Earthing calculator: soil resistivity, grid, step and touch voltage

What this calculator does

Six steps of the same job: turn earth tester readings into soil resistivity and, with several spacings, into a two-layer model; estimate the resistance of rods, conductors and plates; calculate that of a grid with rods; check step and touch voltages against IEEE 80 and the Spanish high-voltage regulation ITC-RAT 13 (RD 337/2014); size the earthing conductor; and calculate the separation between a substation earth and the low-voltage installation earth under the Spanish ITC-BT-18.

Resistivity by the Wenner method

Four probes in line, equally spaced, current through the outer two and voltage between the inner two. Apparent resistivity follows from

ρ = 2 π a R

with spacing a in metres and reading R in ohms. Each measurement averages the soil down to a depth of the order of that spacing, so sweeping several spacings is what reveals the structure: if resistivity grows with a, there is a more resistive layer underneath; if it falls, there is water or a more conductive layer.

That detail is not minor. Designing a grid with the resistivity measured at 2 metres, when the grid will work against the first 10 metres of soil, is the classic mistake that ends in an earthing system that does not comply.

Vertical rod

For a cylindrical rod driven vertically, Dwight’s formula:

R = ρ / (2 π L) × (ln(4L/r) − 1)

with L the buried length and r the radius. Length dominates the result; diameter sits inside a logarithm and hardly matters. Doubling the diameter of a 3-metre rod lowers its resistance by about 10%; doubling the length, by nearly 45%.

For several rods the calculator applies an efficiency factor that you control, 0.8 by default: two rods close together do not halve the resistance, because their zones of influence overlap. The further apart — at least one rod length — the higher the efficiency.

Earthing grid

For a buried grid, Sverak’s equation as given in IEEE 80:

R = ρ [ 1/L_T + 1/√(20A) × (1 + 1/(1 + h√(20/A))) ]

where A is the enclosed area, L_T the total length of buried conductor and rods, and h the depth. It gives the resistance of the whole and, with the fault current, the ground potential rise (GPR), which is the figure that drives the insulation of the telecoms and control circuits entering the substation.

When the grid has rods, the calculator also gives the resistance by the Schwarz method, which combines grid and rods allowing for their mutual resistance, and the simplified formula of table 3 of ITC-RAT 13, R = ρ/(4r) + ρ/L, with r the radius of the circle of equal area. The three usually agree within 5%; if they do not, the geometry is unusual and a numerical calculation is advisable.

Two-layer soil

If resistivity changes with spacing, a single figure does not describe the soil. With three or more different spacings, the calculator finds the upper-layer resistivity ρ1, the lower-layer resistivity ρ2 and the thickness h that best reproduce the readings, using the image-series expression for apparent resistivity over two layers and a least-squares fit. It works with Wenner and with Schlumberger (ρ = π c (c + d) R / d), where only the current rods move between readings.

The sign of the reflection factor K = (ρ2 − ρ1)/(ρ2 + ρ1) sets the strategy: with negative K the lower layer conducts better and rods that reach it are very effective; with positive K long rods do little and more horizontal conductor pays off.

Electrodes and the low-voltage check

Besides the vertical rod (Dwight), the calculator handles the buried horizontal conductor with Sunde’s formula, R = ρ/(πL) × (ln(2L/√(2ah)) − 1), and the buried plate. Next to each result it shows the simplified formula of the Spanish regulations (ITC-BT-18 table 5, ITC-RAT 13 table 3): R = ρ/L for a rod, R = 2ρ/L for a conductor and R = 0.8ρ/P for a plate (1.6ρ/P if vertical). Without a measurement, resistivity can be estimated from the type of soil with table 2 of ITC-RAT 13.

With the RCD sensitivity it checks the TT-system condition of IEC 60364-4-41 (ITC-BT-24 in Spain): RA × IΔn ≤ 50 V (24 V in conductive locations). With a 300 mA RCD, the earth must not exceed 167 Ω.

Step and touch voltages

This is what decides personnel safety. The calculator takes the grid from the Grid tab and calculates, with the IEEE 80 equations:

• the grid current IG = Df · Sf · 3I0, with the decrement factor Df for the DC offset (X/R ratio) and the split factor Sf (the reduction factor r in ITC-RAT 13), which discounts what returns through earth wires and cable screens;
• the mesh voltage Em, the touch voltage at the worst point, and the step voltage Es.

It compares them with two limits. The IEEE 80 tolerable values, (1000 + 1.5 Cs ρs) · 0.116/√t for a 50 kg person. And the ITC-RAT 13 maximum permissible values: the applied touch voltage Uca from its table 1 (204 V at 0.5 s, 107 V at 1 s) extended with the resistance of footwear and of the feet on the ground:

Uc = Uca [1 + (Ra1/2 + 1.5 ρs)/1000] Up = 10 Uca [1 + (2 Ra1 + 6 ρs)/1000]

In both cases ρs is the apparent surface resistivity: that of the gravel or concrete layer multiplied by the factor Cs, which IEEE 80 calculates with 0.09 and ITC-RAT 13 with 0.106. If the grid potential rise is already below the permissible touch voltage, the installation is safe without further analysis.

Earthing conductor cross-section

By IEEE 80 (Onderdonk’s equation), from the current, duration, ambient temperature and maximum allowable temperature, with the constants of the nine materials in its table 1. By ITC-RAT 13, with current densities for at least 1 s: 160 A/mm² for copper, 100 for aluminium and 60 for steel, divided by 1.2 if 300 °C is allowed. The calculator takes the larger of the two, respects the regulatory minimums (25 mm² copper for earthing conductors, 50 mm² for electrodes) and rounds up to the standard size.

Separation between substation and low-voltage earths

The Spanish ITC-BT-18 treats the earth of a substation’s exposed parts and that of the low-voltage installation as independent if they are 15 m apart in soil below 100 Ω·m, or D = ρ · Id / (2π U) apart in worse soil, with U = 1200 V in a TT system if the fault clears in 5 s or less. To join them into a common earth, the fault voltage Id · Rt must not exceed the permissible applied touch voltage of ITC-RAT 13.

Worked example

Soil of 100 Ω·m measured by Wenner at 4 m (reading 3.98 Ω → ρ = 100 Ω·m).

• One rod 3 m long and 16 mm in diameter: 33.5 Ω
• Four rods with 0.8 efficiency: 10.5 Ω

Grid from example 2 of IEEE 80 annex B, which the calculator loads by default: 70 × 70 m with 7 m mesh, 10 mm conductor at 0.5 m, 20 rods of 7.5 m on the perimeter, 400 Ω·m soil, 3I0 = 3180 A with Sf = 0.6, clearing in 0.5 s and 10 cm of 2500 Ω·m gravel.

• Grid resistance: 2.75 Ω and ground potential rise: 5,252 V
• Mesh voltage Em: 749 V and step voltage Es: 549 V, the values in the standard
• IEEE 80 tolerable touch for 70 kg: 840 V, so it passes; for 50 kg, 621 V, and it fails
• ITC-RAT 13 permissible touch with 2000 Ω footwear: 953 V, and it passes

What this calculator does not do

It does not solve irregular grids or soils with more than two layers, which need a numerical calculation. It does not calculate voltages transferred by fences, pipes or cable screens, which ITC-RAT 13 requires to be studied separately. And it does not replace measurement: resistivity varies with moisture and temperature, so the design value must be taken in the least favourable season, and step and touch voltages must be measured once the installation is built.

What measures it

Resistivity and earth resistance are measured with a four-terminal earth tester, which is what makes the Wenner method possible. For low-voltage installation testing, multifunction testers include earth and loop measurement.