#### Transcript EEE 302 Lecture 22

EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001 Lecture 22 1 Resonant 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 1 0 LC • At resonance the voltage and current are in phase, (i.e., zero phase angle) and the power factor is unity Lecture 22 2 Quality 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: WS Max Energy Stored at 0 Q 2 2 WD Energy Dissipated per Cycle • The quality factor is a measure of the sharpness of the resonance peak; the larger the Q value, the sharper the peak 0 Q where BW=bandwidth BW Lecture 22 3 Bandwidth (BW) • The bandwidth (BW) is the difference between the two half-power frequencies BW = ωHI – ωLO = 0 / Q • Hence, a high-Q circuit has a small bandwidth • Note that: 02 = ωLO ωHI LO & HI 1 0 2Q 1 1 2 2Q • See Figs. 12.23 and 12.24 in textbook (p. 692 & 694) Lecture 22 4 Series RLC Circuit • For a series RLC circuit the quality factor is Q 0 BW Qseries 0 L 1 1 L R 0 CR R C Lecture 22 5 Class Examples • • • • • Extension Exercise E12.8 Extension Exercise E12.9 Extension Exercise E12.10 Extension Exercise E12.11 Extension Exercise E12.12 Lecture 22 6 Parallel RLC Circuit • For a parallel RLC circuit, the quality factor is Q 0 BW Q parallel R C 0 CR R 0 L L Lecture 22 7 Class Example • Extension Exercise E12.13 Lecture 22 8 Scaling • 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 affected Lecture 22 9 Magnitude Scaling • Magnitude scaling multiplies the impedance by a scalar, that is, Znew = Zold KM • Resistor: ZR’ = KM ZR = KM R R’ = KM R • Inductor: ZL’ = KM ZL = KM jL L’ = KM L • Capacitor: ZC’ = KM ZC = KM / (jC) C’ = C / KM Lecture 22 10 Frequency Scaling • Frequency scaling multiplies the frequency by a scalar, that is, ωnew = ωold KF but Znew=Zold • Resistor: R” = ZR = R R” = R • Inductor: j(KF)L = ZL = jL L” = L / KF • Capacitor: 1 / [j (KF) C] = ZC = 1 / (jC) C” = C / KF Lecture 22 11 Class Example • Extension Exercise E12.15 Lecture 22 12