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Circuit Theory
Chapter 14
Frequency Response
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Frequency Response
Chapter 14
14.1
14.2
14.3
14.4
14.5
Introduction
Transfer Function
Series Resonance
Parallel Resonance
Passive Filters
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14.1 Introduction (1)
What is FrequencyResponse of a Circuit?
It is the variation in a circuit’s
behavior with change in signal
frequency and may also be
considered as the variation of the gain
and phase with frequency.
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14.2 Transfer Function (1)
• The transfer function H(ω) of a circuit is the
frequency-dependent ratio of a phasor output
Y(ω) (an element voltage or current ) to a phasor
input X(ω) (source voltage or current).
Y( )
H( ) 
 | H( ) | 
X( )
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14.2 Transfer Function (2)
• Four possible transfer functions:
H( )  Voltage gain 
Vo ( )
Vi ( )
H( ) 
H( )  Current gain 
I o ( )
Ii ( )
H( )  Transfer Impedance 
Vo ( )
Ii ( )
Y( )
 | H( ) | 
X( )
H( )  Transfer Admittance 
I o ( )
Vi ( )
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14.2 Transfer Function (3)
Example 1
For the RC circuit shown below, obtain the transfer
function Vo/Vs and its frequency response.
Let vs = Vmcosωt.
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14.2 Transfer Function (4)
Solution:
The transfer function is
1
V
1
jC
H( )  o 

Vs R  1/ j C 1  j RC
,
The magnitude is
H( ) 
The phase is    tan 1

o
o  1/RC
1
1  ( / o ) 2
Low Pass Filter
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14.2 Transfer Function (5)
Example 2
Obtain the transfer function Vo/Vs of the RL circuit
shown below, assuming vs = Vmcosωt. Sketch its
frequency response.
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14.2 Transfer Function (6)
Solution:
Vo
j L
1
H( ) 


Vs R  j L 1  R
j L
The transfer function is
High Pass Filter
,
The magnitude is
1
H ( ) 
The phase is   90  tan 1
o  R/L
1 (
o 2
)


o
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14.3 Series Resonance (1)
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.
Resonance frequency:
1
rad/s
LC
1
fo 
Hz
2 LC
o 
1
Z  R  j ( L 
)
C
or
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14.3 Series Resonance (2)
The features of series resonance:
Z  R  j ( L 
1
)
C
The impedance is purely resistive, Z = R;
• The supply voltage Vs and the current I are in phase, so
cos q = 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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14.3 Series Resonance (3)
Bandwidth B
The frequency response of the
resonance circuit current is
I | I |
Z  R  j ( L 
1
)
C
Vm
R 2  ( L  1 /  C) 2
The average power absorbed
by the RLC circuit is
P( ) 
The highest power dissipated
occurs at resonance:
1 2
IR
2
1 Vm2
P(o ) 
2 R
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14 3 Series Resonance (4)
Half-power frequencies ω1 and ω2 are frequencies at which the
dissipated power is half the maximum value:
1 (Vm / 2 ) 2 Vm2
P(1 )  P(2 ) 

2
R
4R
The half-power frequencies can be obtained by setting Z
equal to √2 R.
1  
R
R
1
 ( )2 
2L
2L
LC
Bandwidth B
2 
R
R
1
 ( )2 
2L
2L
LC
o  12
B  2  1
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14.3 Series Resonance (5)
Quality factor,
Q
 L
Peak energy stored in the circuit
1
 o 
Energy dissipated by the circuit
R
o CR
in one period at resonance
The relationship
between the B, Q
and ωo:
B
R o

 o2 CR
L Q
• The quality factor is the ratio of its
resonant frequency to its bandwidth.
• If the bandwidth is narrow, the
quality factor of the resonant circuit
must be high.
• If the band of frequencies is wide,
the quality factor must be low.
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14.3 Series Resonance (6)
Example 3
A series-connected circuit has R = 4 Ω
and L = 25 mH.
a. Calculate the value of C that will produce a
quality factor of 50.
b. Find ω1 and ω2, and B.
c. Determine the average power dissipated at ω
= ωo, ω1, ω2. Take Vm= 100V.
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14.4 Parallel Resonance (1)
It occurs when imaginary part of Y is zero
1
1
Y   j ( C 
)
R
L
Resonance frequency:
1
1
o 
rad/s or f o 
Hz
LC
2 LC
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14.4 Parallel Resonance (2)
Summary of series and parallel resonance circuits:
characteristic
Series circuit
ωo
1
LC
1
LC
Q
ωo L
1
or
R
ωo RC
R
or o RC
o L
B
ω1, ω2
Q ≥ 10, ω1, ω2
Parallel circuit
o
Q
o
Q
o 1  (

1 2
)  o
2Q
2Q
o 
B
2
o 1  (
1 2 o
) 
2Q
2Q
o 
B
2
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14.4 Parallel Resonance (3)
Example 4
Calculate the resonant frequency of the
circuit in the figure shown below.
Answer:   19  2.179 rad/s
2
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14.7 Passive Filters (1)
• A filter is a circuit
that is designed to
pass signals with
desired frequencies
and reject or
attenuate others.
Low Pass
High Pass
• Passive filter consists
of only passive
element R, L and C.
Band Pass
• There are four types
of filters.
Band Stop
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14.7 Passive Filters (2)
Example 5
For the circuit in the figure below, obtain the transfer function
Vo(ω)/Vi(ω). Identify the type of filter the circuit represents
and determine the corner frequency. Take R1=100W =R2
and L =2mH.
Answer:
  25 krad/s
HW18 Ch14: 47, 55, 57, 59
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