What the calculator does
An earth electrode is a rod, a plate, a strip or a mesh in contact with the ground, and its resistance to the general mass of earth is the number a specification asks for, a tester measures and an installer argues about. Before the rod is driven you can only estimate it, and the estimate depends on one thing above all: the resistivity of the soil, which varies a hundredfold between a wet clay field and a dry gravel bank. This calculator takes that resistivity, the rod's length and diameter, and gives the resistance of one rod by the standard formula.
It then answers the questions that follow. What does a second, third or fourth rod in a line do, and how much of each extra rod is wasted by its neighbours? How many rods would a target of 5 Ω or 1 Ω take at the spacing you have room for, and is that even a sensible route? What would one rod of twice the length do instead? And for a TT installation, where the consumer's own electrode carries the fault current, does the electrode resistance satisfy the 50 V touch-voltage rule for the RCD fitted? The earthing guide explains what the ohms mean and which figure you actually need.
Formula
n rods in a line at spacing s Rn = (R1 + (n − 1) × ρ ÷ (2πs)) ÷ n
Utilisation = (R1 ÷ n) ÷ Rn
TT touch-voltage limit RA × IΔn ≤ 50 V → RA ≤ 50 ÷ IΔn
where ρ is the soil resistivity in Ω·m, L the driven length and d the diameter of the rod, both in metres, and s the spacing between rods. The single-rod expression is the one derived by Dwight (1936) and Sunde (1949) and reproduced in IEEE Std 142 and BS 7430; it treats the soil as uniform and the rod as a thin cylinder. The multi-rod expression adds, for each extra rod, a mutual term ρ ÷ (2πs) that represents the potential each rod raises at its neighbours: rods in the same patch of soil share the same current paths and do not act as independent resistors in parallel. As n grows the group resistance tends to ρ ÷ (2πs), not to zero, which is why the calculator reports the number of rods a target needs and, if that number is silly, says so. The count is the first n whose resistance, rounded to 0.01 Ω, is at or below the target.
Worked example
A single copper-bonded rod, 2.4 m long and 16 mm in diameter, driven into loam of 100 Ω·m resistivity for a small commercial building in Lahore whose specification asks for 5 Ω or less. A 30 mA RCD protects the final circuits.
- ln(8L ÷ d) = ln(8 × 2.4 ÷ 0.016) = ln(1 200) = 7.090, so the bracket is 6.090.
- R1 = 100 ÷ (2π × 2.4) × 6.090 = 6.631 × 6.090 = 40.4 Ω. One rod is nowhere near the target.
- A second rod 4.8 m away adds a mutual term of 100 ÷ (2π × 4.8) = 3.32 Ω: R2 = (40.4 + 3.32) ÷ 2 = 21.9 Ω, a 92 % utilisation of two independent rods. Four rods: (40.4 + 3 × 3.32) ÷ 4 = 12.6 Ω, utilisation 80 %.
- Rods needed for 5 Ω at this spacing: the group resistance tends to 3.32 Ω as rods are added, so 5 Ω is reached only at 22 rods, in a line 100 m long. That is the wrong route.
- One rod of double length, 4.8 m: R = 100 ÷ (2π × 4.8) × (ln(2 400) − 1) = 3.316 × 6.783 = 22.5 Ω, about the same as two rods, from one hole and with a chance of reaching wetter soil below.
- TT check: 50 V ÷ 0.030 A = 1 667 Ω. The 40.4 Ω electrode meets it with room to spare, as almost any electrode does; the 5 Ω figure in the specification comes from stability, lightning and fault-current considerations, not from the RCD.
The practical answer for this site is to look for lower-resistivity ground (a Wenner test at a few spots, or a deep-driven rod to reach the water table), and to use a bentonite backfill in an augered hole rather than to multiply rods across the car park.
Typical soil resistivities
| Soil | Typical resistivity, Ω·m |
|---|---|
| Wet organic soil, marsh | 10–30 |
| Clay, clayey loam | 20–100 |
| Loam, agricultural soil | 50–150 |
| Sand, sandy loam | 200–1 000 |
| Gravel, dry sand | 500–5 000 |
| Rock, dry stony ground | 1 000–10 000 |
Assumptions and limitations
- Uniform soil. The formula assumes one resistivity all the way down. Real ground is layered, and a rod that reaches a wet layer below dry topsoil will measure far lower than the estimate, which is the main reason to drive deep. Layered-soil models exist (IEEE Std 80 and Std 81 describe them) but need resistivity data at several pin spacings.
- The multi-rod formula is an approximation. It overestimates the resistance of widely spaced rods (where the mutual term should fall off faster than ρ ÷ (2πs)) and underestimates it for closely packed ones; the calculator warns when the spacing is less than the rod length. It is for rods in a line; a ring or a grid of rods behaves differently and is best designed to IEEE Std 80.
- Rods only. Plates, horizontal strips, rings, bare buried cable and concrete-encased electrodes (foundation earths) all have their own formulae and are not covered. A building's steel reinforcement, bonded properly, is often a better electrode than any rod.
- Seasonal variation. Expect the measured value to move by a factor of two or three between the monsoon and the dry season; a specification is usually meant to be met in the worst season, so measure then or allow for it.
- The value must be measured. After installation, measure the electrode with a three-terminal fall-of-potential test (IEEE Std 81) and record it; the calculator is a design estimate, not a test result.
- Corrosion and bonding not covered. Copper-bonded steel, galvanised steel and stainless rods corrode at different rates in different soils, and a corroded clamp or a broken conductor makes the best electrode useless. Use exothermic welds or approved clamps, and keep dissimilar metals apart.
- Substation and lightning earths need a mesh. A substation earth must limit step and touch voltages during a high-current fault, and a lightning earth must handle a steep impulse; both are grid designs to IEEE Std 80 and IEC 62305, not a rod count.
Frequently asked questions
What value do I actually need?
There is no single number. For a TT installation protected by a 30 mA RCD, the touch-voltage rule of IEC 60364-4-41 (RA × IΔn ≤ 50 V) allows up to 1 667 Ω, which any rod in the ground meets; what the rule really asks is that the electrode be stable, so that it does not drift up to thousands of ohms in a dry summer. Specifications therefore set a practical ceiling: 10 Ω or 5 Ω for a building is common, 1 Ω for substations, telecoms and data centres, and IEC 62305 suggests 10 Ω for a lightning protection system. Ask what the electrode is for before chasing a number.
Why does doubling the rods not halve the resistance?
Because most of a rod's resistance is in the soil within a metre or so of it, and two rods close together share that soil. Each rod raises the potential of the ground around its neighbour, so the second rod starts from a raised potential and contributes less than a fresh rod would. At a spacing equal to the rod length the second rod gives roughly 85 % of its stand-alone benefit; at twice the rod length over 90 %; and as rods are added in a line the group resistance settles towards ρ ÷ (2πs) rather than towards zero.
Does a longer rod beat more rods?
Usually. The resistance falls roughly in proportion to the length (a 4.8 m rod is about 56 % of a 2.4 m rod in uniform soil, close to what two 2.4 m rods 4.8 m apart give), it takes one hole instead of two and one clamp instead of two, and in most ground the deeper soil is wetter and lower in resistivity, which the formula does not even credit. Couplable rods let you keep driving until the resistance stops falling. The exception is rock or a hard layer at shallow depth, where rods cannot be driven and horizontal strips or a plate in an excavated trench are used instead.
What about bentonite, chemical backfill and salt?
Bentonite clay in an augered hole around the rod works because it holds moisture and stays in contact with the rod and the soil; it lowers the resistance of the first few centimetres, where most of the resistance sits, and it is stable for years. Proprietary conductive backfills do the same with a carbon or conductive-cement base. Salt or charcoal poured around the rod, an old practice still seen on sites, lowers the reading for a season, then washes away and leaves a corroded rod; it is not a design.
How do I measure it?
With a three-terminal earth tester by the fall-of-potential method (IEEE Std 81): the electrode under test, a current spike driven at a distance of several times the electrode's size, and a potential spike moved between them; the reading at 62 % of the distance to the current spike is the electrode resistance, and readings that change quickly with the spike position mean the current spike is too close. The electrode must be disconnected from the installation for the test, so make sure the supply is off first. A clamp-on tester works only where the electrode is part of a multiply earthed system, such as a TN-C-S neutral or a lightning ring with several rods.
References
- IEEE Std 142-2007, IEEE Recommended Practice for Grounding of Industrial and Commercial Power Systems (Green Book) — electrode formulae, soil resistivity tables, multiple-rod behaviour
- IEEE Std 81-2012, IEEE Guide for Measuring Earth Resistivity, Ground Impedance, and Earth Surface Potentials of a Grounding System — Wenner four-pin and fall-of-potential methods
- BS 7430:2011+A1:2015, Code of practice for protective earthing of electrical installations — rod, strip and plate formulae; resistivity by soil type; seasonal variation
- IEC 60364-5-54:2011, Low-voltage electrical installations — Part 5-54: Earthing arrangements and protective conductors
- IEC 60364-4-41:2005, Low-voltage electrical installations — Part 4-41: Protection for safety — Protection against electric shock — clause 411.5, TT systems: RA × IΔn ≤ 50 V
- E. D. Sunde, Earth Conduction Effects in Transmission Systems, Van Nostrand, 1949 — the driven-rod resistance formula
- H. B. Dwight, "Calculation of resistances to ground", Electrical Engineering, vol. 55, 1936, pp. 1319–1328