Transcript Lecture 4 Active Filter (Part I) - Department of EE

Lecture 4 Active Filter (Part I)
• Introduction of passive and active filter
• Categories of filter
– Low pass, high pass, band-pass, band stop (notch)
•
•
•
•
Butterworth/chebyshev/Bessel response
Poles and multiple stages
Transfer Function
Bode Plot
Ref:080222HKN
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Book references
• Microelectronic Circuits Analysis and Design, By
Muhammad H. Rashid (PWS Publishing Company)
• Microelectronic Circuit Design, By Richard C.
Jaeger and Travis N. Blalock (Mc Graw Hill)
• Introduction to Filter Theory, By David E.
Johnson (Prentice Hall)
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Passive Filters
• made up of passive components - resistors, capacitors and
inductors
• no amplifying elements (- transistors, op-amps, etc)
• no signal gain
• 1st order - design is simple (just use standard equations to
find resonant frequency of the circuit)
• 2nd order - complex equations
• require no power supplies
• not restricted by the bandwidth limitations of the op-amps
• can be used at very high frequencies
• can handle larger current or voltage levels than active
devices
• buffer amplifiers might be required
Ref:080222HKN
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Passive elements : Inductor
BIG PROBLEM!
• high accuracy (1% or 2%), small physical size, or large
inductance values are required ??
• standard values of inductors are not very closely spaced
• difficult to find an off-the-shelf inductor within 10 percent
of any arbitrary value
• adjustable inductors are used
• tuning such inductors to the required values is timeconsuming and expensive for larger quantities of filters
• inductors are often prohibitively expensive
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Active Filter
•
•
•
•
•
no inductors
made up of op-amps, resistors and capacitors
provides virtually any arbitrary gain
generally easier to design
high input impedance prevents excessive loading of the
driving source
• low output impedance prevents the filter from being
affected by the load
• at high frequencies is limited by the gain-bandwidth of the
op-amps
• easy to adjust over a wide frequency range without altering
the desired response
Ref:080222HKN
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Categories of Filters
Low Pass Filters:
High Pass Filters:
pass all frequencies from dc
up to the upper cutoff
frequency.
pass all frequencies that are
above its lower cutoff
frequency
Av(dB)
Av(dB)
-3dB
{
-3dB
f
f2
Low-pass response
Ref:080222HKN
{
f1
f
High-pass response
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Categories of Filters
Band Pass Filters:
Band Stop (Notch) Filters:
pass only the frequencies
that fall between its values
of the lower and upper
cutoff frequencies.
eliminate all signals within
the stop band while passing
all frequencies outside this
band.
Av(dB)
Av(dB)
{
-3dB
-3dB
f1
f
f2
f1
Band Pass Response
Ref:080222HKN
{
f2
f
Band Stop Response
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Filter Response Characteristics
Av
Butterworth
Bessel
Chebyshev
f
Ref:080222HKN
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Bessel Characteristic
• Flat response in the
passband.
• Role-off rate less than
20dB/decade/pole.
• Phase response is linear.
• Used for filtering pulse
waveforms without
distorting the shape of the
waveform.
Ref:080222HKN
Av
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Butterworth Characteristic
• Very flat amplitude, Av(dB) ,
response in the passband.
• Role-off rate is
20dB/decade/pole.
• Phase response is not linear.
• Used when all frequencies
in the passband must have
the same gain.
• Often referred to as a
maximally flat response.
Ref:080222HKN
Av
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Chebyshev Characteristic
• Overshoot or ripples in
the passband.
• Role-off rate greater than
20dB/decade/pole.
• Phase response is not
linear - worse than
Butterworth.
• Used when a rapid rolloff is required.
Ref:080222HKN
Av
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Pole
• A pole is nothing more than an RC circuit –
• n-pole filter  contains n-RC circuit.
Ref:080222HKN
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Single-Pole Low/High-Pass Filter
+V
R1
+V
C1
+
vin
vin
C1
+
R1
vout
vout
-
Rf1
-
Rf1
-V
-V
Rf2
Rf2
High Pass Filter
Low Pass Filter
Ref:080222HKN
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Two-Pole (Sallen-Key) Filters
C2
R2
+V
R2
R1
+V
C2
+
vin
C1
+
vin
C1
vout
-
Rf1
-V
R1
vout
-
Rf1
-V
Rf2
High Pass Filter
Low Pass Filter
Ref:080222HKN
Rf2
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Three-Pole Low-Pass Filter
Stage 1
Stage 2
C2
+V
R2
R1
+
+V
vin
R3
C1
-
Rf1
+
C3
vout
-
-V
Rf3
-V
Rf2
Rf4
Ref:080222HKN
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Two-Stage Band-Pass Filter
C2
R4
+V
R2
R1
+V
+
C4
C3
+
vin
C1
-
R3
-
Rf1
-V
vout
-V
Rf3
Rf2
Rf4
Stage 1
Two-pole low-pass
Stage 2
Two-pole high-pass
Av
Stage 2
response
BW
f1
Ref:080222HKN
Stage 1
response
fo
BW = f2 – f1
Q = f0 / BW
f
f2
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Multiple-Feedback Band-Pass Filter
C2
Rf
+V
R1
vin
C1
-
R2
vout
+
-V
Ref:080222HKN
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Band-Stop (Notch) Filter
The notch filter is designed to block all frequencies that fall within its bandwidth.
The circuit is made up of a high pass filter, a low-pass filter and a summing
amplifier. The summing amplifier will have an output that is equal to the sum of
the filter output voltages.
Av(dB)
Low pass
filter
f1
vin
High pass
filter
low-pass
high-pass
{
Summing
amplifier
-3dB

vout
f2
f
f1
Frequency response
Block diagram
Ref:080222HKN
f2
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Notch filter
Ref:080222HKN
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Transfer function H(j)
Vi
Transfer
Function
Vo
H ( j )
Vo ( j )
H ( j ) 
Vi ( j )
H  Re( H ) 2  Im( H ) 2
H  Re(H )  j Im(H )
 Im(H ) 

H  tan 
 Re(H ) 
Re(H )  0
 Im(H ) 

H  180 o  tan 1 
 Re( H ) 
Re(H )  0
Ref:080222HKN
1
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Frequency transfer function of filter H(j)
(I) Low - P ass Filter
(IV) Band - Stop (Notch)Filter
H ( j )  1
f  fo
H ( j )  0
fL  f  fH
H ( j )  0
f  fo
H ( j )  1
f  f L and f  f H
(II) High - P ass Filter
(V) All - P ass (or phase- shift)Filter
H ( j )  0
f  fo
H ( j )  1
H ( j )  1
f  fo
has a specificphase response
for all f
(III) Band - P ass Filter
H ( j )  1
fL  f  fH
H ( j )  0
f  f L and f  f H
Ref:080222HKN
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Passive single pole low pass filter
R
Vi
Vo 
C
Vo
or
H ( s) 
XC
Vi
XC  R
where
s  j
1
1
jC
Vo 
Vi 
Vi
1
1  jCR
R
jC
1
H ( j ) 
1 j
Ref:080222HKN

0
0
s  0
where  o 



 0
 ( )   tan 1 
1
RC
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H ( j ) 
Vo 
1
1  jCR
  0  Vo = Vi  max. value
  ∞  Vo = 0  min. value
1
Vi
1  jCR
vo
vo
1

RC
vo
 Vo = ??
max
2
c
1
Vo 
Vi
1 j
1
1
2
1
(cut - off frequency)
RC
Ref:080222HKN

H ( j )
1
1
Vo  2 2 Vi 
Vi
2
1 1
c  o 
max
EE3110 Active Filter (Part 1)

c
23
Decibel (dB)
(2) Voltage Gain in dB: (P=V2/R)
(1) Power Gain in dB :
Pin
 P2 

 P1 
By Definition: dB  10 log 10 
Pout
vin
P 
Ap ( dB )  10 log 10  o 
 Pin 
v 
Av ( dB )  20 log 10  o 
 vin 
P 
0dB  10 log 10  in 
 Pin 
 vin 
0dB  20 log 10  
 vin 
vout
1 
 Pin 
 3dB  10 log 10  2 
 Pin 


1 
 vin 
 6dB  20 log 10  2 
 vin 


 2P 
 3dB  10 log 10  in 
 Pin 
Ref:080222HKN
 2v 
 6dB  20 log 10  in 
 vin 
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Cascaded System
vin
Av1
Av2
Av3
x10
x10
x10
20dB
20dB
20dB
vout
Av  Av1  Av 2  Av 3
Av  10  10  10  103
Av ( dB )  20 log 10  Av1  Av 2  Av 3 
Av ( dB )  20 log 10  Av1   20 log 10  Av 2   20 log 10  Av 3 
Av ( dB )  Av1 dB   Av 2 dB   Av 3 dB 
Av ( dB )  20dB  20dB  20dB
Av ( dB )  60dB
Ref:080222HKN
EE3110 Active Filter (Part 1)
20 log 10 10  20dB
20 log 10 103   60dB
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Bode Plot (single pole)
H ( j ) 
1

1  jCR
H ( j ) 

H ( j ) dB
1

1  j  
 o 
1

1   
 o 
R
Vi
2

 20 log 10 H ( j )  20 log 10  1


C
Vo
Single pole low-pass filter

1   
 o 
2




For >>o
H ( j ) dB

 20 log 10  
 o 
Ref:080222HKN
EE3110 Active Filter (Part 1)
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
H ( j )  20 log 10  
 o 
 2

For octave apart,
o 1
 10

For decade apart,
o 1
x 2x
H ( j)  20dB
10x
 (log)
6dB
H ( j ) dB
H ( j )  6dB
20dB
slope
-6dB/octave
-20dB/decade
Ref:080222HKN
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Bode plot (Two-pole)
R1
vi
R2
C1
C2
vo
o1  o 2


H ( j )  20 log 10 1


Ref:080222HKN
2
2








 1   1 


 
   

 o1 
 o 2  


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