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Analog Circuits and Systems Prof. K Radhakrishna Rao
Lecture 25: Active Filters
1
Review
� Inductor Simulation � To convert RLC filters to Active RC filters � Gyrator – Inductor Simulator (L=CR2) � Active and Passive - Parameter Sensitivities in Active RC filters � Effect of finite gain and finite gain bandwidth product on inductor
simulated � f0Q<<Gain Bandwidth Product
2
Review (contd.,)
3
of the inductor simulated =
of the filter simulated =
0
0 0
a
a 0 a
0
AQ2 A1GBQQ
Q 2 Q1A GB
ω⎛ ⎞−⎜ ⎟⎝ ⎠
⎛ ⎞ω+ −⎜ ⎟⎝ ⎠
Q-enhancement- Sallen and Key
� The quality factor Qp of a second order passive RC filter is always less than 0.5
� Qp < 0.5 is unacceptable for a general filter design � Sallen and Key proposed use of negative and positive feedback, and
active devices to enhance Q � Several topologies similar to Sallen and Key filters are possible
4
Second Order Passive filter
� Transfer function of second order passive filter =
� D(s) and N(s) are second order polynomials with D(s) having Q <0.5
5
( )( )N sD s
Use of feedback to enhance Q
6
( ) ( )( ) ( )
( )( ) ( )
o
i
-K N s D s -KN sVV D s KN s1 K N s D s
⎡ ⎤⎣ ⎦= =+⎡ ⎤+ ⎣ ⎦
where K is the gain of the active device
Use of feedback to enhance Q (contd.,)
7
p
For a general second-order passive RC/RL filter
where is the natural frequency of the passive RC filter
is quality factor of the passive second-order RC/RL
2
2pp
2
2p pp
p
s sm n pN(s)D(s) s s 1
Q
Q
⎡ ⎤+ +⎢ ⎥ωω⎢ ⎥⎣ ⎦=
⎡ ⎤+ +⎢ ⎥ωω⎢ ⎥⎣ ⎦
ω
filter
Quality Factor
8
is always < 0.5p
2
2p po
2i
2p p p
0 p
pa
Q
s s-K m n pVV s s(1 mK) (1 nK) (1 pK)
Q
1 pK1 mKQ
Q (1 pK) (1 mK)(1 nK)
⎡ ⎤+ +⎢ ⎥ω ω⎢ ⎥⎣ ⎦=
⎡ ⎤+ + + + +⎢ ⎥ω ω⎢ ⎥⎣ ⎦+ω = ω+
= + ++
If K is positive, m and p are positive, and for all values of n it is a negative feedback system If K is negative m and p are positive, all positive values of n it is a positive feedback system
Enhancement of Qa
� Qa can be enhanced by increasing pK >0 with m = n =0, mK>0 with p=n=0, pK>0 and mK>0 with n =0. These make use of negative feedback.
� Qa can also be enhanced by making nK<0 and 0<|nK|<1. This constitutes using positive feedback.
� All types of filters can be designed using any of the Q-enhancement methods.
9
Second-order low-pass RC filter
10
with and
and
p1 1 2 2
p1 1 2 2 1
2 2 1 1 2
1 2 1 2
p p
1R C R C
1QC R C R R1C R C R R
R R R C C C1 1QRC 3
ω =
=⎛ ⎞
+ +⎜ ⎟⎝ ⎠
= = = =
ω = =
( ) ( )( )21 1 2 2 1 1 2 1 2
N(s) 1D(s) C R C R s C R C R R s 1
=+ + + +
Active Low Pass Filter
11
The natural frequency of the active filter is now higher
can be increased to the required value through
suitable selection of .Low-
o2
i2
p pp
0 p a p
a
m 0, n 0 and p 1V -KV s s (1 K)
Q
1 K ; Q Q 1 KQ
K
= = =
=⎡ ⎤
+ + +⎢ ⎥ωω⎢ ⎥⎣ ⎦
ω = ω + = +
pass passive filter with amplifier gain - and feedbackK
and 1 2 1 2R R R C C C= = = =
Structure of Active Low Pass filter
� Addition required between feedback signal and the input � In order to get 2Vo the gain of VCVS will have to be made 2K � A VCVS with gain -K can be realized by having buffer stage
followed by inverting amplifier
12
Structure of Active Low Pass filter (contd.,)
13
Second order Butterworth low-pass filter
14
Bandwidth = 40Hz, .
With and
then
a
1 2
1 2 p
a p
1Q2
R R R1C C C Q3
1 K 1Q Q 1 K3 2
=
= =
= = =
+= + = =
and
For
0
K 3.5
1 K 4.52 40RC RC
4.5RC2 40R 100k ;
4.5C F 84nF2 40
=
+ω = π × = =
=π ×= Ω
= µ =π ×
Frequency Response of the Butterworth LP filter
15
Transient Response of the Butterworth LP filter
16
Frequency response of Low Pass Filter
17
with Q =5 and f0 = 40 Hz R=100k; C=0.6mF; K=224
Frequency response of Low Pass Filter
18
with Q =5 and f0 = 400 Hz C = 60 nF Finite GB of the Op.Amp
Finite GB of the Op.Amp
Transient response of Low-pass Filter
19
with Q =5 and f0 = 400 Hz C = 60 nF
Q-Enhancement due to Finite GB
Low-pass Filter
20
with Q =5 and f0 = 600 Hz C = 40 nF
Q-Enhancement due to Finite GB
Observations
� Q increases from the specified value � The natural frequency reduces slightly from the specified value � At higher natural frequencies the transient responses are more
oscillatory indicating Q enhancement � Beyond a certain natural frequency the system becomes unstable
and goes into oscillations at the natural frequency � These deviations from the expected behavior are due to finite gain
bandwidth product of the active devices used.
21
Effect of Gain Bandwidth Product of Op Amp
( )
( )
⎛ ⎞+= ⎜ ⎟⎜ ⎟+⎛ ⎞ ⎛ ⎞ ⎝ ⎠+ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
+
=
Amplifier using a buffer and an inverting
amplifier of gain K has a transfer function
Transfer function of the active low-pass filter
o
i
o
i
2 1 K sV K K 1-(1 2K)s sV GB1 1GB GB
2 1 K sK 1-
GVV
;
( )
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞++ + + ⎜ ⎟⎜ ⎟ωω ⎝ ⎠
2
2p pp
B2 1 K ss s 1 K 1-
Q GB
22
Effect of GB Product of Op Amp (contd.,)
( )
( )
( )
Normalizing
(due to GB)
o2
i20 0 p
pa
0 p
2 1 K sK 1-1 K GBV
V 2K 1 K ss s - 1GB(1 K)Q 1 K
Q 1 KQ
2K 1 K Q1-
1 KGB
⎛ ⎞+⎜ ⎟⎜ ⎟+ ⎝ ⎠=
⎛ ⎞++ +⎜ ⎟⎜ ⎟+ω ω + ⎝ ⎠
+=⎛ ⎞+ ω⎜ ⎟⎜ ⎟+⎝ ⎠
23
( )GB should be large enough
to make 0 p
0 a
2K 1 K Q1 KGB
2K Q 1GB
+ ω
+ω
= =
Examples
24
Ex:1
(specified) and Hz and
(due to GB)
a 0
pa
0 a
Q 5 f 40 K 3.5
Q 1 KQ 5.23
2K Q1-GB
= = =
+= =
ω⎛ ⎞⎜ ⎟⎝ ⎠
Ex:2
(specified)=5 and Hz and
(due to GB)
a 0
pa
0 a
Q f 400 K 224
Q 1 KQ 48
2K Q1-GB
= =
+= =
ω⎛ ⎞⎜ ⎟⎝ ⎠
Limitations of GB
for the filter to be stable in case inductance simulation
for the filter to be stable in case of filter using feedback
0 a
0 a
2f Q 1GB
2Kf Q 1GB
=
=
25
Fourth-order Butterworth Low-pass Filter
26
2 2
2 20 00 0
1s s s s1 0.765 1 1.848
⎡ ⎤ ⎡ ⎤+ + + +⎢ ⎥ ⎢ ⎥ω ωω ω⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
Effect of finite GB
27
� Taking GB into account with f0 = 3.3 kHz (speech filter)
� Using 741 Op Amp having a GB of 1 MHz
� Q of the 2nd second-order filter changes by 1%
� Q of the first second-order filter changes by about 12.4%
High Pass Filter
28
( )
( )
and p
Natural frequency of the
active filter
2
2po
2i
2p pp
p0
m 1, n 0 0s-K
VV s s1 K 1
Q
1 K
= = =
ω=⎡ ⎤
+ + +⎢ ⎥ωω⎢ ⎥⎣ ⎦
ωω =
+
Natural frequency of the
active filter decreases by
a factor
The quality factor of the
active filter can be increased
to the required value through
suitable selection of
a p
1 K
Q Q 1 KQ
K.
+
= +
Passive HP filter
29
Active HP filter
30
Active HP Filter (contd.,)
31
( )
( ) ( )( ) ( )( )
where and
Required is obtained by selecting .
is determined for a specified and
2
2 2 po 0p 02
i 1 1 2 22
00
a p
1 1 2 2 2 2 1 1 1 2
a
p 0
s-KV 1 ;V C R C Rs s 1 K1
Q
1 KQ Q 1 KC R C R C R C R 1 R R
Q KK.
ωω= ω = ω =
++ +ωω
+= = ++ +
ω ω
Topology of active HP filter
32
Example If the lower cut off frequency is selected as 0.4 Hz.
Assuming and
for maximally flat response
For
1 2 1 2 p
a
a p
0
1C C C R R R; Q3
1Q2
1 K 1Q Q 1 K ;K 3.53 21 1 12 0.4 ;RC
RC 1 K RC 4.5 2 0.4 4.51R 100k ;C
0.8
= = = = =
=
+= + = = =
ω = π × = = =+ π ×
= Ω =π
1.8 F4.5
= µ
33
Simulation
34
Single Op Amp Topology
� Buffer amplifier can be removed by suitable adjustment of the resistances
35
R1 = R = R3//R4 and C1 = C2 = C
Q of active filters
36
( )
of the circuit gets enhanced by a factor of
in case of low-pass and high-pass active filters
Natural frequency of the active low-pass filter
Natural frequency of the active high-pass filt
0 p
Q 1 K
1 K
+
ω = + ω
( )er p0
1 K
ωω =
+
Effect of finite GB
� High Pass filter with f0 = 400 Hz and Q = 5 � Q = 5 gives K = 224 � For f0=400 Hz and R = 100 kW gives C = 265 pF
37
Effect of finite GB (contd.,)
( )
( )
( )
( )
⎡ ⎤+⇒ ⎢ ⎥
⎢ ⎥⎣ ⎦⎡ ⎤+⎢ ⎥ ω⎣ ⎦=
⎡ ⎤⎛ ⎞+⎧ ⎫+ + +⎢ ⎥⎨ ⎬⎜ ⎟ ωω⎩ ⎭⎢⎝ ⎠ ⎥⎣ ⎦⎡ ⎤+⎢ ⎥ ω⎣ ⎦=
⎡ ⎤ω⎛ ⎞+ + + +⎢ ⎥⎜ ⎟ωω ⎝ ⎠⎢ ⎥⎣ ⎦
K changes because of finite GB to
2
2po
2i
2p pp
2
2po
2i 0 a
2p pp
2 1 K sK K 1-
GB
2 1 K s s-K 1-GBV
V 2(1 K)s s s1 K 1- 1GB Q
2 1 K s s-K 1-GBV
V 2K Qs s1 K 1 1Q GB
38
Q of the high-pass filter simulated using 741 Op Amp having a GB of 1 MHz changes to 2.63 that is by 48%
Conclusion
39
Conclusion
40
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