Thank you for your excellent question regarding the Sphere of Influence for horizontal earth rings and conductors. The Sphere of Influence is an electrical theory that claims that an earthing electrode will dissipate electrical energy into a volume of soil equal to the length of the electrode in all directions.
It is fairly easy to imagine a 10-ft driven earthing rod placed vertically into the earth. The Sphere of Influence of this rod would have a radius extending 10-ft out in all directions (20-ft diameter), including down into the earth. It is important to understand that the electrode is also trying to dissipate the electrical energy up into the air, but the strong dielectric properties of air typically prevent such action. The formula in this case is: Volume = (5 * Pi * Length (cubed))/3. If we cross cancel Pi and 3, we get the simple equation of V = 5L cubed. Thus the Sphere of Influence for a 10-ft driven rod is 5,000 cubic feet of soil.
A similar formula can be used for calculating the Sphere of Influence of each of the individual conductor segments of your ring. The Key issue being to remember that this is a volumetric equation designed to figure out how many cubic feet of soil is being utilized by your earthing system. With horizontal conductors, you will need to remove the volume of the Sphere of Influence that extends into the air.
But is Sphere of Influence really your question, or are you wanting to calculate resistance to earth? This will require an understanding of the soil resistivity at your site, and the calculations are far more complex. We recommend the Handbook of Electric Power Calculations by McGraw-Hill, Section 14. It has a complete list of equations for uniform (single layer) soil resistivity models. Unfortunately, uniform soil doesn’t really exist and multi-layer calculations require a computer. Throw in impedance issues caused by AC power and the complexity sky rockets.
The Engineering Team at E&S Earthing Solutions