# Lightning Surge Analysis for Overhead Lines Considering Corona Effect
## 5 Calculated Results and Discussions  
A test overhead line is considered here [17]. Its line conductor is a single-core copper wire with a cross-section of 16 mm\({}^{2}\). The average line height is 10 m above the ground and total length \(l\) is 3084 m. The geometrical capacitance \(C_{0}\) is 6.41 pF/m and the soil resistivity is about 50 \(\Omega\cdot\)m. The corona onset voltages under positive and negative polarities are taken as 70 kV and 80 kV, respectively. The damped oscillation surge voltages with different amplitudes and polarities are applied to the sending end of the line. Two capacitive voltage dividers are installed at the sending end and at 1674 m from the sending end, respectively, to measure the voltage waveshapes. The parameter values for describing the \(q\)-\(u\) curves are given in Table 1, where the signs (+) and (-) denote positive and negative polarities. Using the method proposed above, the voltage waveshapes are calculated at a \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{Applied voltage amplitude} & 156 & 178 & 199 & 226 & 254 & 275 & 319 & 328 \\ \hline The first & Calculated & 87.4 & 114.2 & 116.5 & 117.8 & 120.6 & 127.3 & 136.7 & 142.8 \\ wave crest & Field test & 84.5 & 112.5 & 114.7 & 120.3 & 126.5 & 131.7 & 140.8 & 144.7 \\ \hline \multirow{4}{*}{The first wave trough} & Calculated (in the & & & & & & \\ presence of minor & 55.2 & 60.3 & 70.2 & 82.2 & 84.0 & 92.8 & 109.6 & 110.2 \\  & & Loops) & & & & & & \\ \cline{1-1}  & Calculated (in the & & & & & & \\ absence of minor & 52.6 & 55.2 & 68.1 & 78.9 & 82.2 & 91.3 & 104.7 & 106.8 \\  & & Loops) & & & & & & \\ \cline{1-1}  & Field test & 55.1 & 56.8 & 67.9 & 82.0 & 86.4 & 94.4 & 110.6 & 117.1 \\ \hline \multirow{4}{*}{The second wave crest} & Calculated (in the & & & & & & & \\ presence of minor & 104.1 & 117.1 & 134.9 & 153.6 & 158.0 & 170.0 & 207.9 & 237.8 \\  & & Loops) & & & & & & \\ \cline{1-1}  & Calculated (in the & & & & & & & \\ absence of minor & 106.8 & 118.2 & 136.3 & 157.4 & 165.3 & 171.7 & 215.5 & 241.2 \\  & & Loops) & & & & & & \\ \cline{1-1}  & Field test & 105.2 & 114.4 & 135.8 & 155.7 & 157.1 & 170.4 & 212.3 & 227.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Voltage values under positive polarity (kV). Figure 16: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{Applied voltage amplitude} & 175 & 188 & 195 & 226 & 258 & 292 & 340 & 358 \\ \hline The first & Calculated & 128.3 & 149.2 & 151.1 & 186.1 & 200.3 & 224.6 & 263.4 & 271.9 \\ wave crest & Field test & 125.7 & 147.5 & 149.7 & 181.8 & 197.2 & 222.5 & 261.6 & 269.5 \\ \hline \multirow{4}{*}{The first wave trough} & Calculated in the & & & & & & & \\ presence of minor & 58.7 & 64.9 & 67.3 & 86.7 & 99.1 & 104.5 & 117.5 & 134.8 \\  & & Loops) & & & & & & \\ \cline{1-1}  & Calculated (in the & & & & & & & \\ absence of minor & 56.5 & 62.3 & 65.8 & 79.4 & 90.7 & 98.6 & 105.8 & 119.5 \\  & & Loops) & & & & & & \\ \cline{1-1}  & Field test & 59.6 & 68.1 & 70.7 & 89.4 & 104.2 & 106.7 & 116.5 & 133.0 \\ \hline \hline \multirow{4}{*}{The second wave crest} & Calculated in the & & & & & & & \\ presence of minor & 129.5 & 139.3 & 142.9 & 166.9 & 193.8 & 214.9 & 230.1 & 250.5 \\  & & Loops) & & & & & & \\ absence of minor & 134.8 & 143.1 & 145.7 & 170.1 & 198.4 & 225.2 & 241.3 & 261.4 \\  & & Loops) & & & & & & \\ Field test & 128.3 & 136.0 & 136.7 & 163.2 & 190.1 & 215.2 & 218.2 & 242.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Voltage values under negative polarity (kV). Figure 16: Calculated and field test voltage waveshapes under positive polarity (\(U_{\rm m}\) is the voltage amplitude at the sending end; \(\ldots\) sending end; \(\ldots\) sending end; \(\ldots\) field test; calculated). (**a**) In the presence of minor loops, (**b**) in the absence of minor loops.