chen 2011

Finite-Difference Analysis of the Potential Distribution around an L-shape Building Base under

the Measurement of Grounding Resistance

Jiaqing Chen National Key Laboratory on

Electromagnetic Environmental Effects and Electro-optical Engineering PLA Univ. of Sci. & Tech.

Nanjing, China [email protected]

Wenchun Liao National Key Laboratory on


No.60, PLA 94701 Nanjing, China

[email protected]

Yingqiang Wang National Key Laboratory on



Xiangyu Liu National Key Laboratory on



Abstract—For the first time, a three-dimensional finite difference method 3D FDM is adopted to simulate the potential distribution during grounding resistance measurements around an L-shape building under different directions and electrodes arrangements. The results show that the distributions of zero potential region with different measurement points and measurement electrodes arrangements are strikingly different when using a straight-line-three-probes method. To obtain more accurate measurement results, scientific and suitable measurement point locations should be properly chosen for the grounding resistance measurement of L-shape buildings according to the surrounding conditions in practical measurements.

Keywords- large-scale irregular grounding system; grounding resistance measurement; potential distribution; three-dimensional finite difference method

I. INTRODUCTION The measurement of grounding resistance is one of the most

important questions in the test and acceptance of lightning protection projects. It is applied to determine whether lightning grounding systems meet standards and design requirements or not in practical lightning protection measurement; the value of impulse grounding resistance is calculated based on tested power-frequency grounding resistance with a certain ratio [1, 2]. Thus, how to measure grounding resistance accurately is a very important issue in the test and acceptance of lightning protection. The fall-of-potential method is widely used for grounding resistance measurements in the world, especially the 0.618 method which is derived from the straight-line-three-probes method [3-5]. However, this method is based on several ideal hypothesises: the grounding device, potential probe and current electrode are all assumed to be hemispherical; the terrain above the grounding device is flat and open; the soil is uniform around the grounding device [2-5].In practice, we often encounter irregular building base grounding systems such as L-shape buildings, and measuring electrodes are

usually claviform; both of them are far away from regular hemisphere. In this case, errors will be produced by using 0.618 method. It will cause greater errors if we measure the resistance of large-scale irregular grounding system with the methods used for small-scale regular grounding system without change.

Nowadays, research for accurate grounding resistance measurement mainly focuses on grounding resistance measurements of regular grounding systems in complex soil conditions, electrodes arranging distance and angle, multilayer soils and influence of underground metal object [6-9]. However, there are still fewer studies on the influence of irregular shape of building foundation on grounding resistance measurement.

Finite difference method is one of the earliest methods applied in numerical simulation calculation; it is prevalently applied in electromagnetic numerical calculation fields because its concepts and methods are clear, direct and simple [10].Three-dimensional finite difference method 3D FDMhas been widely used in exploration, seismology and other fields since the late 1980s [11]. However, it hasn’t been applied in grounding resistance measurements simulation research in the open literature. This paper adopts 3D FDM to simulate the measurement of grounding resistance of an L-shape building foundation. In this simulation, we can obtain the whole potential distribution around a large-scale irregular grounding system.

II. CALCULATION MODEL The steps of 3D FDM are as follows: (1) set certain

calculation space; (2) adopt proper discrete grids; (3) construct a difference format; (4) choose an appropriate algebra equation solution from the difference format and compile corresponding programs to work out the discrete solutions [11, 12]. The issue of power-frequency grounding resistence can be treated as a question of constant electric field. We take the potential φ in soil as variable and set φ as the potential of any

7th Asia-Pacific International Conference on Lightning, November 1-4, 2011, Chengdu, China

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2011

point except the grounding system and loop electrode; then, φ satisfies the Laplace equations:

2 2 20

2 2 2x y z

φ φ φ∂ ∂ ∂+ + =

∂ ∂ ∂ (1) As shown in Fig. 1, the grid partition is square 3D grid

nodes division in Cartesian coordinates in the model. Then, we can deduce the Jacobian iterative formula of formula (1) by Taylor expansion equations:

[

]

1( , , ) ( 1, , ) ( 1, , ) ( , 1, )6

( , 1, ) ( , , 1) ( , , 1)

i j k i j k i j k i j k

i j k i j k i j k

φ φ φ φ

φ φ φ

= − + + + −

+ + + − + + (2)

where, ( , , )i j kφ is the potential of node.

Figure 1. Three-dimensional finite difference grids division.

The successive over-relaxation method (SOR) is used to process the numerical calculation [10, 11].The iterative formula of SOR is:

( 1) ( ) ( 1) ( )

( 1) ( ) ( 1)

( ) ( )

ω( , , ) ( , , ) [ ( 1, , ) ( 1, , )6

( , 1, ) ( , 1, ) ( , , 1)( , , 1) 6 ( , , )]

n n n n

n n n

n n

i j k i j k i j k i j k

i j k i j k i j ki j k i j k

φ φ φ φ

φ φ φ

φ φ

+ +

+ +

= + − + +

+ − + + + −

+ + − (3) where, ( ) ( , , )n i j kφ and ( 1) ( , , )n i j kφ +

are the iterative solutions of No.n and No.n+1, respectively; ω is the relaxation factor and is set as ω≈1.94 in this study.

The finite difference space model of grounding resistance measurement is shown in Fig. 2. The grounding system, potential probe and loop electrode are represented by G, P and C, respectively; they are all along a straight line. The calculated region is limited in a three-dimensional space with 200 m long,200 m wide and 20 m high; its upper surface is the boundary between air and soil. According to the definition of resistance, the measured results will be the same whether we use voltage as encouraging source on grounding system to obtain corresponding current response or we inject current encouraging source to get corresponding voltage response. Because the grounding resistance of loop electrode is much larger than that of measured grounding system, we set the absolute potential of auxiliary loop electrode much higher than

that of measured grounding system. For instance, the potentials of G and C are set as 12 V and -120 V, respectively, to produce current in the soil between G and C. Then we can deduce the three-dimensional potential of any position in the calculation space. We approximately treat the steel as an ideal conductor and assume the whole base steel frame be buried in homogeneous soil with the soil resistivity of 50 mΩ⋅ . The grid size is 0.5 m × 0.5 m × 0.5 m. All the sides of the calculation space are fixed at zero value (the first boundary condition) except the upper side (i.e. the earth surface) whose normal potential derivative is set as zero (the second boundary condition).

Figure 2. Finite difference space model.

The L-shape building foundation and the measurement loop are shown in Fig. 3. The burial depth of the steel frame of the base is 0.5 m; the steel grid size is 0.5 m × 0.5 m × 0.5 m; the diameter of the steel bar is 10 mm and the size of its exterior frame is labeled in the figure.

Figure 3. Exterior steel frame of the L-shape building base.

Two different test points and four different loop electrode locations are shown in Fig. 4. In order to compare the maximum coupling with the minimum coupling between measurement loop and L-shape building base, in one case the inner concave corner G1 is set as the test point and the loop electrodes is located at from C1 to C3, respectively; in the other case the exterior convex corner G2 is set as the test point and the loop electrode is located at C4. The distances between the test points and the loop electrodes are given in the figure.

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2011

Figure 4. Top view of test point and loop electrode locations.

III. RESULTS AND DISCUSSION Figure 5 shows the potential distributions at earth surface

when the test points and loop electrode are at different positions in Fig. 4. The equipotential line closer to the building has a shape more similar to that of the L-shape building base; while the equipotential line far away from the building has a shape like an ellipse. Moreover, the distribution density of the equipotential lines near the building is high with a great potential gradient; the maximum of potential gradient appears near the exterior convex corner.This result is basically consistent with the conclusion of Mukhedkar et al about surface potential distribution around the miniatures of simple grounding systems under the potential measurement in experimental environment [13].

It can be seen in Fig. 5 that the potential near the green line is approximately equal to zero. In the straight-line-three-probes method, there is a blank region between the outermost green equipotential line A around the building base and the

(a)

X(m)

Y(m

)

25

100

17525 100 175

Y(m

)

X(m)(b)

25

100

17525 100 175

Y(m

)

X(m)(c)

25

100

17525 100 175

Y(m

)

X(m)(d)

25

100

17525 100 175

-120

-100

-80

-60

-40

-20

0

12 (V)

Figure 5. Earth potential distributions under four different electrode arrangements :( a) current loop G1-C1; (b) current loop G1-C2;(c) current loop G1-C3;(d) current loop G2-C4.

outermost green equipotential line B around the loop electrode along the straight-line direction; we name it as zero potential region. If the zero potential region is large, the measurement results will change slightly when the potential sampling probe moves in the zero potential region back and forth. Generally, larger zero potential region enables more accurate grounding resistance measurement.

When the inner concave corner of the L-shape building (point G1 in Fig. 4) is chosen as the test point, we compare Fig. 5 (a) with (b) and (c). It can be seen that the distributions of ground potential are different with the change of distance between the test point and the loop electrode. In Fig. 5 (a), the distance between G1 and C1 is 40 m and there is almost no zero potential region; this may bring an obvious change of

978-1-4577-1466-5/11/$26.00 ©2011 IEEE 484

2011

measured value when the potential sampling probe moves only a very short distance. However, the zero potential region in Fig. 5(b) becomes obvious when the distance between G1 and C2 increases to 80m. Furthermore, as shown in Fig. 5(c), the zero potential region is larger when the distance between G1 and C3 extends to 100 m. Thus, 40 m long loop lead can’t meet the requirement for an accurate measurement of grounding resistance; to make the zero potential region large enough, we have to lengthen the test lead.

As shown in Fig.4 and Fig. 5(d), the exterior convex corner of the L-shape building (point G2 in Fig. 4) is selected as the test point. The distance between G2 and C4 is also 40 m. The distribution of ground potential is strikingly different from the situation in Fig.5(a); although the loop lead length is still 40 m, we can obtain a certain zero potential region.

In other point of view, the zero potential region reflects the degree of overlapping between the effective resistance region of measured grounding system and that of loop electrode in soil. If the distance between loop electrode and measured grounding system is not far enough, the above two effective resistance regions will overlap each other; thus, the zero potential region becomes quite narrow and even almost doesn’t exist [1, 3, 4]. It can be shown that long measurement leads can produce obvious zero potential region, the choice of potential measurement points is relatively easy and the measurement error will be smaller. However, long measurement leads also cause difficult leads arrangement, increased signal disturbance and enhanced inductance coupling [4, 14]. Therefore, the theoretical analyses of both electromagnetic disturbance and the engineering achievement of field measurement show the preference of shorter measurement leads [14]. Furthermore, measurement instrument only provides a fixed measurement lead, thus we should take account of selecting the exterior convex corner of L-shape building base for measuring and locate the loop electrode at near C4 in Fig. 4. The effective resistance region around the measured grounding system is the narrowest along this direction.

IV. CONCLUSIONS In this study, the 3D FDM is adopted to study the potential

distribution during the grounding resistance measurement of an L-shape building foundation. From the calculation results, the following conclusions can be drawn:

(1) The equipotential line closer to the building has a shape more similar to that of the L-shape building base, and the maximum potential gradient appears at the exterior convex corner.

(2) When the straight-line-three-probes method is adopted and the length of loop electrode lead is fixed, the potential distribution on ground surface is changed with different test point and measurement direction, and the size of zero potential region is also different.

(3) When measuring the grounding resistance at the inner concave corner of the L-shape building foundation, it is difficult to obtain large enough zero potential region if the lead of loop electrode is short; the lead length has to be

increased to obtain accurate measurement result. However, the longer leads also bring some problems: leads arrangement difficulty, signal disturbance, inductance coupling and other problems. On the contrary, obvious and large enough zero potential region is relatively easier to be obtained if the exterior convex corner of L-shape building base is chosen as the measurement point.

(4) In practical measurements, we should give priority to choose the exterior convex corner instead of the inner concave corner of L-shape building base for measuring on account of following reasons: when the exterior convex corner is chosen for measuring, the produced zero potential region is larger, the location of potential sampling probe is relatively easier and the probability of producing measurement errors will be smaller.

In general, 3D FDM analysis makes the potential distribution around large-scale irregular grounding system become clear at a glance; the exact measurement directions and positions will be easily found and the more accurate measurement will be more easily obtained according to the potential distributions. The results presented in this study will serve as a strong directive significance in applying straight-line-three-probes method to measure grounding resistance of L-shape building in practical measurement.

REFERENCES [1] Yang Zhongjiang, Test and acceptance of lightning protection

engineering, Beijing, China: Meteorology Press, 2005. [2] Technical specifications for inspection of lightning protection system in

building, GB/T21431-2008, 2008. [3] IEEE Guide for safety in AC substation grounding, IEEE Std 80-2000. [4] G.F. Tagg “Measurement of earth-electrode resistance with particular

reference to earth-electrode systems covering a large area,” PROC. IEE, vol.3,pp. 2118~2130,Dec.1964

[5] Guide for measuring earth resistivity, ground impedance and earth surface potentials of a ground system Part : Normal measurements GB/T 17949.1-2000, 2000.

[6] Jinxi Ma, Farid P. Dawalibi, “Extended Analysis of Ground Impedance Measurement Using the Fall-of-Potential Method,” IEEE Transactions on Power Dilivery, vol.17,pp. 881-885,Apr.2002.

[7] Zhong Jianling, Wang Jingjing, Zhang Bo, et al, “Grounding resistance measurement by angle-offset method for urban substations,” Power System Technology, vol.34, pp. 34-37, Jun.2010.

[8] Zeng Rong, He Jinliang, Gao Yanqing, et al,“Effects of electrode placement on grounding resisitance of grounding system measured in vertical three-layer soil,” Journal of Tsinghua University (Science and Technology),vol.41,pp. 28-31,Mar.2001.

[9] He Jinliang, Yu Shaofeng, Zeng Rong, et al,“Influence of neighboring underground metal object on measurement of substation grounding resistance,” High Voltage Apparatus,vol.40,pp.1-2,Feb.2004.

[10] Ni GuangZheng, Yang Shiyou Qiu Jie. Engineering electromagnetic numerical calculation, 2nd ed, Beijing, China: Machinery Industry Press, 2010.

[11] Shi Fengfeng, “Study of magnetometric electric method based on 3D finite difference method,” A dissertation submitted to China University of Geosciences for master degree, pp.7-26,May.2010.

[12] Sefer Avdiaj ,Janez Setina, “Numerical solving of Poisson equation in 3D using finite difference method,” Journal of Engineering and Applied Sciences ,vol.5,pp.14-18,Jan.2010

[13] D. Mukhedkar, Y. Gervais, F. P. Dawalibi, “Modelling of potential distribution around a grounding electrode,”IEEE Transactions, vol.2, PP.1455-1459, May.1973.

[14] Rao Hong, “Influence of length of current electrode in grounding resistance measurement by fall-of-potential method,” Power System Technology, vol.30, pp.41-49, Apr.2006.

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2011

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