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Magnetic Field Generated

by Sagging Conductors ofOverhead Power Lines

JORDI-ROGER RIBA RUIZ,1 ANTONIO GARCIA ESPINOSA2

1EUETII, Universitat Polite cnica de Catalunya, Electrical Engineering Department, Catalunya, Spain

2Universitat Polite cnica de Catalunya, Electrical Engineering Department, Catalunya, Spain

Received 3 December 2008; accepted 6 April 2009

ABSTRACT: This article describes the implementation of the equations that govern thebehavior of magnetic fields generated by three-phase power lines. From the equations

obtained in this work, the profiles of the magnetic fields generated by different practical power

lines configurations have been analyzed. Results from simulations have been contrasted with

experimental data showing a close agreement between them. These values have been also

compared with current international regulations showing that in all the situations their values

fulfill the reference levels. Additionally, the effect of a conducting ground has been discussed.

It has been proved that in practical cases the effects from the earth currents can be neglectedas compared with effects from line currents. Furthermore special attention is paid to the

catenary shape of overhead conductors. The magnetic field generated by sagging conductors

has been calculated and compared with the one generated by straight equivalent conductors.

2009 Wiley Periodicals, Inc. Comput Appl Eng Educ; Published online in Wiley InterScience

(www.interscience.wiley.com); DOI 10.1002/cae.20365

Keywords: education; magnetic field; sagging conductors; conducting ground; simulation

INTRODUCTION

Nowadays, many laboratory experiments and class-

room lectures in the electrical engineering courses in

universities around the world are being assisted by

computer simulations [16]. The modeling of elec-trical systems has acquired great importance in

engineering education because of the huge increase

of importance of the computer-based systems inlearning methodologies. As a result, most technical

universities are including modeling sessions in many

courses of their curricula. The present article deals

with the Matlab package because it is widely used in

electrical engineering courses [711].It is well known that processes related to gen-

eration, transmission, distribution, and consumption

of electric power are associated with the generation of

low-frequency electric and magnetic fields (5060HzCorrespondence to J. R. Riba Ruiz

([email protected]).

2009 Wiley Periodicals Inc.

1


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and harmonic frequencies). The constant increase

in the worlds demand for electricity requires the

permanent construction of new electric power stations

and the extension of the existing transmission and

distribution power lines. Thus, it is very important that

electrical engineers acquire knowledge about the

possible effects of electric and magnetic fields aswell as providing them tools for computing and

predicting the values of the electric and magnetic

fields in the vicinity of high-voltage power lines.

In this work a fast method to simulate the

magnetic fields created by different geometries of

three-phase power lines is presented, including the

effects of a conducting ground and the effects of

sagging conductors. The objective of the proposed

system is not only to focus students in the interpre-

tation of a standard executable programs output

results, but also encourage them to understand the

physical and electrical laws involved in the compu-

tation of magnetic field. To meet these objectives it isvery useful that students are able to write the source

code of the program, because in this task their effort is

oriented towards analyzing and thoroughly under-

standing the steps involved in the computation of the

magnetic field generated by overhead power lines.

Although the electricity supply generally has

electric and magnetic fields associated with it, high-

voltage power lines are a particular source of exposure

to elevated levels of these fields.

It is well known that at 5060 Hz we will be inthe condition of near field, given that the distance

in which we find ourselves in relation to the power

line will always be inferior to the wavelength

corresponding to a frequency of 5060 Hz, which is6,0005,000 km [12]. According to the Maxwellequations that govern the behavior of fields and

electromagnetic waves, at 5060 Hz, on the one handthere will be practically no radiation in the form of

electromagnetic waves and, on the other hand, the

electrical and magnetic fields will be decoupled, that

is to say, they will be independent of each other. This

indicates that one cannot be deduced from the other

but rather that they have to be calculated separately. It

is also necessary to take into account that the

electrical field is due to the difference in voltagebetween the power line and the ground and that the

magnetic field is generated by the current transported

by the line.

The methodology proposed for computing the

magnetic field has been validated by the authors of

this work by comparing the results of the simulations

with experimental data and alternative computing

methods reported by Garrido et al. [13], resulting a

close similitude between them.

Note that most of the references that deals with

the computation of magnetic field in the vicinities of

power lines [1214] do not take into account thecatenary shape of sagging conductors. These studies

suppose that the overhead conductors are perfectly

flat, thereby the geometry of the problem is greatly

simplified. This simplification leads to a problem witha high degree of symmetry. Thus, under these condi-

tions the Amperes law can be applied, simplifying

significantly the problem which is the object of this

study. Moreover, so far, a detailed study of the

magnetic field generated by sagging conductors with

catenary shape has not been reported for educational

purposes.

MAGNETIC FIELD GENERATEDBY OVERHEAD POWER LINES

Mostly high-voltage power lines use a three-phasesystem in order to transport electric energy. This

means that they consist of three conductors with

alternating sinusoidal voltages of 50 Hz with equal

amplitude but out of phase with each other at an

electric angle of 1208. Therefore, this angle must be

taken into account in the calculation of the magnetic

field.

Calculation of the Magnetic FieldGenerated by an Infinite and IsolatedStraight Conductor

In this section a formula to determine the magnetic

field generated by an infinite and isolated straight

conductor is deduced. Figure 1 shows a straight

conductor along which a current Ii circulates, which

generates a magnetic field Bi in the space that

surrounds it.

The magnetic field generated by an infinite

straight conductor placed in position (xi,yi,zi) can be

determined by applying BiotSavart law [15],

Figure 1 Vectorial magnetic field created by a single

conductor.

2 RIBA RUIZ AND GARCIA ESPINOSA


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d~Bi moIi4p

d~l ^~rr3

1

being ~r x xi i y yi j 0k, mo 4p 107 N/A2 and d~l dzk has the same direction as Ii.Therefore, as deduced in Ref. [15] the result is

Bi mo2pi

Ii

ri2

From both, Figure 1 and Equation (2), it can also

be deduced that the vectorial magnetic field expres-

sion is given by the following formula:

where j is the phase angle of the current. It isnecessary to take into account that Ii and Bi are

sinusoidal alternating physical variables and it is

convenient to express their magnitudes in root mean

square (RMS) value.

Ground Presence: Image Methodology

The image methodology is useful to take into account

the effects of the presence of a conducting ground

[16]. The alternating magnetic field that the line

generates induces return currents in the ground and

these, at the same time, generate a magnetic field that

is superimposed upon that produced by the line.

The image theory states that the field generated

by a current-carrying wire, when placed at height yiabove a perfectly conducting ground, can be repre-

sented by the combined fields of the wire and its

image [17]. When the finitely conducting ground is

replaced with a perfectly conducting ground, standard

image theory can be used to locate the wire image

depth. As depicted in Figure 2, the image wire is

placed at a depth given by yi a, being a a complexdistance [18] which value depends on the ground

electrical conductivity as expressed in Equation (4).

a ffiffiffi

2p

d45 4

Being d the depth of penetration, given by

Darveniza [18]:

d ffiffiffiffiffiffiffiffir

pmf

r5

where r is the electrical resistivity of the medium

expressed as Om, m the magnetic permeability of the

medium expressed as N/A2, and f the magnetic fieldoscillation frequency in Hz. In the case of a perfectly

conducting ground it results d 0 and, thereforea 0. However, terrains of low conductivity give avery large penetration depth d and, therefore, a is also

very high. In the case of a perfect insulator the result is

a1.Practical ground resistivities fluctuate from

r< 50Om for especially conducting ground tor> 3,000Om for very poor conducting ground[19]. In order to calculate the resistance of

ground, r 100Om has been used. By substitutingin Equation (5) f

50 Hz, r

100Om, and

mmo 4p 107 N/A2 gives a result of d&711.76 m. This value is way superior to the distance

between overhead current carrying conductors and the

ground.

Calculation of the Magnetic FieldCreated by an Infinite and InsulatedStraight Conductor Taking Into Accountthe Effect of a Conducting Ground

We will work with the hypothesis that the

terrain is flat and homogeneous, that is tosay one with a uniform electrical resistivity.

As previously explained and shown in Figure 2,

the effect of the conducting ground is simulated

by another current in the opposite direction and of

the same intensity to the original current, that

circulates at a depth yi a under the airgroundinterface.

Therefore, the resulting vectorial expression of

the total magnetic field is the following:

~Bi moI

ji

2p y yix xi2 y yi2

;x xi

x xi2 y yi2; 0

!3

Figure 2 Real current and image current.

SAGGING CONDUCTORS OF OVERHEAD POWER LINES 3


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The terms (a) of the expression above are due to the

current-carrying conductor, while the terms (b) are

due to the image conductor.

When dealing with n parallel straight conductors

(it is the case in the majority of lines) the resultingmagnetic field would be determined using the

principle of superimposition:

~Bres Xni1

~Bi 7

EXPERIMENTAL VALIDATION OFTHE METHODOLOGY

The results of the method explained in this work have

been validated with experimental data and also havebeen compared with simulated results from other

authors. Using the method explained in Magnetic

Field Generated by Overhead Power Lines Section,

we proceed to study the distribution of the resultant

magnetic field in the proximity of two geometries of

overhead power lines. Geometric and electrical data

of the simulated high-voltage power lines shown in

Figures 3 and 4 have been collected from Ref. [13].

Note that Figure 4 shows two power lines next to

each other, both with the conductors in horizontal

arrangement.

Figure 5 shows the transverse profile at height 1 m

of the resultant magnetic field for a 132 kV horizontalline of a single-circuit with three conductors (config-

uration 1 in Fig. 3). The mean current in the

conductors is 482 A with a light unbalance between

phases (485, 472, and 488 A for phases R, S, and T,

respectively).

Figure 6 shows the transverse profile at height 1 m

of the resultant magnetic field for two horizontal lines

next to each other (configuration 2 in Fig. 4). The

current of the left line is 246 A whereas the current of

the right line is 226 A.

As shown in Figures 5 and 6, results from

simulations are in close agreement with those obtain-

ed from measurements by Garrido et al. [13].

Furthermore, the two simulation methods (conducting

and non-conducting ground) differs only in average

0.0628% for configuration 1 and 0.0164% for con-

figuration 2. Consequently, for most of the practical

applications the effect of the conductivity of the

ground can be neglected.

Bi;x moIji

2p y yix xi2 y yi2

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}a y yi ax xi2 y yi a2

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}b

0BBBB@

1CCCCA

Bi;y moIji

2p

x xix xi2 y yi2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflffl}

a

x xix xi2 y yi a2|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}b

0BBBB@1CCCCA

Bi;z 0

6

Figure 3 Geometric arrangement for configuration 1. Figure 4 Geometric arrangement for configuration 2.



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MAGNETIC FIELD OF SAGGINGOVERHEAD POWER LINES

Conductors of overhead power lines hang over the

earth surface acted upon by its own weight, being the

catenary the theoretical shape of a hanging flexible

cable [20]. Expressions of the magnetic field deduced

in Magnetic Field Generated by Overhead Power

Lines Section do not take into account this effect. In

this section the magnetic field generated by sagging

overhead conductors is deduced.

The equation of the catenary shape of conductor i

placed in the yz plane is given by,

yi a cos h zia

C 8Being a and C constants which are determined

from the boundary conditions applicable in each

conductor.

The magnetic field generated by a sagging

conductor of an overhead power line with span L

between pylons can be determined similarly as done

in Calculation of the Magnetic Field Generated by an

Infinite and Isolated Straight Conductor Section.

Figure 7 shows the profile of the catenary of an

overhead conductor.

From the geometry shown in Figure 7 it results,

d~l dyij dzik sinh zia

dzij dzik 9

By applying the BiotSavart law described inEquation (1), the magnetic field generated by a

sagging conductor in any point placed at mid-span

plane (lowest height of the conductors above the

ground) is given by,

Bi;x moI

ji

4p

ZL=2L=2

sin hzi=az zi y yix xi2 y yi2 z zi23=2

dzi

Bi;y m

oIj

i

4pZL=2

L=2x

x

ix xi2 y yi2 z zi23=2 dzi

Bi;z moI

ji

4p

ZL=2L=2

sinhzi=ax xix xi2 y yi2 z zi23=2

dzi

10(xi,yi,zi) being the coordinates of the conductor and Lthe

distance between two adjacent towers (span).

Figure 5 Measured and simulated resultant magnetic field

at a height y 1 m for configuration 1. The current carryingconductors are placed 12.12 m above the ground.

Figure 6 Measured and simulated resultant magnetic field

at a height y 1 m for configuration 2.

Figure 7 Sagging conductor of an overhead power linewith span L.



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The integrals of Equation (10) are according to

Figure 8a and must be computed numerically (Matlab

provides functions for easily computing these inte-

grals). To calculate the magnetic field in any point

which is not placed at mid-span, it should proceed as

explained in Figure 8b, where the catenary of the rightside is the same as described in Equation (8) but

shifted a distance L.

As explained in Calculation of the Magnetic Field

Created by an Infinite and Insulated Straight Con-

ductor Taking Into Account the Effect of a Conduct-

ing Ground Section, in the case of there being n

parallel sagging conductors, the resulting magnetic

field would be determined applying the superimposi-

tion principle. Additionally, the effect of a conducting

ground can be modeled by adding in Equation (10) the

magnetic field due to the image conductors, similarly

as done in Calculation of the Magnetic Field Createdby an Infinite and Insulated Straight Conductor

Taking Into Account the Effect of a Conducting

Ground Section.

Figure 9 shows the pseudocode for implementing

the proposed computation system in Matlab.

In order to compare both methodsstraight and

sagging conductorsa practical 500 kV power line

shown in Figure 10 has been simulated [21]. The

current per phase is 1,000 A and the spandistance

between two adjacent pylonsis 400 m. For the con-

figuration shown in Figure 10, it results the parameters

a

2001.66 m and C

hi

a, being hi the lowest

height of the ith conductor above the ground (height atmid-span).

Figure 11 shows the mid-span lateral profile

along x-axis of the magnetic field generated by

configuration 3.

Figure 12 shows the longitudinal profile along

z-axis of the magnetic field generated by line 3, which

has been computed at a height y 1 m.As deduced from Figures 11 and 12, the effect of

the catenary shape of the sagging conductors is very

little. The magnetic field due to both methods of

simulation, it is to say, straight and sagging con-

ductors lead to nearly the same results.

Thus, it can be concluded that in practical

overhead power lines, in most of the points of interest,

experimental measuring errors can be greater than the

difference between the results of the two methods.

Figure 8 Sagging conductor of an overhead power line.

The horizontal distance between two towers is L.

Figure 9 Pseudocode of the basic program for computing

the magnetic field generated by n sagging conductors.

Figure 10 Geometric arrangement for configura-

tion 3.



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INTERNATIONAL REGULATIONS

In this section different international regulations

regarding the exposure of workers and the general

public to low-frequency magnetic fields are studied.

According to International Commission on Non-

Ionizing Radiation Protection (ICNIRP) [22], the

occupationally exposed population consists of adults

who are generally exposed under known conditions

and are trained to be aware of the potential risk and to

take appropriate precautions. By contrast, the general

public comprises of individuals of all ages and of

varying health status, and may include particularly

susceptible groups or individuals.

As regards to the exposure of the general public toextremely low-frequency magnetic fields in Europe,

there are the Recommendations of the European

Council 1999/519/EC [23].

In order to limit the exposure of workers to

magnetic field effects, it is important to take into

account the recommendations of the European

Directive 2004/40/EC [24] and the Threshold Limit

Values established by the American Conference of

Governmental Hygienists (ACGIH) [25]. According

to the European Directive 2004/40/EC it is nowconsidered necessary to introduce measures protect-

ing workers from the risks associated with electro-

magnetic fields, owing to their effects on the health

and safety of workers.

Table 1 summarizes the limit values of the

magnetic field established by the different regulations

cited previously. It should be noted that the values

given in Table 1 are related to 50 Hz.

When analyzing the results shown in this article

we find that the maximum values of the magnetic field

generated by the analyzed power lines configurations

and calculated at 1 m above the ground are noticeably

inferior to the limits proposed by the differentinternational regulations.

COURSE DESCRIPTION ANDSTUDENTS ASSESSMENT

This article is useful and suitable in electric and

physics education since the proposed method enables

students to understand the laws involving magnetic

fields as well as acquiring knowledge about the

profiles and practical values of these fields in the

vicinity of power lines.

The proposed method and program were taught in

the Electrotecnia course, undertaken by the

authors, to 78 undergraduate students, in the first

semester of the 20072008 academic year, in theUPC (Universitat Politecnica de Catalunya, Spain).

Electrotecnia is an undergraduate course in the fifth

semester of a 5-year degree course in Industrial

Engineering at the UPC. The duration of the course is

5 lecture hours a week and presents the following

contents:

Figure 11 Mid-span lateral profile of the resultant

magnetic field at a height y 1 m for configuration 3.

Figure 12 Longitudinal profile of the resultant magnetic

field along z-axis (x 0 m ) at a height y 1 m forconfiguration 3.

Table 1 Limit Values of the Magnetic Field Proposed

by the Different International Regulations

Regulation

Magnetic field of 50 Hz (lT)

General public Occupational

ICNIRP 100 500

1999/519/EC 100

2004/40/EC 500

ACGIH (TLV) 1200



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* Module 1. Three-phase systems.* Module 2. Fundamentals of electrical machines.* Module 3. Transformers.* Module 4. Induction machines.* Module 4. Direct current machines

Note that the theoretical concepts related topower lines are included in module 1.

The students of Electrotecnia course have

previously completed the following courses, whose

contents are needed in order to better understand the

proposed practical:

* Electricity and Magnetism (third semester).* Circuits and Systems Theory (fourth semester).* Electrical Engineering Fundamentals (fifth

semester).

In this subject, students have eight 2-h practical

laboratory sessions, one of which is that presented inthis article. The study presented here is done in one

2-h session. During the session the lecturer guides the

students so that they can program the calculation of

magnetic field in the Matlab code. The students work

in groups of 23 and in the following weeks practicalsession they must hand in a 4- to 5-page report which

includes a brief introduction, the program code, the

graphic results of the magnetic fields generated by

each one of the high-voltage power lines and also their

final conclusions, outlining if the configurations

studied comply with current regulations. In this way,

as well as technical knowledge an environmental

point of view is given to students in order to stimulatetheir interest in such topics, which knowledge will be

needed when they become engineers.

A survey was undertaken afterwards in which

students were able to assess different aspects with

reference to their satisfaction of the practical

simulation of magnetic fields and its usefulness in

consolidating their knowledge of this material. The

survey consisted of five questions shown in Table 2.

The students should grade them from 1 (very poor) to

5 (excellent).

Whereas the first two questions make reference to

the students previous knowledge, questions 35make reference to their degree of satisfaction with the

proposed methodology.

Figure 13 shows the global results obtained fromthe students assessment. From these results we

concluded that the practical session was well accepted

by students. This result means that this system moti-

vates students and also helps them to better under-

stand the undergraduate Electrotecnia course.

Table 3 shows the average scores for each

question. The average response for the last three

questions was 4.09. This overall mark indicates a

satisfactory degree of student approval for the metho-

dology presented in this work.

CONCLUSION

In this article a realistic method has been shown with a

low computational burden to simulate the magnetic

field that electrical power lines generate. This system

enables the simulation of the magnetic field created by

the majority of overhead lines and presents a series of

advantages such as the ease of being able to study

different types of lines, as well as the possibility of

predicting the magnetic field that projected lines will

generate.

Special attention has been paid to simulate the

effects of a conducting ground as well as the effects of

the catenary shape of sagging overhead conductors.

As proved in Experimental Validation of the Method-

ology and Magnetic Field of Sagging Overhead Power

Lines Sections, these effects can be neglected in most

of the situations with no noticeable error.

Table 2 Questionnaire Answered for the Students

Questions Assessment

1. I had a previous interest about themagnetic fields generated by power lines

2. I had a previous knowledge of Matlab

3. I think that these type of simulations help

to understand theoretical concepts

4. I have understood the procedure to

calculate the magnetic field generated by

power lines

5. I think that this practical is profitable for

an engineer Figure 13 Results of the students assessment.



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The aim of the proposed system is to encourage

students to understand the physical and electrical laws

involved in the computation of magnetic field as well

as interpreting the programs output results. To meet

these objectives it is very useful that students are able

to write the source code of the Matlab program,

because in this task their effort is oriented towards

analyzing and understanding thoroughly the steps

involved in the computation of magnetic field.Results from simulations through applying the

method explained in this work have been compared

with experimental data available in the technical

literature, showing a close agreement between the

two.

In all the case studies the magnetic field

calculated is way inferior to the limits imposed by

the various regulations studied.

Overhead high-voltage power lines usually have

conductors that act as lightning rods, which are

earthed (generally this connection is undertaken in

every tower). These conductors are not active (they

do not carry electrical power), but some currents

are introduced which in accordance with section 5.3.1

of Ref. [26] can be of 50 mA for every km of

line (supposing that there is no earth connection in

between). Therefore, these currents are very weak

and the magnetic fields that they generate can be

neglected.

The methodology proposed in this work has been

applied to the Electrotecnia course at the UPC

(Universitat Politecnica de Catalunya, Spain).

By means of a questionnaire answered by the

students on the Electrotecnia course, a successful

degree of satisfaction was expressed with the method-ology proposed in this work.

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Question 2 2.94

Question 3 4.06

Question 4 4.10

Question 5 4.12



10/10

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[26] IEEE Standard 524a-1993. IEEE Guide to Grounding

During the Installation of Overhead Transmission Line

Conductors. Supplement to IEEE Std. 524-1992.

Transmission and Distribution Committee of the IEEE

Power Engineering Society.

BIOGRAPHIES

Jordi-Roger Riba Ruiz received the M.S. in

Physics and Ph.D. degrees from the Uni-

versitat de Barcelona (UB) in 1990 and 2000,

respectively. In 1992, he joined the College

of Industrial Engineering of Igualada (Uni-

versitat Politecnica de Catalunya, UPC) as a

full time Lecturer and in 2001 he joined the

Electric Engineering Department of the UPC.His research interests include signal process-

ing, electromagnetic devices, electric machines, variable-speed

drive systems and fault detection algorithms. He belongs to the

Motion and Industrial Control Group (MCIA). The Groups major

research activities concern induction and permanent magnet motor

drives, enhanced efficiency drives, fault detection and diagnosis of

electrical motor drives and improvement of educational tools.

Antonio Garcia Espinosa (M05). He

received his electrical engineering degree

and the Ph.D. degree from the Universitat

Politecnica de Catalunya (UPC) in 2000 and

2005 respectively. In 2000, he joined the

Electric Engineering Department of the UPC,

where he is currently Lecturer. His research

interests include electromagnetic devices,electric machines, variable-speed drive sys-

tems and fault detection algorithms. He belongs to the Motion and

Industrial Control Group (MCIA).


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