# Optimal Design of the Vertical Earthing with Electrodes Arranged in Line
## 2 Methodological Aspects  
### Basic Sizes  
The earthing components of a VEG with electrodes arranged in line (VEGL), as well as the main characteristic dimensions, are shown in Figure 1. Only two vertical adjacent electrodes (rod or pipe, 1) are represented (Figure 1a), tied at their top by a horizontal strip (2). The limit of the footprint area on the earth (4) is considered on both sides of the horizontal strip, at a distance equal to half of the distance between two adjacent electrodes (Figure 1b).  
In a synthetic view, the following 8 physical sizes can be considered as characteristic for VEGL:  
\[\left\{\ell,\;\text{a},\;\text{d},\;\rho_{\text{P}},\;\text{q},\;\text{g},\; \text{R}_{\text{Pn}},\;\text{n}_{\text{e}}\right\},\]  
where \(\ell\) is the length of the electrodes, a--the distance between adjacent electrodes, d--the external diameter of the cylindrical electrodes, \(\rho_{\text{P}}\)--the resistivity of the soil, assumed uniform, q--the burial depth of the electrode's upper end, g--the thickness of the pipe wall from which the electrode is realized, R\({}_{\text{Pn}}\)--the design value resistance and n\({}_{\text{e}}\)--the number of electrodes.  
Concerning these sizes, the following aspects were highlighted [12]:  
* In all practical cases, the variables {\(\rho_{\text{P}}\), q, R\({}_{\text{Pn}}\)} have determined or imposed values. Thus, the resistivity of the soil \(\rho_{\text{P}}\) is experimentally determined by measurements on the soil where the VEGL is to be made. The burial depth, q, is imposed by the maximum freezing depth, according to the geographical position of the objective, q \(\in\) {0.8-0.9 m} (in Romania and other European countries); R\({}_{\text{Pn}}\)\(\in\) {1, 4, 5, 10} \(\Omega\), depending on the VEGL's concrete destination [1; 3; 4; 5] when there is no natural earthing system. When there is a natural earthing system, the design value resistance will be calculated so that the resistance to earth of combined systems has one of the above values.
* The wall thickness, g, of the pipe electrode has a reduced influence on the VEGL resistance to earth, with a recommended value of g \(\geq\) 3.5 mm for a longer lifetime of the earthing system.
* The electrode's number, which can only be a natural number, n\({}_{\text{e}}\)\(\in\) {1, 2, 3,...}.  
In the present study, the classification of the basic sizes that intervene in the VEGL design is as follows:  
* The sizes \(\ell\) and n\({}_{\text{e}}\) are considered as independent variables, with the ranges of interest values \(\ell\)\(\in\) [0.5-4] m and n\({}_{\text{e}}\)\(\in\) {1, 2, 3,...}, increasing the electrodes' numbers up to the limits of the technical solution.  
Figure 1: Multiple earthing system with vertical electrodes (VEG) with vertical electrodes placed after a straight line: (**a**) partial section, with the main characteristic dimensions; (**b**) footprint; 1—electrode; 2—connecting conductor; 3—the soil; 4—footprint contour on the ground.  
* The variable a is considered an intrinsic one because its value is determined by the mathematical model based on the other sizes. The lower technical limit is a \(\approx\) 0.2 m.
* The sizes {d, \(\rho_{\text{P}}\), q, g, \(\text{R}_{\text{Pn}}\)} are considered as parameters that justifiably change from one case to another.