# PD-Free Design of Insulation Systems: An Application to Laminated Busbars
## 5 LBB DC Subsystem  
The values of conductivity parameters for (3) used in the COMSOL simulation are reported in Table 3. They were obtained by surface and bulk conductivity testing performed on flat specimens made by an epoxy resin similar to that used for the LBB insulation, at three different fields and temperatures, by a procedure described in [27]. Notably, the bulk conductivity is expressed in S/m, while the surface conductance is expressed in S (the information provided by surface current measurements is indeed lacking that of the thickness of the surface layer where the current flows. Air conductivity was taken equal to \(10^{-14}\) S/m.  
FEM simulations were performed referring to the section between one layer of Figure 1 and the closest grounded terminal [24]. Figure 5 shows the DC electric field profile at the value of voltage that corresponds to the intersection of the maximum value of the field profile with those of the surface PD inception field given by (4) and using (5) with \(x_{1}=0\) (i.e., coinciding with the triple point). This provides the DC PDIV estimate.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Excited Layer** & \begin{tabular}{c} **Theoretical PDIV at 25 \({}^{\circ}\)C** \\ **and 60 \({}^{\circ}\)C [kV]** \\ \end{tabular} & \begin{tabular}{c} **Measured PDIV** \\ **at 25 \({}^{\circ}\)C [kV]** \\ \end{tabular} & \
\begin{tabular}{c} **Measured PDIV at 60 \({}^{\circ}\)C [kV]** \\ \end{tabular} &
\begin{tabular}{c} **Error at 25 \({}^{\circ}\)C** \\ \end{tabular} \\ \hline Layer 1 & 21.0 & 22.0 & 20.5 & 4.5\% \\ Layer 2 & 21.0 & 22.0 & 20.5 & 4.5\% \\ Layer 3 & 20.5 & 21.5 & 20.5 & 4.7\% \\ Layer 4 & 21.0 & 22.0 & 20.5 & 4.5\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Theoretical and measured PDIV, PD extinction voltages, and estimation error for DC measurements.  
Figure 5: DC tangential electric field profile vs. distance between high-voltage and ground electrodes (Figure 1) and PD inception field from (4) at 21 kV, i.e., estimated DC PDIV for layer 1 (Table 4).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\sigma_{0}\) [S/m] & \(\alpha\) [K\({}^{-1}\)] & \(\beta_{E}\) [mm/kV] & \(E_{0}\) [kV/mm] & \(T_{0}\) [K] \\ \hline Bulk & \(4.4\times 10^{-16}\) & 0.07 & 0.13 & 2.5 & 298 \\ Surface & \(2.8\times 10^{-14}\) & 0.066 & 1.3 & 0.1 & 293 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Physical parameters of epoxy for (3) needed for simulations.  
As in AC, large field gradients can be observed at the triple point, especially at the high-voltage terminals (thus, \(k_{s}\) is very low, i.e., \(1.1\times 10^{-3}\)). Asymmetry is caused by the conductivity dependence on the electric field (through the \(\beta\) coefficient).  
The conductivity temperature dependence (through coefficient \(\alpha\)) is expected to play a role in the inception field, and consequently on the PDIV, through the relevant electric field variation. However, for the parameters in Table 3 and considering the highly divergent field at the triple point, Figure 6 shows that there is very little variation in DC PDIV between 25 and 60 \({}^{\circ}\)C (see also Table 4). It can be speculated that such a result can be addressed to the dependence of conductivity on the field at the triple point (\(\beta\)) prevailing on the electric field variation driven by the temperature-dependence (\(\alpha\)). Note that the DC electric field distribution depends on conductivity which, in turn, changes with the field itself.