Download Analog Filters

Analog Filters:
Basics of OP AMP-RC
Circuits
Stefano Gregori
The University of Texas at Dallas
Introduction

So far we have considered the theory and basic methods of realizing
filters that use passive elements (inductors and capacitors)

Another type of filters, the active filters, are in very common use

They were originally motivated by the desire to realize inductorless
filters, because of the three passive RLC elements the inductor is the
most non-ideal one (especially for low-frequency applications of
filters in which inductors are too costly or bulky)

When low-cost, low-voltage solid-state devices became available,
active filters became applicable over a much wider frequency range
and competitive with passive ones

Now both types of filters have their appropriate applications
Stefano Gregori
Basics of OP AMP-RC Circuits
2
Active-RC filters
In this lesson we concentrate on active-RC filters. They make
use of active devices as well as RC components.
Active filters
 are usually designed without
regard to the load or source
impedance; the terminating
impedance may not affect the
performance of the filter
 it is possible to interconnect
simple standard blocks to form
complicated filters
 are noisy, have limited dynamic
ranges and are prone to
instability
 can be fabricated by integrated
circuits
Stefano Gregori
Passive filters
 the terminating impedance is an
integral part of the filter: this is a
restriction on the synthesis
procedure and reduces the
number of possible circuits
 are less sensitive to element
value variations
 are generally produced in
discrete or hybrid form
Basics of OP AMP-RC Circuits
3
Operational Amplifier
symbol
equivalent circuit
i1
ee+
e2
Ro
e+
e2
Ri
A(e+-e-)
e-
In an ideal op-amp we assume:

input resistance Ri approaches infinity, thus i1 = 0

output resistance Ro approaches zero

amplifier gain A approaches infinity
Stefano Gregori
Basics of OP AMP-RC Circuits
4
Inverting voltage amplifier
R2
R1
i (t ) 
i
vin
vout
i
vin (t )
R1
vout (t )   R2i(t )  
Example:
R2
vin (t )
R1
vin(t)
given
vin (t )  V0 sin2πft 
R1 = 1 kΩ
R2 = 2 kΩ
V0 = 1 V
f = 1 MHz
vout(t)
we have
vout (t )  
R2
V0 sin2πft 
R1
Stefano Gregori
Basics of OP AMP-RC Circuits
5
Weighted summer
Rf
vk (t )
ik (t ) 
Rk
R1
v1
R2
v2
Rn
vn
vout
vout (t )   R f i1 (t )   in (t ) 
 v (t )
v (t ) 
  R f  1    n  
Rn 
 R1
  Rf
Stefano Gregori
Basics of OP AMP-RC Circuits
n
vk (t )
 R
k
k 1
6
Noninverting voltage amplifier
R2
R1
i
i (t ) 
i
vout
vin
Example:
vin (t )
R1
 R2 
 vin (t )
vout (t )  R1  R2  i(t )  1 
R1 

vin(t)
given
vin (t )  V0 sin2πft 
R1 = 1 kΩ
R2 = 1 kΩ
V0 = 1 V
f = 1 MHz
vout(t)
we have
 R2 
V0 sin2πft 
vout (t )  1 
R1 

Stefano Gregori
Basics of OP AMP-RC Circuits
7
Buffer amplifier
vout
vin
vout (t )  vin (t )
Stefano Gregori
Basics of OP AMP-RC Circuits
8
Inverting or Miller integrator
C
R
vin
vout
i
R = 1 kΩ
C = 1 nF
V0 = 1 V
f = 1 MHz
Example:
given
vin (t )
i (t ) 
R
V
Vout   in
dv (t )
s RC
i (t )  C out
dt
t
1
vout (t )  vout (0) 
vin (t ) dt

RC 0
vin(t)
t0
0
vin (t )  
V0 sin 2πft  t  0
vout (0)  0
vout(t)
we have
vout (t ) 
V0
cos2πft   1
2πfRC
Stefano Gregori
Basics of OP AMP-RC Circuits
9
Inverting differentiator (1)
R
C
vin
i
i (t )  C
vout
dvin (t )
dt
vout (t )   Ri (t )   RC
Example:
given
R = 1 kΩ
C = 100 pF
V0 = 1 V
f = 1 MHz
t0
0
vin (t )  
V0 sin 2πft  t  0
Vout   s RC Vin
dvin (t )
dt
sC
vin(t)
vout(t)
we have
t0
0
vout (t )  
 2πfRCV0 cos2πft  t  0
Stefano Gregori
Basics of OP AMP-RC Circuits
10
Inverting differentiator (2)
R = 22 kΩ
C = 47 pF
vin(t) is a triangular waveform with:
- vin max 2 V
- vin min 0 V
- frequency 500 kHz
R
C
vin
vout
sC
vin(t)
vout(t) is a square waveform with:
- vout max 2,068 V
- vout min -2,068 V
- frequency 500 kHz
Stefano Gregori
vout(t)
Basics of OP AMP-RC Circuits
11
Inverting lossy integrator
C
R2
R1
vin
vout
Vout  
Stefano Gregori
1

1 

R1  sC 
R2 

Vin
Basics of OP AMP-RC Circuits
12
Inverting weighted summing integrator
C
R1
v1
vout
R2
v2
Rn
vn
Vout
Stefano Gregori
1
 
sC
n
Vk
R
k 1 k
Basics of OP AMP-RC Circuits
13
Subtractor
R0
R1
v1
vout
v2
R2
Vout  
Stefano Gregori
R3
R0
R R  R1 
V1  3 0
V2
R1
R1 R2  R3 
Basics of OP AMP-RC Circuits
14
Integrator and differentiator
frequency behavior
integrator
integrator
C
differentiator
R
vin
vout
AV 
1
2πfRC
differentiator
R
C
vin
vout
AV  2πfRC
R = 1 kΩ
C = 1 nF
Stefano Gregori
vin(t) is a sinewave with frequency f.
Figure shows how circuit gain AV changes with the
frequency f
AV is the ratio between the amplitude of the output sinewave
vout(t) and the amplitude of the input sinewave vin(t)
Basics of OP AMP-RC Circuits
15
Low-pass and high-pass circuits
low-pass circuit
frequency behavior
R
vin
voutlp
low-pass
high-pass
C
high-pass circuit
C
vouthp
vin
R
R = 1 kΩ
C = 1 nF
Stefano Gregori
Basics of OP AMP-RC Circuits
16
Inverting first-order section
Z2
Z1
v1
v2
V2
Z2
Y1
 

V1
Z1
Y2
R2
R1
C2
v1
v2
V2
R 1  sR1C1
  2
V1
R1 1  sR2C2
C1
inverting lossing integrator
Stefano Gregori
Basics of OP AMP-RC Circuits
17
Noninverting first-order section
Z2
v2
V2
Z2
Y1
 1
 1
V1
Z1
Y2
v2
V2
R 1  sR1C1
 1 2
V1
R1 1  sR2C2
Z1
v1
R2
R1
C2
C1
v1
noninverting lossing integrator
Stefano Gregori
Basics of OP AMP-RC Circuits
18
Finite-gain single op-amp configuration
Many second-order or biquadratic filter circuits use a
combination of a grounded RC threeport and an op-amp
i2
V3
RC
threeport
V1
i1

V2
i3
I1  y11V1  y12V2  y13V3
I3  0
I 2  y 21V1  y 22V2  y 23V3
E3  E 2 / μ
E2

E1
I 3  y31V1  y32V2  y33V3
Stefano Gregori
Basics of OP AMP-RC Circuits
y31
y33
y32 
μ
19
Infinite-gain single op-amp configuration
RC
threeport
V3
V2
V1
y31
E2

E1
y32
Stefano Gregori
Basics of OP AMP-RC Circuits
20
Gain reduction
Z
Z1
V2
V1'
N
V2
N
V1
Z2
V1
To reduce the gain to α times its original value (α < 1) we make
V1
Z2
α
V1
Z1  Z 2
and
Z1 Z 2
Z
Z1  Z 2
solving for Z1 and Z2, we get
Z1 
Z
α
Stefano Gregori
and
Z2 
1
Z
1 α
Basics of OP AMP-RC Circuits
21
Gain enhancement
V2/K
RC
threeport
1/K
V3
K
V2
V1
A simple scheme is to increase the amplifier gain and decrease
the feedback of the same amount
E2

E1
Stefano Gregori
Ky31
y
y32  33
μ
Basics of OP AMP-RC Circuits
22
RC-CR transformation (1)
is applicable to a network N that contains resistors, capacitors, and
dimensionless controlled sources
conductance of Gi [S] → capacitance of Gi [F]
capacitance of Cj [F] → conductance of Cj [S]
the corresponding network functions with the dimension of the
impedance must satisfy

Z ( s ) 

1 1
Z 
s s
the corresponding network functions with the dimension of the
admittance must satisfy
1
Y ( s )  sY  
s

the corresponding network functions that are dimensionless must
satisfy
1
 
H ( s )  H  
s
Stefano Gregori
Basics of OP AMP-RC Circuits
23
RC-CR transformation (2)
N
N’
1/3 F
1
1F
2
2
v1
3
v2
1/2 F
2
v1'
2
1/2 F
V
12
H ( s)  2  2
V1 2s  7 s  6
V2
12 s 2
H (s) 
 2

V1 6s  7s  2
V1 2s 2  7s  6
Z11 ( s) 

I1
s(2s  1)
V1 6s 2  7s  2
 ( s) 
Z11


I1
s( s  2)
Stefano Gregori
v2'
Basics of OP AMP-RC Circuits
24
Sallen-Key filters
lowpass filter
frequency behavior
C
lowpass
R
R
highpass
vout
vin
C
bandpass
highpass filter
R
C
C
vout
vin
R
Stefano Gregori
R = 1 kΩ
C = 1 nF
Basics of OP AMP-RC Circuits
25
Types of biquadratic filters
lowpass

highpass
Gb0

s  b1s  b0
2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
6
8
10
Stefano Gregori
bandpass
Gs 2

s  b1s  b0
2
2
4
6
8
10
bandreject
Gb1 s

s 2  b1s  b0
allpass
a 2 s 2  a0
G
s 2  b1s  b0
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
2
4
6
8
10
2
4
Basics of OP AMP-RC Circuits
6
8
10
s 2  b1s  b0
s 2  b1s  b0
2
4
6
8
10
26