analyzing the transmission line parameters in frequency domain
TRANSCRIPT
7/23/2019 Analyzing the transmission line parameters in frequency domain
http://slidepdf.com/reader/full/analyzing-the-transmission-line-parameters-in-frequency-domain 1/6
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
7/23/2019 Analyzing the transmission line parameters in frequency domain
http://slidepdf.com/reader/full/analyzing-the-transmission-line-parameters-in-frequency-domain 2/6
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
7/23/2019 Analyzing the transmission line parameters in frequency domain
http://slidepdf.com/reader/full/analyzing-the-transmission-line-parameters-in-frequency-domain 3/6
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
7/23/2019 Analyzing the transmission line parameters in frequency domain
http://slidepdf.com/reader/full/analyzing-the-transmission-line-parameters-in-frequency-domain 4/6
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.
Making the reduction of the ground wires, the reduced
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
influence of the external impedance is
considered.
[zPF]..t [ZPF] matrix of the line when only the
influence of the external impedance is
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)
Making the reduction of the ground wires, the reduced
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
7/23/2019 Analyzing the transmission line parameters in frequency domain
http://slidepdf.com/reader/full/analyzing-the-transmission-line-parameters-in-frequency-domain 5/6
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
7/23/2019 Analyzing the transmission line parameters in frequency domain
http://slidepdf.com/reader/full/analyzing-the-transmission-line-parameters-in-frequency-domain 6/6
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
Utilizando Qume Modos”, Tese de Doutorado, Campinas, S50
Paul o, B razi l, 19 98.
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
Ground Return Plane a Simplified Model for Homogeneous and
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
for Ground Return Impedance of Underground Cables”, IEEE
Trans. on Power Delivery, Vol. 11, No 3, Julho, 1981.
M. D’Amore e M. S. Sarto, “A New Formulation of Lossy Ground
Return Parameters for Transient Analysis of Multiconductor
Dissipative Lines”, IEEE Trans. on Power Delivery, Vol. 12, No 1,
Janeiro , 1997.
F. Rachid, C. A. Nucci e M. Ianoz, “Transient Analysis of
Multiconductor Lines Above a LQssy Ground”, IEEE Trans. on
Power Delivery, Vol. 14, No 1, Janeiro, 1999.
R. D, Fuchs, “TransrnissZo de Energia El&trica – Lirrhas A&reas”,
Livros T4cnicos e Cientificos Editors S. A., 1977.
B. Gu st avsen, A. Seml yen,
“Combined Phase and Modal Domain
Calculation of Transmission Line Transients Based on Vector
Fitting”, IEEE Trans. on Power Delivery, Vol. 13, No 2, Abril.
W. H. Hayt Jr, “Engineering Electromagnetic”, Mcgraw-Hill,
four th edi ti on, 1981.
0-7803-6672-7/01/$10.00 (C) 2001 IEEE 883