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GORESET
TheSeriesResonanceCircuit
ThusfarwehaveanalysedthebehaviourofaseriesRLCcircuitwhosesourcevoltageisafixedfrequencysteadystate
sinusoidalsupply.wehavealsoseenthattwoormoresinusoidalsignalscanbecombinedusingphasorsproviding
thattheyhavethesamefrequencysupply.Butwhatwouldhappentothecharacteristicsofthecircuitifasupplyvoltageof
fixedamplitudebutofdifferentfrequencieswasappliedtothecircuit.Alsowhatwouldthecircuits"frequencyresponse"
behaviourbeuponthetworeactivecomponentsduetothisvaryingfrequency.
InaseriesRLCcircuittherebecomesafrequencypointweretheinductivereactanceoftheinductorbecomesequalin
valuetothecapacitivereactanceofthecapacitor.Inotherwords,X=X.Thepointatwhichthisoccursiscalled
theResonantFrequencypoint,(?)ofthecircuit,andasweareanalysingaseriesRLCcircuitthisresonancefrequency
producesaSeriesResonance.
SeriesResonancecircuitsareoneofthemostimportantcircuitsusedelectricalandelectroniccircuits.Theycanbe
foundinvariousformssuchasinACmainsfilters,noisefiltersandalsoinradioandtelevisiontunningcircuits
producingaveryselectivetuningcircuitforthereceivingofthedifferentfrequencychannels.Considerthesimpleseries
RLCcircuitbelow.
SeriesRLCCircuit
Firstly,letusdefinewhatwealreadyknowaboutseriesRLCcircuits.
Fromtheaboveequationforinductivereactance,ifeithertheFrequencyortheInductanceisincreasedtheoverall
inductivereactancevalueoftheinductorwouldalsoincrease.Asthefrequencyapproachesinfinitytheinductors
reactancewouldalsoincreasetowardsinfinitywiththecircuitelementactinglikeanopencircuit.However,asthe
frequencyapproacheszeroorDC,theinductorsreactancewoulddecreasetozero,causingtheoppositeeffectacting
LC
r
likeashortcircuit.Thismeansthenthatinductivereactanceis"Proportional"tofrequencyandissmallatlow
frequenciesandhighathigherfrequenciesandthisdemonstratedinthefollowingcurve:
InductiveReactanceagainstFrequency
Thegraphofinductivereactanceagainstfrequencyisa
straightlinelinearcurve.Theinductivereactancevalue
ofaninductorincreaseslinearlyasthefrequency
acrossitincreases.Therefore,inductivereactanceis
positiveandisdirectlyproportionaltofrequency
(X∝?)
Thesameisalsotrueforthecapacitivereactanceformulaabovebutinreverse.IfeithertheFrequencyor
theCapacitanceisincreasedtheoverallcapacitivereactancewoulddecrease.Asthefrequencyapproachesinfinitythe
capacitorsreactancewouldreducetozerocausingthecircuitelementtoactlikeaperfectconductorof0Ω''s.However,as
thefrequencyapproacheszeroorDClevel,thecapacitorsreactancewouldrapidlyincreaseuptoinfinitycausingittoact
likeaverylargeresistanceactinglikeanopencircuitcondition.Thismeansthenthatcapacitivereactanceis"Inversely
proportional"tofrequencyforanygivenvalueofcapacitanceandthisshownbelow:
CapacitiveReactanceagainstFrequency
Thegraphofcapacitivereactanceagainstfrequencyis
ahyperboliccurve.TheReactancevalueofacapacitor
hasaveryhighvalueatlowfrequenciesbutquickly
decreasesasthefrequencyacrossitincreases.
Therefore,capacitivereactanceisnegativeandis
inverselyproportionaltofrequency(X∝?)
Wecanseethatthevaluesoftheseresistancesdependsuponthefrequencyofthesupply.AtahigherfrequencyXis
highandatalowfrequencyXishigh.ThentheremustbeafrequencypointwerethevalueofXisthesameasthe
valueofXandthereis.Ifwenowplacethecurveforinductivereactanceontopofthecurveforcapacitivereactanceso
thatbothcurvesareonthesameaxes,thepointofintersectionwillgiveustheseriesresonancefrequencypoint,
(?orω)asshownbelow.
SeriesResonanceFrequency
L
C-1
L
CL
C
rr
where:?isinHertz,LisinHenriesandCisinFarads.
ElectricalresonanceoccursinanACcircuitwhenthetworeactanceswhichareoppositeandequalcanceleachother
outasX=Xandthepointonthegraphatwhichthishappensiswerethetworeactancecurvescrosseachother.Ina
seriesresonantcircuit,theresonantfrequency,?pointcanbecalculatedasfollows.
Wecanseethenthatatresonance,thetworeactancescanceleachotherouttherebymakingaseriesLCcombination
actasashortcircuitwiththeonlyoppositiontocurrentflowinaseriesresonancecircuitbeingtheresistance,R.In
complexform,theresonantfrequencyisthefrequencyatwhichthetotalimpedanceofaseriesRLCcircuitbecomes
purely"real",thatisnoimaginaryimpedancesexist.Thisisbecauseatresonancetheyarecancelledout.Sothetotal
impedanceoftheseriescircuitbecomesjustthevalueoftheresistanceandtherefore:Z=R.
Thenatresonancetheimpedanceoftheseriescircuitisatitsminimumvalueandequalonlytotheresistance,Rofthe
circuit.Thecircuitimpedanceatresonanceiscalledthe"dynamicimpedance"ofthecircuitanddependinguponthe
frequency,X(typicallyathighfrequencies)orX(typicallyatlowfrequencies)willdominateeithersideofresonanceas
shownbelow.
ImpedanceinaSeriesResonanceCircuit
r
LC
r
CL
Notethatwhenthecapacitivereactancedominatesthecircuittheimpedancecurvehasahyperbolicshapetoitself,but
whentheinductivereactancedominatesthecircuitthecurveisnon-symmetricalduetothelinearresponseofX.You
mayalsonotethatifthecircuitsimpedanceisatitsminimumatresonancethenconsequently,the
circuitsadmittancemustbeatitsmaximumandoneofthecharacteristicsofaseriesresonancecircuitisthat
admittanceisveryhigh.Butthiscanbeabadthingbecauseaverylowvalueofresistanceatresonancemeansthatthe
circuitscurrentmaybedangerouslyhigh.
WerecallfromtheprevioustutorialaboutseriesRLCcircuitsthatthevoltageacrossaseriescombinationisthephasor
sumofV,VandV.Thenifatresonancethetworeactancesareequalandcancelling,thetwovoltages
representingVandVmustalsobeoppositeandequalinvaluetherebycancellingeachotheroutbecausewithpure
componentsthephasorvoltagesaredrawnat+90and-90respectively.TheninaseriesresonancecircuitV=-
Vtherefore,V=V.
SeriesRLCCircuitatResonance
Sincethecurrentflowingthroughaseriesresonancecircuitistheproductofvoltagedividedbyimpedance,atresonance
theimpedance,Zisatitsminimumvalue,(=R).Therefore,thecircuitcurrentatthisfrequencywillbeatitsmaximum
valueofV/Rasshownbelow.
SeriesCircuitCurrentatResonance
L
RLC
LCoo
L
CR
Thefrequencyresponsecurveofaseriesresonancecircuitshowsthatthemagnitudeofthecurrentisafunctionof
frequencyandplottingthisontoagraphshowsusthattheresponsestartsatneartozero,reachesmaximumvalueat
theresonancefrequencywhenI=Iandthendropsagaintonearlyzeroas?becomesinfinite.Theresultofthisis
thatthemagnitudesofthevoltagesacrosstheinductor,Landthecapacitor,Ccanbecomemanytimeslargerthanthe
supplyvoltage,evenatresonancebutastheyareequalandatoppositiontheycanceleachotherout.
Asaseriesresonancecircuitonlyfunctionsonresonantfrequency,thistypeofcircuitisalsoknownasanAcceptor
Circuitbecauseatresonance,theimpedanceofthecircuitisatitsminimumsoeasilyacceptsthecurrentwhose
frequencyisequaltoitsresonantfrequency.Theeffectofresonanceinaseriescircuitisalsocalled"voltage
resonance".
Youmayalsonoticethatasthemaximumcurrentthroughthecircuitatresonanceislimitedonlybythevalueofthe
resistance(apureandrealvalue),thesourcevoltageandcircuitcurrentmustthereforebeinphasewitheachotherat
thisfrequency.Thenthephaseanglebetweenthevoltageandcurrentofaseriesresonancecircuitisalsoafunctionof
frequencyforafixedsupplyvoltageandwhichiszeroattheresonantfrequencypointwhen:V,IandVareallinphase
witheachotherasshownbelow.Consequently,ifthephaseangleiszerothenthepowerfactormustthereforebeunity.
PhaseAngleofaSeriesResonanceCircuit
MAXR
R
Noticealso,thatthephaseangleispositiveforfrequenciesabove?andnegativeforfrequenciesbelow?andthiscan
beprovenby,
BandwidthofaSeriesResonanceCircuit
IftheseriesRLCcircuitisdrivenbyavariablefrequencyataconstantvoltage,thenthemagnitudeofthecurrent,Iis
proportionaltotheimpedance,Z,thereforeatresonancethepowerabsorbedbythecircuitmustbeatitsmaximumvalue
asP=IZ.Ifwenowreduceorincreasethefrequencyuntiltheaveragepowerabsorbedbytheresistorintheseries
resonancecircuitishalfthatofitsmaximumvalueatresonance,weproducetwofrequencypointscalledthehalf-power
pointswhichare-3dBdownfrommaximum,taking0dBasthemaximumcurrentreference.
These-3dBpointsgiveusacurrentvaluethatis70.7%ofitsmaximumresonantvalueas:0.5(IR)=(0.707xI)R.
Thenthepointcorrespondingtothelowerfrequencyathalfthepoweriscalledthe"lowercut-offfrequency",
labelled?withthepointcorrespondingtotheupperfrequencyathalfpowerbeingcalledthe"uppercut-offfrequency",
labelled?.Thedistancebetweenthesetwopoints,i.e.(?-?)iscalledtheBandwidth,(BW)andistherangeof
frequenciesoverwhichatleasthalfofthemaximumpowerandcurrentisprovidedasshown.
BandwidthofaSeriesResonanceCircuit
rr
2
22
L
HHL
Thefrequencyresponseofthecircuitscurrentmagnitudeabove,relatestothe"sharpness"oftheresonanceinaseries
resonancecircuit.ThesharpnessofthepeakismeasuredquantitativelyandiscalledtheQualityfactor,Qofthecircuit.
Thequalityfactorrelatesthemaximumorpeakenergystoredinthecircuit(thereactance)totheenergydissipated(the
resistance)duringeachcycleofoscillationmeaningthatitisaratioofresonantfrequencytobandwidthandthehigher
thecircuitQ,thesmallerthebandwidth,Q=?/BW.
Asthebandwidthistakenbetweenthetwo-3dBpoints,theselectivityofthecircuitisameasureofitsabilitytorejectany
frequencieseithersideofthesepoints.Amoreselectivecircuitwillhaveanarrowerbandwidthwhereasalessselective
circuitwillhaveawiderbandwidth.Theselectivityofaseriesresonancecircuitcanbecontrolledbyadjustingthevalueof
theresistanceonly,keepingalltheothercomponentsthesame,sinceQ=(XorX)/R.
BandwidthofaSeriesResonanceCircuit
Thentherelationshipbetweenresonance,bandwidth,selectivityandqualityfactorforaseriesresonancecircuitbeing
definedas:
1).ResonantFrequency,(?)
r
LC
r
2).Current,(I)
3).Lowercut-offfrequency,(?)
4).Uppercut-offfrequency,(?)
L
H
5).Bandwidth,(BW)
6).QualityFactor,(Q)
ExampleNo1
Aseriesresonancenetworkconsistingofaresistorof30Ω,acapacitorof2uFandaninductorof20mHisconnected
acrossasinusoidalsupplyvoltagewhichhasaconstantoutputof9voltsatallfrequencies.Calculate,theresonant
frequency,thecurrentatresonance,thevoltageacrosstheinductorandcapacitoratresonance,thequalityfactorandthe
bandwidthofthecircuit.Alsosketchthecorrespondingcurrentwaveformforallfrequencies.
ResonantFrequency,?
CircuitCurrentatResonance,I
InductiveReactanceatResonance,X
Voltagesacrosstheinductorandthecapacitor,V,V
(Note:thesupplyvoltageisonly9volts,butatresonancethereactivevoltagesare30voltspeak!)
r
m
L
LC
Qualityfactor,Q
Bandwidth,BW
Theupperandlower-3dBfrequencypoints,?and?
CurrentWaveform
ExampleNo2
Aseriescircuitconsistsofaresistanceof4Ω,aninductanceof500mHandavariablecapacitanceconnectedacrossa
100V,50Hzsupply.Calculatethecapacitancerequiretogiveseriesresonanceandthevoltagesgeneratedacrossboth
theinductorandthecapacitor.
ResonantFrequency,?
HL
r
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Voltagesacrosstheinductorandthecapacitor,V,V
SeriesResonanceSummary
Youmaynoticethatduringtheanalysisofseriesresonancecircuitsinthistutorial,wehave
lookedatbandwidth,upperandlowerfrequencies,-3dBpointsandqualityorQ-factor.Allthese
aretermsusedindesigningandbuildingofBandpassFiltersandindeed,resonanceis
usedin3-elementmainsfilterdesigntopassallfrequencieswithinthe"passband"range
whilerejectingallothers.
However,themainaimofthistutorialistoanalyseandunderstandtheconceptofhowSeries
ResonanceoccursinpassiveRLCseriescircuits.TheiruseinRLCfilternetworksanddesigns
isoutsidethescopeofthistutorial,andsowillnotbelookedathere,sorry.
Forresonancetooccurinanycircuititmusthaveatleastoneinductorandonecapacitor.
Resonanceistheresultofoscillationsinacircuitasstoredenergyispassedfromthe
inductortothecapacitor.
ResonanceoccurswhenX=Xandtheimaginarypartofthetransferfunctioniszero.
AtresonancetheimpedanceofthecircuitisequaltotheresistancevalueasZ=R.
Atlowfrequenciestheseriescircuitiscapacitiveas:X>X,thisgivesthecircuitaleadingpowerfactor.
Atlowfrequenciestheseriescircuitisinductiveas:X>X,thisgivesthecircuitalaggingpowerfactor.
Thehighvalueofcurrentatresonanceproducesveryhighvaluesofvoltageacrosstheinductorandcapacitor.
Seriesresonancecircuitsareusefulforconstructinghighlyfrequencyselectivefilters.However,itshighcurrent
andveryhighcomponentvoltagevaluescancausedamagetothecircuit.
Themostprominentfeatureofthefrequencyresponseofaresonantcircuitisasharpresonantpeakinits
amplitudecharacteristics.
Becauseimpedanceisminimumandcurrentismaximum,seriesresonancecircuitsarealsocalledAcceptor
Circuits.
LC
LC
CL
LC
InthenexttutorialaboutParallelResonancewewilllookathowfrequencyaffectsthecharacteristicsofaparallel
connectedRLCcircuitandhowthistimetheQ-factorofaparallelresonantcircuitdeterminesitscurrentmagnification.
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