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31 — Calculators

Earth resistance of a vertical rod electrode

Earth resistance depends almost entirely on the soil and the length of the electrode — diameter enters only logarithmically. Measure too high a value and a thicker rod barely helps, whereas a longer one does.

Input

From a measurement on site. Moist loam is low, sand and rock very high — the range is wide.

Several electrodes act in parallel, but not fully — they influence one another.

Takes effect from two electrodes upwards. Close to 1 at sufficient spacing, markedly below with close spacing — spacing of at least the electrode length is the rule.

The value of the intended residual current device, against which the check is made.

Result

Earth resistance of the arrangement
—Ω
Earth resistance of a single electrode
—Ω
At double electrode lengthLength works harder than diameter — go deeper, not thicker.
—Ω
Largest permissible earth resistanceFrom the condition R_A · I_Δn ≤ 50 V.
—Ω
Is the earthing sufficient for this device?
—

A guide, not a design to standard. We give no warranty for the correctness of the results, the applicable standards and case-by-case verification govern.

Worked example

The default case the calculator starts with, worked through once. Change the values above and it recalculates immediately.

Inputs

Soil resistivity
150 Ω·m
Length of the electrode
9 m
Diameter of the electrode
20 mm
Number of parallel electrodes
1
Utilisation factor for several electrodes
0.8
Rated residual current of the protective device
300 mA

Result

Earth resistance of the arrangement
19.9 Ω
Earth resistance of a single electrode
19.9 Ω
At double electrode length
10.9 Ω
Largest permissible earth resistance
167 Ω
Is the earthing sufficient for this device?
Yes, with margin. The earth resistance is well below the permissible value.

Formula

  • R = ρ / (2π · L) · ln(4L / d)
  • For n electrodes: R_total = R / (n · utilisation factor)
  • Condition for a residual current device: R_A · I_Δn ≤ 50 V

Assumptions and standards

  • The formula applies to a rod driven vertically into uniform soil. Layered soil, rock or groundwater change the result considerably.
  • Soil resistivity has to be measured. It varies strongly with the season — in summer and under frost it is markedly higher than in a wet spring.
  • The utilisation factor represents the mutual influence of several electrodes and is entered here. At spacing below the electrode length it drops sharply.
  • Strip and ring electrodes follow a different formula. This calculator covers the rod electrode only.
  • The utilisation factor takes effect from two electrodes upwards. For a single electrode the calculated value stays unchanged.

Frequently asked

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

Because length sits in the denominator while diameter appears only inside the logarithm. Doubling the length nearly halves the resistance. Doubling the diameter gains only a few percent depending on the ratio — and costs considerably more material.

Why do two electrodes not behave like two parallel resistors?

Because they share the same volume of earth. Placed close together their fields overlap and each performs worse than it would alone. Hence the utilisation factor — and hence the rule to keep at least the electrode length as spacing.

How does this relate to the residual current device?

In a TT system the fault current returns through earth. The touch voltage follows from earth resistance times operating current and has to stay below the permitted limit. That is why the largest permissible earth resistance is shown here alongside.

Calculating is the easy part.

A formula gives you a number. Designing a plant also demands installation method, grouping, discrimination, the standards in force and a look at the installed base. That is what we take on.

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