Calculation of the assembled grounding resistance from complex grounding systems by using analytical considerations only

Today many simple equations exist to estimate the low frequency grounding resistances of a grounding system [1], [2], [4], [9], [10]. All of these equations have in common, that they are based on fixed electrode arrangements. Most of a time only single grounding elements are taken into account. When unsymmetrical or complex grounding systems are at hand, time-consuming simulations are conducted [5]. The major problem, which arises is, that all elements of a grounding system are often very thin and small in contrast to its surrounding environment, and each of them has an impact on the distribution of current density of the whole grounding system. To cover this issue, simulations often need a fine meshing technique or have to neglect some characteristics.This work describes an approach with analytical assumptions only. Hereby all calculations originate from the consideration of line current sources. All elements of a depicted grounding system are represented by rotated and shifted copies of this mathematical expression. In order to fulfill the boundary conditions at ground level, mirror sources are depicted. To obtain an assembled grounding resistance of all elements, the mathematical expressions of all sources are superimposed. A constant value for the electrical potential of the grounding system itself is used as a boundary condition. Hereafter the resulting potential and all the exact currents leaving the grounding system are calculated, along with their partial resistances. With this assumption, one is able to calculate the assembled grounding resistance of a whole complex grounding system. The system can combine many line elements with different radii in any direction forming grids, irregular paths or other very complex arrangements. However, some effects, which are most of the time neglected in grounding estimations are neglected, like the soil ionization or the effective length of a grounding rod. The methodology expresses an exact analytical formulation for the electrical potential, the electrical field and as a result the current distribution, from which the assembled grounding resistance. The methodology works very fast, which makes it easy to investigate and optimize complex grounding systems in real time. The methodology is validated and discussed with equations known from the literature. Different grounding resistances of complex and none symmetrical grounding systems are presented.