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

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