calabrese symmetrical components

8/21/2019 Calabrese Symmetrical Components

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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

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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

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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 ,

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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


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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.


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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)


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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


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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


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ENTS

alledimpedancephasoror,simply,theimpedance

eration.Thecurrentimay berepresentedby

K/27orV*/2Z*,orby eitherV/ZotV*/Z* inthe

dforthevoltage.Whateverphasorisused to

hasorrepresentingthevoltagev bytheangle


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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.


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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


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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—


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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?


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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,


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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.


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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-


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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


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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


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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


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CTRICALNETWORKS23

forthiscase become

3fromFig.2-3-2 andsubstitutioninEqs.2-4-1

, Y23,ascalculatedfromFig.2-4-1,are


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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


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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


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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


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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.


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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


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ECTRICALNETWORKS

pressIi,I2,El, E2intermsof theso-called

D.Notingthat

eobtain


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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


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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


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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-


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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


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36/310

ENTS

eequivalentmeshof Fig.2-12-2inthiscase

wninFig. 2-13-3.


/ h t t p : / / w w w . h



37/310

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),


/ h t t p : / / w w w . h



38/310

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.


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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


/ h t t p : / / w w w . h




40/310

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.


/ h t t p : / / w w w . h



41/310

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+ |


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42/310

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


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43/310

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).


/ h t t p : / / w w w . h



44/310

NENTS

E,,I,,

9Jr

thehyperboliccosineandsine,respectively,of

ndedbythe line.

olicfunctionsof81aregivenbyf

+...

17e'74

— -— + •• -

8-3,

oftheline,atthe operatingfrequency/.

meter)

alsoattheoperatingfrequency/, ofthe

spermeter)


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45/310

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)


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46/310

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.


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47/310

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


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48/310

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.


/ h t t p : / / w w w . h




49/310

ECTRICALNETWORKS

atunder thesameconditionsZ0isgiven

-'


50/310

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.


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51/310

ECTRICALNETWORKS

sdirectly,asindicatedin Art.2-17,noting

thesymmetryofthelinein Fig.2-17-1,

nbe determinedfromtestasfollows.With

= 0and

81

rt-circuited,Er=0 and


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52/310

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.


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53/310

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.


/ h t t p : / / w w w . h



54/310

NTS

mpedancesofFig.2-17-1to obtainthe

2-18-5,thatis,

ecalculatedwiththeseriesexpressionsinEqs.

ptthefirst twoareneglected;thatis,

formations

dto bezero,andthus


/ h t t p : / / w w w . h




55/310

ECTRICALNETWORKS

tivereactanceoftheline inohmsandin meg-

y.

s

s

mile

mile

m1

dsonthe

andon

curves

leinall

exceptfor

llequiva-

8-8,

theline,

ce7,,in

mpedanceZandofthetotalshunt admittanceY


/ h t t p : / / w w w . h




56/310

NTS


/ h t t p : / / w w w . h




57/310

CTRICALNETWORKS


/ h t t p : / / w w w . h




58/310

NTS


/ h t t p : / / w w w . h




59/310

CTRICALNETWORKS


/ h t t p : / / w w w . h




60/310

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).


/ h t t p : / / w w w . h




61/310

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.


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62/310

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)


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63/310

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.


/ h t t p : / / w w w . h




64/310

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.


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65/310

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.


/ h t t p : / / w w w . h




66/310

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.


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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.


/ h t t p : / / w w w . h




68/310

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


/ h t t p : / / w w w . h




69/310

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


/ h t t p : / / w w w . h




70/310

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.


/ h t t p : / / w w w . h



71/310

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.


/ h t t p : / / w w w . h



72/310

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.


/ h t t p : / / w w w . h



73/310

e-PhaseSynchronousMachines

alues,timeconstantsare inseconds.Valuesbelowthelines givearangeofvalues;

gevalue.)


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e


74/310

NTS

nceof60-CycleTwo-Winding


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e


75/310

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.


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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.


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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


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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,


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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


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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


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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


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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-


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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


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ENTS

4°65="57

the normalreactanceXsnwillbegivenby

singEq.3-5-4,and notingthat/5=aes/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 ££


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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


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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


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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,


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UANTITIES

ntitiesbecomeofgeneral use?Whataretheir

alues?

teminterconnectionsitoftenbecomesnecessaryto

ormationofthetransformersonthe interconnecting

thesechangesaccountedforonboardstudies?

.2-9showthat,ifa 2-windingtransformerisused

eratioof(Zu)au,theper unitimpedanceasanauto-

eperu

calabrese symmetrical components

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