Sunde [2] gives an equation for the voltage produced at a point P in the earth by a point source of I amps. He states that the voltage produced is:

where r is the resistivity of the soil, r is the distance between P and the source of current and r’ is the distance between P and the image of the source of current. The image is an imaginary second source of current located directly over the current source as far above the surface as the current source is below the surface.

This equation forms the basis of the program.

A technical paper by R.J. Heppe [1] expands the basic equation. It develops equations for:

  • Calculating the voltage at points in the soil due to a thin conductor segment that discharges current uniformly along its length.

  • Calculating the mutual resistance between two conductor segments. The term "mutual resistance" means that for a pair of conductor segments with a given mutual resistance R, the flow of 1.0 amp into the soil, from one conductor segment in the pair, will produce a potential of R x 1.0 volts on the second one. The mutual resistance calculation considers the first conductor as an infinite number of small current sources. Each current source produces a potential at every point along the second conductor. Double integration is used to integrate the potentials along the second conductor for all the current sources on the first.

  • Accounting for the effect of soil layering where there are distinct layers of soil with differing resistivity. The mutual resistance calculation becomes the summation of an infinite series of mutual resistances between the segment and images of the segment. The image locations are determined by the interface between the soil layers. Fortunately the series converges and it is possible to predict the error due to only summing a certain number of terms in the series. The summation can be stopped when enough terms have been evaluated and calculation time is reduced.

The equations developed by Heppe closely simulate the real world providing the conductors segments are thin compared with their length. This is normally the case with ground electrode wires and rods. However, Heppe's  paper only develops equations for horizontal buried conductors that are rectilinear i.e. at right angles to each other. We have derived the equations required to model conductors at any angle and also to calculate soil potentials on or below the soil surface.

The equations assume that the flow of current from a conductor segment is uniform along its length. This is referred to as a current density in units of amps/metre. In the real world, the current flow is not uniform. As an example, if a single horizontal conductor segment is considered, the current density will be higher at the ends than in the middle. The calculation can be made to compensate for this effect to some extent by sub-dividing the conductor segments into smaller pieces. If this were done for the single conductor example, the parts that are near the ends will have a higher current density.

KWIKGRID models ground electrodes as a set of conductor segments that are related by their mutual resistances. In the KWIKGRID analysis, every conductor segment has a mutual resistance with every other segment. When current flows into (or out of) the soil from the conductor segments, it produces a potential rise at all segments, including itself, that is determined by the mutual resistances between the segment pairs.

The analysis is done by developing equations for each conductor segment and solving them under certain constraints. This determines the potential rise and ground resistance of the electrode and the current flow from every conductor segment. With the conductor segment currents known, potentials are calculated at any location on or below the soil surface by summing the effect of the current flow from every conductor segment.

There are two versions of KWIKGRID. One models a ground system that is purely resistive. The other models a ground system that can have ac impedance components. While both versions can be used to model separate electrodes that are close and influenced by each other, the ac version can include the effect of impedance (R + jX) connections between electrodes and connection of lumped remote ground impedances. When there are ac impedances between separate electrodes, the calculation is carried out using complex math. Injected currents can be specified in complex format with implied phase angle. The software is configured to enable current to be injected into any electrode of the model. Conductor segment currents, electrode potential rise and soil potentials, will have complex values.

KWIKGRID can plot contour maps of step, touch or absolute potentials. The method used is:

  • Define an area for plotting.

  • KWIKGRID calculates an array of soil potentials over the area.

  • The contours are interpolated from the soil potential points.

Absolute potentials are the actual soil potentials that will develop when current flows to ground through the ground electrode system. Touch potentials are the difference in potential between a ground electrode and the soil surface. If there are several interconnected electrodes in the model, the potential rise of the correct one must be selected if touch potential calculations are required.

Step potentials are calculated by finding the highest potential difference over a distance of 1 metre in all directions at all points in the area being examined. Human feet are represented by 8 cm radius discs. The step potential is the difference between the average potential of each 8 cm radius foot. This gives a better result than the alternative method of examining the worst voltage gradient in volts/metre at each location, because the voltage gradient can be much higher than the step potential close to buried objects.

KWIKGRID analyses assume the following:

  1. Conductor segments are thin compared with their length.

  2. Parallel conductor segments are not too close to one another. Crossing conductor segments can touch and ends of segments can meet and can be co-linear or at angles.

  3. Current density is constant along each conductor segment.

  4. Soil is perfectly uniform in resistivity or perfectly layered.

Of the above items, a reasonable model can usually be created to satisfy Items 1 and 2. Item 3 can usually be accepted by increasing subdivision of long conductor segments. Item 4 is usually the most difficult to determine with accuracy.

Touch potentials are usually highest around the edges of a conventional ground grid. Experimentation will show that touch potentials around the edges will be higher if no sub-division of the conductors is done. Step potentials will be lower. Therefore if no sub-division is done, the calculation of touch potentials at the grid edges may be overly pessimistic, but step potentials may be under-estimated.


  1. R.J. Heppe, "Computation of Potential at Surface Above an Energized Grid or Other Electrode Allowing for Non-Uniform Current Distribution", IEEE Trans. on Power Apparatus and Systems Vol. PAS-98, No. 6 Nov/Dec 1979.

  2. E.D. Sunde, "Earth Conduction Effects in Transmission Systems", D. van Nostrand Co. Inc., 1949.


This page was last updated on October 24, 2006