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VoltageinducedinasinglephaseTherotorinthegeneratorillustratedaboveproducesafluxwhichpassesradiallyfromtherotorsurfaceintothestator.Thecircumferentialvariationofthisradialfluxdensityaroundtherotorcanbedescribedusing

whereistheanglearoundthesurface.Iftherotorofthemachinerotatesatcon

stantspeedtthenthefluxdensityseenataposition0onthestatorwillbegivenby

Usingthegeneratorlaw

thevoltageinducedineachsideofthecoilcanbeobtained.Consideringtheconductorat0=0,thevelocityoftheconductorwithrespecttothefluxisdown(thefluxdensityismovinguprelativetotheconductor),thefluxdensityislefttoright,thecrossproductisparallelwiththeconductor.Ifwetaketheconductortobeorientedwithpositvedirectionoutofthepagethentheinducedvol

tageisgivenby

Relatinglinearvelocitytoangularvelocity

theinducedvoltagebecomes

Forthereturnsideoftheconductorat0=,

Therfore,thetotalvoltageinducedintheloopformedbythetwoconductorsis

Now,ifthereareNTloopsformingacoil,

Notingthattheareaenclosedbythecoilisgivenby2rlthevoltagecanalsobewrittenas

Amulti-coilmachineInthesimpleexamplewithonlyonecoil,thereisasignificantwasteofspace

inthemachine.Itispossibletoaddadditionalcoilstothestatorandmakefulluseoftheavailablespace.IfthereareNcoilscoils,allcapableofcarryingequalcurrent,Ithemaximumavailablepowerisgivenby

.Unfortunately,toaccessthispower,eachcoilwouldneedtobeconnectedtoaloadbyitsowntransmissionline.Thisisunacceptablyexpensiveforatransmissionanddistributionsystem.Asanalternative,allthecoilscouldbeconnectedtogtetherinseriestoformasinglephasevoltage.Ifallthecoilsareconnectedinseriesthetotalavailablevoltageisgivenbythephasorsummationoftheindividalcoilvoltages:

andtheavailablepoweris

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InthecasewhereNcoilsislarge,theseriesconnectedpoweris2/timestheavailblemaxiumumpower(about64%).

AThree-phasemachineClearlysomecompromisebetweengeneratorutilizationandpowerdeliverycostmustbereached.Thiscompromiseisthethree-phasepowersystem.Inathree-phase

generator,therearestillmanycoilsinthemachine,butthecoilsarenowconnectedinthreedistinctgroups,calledphases.Notingthatthesumofallthecoilvoltagesmustequalthesumofthephases,thermsphasevoltagecanbeobtained:

Now,ifeachphaseisconnectedtoaloadbyitsowntransmissionline,thetotalavailablepowerisgivenby

andifthenumberofcoilsislarge:

General3-phasevoltagesIngeneralinthecoursewewillthinkofathree-phasewindingasbeingmadeupofonlythreecoils,separatedby120degrees.However,itisimportanttorealisethatarealmachinehasmanycoilswhicharegroupedintothreephases.Asshownabove,thefundamentalvoltageinducedinaphasecomprisedofanumberofdistributedcoilswillbeslightlylowerthanthatproducedbyasinglecoilwiththesametotalnumberofturns.Distributedwindingscanalsoreducethemagnitudeofhigherharmonicswhichmaybeinducedinacoil.Detailedinfomrationaboutdistributedwindingsisbeyondthescopeofthecourse,moreinformationcanbefoundintheappendicesofthetextbook.