mitigation plates for the magnetic ﬁeld of underground cables · mitigation plates for the...

Mitigation plates for the magnetic field of undergroundcables

Pedro Daniel Vieira Morgado

Thesis to obtain the Master of Science Degree in

Electrical and Computing Engineering

Supervisor(s): Professor Vítor Manuel de Oliveira Maló MachadoProfessor Maria Eduarda de Sampaio Pinto de Almeida Pedro

Examination Committee

Chairperson: Professor João Manuel Torres Caldinhas Simões VazSupervisor: Professor Maria Eduarda de Sampaio Pinto de Almeida Pedro

Member of the Committee: Professor Paulo José da Costa Branco

November 2014

Dedicated to Ana Gomes.

iii

Acknowledgments

I would like to thank my supervisors, Prof. Vitor Malo Machado and Prof. Maria Eduarda Pedro, for all

the support and all their dedication.

I am grateful to my parents and sisters who were always by my side, for all the support and important

advices over the college journey.

v

Resumo

O objectivo deste trabalho e estudar o efeito da utilizacao de chapas enterradas horizontalmente entre

a superfıcie do solo e cabos subterraneos, na mitigacao do campo magnetico a superfıcie. O modelo

proposto utilizado permite o calculo do valor do campo de inducao magnetica, induzido pela corrente

que percorre os cabos subterraneos. Este modelo leva em conta a permeabilidade magnetica e con-

dutividade eletrica do solo e das chapas utilizadas, bem como da frequencia de transporte e espessura

das chapas.

A validacao do modelo e feita com base na comparacao com o programa informatico ”Finite Element

Method Magnetics” (FEMM) , que permite recriar um problema semelhante ao analisado mas com um

tempo de resolucao consideravelmente maior.

O modelo em estudo e formulado partindo da solucao geral para o potencial vetor magnetico asso-

ciado a cada regiao (ar , solo e chapa condutora). A imposicao das condicoes fronteira nas superfıcies

de separacao entre os diversos meios permite a obtencao da solucao final do modelo. A obtencao

de resultados e feita com base na implementacao de um programa informatico com vista a resolucao

numerica do problema.

Com este estudo pode concluir-se que a profundidade da chapa no subsolo nao influencia o valor do

campo magnetico a superfıcie, para baixos valores da condutividade do solo. A frequencia e a espes-

sura da chapa tem influencias diferentes dependendo da geometria da chapa e das caracterısticas do

material. Existe a possibilidade de alcancar um determinado valor do campo manipulando a geometria

da chapa e as caracterısticas do material que a compoe.

Palavras-chave: mitigacao do campo magnetico, cabos subterraneos, campo inducao magnetica,

potencial vetor magnetico

vii

Abstract

The purpose of this work is to study the effects in the magnetic field mitigation of using horizontally buried

plates between the soil/air interface and underground power cables. The proposed model implemented

allows the calculation of the induction magnetic field induced by the current flowing within underground

power cables. This model takes into account the magnetic permeability and electrical conductivity of the

soil and used plate. The frequency and plate thickness are also considered.

The model validation is made by comparison with a computer program called ”Finite Element Method

Magnetics” (FEMM) which allows the formulation of a problem similar to the analysed one but being

slower then the proposed model.

The studied model is formulated beginning with the general solution for the magnetic vector potential

inside each region (air , soil and conducting plate). The final solution is achieved by using the boundary

conditions on the interfaces between each two different regions. The numerical results are obtained by

implementing a computational program in order to solve the problem numerically.

With this study it was possible to conclude that the depth in which the plate is buried does not

influence the magnetic field value at the surface, due to the small value of the soil electrical conductivity.

The frequency and plate thickness influence differently the magnetic field value, depending on the plate

material characteristics. It is possible to achieve a certain magnetic field value by manipulating the plate

geometry and material characteristics.

Keywords: magnetic field mitigation, underground power cables, magnetic induction field, mag-

netic vector potential

ix

Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Introduction to the text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Methodology 3

2.1 Magnetic field problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Region 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Region 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.3 Region 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.4 Region 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Magnetic field problem solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Integrals calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Simpson’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Fourier integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Model Validation 14

4 Results 19

4.1 Influence of the plate Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Influence of the plate thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3 Influence of the frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 Influence of the ratio t/δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.5 Influence of the height to earth surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.6 Horizontal vs Triangular configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Conclusions 37

xi

Bibliography 39

A Plate depth demonstration 41

xii

List of Tables

3.1 Brms deviation at x = 0m and y = 0m using the detailed model and FEMM. . . . . . . . . 18

4.1 Penetration depth δ calculated for all three materials and f = 50Hz . . . . . . . . . . . . 23

4.2 Penetration depth δ calculated for various frequencies and materials. . . . . . . . . . . . . 26

4.3 Plate thickness t calculated for a desired Fr by using the ratio t/δ. . . . . . . . . . . . . . 28

xiii

List of Figures

2.1 Illustrative figure of a system consisting in three underground cables with a plate buried

between it and the surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1 Plate buried over a three phased underground cable in flat configuration. . . . . . . . . . 15

3.2 FEMM schematic used to simulate the system on figure 3.1 . . . . . . . . . . . . . . . . . 15

3.3 Close up view of the schematic used in FEMM. . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Brms calculated without plate for y = 0 using the proposed model and FEMM; f = 50Hz . 16

3.5 Brms with an Aluminium plate obtained for y = 0 using the proposed model and FEMM;

f = 50Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.6 Brms with a Steel100 plate obtained for y = 0 using the proposed model and FEMM;

f = 50Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Brms profile for different aluminium plate depths calculated for f = 50Hz and t = 3mm . . 20

4.2 Magnetic field factor of reduction Fr calculated for plate thickness between 0 and 10mm

using an aluminium plate with f = 50Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21


using a Steel100 plate with f = 50Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21


using a Steel500 plate with f = 50Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22


for all three materials and f = 50Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.6 Magnetic field factor of reduction Fr calculated for plate depths between 0 and 10mm

using an Aluminium plate and a Steel100 plate with f = 50Hz . . . . . . . . . . . . . . . . 23

4.7 Magnetic field factor of reduction Fr calculated for frequencies between 0 and 10kHz

using an Aluminium plate with t = 3mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.8 Magnetic field factor of reduction Fr calculated for frequencies between 0 and 1.6kHz

using a Steel100 plate with t = 3mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.9 Magnetic field factor of reduction Fr calculated for frequencies between 0 and 400Hz

using a Steel500 plate with t = 3mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.10 Magnetic field factor of reduction Fr calculated for frequencies between 0 and 100Hz for

all materials with t = 3mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

xv

4.11 Magnetic field factor of reduction Fr calculated for frequencies between 0 and 1000Hz

using an Aluminium plate and a Steel100 plate with t = 3mm . . . . . . . . . . . . . . . . 26

4.12 Magnetic field factor of reduction Fr calculated for tδ between 0 and 1 for all materials with

t = 3mm; y = 0m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.13 Magnetic field factor of reduction Fr calculated for tδ between 0 and 1 using a Steel100

plate and a Steel500 plate with t = 3mm; y = 0m . . . . . . . . . . . . . . . . . . . . . . . 27

4.14 Magnetic field factor of reduction Fr calculated for tδ between 0 and 1 using an Aluminium

plate and a Copper plate with t = 3mm; y = 0m . . . . . . . . . . . . . . . . . . . . . . . . 28

4.15 Fr′ calculated for y between 0 and 4m for all materials. t = 1mm;f = 50Hz . . . . . . . . 29



4.18 Fr′′ calculated for y between 0 and 4m for all materials. t = 1mm;f = 50Hz . . . . . . . . 31



4.21 Fr′ calculated for y between 0 and 4m for all materials. t = 3mm and f = 100Hz . . . . . 33

4.22 Fr′′ calculated for y between 0 and 4m for all materials. t = 3mm and f = 100Hz . . . . . 33

4.23 Plate buried over a three phased underground cable in triangular configuration. . . . . . . 34

4.24 Magnetic field factor of reduction Fr calculated for plate depths between 0 and 3mm for

all three materials and flat and triangular(line with triangles on the plot) configurations;f =

50Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.25 Magnetic field factor of reduction Fr calculated for frequencies between 0 and 100Hz for

all materials and flat and triangular(line with triangles on the plot) configurations;t = 3mm 35

4.26 Fr′ calculated for y between 0 and 4m for all materials and flat and triangular(line with

triangles on the plot) configurations;t = 3mm;f = 50Hz . . . . . . . . . . . . . . . . . . . 36

4.27 Fr′ calculated for y between 0 and 4m for all materials and flat and triangular(line with

triangles on the plot) configurations;t = 3mm;f = 50Hz . . . . . . . . . . . . . . . . . . . 36

xvi

Nomenclature

δ Penetration depth.

δs Soil penetration depth.

µ Magnetic permeability.

µ0 Vacuum magnetic permeability.

µs Soil magnetic permeability.

ω Angular frequency.

σ Electrical conductivity.

σs Soil electrical conductivity.

A Magnetic vector potential.

B Magnetic induction field.

Brms root mean square value of the magnetic induction field.

h Buried cable depth.

H(2)1 Hankel function of second kind of order one.

Ik Conductor k current.

J1 Bessel function of the first kind of order one.

nc Number of power cables.

re External cable radius.

s Distance between the center of the cables.

x Cartesian coordinate representing an horizontal distance.

y Cartesian coordinate representing a vertical distance.

xvii

Chapter 1

Introduction

1.1 Motivation

Magnetic fields are a common reality since men started to make use of electrical energy. Nowadays

all developed countries make extreme use of electricity. In order to supply every need it becomes

necessary to built up power lines to transport the energy from the source to the consumer. The final

consumers make use of the energy using several kinds of appliances and domestic devices. From the

power generation until the consumer magnetic fields are induced by the electrical currents.

Magnetic fields have not the same value in every place. Lines transporting more current induce

higher fields. The same happens to the domestic appliances. In the last decades, the concern about

the effects of magnetic fields in the human health has been growing. Several studies have been made

and today many countries impose limits to the value of the magnetic induction field being this value in

permanent discussion to be reduced in the future.

Overhead power lines are more common then underground ones, although underground cables are

used often in places where the landscape must be preserved or in urban places. The necessity to

diminish the magnetic field of underground cables becomes a matter of concern, therefore in this thesis

it is studied an effective way to mitigate the magnetic fields induced by underground cables. References

[1] , [2] are examples of studies where the magnetic field in the air induced by underground cables is

calculated, yet in this studies there is no plate between the cables and the surface. In [3] the magnetic

field is calculated when using a non-infinite plate buried above the cables. In the present work the plate

is considered to be infinite in order to solve the problem analytically.

1

1.2 Introduction to the text

This thesis consists in 5 chapters. In the first chapter it is made an introduction to the magnetic field

values induced by overhead lines and underground cables.

In the second chapter the problem consisting on the mitigation of the magnetic induction field at the

surface caused by underground power cables using a plate is formulated. It is proposed a model in order

to solve the problem and it is explained how the integrals in the model were calculated.

In chapter three the proposed model is validated and in chapter four the magnetic induction field

in the air is calculated when using a plate buried over the underground cables and changing various

parameters. The obtained results are presented and commented.

In the final chapter the conclusions formulated with this work are presented.

2

Chapter 2

Methodology

2.1 Magnetic field problem formulation

Figure 2.1 illustrates a scenario where three underground cables are buried at a depth h carrying an

alternate current with phasor I. Between the cable and the surface is a plate made of a certain material.

Each region is associated to a specific magnetic permeability µk and electrical conductivity σk.

In order to reach the final solution for the magnetic field at the surface constituted by the earth air

interface it will be used the magnetic vector potential solution A inside each region.

The final solution will be achieved considering that:

- each region is a linear and homogeneous space separated from each other by a plane surface;

- the cables are buried at a constant depth;

- the magnetic field is originated by a three-phased axial current flowing through a system of three

underground cables;

- the magnetic field is only dependent on the transversal coordinates;

- the quasi-static field approximation is considered;

3

Figure 2.1: Illustrative figure of a system consisting in three underground cables with a plate buriedbetween it and the surface.

To simplify the problem the three cables were considered to be only one and that there is no plate

applying afterwards the superposition of the results associated to each cable.

Now based on the previous assumptions and [1] ,the resulting phasor of the magnetic vector potential

A is given by:

A = A−→u z, A = A(x, y) (2.1)

Within the homogeneous soil without any mitigation plate and due to the quasi-static approximation,

A must satisfy :

∇2A− jωµsσsA = 0 (2.2)

and is obtained by the linear combination of two linear independent terms A′

and A′′

. A′

takes into

consideration the presence of the soil/air plane and A′′

takes into account the presence of the soil/cable

cylindrical surface.

4

From [1] A′

and A′′

may be written as:

A′=

+∞∫−∞

F (a)ey√a2−q2sejax da, y < 0 (2.3)

A′′=

1

π

+∞∫−∞

e−|y+h|√a2−q2sG0W0(a)Ike

jax da (2.4)

F (a) is a function to be determined and a represents the integration variable with the meaning of a

space frequency, with qs and W0(a) being given by:

qs =

√2

δse−j

π4 , δs =

√2

ωµsσs(2.5)

where δs is the penetration depth inside the soil,

W0(a) =j√

a2 − q2s, y > −h (2.6)

G0 is achieved by applying the boundary conditions to the cable surface as indicated in [1] :

Ac = A1

1

µc∂Ac∂n =

1

µ1

∂A1

∂n

(2.7)

where Ac and A1 are the magnetic vector potential solutions inside the cable and region 1 respec-

tively, and n is the coordinate normal to the surface.

G0 =µs

2πqsre

1

H(2)1 (qsre) + J1(qsre)b0,0

(2.8)

with

b0,0 =2j

π

+∞∫0

e−j2qshcoshae−2a da (2.9)

and where H(2)1 and J1 are the Hankel function of the second kind of order one and the Bessel

function of the first kind of order one respectively.

5

Therefore the generic solution for the magnetic vector potential A becomes:

A =

+∞∫−∞

(F (a)ey√a2−q2s +

1

πe−|y+h|

√a2−q2sG0W0(a)Ik)e

jax da (2.10)

where Ik is the phasor of the current flowing trough cable k. From the previous result as a general-

ization is now possible to deduct the magnetic potential solution inside each region represented in figure

2.1 .

2.1.1 Region 1

Region 1 is the region that contains the cable, the boundary plane with region 2 is located at y = −(d+ t2 )

and the cable at y = −h , therefore, from (2.10):

A1 =

+∞∫−∞

(F1(a)e(y+(d+ t

2 ))√a2−q21 +

1

πe−|y+h|

√a2−q21G0W0(a)I)e

jax da (2.11)

where q1 has the same meaning of qs but now it is calculated assigning the parameters of region 1:

q1 =√ωµ1σ1e

−j π4 (2.12)

2.1.2 Region 2

Region 2 represents the mitigation plate where the boundary plane with region 3 is located at y =

−(d− t2 ), so, according to the presence of the upper and lower boundary planes the following result can

be obtained:

A2 =

+∞∫−∞

[D1(a)e(y+(d− t2 ))

√a2−q22 +D2(a)e

−(y+(d− t2 ))√a2−q22 ]ejax da (2.13)


q2 =√ωµ2σ2e

−j π4 (2.14)

6

2.1.3 Region 3

Region 3 represents the soil, although different electrical and magnetic characteristics may be assigned

to it. The boundary plane with region 4 is located at y = 0 , and an analogous result as (2.13) may be

found :

A3 =

+∞∫−∞

[R1(a)ey√a2−q23 +R2(a)e

−y√a2−q23 ]ejax da (2.15)


q3 =√ωµ3σ3e

−j π4 (2.16)

2.1.4 Region 4

The magnetic potential solution in the air is achieved by solving now a new fundamental field equation,

as a consequence of a null conductivity:

∇2A4 = 0 (2.17)

where the solution may be given by:

A4 =

+∞∫−∞

U(a)e−|a|yejax da (2.18)

With U(a) being a function to be determined by imposing the appropriate boundary conditions.

7

2.2 Magnetic field problem solution

The magnetic vector potential solution for each region is now possible to achieve applying the boundary

conditions on each interface.

From references [1] , [2] the boundary conditions correspond to the following two conditions on each

interface:

- the continuity of the magnetic vector potential;

- the continuity of the tangential component of the magnetic field strength;

Previous conditions may be represented as:

Ak = Ak+1

1

µk

∂Ak∂y =

1

µk+1

∂Ak+1

∂y

(2.19)

with k = 1, 2, 3 .

For plane y = 0 the following equation system can be obtained applying (2.19) to equations (2.15)

and (2.18):

R1(a) +R2(a) = U(a)√a2 − q23µ3

[R1(a)−R2(a)] = −|a|µ0U(a)

(2.20)

getting to:

R1(a) =

1

2

(1− |a|

µ0

µ3√a2 − q23

)U(a)

R2(a) =1

2

(1 +|a|µ0

µ3√a2 − q23

)U(a)

(2.21)

For plane y = −(d− t2 ) applying (2.19) to equations (2.13) and (2.15):

D1(a) +D2(a) = R1(a)e

−(d− t2 )√a2−q23 +R2(a)e

(d− t2 )√a2−q23√

a2 − q22µ2

[D1(a)−D2(a)] =

√a2 − q23µ3

[R1(a)e

−(d− t2 )√a2−q23 −R2(a)e

(d− t2 )√a2−q23

] (2.22)

8

replacing (2.21) in (2.22) :

D1(a) +D2(a) =

[ch(ξ) +

µ3|a|µ0

√a2 − q23

sh(ξ)

]U(a)

D1(a)−D2(a) = −µ2

√a2 − q23

µ3

√a2 − q22

[sh(ξ) +

µ3|a|µ0

√a2 − q23

ch(ξ)

]U(a)

(2.23)

with

ξ = (d− t

2)√a2 − q23 (2.24)

from (2.23) :

D1(a) =

1

2

[(1− µ2|a|

µ0

√a2 − q22

)ch(ξ) +

(µ3|a|

µ0

√a2 − q23

− µ2

√a2 − q23

µ3

√a2 − q22

)sh(ξ)

]U(a)

D2(a) =1

2

[(1 +

µ2|a|µ0

√a2 − q22

)ch(ξ) +

(µ3|a|

µ0

√a2 − q23

+µ2

√a2 − q23

µ3

√a2 − q22

)sh(ξ)

]U(a)

(2.25)

for plane y = −(d+ t2 ) applying (2.19) to equations (2.11) and (2.13):

F1(a) +G1(a) = D1(a)e

−t√a2−q22 +D2(a)e

t√a2−q22√

a2 − q21µ1

[F1(a)−G1(a)] =

√a2 − q22µ2

[D1(a)e

−t√a2−q22 −D2(a)e

t√a2−q22

] (2.26)

with

G1(a) =1

πe−(h−(d+

t2 ))√a2−q21G0W0(a)I (2.27)

replacing (2.25) in (2.26) :

F1(a) +G1(a) = X(a)U(a)√a2 − q21µ1

[F1(a)−G1(a)] = −√a2 − q22µ2

X ′(a)U(a)(2.28)

9

with

X(a) =

[ch(ν) +

µ2|a|µ0

√a2 − q22

sh(ν)

]ch(ξ) +

[µ3|a|

µ0

√a2 − q23

ch(ν) +µ2

√a2 − q23

µ3

√a2 − q22

sh(ν)

]sh(ξ) (2.29)

X ′(a) =

[sh(ν) +

µ2|a|µ0

√a2 − q22

ch(ν)

]ch(ξ) +

[µ3|a|

µ0

√a2 − q23

sh(ν) +µ2

√a2 − q23

µ3

√a2 − q22

ch(ν)

]sh(ξ) (2.30)

where

ν = t√a2 − q22 (2.31)

From (2.28) it can be deducted :

F1(a) =

√a2 − q21µ1

X(a)−√a2 − q22µ2

X ′(a)√a2 − q21µ1

X(a) +

√a2 − q22µ2

X ′(a)

G1(a)

U(a) =

2

√a2 − q21µ1√

a2 − q21µ1

X(a) +

√a2 − q22µ2

X ′(a)

G1(a)

(2.32)

Replacing U(a) of (2.32) in (2.18) it is possible to obtain the final solution for the magnetic induction

field in the air due to nc cables, as it is done analogously in [1] and being given as:

Bx =

nc∑k=1

−j 2G0Ikπ

+∞∫−∞

|a|e−(hk−(d+ t2 ))√a2−q21√

a2 − q21X(a) +µ1

µ2

√a2 − q22X ′(a)

e−|a|yeja(x−xk)da

(2.33)

By =

nc∑k=1

2G0Ikπ

+∞∫−∞

ae−(hk−(d+t2 ))√a2−q21√

a2 − q21X(a) +µ1

µ2

√a2 − q22X ′(a)

e−|a|yeja(x−xk)da

(2.34)

where (xk,−hk) are the coordinates of the cable k and Ik the current flowing in cable k . Bx and By

are the axial and transversal components of the magnetic induction field B respectively.

10

Finally, the root mean square of the magnetic induction field Brms in the air is given by:

Brms =

√B ·B∗

2=√B2xrms +B2

yrms (2.35)

11

2.3 Integrals calculation

2.3.1 Simpson’s rule

In order to solve the integral on ( 2.9 ) it was used the composite Simpson’s rule. This rule may be

written as in [4] :

c∫b

f(a) da ≈ 3

h

f(a0) + 2

n2−1∑j=1

f(a2j) + 4

n2∑j=1

f(a2j−1) + f(an)

(2.36)

where

h = (c− b)/n , aj = b+ jh for j = 0, 1, ..., n− 1, n and a0 = b , an = c

The function f(a) tends to 0 when a tends to infinite which means that the value of the integralc∫b

f(a) da will not practically change if the upper limit c rises. This means that it is possible to use a small

value ε that maximizes f(a) for a certain a which will be afterwards assigned to c. Using ε the upper limit

c may be found as follows:

f(a) = e−j2qshcoshae−2a

=

e−j2hδs

(1−j) ea+e−a

2

=

e−jhδs

ea+e−a1 e−

hδs

ea+e−a1 e−2a

|e−jhδs

ea+e−a1 e−

hδs

ea+e−a1 e−2a| ≈ e−

hδseae−2a

The function absolute value must be smaller than ε :

e−hδseae−2a < ε

⇔

e−hδsea−2a < ε

neglecting e−2a:

ehδsea > 1

ε

⇔hδsea > ln

(1ε

)⇔

ea > δsh ln

(1ε

)⇔

a > ln(δsh ln

(1ε

))

12

Finally the upper limit of the integral is given by:

c = ln(δsh ln

(1ε

))The calculations in this work were made with ε = 10−10.

2.3.2 Fourier integrals

The integrals on equations 2.33 and 2.34 are Fourier integrals and can be written as:

amax∫−amax

|a|e−(hk−(d+ t2 ))√a2−q21√

a2 − q21X(a) +µ1

µ2

√a2 − q22X ′(a)

e−|a|ye−jaxkejaxda (2.37)

amax∫−amax

ae−(hk−(d+t2 ))√a2−q21√

a2 − q21X(a) +µ1

µ2

√a2 − q22X ′(a)

e−|a|ye−jaxkejaxda (2.38)

Using a function IFFT from a computational program [5] it was possible to calculate the inverse

discrete Fourier transform while paying attention to the following relations:

xmax = 2πN4amax

(i) , δa = 2amaxN (ii) , δx = xmax

N/2 (iii)

The IFFT output function will be defined between −xmax and xmax and the space between these

points will be determined by N with the relation (iii). Having xmax and N it is possible to determine amax

from (i). Then, from (ii) δa is easily calculated.Being δa the descretization of the Fourier integral between

−amax and amax. For these particular problem the maximum distance xmax of interest is relatively small,

therefore from (i) and (iii) amax or N can be changed in order to obtain a better discretization. As long

as xmax does not get to small the better option is to reduce amax and fix N so that the calculation time

does not rise.

13

Chapter 3

Model Validation

In order to use the proposed model to perform the desired studies, it was necessary to validate it. The

validation was made using an application called FEMM (Finite Element Method Magnetics) which allows

solving several electromagnetic problems.

To perform the validation it was considered a 400kV cable which configuration is displayed in figure

3.1. The used parameters were:

- s = 21.8cm;

- re = 7.4cm

- h = 1.5m;

- t = 3mm;

- d = 1.352m;

- f = 50Hz;

- µ1 = µ0 , µ2 = µ0 (Aluminium) or µ2 = 100µ0 (Steel100) , µ3 = µ0;

- σ1 = σ3 = 0.01Sm−1 , σ2 = 3.5× 107Sm−1 (Aluminium) or σ2 = 1× 107Sm−1 (Steel100);

- the currents flowing through cable are given as:

Ik =√2Iefe

−j(k−1) 2π3

Ief = 1995A

with k = 1, 2, 3 .

14

Figure 3.1: Plate buried over a three phased underground cable in flat configuration.

On FEMM the problem was created using a non infinite border where the region representing the air

is 98m long and 60m high. Analogously the region representing the soil is 98m long and 60m high. The

boundary between the soil and air is represented exactly between them as depicted in figure 3.2. All the

remaining parameters were assigned and represented exactly as in the proposed model.

Figure 3.2: FEMM schematic used to simulate the system on figure 3.1

Figure 3.3 shows a closer view of the FEMM schematic.

15

Figure 3.3: Close up view of the schematic used in FEMM.

Three different cases were calculated using both the proposed model and FEMM. The case where

there is no mitigation plate, the case with an aluminium plate and the case with a steel100 plate, all cal-

culated for y = 0m. The results for the three cases are displayed in figures 3.4 , 3.5 and 3.6 respectively.

Figure 3.4: Brms calculated without plate for y = 0 using the proposed model and FEMM; f = 50Hz

16

Figure 3.5: Brms with an Aluminium plate obtained for y = 0 using the proposed model and FEMM;f = 50Hz

Figure 3.6: Brms with a Steel100 plate obtained for y = 0 using the proposed model and FEMM;f = 50Hz

To facilitate the analysis of the different plots the deviation between Brms at x = 0m and y = 0m in

the two applications was calculated and grouped in table 3.1 .

17

without plate Aluminium Steel100deviation(%) 0.2000 0.0874 0.8600

Table 3.1: Brms deviation at x = 0m and y = 0m using the detailed model and FEMM.

The results using the two programs are very similar and the deviation between the two Brms values

sustain this conclusion, although even the small deviation can my be explained by two facts:

- FEMM results are highly dependent on the chosen mesh size that can be applied to every region

on the schematic;

- FEMM does not work with semi-infinite regions;

Due to the previous results it is possible to say that the detailed Model was successfully validated.

18

Chapter 4

Results

In this chapter it is made a presentation and analysis of the results obtained using the Model presented

in chapter 2 .

Several aspects such as plate positioning, plate thickness effect, frequency and plate electrical and

magnetic characteristics were studied in order to understand the behaviour of the magnetic field in the

surface, when a plate is buried between the power cables and the surface.

The assigned parameters are the same used in chapter 3 but adding Steel500 with (σ2 = 1×107Sm−1

; µ2 = 500µ0) and Copper with (σ2 = 5.9× 107Sm−1 ; µ2 = µ0) to the studied plate materials.

19

4.1 Influence of the plate Depth

In appendix it is shown using the proposed model that the plate’s depth does not affect the Brms value

at the surface, for small values of soil electrical conductivity. The plot in figure 4.1 verifies that same

conclusion for a soil conductivity equal to 10−2Sm−1.

Figure 4.1: Brms profile for different aluminium plate depths calculated for f = 50Hz and t = 3mm

The plots in figure 4.1 are perfectly overlapped showing that the plate positioning has no influence

on Brms for small values of soil electrical conductivity.

20

4.2 Influence of the plate thickness

It is expectable that the thickness of the plate used to mitigate the absolute value of the magnetic field

influences the obtained results. Figures 4.2, 4.3 and 4.4 illustrate the variance of Brms at x = 0m and

y = 0m for different plate materials when the plate thickness t changes. This variance is implicit in Fr

which is equal to the ratio between Brms without plate and Brms with plate. As d does not affect the

results the variation of t is made by fixing d and changing the upper and lower side of the plate at the

same time.

Figure 4.2: Magnetic field factor of reduction Fr calculated for plate thickness between 0 and 10mmusing an aluminium plate with f = 50Hz

Figure 4.3: Magnetic field factor of reduction Fr calculated for plate thickness between 0 and 10mmusing a Steel100 plate with f = 50Hz

21

Figure 4.4: Magnetic field factor of reduction Fr calculated for plate thickness between 0 and 5mm usinga Steel500 plate with f = 50Hz

Looking at each particular case it is possible to acknowledge that the steel with µ2 = 500µ0 is the

most effective when the plate thickness increases, followed by the µ2 = 100µ0 steel. Fr has an almost

linear behaviour with an Aluminium plate until t = 10mm although with both steels it appears to be

clearly nonlinear.

Adding all results together in a plot as in figure 4.5 helps seeing the different reduction factors.

Figure 4.5: Magnetic field factor of reduction Fr calculated for plate thickness between 0 and 5mm forall three materials and f = 50Hz

Aluminium plate has the greater Fr of the three materials until a thickness near 2.4mm but the

steel500 plate has the highest Fr after that thickness starting to grow faster. Until 5mm the steel100

has the worst results, although as we have seen above it surpasses Aluminium at a certain point. This

behaviour becomes more clearer in figure 4.6.

22

Figure 4.6: Magnetic field factor of reduction Fr calculated for plate depths between 0 and 10mm usingan Aluminium plate and a Steel100 plate with f = 50Hz .

The magnetic field behaviour for the same frequency but different thicknesses is directly related to

the ”Penetration Depth” introduced in ( 2.5 ) of chapter 2.

Only the material properties and the working frequency affect the value of δ ,although if δ does

not change but thickness t grows, the field capability to penetrate the plate decreases. This capability

decreases faster or slower depending on the ratio between t and δ. Various penetration depths were

calculated and grouped in table 4.1 .

Aluminium Steel100 Steel500δ(mm) 12 2.3 1

Table 4.1: Penetration depth δ calculated for all three materials and f = 50Hz

Both steels have reduced penetration depths compared to aluminium but their magnetic permeability

is higher which leads to contradictory behaviours.

23

4.3 Influence of the frequency

The frequency has an important roll in the way the magnetic field manifests itself at the surface.

In order to analyse the frequency effect, the plate thickness was fixed at 3mm and the depth was the

same used in section 4.2. Then, a frequency variation was applied being the obtained results displayed

in figures 4.7, 4.8 and 4.9.

Figure 4.7: Magnetic field factor of reduction Fr calculated for frequencies between 0 and 10kHz usingan Aluminium plate with t = 3mm

Figure 4.8: Magnetic field factor of reduction Fr calculated for frequencies between 0 and 1.6kHz usinga Steel100 plate with t = 3mm

24

Figure 4.9: Magnetic field factor of reduction Fr calculated for frequencies between 0 and 400Hz usinga Steel500 plate with t = 3mm

All the three materials apparently have the same behaviour, although the frequency range to achieve

those high Fr values are not so similar. Aluminium reaches an Fr near 11000 for f = 10kHz, Steel100

just needs 1.6kHz to achieve almost Fr = 14000 while Steel500 gets Fr = 10000 when f = 400Hz. All

materials can be compared in fig 4.10.

Figure 4.10: Magnetic field factor of reduction Fr calculated for frequencies between 0 and 100Hz forall materials with t = 3mm

25

The results of Aluminium and Steel100 can be compared in figure 4.11.

Figure 4.11: Magnetic field factor of reduction Fr calculated for frequencies between 0 and 1000Hzusing an Aluminium plate and a Steel100 plate with t = 3mm

For t = 3mm Steel500 has the higher Fr value even at a low frequency such as 50Hz as seen in

section 4.2. Aluminium has a better Fr then Steel100 until 100Hz, even though with the enlargement of

the frequency interval Steel100 starts to achieve better results.

Penetration depth δ is directly influenced by the frequency. On table 4.2 are displayed some δ values

for three different frequencies calculated using the expression in Section 4.2.

Aluminium Steel100 Steel500 f (Hz)δ(mm) 12 2.3 1 50δ(mm) 2.7 0.503 0.225 1000δ(mm) 0.85 0.159 0.0712 10000

Table 4.2: Penetration depth δ calculated for various frequencies and materials.

Penetration depth rapidly decreases when frequency increases. Even testing a 3mm thick plate

which is in general a small thickness value shows that the magnetic field is highly reduced. As in section

4.2 the ration between t and δ is determinant on the mitigation of the magnetic field, yet this time the

parameter changing is δ and not t.

26

4.4 Influence of the ratio t/δ

In sections 4.2 and 4.3 it became clear that the magnetic field mitigation depends on the ratio between

the thickness and penetration depth of the used plate. In order to study the impact of tδ on the field

mitigation, Fr was calculated for all three materials varying tδ . The obtained results are displayed in

figures 4.12, 4.13 and 4.14.

Figure 4.12: Magnetic field factor of reduction Fr calculated for tδ between 0 and 1 for all materials with

t = 3mm; y = 0m

Figure 4.13: Magnetic field factor of reduction Fr calculated for tδ between 0 and 1 using a Steel100

plate and a Steel500 plate with t = 3mm; y = 0m

27

Figure 4.14: Magnetic field factor of reduction Fr calculated for tδ between 0 and 1 using an Aluminium

plate and a Copper plate with t = 3mm; y = 0m

From figure 4.12 it becomes clear that aluminium makes a better mitigation for tδ higher than a few

hundredths. Steel500 as a higher Fr than steel100 until a value near 0.6 but the behaviours invert after

that. In figure 4.13 the results show that for high penetration depths the materials with higher magnetic

permeabilities are better for mitigation and for small δ values the mitigation is higher for small µmaterials.

This results show that for a high µ and δ the magnetic field lines tend to penetrate de plate but do not

cross it. For small µ and δ the field lines tend to stay bellow the plate.

The plot in figure 4.14 was obtained for an Aluminium plate and a Copper plate. The difference

between these two materials is the electrical conductivity. The curves are overlapped showing that the

plate σ does not affect the results in terms of t/δ.

These results are relevant to size the plate thickness, for instance table 4.3 shows the thickness that

should be used to achieve Fr = 10, with frequency equal to 50Hz, for different plate materials.

Aluminium Copper Steel100 Steel500f (Hz) 50 50 50 50t/δ 0.196 0.196 1.584 2.214t(mm) 2.4 1.8 3.6 2.2

Table 4.3: Plate thickness t calculated for a desired Fr by using the ratio t/δ.

28

4.5 Influence of the height to earth surface

When trying to ensure a certain Brms value for some specific height, the reduction factor Fr along the

height y becomes an important aspect.

Two different reduction factors were calculated. Fr′ which is the ratio between Brms without plate at

(x = 0 , y = 0) and Brms with plate at (x = 0 , y ≥ 0). Fr′′ is the ratio between Brms without plate at

(x = 0 , y ≥ 0) and Brms with plate at that precise location.

Fr′ and Fr′′ were calculated for different materials and thickness where y varies from 0m to 4m.

Frequency was maintained constant at 50Hz. Results are displayed in figures 4.15, 4.16, 4.17, 4.18,

4.19 and 4.20.

Figure 4.15: Fr′ calculated for y between 0 and 4m for all materials. t = 1mm;f = 50Hz

29



30

Figure 4.18: Fr′′ calculated for y between 0 and 4m for all materials. t = 1mm;f = 50Hz


31


Figures 4.15 , 4.16 and 4.17 show that the magnetic field value for a particular y is influenced by the

plate’s material and thickness. In this particular case δ does not change for each material as frequency

stays at 50Hz, a fact that explains the variance between results when t changes as explained in sections

4.2 and 4.3 .

Considering figures 4.18 , 4.19 and 4.20, it is discernible that the Fr′′ crossing points are the same

when material changes. The relative positioning between curves is also preserved. The only difference

is that Fr′′ has a linear grow unlike Fr′ which grow is undoubtedly nonlinear.

Figures 4.21 and 4.22 illustrate the changes on Fr′ and Fr′′ for f = 100Hz and t = 3mm to be

compared with the results in figure 4.16 and figure 4.19 , respectively.

32

Figure 4.21: Fr′ calculated for y between 0 and 4m for all materials. t = 3mm and f = 100Hz

Figure 4.22: Fr′′ calculated for y between 0 and 4m for all materials. t = 3mm and f = 100Hz

33

4.6 Horizontal vs Triangular configuration

In order to discern if the cable configuration impacts the previous obtained results, it was used a trian-

gular configuration displayed in figure 4.23 .

Figure 4.23: Plate buried over a three phased underground cable in triangular configuration.

The used currents and distance s are the same, although h is now the depth of the triangular central

point and is maintained at 1.5m.

Every preceding tests were repeated using all three materials and opposing the two different config-

urations. The results are displayed in figures 4.24, 4.25, 4.26 and 4.27.

34

Figure 4.24: Magnetic field factor of reduction Fr calculated for plate depths between 0 and 3mm for allthree materials and flat and triangular(line with triangles on the plot) configurations;f = 50Hz

Figure 4.25: Magnetic field factor of reduction Fr calculated for frequencies between 0 and 100Hz forall materials and flat and triangular(line with triangles on the plot) configurations;t = 3mm

35

Figure 4.26: Fr′ calculated for y between 0 and 4m for all materials and flat and triangular(line withtriangles on the plot) configurations;t = 3mm;f = 50Hz

Figure 4.27: Fr′ calculated for y between 0 and 4m for all materials and flat and triangular(line withtriangles on the plot) configurations;t = 3mm;f = 50Hz

Figures 4.24 , 4.25 , 4.26 and 4.27 show that the cable configuration does not affect the curve

behaviour although it is possible to notice a curve shifting when comparing the two geometries due to

the different magnetic field value, affected by the different configuration itself.

36

Chapter 5

Conclusions

With this work it was possible to study and analyse a procedure that allows the mitigation of the magnetic

induction field at the surface, created by underground cables carrying a certain amount of electrical

current. It becomes clear that the effectiveness of the procedure, which consists on burying a plate

between the underground power cables and the surface is directly related to several aspects such as:

plate thickness; plate magnetic permeability and electrical conductivity; frequency;

As expected and was afterwards confirmed the plate thickness has a major role on the field mitigation.

The mitigation of the magnetic field becomes higher when the plate thickness grows. Although the

mitigation is not the same in all the cases. It was possible to conclude that if the frequency for a certain

material remains constant the penetration depth does not change. So for small thickness values the

magnetic permeability of the plate is determinant and the materials with lower magnetic permeabilities

are better to mitigate the field. On the other side increasing thickness and maintaining f reveals that

materials with higher permeabilities are better.

Frequency is directly related to the penetration depth of the magnetic field into a certain material.

When the frequency increases the mitigation of the field becomes higher for all materials, yet it grows

faster in the materials with higher µ.

Varying the ratio between the plate thickness and penetration depth revealed that for small ratios the

higher the magnetic permeability the better the material, although as the value tδ starts to increase this

behaviour changes and the materials with small permeabilities become better for mitigation.

Another conclusion concerns the possibility to manipulate the plate characteristics in order to achieve

a certain magnetic induction field value Brms . This aspect does not apply only to 0 meters but to any

height above ground.

Not less important is the fact that when comparing the horizontal configuration with the triangular one,

the results do not differ to much, despite the fact of a small curve shift, due to the smaller magnetic field

value on the triangular geometry compared to the horizontal one considering that the used configurations

on this theses are equivalent.

It was also concluded that the depth at which the plate is buried does not influence the mitigation

result. It was demonstrated that this statement is valid for small values of soil electrical conductivity. This

37

is an important result since it is cheaper to bury closer to the surface then deeper.

38

Bibliography

[1] V. M. Machado, M. E. Almeida, and M. G. das Neves, “Accurate magnetic field evaluation due to

underground power cables,” EUROPEAN TRANSACTIONS ON ELECTRICAL POWER, pp. 1153–

1160, Nov. 2008. doi:10.1002/etep.296.

[2] P. Marchante, “Calculo do campo magnetico originado por cabos subterraneos de transito de ener-

gia,” Master’s thesis, IST, 2008.

[3] V. M. Machado, “Magnetic field mitigation shielding of underground power cables,” IEEE Transactions

on Magnetics, vol. 48, no. 2, pp. 707–710, 2012. doi:10.1109/TMAG.2011.2174775.

[4] E. W. Weisstein, “”simpson’s rule.” from mathworld–a wolfram web resource.”

[5] MATLAB, version 7.14.0.739 (R2012a). Natick, Massachusetts: The MathWorks Inc., 2012.

[6] D. Meeker, femm 4.2. http://www.femm.info/wiki/HomePage, 2013.

39

Appendix A

Plate depth demonstration

The regions representing soil are now assumed to be air. Therefore:

µ1 = µ0 , σ1 = 0

µ3 = µ0 , σ3 = 0

This changes make q1 = 0 and q3 = 0 and equations (2.29) and (2.30) may be rewritten as:

X(a) =

[ch(ν) +

µ2|a|µ0

√a2 − q22

sh(ν)

]ch(ξ) +

[µ0|a|µ0|a|

ch(ν) +µ2|a|

µ0

√a2 − q22

sh(ν)

]sh(ξ) (A.1)

X ′(a) =

[sh(ν) +

µ2|a|µ0

√a2 − q22

ch(ν)

]ch(ξ) +

[µ0|a|µ0|a|

sh(ν) +µ2|a|

µ0

√a2 − q22

ch(ν)

]sh(ξ) (A.2)

⇔

X(a) =

[ch(ν) +

µ2|a|µ0

√a2 − q22

sh(ν)

][ch(ξ) + sh(ξ)] (A.3)

X ′(a) =

[sh(ν) +

µ2|a|µ0

√a2 − q22

ch(ν)

][ch(ξ) + sh(ξ)] (A.4)

⇔

41

X(a) =

[ch(ν) +

µ2|a|µ0

√a2 − q22

sh(ν)

]eξ (A.5)

X ′(a) =

[sh(ν) +

µ2|a|µ0

√a2 − q22

ch(ν)

]eξ (A.6)

Replacing (A.5) and (A.6) in the integrating functions from (2.32) and (2.33):

|a|e−(hk−(d+ t2 ))|a|

|a|

[ch(ν) +

µ2|a|µ0

√a2 − q22

sh(ν)

]+µ1

µ2

√a2 − q22

[sh(ν) +

µ2|a|µ0

√a2 − q22

ch(ν)

]e−|a|yeja(x−xk)e−ξ (A.7)

ae−(hk−(d+t2 ))|a|

|a|

[ch(ν) +

µ2|a|µ0

√a2 − q22

sh(ν)

]+µ1

µ2

√a2 − q22

[sh(ν) +

µ2|a|µ0

√a2 − q22

ch(ν)

]e−|a|yeja(x−xk)e−ξ (A.8)

Replacing ξ for (d− t2 )|a| :

|a|e−(hk−t)|a|

|a|

[ch(ν) +

µ2|a|µ0

√a2 − q22

sh(ν)

]+µ1

µ2

√a2 − q22

[sh(ν) +

µ2|a|µ0

√a2 − q22

ch(ν)

]e−|a|yeja(x−xk) (A.9)

ae−(hk−t)|a|

|a|

[ch(ν) +

µ2|a|µ0

√a2 − q22

sh(ν)

]+µ1

µ2

√a2 − q22

[sh(ν) +

µ2|a|µ0

√a2 − q22

ch(ν)

]e−|a|yeja(x−xk) (A.10)

The final solution is now only dependent on the plate thickness t. For small soil conductivities the

results are an approximation to the above ones. It is possible to conclude that the plate depth doesn’t

affect the results for small soil electrical conductivities.

42

mitigation plates for the magnetic ﬁeld of underground cables · mitigation plates for the...

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