## Ground Grid Additional Background Information

### The Earth Effects

The earth reacts to the AC electromagnetic fields and the reaction effects grow with frequency and soil conductivity. The exact solution of this problem was found by Sommerfed and involves integrals (known as Sommerfeld integrals) that represent the solution of the Maxwell equations related to infinitesimal horizontal or vertical current elements radiating in the presence of a lossy half space. Sommerfeld integrals take into account the boundary conditions on the tangential components of the electromagnetic fields at the half space interface.

These integrals usually cannot be solved in closed form and in general are quite difficult to calculate also with a numerical approach because contain very oscillating Bessel functions.

The earth reaction to the AC electromagnetic fields grows with frequency and soil conductivity and is different for horizontal and vertical and moreover buried or aerial sources.

In order to display the earth reaction in an effective way, in the following, is represented the cross section of the magnetic field close an horizontal or vertical source above the soil surface. In DC condition there is no earth reaction.

At low frequency the earth reaction is negligible for horizontal sources but significant for vertical sources.

At high frequency, the earth reaction becomes always relevant and the earth acts as a mirror for the magnetic field. With vertical sources this happen also at relatively low frequency. Far from the source, the earth reaction is significant also at low frequency.

### The Great Thinkers

*XGSLab is based on Maxwell equations and Sommerfeld integrals.
Most people know that the electromagnetic field are governed by a set of experimental
laws known as Maxwell equations.*

*James Clerk Maxwell (Edinburgh 1831 – Cambridge 1879)*

On the other hand, not many people know about the fundamental studies carried out by Sommerfeld. Among other things, Sommerfeld studied the effect of the earth reaction to the electromagnetic field and the rigorous solutions of the half space problem are known as Sommerfeld integrals.

Without Sommerfeld studies would not have been possible to develop XGSLab.

*Arnold Johannes Wilhelm Sommerfeld (Konigsberg 1868 – Munich 1951)*

Furthermore, the calculation in the time domain were been possible by using the Fourier transform. Fourier transform allows moving from the time domain and viceversa.

*Jean-Baptiste Joseph Fourier (Auxerre 1768 – Paris 1830)*

XGSLab is based on the research of many other scientists, but it is important to remember these great thinkers of the past.