JET 45
JET Volume 16 (2023) p.p. 45-52
Issue 3, 2023
Type of article: 1.01 
http://www.fe.um.si/si/jet.htm
AN APPROXIMATE MODEL FOR 
DETERMINING THE RESISTANCE OF A 
HEMISPHERICAL GROUND ELECTRODE 
PLACED AT THE TOP OF A NON-
HOMOGENEOUS TRUNCATED CONE
PRIBLIŽNI MODEL ZA DOLOČANJE 
UPORA POLOKROGLE OZEMLJENE 
ELEKTRODE, NAMEŠČENE NA VRH 
NEHOMOGENEGA PRISEKANEGA 
STOŽCA
Dragan Vučković
1R
,
1
Dejan Jovanović
2
, Nenad Cvetković
2
, Miodrag Stojanović
1
, Dragan Tasić
1
Keywords: Estimation method, grounding system, non-homogeneous soil, resistance, 
hemispherical electrode
Abstract
The procedure for analysis of a hemispherical ground electrode placed at the top of a hill is 
presented in the paper. The mountain is modelled as a truncated cone of finite height consisting 
of two homogeneous areas, each having different electrical characteristics. The applied 
approach is a combination of several recently proposed procedures. It includes the application of 
the Estimation method and the use of an approximate expression in closed form. The obtained 
results are validated and compared with those obtained using the COMSOL program package.
1R Corresponding author: M.Sc Dragan Vučković, University of Nis, Faculty of Electronic Engineering,  
 Department of Power Engineering, A. Medvedeva 14, 18000 Nis, Serbia, Tel.: +381 18 529 305, E-mail  
 address: dragan.vuckovic@elfak.ni.ac.rs
1  University of Nis, Faculty of Electronic Engineering, Department of Power Engineering, A. Medvedeva 14,  
 18000 Nis, Serbia
2  University of Nis, Faculty of Electronic Engineering, Department of Theoretical Electrical Engineering, A.  
 Medvedeva 14, 18000 Nis, Serbia

46  JET
Povzetek
V prispevku je predstavljen postopek analize polkrogle ozemljitvene elektrode, nameščene na 
vrhu hriba. Gora je modelirana kot prisekan stožec končne višine, sestavljen iz dveh homogenih 
območij, od katerih ima vsako različne električne značilnosti. Uporabljeni pristop je kombinacija 
več nedavno predlaganih postopkov. Vključuje uporabo metode ocenjevanja in uporabo 
približnega izraza v zaprti obliki. Dobljene rezultate smo validirali in primerjali z rezultati, 
pridobljenimi s programskim paketom COMSOL.
1 INTRODUCTION
The most important feature of the grounding electrode, the resistance value, is conditioned 
by electrode geometry, conductor characteristics, soil structure and the physical shape of the 
surrounding ground. Various procedures are based on different numerical and semi-numerical 
methods, depending on the soil structure and corresponding geometrical model. For example, in 
[1-2], the ground is modelled as a homogeneous half space of a flat surface. Very often, the non-
homogeneous ground is modelled as multi-layered [3-4], sectoral [5], a semi-spherically [6-7] or 
semi-cylindrically shaped domain [8].
The problem of determining the resistance of a hemispherical ground electrode placed at the 
top of a non-homogeneous truncated cone modelled with three homogeneous domains having 
different electrical characteristics is investigated in this paper. The procedure applied in this 
paper is based on approaches from [9-11]. The procedure for analysis of a hemispherical ground 
electrode placed at the top of a hill was proposed in [9]. This approach was improved in [10] 
by assuming current density distribution in two different forms, depending on the observed 
domain. The method applied in this paper is an extended application of the approach based on 
procedures from [9-10] and presented in [11]. The previously mentioned approach offers the 
possibility of modelling a truncated cone as a non-homogeneous domain consisting of different 
homogeneous areas.
The applied procedure does not require integration, and all the used approximate expressions 
are given in closed form. The COMSOL program package based on the Finite Element Method 
was used to validate the obtained results.
2 PROBLEM DESCRIPTION
The hemispherical ground electrode is placed at the top of a non-homogeneous truncated 
cone consisting of three homogeneous domains having specific conductivities 
1
σ , 
3
σ and 
3
σ 
respectively. The boundary surfaces between the domains are flat.
The radius of the electrode is 
1
r
 and the cone base radius is 
1
d
. The depth of the boundary 
surfaces is labelled with 
1
d , and the cone slope is defined with angle α. In this paper, the 
proposed solution for the structure from Fig. 1 is based, as has already been emphasised, on the 
approaches from [9-11]. However, the chosen model of a non-homogeneous truncated cone is a 
more complex structure related to those from [9-11]. One could expect that, in general, the case 
ground structure is non-homogeneous, and therefore, the model from Figure 1 can be used for 
approximating ground non-homogeneity.
Dragan Vučković, Dejan Jovanović, Nenad Cvetković,  
Miodrag Stojanović, Dragan Tasić
JET Volume 16 (2023) p.p. 
Issue 3, 2023

  JET 47
Figure 1: The hemispherical ground electrode and truncated cone approximated  
with three homogeneous domains (flat boundary surface).
3 PROCEDURE DESCRIPTION
The procedure for determination of the resistance of hemispherical ground electrode placed at 
the top of the truncated cone approximated with three homogeneous domains (Figure 2) will 
be derived, in order to generate the procedure for determination of the approximate resistance 
of the electrode from Figure 1. It is based on the approaches proposed in [11]. The boundary 
surfaces between domains are calotte surfaces having the radii 3
r
 and 3
r
, and the rest of the 
geometry parameters can be seen in Figure 2.
Figure 2: The hemispherical ground electrode and truncated cone approximated  
with three homogeneous domains (oval boundary surface).
Based on the procedure described in [11], the current densities in the cone domain can be 
expressed as,
1 1t
2
ˆ,
2
I
J rr r r
r
= <<
π

 
, and              (3.1a)
( )
21
2
ˆ
,
2 1 cos
I
J RR R
R
= < <∞
π− α

             
(3.1b)
An approximate model for determining the resistance of a hemispherical ground electrode  
placed at the top of a non-homogeneous truncated cone

48  JET
In previous expressions, r is a radial
 
coordinate, while ˆ r is the corresponding spot of the 
coordinate system having origin at the middle point of the upper surface of the truncated cone. 
Similarly, 
ˆ
R and 
ˆ
R
 
are the radial
 
coordinate and corresponding spot of the coordinate system 
having origin at the imaginary top point of the cone [11]. 
Using a local form of Ohm’s law, for the electric field is obtained,
1
11 1 t
2
1
1
ˆ,
2
J I
E rr r r
r
= = <<
σ
πσ


      
        (3.2a)
( )
2
12 1 2
2
1
1
ˆ
,
2 1 cos
J I
E RR R R
R
= = <<
σ
πσ − α


       
      (3.2b)
( )
2
2 23
2
2
2
ˆ
,
2 1 cos
J I
E RR R R
R
= = <<
σ
πσ − α


             
(3.2c)
( )
2
33
2
3
3
ˆ
,
2 1 cos
J I
E RR R
R
= = < <∞
σ
πσ − α


            
(3.2d)
The potential 
s
ϕ
 
of the electrode surface from Fig 2 can be determined approximately as [9, 12],
t3 2
11 23
s 11 12 2 3
d dd d
rR R
r RR R
E r E rE R ER
∞
ϕ= + +
∫∫∫∫
               
(3.3)
Now, for the potential 
s
ϕ
 obtains,
( ) ( )
( )
s
1 1 t 12 2 12
33
1 1 1 11 1 11
2 1 cos 2 1 cos
11
.
2 1 cos
II
rr RR RR
I
R
  
ϕ= − + − + − +
  
πσ − α πσ − α
  

+
πσ − α
         
(3.4)
The electrode resistance value is
( )
( ) ( )
s
g
11 t 1 2
2 12 3 3
1 11 1 1 1
2 1 cos
1 1 11 1 1 1
.
2 1 cos 2 1 cos
R
I rr RR
RR R
  ϕ
== −+ − +
 
πσ − α
  

+ −+

πσ − α πσ − α

              
(3.5)
There is interest to emphasise that in the previous expression
t
11
tan
r
Rr = +
α
                  
(3.6)
Dragan Vučković, Dejan Jovanović, Nenad Cvetković,  
Miodrag Stojanović, Dragan Tasić
JET Volume 16 (2023) p.p. 
Issue 3, 2023

  JET 49
4 ESTIMATION METHOD
In order to use the previously described procedure for the determination of the approximate 
resistance of the hemispherical grounding electrode from Figure 1, the Estimation method is 
introduced into the procedure.
From Figure 3 can be noticed the upper (
2e
R , 
3e
R ) and lower (
3i
R , 
3i
R ) values of the boundary 
surface radii. The approximate resistance of the system can be determined as the arithmetic 
mean of the resistance values calculated for the upper and lower radii values.
Figure 3: Estimation method application.
The geometrical parameters from Figure 3 can be determined as
1
2i 1 2e
2
3i 2 3e
cot , , ,
cos
,.
cos
t
dd
d r R d dR
dd
R ddR
+
=α= + =
α
+
= +=
α
               
 (4.1)
Now, after determination of the resistance values for 
2 2i
RR = ,
 
3 3i
RR =
 
(labels as 
gi
R ), i.e. 
for
 
2 2e
RR = ,
 
3 3e
RR =
 
(labelled as
ge
R ), the approximate value of the grounding electrode 
resistance can be determined as the arithmetic mean
2e 2i
g
2
RR
R
+
=
 
, i.e.               (4.2a)
( )
( ) ( )
2e 2i
g
1 1 t 1 t 2e 2i
2e 2i 3e 3i 3e 3i
2 2e 2i 3e 3i 3 3e 3i
1 11 1 1
2 2 1 cos 2
11
.
2 1 cos 2 2 1 cos 2
RR
R
r r rd RR
RRRR RR
R rR RR RR
   +
= −+ − +
  
πσ πσ − α +
  
 ++ +
+ −+

πσ − α πσ − α

          
(4.2b)
An approximate model for determining the resistance of a hemispherical ground electrode  
placed at the top of a non-homogeneous truncated cone

50  JET
Dragan Vučković, Dejan Jovanović, Nenad Cvetković,  
Miodrag Stojanović, Dragan Tasić
JET Volume 16 (2023) p.p. 
Issue 3, 2023
5 NUMERICAL RESULTS
The obtained results and relative errors obtained by the described method are shown in Table 1 
for { }
21 31
5, 10 σ σ= σ σ= and Table 2 for { }
21 31
5, 10 σ σ= σ σ=
.
The set of other parameters` values is 
1
0.001S/ m σ= , 
1
10m d = ,
 
1
10m d = , 
 
{ }
0000
45 ,50 ,55 ,60 α∈ and { }
t
10m ,20m r ∈
. 
The values of the parameters have been selected based on [11]. The obtained results (R
g ap
) were 
validated with the values obtained from the COMSOL program package application (R
g
).
From the results shown in Tables 1-2, one can conclude that the relative error was not larger 
than 15,4%, which can be assumed as a satisfactory result. A more detailed analysis requires a 
more extensive data set, but at this research level, it can be indicated that the proposed simple 
approach can be assumed as valuable and applicable in practice.
Table 1: Grounding resistance for 
1
0.001S/ m σ= ,
1
5m r = , 
1
10m d = , 
2
20 m d = ,   
                                   
21
5 σ σ=  and 
31
10 σ σ=
α=45
0
r
t
[m] R
g ap
 [Ω] R
g
[Ω] Relative error [%]
10 22,2134 24,066 7,697997
20 23,5436 22,348 5,349919
α=50
0
r
t
[m] R
g ap
 [Ω] R
g
[Ω] Relative error [%]
10 22,2324 23,735 6,330735
20 23,6881 22,327 6,096206
α=55
0
r
t
[m] R
g ap
 [Ω] R
g
[Ω] Relative error [%]
10 22,3231 23,445 4,785242
20 23,8392 22,310 6,854325
α=60
0
r
t
[m] R
g ap
 [Ω] R
g
[Ω] Relative error [%]
10 22,4740 23,198 3,120959
20 23,9916 22,295 7,609778

  JET 51
An approximate model for determining the resistance of a hemispherical ground electrode  
placed at the top of a non-homogeneous truncated cone
Table 2: Grounding resistance for 
1
0.001S/ m σ= , 
1
5m r = , 
1
10m d = , 
2
20 m d = ,  
                       
21
10 σ σ= and 
31
50 σ σ= .
α=45
0
r
t
[m] R
g ap
 [Ω] R
g
[Ω] Relative error [%]
10 20,3582 24,066 15,4068
20 22,3068 22,348 0,184357
α=50
0
r
t
[m] R
g ap
 [Ω] R
g
[Ω] Relative error [%]
10 20,6405 23,735 13,03771
20 22,5950 22,327 1,20034
α=55
0
r
t
[m] R
g ap
 [Ω] R
g
[Ω] Relative error [%]
10 20,9414 23,445 10,67861
20 22,8605 22,310 2,467503
α=60
0
r
t
[m] R
g ap
 [Ω] R
g
[Ω] Relative error [%]
10 21,2632 23,198 8,340374
20 23,1052 22,294 3,638647
6 CONCLUSIONS
The procedure, based on several recently proposed approaches for approximate determination 
of the resistance value of a hemispherical ground electrode placed at the top of the truncated 
cone, is presented in the paper. The obtained results indicate that this simple approach can be 
used for approximating the described ground non-homogeneity.
ACKNOWLEDGEMENTS
The research described in the paper was partially supported by the Ministry of Science, 
Technological Development and Innovation of the Republic of Serbia.

52  JET
Dragan Vučković, Dejan Jovanović, Nenad Cvetković,  
Miodrag Stojanović, Dragan Tasić
JET Volume 16 (2023) p.p. 
Issue 3, 2023
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