Resonance in electrical circuits – series resonance

Series Resonance, bandwidth, quality factor

1. T.Mrunalini
Resonance in Electrical
Circuits –Series Resonance

2. Electrical Resonance
Resonancedescribesthephenomenonofincreased
amplitudethatoccurswhenthefrequencyofaperiodically
appliedforce(oraFouriercomponentofit)isequalor
closetoanaturalfrequencyofthesystemonwhichitacts.
Electricalresonanceoccursinanelectriccircuitat
aparticularresonantfrequencywhentheimpedances
oradmittancesofcircuitelementscanceleachother.In
somecircuits,thishappenswhentheimpedancebetween
theinputandoutputofthecircuitisalmostzeroandthe
transferfunctionisclosetoone.

3. TheresonanceofaseriesRLCcircuit
occurswhentheinductiveand
capacitivereactanceareequalin
magnitudebutcanceleachother
becausetheyare180degreesapartin
phase.
Thesharpminimuminimpedance
whichoccursisusefulintuning
applications.

4. Application
Oneapplicationforresonanceistoestablisha
conditionofstablefrequencyincircuitsdesignedto
produceACsignals.Usually,aparallel(tank)circuit
isusedforthispurpose,withthecapacitorand
inductordirectlyconnectedtogether,exchanging
energybetweeneachother.

5. Working
Aresonantcircuitisformedwhenacapacitorand
inductor(coil)areinparallelorinseries.Thetwo
circuitelementswillblockorpassasinglespecific
frequencyoutofadiversmix.Forthisreason,resonant
circuitsmakepossibleradioandTVtransmissionand
receptionandperformmanyotherusefultasks.

6. Series Resonance

7. Series Resonance
SeriesResonancecircuitsareoneofthemostimportant
circuitsusedelectricalandelectroniccircuits.Theycanbe
foundinvariousformssuchasinACmainsfilters,noise
filtersandalsoinradioandtelevisiontuningcircuits
producingaveryselectivetuningcircuitforthereceiving
ofthedifferentfrequencychannels.

8. In a series RLC circuit there becomes a frequency point
were the inductive reactance of the inductor becomes
equal in value to the capacitive reactance of the capacitor.
In other words, X
L=X
C. The point at which this occurs is
calledthe Resonant Frequencypoint, (ƒ
r) of the circuit.

10. •ThismakestheseriesLCcombinationactasashort
circuitwiththeonlyoppositiontocurrentflowinaseries
resonancecircuitbeingtheresistance,R.
•Incomplexform,theresonantfrequencyisthefrequency
atwhichthetotalimpedanceofaseriesRLCcircuit
becomespurely“real”,thatisnoimaginaryimpedance’s
exist.Thisisbecauseatresonancetheyarecancelledout.
Sothetotalimpedanceoftheseriescircuitbecomesjust
thevalueoftheresistanceandtherefore:Z=R.

11. Series Circuit Current at Resonance

13. Bandwidth of a Series Resonance Circuit
•IftheseriesRLCcircuitisdrivenbyavariable
frequencyataconstantvoltage,thenthemagnitudeof
thecurrent,Iisproportionaltotheimpedance,Z,
thereforeatresonancethepowerabsorbedbythe
circuitmustbeatitsmaximumvalueasP=I
2
Z.
•Uponreducingorincreasingthefrequencyuntilthe
averagepowerabsorbedbytheresistorintheseries
resonancecircuitishalfthatofitsmaximumvalueat
resonance,weproducetwofrequencypointscalledthe
half-powerpointswhichare-3dBdownfrom
maximum,taking0dBasthemaximumcurrent
reference.

14. These-3dBpointsprovidethecurrentvaluethatis70.7%of
itsmaximumresonantvaluewhichisdefinedas:0.5(I
2
R)=(0.707
xI)
2
R.Thenthepointcorrespondingtothelowerfrequencyathalf
thepoweriscalledthe“lowercut-offfrequency”,labelledƒ
Lwith
thepointcorrespondingtotheupperfrequencyathalfpowerbeing
calledthe“uppercut-offfrequency”,labelledƒ
H.Thedistance
betweenthesetwopoints,i.e.(ƒ
H–ƒ
L)iscalledtheBandwidth,
(BW)andistherangeoffrequenciesoverwhichatleasthalfofthe
maximumpowerandcurrentisprovidedasshown.

16. Quality Factor
•Thefrequencyresponseofthecircuitscurrent
magnitudeshowninthepreviousslide,relatestothe
“sharpness”oftheresonanceinaseriesresonance
circuit.Thesharpnessofthepeakismeasured
quantitativelyandiscalledtheQualityfactor,Qof
thecircuit.Thequalityfactorrelatesthemaximumor
peakenergystoredinthecircuit(thereactance)tothe
energydissipated(theresistance)duringeachcycleof
oscillationmeaningthatitisaratioofresonant
frequencytobandwidthandthehigherthecircuitQ,
thesmallerthebandwidth,Q=ƒ
r/BW.

18. Thank you