The following table summarizes the main assumptions on which GSA and GSA_FD modules are based.

The following diagram represents the application domain of the two modules.

The highlighted central area indicates the usual condition at power frequency.

The diagram has been obtained from a parametric analysis using square well-meshed copper grids energized with a current injected in a corner. The analysed parameters were the grid diagonal “D”, the soil resistivity and the frequency.

In its application dominion as defined by the red solid line, the error made by GSA in the GPR and touch voltages calculation is lower than about 10%.

Application domain of GSA and GSA_FD (Log scale)

In practice, in case of well-meshed grids, application limits of GSA can be defined as a function of the wavelength of the electromagnetic field in the earth as follows:

where λ (m) = wavelength, ρ (Ωm) = soil resistivity and f (Hz) = frequency.

The previous diagram indicates that GSA can be used if “D < λ/10”.

This result is in agreement with the simultaneity concept of Albert Einstein.

GSA also requires “D < 500 m” as reasonable limit.

The application limits will be lower if the grid shape is not regular, if the meshes are sparse and if the grid is made of steel or other high resistivity metal. In all these cases, the limit related to poor-meshed grids as defined by the red dashed line should be considered.

The following three figures show the earth surface potential distribution calculated by applying GSA and GSA_FD to a 100 m x 100 m grid with the same injected current, the same frequency (50 Hz), the same injection point (marked with arrow) and the same soil model.

The qualitative difference between results is evident. GPR and impedance to earth tend to grow whether self impedance and mutual impedance are taken into account. High frequency or low soil resistivity can make this difference even more evident.

Of course, a difference in the earth surface potential distribution corresponds to a difference in touch and step voltage distribution.

In brief, in grounding system analysis at power frequency, GSA can be used in many practical situations but it tends to underestimate the results if the grid size is greater than one tenth of the wavelength of the electromagnetic field, while GSA_FD may be applied in all conditions.

At high frequency, GSA can be applied only to grids with a maximum size of about some tens of meters. In general, at high frequency GSA_FD should be used.

In electromagnetic interference analysis, GSA and GSA_FD can be used respectively for only resistive and resistive + capacitive + inductive coupling evaluation.

Earth surface potential distribution - GSA (equipotential condition)

Earth surface potential distribution - GSA_FD (only self impedances)

Earth surface potential distribution - GSA_FD (self and mutual impedances)

After these conclusions a question could arise: why not just GSA_FD?

The answer is simple but not trivial.

GSA requires an easier data entry, accept rough layouts, is cheaper and faster than GSA_FD and requires fewer computer resources.

GSA_FD requires additional information about the topology of the conductors system and in order to calculate their self and mutual impedances and a well finished layout.

Moreover GSA_FD requires more experience in the evaluation of results.

If GSA cannot be used, GSA_FD is the right solution.

XGSA_FD is based on the same model of GSA_FD but extended to overhead conductors. The application limits of XGSA_FD can be assumed from DC to about 100 MHz. XGSA_FD greatly expands the application field of XGS and makes it a real laboratory for engineering applications and for research.

XGSA_FD is an irreplaceable tool when conductors are partly overhead and partly underground. This situation is usual in electromagnetic fields evaluation (where sources may be underground cables or overhead wires) or interference analysis (where often the inductor is overhead and the induced is underground).

Anyway XGSA_FD operates to a single frequency.

 XGSA_TD can calculate the response in the time domain of a conductors network energized with  current or voltage transients.

As known, the methods to calculate the transient behaviour of conductors network in the time domain can be divided into two main categories: those based on the calculation of the solution directly in the time domain and those based on frequency domain calculations and then using the forward and inverse Fourier transforms.

Methods of the first category require low frequency and quasi-static approximations and in addition do not allow considering the frequency dependent characteristics of the grounding system.

Methods of the second category use an electromagnetic field approach for the calculation of the response of the grounding system in a wide range of frequencies and have a good accuracy because they are based strictly on the principles of electromagnetism. On the other hand, in these methods, a system of equations has to be solved for every particular frequency, and a large number of discrete frequency points over the frequency band are chosen to satisfy the frequency sampling theorem.

XGSA_TD is based on the second category methods and uses XGSA_FD as solver in the frequency domain. Then the application limits of XGSA_TD can be assumed as the same of XGSA_FD and in particular the maximum bandwidth of the input transient should be lower than 100 MHz.

This means that XGSA_TD can consider transient input as switching transients, standard lightning currents and also fault transients in GIS.

The simulation of lightning represents the most typical application of XGSA_TD.

The lightning current can be simulated by using the standard short stroke wave form IEC 62305: first positive; first negative; subsequent negative.

With the direct Fourier transform, the time domain input transient is decomposed in the frequency domain.

In the following figures the normalized wave shape of the subsequent negative standard lightning current and their normalized frequency spectrum. The spectrum can be neglected when normalized values are lower than about 10^-3 - 10^-4. The standard lightning bandwidth is lower than a few MHz also for the fastest lighting, the subsequent negative ones.

Normalized subsequent negative standard lightning (Linear scale)

Normalized frequency spectrum – Magnitude, real and imaginary parts (Linear scale)

Normalized frequency spectrum – Magnitude (Log scale)

Normalized frequency spectrum – Argument (Log Linear scale)

After the calculation in the frequency domain (taking into account a suitable number of representative frequencies in order to limits the calculation time), the response in the time domain is obtained with interpolation of results and the inverse Fourier transform.

The evaluation of lightning effects is important in many practical applications.

For instance, current generated by a stroke flows in the LPS conductors and dissipate in the soil. The electric and magnetic field generated by such high voltages and currents can cause internal discharges, fires and explosions, may cause damage of equipment and buildings and may be dangerous for people.

In conclusion, XGSA_TD uses XGSA_FD as a calculation engine and Fourier transforms in order to move from time to frequency domain and vice versa.

The choice of the soil model is crucial in electromagnetic simulations of systems close to the soil surface and in particular in the grounding systems analysis. There is much literature about the criteria to set an appropriate soil model which can be used to predict the performances of a grounding system. XGS allows to use uniform, multilayer and multizone soil models.

A typical soil cross section including ground water

A uniform soil model should be used only when there is a moderate variation in apparent measured resistivity both in vertical and horizontal direction but, for the majority of the soils, this assumption is not valid. A uniform soil model can also be used at high frequency because in that case, the skin effect limits the penetration depth of the electromagnetic field to a few meters and so, the soil resistivity of the depth layers do not affect the results.

The soil structure in general changes both in vertical and horizontal direction and the presence of ground water further complicates things. The vertical changings are usually predominant on the horizontal ones, but about this aspect, is essential to consider also the grounding system size.

In case of small grounding systems (maximum size up to a few hundred meters), soil model is not significantly affected by horizontal changings in soil resistivity and usually a multilayer soil model is appropriate. The layer number depends on the soil resistivity variations in vertical direction and three, four or five layers can be sufficient for most cases. Sometime, in order to consider seasonal effects on soil model like frozen soil, a bigger layers number can be necessary. For this reason, XGS allows to consider up to 20 layers (limited only in order to avoid unappropriated usage).

In case of grounding systems of intermediate size, soil model is affected by both horizontal and vertical changings in soil resistivity and usually an equivalent double or triple layer soil model is appropriate. This is the most important case in practical applications.

In case of large grounding systems (maximum size over a few kilometres), soil model is significantly affected by horizontal changings in soil resistivity and usually a multizone soil model is appropriate. The zones number depends on the systems size and soil resistivity variations in horizontal direction.

The earth reacts to the AC electromagnetic fields.

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 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 on or 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 following figures show the electric and magnetic fields distributions related to a single and long overhead line at different frequencies. The calculation area is vertical and across the soil surface.