Step and Touch Voltage Formulas

Substation earthing formulas: the tolerable touch and step voltage for a fault duration, the surface layer factor and the ground potential rise.

The formulas

ρ is the resistivity of the soil and ρs that of the surface layer, both in Ω·m, hs the thickness of the surface layer in metres, ts the duration of the shock (the fault clearing time) in seconds, Ig the earth fault current into the earth grid in amperes and Rg the grid resistance in ohms. The factors are those of IEEE Std 80 for a 70 kg body, and the same standard gives 0.116 in place of 0.157 for a 50 kg body. Without a surface layer, Cs = 1 and ρs = ρ. The page is an estimate for a first check. The standard and the local rules decide the design, and the actual step and touch voltages of a grid come from a grid analysis. All values here are examples.

Worked example: soil of 100 Ω·m, 0.1 m of 2,500 Ω·m crushed rock, 0.5 s

1. Cs = 1 − 0.09 × (1 − 100 / 2500) / (2 × 0.1 + 0.09) = 0.7021
2. Etouch = (1000 + 1.5 × 0.7021 × 2500) × 0.157 / √0.5 = 806.6 V
3. Estep = (1000 + 6 × 0.7021 × 2500) × 0.157 / √0.5 = 2,560 V

Worked example: the same soil with no surface layer

1. Etouch = (1000 + 1.5 × 100) × 0.157 / √0.5 = 255.3 V
2. Estep = (1000 + 6 × 100) × 0.157 / √0.5 = 355.3 V

Worked example: ground potential rise

1. GPR = 3000 × 0.8 = 2,400 V

Tolerable voltages with the crushed rock layer, by fault duration

Frequently asked questions

Why does the limit fall as the fault lasts longer?

The body tolerates less energy from a long shock, so the limit goes with 1 / √ts. A fast protection trip allows a higher voltage.

Why does a layer of crushed rock help?

It has a high resistivity, so it adds resistance between the foot and the earth and the current through the body falls. Cs is the factor that counts for it.

What is the difference between step and touch voltage?

Touch voltage is between a hand on an earthed object and the feet. Step voltage is between the two feet, about one metre apart. The step limit is higher because the current does not pass through the chest.

Is the ground potential rise a limit for people?

Not in itself. It is the voltage of the grid against remote earth, and it is compared with the touch limit to see whether the design needs a closer study.

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