# An Estimator of the Resistance of Large Grounding Electrodes from Its Geometric Characterization
## 2 Basics of the Method  
As mentioned in the introduction, an electrode of length L buried in a conductive soil and excited by a current I, presents an electrical resistance that depends significantly on its geometric shape. As the electrode is stretched its resistance decreases. This fact is explained by the decrease (increase) in the electric potential of the mutually interacting electrodes when they move away (get closer). To quantify how packed or stretched its profile is, the so-called stretching index \(\mathrm{i_{s}}\) is introduced here, which is defined by the following relationship  
\[\mathrm{i_{s}}=\frac{\sum\limits_{\mathrm{i,j}\geq\mathrm{i}=\mathrm{i}}^{ \mathrm{N-L}}\left|\overset{\rightarrow}{\mathrm{r}}_{\mathrm{i}}-\overset{ \rightarrow}{\mathrm{r}}_{\mathrm{j}}\right|}{\mathrm{N}^{2}\mathrm{L}^{3}} \tag{1}\]where N is the number of segments per meter of conductor, L is the total length of the electrode and \(\left|\overrightarrow{r}_{i}-\overrightarrow{r}_{j}\right|\) is the distance between segment i and segment j. From Equation (1) it is observed that a stretched electrode will present an index i\({}_{\text{s}}\) greater than a more compacted one of the same length and number of segments per meter. The dimensionless property of Equation (1) guarantees that i\({}_{\text{s}}\) will not change when considering equally stretched electrodes but of different L and N.  
In order to establish a functional relationship between the grounding resistance R, the total length of the electrode L and the stretching index i\({}_{\text{s}}\), numerical experiments with electrodes of lengths L = 1 m, 4 m, 7 m, 10 m and 20 m will be carried out. For each length, several geometric shapes of different folding will be considered to numerically establish the relationship \(\text{R}=\text{f}(\text{L},\text{i}_{\text{s}})\). In these experiments a current I = 1A will be injected to the electrodes, which are buried in a homogeneous soil with resistivity \(\rho=1\)\(\Omega\)m. The electrodes are made up of perfect conductive thin wires of a radius set to r = 6 mm and are assumed to be buried horizontally at a depth h = 0.5 m. For each electrode considered with a length and shape preset, the grounding resistance is calculated by means of the boundary elements method combined with the moments method using a code written by the authors [11], which is inspired by works such as those already cited in the introduction. In a very concise way, the theoretical calculation of the resistance is carried out by calculating the potential acquired by the electrode with respect to the potential of infinity, which is assumed to be zero. Regarding the numerical calculation of the potential, the electrode is divided into smaller portions that filter a part of the total injected current to the ground so that between they all generate a total potential that is the same in all those conductor portions. This procedure leads to a linear system of equations from which the currents filtered by all the portions and also the common potential of all of them, which is the potential of the electrode itself, are obtained. For different soil resistivity and current intensity, R has to be multiplied by the factor \(\rho\)I, while the dependence of R with r and h attenuates as L increases, as will be seen later. The geometric shapes considered here are shown in Figure 1. The common length of all these shapes is L and they range from the straight line, with the maximum stretching, to the capital letter M shape, with the minimum stretching in the series. Except for the rectilinear shape, the total length L has been divided into rods of equal size. Figure 1: Electrode shapes numbered from 1 to 7, with common length L and sorted by the stretching index i\({}_{\text{s}}\). The exponent \(\beta\) shows a certain dispersion, so the mean value of \(\beta=-0.307\) and a 95% confidence interval [\(-0.319\), \(-0.295\)] will be taken for this parameter.