**3. Evaluation Procedures**

### *3.1. Breakdown Probability Distributions*

From the recorded breakdown fields, the empirical cumulative breakdown probability distributions were determined by the Turnbull algorithm [44] and fitted with three-parameter Weibull distributions [41] as proposed in [6,42]. The zero crossing *E*0 of the Weibull distribution was set to the streamer inception field, calculated for each protrusion length as will be described below in Section 3.2.2. From the distributions, the 50% BD probability field *E*50 and the standard deviation σ was determined. Note that these values refer to the background field, i.e., *E*50 = U50/D with 50% breakdown probability voltage U50 and the gap length D. Generally, for the identical protrusions of *m*, the enlargement law based on the Weibull distributions generally follows from [6]

$$p(E) = 1 - \varepsilon^{-m \cdot \left[\frac{E - E\_0}{E\_{\rm cl3} - E\_0}\right]^\gamma} \tag{1}$$

With the shape parameter γ and the 63% breakdown probability field *E*63. In the present case for γ and *E*63 the values from the fits to the empirical cumulative breakdown probability distributions were used. An example of the enlargement scaling is shown in Figure 4. With increasing number of protrusions, the breakdown probability distribution becomes steeper and shifts towards *E*0. Note, that such scaling is purely a statistical process and does not necessarily describe the scaling of any physical processes. This can be interpreted as follows: if the breakdown probability distribution follows the scaling from (1), it can be expected that the underlying processes are of statistical nature, e.g., by the availability of a first electron to start an avalanche. For electrodes with surface roughness, *m* is the ratio of the surface areas.

**Figure 4.** Example of Weibull distributions for increasing number of protrusions *m*. In this example, *E*63 = 250 kV/cm, *E*0 = 100 kV/cm, γ = 7 was chosen arbitrarily.

### *3.2. Discharge Inception and Breakdown Models*

In the following subsections, models for the description of first electron, streamer inception and leader propagation will be described. The models are the same as presented in [38] for SF6 and will only be briefly summarized here with the specific adaptions for CO2. For these models, the decay of the electric field at the protrusion and at the electrode surface is needed. The field enhancement in the axis of symmetry in front of a protrusion was calculated by a multipole approximation

$$\frac{E(\mathbf{x})}{E\_b} \approx 1 + \frac{l/r - 1}{\left(\mathbf{x}/r + 1\right)^2} + \frac{2}{\left(\mathbf{x}/r + 1\right)^3} \tag{2}$$

with the protrusion length and radius *l* and *r*, respectively, as used in [36]. *Eb* is the undisturbed background field. It was checked by 3D electric field calculations with COMSOL for some cases that this approximation was sufficiently precise for the present purpose. With *r* = 100 μm as assumed protrusion radius, average deviations in the field decay were within 10%. The formula is only correct until *l* = *r*, i.e., *l* = 100 μm. Therefore, predictions will only be shown for 100 μm length and higher. For rough surfaces l/*r* = 1 with *l* = 20 μm as a typical peak height of roughness structures was assumed. This is similar to the approach of [14]. The chosen length is deduced from the measured *Rz* by assuming *l* = *Rz*/3. Note that *Rz* describes the average difference between highest peaks and lowest valleys of the surface.

### 3.2.1. First Electron

The electric field needed for a start electron at positive polarity in SF6 can be estimated from the volume where the electric field is above the critical field and where an electron can be detached from a negative ion in the available time of voltage application, i.e., in the present case within 10 μs—see [38] for details of the model. All parameters for SF6 were taken from [34].

At negative polarity, a start electron is assumed to be delivered from the electrode surface. For SF6, this could be described reasonably well by using the Fowler–Nordheim (FN) equation in [34], which gives the electron production rate emitted from a surface area at given electric field. The statistical time lag at negative polarity is then

$$t\_s = (A\_{eff} \cdot 10^{4.52/\sqrt{\Phi}} \cdot 1.54 \cdot 10^{-6} \cdot \frac{(\beta \cdot E\_b)}{\Phi} \cdot \exp\left[-\left(\mathcal{O}^{1.5} \cdot 2.84 \cdot 10^9\right)/\left(\beta \cdot E\_b\right)\right] / e\right)^{-1} \tag{3}$$

where Φ ≈ 4.5 eV is the work function for steel, e the elementary charge, β = (2 + *<sup>l</sup>*/*r*)·β<sup>2</sup> a field enhancement factor at to the protrusion tip (see also (2)) and due to micro-surface roughness and *Ae*ff the effective electron emitting area. For SF6 values for β2 and *Aeff* were taken from the fits in [34] with β2 = 20 and *Aeff* = 10−<sup>16</sup> m2. From this, the necessary electric field for a start electron within the available time can be deduced.

For CO2, the same approach and models as for SF6 were used but with CO2 specific adaptions and empirical approximations for the relevant parameters. For the statistical time lags, the field dependence of the electron detachment rate, the equilibrium negative ion concentration and the parameters in the FN equation were adapted to experimental data from [37]. At positive polarity, best agreement, within a large scatter, was achieved by empirically using an equilibrium concentration of 10<sup>4</sup> ions/m<sup>3</sup> at 0.1 MPa in the pressure-dependent ion concentration, as given in [38], and a field dependence of the electron detachment rate coefficient of

$$\delta = 10 \cdot \left(\frac{E}{E\_{cr}}\right)^{13} \left[1/\text{s}\right] \tag{4}$$

with the critical electric field of CO2 at pressure p: *Ecr* = *<sup>p</sup>*·(*E*/*p*)*cr*,0, with (*E*/*p*)*cr*,<sup>0</sup> = 23 V/(<sup>m</sup>·Pa) [37]. To our knowledge, no experimental data are available to verify this scaling of the electron detachment rate. At negative polarity, the field enhancement parameter β2 in (3) was set to β2 = 70 and the effective area needed to be set to *A*eff = 10−<sup>24</sup> m<sup>2</sup> to achieve best agreemen<sup>t</sup> with the experimental data. Such small value for *A*eff is probably not realistic, but one has to consider the simplicity of the FN equation. The approximations have to be seen, therefore, just as physically motivated fits to the experimental data which are valid in the given parameter range of protrusions up to a 4.5 mm length. Examples for statistical time lags for negative and positive polarity with the data from [37] are shown in Figure 5. For surface roughness, values from the rod plane experiment in [37] are used. The field enhancement factor at the surface of the rod (=*E*surface/(1/D)) of the rod in these experiments was about 17. The background field in the plots is the undisturbed electric field U/D without field enhancement by a protrusion or by the rod.

**Figure 5.** Statistical time lags in CO2 at negative (**left**) and positive (**right**) polarity vs. background electric field for technical surface roughness and protrusions with 1.5 and 4.5 mm lengths. The curves are the model predictions and the symbols result from measurements given in [37]. For the technical surface roughness estimates, a 20-μm hemispherical protrusion on the surface was assumed [36]. Note that, for positive polarity, the electric field is normalized to the critical field *Ecr*, which yields a less scattered representation for the different pressures used.
