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InductiveReactance
SofarwehavelookedatinductorsconnectedtoDCnetworksandhopefullybynowweknowthatwhenaDCvoltageis
appliedacrossaninductor,thegrowthofthecurrentthroughitisnotinstantbutisdeterminedbytheinductorsself-
inducedorbackemfvalue.Alsothatthecurrentcontinuestoriseuntilitreachesitsmaximumsteadystateconditionafter
fivetimeconstants.Thenthemaximumcurrentflowingthroughacoilislimitedonlybytheresistiveelementofthecoils
windingsinOhms,andasweknowfromOhmslawisdeterminedbytheratioofvoltageovercurrent,V/R.
WhenanalternatingorACvoltageisappliedacrossaninductortheflowofcurrentthroughitbehavesverydifferentlyto
thatofanappliedDCvoltage.Theeffectofasinusoidalsupplyproducesaphasedifferencebetweenthevoltageandthe
currentwaveforms.NowinanACcircuit,theoppositiontocurrentflowthroughthecoilswindingsnotonlydependsupon
theinductanceofthecoilbutalsothefrequencyoftheACwaveform.
TheoppositiontocurrentflowingthroughacoilinanACcircuitisdeterminedbytheACresistance,morecommonly
knownasImpedance(Z),ofthecircuit.ButresistanceisalwaysassociatedwithDCcircuitssotodistinguishDC
resistancefromACresistancethetermReactanceisgenerallyused.
Justlikeresistance,thevalueofreactanceisalsomeasuredinOhm''sbutisgiventhesymbolX,(uppercaseletter"X"),
todistinguishitfromapurelyresistivevalue.Asthecomponentweareinterestedinisaninductor,thereactanceofan
inductoristhereforecalled"InductiveReactance".Inotherwords,aninductorselectricalresistancewhenusedinanAC
circuitiscalledInductiveReactance.
InductiveReactancewhichisgiventhesymbolX,isthepropertyinanACcircuitwhichopposesthechangeinthe
current.InourtutorialsaboutCapacitorsinACCircuits,wesawthatinapurelycapacitivecircuit,the
currentI"LEADS"thevoltageby90.InapurelyinductiveACcircuittheexactoppositeistrue,thecurrentI"LAGS"the
appliedvoltageby90,or(π/2rads).
ACInductorCircuit
Inthepurelyinductivecircuitabove,theinductorisconnecteddirectlyacrosstheACsupplyvoltage.Asthesupplyvoltage
increasesanddecreaseswiththefrequency,theself-inducedbackemfalsoincreasesanddecreasesinthecoilwith
respecttothischange.Weknowthatthisself-inducedemfisdirectlyproportionaltotherateofchangeofthecurrent
throughthecoilandisatitsgreatestasthesupplyvoltagecrossesoverfromitspositivehalfcycletoitsnegativehalf
cycleorviceversaatpoints,0and180alongthesinewave.
Consequently,theminimumrateofchangeofthevoltageoccurswhentheACsinewavecrossesoveratitsmaximumor
minimumpeakvoltagelevel.Atthesepositionsinthecyclethemaximumorminimumcurrentsareflowingthroughthe
inductorcircuitandthisisshownbelow.
ACInductorPhasorDiagram
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CoLo
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Thesevoltageandcurrentwaveformsshowthatforapurelyinductivecircuitthecurrentlagsthevoltageby90.Likewise,
wecanalsosaythatthevoltageleadsthecurrentby90.Eitherwaythegeneralexpressionisthatthecurrentlagsas
showninthevectordiagram.Herethecurrentvectorandthevoltagevectorareshowndisplacedby90.Thecurrentlags
thevoltage.
Wecanalsowritethisstatementas,V=0andI=-90withrespecttothevoltage,V.Ifthevoltagewaveformis
classedasasinewavethenthecurrent,Icanbeclassedasanegativecosineandwecandefinethevalueofthe
currentatanypointintimeasbeing:
Where:ωisinradianspersecondandtisinseconds.
Sincethecurrentalwayslagsthevoltageby90inapurelyinductivecircuit,wecanfindthephaseofthecurrentby
knowingthephaseofthevoltageorviceversa.SoifweknowthevalueofV,thenImustlagby90.Likewise,ifwe
knowthevalueofIthenVmustthereforeleadby90.Thenthisratioofvoltagetocurrentinaninductivecircuitwill
produceanequationthatdefinestheInductiveReactance,Xofthecoil.
InductiveReactance
Wecanrewritetheaboveequationforinductivereactanceintoamorefamiliarformthatusestheordinaryfrequencyof
thesupplyinsteadoftheangularfrequencyinradians,ωandthisisgivenas:
Where:?istheFrequencyandListheInductanceoftheCoiland2π?=ω.
Fromtheaboveequationforinductivereactance,itcanbeseenthatifeitheroftheFrequencyorInductancewas
increasedtheoverallinductivereactancevaluewouldalsoincrease.Asthefrequencyapproachesinfinitytheinductors
reactancewouldalsoincreasetoinfinityactinglikeanopencircuit.
However,asthefrequencyapproacheszeroorDC,theinductorsreactancewoulddecreasetozero,actinglikeashort
circuit.Thismeansthenthatinductivereactanceis"Proportional"tofrequencyandissmallatlowfrequenciesandhigh
athigherfrequenciesandthisdemonstratedinthefollowinggraph:
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LoLoL
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LLo
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InductiveReactanceagainstFrequency
Theslopeshowsthatthe"InductiveReactance"ofan
inductorincreasesasthesupplyfrequencyacrossit
increases.
ThereforeInductiveReactanceisproportionalto
frequency.(Xα?)
ThenwecanseethatatDCaninductorhaszeroreactance(short-circuit),athighfrequenciesaninductorhasininite
reactance(open-circuit).
ExampleNo1
Acoilofinductance150mHandzeroresistanceisconnectedacrossa100V,50Hzsupply.Calculatetheinductive
reactanceofthecoilandthecurrentflowingthroughit.
ACSupplythroughanRLSeriesCircuit
Sofarwehaveconsideredapurelyinductivecoil,butitisimpossibletohaveapureinductanceasallcoils,relaysor
solenoidswillhaveacertainamountofresistancenomatterhowsmallassociatedwiththecoilsturnsofwirebeing
used.Thenwecanconsideroursimplecoilasbeingaresistanceinserieswithaninductance.
InanACcircuitthatcontainsbothinductance,Landresistance,Rthevoltage,Vwillbethephasorsumofthetwo
componentvoltages,VandV.Thismeansthenthatthecurrentflowingthroughthecoilwillstilllagthevoltage,butby
anamountlessthan90dependinguponthevaluesofVandV.Thenewphaseanglebetweenthevoltageandthe
currentisknownasthephaseangleofthecircuitandisgiventheGreeksymbolphi,Φ.
Tobeabletoproduceavectordiagramoftherelationshipbetweenthevoltageandthecurrent,areferenceorcommon
componentmustbefound.InaseriesconnectedR-Lcircuitthecurrentiscommonasthesamecurrentflowsthrough
eachcomponent.Thevectorofthisreferencequantityisgenerallydrawnhorizontallyfromlefttoright.
FromourtutorialsaboutresistorsandcapacitorsweknowthatthecurrentandvoltageinaresistiveACcircuitareboth
"in-phase"andthereforevector,Visdrawnsuperimposedtoscaleonthecurrentorreferenceline.
Wealsoknowfromabove,thatthecurrent"lags"thevoltageinapurelyinductivecircuitandthereforevector,Visdrawn
90infrontofthecurrentreferenceandtothesamescaleasVandthisisshownbelow.
RLSeriesACCircuit
L
RLo
RL
R
Lo
R
InthevectordiagramaboveitcanbeseenthatlineOBrepresentsthecurrentreferenceline,lineOAisthevoltageofthe
resistivecomponentandwhichisin-phasewiththecurrent.LineOCshowstheinductivevoltagewhichis90infrontof
thecurrent,thereforeitcanbeseenthatthecurrentlagsthevoltageby90.LineODgivesustheresultantorsupply
voltageacrossthecircuit.ThevoltagetriangleisderivedfromPythagoras''stheoremandisgivenas:
InaDCcircuit,theratioofvoltagetocurrentiscalledresistance.However,inanACcircuitthisratioisknown
asImpedance,ZwithunitsagaininOhms.Impedanceisthetotalresistancetocurrentflowinan"ACcircuit"containing
bothresistanceandinductivereactance.Ifwedividethesidesofthevoltagetriangleabovebythecurrent,another
triangleisobtainedwhosesidesrepresenttheresistance,reactanceandimpedanceofthecoil.Thisnewtriangleis
calledan"ImpedanceTriangle"
TheImpedanceTriangle
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ExampleNo2
Asolenoidcoilhasaresistanceof30Ohmsandaninductanceof0.5H.Ifthecurrentflowingthroughthecoilis4amps.
Calculate,
a)Thevoltageofthesupplyifthefrequencyis50Hz.
b)Thephaseanglebetweenthevoltageandthecurrent.
PowerTriangleofanACInductor
Thereisoneothertypeoftriangleconfigurationthatwecanuseforaninductivecircuitandthatisofthe"PowerTriangle".
ThepowerinaninductivecircuitisknownasReactivePowerorvolt-ampsreactive,symbolVarwhichismeasuredin
volt-amps.InaRLseriesACcircuit,thecurrentlagsthesupplyvoltagebyanangleofΦ.
InapurelyinductiveACcircuitthepowerwillbeoutofphaseby90,becauseofthecurrentbutthetotalpowerofthecoil
willbeequaltozeroasanyconsumedpoweriscancelledoutthegeneratedself-inducedemfpower.Inotherwords,
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energyisbothtakenfromthesupplyandreturnedtoit.
TheReactivePower,(Q)ofacoilcanbegivenas:IxX(similartoIRinaDCcircuit).Thenthethreesidesofa
powertriangleinanACcircuitarerepresentedbyapparentpower,(S),realpower,(P)andthereactivepower,(Q)as
shown.
PowerTriangle
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