CALCULATING RESISTANCE OF SIMPLE GROUNDING FORMS WITH OR WITHOUT

THE SOIL IMPROVED CHEMICAL SUBSTANCE

Chuong Ho Van Nhat

Faculty of Electrical and Electronics Engineering HCM City University of Technology, Vietnam

Abstract.- Solution of chemical substance application for

decreasing resistance of grounding system is a current problem that is interested by scientists [1]-[2]. In present time, some of companies introduced and used the chemical substances for improving soil and enclosed the software for calculating the experimental grounding resistance but there is not any theoretic document that explains it. This article suggested some common formulas for calculating resistance of simple grounding forms with or without the soil improved chemical substance. Calculating results that were compared with results of GEM software of ERICO Company [3] were realized.

1. CALCULATING METHOD 1.1 Grounding Resistance of single vertical rod 1.1.1 Soil improved chemical substance layer of cylinder form

Single vertical rod with diameter d , length l and

resistivity ρ3 is fixed in the ground with depth t , surround it the chemical environment with resistivity ρ1 and the soil environment with resistivity ρ2 (see Fig .1)

Choosing coordinates system as in Fig.1 and considering

the potential at point (x,z), according to [4], we have

∫ +=

224),(

xzldzIxz

πρϕ (1)

Where I is current flows through the rod, ρ is resistivity

of environment at point (x, z) and l is the length of rod. For simplifying the calculation, we can use the reflect

method and the soil is a uniformed environment with resistivity ρ. So, the potential along the rod is calculated as follows:

Fig .1 Vertical section of grounding rod and its image in uniformed environment

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛=

+== ∫ ∫∫−

+

−

+

−

xtArsh

xtArsh

xlArsh

lI

xzdxzdxzdxl

l

t

t

lt

l

2/122/122

24

),(),(),()(2/

2/

2/12

2/12

2/2

2/

πρ

ϕϕϕϕ (2)

with ( ) ( )[ ]1ln 2 ++= xxxArsh . After some

transformations, we receive:

( ) ( )( ) ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−++−

+++++⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ++=

ltxlt

ltxltx

lxll

Ix444

444ln

24

ln24 22

2222

πρϕ (3)

1.1.1.1 Grounding resistance of vertical rod with the soil

improved chemical substance This resistance consists of three components (see Fig. 2):

Fig.2 Grounding rod with chemical substance

978-1-4244-4813-5/10/$25.00 ©2010 IEEE

a. The component in the metal environment The potential function )2/()0(1 dϕϕϕ −=Δ with ρ = ρ3. From (3), we have:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

ltlt

lI

44ln

4)0( 3

πρϕ (4)

and ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−++⎟

⎠⎞

⎜⎝⎛=

ltlt

dl

lI

d44ln

212ln

2)2/( 3

πρϕ (5)

From (4) and (5), we receive

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=Δ

dl

lI 2ln2

31 π

ρϕ (6)

So, the resistance value of this component

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=Δ=

dl

lIR 2ln

231

1 πρϕ (7)

b. The component in the chemical environment The potential function

)()2/(2 Cd ϕϕϕ −=Δ with ρ = ρ2. when x= d /2, x=C and note that ( )2

2

42

ltd +<<⎟⎠⎞

⎜⎝⎛ , we have Δϕ2

( ) ( )( ) ⎥

⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−++−

++++−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ++

⎢⎣

⎡ −⎟⎠⎞

⎜⎝⎛

−++⎟

⎠⎞

⎜⎝⎛=Δ

ltClt

ltCltC

lCl

ltlt

dl

lI

444

444ln

21

24

ln

44ln

212ln

2

22

2222

22 π

ρϕ (8)

IR 2

2ϕΔ=

( ) ( )( ) ⎥

⎥

⎦

⎤

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−++−

++++−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ++

⎢⎣

⎡ −⎟⎠⎞

⎜⎝⎛

−++⎟

⎠⎞

⎜⎝⎛=

ltClt

ltCltC

lCl

ltlt

dl

lR

444

444ln

21

24

ln

44ln

212ln

2

22

2222

22 π

ρ (9)

c. The component in the soil environment

The potential function )()(3 ∞−=Δ ϕϕϕ C with ρ =

ρ1. Because of ϕ(∞) = 0, )(3 Cϕϕ =Δ . So:

( )( ) ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−++−

+++++⎟⎟⎠

⎞⎜⎜⎝

⎛ ++==ΔltClt

ltCltC

lCll

IC444

444ln

24ln2

4)(

22

22221

3 πρϕϕ

(10)

( )( ) ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−++−

+++++⎟⎟⎠

⎞⎜⎜⎝

⎛ ++=Δ

=ltClt

ltCltC

lCllI

R444

444ln

21

24ln

2 22

222213

3 πρϕ (11)

At last, the total resistance value:

321 RRRR C ++=

In reality, a thickness of the chemical layer C is very smaller than the rod length l (C<< l ). Therefore, from (7), (9) and (11), we have:

( )

⎥⎦

⎤⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛+

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

dl

dC

ltCltl

lR C

2ln2ln

44ln

21

32

1

ρρ

ρπ

(12)

1.1.1.2 Grounding resistance value of equivalent vertical rod

We wish that for calculating grounding resistance value with or without the chemical substance we only use a form of a common formula. So, the formula for calculating such equivalent grounding resistance value will be recommended in this paper.

The single vertical rod with the chemical layer is converted into equivalent one that has resistivity of metal made up rod 3ρ and equivalent diameter DC. This resistance value consists of only two components (see Fig. H3).

Calculating similarly as the points (1.1.1.1a, c), we have: a. The component in the metal environment

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=

CDl

lR 2ln

2' 3

1 πρ (13)

b. The component in the soil environment

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+= 2

21'

3 )4(4)4(ln

4 CDltllt

lR

πρ (14)

Fig .3 Grounding rod and its image

Total resistance: RC = R’1 + R’3

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+=

CCC D

lDlt

lltl

R 2ln)4(4)4(ln2/

21

32

2

1 ρρπ

(15)

From (12) and (15), we receive

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

⎟⎟⎠

⎞⎜⎜⎝

⎛++

= 13

21

13

32

)2( ρρρρ

ρρρρ

CdDC (16)

Hence, we can convert single grounding vertical rod ( d , l ,ρ3) with the chemical substance (C, l , ρ2) into another (see Fig.4) with the length l , the resistivity ρ3 and the diameter DC is calculated by equation (16).

Fig .4 Equivalent grounding rod

In reality, 03 ≈ρ , we have:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−++=

)4()4(ln

212ln

21

ltlt

Dl

lR

CC π

ρ (17)

With ⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛

= 1

21

1

2

)2( ρρρ

ρρ

CdD C (18)

If there is not the chemical substance layer, then 12 ρρ = and (15), (16) will become [5]-[7]:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−++=

)4()4(ln

212ln

21

ltlt

dl

lR C π

ρ (19)

With dD C = 1.1.2 Soil improved chemical substance layer of rectangular

prism form We consider the single vertical rod with diameter d ,

length l and resistivity ρ3. Surround it the chemical substance layer with resistivity ρ1 and the soil environment with resistivity ρ2 . It is fixed in the ground with the depth t and the chemical substance layer has the dimensions lba ×× in Fig.5

Fig.5 Single grounding vertical rod with the chemical

substance layer In this case, we can convert the rectangular prism form

into a cylinder form with its radius C [8]

πabC 4=

(20)

and we will receive the formulas as in the case of the cylinder form soil improved chemical substance layer. 1.2 Grounding resistance of single horizontal conductor 1.2.1 Solid improved chemical substance layer cylinder form:

Single horizontal conductor with diameter d , length l and resistivity ρ3 is fixed in the ground with the depth t . Surround it the chemical environment with resistivity ρ1 and the soil environment with resistivity ρ2 in Fig.6

Fig.6 Single grounding horizontal conductor with the chemical

substance layer Applying (1) and after some transformations, we

found the magnetic field that is created by the horizontal conductor at the point (x, z)

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟

⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛= 1

22ln

2)(

2

1 xl

xl

lIxπρϕ

(21)

and the magnetic field that is created by the image of the horizontal conductor at the point (x, z):

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

= 1)2(2)2(2

ln2

)(2

2 xtl

xxtl

lIxπρϕ

(22)

1.2.1.1 Grounding resistance of horizontal conductor with the soil improved chemical substance

We use the calculating method as in the point (1.1.1.1) and receive result as follows : 321 RRRRT ++= RT

( )

⎥⎦

⎤+⎟⎠⎞

⎜⎝⎛ −+

⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

ld

dtCtC

CtCl

lRT

2ln)2(ln

)2(ln

21

32

2

1

ρρ

ρπ

(26)

1.2.1.2 Grounding resistance value of equivalent horizontal conductor We use the calculating method as in the point (1.1.1.2)

and receive the result as follows:

'2

'1 RRRT +=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

lD

tDl

lR T

TT 2

lnln2

13

2

1 ρρπ

(29)

From formulas (26) and (29), we have:

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−= 13

12

13

21

13

23

)]2([ ρρρρ

ρρρρ

ρρρρ

CtCtdD T (30)

Fig.7. Equivalent grounding horizontal conductor

Hence, we can convert single horizontal rod ( d , l , ρ3) with the chemical substance (C, l , ρ2) into another (Fig.7) with the length l , the resistivity ρ3 and the diameter DC is calculated by the equation (30). In reality, 03 ≈ρ , we have:

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=

TT tD

ll

R2

1 ln2 πρ

(31)

⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛ −⎟⎟⎠

⎞⎜⎜⎝

⎛

−= 1

21

1

12

1

2

)]2([ ρρρ

ρρρ

ρρ

CtCtdD T (32)

When without the chemical substance, 12 ρρ = and the formulas (31), (32) will become [9]-[12]

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=

dtl

lR T

21 ln

2 πρ

(33)

dD T = (34)

1.2.2 Soil improved chemical substance layer of rectangular prism form

Fig.8. Single horizontal conductor with the chemical substance

Similar as above, we can convert a conductor of the rectangular prism form (see Fig.8) into a cylinder with radius C is calculated by formula (20) and receive the formulas as in the case of the cylinder form soil improved chemical substance layer.

2. CALCULATING APPLICATION

2.1 Calculating data We use the following data: resistivity of ground 10.000

Ωcm ; resistivity of chemical substance 12 Ωcm ; resistivity of metal rod 0.000178 Ωcm

a. In table B1: the diameter of a rod 2 cm ; the diameter of a cylinder hole 22 cm

b. In table B2: the diameter of a rod 2 cm ; the diameter of a cylinder hole 22cm ; the number of rods 4 ; the depth of a cylinder hole 50 cm.

c. In table B3 and B4: the diameter of horizontal conductor 4cm ; the length and width of a hole section 100cm×150cm ; the depth of a hole 50 cm. 2.2 Compared results between this paper and GEM software of ERCO Company

TABLE 01. CALCULATING RESULTS OF SINGLE VERTICAL ROD

TABLE 02. CALCULATING RESULTS OF SYSTEM OF VERTICAL ROD

Note:”spacing” is distance between two vertical rods

TABLE 03. CALCULATING RESULTS OF SINGLE HORIZONTAL CONDUCTOR

Note:”a, b are the length, width of a hole section

TABLE 04. CALCULATING RESULTS OF SINGLE HORIZONTAL CONDUCTOR

Comment: From data in 2 tables B1 and B2, calculating results of resistance values for single vertical rod and system of vertical rods showed that the results of this paper and GEM software are approximate. The data from B3 and B4, the error between two calculating results is not great with short length of horizontal conductor and is approximate with long.

3. CONCLUSIONS 1). We received formulas for calculating the resistance of simple grounding forms with influence of the soil improved chemical substance. 2). We received formulas for calculating the equivalent diameter of simple grounding forms with influence of the soil improved chemical substance. 3). We can use the common formulas for calculating the resistance of simple grounding forms with or without the soil improved chemical substance.

4. REFERENCES [1]. Lightning protection international PTY, LTD. Global Lightning Technologies. Hobrut,Tasmania, Australia 7000 [2]. Van Dinh An and the Others. Researching and suggest- ing some solution for improving of lightning, grounding system of the line and transformer with high voltage of HoChiMinh City power company, HoChiMinh 2004. [3]. ERICO Company LTD. The GEM software for Cal- culating grounding resistance. [4]. Nathan Ida. Engineering Electromagnetics. Hamilton Printing Co., Rensselaser, NY., 2000. [5]. ANSI/IEEE Std 80-1986. IEEE guide for safety in AC substation grounding. [6]. L. F. Dmoxovskaia and the Others. High Voltage Engineer ing. Publisher “Energy”, Moscow, 1976. [7]. Ho Van Nhat Chuong. Problems in high voltage Engineering. Publisher “Ho Chi Minh City National University”, 2003. [8]. M. E. Ilrusalimov, N. N. Orlov. High Voltage Engineering. Publisher “Ralianskaia”, Kiev, 1967. [9]. A. Y. Dolginov. High Voltage Engineering in Energy System. Publisher “Energy”, Moscow, 1968. [10]. A. P. Sakis Meliopoulos. Power system grounding and transients. New York and Basel. [11]. V.V.Bazitkie. High Voltage Engineering: Insulators And overvoltage in power system. Publisher “Energo- automatic”, Moscow, 1986. [12]. M. A. Babakov and the Others. High Voltage Engineering. Publisher “Energy”, Moscow, 1963.