# An Estimator of the Resistance of Large Grounding Electrodes from Its Geometric Characterization
## 2 Basics of the Method
Fitting to the power law \(\mathrm{R_{L}}=\mathrm{R_{0}}\mathrm{\rho L^{\alpha}}\) with \(\rho=1\)\(\Omega\mathrm{m}\), the results for the value of \(\mathrm{R_{0}}\) and \(\alpha\) parameters together with a 95% confidence interval are \(\mathrm{R_{0}}=0.457\pm 0.014\) and \(\alpha=-0.731\pm 0.038\), from which the desired relationship Equation (4) is obtained  
\[\mathrm{R}=\mathrm{R_{0}}\mathrm{\rho L^{\alpha}i_{s}}^{\beta}\qquad\left\{ \begin{array}{l}\mathrm{R_{0}}=0.457\\ \alpha=-0.731\\ \beta=-0.307\end{array} \right. \tag{4}\]  
To justify that the dependence with \(\mathrm{h}\) and \(\mathrm{r}\) is not considered, a numerical experiment will be carried out in which the resistance of electrodes will be theoretically calculated as a function of the length L, burial depth \(\mathrm{h}\) and radius \(\mathrm{r}\). A horizontal rod type electrode of variable length L with values 1, 3, 5, 7, 10, 15, 20 and 30 m was considered. For each length, a range of values of the radius \(\mathrm{r}\) among 2, 4, 6, 8 and 10 mm were used and for each radius, a range of values of the burial depth \(\mathrm{h}\) among 0.1, 0.3, 0.5, 0.7 and 0.9 m were taken into the calculations. Figure 3 shows the variation of the grounding resistance \(\mathrm{R}\) as a function of the length of the electrode L when \(\mathrm{r}\) varies for a fixed value of \(\mathrm{h}\), on the left panel, and when \(\mathrm{h}\) varies for a fixed value of \(\mathrm{r}\), on the right panel. From the mentioned figure, it follows that as the length L of the electrode increases, the variation of \(\mathrm{R}\) with \(\mathrm{h}\) and \(\mathrm{r}\) is clearly the more attenuated the longer the length of the electrode is. Regarding the uncertainty in determining the estimator of \(\mathrm{R}\), assuming no correlation between the variables \(\mathrm{R_{0}}\), \(\alpha\) and \(\beta\), the standard deviation of \(\mathrm{R}\) is calculated by  
\[\sigma_{\mathrm{R}}=\mathrm{R}\cdot\sqrt{\frac{\sigma_{\mathrm{R_{0}}}^{2}}{ \mathrm{R_{0}^{2}}}+\left(\mathrm{log(L)}\right)^{2}\cdot\sigma_{\alpha}^{2}+ \left(\mathrm{log(i_{s})}\right)^{2}\cdot\sigma_{\beta}^{2}} \tag{5}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**L (m)** & \(\mathrm{R_{L}}\) & \(\beta\) \\ \hline
1 & 0.456 & \(-0.324\) \\ \hline
4 & 0.171 & \(-0.296\) \\ \hline
7 & 0.110 & \(-0.302\) \\ \hline
10 & 0.083 & \(-0.298\) \\ \hline
20 & 0.046 & \(-0.314\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical fitting to the power law Equation (2) from data of Table 1.  
where \(\sigma_{i}^{2}\) are the variances of the R\({}_{0}\), \(\alpha\) and \(\beta\) variables. From Equation (5), a 95% confidence interval could be obtained for the estimator of R by \(\left[\text{R}-\text{Z}\frac{\sigma_{\text{R}}}{\sqrt{\text{N}}},\text{R}+\text {Z}\frac{\sigma_{\text{R}}}{\sqrt{\text{N}}}\right]\) with \(Z=1.96\). As a conclusion, it can be stated that Equation (4) can be accepted as an approximation to the resistance with better results as the length of the electrode increases. The possible variation of the numerical parameters of Equation (4) with the burial depth h and the electrode radius r can be ignored, always within the limits imposed by practical applications. The Estimation of \(R\) on Synthetic Models. Estimation of the Resistance of a Real Operating Electrode  
In this section, Equation (4) will be tested on some electrode models already used and on other electrodes that will be fully defined in the text. As a first test, the result of the estimation of the resistance R of the type 7 electrode, with a total length of L = 20 m and with a segmentation of N = 20 segments per meter in a soil of resistivity  
Figure 3: Grounding resistance vs. the electrode length L when h and r take extreme values according to their ranges of variation, as indicated in the text. \(\rho=1\)\(\Omega\)m, was tested.