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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
ii
Dedicated to Ana Gomes.
iii
iv
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
vi
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
viii
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
x
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
xiv
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
4.3 Magnetic field factor of reduction Fr calculated for plate thickness between 0 and 10mm
using a Steel100 plate with f = 50Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 Magnetic field factor of reduction Fr calculated for plate thickness between 0 and 5mm
using a Steel500 plate with f = 50Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.5 Magnetic field factor of reduction Fr calculated for plate thickness between 0 and 5mm
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
4.16 Fr′ calculated for y between 0 and 4m for all materials. t = 3mm;f = 50Hz . . . . . . . . 30
4.17 Fr′ calculated for y between 0 and 4m for all materials. t = 5mm;f = 50Hz . . . . . . . . 30
4.18 Fr′′ calculated for y between 0 and 4m for all materials. t = 1mm;f = 50Hz . . . . . . . . 31
4.19 Fr′′ calculated for y between 0 and 4m for all materials. t = 3mm;f = 50Hz . . . . . . . . 31
4.20 Fr′′ calculated for y between 0 and 4m for all materials. t = 5mm;f = 50Hz . . . . . . . . 32
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
xviii
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)
where q2 has the same meaning of qs but now it is calculated assigning the parameters of region 2:
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)
where q3 has the same meaning of qs but now it is calculated assigning the parameters of region 3:
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
Figure 4.16: Fr′ calculated for y between 0 and 4m for all materials. t = 3mm;f = 50Hz
Figure 4.17: Fr′ calculated for y between 0 and 4m for all materials. t = 5mm;f = 50Hz
30
Figure 4.18: Fr′′ calculated for y between 0 and 4m for all materials. t = 1mm;f = 50Hz
Figure 4.19: Fr′′ calculated for y between 0 and 4m for all materials. t = 3mm;f = 50Hz
31
Figure 4.20: Fr′′ calculated for y between 0 and 4m for all materials. t = 5mm;f = 50Hz
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
40
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