Journal of ELECTRICAL ENGINEERING, VOL. 54, NO. 5-6, 2003, 113–117
ELECTRIC FIELD NEAR BUNDLE CONDUCTORS
Daniel Mayer — Vı́t Veselý
∗
The paper deals with computation of electric field distribution along the surfaces of a system of parallel conductors
with various potentials. The method starts from integral equations and the elaborated algorithm is applied to hv and uhv
overhead lines with bundle conductors. The results allow evaluating the danger of giving rise to corona (even with respecting
the influence of a rough surface of the cable) with all its interference effects. The algorithm was compared with other already
published methods with remarkable agreement.
K e y w o r d s: hv and uhv overhead lines, bundle conductors, corona, method of integral equations.
1 INTRODUCTION
Hv and uhv overhead lines are mostly realised by
bundle conductors. Electric field strength on their surfaces depends on the potential of particular conductors,
their radii, mutual distances and quality of their surface.
Exceeding the critical value Ecrit ≈ 21 kV/cm gives rise
to a corona. One of the advantages of using bundle conductors is reduction of this discharge that, as it is well
known, increases losses along the line, unfavourably affects near telecommunication devices and produces acoustic noise. A surprising amount of papers were published
on the topic, see the bibliographic list [6] compiled already
in 1964.
Various methods have been used for determining the
electric field near the bundle. Older papers, see, for instance, [2], [15] started from an analytical solution of the
electric field produced by one conductor and the resultant electric field of a bundle was calculated using superposition. The results obtained in this way are, however,
only approximate. Other authors [3], [4] and [8] replace
the particular conductors in the bundle by a system of
suitably located current filaments and solve their electric field. Conformal mapping was used in [13]. Simulated
charge method was successfully applied in [11] and [12].
Numerical calculation based on standard finite element
techniques (and realised by means of professional codes)
would be comfortable, nevertheless, problems with geometrical incommensurability (small cross-sections of the
conductors versus their large mutual distances and domain containing air) would lead to using a strongly nonuniform mesh and enormous number of equations. The
task represents, moreover, an open boundary problem,
which is often a source of further errors.
That is why the method of integral equations (see [5])
may appear advantageous for solving this case; its fundamentals can also be found in [6] or [14]. The unknown
quantity is now the charge density σ along the surface of
particular conductors. We formulate the corresponding
equation for σ and solve it numerically. We are, however, mainly interested in the vector of the electric field
strength on the surface of particular conductors. It has
only the normal component En that is given by a simple
(Coulomb) formula
σ
.
En =
ε0
The principal advantage of this approach consists in
the fact that the numerical computation of the charge
density is much simpler than computation of the distribution of the potential near the conductors (which would,
of course, also provide the values of En ). While the distribution of the potential in the solved arrangement represents a 2D problem, solution of the charge density is only
a 1D problem. Reduction of the dimension by 1 leads
to a significant simplification of the task. Unlike in the
open boundary problem associated with computation of
the potential, the charge is distributed along surfaces of
the conductors, ie in a spatially bounded domain.
2 MATHEMATICAL MODEL
OF THE PROBLEM
Considered is homogeneous, linear and isotropic dielectric (air) of permittivity ε0 with a set of n direct
parallel conductors with constant potentials ϕk , k =
1, . . . , n .. The electric charge on their surfaces is supposed to be distributed continuously, with charge density
σk that does not change along their lengths. The electric
field near the conductors is obviously two-dimensional.
Potential at a general point B of this field is given by
expression [7]
n I
1 X
1
ϕ(rB ) =
σk (r (k) ln (k)
dl(k) ,
2πε0
|r
−
r
|
B
(1)
k=1 (k)
l
k = 1, . . . , n
∗ Department of Theory of Electrical Engineering, Faculty of Electrical Engineering, University of West Bohemia in Pilsen, Sady
Pětatřicátnı́ků 14, 306 14 Pilsen, Czech Republic, E-mail: mayer@kte.zcu.cz, vvesely@kte.zcu.cz
c 2003 FEI STU
ISSN 1335-3632 °
114
D. Mayer — V. Veselý: ELECTRIC FIELD NEAR BUNDLE CONDUCTORS
The potential ϕ would be determined from (1) while the
electric field strength from relation
dlk
(k)
lk
e0
jk(k) = const.
E(rB ) = − grad ϕ(rB ) .
A
y
r
(k)
This is, however, beyond our interest.
Let us return to our case, when B ∈ l (k) (Fig. 2). The
electric field strength at such a point has only a normal
component and may be determined from formula
- rB
(k)
r
B
rB
En(k) (r (k) ) =
0
(3)
σk (r (k) )
.
ε0
(4)
x
Fig. 1. The k -th conductor in the set of n parallel conductors.
A(xi,yj)
e0
(k)
lk
jk(k) = const.
A
y
In this way we solved our problem because the value
of the electric field strength decides about giving rise to
corona on the surface of any conductor. Now it remains
to carry out numerical solution of (2). First we divide the
contour lines l(k) of cross-sections of the particular con(k)
ductors into N (k) parts of lengths ∆li , i = 1, . . . , Nk .
When this division is sufficiently fine, each part may be
supposed to carry a constant value of the charge density
ri - rj
(k)
σi
B(xi,yj)
r
∼
= const (i = 1, . . . , Nk , k = 1, . . . , n) .
(k)
The distance between the midpoints of parts ∆li
(k)
(l)
∆lj
rB
is given as
(k)
|ri
x
0
Fig. 2. To the calculation of the charge density on the surface of
the k -th conductor.
where l(k) is a simply connected curve in which the crosssection S (k) intersects the plane of the field perpendicular
to the conductors, r (k) is the radiusvector of the element
dl(k) , rB is the radiusvector of point B and |r (k) − rB |
is the distance of a variable point A of the planar curve
l(k) from point B outside the conductors (see Fig. 1). We
put, moreover, that
and
(l)
− rj | =
q
(k)
(l)
(k)
− xj )2 + (yi
(xi
(l)
− y j )2 .
(5)
Equation (2) for the discretised model now reads (in
order to avoid dividing by zero we leave the term for i = j
and k = l in the form of the definite integral)
2πε0 ϕk =
n
X
l=1
(l)
N
X
i=1
i6=j∨k6=l
(k)
+
(k)
2σi
∆lZi
(k)
|ri
−
(l)
(l)
rj |
∆li
/2
ln
0
lim ϕ(rB ) = 0 .
1
(l)
σi ln
1
(k)
dl , k = 1, . . . , n .
|rii | i
(6)
|rB |→∞
(k)
(k)
Let point B ∈ l . Now (1) transforms into the firstkind Fredholm integral equation with an unknown distribution of the charge density σk .
2πε0 ϕk =
n
X
I
σk (r (k) ) ln
k=1 (k)
l
1
dl(k) ,
(k)
|r − rB |
As far as ∆li is a line segment, calculation of the integral in (6) is easy and its result is
(k)
∆lZi
0
/2
ln
1
(k)
dl
(k) i
rii
=
(k)
(k)
∆l ´
∆li ³
1 − ln i
.
2
2
(7)
(2)
k = 1, . . . , n .
Solution of system (2) provides σk at an arbitrary point
B ∈ l(k) . This knowledge allows consequent computing of
the field quantities at any point outside the conductors.
In this manner we obtain a system of algebraic equations
whose number is
m=
n
X
k=1
N (k) .
(8)
115
Journal of ELECTRICAL ENGINEERING VOL. 54, NO. 5-6, 2003
En min=10.58 kV/cm
120
r
En =(kV/cm)
En max=14.30 kV/cm
90
15
60
10
R
30
150
5
0
180
330
210
240
Fig. 3. A bundle conductor for n = 4 .
270
300
Fig. 4. Distribution of the normal component of the electric field
strength along the surface of one conductor of the bundle ( n = 4 ).
En min=10.58 kV/cm
r
120
R
En =(kV/cm)
En max=14.30 kV/cm
90
10
60
8
6
4
2
150
30
180
0
330
210
240
Fig. 5. A bundle conductor for n = 8 .
3 ILLUSTRATIVE EXAMPLES
(9)
where
A(m, n) = [apq ] , p, q = 1, . . . , m ,

 ∆lp ln 1|rp − rq | , for p 6= q
³
´
apq =
 ∆lp 1 − ln ∆l2 p , for p = q .
300
Fig. 6. Distribution of the normal component of the electric field
strength along the surface of one conductor of the bundle ( n = 8 ).
The system can be rewritten into a matrix form
Aσ = 2πε0 ϕ
270
Here σ(m, 1) is the column vector of charge densities in
the particular segments and ϕ(m, l) their potentials.
The presented relations respect the influence of all
conductors in the system (all bundle conductors in all
phases) and possibly the influence of the earth.
The next examples solve, however, only the case of
conductors in a one phase bundle (ie we consider neither
the influence of other phases nor of the earth). Determined is the distribution of the electric field strength on
the surface of one conductor in the bundle.
a. A bundle conductor n = 4
For the conductors placed in vertices of a regular
square (see Fig. 3) we calculated the distribution of the
normal component of En along the surface of one conductor. All conductors in the bundle have the same potential
ϕ = 100 kV. Perimeters of all conductors were discretised
116
D. Mayer — V. Veselý: ELECTRIC FIELD NEAR BUNDLE CONDUCTORS
If we replace the bundle conductor by a single conductor of radius r = 2r0 (its cross-section being the same
as the total cross-section of the bundle conductor), the
electric field strength En would be distributed uniformly
and its value would be
RE
R
En =
r
4 · 6.8783 × 10−7 1
nQ 1
=
= 24.73 kV/m.
2πε0 r
2πε0
0.02
If we use a bundle conductor where the distance between
individual conductors is very large (their mutual electrostatic interaction is negligible), the electric field strength
En will again be uniform and its value would be
Fig. 7. A massive conductor and equivalent transmission cable for
n = 8.
En =
Q 1
6.8783 × 10−7 1
=
= 12.36 kV/m.
2πε0 ri
2πε0
0.01
b. A bundle conductor n = 8
En =(kV/cm)
En max=11.10 kV/cm
90 15
60
120
10
30
150
c. Influence of rough surface of the transmission
cable
5
0
180
330
210
240
270
300
Fig. 8. Distribution of the normal component of the electric field
strength along the surface of one conductor of the transmission
cable.
Table 1. Values of En for various arrangements of conductors.
arrangement
En (kV/cm) En /En1
bundle conductor
En max = 9.858 0.339
n = 8 (Fig. 5)
En min = 4.837 0.166
one equivalent conductor En1 = 29.072
—
bundle conductor,
large distance among
En = 7.268
0.250
particular conductors
into N (1) = · · · = N (4) = 50 line segments. The results
are depicted in the polar co-ordinates in Fig. 4. The maximum and minimum values of the electric field strength
are En max = 14.30 kV/cm, En min = 10.58 kV/cm. The
total electric charge per unit length of one conductor of
the bundle is
Z
N
X
Q = σdS ∼
σi ∆li = 6.8783 × 10−7 C/m.
=
S
We calculated En on the surface of one conductor
arranged at vertices of a regular octagon (Fig. 5). Its
distribution is in Fig. 6. Other results analogous to the
previous case are summarised in Tab. 1.
i=1
The transmission (for instance aluminium-steel) cable
may be taken as an extreme case of the bundle conductor, whose partial conductors are placed very close one to
another. While on the surface of a massive conductor the
normal component En is distributed uniformly, in the
case of the transmission cable this component changes
from zero on the internal surface (influence of shielding)
to its maximum En max on the external surface. Obviously En max > En . Figure 7 depicts both the crosssection of the massive conductor and cross-section of the
cable with 8 conductors. Figure 8 contains a graph in
polar co-ordinates showing the distribution of the electric field strength on the surface of one cable conductor.
Calculations were performed for potential ϕ = 100 kV,
perimeter of each conductor being divided into N = 50
line segments. Analogously to previous cases we obtain
• for a single conductor of the cable
En max = 11.103 kV/cm, En min ≈ 0 ,
N
X
Q∼
σi ∆li = 2.089 × 10−7 C/m,
=
i=1
• for an equivalent bundle conductor
nQ
√
= 10.077 kV/cm.
En =
2πε0 8ri
On the surface of the cable (in comparison with the
equivalent massive conductor with smooth surface) the
dielectric stress is higher. The measure of this increase
may be given by the ratio between the maximal value of
its normal component to the magnetic field strength of
the massive conductor: En max /En = 1.102 .
Let us remark that microscopic roughness of the surface caused, for example, by corrosion or small water
drops gives also rise to corona.
Journal of ELECTRICAL ENGINEERING VOL. 54, NO. 5-6, 2003
4 CONCLUSION
[7] MAYER, D.—POLÁK, J. : Methods of Solution of Electric and
Magnetic Fields, SNTL/ALFA, Praha, 1983. (in Czech)
The presented algorithm for computation of dielectric
stress on the surface of a conductor in a bundle was compared with results obtained by other methods. Table 2
contains some results obtained by different algorithms for
a bundle containing n = 4 , 8 and 12 conductors characterised by d/r = 26.099 ( d being the distance of axes of
two neighbouring conductors in the bundle and r their radius). The perimeter of each conductor was then divided
into N = 32 line segments. The agreement between particular methods is outstanding.
n=4
0.7217
0.7203
0.7222
n=8
0.6624
0.6593
0.6632
[8] MIROLJUBOV, N. N.—KOSTENKO, M. V.—LEVINŠTEJN,
M. L.—TICHODEJEV, N. N. : Methods of Calculation of Electrostatic Fields, Vysshaja shkola, Moskva, 1963. (in Russian)
[9] SANDELL, D. H.—SHEALY, A. N.—WHITE, H. B. : Bibliography on Bundled Conductors, IEEE Trans. on Power Apparatus and Systems, December 1963, No. 69, pp. 1115–1128.
[10] SARMA, M. P.—JANISCHEWSKYJ, W. : Electrostatic Field
of a System of Parallel Cylindrical Conductors, Trans IEEE
PAS-88 (1969), 1069–1079.
[11] SINGER, H.—STEINBIGLER, H.—WEISS, P. : A Charge Simulation Method for the Calculation of High Voltage Fields, IEEE
Trans, paper T74 085–7, 1973.
[12] STEINBIGLER, H.—SINGER, H.—BERGER, S. : Berechnung
der Randfeldstärken von Bundelleitern in Drehstromsystemen.
ETZ-A, 92, 1971, pp. 612–617.
Table 2. Comparison of results.
Emin /Emax
the described method
by Cahen [3]
by Timascheff [13]
117
n = 12
0.6454
0.6418
0.6463
[13] TIMASCHEFF, A. S. : Fast Calculation of Gradients of a
Three-Phase Bundle Conductor Line with any Number of Subconductors, AIEE Trans., Vol. PAS-90, 1971, pp. 157–164;
PAS-94, 1975, pp. 104–107.
[14] TOZONI, O. V. : Method of Secondary Sources in Electrical
Engineering, Energija, Moskva, 1975. (in Russian)
[15] VEVERKA, A. :
High-Voltage Engineering, SNTL/ALFA,
Praha, 1976. (in Czech)
Acknowledgement
This work has been financially supported from the
grant No. 102/01/1401 of the Grant Agency of the Czech
Republic.
References
[1] ALLAN, R. N.—SALMAN, S. K. : Electrostatic Fields Underneath Power Lines Operated at Very High Voltages, Proc. IEE
121 (1974), 1404–1408.
[2] BRÜDERLINK, R. : Induktivität und Kapazität der Statrkstrom-Freileitungen, G. Braun, Karlsruhe, 1954.
[3] CAHEN, F.—PELISSEUR, R. : L’emploi de conducteurs en
faisceaus pour l’armement des linges a tres haute tension, Bull.
Soc. Fr. Electr., March 1948, pp. 111–160.
[4] KING, S. Y. : The Electric Field near Bundle Conductors, Proc.
IEE 106C (1959), 200–206.
[5] FOO, P. Y.—KING, S. Y. : Bundle-Conductor Electric Field
by Integral-Equation Method, Proc. IEE 123 (1976), 702–706.
[6] MAYER, D.—ULRYCH, B. : Solution of 2D and 3D Electrostatic and Magnetostatic Fields, Elektrotechn. obzor 69 No. 8
(1980), 458–463. (in Czech)
Received 18 March 2003
Daniel Mayer (Prof, Ing, DrSc) was born in Pilsen, Czech
Republic in 1930. He received the Ing, PhD and DrSc degrees
in electrical engineering from Technical University of Prague
in 1952, 1958 and 1979, respectively. In 1956 he began his professional career as a Senior Lecturer and later as a Associate
Professor at the University of West Bohemia in Pilsen. In 1968
he was appointed Full Professor of the Theory of Electrical
Engineering. Many years he has been head of the Institution
of Theory of Electrical Engineering. His main teaching and
research interests include circuit theory, electromagnetic field
theory, electrical machines and history of electrical engineering. He has published 6 books and more than 200 scientific
papers. He is a Fellow of the IEE, member ICS, ISTET and
UICEE, member of editorial boards of several international
journals and leader of many grant projects.
Vı́t Veselý (Ing) was born in Domažlice, Czech Republic
in 1979. He graduated in electronics engineering from the
Faculty of Electrical Engineering, University of West Bohemia
in Pilsen in 2002. His research interests lie mainly in the area
of computer sciences.