# Earthing Calculation

## Contents

- 1 Introduction
- 2 Calculation Methodology
- 2.1 Prerequisites
- 2.2 Earthing Grid Conductor Sizing
- 2.3 Touch and Step Potential Calculations
- 2.3.1 Step 1: Soil Resistivity
- 2.3.2 Step 2: Surface Layer Materials
- 2.3.3 Step 3: Earthing Grid Resistance
- 2.3.4 Step 4: Maximum Grid Current
- 2.3.5 Step 5: Touch and Step Potential Criteria
- 2.3.6 Step 6: Ground Potential Rise (GPR)
- 2.3.7 Step 7: Earthing Grid Design Verification

- 3 Worked Example
- 4 Computer Based Tools
- 5 What next?

## Introduction

The earthing system in a plant / facility is very important for a few reasons, all of which are related to either the protection of people and equipment and/or the optimal operation of the electrical system. These include:

- Equipotential bonding of conductive objects (e.g. metallic equipment, buildings, piping etc) to the earthing system prevent the presence of dangerous voltages between objects (and earth).
- The earthing system provides a low resistance return path for earth faults within the plant, which protects both personnel and equipment
- For earth faults with return paths to offsite generation sources, a low resistance earthing grid relative to remote earth prevents dangerous ground potential rises (touch and step potentials)
- The earthing system provides a low resistance path (relative to remote earth) for voltage transients such as lightning and surges / overvoltages
- Equipotential bonding helps prevent electrostatic buildup and discharge, which can cause sparks with enough energy to ignite flammable atmospheres
- The earthing system provides a reference potential for electronic circuits and helps reduce electrical noise for electronic, instrumentation and communication systems

This calculation is based primarily on the guidelines provided by IEEE Std 80 (2000), "Guide for safety in AC substation grounding". Lightning protection is excluded from the scope of this calculation (refer to the specific lightning protection calculation for more details).

### Why do the calculation?

The earthing calculation aids in the proper design of the earthing system. Using the results of this calculation, you can:

- Determine the minimum size of the earthing conductors required for the main earth grid
- Ensure that the earthing design is appropriate to prevent dangerous step and touch potentials (if this is necessary)

### When to do the calculation?

This calculation should be performed when the earthing system is being designed. It could also be done after the preliminary design has been completed to confirm that the earthing system is adequate, or highlight the need for improvement / redesign. Ideally, soil resistivity test results from the site will be available for use in touch and step potential calculations (if necessary).

### When is the calculation unnecessary?

The sizing of earthing conductors should always be performed, but touch and step potential calculations (per IEEE Std 80 for earth faults with a return path through remote earth) are not always necessary.

For example, when all electricity is generated on-site and the HV/MV/LV earthing systems are interconnected, then there is no need to do a touch and step potential calculation. In such a case, all earth faults would return to the source via the earthing system (notwithstanding some small leakage through earth).

However, where there are decoupled networks (e.g. long transmission lines to remote areas of the plant), then touch and step potential calculations should be performed for the remote area only.

## Calculation Methodology

This calculation is based on IEEE Std 80 (2000), "Guide for safety in AC substation grounding". There are two main parts to this calculation:

- Earthing grid conductor sizing
- Touch and step potential calculations

IEEE Std 80 is quite descriptive, detailed and easy to follow, so only an overview will be presented here and IEEE Std 80 should be consulted for further details (although references will be given herein).

### Prerequisites

The following information is required / desirable before starting the calculation:

- A layout of the site
- Maximum earth fault current into the earthing grid
- Maximum fault clearing time
- Ambient (or soil) temperature at the site
- Soil resistivity measurements at the site (for touch and step only)
- Resistivity of any surface layers intended to be laid (for touch and step only)

### Earthing Grid Conductor Sizing

Determining the minimum size of the earthing grid conductors is necessary to ensure that the earthing grid will be able to withstand the maximum earth fault current. Like a normal power cable under fault, the earthing grid conductors experience an adiabatic short circuit temperature rise. However unlike a fault on a normal cable, where the limiting temperature is that which would cause permanent damage to the cable's insulation, the temperature limit for earthing grid conductors is the melting point of the conductor. In other words, during the worst case earth fault, we don't want the earthing grid conductors to start melting!

The minimum conductor size capable of withstanding the adiabatic temperature rise associated with an earth fault is given by re-arranging IEEE Std 80 Equation 37:

- [math] A = \sqrt{ i^{2}t \left( \frac{\frac{\alpha_{r} \rho_{r} 10^{4}}{TCAP}}{\ln \left[ 1+ \left( \frac{T_{m}-T_{a}}{K_{0}+T_{a}} \right) \right]} \right)} \, [/math]

Where [math]A \,[/math] is the minimum cross-sectional area of the earthing grid conductor ([math]mm^{2}[/math])

- [math]i^{2}t \,[/math] is the energy of the maximum earth fault ([math]A^{2}s[/math])
- [math]T_{m} \,[/math] is the maximum allowable (fusing) temperature (ºC)
- [math]T_{a} \,[/math] is the ambient temperature (ºC)
- [math]\alpha_{r} \,[/math] is the thermal coefficient of resistivity (º[math]C^{-1}[/math])
- [math]\rho_{r} \,[/math] is the resistivity of the earthing conductor ([math]\mu \Omega . cm [/math])
- [math]K_{0} \,[/math] is [math] \left( \frac{1}{\alpha_{r}} - 20 \deg C \right)[/math]
- [math]TCAP \,[/math] is the thermal capacity of the conductor per unit volume([math]J cm^{-3}[/math]º[math]C^{-1}[/math])

The material constants [math]T_{m}[/math], [math]\alpha_{r}[/math], [math]\rho_{r}[/math] and [math]TCAP[/math] for common conductor materials can be found in IEEE Std 80 Table 1. For example. commercial hard-drawn copper has material constants:

- [math]T_{m}=[/math]1084 ºC
- [math]\alpha_{r}=[/math]0.00381 º[math]C^{-1}[/math]
- [math]\rho_{r}=[/math]1.78 [math] \mu \Omega . cm [/math]
- [math]TCAP=[/math]3.42 [math] J cm^{-3}[/math]º[math]C^{-1}[/math].

As described in IEEE Std 80 Section 11.3.1.1, there are alternative methods to formulate this equation, all of which can also be derived from first principles).

There are also additional factors that should be considered (e.g. taking into account future growth in fault levels), as discussed in IEEE Std 80 Section 11.3.3.

### Touch and Step Potential Calculations

When electricity is generated remotely and there are no return paths for earth faults other than the earth itself, then there is a risk that earth faults can cause dangerous voltage gradients in the earth around the site of the fault (called ground potential rises). This means that someone standing near the fault can receive a dangerous electrical shock due to:

- Touch voltages - there is a dangerous potential difference between the earth and a metallic object that a person is touching
- Step voltages - there is a dangerous voltage gradient between the feet of a person standing on earth

The earthing grid can be used to dissipate fault currents to remote earth and reduce the voltage gradients in the earth. The touch and step potential calculations are performed in order to assess whether the earthing grid can dissipate the fault currents so that dangerous touch and step voltages cannot exist.

#### Step 1: Soil Resistivity

The resistivity properties of the soil where the earthing grid will be laid is an important factor in determining the earthing grid's resistance with respect to remote earth. Soils with lower resistivity lead to lower overall grid resistances and potentially smaller earthing grid configurations can be designed (i.e. that comply with safe step and touch potentials).

It is good practice to perform soil resistivity tests on the site. There are a few standard methods for measuring soil resistivity (e.g. Wenner four-pin method). A good discussion on the interpretation of soil resistivity test measurements is found in IEEE Std 80 Section 13.4.

Sometimes it isn't possible to conduct soil resistivity tests and an estimate must suffice. When estimating soil resistivity, it goes without saying that one should err on the side of caution and select a higher resistivity. IEEE Std 80 Table 8 gives some guidance on range of soil resistivities based on the general characteristics of the soil (i.e. wet organic soil = 10 [math]\Omega[/math].m, moist soil = 100 [math]\Omega[/math].m, dry soil = 1,000 [math]\Omega[/math].m and bedrock = 10,000 [math]\Omega[/math].m).

#### Step 2: Surface Layer Materials

Applying a thin layer (0.08m - 0.15m) of high resistivity material (such as gravel, blue metal, crushed rock, etc) over the surface of the ground is commonly used to help protect against dangerous touch and step voltages. This is because the surface layer material increases the contact resistance between the soil (i.e. earth) and the feet of a person standing on it, thereby lowering the current flowing through the person in the event of a fault.

IEEE Std 80 Table 7 gives typical values for surface layer material resistivity in dry and wet conditions (e.g. 40mm crushed granite = 4,000 [math]\Omega[/math].m (dry) and 1,200 [math]\Omega[/math].m (wet)).

The effective resistance of a person's feet (with respect to earth) when standing on a surface layer is not the same as the surface layer resistance because the layer is not thick enough to have uniform resistivity in all directions. A surface layer derating factor needs to be applied in order to compute the effective foot resistance (with respect to earth) in the presence of a finite thickness of surface layer material. This derating factor can be approximated by an empirical formula as per IEEE Std 80 Equation 27:

- [math] C_{s} = 1 - \frac{0.09 \left(1 - \frac{\rho}{\rho_{s}} \right)}{2h_{s} + 0.09} \,[/math]

Where [math]C_{s} \,[/math] is the surface layer derating factor

- [math]\rho \,[/math] is the soil resistivity ([math]\Omega[/math].m)
- [math]\rho_{s} \,[/math] is the resistivity of the surface layer material ([math]\Omega[/math].m)
- [math]h_{s} \,[/math] is the thickness of the surface layer (m)

This derating factor will be used later in Step 5 when calculating the maximum allowable touch and step voltages.

#### Step 3: Earthing Grid Resistance

A good earthing grid has low resistance (with respect to remote earth) to minimise ground potential rise (GPR) and consequently avoid dangerous touch and step voltages. Calculating the earthing grid resistance usually goes hand in hand with earthing grid design - that is, you design the earthing grid to minimise grid resistance. The earthing grid resistance mainly depends on the area taken up by the earthing grid, the total length of buried earthing conductors and the number of earthing rods / electrodes.

IEEE Std 80 offers two alternative options for calculating the earthing grid resistance (with respect to remote earth) - 1) the simplified method (Section 14.2) and 2) the Schwarz equations (Section 14.3), both of which are outlined briefly below. IEEE Std 80 also includes methods for reducing soil resistivity (in Section 14.5) and a treatment for concrete-encased earthing electrodes (in Section 14.6).

##### Simplified Method

IEEE Std 80 Equation 52 gives the simplified method as modified by Sverak to include the effect of earthing grid depth:

- [math] R_{g} = \rho \left[ \frac{1}{L_{T}} + \frac{1}{\sqrt{20A}} \left( 1 + \frac{1}{1 + h \sqrt{20 / A}} \right) \right] \,[/math]

Where [math] R_{g} \,[/math] is the earthing grid resistance with respect to remote earth ([math]\Omega[/math])

- [math] \rho \,[/math] is the soil resistivitiy ([math]\Omega[/math].m)
- [math] L_{T} \,[/math] is the total length of buried conductors (m)
- [math] A \,[/math] is the total area occupied by the earthing grid ([math]m^{2}[/math])
- [math] h \,[/math] is the depth of the earthing grid (m)

##### Schwarz Equations

The Schwarz equations are a series of equations that are more accurate in modelling the effect of earthing rods / electrodes. The equations are found in IEEE Std 80 Equations 53, 54, 55^{(footnote)} and 56, as follows:

- [math] R_{g} = \frac{R_{1}R_{2} - R_{m}^{2}}{R_{1} + R_{2} - 2R_{m}} \,[/math]

Where [math] R_{g} \, [/math] is the earthing grid resistance with respect to remote earth ([math]\Omega[/math])

- [math] R_{1} \, [/math] is the earth resistance of the grid conductors ([math]\Omega[/math])
- [math] R_{2} \, [/math] is the earth resistance of the earthing electrodes ([math]\Omega[/math])
- [math] R_{m} \, [/math] is the mutual earth resistance between the grid conductors and earthing electrodes ([math]\Omega[/math])

And the grid, earthing electrode and mutual earth resistances are:

- [math] R_{1} = \frac{\rho}{\pi L_{c}} \left[ \ln \left( \frac{2L_{c}}{a'} \right) + \frac{k_{1} L_{c}}{\sqrt{A}} - k_{2} \right] \, [/math]

- [math] R_{2} = \frac{\rho}{2 \pi n_{r} L_{r}} \left[ \ln \left( \frac{4L_{r}}{b} \right) -1 + \frac{2k_{1} L_{r}}{\sqrt{A}} \left( \sqrt{n_{r}} - 1 \right)^{2} \right] \, [/math]

- [math] R_{m} = \frac{\rho}{\pi L_{c}} \left[ \ln \left( \frac{2L_{c}}{L_{r}} \right) + \frac{k_{1} L_{c}}{\sqrt{A}} - k_{2} + 1 \right] \, [/math]

Where [math] \rho \, [/math] is the soil resistivity ([math]\Omega[/math].m)

- [math] L_{c} \, [/math] is the total length of buried grid conductors (m)
- [math] a' \, [/math] is [math] \sqrt{r . 2h} \,[/math] for conductors buried at depth [math] h \, [/math] metres and with cross-sectional radius [math] r \, [/math] metres, or simply [math] r \, [/math] for grid conductors on the surface
- [math] A \, [/math] is the total area covered by the grid conductors ([math]m^{2}[/math])
- [math] L_{r} \, [/math] is the length of each earthing electrode (m)
- [math] n_{r} \, [/math] is number of earthing electrodes in area [math] A \, [/math]
- [math] b \, [/math] is the cross-sectional radius of an earthing electrode (m)
- [math] k_{1} \, [/math] and [math] k_{2} \, [/math] are constant coefficients depending on the geometry of the grid

The coefficient [math] k_{1} \, [/math] can be approximated by the following:

- (1) For depth [math] h = 0 \, [/math]: [math] k_{1} = -0.04L/R + 1.41 \, [/math]
- (2) For depth [math] h = \frac{1}{10} \sqrt{A} \, [/math]: [math] k_{1} = -0.05L/R + 1.20 \, [/math]
- (3) For depth [math] h = \frac{1}{6} \sqrt{A} \, [/math]: [math] k_{1} = -0.05L/R + 1.13 \, [/math]

The coefficient [math] k_{2} \, [/math] can be approximated by the following:

- (1) For depth [math] h = 0 \, [/math]: [math] k_{2} = 0.15L/R + 5.50 \, [/math]
- (2) For depth [math] h = \frac{1}{10} \sqrt{A} \, [/math]: [math] k_{2} = 0.10L/R + 4.68 \, [/math]
- (3) For depth [math] h = \frac{1}{6} \sqrt{A} \, [/math]: [math] k_{2} = 0.05L/R + 4.40 \, [/math]

Where in both cases, [math] L/R \, [/math] is the length-to-width ratio of the earthing grid.

#### Step 4: Maximum Grid Current

The maximum grid current is the worst case earth fault current that would flow via the earthing grid back to remote earth. To calculate the maximum grid current, you firstly need to calculate the worst case symmetrical earth fault current at the facility that would have a return path through remote earth (call this [math]I_{k,e} \,[/math]). This can be found from the power systems studies or from manual calculation. Generally speaking, the highest relevant earth fault level will be on the primary side of the largest distribution transformer (i.e. either the terminals or the delta windings).

##### Current Division Factor

Not all of the earth fault current will flow back through remote earth. A portion of the earth fault current may have local return paths (e.g. local generation) or there could be alternative return paths other than remote earth (e.g. overhead earth return cables, buried pipes and cables, etc). Therefore a current division factor [math]S_{f} \,[/math] must be applied to account for the proportion of the fault current flowing back through remote earth.

Computing the current division factor is a task that is specific to each project and the fault location and it may incorporate some subjectivity (i.e. "engineeing judgement"). In any case, IEEE Std 80 Section 15.9 has a good discussion on calculating the current division factor. In the most conservative case, a current division factor of [math]S_{f} = 1 \,[/math] can be applied, meaning that 100% of earth fault current flows back through remote earth.

The symmetrical grid current [math]I_{g} \,[/math] is calculated by:

- [math] I_{g} = I_{k,e} S_{f} \, [/math]

##### Decrement Factor

The symmetrical grid current is not the maximum grid current because of asymmetry in short circuits, namely a dc current offset. This is captured by the decrement factor, which can be calculated from IEEE Std 80 Equation 79:

- [math] D_{f} = \sqrt{1 + \frac{T_{a}}{t_{f}} \left( 1 - e^{\frac{-2t_{f}}{T_{a}}} \right)} \, [/math]

Where [math] D_{f} \, [/math] is the decrement factor

- [math] t_{f} \, [/math] is the duration of the fault (s)
- [math] T_{A} \, [/math] is the dc time offset constant (see below)

The dc time offset constant is derived from IEEE Std 80 Equation 74:

- [math] T_{A} = \frac{X}{R} . \frac{1}{2 \pi f} \, [/math]

Where [math] \frac{X}{R} \, [/math] is the X/R ratio at the fault location

- [math] f \, [/math] is the system frequency (Hz)

The maximum grid current [math]I_{G} \,[/math] is lastly calculated by:

- [math] I_{G} = I_{g} D_{f} \, [/math]

#### Step 5: Touch and Step Potential Criteria

One of the goals of a safe earthing grid is to protect people against lethal electric shocks in the event of an earth fault. The magnitude of ac electric current (at 50Hz or 60Hz) that a human body can withstand is typically in the range of 60 to 100mA, when ventricular fibrillation and heart stoppage can occur. The duration of an electric shock also contributes to the risk of mortality, so the speed at which faults are cleared is also vital. Given this, we need to prescribe maximum tolerable limits for touch and step voltages that do not lead to lethal shocks.

The maximum tolerable voltages for step and touch scenarios can be calculated empirically from IEEE Std Section 8.3 for body weights of 50kg and 70kg:

Touch voltage limit - the maximum potential difference between the surface potential and the potential of an earthed conducting structure during a fault (due to ground potential rise):

- 50kg person: [math] E_{touch,50} = \left( 1000 + 1.5C_{s} \rho_{s} \right) \frac{0.116}{\sqrt{t_{s}}} \,[/math]
- 70kg person: [math] E_{touch,70} = \left( 1000 + 1.5C_{s} \rho_{s} \right) \frac{0.157}{\sqrt{t_{s}}} \,[/math]

Step voltage limit - is the maximum difference in surface potential experience by a person bridging a distance of 1m with the feet without contact to any earthed object:

- 50kg person: [math] E_{step,50} = \left( 1000 + 6C_{s} \rho_{s} \right) \frac{0.116}{\sqrt{t_{s}}} \,[/math]
- 70kg person: [math] E_{step,70} = \left( 1000 + 6C_{s} \rho_{s} \right) \frac{0.157}{\sqrt{t_{s}}} \,[/math]

Where [math] E_{touch} \,[/math] is the touch voltage limit (V)

- [math] E_{step} \,[/math] is the step voltage limit (V)
- [math] C_{s} \,[/math] is the surface layer derating factor (as calculated in Step 2)
- [math] \rho_{s} \,[/math] is the soil resistivity ([math]\Omega[/math].m)
- [math] t_{s} \,[/math] is the maximum fault clearing time (s)

The choice of body weight (50kg or 70kg) depends on the expected weight of the personnel at the site. Typically, where women are expected to be on site, the conservative option is to choose 50kg.

#### Step 6: Ground Potential Rise (GPR)

Normally, the potential difference between the local earth around the site and remote earth is considered to be zero (i.e. they are at the same potential). However an earth fault (where the fault current flows back through remote earth), the flow of current through the earth causes local potential gradients in and around the site. The maximum potential difference between the site and remote earth is known as the ground potential rise (GPR). It is important to note that this is a **maximum** potential potential difference and that earth potentials around the site will vary relative to the point of fault.

The maximum GPR is calculated by:

- [math] GPR = I_{G} R_{g}\, [/math]

Where [math] GPR \, [/math] is the maximum ground potential rise (V)

#### Step 7: Earthing Grid Design Verification

Now we just need to verify that the earthing grid design is safe for touch and step potential. If the maximum GPR calculated above does not exceed either of the touch and step voltage limits (from Step 5), then the grid design is safe.

However if it **does exceed** the touch and step voltage limits, then some further analysis is required to verify the design, namely the calculation of the maximum mesh and step voltages as per IEEE Std 80 Section 16.5.

##### Mesh Voltage Calculation

The mesh voltage is the maximum touch voltage within a mesh of an earthing grid and is derived from IEEE Std 80 Equation 80:

- [math] E_{m} = \frac{\rho_{s} K_{m} K_{i} I_{G}} {L_{M}} \, [/math]

Where :: [math] \rho_{s} \, [/math] is the soil resistivity ([math]\Omega[/math].m)

- [math] I_{G}\, [/math] is the maximum grid current found earlier in Step 4 (A)
- [math] K_{m}\, [/math] is the geometric spacing factor (see below)
- [math] K_{i}\, [/math] is the irregularity factor (see below)
- [math] L_{M}\, [/math] is the effective buried length of the grid (see below)

###### Geometric Spacing Factor [math]K_{m}[/math]

The geometric spacing factor [math] K_{m}\, [/math] is calculated from IEEE Std 80 Equation 81:

- [math] K_{m} = \frac{1}{2\pi} \left( \ln \left[ \frac{D^{2}}{16h \times d} + \frac{\left( D + 2h \right)^{2}}{8D \times d} - \frac{h}{4d} \right] + \frac{K_{ii}}{K_{h}} \ln \left[ \frac{8}{\pi (2n-1)} \right] \right) \,[/math]

Where [math] D \, [/math] is the spacing between parallel grid conductors (m)

- [math] h \, [/math] is the depth of buried grid conductors (m)
- [math] d \, [/math] is the cross-sectional diameter of a grid conductor (m)
- [math] K_{h} \, [/math] is a weighting factor for depth of burial = [math] \sqrt{1+h} \, [/math]
- [math] K_{ii} \, [/math] is a weighting factor for earth electrodes /rods on the corner mesh

- [math] K_{ii} = 1 \, [/math] for grids with earth electrodes along the grid perimeter or corners
- [math] K_{ii} = \frac{1}{2n^{n/2}} \, [/math] for grids with no earth electrodes on the corners or on the perimeter

- [math] n \, [/math] is a geometric factor (see below)

###### Geometric Factor [math]n[/math]

The geometric factor [math]n \,[/math] is calculated from IEEE Std 80 Equation 85:

- [math] n = n_{a} \times n_{b} \times n_{c} \times n_{d} [/math]

With [math] n_{a} = \frac{2L_{c}}{L_{p}} \, [/math]

- [math] n_{b} = 1 \, [/math] for square grids, or otherwise [math] n_{b} = \sqrt{\frac{L_{p}}{4\sqrt{A}}} \, [/math]
- [math] n_{c} = 1 \, [/math] for square and rectangular grids, or otherwise [math] n_{c} = \left[ \frac{L_{x}L_{y}}{A} \right]^{\frac{0.7A}{L_{x}L_{y}}} \, [/math]
- [math] n_{d} = 1 \, [/math] for square, rectangular and L-shaped grids, or otherwise [math] n_{d} = \frac{D_{m}}{\sqrt{L_{x}^{2} + L_{y}^{2}}} \, [/math]

Where [math] L_{c} \, [/math] is the total length of horizontal grid conductors (m)

- [math] L_{p} \, [/math] is the length of grid conductors on the perimeter (m)
- [math] A \, [/math] is the total area of the grid ([math]m^{2}[/math])
- [math] L_{x} \, [/math] and [math] L_{y} \, [/math] are the maximum length of the grids in the x and y directions (m)
- [math] D_{m} \, [/math] is the maximum distance between any two points on the grid (m)

###### Irregularity Factor [math]K_{i}[/math]

The irregularity factor [math] K_{i}\, [/math] is calculated from IEEE Std 80 Equation 89:

- [math] K_{i} = 0.644 + 0.148n \, [/math]

Where [math] n \, [/math] is the geometric factor derived above

###### Effective Buried Length [math]L_{M}[/math]

The effective buried length [math]L_{M} \,[/math] is found as follows:

- For grids with few or no earthing electrodes (and none on corners or along the perimeter):

- [math]L_{M} = L_{c} + L_{R} \,[/math]

Where [math] L_{c} \, [/math] is the total length of horizontal grid conductors (m)

- [math] L_{R} \, [/math] is the total length of earthing electrodes / rods (m)

- For grids with earthing electrodes on the corners and along the perimeter:

- [math]L_{M} = L_{c} + \left[ 1.55 + 1.22 \left( \frac{L_{r}}{\sqrt{L_{x}^{2} + L_{y}^{2}}} \right) \right] L_{R} \,[/math]

Where [math] L_{c} \, [/math] is the total length of horizontal grid conductors (m)

- [math] L_{R} \, [/math] is the total length of earthing electrodes / rods (m)
- [math] L_{r} \, [/math] is the length of each earthing electrode / rod (m)
- [math] L_{x} \, [/math] and [math] L_{y} \, [/math] are the maximum length of the grids in the x and y directions (m)

##### Step Voltage Calculation

The maximum allowable step voltage is calculated from IEEE Std 80 Equation 92:

- [math] E_{s} = \frac{\rho_{s} K_{s} K_{i} I_{G}} {L_{S}} \, [/math]

Where :: [math] \rho_{s} \, [/math] is the soil resistivity ([math]\Omega[/math].m)

- [math] I_{G}\, [/math] is the maximum grid current found earlier in Step 4 (A)
- [math] K_{s}\, [/math] is the geometric spacing factor (see below)
- [math] K_{i}\, [/math] is the irregularity factor (as derived above in the mesh voltage calculation)
- [math] L_{S}\, [/math] is the effective buried length of the grid (see below)

###### Geometric Spacing Factor [math]K_{s}[/math]

The geometric spacing factor [math] K_{s}\, [/math] based on IEEE Std 80 Equation 81 is applicable for burial depths between 0.25m and 2.5m:

- [math] K_{s} = \frac{1}{\pi} \left[ \frac{1}{2h} + \frac{1}{D+h} + \frac{1}{D} \left( 1 - 0.5^{n-2} \right) \right] \, [/math]

Where [math] D \, [/math] is the spacing between parallel grid conductors (m)

- [math] h \, [/math] is the depth of buried grid conductors (m)
- [math] n \, [/math] is a geometric factor (as derived above in the mesh voltage calculation)

###### Effective Buried Length [math]L_{S}[/math]

The effective buried length [math] L_{S}\, [/math] for all cases can be calculated by IEEE Std 80 Equation 93:

- [math] L_{S} = 0.75L_{c} + 0.85L_{R} \, [/math]

Where [math] L_{c} \, [/math] is the total length of horizontal grid conductors (m)

- [math] L_{R} \, [/math] is the total length of earthing electrodes / rods (m)

##### What Now?

Now that the mesh and step voltages are calculated, compare them to the maximum tolerable touch and step voltages respectively. If:

- [math] E_{m} \lt E_{touch} \,[/math], and
- [math] E_{s} \lt E_{step} \,[/math]

then the earthing grid design is safe.

If not, however, then further work needs to be done. Some of the things that can be done to make the earthing grid design safe:

- Redesign the earthing grid to lower the grid resistance (e.g. more grid conductors, more earthing electrodes, increasing cross-sectional area of conductors, etc). Once this is done, re-compute the earthing grid resistance (see Step 3) and re-do the touch and step potential calculations.

- Limit the total earth fault current or create alternative earth fault return paths

- Consider soil treatments to lower the resistivity of the soil

- Greater use of high resistivity surface layer materials

## Worked Example

In this example, the touch and step potential calculations for an earthing grid design will be performed. The proposed site is a small industrial facility with a network connection via a transmission line and a delta-wye connected transformer.

### Step 1: Soil Resistivity

The soil resistivity around the site was measured with a Wenner four-pin probe and found to be approximately 300 [math] \Omega [/math].m.

### Step 2: Surface Layer Materials

A thin 100mm layer of blue metal (3,000 [math] \Omega [/math].m) is proposed to be installed on the site. The surface layer derating factor is:

- [math] C_{s} = 1 - \frac{0.09 \left(1 - \frac{\rho}{\rho_{s}} \right)}{2h_{s} + 0.09} \,[/math]
- [math] = 1 - \frac{0.09 \left(1 - \frac{300}{3,000} \right)}{2 \times 0.1 + 0.09} \, [/math]
- [math] = 0.7207 \, [/math]

### Step 3: Earthing Grid Resistance

A rectangular earthing grid (see the figure right) with the following parameters is proposed:

- Length of 90m and a width of 50m
- 6 parallel rows and 7 parallel columns
- Grid conductors will be 120 [math]mm^{2}[/math] and buried at a depth of 600mm
- 22 earthing rods will be installed on the corners and perimeter of the grid
- Each earthing rod will be 3m long

Using the simplified equation, the resistance of the earthing grid with respect to remote earth is:

- [math] R_{g} = \rho \left[ \frac{1}{L_{T}} + \frac{1}{\sqrt{20A}} \left( 1 + \frac{1}{1 + h \sqrt{20 / A}} \right) \right] \,[/math]
- [math] = 300 \left[ \frac{1}{956} + \frac{1}{\sqrt{20 \times 4,500}} \left( 1 + \frac{1}{1 + 0.6 \sqrt{20 / 4,500}} \right) \right] \,[/math]
- [math] = 2.2753 \Omega \, [/math]

### Step 4: Maximum Grid Current

Suppose that the maximum single phase to earth fault at the HV winding of the transformer is 3.1kA and that the current division factor is 1 (all the fault current flows back to remote earth).

The X/R ratio at the fault is approximately 15, the maximum fault duration 150ms and the system nominal frequency is 50Hz. The DC time offset is therefore:

- [math] T_{A} = \frac{X}{R} . \frac{1}{2 \pi f} \, [/math]
- [math] = 15 \times \frac{1}{2 \pi 50} \, [/math]
- [math] = 0.04774 \, [/math]

The decrement factor is then:

- [math] D_{f} = \sqrt{1 + \frac{T_{a}}{t_{f}} \left( 1 - e^{\frac{-2t_{f}}{T_{a}}} \right)} \, [/math]
- [math] = \sqrt{1 + \frac{0.04774}{0.15} \left( 1 - e^{\frac{-2 \times 0.15}{0.04774}} \right)} \, [/math]
- [math] = 1.1479 \, [/math]

Fianlly, the maximum grid current is:

- [math] I_{G} = I_{g} D_{f} \, [/math]
- [math] = 3.1 \times 1.1479 \, [/math]
- [math] = 3.559 \, [/math] kA

### Step 5: Touch and Step Potential Criteria

Based on the average weight of the workers on the site, a body weight of 70kg is assumed for the maximum touch and step potential. A maximum fault clearing time of 150ms is also assumed.

The maximum allowable touch potential is:

- [math] E_{touch,70} = \left( 1000 + 1.5C_{s} \rho_{s} \right) \frac{0.157}{\sqrt{t_{s}}} \,[/math]
- [math] = \left( 1000 + 1.5 \times 0.7207 \times 3,000 \right) \frac{0.157}{\sqrt{0.15}} \,[/math]
- [math] = 1,720.04 \, [/math] V

The maximum allowable step potential is:

- [math] E_{step,70} = \left( 1000 + 6C_{s} \rho_{s} \right) \frac{0.157}{\sqrt{t_{s}}} \,[/math]
- [math] = \left( 1000 + 6 \times 0.7207 \times 3,000 \right) \frac{0.157}{\sqrt{0.15}} \,[/math]
- [math] = 5,664.03 \, [/math] V

### Step 6: Ground Potential Rise (GPR)

The maximum ground potential rise is:

- [math] GPR = I_{G} R_{g}\, [/math]
- [math] = 3,559 \times 2.2753 \, [/math]
- [math] = 8,097 \, [/math] V

The GPR far exceeds the maximum allowable touch and step potentials, and further analysis of mesh and step voltages need to be performed.

### Step 7: Earthing Grid Design Verification

#### Mesh Voltage Calculation

The components of the geometric factor [math]n_{a} \,[/math], [math]n_{b} \,[/math], [math]n_{c} \,[/math] and [math]n_{d} \,[/math] for the rectangular grid are:

- [math] n_{a} = \frac{2L_{c}}{L_{p}} \, [/math]
- [math] = \frac{2 \times 890}{280} = 6.357 \, [/math]

- [math] n_{b} = \sqrt{\frac{L_{p}}{4\sqrt{A}}} \, [/math]
- [math] = \sqrt{\frac{280}{4\sqrt{4500}}} = 1.022 \, [/math]

- [math] n_{c} = n_{d} = 1 \, [/math]

Therefore the geometric factor [math]n \,[/math] is:

- [math] n = n_{a} \times n_{b} \times n_{c} \times n_{d} \, [/math]
- [math] = 6.357 \times 1.022 \times 1 \times 1 \, [/math]
- [math] = 6.4939 \, [/math]

The average spacing between parallel grid conductors [math] D \, [/math] is:

- [math] D = \frac{1}{2} \left( \frac{W_{g}}{n_{r} - 1} + \frac{L_{g}}{n_{c} - 1} \right) \, [/math]
- [math] = \frac{1}{2} \left( \frac{50}{6 - 1} + \frac{90}{7 - 1} \right) \, [/math]
- [math] = 12.5 \, [/math]

where [math] W_{g} \, [/math] and [math] L_{g} \, [/math] are the width and length of the grid respectively (e.g. 50m and 90m)

- [math] n_{r} \, [/math] and [math] n_{c} \, [/math] is the number of parallel rows and columns respectively (e.g. 6 and 7)

The geometric spacing factor [math] K_{m}\, [/math] is:

- [math] K_{m} = \frac{1}{2\pi} \left( \ln \left[ \frac{D^{2}}{16h \times d} + \frac{\left( D + 2h \right)^{2}}{8D \times d} - \frac{h}{4d} \right] + \frac{K_{ii}}{K_{h}} \ln \left[ \frac{8}{\pi (2n-1)} \right] \right) \,[/math]
- [math] K_{m} = \frac{1}{2\pi} \left( \ln \left[ \frac{12.5^{2}}{16 \times 0.6 \times 0.01236} + \frac{\left( 12.5 + 2 \times 0.6 \right)^{2}}{8 \times 12.5 \times 0.01236} - \frac{0.6}{4 \times 0.01236} \right] + \frac{1}{1.26} \ln \left[ \frac{8}{\pi (2 \times 6.4939 -1)} \right] \right) \,[/math]
- [math] = 0.964 \,[/math]

The irregularity factor [math] K_{i}\, [/math] is:

- [math] K_{i} = 0.644 + 0.148n \, [/math]
- [math] = 0.644 + 0.148 \times 6.4939 \, [/math]
- [math] = 1.605 \,[/math]

The effective buried length [math]L_{M} \,[/math] is:

- [math]L_{M} = L_{c} + \left[ 1.55 + 1.22 \left( \frac{L_{r}}{\sqrt{L_{x}^{2} + L_{y}^{2}}} \right) \right] L_{R} \,[/math]
- [math] = 890 + \left[ 1.55 + 1.22 \left( \frac{3}{\sqrt{90^{2} + 50^{2}}} \right) \right] \times 66 \,[/math]
- [math] = 994.65 \,[/math]m

Finally, the maximum mesh voltage is:

- [math] E_{m} = \frac{\rho_{s} K_{m} K_{i} I_{G}} {L_{M}} \, [/math]
- [math] = \frac{300 \times 0.964 \times 1.605 \times 3,559} {994.65} \, [/math]
- [math] = 1,661 \,[/math]V

The maximum allowable touch potential is 1,720V, which exceeds the mesh voltage calculated above and the earthing system passes the touch potential criteria (although it is quite marginal).

#### Step Voltage Calculation

The geometric spacing factor [math] K_{s}\, [/math] is:

- [math] K_{s} = \frac{1}{\pi} \left[ \frac{1}{2h} + \frac{1}{D+h} + \frac{1}{D} \left( 1 - 0.5^{n-2} \right) \right] \, [/math]
- [math] = \frac{1}{\pi} \left[ \frac{1}{2 \times 0.6} + \frac{1}{12.5 + 0.6} + \frac{1}{12.5} \left( 1 - 0.5^{6.4939 - 2} \right) \right] \, [/math]
- [math] = 0.314 \,[/math]

The effective buried length [math] L_{S}\, [/math] is:

- [math] L_{S} = 0.75L_{c} + 0.85L_{R} \, [/math]
- [math] = 0.75 \times 890 + 0.85 \times 66 \, [/math]
- [math] = 723.6 \,[/math] m

Finally, the maximum allowable step voltage is:

- [math] E_{s} = \frac{\rho_{s} K_{s} K_{i} I_{G}} {L_{S}} \, [/math]
- [math] = \frac{300 \times 0.314 \times 1.605 \times 3,559} {723.6} \, [/math]
- [math] = 743.6 \,[/math] V

The maximum allowable step potential is 5,664V, which exceeds the step voltage calculated above and the earthing system passes the step potential criteria. Having passed both touch and step potential criteria, we can conclude that the earthing system is safe.

## Computer Based Tools

As can be seen from above, touch and step potential calculations can be quite a tedious and laborious task, and one that could conceivably be done much quicker by a computer. Even IEEE Std 80 recommends the use of computer software to calculate grid resistances, and mesh and step voltages, and also to create potential gradient visualisations of the site.

Computer software packages can be used to assist in earthing grid design by modeling and simulation of different earthing grid configurations. The tools either come as standalone packages or plug-in modules to power system analysis software (such as PTW's GroundMat or ETAP's Ground Grid Design Assessment. Examples of standalone packages include SES Autogrid and SafeGrid.

## What next?

The minimum size for the earthing grid conductors can be used to specify the earthing grid conductor sizes in the material take-offs and earthing drawings. The touch and step potential calculations (where necessary) verify that the earthing grid design is safe for the worst earth faults to remote earth. The earthing drawings can therefore be approved for the next stage of reviews.