Faculty of Electrical Engineering

University of Ljubljana

Distribution and industrial networks

Laboratory for lighting and photometry

Seminar paper

Earth resistance and calculation of earthing

Author: Supervisor:

Amar Bičo prof. dr. Grega Bizjak

Ljubljana, March, 2019.

Contents 1. INTRODUCTION .................................................................................................................................................7

2. EARTH RESISTANCE........................................................................................................................................8

2.1. Earthing materials ................................................................................................................................... 10

2.1.1. Conductors .............................................................................................................................................. 10

2.1.2. Earth electrode ..................................................................................................................................... 10

2.2. Measurements of soil resistivity ..................................................................................................... 10

2.2.1. Wenner method.................................................................................................................................... 11

2.2.2. Sclumberger method.......................................................................................................................... 12

2.3. Types of ground electrodes ................................................................................................................ 13

2.3.1. Driven rod ............................................................................................................................................... 13

2.3.2. Grounding plates.................................................................................................................................. 14

2.3.3. Grounding strip .................................................................................................................................... 15

2.3.4. Mesh grounding.................................................................................................................................... 16

2.3.5. Semi spherical grounding electrode ........................................................................................... 17

2.3.6. Foundation earth electrodes .......................................................................................................... 17

2.3.7. Buried ring .............................................................................................................................................. 18

2.4. Measuring earth resistance on a existing earth electrode ............................................. 19

2.4.1. Earth resistance measurements on installations with a single earth electrode ... 20

2.4.2. 3-pole measurement method (62 % method) ....................................................................... 20

2.4.3. The triangle measurement method ............................................................................................ 21

2.4.4. 4-pole earth resistance measurements method ................................................................... 22

2.4.5. The variant 62 % method (one stake, only on TT or impedant systems) ................ 23

2.4.6. Selective 4-pole earth resistance measurement................................................................... 24

3. Addition ................................................................................................................................................................... 25

3.1. Questions .......................................................................................................................................................... 25

3.2. Example............................................................................................................................................................. 26

List of figures Figure 1. Soil resistivity in function of moisture. ..........................................................................................9

Figure 2. Wenner method...................................................................................................................................... 11

Figure 3. Sclumberger method. .......................................................................................................................... 12

Figure 4. Driven rod. ................................................................................................................................................ 13

Figure 5. Grounding plate. .................................................................................................................................... 14

Figure 6. Grounding strip. .................................................................................................................................... 15

Figure 7. Grounding with mesh. ........................................................................................................................ 16

Figure 8. Semi spherical electrode. ................................................................................................................... 17

Figure 9. Foundation earth electrode. ............................................................................................................. 18

Figure 10. Buried ring around house. ............................................................................................................. 19

Figure 11. Measurements with single earth electrode. ............................................................................ 20

Figure 12. 3-pole measurement method. ........................................................................................................ 21

Figure 13. Triangle measurement method. ................................................................................................... 22

Figure 14. 4-pole measurement method. ........................................................................................................ 22

Figure 15. The variatn 62% method (one stake). ...................................................................................... 23

Figure 16. Selective 4-pole earth resistance measurement. ................................................................... 24

1. INTRODUCTION

Nothing is quite so common or abundantly available throughout the world as the earth’s soil. We

are more apt to think of earth as something to be tilled for planting or to be excavated for a building

foundation. Yet, it also has an electrical property - conductivity (or low resistance) - that is put to

practical use every day in industrial plants and utilities.

Broadly speaking, “earth resistance” is the resistance of soil to the passage of electric current.

Actually, the earth is a relatively poor conductor of electricity compared to normal conductors like

copper wire. But, if the area of a path for current is large enough, resistance can be quite low and

the earth can be a good conductor. It is the earth’s abundance and availability that make it an

indispensable component of a properly functioning electrical system.

2. EARTH RESISTANCE

The resistance offered by the earth electrode to the flow of current into the ground is known as the

earth resistance or resistance to earth. The earth resistance mainly implies the resistance between

the electrode and the point of zero potential. Numerically, it is equal to the ratio of the potential of

the earth electrode to the current dissipated by it.

The resistance of the earthing electrode is not concentrated at one point, but it is distributed over

the soil around the electrode. Mathematically, the earth resistance is given as the ratio of the

voltage and the current shown below at (1) and (2):

𝐸𝐴𝑅𝑇𝐻 𝑅𝐸𝑆𝐼𝑆𝑇𝐴𝑁𝐶𝐸 =𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑡𝑜 𝑒𝑎𝑟𝑡ℎ 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑑𝑒

𝐶𝑢𝑟𝑟𝑒𝑛𝑡

(1)

𝑅𝐸𝑎𝑟𝑡ℎ(𝛺) =𝑉 (𝑉)

𝐼 (𝐴)

(2)

For correct design and construction of a grounding system it is necessary to know the properties

of soil on which is a protected substation or device.

The value of earth resistance for different power stations is shown below in Table 1:

Table 1. Value of earth resistance for different power station.

Large Power Station 0,5 (𝛺)

Major Power Station 1,0 (𝛺)

Small Substation 2,0 (𝛺)

The region around the earth in which the electrode is driven is known as the resistance area or

potential area of the ground. The fault current which is injected from the earth electrode is passing

away from the electrode in all directions shown below in the figure. The flow of current into the

grounds depends on the resistivity of the soil in which the earth electrode is placed. The resistivity

of the soil may vary from 1 to 1000 𝛺𝑚 depends on the nature of the soil. Specific soil resistivity

𝜌 is in function of geological properties of soil (moisture, chemical composition of soil). The

specific soil resistivity in function of moisture is shown below on Figure 1.

Figure 1. Soil resistivity in function of moisture.

Conduct of soil is changing for months, if temperature drops below the 0°C there is a chance of

freezing. In that case when the soil is frozen then the soil resistance will increasing. If we go deeper

with ground electrode in earth the changes of soil resistivity are less. Problem is there because the

soil is not uniform in all direction. Soil resistivity for different ground type is given below in Table

2:

Table 2. Typical ground resistivity values.

Ground type Resistivity (𝜴𝒎)

Loams, garden soils, etc. 5-50

Clays 10-100

Chalk 30-100

Clay, sand and gravel mixture 40-250

Marsh, peat 150-300

Sand 250-500

Slates and shales 300-3000

Rock 1000-10 000

Soil resistivity (ρ) is expressed in Ohm metres (Ω.m). This corresponds to the theoretical resistance

in Ohms of a cylinder of earth with a cross-section area of 1 m2 and a length of 1 m. By measuring

it, you can find out how well the soil conducts electric currents. So the lower the resistivity, the

lower the earth electrode resistance required at that location. Resistivity varies significantly

according to the region and the type of soil because it depends on the level of humidity and the

temperature (frost or drought increase it). This is why earth resistance may vary according to the

season or the measurement conditions. As temperature and humidity levels become more stable

the further you go from the ground surface, the deeper the earthing system, the less sensitive it is

to environmental variations. It is advisable to bury your earth electrode as deep as possible.

2.1. Earthing materials

2.1.1. Conductors

The conductors used must be capable of carrying the anticipated fault current and cope with

corrosion over the lifetime of the installation. Bare copper is normally used for a substation

earthing grid, being buried at depths between 0.6 and 1.0 m in rectangles of between 3 and 7 m

side length. Equipment connections are generally laid at a shallower depth of about 0.2 m. Because

of mechanical and thermal criteria, it is unusual for copper of less than 70mm2 to be used.

Aluminium is often used for above ground connections and could be used below ground if it is

certain that the soil will not cause corrosion problems, but most standards prohibit this. Some

protection, such as painting with bitumastic paint, is recommended in the area where the conductor

emerges from the ground, as corrosion may occur here and just below. Where not connected

directly to the electrode, all metallic substation plant is bonded via above ground conductors.

2.1.2. Earth electrode

The earth grid’s horizontal electrodes may be supplemented by vertical rods to assist the

dissipation of earth fault current, further reduce the overall substation earthing resistance and

provide some stability against seasonal changes. This is especially useful for small area substation

sites (such as GIS substations) or where the rod is of sufficient length to enter the water table. Rods

may be of solid copper or copper clad steel and are usually of 1.2 m length with screw threads and

joints for connecting together in order to obtain the required length for installation in the soil. The

formula for the effective resistance, 𝑅𝑟𝑜𝑑𝑠(𝛺) of a single earth rod is shown below (3):

𝑅𝑟𝑜𝑑𝑠 =𝜌

2𝜋𝑙· [ln (

8𝑙

𝑑) − 1]

(3)

Where is:

𝑅𝑟𝑜𝑑𝑠 – vertical earthing rod effective resistance (𝛺)

𝜌 – resistivity of soil (𝛺𝑚)

𝑙 – length of earthing rods (𝑚)

𝑑 −diameter of earthing rods (𝑚)

2.2. Measurements of soil resistivity

Because soil quality may vary greatly with depth and over a wide lateral area, estimation of soil

resistivity based on soil classification provide only a rough approximation. Actual resistivity

measurements are required to fully qualify the resistivity and its effects on the overall transmission

system.

2.2.1. Wenner method

This method known as Wenner four-pin method, developed by Dr. Frank Wenner. On these

methods used four electrodes, two for current injection and two for voltage measurement. The

four electrodes embedded to the ground in straight line, the two outer electrodes are current

electrode and two inner electrodes to measure voltage drop due to resistance of soil path when

current passed between the outer electrodes. The arrangement of Wenner method is shown on

Figure 2.

Figure 2. Wenner method.

The Wenner four-pin method, as shown in figure above, is the most commonly used

technique for soil resistivity measurements. Using the Wenner method, the apparent soil

resistivity value is:

𝜌 =4 · 𝜋 · 𝑎 · 𝑅𝑤

1 +2 · 𝑎

√𝑎2 + 4 · 𝑏2−

𝑎

√𝑎2 + 𝑏2

(4)

Where is:

𝜌 = measured apparent soil resistivity (𝛺𝑚)

𝑎 = electrode spacing (𝑚)

𝑏 = depth of the electrodes (𝑚)

𝑅𝑤 = Wenner resistance measures as (“𝑉

𝐼”) in figure 2, if b is small compared to a, the

previous equation can be reduced to (5):

𝜌 = 2 · 𝜋 · 𝑎 · 𝑅𝑤 (5)

2.2.2. Sclumberger method

In the Schlumberger method the distance between the voltages probe is a and the distances

from voltages probe and currents probe are c (Figure 3.), that’s the difference between this

method and Wanner method.

Figure 3. Sclumberger method.

Using the Schlumberger method, if b is small compered to a and c, and c>2a, the apparent

soil resistivity value is (6):

𝜌 =𝜋 · 𝑐 · (𝑐 + 𝑎)

𝑎· 𝑅𝑠

(6)

Where is:

𝜌 = measured apparent soil resistance (𝛺𝑚)

𝑎 = electrode spacing (𝑚)

𝑏 = depth of the electrodes (𝑚)

𝑐 = electrode spacing (𝑚)

𝑅𝑠 = Schlumberger resistance measures as (“𝑉

𝐼”), (𝛺)

2.3. Types of ground electrodes

2.3.1. Driven rod

This is by far the most common grounding device used in the field today. Driven rods

are vertically buried into the ground, their length is form 1 up to 3 meters. The top of

the driven rods must be buried under the surface of earth as much how soil is freezing

because the temperature has big impact on earth resistance. Driven rods also have a very small surface area, which is not always conducive to good contact with the soil.

This is especially true in rocky soils, in which the rod will only make contact on the

edges of the surrounding rock.

Figure 4. Driven rod.

The resistance of driven rod we can calculate using formula (7):

𝑅𝑑𝑟 =𝜌

2𝜋𝑙[𝑙𝑛

2𝑙

𝑑+

1

2ln (

4𝑡 + 1

4𝑡 − 1)]

(7)

Where is:

𝑅𝑑𝑟 = earth resistance of driven rod (Ω)

𝑙 = length of driven rod (𝑚)

𝑑 = diameter of driven rod (𝑚)

𝜌 = soil resistance (Ω𝑚)

𝑡 = depth from surface to middle driven rod (𝑚)

2.3.2. Grounding plates

Grounding plates are typically thin copper plates buried in direct contact with the earth

(Figure 5).

Figure 5. Grounding plate.

Grounding plate are buried vertically into the ground to achieve better contact with the soil,

because soil can be frizzed grounding plates are buried into the soil on depth of 0,5 m. The

earth resistance of grounding plates we can calculate using (8):

𝑅 =𝜌

4√

𝜋

𝐴

(8)

Where is:

𝑅 = earth resistance of ground plate (Ω)

𝐴 = surface of grounding plate (𝑚2)

𝜌 = soil resistance (Ω𝑚)

In case that we using more then one grounding plate, it is recommending that the space

between grounding plates are 3 m, in that case the earth resistance we can calculate using

(9):

𝑅𝑡 = 0.23 ·𝜌

𝑎 · 𝑛· 𝑘 · µ

(9)

Where is:

𝑅𝑡 = total earth resistance of grounding plates (Ω)

𝑎 = page length of grounding plate (𝑚)

𝑘 = correction factor (depth of soil)

𝜌 = soil resistance (Ω𝑚)

𝑛 = number of grounding plates

µ = coefficient of interaction between the grounding plates

2.3.3. Grounding strip

The grounding strips are horizontally buried into the ground in depth of 0,5 m up to 1 m,

the strip is shown on Figure 6:

Figure 6. Grounding strip.

The resistance of grounding strip with condition that is length of strip is much longer then

the depth on which is buried grounding strip (L >> h) is shown below (10):

𝑅 =𝐾 · 𝜌

2 · 𝜋 · 𝑙ln (

𝐿2

𝑑 · ℎ)

(10)

Where is:

𝑅 = earth resistance of grounding strip (Ω)

𝐿 = length of grounding strip (𝑚)

𝑑 = diameter of grounding strip (𝑚)

𝜌 = soil resistance (Ω𝑚)

ℎ = depth of grounding strip (𝑚)

𝐾 = correction factor (1 – 1,5)

In case that is non circular it is necessary to calculate the value of diameter using formula

(11):

𝑑 = √4 · 𝐴

𝜋

(11)

𝐴 = cross section area of grounding strip (𝑚2)

2.3.4. Mesh grounding

Mesh type of grounding is most commonly using for protection of transformer stations with high voltage with directly grounded neutral point (Figure 7).

Figure 7. Grounding with mesh.

Simplified formula for calculation earth resistance in this case (12):

𝑅 =0,55 · 𝜌

√𝐴

(12)

A = area of mesh (𝑚2); 𝜌 = soil resistivity (Ω𝑚)

2.3.5. Semi spherical grounding electrode

Semi spherical type of grounding electrode is rarely used (Figure 8), but we can use

them for describing foundation earth electrodes like most commonly used type of

grounding electrode.

Figure 8. Semi spherical electrode.

The resistance for semi spherical grounding electrode is (13):

𝑅 =𝜌

2 · 𝜋 · 𝑟0 (13)

2.3.6. Foundation earth electrodes

With a foundation earth electrode, a functioning and maintenance-free earth-termination

system is installed throughout the building’s lifecycle. Foundation earth electrodes are

embedded in the concrete foundation and covered by a concrete layer of at least 5 cm.

Consequently, two requirements are fulfilled:

The concrete conserves the earthing material, corrosion effects are not to be expected

The typically moist concrete on the outside of the foundation establishes a

conductive connection between the systems mentioned above and the ground

Figure 9. Foundation earth electrode.

Earth resistance of foundation earth electrode is (14):

𝑅 =𝜌

𝜋 · 𝑑 (14)

Where is:

𝜌 = soil resistivity (Ω𝑚), diameter d we can calculate 𝑑 = 1,57 · √𝑉3

, here V is volume of foundation (𝑚3).

2.3.7. Buried ring

This solution is strongly recommended, particularly in the case of a new building.

The electrode should be buried around the perimeter of the excavation made for the foundations.

It is important that the bare conductor be in intimate contact with the soil (and not placed in the

gravel or aggregate hard-core, often forming a base for concrete). At least four (widely-spaced)

vertically arranged conductors from the electrode should be provided for the installation

connections and, where possible, any reinforcing rods in concrete work should be connected to the

electrode. The conductor forming the earth electrode, particularly when it is laid in an excavation

for foundations, must be in the earth, at least 50 cm below the hard-core or aggregate base for the

concrete foundation. Neither the electrode nor the vertical rising conductors to the ground floor,

should ever be in contact with the foundation concrete.

Figure 10. Buried ring around house.

The approximate resistance R of the electrode (15):

𝑅 =2 · 𝜌

𝐿

(15)

Where is:

L = length of conductors (𝑚)

𝜌 = resistivity of soil (Ω𝑚)

2.4. Measuring earth resistance on a existing earth electrode

The soil resistivity measurement methods presented so far can only be used when installing a new

earth electrode: they can be used to check the resistance value in advance and adjust the electrode

according to the earth value required. For existing earth electrodes, the method involves checking

that they comply with the safety standards in terms of their construction and resistance value.

Various measurement methods may be used, however, depending on the installation's

characteristics: whether it is possible to cut off the installation's power supply or disconnect the

earth electrode, whether the electrode to be tested is the only one or is connected to others, what

level of measurement accuracy is required, where the installation is located (urban or rural

environment).

The region around the earth in which the electrode is driven is known as the resistance area or

potential area of the ground. The fault current which is injected from the earth electrode is passing

away from the electrode in all directions. The flow of current into the grounds depends on the

resistivity of the soil in which the earth electrode is placed. The resistivity of the soil may vary

from 1 to 1000 𝛺𝑚 depends on the nature of the soil. Specific soil resistivity 𝜌 is in function of

geological properties of soil (moisture, chemical composition of soil). Characteristic values of

earth resistance are: 0.5𝛺 (Larger station), 1,0𝛺 (Major station), 2,0𝛺 (Small station).

2.4.1. Earth resistance measurements on installations with a single earth electrode

It is important to point out that the earth resistance measurement of reference is the 2-stake method.

This method is referenced in all the electrical installation testing standards and can be used to

measure the earth resistance both accurately and safely. The measurement principle involves using

an appropriate generator G to inject an alternating current (i) through the auxiliary electrode H and

back through the earth electrode E.

The voltage V between the earth electrode E and the point in the earth where the potential is zero

is measured using another auxiliary electrode S. The resistance can then be calculated by dividing

the voltage measured by the constant current injected (i), thus (16):

𝑅𝐸 =𝑈𝐸𝑆

𝐼𝐸𝐻

(16)

Figure 11. Measurements with single earth electrode.

2.4.2. 3-pole measurement method (62 % method)

This method requires the use of two auxiliary electrodes (or "stakes") to inject the current and to

provide the 0 V potential reference. The positioning of the two auxiliary electrodes in relation to

the earth electrode to be measured E(X) is crucial.

For correct measurement, the "0 V potential auxiliary electrode" must not be set up in the zones of

influence of the earths E & H caused by the current (i) flowing.

Statistics from the field have shown that the best method for ensuring high measurement accuracy

is to place stake S at a position 62 % of the distance from E on the straight line EH. You then need

to make sure that the measurement does not vary or only varies slightly when stake S is moved by

± 10 % (S’ and S”) on either side of its initial position on the line EH. If the measurement does

vary, it means that (S) is in an influence zone, so you must increase the distances and then repeat

the measurements.

Figure 12. 3-pole measurement method.

2.4.3. The triangle measurement method

This method requires two auxiliary electrodes (stakes). It is used when the method described above

(Figure 11) is not suitable (alignment not possible or obstacle preventing a sufficient distance from

H).

It involves:

Setting up the stakes S and H so that the earth electrode E and the stakes S and H form an

equilateral triangle

First measuring with S on one side and then measuring with S on the other side.

If the values found differ significantly, it means stake S is in a zone of influence. You must then

increase the distances and repeat the measurements. If the values obtained are within a few percent

of one another, the measurement can be considered valid. The results of this method may be

uncertain, however, because even when the values found are similar, the zones of influence may

overlap. To make sure, repeat the measurements after increasing the distances.

Figure 13. Triangle measurement method.

2.4.4. 4-pole earth resistance measurements method

The 4-pole earth resistance measurement method is based on the same principle as 3-pole

measurement, but with an additional connection between the earth to be measured E and the

measurement instrument. This method offers better resolution (10 times better than the 3-pole

method) and means that the resistance of the measurement leads no longer needs to be taken into

account. This function is ideal for measuring very low earth resistance values, so it is particularly

prized by power transmission and distribution companies who need to measure earth resistance

values of just a few Ohms.

Figure 14. 4-pole measurement method.

The advantage of 3-pole and 4-pole earth resistance measurements is that they can be performed

on an installation with the power off, so the earth can be tested even if the house or building

involved has not yet been connected to the power distribution network or has been disconnected

from it.

For these two types of measurement, you are advised to open the earth bar in order to isolate the

earth electrode to be measured, thus making sure that the earth resistance measured really is the

resistance of the earth electrode. Otherwise, there may be a de facto bond between the earthing

installation and an earth electrode due, for example, to the metal ducts of a water or gas distribution

network. Earth resistance measurements with the bar closed will then be incorrect due to the

presence of this de facto earth electrode. This may lead to an excessively high earth resistance

value later on (if a metal duct is replaced with an insulating material, for example). Consequently,

unless you are sure that there is no de facto earth electrode, you must open the earth bar for any

earth resistance measurements. To detect any de facto earth electrodes, it may be useful to measure

the earth electrodes with the bar open and with the bar closed so that you can check whether the

"closed-bar" value is due to the installed earth electrode or to de facto

earth electrodes.

2.4.5. The variant 62 % method (one stake, only on TT or impedant systems)

This method does not require disconnection of the earth bar and only one auxiliary stake (S) is

necessary. With this method, the earthing system of the distribution transformer acts as the H stake

and the PE conductor accessible on the protective conductor (or earth bar) acts as the E stake.

Figure 15. The variatn 62% method (one stake).

The measurement principle is the same as for the normal 62 % method. The S stake will be

positioned so that the distance S-E is equal to 62 % of the total distance (distance between E and

H). As a result, S will normally be located in the neutral "0 V reference earth" zone. The earth

resistance is calculated by dividing the measured voltage by the current injected.

Differences compared with the normal 62 % method:

The power supply for measurement comes from the mains instead of from batteries.

A single auxiliary stake is required (stake S) so the measurement can be set up more

quickly.

It is not necessary to disconnect the building’s earth bar. This also saves time and makes

sure that safety is maintained on the installation during measurement.

2.4.6. Selective 4-pole earth resistance measurement

When a classic 3-pole or 4-pole measurement method is used on a system with parallel earthing,

the measurement current injected into the system is divided between the different earths. This

means it is impossible to determine the amount of current in a given earth electrode, so its

resistance cannot be determined either. In such cases, it is the total current flowing in the earthing

system which is measured, giving the overall earth resistance equivalent to the resistances of all

the earth electrodes set up in parallel.

To neutralize the influence of the parallel earth electrodes, there is a selective variant of the 4-pole

measurement method. The principle is the same except that a current clamp is added to measure

the exact current flowing in the earth to be measured, so that its precise value can be determined.

Due to the use of auxiliary stakes and more particularly the 0 V reference with the S stake, this

method ensures accurate measurement of the earth resistance.

Figure 16. Selective 4-pole earth resistance measurement.

3. Addition

3.1. Questions

1.) What is earth resistance and values for different types of power station ?

The resistance offered by the earth electrode to the flow of current into the ground is known as the

earth resistance or resistance to earth. The earth resistance mainly implies the resistance between

the electrode and the point of zero potential. Numerically, it is equal to the ratio of the potential of

the earth electrode to the current dissipated by it. For correct design and construction of a

grounding system it is necessary to know the properties of soil on which is a protected substation

or device.

2.) Describe earth resistance measurements on installations with a single earth electrode?

It is important to point out that the earth resistance measurement of reference is the 2-stake method.

This method is referenced in all the electrical installation testing standards and can be used to

measure the earth resistance both accurately and safely.

The measurement principle involves using an appropriate generator G to inject an alternating

current (i) through the auxiliary electrode H and back through the earth electrode E.

The voltage V between the earth electrode E and the point in the earth where the potential is zero

is measured using another auxiliary electrode S.

3.) Describe the Wenner method for measuring soil resistivity ?

This method known as Wenner four-pin method, developed by Dr. Frank Wenner. On these

methods used four electrodes, two for current injection and two for voltage measurement. The

four electrodes embedded to the ground in straight line, the two outer electrodes are current

electrode and two inner electrodes to measure voltage drop due to resistance of soil path when

current passed between the outer electrodes.

3.2. Example

Calculate earth resistance for grounding electrodes:

- Length of rod: l = 4,5 m

- Specific earth resistance 𝜌 = 250 Ωm

- Diameter d = 25,4 mm

- n (number of rods)

a) Configuration 1:

𝑅 =𝜌

2𝜋𝑙𝑙𝑛

4𝑙

𝑑=

250

2 · 𝜋 · 4,5𝑙𝑛

4 · 4,5

25,5 · 10−3= 58,03 (Ω)

b) Configuration 2: two rods with space between of 2m

𝑅 =𝜌

2𝜋𝑙𝑙𝑛

4𝑙

𝐴

𝐴 = 2 [𝑑

2· 𝑘𝑛−1]

1𝑛

= 0,7407

𝑅 = 22,9 (Ω)

c) Configuration 3: four rods with space between of 2m

𝑅 =𝜌

2𝜋𝑙𝑙𝑛

4𝑙

𝐴

𝐴 = 2 [𝑑

2· 𝑘𝑛−1]

1𝑛

= 1,129

𝑅 = 24,48 (Ω)