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8/8/2019 Slide07 Chapter04
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BEE3133
Electrical Power SystemsChapter 3
Transmission Line Parameters
Rahmatul Hidayah Salimin
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RESISTANCE
Important in transmission
efficiency evaluation and
economic studies.
Significant effect Generation ofI2R loss in
transmission line.
ProducesIR-type voltage dropwhich affect voltage regulation.
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RESISTANCE
The dc resistance of a solid roundconductorat a specified temperatureis
Where :
= conductor resistivity (-m),
l = conductor length (m) ; and
A = conductor cross-sectional area (m2)
dc
lR
AV!
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RESISTANCE
Conductor resistance isaffected by three factors:-
Frequency (skin effect)
Spiraling
Temperature
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RESISTANCE
Frequency Skin Effect
When ac flows in a conductor, the
current distribution is not uniform over
the conductor cross-sectional area andthe current density is greatest at the
surface of the conductor.
This causes the ac resistance to be
somewhat higher than the dcresistance. The behavior is known as
skin effect.
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RESISTANCE
The skin effect is where alternating
current 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
Skin effect correction factoraredefined as
Where
R = AC resistance ; and
Ro= DC resistance.
O
R
R
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RESISTANCE
Spiraling
For stranded conductors, alternatelayers of strands are spiraled inopposite directions to hold the strands
together. Spiraling makes the strands 1 2%
longer than the actual conductorlength.
DC resistance of a stranded conductoris 1 2% larger than the calculatedvalue.
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RESISTANCE
Temperature
The conductor resistance increasesas temperature increases. Thischange can be considered linear overthe range of temperature normally
encountered and may be calculatedfrom :
Where:R1 = conductor resistances at t1 in C
R2 = conductor resistances at t2 in C
T = temperature constant (depends on
the conductor material)
22 1
1
T tR R
T t
!
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RESISTANCE
The conductor resistance is best
determined from manufacturers
data.
Some conversion used incalculating line resistance:-
1 cmil = 5.067x10-4 mm2
=5.067x10
-6
cm
2
= 5.067x10-10 m2
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RESISTANCE
Example:-
A solid cylindrical aluminum
conductor 25km long has an area
of 336,400 circular mils. Obtain theconductor resistance at
(a) 20C and
(b) 50C.
The resistivity of aluminum at 20C is
= 2.8x10-8-m.
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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
V!
!
v v v!
v v
! v ;
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RESISTANCE
Answer (b)
5050 20
20
6
6
228 504.0994 10
228 204.5953 10
C
C C
C
T tR R T t
rr r
r
!
! v
! v ;
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RESISTANCE
Exercise 1
A transmission-line cable consists of
12 identical strands of aluminum,
each 3mm in diameter. Theresistivity of aluminum strand at
20C is 2.8x10-8-m. Find the 50C
ac resistance per km of the cable.
Assume a skin-effect correctionfactor of 1.02 at 50Hz.
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RESISTANCE
Exercise 2:-A solid cylindrical aluminum
conductor 115km long has an area
of 336,400 circular mils. Obtain the
conductor resistance at:
(a) 20C
(b) 40C
(c) 70C
The resistivity of aluminum at 20C is
= 2.8x10-8-m.
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RESISTANCE
Exercise 3
A transmission-line cable consists of
15 identical strands of aluminum,
each 2.5mm in diameter. Theresistivity of aluminum strand at
20C is 2.8x10-8-m. Find the 50C
ac resistance per km of the cable.
Assume a skin-effect correctionfactor of 1.015 at 50Hz.
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INDUCTANCE :
A SINGLE CONDUCTOR
A current-carrying conductor producesa magnetic field around the conductor.
The magnetic flux can be determined
by using the right hand rule.
For nonmagnetic material, the
inductance L is the ratio of its total
magnetic flux linkage to the currentI,
given by
where=flux linkages, in Weber turns.
LI
P!
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INDUCTANCE :
A SINGLE CONDUCTOR
For illustrativeexample, considera long roundconductor with
radius r, carryinga currentIasshown.
The magnetic
field intensityHx,around a circle ofradiusx, isconstant andtangent to thecircle.
2
xx
IH
xT!
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INDUCTANCE :
A SINGLE 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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Flux Linkage
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INDUCTANCE :
A SINGLE CONDUCTOR
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INDUCTANCE :
A SINGLE PHASE LINES
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
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What and How to Calculate:-
Lint , Lext @ L?
L1 , L2 @ L?
L11 , L12 @ L22? GMR?
GMD?
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INDUCTANCE :
A SINGLE CONDUCTOR
INTERNAL INDUCTANCE
Internal inductance can be express as
follows:-
Where
o = permeability of air (4 x 10-7 H/m)
The internal inductance is independent of
the conductor radius r
70int 1 10 /
8 2L H mQ
T! ! v
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INDUCTANCE :
A SINGLE CONDUCTOR
INDUCTANCE DUETO EXTERNAL
FLUX LINKAGE
External
inductancebetween to point
D2 and D1 can be
express as
follows:
7 2
1
2 10 ln /extD
L H mD
! v
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INDUCTANCE :
A SINGLE PHASE LINES
A single phase lines consist of asingle current carrying line with a
return line which is in opposite
direction. This can be illustrated as:
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INDUCTANCE :
A SINGLE PHASE LINES
Inductance of a single-phaselines can be expressed as
below with an assumption
that the radius of r1=r2=r.7 7 2
int
1
7 7 7
1
7 741
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
D L L L H m
D
D DH m H m
r r
D De H m H m
r re
DH m
re
! ! v v
! v v ! v
! v ! v
! v
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SELF AND MUTUAL
INDUCTANCES
The series inductance per phase canbe express in terms of self-inductanceof each conductor and their mutualinductance.
Consider the one meter length single-phase circuit in figure below:-
Where L11 and L22 are self-inductanceand the mutual inductance L
12
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SELF AND MUTUAL
INDUCTANCES
D
xD
xL
DxL
erxL
ILLID
xer
xIL
ILL
ILL
mHD
xer
xL
mH
D
xerxL
1ln102
1
ln102
1ln102
1
ln102
1ln102
1ln102
/1
ln1021
ln102
/1ln102
1
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
!
!
!
!
!
!!
!
!
!
!
P
P
P
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SELF AND MUTUAL
INDUCTANCES
L11, L22 and L12 can be expressed asbelow:-
7
11 0.25
1
7
22 0.25
2
7
12 21
1
2 10 ln
12 10 ln
12 10 ln
L r e
L
r e
L LD
! v
! v
! ! v
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SELF AND MUTUAL
INDUCTANCES
Flux linkage of conductor i
ijDIerIx
n
j ij
j
i
ii {
! !
1
ln1
ln1021
25.0
7
P
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Symmetrical Spacing Consider 1 meter length of a three-phase
line with three conductors, each radius r,
symmetrically spaced in a triangular
configuration.
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Assume balance 3-phase current
Ia+ Ib+ Ic = 0
The total flux linkage of phase a
conductor
Substitute for Ib + Ic=-Ia
!
DI
DI
erIx
cb
a
aa
1ln
1ln
1ln102
25.0
7P
25.0
7
25.0
7 ln1021
ln1
ln102
!
!
er
DIx
DI
erIx
a
aa
a
aaP
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Because of symmetry, a=b=c
The inductance per phase per
kilometer length
kmmH
re
Dx
I
L /ln10225.0
7
!!P
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Asymmetrical Spacing Practical transmission lines cannot maintain
symmetrical spacing of conductors because of
construction considerations.
Consider one meter length of three-phase line with
three conductors, each with radius r. Theconductor are asymmetrically spaced with
distances as shown.
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
The flux linkages are:-
v!
v!
v!
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
P
P
P
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
For balanced three-phase currentwithI
aas reference, we have:-
a
o
ac
a
o
ab
aIIII
aII
!!!!
120240
2
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Thus La,Lb and Lc can be foundusing the following equation:-
v!!
23
2
25.0
12
7 1ln1
ln1
ln102
D
a
reD
a
I
L
b
b
b
P
v!!
1312
2
25.0
7 1ln1
ln1
ln102D
aD
areI
La
a
a
P
v!!
25.0
2313
27 1ln1
ln1
ln102reD
aD
aI
Lc
c
c
P
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Transpose Line Transposition is used to regain symmetry
in good measures and obtain a per-phaseanalysis.
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
This consists of interchanging the phaseconfiguration every one-third the length sothat each conductor is moved to occupy thenext physical position in a regular sequence.
Transposition arrangement are shown in the
figure
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Since in a transposed line eachphase takes all three positions,
the inductance per phase can be
obtained by finding the averagevalue.
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0.25
12 13
7
0.25
23 12
0.25
13 23
7
0.25
12
3
1 1 1ln 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 c L L L
L
re D D
re D D
re D D
re D D
!
r r
v ! r r
r r
-
v!
R R
R R
R R
23 13
312 23 137
0.25
1ln
2 10 ln
D
D D D
re
! v
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Since in a transposed line each phasetakes all three positions, the
inductance per phase can be obtained
by finding the average value.
3
cba
a
LLL
L
!
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Noting a + a2 = -1
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
v!
v!
v!
re
DDD
DDDre
DDDreL
kmmH
re
DDDL /ln2.0
25.0
3
1
132312
!
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What and How to Calculate:-
Lint , Lext @ L?
L1 , L2 @ L?
L11 , L12 @ L22? GMR?
GMD?
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Inductance of Composite
Conductors
In evaluationofinductance, solid round
conductors were considered.However,in
practicaltransmissionlines, stranded
conductors are used.
Consider a single-phase line consistingof
twocomposite conductorsx and y as shown
in Figure 1. The currentinx isIreferenced
intothe page, andthe returnin y is I.
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Inductance of Composite
Conductors
Conductorx consistofn identical strands or
subconductors, each with radius rx.
Conductory consistofm identical strands or
subconductors, each with radius ry
.
The currentis assumedtobe equally divided
amonthe subconductors. The current per
strands isI/n inx andI/m iny.
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Inductance of Composite
Conductors
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
v!!
v!!
v!
v
v!
P
P
P
P
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'...
)...)...(...(
)...)...(...(
/ln102
2
''''
7
xnnbbaa
nnnnbnaanabaax
mnnmnbnaamabaa
x
x
rDDD
where
DDDDDDGMR
DDDDDDGMD
where
mHGMR
GMDL
!!!
!
!
v!
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GMR of Bundled Conductors
Extra high voltage transmissionlines are
usually constructed with bundledconductors.
Bundling reduces the line reactance, which
improves the line performance andincreases
the powercapability ofthe line.
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GMR of Bundled Conductors
4 316 42/1
3 29 3
09.1)2(
)(
dDdddDD
bundleorsubconductfourthefor
dDddDD
bundleorsubconductthreethefor
ss
b
s
ss
b
s
v!vvvv!
v!vv!
dD
dDD
bundleorsubconducttwothefor
DDDDDDGMR
ss
b
s
nnnnbnaanabaax
v!v!
!
4 2
)(
)...)...(...(2
I d t f Th h
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Inductance of Three-phase
Double Circuit Lines
Athree-phase double-circuittransmission
line consists oftwoidenticalthree-phase
circuits. To achievebalance, each phase
conductormustbe transposed withinitgroup
and with respecttothe parallelthree-phase
line.
Consider a three-phase double-circuitline
with relative phase positions a1
b1
c1
-c2
b2
a2
.
I d t f Th h
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Inductance of Three-phase
Double Circuit Lines
c
GMDbetween each phase group
422122111
422122111
422122111
cacacacaC
cbcbcbcbBC
babababaB
DDDDD
DDDDD
DDDDD
!
!
!
I d t f Th h
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Inductance of Three-phase
Double Circuit Lines
The equivalentGMD per phase is then
3ACBCAB
DDDGMD !
Similarly,GMRofeach phase group is
214 2
21
214 2
21
214 2
21
)(
)(
)(
cc
b
cc
b
SC
bb
b
bb
b
SB
aa
b
aa
b
SA
DDDDD
DDDDD
DDDDD
ss
ss
ss
!!
!!
!!
where is the geometricmean radius of
bundledconductors.
b
sD
I d t f Th h
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Inductance of Three-phase
Double Circuit Lines
The equivalentGMR per phase is then
3SCSSAL DDDGMR !
The inductance per-phase is
mHGMR
GMDL
L
x /ln1027v!
INDUCTANCE :
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Question 4
A three-phase, 50 Hz transmission line has a
reactance 0.5 per kilometer. The conductor
geometric mean radius is 2 cm. Determine thephase spacing D in meter.
INDUCTANCE :
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Question 4
A three-phase, 60 Hz transmission line has a
reactance 0.25 per kilometer. The conductor
geometric mean radius is 5 cm. Determine thephase spacing D in meter.
INDUCTANCE :
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INDUCTANCE :
3-PHASE TRANSMISSION LINES
Question 4
A three-phase, 50 Hz transmission line has
Xc = 0.5 per kilometer. The conductor geometric
mean radius is 2 cm. Determine the phase spacingD in meter.
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CAPACITANCE
Transmission line conductorsexhibit capacitance with respect
to each other due to the potential
difference between them.
The amount of capacitancebetween conductors is a function
ofconductor size, spacing, and
height above ground.
Capacitance C is:-
qC
V!
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LINE CAPACITANCE
Consider a longround conductor
with radius r,
carrying a
charge ofqcoulombs per
meter length as
shown.
The electricalflux density at a
cylinder of radius
x is given by: 2
q qD
A xT! !
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LINE CAPACITANCE
The electric field intensity E is:-
Where permittivity of free space, 0 = 8.85x10-12 F/m.
The potential difference between cylinders
from position D1 to D2 is defined as:-
The notation V12 implies the voltage drop from 1
relative to 2.
0 02
D qE
xI TI! !
212
0 1
ln2
q DVDTI
!
CAPACITANCE OF SINGLE
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CAPACITANCE OF SINGLE-
PHASE LINES
Consider one meter length of a single-phase line consisting of two long solid
round conductors each having a
radius r as shown.
For a single phase, voltage betweenconductor 1 and 2 is:-
12
0
ln /q D
V F mrTI
!
CAPACITANCE OF SINGLE
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CAPACITANCE OF SINGLE-
PHASE LINES
The capacitance between theconductors:-
012 /
ln
C F mD
r
TI!
CAPACITANCE OF SINGLE
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CAPACITANCE OF SINGLE-
PHASE LINES
The equation gives the line-to-line capacitance between the
conductors
For the purpose of transmission
line modeling, we find itconvenient to define a
capacitance Cbetween each
conductor and a neutral line as
illustrated.
CAPACITANCE OF SINGLE
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CAPACITANCE OF SINGLE-
PHASE LINES
Voltage to neutral is half ofV12 and the capacitance to
neutral is C=2C12 or:-
02 /
ln
C F mD
r
TI!
Potential Difference in a
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Potential Difference in a
Multiconductor configuration
Consider n parallel long conductorswith charges q1, q2,,qncoulombs/meter as shown below.
ki
kj
n
k
kijD
DqV ln
2
1
10!! TI
Potential difference between conductori and j due to the presence of all
charges is
CAPACITANCE OF THREE-
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CAPACITANCE OF THREE-
PHASE LINES
Consider one meter length of 3-phaseline with three long conductors, each
with radius r, with conductor spacing
as shown below:
CAPACITANCE OF THREE
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CAPACITANCE OF THREE-
PHASE LINES
For balanced 3-phase system, the
capacitance per phase to neutral is:
1/3
12 23 13
2F/m
ln
a o
an
qC
VD D D
r
TI! !
CAPACITANCE OF THREE-
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CAPACITANCE OF THREE-
PHASE LINES
1/312 23 130.0556 F/km
ln
C
D D D
r
Q!
The capacitance to neutral in F perkilometer is:
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Effect of bundling
mF
r
GMDCb
/ln
2 0TI
!
The effect of bundling is introduce an
equivalent radius rb. The radius rb issimilar to GMRcalculate earlier for the
inductance with the exception that
radius rof each subconductor is used
instead ofDs.
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Effect of bundling
Ifd is the bundle spacing, we obtain for
the two-subconductor bundle
drrb v!
For the three-subconductor bundle
3 2drrb v!
For the four-subconductor bundle4 309.1 drrb v!
Capacitance of Three-phase
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Capacitance of Three phase
Double Circuit Lines
mF
GMRGMD
C
c
/
ln
2 0TI!
The per-phase equivalent capacitanceto neutral is obtained to
GMD is the same as was found for
inductance calculation
4
22122111
422122111
422122111
cacacacaAC
cbcbcbcbC
babababaA
DDDDD
DDDDD
DDDDD
!
!
!
Capacitance of Three-phase
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Capacitance of Three phase
Double Circuit Lines
The equivalentGMD per phase is then
3ACCA
DDDGMD !
The GMRC of each phase is similar to
the GMRL, with the exception that rb isused instead of b
sD
This will results in the following equ
21
21
21
cc
b
C
bb
b
B
aab
A
Drr
Drr
Drr
!
!
! 3CBAC rrrGMR !
EFFECT OF EARTH ON THE
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EFFECT OF EARTH ON THE
CAPACITANCE
For isolated charged conductor theelectric flux lines are radial andorthogonal to cylindrical equipotentialsurfaces, which will change theeffective capacitance of the line.
The earth level is an equipotentialsurface. Therefore flux lines are forcedto cut the surface of the earthorthogonally.
The effect of the earth is to increasethe capacitance.
EFFECT OF EARTH ON THE
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EFFECT OF EARTH ON THE
CAPACITANCE
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.
However, for unbalance analysis suchas unbalance faults, the earths effectand shield wires should be considered.
MAGNETIC FIELD
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MAGNETIC FIELD
INDUCTION
Transmission line magnetic fieldsaffect objects in the proximity of
the line.
Produced by the currents in theline.
It induces voltage in objects that
have a considerable length
parallel to the line (Ex: telephone
wires, pipelines etc.).
MAGNETIC FIELD
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MAGNETIC FIELD
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.
ELECTROSTATIC
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ELECTROSTATIC
INDUCTION
Transmission line electric fields affectobjects in the proximity of the line.
It produced by high voltage in thelines.
Electric field induces current inobjects which are in the area of theelectric fields.
The effect of electric fields becomes
more concern at higher voltages.
ELECTROSTATIC
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ELECTROSTATIC
INDUCTION
Primary cause of induction to vehicles,buildings, and object of comparable
size.
Human body is effected to electric
discharges from charged objects in thefield of the line.
The current densities in human cause
by electric fields of transmission lines
are much higher than those induced by
magnetic fields!
CORONA
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CORONA
When surface potential gradientexceeds the dielectric strength ofsurrounding air, ionization occursin the area close to conductor
surface. This partial ionization is known as
corona.
Corona generate by atmosphericconditions (i.e. air density,humidity, wind)
CORONA
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CORONA
Corona produces power loss andaudible noise (Ex: radio
interference).
Corona can be reduced by: Increase the conductor size.
Use of conductor bundling.
Re ie
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Review
Transmission Line Parameters: Resistance
Skin effect
Inductance Single phase line
3 phase line equal & unequal spacing
Capacitance Single phase line
3 phase line equal & unequal spacing
Conductance Neglected Corona
Review
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Review
Effect of Earth on theCapacitance
Magnetic Field Induction
Electrostatic Induction Corona