Measurement of earth resistance of spread earth system using the slope method Previous publications have presented a solution to the problem related to the measurement of large ground system using an arbitrarily selected method of calculating this resistance for specifically calculated coefficients and three physical measurements of the earth resistance R1, R2, R3 at distances of 0.4, 0.6 and 0.8 respectively to the voltage probe from the tested object. =
∙
+
∙
+ ∙
where: −
ℎ
ℎ
, , − 0.4 , 0.6 , 0.8 , ℎ ℎ
, ℎ ℎ
ℎ .
ℎ ℎ
= −1,335;
:
= 3,041; = −0,7057
This method is correct, although not without uncertainty due to the lack of unambiguity in the proper determination of the distance to the current probe H. Additionally, at small distances to current probe H, high precision is required in determining the position of the voltage probe S. This study presents the slope method. It is the starting point for deriving the above formula. It enables testing of extensive earthing systems and evaluation of the correctness of the obtained results. The difference is in carrying out three measurements to determine the position of the voltage probe and physically carry out the actual measurement, not just to calculate its value from a formula. Introduction. Technical method The slope method, proposed by Dr. G. F. Tagg, is based on the analysis of the rate of changes in the slope of the earth resistance curve, which in turn allows to determine the appropriate earth resistance for large objects. Moreover, it is possible to verify the obtained results by repeating the measurement procedure for the changed positions of the current probe. In order to discuss the method, it was assumed that the considered earthing - for mathematical reasons - would be a halfsphere in the example.
Fig. 1 - Half-sphere grounding d – distance to the point in relation to which the earth resistance is calculated r – radius of the half-sphere
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We also assume that the ground has a uniform resistivity. In fact, this is virtually unheard of, but the ability to verify measurements largely reduces this inconvenience.
The theoretical ground resistance in this case will be:
=
→
In fact, the variable d - that is, the distance from earth to the current probe - has a certain, specific value. Additionally, we must also take into account one more parameter: the distance of the voltage probe p, which enables the measurement of the voltage drop across the tested earthing (Fig. 2). Therefore, the formula for the resistance will take the form:
=
Fig. 2 - Technical method. S and H auxiliary probes
In the case presented in Fig. 2, the optimal position of the probe S is determined by the formula:
=
∙
√5 − 1 ≅ 0,618 2
Assuming that the ground is electrically homogeneous, this position is actually independent of the distance d to the current probe. Measurement of concentrated earthing should not cause any problems if we use the above method. The situation changes when we consider the large earthing systems.
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Large Ground System. Slope method
Fig. 3 - Large half-spherical earthing
It is assumed that the measurement and determination of distances d and p will be made from the center of the hemisphere. In practice, however, such a procedure is not applicable. Measurements are made for any convenient point on the half-sphere, which introduces an error in determining the distances p and d. In fact, p will be (p + x), and d (d + x).
Fig 4. An example of a place from which we determine p and d
Thus, the resistance in such a system:
= = =
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1 2
2
− −
−
2
1 + 1 + 1 +
− − −
1 + 1 + 1 +
+ + +
1 − 1 − 1 − www.sonel.com
where: −
ℎ
−
,
ℎ
−
,
ℎ ℎ
−
ℎ
,
ℎ
−
,
−
ℎ
ℎ ℎ
ℎ ℎ
−
ℎ
ℎ
,
, ℎ
ℎ
ℎ
ℎ ℎ
ℎ
,
,
= ⁄2 −
ℎ
.
Determining the change in resistance for any point is difficult. On the other hand, it is quite easy to determine the speed of these changes. This can be done by measuring the resistance for three points of the voltage probe position - that is, three different p distances (0.2d, 0.4d and 0.6d) obtaining the values of R1, R2 and R3. The rate of change of resistance μ is calculated from the formula below. =
− −
The resistance for each distance p (0.2d, 0.4d and 0.6d) will be respectively:
=
−
1 +
−
1 0,2 +
+
1 0,8 −
=
−
1 +
−
1 0,4 +
+
1 0,6 −
=
−
1 +
−
1 0,6 +
+
1 0,4 −
Solving the system of the above three equations we get: (
−
)+
(
−
)+ ( where
−
)+(
−
)=
=
This relationship can be solved for α for any value of μ. The proper distance to the current probe d will be d(1+ α), and the distance to the voltage probe p, taking into account the previously established rule of 61.8%, will be 0.6180d(1+ α).
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The distance from the starting point for the actual earth resistance measurement pT will be: = 0,6180 ∙ (1 + ) − =
∙
∙ (0,6180 − 0,3820 ∙ )
therefore: = 0,6180 − 0,3820 ∙ We need to calculate and put in the table the values of the formula: (
−
)+
(
−
)+ (
for any value of μ using )+(
−
)=
−
Tables with calculated data are available in Appendix 1.
Slope method. Procedure 1. Choose a convenient place to connect the meter to the tested earthing (starting point). Place the current probe at a suitable distance d from this point. 2. Align the voltage probe. Perform measurements at the following distances from the tested earthing: 0.2d, 0.4d and 0.6d. 3. For each of these three positions we will get a resistance result R1, R2 and R3. . 4. Calculate the coefficient from the relationship = 5. Read the
value from the table (Appendix 1) for the calculated .
6. Calculate the pT value by multiplying the ratio
by .
7. Place the voltage probe within the calculated distance pT. 8. Carry out a resistance measurement. The result obtained in this way should be the true value (R∞) of the resistance of the tested earthing. 9. Verify the correctness of the measurement for other values d (distance to the current probe). 10. Correctly performed measurement should give convergent values of resistance R for different distances d.
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Slope method. Practical examples Example no. 1. Effective measurement without unforeseen difficulties The object on which the measurements were carried out was a small substation with dimensions of approx. 30.5 m x 24.5 m, surrounded by a fence connected to the earthing system. The earthing consists of an earthing net, earthing tape, etc. The current probe was placed at a distance of d = 116 m, then d = 91.5 m. The last measurement was made at a distance of d = 61 m. The place of connecting the meter to the ground is approx. 30 cm from the grid on the short side. Distance D M 61 91,5 116
Resistance measured for p=0,2d p=0,4d p=0,6d Ω Ω Ω 0,169 0,197 0,231 0,190 0,204 0,221 0,186 0,203 0,215
1,253 1,178 0,757
0,4788 0,4997 0,5877
pT
R
m 29,21 45,72 68,17
Ω 0,209 0,210 0,214
The average resistance of the three measurements is 0.211 Ω, so the possible error in relation to the average is approx. 1.4%. The measurement results can be considered correct. Example no. 2. Large substation, longer distances d required The object in this example is a larger station with dimensions of approx. 76 m x 91 m. The grounding infrastructure is as in the previous example, the conductive elements are fastened with a copper cable. The connection point of the meter is the middle of one side. Distance d m 122 183 244 308,5
Resistance measured for p=0,2d p=0,4d p=0,6d Ω Ω Ω 0,120 0,183 0,258 0,098 0,152 0,223 0,090 0,133 0,184 0,776 0,121 0,157
pT 1,183 1,338 1,212 0,849
0,4985 0,4522 0,4907 0,5712
m 60,82 82,75 119,73 176,22
R Ω 0,215 0,166 0,152 0,151
Changes in the earthing resistance can be observed: the value decreases with increasing distance d to the current probe. However, they are asymptotically up to 0.150 Ω. The results of the measurements made for the current probe at the distances of 122 m and 183 m are noticeably low. Thus, the method itself also allows us to determine whether the chosen distances d are appropriate. The above procedure allows to determine the proper grounding resistance of the tested station. In the discussed case, however, it required a certain commitment and time to properly perform the study.
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Chart 1 shows the distribution of resistance for individual tests. It confirms the correctness of the conclusion that the resistance of this object is approximately 0.150 Ω. It should be noted that in the discussed cases the soil resistivity was very low, and the earthing values were of considerable size. Therefore, the resistance values turned out to be small.
Graph 1. Change of resistance as a function of distance to the current court d
Summary The maximum possible value of the coefficient µ is 2. If the measurements result in resistances greater than 2, then the conditions do not correspond to those on which this method is based - it is necessary to increase the distance d to the current probe. However, practice shows that the coefficients should be limited to the value of µ: 0.4 <µ <1.6. The slope method is therefore satisfactory. It can be successfully used to check the earthing resistance of objects with extensive earthing systems, which was demonstrated in the presented two practical cases. The results also meet expectations when the ground is heterogeneous. The method itself will make it possible to determine when the distance d to the current probe is sufficient. Literature and reference materials: 1.
G. F. Tagg , Measurement of the resistance of physically large earth-electrode systems, PROC. IEE, Vol. 117, No. 11, NOVEMBER 1970.
2.
G. F. Tagg, Earth resistances. New York, Pitman Pub. Corp. [1964]
3. 4.
Roman Domański, Jacek Osiecki – Metoda techniczna pomiaru rezystancji uziemienia. Elektro Info nr. 11/2019 (179) Own materials of SONEL S.A
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