Week 6 - Analog Filters.pdf for analog electronics

Analog Circuits and Systems
Prof. K Radhakrishna Rao
Lecture 21: Filters
1

Review
 Integrators as building blocks of filters
 Frequency compensation in negative feedback systems
 Opamp and LDO frequency compensation
 Log-Antilog amplifier frequency compensation
 Analog filters
 Digital filters
2

What are Filters?
 Characteristics of an ideal
band pass filter
3
 Characteristics of an ideal
band stop filter

What are Filters?
 Characteristics of an ideal low
pass filter
4
 Characteristics of an ideal high
pass filter

What are Filters?
 Characteristics
of an ideal all
pass filter
5
 Characteristics
of a practical
filter

Where filters are used?
 FM Receiver
 Electrocardiograph (ECG)
 Music systems and hearing aids
◦ Graphic equalizer
◦ Parametric equalizer
6

FM Receiver
 Much of the radio transmission is done through frequency
modulation (FM)
 Spectrum allocated for FM transmission is 87.9 MHz to 107.9 MHz
 Several radio stations operate within this spectrum spaced 200 KHz
apart
 FM 102.9 means the carrier frequency used by this station is
102.9 MHz
 The radio station has to filter signals outside 102.9 MHz + 75 KHz
using a band pass filter before transmitting
 FM receiver should have a tuned circuit, which is a band pass filter,
associated with its antenna to select the station FM 102.9
7

FM Receiver (contd.,)
 Tuned circuit must have a Q (center frequency/band width) of
686 = 102.9/0.15
 Receiver uses an intermediate frequency of 10.7 MHz
 The local oscillator is adjusted to produce 113.6 MHz
(102.9 + 10.7)
 The mixer produces output in the frequency bands of (10.7+0.075)
MHz and (102.9+113.6+0.075) MHz
 A band pass filter with centre frequency of 10.7 MHz and a band
width of 150 KHz is used to select signal
8

Electrocardiograph (ECG)
 Instrument for recording the electrical activity of the heart
 ECG electrodes produce signals in the range of 0.05 Hz to 1 kHz
and 0.1 to 300 mV
 Interference signals include
◦ 50 Hz interference from the power supplies
◦ motion artifacts due to patient movement
◦ radio frequency interference from electro-surgery equipments
◦ defibrillation pulses, pace maker pulses, other monitoring
equipment, etc.
9

Electrocardiograph (ECG) (contd.,)
 Modern ECG will have monitor mode and diagnostic mode.
◦ Monitor mode: high pass filter is set at either 0.5 Hz or 1 Hz and
the low pass filter is set at 40 Hz. This limits artifacts for routine
cardiac rhythm monitoring. High-pass filter helps reduce
wandering baseline and the low-pass filter helps reduce 50 Hz
power line noise.
◦ Diagnostic mode: high-pass filter is set at 0.05 Hz, which allows
accurate ST segments to be recorded.The low-pass filter is set to
1000 Hz, in which case a notch filter becomes necessary at
50 Hz.
10

Music systems and hearing aids
 Necessary to adjust amplification of signals differently in different
frequency bands
 Equalization: Compensates for the acoustical properties of the
environment and characteristics of receptor (loud speaker and ear)
 Equalization may require
◦ Low frequency shelf filter (bass level controller): the gain is unity
above a certain critical frequency
◦ High frequency shelf filter (treble level controller): the gain is
unity below a certain critical frequency (shelf frequency)
◦ Graphic equalizer
11

Music systems and hearing aids (contd.,)
 When a low frequency shelf filter is combined with a high frequency
shelf filter, it can act as a versatile tone controller.
 Graphic equalizer
◦ Adjusts the relative loudness of audio signals in various
frequencies
◦ Permits a very detailed control of amplitude vs frequency control
◦ Requires several overlapping band pass filters with independent
gain controls over these bands
 Parametric equalizer
◦ graphic equalizer which provides independent control over the
gain, center frequency, bandwidth, and skirt slopes for each filter.
12

Filters
 Can be passive electrical networks or active electronic circuits.
 Historically all filtering functions used to be realized using passive
filters using R, L and C
 With the advent of transistors and integrated circuits there has
been requirement for size reduction of filters.
 This led to the development of active RC filters.
 LC filters are the most reliable units in the microwave range as the
size of the passive components in this frequency range become
small.
13

Filter functions
 Ideal filters (box like behavior) cannot be realized because of
requirement of multiple values at the edge of pass band.
 Electronic circuits can only realize single valued functions.
 Single valued functions that are approximations to the ideal multi
valued filter functions
14

Ideal band pass filtering function
 x = x0 is the center point of the band and δx is the band width
 Normalized filter function
 X = x – x0 and (δx/2) = 1.
 Bandwidth of the normalized function is 2.
15
0 0
0 0
δx δx
T =1 x - <x< x +
2 2
δx δx
T =0 x<x - x>x +
2 2
   
   
   
for
for and
1
T =1 -1<X<+1
T =0 X< X>1
−
for
for and

Practical normalized function
17

Physically realizable functions
 Order of D(X) must be higher than that of N(X) so as to make the
function go to zero as X increases to + ∞.
 As the function is symmetric around X = 0 it has to be an even
function of X.
 The slope at X = 0 is zero (flatness).
 A function with these three properties is called a flat function.
18
N(X)
T
D(X)
=

Physically realizable functions (contd.,)
 Flat Function
19
2 4 2m
0 1 2 m
2 4 2n
1 2 n
H 1 N X N X .. N X
T
1 K X K X .. K X
m n
 
+ + +
 
=
 
+ + +
 
<
where

Maximally flat function
 A flat function that has all its (n-1) derivatives at X = 0 should be
zero
 This requires
◦ N1 = K1; N2 = K2 . . . Nm = Km and Km+1 = Km+2 . . . = Kn-1 = 0 and
Kn ≠ 0.
 T will have a value between (1+ε1) and (1-ε2) , where ε1 and ε2 are
small (<<1) positive values, within the band defined by -1<X<+1 .
20

Maximally Flat Function based on Taylor Series
21
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
2
o o
o o o
n
3 n
o o
o o
f x f x
f x f x x x x x
1! 2!
f x f x
x x x x
3! n!
′ ′′
= + − + −
′′′
+ − −


Second order maximally flat function
22
ε2 = 0 and ε1 = 0.05, 0.1, 0.2.
1
4
1
1
T
1 X
+ ε
=
+ ε
These Maximally
Flat Functions are
also known as
Butterworth
functions

Butterworth functions
23
First, second and third order for ε1 = 0.1 and ε2 = 0
1
2
1
1
4
1
1
6
1
1
T
1 X
1
T
1 X
1
T
1 X
+ ε
=
+ ε
+ ε
=
+ ε
+ ε
=
+ ε

Butterworth functions (contd.,)
 Higher order Butterworth functions have better pass band
response and higher rates of attenuation in the stop band.
 Rate of attenuation close to the edge of the pass band of
Butterworth functions may not always be acceptable.
24

Functions to improve response at pass band edge
25
2 4
1 2
2
1
2 4
1 2
1
T
1 K X K X
1 N X
T
1 K X K X
=
+ +
+
=
+ +
2
1 1 2
where K >0
where N > K and K > 0

Chebyshev Function
 This function can approximate a box like
behavior by having
 K2 to be positive
 Choosing K1 and K2 to have T =1 at X =1
 This requires K1 = -K2.
 T peaks at X2 = 0.5 and attains a value of
(1+K2/4) leading to K2=4ε1 (1+ε1)
 For a specified variation in the pass band
(ε1) the parameters of the flat function can
be chosen
26
2 4
1 2
1
T
1 K X K X
=
+ +

Example
 ε1=0.05 and K2 = 0.19 ; ε1=0.1 and K2=0.0.367 ; ε1=0.2 and K2=0.0.667
 Second order Butterworth function with ε1=0.05
27
With higher ε1 it is
possible to have
faster rate of
attenuation at the
edge of pass band.

Chebyshev Function
 This function will have value 1+ε1 at X = 0.
 N1, K1 and K2 are selected to have
T=1 at X = 1.
 K2 will be positive and N1 = K1 for the
function to be maximally flat.
 K2 = (1+N1) ε1
28
( ) 2
1 1
2 4
1 2
1 (1 N X )
T
1 K X K X
+ ε +
=
+ +

Example
 Consider positive values of N1/K1 with N1 = K1 = 0.5 and 1
 ε1 = 0.1
 K2 will then be 0.15 and 0.2.
 Flat function with numerator polynomial in comparison to second
order Butterworth function
 It is observed that with positive values of N1 the response of flat
function with numerator polynomial is inferior to the second order
Butterworth function.
29

Inverse Chebyshev Function
 Negative values of N1 can make the function go to zero at X>1 if
|N1|<1.
 If ε1 = 0.1 at X =0 and ε2 = 0 at X = 1, then K2 is positive
 N1 = K1 and K2 = (N1+1) ε1
 If the function is to become zero at X = 2, then N1 = K1 = -0.25
and K2 = 0.075.
 If ε1 = 0.1 at X =0 and ε2 = 0.5 at X = 1, then K2 is positive, N1 = K1
and K2 = (N+1) ε1.
 If the function is to become zero at X = 2, then N1 = K1 = -0.25
and K2 = 0.9.
31

Inverse Chebyshev Function (contd.,)
32
•The functions with different values of ε2 pass through zero at X =2.
•The behavior of the function in the stop band beyond X = 2 is better
for function with ε2 = 0.5 while the
function has much better behavior
in the pass band with ε2 = 0.

Elliptic Function
 If |K1| > |N1| the response will slightly peak within the pass band
and a better rate of attenuation at the edge of pass band.
 ε1 = 0.1, N1 = -0.25 and K1 = -0.35.
 K2 = 0.175 for ε2 = 0 and K2 =0.675 for ε2 = 0.5.
 The behavior of the function in the stop band beyond X = 2 is
better for function with ε2 = 0.5 while the function has much better
behavior in the pass band and at the edge of the pass band with
ε2 = 0.
33

Elliptic Function (contd.,)
34
The functions with different values of ε2 pass through zero at X =2.

Wideband Using Staggered Narrowbands
35

Lecture 22: Passive Filters
1

Review
 Second Order Filters
2

Review (contd.,)
 Higher order wide band filter with stagger tuned narrow band
filters of lower order
3

What are passive filters?
 Filters that use only passive components R, L, C and
transformer are known as passive filters.
 Before the commercial availability of Op Amps, all base band
filters were mainly passive in nature because of reliability,
precision, and low sensitivity to temperature variations and
aging.
 Transformers were mainly used for impedance matching.
 Present day base band filters no longer use discrete
transformers.
4

Passive base band filters
 Passive filters are still used in microwave region
 Interconnect models also are low pass passive filters
 Passive filters are mainly designed as first or second
order filters
 In higher order passive filters the coefficients in the filter
functions can become very complex functions of passive
component values.
5

First Order Passive Low Pass Filters
 A first order RC-network
6
( )
o
i
o
i
2 *
o o o
i i i
2 2
V 1
V 1 sCR
V 1
V 1 j CR
V V V
V V V
1 1
1
1 CR
CR
=
+
=
+ ω
   
⋅
   
   
= =
+ Ω
+ ω
Ω = ω

For sinusoidal excitation
where

First Order MFM
 It is similar to a Maximally Flat Magnitude (MFM) (Butterworth)
function
 Response is similar to that of low pass filter.
 Square of magnitude and the delay are frequency dependent in
the pass band.
 We always consider the filter response in the region Ω2>0
7
( )
1
o
i
0 0
2 2
0
V
tan CR ; delay
V
CR 1 1 1
;
CR
1
1 CR
− ∂φ
φ = = − ω = φ − τ
∂ω
 
∂φ  
− = τ = = ω τ ω = =
∂ω τ
+ Ω
+ ω
 
where

Magnitude and Delay Plots
8
The normalized magnitude
and delay
plots of this filter are
relevant for >0.
o
2
i
0
2
V 1
,
V 1
T
 
 
 
+ Ω
 
 
τ
=
 
τ
 
Ω


Magnitude and Delay Plots (contd.,)
 Ω = 1 is recognized as the (half-power) bandwidth of the filter.
 Filters with maximally flat magnitude function are called
Butterworth filters
 Filters with maximally flat delay characteristics are called Bessel or
Thompson filters.
 Rate of attenuation at the edge of pass band (Ω = 1) is -0.5
9

First Order Low Pass R L Filter
10
( )
o
i
0
0
V 1
L
V
1 s
R
1 R
RC L
=
+
ω= =
ω ω

First order RC and RL low pass filters
have a bandwidth of
The magnitude decreases in the
stop band at the rate of
20 dB/decade or 6dB/Octave).

Second Order Butterworth Passive Low Pass Filter
 The second order Butterworth filter will have a magnitude function
similar to
where ε2 indicates the deviation from 1 in magnitude at X = 1
11
o
2
i
V 1
V 1 sCR s LC
=
+ +
4
2
1
1 X
+ ε

Second Order Butterworth Passive Low Pass Filter
12
( )
ω
= =
ω
=
ω =
+ ω − ω
Define
Substitute
2
2
o 2
0
o
2
i
1 s
s LC
LC
V 1
s j
V 1 j CR LC
( )
=
 Ω 
− Ω +  
 
=
 
+ − Ω + Ω
 
 
2
o
2
2
i 2
2 4
2
V 1
V
1
Q
1
1
1 2
Q
ω
Ω
= ω
= = =
ω ω
= =
+ Ω
where where and
is known as quality factor.If then
0
0 0
o
4
i
1 1 L 1
Q
CR C R
LC
V
1 1
Q Q
V
2 1

Phase of the second-order filter
13
=
+ ω − ω
o
2
i
V 1
V 1 j CR LC
=
Ω
− Ω +
o
2
i
V 1
V
1 j
Q
( )
Ω
φ
Ω
-1
0
2
i
Q
V
Phase of = = -tan
V 1-
∂Φ
Τ = −
∂Ω
Delay
2
2 4
2
1 1
1
1 ( 2 )
 
 
+ Ω
 
Τ =  
 
 
+ − + Ω + Ω
 
 
 
 
Q
Q
Tmax = 2Q at Ω = 1

Second Order Low Pass RLC Filter
14

Band Pass Filter – Fourth Order
17

Maximally Flat Function
18
For delay (Q = 1/sqrt(3))
Thomson’s/Bessel’s filter

Chebyschev or Equi-ripple Low Pass Filter
19
( )
1
peak 1 1
2 2
1
1
If then where
K 1 1
= where K = 2- and the peak 1+ =
2 Q K
1-
4
for =0.1
o
1 2
2 4
i
1
1
2
1 1
V
1 1 1
Q K 2
V Q
2 1 K
2
K 0.83
1
1
2
> = =
−
− Ω + Ω
Ω ε
= = ε
+
ε + ε

Chebyschev or Equi-ripple Low Pass Filter (contd.,)
20
 Filter with gives a better
performance at the pass band edge
(faster rate of attenuation).Achieved at
the cost of deviation from flatness
(ripple) in the pass band
 Functions of the type
are known as second order Chebyshev
functions.
1
2
Q >
( )
2 4
1
1
1 K
− Ω + Ω

Inverse Chebyshev Low Pass Filter
 Addition of a zero to a Chebyshev function improves the response
at the pass band edge
 It is known as inverse Chebyshev function
21

Elliptic Filter
22
R=40, L2 = 0.9m, L1=0.1m, C=0.1micro

Inverse Chebyshev Low Pass Filter (contd.,)
23
( )
( )
( )
( ) ( )
( )
and
2
2
1
o o
1
2 2 2
2
i i
1 2
1 2
2
2 1 2
1
2
p
z
1 L C
V V
1 s L C
;
V V
1 s L L C sCR
1 L L C CR
L L
1 1
K Q
CR C R
− ω
+
=
+ + +
− ω + + ω
+
ω
=
Ω = =
ω
ω
( )
( ) ( )
where
becomes zero when
2
2 2
z 1 2 p
2
1 1 2
p
2
1
o o
i i 1
2 4
2
1 1
; L L C ;
L C L L C
1 N
V V 1
;
V V N
1
1 2
Q
ω
ω = ω + = =Ω ω =
+
ω
− Ω
Ω
=
 
+ Ω − + + Ω
 
 

Inverse Chebyshev Low Pass Filter (contd.,)
24
If is selected to be less than 1, zero occurs outside the pass band
becomes zero for
is of the type
For this function to become a maximally flat
1
o
i
2 2 2 4
o 1 1 1
2 4
2 4
i 1
1
1 1
N
V
1
V
V 1 N X 1 2N X N X
V 1 K X X
1 K X X
2N K .
Ω >
− − +
=
− +
− +
= For
the zero will occur at For the response will peak in
the pass band and will have higher rate of attenuation at the edge of the pass band.
1 1
1 1
1
2N K 0.5
1
X 2. K 2N
N
= =
= = >

Elliptic Low Pass filter
 Filter with a zero(s) in stop
band and peak(s) in the pass
band
 For N1=0.25 and K1=0.7 the
response of the Elliptic filter in
comparison with the inverse
Chebyshev and second order
Butterworth filters.
25

Lecture 23: Passive Filters
1

Review
 First order and second order low-pass filters
 Butterworth (MFM) Filters
 Chebyshev and inverse Chebyshev Filters
 Elliptic Filters
 Bessel’s Filters (Maximally Flat Delay)
 Passive Filters (RC, RL and RLC)
2

First Order High Pass Filters
4

First Order High Pass Filters (contd.,)
5
 
 
o o
i i
2 * 2
2
o o o
2 2
i i i
0
0
sL
V V
sCR R
L
V 1 sCR V
1 s
R
CR
V V V X
V V V 1 X
1 CR
1
X is the Bandwidth(BW)
CR
 



   
   
   

 
   

  

where and

Magnitude and delay response
of the high pass filter (maximally
flat functions of first order)
Rate of attenuation of magnitude
at the edge of pass band
(X = 1) is 0.25
6

Second order RC low pass filter
7
 
 
o
2
i 1 1 2 1 2 1 1 2 2
1 1 2 2 1
0
2 2 1 1 2
1 1 2 2
V 1
V 1 s C R C R R s C R C R
C R C R R
1 1
; 1
Q C R C R R
R C R C

 
   
 
 
   
 
 
For maximizing
occurs when
and
As is a
parameter that determines
the characteristics of the
filter a maximum value
of 0.5 leaves very little
scope for general design
of l
2 1
max
1 1 2 2
max
Q
R R
Q
C R C R
1
Q . Q
2


ow-pass filters

Second order low pass RLC filter
8
where where
2
o
2 4
i
2
0
0
V 1
V 1
1 - 2 X X
Q
1
X
LC

 
 
 
 

  

and the quality factor Q
where
0
0 0
0
1 L 1
Q
CR C R
X LC 2 f
 


     


Example: Second Order Low-pass filter
While the C and R values are reasonable the inductor with 15.6 H will
become too big to be accommodated
9
 
Bandwidth = 40 Hz;
1
For maximally flat response Q=
2
This determines
0 80
If C 1 F; L 15.8H !
1 L
R 1000 31.2 5.62k
Q C
  
  
   

FrequencyTransformation
 Design of high pass, band pass and band stop filters
can be done starting from the corresponding low
pass prototype.
 Low-pass filter function is expressed as a function of
X2
10

Low-pass to High-pass transformation
11
Low-pass High-pass
2
o
0
2
i 0
V 1 1
where X
V 1 X CR

   
 

Low-pass to High-pass transformation (contd.,)
12
0
when =BW Bandwidth of the filter
Magnitude function of the high-pass filter
Replace
and
2
2
o
2
i
2
X 1
V
1 X
Xby
X V 1 X
R 1
L CR BW
L CR
 


  

LP-HP Transformation
13
Low-pass High-pass
where
2
o
0
2
i 0
V 1 R
X ;
V 1 X L

   
 

LP-HP Transformation (contd.,)
14
when Bandwidth of the filter
Replace by
and
0
2
2
o
2
i
2
X 1 BW
V
1 X
X
X V 1 X
L R 1
C BW
R L CR
  


  

Low-pass to High-pass transformation
If X2 is replaced by 1/X2 then low-
pass filter gets transformed to a
high-pass filter of the same band
width.The range of X2 from 0 to
∞ gets transformed to the range
∞ to 0 for 1/X2.
15
Square of magnitude vs frequency (First-order filter)

Example
 Design a first-order high-pass filter with f0 = 40Hz; 0 = 80
 Let us design first-order a low-pass filter prototype with 0 = 80
 RC = /80, If C = 1 F, R = 5.7 k, L=CR2= 15.7 H (!)
16

Low-pass to Band-pass Filter
17
where
Replace by
where the center frequency of
band-pass filter isK(BW)
2
o
2
i 0
2
2
2
4
o
2 4
i
2 2 4
V 1 1
X ; BW
V CR
1 X
K
X (X )
X
X
V K
V 1 X X
1 2
K K K

  




 
 
  
 
 
 
 

Low-pass to Band-pass Filter (contd.,)
 If C replaced by a parallel
resonant circuit with
resonance frequency K(BW)
18
Band-width of the band-pass filter
remains the same as that of
low-pass filter
and hence
2 2
1
BW
RC
1
K(BW)
LC
1
L
K (BW) C




Low-pass to Band-pass Transformation
19
Square of magnitude vs frequency

Example
Design a second-order band-pass filter with center
frequency = 5 kHz and a band-width of 1 kHz.
 Start with the design of first-order a low-pass filter
prototype for a bandwidth = bandwidth of the band-
pass filter (1kHz)
 For a C of 1 F R = 159 
 L for a resonance of 5 kHz = 0.987 mH
20

Example: Low-pass to Band-pass Transformation
21

Low-pass to Band-stop Filter
22
where
Replace by
where the center frequency of
band-stop filter isK(BW)
2
o
2
i 0
1
2
2 4
2
2 4
o
2 4
i
2 2 4
V 1 1
X ; BW
V CR
1 X
K
X X
X
X X
1 2
V K K
V 1 X X
1 2
K K K


  


 

 
 
 

 
 
  
 
 
 
 

Low-pass to Band-stop Filter (contd.,)
 If C replaced by a series
resonant circuit with
resonance frequency K(BW)
23
Band-width of the band-stop filter remains
the same as that of low-pass filter
2
2
2 2 2
1
BW
RC
1
L CR
(BW) C
1
As K(BW)
L C
1 C
C
K (BW) L K

  

 
  


Example
 Design a 4th-order Chebyschev band-pass filter centred around
10 k Radians/sec with a bandwidth of 1000 Radians/sec.
 The proposed BP filter can be designed from second-order
Chebyschev LP filter prototype with a bandwidth of 1000
Radians/sec. It will have a Q of 1(> 1/2).
 The second-order Chebyschev LP filter will be a RLC filter. If we
assume a values of 1 F for C, from the relation: bandwidth = 1/RC,
the value of resistance is 1 k
 L = CR2=1 H
24

Frequency response of the LPF
25

Frequency response of the LPF (contd.,)
 The corresponding BPF
 L is replaced by L forming a series resonant circuit at 10 k
Radians/sec with a capacitor C/= 1/0
2L = 10 nF for L = 1 H.
 C is replaced by C forming a parallel resonance at the same
frequency 10 k Radians/sec. C of 1 F results in an inductance
L/=1/0
2C = 10 mH.
26

Example
 Design a 4th-order Chebyschev band-stop filter centred around 10
k Radians/sec with a bandwidth of 1000 Radians/sec.
 The proposed BS filter can be designed from second-order
Chebyschev LP filter prototype with a bandwidth of 1000
Radians/sec. It will have a Q of 1(> 1/2).
 The second-order Chebyschev LP filter will be a RLC filter. If we
assume a values of 1 F for C, from the relation: bandwidth = 1/RC,
the value of resistance is 1 k
 L = CR2=1 H
28

Frequency response of the LPF
29

 L is replaced by L forming a parallel resonant circuit at 10 k
Radians/sec with a capacitor C/= 1F for L/ = 10 mH.
 C is replaced by C forming a series resonance at the same
frequency 10 k Radians/sec. C// of 10 nF results in an inductance
L//=1/0
2C = 1H.
30

31

Example
32
 Let us consider using inverse Chebyshev filter as given in the figure
 
 
2
1
o
i 2 4
2
p
p 1 2
1 2
0
1 N
V
V 1
1 2
Q
1
L L C
L L
1 1
Q
CR C R
 

 
     
 
 

   
 

 

where where
and the quality factor

Example (contd.,)
33
 
where with
where is chosen as
where (interferance)
1 2
p
p
2
p
1 z
z
z z
1 2
L L C
2 f f 40Hz
N
2 f f 50Hz
1
2N 2
Q

    

   

 
 
 

 
 
 
 
If
For maximally flat response
with a zero
which determines
1 2
1 2
C 1 F
L L 15.6H
Q 1.18
L L
1
R 3.34k
Q C
 
 


  

Responses of the four low pass filters
 While the behavior of second order RLC filters is better, in
view of the large inductance values they are impractical at
these lower frequencies.Active RC filters, having active
elements like transistors, Op Amps and transconductors and
RC elements, can overcome the limitations of passive filters
34

Lecture 24:Active Filters
1

Review
 RC and RL low pass filters
 First order and second order filters
 Q of second order filters less than half
 RLC second order filters of any Q
 Low pass RLC Butterworth, Chebyschev and inverse Chebyschev
and Elliptic filter designs
2

Active Filters
 Limitations of passive RC filters can be addressed using active
elements
 Approaches in using active elements in designing filters
 Inductor simulation
 The problem of large size of inductor can be resolved using active
devices and RC elements to simulate the inductor in a traditional
RLC filter.
3

Active Filters (contd.,)
 Q enhancement by feedback
 Q of a passive second order RC Filter can be enhanced using
feedback and amplification
 Biquad
 Simulate nth order differential equations using n-integrators and
summing amplifiers. A simulator of a second order differential
equation is popularly known as Biquad.
 The traditional approaches to filter design through Q-enhancement
and inductor simulation are increasingly replaced by Biquad method
because of commercial availability of universal active filter blocks
(UAF 42 and UF 10).
4

Active Filters: Inductor Simulation
 All filters used in base-band applications, particularly in telephony,
require large valued inductances resulting in large sizes.
 These filters needed to be designed as active filters simulating large
inductances using active devices.
5

Miller’sTheorem
 A voltage amplifier with gain
G and an impedance, Z,
connected between input
and output terminals
simulates an impedance
at its input port
6
i i i
i in
i in
V GV I 1 Z
I; ; Z
Z V Z 1 G
−
= = =
−
Z
1 G
−

Simulation of Inductance in series with a Resistance
7
Series resistance be Inductance L be
which represents a first order high-pass filter
1 1 2
1
in 1 1 1 2
2
2
R ; CR R
R
Z R sL R sCR R
1 G
sCR
G
1 sCR
= = + = +
−
=
+

Modified L-simulator with only one buffer
 When the buffer 2 is shorted
 It simulates the same inductance in series with R1+R2
8

Bootstrapping Circuit Function
9

Simulation of Inductance // Resistance
10
1 1
2
2
R R
1
1 G
1
sCR
1
G
sCR
=
−
+
= −
The circuit simulating inductance in
parallel with resistance

Simulation of Inductance // Resistance (contd.,)
 If the first buffer is shorted the resultant circuit
11
It simulates the same inductance in parallel with R1 and R2.

Simulation of Ideal Inductor (Gyrator)
12
1
1 2
2
R
sCR R
1 G
1
G 1
sCR
=
−
= −

Band Pass Filter
 Design a second-order band-pass filter with
center frequency = 5 kHz and a band-width of 1 kHz
 For a C of 0.1 µF R = 1590 Ω
 L for a resonance of 5 kHz = 9.87 mH
13
R
Q 5
L
C
= =

BP Filter with simulated inductance
 L= CR1R2 = 9.87 mH
Let R1 = R2
 R1R2= 9.87 x 104
 R1=R2= 314 Ω
14

Frequency response of the BP filter
15

Transient response of BP filter for Q = 5
16

Transient response of BP filter for Q = 10 (R =3180 Ω)
17

Increasing Q by Negative Resistance
 Negative resistance is simulated across the simulated inductance
 As the gain of the first amplifier is 2, and a resistance of RP is
connected between its input and output, according to Miller’s theorem,
negative resistance gets simulated in shunt with simulated inductance
18

Increasing Q by Negative Resistance (contd.,)
19
P
For R = 3140 the system becomes unstable
P
P
R
R ;
1 2
=
− Ω
−

Increasing the resonant frequency
 Frequency is increased by decreasing the value of simulated inductance
 Simulated inductance can be decreased by reducing the values of R1, R2
and/or C with C =0.01 mF and Op Amp 741 with GB of 1 MHz
20
The amplitude of
oscillation is now
limited by slew
rate (1V/µ sec) and
not by saturation

Effect of Active Device Parameters
 Simulated inductance is influenced by the parameters, DC Gain
(A0) and Gain-Bandwidth Product (GB)
 Gyrator circuit uses non-inverting amplifier of gain (=2) followed
by an integrator
 Ideal value for G
21
0
G 1
s
ω
 
= −
 
 

Effect of Active Device Parameters (contd.,)
22
With finite gain A and DC gain of of Op Amps
0
0
0 0
0
0 0 0
A
1
s 3
G 1 1
s A sA
1
2 s
1 1
A A
3 3
1 1
A sA s A sA
ω
 
−
  ω ω
   
  − − −
   
ω
     
+
 
 
+ +
 
 
   
 
 
ω ω ω
 
= − − − − −
 
 


23
where
2
0 0 0 0
2
2
0 0
2
0 0
0 0
0
0 0
in 0 0
0 0
0
0
0 0
3
G 1
s sA sA s A
1 3
1 1 atDC
s A A
A
1 2
1 1
A s A
R R 1 2
Z 1
1
1 G A
1 2
1
RA sR
A s A
ω ω ω ω
= − − + +
 
ω ω
= − − + −
 
ω  
   
ω
= − − −
   
   
 
′
= = = ω =ω −
 
′
ω
−  
ω  
+
+ −
 
 

24
Inductance is shunted by a
negative resistance
0
0
0
2
0 0 0 0
2
0 0
Q A
R
L
RA
3
G 1
s sA sA s A
2
s
1
s GB GB
=
=
′
ω
ω ω ω ω
= − − + +
ω ω
− − +

Effect of finite gain bandwidth
product is to slightly increase
the inductance and add a
negative resistance
in shunt with the Inductance
0
0
0
R
L
1
GB
RGB
2
=
ω
 
ω −
 
 
ω

Example
 Band-pass filter with simulated inductance
 For C = 0.1 µF, R = 1 kΩ
25
0
i
V
1
V
=

Effect of finite GB
 Q =10 f0=1.59 kHz ;With GB = 1 MHz
the negative resistance = 314 kΩ; Gain changes to 1.033
26

Effect of finite GB (contd.,)
 BP filter with Q = 100
 Q =100 f0=1.59 kHz;With GB = 1 MHz
the negative resistance = 314 kΩ; Gain changes to 1.47
27

Effect of increased frequency
 Q =100 f0=15.9 kHz
 The circuit oscillates. Negative resistance simulated is
31.4 kΩ < positive resistance of 100 kΩ used in the circuit
28

Effect of increased frequency (contd.,)
 Amplitude of
oscillations gets
limited by the slew
rate of the Op Amp
which is 1V/µ sec
 Filter designed with
simulated inductor
will require usage of
an Op Amp with
GB >>f0Q
29
Sensitivity Sensitivity 0
A
A
0
Q 0 0
GB GB
Q
Q
2 Q
1
GB
2 Q
S ; S
GB GB
ω
=
ω
 
−
 
 
ω ω
=
− =
−

Q-enhancement
 due to finite GB of the active device
30
Sensitivity
Sensitivity
A
0
0
A 0A
0
0
Q 0
GB
0
GB
Q
Q ;
2 Q
1
1
GB
GB
2 Q
S
GB
S
GB
ω
ω
= ω
=
ω
ω
 
−
−
 
 
ω
= −
ω
= −

Generalization of Gyrator
 Positive Impedance Inverter
31
1 3 5
in
2 4
Z Z Z
Z
Z Z
=

Generalization of Gyrator (contd.,)
 When Z1=Z2=Z3=Z5=R and Z4 = 1/sC the resultant
inductance simulator
32

Generalization of Gyrator (contd.,)
 When Z1=Z3=Z4=Z5=R and Z2 = 1/sC the resultant
inductance simulator
33

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
A
Q
2 A
1
GB
Q
Q
Q 2 Q
1
A 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 s
D s

Use of feedback to enhance Q
6
( ) ( )
( ) ( )
( )
( ) ( )
o
i
-K N s D s -KN s
V
V D s KN s
1 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
2
p
p
2
2
p p
p
p
s s
m n p
N(s)
D(s) s s
1
Q
Q
 
+ +
 
ω
ω
 
 
=
 
+ +
 
ω
ω
 
 
ω
filter

Quality Factor
8
is always < 0.5
p
2
2
p p
o
2
i
p
2
p p p
0 p
p
a
p
Q
s s
-K m n p
V
V s s
(1 mK) (1 nKQ ) (1 pK)
Q
1 pK
1 mK
Q
Q (1 pK) (1 mK)
(1 nKQ )
 
+ +
 
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<|nKQp|<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
p
1 1 2 2
p
1 1 2 2 1
2 2 1 1 2
1 2 1 2
p p
1
R C R C
1
Q
C R C R R
1
C R C R R
R R R C C C
1 1
Q
RC 3
ω =
=
 
+ +
 
 
= = = =
ω
= =
( ) ( )
( )
2
1 1 2 2 1 1 2 1 2
N(s) 1
D(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-
o
2
i
2
p p
p
0 p a p
a
m 0, n 0 and p 1
V -K
V s s
(1 K)
Q
1 K ; Q Q 1 K
Q
K
= = =
=
 
+ + +
 
ω
ω
 
 
ω = ω + = +
pass passive filter with amplifier gain - and feedback
K
and
1 2 1 2
R 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 ofVCVS will have to be made 2K
 AVCVS 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
1
Q
2
R R R
1
C C C Q
3
1 K 1
Q Q 1 K
3 2
=
= =
= = =
+
= + = =
and
For
0
K 3.5
1 K 4.5
2 40
RC RC
4.5
RC
2 40
R 100k ;
4.5
C F 84nF
2 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=0.6µF; K=224

Frequency response of Low Pass Filter
18
with Q =5 and f0 = 400 Hz C = 60 nF

Transient response of Low-pass Filter
19

Low-pass Filter
20

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 s
V K
K 1 -
(1 2K)s s
V GB
1 1
GB GB
2 1 K s
K 1 -
G
V
V
 
+
=  
 
+
     
+ +
   
   
+
=

( )
2
2
p p
p
B
2 1 K s
s s
1 K 1 -
Q GB
 
 
 
 
 
+
+ + +  
 
ω
ω  
22

Effect of GB Product of Op Amp (contd.,)
( )
( )
( )
Normalizing
(due to GB)
o
2
i
2
0 0 p
p
a
0 p
2 1 K s
K
1 -
1 K GB
V
V 2K 1 K s
s s
- 1
GB(1 K)
Q 1 K
Q 1 K
Q
2K 1 K Q
1 -
1 KGB
 
+
 
 
+  
=
 
+
+ +
 
 
+
ω ω +  
+
=
 
+ ω
 
 
+
 
23
( )
GB should be large enough
to make 0 p
0 a
2K 1 K Q
1 KGB
2K Q
1
GB
+ ω
+
ω
= 

Examples
24
Ex:1
(specified) and Hz and
(due to GB)
a 0
p
a
0 a
Q 5 f 40 K 3.5
Q 1 K
Q 5.55
2K Q
1 -
GB
= = =
+
= =
ω
 
 
 
Ex:2
(specified)=5 and Hz and
(due to GB)
a 0
p
a
0 a
Q f 400 K 224
Q 1 K
Q 48
2K Q
1 -
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
1
GB
2Kf Q
1
GB


25

Fourth-order Butterworth Low-pass Filter
26
2 2
2 2
0 0
0 0
1
s s s s
1 0.765 1 1.848
   
+ + + +
   
ω ω
ω 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
2
p
o
2
i
2
p p
p
p
0
m 1, n 0 0
s
-K
V
V s s
1 K 1
Q
1 K
= = =
ω
=
 
+ + +
 
ω
ω
 
 
ω
ω =
+
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 K
Q
K.
+
= +

Active HP Filter (contd.,)
31
( )
( ) ( )
( ) ( )
( )
where and
Required is obtained by selecting .
is determined for a specified and
2
2
2 p
o 0
p 0
2
i 1 1 2 2
2
0
0
a p
1 1 2 2 2 2 1 1 1 2
a
p 0
s
-K
V 1
;
V C R C R
s s 1 K
1
Q
1 K
Q Q 1 K
C R C R C R C R 1 R R
Q K
K.
ω
ω
= ω
= ω
=
+
+ +
ω
ω
+
= = +
+ +
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
1
C C C R R R; Q
3
1
Q
2
1 K 1
Q Q 1 K ;K 3.5
3 2
1 1 1
2 0.4 ;RC
RC 1 K RC 4.5 2 0.4 4.5
1
R 100k ;C
0.8
= = = = =
=
+
= + = = =
ω = π × = = =
+ π ×
= Ω =
π
1.8 F
4.5
= µ
33

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 p
0
1 K
ω
ω =
+

Effect of finite GB
 High Pass filter with f0 = 400 Hz and Q = 5 gives K = 224
 For f0=400 Hz and R = 100 kΩ gives C = 265 pF
37

Effect of finite GB (contd.,)
( )
( )
( )
( )
K changes because of finite GB to
2
2
p
o
2
i
2
p p
p
2
2
p
o
2
i o a
2
p p
p
2 1 K s
K K 1 -
GB
2 1 K s s
-K 1 -
GB
V
V 2(1 K)s s s
1 K 1 - 1
GB Q
2 1 K s s
-K 1 -
GB
V
V 2K Q
s s
1 K 1 1
Q GB
 
+
⇒  
 
 
 
+
 
ω
 
=
 
 
+
 
+ + +
 
 
 
ω
ω
 
  
 
 
+
 
ω
 
=
 
ω
 
+ + + +
 
 
ω
ω  
 
 
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%

Lecture 26:Active Filters: Q-enhancement
1

Review
2
( )
( )
( )
( )
Second-order low-pass filter design
0 p
a p
0 p
0 a
1 K
Q Q 1 K
GB Q 2K 1 K
Q 2K
ω =ω +
= +
ω +
ω


Review (contd.,)
3
( )
( )
Second-order high-pass filter design
p
0
a p
0 a
1 K
Q Q 1 K
GB 2K Q
ω
ω =
+
= +
ω


Second Order Active Notch Filter
 The quality factor Q of
the active filter can be
increased to the required
value through suitable
selection of K.
 Qa can be changed
without altering the
natural frequency
 Qa is directly proportional
to the gain factor K.
4
( ) ( )
( )
Notch Filter and
Natural frequency of the active filter
2
2
p
o
2
i
2
p p
p
0 p
a p
m p n 0
s
-K 1
V
V s s
1 K 1 K
Q
Q Q 1 K
=
=
α =
 
α +
 
 
ω
 
=
 
+ α + + + α
 
ω
ω
 
 
ω =
ω
= + α

Second order passive notch filter
5
( ) ( )
( )
where
For the coefficient of ' ' in to
become zero
If and then
and
b
p
a b
2
2
p
o
2
i
2
p p
p
1 2 1 2
p p
R
Q
R R
s
-K 1
V
V s s
1 K 1 K
Q
s N s
R R R C C C
1 1
Q
RC 3
α
= =
+
 
α +
 
 
ω
 
=
 
+ α + + + α
 
ω
ω
 
 
= = = =
ω
= =

Active second order notch filter
6
with negative feedback
( )
a p
Q Q 1 K
1 K
1
3 3
= + α
 
= +
 
 

Example
7
Centre frequency of a notch filter is to be 50Hz and the to be 100.
If then
a
a
1 1
p
1 2 1 2 7
Q
1 K 1
Q 100 1 ;
3 3 3
1 1
K 897;R C
100
1 100
R R 100 k C C nF
10
 
= = + α
=
 
 
= = =
ω π
== Ω == =
π
π

Example (contd.,)
8
The 50Hz notch filter circuit; Rs=0.5k
R=25k; C≅32nF with a Qa of 100.

Example (contd.,)
9
( ) ( )
50 Hz notch filter where the instrumentation amplifier
(shown in dotted lines)
the gain
Transfer function now becomes
2
p 2
p
o a b
2
i 1 a b
2
p p
p
50k
K 1
R
s
-KQ 1
V R R
V R R R
s s
1 K 1 K
Q
Ω
= +
′
 
+
 
 
ω  
+
 
=  
+ +
   
+ α + + + α
 
ω
ω
 
 

Band Pass Filter
 Second-order Passive
Band-Pass filter
11
 Second-order Active
Band-Pass Filter

Band Pass Filter (contd.,)
o 0
2 2
i
0 0 a
a p
0 p
K s
-
V KCRs 1 K
V (CRs) 3CRs (1 K) s s
1
Q
1 K
Q Q 1 K
3
1 K
1 K;
3
ω
− +
=
   
+ + +  
   
+ +
 
ω ω
 
 
 
+
= + =
+
ω = + ω =
12

Example
13
Band-pass filter with and centre frequency of 50 Hz
Gain at the centre frequency
a
a
0
a
Q 5
1 K
Q 5
3
K 224
15
f 50
2 RC
for R 100 k ; C 0.477 F
KQ
75
1 K
=
+
= =
=
= =
π
= Ω = µ
=
+


Effect of finite GB
 Qa(due to finite GB) =
 Using 741 Op Amp with
GB of 1 MHz the gain increases from 75 to 79
15
0
1
a
a
Q
KQ
-
GB
ω

Q-enhancement using positive feedback
16
( )
( ) ( )
where is the natural
frequency of the passive
RC filter
o
i
2
2
p
p
2
2
p p
p
p
KN s
V
V D s - KN s
s s
m n p
N(s)
D(s) s s
1
Q
=
 
+ +
 
ω
ω
 
 
=
 
+ +
 
ω
ω
 
 
ω

Band-Pass Filter
17
( )
and
p
2
2
p p
p
0
o
2
i p
2
0 p
0
n 1, m 0 p 0
s
N(s)
D(s) s s
1
Q
s
K
V
V 1 - KQ s
s
1
Q
= = =
 
 
ω
 
 
=
 
+ +
 
ω
ω
 
 
 
 
ω
 
=
 
 
+ +
ω
ω
 
 
The natural frequency
of resultant
active filter is
0 p
p
a
p
Q
Q
1 - KQ
ω = ω
=

Active band-pass filter with positive feedback
18
0
o
2
i p
2
0 p
0
s
K
V
V (1 - KQ )s
s
1
Q
 
 
ω
 
=
 
+ +
 
ω
ω
 
 
a p
p
a
1
RC
Q
Q
K
1 -
3
1
3
K
1 -
3
ω = =ω
=
 
 
 
=
 
 
 
1 2
Assuming and C =C =C
1 2
R R R
= =

Example
1 1 2
5 2 8
1
2 40
100 39 7
2
Band-pass filter with centre frequency of 40Hz is and a of 5
leads to
and
If R = 100 k ; C=39.7 nF
40 Hz band pass active filter with and
a
p
Q
Q K .
C C C R R R
RC
R k C . n
= =
= = = =
ω = = π ×
Ω
= Ω = F
19

Example (contd.,)
 40 Hz band pass active
filter with R=100k ohm;
C=39.7nF
20
and
b
a
b
a
a
b
R
1 2.8x2 5.6
R
R
4.6; R 1 k
R
R 4.6k
+ = =
= = Ω
= Ω

Simulation
 Response of 40 Hz active band-pass filter with Q = 5
21

Q - Sensitivity to K
( )
{ }
{ }
( )
( )
1 3
1 3
1
1
a
p
a
p
p
Q
K a
p
Q
Q
- K
- KQ
Q K
S Q K
- KQ
= =
= =
23

Simulation
24
 For K=2.8, 2.825 and 2.85

Q – Sensitivity to GB in the case of positive feedback
( )
{ }
0
2
2
0
0
1
1
2 1
1
with replaced by
o
i p
p
s
K
V
V - KQ s
s
Q
K
K s
K
GB
 
 
ω
 
=
 
 
+ +
ω
ω
 
 
 
+
−
 
 
 
25

Observations on Q-enhancement
 All higher-order filters require higher Q at higher frequencies as
second-order building blocks
 Filters with high Q should not be designed using positive feedback
 Positive feedback can be used for Q enhancement for designing
Low Q filters
 Positive feedback permits independent adjustment of Q and f0
 Sensitivity of Q to passive parameter variation in filters using
positive feedback is as high as Q
 Filters with high Q should be designed using only negative feedback
 Sensitivity of Q to passive parameter variation in filters using
negative feedback is always less than one
26

Lecture 27: State Space Filters
1

Review
2
 Q enhancement of passive RC using negative and positive feedback
 Effect of finite GB of the active device on filter parameters
 LP Passive RC second order
where is the passive filter normalizing frequency
where is the passive filter Q;
K is the gain of the inverting amplifier
used in the negative feedback loop
p
p
Q
ω
0 1
p K
ω =ω + 1
a
Q p
Q K
→ +

Review (contd.,)
3
 HP passive RC second order
 LP + HP (notch) passive
 BP passive filter (RC) passive feedback
0
1
p
p a
p
Q
,Q
KQ
ω → ω =
−
0 1
1
p
a p
,Q Q K
K
ω
ω → = +
+
( )
0 1
p a p p
,Q Q KQ
ω → ω = +

Review (contd.,)
 Positive feedback because of high sensitivity to K used for only low
Qa
 Negative feedback because of low sensitivity to K but sensitivity to
GB in case of 3 and 4
 Independent Q adjustment and ω0 adjustment not possible
4
( )
( )
( )
0
1
1
total phase lag error in the loop
product is the criteria for low sensitivity to GB of
a
GB
a
a a
a
Q
Q
Q
f Q Q
Q
=
−
∆φ 

StateVariable Filters
 Are also known as
◦ Biquad filters (use two integrators)
◦ KHN filters (Kervin, Heulessman and Newcomb of Burr-Brown)
◦ Universal Active filters (UAF)
5

Active filter design as solution of differential
equation
6
( )
1
1 0
1
1
1 0
1
1
order linear differential equation order filter design
from state variable derive state variables using inte
th th
n n
o o
n o i i
n n
n n
o o
n o i i
n n
th
n n
d V d V
k k V k V
dt dt
d V d V
k k V k V
dt dt
n n -
−
− −
−
− −
→
+ + =
 
=
− + −
 
 
 

grators
Then sumup these with input.
Connect the summer output to input
results in the solution of nth order differential equation.

First-Order Filter
7
0
0
i
Is represented by a
first order differential equation
rewritten as
o
o i
o
o i i
dV
K V K V
dt
dV
- K V K V
dt
+ =
= +

Simulation of LP and HP filters
 using ideal integrators
8

First-order filter using Op Amps
9

Second-order filter
10
2
1 0
2
2
2 2
0
0 0 0
2
can be represented by a
second order differential equation
rewritten as
o o
o i i
o o
o i
d V dV
K K V K V
dt
dt
d V dV
- - V H V
Q dt
dt
+ + =
ω
= ω + ω

Simulation of LP, HP and BP filters
 using ideal integrators; f0=1.59kHz, Q=5
11

Phase Plot
12

Transient Plot
13

Second-order filter using Op Amps
14

Simulation Second-order filter with Op Amps
 (where effect of GB is minimal and f0 is 1.59 kHz; Q=5)
15

Simulation – Phase Plot
 (where effect of GB is minimal and f0 is 1.59 kHz; Q=5 changed
from 1, 5 and 9)
16

Simulation – Transient
 (where effect of GB is minimal and f0 is 1.59 kHz; Q=5)
17

Outputs of UAF for a square-wave input at f0
18

Simulation of Second-order filter with Op Amps
 (where effect of GB is significant and f0 is 15.9 kHz; Q=5)
 The effect of finite GB is on the peak and notch
19

Simulation – Transient
 (where effect of GB is significant and f0 is 15.9 kHz; Q=5)
20

Third-order filter
21
3 2
2 1 0
3 2
3 2
2 1 0
3 2
rewritten as
o o o
o i i
o o o
o i i
d V d V dV
K K K V K V
dt
dt dt
d V d V dV
-K - K - K V K V
dt
dt dt
+ + + =
= +

Third-order filter using Ideal Integrators
22
3
3
0
2 3 2 3
2 3 2 3
0 0
0 0 0 0
1
1 2 2 1 2 2
and
s
s s s s s s
ω
+ + + + + +
ω ω
ω ω ω ω

Third-order filter using Ideal Integrators
23

Third-order Butterworth using LF353 or TL082
 (where the effect of GB is minimal and f0 is 1.59 kHz )
24

Third-order Butterworth using LF353 or TL082
 (where the effect of GB is significant and f0 is 15.9 kHz)
25

Butterworth Low-Pass Filter
 Synthesis as third-order filter (Using LM741)
26

Butterworth Low-Pass Filter
 Synthesis as second-order filter followed by first-order filter
(Using LM 741)
27

Observations
 Higher even-order filters can be realized by cascading second order
filters functions.
 Higher odd-order filters is can be realized cascading one first-order
filter with required number of second order filters.
 Direct realization of higher order (> 3) using any of the Op Amps
will lead to inferior performance due to cumulative phase error in
the feedback loop
28

Outputs at different points in a second-order filter
29
( )
( ) ( )
Output can be taken at several points in the circuit:
, , and
Input, output relationships
- High Pass Filter
o1 o2 o3 o4
2 2
0 0
o1
2 2
i 0 0
V V V V
H s
V
V s s Q 1
ω
=
ω + ω +

Simulation
30
0 0 0
Gain at is H and at is H Q; Q=5; H
0 0 0
1;f 1.59kHz
ω ω ω = ω = =


31
( )
( ) ( )
0
- Band Pass Filter
Gain at is H Q
0 0
o2
2 2
i 0 0
0
H s
V
V s s Q 1
− ω
=
ω + ω +
ω = ω

Simulation
32
0
Q=5; H 0
1;f 1.59kHz
= =

33
( ) ( )
0 0
- Low Pass Filter
Gain at is H and at is H Q
o3 0
2 2
i 0 0
0 0
V H
V s s Q 1
=
ω + ω +
ω ω ω = ω


Simulation
34
0
Q=5; H 0
1;f 1.59kHz
= =

35
( )
( ) ( )
0
- Band Stop Filter
Gain at and is H , and at is zero
2 2
0 0
o4
2 2
i 0 0
0 0 0
H 1 s
V
V s s Q 1
 
+ ω
 
=
ω + ω +
ω ω ω ω ω =
ω
 

Simulation
36
0
Q=5; H 0
1;f 1.59kHz
= =

AddingVo1,Vo2 andVo3
 It is possible to realize any second order filter function
◦ where α1, α2 and α3 can be negative, positive or zero
◦ and a, b and c can be positive or negative and of zero or any
non-zero value
37
( ) ( )
2
1 o1 2 o2 3 o3
2 2
i 0 0
V V V as bs c
V s s Q 1
α + α + α + +
=
ω + ω +

All pass filter design
38
( ) ( )
( ) ( )
( ) ( )
2
1 o1 2 o2 3 o3
2 2
i 0 0
2 2
0 0
0
2 2
0 0
1 0
2
0
V V V as bs c
V s s Q 1
s s Q 1
H
s s Q 1
Q
2 tan
1
−
α + α + α + +
=
ω + ω +
ω − ω +
=
ω + ω +
ω
ω
φ = −
 
ω
−  
ω
 

Simulation - All pass filter design
 H0=1;Q=1;f0=1.59kHz; α1= α2 = α3 =1
39

Lecture 28: Universal Active Filter – Effect of Active
Device GB
1

Review
 First , second and third order state space filter realizations
 Universal active filter block
 Low-pass, high-pass, band-pass and band-stop filters
 Magnitude and phase plot
 Transient response
 UAF as a resonator block
2

Simulation – Resonator Block
4

UAF42
 It is an analog IC that serves as a second order active filter building
block that can be used to realize any filter.
 It is a four Op Amp circuit with two integrators
 Realization of any filter using UAF42 requires use of resistors
external to the IC to design the filter with specified Q, ω0 and H0.
5

Schematic diagram of UAF 42
 Universal Active Filter (courtesy:Texas Instruments)
6

Butterworth Low Pass filter
 Design a Butterworth low-pass filter with cut-off frequency of 4kHz
and a gain of 2 using Biquad
 The quality factor of a Butterworth filter is
 If we use UAF42, the integrating capacitor is 1 nF
7
1 / 2
( )
and 1 -9 3
1 0
1 1
R 1 k R 39.8k
2 C f 2 10 4 10
=Ω = = = Ω
π π ×

Low-pass filter circuit using UAF42
8

Response of the low-pass (4 kHz) filter
9
TINA-TI Simulation

Band Pass filter
 Design a band-pass filter using Biquad with a bandwidth of 120 Hz
and a center frequency of 1.2 kHz and a gain of 100 at center
frequency
 The quality factor of the filter is 10 and H0 =10 (Center frequency
gain/Q).
 For a feedback resistance of 10 kΩ
 Q determining resistance R = 100 kΩ
 Gain determining resistance is 1 kΩ
 If we use UAF42, the integrating capacitor C1 is 1 nF
10
( )
1 -9 3
1 0
1 1
R 132k
2 C f 2 10 1.2 10
= = = Ω
π π ×

Band-pass filter circuit using UAF42
11

Response of the band-pass filter
12
TINA-TI Simulation

Notch Filter
 Design a notch filter with a Q of 100 and a center frequency of
50 Hz and a H0 of 1 using Biquad
 Quality factor of the filter is 100 and H0 = 1.
 For a feedback resistance of 1 kΩ
◦ Q determining resistance R = 100 kΩ
◦ Gain determining resistance = 100 kΩ.
 If we use UAF42, the integrating capacitor C1 is 101 nF
 The Butterworth notch filter circuit using UAF42
13
1 -9
1 0
1 1
R 31.5k
2 C f 2 101x10 x50
= = = Ω
π π

Notch Filter at 50Hz; Q=100
14

Simulation
 Response of the 50Hz notch filter
15
TINA-TI Simulation

Biquads
 Commercially available Biquads
◦ consist of four Op Amps and several precision resistors and
capacitors,
◦ are more expensive.
 UAF 42AP costs $8.25 per unit when procured in quantities of
1000
 Several discrete passive components have to be used to achieve the
required filter parameters
16

Alternative to UAF42
 A second order Biquad filter can also be realized using Quad Op
Amps like LM324 and TL084.
 Quad Op Amp like TI LM348, which costs a $0.2 per unit when
procured in quantities of 1000 (as perTI web site in 2014).
17

Butterworth second-order high-pass filter
 Design a HP filter with lower cutoff frequency of 4 kHz using quad
Op Amp.
 The quality factor of a Butterworth filter is
 We choose C1 = 1 nF
18
1 / 2
( )
1 -9 3
1 0
1 1
R 39.8 k
2 C f 2 10 4 10
= = = Ω
π π ×

HP filter circuit with Op Amp 741/348
19

Filter response
 High-pass filter response is
satisfactory only up to
150 kHz after which the gain
falls. This is mainly due to the
limited gain-bandwidth
product (1 MHz) of LM741.
 If LM741 is replaced by an Op
Amp with better gain-band
width product (5 MHz) the
response is satisfactory up to
1.2 MHz.
20
TINA-TI Simulation

Response of the high pass filter using LF356
21
TINA-TI Simulation

Biquad Filter: Limitations due to finite GB
 High-pass filter
22

Biquad Filter: Limitations due to finite GB (contd.,)
23
The Biquad filter has two feedback loops.
The feedback loop formed by two integrators and
an inverting amplifier (ABC) is known as pole forming loop,
and has a loop gain of .
The second feedbac
0
2
2
-
s
ω
k loop formed by ADC, one integrator
and two inverters is known as Q-forming loop .
As C is an adder the overall loop gain of the filter is 0
0
2
0
2
-
Qs
- -
s sQ
ω
ω ω

24
If the Op Amps used are ideal, the loop gain
of the feedback loop is .
The characteristic equation of the
ideal filter becomes
2
0 0
l 2
2
0 0
l 2
g - -
s sQ
1 - g 1
sQ s
ω ω
=
ω ω
= + +

25
When the Op Amp has finite gain-bandwidth product - GB,
the characteristic equation
0
2
0 0 0
2
0
2
0 0 0
2
1 -
GB
s
1 1 - 2 - 5
1 s
s GB GB sQ
- H 2
Q GB
s
1 1 - 2 - 5
s GB GB sQ
ω
 
 
ω ω ω
   
+ +
     
  + +
 
 
 
 
ω ω ω
 
+ +
 
 


The new Q-factor
26
{ } { }
The new natural frequency of the filter with
Op Amps having finite GBs
0
a 0
0 0
a
0
1
1 - 2 GB 1 - GB
Q
Q
5 Q
1 -
GB
ω
ω =ω =
ω  
ω
 
ω
 
 
 


Sensitivity of Q for variations GB
27
ω
δ
= =
ω
δ
ω
 for the topology of the given filter
The manufacturers of Biquads usually specify
product up to which the filter can effectively used.
of UAF42 is 100 kH
0
Q a
GB
0
a
0
0
0
5 Q
Q GB GB
S
5 Q
GB Q
1 -
GB
5 Q
1
GB
f Q
f Q z

Simulation – Q Sensitivity to GB = 1MHz
28
0
Q=5; H 0
1;f 1.59kHz
= =
3
6
5
5 12
5 3 1 59 10
1
10
.
.
=
 
× × ×
−
 
 

Simulation – Q Sensitivity to GB = 1MHz
29
0
Q=5; H 0
1;f 15.9kHz
= =
3
6
5
6 57
5 3 15 9 10
1
10
.
.
=
 
× × ×
−
 
 

Notch Filter: Finite GB
30
( )
( ) ( )
2 2
0 0
4
2 2
0 0
2 2
0 0
2 2
0 0
4
2 2
0
2 2
0 0
0 0
0
0
0
0
1
1
5 5
1 1 1
5
5
1 1 1 1
5
5
1
Ideal
Gain at
o
i
o
i
H s
V
V s s Q
s s s s
H ( - ) H
GB GB
V
V Q
s s s s s
( - ) ( - )
GB Q Q GB
Q
H
GB
Q
-
GB
 
+ ω
 
=
ω + ω +
   
+ + +
   
ω ω
   
= =
   
ω
+ + + +
   
ω ω
ω ω
   
ω
ω =
ω
 
 
 

Simulation
31
0
Q=5; H 0
1;f 1.59kHz
= = 0
0
0
3
0 024
0 025
3 1 0 024
1
Q
H
.
GB .
Q .
-
GB
ω
= =
ω −
 
 
 

Simulation
32
0
Q=5; H 0
1;f 15.9kHz
= = 0
0
0
3
0 24
0 32
3 1 0 24
1
Q
H
.
GB .
Q .
-
GB
ω
= =
ω −
 
 
 

State Space Filters
 The double-integrator loop is called resonator block as it simulates
a second-order system with the coefficient of first-order term is
zero
 Normalizing frequency ω0 is solely determined by the integrator
time constants and inverter gain in resonator block
 Q determining loop is separately added
 Q and ω0 can be adjusted separately
33

State Space Filters (contd.)
 H0 (gain at low frequency for low-pass, gain at high frequency for
high-pass, gain at center frequency of band-pass, and gain at low and
high frequencies of notch) can also be independently set
 These independent adjustments of filter parameters are not
possible with other active filter realizations
 These constitute the basic building blocks of all IC filters
34

Voltage Controlled State Space Filters (VCF)
 Integrators in Biquad have fixed resistance and capacitance
 Filter is tuned by changing the resistances, capacitance or both
 By replacing the integrator with a fixed integrator and a multiplier it
is possible to create a voltage controlled filter
 Q remains unaffected by
this arrangement
35
0
10
C
V
RC
ω =
10 10
C C
i i
V V
Vdt V
RC sRC
− ⇒ −
∫

Simulation
36
0
10
C
V
RC
ω =

Lecture 29: State-Space Filters
1

Review
 Effect of gain bandwidth product on State-space filter performance
 Q enhancement due to gain bandwidth product
2
a
GB
a
Q
Q
1 - Q
∆φ
 
 


Frequency Compensation
 Basic building blocks of active filters are second-order negative
feedback systems
 Second-order active filter consists of amplifier and two integrators
which have frequency dependence
 If the amplifier is internally compensated it may be modeled as a
first order system as the second pole is normally beyond gain
bandwidth product
3
( )
1 K
1 K s
1
GB
+
 
+
+
 
 
Non-inverting amplifier
( )
K
1 K s
1
GB
−
 
+
+
 
 
Inverting amplifier

Frequency Compensation (contd.,)
4
( ) 0
0
0 0
0
0
1
1
1
1
1
Loop gain of these negative feedback systems is
Phase error in the amplifier at is
Integrator
Phase error in the integrator
GB
s K
K
GB
s s
s
s
GB
GB GB
GB
+
ω
ω − +
ω ω
− −
ω
   
+
+ +  
 
 
 
ω
−


State-Space Filters
 Use two inverting integrators and one summing amplifier in a loop
 The cumulative phase error in the loop
5
0
2
5
1 1
3s GB
s 1
1
GB
GB
ω
−
−
 
  +
+  
   
 


State-Space Filters (contd.,)
 Single-stage summing amplifier modified by replacing Op Amp by an
Op Amp and a buffer
6

Input-output Relations
2
2
2
1
3
1
3
1
1
3 9
1
o1
i
Single stage summing amplifier
Composite summing amplifier
Phase error becomes zero
V 3s
= 1+ which has a positive phase (lead) error
V GB
o
i
s
GB
s
V GB
V s s
GB GB
−
 
+
 
 
 
− +
 
 
= −
 
+ +
 
 
 
 
 

7

Composite Integrator (contd.,)
9
0
0
2
1
1
o
i
s
V s GB
V s
s s
GB GB
ω  
− +
  ω
  −
 
+ +  
 


Composite Integrator and Summer
10

Composite Integrator and Summer (contd.,)
11
0
2
0
3
1
3 3
1
o
i
s
V s GB
V s s
GB GB
s
ω  
+
 
 
=
 
+ +  
 
ω
−


CompensatedVCF (Ackerberg-Mossberg Circuit)
 Uses composite integrator-summer
12
0
2
s
1 3
s GB
s s
1 2 3
GB GB
−ω  
+
 
 
 
+ +  
 

Uncompensated and compensated filters
13
Q=5

Switched capacitor filter
 Any filter using LC or RC has its pole-frequency or
 Tolerance of components has great influence on the accuracy with
which it is fixed.
 Resistors and capacitors have poor tolerance and large
temperature coefficients in integrated circuits
 Ratios of capacitors or resistors have very good tolerance (one
order of magnitude better than absolute values).
14
0
1
RC
ω =
1
LC

Switched Capacitors
15
1
1 1
1
1
o C
i
V i dt
C
Vdt
C R
= −
= −
∫
∫

Switched capacitor replaces the R1
16

Switched capacitor replaces the R1 (contd.,)
17
1
i
1
2
The capacitor is connected
to the input initially during
the period of the clock 1.
It collects a charge of CV.
This charge gets transferred
to the capacitor C during
the of the clock 2, when
φ
φ
it gets connected to 2, the
virtual ground

Switched capacitor replaces the R1 (contd.,)
 ω0 of the filter is now
dependent on the ratio of
capacitors and clock
frequency. Precision filters can
now be realized in monolithic
form.
 Frequency of the clock will
have to be higher than 2fmax
where fmax is frequency of
highest desirable signal in the
input to the filter
18
i i
eq
eq
0 C
1 1 1 1
Charge per unit time
V CV
= current
R T
T
R
C
1 C 1 C
f
R C C T C
= =
=
   
ω
= = =
   
   

Features of Switched Capacitor Filters
 It is programmable filter as the clock frequency can be changed
over a wide range
 Ratio of capacitors have a tolerance one order of magnitude than
the absolute values
 Temperature coefficient of capacitances is very close to zero
 Switches introduce switching noise into the entire system
 Require additional analog filters for band limiting (pre-filters) and
smoothening (post-filters)
 With supply voltage scaling down the switches become more leaky
reducing the performance
19

Biquad based switched capacitor filter is MF10
 Cost ~ $ 2 for more than1 k units
 Center frequency (f0) = 2 Hz to 20 kHz
 Clock frequency (fc) = 10 Hz to 1 MHz (f0:fc:: 1:50)
 GBW of the Op Amp = 2.5 MHz
 Slew Rate of Op Amp = 7V/µsec
20

Tuning of Filters
 Need for tuning
◦ Key parameters characterizing filters are Q (quality factor),
f0 (normalizing frequency) and H0 (factor determining the
gain at f0)
◦ Q and H0 are dimensionless quantities and are ratios of
capacitances and resistances in active RC filters
◦ f0 is inversely proportional to resistance and capacitances
◦ Precision Rs and Cs are necessary to have specified f0
21

Tuning of Filters (contd.,)
◦ Values of R and C should be independent of temperature and
time
◦ Resistances and capacitances in ICs have very poor tolerances
◦ Resistances in ICs have very high temperature dependency where
as the capacitances in ICs have acceptable temperature
sensitivities
◦ Tuning becomes necessary to achieve the required specifications
22

Tuning of f0
 R, C or R and C are adjusted to get the precise f0
 Magnitude or phase of the filter at f0 can be used for tuning
 As R and C values drift RC tuning is not the best choice
 Voltage controlled tuning is preferable
 Digitally programmable analog reconfigurable front-end and back-
end filters
23

Voltage Controlled Filter (VCF)
24
0
10
C
V
RC
ω =

GeneralVCF
α1 = Η0 α2 = 0 α3 = 0 High-Pass
α1 = 0 α2 = -H0 α3 = 0 Band-Pass
α1 = 0 α2 = 0 α3 = H0 Low-Pass
α1 = Η0 α2 = 0 α3 = H0 Band-Stop
α1 = Η0 α2 = -H0/Q α3= H0 All Pass
25
2
1 2 3
2
0
0
o
2
i
2
0
0
s s
+ +
ω
ω
V
=
V s s
+ +1
ω Q
ω
 
α α α
 
 
 
 
 
 
 

ManualTuning of a second-order filter
 f0 of the filter to be tuned to fref, specified frequency, by trial
and error adjustment of Vc to makeVav = 0
 Use an oscillator with frequency fref
26

ManualTuning of a second-order filter (contd.,)
27
( )
1
20
ref
L L
o p ref
p p
av
R C
V V sin t
V V
V cos
ω
′
= ω + φ
′
= φ


ManualTuning of a second-order filter (contd.,)
 Sensitivity ofVav to changes in phase shift is maximum at φ = π/2 at
whichVav = 0
 In case of LP and HP filters phase shift at f0 should be π /2 at which
Vav = 0 when the filter is tuned to fref
 In case of BP and BS filtersVav is maximum when tuned to fref
 As sensitivity is zero whenVav is maximum, BP and BS filters must
be tuned using the LP or HP outputs of theVCF
28

Example
 High-pass filter with fref = 1 KHz; R = 1 kΩ and C = 0.1 µF in the
VCF; f0 = 1.592(VC/10)103 Hz; RL = 100 kΩ, CL= 1 µF
 WhenVav = 0,VC = 6.3
29

Multiplying DAC used as Multiplier
 DAC converts a digital input to an analog output if the input is a
fixed analog voltageVref
where bi = 0 or 1
 If Vref is variable then the device becomes multiplying DAC and can
be used as a multiplier
30
i
0 ref i
i 0..n 1
V V b 2−
= −
= ∑

Multiplying DAC used as Multiplier (contd.,)
 12-bit DAC 7821 costs about $ 3.15 for >1k units where as the
multiplier MPY 634 costs $ 13.25 for > 1k units
31

Lecture 30:Automatic Tuning of Filters (PLL) and
Review of Filter Design
1

Review
 Frequency Compensation
2

Review (contd.,)
 Switched Capacitor Filters
 Manual Tuning of a second-order filter
◦ f0 of the filter to be tuned to fref, specified frequency, by trial
and error adjustment of Vc to makeVav = 0
◦ Use an oscillator with frequency fref
3

AutomaticTuning of Filters (Phase Locked Loop)
4
c
C
ref
ref
V adjusts itself to make
input to the comparator
(integrator) go to zero
V
=
10RC
where is the frequency
of theinput.
This is known as
Phase Locked Loop (PLL)
ω
ω
0
C 0.1 ;R 1k;Q 5;H 1
= µ = = =

Phase Locked Loop (PLL)
5
0
VCF PD
Phase Follower
Dynamic Characteristic
Loop Gain K K
s
ω
=
o
i
VCF PD 0
VCF PD 0
1
s
1
K K
The Bandwidth of the PLL=K K rad/sec
∂φ
=
∂φ
+
ω
ω

Simulation 1
 Sine wave input; 1V, fref=1 kHz
6

Simulation 2
 Sine wave input; 1V, fref=2 kHz
7

Simulation 3
 Square wave input; 1V, fref=1 kHz
8

Simulation 4
 Square wave input; 1V, fref=2 kHz
9

Phase Locked Loop
 Static Characteristics:The phase is locked to 90O independent of
the frequency of the input signal
 Lock Range:The system has a lock range that is decided by the
range of control voltage
 Dynamics of PLL: Capture Range: Range of frequencies over which
the loop can have loop gain much greater than 1 while starting
10

Time Multiplexing for Calibration and Use
11
0
C 0.1 ;R 1k;Q 5;H 1
= µ = = =

Master-SlaveTuning (Continuous-time filters)
12
Cm Cs
ref s
m m s s
ref s s
Cs Cm
s m m
V V
= =
10R C 10R C
R C
= V V
R C
;
as
ω ω
ω
=
ω

Design of 4th order Band-pass and Band-stop Filter
 Centre Frequency: 5.3 kHz
 Maximally flat magnitude
(Butterworth)
 Second order state-space
filter will have for R = 30 kΩ
and C=1nF
 Use UAF 42
13

2nd Order BPF
 Q =10; H0 = 1
14

Cascading two 2nd order BPFs
 Bandwidth gets reduced and gain at centre frequency is (H0Q)2
15

Wideband Amplifier
 Cascading BPFs with staggered centre frequencies known as
Distributed Amplifiers
16

Wideband Amplifier
 Two 2nd
order BPFs
cascaded
 Centre
frequencies
of the two
filters are
staggered by
10%
17

Wideband Amplifier with staggering >10%
18

Notch filter 2nd Order and 4th Order
 4th order
filter has
narrower
stop band
19

Broad Band band-stop filter
 Centre frequencies are staggered by 5%
20

Review of Filter Structures
21

Butterworth and Chebyschev Filters
 All pole filters
 Useful when white noise dominates over signal
 Rates of attenuation at thee pass band edge are slow
 Dominant coloured noise is not effectively removed by these filters
22

Inverse Chebyschev and Elliptic Filters
 Have poles and zeros. Presence of zeros helps in eliminating narrow
band dominant noise components in the stop band
 Attenuation in the stop band is decided by n-m ( number of poles –
number of zeros)
 When white noise dominant signal-to-noise ratio improvement is
not as much as that of all pole filters
23

Second-order Filter
( )
( )
( )
( )
2
2
2
2
2
2
p
2
2
2
2
p
2
2
p
p
Input-output relationship of a second order filter
1 X
with a zero
X
1 X
Q
1
Input-output relationship of all-pole filter
X
1 X
Q
Zero: 1 0.5X
X= where is the normalizing frequency.
Zero to
− α
− +
− +
−
ω
ω
ω
be located beyond the pass band
24

Responses
 For Qp=2 and α=0.5
25

Responses
 For Qp=1 and 2 with α=0.5
26
( )
2
p
1
2 1
Q
for maximum flatness
in the pass-band
= − α
White noise still comes through in our
attempt to remove the coloured noise

Addition of another first order filter
 For Qp=1/√2 with α=β=0.5
27
Both coloured noise and
white noise are attenuated
( )
2
p
2
1
2 1
Q
where is the scalling
factor in the first order
1
low pass=
1+ X
= − α − β
β
β

How should filters be designed at present?
 Present day electronic systems have both digital and analog
subsystems
 Many of the present systems are portable and hence battery
operated
 Analog sub-systems have to be designed using the digital device
technologies for single chip solutions
 Analog sub-systems have to be designed using low voltage (3V at
present)
28

How should filters be designed at present? (contd.,)
 With leaky switches and switching noise switched-capacitor filter is
not a viable option
 While L-replacement method provides a reliable filter it is less
flexible (in terms of selecting the parameters Q, f0 and H0
independently) beside high component count
 Q-enhancement method can lead to active filter with single active
device, but less reliable than multi-active device based filter. It is also
less flexible.
29

How should filters be designed at present?
 State-space filters offer the best solution in terms of reliability and
flexibility.
 While Biquad IC based state-space filter is convenient but more
expensive at present compared to state-space filter designed using
quad Op Amp IC
 State-space filters can be tuned precisely using either a multiplier
and multiplying DAC.
 State-space filters with provision for tuning offer the best solution
to filtering
30

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