E283C4Word文档格式.docx
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E283C4Word文档格式.docx
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-Theinducedvoltageinathree-phasesetofcoils
-TheRMSvoltageinaThree-PhaseStator
5.InducedTorqueinanACMachines
Thefigurebelowshowsasimplerotatingloopinauniformmagneticfield.(a)isthefrontviewand(b)istheviewofthecoil.Therotatingpartiscalledtherotor,andthestationarypartiscalledthestator.
Thiscaseinnotrepresentativeofrealacmachines(fluxinrealacmachinesisnotconstantineithermagnitudeordirection).However,thefactorsthatcontrolthevoltageandtorqueonthelooparethesameasthefactorsthatcontrolthevoltageandtorqueinrealacmachines.
Thevoltageinducedinasimplerotatingloop
Iftherotor(loop)isrotated,avoltagewillbeinducedinthewireloop.Todeterminethemagnitudeandshape,examinethephasorsbelow:
Todeterminethetotalvoltageinducedetotontheloop,examineeachsegmentoftheloopseparatelyandsumalltheresultingvoltages.Thevoltageoneachsegmentisgivenbyequation
eind=(vxB).l
(rememberthattheseideasallrevertbacktothelinearDCmachineconceptsinChapter1).
1.Segmentab
Thevelocityofthewireistangentialtothepathofrotation,whilethemagneticfieldBpointstotheright.ThequantityvxBpointsintothepage,whichisthesamedirectionassegmentab.Thus,theinducedvoltageonthissegmentis:
eba=(vxB).l
=vBlsinθabintothepage
2.Segmentbc
Inthefirsthalfofthissegment,thequantityvxBpointsintothepage,andinthesecondhalfofthissegment,thequantityvxBpointsoutofthepage.Sincethelengthlisintheplaneofthepage,vxBisperpendiculartolforbothportionsofthesegment.Thus,
ecb=0
3.Segmentcd
Thevelocityofthewireistangentialtothepathofrotation,whileBpointstotheright.ThequantityvxBpointsintothepage,whichisthesamedirectionassegmentcd.Thus,
ecd=(vxB).l
=vBlsinθcdoutofthepage
4.Segmentda
sameassegmentbc,vxBisperpendiculartol.Thus,
eda=0
Totalinducedvoltageontheloopeind=eba+ecb+edc+ead
=vBlsinθab+vBlsinθcd
=2vBLsinθ
sinceθab=180º
-θcdandsinθ=sin(180º
-θ)
Alternativewaytoexpresseind:
Iftheloopisrotatingataconstantangularvelocityω,thentheangleθoftheloopwillincreaselinearlywithtime.
θ=ωt
also,thetangentialvelocityvoftheedgesoftheloopis:
v=rω
whereristheradiusfromaxisofrotationouttotheedgeoftheloopandωistheangularvelocityoftheloop.Hence,
eind=2rωBlsinωt
sincearea,A=2rl,
eind=ABωsinωt
Finally,sincemaximumfluxthroughtheloopoccurswhentheloopisperpendiculartothemagneticfluxdensitylines,so
Thus,
Fromherewemayconcludethattheinducedvoltageisdependentupon:
∙Fluxlevel(theBcomponent)
∙SpeedofRotation(thevcomponent)
∙MachineConstants(thelcomponentandmachinematerials)
TheTorqueInducedinaCurrent-CarryingLoop
Assumethattherotorloopisatsomearbitraryangleθwrtthemagneticfield,andthatcurrentisflowingintheloop.
Todeterminethemagnitudeanddirectionofthetorque,examinethephasorsbelow:
Theforceoneachsegmentoftheloopisgivenby:
F=i(lxB)
Torqueonthatsegment,
Thedirectionofthecurrentisintothepage,whilethemagneticfieldBpointstotheright.(lxB)pointsdown.Thus,
=ilBdown
Resultingtorque,
clockwise
Thedirectionofthecurrentisintheplaneofthepage,whilethemagneticfieldBpointstotheright.(lxB)pointsintothepage.Thus,
=ilBintothepage
Resultingtorqueiszero,sincevectorrandlareparallelandtheangleθbcis0.
=0
Thedirectionofthecurrentisoutofthepage,whilethemagneticfieldBpointstotheright.(lxB)pointsup.Thus,
=ilBup
Thedirectionofthecurrentisintheplaneofthepage,whilethemagneticfieldBpointstotheright.(lxB)pointsoutofthepage.Thus,
=ilBoutofthepage
Resultingtorqueiszero,sincevectorrandlareparallelandtheangleθdais0.
Thetotalinducedtorqueontheloop:
Note:
thetorqueismaximumwhentheplaneoftheloopisparalleltothemagneticfield,andthetorqueiszerowhentheplaneoftheloopisperpendiculartothemagneticfield.
Analternativewaytoexpressthetorqueequationcanbedonewhichclearlyrelatesthebehaviourofthesinglelooptothebehaviouroflargeracmachines.Examinethephasorsbelow:
Ifthecurrentintheloopisasshown,thatcurrentwillgenerateamagneticfluxdensityBloopwiththedirectionshown.ThemagnitudeofBloopis:
WhereGisafactorthatdependsonthegeometryoftheloop.
TheareaoftheloopAis2rlandsubstitutingthesetwoequationsintothetorqueequationearlieryields:
Wherek=AG/µ
isafactordependingontheconstructionofthemachine,BSisusedforthestatormagneticfieldtodistinguishitfromthemagneticfieldgeneratedbytherotor,andθistheanglebetweenBloopandBS.
Thus,
Fromhere,wemayconcludethattorqueisdependentupon:
∙Strengthofrotormagneticfield
∙Strengthofstatormagneticfield
∙Anglebetweenthe2fields
∙Machineconstants
Beforewehavelookedathowiftwomagneticfieldsarepresentinamachine,thenatorquewillbecreatedwhichwilltendtolineupthetwomagneticfields.Ifonemagneticfieldisproducedbythestatorofanacmachineandtheotherbytherotor,thenatorquewillbeinducedintherotorwhichwillcausetherotortoturnandalignitselfwiththestatormagneticfield.
Ifthereweresomewaytomakethestatormagneticfieldrotate,thentheinducedtorqueintherotorwouldcauseitto‘chase’thestatormagneticfield.
Howdowemakethestatormagneticfieldtorotate?
Fundamentalprinciple–a3-phasesetofcurrents,eachofequalmagnitudeanddifferinginphaseby120º
flowsina3-phasewinding,thenitwillproducearotatingmagneticfieldofconstantmagnitude.
Therotatingmagneticfieldconceptisillustratedbelow–emptystatorcontaining3coils120º
apart.Itisa2-polewinding(onenorthandonesouth).
(b)ThemagnetizingintensityvectorHaa’(t)producedbyacurrentflowingincoilaa’.
(a)Asimplethreephasestator.Currentsinthisstatorareassumedpositiveiftheyflowintotheunprimedendandouttheprimedendofthecoils.TheHproducedbyeachcoilarealsoshown.
Let’sapplyasetofcurrentstothestatoraboveandseewhathappensatspecificinstantsoftime.Assumecurrentsinthe3coilsare:
Thecurrentincoilaa’flowsintotheaendofthecoilandoutthea’endofthecoil.Itproducesthemagneticfieldintensity:
Thefluxdensitiesequationsare:
WhereBM=µ
HM.
Attime
Thetotalmagneticfieldfromallthreecoilsaddedtogetherwillbe
Bnet=Baa’+Bbb’+Bcc’
Theresultingmagneticfluxisasshownbelow:
ProofofRotatingMagneticFieldConcept
Atanytimet,themagneticfieldwillhavethesamemagnitude1.5BManditwillcontinuetorotateatangularvelocityω.
Proof:
ReferagaintothestatorinFigure4.1.xdirectionistotherightandydirectionisupward.
Assumethatwerepresentthedirectionofthemagneticfielddensitiesintheformof:
Tofindthetotalmagneticfluxdensityinthestator,simplyaddvectoriallythethreecomponentmagneticfieldsanddeterminetheirsum.
Weknowthat:
Wemayconvertthetotalfluxdensityintounitvectorformstogive:
Noticethatthemagnitudeofthefieldisaconstant1.5BMandtheanglechangescontinuallyinacounterclockwisedirectionatangularvelocityω.Also,atωt=0°
Bnet=1.5BM-90°
andatωt=90°
Bnet=1.5BM0°
.
TheRelationshipbetweenElectricalFrequencyandtheSpeedofMagneticFieldRotation
Thefigureaboveshowsthattherotatingmagneticfieldinthisstatorcanberepresentedasanorthpole(thefluxleavesthestator)andasouthpole(fluxentersthestator).
Thesemagneticpolescompleteonemechanicalrotationaroundthestatorsurfaceforeachelectricalcycleoftheappliedcurrent.Themechanicalspeedofrotationofthemagneticfieldinrevolutionspersecondisequaltoelectricfrequencyinhertz:
fe(hertz)=fm(revolutionspersecond)twopoles
ωe(radianspersecond)=ωm(radianspersecond)twopoles
Thewindingsonthe2-polestatoraboveoccurintheordera–c’–b–a’–c–b’
Ifweweretodoubletheamountofwindings,hencethesequenceofwindingswillbeasfollows:
a1–c1’–b1–a1’–c1–b1’–a2–c2’–b2–a2’–c2–b2’
Forathree-phasesetofcurrents,thisstatorwillhave2northpolesand2southpolesproducedinthestatorwinding,(referfigure(b)below):
(a)Asimplefour-polestatorwinding.(b)Theresultingstatormagnetic
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