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 )
Franco Maloberti
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
Franco Maloberti
Analog Filters: Sensitivity
3
Single Amplifier Configurations
RC
+
-
RC
RC
+
R(k-1)
+
R
R
R(k-1)
Enhanced positive or negative feedback
Franco Maloberti
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
Franco Maloberti
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
Franco Maloberti
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
Franco Maloberti
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/Q0 R2=1/31/20
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
Franco Maloberti




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
Franco Maloberti
R2
E2
C2
Analog Filters: Sensitivity
9
Generic Sallen-Key
E2 Vou t
Z1'Z '2


E1 Vin (Z1  Z2  Z2' )Z1'  Z1 Z2
Franco Maloberti
Analog Filters: Sensitivity
10
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
Franco Maloberti
Analog Filters: Sensitivity
11
Sallen-Key in IC
R1
C1
R2
E1
E3
E2
E4
C2
Rb
Ra
R1
R2
E1
E3
Franco Maloberti
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
Franco Maloberti
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.
Franco Maloberti
Analog Filters: Sensitivity
14
LP Sallen-Key with real op-amp (iii)
Possible responses
Franco Maloberti
Analog Filters: Sensitivity
15
Sallen-Key IC Implementations
Y (s) 
s C/ R
s in h s RC
Y p (s)  Y (co sh ( s RC 1))
Franco Maloberti
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
Franco Maloberti
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
Franco Maloberti
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

Franco Maloberti
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
Franco Maloberti
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
Franco Maloberti
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

Franco Maloberti
Analog Filters: Sensitivity
22
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
Franco Maloberti
Analog Filters: Sensitivity
23
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

Franco Maloberti
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)
Franco Maloberti
Analog Filters: Sensitivity
25
Basic Blocks
K1V1  K 2V2  Vout

sVout  V1

(s  K1)Vout1  V1
Franco Maloberti

Analog Filters: Sensitivity
26


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
Franco Maloberti
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

Franco Maloberti
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


Franco Maloberti
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'
Franco Maloberti
Analog Filters: Sensitivity
30
Implementations





Kervin-Huelsman-Newcomb
Tow-Thomson
Fleischer-Tow
….
Fleischer-Laker
Franco Maloberti
Analog Filters: Sensitivity
31
Implementations (ii)
Franco Maloberti
Analog Filters: Sensitivity
32