magnetic field calculation underehvtransmission lines for more

2008 5th International Multi-Conference on Systems, Signals and Devices

Magnetic Field Calculation under EHV Transmission Lines for MoreRealistic Cases

Adel Zein E. M.

Department of Electrical EngineeringHigh Institute of Energy, South Valley University

Aswan, Egypt

Abstract-Ground level electric and magnetic fields from overhead power transmission lines are of increasingly importantconsiderations in several research areas. Common methods for the calculation of the magnetic fields created by power transmissionlines assume straight horizontal lines parallel to a flat ground and parallel with each other. The influence of the sag due to the lineweight is neglected or modeled by introducing an effective height for the horizontal line in between the maximum and minimumheights of the line. Also, the influences of the different heights of the towers, the different distances of the power transmission linesspans and the different angles between the power transmission lines' spans are neglected. These assumptions result in a model wheremagnetic fields are distorted from those produced in reality. This paper investigates the effects of the sag in case of different heights ofthe towers and when the power transmission lines' spans are not parallel to each other.

Index Terms- OHTL, Magnetic Field

(1)

where

1 a parametric position along the current path,

I (1) the line current,

r" (1) a vector from the source point (x,Y,z) to the field point(xo,yo,zo),

a(I). . h d· . r (I) do unIt vector In t e IrectIon 0 , an

dl a differential element at the direction of the current.

_.....-....._--....~......--_..........--.z

B. The 3-D Integration Technique

In fact, the power transmission lines are nearly periodiccatenaries, the sag of each depends on individualcharacteristics of the line and an environmental conditions.The integration technique is a three-dimensional techniquewhich views the power transmission conductor as a catenary.In the integration technique, if the currents induced in theearth are ignored, then the magnetic field of a single currentcarrying conductor at any point P(xo,yo,zo) shown in Fig. (1)can be obtained by using the Biot-Savart law [2-4], as:

A. The 2-D Straight Line Technique

The common practice is to assume that power transmissionlines are straight horizontal wires of infinite length, parallel toa flat ground and parallel with each other. This is a 2DStraight line Technique, which can be found in manyreferences [2-5].

II. MAGETIC FIELD CALCULATIONS

I. INTRODUCTION

PRECISE analytical modeling and quantization of electricand magnetic fields produced by overhead power

transmission lines are important in several research areas.Considerable research and public attention are concentrated onpossible health effects of extremely low frequency (ELF)electric and magnetic fields [1]. An analytical calculation ofthe magnetic field produced by electric power lines isproduced in [2], which is suitable for flat, vertical, or deltaarrangement, as well as for hexagonal lines. Also theestimation of the magnetic field density at locations under andfar from the two parallel tran~mission lines with differentdesign arrangements is presented in [3]. The effects ofconductors sag on the spatial distribution of the magnetic fieldare presented in [4], in case of equal heights of the towers,equal spans between towers and the power transmission lines'spans are always parallel to each others.

In this paper, the magnetic field is calculated by twodifferent techniques; Two-Dimensions Straight line Techniqueand Three-Dimensions Integration Technique, where the effectof the sag in the magnetic fields calculation, and the effects ofunequal span distances between the towers, unequal towersheights, and when the power transmission lines' spans are notparallel to each other are investigated.

Fig. 1. Application of the Biot-Savart law.

The exact shape of a conductor suspended between twotowers of equal height can be described by such parameters; as

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(10)

(11 )

d;

H = I;(x-xo)y d.

1

I;[(z - Zo + kl)sinh(~) - (y - Yo)]H = a

x

d=[(X-xo)2+(Y_Yo)2+(Z-ZO)2]3/2 (8)

This result can be extended to account for the multiphaseconductors in the support structures. For (M) individualconductors on the support structures, the expression for thetotal magnetic field becomes:

_ 1 M N 1/2 (9)

Ho =-I I f(Hxox +HyOy + Hzoz)dz4Jr ;=1 k=-N -1/2

where:

where:

(2)

tvt

y

"'! ZY =h+2a sinh L (-)

2awhere a is the solution of the transcendental equation:

h -h L2-"_1-u = sinh 2 (u) U=-

L ·w~ 4a,

the distance between the points of suspension span, L, the sagof the conductor, S, the height of the lowest point above theground, h, and the height of the highest point above theground, hm• These parameters can be used in differentcombinations. Only two paramt~ters are needed in order todefine the shape of the catenary (S and L), and the third one (hor hm), determines its location in relation to the groundsurface. Figure (2) depicts the basic catenary geometry for asingle-conductor line, this geometry is described by:

(19)

(18)

(17)

(15)

(14)

(12)

the skin depth of the earth represented by[5];

t5=503Jpl f

- Ii(x-xo)sinh(~) Ii(x-xo)sinh(~)H = a + a

Z d; d;

di' = [(x - x0) 2 + (y +Yo + S)2 + (z - z0 + kl) 2

] 3/2

where;g

p the resistivity of the earth in n.m,

f the frequency of the source current in Hz.

The resultant magnetic field with the image currents takeninto account is also represented by equation (9), but itscomponents will change and take the following formulas:

z (16)Ii [(z - Zo + kl)sinh(-) - (y - Yo)]

Hx

= adi

Ii [z - Zo + kl)sinh(~) - (Yo +Y + ()]a

- I; (x - xo)sinh(~)H

z= a

d;

d; =[(X-xo)2 +(y-Yo)2 +(z-zo +kl)2]3/2 (13)

The parameter (N) in equation (9) represents the number ofspans to the right and to the left from the generic one, asexplained in Fig.(2). One can take into account part of themagnetic field caused by the image currents. The complexdepth ~ of each conductor image current can be found as givenin [4].

(7)

(5)

(3)

(4)

dl = dYGy +dzGz

dT =dz(: Oy +oJ

dT = dz(sinh(~)iiy + oJa

~ = (xo- x)iix + (Yo - Y)Gy + (zo - z)iiz (6)

where point (xo,yo,zo) is the field point at which the field willbe calculated, and point (x,y,z) is any point on the conductorcatenary. Now, by substituting equations (5) and (6) intoequation (1), and carrying out the cross product, the result atany point (xo,yo,zo) is :

x~

Fig. 2. Linear dimensions which determine parameters of thecatenary.

The parameter a is also associated with the mechanical

f h 1· a = Th I W h ~. h dparameters 0 t e Ine: were IS t e con uctor

tension at mid-span and W is the weight per unit length of theline.1) Case (A)

In Case A, the power transmission lines specified by; equalheights of the towers, equal spans between towers and thepower transmission lines' spans are always parallel to eachothers (0=0). For a single span single conductor catenary,represented by equation (2), since the modeled curve islocated in the y-z plane, the differential element of thecatenary can be written as:



~•••••••.•. Hx_._._ .• Hy

25

o 5 10 15 20 25 30 35 40Distance from the center phase (m)

Fig. 4. The computed magnetic field intensity by using the 2DStraight Line Technique.

o _~----L-__----.L-__ ----.L-~_~


III. ANALYSIS OF MAGNETIC FIELDS TECHNIQUES

To calculate the Magnetic field intensity at points one meterabove ground level, under 500kV TL single circuit, the data inappendix (A) are used.

Figure (4) shows the computed magnetic field intensity andits components with and without the effect of the imagecurrents, by using the 2-D Straight Line Technique, where theaverage heights of the transmission lines are used, sincetypical values for the resistivity of earth range from 10 toIOOOn.m, the image currents are normally located at hundredsof meters below the ground [6], and do not effect the magneticfield intensity levels especially in areas close to theconductors. Figure (5) shows the computed magnetic fieldintensity and its components under a single span with theeffect of the image currents, at the mid-span ( where themaximum sag, point PI in Fig. (2)), by using the 3-Dintegration technique (Case A).

h -h Ll+L'2 m2 U =sinh 2 (u) with u= and the same

LI+L' ' 4aequations as in case (A) is used, with the integration limits

-LI-L' , LI+L'from +L to ---

2 2

Fig. 3. The presentation of Case (C)

ty

2) Case (B)In Case B, the power transmission lines specified by; equal

heights of the towers, equal spans between towers and thepower transmission lines' spans are not parallel to each others.The two catenaries Land L2, in Fig.(2), each have its originalpoint and coordinate system. The field points are located onaxis X of system (X,Y,Z) of L catenary. This field pointsshould be transferred to the coordinate system of the cataneryunder calculation. By applying this rule on field points andcaterany L, it is seen that the same equations of case (A) areused, where the field points are already in caterany L system.But for caterany L2, the field points should be transferred tothe caterany L2 system.

For any field point (xl,yl,zl), that can be done in threesteps:1- Transfer the original of caterany L2 to the field point

system. From Fig.(3), for - 90 < B < 90

z =~ + L2 cos(()), x = - L2 sin(()), and y =0c 2 2 c 2 c

2- Transfer the field point (xl,yl,zl) from its system to thesystem (U,V,W) of the caterany under calculation L2, fromappendix (B):

zI-z xI-xWI =. C sin(p) ,uI = C cos(P) , and

sln(p +8) cos(P +8)

vI =yI- YC' where (xc,yc,zc) is the original point of system

(U,V,W) refer to system (X,Y,Z), which calculated in step (I),

P-1 zI-z

and =tan ---8xI-x

3- Finally use this point (ul,vl,wl) in the same equations ofcase A.

By the superposition technique, the magnetic field at anyfield point from many catenaries can be calculated.ReviewStage3) Case (C)

In Case C, the power transmission lines specified by;unequal heights of the towers, unequal spans between towersand the power transmission lines' spans are always parallel toeach others.

Figure (3) presents a catenary L1, which have unequalheights of its towers (hmhhm2). In this case a is the solutionof the transcendental equation:

Fig. 5. The computed magnetic field intensity by using the 3DIntegration Technique (point PI).

Figure (6) shows the computed magnetic field intensity andits components under a single span with the effect of the



image currents, at maximum tower height (point P2 in Fig.(2)), by using the 3-D integration technique (Case A).

10 15 20 25 30Distance from the center phase (m)

Distancebetweenthetwotowers(m)Distance from 1I1e center phase (m)

i 15

t-l =~ =-I - " ":"

~ 10 - I

5 -

o

Fig. 8. The presentation of the 3D magnetic field intensity distributionat 1m above ground level under 500kV TL with the effect of image

currents, by using the 3D Integration Technique.4035

\\

\~/

/

/

II

I

II

I

45 ~-~--~.~_.~~-~-----~-~~-_. __._-~-~--_._--

~~I--theta=O

----- theta=5•• ..00 ..• theta=10

.......... theta=20...._._._ •• theta=40

--theta=O

----- theta=5.......... theta=10

-.-.-.• theta=20

····00··· theta=40~5

Fig. 10. The effect of the angle eon the magnetic field intensitycalculated under mid-span.

o __L_~ , ,~,Gao)o 5 10 15 20 25 30 35 40

Distance from the center phase (m)

Fig. 9. The effect of the angle eon the magnetic field intensitycalculated under tower height.

Figure (11) shows the effect of the span length on thecalculated magnetic field intensity under a single span, attower height. It is seen that as the span length decreased, themagnetic field intensity decreases.

Figure (10) shows the same results as in Fig.(9), except that,the calculation points are at mid-span, it is noticed that, theeffect of angle e is higher in this case because both the twoends of the span go far from the calculation points as the anglee increased.

::-~~-~~~::~:~m.~:mlf: ----- al po;"1 P2, span <2,

3AiII,

Ql 25E~ 20'u..(,)

J:: -------------------------------- _

Fig. 7. The effect of the spans' n Jmbers on the magnetic fieldintensity.

Fig. 6. The computed magnetic field intensity by using the 3DIntegration Technique (point P2).

Figure (7) shows the effect of the number of spans (N) onthe calculated magnetic field intensity. It is noticed that, whenthe magnetic field intensity calculated at point PI (Fig.2) and adistance a way from the center phase, the effect of the spans'number is very small due to the symmetry of the spans aroundthe calculation points. Also it was seen that as the number ofthe spans (N) is greater than 2 the result of the calculatedmagnetic field intensity is the same, that due to the fardistance between the current source and the field points. Forthis reason the number of spans does not exceed 4. Figure (8)shows the presentation of 3D computed magnetic fieldintensity, with the effect of the image currents, by using the 3D integration technique (with span number N=4). It is noticedthat, the magnetic field intensity varies with the position of thefield points between the two towers and also with theirdistance from the center phase, where in the 2D straight linetechnique; it varies only with the field points' distance fromthe center phase. Figure (9) shows the effect of the angle e asexplained in case (B) on the calculated magnetic field intensityof a single span under a tower height and a way from thecenter phase. It is seen that aj the angle e increased themagnetic field intensity decreases, that due to the increases ofthe distance between the current source and the field points.

:(""=.~=~..==~='=."== .."=.,"':',==~'.o 5 10 15 20 25 30 35 40

Distance from the center phase (m)



45 -~,--~--,-------r-------,----------r--------,-~-r-----l s~n=4OOm40 ----- span=200m

.::~= span=100m35 1

~J

!25

1

~ 20

1

'~ 15~

~ ':1 ~... ----------'--------------"--~________L_ -----l-__~__


Fig. 12. The effect of the span length on the magnetic field intensitycalculated under mid-span

lengths, and various difference between the towers' heights,that at tower height and mid-span respectively. From both twotables it seen that the difference between the towers' heightshave a small effect, when the magnetic field intensitycalculated at tower height, but when the magnetic fieldintensity calculated at mid-span it have a greater effect,especially when this difference is equal to the sag itself.

Fig. 11. The effect of the span length on the magnetic field intensitycalculated under :ower height.

Figure (12) shows the same results as in Fig.(ll), exceptthat, the calculation points are at mid-span, it is noticed that,the effect of span length is very small in this case because theeffect of the conductor height is greater than its span lengtheffect. Tables I and II present a comparison between themagnetic field intensity calculated with both 20 straight linetechnique, where the average conductors' heights are used, and3D integration technique, with various angles 0, various span

TABLE I

COMPARISON BETWEEN THE RESULTS OF 30 INTEGRATION TECHNIQUE WITH VARIOUS PARAMETERS AT TOWER HEIGHT AND 20 STRAIGHT LINE TECHNIQUE

Distance 2D straight 3D integration technique Single span at point P2 (tower height) (Aim)from the line technique

Angle (8) (deg.) Span (L) (m) Different between towers'center with averagephase heights With: L=400m, LL=Om With: 8 =Odeg, LL=Om heights (LL) (m);

(m) (Aim) With: 8 =Odeg, L=400m8=0 8=10 8=40 L=400 L=350 L=300 LL=O LL=10 LL=S

0 25.236 6.824 6.824 6.824 6.824 1.824 0.630 6.824 6.808 6.79210 23.619 6.337 5.817 4.674 6.337 1.633 0.592 6.337 6.324 6.31320 15.218 4.852 4.044 2.660 4.852 1.206 0.494 4.852 4.849 4.84630 7.957 3.202 2.482 1.399 3.202 0.820 0.372 3.202 3.207 3.21040 4.584 2.081 1.547 0.765 2.081 0.587 0.262 2.081 2.090 2.097

TABLE IICOMPARISON BETWEEN THE RESULTS OF 3D INTEGRATION TECHNIQUE WITH VARIOUS PARAMETERS AT MID-SPAN AND 2D STRAIGHT LINE TECHNIQUE

Distance 2D straight 3D integration technique Single span at point PI (mid-span) (Aim)from the line technique

Angle (8)(deg) Span (L) (m) Different between towers'center with averagephase heights With: L=400m, LL=Om With: e=Odeg, LL=Om heights (LL) (m);

(m) (Aim) With: 8 =Odeg, L=400m8=0 8=10 8=40 L=400 L=350 L=300 LL=O LL=10 LL=S

0 25.236 40.796 6.690 0.476 40.796 40.718 40.702 40.796 40.335 20.39810 23.619 39.499 3.953 0.422 39.499 39.433 39.371 39.499 39.152 19.75020 15.218 21.381 2.624 0.375 21.381 21.350 21.321 21.381 21.534 10.69130 7.957 9.164 1.877 0.335 9.164 9.164 9.172 9.164 9.357 4.58240 4.584 4.959 1.414 0.300 4.959 4.969 4.986 4.959 5.061 2.479

IV. CONCLUSIONS

The 2-D Straight Line and 3-I) Integration Techniques givetwo choices for calculating magnetic field. The 2-D StraightLine is a rough approximation, and the 3-D Integration is anexact solution, however it requires integration over the threephase spans which results in a large computation time. It isseen that by using the 3D Integration Technique the Zcomponent of the magnetic field intensity appears, where thiscomponent is always equal zero in the 20 Straight Line

Technique. Under 3D Integration Technique, the paper presenta multi-special cases to calculate the magnetic field intensity,by using these cases, it is possible to calculate the magneticfield intensity at any point under a complex configurations ofa power transmission lines. Also it is possible to use the sametechnique, with some treatment, in the calculation of theelectric field under overhead transmission lines.



REFERENCES

[1] Hanaa Karawia, Kamelia Youssef and Ahmed Hossam-Eldin"Measurements and Evaluation of Adverse Health Effects ofElectromagnetic Fields from Low Voltage Equipments" MEPCON2008, Aswan, Egypt, March 12-15 ,PP. 436-440.

[2] George Filippopoulos, and Dimitris Tsanakas " Analytical Calculationof the Magnetic Field Produced by Electric Power Lines" IEEETransactions on Power Delivery, Vol. 20, No.2, pp. 1474-1482, April2005.

[3] A. A. Dahab, F. K. Amoura, and W. S. Abu-Elhaija "Comparison ofMagnetic-Field Distribution of Noncompact and Compact ParallelTransmission-Line Configurations" IEEE Transactions on PowerDelivery, Vol. 20, No.3, pp. 2114-2118, July 2005.

[4] A. V. Mamishev, R. D. Nevels, and B. D. Russell "Effects of ConductorSag on Spatial Distribution of Power Line Magnetic Field" IEEETransactions on Power Delivery, Vol. 11, No.3, pp. 1571-1576, July1996.

[5] Rakosk Das Begamudre,"Extra High Voltage AC. TransmissionEngineering" third Edition, Book, Chapter 7, pp.172-205, 2006 WileyEastern Limited.

[6] G. 1. Anders, G. L. Ford and D. 1. Horrocks" The Effect of MagneticField on Optimal Design of a Ring-Bus Substaion" IEEE Transactionson Power Delivery, Vol. 9, No.3, July 1994.

(B.11) I Xl = Xc +XX

vy

Figure B.l Cartesian coordinates of two systems in space

(8.12) I

ApPENDIX (A)

To calculate the Magnetic field intensity under 500kV TLsingle circuit, the following data are used.Tower span 400mNumber of subconductor per phase 3Diameter of a subconductor 30.6mmSpacing between subconductor 45cmMinimum clearance to ground 9mOuter phase Maximum height 22mInner phase Maximum height 24.35mDistance between adjacent two phases 13.2m

(B.18)

(B.20)

(B.17)

(B.19)

(B.16)

(B.14)

(B. 15)

(B.13)

By substituting (B.6) and (B.9) into (B. 1):

zl-zWI = c sin(fJ)

sin(fJ + 8)By substituting (B.8) and (B. 10) into (B.3):

xl-xuI = c cos(fJ)

cos(fJ + B)

and; vI == yl- Yc

1 ZZ 1 zl-zwhere: fJ == tan - -() == tan- __c -()

xx xl-xc

and; Yl = Yc +v1

P-1 wI

where: = tan -ul

B2: To transfer any point (x1,y1,z1) in (Xr:Z) system to a point(u1, v1,w1) in (u, V, W) system;

By substituting (B.5) and (B.2) into (B.11):

ZI =Zc + --;L- sin(fJ + 8)sln(fJ)

By substituting (B.7) and (B.4) into (B.12):

uXl = Xc + 1 cos(fJ + 8)

cos(fJ)

B1: To transfer any point (u1,v1,w1) in (u, V, W) system to a point(x1,y1,=1) in (X r:Z) system;

(B.4)

(B.2)

(B.3)

(B.l)

~ =L' cosf/J)

ApPENDIX (B)

Assume two coordinates' systems (X,Y,X) and (U,V,W) ina space, where axis U and axis W in system (U,V,W) make anangle ewith axis X and axis Z in system (X,Y,Z) respectively,while axis V and axis Yare parallel to each other, and originalof the system (U,V,W) located at point (xc,yc,zc) referred tosystem (X,Y,Z), as indicated in Fig.(B.1). Any point P inspace can be presented by the two system as (x1,y1,z1) insystem (X,Y,Z) and (ul,vl,wl) in system (U,V,W).

From Fig. (B. 1), it is seen:

zz=L'sin(j3+8)

xx=L" cos(j3+ 8)

(B.5)

(B.7)

zzL"==-----

sin(j3+8)

r' == X_X__

cos(/J+8)

(B.6)

(B.8)

Adel Zein E. M. was born in Egypt 1971. He received his B. Sc., M. Sc andPh. D. degrees in electric engineering from the High Institute of Energy,Aswan, Egypt in 1995, 2000 and from Kazan State Technical University,Kazan, Russia in 2005, respectively. His fields of interest include electric andmagnetic fields, Comparison between the Numerical techniques inElectromagnetic, and Calculation of SAR in the Human Body.

zz= zl-zc (B.9) xx=xl-xc (B.I0)


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