Download Lecture 11 AC Circuits Frequency Response

Lecture 11 AC Circuits
Frequency Response
Contents
•
•
•
•
•
Introduction
Transfer Function
Series Resonance
Parallel Resonance
Passive Filters
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Introduction
What is Frequency Response 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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Transfer Function
• 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).
H( )  Voltage gain 
Vo ( )
Vi ( )
H( ) 
I ( )
H( )  Current gain  o
Ii ( )
H( )  Transfer Impedance 
Vo ( )
Ii ( )
Y( )
 | H( ) | 
X( )
H( )  Transfer Admittance 
I o ( )
Vi ( )
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Transfer Function: Example
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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Solution (Example 1)
Solution:
The transfer function is
1
V
1
jC
H( )  o 

Vs R  1/ j C 1  j RC
,
The magnitude is
The phase is
H( ) 
   tan 1
1
1  ( / o ) 2

o
o  1/RC
Low Pass Filter
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Transfer Function: Example
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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Solution (Example 2)
Solution:
The transfer function is
H( ) 
Vo
j L
1


Vs R  j L 1  R
j L
High Pass Filter
The magnitude is
H( ) 
,
1
1 (
o 2
)

The phase is   90  tan 1 
o  R/L
o
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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.
Resonance frequency:
1
rad/s
LC
1
fo 
Hz
2 LC
o 
1 

Z  R  j ω L 

ω C

or
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Series Resonance
The features of series resonance:
1 

Z  R  j ω L 

ω
C


The impedance is purely resistive, Z = R;
• The supply voltage Vs and the current I are in phase, so, cos  = 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
Bandwidth B
The frequency response of the resonance
circuit current is
Vm
I | I |
R 2  ( L  1 /  C) 2
The average power absorbed by the
RLC circuit is
P( ) 
The highest power dissipated occurs at
resonance:
1 

Z  R  j ω L 

ω
C


1 2
IR
2
1 Vm2
P(o ) 
2 R
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Series Resonance
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
1 Vm2
P(o ) 
2 R
Highest power
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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Series Resonance: Quality factor
Q
Peak energy stored in the circuit o L
1


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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Series Resonance
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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Parallel Resonance
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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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
o
Q
o 1  (
Q

1 2
)  o
2Q
2Q
o 
B
2
o 1  (
1 2 o
) 
2Q
2Q
o 
B
2
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Parallel Resonance
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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Passive Filters
• 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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Passive Filters
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
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