Power System Slide07
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Transcript of Power System Slide07
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INTRODUCTION
All transmission lines in a power system exhibit the
electrical properties of resistance, inductance,capacitance andconductance.
Inductance and capacitance are due to the effectsof magnetic and electric fieldsaround theconductor.
These parameters are essential for thedevelopment of the transmission line modelsusedin power system analysis.
The shunt conductanceaccounts for leakagecurrents flowing across insulators and ionized
pathways in the air. The leakage currents are negligiblecompared to
the current flowing in the transmission lines andmay be neglected.
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RESISTANCE
Important in transmissionefficiency evaluation and
economic studies.
ignificant effect! "eneration ofI2Rloss in
transmission line.
! #roducesIR$type voltage dropwhich affect voltage regulation.
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RESISTANCE
The dc resistance of a solid roundconductorat a specified temperature is
%here &
' conductor resistivity ()$m*,
l ' conductor length (m* + andA ' conductor cross$sectional area (m*
dc
lR A
=
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RESISTANCE
-onductor resistance isaffected by three factors&$
re/uency (0skin effect1*
piraling
Temperature
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RESISTANCE
re/uency ! kin 2ffect %hen ac flows in a conductor, the current
distribution is not uniformover the
conductor cross$sectional area and the
current density is greatest at the surfaceof the conductor.
This causes the ac resistance to be
somewhat higher than the dc resistance.
The behavior is known as skin effect.
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RESISTANCE
The skin effect is where alternatingcurrent tends to avoid travel through
the center of a solid conductor, limiting
itself to conduction near the surface.
This effectively limits the cross$
sectional conductor area available to
carry alternating electron flow,
increasing the resistance of thatconductor above what it would
normally be for direct current
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RESISTANCE
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RESISTANCE
kin effect correction factoraredefined as
%here
3 ' A- resistance + and3o' 4- resistance.
O
R
R
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RESISTANCE
piraling or stranded conductors, alternate layers
of strands are spiraled in oppositedirections to hold the strands together.
piraling makes the strands 5 ! 6longer than the actual conductor length.
4- resistance of a stranded conductor is5 ! 6 larger than the calculated value.
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RESISTANCE
Temperature
The conductor resistance increases as temperatureincreases. This change can be considered linearover the range of temperature normally encounteredand may be calculated from &
%here&35' conductor resistances at t5in 7-
3' conductor resistances at t in 7-
T ' temperature constant (depends on theconductor material*
22 1
1
T tR R
T t
+=
+
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RESISTANCE
The conductor resistance is bestdetermined from manufacturer1s
data.
ome conversion used incalculating line resistance&$
5 cmil ' 8.9:;x59$
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Resistivity & Temparature
Constant of Conductor Metals
=aterial 20C T
3esistivity at 9>- Temperature -onstant
m10-8 cmil/ft >-
-opper
Annealed 5.; 59.?; ?
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RESISTANCE
2xample&$A solid cylindrical aluminum
conductor 8km long has an area of
??:,
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RESISTANCE
Answer (a*
( ) ( )
( )
25
8 3
4
6
2.8 10 25 10
336,400 5.076 10
4.0994 10
l km
l
R A
=
=
=
=
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RESISTANCE
Answer (b*
( )
50
50 2020
6
6
228 504.0994 10
228 204.5953 10
C
C CC
T t
R R T t
+
= ++
=
+=
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RESISTANCE
2xercise 5A transmission$line cable consists
of 5 identical strands of
aluminum, each ?mm in diameter.The resistivity of aluminum strand
at 97- is .x59$)$m. ind the
897- ac resistance per km of the
cable. Assume a skin$effectcorrection factor of 5.9 at 89@z.
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RESISTANCE
2xercise &$
A solid cylindrical aluminum conductor
558km long has an area of ??:,
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RESISTANCE
2xercise ?A transmission$line cable consists
of 58 identical strands of
aluminum, each .8mm indiameter. The resistivity of
aluminum strand at 97- is
.x59$)$m. ind the 897- ac
resistance per km of the cable.Assume a skin$effect correction
factor of 5.958 at 89@z.
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INDUCTANCE
A SIN!"E CONDUCTOR
A current$carrying conductor produces amagnetic field around the conductor.
The magnetic flux can be determined by
using the right hand rule.
or nonmagnetic material, the inductanceLis the ratio of its total magnetic flux linkage
to the currentI, given by
where'flux linkages, in %eber turns.LI
=
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INDUCTANCE
A SIN!"E CONDUCTOR
or illustrativeexample, consider along round conductorwith radius r, carryinga currentIas shown.
The magnetic fieldintensityHx, arounda circle of radiusx, isconstant and tangentto the circle.
2
xx
IH
x=
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INDUCTANCE
A SIN!"E CONDUCTOR
The inductance of the conductorcan be defined as the sum of
contributions from flux linkages
internal and external to theconductor.
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#lu$ "in%ae
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INDUCTANCE
A SIN!"E CONDUCTOR
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INDUCTANCE
A SIN!"E '(ASE "INES
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
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What and How to Calculate:- Fint , Fext G FH
F5, FG FH
F55 , F5 G F H "=3H
"=4H
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INDUCTANCE
A SIN!"E CONDUCTOR
IT23AF I4J-TA-2! Internal inductance can be express as
follows&$
! %here
o' permeability of air (
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INDUCTANCE
A SIN!"E CONDUCTOR
I4J-TA-2 4J2TD 2KT23AF
FJK FILA"2
! 2xternal
inductance
between to point
4,and 4
5can be
express as
follows&
7 2
1
2 10 ln /extD
L H mD
=
INDUCTANCE
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INDUCTANCE
A SIN!"E '(ASE "INES
A single phase lines consist of asingle current carrying line with a
return line which is in opposite
direction. This can be illustrated as&
INDUCTANCE
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INDUCTANCE
A SIN!"E '(ASE "INES
Inductance of a single$phaselines can be expressed as
below with an assumption
that the radius of r5'r,'r.7 7 2
int
1
7 7 7
1
7 74
1
4
7
0.25
110 2 10 ln /
2
1 110 2 10 ln / 2 10 ln /
2 4
12 10 ln ln / 2 10 ln ln /
2 10 ln /
ext
DL L L H m
D
D DH m H m
r r
D De H m H m
r re
D
H mre
= + = +
= + = + = + = +
=
SE"# AND MUTUA"
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SE"# AND MUTUA"
INDUCTANCES
The series inductance per phase can be express in terms
of self$inductance of each conductor and their mutualinductance.
-onsider the one meter length single$phase circuit infigure below&$
! %here F55 and F are self$inductance and the mutual inductanceF5
SE"# AND MUTUA"
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SE"# AND MUTUA"
INDUCTANCES
( )
( )
( )
D
xD
xL
DxL
erxL
ILLID
xer
xIL
ILL
ILL
mHD
xer
xL
mHD
x
er
xL
1ln102
1
ln102
1ln102
1ln102
1ln102
1ln102
/1
ln1021
ln102
/
1
ln1021
ln102
77
12
7
12
25.01
7
11
112111
7
25.0
1
7
111
222212
112111
7
25.0
2
7
2
7
25.0
1
7
1
=
=
=
=
=
+==
+=
=
+=
+=
SE"# AND MUTUA"
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SE"# AND MUTUA"
INDUCTANCES
F55, F,,and F5,can be expressed asbelow&$
7
11 0.251
7
22 0.25
2
7
12 21
1
2 10 ln
12 10 ln
12 10 ln
L re
L
r e
L LD
=
=
= =
SE"# AND MUTUA"
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SE"# AND MUTUA"
INDUCTANCES
lux linkage of conductor i
ijDIerIx
n
j ijj
iii
+= =
1
ln
1
ln1021
25.0
7
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
ymmetrical pacing! -onsider 5 meter length of a three$phase
line with three conductors, each radius r,
symmetrically spaced in a triangular
configuration.
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
Assume balance ?$phase currentIaM I
bM I
c' 9
The total flux linkage of phase a
conductor
ubstitute for IbM I
c'$I
a
++=
DI
DI
erIx
cb
a
aa
1ln
1ln
1ln102
25.0
7
25.0
7
25.0
7ln102
1ln
1ln102
=
=
er
DIx
DI
erIx
a
aa
a
aa
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
Because of symmetry, Na'Nb'Nc
The inductance per phase per
kilometer length
kmmHre
Dx
IL /ln102
25.0
7
==
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
Asymmetrical pacing! #ractical transmission lines cannot maintainsymmetrical spacing of conductors because of
construction considerations.
! -onsider one meter length of three$phase line with
three conductors, each with radius r. The conductor
are asymmetrically spaced with distances as shown.
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
! The flux linkages are&$
++=
++=
++=
2313
25.0
7
2312
25.0
7
1312
25.0
7
1ln
1ln
1ln102
1ln
1ln
1ln102
1ln
1ln
1ln102
DI
DI
reI
DI
DI
reI
DI
DI
reI
bacc
cabb
cbaa
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
! or balanced three$phase currentwithI
aas reference, we have&$
a
o
ac
a
o
ab
aIII
IaII
== == 120
240 2
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
Thus La, Lband Lccan be foundusing the following e/uation&$
++==
2 3
2
2 5.0
1 2
7 1ln
1ln
1ln1 02
D
a
r eD
a
I
L
b
bb
++==
1312
2
25.0
7 1ln
1ln
1ln102
D
a
D
a
r eI
L
a
a
a
++==
25.0
2313
27 1ln
1ln
1ln102
reDa
Da
IL
c
c
c
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
Transpose Fine! Transposition is used to regain symmetry
in good measures and obtain a per$phaseanalysis.
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
This consists of interchanging the phaseconfiguration every one$third the length so thateach conductor is moved to occupy the nextphysical position in a regular se/uence.
Transposition arrangement are shown in the figure
INDUCTANCE
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INDUCTANCE
)*'(ASE TRANSMISSION "INES
ince in a transposed line eachphase takes all three positions,
the inductance per phase can be
obtained by finding the average
value.
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0.25
12 13
7
0.25
23 12
0.25
13 23
7
0.25
12
3
1 1 1
ln 1 240 ln 1 120 ln
2 10 1 1 1ln 1 240 ln 1 120 ln
3
1 1 1ln 1 240 ln 1 120 ln
2 10 1 1 13ln ln ln
3
a b cL L LL
re D D
re D D
re D D
re D D
+ +=
+ + = + + +
+ + +
=
R R
R R
R R
23 13
312 23 137
0.25
1ln
2 10 ln
DD D D
re
=
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ince in a transposed line each phase
takes all three positions, the
inductance per phase can be obtained
by finding the average value.
3
cba
a
LLL
L
++
=
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oting a M a ' $5
Inductance per phase per kilometer length
( )
( )25.0
3
1
1323127
3
1
132312
25.0
7
132312
25.0
7
ln102
1ln1ln102
1ln
1ln
1ln
1ln3
3
102
=
=
=
re
DDD
DDDre
DDDreL
( )kmmH
re
DDDL /ln2.0
25.0
3
1
132312
=
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What and How to Calculate:- Fint , Fext G FH
F5, FG FH
F55 , F5 G F H "=3H
"=4H
Inductance of Composite
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Inductance of Composite
Conductors
n !"#l$#ti%n %f in&$ct#nc!, '%li& (%$n&c%n&$ct%(' !(! c%n'i&!(!&. H%!"!(, in
*(#ctic#l t(#n'mi''i%n lin!', 't(#n&!&
c%n&$ct%(' #(! $'!&.
C%n'i&!( # 'in+l!-*#'! lin! c%n'i'tin+ %f
t% c%m*%'it! c%n&$ct%('x#n& #' '%n
in i+$(! 1. ! c$((!nt inxi'I(!f!(!nc!&
int% t! *#+!, #n& t! (!t$(n in i' I.
Inductance of Composite
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Inductance of Composite
Conductors
C%n&$ct%(x c%n'i't %f ni&!ntic#l 't(#n&' %('$c%n&$ct%(', !#c it (#&i$' (x.
C%n&$ct%(y c%n'i't %f mi&!ntic#l 't(#n&' %(
'$c%n&$ct%(', !#c it (#&i$' (.
! c$((!nt i' #''$m!& t% ! !$#ll &i"i&!&
#m%n t! '$c%n&$ct%('. ! c$((!nt *!(
't(#n&' i'I/ninx#n&I/miny.
Inductance of Composite
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Inductance of Composite
Conductors
&
#
c
&
n
#
c
m
x y
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nncnbnax
mnmncnbnan
n
nanacabx
mamacabaaa
a
nanacabx
mamacabaa
a
amacabaa
anacabx
a
DDDr
DDDDn
nIL
DDDr
DDDD
nnIL
DDDr
DDDDI
or
DDDDm
I
DDDrn
I
...
...ln102
/
...
...ln102
/
...
...ln102
1ln...
1ln
1ln
1ln102
1ln...
1ln
1ln
1ln102
7
7
7
7
7
==
==
=
++++
++++=
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...
)...)......
)...)......
/ln102
2
7
xnnbbaa
nnnnbnaanabaax
mnnmnbnaamabaa
x
x
rDDD
where
DDDDDD!R
DDDDDD!Dwhere
mH!R
!DL
===
=
=
=
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!MR of +undled Conductors
&
& &
&
&
&
&
&
5xt(# i+ "%lt#+! t(#n'mi''i%n lin!' #(!$'$#ll c%n't($ct!& it $n&l!& c%n&$ct%('.
6$n&lin+ (!&$c!' t! lin! (!#ct#nc!, ic
im*(%"!' t! lin! *!(f%(m#nc! #n& inc(!#'!'
t! *%!( c#*#ilit %f t! lin!.
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!MR of +undled Conductors
4 316 42/1
3 29 3
09.1)2
)
dDdddDD
b"ndleor#"bcond"ct$o"rthe$or
dDddDD
b"ndleor#"bcond"ctthreethe$or
##
b
#
##
b
#
==
==
dDdDD
b"ndleor#"bcond"cttwothe$or
DDDDDD!R
##
b
#
nnnnbnaanabaax
==
=
4 2)
)...)......2
Inductance of T,ree*p,ase
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Inductance of T,ree p,ase
Dou-le Circuit "ines
7 t(!!-*#'! &%$l!-ci(c$it t(#n'mi''i%nlin! c%n'i't' %f t% i&!ntic#l t(!!-*#'!
ci(c$it'. % #ci!"! #l#nc!, !#c *#'!
c%n&$ct%( m$'t ! t(#n'*%'!& itin it +(%$*
#n& it (!'*!ct t% t! *#(#ll!l t(!!-*#'!lin!.
C%n'i&!( # t(!!-*#'! &%$l!-ci(c$it lin!
it (!l#ti"! *#'! *%'iti%n' #11c1-c22#2.
Inductance of T,ree*p,ase
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Inductance of T,ree p,ase
Dou-le Circuit "ines
c1
#1
#2
1 2
c211
22
33
9:; !t!!n !#c *#'! +(%$*
4 22122111
422122111
422122111
cacacacaAC
cbcbcbcb%C
babababaA%
DDDDD
DDDDD
DDDDD
=
=
=
Inductance of T,ree*p,ase
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Inductance of T,ree p,ase
Dou-le Circuit "ines
! !$i"#l!nt 9:; *!( *#'! i' t!n
3AC%CA% DDD!D =
imil#(l, 9:< %f !#c *#'! +(%$* i'
214 2
21
214 2
21
214 2
21
)
)
)
ccb
ccb
&C
bb
b
bb
b
&%
aa
b
aa
b
&A
DDDDD
DDDDD
DDDDD
##
##
##
==
==
==
!(! i' t! +!%m!t(ic m!#n (#&i$' %f
$n&l!& c%n&$ct%('.
b
#D
Inductance of T,ree*p,ase
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Inductance of T,ree p,ase
Dou-le Circuit "ines
! !$i"#l!nt 9:< *!( *#'! i' t!n
3&C&%&AL DDD!R =
! in&$ct#nc! *!(-*#'! i'
mH!R
!DL
L
x /ln102 7
=
INDUCTANCE
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)*'(ASE TRANSMISSION "INES
Ouestion ! !$i"#l!nt 9:; *!( *#'! i' t!n
3AC%CA% DDD!D =
The "=3-of each phase is similar to
the "=3F, with the exception that rbisused instead of
b
#D
This will results in the following e/uS
21
21
21
cc
b
C
bb
b
%
aa
b
A
Drr
Drr
Drr
=
=
= 3C%AC rrr!R =
E##ECT O# EART( ON T(E
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CA'ACITANCE
or isolated charged conductor the
electric flux lines are radial andorthogonal to cylindrical e/uipotentialsurfaces, which will change the effectivecapacitance of the line.
The earth level is an e/uipotentialsurface. Therefore flux lines are forcedto cut the surface of the earthorthogonally.
The effect of the earth is to increase thecapacitance.
E##ECT O# EART( ON T(E
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CA'ACITANCE
But, normally, the height of theconductor is large compared to thedistance between the conductors, andthe earth effect is negligible.
Therefore, for all line models used forbalanced steady$state analysis, theeffect of earth on the capacitance canbe negligible.
@owever, for unbalance analysis suchas unbalance faults, the earth1s effectand shield wires should be considered.
MA!NETIC #IE"D
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INDUCTION
Transmission line magnetic fieldsaffect obUects in the proximity of the
line.
#roduced by the currents in the line.
It induces voltage in obUects that have
a considerable length parallel to the
line (2x& telephone wires, pipelines
etc.*.
MA!NETIC #IE"D
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INDUCTION
The magnetic field is effected bythe presence of earth return
currents.
There are general concernsregarding the biological effects of
electromagnetic and electrostatic
fields on people.
E"ECTROSTATIC
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INDUCTION
Transmission line electric fields affectobUects in the proximity of the line.
It produced by high voltage in thelines.
2lectric field induces current inobUects which are in the area of theelectric fields.
The effect of electric fields becomes
more concern at higher voltages.
E"ECTROSTATIC
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INDUCTION
#rimary cause of induction to vehicles,
buildings, and obUect of comparable size.
@uman body is effected to electric
discharges from charged obUects in the
field of the line. The current densities in human cause by
electric fields of transmission lines are
much higher than those induced by
magnetic fieldsV
CORONA
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CORONA
%hen surface potential gradientexceeds the dielectric strength ofsurrounding air, ionization occurs inthe area close to conductor surface.
This partial ionization is known ascorona.
-orona generate by atmosphericconditions (i.e. air density, humidity,
wind*
CORONA
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CORONA
-orona produces power loss andaudible noise (2x& radio
interference*.
-orona can be reduced by&! Increase the conductor size.
! Jse of conductor bundling.
Revie.
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Revie.
Transmission Fine #arameters&! 3esistance
kin effect
! Inductance ingle phase line
? phase line e/ual W une/ual spacing
! -apacitance ingle phase line
? phase line e/ual W une/ual spacing
! -onductance eglected -orona
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Revie.
2ffect of 2arth on the-apacitance
=agnetic ield Induction
2lectrostatic Induction -orona