EEE 302 Electrical Networks II

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EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001 Lecture 22

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 At resonance the voltage and current are in phase, (i.e., zero phase angle) and the power factor is unity Lecture 22

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: The quality factor is a measure of the sharpness of the resonance peak; the larger the Q value, the sharper the peak where BW=bandwidth Lecture 22

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 See Figs. 12.23 and 12.24 in textbook (p. 692 & 694) Lecture 22

Series RLC Circuit For a series RLC circuit the quality factor is Lecture 22

Class Examples Extension Exercise E12.8 Extension Exercise E12.9 Lecture 22

Parallel RLC Circuit For a parallel RLC circuit, the quality factor is Lecture 22

Class Example Extension Exercise E12.13 Lecture 22

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

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 jL L’ = KM L Capacitor: ZC’ = KM ZC = KM / (jC) C’ = C / KM Lecture 22

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 = jL L” = L / KF Capacitor: 1 / [j (KF) C] = ZC = 1 / (jC) C” = C / KF Lecture 22

Class Example Extension Exercise E12.15 Lecture 22