Transcript bee1113: electric circuit i chapter 1: basic concept

CHAPTER 6: INTRODUCTION
TO PASSIVE FILTERS
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AHBMH
Series & Parallel Resonance
Passive Filter
DEE2113 : Chap 6 - Introduction to Passive Filters
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Resonance
Resonance is a condition in an RLC circuit in which the
capacitive and inductive reactances are equal in
magnitude, thereby resulting in a purely resistive
impedance.
The series resonant circuit
AHBMH
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Series Resonance
Input impedance:
Vs
1
Z  H()   R  jL 
I
jC
1 

Z  R  j L 

C 

Resonance
occurs when
imaginary part
is 0
Resonant/center frequency:
1
0 
rad / s
LC
AHBMH
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Series Resonance
At resonance:
1. The impedance is purely resistive, Z = R
2. The voltage and the current are in phase, pf=1
3. The magnitude of transfer function H(w) = Z(w) is
minimum
4. The inductor voltage and capacitor voltage can be much
more than the source voltage
AHBMH
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Series Resonance
Average power dissipated by the RLC circuit:
1 2
P()  I R
2
Where:
I 
Vm
R  L  1 / C 
2
AHBMH
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Series Resonance
The current amplitude vs. frequency for the series resonant circuit
Maximum power:
2
m
1V
P(0 ) 
2 R
Power at certain frequency:
2
m
V
P(1 )  P(2 ) 
4R
AHBMH
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Series Resonance
Half power frequency:
2
R
1
R 
1  
   
2L
 2L  LC
2
R
1
R 
2 
   
2L
 2L  LC
0  12
AHBMH
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Series Resonance
The “sharpness” of the resonance in a resonant
circuit is measured quantitatively by the quality
factor Q
0 L
0
1
Q


R
0 CR B
The quality factor of a resonant circuits is the
ratio of its resonant frequency to its bandwidth
AHBMH
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Series Resonance
Relation between Q and bandwidth B:
R 0
B   2  1  
L Q
The higher the circuit Q, the smaller the bandwidth
AHBMH
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Series Resonance
High Q circuit if,
Q  10
and half power frequency can be approximated as:
B
1  0 
2
B
2  0 
2
AHBMH
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Example 1
R=2Ω, L=1mH, C=0.4μF. Determine :
a) The resonant frequency and the half-power frequency
b) The quality factor and bandwidth
c) The amplitude of the current at ω0, ω1 and ω2
AHBMH
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Parallel Resonance
The parallel-resonant circuit
AHBMH
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Parallel Resonance
Input admittance:
I 1
1
Y  H()    jC 
V R
jL
1 
1 
Y   j C 

R 
L 
Resonant frequency:
Resonance
occurs when
imaginary part
is 0
1
0 
rad / s
LC
AHBMH
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Parallel Resonance
Half power frequency:
2
1
1
 1 
1  
 
 
2RC
 2RC  LC
2
1
1
 1 
2 
 
 
2RC
 2RC  LC
AHBMH
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Parallel Resonance
1
B  2  1 
RC
0
R
Q
 0 RC 
B
0 L
AHBMH
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Parallel Resonance
High Q circuit if,
Q  10
and half power frequency can be approximated as:
B
1  0 
2
B
2  0 
2
AHBMH
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Example 2
R=8 kΩ, L=0.2 mH, C=8 μF. Determine :
a) The resonant frequency, quality factor and bandwidth
b) The half-power frequencies
c) The power dissipated at ω0, ω1 and ω2
AHBMH
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AHBMH
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Filters
A filter is a circuit that is designed to pass signals with desired
frequencies and reject or attenuate others.
4 types of filters:
1. Lowpass filter: passes low frequencies and stops high
frequencies
2. Highpass filter: passes high frequencies and rejects low
frequencies
3. Bandpass filter: passes frequencies within a frequency band and
blocks or attenuates frequencies outside the band
4. Bandstop filter: passes frequencies outside a frequency band and
blocks or attenuates frequencies within the band
AHBMH
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Filters
Ideal frequency response of four types of filters:
a) lowpass
c) bandpass
AHBMH
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b) highpass
d) bandstop
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Lowpass Filters
A lowpass filter is designed to pass only frequencies
from dc up to the cutoff frequency ωc
AHBMH
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Lowpass Filters
Transfer function:
V0
1/ jC
1
H() 


Vi R  1/ jC 1  jRC
1
1
H(C ) 

2
2 2
2
1  C R C
Cutoff frequency:
AHBMH
1
C 
RC
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Highpass Filter
A highpass filter is designed to pass all frequencies
above its cutoff frequency ωc
AHBMH
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Highpass Filters
Transfer function:
V0
R
jRC
1
H ( )  


Vi R  1 / jC 1  jRC 1  1
jRC
1
1
H (C ) 

1
2
1 2 2 2
C R C
Cutoff frequency:
AHBMH
1
C 
RC
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Bandpass Filter
A bandpass filter is designed to pass all frequencies
within a band of frequencies, ω1 < ω0 < ω2
AHBMH
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Bandpass Filters
Transfer function:
V0
R
H() 

Vi R  jL  1/ C
Center frequency:
AHBMH
1
0 
LC
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Bandstop Filter
A bandstop filter is designed to stop or eliminate all
frequencies within a band of frequencies, ω1 < ω0 < ω2
AHBMH
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Bandstop Filters
Transfer function:
V0
jL  1/ C
H() 

Vi R  jL  1/ C
Center frequency:
1
0 
LC
AHBMH
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Example 3
Bandstop filter rejects 200 Hz while passing other
frequencies. For R=150 Ω and bandwidth 100 Hz,
determine:
a) L
b) C
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Exercise 1
For a series RLC bandstop filter, R=2 kΩ, L=0.1 mH,
C=40 pF. Determine :
a) The center frequency
b) The bandwidth
c) The half-power frequencies
d) The quality factor
AHBMH
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