E283C2.docx

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