Improper earthing and unsafe step/touch potential

When an earth fault occurs, fault current $I_g$ enters the soil through the earth electrode or substation grid. The grid rises to the ground potential rise

$$GPR = I_g R_g$$

and the ground surface potential falls with distance from the electrode. A person bridging part of this potential gets a shock. Improper earthing (high $R_g$, too few electrodes, wide mesh, no bonding, no surface layer, slow protection) makes these potentials exceed the safe values.

 V
 |\_ GPR = Ig Rg
 |  \       steep gradient
 |   \__    (poor earthing)
 |      \___
 |  Et      \____ Es
 +---|--|-----|-|------> distance
   structure  feet 1 m apart

Touch potential

Voltage between the hand on an earthed structure and the feet standing 1 m away. Equivalent circuit: body resistance $R_b$ in series with two feet in parallel ($R_f/2$), where $R_f \approx 3\rho_s$:

$$I_b = \frac{E_{touch}}{R_b + 1.5\rho_s}$$

Step potential

Voltage between two feet 1 m apart. Body $R_b$ in series with two feet ($2R_f$):

$$I_b = \frac{E_{step}}{R_b + 6\rho_s}$$

Safe (tolerable) limits (IEEE 80, 50 kg person)

Using Dalziel's current $I_b = \frac{0.116}{\sqrt{t_s}}$ and $R_b = 1000\ \Omega$:

$$E_{touch,50} = (1000 + 1.5C_s\rho_s)\frac{0.116}{\sqrt{t_s}}, \qquad E_{step,50} = (1000 + 6C_s\rho_s)\frac{0.116}{\sqrt{t_s}}$$

$C_s$ is the surface-layer derating factor and $t_s$ the fault duration.

How improper earthing makes them unsafe

1. High grid resistance $R_g$: GPR $= I_gR_g$ is large, so actual touch voltage (a fraction of GPR) is large. For example, $I_g = 2$ kA and $R_g = 5\ \Omega$ gives GPR = 10 kV; even 30% of this (3 kV) as touch voltage is far above the safe value of a few hundred volts.
2. Wide mesh or no grid: surface potential is not uniform; mesh (touch) voltage

$$E_m = \frac{\rho K_m K_i I_g}{L_M}$$

rises when the total buried length $L_M$ is small. 3. Steep gradient at the edge (no grading ring): high step voltage

$$E_s = \frac{\rho K_s K_i I_g}{L_S}$$

4. No surface layer of crushed rock: low $\rho_s$, so tolerable voltages drop.
5. Unbonded equipment or fences: the full GPR may appear as touch voltage.
6. High $R_g$ reduces fault current so relays may be slow; longer $t_s$ lowers the tolerable limits.

Remedies

Lower $R_g$ (more rods, larger grid, deep electrodes, chemical backfill), closer mesh spacing, bonding of all metal parts, a crushed-rock surface layer, grading rings, and fast fault clearing so that $E_{actual} < E_{tolerable}$.