## 2 Theoretical Background  
The three-pin method is represented in Figure 1. The electrode to be measured is connected to a distant auxiliary electrode through a current source with intensity I.  
Figure 1: Illustration of the three-pin method for measuring the grounding resistance \(R_{G}\). The circuit is completed through the ground. The potential difference between points \(A\) and \(B\) is  
\[ \begin{array}{l}V_{A}=V_{e}-\frac{\rho I}{2\pi D}\\ V_{B}(x)=\frac{\rho I}{2\pi x}-\frac{\rho I}{2\pi(D-x)}\\ V_{A}-V_{B}(x)=V_{e}-\frac{\rho I}{2\pi}(\frac{1}{D}+\frac{1}{x}-\frac{1}{D-x} )\end{array}  \tag{1}\]  
In expression (1), \(V_{e}\) is the potential of the electrode when it is isolated, i.e., the grounding potential. Additionally, it is implicitly assumed that at point \(x\), both the grounding electrode and the auxiliary current electrode behave as point sources of current at ground level. The same applies to the auxiliary electrode with respect to point \(A\). This question should be discussed in the case of long electrodes or not very large distances \(D\). Based on expression (1), the measured grounding resistance \(R_{G}^{meas}\) is defined as  
\[R_{G}^{meas}(x)=\frac{V_{A}-V_{B}(x)}{I}=R_{G}-\frac{\rho}{2\pi}(\frac{1}{D}+ \frac{1}{x}-\frac{1}{D-x}) \tag{2}\]  
where \(R_{G}\) is the true grounding resistance. It is clear that \(R_{G}^{meas}=R_{G}\) for any given value of \(x\), provided that the quantity in brackets in (2) \((\frac{1}{D}+\frac{1}{x}-\frac{1}{D-x})=0\) is zero. Thus, solving for the value of \(x\), it gives \(x=0.618\cdot D\), known as the 62% rule, provided that the assumptions mentioned above are satisfied. In summary, by placing the auxiliary electrode at a sufficiently distant location \(D\) from the grounding point, this rule can be applied to obtain values that are very close to the true grounding resistance. It is evident that as the implicit assumptions described above are no longer valid, the analysis presented here cannot be applied when measuring an extensive electrode or also when the distance \(D\) is not large enough to ensure that the potential generated by the auxiliary current electrode on the electrode to be measured is \(\frac{\rho I}{2\pi D}\). An extensive mesh cannot behave like a point source of current unless viewed from an immensely large distance. It also cannot be applied if there is not enough space to place the auxiliary electrode sufficiently far away, even if the grounding system is not very extensive. Since the electrode can no longer be assimilated to a current point,  
\[ \begin{array}{l}V_{A}=V_{e}-\frac{\rho I}{2\pi D}\\ V_{B}(x)=V_{e}(x)-\frac{\rho I}{2\pi(D-x)}\\ V_{A}-V_{B}(x)=V_{e}-V_{e}(x)-\frac{\rho I}{2\pi D}+\frac{\rho I}{2\pi(D-x) }\end{array}  \tag{3}\]  
where \(V_{e}(x)\) is the potential created by the electrode at point \(x\), considered as isolated, and the distance \(D\) is not well defined since it must correspond to the distance at which \(V_{A}\) coincides with the true potential of the electrode. Dividing by the injected current \(I\),  
\[R_{G}^{meas}(x)=\frac{V_{A}-V_{B}(x)}{I}=R_{G}-\frac{V_{e}(x)}{I}+\frac{\rho} {2\pi}(\frac{1}{D-x}-\frac{1}{D}) \tag{4}\]  
Note that from expression (4), \(R_{G}^{meas}=R_{G}\), if and only if \(V_{e}(x)=\frac{\rho I}{2\pi}(\frac{1}{D-x}-\frac{1}{x})\) for some value of \(x\), and otherwise, it will be above or below \(R_{G}\) depending on the value of \(x\). This condition is equivalent to the 62% rule when it is not possible to assume that the electrode to be measured behaves as a current point. Next, the so-called correction factor \(k_{m}(x)\) will be introduced. It is defined as the ratio between the true grounding resistance and the resistance measured in the field \(k_{m}(x)=\frac{R_{G}}{R_{G}^{meas}(x)}\).