Earthing / Grounding Resistance Calculator
Earthing / Grounding Resistance Calculator
The resistance of a driven rod by Dwight’s formula, and of a row of rods with the interaction between them worked out properly — with the efficiency you actually get, and a plain account of why the answer must still be measured on site.
Earth electrode resistance
Three 3 m rods of 16 mm diameter, 3 m apart, in 100 Ω·m soil
Dwight’s rod, and a row of them
Rn = (R₁ ÷ n) × (1 + λα) α = ρ ÷ (2π s R₁) λ = 2Hn−1 − 2(n−1)/n
η = R₁ ÷ (n Rn)
- ρ
- soil resistivity, in ohm-metres
- L, d
- length of rod IN the soil and its diameter, both in metres. IEEE Std 142 writes the same formula with the radius a, as ln(4L/a) − 1; d = 2a, so they are identical
- s
- spacing between adjacent rods, in metres
- λ
- the mutual-coupling factor for n rods in a straight row, with Hn−1 the harmonic number 1 + ½ + … + 1/(n−1). It is 1 for two rods and 3.858 for ten — the same values BS 7430 tabulates
- η
- efficiency: how close the group gets to the R₁/n it would reach if the rods were infinitely far apart
Worked example
Three 3 m rods of 16 mm diameter, 3 m apart, in 100 Ω·m soil
8L/d = (8 × 3) ÷ 0.016 = 1,500, and ln(1,500) = 7.3132
One rod: R₁ = 100 ÷ (2π × 3) × (7.3132 − 1) = 5.3052 × 6.3132 = 33.49 Ω
Coupling: α = 100 ÷ (2π × 3 × 33.4927) = 0.15840; for three rods H₂ = 1 + ½ = 1.5, so λ = 3 − 4/3 = 1.6667
Three rods: R₃ = (33.4927 ÷ 3) × (1 + 1.6667 × 0.15840) = 11.1642 × 1.26400 = 14.11 Ω
Efficiency: 33.49 ÷ (3 × 14.1115) = 79.1%. Three rods behave like 2.37 ideal ones, not three
Typical soil resistivity — IS 3043 Table 3, normal rainfall
| Soil | ρ (Ω·m) | One 3 m × 16 mm rod |
|---|---|---|
| Alluvium, lighter clays | 5 | 1.675 Ω |
| Clays | 10 | 3.349 Ω |
| Marls | 20 | 6.699 Ω |
| Porous limestone | 50 | 16.75 Ω |
| Porous sandstone | 100 | 33.49 Ω |
| Compact limestone, granite | 1,000 | 334.9 Ω |
Length beats number: one rod, in 100 Ω·m soil
| Rod length in soil | Resistance | vs a 1.2 m rod |
|---|---|---|
| 1.2 m | 71.58 Ω | 100% |
| 1.8 m | 51.3 Ω | 72% |
| 2.4 m | 40.39 Ω | 56% |
| 3.0 m | 33.49 Ω | 47% |
| 4.5 m | 23.76 Ω | 33% |
| 6.0 m | 18.58 Ω | 26% |
| 9.0 m | 13.11 Ω | 18% |
What earth electrode resistance really is
The resistance of an earth electrode is not the resistance of the rod. Copper carries current perfectly well; what resists is the soil, and almost all of the resistance is in the first metre or so around the rod, where the current has to squeeze through a small cross-section before it spreads out. That is why a rod’s diameter barely matters — it enters through a logarithm — and why its length matters a great deal.
Dwight’s formula. Treat the rod as a line of current sources in a half-space of uniform resistivity, integrate the potential they produce at the rod’s own surface, and you get R = ρ/(2πL) × (ln(8L/d) − 1). IEEE Std 142 writes it with the radius, ln(4L/a) − 1; since d = 2a the two are the same expression. A 3 m rod of 16 mm diameter in 100 Ω·m soil comes out at 33.49 Ω. Halve the diameter and you get 37.17 Ω — only 11% more for a rod half as thick. Double the length and it drops by nearly half.
Why rods in a row do not simply divide. Each rod raises the potential of the soil around it, including the soil around its neighbours, so a second rod has to work against a ground the first has already lifted. Superposing the rods’ potentials and assuming they share the current equally gives the group resistance as the average of the whole matrix of self and mutual terms, and for a straight row with spacing s that sums in closed form to Rₙ = (R₁/n)(1 + λα), with α = ρ/(2πsR₁) and λ = 2Hₙ₋₁ − 2(n−1)/n. The λ values that fall out — 1.0 for two rods, 3.858 for ten — are the ones BS 7430 tabulates. Three rods at 3 m spacing here reach 14.11 Ω rather than the 11.16 Ω you would hope for: 79.1% efficiency.
What this page deliberately does not do. It does not do plate electrodes. IS 3043 clause 9.2.1 gives a plate formula in terms of the area of both sides of the plate, but in every source available here the expression itself is reproduced as an image rather than as text, so it could not be verified — and a formula for a safety earth is not something to reconstruct from memory. It does not do strip or ring electrodes, or a mesh. And it does not tell you what resistance you are allowed: that is set by your wiring code, the earthing system in use, and in a TT installation by the RCD you are relying on, so the target here is an input with an example in it.
Measure it. Resistivity varies with moisture, temperature and the season — the same ground can read several times higher at the end of a dry season than it does after rain, and frozen soil is effectively an insulator. Every calculated figure is a starting point for choosing how many rods to order. The installed electrode must be tested, by fall-of-potential or with a clamp-on tester on a bonded system, and the test repeated in the driest part of the year. For the circuit protection that earthing exists to serve, see the MCB size calculator; for the supply it protects, the voltage drop calculator and the cable ampacity calculator.
Frequently asked questions
How do I calculate earth rod resistance?
Dwight’s formula: R = ρ/(2πL) × (ln(8L/d) − 1), with ρ in ohm-metres and L and d the buried length and diameter in metres. A 3 m rod of 16 mm diameter in 100 Ω·m soil gives 33.49 Ω.
Does adding a second earth rod halve the resistance?
No. Each rod raises the potential of the soil its neighbours sit in, so the group is always worse than R₁/n. Three rods 3 m apart in the default case give 14.11 Ω instead of 11.16 Ω — 79.1% efficiency. Spacing them further apart recovers most of the loss.
How far apart should earth rods be?
At least the driven depth, which is IS 3043’s rule, and preferably two to three times it. Closer than that and the rods sit inside one another’s resistance area, so you buy copper and get very little resistance back.
Is a longer rod better than more rods?
Usually, yes. Resistance falls roughly as 1/L with length, a deeper rod reaches damper and more stable soil, and there is no spacing penalty. Two coupled 3 m rods driven as one 6 m electrode typically beat two separate 3 m rods.
What earth resistance is acceptable?
Whatever your code and your distribution company require, which depends on the earthing system and the country — IS 3043, IEC 60364 and BS 7671 all set different limits, and a TT installation’s figure follows from the trip current of the RCD relied on. This page takes the target as an input for that reason.
Related calculators
References
- IEEE Std 142-2007, IEEE Recommended Practice for Grounding of Industrial and Commercial Power Systems (Green Book), Table 4-5 — Dwight’s formula for a driven rod, R = ρ/(2πL) × (ln(4L/a) − 1) with a the rod radius. The table itself is behind the IEEE paywall; the form above is as quoted, with the same attribution, by Industrial Monitor Direct, Earth resistance calculation formulas for grounding electrodes.
- BS 7430, Code of practice for protective earthing of electrical installations — rod electrode resistance written with the diameter, R = ρ/(2πL) × (ln(8L/d) − 1), as quoted by Electrical Engineering Portal, How to determine correct number of earthing electrodes, part 1. Identical to the IEEE form, since d = 2a.
- IS 3043:1987, Code of practice for earthing (Bureau of Indian Standards), clause 9.2 and Table 3: typical soil resistivity under normal rainfall — alluvium and lighter clays 5 Ω·m, clays 10, marls 20, porous limestone 50, porous sandstone 100, compact limestone and granite 1,000 Ω·m; and the rule that rods act independently only when separated by at least their driven depth.
- Tagg GF. Earth Resistances. George Newnes, 1964 — the superposition treatment of groups of electrodes that the λ factor above is derived from. Cited from the secondary literature; the book itself was not consulted here.