IEEE TRANSACTIONS ON MAGNETICS, VOL. 28, N0.2, MARCH 1992 1497

A New Model of the Vertical Ground Rod in Two-Layer Earth

P. D. RanCiC, L. V. StefanoviC Departments of Energetics and Mathematics

Faculty of Electronic Engineering P.O.Box 73, 18000 NiS, Yugoslavia

Abstract-A new system of integral equations (SIE) for current distribution function in a vertical ground rod in two-layer earth is developed in this paper. The system of Integral equations is numerically solved by the point match- ing method using polynomial approximation of current dis- tribution function. The supply zone is approximated by the electric fleld of a point source.

I. INTRODUCTION

A grounding system is most commonly made of vertical, horizontal, and inclined linear conductors. A nonhomogeneous earth structure is approximated by several homogeneous isotropic layers. The anal- yses of grounding system up to now have used the following mathematical model:

1) The source of the electric field is an unknown current density on the grounding electrode surface.

2) Due to the structure of conductor, a line cur- rent density (in A/m) has been introduced.

3) The structure of electric field and potential are determined by the analogy to electrostatic field.

4) The unknown line current density function is determined from the condition that the grounding conductor surface potential is constant.

51 To solve this Droblem several mathematical methids have been uied (see [l], [2], [5], [6], [7], [8],

This DaDer is the continuation of the research on [111, [121, ~ 3 1 ) .

A A

a new mathematical model of line groundings, given in [14] and [15]. The presented model is a modified mathematical model of linear antennas ([9], [lo]), ap- plied to the linear groundings analysis. This model is composed from the following:

1) The source of the electric field is an unknown current distribution (in A) localized along the axis of

Manuscript received July 7, 1991.

Dj. R. DjordjeviC Department of Computer Science

Faculty of Civil Engineering P.O.Box 115, 18000 NiS, Yugoslavia

the linear conductor ([lo], [14]). One can easily show that the negative first derivative of this current is exactly proportional to the line current density.

2) The structure of electric field is determined by integration of Maxwell’s equations. This structure is a special case of the structure determined in [lo].

3) The unknown current distribution function is determined by solving the system of integral equa- tions (SIE). SIE is obtained by satisfying the bound- ary condition that the resulting tangential electric field on the ideal conductor grounding surface is equal to zero. Using this boundary condition i t is possible to obtain several forms of SIE for the solution of the same problem ([9], [14]).

4) The supply zone is approximated by the elec- tric field of a point source as in [ 141 and [ 151.

5) Finally, SIE is numerically solved by the point matching method ([3]), using polynomial approxim& tion of the current distribution ([4]).

In this paper, the described mathematical model is applied to the analysis of the vertical ground rod in a two-layer earth ([ 151).

11. THEORETICAL BACKGROUND

A vertical straight ground rod of length 12 and circular cross section with radius a (a << l 2 ) is con- sidered. The groucnd rod is fed by power frequency current source of the r.m.s. value I, = 1A. The rod is made of ideal conducting material and located in nonhomogeneous earth approximated by two homo- geneous and isotropic layers (Fig. 1).

The elementary Hertz’s vector values, d nii= dII,ijf (i = 0 , 1 , 2 ; j = 1,2) , originating from the cur- rent element F e j (d) = I3(z f )dz’9, are derived as so- lutions of a system of Helmholtz’s differential equa- tions, which are reduced, due to l$[ M 0 (2; = j w p i s - complex propagation constant, gi = a i+ jws i = ai - complex conductivity), to a system of Laplace’s and

+

0018-9464/!22$03.00 0 1992 EEE

1498

Poisson’s differential equations of the form

A(dIT,;,) = -(l/ai)p,j(z’)6(T - ;‘)6;j , (1)

where 6(; - ;’) - Dirac’s space 6-function, T - vec- tor of point coordinates in which the z-component of Hertz’s vector is calculated, T‘ - vector of point coordinates in which the current element is located, 6;j - Kronecker’s symbol, i - index of the medium in which the Hertz’s vector is calculated, j - index of the medium in which the current element is located.

Cl’ C l ’ Pl

Fig. 1. Geometry of problem

By integrating (1) and determining the integra- tion constants on the basis of general boundary con- ditions, for z-components of Hertz vector in the first and the second medium, the following expressions are obtained ([15]):

+ ITA ( r ; 0 5 z 5 hl) = L l + K 1 2

9 3 ( 4 = e - a ( z - h l ) f l ( ~ ) J o ( ~ P o ) 7

g4(a) = [e -a(z -2hl ) - R201e-a(z+2hl) f 2 ( a ) I . The denotations introduced in (2) and (3) are as fol- lows: J o ( a p 0 ) - Bessel’s function, PO = d-,

&;-I,; = -&;,;-I = Tz;-l,; - 1 = -Tz;,;-l + 1 - - (Ei-1 - %)/(%-I + E;) , i = 1 , 2 ,

fo(ly) = eahl + ~ ~ O l ~ ~ l 2 e - ~ ~ 1 , f l ( ly ) = (ecrz’ + RzOle-az’) / fo(a) , f 2 ( l y ) = e-a(z ’ -h l ) / f o ( 4

(4)

(5)

(6)

(7) +

The electric field E; and the potential p; are cal- culated from the defining expressions E;= - grad pi + ---z 7?IT;, p;=-d ivI I ; fora=cr i , 132,21=O7andi=1,2.

Satisfying the total tangential electric field boun- dary condition on the ground rod surface,

-*

-.+ +

Ezi(2) + Ezez(2) = 0 PO = a 7 k - 1 I 2 5 4 (8)

for i = 1,2, and substituting (2)-(7) in the defining expressions for Ei, p;, and then in (8), SIE for cur- rents are obtained. Shifting the second derivative from the kernel to the current, one may finally ob- tain:

-*

1 (9) 13

+ 1 I ; ! ( Z ’ ) E j ( Z , 2’)dZ’ = -4SU;E,,,(z) 2 ’ d j - l

for i = 1,2, and where Ej(z , z ’ ) denotes the kernel values of the subintegral functions (terms in braces in (2) and (3)). E,,, denotes the electric field of the external source. The feeding zone (external source) is approximated by the electric field of the point source ([14], [15]). For the numerical solution of SIE, the following current conditions are to be used:

and 12(12) = 0. I, = Il(l0) = 1A, Il(l1) = IZ(l l ) , I:(ll) = (g l /U2)I; ( l l )>

The grounding resistance is defined in [14] as

Ruz = (PI + p e z ) / L 7 PO = a 7 z = 0 * (10)

111. NUMERICAL RESULTS

Equations (9) were numerically solved using the point matching method ([3]) with the polynomial ap- proximation of an unknown current distribution ([4]).

The validity of the given procedure is tested by numerical experiment. The grounding resistances in 0, for the case of a = 0.025m7 11 = 12 = 6m, hl = 10m and cr1 = u2 = O.OlS/m are given in TABLE I.

TABLE I GROUNDING RESISTANCE IN n AS A FUNCTION

OF DEGREE

MI I1 I11 IV V 1 16.376 12.633 16.376 15.995 2 16.572 13.556 15.883 15.886 S 16.833 14.309 15.779 15.692

4 16.636 14.601 15.672 16.659 6 16.676 14.806 16.638 15.602

10 15.479 15.166 15.627 15.523 20 16.460 15.318 15.472 15.471

R,,lsl = 15.95fI, RuZlp~ = 15.5n, EuElll = 16.376n

The convergence of the results obtained are repre- sented as a function of increasing degree Ml of an approximating polynomial, by using the condition I l ( l 1 ) = 0 (column 11), and without it (column 111). The values obtained by solving SIE which are formed by using the boundary conditions for the electric po- tential (column IV) and for the Hertz’s vector (col- umn V) are given also in TABLE I. The last row shows the comparison of the theoretical ([ 11, [6]) and exper- imental values ([2]).

On Fig. 2, the distribution of the line current density -Ii(z‘), for several degrees of the polynomial in the approximation of the current 4 , is shown. In order to compare, the corresponding measured values ([8]) are marked by the symbol ” 0 ” . The structure of grounding is: 11 = I2 = hl = 20m, a = 0.16m, p1 = pa = 100nm and I , = 300A.

Fig. 3 shows the variation of grounding resis- tance as a function of the rod length calculated us- ing (9) and (lo), and Ml = M2 = 2 for the approx- imating polynomial degrees. The quotient p 1 / p 2 is a parameter. In the same figure, the corresponding theoretical values taken from [12] are marked by the symbols ”+ * 0 ” .

The influence of the first layer depth, hl, and the resistivity of the second layer, p 2 = 1 /02 , on the grounding resistance (in n) are shown in TABLE 11. The values are obtained by solving SIE that are formed by using the boundary condition for poten- tial and for: 2, = 6m, a = 0.025m, p1 = l/al = 100nm, and, for hl < 6m, MI = 2, M2 = 3, and Ml = 3 for hl 2 6 m + 0.

IV. CONCLUSION

From the given theoretical procedure and numer- ical results one can conclude that the introduction of the current distribution along the grounding axis in- stead of the current density distribution flowing off

1499

from the grounding surface, a simple, reliable, and usable mathematical model of a linear grounding could be obtained. The model of this kind needs the determination of a grounding feed point in ad- vance and the corresponding mathematical model of the feeding zone. This conclusion is supported by the results presented and the numerical experiments and by a good agreement of theoretical and measured results given by other authors.

Generally, the method presented is the modified mathematical model of linear antennas ([9],[10]) app- lied to the linear ground rod in twdayer earth ([ 151).

24 -

22 - -

o o o Exp.[8]

I M1=ll

I

0 . 0 0 . 2 d. _i 0.6 0.a 1. _ / I

10 I

normalized z-axis

Fig. 2. Distribution of line current density on ground rod ( - I ~ ( z ‘ ) )

r- +

t-100

k 2a=0.0635m I I , , I I . * . . I . . . .

0 1 2 3 4 length of rod (m)

Fig. 3. Grounding resistance as a function of rod length

1500 TABLE I1

GROUNDING RESISTANCE IN fl AS A FUNCTION OF hi AND p2

o&ml hl Im]

1 2 3 4 5

6+ 0 10 60 100

0.1 0.017 0.022 0.027 0.039 0.072

13.194 14.523 15.595 15.669

50 8.509 9.983

10.025 10.651 12.400 14.741 15.284 15.703 15.734

2 00 100 15.809 27.732 15.759 24.678 15.702 22.197 15.637 20.169 15.571 18.454 15.779 17.138 15.779 16.465 15.779 15.887 15.779 15.844

500 1000 51.071 71.939 38.063 47.617 30.479 35.908 25.472 28.951 21.894 24.311 19.273 21.022 17.612 18.598 16.071 16.226 15.954 16.051

ACKNOWLEDGMENTS

The authors would like to thank to Faculty of Electronic Engeneering of University of NiS for its financial support given for this research activity.

REFERENCES

[l] H. B. Dwight, "Calculation of Resistances to Ground", AIEE Trans., pp. 1319-1328, Dec. 1936.

[2] E. Sunde, Earth Conduction Effects in Transmission Sys- tems, Dover Publ., New York, 1968.

[3] R. F . Harrington, Field Computation b y Moment Methods, Mc Millan, New York, 1968.

[4] B. D. Popovit, "Polynomial Approximation of Current Along Thin Symmetrical Cylindrical Dipoles", Proc. IEE, Vol. 117, NO. 5, pp. 873-878, 1970.

151 F . Dawalibi and D. Mukhedkar, "Optimum Design of Substation Grounding in a Two Layer Earth Structure",

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[9] B. D. Popovit, M. B. Dragovit and A. R. Djordjevit, Analysis and Syntesis of Wire Antennas, RSP, John Wiley & Sons LTD, Chichester, England, 1982.

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[ll] Z. Haenadar and S. Berberovit, "The Advanced Numer- ical Procedure for Analysis and Design of Earthing Sys- tems", Elektrotehnika ELTHBI, Vol. 29, No. 1-2, pp. 3-7, 1986. (in Serbian)

[12] J . Nahman and D. Salamon, "Interpretation of Ground Resistivity Data Obtained from Single Driven Rod Tests", Elektrotehnika ELTHBI, Vol. 29, No. 3, pp. 175- 180, 1986. (in Serbian)

[13] V. Bugsdorf and A. Yakobs, Zazemlyayushchie Ustroisfua Elektroustanouok, Energoatomizdat, Moskva, 1987. (in Russian)

[14] P. D. Rantif , " Quasistationary and Stationary Current Field of Linear Ground Electrodes", Proc. of XXxIII Yug. Conf. ETAN, Vol. V, pp. V.109-V.115, Novi Sad, June 1989. (in Serbian)

[15] P. D. Rant i t , "Vertical Ground Rod Electrode in Two- Layer Earth - Theoretical Approache", III Int. Symp. PES'SU, pp.145-151, Nix, October 1990.

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