Franco Maloberti
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Transcript Franco Maloberti
Analog Filters: Biquad Circuits
Franco Maloberti
Introduction
Active filters which realize the biquadratic transfer
function
a2 s2 a1s a 0 a2 s2 a1 s a0
H(s) 2
s b1 s b0
s 2 0 s 02
Q
are important building blocks
(biquad)
2
2
0 p 2p
a 2 s 2 a 1s a 0
H(s)
(s s p )(s s*p )
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0
Q
p
0
p
2 p
Analog Filters: Biquad Circuits
2
Introduction
Biquads can build high-order filters
P(s)
H(s)
Q(s)
n
1
m
1
(s si )
Poles and zeros are
Real or complex conjugate
(s s j )
(s sz ,1 )(s sz ,1 ) (s sz ,2)(s sz ,2 ) (s sz ,3)(s sz ,3 )
H(s)
(s s p,1 )(s s*p ,1 ) (s s p ,2)(s s*p ,2 ) (s s p ,3)(s s*p ,3 )
*
*
*
s or 1/s
B1
B2
B3
Problem: how to properly pair poles and zeros
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Analog Filters: Sensitivity
3
Single Amplifier Configurations
RC
+
-
RC
RC
+
R(k-1)
+
R
R
R(k-1)
Enhanced positive or negative feedback
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Analog Filters: Sensitivity
4
Sallen-Key Biquad
R1
R2
E2
E1
C1
C2
E1 E2 (1 sR1C1 )(1 sR 2C 2)
Only real poles (or zeros)
E1
E2
C1
C2
The feedback permits us to achieve complex poles
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Analog Filters: Sensitivity
5
Sallen-Key Biquad (ii)
C1
R1
R2
E2
E1
E3
R1R2C 1C 2
E4
C2
1
Ra
Rb
E2
R1R 2C1C2
E1 s2 ( 1 1 1 )s
R1C1 R2C1 R2C2
R1R 2C1C2
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1
0
Analog Filters: Sensitivity
Q
R1R2C 1C 2
1
1
1
R1C1 R2C 1 R2C 2
G
6
Sallen-Key Biquad (ii)
0
Five design elements, two properties (G is not important)
1
R1R2C 1C 2
1
R1R2C 1C 2
Q
1
1
1
R1C1 R2C 1 R2C 2
G
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Case 1: C1=C2; R1=R2=R
R=1/ 0 =3-1/Q
Case 2: C1=C2; Ra=Rb
R1=Q/ 0 R2=1/Q 0
Case 3: R1=R2; =1
C1=2Q/ 0 C2=1/2Q 0
Case 4: C1=31/2Q C2; =4/3
R1=1/Q0 R2=1/31/20
Analog Filters: Sensitivity
7
Sallen-Key Biquad (iii)
Sensitivities
0
1
R1R2C 1C 2
1
R1R2C 1C 2
Q
1
1
1
R1C1 R2C 1 R2C 2
G
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SR10 SR20 SC10 SC20
SQR1
1
2
1
RC
Q 2 2
2
R1C1
R C
1
R1C1
1 2
S Q
(1 )
2
R
C
R
C
2 1
2 2
Q
R2
R1C 2
1
R1R2C1
S Q
2
R
C
R
C
2 1
2 2
Q
C1
1
RC
SCQ2 (1 )Q 1 1
2
R2C 2
Analog Filters: Sensitivity
8
Sallen-Key High- and Band-pass
R1
R2
E2
E1
LP
C2
C1
C1
E2
E1
C1 R1
C2
R2
HP
E1
BP
C2
C1
R1
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R2
E2
C2
Analog Filters: Sensitivity
9
Generic Sallen-Key
E2 Vou t
Z1'Z '2
E1 Vin (Z1 Z2 Z2' )Z1' Z1 Z2
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Analog Filters: Sensitivity
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Sallen-Key: finite op-amp gain
The inverting and non-inverting terminals are not
virtually shorted
C1
R1
R2
E1
E3
E2
E4
C2
Ra
E4
Rb
E2
E
Ra 2
R a Rb
A0
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Analog Filters: Sensitivity
11
Sallen-Key in IC
R1
C1
R2
E1
E3
E2
E4
C2
Rb
Ra
R1
R2
E1
E3
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C1
E2
E4
C2
Analog Filters: Sensitivity
12
LP Sallen-Key with real op-amp
a
2C
gm
RC2
b
gm
1
A0
1
2RC
R0C 0 2R0C 4RC
A0
4RR0C (C C 0 ) R C
2R C
A0
2
2
1 as bs2
H(s)
s s2 s 3
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2
2
R 2C 2
2
g
C0 m
Analog Filters: Sensitivity
13
LP Sallen-Key with real op-amp (ii)
1 as bs2
H(s)
s s2 s 3
The transfer function has two zeros and three poles.
If k = Rgm >> 1 the zeros are practically complex
conjugates and are located at
gm
0,p
pK
2
2RC
The extra-pole is real and is located around the GBW
of the op-amp.
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Analog Filters: Sensitivity
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LP Sallen-Key with real op-amp (iii)
Possible responses
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Analog Filters: Sensitivity
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Sallen-Key IC Implementations
Y (s)
s C/ R
s in h s RC
Y p (s) Y (co sh ( s RC 1))
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Analog Filters: Sensitivity
16
Band-reject Biquad
A band-reject response requires zeros on the
immaginary axis
It can be obtained with the generic SK
implementation
Another option is to use a twin-T network
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Analog Filters: Sensitivity
17
Band-reject Biquad (ii)
Using complementary values
2
1
s 2 2
E2
R C
E1 s2 4(1 ) s 1
RC
R 2C 2
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Q
Analog Filters: Sensitivity
1
4(1 )
18
Use of Feed-forward
k
P(s)
Q(s)
E 2 P(s) kQ(s)
E1
Q(s)
+
Assume
P(s) a1s
E2
k(s2 b0 )
2
E1
s b1s b0
Q(s) s2 b1s b0
k b1 /a1
High-pass
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Band-pass
Analog Filters: Sensitivity
19
Infinite-Gain Feedback Biquad
Sallen-Key architectures require input common
mode range.
Input parasitic capacitance of the op-amp can affect
the filter response
Keep the inputs of the op-amp at ground or virtual
ground
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Analog Filters: Sensitivity
20
Infinite-Gain Multi-Feedback Biquad
A conventional op-amp amplifier is not able to
realize complex-conjugate poles
Two or more feedback connections achieve the
result
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Analog Filters: Sensitivity
21
Low-Pass MFB
C1
Q
C2
1
R2 R3
R1
0
1
E2
R1R3C1C2
1
E1
1
1
1
2
s
s
R1 R2 R3 R2 R3C1C2
R3
R2
R2
R3
1
R2 R3C1C2
G
R2
R1
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Analog Filters: Sensitivity
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Design and Sensitivity
Five elements and three equations
“Arbitrarily choose two of them and determine the
remaining three parameters
Assess the “quality of design”
Sensitivity on relevant design element
Spread of components
Cost of the implementation
Linearity of components
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Analog Filters: Sensitivity
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High-pass and Band-pass
C1 2
s
E2
C2
C1 C2 C3
1
E1
2
s
s
R1C1C2
R1R2C2C3
1
s
E2
R1C2
1
E1
1
1
2
s
s
R2C1 R2C2 R1R2C2C3
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Analog Filters: Sensitivity
24
Two-Integrators Biquad
Use of state-variable method
Derive the block diagram
Translate the block diagram into an active
implementation
Addition or subtraction
Integration
Dumped integration (integration plus addition)
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Analog Filters: Sensitivity
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Basic Blocks
K1V1 K 2V2 Vout
sVout V1
(s K1)Vout1 V1
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Analog Filters: Sensitivity
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State Variables
The state variable are relevant voltages of the network
E2
a0
G 2
E1
s b1s b0
E2 2
s b1s b0 GE1
E1
E2
a0
G
a0
2
E6 s b1s b0 E5
E1
E
E5
E6
2
E 4 E 6b0 E 5
E6 s2 b1s E4
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E5
E6
E4
Analog Filters: Sensitivity
27
State Variables (ii)
E6 s2 b1s E4
E 3 s b1 E 4
E3
E4
E 3 sE 6
E3
E6
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Analog Filters: Sensitivity
28
State Variables (iii)
E6 s2 b1s E4
E4
E3
E 4 sE 3
E 6 s b1 E 3
E3
E6
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Analog Filters: Sensitivity
29
State Variables (iv)
E 2'
1
G 2
E1
s b1s b0
E2
a2 s a1s a0
G 2
E1
s b1s b0
2
E2' a2s2 a1s a0 E2
a0
E 3 sE 2'
a2
-a1
+
E2
E 7 s2 E 2'
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Analog Filters: Sensitivity
30
Implementations
Kervin-Huelsman-Newcomb
Tow-Thomson
Fleischer-Tow
….
Fleischer-Laker
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Analog Filters: Sensitivity
31
Implementations (ii)
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Analog Filters: Sensitivity
32