## 2 Modeling Grounding Electrodes with Internal Resistance: Backgrounds
In this case, the electrode potential is commonly measured at the current injection point which is the point of the electrode where the fault current to be leaked to the ground enters. Usually, for electrodes with non-negligible internal resistance, both the current injection point and the measurement point are important parameters since the potential value is strongly dependent on the internal distribution of leakage currents to the ground. With respect to the model used to take into account the internal resistance of the electrodes, the one used here is the Circuit-Based Model (CBM) [17; 18]. According to this model, the electrodes are broken down into branches and nodes to which circuit theory applies. Figure 2 shows in the upper part a rod like the ones used in the driven-rod test, which is divided into N branches. The potential \(U_{k}\) and the leakage current \(I_{k}\) in each branch are displayed. At the bottom, the circuit-based model is shown. At each node of the circuit, the potential \(V_{t}\) and the source of nodal leakage current \(I_{t}\) are located. The internal currents \(i_{k}\) that move inside the conductor are also shown. Each branch obeys a general equation for the potential difference between the ends (nodes) that for the generic branch k takes the form,  
\[V_{k^{+}}-V_{k^{-}}=i_{k}\cdot z_{k}+\sum\limits_{r=1}^{B}i_{r}\cdot Z_{kr}^{L} \tag{6}\]  
where \(B\) is the number of branches of the electrode system containing \(N\) nodes, \(i_{r}\) is the current through the branch \(r\), \(z_{k}\) is the internal resistance of the branch \(k\) and \(Z_{kr}^{L}\) is the inductive impedance of the branch \(k\) due to the branch \(r\) (sometimes called longitudinal impedance), which can be neglected for stationary or low frequency currents. Figure 3 shows a diagram of the magnitudes introduced for a branch of the circuit. Figure 2: A picture of the circuit-based model of the rod driven into the soil. The branch currents \(i_{k}\) and the node currents \(J_{n}\) are related by Kirchhoff's law: for the node \(n\), the algebraic sum of all currents that concur in the node is zero, \(\sum\limits_{n}i_{k}+J_{n}+F_{n}=0\), where the possible source or injection currents \(F_{n}\) on the node \(n\) have been introduced. This expression must be introduced in (6) so that only the currents and potentials of node, \(J_{n}\) and \(V_{n}\) respectively, together with the supply currents \(F_{n}\), are considered as the main variables. To establish the relationship between the currents in the nodes, it is necessary to agree on a consistent sign criterion. It will be agreed to consider positive the outgoing currents of the nodes either towards the branches, \(i_{k}\), or towards the ground, \(J_{n}\), and the potential differences between the nodes of each branch will be evaluated from the node through which the current leaves until the node through which it enters. Using this convention, it can be written:  
\[J_{k}+\sum\limits_{l=1}^{N}G_{lk}\cdot V_{l}=F_{k}\ (k=1\cdots N) \tag{7}\]  
The square matrix \(G\) is constructed from [PERSON]'s law in each node, [PERSON]'s law in each branch and the branch-node connection matrices, all together with the coherent criterion of signs for the currents. At the same time that a potential difference due to the internal circulation of branch currents is generated in each branch, an absolute potential (with respect to the null potential of infinity) is also generated in each branch due to the leakage currents towards the surrounding conductive soil. Thus, each branch acquires a potential \(U_{k}\), which depends on the currents filtered to the ground by all other branches \(I_{m}\), including the branch \(k\) itself.