## 3 Impact of the Electrode Shape on the Electric Field Computation and PD Inception Field Estimation
### Influence of Electrode-Specimen Contact Area  
The electrode configuration used to trigger surface PD, Figures 1 and 2, is geometrically modeled by considering its zoomed image taken by a microscope (see Figure 5a); this allows the accurate geometrical profile sketched in Figure 5b to be achieved. In particular, it is noteworthy that the contour of the base touching the tested specimen would present a curved shape (which can be modeled by a quadratic law) rather than straight. The cause of this curvature could be a design purposely developed to reduce electric field concentration along its contour (as for pseudo-Rogowski electrodes) or, practically, a manufacturing defect or usage deterioration. Ideally, in fact, such electrodes should be thought to have a  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Radius R (mm)**} & **Surface Inception Field (kV/mm)** & **Surface PDIV (kV)** & **Gas Inception Field (kV/mm)** & **Gas PDIV (kV)** \\ \hline
0.600 & 1.64 & 1.54 & 9.1 & 3.3 \\
0.625 & 1.64 & 1.56 & 10.1 & 3.4 \\
0.650 & 1.64 & 1.58 & 9.5 & 3.3 \\
0.675 & 1.62 & 1.60 & 9.4 & 3.3 \\
0.700 & 1.61 & 1.61 & 9.9 & 3.3 \\
0.725 & 1.60 & 1.63 & 9.8 & 3.  
straight, flat base, as in the example represented in Figure 6, where both a particular HV electrode (Figure 6a) and the derived accurate geometrical profile (Figure 6b) are shown.  
From now on, let us define the electrode with the curve-shaped contact surface as electrode A and the electrode with the perfect flat-shaped contact surface as electrode B.  
Calculations shall then be made to evaluate whether electrode A provides significantly different values of the PD inception field compared to electrode B. This, indeed, could affect the discharge model validation and, thus, the application of the three-leg approach for the insulation system's PD-free design. Eventually (see Section 4), an experimental validation will be performed to verify the validity of the computations made with electrodes A and B.  
As regards simulations relevant to electrodes A and B, the electric field computations start at the intersection of the electrode profile with the horizontal line at 40 \(\upmu\)m (at 2.1 mm from the symmetry axis for electrode A and 0.24 mm for electrode B, Figure 7). This choice comes to avoid unrealistic electric field values at the triple points, and it is supported by  
Figure 4: Parametric profiles of electric field at the HV electrode of the cell in Figure 1, with variation of the curvature radius, \(R\). The \(x\)-axis zero is the triple point \(P\). (**a**) Tangential surface field: peak field decreases and the value of \(k\)s increases as long as \(R\) goes up (this causes a slight increase in surface PDIV); and (**b**) Orthogonal electric field: the higher curvature radius, the larger electric field, above 0.05 mm (and the opposite below 0.05 mm).  
Table 3 summarizes the estimates of PD inception field and voltage for gas and tangential discharges, referring to electrodes A and B. Computations made by means of (1)-(3) show there is a significant incremental factor of 2.5 between the gas PDIV values for electrodes A and B, while the surface PDIV is not affected as much by the slight rounding of the contact surface (A vs. B). In practice, this means that for electrode A, gas discharges have a lower PDIV value compared to surface ones; thus, the cell designed to generate surface discharges (Figures 1 and 2), with the purpose of establishing criteria for a PD-free design of the insulation surface, might not work properly if electrode B is replaced by electrode A.  
To go more in detail, Figures 8 and 9 display the orthogonal electric fields computed at \(V_{sim}=\)1 kV AC and at the PDIV values, together with the gas PD inception field from (1), for electrodes A and B. It can be seen (from curve intersections) that the gas discharge PDIV is 1.3 kV for electrode A (Figure 8b), but 3.3 kV for B (Figure 9b), which supports the estimates in Table 3. Noticeably, the intersection of the electric field profile and gas discharge model