Transcript Lecture Notes - Resonance Circuits and Characteristics File

Chapter 14
Resonance Circuits
Chapter Objectives:
 Understand the Concept of Transfer Functions.
 Be Familiar with the Decibel Scale.
 Learn how to make Bode Magnitude and Phase plots.
 Learn about series and parallel resonant RLC circuits.
 Know Different Types of Passive and Active Filters and their
Characteristics.
 Understand the use of scaling in circuit analysis.
 Be Able to use PSpice to obtain frequency response.
 Apply what is learnt to radio receiver and touch-tone telephone.
Huseyin Bilgekul
Eeng 224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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SERIES RESONANCE
 Resonance is a condition in an RLC circuit in which the
capacitive and inductive reactance are equal in magnitude,
thereby resulting in purely resistive impedance.
 The features of series resonance:
The impedance is purely resistive, Z = R;
• The supply voltage Vs and the current I are in phase (cosq = 1)
• The magnitude of the transfer function H(ω) = Z(ω) is minimum;
• The inductor voltage and capacitor voltage can be much more
than the source voltage.
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SERIES RESONANCE
Vs ( )
1
1 

 R  j L 
 R  j L 
I ( )
jC
C 

Resonance occurs when circuit is purely resistive
1
1
Im( Z )   L 
 0  o L 
C
oC
Z ( )  H ( ) 
o 
o 
1
LC
1
,
LC
Resonance Frequency
fo 
1
2 LC
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SERIES RESONANCE
 VR, VL, VC, and I versus frequency for a series resonant circuit.
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SERIES RESONANCE
Inductive reactance versus frequency.
Placing the frequency response of the
inductive and capacitive reactance of a
series R-L-C circuit on the same set of
axes.
Capacitive reactance versus frequency.
ZT (total impedance) versus frequency for the
series resonant circuit.
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PHASE OF SERIES RESONANCE CIRCUIT
fo
f  f o : Network Capacitive
f  f o : Network Inductive
f  f o : Network Resistive
Phase plot for the series resonant circuit.
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SERIES RESONANCE
 Resonance occurs when the circuit has a complex conjugate pair of
poles.
 Resonance allows frequency discrimination in circuits.
 Resonance occurs in a circuit that has at least one inductor and one
capacitor.
At Resonance:
1) Impedance is purely resistive.
2) The voltage and current are in phase.
3) The transfer function H()= Z() is Minimum
4) Inductor and capacitor voltages can be much more than (Q times) source
voltage.
Vm
VL  o L  QVm
R
V 1
VC  m
 QVm
R oC
Q
o L
R

1
oCR
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BANDWIDTH of SERIES RESONANCE
 Current versus frequency for the series resonant circuit.
Vm
II 
R 2  ( L  1 )2
C
Half Power Frequencies
Dissipated power is half of the
maximum value.
• The half-power frequencies 1 and 2 can be obtained by setting,
Z (1 )  Z (2 )  R 2  ( L  1
C
 Vm



2

P(1 )  P(2 ) 
2R
2
)2  2 R
2
2
R
1
R
1
 R 
 R 
1  
 
, 2  
 
 
 
2L
2L
 2 L  LC
 2 L  LC
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Selectivity
 The frequencies corresponding to 0.707 of the maximum current are called the band
frequencies, cutoff frequencies, or half-power frequencies (ƒ1, ƒ2).
 Half-power frequencies are those frequencies at which the power delivered is onehalf that delivered at resonant frequency.
 The range of frequencies between the two are referred to as bandwidth (abbreviated
BW) of the resonant circuit.
 Since the resonant circuit is adjusted to select a band of frequencies it is called a
selectivity curve.
 The shape of the curve depends on each element of the series R-L-C circuit.
 If resistance is made smaller with a fixed inductance and capacitance, the bandwidth
decreases and the selectivity increases.
 If the ratio L/C increases with fixed resistance, the bandwidth again decreases with
an increase in selectivity.
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BANDWIDTH OF SERIES RESONANCE
 The width of the response is measured by the BANDWIDTH.
 BANDWIDTH is the difference between the half-power
frequencies.
B  2  1
 Resonance frequency can be obtained from the half-power
frequencies.
o  12 , B  2  1
 The SHARPNESS of the resonance is measured by the QUALITY
FACTOR.
 QUALITY FACTOR is the ratio of the resonance frequency to the
bandwidth. The higher the Q the smaller is the bandwidth.
Q
o
B
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QUALITY FACTOR OF SERIES RESONANCE
Q
Q  2
o
B
Peak Energy Stored
Energy Dissipated in one Period at Resonance
L
1
Q o 
R
o RC
Q
o L
R

1
o RC
R o
B 
L Q
B
B
1  o  ,  2   o 
2
2
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Effect on Selectivity of R, L, C for Series Resonance
Effect of R on selectivity
Effect of L and C on selectivity
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PARALLEL RESONANCE
 Resonance is a condition in an RLC circuit in which the capacitive and inductive
reactances are equal in magnitude, resulting in a purely resistive impedance.
 Parallel resonance circuit behaves similarly but in opposite fashion compared to
series resonant circuit.
 The admitance is minimum at resonance or impedance is maximum.
o 
1
LC
Parallel resonant circuit.
I 1
1
1
1 

  jC 
  j  C 
V R
j L R
 L 

Resonance occurs when admitance is purely resistive
1
1
1
Im(Y )   L 
 0  o L 
o 
rad/sec
C
oC
LC
Y  H ( ) 
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PARALLEL RESONANCE
 At Resonance frequency:
1) Admitance is purely resistive.
2) The voltage and current are in phase.
3) The transfer function H()= Y() is Minimum.
4) Inductor and capacitor currents can be much more than the source current.
IL 
Im R
 QI m
o L
I C  oCI m R  QVm
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PARALLEL RESONANCE
VV 
Im
2
1
1 )2

(

C

 
L
R
Voltage versus frequency for the parallel resonant circuit.
 The half-power frequencies can be obtained as:
2
1
1
 1 
1  
 


2 RC
 2 RC  LC
2
1
1
 1 
2  
 


2 RC
 2 RC  LC
1
o  1 2 , B   2  1 
RC
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Summary of series and parallel resonance circuits:
Characteristic
Series circuit
Parallel circuit
ωo
1
LC
1
LC
ωo L
1
or
R
ωo RC
Q
Q ≥ 10, ω1, ω2
o
Q
o
Q
B
ω1, ω2
R
or o RC
o L
o
1  (
o
1 2
) 
2Q
2Q
o 
B
2
o
1  (

1 2
)  o
2Q
2Q
o 
B
2
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Dual-channel 15-band
“Constant Q” Graphic
Equalizer
The equalizer changes
the contribution of the
different frequency
components of the music
signal according to the
listeners’s wish.
Gains given to
different frequency
components of the
music signal.
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