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

2. 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

3. 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

4. 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 and in textbook (p. 692 & 694)
Lecture 22

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

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

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

8. Class Example
Extension Exercise E12.13
Lecture 22

9. 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

10. 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

11. 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

12. Class Example
Extension Exercise E12.15
Lecture 22