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C I R E D
20
th
International Conference on Electricity Distribution Prague, 8-11 June 2009

Paper 0628


CIRED2009 Session 2 Paper No 0628
IMPACT OF THE PHASE POSITIONS ON THE ELECTRIC AND MAGNETIC FIELD OF
HIGH-VOLTAGE OVERHEAD LINES


Katrin FRIEDL Ernst SCHMAUTZER
TU Graz - Austria TU Graz - Austria
Katrin.friedl@tugraz.at Schmautzer@tugraz.at

ABSTRACT
The magnetic and the electric field of a three phase double
circuit high-voltage overhead power line depend on the
geometrical arrangement of the conductors and on the
allocation of the phases. This paper shows the resulting
magnetic and electric fields for three different conductor
arrangements (i.e. given tower geometries) often found in
Europe - for all possible phase allocations respectively. In
addition, a method is derived to identify exemplarily the
best phase allocation for specific immission of a 3-level-
tower, as well as an evaluation method of the highest field
value without knowing the actual phase allocation.
INTRODUCTION
Approval processes for the construction and/or modification
of high-voltage overhead lines have led to an increasing
demand for detailed investigation of magnetic and electric
fields. For this paper three common European tower designs
for double circuit 400 kV overhead lines were chosen to
demonstrate the influence of the positions of the phases on
the magnetic and electric field. The positions of the
conductors for a assumed 3-level, 2-level and 1-level-tower
are given in Fig.1. The distance from the ground to the
lowest conductors has been set to 10 m (i.e. the minimum
distance considering wire-sag), in order to enable
comparison of the results for different tower-geometries.

P1
P2
P3
P1’
P2’
P3’
P1P2
P3
P1’ P2’
P3’
E
E
P1P2P3 P1’ P2’ P3’
3-level-tower 2-level-tower 1-level-tower
P3 P3'
P2 P2'
P1 P1'
No.x
x y x y x y x y x y x y x y
3-level -8,5 10,0 -11,5 18,0 -7,5 27,5 8,5 10,0 11,5 18,0 7,5 27,5 0,0 39,0
2-level -7,5 10,0 -13,0 10,0 -10,0 19,0 7,5 10,0 13,0 10,0 10,0 19,0 0,0 32,5
1-level -7,5 10,0 -13,0 10,0 -21,0 10,0 7,5 10,0 13,0 10,0 21,0 10,0 - -
EP2'P3'P1 P2 P3 P1'

Fig. 1 Conductor positions P1…P3’ (x = distance from
center axis, y =height) for 1-, 2-, and 3-level-towers in m

A double circuit line offers 36 (6 times 6) different
possibilities of allocating the phases (L1, L2, L3, L1’, L2’,
L3’) for a given geometrical conductor arrangement (P1,
P2, P3, P1’, P2’, P3’) as shown in Tab. 1. These
combinations result in 12 different field configurations –
Var. 1, 2 and 3 all share the same electric and magnetic
field and can be obtained by cyclically exchanging the
phases. In Tab. 1 these 12 different cases are shown (No.1
to No.12) as well as the 3 variations obtained by cyclical
exchanges (Var. 1-Var. 3). Cases No.1 to No.6 differ with
cases No.7 to No.12 only in the fact that L2 and L3 are
switched.
L3 L3'L2 L3'L3 L2'L1 L1'L3 L2'L3 L2'
L2 L2'L3 L2'L1 L1'L2 L3'L2 L1'L1 L3'
L1 L1'L1 L1'L2 L3'L3 L2'L1 L3'L2 L1'
L1 L1'L3 L1'L1 L3'L2 L2'L1 L3'L1 L3'
L3 L3'L1 L3'L2 L2'L3 L1'L3 L2'L2 L1'
L2 L2'L2 L2'L3 L1'L1 L3'L2 L1'L3 L2'
L2 L2'L1 L2'L2 L1'L3 L3'L2 L1'L2 L1'
L1 L1'L2 L1'L3 L3'L1 L2'L1 L3'L3 L2'
L3 L3'L3 L3'L1 L2'L2 L1'L3 L2'L1 L3'
L2 L2'L3 L2'L2 L3'L1 L1'L2 L3'L2 L3'
L3 L3'L2 L3'L1 L1'L3 L2'L3 L1'L1 L2'
L1 L1'L1 L1'L3 L2'L2 L3'L1 L2'L3 L1'
L1 L1'L2 L1'L1 L2'L3 L3'L1 L2'L1 L2'
L2 L2'L1 L2'L3 L3'L2 L1'L2 L3'L3 L1'
L3 L3'L3 L3'L2 L1'L1 L2'L3 L1'L2 L3'
L3 L3'L1 L3'L3 L1'L2 L2'L3 L1'L3 L1'
L1 L1'L3 L1'L2 L2'L1 L3'L1 L2'L2 L3'
L2 L2'L2 L2'L1 L3'L3 L1'L2 L3'L1 L2'
Var.2Var.3Var.1Var.2Var.3Var.1
No.5 No.6
No.9 No.10 No.11 No.12
No.1 No.2
No.8
No.3 No.4
No.7

Tab. 1 12 cases, 3 variations respectively, for allocating
the phases of a given double circuit tower geometry
CALCULATION
Calculation of the Magnetic Field

The calculation of the magnetic flux density follows the
theory of Biot and Savart using a two-dimensional vertical
model. For an infinite straight thin conductor carrying a
current I the magnetic flux density can be calculated
applying the following expression (1).

0
B
μ I(t)
B(t) e
4 r
= ⋅
π


(1)

B(t)

time-dependent magnetic flux density in Vs/m² (T)
I(t)
time-dependent current, a sinusoidal current with a
frequency of 50 Hz in A
0
μ
permeability of vacuum in Vs/Am
r
distance from the thin conductor in m
B
e

unit vector in circumferential direction
The magnetic flux density in a space free of magnetic
materials obeys the principle of superposition. Therefore the
resulting magnetic flux density of a given geometrical
C I R E D
20
th
International Conference on Electricity Distribution Prague, 8-11 June 2009

Paper 0628


CIRED2009 Session 2 Paper No 0628
arrangement can be calculated by vectorially adding the
contribution of each line conductor respectively. For
sinusoidal currents I(t), the magnetic flux density and the
components in x- and y-direction B
x
and B
y
are also
sinusoidal. If there is more than one conductor and the
currents in the conductors have different phasing, a rotating
field occurs. The RMS of the magnetic flux density (i.e. the
equivalent magnetic flux density) can be calculated with the
root mean square of the RMS values of the sinusoidal
components B
x
and B
y
(2):

2 2
rms xrms yrms
B B B= +
(2)

rms
B
RMS of the rotating magnetic flux
density
xrms yrms
B, B
RMS of the components in x- and y-
direction

In the following always the equivalent flux density B
rms
is
applied.

Calculation of the Electric Field

The electric field is calculated using the method of mirror
charges (or image charges), using a mirror-plane - the
conducting soil. The line charges
τ
of the conductors are
calculated by using the potential coefficient matrix of the
power line. Afterwards the electrical field strength can be
calculated with following expression derived from
Coulomb’s law (3):
r
0
E e
4 r
τ
= ⋅
πε


(3)
E

electric field strength in V/m
τ
line charge in As/m
0
ε
permittivity of vacuum in As/Vm
r
distance from line conductor in m
r
e

unit vector in radial direction
The electric field also obeys the superposition principle and
the equivalent electric field strength E
rms
can be calculated
analogously to B
rms
.
Calculation of the Current in the Earth Wire

The current in the overhead earth wire I
e
is calculated with
the impedance-formulae by Cason and Pollaczek [1, 2] as
shown in [3] using equation (4):
p
pp pep
ep ee
e
I
U Z Z
=
Z Z I
0
⎛ ⎞
⎛ ⎞ ⎛ ⎞
⋅ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟⎜ ⎟
⎜ ⎟
⎝ ⎠⎝ ⎠
⎝ ⎠
(4)
U
p
voltage of active phase conductors
U
e
=0 voltage of earth conductor(s)
Z
pp
, Z
ee
impedance matrix of system p, e
Z
pe
= Z
ep
impedance matrix between systems p
(phases) and e (earth wires)
The resulting currents I
e
of the earth wires can be calculated
as follows (5).
1
e ee pe p
I = -Z Z I

⋅ ⋅
(5)
The currents in the earth wire(s) cause an alternating
magnetic field which has to be vectorially added to the
rotary magnetic field of the currents in the phase
conductors. The direction and amplitude of the current in
the earth wire strongly depends on phase allocation and
conductor arrangement. Therefore the current in the earth
wire must be calculated for each case of phase arrangement
of Tab. 1.
RESULTS
In the following the calculation results for electric and
magnetic fields are presented for tower designs according to
Fig. 1. Phase currents of 2300 A and a maximum voltage of
420 kV (phase-to-phase) were assumed, together with a
distance of 10 m from the ground to the lowest conductors.
The following figures show the resulting fields in a height
of 1 m above ground.
3-Level-Tower

In the following Fig. 2 the magnetic flux density in 1 m
above ground of a 3-level-tower for all 12 cases of
allocating the phases are shown. As can be seen, it is not
possible to point out a clearly best or worst case. Cases
No.3 and No.9 provide the maximum values of the magnetic
field in a region with a distance less than approx. 10 m from
the line axis. On the other hand, these cases are the ones
with the lowest values at a distance more than 18 m from
the line axis. Cases No.4 and No.10 contribute the lowest
maximum value but cause higher magnetic flux densities in
more distant points. This means that by choosing a phase
allocation with low magnetic flux density values directly
under the wires, higher values in an outer area will occur.
In Fig. 3 a zoomed area of Fig. 2 is presented to show that
there are 12 different characteristics. Two characteristics are
always very similar. The small deviation is due to the
contribution of the field caused by the current in the earth
wire to the field caused by the currents in the phase
conductors.
B
rms
in µT
x in m
-30
-20
-10
0
10
20
30
0
10
20
30
40
50
60
No.1
No.2
No.3
No.4
No.5
No.6
No.7
No.8
No.9
No.10
No.11
No.12

Fig. 2 Magnetic flux density of the 3-level-tower
C I R E D
20
th
International Conference on Electricity Distribution Prague, 8-11 June 2009

Paper 0628


CIRED2009 Session 2 Paper No 0628
B
e
in µT
x in m
-20
-18
-16
-14
-12
-10
20
25
30
35
40
No.1
No.2
No.3
No.4
No.5
No.6
No.7
No.8
No.9
No.10
No.11
No.12

Fig. 3 Zoomed area of Fig. 2

For the electrical field the maximum field strength directly
under the high-voltage-line (inner section) is interesting,
because e.g. trees and buildings may mitigate the electric
field. The highest maximum values under the high-voltage
line are caused by case No.1 and No.7, the lowest maximum
values by No.3 and No.9.
Contrary to the magnetic field there are only 6 different
electric field patterns for the 12 phase allocation cases. This
is because the earth wire doesn’t have the same effect it has
on the magnetic flux density.
Erms in kV/m
x in m
-30
-20
-10
0
10
20
30
0
1
2
3
4
5
6
No.1
No.2
No.3
No.4
No.5
No.6
No.7
No.8
No.9
No.10
No.11
No.12

Fig. 4 Electric field strength of the 3-level-tower

In the following the calculation is expanded from evaluation
of the field in one height to the evaluation of the whole
cross section. In Fig. 5 the cases with the highest values at
specific points can be identified.
2
10
1
3
4
7
9
10
1
3
4
7
8
9
x in
m
y in m

Fig. 5 Areas, where the different phase allocation cases
No. 1 to No. 12 cause the highest
values for magnetic
flux density
For example for a point 10 m above ground, in a distance of
40 m from the line axis, case No.7 provides the highest
values of magnetic flux density. For each point in a cross
section the maximum values can be calculated and
visualized in one graph, as done in Fig.6. By means of this
figure the highest value of the flux density can be evaluated
without knowing the exact position of the phases.
x in m
y in m
-50
-40
-30
-20
-10
0
10
20
30
40
50
-10
0
10
20
30
40
50

Fig. 6 Contour plot for the highest values of the
magnetic flux density of all 12 phase allocations cases for
a 3-level-tower in µT

For choosing an optimal phase position, the phase allocating
cases which are providing the lowest flux densities are of
more interest, thus a similar chart to Fig. 5 for the lowest
values is provided in Fig. 7. These results can be a useful
basis for designing overhead power lines to choose the
positions of the phases for a specific immission point.
10
3
4
9
11
1
6
7
3
9
x in m
y in m
2
8
11
11
11
12
5
2
8
6

Fig. 7 Areas where the different phase allocation cases
No.1 to No.12 provide the lowest
values of magnetic flux
density
2-Level-Tower

For the assumed 2-level-tower the phase allocation No.3
and No.9 causes the lowest maximum magnetic flux density
as well the lowest values in a distant area (distance to the
line axis larger than approx. 20 m) as shown in Fig. 8. Also
for the electrical field, phase allocating No.3 and No.9 are
the best choices (Fig. 9). No.2 and No.8 provide the highest
maximum values of magnetic flux density and electrical
field strength in the inner area.
C I R E D
20
th
International Conference on Electricity Distribution Prague, 8-11 June 2009

Paper 0628


CIRED2009 Session 2 Paper No 0628
Brms
in µT
x in m
-30
-20
-10
0
10
20
30
0
10
20
30
40
50
60
No.1
No.2
No.3
No.4
No.5
No.6
No.7
No.8
No.9
No.10
No.11
No.12

Fig. 8 Magnetic flux density of a 2-level-tower
E
rms
in kV/m
x in m
-30
-20
-10
0
10
20
30
0
1
2
3
4
5
6
No.1
No.2
No.3
No.4
No.5
No.6
No.7
No.8
No.9
No.10
No.11
No.12

Fig. 9 Electric field strength of a 2-level-tower
1-Level-Tower

In Fig. 10 and Fig. 11 calculation results for the 1-level-
tower are shown.
B
rms
in µT
x in m
-30
-20
-10
0
10
20
30
0
10
20
30
40
50
60
No.1
No.2
No.3
No.4
No.5
No.6
No.7
No.8
No.9
No.10
No.11
No.12

Fig. 10: Magnetic flux density of a 1-level-tower
Erms in kV/m
x in m
-30
-20
-10
0
10
20
30
0
1
2
3
4
5
6
No.1
No.2
No.3
No.4
No.5
No.6
No.7
No.8
No.9
No.10
No.11
No.12

Fig. 11: Electric field strength of a 1-level-tower
SUMMARY AND CONCLUSION
In Tab. 2 a summary of the calculation results is provided.
For each tower design in Fig. 1 the
• maximum value
• and the value in 50 m distance from the line axis
for the best and the worst case of phase allocation (lowest
and highest immission value) for the magnetic flux density
and the electric field strength in a height 1 m above ground
are presented.

No.Brms No.Brms No.Erms No.Erms
- µT - µT - kV/m - kV/m
best case 4/10 31,3 3/9 1,9 3/9 4,7 5/11 0,1
worst case 3/9 36,9 1/7 5,3 7 5,4 2/8 0,3
difference +18% +178% +15% +161%
best case 3/9 30,8 3/9 2,2 3/9 3,5 4/10 0,2
worst case 2/8 41,0 4/10 4,0 2/8 4,5 5/11 0,3
difference +33% +77% +27% +101%
best case 3/9 43,1 3/9 2,9 4/10 4,7 1/7 0,4
worst case 1/7 51,8 1/7 5,4 1/7 5,7 3/9 0,4
difference +20% +84% +21% +18%
3-level2-level1-level
maximum
magnetic flux density electric field strength
value in 50m maximum value in 50m

Tab. 2 Summary of the calculation

As can be seen in Tab. 2 there are great differences due to
different allocation of the phases – e.g. the maximum B
rms

caused by the worst cases of phase allocation (No.2 and
No.8) of the 2-level-tower is 33% higher than the maximum
B
rms
caused by the best cases of phase allocation (No.3 and
No.9) of this tower.
Furthermore one fact should be pointed out: The phase
allocation cases No.3 or No.9 for the 3-level-tower have the
highest maximum values of the magnetic flux density and
the lowest one in 50 m distance. That means if the phase
allocation is optimized for a distant point, stronger fields
have to be taken into account directly under the power lines.

REFERENCES

[1] J. Carson, 1929, "Wave Propagation in Overhead
Wires with Ground return", Bell System Technical
Journal, vol. 5, 539 - 554
[2] F. Pollaczek, 1926, "Über das Feld einer unendlich
langen wechselstromdurchflossenen Einfachleitung",
Elektrische Nachrichtentechnik, Heft 9, Band 3
[3] W. Friedl, E. Schmautzer, G. Rechberger, A. Gaun,
2007, "Constructional Magnetic Field Reducing
Measures Of High-voltage Overhead Transmission
Lines", CIRED 2007, Vienna, Paper 0597
[4] W. Friedl, E. Schmautzer, G. Rechberger, A. Gaun,
2007, "Aspects Concerning Electromagnetic Fields
with Conventional and Field-reduced High-voltage
Transmission Lines", ISH 2007, Ljubljana, Paper T1-
370