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.