Earthing Formulas: Earth Rod and Fault Loop

Earthing formulas for the resistance of a vertical earth rod, rods in parallel, earth fault loop impedance and fault current, with worked examples and a table.

The formulas

ρ is the soil resistivity in Ω·m, L the length of the rod in the ground and d its diameter, both in metres. n is the number of rods and u a utilisation factor below 1 that allows for the rods affecting each other. U0 is the voltage to earth, Ia the current that makes the protective device trip in the required time, Ze the external loop impedance and (R1 + R2) the resistance of the phase and protective conductors. The rod formula assumes uniform soil.

Worked example: one 3 m rod in 100 Ω·m soil

The rod diameter is 16 mm.

1. ln(8 × 3 / 0.016) = ln(1,500) = 7.313
2. R = 100 / (2π × 3) × (7.313 − 1) = 33.49 Ω

Worked example: four rods

Four of these rods, spaced well apart, with a utilisation factor of 0.85.

1. Rn = 33.49 / (4 × 0.85) = 9.851 Ω

Worked example: fault current and the limit on Zs

1. With Zs = 0.8 Ω on 230 V: If = 230 / 0.8 = 287.5 A
2. A breaker that must trip at 160 A (for example a 32 A device with a 5 × In magnetic trip) needs Zs ≤ 230 / 160 = 1.438 Ω

Resistance of a 16 mm rod by length and soil

Frequently asked questions

Why does a longer rod help more than a thicker one?

The resistance falls roughly in proportion to the length, but only with the logarithm of the diameter, so a longer or deeper rod is far more effective.

Why do four rods not give a quarter of the resistance?

The current spreading from each rod overlaps the others. Rods spaced at least their own length apart come close to the ideal, and the utilisation factor allows for the rest.

Does the soil resistivity stay constant?

No. Moisture, temperature and season change it, sometimes several times over. Measure the resistance on site, in the driest season if you can.

Is a low earth resistance enough for safety?

No. Protection also depends on the loop impedance, the protective device and the touch voltage. The earth electrode is one part of the system.

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