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1
Frequency-dependent underground cable model for
electromagnetic transient simulation
Vasco José Realista Medeiros Soeiro
Abstract – A computation method for transients
in a three-phase underground power-transmission
system is presented in this paper. Two methods
will be used for this purpose: The first using the
Fourier Transform, allowing the transient analysis
in linear time-invariant systems. The second is an
equivalent network with lumped parameters
whose behaviour, within a given frequency band,
is similar to the transmission line itself. Transient
waveforms are evaluated using a software for
mathematical applications, MATLAB, and in
particular one of its tools, SIMULINK.
The main interest in the use of a method to make a
time analysis is the introduction possibility of non-
linear elements in the network.
Nomenclature
Voltage vector Current vector Impedance matrix Admittance matrix Bessel Function Bessel functions H Hankel function Radius Radius of central conductor Radius over main insulation Radius over conducting sheath Outer radius cable Angular frequency Resistivity or charge density Conductivity Permitivity Permeability Soil penetration
I – Introduction
The social and economic development that occurred
in the past years led to an urban and industrial centre
growth, increasing the electrical power demand and leading to the use of relatively long cable circuits
operating at high voltage. In these conditions it is
expected to overcome transient overvoltages induced in
the conductors of the underground system. For the
stated reasons and the recent interest in underground
transmission systems, researches on the viability of the
underground cable models became necessary.
The magnetic field based on Maxwell’s equations is
calculated for the underground cable model system.
The general solution for the magnetic field on the soil is
developed using arbitrary boundary conditions in a
cylindrical surface, at a finite depth under the plane earth/air surface. The general Polaczek solution is
developed. Finally the system constitutive parameters
are evaluated.
Two methods for the electromagnetic transient
simulation on underground cable systems are studied in this paper. The first is the Fourier Transform that will
be implemented using the Fast Fourier Transform
(FFT) and the Inverse Fast Fourier Transform (IFFT).
The second is the equivalent network with lumped
parameters. Transient regimes obtained by both methods are compared showing an excellent result
accuracy.
II – Magnetic field in underground power-
transmission systems
The problem of an infinitely long cylindrical
conductor can be treated as a 2D problem
which is easier to analyze. When the conductors are
displayed with an axial symmetry the field also
satisfies this symmetry and the solution becomes
considerably easier.
The calculation of the magnetic field due to an
underground cable of finite radius with cylindrical
boundary will be made taking into account several
assumptions:
1. The earth is a semi-infinite surface where
the Earth / Air is a plane.
2. The geometry is considered infinitely
long in the z coordinate (axial).
3. The cable is cylindrical and it is buried at a constant depth.
4. Earth and air are considered
homogeneous, the air with a permeability and earth with a permeability and
conductivity .
5. The hypothesis of a quasi-static regime is
considered, neglecting the capacitive
effects, which for the case of the earth is
an adequate approximation for
frequencies up to 1 MHz.
2
Figure 1. Underground cylindrical surface of finite radius.
The formulation of the electromagnetic field in a
power transmission system is based on the magnetic
vector potential, , which satisfies in the frequency
domain:
!, #$ %&' ()$*+(%), -- 0, #$ %&' ,)#/0 (1)
Where η represents the voltage drop per unit of axial
length.
Magnetic vector potential inside earth
The general solution for the magnetic field in the
soil is developed satisfying arbitrary boundary
conditions in a cylindrical surface buried at a finite
depth under a plain Earth/Air surface. It is considered
a generalization of the Pollaczec solution for
cylindrical underground cables with circular sheath
and finite radius, taking into account the proximity of
the magnetic field. The corrections for the
longitudinal impedance due to the return path of the
earth are determined at the expense of an approximation equivalent to the solution of
Pollaczek.
The solution of the field in the soil can be made by
the linear combination of two linearly independent
terms. The first, 12, to consider the boundary
conditions on the earth / air surface, Sa. The second, 122, in turn, allows the boundary conditions to be
considered on the surface 3. The solution can then
be written in Cartesian coordinates (x, y):
A5 6x, y9 : N6a, y9e>?@daBCDC (2)
Where
N6a, y9 F6a9eFG?HDIJKKKHda , y L 0 (3)
Re6NaO qKKKO9 Q 0 (4)
qKKK √OSTUVWXJ , δ G OZµ\]J (5)
F6a9 is a function to be determined by the boundary
conditions of the problem.
The 122 solution is written in Fourier series:
A22KKKK ∑ RBC_DC 6r9e>a (6)
Considering the Bessel functions we obtain:
b c6d- 9 e cO6d- 9 (7)
Or:
b fgh6g96d- 9 e fh6O96d- 9 (8)
Where 6d- 9 is a Bessel function of the first kind
of order m and argument d- . 6d- 9 is a Bessel
function of the second kind of order m and argument d- . h6g9 and h6O9
are the Hankel functions of the
first and second kind respectively of order m and d-
argument.
For a hollow conductor, the case in study, it is used
the Hankel equation of the second kind that is regular
for i ∞.
b fh6O96d- 9 (9)
Thus, the term A22KKKK can be written in cylindrical
coordinates 6r, φ9 around l2 by: 155KKKK6, m9 ∑ fBC_DC h6O96d- 9'no, Q (10)
Where h6O9 is the Hankel function of the second kind
and order m with argument 6d- 9 and f, p 0, q1, q2, … are coefficients to be determined. In
order to impose boundary conditions on the surface 3u is convenient to write 6129 in Cartesian
coordinates.
Since the Hankel function of the second type is
defined by [2]:
h6O96d- 9 gv : w6x9*xyO (11)
Considering the integration paths defined in [3] we
obtain for the case study: zh6O96d- 9'no
: 69'D|B~|GuHDKKKH'nu*BCDC
(12)
Where:
69 uDGuHDKKKHnKKK nGuHDKKKH , Q & (13)
3
Thus the solution of the vector potential is:
1!6, 9 15 e 155KKKK f6, 9'nu*BCDC , & L L 0
(14)
With:
f6, 9 c69'G2d,KKKK2 e e gv 'D|B~|G2d,KKKK2 ∑ f69BC_DC (15)
Potential vector of the magnetic field in air
Assuming that the solution is: 1 : 6, 9'nuBCDC * (16)
And
H6u,9H O6, 9 0 (17)
Where
6, 9 69'D|u| , Q 0 (18)
The following result in:
16, 9 : 69'D|u|'nu* , Q 0BCDC (19)
Boundary conditions on the earth/air surface
The boundary conditions are:
g6,9 |_T g\
6,9 |_ 16, 0D9 16, 0B9 0 (20)
In the earth, taking into account the earth/air and
earth/conductor boundaries:
69 gv OGuHDKKKH\|u|BGuHDKKKH ∑ fpp69e∞p∞ (21)
c69 gv\|u|DGuHDKKKH\|u|BGuHDKKKH '&GuHDKKKH ∑ fpp69e∞p∞
(22)
Magnetic vector potential in the air, taking into
account the two boundary conditions:
1!6, 9 : gv OGuHDKKKH
\|u|BGuHDKKKHBCDC '||' ∑ fpp69e∞p∞ * (23)
Under certain conditions it is possible to use the
Pollaczec solution where: f 0, , p 0
valid for d- 1, d- |d- | and d-& 1.
122KKKK ∑ fh6O96d- 9BC_DC 'no , Q (24)
In the surface of the conductor and taking in
consideration the boundaries:
122KKKK fh6O96d- 9 , (25)
Boundary conditions on the surface cable/earth
Due to the geometric shape of the surface of the
cable, the solution is written in cylindrical
coordinates.
Figure 2. Representation of the earth/cable surface 3, with
indication of the cable axis and the exterior radius
Given the Pollaczec approximation:
1!|_ 6d- 9f, e fh6O96d- 9 (26)
Now taking into account the boundary conditions at (earth/conductor):
g6,o9 |_ g\
6,o9 |_T 16B, m9 16D, m9 0 (27)
Inside the cable sheath with dielectric characteristics,
being the radius shown in Figure 3:
1 g/$ g e O , L L (28)
Then
f h6O96d- 9 e 6d- 9, g/$ g e O
g f hg6O96d- 9 e g6d- 9, g\ g gKKK0(29)
4
f \ ¡¢KKK g£\6H96KKK9B¤\6KKK9\,\g \6¥¦B¥§9O/vO g/$ g e \ g©
0 (30)
Where
© gKKK £\6H96KKK9B¤\6KKK90,0£1
6H96KKK9B¤16KKK90,0 (31)
ª« and ª¬are complex current amplitudes that flow in
the phase conductor and the conducting sheath of the
cable.
III – Constitutive parameters of
underground power-transmission systems
The three phase underground cables can be
monopolar or tripolar. The conductors are isolated
and surrounded by a sheath, with mechanical and
chemical protection function, connected to the earth.
Each cable has two metallic conductors, one is the central conductor and the other the conductor sheath.
This is the basic configuration normally used for high
voltage cables. In this work it was considered
monopolar cables.
Figure 3. Cross section of an underground cable.
Longitudinal Impedance
Assuming a three-phase system consisting of three
equal cables:
6 g96 O96 ®9¯ °±gg² °±gO² °±g®²°±Og² °±OO² °±O®²°±®g² °±O®² °±®®²¯ °ªg²°ªO²°ª®²¯ (32)
The matrices of the diagonal are then:
°± ² ³±´´ ±´´±´´ ±´´µ (33)
The impedance of the sheath itself is then: ± ±- e ±¶ e ±· (34)
Where ±- is the impedance related to the earth, ±¶ is
due to the variation of the flux in the outer insulation,
and ±· is the outer sheath internal impedance given
from the voltage drop along the outer surface of the
sheath [4].
±- ¸Ov © (35)
±¶ \Ov /$ § (36)
±· \¹v º£16¢9»q§KKKr§´¼£\6H9»q§KKKr½¼D£\6¢9»q§KKKr§¼£1
6H9»q§KKKr½¾¼¿q§KKKr½À
(37)
With:
Á gOn ºh16g9»qKKKr¼h1
6O9»qKKKr¬ ¼ h16g9»qKKKr ¼h1
6O9»qKKKr¬S¼¿ (40)
The mutual impedance between the cable and the
sheath is:
± ± ± ±¹ (38)
±¹ is the sheath mutual impedance given by [4]:
±¹ §vHqs H½Ã½¾À (39)
The impedance of the conductor itself is given by:
± ± e ±g e ±O e ±® ±¹ (40)
±g is the internal impedance of the inner
conductor, ±O the impedance due to the time-varying
magnetic field in the inner insulation and ±® is the
inner sheath internal impedance calculated from the
voltage drop on the inner surface of the sheath [4].
±g \Ov /$ §´¦ (41)
±O ¦Ov J\»q¦KKKrļq¦KKKrÄJ¢»q¦KKKrļ (42)
±® §¹v ºh1619»q r'¼h0629»q rb#¼h0619»q r#¼h1
629»q rbe¼¿q§KKKr½´À
(43)
In the calculation of the impedance for elements
outside the diagonal, the provision of the different
cables must be taken into account. In this case the
cables are in flat configuration.
Figure 4. Geometry of the system. Cables in line.
5
Thus the impedance between cable # and is given
by:
º± n¿ Ʊ´Ç ±´Ç±´Ç ±´ÇÈ (44)
±´Ç is the impedance between the phase conductor
of cable # and the phase conductor of cable . ±´Ç is the impedance between the sheath of cable # and the sheath of cable .
±´Ç ±´Ç p n (45)
The field outside the cable is taken into account for
the calculation of p n. For the mutual impedance
between the cable sheath and the conductor, the cable
is seen as driven by a conductor running in a
homogeneous soil. p n may be evaluated by using a
closed form with appropriate Bessel and Henkel
functions:
p n ºfh6O9»q,KKKxij¼ e »q,KKKxij¼f,¿ (46)
Longitudinal Admittance
The complex amplitude and are now just
function of the longitudinal coordinate Ë.
ÌÌÍÎÏÏÏÏÐgO®gO®ÑÒÒ
ÒÒÓ ³°ÔÕÕ² °ÔÕÖ²°ÔÖÕ² °ÔÖÖ²µÎÏÏÏÏÐgO®gO®ÑÒÒ
ÒÒÓ (47)
Capacity coefficients matrix
The capacitance matrix allows the conductor to
hold the potential throughout the insulation. In order
for this matrix to be determined, it is considered its
inverse, the potential coefficients matrix. It is only considered the potential vector inside the cables,
being the non diagonal elements, 3 n, null. For the
main diagonal elements, 3 , we have:
× Æº×´´¿ º×´´¿º×´´¿ º×´´¿È (48)
With
×´´ gOvئ´ /$ §´§¦ (49)
×´´ gOvئ´Ø§´ /$ § ×´´ (50)
×´´ gOvا´ /$ § (51)
In matrix form:
× gOvØ /$ §´§¦ /$ §/$ § /$ §¯ (52)
Finally resulting in the longitudinal admittance
matrix, inverting the °3² matrix in order to obtain °².
Ù °² ÎÏÏÏÏÏÐ ¢¢ 0 00 HH 00 0 ÚÚ
¢¢ 0 00 HH 00 0 ÚÚ¢¢ 0 00 HH 00 0 ÚÚ ¢¢ 0 00 HH 00 0 ÚÚÑÒÒ
ÒÒÒÓ
(60)
Frequency domain propagation
According to the linearity problem, the propagation
problem is entirely formulated in the frequency
domain. So the following systems can be written:
Ì°Û²ÌÍ °Ü²°²Ì°¥²ÌÍ °Ù²°²0 (53)
ÌH°Û²ÌÍH °Ü²°Ù²°²
ÌH°¥²ÌÍH °Ù²°Ü²°² 0 (54)
The 6°Ü²°Ù²9 product can be transformed in a
diagonal matrix using the transformation matrix °Ý². The transformation matrix °² can transform the 6°Ù²°Ü²9 product in a diagonal matrix. Matrix °Ý² is
obtained by the eigenvectors of 6°Ü²°Ù²9 and matrix °² by the eigenvectors of 6°Ù²°Ü²9 .
The diagonal matrix of the eigenvalues of 6°Ü²°Ù²9 is °Þ²O and is obtained by:
°Þ²O °Ý²Dg»°²°Ý²¼°Ý² (55)
°Þ²O °²Dg6°Ù²°Ü²9°² (56)
Can be decomposed into a product of two diagonal
matrices °ß ܲ and °àÙ² by:
°á² °Ý²Dg°Ü²°²°áݲ °²Dg°Ù²°Ý²0 (57)
For the calculation of the matrix °², it is necessary
to build a matrix °áݲ so it verifies the equation °á²°áݲ °Þ²O, then:
6
°² °Ù²°Ý²°áݲ°²Dg °áݲ°Ý²Dg°Ý²°áݲDg 0 (58)
Introducing °Ý² and °² to the system and applying
the necessary simplifications:
°â² °Ý²Dg°²°² °²Dg°² 0 (59)
There is now a system of equations, which can be
written in the following matrix form:
°â² '6°Þ²Ë9°â1² e '6°Þ²Ë9°â2²°² '6°Þ²Ë9°1² e '6°Þ²Ë9°2² 0
(60)
ºßg¿, ºßO¿, ºäg¿ and ºäO¿ are column vectors for modal
quantities. For the voltage the expression is:
°² '6°Γ²Ë9°g² e '6°Γ²Ë9°O²
(61)
And for the current:
°² °Ü²Dg°Γ²'6°Γ²Ë9°g² °Ü²Dg°Γ²'6°Γ²Ë9°O² (62)
Assuming:
°² °Ù²°Γ²Dg'6°Γ²Ë9°g² °Ù²°Γ²Dg'6°Γ²Ë9°O² (68)
Where:
°Γ²1 °Ý²°Þ²1°Ý²1'6q°Γ²Ë9 °Ý²'6q°Þ²Ë9°Ý²1 0
(63)
Note that °Γ² is a non-diagonal matrix.
IV-Transient analysis of underground
power-transmission systems
The tools used in this work are the Fast Fourier Transform (FFT) and the Inverse Fast Fourier
Transform (IFFT). These are fast algorithms for
implementing a number of samples where the input
signal is transformed in the same number of
frequency points. The calculations performed by
these algorithms are gO /)æOç multiplications and /)æOç additions for 2è samples [7].
Equations of the line ended with a three-phase
load
Considering a three-phase generator and the
frequency propagation equations:
°609² ºé¿ ºé¿°609² (64)
Where:
°609² - Column vector of complex amplitude
voltages at the beginning of the line, Ë 0.
ºé¿ - Column vector of complex amplitude voltages
for the three-phase generator
ºé¿ – 3x3 matrix with each element being an
impedance
°609² - Column vector of complex amplitude
currents at the beginning of the line, Ë 0.
Now considering the three-phase load at the end of
the line:
°6/9² °²°6/9² (65)
Where:
°6/9² - Column vector of complex amplitude
voltages at the end of the line, Ë /. °² – 3x3 matrix with each element being an
impedance
°6/9² - Column vector of complex amplitude
currents at the end of the line, Ë /.
It will be established matrix transfer functions in
order to relate the voltage and current complex
amplitudes on a determined place of the line. The
generator is considered to be of 230 © amplitude.
Transfer Functions
For Ë 0 and Ë /:
°609² °g² e °O² °609² °Ù²°Γ²Dg°g² °Ù²°Γ²Dg°O² °6/9² '6°Γ²/9°g² e '6°Γ²/9°O² °6/9² °Ù²°Γ²Dg'6°Γ²/9°g² °Ù²°Γ²Dg'6°Γ²/9°O²
0 (66)
Where:
°g² ë1Dgºé¿ (67)
°O² 1Dgºé¿ (68)
And the transfer functions that allow the calculation
of voltages and currents at a generic point Ë:
°Í² ì'6°Γ²Ë9ë1Dg e '6°Γ²Ë91Dgíºé¿°Í² Ù°Γ²Dgì'6°Γ²Ë9ë1Dg '6°Γ²Ë91Dgíºé¿0
(69)
7
With:
1 鿰ٲ°Γ²Dg e °E²¼'6°Γ²/9»ºé¿°Ù²°Γ²Dg °E²¼Dg»ºé¿°Ù²°Γ²Dg e °E²¼'6°Γ²/9 e »°ð² ºé¿°Ù²°Γ²Dg¼ñ (70)
ë '6°Γ²/9ì6°²°Ù²°Γ²Dg °E²9Dg6°²°Ù²°Γ²Dg e°E²9í (71)
Frequency domain
To analyze the electromagnetic transient in a
power-transmission system, it is necessary to pass the
time domain voltages of the generator to the
frequency domain. The FFT algorithm was used, the
real part is represented in blue with the imaginary in red.
Figure 5. Fourier transform of the phase generator voltages.
Transient analysis of a three-phase line with load
To study the electromagnetic transients it is
necessary to know the parameters of the three-phase
line:
Table 1. Parameters of the system.
The permittivity and permeability represented in the above table refers to the insulating layer surrounding
the phase conductor and conducting sheath.
No Load
Phase impedance °² ∞, sheath impedance °² 0Ω. The generator phases are in short-circuit
and the sheaths with a 1 Ω impedance. The system
is 5km in length. Real part is represented in blue with
the imaginary in red.
Figure 6. Phase Voltage at z=5km.
Figure 7. Sheath current at z=5km.
The other voltages and currents are not represented
because the results gave a null amplitude.
Short-Circuit Load
Phase impedance °² 0Ω, sheath impedance °² 0Ω. The generator phases are in short-circuit
and the sheaths with a 1 Ω impedance. The system
is 5km in length. Real part is represented in blue with
the imaginary in red.
Figure 8. Phase current at z=5km.
Figure 9. Sheath current at z=5km.
8
The other voltages and currents are not represented
because the results gave a null amplitude.
Adapted Load
Phase impedance °g² °®² 66.4Ω, °O² 63.14Ω , sheath impedance °² 0Ω. The generator
phases are in short-circuit and the sheaths with a 1 Ω impedance. The system is 5km in length. Real
part is represented in blue with the imaginary in red.
Figure 10. Phase voltage at z=5km.
Figure 11. Phase current at z=5km.
Figure 12. Sheath current at z=5km.
The other voltages and currents are not represented
because the results gave a null amplitude.
V – Equivalent network with lumped
parameters
An equivalent network with lumped parameters is a
system that is close to the behavior of a power
transmission line in a determined frequency band.
In this work the conducting sheaths are connected
to the earth. This allows the reduction of the matrix
dimensions from 6x6 to 3x3, obtaining three
independent modes.
Propagation mode parameters
From the equivalent quadripole of a mono-phase
line section:
ö÷609 +á÷609 áøO e ùâú69Dùâú6û9Íü
ö÷6/9 +á÷6/9 áøO e ùâú6û9Dùâú69Íü0 (72)
Where Ëû and áý are the longitudinal modal
impedance and transversal modal admittance:
Ëû þúáú ,#$&6Þ÷/9
áý O-~6þúû9DOúâú- ~6þúû9 2 áúþú %æ& þúûO 0
(73)
If Þ÷/ 1, then ,#$&6Þ÷/9 Þ÷/, %æ& þúûO þúûO
and:
Ëû Ë÷/áý á÷/0 (74)
Where Ë÷ is the k mode longitudinal impedance in
modal coordinates. Ë÷ and á÷ are the diagonal
elements of ºä¿ and ºà¿. In order to simplify the
problem is is considered that the sheaths are
connected to the earth, this way only the phases are
considered.
ºà¿ e 2p 0 00 p 00 0 p¯
(75) u 6¢¢BHHBÚÚ9® (76)
6¢HB¢ÚBHÚ9® (77)
Transformation matrix °Ý² is the eigenvectors matrix
that diagonalizes the 6°Ü²°Ù²9 product.
Transformation matrix °² is:
°² °² °Ý²ºà¿Dg (78)
Modal impedance matrix is obtained by:
ºä¿ °Þ²Oºà¿Dg (79)
Figure 13. Real part of the longitudinal impedance.
9
Figure 14. Imaginary part of the longitudinal impedance
Longitudinal impedance synthesis
Each element of ºä¿ will be approximate by the 5
parallel b branches of the circuit impedance.
Zà6©9 1∑ 1b#e©#$#1 (80)
For n frequencies (86) represents a system of n
equations. It is resolved by the following interactive
process:
b÷ e ÷÷ g¢ZáiiD∑ ¢
´Çú´´¢,´ú (81)
The process is initiated with b÷ and ÷ values that
correspond to 5 times the longitudinal impedance. This
values are changed at each interaction, and it is applied
to all ©. The process is repeated until there is no
significant change in the parameters values. The
adjustment frequencies where chosen in geometric
progression so no negative elements would occur. The
following figures represent the results for 3 propagation
modes at a frequency band, the equivalent system is
represented by lines and the equivalent network by
dots. The line has 5 ©p divided by 20 sections 250 p
each.
Figure 15. Comparison of the longitudinal impedance at 50Hz.
Figure 16. Comparison of the longitudinal impedance at
100kHz.
Transformation matrices of voltages and currents
It is implemented in this section a circuit with the
objective to simulate the behavior of an underground
system with a equivalent network. With the 20 section
circuit there exist two adaptation nets, one at the generator and the other at the load that transform modal
magnitudes into real ones and vice-versa. It is
considered the normalized °Ý² and °² matrices, and
their inverse. No frequency variation is considered.
No Load
Figure 17. Current by the Fourier Transform for z=0km
Figure 18. Current by the equivalent network for z=0km
Figure 19. Voltage by the Fourier Transform for z=5km
Figure 20. Voltage by the equivalent network for z=5km
10
Short-Circuit Load
Figure 21. Current by the Fourier Transform for z=0km
Figure 22. Current by the equivalent network for z=0km
Figure 23. Voltage by the Fourier Transform for z=5km
Figure 24. Voltage by the equivalent network for z=5km
VI – Conclusions
The equivalent network with lumped parameters is a
more complete method for the system analysis, but
requires more memory and processor capacity. After applying some simplifications, such as the sheath
connection to the earth, the phase modes have similar
attenuations, converging at high frequencies. The
modal method and the longitudinal impedance
synthesis converge at both low and high frequencies.
The results are similar, obtaining a small difference in the oscillations of the transients for both voltage and
current. For a possible investigation topic in this
subject, it is recommended a development of a
program able to solve the set of differential equations
to treat this transmission system, as well as the
problem of the complete set of six modes.
VII – References
[1] Simonyi, K. 1963. Foundations of Electrical
Engineering. Pergamon Press.
[2] Sommerfeld, A. 1949. Partial Differential Equations in
Physics. Academic Press
[3] Machado, V. Maló & da Silva, J. F. Borges 1988.
‘Series-Impedance of Underground Transmission
Systems’, in IEEE Transactions on Power Delivery,
Vol. 3, No.2, pp.417-424
[4] Wedepohl, L. M. & Wilcox, D. J. 1973 ‘Transient
analysis of underground power-transmission systems:
System model and wave propagation characteristics’,
Proceedings Inst. Elec. Eng., vol. 120, pp. 253-260.
[5] Guerreiro Neves, M. V. 1990. Cálculo de Transitórios
em Linhas de Transmissão de Energia Baseado no
Emprego dum Esquema Equivalente por Troços –
Comparação com o Método da Transformada de
Laplace, Departamento de Engenharia Electrotécnica e
de Computadores do Instituto Superior Tecnico,
Universidade Técnica de Lisboa.
[6] Da Silva, J. F. Borges 1995. Electrotecnia Teórica,
Departamento de Engenharia Electrotécnica e de
Computadores do Instituto Superior Técnico,
Universidade Técnica de Lisboa.
[7] Shenkman, Arieh L. 2005. Transient Analysis of
Electric Power Circuits Handbook. Holon Academic
Institute of Technology.