E283C2.docx
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E283C2.docx
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E283C2
CHAPTER2–TRANSFORMERS
Summary:
1.TypesandConstructionofTransformers
2.TheIdealTransformer
▪PowerinanIdealTransformer
▪Impedancetransformationthroughatransformer
▪Analysisofcircuitscontainingidealtransformer
3.Theoryofoperationofrealsingle-phasetransformers.
▪Thevoltageratioacrossatransformer
▪ThemagnetizationcurrentinaRealTransformer
▪ThecurrentratioonatransformerandtheDotConvention
4.TheEquivalentCircuitofaTransformer.
▪Exactequivalentcircuit
▪Approximateequivalentcircuit
▪Determiningthevaluespfcomponentsinthetransformermodel
5.ThePer-UnitSystemofMeasurement
6.Transformervoltageregulationandefficiency
▪Thetransformerphasordiagram
▪Transformerefficiency
7.Threephasetransformers
1.TypesandConstructionofTransformers
Typesofcoresforpowertransformer(bothtypesareconstructedfromthinlaminationselectricallyisolatedfromeachother–minimizeeddycurrents)
i)
CoreForm:
asimplerectangularlaminatedpieceofsteelwiththetransformerwindingswrappedaroundtwosidesoftherectangle.
ii)ShellForm:
athreeleggedlaminatedcorewiththewindingswrappedaroundthecentreleg.
Theprimaryandsecondarywindingsarewrappedoneontopoftheotherwiththelow-voltagewindinginnermost,dueto2purposes:
i)Itsimplifiestheproblemofinsulatingthehigh-voltagewindingfromthecore.
ii)Itresultsinmuchlessleakageflux
Typesoftransformers:
i)Stepup/Unittransformers–Usuallylocatedattheoutputofagenerator.Itsfunctionistostepupthevoltagelevelsothattransmissionofpowerispossible.
ii)Stepdown/Substationtransformers–Locatedatmaindistributionorsecondaryleveltransmissionsubstations.Itsfunctionistolowerthevoltagelevelsfordistribution1stlevelpurposes.
iii)DistributionTransformers–locatedatsmalldistributionsubstation.Itlowersthevoltagelevelsfor2ndleveldistributionpurposes.
iv)SpecialPurposeTransformers-E.g.PotentialTransformer(PT),CurrentTransformer(CT)
2.TheIdealTransformer
1.Definition–alosslessdevicewithaninputwindingandanoutputwinding.
2.Figuresbelowshowanidealtransformerandschematicsymbolsofatransformer.
3.ThetransformerhasNpturnsofwireonitsprimarysideandNsturnsofwireonitssecondarysides.Therelationshipbetweentheprimaryandsecondaryvoltageisasfollows:
whereaistheturnsratioofthetransformer.
4.Therelationshipbetweenprimaryandsecondarycurrentis:
Npip(t)=Nsis(t)
5.Notethatsincebothtypeofrelationsgivesaconstantratio,hencethetransformeronlychangesONLYthemagnitudevalueofcurrentandvoltage.Phaseanglesarenotaffected.
6.Thedotconventioninschematicdiagramfortransformershasthefollowingrelationship:
i)Iftheprimaryvoltageis+veatthedottedendofthewindingwrttheundottedend,thenthesecondaryvoltagewillbepositiveatthedottedendalso.Voltagepolaritiesarethesamewrtthedotsoneachsideofthecore.
ii)Iftheprimarycurrentofthetransformerflowsintothedottedendoftheprimarywinding,thesecondarycurrentwillflowoutofthedottedendofthesecondarywinding.
PowerinanIdealTransformer
1.Thepowersuppliedtothetransformerbytheprimarycircuit:
Pin=VpIpcosθp
Whereθp=theanglebetweentheprimaryvoltageandtheprimarycurrent.Thepowersuppliedbythetransformersecondarycircuittoitsloadsisgivenby:
Pout=VsIscosθs
Whereθs=theanglebetweenthesecondaryvoltageandthesecondarycurrent.
2.TheprimaryandsecondarywindingsofanidealtransformerhavetheSAMEpowerfactor–becausevoltageandcurrentanglesareunaffectedθp-θs=θ
3.Howdoespowergoingintotheprimarycircuitcomparetothepowercomingout?
Pout=VsIscosθ
Also,Vs=Vp/aandIs=aIp
So,
Pout=VpIpcosθ=Pin
ThesameideacanbeappliedforreactivepowerQandapparentpowerS.
Outputpower=Inputpower
ImpedanceTransformationthroughaTransformer
1.Theimpedanceofadeviceoranelementisdefinedastheratioofthephasorvoltageacrossittothephasorcurrentflowingthroughit:
2.
Definitionofimpedanceandimpedancescalingthroughatransformer:
3.
Hence,theimpedanceoftheloadis:
4.Theapparentimpedanceoftheprimarycircuitofthetransformeris:
5.SinceprimaryvoltagecanbeexpressedasVP=aVS,andprimarycurrentasIP=IS/a,thustheapparentimpedanceoftheprimaryis
ZL’=a2ZL
AnalysisofCircuitscontainingIdealTransformers
Theeasiestwayforcircuitanalysisthathasatransformerincorporatedisbysimplifyingthetransformerintoanequivalentcircuit.
Example2.1
Ageneratorratedat480V,60Hzisconnectedatransmissionlinewithanimpedanceof0.18+j0.24Ω.Attheendofthetransmissionlinethereisaloadof4+j3Ω.
(a)IfthepowersystemisexactlyasdescribedaboveinFigure(a),whatwillthevoltageattheloadbe?
Whatwillthetransmissionlinelossesbe?
(b)Supposea1:
10step-uptransformerisplacedatthegeneratorendofthetransmissionlineanda10:
1step-downtransformerisplacedattheloadendoftheline(Figure(b)).Whatwilltheloadvoltagebenow?
Whatwillthetransmissionlinelossesbenow?
3.TheoryofOperationofRealSingle-PhaseTransformers
Idealtransformersmayneverexistduetothefactthattherearelossesassociatedtotheoperationoftransformers.Hencethereisaneedtoactuallylookintolossesandcalculationofrealsinglephasetransformers.
Assumethatthereisatransformerwithitsprimarywindingsconnectedtoavaryingsinglephasevoltagesupply,andtheoutputisopencircuit.
Rightafterweactivatethepowersupply,fluxwillbegeneratedintheprimarycoils,baseduponFaraday’slaw,
whereλisthefluxlinkageinthecoilacrosswhichthevoltageisbeinginduced.Thefluxlinkageλisthesumofthefluxpassingthrougheachturninthecoiladdedoveralltheturnsofthecoil.
Thisrelationistrueprovidedontheassumptionthatthefluxinducedateachturnisatthesamemagnitudeanddirection.Butinreality,thefluxvalueateachturnmayvaryduetothepositionofthecoilitself,atcertainpositions,theremaybeahigherfluxlevelduetocombinationofotherfluxfromotherturnsoftheprimarywinding.
Hencethemostsuitableapproachistoactuallyaveragethefluxlevelas
HenceFaraday’slawmayberewrittenas:
ThevoltageratioacrossaTransformer
fthevoltagesourceisvp(t),howwillthetransformerreacttothisappliedvoltage?
BaseduponFaraday’sLaw,lookingattheprimarysideofthetransformer,wecandeterminetheaveragefluxlevelbaseduponthenumberofturns;where,
Thisrelationmeansthattheaveragefluxattheprimarywindingisproportionaltothevoltagelevelattheprimarysidedividedbythenumberofturnsattheprimarywinding.Thisgeneratedfluxwilltraveltothesecondarysidehenceinducingpotentialacrossthesecondaryterminal.
Foranidealtransformer,weassumethat100%offluxwouldtraveltothesecondarywindings.However,inreality,therearefluxwhichdoesnotreachthesecondarycoil,inthiscasethefluxleaksoutofthetransformercoreintothesurrounding.Thisleakistermedasfluxleakage.
Takingintoaccounttheleakageflux,thefluxthatreachesthesecondarysideistermedasmutualflux.
Lookingatthesecondaryside,therearesimilardivisionofflux;hencetheoverallpictureoffluxflowmaybeseenasbelow:
PrimarySide:
=totalaverageprimaryflux
=fluxcomponentlinkingbothrpimaryandsecondarycoils
=primaryleakageflux
Forthesecondaryside,similardivisionapplies.
Hence,lookingbackatFaraday’sLaw,
Orthisequationmayberewritteninto:
Thesamemaybewrittenforthesecondaryvoltage.
Theprimaryvoltageduetothemutualfluxisgivenby
Andthesamegoesforthesecondary(justreplace‘P’with‘S’)
Fromthesetworelationships(primaryandsecondaryvoltage),wehave
Therefore,
MagnetizationCurrentinaRealtransformer
Althoughtheoutputofthetransformerisopencircuit,therewillstillbecurrentflowintheprimarywindings.Thecurrentcomponentsmaybedividedinto2components:
1)Magnetizationcurrent,iM–currentrequiredtoproducefluxinthecore.
2)Core-losscurrent,ih+e–currentrequiredtocompensatehysteresisandeddycurrentlosses.
Weknowthattherelationbetweencurrentandfluxisproportionalsince,
Therefore,intheory,ifthefluxproduceincoreissinusoidal,thereforethecurrentshouldalsobeaperfectsinusoidal.Unfortunately,thisisnottruesincethetransformerwillreachtoastateofnearsaturationatthetopofthefluxcycle.Henceatthispoint,morecurrentisrequiredtoproduceacertainamountofflux.
Ifthevaluesofcurrentrequiredtoproduceagivenfluxarecomparedtothefluxinthecoreatdifferenttimes,itispossibletoconstructasketchofthemagnetizationcurrentinthewindingonthecore.Thisisshownbelow:
Hencewecansaythatcurrentinatransformerhasthefollowingcharacteristics:
1.Itisnotsinusoidalbutacombinationofhighfrequencyoscillationontopofthefundamentalfrequencyduetomagneticsaturation.
2.Thecurrentlagsthevoltageat90o
3.Atsaturation,thehighfrequencycomponentswillbeextremeassuchthatharmonicproblemswilloccur.
Lookingatthecore-losscurrent,itagainisdependentuponhysteresisandeddycurrentflow.SinceEddycurrentisdependentupontherateofchangeofflux,hencewecanalsosaythatthecore-losscurrentisgreat
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