Analyzing the transmission line parameters in frequency domain

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ANALYZING THE TRANSMISSION LINE PARAMETERS IN FREQUENCY

DOMAIN

S. Kurokawai

J. Pissolato Filho

M. C. Tavares

C. M. Portela

[email protected]. unesp.br

[email protected]. unicamp.br

[email protected] .sc.usp.br [email protected]

[email protected]. unicamp.br

UNICAMP – State University of Campinas

S50 Carlos Engineering

COPPE - Federal

Campinas, SP

School - USP

University of Rio de

S50 Carlos, SP

Janeiro

Rio de Janeiro, RJ

BRAZIL

1– UNESP - Ilha Solteira : Electrical Engineering Department

Abstract : The objective of this paper is to analyze the influence

that the ground wires and soil and skin effects have on the R and

L transmission line parameters. Initially a theoretical analysis of

the influence which the ground wires have on the R and L

parameters is presented. Some results obtained of a hypothetical

line with frequency constant parameters and of a line with

variable parameters are presented. The individual influence of the

skin and ground effects on the R and L parameters is analyzed.

Keywords : Transmission lines, R and L parameters, ground

wires, ground effect, skin effect, external impedance, frequency

dependency.

I

INTRODUCTION

Several factors are related to the fi-equency variation

of the longitudinal parameters. We can mention as factors

that influence the behavior of the longitudinal parameters,

the ground effect, the skin effect and the external

impedance of the cables [1]. The presence of ground wires

grounded in all the structures also alter the R and L

parameters.

The skin effect is related to the fhct that in conductor

materials the electromagnetic power is transmitted only in

the superficial area [2,9]. The skin effect is calculated using

the Bessel function.

Several authors have studied the parameter line

considering the ground return and it has been verified that

the more known and accepted expressions for calculation

of the ground return parameters are, for overhead cables,

Carson’s expressions and, for underground cables,

Pollaczeck’s expressions [3,4,5,6].

Carson’s and Pollaczeck’s models are valid for

homogeneous

semi- infinite earth,

neglecting the

displacement current, and the wave length is sufficiently

long compared to the transversal geometric dimensions [4] .

The calculation of line impedance according to

Carson’s and Pollaczeck’s models are given by expressions

containing complex infinite integrals. Traditionally, these

integrals have been evaluated by algebraic infinite series.

For overhead lines, Carson has proposed infinite series and

also some convenient approximations for low and high

fi-equencies. While these approximations are relatively

simple they are each valid for a lirnhed range of

f requencies only, and medium li-equencies are not covered

[3,4].

The ground return cables, presented in overhead cables,

can be grounded in all the structures, or they can be

isolated. When the overhead cables are isolated can be used

as telecommunications circuits. The employed insulators

are of low disruptive voltage [7].

Initially we define the per unit length longitudinal

impedance and the shunt admittance matrices for a generic

multiphase transmission line and then we define the

longitudinal impedance matrix for a three-phase

transmission line with two ground wires and define the

appropriate boundary condition to represent the ground

wires. In thk way , we have the equations that show how

the longitudinal impedance matrix elements are written

considering the reduct ion of the ground wires.

The equations above mentioned are used to show how

the ground wires change the R and L parameters of a

hypothetical transmission line that has constant parameters

and in a transmission l ine that has the tlequency-dependent

parameters.

An individual analysis of the skin and ground effects

behavior is made and the external impedance behavior in

fimction of the frequency. It is also analyzed how the

ground wires change the skin and ground effects and the

external impedance.

Concluding, the individual contribution of the skin and

ground effects and the external impedance in the R and L

parameters of the transmission lines with and without

ground wires, are shown.

II THEORETICAL ANALYSIS OF THE

INFLUENCE OF GROUND WIRES ON THE

PARAMETERS OF THE TRANSMISSION LINE

The fundamental telegrafer’s equations for a multiphase

transmission l ine are:

d2[V] d2[I]

—= [Z][Y][V] ; -&j- = [Y][z][g

dx’

(1)

[Z] and [Y] are the per unit length longitudinal

impedance and the shunt admittance matrices respectively.

The [Z] and [Y] matrices are ffequency-dependent.

The [Z] matrix can be written as:

[Z] = [Z]skin + [Z]ext + [Z]soil

Where:

(2)

[Z],,,. Longitudinal impedance caused by skin effect

(internal impedance) considering the ground

with infinite conductivity

[Z]e,[ Longitudinal impedance considering the ground

with infinite conductivity (external impedance);

[Z],Oil Ground contribution considering the air and

ground magnetic permeability equal to vacuum

permeability.

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 878



The [Z] matrix can be decomposed in real and

imaginary part, as shown in equation 3.

[Z] = [R]+ j@L]

(3)

The [R] matrix is the real part of the [Z] matrix, [L]

matrix is is the imaginary part of the [Z] matrix and o is

the angular frequency. The [R] and [L] matrices are the per

unit longitudinal resistance matrix and the per unit

longitudinal inductance matrix respectively and are

frequency-dependent.

[V] and [1] are column voltage and current matrices.

The column matrix shows the phase- earth and the ground

wires- earth t ransverse voltages. The column current matrix

shows the phase and the ground wires longitudinal

currents.

Being a transmission line with three phases and two

ground wires. The impedance matrix for this transmission

line is :

z=

‘zll 212 213 214 215

‘Z21 =22

’23

z 24 z 25

Z31 ’32 233 ‘Z34 Z35

41 ’42

=43 =44

245

Z5,

’52 253 254 ‘Z55

(4)

In fimction of the geometric characteristics of the line,

we can write the equat;n 1 as [1]:

1]

AD DEE

DBHFG

Z= DHBGF (5)

EFGCI

EGFIC

The longitudinal impedance matrix shown in equations

1 and 2 will be denominated the primitive matrix. It

contains the series impedance of a line with 5 cables.

Observe that the Z matrix does not show if a cable of the

line is a phase cable or a ground wire. Therefore, the

impedance matrix can represent a line with 5 phases

without ground wires or three-phase lie with 2 ground

wires.

Then we apply a boundary condition to represent a Z

matrix for a three-phase line with 2 ground wires.

Considering that the phase-earth voltage in the ground

wires is zero, the matrix of the equation 5 can be reduced to

an order 3 matrix.

The impedance matrix considering the reduction of the

ground wires can be written as [7,8]:

[z’] = [ZFF] - [ZFP] [ZPP]-’[ZPF]

(6)

The Z’ matrix shown in equation 6 will be denominated

reduced matrix.

In equation 6, the [ZFF] , [ZFP], [ZPF] and [ZPP] matrices

are expressed by

z,, =

ADD

DBH

DHB

(7)

E

Z,P=F G

GF

(8)

[1

FG

z,, =

EGF

(9)

[1

CI

z,, =

IC

(

10)

The Z’ matrix is a 3x3 matrix written as:

[1

2’11

2’12 Z’13

z = z

21

z’22 Z’23 (11)

Z’3,

Z’32 Z’33

In equation 11, we have :

2E2

z’,, =A-—

C+ I

(

12)

C(F2 + G2) – 2FIG

Z’2Z= B –

C2 –12

(

13)

( 14)

,,2 = D_

‘(F + ‘)

C+ I

2FGC– I(F2 + G2)

Z’23= H –

C2 -12

(

15)

2’22 = 2’33

( 16)

Z’12 = Z’,3 = Z’2, = Z’3, ( 17)

“23 = “32

( 18)

The equations 12-18 show the longitudinal impedance

matrix elements considering the reduction of the ground

wires.

HI INFLUENCE OF THE GROUND WIRES ON

LINES WITH CONSTANT PARAMETERS

We now consider, in a hypothetical situation, that a

transmission line is over an ideal ground with infinite

conductivity. In this situation, the ground effect is nil. We

disregard the skin effect and consider that the t ransmission

line conductors (phase and ground wires conductors) have

a constant resistance.

The hypothesis previously mentioned does not exist in

real transmission lines but will be used to show the

applications of the equations 12, 13, 14 and 15.

For a line previously mentioned the R and L parameters

are constants. The elements of the matrix shown in

equation 4 are expressed by

Zii = Rii + jr&,

( 19)

Zij = jo3Lij

( 20)

Where :

Rii Resistance of the cables of phase i

Lii Self inductance of phase i

L,, Mutual inductance between phases i and j

The ~1 parameter is calculated for a particular

tlequency.

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 879



The matrix for this line, considering the reduction of

ground wires, is :

[z’] = [z,,] - [Z,,] [z,,]-’ [zp,]

(21)

The elements of the [Z’] matrix are:

Z’ii(~) = Rii(~) + j(oL’ii(~)

( 22)

Z’ij(CO) = R’ij(@) + j(oL’ij(@)

( 23)

Equations 22 and 23 show that the presence of the

ground wires makes the self resistance dependent on the

frequency. The same happens with the self and mutual

inductances. The ground wires also produce mutual

resistance that are dependent of the frequency.

Figure 1 shows the self resistance of the phases 1,

considering the primhive (RI 1) and reduced (R’ l 1)matrix.

, 7 ( Ohr dkm)

0.6

0,5

0.4

0,3

0.2-

0.1

0

<0’

102 103 104 105 106 ,

Hz)

Figura 1- Self resistance of the phases 1considering the primitive (R,,)

and reduced (N,, ) matrix

Figure 2 shows the self inductance of the uhases 1

considering the primitive (LI 1)and reduced (L’11) matrix.

(rrrHerrrys/km)

13

1.28

L I

1.26

1,24

1.22

1.2

1.18-

.16-

1.141

10’

102 103

104 10’ 1[

:Hz)

Figura 2- Sel f inductance of the phase 1 consider ing the primitive (L, , )

and red uc ed (L’ , I ) mat ri x.

Iv INFLUENCE OF THE GROUND WIRES ON

LINES WITH VARIABLE PARAMETERS

Consider a transmission line that is over a non ideal

ground with tinite conductivity and that the distance

between the conductors is much larger than the sum of the

radii of the conductors. For thk line, whose parameters are

variable in fimction of the frequency, the elements of the

impedance matrix shown in equation 4, are expressed by

Zii(~) = Ril(~) + jmLii(~)

( 24)

Z,,(fO) = Rij(@) + j~LiJ(@)

Where :

( 25)

I+(m) Self resistance of the cable of phase i, considering

the soil and skin effects;

~j(m) Mutual resistance between phases i and j,

considering the soil effect;

Lii(~) Self inductance of phase i, considering the ground

and skin effects and the external impedance;

Llj(o)) Mutual inductance between

phases i and j,

considering the ground effect and the external

impedance.

The impedance matrix considering the reduction of the

ground wires for these lines is:

z’= [z,, (0 ] - [z,, (0))][zp, (@)]-’[z,F ((0)]

( 26)

The ground wires alter the self and mutual resistance

and the self and mutual inductance.

The elements of the Z’ matrix are :

Z’li(~) = R’ii(~) + jo)L’ii(@)

( 27)

Z’ij((0) = Rij(~) + jmL’ij(~)

( 28)

The terms Rii(m), R’ij(o)), L’ii(@) and L’ij(@) are the

parameters of the transmission line atler the reduction of

ground wires.

Figure 3 shows the self resistance of phase 1

considering the primitive (Rl I ) and reduced (R’11) matrix.

103

102

10’

10”

1o“

,~.:

Figura 3- Self resistance of t he pha se 1 cons id er ing the pr imi ti ve (R,,)

and r educed (RI I) mat rix

Figure 4 shows the self inductance of phase 1 considering

the primitive (Ll 1) and reduced (L’1 i )

matrix.

(rrrHerrrys/km)

2.5

,

(Hz)

F@Ira 4 Self inductance of the phase 1 c onsidering the primit & (L,, )

red uc e (L’ , 1 )mat ri x.

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 880



v

INDIVIDUAL ANALYSIS OF THE EFFECTS

OF THE EXTERNAL IMPEDANCE AND THE

SOIL AND SKIN EFFECTS

V.1 Skin effect

The skin effect is related to the fact that in conductor

materials the electromagnetic power is transmitted only in

the superficial area [2, 9]. This fact happens because, in

conductor materials, any current density or any intensity of

electric field is only present in the superficial area. This

area is denominated penetration depth or skin depth [9].

The skin depth, in good conductors, are extremely small.

Therefore the resistance of the conductors increases, with

the increase of the frequency, because the skin depth

decreases. The internal inductance is directly related with

the penetration depth. Therefore when the frequency

increases, the internal inductance decreases, In low

frequencies, the internal inductance of the cables of the

transmission line assumes considerable values when

compared with the total distributed inductance of the line.

In higher f requencies, the internal inductance distributed of

a circular cable becomes very small in relation to its value

for low frequencies.

Consider a transmission line where that the distance

among the conductors is much larger than the sum of the

radii of the conductors. Considering only the presence of

the skin effect, the elements of the primitive matrix have

the following formation rule:

Zil = R1i(~) + j(oLii(~) ( 29)

Zij = O ( 30)

Making the reduction of the ground wires, the reduced

matrix that is considered only the action of the skin effect,

is given fo~

Z’S~,. = [Z~~] ~” - [Z~~],~,n [Z~~]-i ,~. [ZF’~],~,”( 3 1)

As the primhive matrix, where only the presence of

the skin effect is considered, there are elements of the type

shown in the equations 29 and 30 and the [ZFP] and [ZPF]

matrixes are nil. This way, Z’,kin matrix is:

z’~kin = [zFF] ki.

( 32)

Where :

~,kin Reduced matrix, considering only the presence

of the skin effect

[zFF]skin [ZFF] matrix of the line where only the

presence of the skin effect is considered.

Equation 32 shows that the presence of the ground

wires does not alter the parameters of the line, when only

the influence of the skin effect is considered. This

affirmation is true when the distance among the conductors

is much larger than the sum of the radii of the conductors.

V.2 External Impedance

Considering only the influence of the external

impedance, the elements of the primitive matrix has the

following formation rule:

Zil = j@Lii

( 33)

Zij = j@Lij

( 34)

In the equations 33 and 34, the external

inductances Lii and Lij are constant.


matrix when only the influence of the external impedance

is considered, is given by:

Z’ext = [ZFF]ex,– [ZFP]e.([ZPP]exJ “ [ZPF]ext35)

Where :

Z’ext Reduced matrix, considering only the

inf luence of the external impedance

[ZFF],Xt [ZFF] Matrix of the lie when only the

influence of the external impedance is

considered

[ZFP],Xt [Z,p] matrix of the line when only the


considered.

[zPF]..t [ZPF] matrix of the line when only the


considered.

The elements of the Z’.., matrix have the following

formation rule:

Z’ii = jmL’ii((o) (

36)

Z’ij = j~L’ij(~) ( 37)

Equations 36 and 37 show that the external inductances

that are fixed in lines without ground wires, become varied

when the presence of the ground wires is considered. The

ground wires do not produce mutual resistances.

V.3 Ground Effect

Considering only the influence of the ground effect, the

elements of the primitive matrix have the following

formation rule:

Zii = Rii((o) + j(oLii(o)) ( 38)

Zij =

Rij(~) + j~LiJ(@)

(

39)


matrix is given by

z’,~il = [zFF],oil – [ZF+’]SO,IzPF’]”i,~il[zPF]SOil( 40)

Where :

Z’soil

Reduced matrix , only the influence of the

ground effect is considering

[zFF]sd [ZFF] rnatriX of the line where only the

influence of the ground effect is considered

[zFP]soil [ZFP] matrix of the line where only the

inf luence of the ground effect is considered

[zp~]~~il [zpE] matrix of

the line where only the

influence of the ground ef fect is considered

The elements of the Z’,Oil matrix have the following

formation rule:

Z’ii = R’ii(~) + j@L’ii(@)

(41)

Z’ij = R’ij(@) + jOL’ij(w)

( 42)

Equations 41 and 42 show that, considering only the

action of the ground effect, then self and mutual resistances

are altered due to the presence of the ground wires. The self

and mutual inductances ae also altered.

w

SUPERPOSITION OF THE EFFECTS ON

LINES WITHOUT GROUND WIRES

Consider a transmission line where distance among the

conductors is much larger than the sum of the radii of the

conductors. Consider that the primitive matrix shown in

equation 1 contains the influence of the ground and skin

effects and also of the external impedance.

Despite the presence of the ground wires, the

impedance matrix of the line that considered the ground

and skin effects and the external impedance, is the [ZFF]

matrix. Therefore:

z=[zw]

( 43)

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 881



The superposition of the ground and skin effects and of

the external impedance for a line without ground wires is :

Z,up = [ZFF] tin + [Z~~]cXt + [zFF] ~il ( 44)

Comparing equations 43 and 44, it is observed that they

are identical. Therefore, the individual analysis of the

~arameters of the line without mound wires can be used so

~hat they are observed in the co~tribution of each one of the

effects in the calculation of the line parameters.

Figure 5 shows the self resistance of the phase 1,

considering each one of the effects separately (R~kiml1,

Rs.illi and &iI) and considering all the effects (RI l).

(Ohrrrs/km)

103

1

,&ll=f J

<0’

10”’---

....’..’

,..3 I

I

10’ 102 10’ 104 105

10’ (Hz)

Figura 5- Self resistance of the phase 1 considering each one of the

effects separately

(R,.,t,,, ILw,

and IG,I J and considering all the effects

(fLI)

Figure 6 shows the self inductance of the phase 1,

considering each one of the effects separately (L~kinl1 ,

LSOill,

and LeXtl, ) and considering all the effects (Ll I ).

, ., (rnHenrys /km)

I

10”1

1

10’ 102

10’ 104

10’

10’ (Hz)

Figura 6- Self inductance of the phase 1 considering all the effects (L I I)

and considering each one of the effects separately (L~,II 1, L sw I and LM I I ).

VII SUPERPOSITION OF THE EFFECTS IN

LINES WITH GROUND WIRES

Consider that the matrix of impedance shown in

equation 1 contains the influence of the ground and skin

effects and also of the external impedance.

In this line, when the presence of the ground wires is

considered, the ser ies impedance matrix is written as:

z ’ = [z~~] – [Z~~][Z~~]- i[ZFF]

( 45)

The [Z~~] , [Z~P], [Zp~] and [ZPP] matrices are calculated

according to equations 7, 8, 9 and 10, respectively.

The superposition of all the effects for the line with

ground wires is given by

z’ ,Up= z’,kin + Z’cXt + z’SOil ( 46)

The [Z’,kin], [Z’..(] and [z’,Oil] matrices are shown in

equations 31, 35 and 40, respectively.

Comparing the equations 45 and 46, it is observed that

they are different. Therefore when the presence of the

ground wires is considered, we can not make the

superposition of the effects.

Figure 7 shows the self resistance of phase 1 of the line

with ground wires, considering each one of the effects

separately (R’,~inl,, R’Will, and R’.X,l,) and considering all

the effects (Rl ,).

. {Ohnw’km)

,...,:

10=

10’ 102

103

10’

10

IO’ (Hz)

Figura 7- Self resistance of the phase 1 of the line with ground return

cables, with each one of the effects separately (N,., ,, ,, R,~~lt and R.,, [ i)

and with a ll the effects (Rt i )

Figure 8 shows the self inductance of the phase 1 of

lime with ground return cables, considering all the effects

(L’ll) and considering each one of the effects separately

(L’S~inl1,L’will, and L’extil).

,

(rnHerrrys/km)

‘0 ~

Figura 8- Self inductance of the phase 1 of the line with ground return

cables, with all the effects (L’, I) and with each one of the effects

separately (L’ ,~~1 I, L ’, .,11I and U.~,1 I)

VIII CONCLUSIONS

The presence of the ground wires grounded in all

s tructures changes the R and L parameters.

We know that the transmission lines with constant

parameters are hypothetical situations but we can use this

0-7803-6672-7/01/$10.00 (C) 2001 IEEE 882



hypothetical example to show the influence of the ground

wires on the R and L parameters. In this situation, in low

frequencies the ground wires do not change the self

resistances and self inductances. For the intermediate

tlequencies the ground wires transform the sel f resistances

and the self inductances in variable parameters. In the high

tlequencies the self resistances are constant but with values

larger than the initial value and the self inductances are

constant but with values lower than the initial value.

The influence of the ground wires on the self

resistances is in fhnction of the frequency range

considered. In the example that was studied in this paper it

was verified that in frequencies between 10 Hz e 1 kHz the

ground wires make self resistance have a larger value and

for fi-equencies above 1 kHz the ground wires make the self

resistance have a lower value.

It was observed, in the example that was studied, that in

frequencies between 10 Hz e 100 Hz, the ground wires do

not change the self inductance. For frequencies higher than

100 Hz, the ground wires make the self inductance to have

a lower value.

If we consider, in a hypothetical situation, a line

without ground wires, we can use the superposition

principle. In this situation it is possible to analyze the

influence that each one of the effects (the skin effect,

ground effect and the external impedance) exercises

separately. In the example that was studied, considering a

t ransmission line where the distance among the conductors

is much larger than the sum of the radii of the conductors,

it was verif ied that in fi-equencies below 100 Hz the ground

and the skin effects produce the self resistance and for

fi-equencies above 100 Hz the ground effect influence on

the self resistance is much larger than the skin effect

influence. At low frequencies, the self inductance is the

result of the sum of the inductances produced by ground

and skin effects and the external inductance, and in high

frequencies the largest influence is of the external

inductance, followed by inductance produced by ground

effect and skin effect respectively.

In transmission lines with ground wires is not possible

to use the superposition principle. The ground and the skin

effects produce the self resistance. The self inductance is

the result of the combination of the inductance produced by

ground and skin ef fects and the external inductance.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

Ix

REFERENCES

M. C. Tavares, “ Modelo de Iinha de TrmrsmissZo Polif%ico

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R. A Chipman, “ Teoria e Problemas de Linhas de Transmiss~o”,

Editora Mc Graw-Hill do Brasil LTDA, 1976.

A Deri, G. Tevan, A. Semlyen e A. Castanheira, “The Complex

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Multi-Layer Eafih Return”, IEEE Trans. on Power Apparatus and

Systems, Vol. PAS-1OO, No 8, Agosto, 1981.

0. Saad, G. Gaba e M. Giroux, “A Closed- Form Approximation

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M. D’Amore e M. S. Sarto, “A New Formulation of Lossy Ground

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Janeiro , 1997.

F. Rachid, C. A. Nucci e M. Ianoz, “Transient Analysis of

Multiconductor Lines Above a LQssy Ground”, IEEE Trans. on

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Livros T4cnicos e Cientificos Editors S. A., 1977.

B. Gu st avsen, A. Seml yen,

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four th edi ti on, 1981.

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Analyzing the transmission line parameters in frequency domain

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