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GORESET
GenerationofaSinusoidalWaveform
InourtutorialsaboutElectromagnetism,wesawhowanelectriccurrentflowingthroughaconductorcanbeusedto
generateamagneticfieldarounditself,andalsoifasinglewireconductorismovedorrotatedwithinastationary
magneticfield,an"EMF",(Electro-MotiveForce)willbeinducedwithintheconductorduetothismovement.Fromthis
tutorialwelearntthatarelationshipexistsbetweenElectricityandMagnetismgivingus,asMichaelFaradaydiscovered
theeffectof"ElectromagneticInduction"anditisthisbasicprincipalthatisusedtogenerateaSinusoidalWaveform.
IntheElectromagneticInduction,tutorialwesaidthatwhenasinglewire
conductormovesthroughapermanentmagneticfieldtherebycuttingitslinesofflux,
anEMFisinducedinit.However,iftheconductormovesinparallelwiththemagnetic
fieldinthecaseofpointsAandB,nolinesoffluxarecutandnoEMFisinducedinto
theconductor,butiftheconductormovesatrightanglestothemagneticfieldasin
thecaseofpointsCandD,themaximumamountofmagneticfluxiscutproducing
themaximumamountofinducedEMF.
Also,astheconductorcutsthemagneticfieldatdifferentanglesbetween
pointsAandC,0and90theamountofinducedEMFwillliesomewherebetween
thiszeroandmaximumvalue.Thentheamountofemfinducedwithinaconductor
dependsontheanglebetweentheconductorandthemagneticfluxaswellasthe
strengthofthemagneticfield.
AnACgeneratorusestheprincipalofFaraday''selectromagneticinductiontoconvert
amechanicalenergysuchasrotation,intoelectricalenergy,aSinusoidalWaveform.Asimplegeneratorconsistsofa
pairofpermanentmagnetsproducingafixedmagneticfieldbetweenanorthandasouthpole.Insidethismagneticfield
isasinglerectangularloopofwirethatcanberotatedaroundafixedaxisallowingittocutthemagneticfluxatvarious
anglesasshownbelow.
BasicSingleCoilACGenerator
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Asthecoilrotatesanticlockwisearoundthecentralaxiswhichisperpendiculartothemagneticfield,thewireloopcuts
thelinesofmagneticforcesetupbetweenthenorthandsouthpolesatdifferentanglesasthelooprotates.Theamount
ofinducedEMFintheloopatanyinstantoftimeisproportionaltotheangleofrotationofthewireloop.Asthiswireloop
rotates,electronsinthewireflowinonedirectionaroundtheloop.Nowwhenthewireloophasrotatedpastthe
180pointandmovesacrossthemagneticlinesofforceintheoppositedirection,theelectronsinthewireloopchange
andflowintheoppositedirection.Thenthedirectionoftheelectronmovementdeterminesthepolarityoftheinduced
voltage.
Sowecanseethatwhenthelooporcoilphysicallyrotatesonecompleterevolution,or360,onefullsinusoidal
waveformisproducedwithonecycleofthewaveformbeingproducedforeachrevolutionofthecoil.Asthecoilrotates
withinthemagneticfield,theelectricalconnectionsaremadetothecoilbymeansofcarbonbrushesandslip-rings
whichareusedtotransfertheelectricalcurrentinducedinthecoil.
TheamountofEMFinducedintoacoilcuttingthemagneticlinesofforceisdeterminedbythefollowingthreefactors.
?Speed–thespeedatwhichthecoilrotatesinsidethemagneticfield.
?Strength–thestrengthofthemagneticfield.
?Length–thelengthofthecoilorconductorpassingthroughthemagneticfield.
Weknowthatthefrequencyofasupplyisthenumberoftimesacycleappearsinonesecondandthatfrequencyis
measuredinHertz.Asonecycleofinducedemfisproducedeachfullrevolutionofthecoilthroughamagneticfield
comprisingofanorthandsouthpoleasshownabove,ifthecoilrotatesataconstantspeedaconstantnumberofcycles
willbeproducedpersecondgivingaconstantfrequency.Sobyincreasingthespeedofrotationofthecoilthefrequency
willalsobeincreased.Therefore,frequencyisproportionaltothespeedofrotation,(?∝Ν)whereΝ=r.p.m.
Also,oursimplesinglecoilgeneratoraboveonlyhastwopoles,onenorthandonesouthpole,givingjustonepairof
poles.Ifweaddmoremagneticpolestothegeneratorabovesothatitnowhasfourpolesintotal,twonorthandtwo
south,thenforeachrevolutionofthecoiltwocycleswillbeproducedforthesamerotationalspeed.Therefore,frequency
isproportionaltothenumberofpairsofmagneticpoles,(?∝P)ofthegeneratorwhereP=isthenumberof"pairsof
poles".
ThenfromthesetwofactswecansaythatthefrequencyoutputfromanACgeneratoris:
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Where:Νisthespeedofrotationinr.p.m.Pisthenumberof"pairsofpoles"and60convertsitintoseconds.
InstantaneousVoltage
TheEMFinducedinthecoilatanyinstantoftimedependsupontherateorspeedatwhichthecoilcutsthelinesof
magneticfluxbetweenthepolesandthisisdependantupontheangleofrotation,Theta(θ)ofthegeneratingdevice.
BecauseanACwaveformisconstantlychangingitsvalueoramplitude,thewaveformatanyinstantintimewillhavea
differentvaluefromitsnextinstantintime.Forexample,thevalueat1mswillbedifferenttothevalueat1.2msandsoon.
ThesevaluesareknowngenerallyastheInstantaneousValues,orVThentheinstantaneousvalueofthewaveform
andalsoitsdirectionwillvaryaccordingtothepositionofthecoilwithinthemagneticfieldasshownbelow.
DisplacementofaCoilwithinaMagneticField
Theinstantaneousvaluesofasinusoidalwaveformisgivenasthe"Instantaneousvalue=Maximumvaluexsinθ"and
thisisgeneralizedbytheformula.
Where,Visthemaximumvoltageinducedinthecoilandθ=ωt,istheangleofcoilrotation.
Ifweknowthemaximumorpeakvalueofthewaveform,byusingtheformulaabovetheinstantaneousvaluesatvarious
pointsalongthewaveformcanbecalculated.Byplottingthesevaluesoutontographpaper,asinusoidalwaveform
shapecanbeconstructed.Inordertokeepthingssimplewewillplottheinstantaneousvaluesforthesinusoidal
waveformatevery45andassumeamaximumvalueof100V.Plottingtheinstantaneousvaluesatshorterintervals,for
exampleatevery30wouldresultinamoreaccuratewaveformconstruction.
SinusoidalWaveformConstruction
CoilAngle(θ)04590135180225270315360
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max
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e=Vmax.sinθ070.7110070.710-70.71-100-70.71-0
Thepointsonthesinusoidalwaveformareobtainedbyprojectingacrossfromthevariouspositionsofrotationbetween
0and360totheordinateofthewaveformthatcorrespondstotheangle,θandwhenthewirelooporcoilrotatesone
completerevolution,or360,onefullwaveformisproduced.Fromtheplotofthesinusoidalwaveformwecanseethat
whenθisequalto0,180or360,thegeneratedEMFiszeroasthecoilcutstheminimumamountoflinesofflux.But
whenθisequalto90and270thegeneratedEMFisatitsmaximumvalueasthemaximumamountoffluxiscut.The
sinusoidalwaveformhasapositivepeakat90andanegativepeakat270.PositionsB,D,FandHgenerateavalue
ofEMFcorrespondingtotheformulae=Vmax.sinθ.
Thenthewaveformshapeproducedbyoursimplesingleloopgeneratoriscommonlyreferred
toasaSineWaveasitissaidtobesinusoidalinitsshape.Thistypeofwaveformiscalleda
sinewavebecauseitisbasedonthetrigonometricsinefunctionusedinmathematics,
(x(t)=Amax.sinθ).
Whendealingwithsinewavesinthetimedomainandespeciallycurrentrelatedsinewavesthe
unitofmeasurementusedalongthehorizontalaxisofthewaveformcanbeeithertime,
degreesorradians.InelectricalengineeringitismorecommontousetheRadianasthe
angularmeasurementoftheanglealongthehorizontalaxisratherthandegrees.For
example,ω=100rad/s,or500rad/s.
Radians
TheRadian,(rad)isdefinedmathematicallyasaquadrantofacirclewherethedistance
subtendedonthecircumferenceequalstheradius(r)ofthecircle.Sincethecircumferenceofa
circleisequalto2πxradius,theremustbe2πradiansarounda360circle,so1radian=
360/2π=57.3.Inelectricalengineeringtheuseofradiansisverycommonsoitisimportanttorememberthe
followingformula.
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Usingradiansastheunitofmeasurementforasinusoidalwaveformwouldgive2πradiansforonefullcycleof360.
Thenhalfasinusoidalwaveformmustbeequalto1πradiansorjustπ(pi).Thenknowingthatpi,πisequal
to3.142or22÷7,therelationshipbetweendegreesandradiansforasinusoidalwaveformisgivenas.
RelationshipbetweenDegreesandRadians
Applyingthesetwoequationstovariouspointsalongthewaveformgivesus.
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Theconversionbetweendegreesandradiansforthemorecommonequivalentsusedinsinusoidalanalysisaregiven
inthefollowingtable.
DegreesRadiansDegreesRadiansDegreesRadians
001353π42703π2
30π61505π63005π3
45π4180π3157π4
60π32107π633011π6
90π22255π43602π
1202π32404π3
Thevelocityatwhichthegeneratorrotatesarounditscentralaxisdeterminesthefrequencyofthesinusoidalwaveform.
Asthefrequencyofthewaveformisgivenas?Hzorcyclespersecond,thewaveformhasangularfrequency,ω,(Greek
letteromega),inradianspersecond.Thentheangularvelocityofasinusoidalwaveformisgivenas.
AngularVelocityofaSinusoidalWaveform
andintheUnitedKingdom,theangularvelocityorfrequencyofthemainssupplyisgivenas:
intheUSAastheirmainssupplyfrequencyis60Hzitistherefore:377rad/s
Sowenowknowthatthevelocityatwhichthegeneratorrotatesarounditscentralaxisdeterminesthefrequencyofthe
sinusoidalwaveformandwhichcanalsobecalleditsangularvelocity,ω.Butweshouldbynowalsoknowthatthetime
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requiredtocompleteonerevolutionisequaltotheperiodictime,(T)ofthesinusoidalwaveform.Asfrequencyis
inverselyproportionaltoitstimeperiod,?=1/Twecanthereforesubstitutethefrequencyquantityintheaboveequation
fortheequivalentperiodictimequantityandsubstitutinggivesus.
Theaboveequationstatesthatforasmallerperiodictimeofthesinusoidalwaveform,thegreatermustbetheangular
velocityofthewaveform.Likewiseintheequationaboveforthefrequencyquantity,thehigherthefrequencythehigherthe
angularvelocity.
ExampleNo1
Asinusoidalwaveformisdefinedas:V=169.8sin(377t)volts.CalculatetheRMSvoltageofthewaveform,its
frequencyandtheinstantaneousvalueofthevoltageafteratimeof6mS.
Weknowfromabovethatthegeneralexpressiongivenforasinusoidalwaveformis:
ThencomparingthistoourgivenexpressionforasinusoidalwaveformaboveofV=169.8sin(377t)willgiveusthe
peakvoltagevalueof169.8voltsforthewaveform.
ThewaveformsRMSvoltageiscalculatedas:
Theangularvelocity(ω)isgivenas377rad/s.Then2π?=377.Sothefrequencyofthewaveformiscalculatedas:
TheinstantaneousvoltageVvalueafteratimeof6mSisgivenas:
Notethatthephaseangleattimet=6mSisgiveninradians.Wecouldquiteeasilyconvertthistodegreesifwewanted
toandusethisvalueinsteadtocalculatetheinstantaneousvoltagevalue.Theangleindegreeswillthereforebegiven
as:
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SinusoidalWaveform
ThenthegeneralisedformatusedforanalysingandcalculatingthevariousvaluesofaSinusoidalWaveformisas
follows:
ASinusoidalWaveform
InthenexttutorialaboutPhaseDifferencewewilllookattherelationshipbetweentwosinusoidalwaveformsthatare
ofthesamefrequencybutpassthroughthehorizontalzeroaxisatdifferenttimeintervals.
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