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8/21/2019 Calabrese Symmetrical Components
1/310
etworks
RATORY
FELECTRICALENGINEERING
PANY•NEWYORK
P u b l
i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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aratus,transmissionlines,andutilizingdevices
esystemsaredesignedandarrangedtoproduce
eversingle-phasedevicesmustbeused,theyare
a minimumunbalance.Engineersarecarefulto
theymustgenerate,transmit,anddistribute
mumlossesandinconvenience.How,then,are
eofelectricalapparatusaffectedbyunbalanced
etricalcomponents,establishedin1918,pro-
eansof answeringthismostimportantquestion
erbetterthananyotherknownmethod.Calcu-
nsofsymmetricalcomponentshas,infact,
hodforcomputingcurrentsandvoltagesinpoly-
tconditions.
dyof unbalancedthree-phasesystemsby
cedthree-phasesystemsandtheirsingle-phase
mentalfactofthepresenceofathree-phase
phenomenaarisingfromitneedbe neglected.
whetherpowersystemstudiesaremadeinlong-
culatingboardoranetworkanalyzer.Hence,
calcomponentsisan essentialpartofthetech-
ersengagedinsuchstudies.
eisto familiarizethestudentofelectrical
cingengineerwiththetheoryofsymmetrical
andcarefullychosenillustrationsandproblems,
tediscussionofthe fieldsinwhichthetheoryis
arilyfor useatthegraduatelevel,but the
udedinthefirstfivechaptersmakesthebook
orcourseswherethecurriculumpermitssucha
hemanyreferenceswillencouragethestudent
er—avaluablepartof histraining.Theinclu-
mentalsoservesthe samepurpose,formany
studyofthe sourcesgiven.
has appliedsymmetricalcomponentsto
stability,andrelayingproblemsofelectricpower
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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hetaughtsymmetricalcomponentstograduate
he outgrowthofthenoteshedevelopedfor
t stressestheuseofmathematicsandthe
terpretationasaprecisemeansofachievingreal
opmentofthematerialiscarriedthroughby
tors,themathematicsofwhichwasdevelopedby
calpaper("MethodofSymmetricalCoordinates
olyphaseNetworks,"Transactions,A.I.E.E..
27-1140)andmoreextensivelyinthe paper
uthor("L'ApplicazionedelleCoordinate
onedelleEquazioniAlgebriche,"AttidelCongresso
atici,Sept.,1928).Inanagewhichseeksnew
aticaltoolsofanalysis,sequenceoperatorsoffer
nelegantshorthandmethodofwritingotherwise
alequipment,publishers,andtheengineering
cooperativeinpermittingthereproductionof
edinthesechapters.Specificacknowledgments
points wherethematerialshavebeenused.
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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ricalNetworks...18
antities.....69
90
itStudies....103
yofSymmetricalComponents
.. .114
calComponentstoThree-Phase
SymmetricalSystemswithOne
04
aseTransmissionLinestoCur-
,andZeroSequence.Zero
nbalanceFactorsofUntrans-
......282
issionLinesinPositive,Negative,
orks.....371
CurrentsofZero,Positive,and
omponents.....433
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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eshall dealextensivelywithsinusoidal
esareusuallyrepresentedbyvectorsina plane.
essof whetheritisin aplaneornot andregard-
tsa force,avelocity,orasinusoidalquantity.
arsithasbecomecustomarytoreservethe term
escharacterizedbymagnitudeanddirection,
sor"and"phasorquantity"foraplanevectoror
planevectorrepresentingasinusoidalquantity
ewillbefollowedinthisbook;thuswe shall
r,admittancephasor,orcurrentphasor,etc.,
operatorwillbe usedextensivelyinthe
oreimportanttoreviewthe lawsitobeysin
mprehensiveunderstandingof
1,xand yaretwoarbitrary
ce.Asacomplexnumberthe
reisgivenbyf
x,weseethatthe effectof
n 6isto rotateitinthe FIG1-2-1.
gle6.The termcos6+j sin6
,whenappliedtoa phasor,rotatesthelatter
vedirection.
expansionsofthecosineandsine:
indicatesaphasor.Thesameletternotin boldfaceindi-
hasor.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ENTS
addtothe resultEq.1-2-1:
j-+-+j'-+•••(1-2-3)
sion
-5)
exponentialei9 isidenticalwiththeoperator
eis itselfanoperatorwhich,whenappliedtoa
gle0inthe positivedirection.
n easilyobtain
-6)
torwhich,whenappliedtoaphasor,rotatesit
vedirection.
s,wesometimesencounteroperatorswhichare
oth,ofoperatorsof theformei9.Canwe,in such
aryexponentialandapply toittheordinarylaws
oanswerthis questionwehavetoprovethat
nentials.Althoughthisproofcan befoundin
ns,itisgivenherefor theconvenienceofthe
uct.Bythelawofexponentials,theproduct
+jsin02)
in02+j(c os0isin02+sin0ic os02)
+02)
asee>0obeysthelawof exponentials.Similarly,
0, -02)
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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,with8positiveor negative,obeysthelawsof
asanordinaryexponential.
8+jc os8=j(c os8+jsin8)=je'
8)d8=sin8— jc os8
also obeysthelawsofderivationandintegration
onofScalarQuantities.FromEqs.1-2-5
orofunitlength makinganangle8in the
Oxaxis.Onthe otherhand,e~jeisaunit phasor
egativedirectionwithOx.Thesetwophasors,
econjugatephasors.e~,eistheconjugateofe,e,
conjugateofe~je.Fromy
e thatasinusoidalquantity
me,canbewrittenin the
and(V/2)e~,e,arealsoconjugatephasors
magnitudeandsymmetricalwithrespecttothe
ppersideofan italicboldfaceletterindicatestheconjugate
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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NTS
OA'=V*/2.Atanyparticularinstant
usequaltothesum ofthetwoconjugatephasors
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sedin Fig.1-3-3,whereOB=V.At thetime
asmovedfrompositionOBto positionOB',
ttoOB.Theinstantaneousvaluev atanytimet
ephasor Vina directionatanegativeangle
V.Thedirectionthus obtainedistheonefrom
d;itmaybe calledtheaxisofinstantaneousvalues.
positionofthephasorV attheinstantt isgiven,
uevis oftheform
aluesisobtainedby drawingalineatan angle
rectionwithrespecttoV.
edtoacircuit ofresistanceR,inductanceL,
renti inthecircuitwill beobtainedbysolving
ionbecomes
bolically,
¥,!)i
+R+LP+rl/ c\
understeadystateconditions,
R - jaL-1/ juC ]
+ j/ uC
wnfactthat,ina linearcircuittowhich asinus-
ecurrent isalsosinusoidal.Let
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ENTS
alledimpedancephasoror,simply,theimpedance
eration.Thecurrentimay berepresentedby
K/27orV*/2Z*,orby eitherV/ZotV*/Z* inthe
dforthevoltage.Whateverphasorisused to
hasorrepresentingthevoltagev bytheangle
8/21/2019 Calabrese Symmetrical Components
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asorsIej9andZe}*1is
sIV* andZej*1is
ej1is
ctof twophasorsisequalto theproductof
hasors.Let thetwophasorsbe
Iare,respectively,
}*'
s
conjugateoftheproductEEl.
rs.Three-phaseoperatorsareusedexten-
eanalysisof three-phasenetworks.Usually
alsymbols:
sentedbythreephasorsofunit lengthdisplaced
g.1-5-1.I tfollowsthat
aseoperatorsrepeatperiodically.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ENTS
tantaneousPower.Theinstantaneous
=Vcos (wt),andacurrent,i =Icos (wt— ip),
angle ipisequalto theworkdoneincarryingthe
tialdifferencev andisgivenby
ip)
ip)
s2
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um oftwoconjugatephasorsOBandOBi ;that
s,notingthat
— sin< p
p
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ONENTS
alues,
-"(1-6-1)
* HgXZ"'-*)
*eff+VMIctt)(1-6-2)
ponentofPissignificanttothe effectofthe
erealcomponentVeffJeffcos
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e,RIttt=Vctt,willbe thesameasforgenerator
generatorsmustbecapableofgeneratingthe
voltageVandofcarryingthesameeffectivecur-
ratingofthe twogeneratorswillbethesame,
ntheaverageperformsno work.Forgenerator
os
ett,andtheinstantaneouspowerfluctuateswith
isvalue.Thisisillustratedin Fig.1-6-^1,which
currentt,andthepowervi. Figure1-6-5gives
nstant/.
ththeresistanceareactanceX.Letit be
to forcethesamecurrent/eftthroughtheheater
nthe sameamountofheatenergy.Thevoltage
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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NENTS
andthe instantaneouspowersuppliedbythe
+ Li—
ntsthepowerabsorbedby theresistance.Itis
causeitdependson thesquareofthecurrent.
representsthe powerabsorbedorreleasedby
yinstantthe signofthispowerdependson the
e instantthevoltagevis maximum,from
t
/cos (ut—
8/21/2019 Calabrese Symmetrical Components
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pableofgeneratingamaximumvoltage
andofcarryingtheeffectivecurrentIeft.
hegeneratorwillbe Ieft2VR2+(uL— 1/«C)2.
ouspoweroutputis
nentsare
-~-(1-c os2uf )
osut= — -^-sin
c osut=— ̂ —
— Isin2ut
ndmemberistheinstantaneouspowerdissipated
Thesecondtermisthepowerflowingbackand
orandthemagneticandelectricfields.Both
quency.
stantaneouspowermaybe representedby
powerisrepresentedby2Pi,averagepoweris
Feff/effcos
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ONENTS
*ef(/eff.Accordingtothisconvention,anin-
eimaginarypowercomponentandacapacitance
powercomponent;thatis,alaggingcurrent
werandleadingcurrentgivesleadingreactive
hthisconvention,thetwocomponentsofthe
taineddirectlyfromtheeffectivecurrentphasor
thevoltageasreference,merelybymultiplying
P\torepresentpowerhas theinconvenienceof
epowerduetoa laggingcurrent.Theconvention
nconvenienceandthereforeseemsamorecon-
EECommitteeondefinitionsfirstproposedthe
ontorepresentpower.Lateron,the Committee
sedthefirstconvention,whichhasnowbeen
edforgeneralusebytheprofession.Accepting
shallrepresentpowerbythephasorP= 2/V
cussiononpower,afew morewordsmustbe
gure1-6-6showsa generatorGsupplyinga
eratesaneffectiveemfE,whichis assumedposi-
o A,thatis,froma
ntof higherpotential.
gthe dropsintheconnec-
oltageappliedto the
thevoltageacrossa
sitivefroma pointofhigher
al.Thecurrent/
edpositivewhenflowingawayfromthegenerator
eseconventionstheinstantaneouspowerofthe
ealcomponentofEI* +EI.Then,whenpower
ession,apositivesign willindicatepowerab-
sign powergeneratedby,theloadL.
onentialoperatorsfindthe expressionof:
ressionsfor:(a)e'60;(b) e'120;(c)e'"°.
e voltageofa3-phasesystemis115 volts.
tralvoltagesinexponentialform.
powerrepresentedphasorially?
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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representedphasorially?
onentsofthe productV,ul*turepresent?
werwhichflowsalternatelytoandfromthe source.
reavailableforrepresentingphasoriallytheaverage
onofa sinusoidalvoltagewithasinusoidalcurrentof
atdotheydiffer? Whichisthemethodadoptedby
eloadhaving85%laggingpfis suppliedfroma
e.
enting(1) theinstantaneouspower;(2)theaverage
antaneouspower.
onventionofphasorpowerrepresentationyou
medin(a),whatwillbe thesignofthe reactive
omposedofresistanceanda capacitorinseries?
owerin a3-phaseungroundedsystemisequalto
espondingtotheinteractionoftwolinevoltages,re-
ent.
es,/„,/&, Icthethreecurrents,E„,Eb,Ec thethree
dif Iaandh arethecurrentsused,whichvoltages
orrectpower?
asecircuit.Letvandi betheinstantaneousvoltage
ughit.Both voltageandcurrentareperfectlysinus-
graphicrecordsofvoltagev,currenti,andinstan-
able.(Itis oftenverydifficulttodeterminethephase
eenvoltageandcurrentdirectlyfromtherecords.)
risgiven by
hevicurveabovethezero line;6=peakvalueof the
.
evolt-amperesandtheaveragepoweraregiven,
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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CTRICALNETWORKSf
worksitoftenhappensthat onlytheterminal
nsuch casesitwouldbehelpful toreplacethe
plerone,equivalenttothe formeronlyinrespect
Iti sthepurposeof
erulesunderwhich
ade.
alenceofElectri-
1representanet-
3,•••, nsupplied
,E3, ••-,Earespectively.I|_ Someoftheseemf'smaybezero.Let Ii,/2,
ntssuppliedtothenetworkthrough
tsbeing takenpositive
mmonneutralintothenetwork.Then
)
cipleofsuperpositionappliestothe network
s,itis assumedthattheresistance,inductance,
erentcircuits areindependentofthevalueofthe
em.Withthisassumption,thecurrent dis-
ybe foundbyconsideringeachemfinde-
nycircuit duetoanyoneof theappliedemf•s
The currentinthesamecircuit duetoall
tothesumofn currentsdueton appliedemf's.
onalityvarieswiththepointof applicationofthe
ents inthen terminalsaregivenby
eEmf
En
nnEn
nthe EquivalenceofElectricalNetworks,"GeneralElectric
3-25.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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CTRICALNETWORKS19
tsofproportionality.Thefollowingrelations
+^ In
+ Y2n
Y3n(2-1-2)
rrentsfi,I2,•••,/nare
— •••— YniEn
'32»*3~ — In2£n
—o)
~ ~*n3*n
3nE3—•••-(-YnnEn
rsofEqs.2-1-2arethe so-calleddrivingadmit-
ely,totheterminals1,2,•••,n, andtheY's
ethe so-calledtransferadmittances.ThusYnn
fthe nthterminal;itisequal tothecurrent
minalwhenEn =1andEI =E2=• ••=
econditionsthecurrentflowinginthe first
inusthe transferadmittancebetweenthenth
ersesofthedrivingadmittances,arecalledthe
minals1,2,• ••,respectively.l/Y,kforaj* k,
admittancebetweenthesth andthekthter-
rimpedancebetweenthoseterminals.
mittancescompletelydefinetheterminal
Ifbothdriving andtransferadmittancesare
onsofthenetworkareknownautomatically.
s.2-1-2,notall drivingandtransferadmittances
definetheterminalconditionsofthe network.
s. 2-1-3arenotallindependent,buteach of
ftheremainingn— 1withthesign changed.
damentalequationsfortheanalysisof the
pearinginthem,otherthan byEqs.2-1-2,are
= Yqpforq^ p.
given,eachhavingnterminals,onesetof
1-3maybe writtenforeachnetwork.In
gI"softhesesets ofequationsaredifferent.In
eymaybeidentical.Thenthenetworksareequiva-
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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ONENTS
lsareconcerned;thatis, itwillbeimpossibleto
rterminalsbecause,whenviewedfromthe
veinexactlythesamemanner.Itfollowsthat
equivalent,asfar astheterminalconditions
rrespondingY'sof thenetworksareequal.
nditionwhichmustbesatisfiedin constructing
ethispointisclearlyunderstood,itbecomeseasy
rwhichonen-terminalnetworkof acertaintype
ern-terminalnetworkofanothertype.It fol-
etworkofonetype maybereplacedbyanetwork
Eqs.2-1-2are sufficienttodetermineallthe
esecondmembersintermsofthe drivingadmit-
s,becauseinsuchcasesit isonlynecessaryto
desiredtypehavingthe samedrivingadmit-
rk.Ingeneral,however,Eqs.2-1-2arenot suffi-
ansfer admittancesintermsofthedriving
otalnumberoftransferadmittancesinan
— l)/2,whereasEqs.2-1-2aren innumber;
— 3)/2additionalequationsarenecessary
pletelythetransferadmittances.Givenanet-
ttancesaredeterminedandthen(n — 3)/2addi-
mongthemarealsodetermined.Besidessatis-
eradmittancesofanyequivalentnetworkmust
ditionalequations.Thusanetworkofagiven
bynetworksofthose typesforwhichitis possible
dditionalequationsexistingamongthetransfer
lnetwork.
ontoallnetworks.Then(n— 3)/2addi-
networktonetwork;thereforetheymaybecon-
equationsofeachnetwork.Then(n— 3)/2
orastarnetworkmay
withan emfapplied
edropin theimped-
o theremainingn— 1
sequently,thenet-
Fig.2-1-1canbe
sandvoltagesin the
)byanequivalent
showninFig.2-1-2onlyif its
idessatisfyingEqs.
eristicequationsofthen-terminalstarnetwork.
oregoingconclusionswillnowbeapplied to
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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CTRICALNETWORKS21
rk.Inatwo-terminalnetwork,n=2and
2-2-1)
nalconditionsareconcerned,anytwo-terminal
ybe replacedbyanothertwo-terminalnetwork
sameadmittanceas theoriginalbetweenthe
pedances,Zi,Z2inparallel,maybe replaced
ce
dancesZi,Z2andmutualimpedanceZm,as
ereplacedby anequivalentimpedanceobtained
Ii, I2,thetwocurrentsthroughZi andZ2,
3which,forthisnetwork,becauseofEq. 2-2-1,
anceis
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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NTS
2•also showthatthegivennetworkmaybe
fFig.2-2-2,whichhastwo lineswithnomutual
helinesof Fig.2-2-2arethesameas thecur-
inesof Fig.2-2-1.
ork.Ifn=3,thenumber n(n— 3)/2of
nsiszero;thatis,Eqs.2-1-2 aresufficientto
otherwords,allthree-terminalnetworkshaving
cesareequivalent.Inparticular,anythree-
-1)maybereplacedbyanequivalentstar
edancesZitZ2,Z3ofthe lattermaybeob-
edances,measuredinanymanner,between
ofthegivenandequivalentnetworks.Forin-
mpedancesbetweenter-
nd3and1of thegiven
vingthethirdterminal
othecorresponding
2+ Z3,Z3+Zi,obtained
ork,theimpedancesZi,
dilyobtained.
ork.Star-Delta
pleofthegeneral
cedingarticleis offered
asfortransformingathree-terminalstar(Fig.
elta(Fig.2-4-1),and viceversa.Fromthe
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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CTRICALNETWORKS23
forthiscase become
3fromFig.2-3-2 andsubstitutioninEqs.2-4-1
, Y23,ascalculatedfromFig.2-4-1,are
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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NTS
espondingvaluesofEqs.2-4-3and 2-4—1,
esofthedeltain termsofthoseofthe star.For
nofadeltainto anequivalentstarfromEqs.
asmaybederived:
ork.TwoLinesMutuallyCoupled,Bussed
1-2.and1-3areshownin Fig.2-5-1./6
eself-impedances,andZmisthemutualimped-
networkisshowninFig. 2-3-2.Inaccord-
nedinArt.2-3, theimpedancesZitZ2,Z3of
ZcandZm,maybefoundbyequatingthe imped-
pairsofcorrespondingterminalsofthetwonet-
eingleftopen ineachinstance.
s
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ECTRICALNETWORKS25
Z2,%3,
of Fig.2-5-2isequivalenttothe networkof
ork.TwoParallelLinesBussedattheTwo
fThem.Thisnetworkisshownin Fig.2-6-1.
s,Cisthe pointoffault,andC'i sthepointon
gtoC.
ofthefaultyline fromAtoC.
eofthegoodline fromAtoC'.
cebetweenACandAC'.
cebetweenBC'andBC.
ofthefaultyline fromBtoC.
eofthegoodline fromBtoC'.
C',withmutualcouplingbetweenAC'and
C,canbereplacedby twoequivalentstarsand
plainedintheprecedingarticle.Theequivalent
6-2.Thenetworkmaybe furthertransformed
workA'B'Cwiththeequivalentstar,as explained
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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NTS
withtheThreeLegsMutuallyCoupled.
wnin Fig.2-7-1,andtheequivalentnetwork
cesZi,Z2,Z3of theequivalentnetworkare
eimpedancesbetweencorrespondingpointsof
2
ork.EquivalentNetworkforaSingle-Phase
otherillustrationoftheequivalenceofthree-
uivalentnetworkofthesingle-phaseinduction
owbe determined.
gulatorsareprovidedwithashuntwinding,
ort-circuitedwinding,asshownintheinsetof
heshort-circuitedwindingisto limitthe
nthelinewhenthe regulatorisinthe neutral
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ECTRICALNETWORKS
sathree-terminalnetworkwithmutualcou-
cewiththeconceptsdevelopedinthepreceding
yanequivalentthree-terminalnetworkwithno
ninFig.2-8-2.The
Zroftheequivalent
gimpedancesbe-
terminals,thethird
ndingsoftheregu-
ofFig.2-8-1.In
ngandtheshort-
estatorat 90de-
other,so thatMIO=0.The serieswindingis
rotatedwithrespecttothestatorso that
eentheaxisoftheshunt windingandtheaxisof
whenthesetwoaxescoincideandthe rotorisin
sition.Thebuckingpositionisfor6 intherange
positionis for6in therange90°,180°,270°.
,forthe boostingposition,weobtain
showninFig.2-8-3.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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NENTS
stingpositionandnegativeforthebucking
weput
position
heneutralposition,M12= 0,Zi2=0.
btainedbyapplyingtheformulasderivedinArt.
lingbetweentheshuntwindingand the
esecondaryplustheshort-circuitedwinding
eimpedanceZ2— Z^/Z^,asshowninFig.2-8-4.
orkwithVoltagesAppliedtoOnlyTwoTer-
s.Fora three-terminalnetwork,Eqs.2-1-3
^3
s,sayE3,iszero,the networkwillassumethe
hiswill berecognizedasthecase ofanetwork
tsources.Theaboveequationsbecome
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ECTRICALNETWORKS
pressIi,I2,El, E2intermsof theso-called
D.Notingthat
eobtain
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ONENTS
quations,respectively,weobtain
ndsendingendcurrentsin termsoftheapplied
constants.
vebeen referredtointheliterature asthe
hat
ositivewhenflowingawayfromthenetwork,
2-9-4 willbecome
ThanThreeTerminalsandTheirEquiva-
at athree-terminalnetworkofagiventype
edbyanequivalentnetworkofanyother
-terminalnetworks,theequivalencewillbe
hedrivingadmittancesoftheequivalentnetwork
ces,fromthecorrespondingpoints,ofthegiven
theoriginalnetworkhasmorethanthree
not sufficetodeterminethetransferadmit-
fagiventypecan bereplacedbyanequivalent
onlyif thetransferadmittancesoftheformer
equationsofthelatter.
ationsonann-terminalnetwork,itoften
inalcurrentsandvoltagesare desired.Insuch
nvenienttoreplacetheoftencomplicatedgiven
networkofasimplertype,suchas astarora mesh
heconditionsthatmustbe satisfiedinorder
equivalentstarnetworkormeshnetworkwill
ts.Forastarnetworkthecharacteristic
dbynotingthat,if aunitvoltageis appliedat
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ECTRICALNETWORKS31
1-1)whilethe voltagesappliedattheother
pfrom thecommonpointtothezerovoltage
I,k+1, •••,n, mustbeequal.
kthereare,inadditionto Eqs.2-1-2,two
whichmustbesatisfied.AssumingEl=1and
.2-11-2,we obtain
nsformations,
ya(2-11-1)
eristicequationsforafour-terminalstar.It
lnetworkmaybereplacedbyanequivalent
mittancesoftheoriginalnetworkhappento
Eq.2-11-1.Forafive-terminalstarthereare
ons:
nalnetworkmaybereplacedbyanequivalent
mittancesoftheoriginalnetworkhappento
Eqs.2-11-2.Othercasesmaybeanalyzedina
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ENTS
n-terminalnetworkcanalwaysbereplaced
inedbyconnectingeachterminalkth toevery
— 1terminalsbymeansofan admittanceY^,.
rmeshasitis oftencalled,hasn(n— l)/2
equationsoftheoriginal networkremain
themannerinwhichthe equivalentnetworkis
workofFig.2-12-1maybereplacedby the
2-12-2.Theimpedancesofthelatter,sixi n
ransferadmittancesof theformerbytherelations
ntmeshmaybeobtainedforanygivennet-
arises,however,becausesomeofthebranch
entnetworkmayrequireanegativeresistance
ormerorTwoParallelTransmissionLines.
stration,letusconsider twocircuitsmutually
otransmissionlines(Fig.2-13-1)or thetwo
Fig.2-13-2).LetZi,Z2, Zmbetheirself-
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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CTRICALNETWORKS33
alimpedancebetweenthem.If
it1-2with 3-4open,
fcircuit1-2with3-4 open,
uit3-4with1-2open,
fcircuit 3-4with1-2open,and
ualinductionbetweenthetwocircuits,
agesappliedto theterminals,
Zm
1-1,2-1-2,2-1-3,forthiscasewill become
^-o^2
Z2
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ENTS
eequivalentmeshof Fig.2-12-2inthiscase
wninFig. 2-13-3.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ECTRICALNETWORKS35
ginalnetworkwill reducetooneofa perfect
anceineitherwinding.Equations2-13-4will
eas showninFig.2-13-4.
.2-13-1and Fig.2-13-2,theself-impedances
xternal,asshownin Fig.2-13-5.Thenetwork
e networkofFig.2-13^4,asshowninFig.
herequivalentnetworkofthoseshownin Figs.
3.f
be assumedforallpracticalpurposesthat
nectedtoeachother.Inthismannerthefour-
13-2becomesthethree-terminalnetworkof
eplacedbyathree-terminalequivalent,as
valentnetworkisgiven inFig.2-13-8.
emeshesofFigs.2-13-3and2-13-6are the
merunderallconditions,whereasthenetworkof
nce,the conditionswith2and4open cannotbe
kofFig.2-13-8but canbereproducedinthenet-
-13-6.
hichgivesthecharacteristicequationsfora
hatthetransformerinits generalrepresentation
ntCircuits:1,"Transactions,AIEE,LI(June1932),
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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NENTS
equivalentfour-terminalstarbecause
ocoupled circuitsistheautotransformer(Fig.
eonlythreetermi-
canbereplacedby
minalnetworkofany
nsformer.fA
risshowninFig.
alnetwork.For
practice itwill
•,and 3•areconnectedasshownin Fig.2-14—2.
rrents12',Iy' flowinginwindings2and3,
whenproducingfluxinoppositiontothat pro-
winginwinding1,thetotal ampere-turnswillbe
ndcurrentof winding1;HZ,/2'arethe turnsand
• aretheturnsand currentofwinding3.70 is
chisusuallynegligible.Thus,disregardingit
ysisof A-CPowerSystems,Vol.1(NewYork:JohnWiley
O.G.C.Dahl,Electric Circuits,TheoryandApplications,Vol.1
ookCo.,Inc.,1928),p.34.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ECTRICALNETWORKS37
ctively,thecurrents/2',I3,expressedinterms
windingsisassumed,thenetworkofFig.
thatof Fig.2-14-3,inwhichtheleakageim-
rofwindingsisequalto theactualleakageim-
msofwinding1.
urrentreturninginthecommonconnection.
showninFig.2-14-4.Thisis athree-terminal
weenwindings,whichcanbereplacedbyan
networkwithnomutualsbetweenwindings,as
a three-windingtransformercanberepre-
networkunderthefollowingassumptions:
lthree windings.
tisdisregarded.
theimpedancesZ\,Z2,Z3oftheequivalentnet-
hownin Fig.2-14-5,areobtainedby
ncesbetween
hortedand3and3'open
hortedand2and2'open
shortedand1andl'open
ncesmeasuredunder(a)tothesame voltage
pter3,thus obtaining,respectively,
),Z23from (a3)
othe impedancesbetweencorresponding
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NENTS
,thevaluesofZi,Z2, Z3canbeobtainedfromthe
sforaThree-TerminalNetworkwithVoltage
nals.In stabilitystudies,powerexpressions
rk,withvoltagesappliedtoonlytwoterminals,
ftheA,B,C, Dconstants.Thevaluesofthese
ree-terminalnetworkarederivedby following
2-9, ortheymaybeobtainedfrompublished
vationwill beillustratedfortwocommon
ythegeneralTandthe generalpinetworks.
sforaTNetwork.Figure2-16-1showsthe
Forthis type,fromEqs.2-4-2,2-4-3,and
ndDistributionReferenceBook,4thEd.,Weetinghouse
9,p.327.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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CTRICALNETWORKS
ainedfromthe expressionforthecurrentsIi and
sEI,E2andthe impedancesZ,,Zr,Z%written
sforaPiNetwork.Figure2-17-1showsthe
Forthis type,fromArt.2-4,theexpressions
heserelationsdirectlyfromFig. 2-17-1,
l+ |
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NENTS
~+-^- - /2
4givesthevaluesofA, B,C,Dshownin
pi canfirstbeconvertedto anequivalentT
-16-1bymeansofEqs.2-4-6.We obtain
yieldsEqs.2-17-1.
sforLongTransmissionLines.ExactT
missionlinesarec omposedofdistributed
ctanceanddistributedshuntresistance and
enerality,letusassumethatthe lineiscom-
elementsZiandshunt impedanceelementsZj,
xbethedistancefrom somepointchosenas
gth,saymeters;thenZiandZ2 areexpressed
bethevoltageat thedistancex,andIx theline
ment,oflengthdx,thevoltageExdecreasesby
reasesbydIx.Thus
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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CTRICALNETWORKS41
andEqs.2-18-1weobtain
ermeter (2-18-3)
nconstantoftheline.t
mponentaiandanimaginarycomponentft,or
alled theattenuationconstant.Itisexpressedin
ginarycomponent3is calledthephaseconstant.
usis expressedincircularradianspermeter.
sexpressedincomplexradianspermeter.
aracteristicimpedanceoftheline
ngleoftheline
ne,in meters.Withtheserelationsasa starting
ssionsareobtainedbetweenthevoltagesE,,E,
eoriginandthe receivingend,respectively,of
8-2:
/,
ohnson,TransmissionLinesandNetworks(NewYork:
,1950).
cationofHyperbolicFunctionstoElectrical-Engineering
raw-HillBookCo.,Inc.,1925).
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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NENTS
E,,I,,
9Jr
thehyperboliccosineandsine,respectively,of
ndedbythe line.
olicfunctionsof81aregivenbyf
+...
17e'74
— -— + •• -
8-3,
oftheline,atthe operatingfrequency/.
meter)
alsoattheoperatingfrequency/, ofthe
spermeter)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ECTRICALNETWORKS
ceofthelineperunitlength (mhosper
eofthelineperunitlength (faradsper
cimpedancetakestheform
polyphasesymmetricallineswithnocurrent
yLH— Maw,whereLiiistheself -inductance
s,whenconsideredisolated,andMaWis theaver-
uctancesbetweenanytwophases.Thecase
n thegroundwillbeconsideredlater.
unt(Z'")impedancesandthetotalcomplex
ectively,
VZY
inesoperatingatfrequencieshigherthan10
are negligibleincomparisontouLanduC,
pproximately,
5thatthe apparentphaseimpedanceofa
line,underbalancedconditions,thatis,with
gnitudeanddisplaced120degrees,is givenby
ohmspermeter(2-18-10)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ONENTS
meanspacingofthethreeconductors
radiusoftheconductorinclusiveof
conductoratthefrequency/
garithm.
wnthatthe capacitanceofthesamecircuit
sisgiven,withgood approximation,by
conductorandkis therelativedielectriccon-
ndingtheconductors.Thus
eter
esresistanceandtheshunt conductancefor
=1),
102log——-log -ohms
mS
R=r,
the base10.
oadjacentconductorsofanoverheadcircuit
entcanbeapproximatedbyf
ers
sandkvisthe linetolinerated voltage,inkilo-
y,
65kv)meters(2-18-14)
ordinationofInsulationandSpacingof Transmission
ctions,AIEE,LXV(Oct,1946),690-93.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ECTRICALNETWORKS45
ssedinmeters.
-3and2-18-4,
ancegisnegligibleandcan beassumedtobe
on,
osa„)(2-18-15)
^)(cosa0- jsina0) ohms(2-18-17)
osedoftwoparts,L,and Lc,corresponding,
nductioninsideandoutside oftheconductor.
erthan Lc,sothatwecanassume,approxi-
actancecomponentofEq.2-18-10byreplacing
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NTS
ermeabilityofthemediumsurroundingthecon-
mptionGMR=r,madein calculatingZ0
ofwithEqs. 2-18-12,correspondstoreplacing
=0,theline isalosslessone.If forsucha
omEqs.2-18-15and2-18-11.
nlines,
. 2-18-15,wecanwrite
ely,foroverheadlinesthepropagationconstant
Xiand thefrequency/.Inpractice,thefre-
inesisusually00 cycles.Correspondingly,y
Me~iaocomplexradianspermeter
=0,0.1,0.2,0.4,0.6are giveninTable2-18-1.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ECTRICALNETWORKS
atunder thesameconditionsZ0isgiven
-'
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ONENTS
replacedbyanyothertypeofthree-terminal
thepi network.Itcanalsobeexpressedin
stants.BydirectcomparisonofEqs.2-18-5
e transmissionline
nswithEqs.2-16-1,weobtainfortheequivalent
e,-1)
22)
Tnetworkofthetransmissionline isasshownin
btainedin asimilarmanner.FromEqs.
piis asshowninFig.2-18-5.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ECTRICALNETWORKS
sdirectly,asindicatedin Art.2-17,noting
thesymmetryofthelinein Fig.2-17-1,
nbe determinedfromtestasfollows.With
= 0and
81
rt-circuited,Er=0 and
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NTS
minethetotal complexangleoftheline.
Eq.2-18-25,weobtain
atethesurgeor characteristicimpedanceofthe
2-18-5,ifforbrevitywelet
,
0inEq.2-18-29,B canbeexpressedintermsof
nalso beexpressedintermsof Z,,,Z*,.
2-18-24,and2-18-25,
eA,B,C,sinceA2— BC=1.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ECTRICALNETWORKS51
,Z,,areobtainedbyapplyingat thehomeend
equalinmagnitudeand120degreesapart.f
iEquivalentNetworksofaTransmission
oConvertApproximateintoExactEquivalent
TwoLinesToHavetheSameCorrection
econstantscannotbedeterminedwithmorethan
Someofthefactorscontributingtothisinac-
geswithtemperatureandlineloading.
withtemperature.
ctedbythecontourofthe surroundingground,
swhichareapproximatewithin0.5percentare
theexactequivalentnetworksinvolvethe
s.Thesearecumbersometocalculate,although
plifiedbypublishedtables.In practice,power
allyshortandfor mostapplicationsapproximate
seareobtainedasfollows:Forthe approximate
areeachtakenequaltoone halfthetotalseries
dZ3is takenequaltothetotalshunt impedance
anh—
ivalentTnetworkwillbeas showninFig.
piisobtainedbytaking,in Fig.2-17-1,
h8i
adof .
ivalentpiwillbeas showninFig.2-19-2.
worksmaybeobtainedfromtheapproximate
ultiplyingthevariousbranchimpedancesby
ctors.Twomethodswhichhavebeensuggested
house,andR.B.Gow,"MeasuredElectricalConstantsof
ionLine,"Transactions,AIEE,LXIII(July1944),538-42.
ndDistributionReferenceBook,4thed.,1950,Ch.9.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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NTS
mpedancesofFig.2-17-1to obtainthe
2-18-5,thatis,
ecalculatedwiththeseriesexpressionsinEqs.
ptthefirst twoareneglected;thatis,
formations
dto bezero,andthus
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ECTRICALNETWORKS
tivereactanceoftheline inohmsandin meg-
y.
s
s
mile
mile
m1
dsonthe
andon
curves
leinall
exceptfor
llequiva-
8-8,
theline,
ce7,,in
mpedanceZandofthetotalshunt admittanceY
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NTS
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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CTRICALNETWORKS
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NTS
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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CTRICALNETWORKS
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NENTS
6,
ximateequivalentTofFig.2-19-1 intothe
-18-4,theseries impedanceZ/2ismultiplied
andtheshuntimpedanceZ'is dividedbythe
rly,
2Z (2-19-12)
(6,/2)JC,
heapproximateequivalentpiofFig.2-19-2
of Fig.2-18-5,theseriesimpedanceZis multi-
or K-,andtheshuntimpedance2Z'is divided
epend entirelyon81,thecomplexangle
terminedfromFigs.2-19-4!and2-19-5.f
nceg,fromEq.2-18-16,81is givenby
K iandK2 arefunctionsof"X/LC,fl,and
out that,forexistingoverheadtrans-
on,fromL.F.Woodruff,PrinciplesofElectric Power
York:JohnWiley4Sons, Inc.,1938).
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ECTRICALNETWORKS59
earlyconstantandequal totheinverseofthe
ivenbyEq.2-18-19.Thusfor existingpower
sKiandKZ
y onfland
000
iandA2/(Ri/Xl)asfunctionsof fl(/being
miles)areshowninFig.2-19-6.f Fromthese
culateKi,K2and
0correspondingtoaline 800mileslongat25
at60cycles.
ertwolines ofdifferentlength,frequency,
hownbelow.Thecorrectionfactorsforthetwo
er)
er)
meter)
eter)
ermeter)
edonlyif
ssionofAIEE,fromEdithClarke,"Steady-StateStability
"Transactions,AIEE,XLV(1926),22-41.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ONENTS
thetwolinesmusthavethesameRi/Xiratio
tingatfrequency/•hasthe sameangleasline1
avingthesameRi/Xiratioandhavinga length
overheadlineoperatedat60cycles,thatis,
heaveragecharacteristicspreviouslyreferredto,
498X10~3 ohmpermeterand6 =5.2X
3X10•"•9mhopermeter,
agecharacteristics,equivalenttothelineof
ce/reactanceratioRi'/Xiandoperatingat
esameresistance/reactanceratiooflength I
',operatingat frequency/'andhavinga
ter,ashuntcapacitivesusceptanceb'mhos
actanceratioRi/Xi,subtendsthesamecomplex
esamecorrectionfactors,KltK2,as a60-cycle
sticsoflengthIhavinga reactanceof0.498X10~3
er mile),ashuntcapacitivesusceptanceof
r (5.2X10~6mhopermile),and thesamere-
i'/Xi',thelengthI beinggivenbyEq.2-19-17a.
quationsinTermsofLoadImpedance.
ewritten:
(2-20-2)
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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CTRICALNETWORKS61
8r+cosh8rsinh81
8rsinh81
h81=—
ingendis
sforOtherTypesofNetworks.Byappro-
,B,C,D constantsofvariousformsofnetworks
canbeobtainedeasily.\
onstantsofTransmissionLines.Aconcept
electricpowertransmissionistheconceptof
ornaturalloading.A transmissionlineissaidto
rnatural load(SIL)whentheapparentload
2-20-2is equaltothesurgeimpedanceZ0 of
nthatin makingpowersystemstudiesitis
essimpedanceasafraction,or perunit,ofan
cuits,TheoryandApplications.
al TransmissionandDistributionReferenceBook.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NENTS
uecalledthenormalreactanceornormalimped-
ndicatedby thesymbolXn.Similarly,admit-
action,orperunit,of areferencevaluegivenby
ladmittance.Inpowerstudiesitis sometimes
magnitudeZQof thesurgeimpedanceZOasthe
Zoasthenormaladmittance.
ofa transmissionline,givenbyEqs.2-18-21'
ns,Bisan impedance,andCisanadmittance.
tof Z0andCis expressedinperunitof
e~}misinh|
=c'a° sinh(ul\/LCJ-1+j —*
Cispracticallyconstantandapproximately
xistingoverheadpowertransmissionlines.
functionofRi/Xi,for agivenfrequencythe
andAi,A2,the twocomponentsofcosh61
1.Ithasbeenshownalso thatontheabove
powertransmissionlines,toVLCis approxi-
~8 radianpermeter(2.04X 10~3radianper
u,Ciu,C2u,Ai, A2havebeenpublishedby
60-cyclepowertransmissionline usingfor
~6radianpermeter (2.06X10~3radianper
.39X10~9 andRi/Xl=0.1,0.2,0.3,0.4.
ooverheadlines oflengthI'operatingat
cycles,havingthe sameresistance/reactance
matevalueVLC= 3.39X10~9if anequivalent
s usedwhichisgivenby
pliedtolinesotherthan overheadonesifa60-cycle
byEquation2-19-17a,isused.
orkofonetypebe transformedintoa3-terminal
nson,loc.tit.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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ECTRICALNETWORKS
scananetworkofonetype havingmorethanthree
ntoanetworkofthe samenumberofterminalsand
rtransforminga
withthe threelegs
formedintoa3-ter-
etweenthe three
eaffirmative,show
networkfora
gulator.
C,10mileslong,
bussedatoneend,S.
f795,000-CMACSR
trands) conductors
wnin thefigure.
dby thetwolinesarebalanced,findthe equivalent
ming:(a)completetranspositionofthetwocircuits;
sfortheA, B,C,DconstantsfortheT networkof
dto1 and2only.ExpressA,B,C, Dintermsof Zi,
entsI, and/-jarepositivewhenflowingas shown,
msof theA,B,C,Dconstantsas obtainedabove.
asin (a),butassumethatthepositive direction
dingtransformerofFig.Ais convertedintothe
byconnectingtogetherterminals2and4.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NTS
minalnetworksofthe2-windingtransformerand
hyistheimpedanceoftheautotransformersmaller
ding2-windingtransformer?
composedofthehigh-tensionportionofthe winding
fthewinding.Subscripth isusedforthe former,and
ortion.ThusRkandXkare,respectively,theresistance
e,inohms,ofthehigh-tensionportion;It,and A"are
kagereactanceofthecommonportionofthe winding;
einhigh-voltagetermscorrespondingtothemagnetizing
sisand eddycurrentlosses.
nt3-terminalnetworkofthe autotransformeras
ageterminalsisasshownin Fig.A,where
erminalnetworkintermsof lowvoltage,asshown
nscana3-windingtransformerbereplacedbyan
e,60-cycletransmissionlinefromBoulderDamto
mileslong,has hollowinterlockingcopperconductors
andan areaof512,000CM.Formostof thelength,the
owers,with32.5-fthorizontalspacingand50-ft average
the abovespacingsandarrangementforthewhole
ails.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ECTRICALNETWORKS65
77777Gr0und
Zandthe shuntadmittanceYoftheline.
xactequivalentpiandT networksandtheA,B,
Kiandtf2 1+>♦ Groundwires
ceZ,at
eceivingend
ceZ„at
eceivingend32,5'
transmis-
ationshownin
e-circuitof
onductors
50,000CMof Prob.2-12.
r.Theground
eel crossmemberandare10ft abovethepowercon-
d18 ftatmidspan.Thefollowingtestsweremadeby
positivesequenceatthesendingend:fapparentim-
ngendwithreceivingendunloaded:
Z„atthesendingendwith receivingendshort-circuited:
o-
.
81.
nceZandshuntadmittanceY.
xactTandpi networks.
KiandK* forthisline.
ow161-kvinterconnectionis288miles long;one
andtheother iscopperconductor,300,000CM.
3ft.
edby theline.
ndexactequivalentpiand Tnetworks.
stants.
0cycles.
288mileslongt hasthefollowingcharacteristics:
833=0.857e'7,-77°
spermile
matepiandT andtheA,B,C,D constants.
houBe,andR.B.Gow,loc.cit.
gaveragevalues.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NTS
onstantsofthecircuitformedbyatransmission
rmeratthe receivingend.
Bu,Ca,Dconstantsofthe transmissionlineofProb.
2-12obtainthevaluesofthe correctionfactorsfrom
dcomparewithvaluescalculatedforProb.2-12.
similartothatof Art.2-13fora2-windingtrans-
tstarnetworkfora polyphaseinductionmotoroperat-
tvoltageisappliedto boththestatorandthe rotor.
inalnetworksinserieswithvoltagesEi,EZ applied
ownin thefigure.Inrelaystudiesit isdesirableto
It,thelineto neutralvoltagesat,andthecurrent
etwonetworks,interms oftheterminalvoltagesE
thenetworks.
and/tandshowthat
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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ECTRICALNETWORKS
rms,showthat
ha
osh* a
oshf trfc c oshasinhf t
c oshf t±sinhasinhf t
anhafora realpositiveornegative.
up ofimpedancesZI,Z* connectedasshownin
onpointsA,B, C,threeequalimpedancesZ'are
d.
nnectedtothethreepoints a,b,c.
ailableformeasuringzerosequencevoltage,as
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_
u s e # p d - g o o g l e
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NTS
tlaterwithrespectto zerosequencequantities,
dtobeatthe samepotential.Underthisassumption
tarof thetwoschemes.
Cu,Cu,CtiareconnectedinA.Findthecapaci-
ivalentY.
sAandB,150 milesapart,areinterconnectedby
ssionline,500,000-CMcopperconductorshaving15-ft
of2000+j'7500 kvaistappedoffthe midpointMof
ure.
13.8kv;forthe
%reactanceon
tance;ratioof
ormation110/13.8.
mpletelytransposed.Neglect
, C,Dconstants,at100,000kva,ofthenetwork:
apacitance;(b)excludingtheeffectofcapacitance.
ransmissionline270mileslonghas thefollowing
ms/mile
mho/mile
Zandthe shuntadmittanceYoftheline.
xactTandpi networksandtheA,B,C,D constants.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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UANTITIES
ance,reactance,andimpedanceofacircuit
Whendealingwith short-circuit,stability,
tisconvenientto expresstheminperunit (frac-
ereferencevalue.Thesemaybeobtainedas
inohms)underconsiderationis partofacircuit
d thenormalvoltage,andfora current/n,called
runit valueZuoftheimpedanceZis obtained
(causedby thenormalcurrentwhen'flowing
thenormalvoltageEn.Thatis,
alueofZis itsvalueasa fractionofXn.Xnis
ceornormalreactance.
inedinasimilarmannerby multiplyingthe
composedofaresistanceRanda reactanceX
ponentscan alsobeexpressedinperunit:
anceandXu=per unitreactance.
hat bothEnand/nmust begiveninorderto
es.IfEn and/nareknown,then3EnIn
basevolt-amperes,isalsoknown.Ordinarily,
asekilovolt-amperes(kva)andthenormalline
en.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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NTS
weobtain
perunitimpedancesvaryproportionallywith
portiontothesquareofthe voltage.Thus,if,
currentnX/nis chosenasnormalcurrent,
maintainedconstant,thekvabase willincrease
pedancewillalsoincreasentimes.If, onthe
keptconstantbutthenormalvoltageis varied,
ary ininverseproportiontothevoltage,the
ryindirectproportiontothe squareofthe
pedancewillvaryin inverseproportiontothe
seconclusionsareevidentfromEqs.3-1^i,
ircuitinohms.
t(kva)ikilovolt-amperesandV\volts.
at(kva)2kilovolt-amperesandV2volts.
omexpressingimpedancesasperunitquanti-
t theimpedanceofgenerators,motors,andtrans-
n,whenexpressedinper unitonratedkvaand
anarrowrangeregardlessof therating,although
inwidelimitswith therating.
ctanceofsolidrotorturbinegeneratorsranges
runitwith anaveragevalueof1.2per unitre-
generator,t
eof variationofthereactancesofsynchronous
ndDistributionReferenceBook,4thed.,Westinghouae
Knowlton,StandardHandbookforElectricalEngineers,
w-HillBookCo.Inc.),p.688.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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e-PhaseSynchronousMachines
alues,timeconstantsare inseconds.Valuesbelowthelines givearangeofvalues;
gevalue.)
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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NTS
nceof60-CycleTwo-Winding
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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UANTITIES
dingtransformersdependsontheinsulationof
nof thetransformer,rangingfromafewpercent
ancedecreasesastheratingincreases,being
tfor5000-kvasingle-phaseand10,000-kvathree-
ftheorderof 0.3percentfor50,000-kvasingle-
phasetransformers.t
1-4give,respectively,theaveragepercent
ngpowertransformers,distributiontransformers,
ationtransformers.
-CycleDistributionTRANSFORMERst
va)
nsformers— distributiontransformers— approx.80%of
ManufacturingCo.
AdCenterUnitSubstAtionTrAnsformers!
va
ManufacturingCo.
anufacturersoftwo-windingtransformers
merimpedanceinpercentbasedon thekva
ndDistributionReferenceBook,p.99;StandardHandbook
gs.6-7aand6-7b,pp.561,562; L.F.Blumeetal,Trans-
.(NewYork:JohnWiley& Sons,Inc.,1951),p.91.
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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NTS
ndon anormalvoltageequaltothe rated
chtheimpedanceisgiven,f
useofper unitimpedancesderivesfromthe
vingtheseriescombinationofimpedancesat
timpedancesareaddeddirectlywithcomplete
esinvoltage,withincertainlimitationsthatwill
ssionofthe followingpages.Toillustratethis
atransformerisinterposedbetweentwostations
reZ\ andZ2aretheimpedancesonthe two
orsimplicityweassumethatthe impedanceof
udedin Z\andZ2andthat thetransformerisa
ofFig.3-1-1canbereplacedby thenetworkof
wherea =ni/n2isthe ratiooftransforma-
esasviewedfrombus1 andbus2,respectively.
enormallinetoneutral voltage,current,react-
bviously,
"GeneralElectricPublication,GET-1285A,p.42.
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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UANTITIES75
termsofthebus1 normalvoltageis
henormalvoltage,current,andreactanceonbus
alkvaon bothbus1and bus2,andin addition
omes
eZi inperunitof Xin,andZ2uisthe impedance
rwise,fromFig.3-1-3,intermsof thebus2
tain
3-1-11)
anceisthesameregardlessof whichsideofthe
.It followsthat,ifimpedancesareexpressed
ferencesinvoltagecanbe disregarded.The
runitis showninFig.3-1-2b,where
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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ENTS
hecircuitis
3-1-12)
Ratioof Transformation.Inpracticeit
oof transformationa=ni/712ischangedwith
ionswhileEiandE2 remainconstant.Howdo
ange?Startagain fromtheconditionsshownin
ditionsspecifiedinthe precedingarticlethe
kisshownin Fig.3-1-2bandtheperunit equa-
Now,assumingthesamevoltagesE\,E2on
eratio fromato01.The equivalentcircuitin
1voltagewillbeas showninFig.3-1-2aexcept
landthe currentwillchangeasshownin Fig.
onthe bus1sideare
alkvaonboth busses,thenormalvaluesonbus2
andontheratio oftransformationaccordingto
10.Thus,accordingtothoseequations,the
sideshouldbe
canbeexpressedin termsofthenormalquan-
2,whichobtainwhenthetransformerratio isa;
-2-2,3-2-3,respectively,withEqs.3-1-8,
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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UANTITIES
ofFig.3-2-1ais
,current,andreactanceon thebus2side are
ntoE2n', I2n',X2n'astheratio oftransformation
perunitnetworkwillbeas showninFig.3-2-1b,
ectively,impedanceandvoltagein perunitof
makinguseofEqs.3-2-4,3-2-5,3-2-6,we
workwillbe asshowninFig.3-2-2b,where
ely,perunit impedanceandvoltageintermsof
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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ONENTS
nandnormalvoltageE2nobtainingwiththe
the correspondingnetworkinohmsasviewed
peres,withtransformerratiosa andaiwillbe
aand3-1-2bwith3-2-2aand3-2-2b,re-
artingwitharatioa andvoltagesEiandE2,
llbenecessarytochangethe voltageonbus2
mpedancesonthesamebus2 sidebytheratio
aybetroublesome.Cantheybeavoided?
erto Figs.3-2-3aand3-2-3b.Hereanauto-
weenZ\anda2Z2.Letit beassumedthatthe
ctoneofrationo/no'.The equationofthe
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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UANTITIES
turnofthe autotransformer.Ifno/no'=Oi/a,
aand3-2-3bbecomeidenticalwith thoseof
pectively.Thus,inmakingstudiesona cal-
smadeonthebasis ofavoltageratioa, whenthe
ditis notnecessarytochangethevariousim-
nautotransformeroftheappropriateratio is
e.
sinSeries.Ifthereare severaltransformers
ninFig.3-3-1,theimpedanceasviewedfromthe
a32Z4)]
sameforallcircuits. Assumethenormalvoltages
columnofTable 3-3-1.Thenormalcurrents
ondcolumnandthenormalreactancesas shown
cuitisequivalenttothe oneshowninFig.3-3-2a
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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NTS
s 1.Theperunit impedanceis
a342Z4)]
*«+Z3u+Z4u(3-3-2)
erunit,is showninFig.3-3-2b,where
eorMoreRatiosof Transformation.This
mannersimilartothatgivenin Art.3-2forthe
,the ratiooftransformer1is changedfrom
f Fig.3-3-2becomethoseofFig.3-4-1aand
ponacalculatingboardis madeonthebasisof a
dthisis changed,thesetupcanbeusedfor
providedthatanautotransformeroftheappro-
thepointwherethetransformeris.Theonly
emadeis thatduetothe impedanceoftheauto-
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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UANTITIES
nsformersinParallel.Ifthereare two
usses asshowninFig.3-5-1,the equivalent
econstructedasviewedfrombus 1.Fromthis,
a342Z4\\
spectively,thenormalvoltage,normalcurrent,
actanceonbus1.Equations3-5-1and 3-5-2
t ofEinasfollows:
B5-=r~(3-5-5)
ienttoexpresseverythinginperunit; how-
conversionofall quantitiestothebus1 base.
thisconversion?Usingthesamenormalkvaon
ethenormalvoltage,normalcurrent,andnormal
onthisbus.Assumingthat ESnisderivedfrom
P u b l i c D o m a i n ,
G o o g l e - d
i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s
_ u s e # p d - g
o o g l e
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ENTS
4°65="57
the normalreactanceXsnwillbegivenby
singEq.3-5-4,and notingthat/5=aes/e.
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
a t h i t r u s t . o r g / a c c e s s_ u s e # p d - g o o g l e
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UANTITIES83
2)
efirst termofthesecondmemberofEq. 3-5-5,
3+ a342£i}]/i
uits 2,3,and4,let E2a,E3n,Einbethe corre-
defined,respectively,bytherelations
currents72a,I3a,/«nandthe normalreact-
tively,by
—2(3-5-15)
=5 ££
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
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ENTS
s
3uI3u+Z\J.Ku— ^1—
utethe perunitequationsofthetwo parallel
erunitequationsof eithercircuit,thenormal
obtainedby startingfromthenormalquantities
y theratiooftransformation,orproductof
fthetransformersinterposedbetweenbus1and
tion.Thus,in settinguptheperunit equivalent
bus1thenormalquantitieson thebus1side
us 5bytakingintoconsiderationthetransformer
nsformersonthe othercircuit.Theperunit
uit arealsoobtainedbystartingfrom bus1and
akingintoconsiderationthevarioustransformers
dingthetransformersoncircuit5-6.The differ-
ngoingfrombus1 tobus5is accountedforby
voltage(1— ai2a23Cl34J£5uTheequivalent
luesareshownin Fig.3-5-3.
(1
makingstudiesonana-ccalculatingboard is
ebymeansofanautotransformerasshownin
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
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UANTITIES85
Z\u
4are
turnofthe autotransformer.Fromtheseequa-
-18)
nsothat
llbe identicalwiththatofFig.3-5-3,since
omeidenticalwithEqs.3-5-17 and3-5-11,
eor MoreRatiosofTransformationon
ofonetransformerischanged(saytrans-
, insteadofchangingtheperunit impedances
insertedasdiscussedin Art.3-4andas shown
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
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NTS
.Perunitsusceptancescanbeobtainedina
impedances.Sincesusceptancebisgiven by
sif /isgivenin amperesandEisin volts.Per
enby
eptance
mperes,volts,andohms,respectively,Bnwill
mtheexpressionfor/n andEnforthree-phase
wealso obtain
itsusceptancevariesproportionallytothesquare
yto thekva.Thesusceptanceisordinarily
nsmissionlines.Forsuchapplicationsthe
unit.Itis moreconvenienttousethemicromho.
mho,
P u b l i c D o m a i n , G o o g l e - d i g i t i z e d
/ h t t p : / / w w w . h
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UANTITIES
ntitiesbecomeofgeneral use?Whataretheir
alues?
teminterconnectionsitoftenbecomesnecessaryto
ormationofthetransformersonthe interconnecting
thesechangesaccountedforonboardstudies?
.2-9showthat,ifa 2-windingtransformerisused
eratioof(Zu)au,theper unitimpedanceasanauto-
eperu