# An Estimator of the Resistance of Large Grounding Electrodes from Its Geometric Characterization
## 1 Introduction  
Determining the theoretical value of the grounding resistance of an electrode is a well-known calculation process that requires knowledge of the geometry of the conductors that make up the electrode, their location in the ground, burial depth, orientation and of the conductive soil model. The electrode must be composed by conductors in the form of thin wires, which must be segmented in the calculation process to build the impedance matrix from which, by an inversion procedure, the distribution of leakage currents to the ground is obtained by all the segments into which the entire electrode has been divided [1; 2; 3]. The calculation is made possible by imposing either the electrode to be equipotential if the conductor is considered perfect, or the distribution of potentials according to the entry point of the total current to be filtered to the ground in the case of real conductors with non-zero internal resistivity [4]. As the electrode grows in size, the impedance matrix gets bigger and bigger with the consequent computational cost to carry out the matrix inversion. A thicker segmentation can help to speed up the calculation, but in very large electrodes this can lead to obtaining theoretical resistance values that could be far from the true value although always above it [5]. For decades, some authors have worked on finding calculation expressions for the grounding resistance of an electrode without carrying out the direct calculation based on the determination of the potential acquired by the electrode itself by means of numerical techniques. Many of these works are focused on grounding grids for substations. Some use properties of these electrodes such as the duality between resistance and capacitance [6]. Others, on the other hand, deal with a calculation of \(R\) based on the equipotential surfaces of the conductive soil around the electrode, which are evaluated using FEM [7]. Another group of works base their estimators on empirical expressions for simple structures, which combined with each other, build a complex electrode [8]. On the other hand, it is not easy to find research that relates the grounding resistance with the geometric properties of the electrodes. In the field of geophysics, mainly related to electrode arrays in electric resistivity tomography (ERT), some work can be found in which an attempt is made to establish a relationship between geometric configurations of electrodes with the electric resistance in order to improve geoelectric measurements of the soil [9]. In the present work, an expression is proposed to estimate the grounding resistance of an electrode buried in a homogeneous soil, which is based on a known fact, although scarcely cited in the literature. An electrode of total length L presents a grounding resistance that is so much lower the more stretched it is, in other words, the further away some parts of the electrode are from others of the electrode itself. The minimum resistance is only achieved by completely stretching the electrode in a straight line. This property is nothing more than a consequence of the interaction between conductors, which causes the potential to decrease with the distance between them [10]. An electrode of length L folded has a higher grounding resistance than the same electrode stretched in a straight line. If the degree of stretching of an electrode, here quantified by the so-called stretching index \(\mathrm{i_{s}}\) of the electrode, is defined so that it is independent of the size of the electrode and the segmentation used to divide it, a relationship could be established between the grounding resistance R, the length L of the electrode and the stretch index \(\mathrm{i_{s}}\). This relationship depends, in principle, on the burial depth \(\mathrm{h}\), on the radius \(\mathrm{r}\) of the thin wire conductors that make up the electrode and on the homogeneous resistivity of the ground \(\mathrm{\rho}\).