EEE 302 Electrical Networks IIResonant CircuitsQuality Factor (Q)Bandwidth (BW)Series RLC CircuitPowerPoint PresentationParallel RLC CircuitSlide 8ScalingMagnitude ScalingFrequency ScalingSlide 12Lecture 22 1EEE 302Electrical Networks IIDr. Keith E. HolbertSummer 2001Lecture 22 2Resonant Circuits•Resonant frequency: the frequency at which the impedance of a series RLC circuit or the admittance of a parallel RLC circuit is purely real, i.e., the imaginary term is zero (ωL=1/ωC)•For both series and parallel RLC circuits, the resonance frequency is•At resonance the voltage and current are in phase, (i.e., zero phase angle) and the power factor is unityC L 10Lecture 22 3Quality Factor (Q)•An energy analysis of a RLC circuit provides a basic definition of the quality factor (Q) that is used across engineering disciplines, specifically:•The quality factor is a measure of the sharpness of the resonance peak; the larger the Q value, the sharper the peakwhere BW=bandwidthCycleperDissipatedEnergyatStoredEnergyMaxWWQDS 022BWQ0Lecture 22 4Bandwidth (BW)•The bandwidth (BW) is the difference between the two half-power frequenciesBW = ωHI – ωLO = 0 / Q•Hence, a high-Q circuit has a small bandwidth•Note that: 02 = ωLO ωHI•See Figs. 12.23 and 12.24 in textbook (p. 692 & 694)  12121&20QQHILOLecture 22 5Series RLC Circuit•For a series RLC circuit the quality factor isCLRCRRLQBWQseries11000Lecture 22 6Class Examples•Extension Exercise E12.8•Extension Exercise E12.9•Extension Exercise E12.10•Extension Exercise E12.11•Extension Exercise E12.12Lecture 22 7Parallel RLC Circuit•For a parallel RLC circuit, the quality factor isLCRCRLRQBWQparallel000Lecture 22 8Class Example•Extension Exercise E12.13Lecture 22 9Scaling•Two methods of scaling: 1) Magnitude (or impedance) scaling multiplies the impedance by a scalar, KM–resonant frequency, bandwidth, quality factor are unaffected 2) Frequency scaling multiplies the frequency by a scalar, ω'=KFω–resonant frequency, bandwidth, quality factor are affectedLecture 22 10Magnitude Scaling•Magnitude scaling multiplies the impedance by a scalar, that is, Znew = Zold KM•Resistor: ZR’ = KM ZR = KM RR’ = KM R•Inductor: ZL’ = KM ZL = KM jLL’ = KM L•Capacitor: ZC’ = KM ZC = KM / (jC)C’ = C / KMLecture 22 11Frequency Scaling•Frequency scaling multiplies the frequency by a scalar, that is, ωnew = ωold KF but Znew=Zold •Resistor: R” = ZR = RR” = R•Inductor: j(KF)L = ZL = jLL” = L / KF•Capacitor: 1 / [j (KF) C] = ZC = 1 / (jC)C” = C / KFLecture 22 12Class Example•Extension Exercise