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GORESET
ThePhasorDiagram
Inthelasttutorial,wesawthatsinusoidalwaveformsofthesamefrequencycanhaveaPhaseDifferencebetween
themselveswhichrepresentstheangulardifferenceofthetwosinusoidalwaveforms.Alsotheterms"lead"and"lag"as
wellas"in-phase"and"out-of-phase"wereusedtoindicatetherelationshipofonewaveformtotheotherwiththe
generalizedsinusoidalexpressiongivenas:A=Asin(ωt±Φ)representingthesinusoidinthetime-domain
form.Butwhenpresentedmathematicallyinthiswayitissometimesdifficulttovisualisethisangularorphase
differencebetweentwoormoresinusoidalwaveformssosinusoidscanalsoberepresentedgraphicallyinthespacial
orphasor-domainformbyaPhasorDiagram,andthisisachievedbyusingtherotatingvectormethod.
Basicallyarotatingvector,simplycalleda"Phasor"isascaledlinewhoselengthrepresentsanACquantitythathas
bothmagnitude("peakamplitude")anddirection("phase")whichis"frozen"atsomepointintime.Aphasorisavector
thathasanarrowheadatoneendwhichsignifiespartlythemaximumvalueofthevectorquantity(VorI)andpartlythe
endofthevectorthatrotates.
Generally,vectorsareassumedtopivotatoneendarounda
fixedzeropointknownasthe"pointoforigin"whilethe
arrowedendrepresentingthequantity,freelyrotatesin
ananti-clockwisedirectionatanangularvelocity,(ω)ofone
fullrevolutionforeverycycle.Thisanti-clockwiserotationof
thevectorisconsideredtobeapositiverotation.Likewise,a
clockwiserotationisconsideredtobeanegativerotation.
Althoughtheboththetermsvectorsandphasorsareusedto
describearotatinglinethatitselfhasbothmagnitudeand
direction,themaindifferencebetweenthetwoisthata
vectorsmagnitudeisthe"peakvalue"ofthesinusoidwhilea
phasorsmagnitudeisthe"rmsvalue"ofthesinusoid.Inboth
casesthephaseangleanddirectionremainsthesame.
Thephaseofanalternatingquantityatanyinstantintimecan
berepresentedbyaphasordiagram,sophasordiagrams
canbethoughtofas"functionsoftime".Acompletesinewavecanbeconstructedbyasinglevectorrotatingatan
angularvelocityofω=2π?,where?isthefrequencyofthewaveform.ThenaPhasorisaquantitythathasboth
"Magnitude"and"Direction".Generally,whenconstructingaphasordiagram,angularvelocityofasinewaveisalways
assumedtobe:ωinrad/s.Considerthephasordiagrambelow.
PhasorDiagramofaSinusoidalWaveform
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Asthesinglevectorrotatesinananti-clockwisedirection,itstipatpointAwillrotateonecompleterevolution
of360or2πrepresentingonecompletecycle.Ifthelengthofitsmovingtipistransferredatdifferentangularintervalsin
timetoagraphasshownabove,asinusoidalwaveformwouldbedrawnstartingattheleftwithzerotime.Eachposition
alongthehorizontalaxisindicatesthetimethathaselapsedsincezerotime,t=0.Whenthevectorishorizontalthetipof
thevectorrepresentstheanglesat0,180andat360.
Likewise,whenthetipofthevectorisverticalitrepresentsthepositivepeakvalue,(+Am)at90orπ/2andthenegative
peakvalue,(-Am)at270or3π/2.Thenthetimeaxisofthewaveformrepresentstheangleeitherindegreesor
radiansthroughwhichthephasorhasmoved.Sowecansaythataphasorrepresentascaledvoltageorcurrentvalueof
arotatingvectorwhichis"frozen"atsomepointintime,(t)andinourexampleabove,thisisatanangleof30.
Sometimeswhenweareanalysingalternatingwaveformswemayneedtoknowthepositionofthephasor,representing
thealternatingquantityatsomeparticularinstantintimeespeciallywhenwewanttocomparetwodifferentwaveforms
onthesameaxis.Forexample,voltageandcurrent.Wehaveassumedinthewaveformabovethatthewaveformstarts
attimet=0withacorrespondingphaseangleineitherdegreesorradians.Butififasecondwaveformstartstotheleft
ortotherightofthiszeropointorwewanttorepresentinphasornotationtherelationshipbetweenthetwowaveforms
thenwewillneedtotakeintoaccountthisphasedifference,Φofthewaveform.Considerthediagrambelowfromthe
previousPhaseDifferencetutorial.
PhaseDifferenceofaSinusoidalWaveform
Thegeneralisedmathematicalexpressiontodefinethesetwosinusoidalquantitieswillbewrittenas:
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Thecurrent,iislaggingthevoltage,vbyangleΦandinourexampleabovethisis30.Sothedifferencebetweenthetwo
phasorsrepresentingthetwosinusoidalquantitiesisangleΦandtheresultingphasordiagramwillbe.
PhasorDiagramofaSinusoidalWaveform
Thephasordiagramisdrawncorrespondingtotimezero(t=0)onthehorizontalaxis.Thelengthsofthephasorsare
proportionaltothevaluesofthevoltage,(V)andthecurrent,(I)attheinstantintimethatthephasordiagramisdrawn.
Thecurrentphasorlagsthevoltagephasorbytheangle,Φ,asthetwophasorsrotateinananticlockwisedirectionas
statedearlier,thereforetheangle,Φisalsomeasuredinthesameanticlockwisedirection.
Ifhowever,thewaveformsarefrozenattimet=30,the
correspondingphasordiagramwouldlookliketheoneshownonthe
right.Onceagainthecurrentphasorlagsbehindthevoltagephasor
asthetwowaveformsareofthesamefrequency.
However,asthecurrentwaveformisnowcrossingthehorizontalzero
axislineatthisinstantintimewecanusethecurrentphasorasour
newreferenceandcorrectlysaythatthevoltagephasoris"leading"
thecurrentphasorbyangle,Φ.Eitherway,onephasorisdesignated
asthereferencephasorandalltheotherphasorswillbeeither
leadingorlaggingwithrespecttothisreference.
PhasorAddition
Sometimesitisnecessarywhenstudyingsinusoidstoaddtogethertwoalternatingwaveforms,forexampleinanAC
seriescircuit,thatarenotin-phasewitheachother.Iftheyarein-phasethatis,thereisnophaseshiftthentheycanbe
addedtogetherinthesamewayasDCvaluestofindthealgebraicsumofthetwovectors.Forexample,twovoltagesin
phaseofsay50voltsand25voltsrespectively,willsumtogetherasone75voltsvoltage.Ifhowever,theyarenotin-
phasethatis,theydonothaveidenticaldirectionsorstartingpointthenthephaseanglebetweenthemneedstobe
takenintoaccountsotheyareaddedtogetherusingphasordiagramstodeterminetheirResultantPhasororVector
Sumbyusingtheparallelogramlaw.
ConsidertwoACvoltages,Vhavingapeakvoltageof20volts,andVhavingapeakvoltageof30volts
whereVleadsVby60.Thetotalvoltage,Vofthetwovoltagescanbefoundbyfirstlydrawingaphasordiagram
representingthetwovectorsandthenconstructingaparallelograminwhichtwoofthesidesarethe
voltages,VandVasshownbelow.
PhasorAdditionoftwoPhasors
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Bydrawingoutthetwophasorstoscaleontographpaper,theirphasorsumV+Vcanbeeasilyfoundbymeasuring
thelengthofthediagonalline,knownasthe"resultantr-vector",fromthezeropointtotheintersectionoftheconstruction
lines0-A.Thedownsideofthisgraphicalmethodisthatitistimeconsumingwhendrawingthephasorstoscale.Also,
whilethisgraphicalmethodgivesananswerwhichisaccurateenoughformostpurposes,itmayproduceanerrorifnot
drawnaccuratelyorcorrectlytoscale.Thenonewaytoensurethatthecorrectanswerisalwaysobtainedisbyan
analyticalmethod.
Mathematicallywecanaddthetwovoltagestogetherbyfirstlyfindingtheir"vertical"and"horizontal"directions,andfrom
thiswecanthencalculateboththe"vertical"and"horizontal"componentsfortheresultant"rvector",V.Thisanalytical
methodwhichusesthecosineandsineruletofindthisresultantvalueiscommonlycalledtheRectangularForm.
Intherectangularform,thephasorisdividedupintoarealpart,xandanimaginarypart,yformingthegeneralised
expressionZ=x±jy.(wewilldiscussthisinmoredetailinthenexttutorial).Thisthengivesusamathematical
expressionthatrepresentsboththemagnitudeandthephaseofthesinusoidalvoltageas:
Sotheadditionoftwovectors,AandBusingthepreviousgeneralisedexpressionisasfollows:
PhasorAdditionusingRectangularForm
Voltage,Vof30voltspointsinthereferencedirectionalongthehorizontalzeroaxis,thenithasahorizontalcomponent
butnoverticalcomponentasfollows.
Horizontalcomponent=30cos0=30volts
Verticalcomponent=30sin0=0volts
ThisthengivesustherectangularexpressionforvoltageVof:30+j0
Voltage,Vof20voltsleadsvoltage,Vby60,thenithasbothhorizontalandverticalcomponentsasfollows.
Horizontalcomponent=20cos60=20x0.5=10volts
Verticalcomponent=20sin60=20x0.866=17.32volts
ThisthengivesustherectangularexpressionforvoltageVof:10+j17.32
Theresultantvoltage,Visfoundbyaddingtogetherthehorizontalandverticalcomponentsasfollows.
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V=sumofrealpartsofVandV=30+10=40volts
V=sumofimaginarypartsofVandV=0+17.32=17.32volts
Nowthatboththerealandimaginaryvalueshavebeenfoundthemagnitudeofvoltage,Visdeterminedbysimply
usingPythagoras''sTheoremfora90triangleasfollows.
Thentheresultingphasordiagramwillbe:
ResultantValueofV
PhasorSubtraction
Phasorsubtractionisverysimilartotheaboverectangularmethodofaddition,exceptthistimethevectordifferenceisthe
otherdiagonaloftheparallelogrambetweenthetwovoltagesofVandVasshown.
VectorSubtractionoftwoPhasors
Thistimeinsteadof"adding"togetherboththehorizontalandverticalcomponentswetakethemaway,subtraction.
Horizontal12
Vertical12
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The3-PhasePhasorDiagram
Previouslywehaveonlylookedatsingle-phaseACwaveformswhereasinglemultiturncoilrotateswithinamagnetic
field.Butifthreeidenticalcoilseachwiththesamenumberofcoilturnsareplacedatanelectricalangleof120toeach
otheronthesamerotorshaft,athree-phasevoltagesupplywouldbegenerated.Abalancedthree-phasevoltagesupply
consistsofthreeindividualsinusoidalvoltagesthatareallequalinmagnitudeandfrequencybutareout-of-phasewith
eachotherbyexactly120electricaldegrees.
StandardpracticeistocolourcodethethreephasesasRed,YellowandBluetoidentifyeachindividualphasewiththe
redphaseasthereferencephase.ThenormalsequenceofrotationforathreephasesupplyisRedfollowed
byYellowfollowedbyBlue,(R,Y,B).
Aswiththesingle-phasephasorsabove,thephasorsrepresentingathree-phasesystemalsorotateinananti-
clockwisedirectionaroundacentralpointasindicatedbythearrowmarkedωinrad/s.Thephasorsforathree-phase
balancedstarordeltaconnectedsystemareshownbelow.
Three-phasePhasorDiagram
Thephasevoltagesareallequalinmagnitudebutonlydifferintheirphaseangle.Thethreewindingsofthecoilsare
connectedtogetheratpoints,a,bandctoproduceacommonneutralconnectionforthethreeindividualphases.
Theniftheredphaseistakenasthereferencephaseeachindividualphasevoltagecanbedefinedwithrespecttothe
commonneutralas.
Three-phaseVoltageEquations
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Iftheredphasevoltage,VistakenasthereferencevoltageasstatedearlierthenthephasesequencewillbeR–Y–
BsothevoltageintheyellowphaselagsVby120,andthevoltageinthebluephaselagsValsoby120.Butwe
canalsosaythebluephasevoltage,Vleadstheredphasevoltage,Vby120.
Onefinalpointaboutathree-phasesystem.Asthethreeindividualsinusoidalvoltageshaveafixedrelationshipbetween
eachotherof120theyaresaidtobe"balanced"therefore,inasetofbalancedthreephasevoltagestheirphasorsum
willalwaysbezeroas:V+V+V=0
PhasorDiagramSummary
ThentosummarizethistutorialaboutPhasorDiagrams.
Intheirsimplestterms,phasordiagramsareaprojectionofarotatingvectorontoahorizontal
axiswhichrepresentstheinstantaneousvalue.Asaphasordiagramcanbedrawntorepresent
anyinstantoftimeandthereforeanyangle,thereferencephasorofanalternatingquantityis
alwaysdrawnalongthepositivex-axisdirection.
Vectors,PhasorsandPhasorDiagramsONLYapplytosinusoidalACwaveforms.
APhasorDiagramcanbeusedtorepresenttwoormorestationarysinusoidalquantitiesat
anyinstantintime.
Generallythereferencephasorisdrawnalongthehorizontalaxisandatthatinstantintime
theotherphasorsaredrawn.Allphasorsaredrawnreferencedtothehorizontalzeroaxis.
Phasordiagramscanbedrawntorepresentmorethantwosinusoids.Theycanbeeither
voltage,currentorsomeotheralternatingquantitybutthefrequencyofallofthemmustbe
thesame.
Allphasorsaredrawnrotatinginananticlockwisedirection.Allthephasorsaheadofthereferencephasorare
saidtobe"leading"whileallthephasorsbehindthereferencephasoraresaidtobe"lagging".
Generally,thelengthofaphasorrepresentstheR.M.S.valueofthesinusoidalquantityratherthanitsmaximum
value.
Sinusoidsofdifferentfrequenciescannotberepresentedonthesamephasordiagramduetothedifferentspeed
ofthevectors.Atanyinstantintimethephaseanglebetweenthemwillbedifferent.
Twoormorevectorscanbeaddedorsubtractedtogetherandbecomeasinglevector,calledaResultantVector.
Thehorizontalsideofavectorisequaltotherealorxvector.Theverticalsideofavectorisequaltotheimaginary
oryvector.Thehypotenuseoftheresultantrightangledtriangleisequivalenttothervector.
Inathree-phasebalancedsystemeachindividualphasorisdisplacedby120.
InthenexttutorialaboutACTheorywewilllookatrepresentingsinusoidalwaveformsasComplexNumbersin
Rectangularform,PolarformandExponentialform.
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RNoYNo
BNRNo
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