streamer channel need a sufficient electric field and time. This field increase was estimated from [46] to roughly 35% compared to the streamer crossing field, which is based on the breakdown voltage ratio for short and long duration waveforms.

### 3.2.4. Leader Propagation and Breakdown

As shown in [35,38], leader propagation through the gap can be associated with breakdown. Leader inception and propagation was calculated using the previous model from [35] for SF6. In this model, the streamer corona charge at the protrusion tip or a propagating leader is calculated and fed into the streamer or leader channel. This leads to stepped heating followed by stepwise leader propagation through the gap. Crossing 90% of the gap distance was defined as sufficient for breakdown. For CO2, the same model but with adapted thermodynamic properties of mass density, enthalpy, velocity of sound and effective ionization coefficients [47] from our own, in-house-developed solver were used.
