Chapter 2 transmission line parameters

power transmissiom

1. CHAPTER-TWO
TRANSMISSIONLINEPARAMETERS
132 kV Double Circuit
Overhead Transmission Line
Kaliti Substation
132 KV Single Circuit
Overhead Transmission Line
Kaliti-Substation

2. 2.1 INTRODUCTION
Generatorsandloadsareconnectedtogetherthroughtransmissionlines
transportingelectricpowerfromoneplacetoanother.Transmissionline
must,therefore,takepowerfromgenerators,transmitittolocationwhere
itwillbeused,andthendistributeittoindividualconsumers.
Transmissionofelectricpowerhasbeenalongtheyearsandwillstill
continueoneofthemostimportantelementsoftoday’selectricpower
systems.Powertransmissionfromgeneratingstationstoindustrialsites
andtosubstationsisthefundamentalobjectiveofthetransmission
systems.
TheTransmissionlinefunctionisaccomplishedbyoverheadACorDC
transmissionlinesorundergroundcablesthatconnectthepowerplantsin
tothetransmissionnetwork,interconnectvariousareasoftransmission
networks,interconnectoneelectricutilitywithanother,ordeliverthe
electricpowerfromvariousareaswithinthetransmissionnetworktothe
distributionsubstations,fromwhichthedistributionsystemssupply
residentialandcommercialconsumers.

3. INTRODUCTION CONT’D
Thevoltagelevelsoftransmissionlines;eitheroverheadorcablesare
selectedaccordingtothepowertobetransmittedtoacertainareaor
suppliedtoacustomer.Theadvantageofhigherlevelsoftransmissionline
voltageclearlyappearswhenconsiderationisgiventothetransmitting
capabilityofaline,whichincreaseswiththesquareofthevoltage.
i.e.P=VI=V
2
/R
Therefore,veryhighvoltagelevelsareusedtotransmitpoweroverlong
distances.Oncethepowerreachestheareawhereitwillbeused,itis
steppeddowntoalowervoltagesindistributionsubstations,andthen
deliveredtocustomersthroughdistributionlines.

4. There two types of transmission lines: overheadlines and buried cables.
Figure 1
Introduction Cont’d

5. Thetransmissionanddistributionofthree-phaseelectricalpoweron
overheadlinesrequirestheuseofatleastthree-phaseconductors.
Mostlowvoltagelinesusethree-phaseconductorsformingasinglethree-
phasecircuit.Manyhighervoltagelinesconsistofasinglethree-phase
circuitortwothree-phasecircuitsstrungorsuspendedfromthesame
towerstructureandusuallycalledadouble-circuitline.
Thetwocircuitsmaybestrunginavarietyofconfigurationssuchas
vertical,horizontalortriangularconfigurations.
Figure 2 illustrates typical single-circuit lines and double-circuit lines in
horizontal, triangular and vertical phase conductor arrangements.
A line may also consist of two circuits running physically in parallel but on
different towers.
In addition, a few lines have been built with three, four or even six three-
phase circuits strung on the same tower structure in various horizontal
and/or triangular formations.
Introduction Cont’d

6. Introduction Cont’d
Figure 2

7. Anoverheadtransmissionlineusuallyconsistsofthreeconductorsor
bundlesofconductorscontainingthethreephasesofthepowersystem.
Theconductorsareusuallyaluminumcablesteelreinforced(ACSR),
whicharesteelcore(forstrength)andaluminumwires(havinglow
resistance)wrappedaroundthecore.
OVERHEADLINE CONT’D

8. OVERHEADLINE CONT’D
In overhead transmission lines, the conductors are suspended from a pole
or a tower via insulators.

9. Inadditiontophaseconductors,atransmissionlineusuallyincludesone
ortwosteelwirescalledground(shield)wires.Thesewiresareelectrically
connectedtothetowerandtotheground,and,therefore,areatground
potential.
In large transmission lines,
these wires are located
above the phase conductors,
shielding them from lightning.
OVERHEADLINE CONT’D

10. CONT…
Bundledphaseconductorsareusuallyusedontransmissionlinesat220
kVandabove.Theseareconstructedwithmorethanoneconductorper
phaseseparatedatregularintervalsalongthespanlengthbetweentwo
towersbymetalspacers.
Conductorbundlesoftwo,three,four,sixandeightareinuseinvarious
countries.
Thepurposeofbundledconductorsistoreducethevoltagegradientsat
thesurfaceoftheconductorsbecausethebundleappearsasan
equivalentconductorofmuchlargerdiameterthanthatofthe
componentconductors.
Thisminimizesactivelossesduetocorona,reducesnoisegeneration,
e.g.radiointerference,reducestheinductivereactanceandincreases
thecapacitivesusceptanceorcapacitanceoftheline.
Thelattertwoeffectsimprovethesteadystatepowertransfercapability
oftheline.

11. Cablelinesaredesignedtobeplaced
undergroundorunderwater.The
conductorsareinsulatedfromoneanother
andsurroundedbyprotectivesheath.
Cablelinesareusuallymoreexpensiveand
hardertomaintain.Theyalsohave
capacitanceproblem–notsuitableforlong
distance.
UNDERGROUND CABLES CONT’D

12. 2.2 TRANSMISSIONLINEPARAMETERS
Anelectrictransmissionlinesarecharacterizedbyfourparameters,
namelyresistance,inductance,capacitanceandshuntconductance.The
electricaldesignandperformanceofalinearedependentonthese
parameters.Thesevaluesdeterminethepower-carryingcapacityofthe
transmissionlineandthevoltagedropacrossitatfullload.
Theseparametersareuniformlydistributedalongthewholeline.Eachline
hasitsownvalue,anditisnotpossibletoconcentrateorlumpthemat
discretepointsontheline.Forthisreasonthelineparametersareknown
asdistributedparameters.Theirvaluesaregiveninperunitlengthofline
andtheyaredenotedasR,L,CandGrespectively.Thelineparameters
arefunctionsoftheline-geometry,constructionmaterialandoperational
frequency.
Thelineresistanceandinductanceformtheseriesimpedanceoftheline.
Whilethecapacitanceandconductanceformtheshuntadmittanceofthe
line.

13. CONT…
Theseriesresistancereliesbasicallyonthephysical
compositionoftheconductoratagiventemperature.
Theseriesinductanceandshuntcapacitanceareproduced
bythepresenceofmagneticandelectricfieldsaroundthe
conductors,anddependontheirgeometricalarrangement.
Theshuntconductanceisduetoleakagecurrentsflowing
acrossinsulatorsandair.Asleakagecurrentisconsiderably
smallcomparedtonominalcurrent,itisusuallyneglected,
andtherefore,shuntconductanceisnormallynotconsidered
forthetransmissionlinemodeling

14. TRANSMISSIONLINEPARAMETERS CONT’D
Transmissionlinesareclassifiedintoshort,mediumandlonggroups
accordingtothelengthofthetransmissionlines.Theperformanceofone
ormoreoftheparametersofalineisgovernedbyitslengthand
conductorconfiguration.Foroverheadlinesupto80kmthecapacitance
CandshuntconductanceGarenegligiblysmallbutforcablelineswhere
thedistancebetweentheconductorsissmall,theeffectofcapacitance
cannotignored.Alllow-voltageoverheadlineshavinglengthsupto80km
aregenerallyclassifiedasShortlines.
Thelinesranginginlengthfrom80to240kmaretermedasMediumor
moderatelylonglines.Fortheselines,thecapacitanceofthelinecannot
beneglectedandit’sconsideredtobelumpedatoneormorepointsofthe
line.Theeffectofcapacitanceismoreathigherfrequency.Theleakage
conductanceorleakanceisneglected.
ThetermLonglinereferstoalinehavingitslengthmorethan240km.
Thelonglinetreatmenttakesallthefourparametersintoaccountand
allowsforthefactthattheyaredistributeduniformlyovertheentirelength
oftheline.

15. Short Transmission line Equivalent
Circuit (Series Impedance
parameters only)
Medium Transmission line Equivalent
Circuit (Series Impedance and Shunt
Capacitance
Long Transmission line
Equivalent Circuit ( All the
four TL Parameters)
TRANSMISSIONLINEPARAMETERS CONT’D

16. 2.2.1 RESISTANCE
The AC resistance of a conductor in a transmission line is based on the
calculation of its DC resistance.
If DC current is flowing along a round cylindrical conductor, the current is
uniformly distributed over its cross-section area and its DC resistance is
evaluated by
R
DC= l /A [Ω]
Wherelisthelengthofconductor;A–cross-sectionalarea,isthe
resistivityoftheconductor.Therefore,theDCresistancepermeterofthe
conductoris
r
DC=/A[Ω/m]
Theresistivityofaconductorisafundamentalpropertyofthematerialthat
theconductorismadefrom.Itvarieswithbothtypeandtemperatureof
thematerial.Atthesametemperature,theresistivityofaluminumishigher
thantheresistivityofcopper.

17. LINERESISTANCE CONT’D
IfACcurrentisflowing,ratherthanDCcurrent,thefollowingfactorsneedto
beconsidered:
1.Frequencyorskineffect2.Temperature
3.Spiralingofstrandedconductors4.Bundleconductorsarrangement
5.Proximityeffect
6.Alsotheresistanceofmagneticconductorvarieswithcurrentmagnitude.
A.SkinEffect
ThefrequencyoftheACvoltageproduceseffectontheconductorresistance
duetothenonuniformdistributionofthecurrent.Thisphenomenonisknown
asskineffectorfrequencyeffect.
Asfrequencyincreases,thecurrenttendstogotowardthesurfaceofthe
conductorandthecurrentdensitydecreasesatthecenter.
Skineffectreducestheeffectivecross-sectionareausedbythecurrent,and
thus,theeffectiveresistanceincreases.
Also,althoughinsmallamount,afurtherresistanceincreaseoccurswhen
othercurrent-carryingconductorsarepresentintheimmediatevicinity.

18. Askincorrectionfactork,obtainedbydifferentialequationsandBessel
functions,isconsideredtoreevaluatetheACresistance.
For60Hz,kisestimatedaround1.02
??????
??????�=�??????
��
B. Temperature Effect
The resistivity of any conductive material varies linearly over an operating
temperature, and therefore, the resistance of any conductor suffers the
same variations.
As temperature rises, the conductor resistance increases linearly, over
normal operating temperatures, according to the following equation:
Where R2 is the resistance at second temperature t2
R1 is the resistance at initial temperature t1
T is the temperature coefficient for the particular material (C°)

19. RESISTANCE CONT’D
Theresistivityincreaseslinearlywithtemperatureovernormalrangeof
temperatures.Iftheresistivityatonetemperatureisknown,theresistivity
atanothertemperaturecanbefoundfrom

T2=(T+t2)/(T+t1)
T1
Wheret
1and
t1aretemperature1in
o
Candtheresistivityatthat
temperature,t
2and
t2aretemperature2in
o
Candtheresistivityatthat
temperature,andTisthetemperatureconstant.
Resistivity(??????)andtemperaturecoefficient(T)constantsdependuponthe
particularconductormaterial
Material Resistivity at 20
o
C
[m]
Temperature constant
[
o
C]
Annealed copper1.7210
-8
234.5
Hard-drawn
copper
1.7710
-8
241.5
Aluminum 2.8310
-8
228.1
Iron 10.0010
-8
180.0

20. Wenoticethatsilverandcopperwouldbeamongthebestconductors.
However,aluminum,beingmuchcheaperandlighter,isusedtomake
mostofthetransmissionlineconductors.Conductorsmadeoutof
aluminumshouldhavebiggerdiameterthancopperconductorstooffset
thehigherresistivityofthematerialand,therefore,supportthenecessary
currents.
AC resistance of a conductor is always higher than its DC resistance due
to the skin effect forcing more current flow near the outer surface of the
conductor. The higher the frequency of current, the more noticeable skin
effect would be.
At frequencies of our interest (50-60 Hz), however, skin effect is not very
strong.
Wire manufacturers usually supply tables of resistance per unit length at
common frequencies (50 and 60 Hz). Therefore, the resistance can be
determined from such tables.
RESISTANCE CONT’D

21. C.SpiralingandBundleConductorEffect
•Overheadconductors,madeofnakedmetalandsuspendedoninsulators,
arepreferredoverundergroundconductorsbecauseofthelowercostand
easymaintenance.Also,overheadtransmissionlinesusealuminum
conductors,becauseofthelowercostandlighterweightcomparedto
copperconductors,althoughmorecross-sectionareaisneededtoconduct
thesameamountofcurrent.
•Therearedifferenttypesofcommerciallyavailablealuminumconductors:
aluminum-conductor-steel-reinforced(ACSR),aluminum-conductor-alloy-
reinforced(ACAR),all-aluminum-conductor(AAC),andall-aluminumalloy-
conductor(AAAC).
Figure 3: Stranded aluminum conductor with stranded steel core (ACSR).

22. ACSRisoneofthemostusedconductorsintransmissionlines.
•Itconsistsofalternatelayersofstrandedconductors,spiraledinopposite
directionstoholdthestrandstogether,surroundingacoreofsteel
strands.
Thepurposeofintroducingasteelcoreinsidethestrandedaluminum
conductorsistoobtainahighstrength-to-weightratio.
Astrandedconductoroffersmoreflexibilityandeasiertomanufacture
thanasolidlargeconductor.However,thetotalresistanceisincreased
becausetheoutsidestrandsarelargerthantheinsidestrandsonaccount
ofthespiraling.
Theresistanceofeachwoundconductoratanylayer,perunitlength,is
basedonitstotallengthasfollows:
Resistance Cont’d

23. where ??????
����: resistance of wound conductor (Ω)
 : : length of wound conductor (m)
??????
����=�
t��n /2�
��??????��relative pitch of wound conductor
�
����: length of one turn of the spiral (m)
2�
��??????��: diameter of the layer (m)
The parallel combination of n conductors, with same diameter
per layer, gives the resistance per layer as follows:
Similarly, the total resistance of the stranded conductor is
evaluated by the parallel combination of resistances per layer.

24. Inhigh-voltagetransmissionlines,theremaybemorethanone
conductorperphase(bundleconfiguration)toincreasethecurrent
capabilityandtoreducecoronaeffectdischarge.
Coronaeffectoccurswhenthesurfacepotentialgradientofaconductor
exceedsthedielectricstrengthofthesurroundingair(30kV/cmduringfair
weather),producingionizationintheareaclosetotheconductor,with
consequentcoronalosses,audiblenoise,andradiointerference.
Ascoronaeffectisafunctionofconductordiameter,lineconfiguration,
andconductorsurfacecondition,thenmeteorologicalconditionsplaya
keyroleinitsevaluation.
Coronalossesunderrainorsnow,forinstance,aremuchhigherthanin
dryweather.

25. Figure:Strandedconductorsarrangedinbundlesperphaseof(a)two,(b)
three,and(c)four.
•Corona,however,canbereducedbyincreasingthetotalconductor
surface.Althoughcoronalossesrelyonmeteorologicalconditions,their
evaluationtakesintoaccounttheconductancebetweenconductorsand
betweenconductorsandground.
•Byincreasingthenumberofconductorsperphase,thetotalcross-section
areaincreases,thecurrentcapacityincreases,andthetotalACresistance
decreasesproportionallytothenumberofconductorsperbundle.
•Conductorbundlesmaybeappliedtoanyvoltagebutarealwaysusedat
345kVandabovetolimitcorona.
•Tomaintainthedistancebetweenbundleconductorsalongtheline,
spacersmadeofsteeloraluminumbarsareused.

26. 26
D. PROXIMITYEFFECTS
Inatransmissionlinethereisanon-uniformityofcurrentdistributioncaused
byahighercurrentdensityintheelementsofadjacentconductorsnearesteach
otherthanintheelementsfartherapart.Thephenomenonisknownas
proximityeffect.
Itispresentforthree-phaseaswellassingle-phasecircuits.
Fortheusualspacingofoverheadlinesat60Hz,theproximityeffectis
neglected.

27. 27
EXAMPLE1
Athreephasetransmissionlineisdesignedtodeliver190.5MVAat220kVover
adistanceof63km.thetotaltransmissionlossisnottoexceed2.5percentofthe
ratedlineMVA.Iftheresistivityoftheconductormaterialat20°cis2.8×10
−8
Ωm
,determinetherequiredconductordiameterandtheconductorsize.
Solution

28. 28
Theinductivereactanceisbyfarthemostdominatingimpedanceelement.
Acurrent-carryingconductorproducesamagneticfieldaroundtheconductor.
Themagneticfluxlinesareconcentricclosedcircleswithdirectiongivenbythe
righthandrule.Withthethumbpointinginthedirectionofthecurrent,the
fingersoftherighthandencircledthewirepointinthedirectionofthemagnetic
field.
Whenthecurrentchanges,thefluxchangesandavoltageisinducedinthe
circuit.
Bydefinition,fornonmagneticmaterial,theinductanceListheratioofitstotal
magneticfluxlinkagetothecurrentI,givenby
2.2.2 Inductance and Inductive Reactance

29. Theseriesinductanceofatransmissionlineconsistsoftwocomponents:
internalandexternalinductances,whichareduethemagneticfluxinside
andoutsidetheconductorrespectively.Theinductanceofatransmission
lineisdefinedasthenumberoffluxlinkages[Wb-turns]producedper
ampereofcurrentflowingthroughtheline:
L=λ/I
1. Internal inductance:
Consider a conductor of radius rcarrying a current
I. At a distance xfrom the center of this conductor,
the magnetic field intensity H
xcan be found from
Ampere’s law:
CONT’D

30. INDUCTANCEANDINDUCTIVEREACTANCE CONT’D
Where H
xis the magnetic field intensity at each point along a closed path,
dlis a unit vector along that path and I
xis the net current enclosed in the
path. For the homogeneous materials and a circular path of radius x, the
magnitude of H
xis constant, and dlis always parallel to H
x. Therefore:
Assuming next that the current is distributed uniformly in the conductor:
Thus, the magnetic intensity at radius xinside the conductor is
I

31. The flux density at a distance xfrom the center of the conductor is
The differential magnetic flux contained in a circular tube of thickness dx
and at a distance xfrom the center of the conductor is
The flux linkages per meter of length due to flux in the tube is the product
of the differential flux and the fraction of current linked:
INDUCTANCEANDINDUCTIVEREACTANCE CONT’D

32. The total internal flux linkages per meter can be found via integration…
Therefore, the internal inductance per meter is: L = λ/I
If the relative permeability of the conductor is 1 (non-ferromagnetic
materials, such as copper and aluminum), the permeability reduces to
INDUCTANCEANDINDUCTIVEREACTANCE CONT’D
Where μ
0= 4л*10
-7

33. 2. EXTERNALINDUCTANCEBETWEEN2 POINTSOUTSIDEOFTHELINE
Tofindtheinductanceexternaltoaconductor,we
needtocalculatethefluxlinkagesofthe
conductordueonlytheportionoffluxbetweentwo
pointsP
1andP
2thatlieatdistancesD
1andD
2
fromthecenteroftheconductor.
Intheexternaltotheconductorregion,the
magneticintensityatadistancexfromthecenter
ofconductoris
since all the current is within the tube, I
x= I.
The flux density at a distance xfrom the center of conductor is

34. The differential magnetic flux contained in a circular tube of thickness dx
and at a distance xfrom the center of the conductor is
The flux links the full current carried by the conductor, therefore:
The total external flux linkages per meter can be found via integration…
The external inductance per meter is
The inductance between two points external to a conductor For non-ferromagnetic
materials is then

35. 3. INDUCTANCE OFASINGLE-PHASE2-WIRETRANSMISSION LINE
We determine next the series inductance of a
single-phase line consisting of two conductors
of radii rspaced by a distance Dand both
carrying currents of magnitude Iflowing into
the page in the left-hand conductor and out of
the page in the right-hand conductor.
Considering two circular integration paths, we
notice that the line integral along x
1produces
a net magnetic intensity since a non-zero net
current is enclosed by x
1. Thus:
Since the path of radius x
2encloses both conductors and the currents are
equal and opposite, the net current enclosed is 0 and, therefore, there are
no contributions to the total inductance from the magnetic fields at
distances greater than D.

36. The total inductance of a wire per unit length in this transmission line is a
sum of the internal inductance and the external inductance between the
conductor surface (r) and the separation distance (D):
By symmetry, the total inductance of the other wire is the same, therefore,
the total inductance of a two-wire transmission line is
Where ris the radius of each conductor and Dis the distance between
conductors.
INDUCTANCE OFASINGLE-PHASE2-WIRETRANSMISSIONLINE CONT’D
The flux beyond D links a net current of zero and does not contribute to
the net magnetic flux linkage in the circuit. Thus, to obtain the inductance
of conductor 1 due to the net external flux linkage, it is necessary to
evaluate external inductance from �1=�1 to �2=�.

37. 37

38. 38

39. 39

40. 40
FLUXLINKAGEINTERMSOFSELFANDMUTUALINDUCTANCE
The series inductance per phase for the above single phase two wire line can
be expressed in terms of self inductance of each conductor and their mutual
inductance. Consider one meter length of the single phase circuit represented
by two coils characterized by the self inductances ??????
11and ??????
22and the mutual
inductance ??????
12. The magnetic polarity is indicated by dot symbols as shown
in the following Figure .
The single phase line viewed as two magnetically coupled coils

41. 41
Comparing the above questions with the external inductance of a conductor
(L
ext= 2x10
-7
ln(D2/D1) H/m), we conclude the following equivalent expressions
for the self and mutual inductances:

42. 42
4. INDUCTANCE OFTHREEPHASETRANSMISSION LINES-SYMMETRICAL SPACING
Consider one meter length of a three phase line with three conductors,
each with radius r, symmetrically spaced in a triangular configuration as
shown in the following Figure.

43. 43

44. 44
5. INDUCTANCE OFTHREEPHASETRANSMISSION LINES-ASYMMETRICAL SPACING
Practicaltransmissionlinescannotmaintainsymmetricalspacingofconductors
becauseofconstructionconsiderations.Withasymmetricalspacing,evenwith
balancedcurrents,thevoltagedropduetothelineinductancewillbe
unbalanced.Consideronemeterlengthofathreephaselinewiththree
conductors,eachwithradiusr.theconductorsareasymmetricallyspacedwith
distancesshowninthefollowingFigure.
Three phase line with asymmetrical spacing.

45. 45
Or in matrix form

46. 46
6. TRANSPOSELINE
Theequilateraltriangularspacingconfigurationisnottheonlyconfiguration
commonlyusedinpractice.Thustheneedexistsforequalizingthemutual
inductances.Onemeansfordoingthisistoconstructtranspositionsor
rotationsofoverheadlinewires.Atranspositionisaphysicalrotationofthe
conductors,arrangedsothateachconductorismovedtooccupythenext
physicalpositioninaregularsequencesuchasa-b-c,b-c-a,c-a-b,etc.Sucha
transpositionarrangementisshowninthefollowingFigure.Ifasectionoflineis
dividedintothreesegmentsofequallengthseparatedbyrotations,wesaythat
thelineis“completelytransposed.”
A transposed three phase line

47. 47

48. 48
INDUCTANCE OF COMPOSITE CONDUCTORS
Intheevaluationofinductance,solidroundconductorswereconsidered.
However,inpracticaltransmissionlines,strandedconductorsareused.Also,
forreasonsofeconomy,mostEHVlinesareconstructedwithbundled
conductors.Inthissectionanexpressionisfoundfortheinductanceof
compositeconductors.TheresultcanbeusedforevaluatingtheGMRof
strandedorbundledconductors.Itisalsousefulinfindingtheequivalent
GMRandGMDofparallelcircuits.

49. 49

50. 50

51. 51

52. 52

53. 53

54. 54
7. INDUCTANCE OF THREE PHASE DOUBLE CIRCUIT LINE
Athreedoublecircuitlineconsistsoftwoidenticalthreephasecircuits.The
circuitsareopenedwith�
1−�
2,�
1−�
2����
1−�
2inparallel.Becauseof
geometricaldifferencesbetweenconductors,voltagedropduetoline
inductancewillbeunbalanced.Toachievebalance,eachphaseconductormust
betransposedwithinitsgroupandwithrespecttoparallelthreephaseline.
Considerathreephasedoublecircuitlinewithrelativephasepositions�
1�
1�
1−�
2
�
2�
2,asshowninthefollowingFigure

55. 55

56. 56

57. INDUCTANCE OFATRANSMISSIONLINE CONT’D
A two-conductor
bundle
A four-conductor
bundle

58. CONCLUSIONSONINDUCTANCE OFATRANSMISSIONLINE
Analysisoftheaboveequationsshowthat:
1.Thegreaterthespacingbetweenthephasesofatransmissionline,the
greatertheinductanceoftheline.Sincethephasesofahigh-voltage
overheadtransmissionlinemustbespacedfurtheraparttoensure
properinsulation,ahigh-voltagelinewillhaveahigherinductancethan
alow-voltageline.Sincethespacingbetweenlinesinburiedcablesis
verysmall,seriesinductanceofcablesismuchsmallerthanthe
inductanceofoverheadlines.
1.Thegreatertheradiusoftheconductorsinatransmissionline,the
lowertheinductanceoftheline.Inpracticaltransmissionlines,instead
ofusingheavyandinflexibleconductorsoflargeradii,twoandmore
conductorsarebundledtogethertoapproximatealargediameter
conductor.Themoreconductorsincludedinthebundle,thebetterthe
approximationbecomes.Bundlesareoftenusedinthehigh-voltage
transmissionlines.

59. INDUCTIVEREACTANCEOFALINE
The series inductive reactance of a transmission line depends on both the
inductanceof the line and the frequencyof the power system. Denoting
the inductance per unit length as l, the inductive reactance per unit length
will be
where fis the power system frequency. Therefore, the total series
inductive reactance of a transmission line can be found as
where dis the length of the line.
x
I = jωɭ = j2лfɭ
X
I = x
Id

60. 2.2.3 CAPACITANCEANDCAPACITIVEREACTANCE
SinceavoltageVisappliedtoapairofconductorsseparatedbya
dielectric(air),chargesofequalmagnitudebutoppositesignwill
accumulateontheconductors.
Transmissionlineconductorsexhibitcapacitancewithrespecttoeach
otherduetothepotentialdifferencebetweenthem.Theamountof
capacitancebetweenconductorsisafunctionofconductorsize,spacing,
andheightaboveground.Bydefinition,thecapacitanceCistheratioof
chargeqtothevoltagev,givenby
InACpowersystems,atransmissionlinecarriesatime-varyingvoltage
differentineachphase.Thistime-varyingvoltagecausesthechangesin
chargesstoredonconductors.Changingchargesproduceachanging
current,whichwillincreasethecurrentthroughthetransmissionlineand
affectthepowerfactorandvoltagedropoftheline.Thischangingcurrent
willflowinatransmissionlineevenifitisopencircuited.

61. Consider a long round conductor with radius r, carrying a charge of q coulombs
per meter length as shown in the following Figure.
Electric field around a long round conductor.
Thechargeontheconductorgivesrisetoanelectricfieldwithradialfluxlines.The
totalelectricfluxisnumericallyequaltothevalueofchargeontheconductor.The
intensityofthefieldatanypointdefinedastheforceperunitchargeandistermed
electricfieldintensitydesignatedasE.Concentriccylinderssurroundingthe
conductorareequipententialsurfacesandhavethesameelectricfluxdensity.

62. The electric flux density D may be found from the relation:
From Gauss’s law for one meter length of the conductor, the electric flux
density at a cylinder of a radius x is given by
where Aspecifies a closed surface; dAis the unit vector normal to the
surface; qis the charge inside the surface; Dis the electric flux density at the
surface:
where Eis the electric field intensity at that point; is the permittivity of the material:
Where 
ris Relative permittivity of the material
The permittivity of free space 
0= 8.8510
-12
F/m
Ԑ =Ԑ
r
Ԑ
o

63. Electricfluxlinesradiateuniformlyoutwards
fromthesurfaceoftheconductorwitha
positivechargeonitssurface.Inthiscase,the
fluxdensityvectorDisalwaysparalleltothe
normalvectordAandisconstantatallpoints
aroundapathofconstantradiusr.Therefore:
were lis the length of conductor; qis the
charge density; Qis the total charge on the
conductor.
Then the flux density is
The electric field intensity is
Capacitance and capacitive reactance Cont’d

64. The potential difference between two points P
1and P
2can be found as
where dlis a differential element tangential to the integration path
between P
1and P
2. The path is irrelevant.
Selection of path can simplify calculations.
For P
1-P
int, vectors Eand dlare parallel;
therefore, Edl = Edx. For P
int–P
2vectors
are orthogonal, therefore Edl = 0.\
CAPACITANCEANDCAPACITIVEREACTANCE CONT’D

65. 2.3.1 CAPACITANCEOFASINGLEPHASETWO-WIRETRANSMISSIONLINE
The potential difference due to the
charge on conductor acan be
found as
Similarly, the potential difference due to the charge on conductor bis
or

66. The total voltage between the lines is
Since q
1= q
2= q, the equation reduces to
The capacitance per unit length between the two conductors of the line is
CAPACITANCEOFASINGLEPHASETWO-WIRETRANSMISSIONLINECONT’D

67. Thus:
Whichisthecapacitanceperunitlengthofasingle-phasetwo-wire
transmissionline.
The above Equation gives the line to line capacitance between
conductors. For the purpose of transmission line modeling, it is convenient
to define a capacitance C between each conductor and a neural as
illustrated in below.
Thepotentialdifferencebetweeneachconductorandtheground(or
neutral)isonehalfofthepotentialdifferencebetweenthetwoconductors.
Therefore,thecapacitancetogroundofthissingle-phasetransmission
linewillbe
CAPACITANCEOFASINGLEPHASETWO-WIRETRANSMISSIONLINECONT’D

68. Recalling ??????
0=8.85×10
−12
??????/�and converting to ????????????per kilometer, we have
Analysisoftheabovefinalequationshowsthat:
1.Thegreaterthespacingbetweenthephasesofatransmissionline,the
lowerthecapacitanceoftheline.Sincethephasesofahigh-voltage
overheadtransmissionlinemustbespacedfurtheraparttoensure
properinsulation,ahigh-voltagelinewillhavealowercapacitancethan
alow-voltageline.Sincethespacingbetweenlinesinburiedcablesis
verysmall,shuntcapacitanceofcablesismuchlargerthanthe
capacitanceofoverheadlines.Cablelinesarenormallyusedforshort
transmissionlines(tomincapacitance)inurbanareas.
2.Thegreatertheradiusoftheconductorsinatransmissionline,the
higherthecapacitanceoftheline.Therefore,bundlingincreasesthe
capacitance.Goodtransmissionlineisacompromiseamongthe
requirementsforlowseriesinductance,lowshuntcapacitance,anda
largeenoughseparationtoprovideinsulationbetweenthephases.

69. SHUNTCAPACITIVEADMITTANCE
The shunt capacitive admittance of a transmission line depends on both
the capacitanceof the line and the frequencyof the power system.
Denoting the capacitance per unit length as c, the shunt admittance per
unit length will be
The total shunt capacitive admittance therefore is
where dis the length of the line. The corresponding capacitive reactance
is the reciprocal to the admittance:11
2
C
C
Zj
Y fcd
  

70. POTENTIAL DIFFERENCE IN A MULTICONDUCTOR CONFIGURATION
Consider n parallel long conductors with charges �1,�2,…,��coulombs/meter as
shown in
Multi-conductor configuration.
Assume that the distortion effect is negligible and charge is uniformly distributed
around the conductor, with the following constraint

71. Using superposition, the potential difference between conductor I and j due to
the presence of all charges is given by:
When k=I, �????????????is the distance between the surface of the conductor and its
center, namely its radius r.
CAPACITANCE OF THREE PHASE LINES
Consider one meter length of a three phase line with three long conductors, each
with radius r, with conductor spacing as shown in the following figure:

72. For a balanced three phase system
Weshallneglecttheeffectofgroundandtheshieldwires.Assumethatthelines
istransposed.Weproceedwiththecalculationofthepotentialdifference
betweenaandbforeachsectionoftransposition.Applying(80)tothefirst
sectionofthetransposition,??????_��is

75. EFFECT OF EARTH ON THE CAPACITANCE
Foranisolatedchargedconductortheelectricfluxlinesareradialandare
orthogonaltothecylindricalequipotentialsurfaces.Thepresenceofearthwill
alterthedistributionofelectricfluxlinesandequipotentialsurfaces,whichwill
changetheeffectivecapacitanceoftheline.
Theearthlevelisanequipotentialsurface,thereforethefluxlinesareforcedto
cutthesurfaceoftheearthorthogonally.Theeffectofthepresenceofearthcan
beaccountedforbythemethodofimagechargesintroducedbyKelvin.

76. Toillustratethismethod,consideraconductorwithachargeqcoulombs/meterata
highHaboveground.Also,imagineacharge–qatadepth–Hbelowthesurfaceof
earth.Thisconfigurationwithoutthepresenceoftheearthsurfacewillproducethe
samefielddistributionasasinglechargeandtheearthsurface.Thus,theearthcan
bereplacedforthecalculationofelectricfieldpotentialbyafictitiouscharged
conductorwithchargeequalandoppositetothechargeontheactualconductorand
atadepthbelowthesurfaceoftheearththesameastheheightoftheactual
conductoraboveearth.This,imaginaryconductoriscalledtheimageoftheactual
conductor.

77. Theeffectoftheearthistoincreasethecapacitance.Butnormallytheheightof
theconductorislargeascomparedtothedistancebetweentheconductors,and
theeartheffectisnegligible.Therefore,foralllinemodelsusedforbalanced
steadystateanalysis,theeffectofearthonthecapacitancecanbeneglected.
However,forunbalancedanalysissuchasunbalancedfaults,theearth’seffectas
wellastheshieldwiresshouldbeconsidered.
Example 1:
If a single phase line has parameters D=5ft, r=0.023 ft, and a flat horizontal spacing
H=18ft average line height, determine the effect of the earth on capacitance.
Assume a perfectly conducting earth plane.

78. Solution:
The earth plane is replaced by a separate image for each overhead conductor, and
the conductors are charged as shown in the above figure. The voltages between
conductors x and y is:

79. Example2:
A500kVthreephasetransposedlineiscomposedofoneACSR1,272,000cmil,
45/7Bitternconductorperphasewithhorizontalconductorconfigurationasshown
inthefollowingFigure.Theconductorshaveadiameterof1.345inandaGMRof
0.5328in.findtheinductanceandcapacitanceperphaseperkilometeroftheline.
Figure: conductor layout for the three phase transmission line

81. EXAMPLE3:
An8000V,60Hz,single-phase,transmissionlineconsistsoftwohard-
drawnaluminumconductorswitharadiusof2cmspaced1.2mapart.If
thetransmissionlineis30kmlongandthetemperatureoftheconductors
is20
0
C,
a.Whatistheseriesresistanceperkilometerofthisline?
b.Whatistheseriesinductanceperkilometerofthisline?
c.Whatistheshuntcapacitanceperkilometerofthisline?
d.Whatisthetotalseriesreactanceofthisline?
e.Whatisthetotalshuntadmittanceofthisline?
Solution:
a. The series resistance of the transmission line isl
R
A


Ignoring the skin effect, the resistivity of the line at 20
0
will be 2.8310
-8
-
m and the resistance per kilometer of the line is8
2
2.83 10 1000
0.0225
0.02
l
r km
A




   


82. SOLUTION CONT’D
b. The series inductance per kilometer of the transmission line is31 1 1.2
ln 1000 ln 1000 1.738 10
4 4 0.02
D
l H km
r


   
        
   
   
c. The shunt capacitance per kilometer of the transmission line is12
98.854 10
1000 1000 6.794 10
1.2
ln ln
0.02
ab
c F km
D
r
 


      
d. The series impedance per kilometer of the transmission line is3
2 0.0225 2 60 1.738 10 0.0225 0.655
se
z r jx r j fl j j km

           
Then the total series impedance of the line is 0.0225 0.655 30 0.675 19.7
se
Z j j     

83. e. The shunt admittance per kilometer of the transmission line is96
2 2 60 6.794 10 2.561 10
C
y j fc j j S m

       
The total shunt admittance will be 
65
2.561 10 30 7.684 10
se
Y j j S

     
The corresponding shunt capacitive reactance is5
11
13.0
7.684 10
sh
sh
Z j k
Yj

     

SOLUTION CONT’D