#### 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 jL
L’ = KM L
• Capacitor:
ZC’ = KM ZC = KM / (jC)
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 = jL
L” = L / KF
• Capacitor:
1 / [j (KF) C] = ZC = 1 / (jC)
C” = C / KF
Lecture 22
11
Class Example
• Extension Exercise E12.15
Lecture 22
12