Technical Article

# Substation Grounding Basics: Maximum Tolerable Step and Touch Voltages

October 16, 2020 by Lorenzo Mari

## Learn about the maximum step and touch voltages that people and animals in substation surroundings can tolerate.

For a grounding grid to fulfill its safety role, it must prevent step voltage and touch voltage values above those considered safe. Thus, we first need to estimate the maximum acceptable figures of step voltage and touch voltage.

## Thevenin’s and Norton’s Theorems

To derive mathematical expressions for the maximum tolerable step and touch voltages, we’ll rely on Thevenin’s and Norton’s theorems. These names are in honor of the French telegraph engineer Charles León Thévenin (1857-1926) and the American engineer E.L. Norton (1898-1983).

Thevenin’s and Norton’s theorems allow the circuit analyst or designer to simplify less significant portions of a circuit and concentrate on the part of greater relevance to the problem at hand.

Thevenin’s and Norton’s theorems separate any electric circuit into two networks: the source network and the load network connected by a single pair of terminals called the load terminals.

Thevenin’s theorem replaces the source network with an equivalent one consisting of an ideal independent voltage source in series with a linear resistance. This equivalent network delivers the same voltage and current to the load terminals as the source network. The voltage source is Thevenin’s voltage Vth, and the resistance is Thevenin’s resistance Rth.

Norton’s theorem is the dual of Thevenin’s theorem. The equivalent circuit is a current source in parallel with a linear resistance. The current source is Norton’s current In, and the resistance is Norton’s resistance Rn.

Norton’s source is the source conversion of Thevenin’s source and vice versa. It is also true that Rth = Rn.

Thevenin’s voltage is empirically measured or analytically calculated at the load terminals with the load network removed: the open-circuit voltage Voc = Vth. See Figure 1.

##### Figure 1. Thevenin’s equivalent circuit with load removed

Norton’s current is the short-circuit current at the load terminals with the load network removed: In = Isc. See Figure 2.

##### Figure 2. Thevenin’s equivalent circuit with load replaced by a short circuit

Figure 2 shows that the short-circuit current, measured or calculated, depends only upon Vth and Rth. Then,

$$Isc = \frac{Vth}{Rth}$$

or

$$Rth = \frac{Vth}{Isc} = \frac{Vth}{In}$$

## The Body as a Circuit Parameter

Under fault conditions, the Earth carries current coming from the grounding electrode. This current generates potential gradients on the Earth’s surface, which, in turn, lead to harmful step and touch voltages.

Figure 3 shows a person standing in the surroundings of a grounded structure. The points on Earth in contact with the feet have different potentials, i.e., a potential difference (voltage) is present. This potential difference generates step voltage Vs.

##### Figure 3. Step voltage at a grounded structure

The total resistance to the ground of the electrode may be split into three sections: R1, R2, and R0.

R1 is the ground resistance from the grounding electrode to the first foot; R2 is the ground resistance between the feet; R0 is the ground resistance from the second foot to infinity.

Other electrical parameters are:

• Rf = contact resistance of one bare foot with the Earth–neglects the resistance of the shoes and socks, Ω
• Rb = body resistance, Ω
• Ib = current through the body, A
• If = current source, modeling the fault current through the grounding electrode, A

Figure 4 shows a simple electrical equivalent network, with the body (Rb) as a circuit parameter.

##### Figure 4. Equivalent network for step voltage

As will be seen later, Vs = Vth (we obtain this by removing Rb).

Figure 5 shows a person standing on the Earth and touching a metallic object connected to the grounding electrode. The potential difference between the hand and the feet generates the touch voltage Vt. The feet are close enough to consider them in parallel, thus R2 = 0.

##### Figure 5. Touch voltage at a grounded structure

Figure 6 shows the equivalent electrical network, with the body (Rb) as a circuit parameter.

##### Figure 6. Equivalent network for touch voltage

As before, Vt = Vth (we obtain this by removing Rb).

## Computing the Maximum Tolerable Step Voltage

The step voltage Vs equals the Thevenin’s voltage Vth.

The load network in Figure 4 is the body resistance Rb. Applying Thevenin’s theorem, Vth will be calculated by separating Rb and solving the open-circuit voltage Voc. See Figure 7.

##### Figure 7. Open circuit voltage

The open-circuit voltage is

$$Voc = Vth = Vs = If \cdot R2$$

Isc = In is the short-circuit current. See Figure 8.

##### Figure 8. Short circuit current

Applying the current divider rule,

$$I_{SC} = In = If \cdot \frac{R2}{R2+2 \cdot Rf}$$

$$Rth = \frac{Voc}{Isc} = \frac{(If \cdot R2)(R2+2 \cdot Rf)}{If \cdot R2} = R2 + 2 \cdot Rf$$

Assuming R2<<2 ∙ Rf,

$$Rth = 2 \cdot Rf$$

This assumption is conservative because the voltage drop would be higher with R2 in the circuit.

Figure 9 shows the Thevenin’s equivalent circuit, where

$$Vs = Vth = Ib \cdot (Rth +Rb) = Ib \cdot (2 \cdot Rf + Rb)$$

##### Figure 9. Thevenin’s equivalent circuit for step voltage

A typical figure for body resistance — hand-to-feet and foot-to-foot — is 1000 Ω.

Buried horizontal round plates can model the feet, with a radius of 8cm. Using P. G. Laurent’s expression for this resistance,

$$Rf = \frac{\varrho}{4 \cdot b} = \frac{\varrho}{4 \cdot 0.08} = 3 \cdot \varrho$$

where

ρ = soil resistivity, Ω∙m0

The maximum tolerable body current for a 70 kg  person, according to C.F. Dalziel’s research, is

$$Ib = \frac{0.157}{\sqrt{ts}}$$

where

ts = duration of current exposure, s

The maximum tolerable step voltage is

$$Vs = \frac{0.157}{\sqrt{ts}} \cdot (2 \cdot 3 \cdot \varrho + 1000) = \frac{0.94 \cdot \varrho + 157}{\sqrt{ts}}$$

## Computing the Maximum Tolerable Touch Voltage

Likewise, the touch voltage Vt equals the Thevenin’s voltage Vth.

From Figure 6, Vth is calculated by separating Rb and solving the open-circuit voltage Voc. See Figure 10.

##### Figure 10. Open circuit voltage

The open-circuit voltage is

$$Voc = Vth = If \cdot R1$$

Isc = In is the short-circuit current. See Figure 11.

##### Figure 11. Short circuit current

Applying the current divider rule,

$$Isc = In = If \cdot \frac{R1}{R1 + \frac{Rf}{2}}$$

$$Rth= \frac{Voc}{Isc} = \frac{(If \cdot R1)(R1 + \frac{Rf}{2})}{If \cdot R1} = R1 + \frac{Rf}{2}$$

Assuming R1<

$$Rth= \frac{Rf}{2}$$

Figure 12 shows the Thevenin’s equivalent circuit, where

$$Vt=Vth=Ib \cdot (Rth+Rb) = Ib \cdot (\frac{Rf}{2} +Rb)$$

##### Figure 12. Thevenin’s equivalent circuit for touch voltage

Then, the maximum tolerable touch voltage is

$$Vt = \frac{0.157}{\sqrt{ts}} \cdot (\frac{3}{2} \cdot \varrho + 1000) = \frac{0.24 + \varrho + 157}{\sqrt{ts}}$$

## A Key Clarification

A fundamental concept is that the potential differences giving rise to the step and touch voltages are the ones on the Earth when the person is not present. The equivalent Thevenin circuits model this fact.

However, when comparing Figure 9 with Figure 4 and Figure 12 with Figure 6, we conclude that by assuming R2 is much less than 2 ∙ Rf and R1 is much less than Rf2, the step voltage is the voltage drop across R2, and the touch voltage is the voltage drop across R1, even with the body resistance in the circuit.

## A Review of Maximum Tolerable Step and Touch Voltages

A fault or lightning current flowing through the substation grounding electrode generates potential gradients on the Earth. People around the substation can be affected by these potential gradients.

Particularly noteworthy are the step voltage and the touch voltage.

Two feet on the ground in points with different potentials are subjected to the step voltage. Similarly, the action of touching a grounded metal object while the feet are on the Earth at another potential subjects a person to the touch voltage.

The step and touch voltages are the ones on the Earth when the person is not present. The Thevenin equivalent circuits model this condition.

The substation’s grounding electrode must ensure appropriate step and touch voltages, keeping the people safe.

This article described an elementary methodology to estimate the maximum tolerable step and touch voltages. 