analog filters: basics of op amp-rc circuits
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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) - PowerPoint PPT PresentationTRANSCRIPT
Analog Filters:Basics of OP AMP-RC Circuits
Stefano GregoriThe University of Texas at Dallas
Stefano Gregori Basics of OP AMP-RC Circuits 2
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 3
Active-RC filters
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
In this lesson we concentrate on active-RC filters. They make use of active devices as well as RC components.
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
Stefano Gregori Basics of OP AMP-RC Circuits 4
Operational Amplifier
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
Ri
Ro
A(e+-e-)
e+
e-
e2
i1
equivalent circuitsymbol
e+
e-e2
Stefano Gregori Basics of OP AMP-RC Circuits 5
Inverting voltage amplifier
R1
R2
vinvout
i
i1
in )()(
Rtv
ti
)()()( in1
22out tv
RR
tiRtv
Example:
ftπVtv 2sin)( 0in
given
we have
R1 = 1 kΩR2 = 2 kΩV0 = 1 Vf = 1 MHz
ftπVRR
tv 2sin)( 01
2out
vin(t)
vout(t)
Stefano Gregori Basics of OP AMP-RC Circuits 6
Weighted summer
k
kk R
tvti
)()(
)()( )( 1out titiRtv nf
R1
Rf
v1voutR2
v2
Rn
vn
n
k k
kf R
tvR
1
)(
n
nf R
tvR
tvR
)()(
1
1
Stefano Gregori Basics of OP AMP-RC Circuits 7
Noninverting voltage amplifier
R1
R2
vin
vouti
i 1
in )()(
Rtv
ti
)( 1)( )( in1
221out tv
RR
tiRRtv
Example:
ftπVtv 2sin)( 0in
given
we have
R1 = 1 kΩR2 = 1 kΩV0 = 1 Vf = 1 MHz
ftπVRR
tv 2sin1)( 01
2out
vin(t)
vout(t)
Stefano Gregori Basics of OP AMP-RC Circuits 8
Buffer amplifier
vin
vout
)()( inout tvtv
Stefano Gregori Basics of OP AMP-RC Circuits 9
Inverting or Miller integrator
Rtv
ti)(
)( in
)(1)0()(0
inoutout t
dttvRC
vtv
R
C
vinvout
i dttdv
Cti)(
)( out
Example:
00
2sin
0)(
0in
tt
ftπVtv
0)0(out v
12cos2
)( 0out πft
fRCπV
tv
given
we have
vin(t)
vout(t)
R = 1 kΩC = 1 nFV0 = 1 Vf = 1 MHz
RCsV
V
inout
Stefano Gregori Basics of OP AMP-RC Circuits 10
Inverting differentiator (1)
dttdv
RCtRitv)(
)()( inout
dttdv
Cti)(
)( in
Example:
00
2sin
0)(
0in
tt
ftπVtv
given
we have
R = 1 kΩC = 100 pFV0 = 1 Vf = 1 MHz
C
R
vinvout
i
vin(t)
vout(t)
00
2cos2
0)(
0out
tt
ftπfRCVπtv
Cs
inout VRCsV
Stefano Gregori Basics of OP AMP-RC Circuits 11
Inverting differentiator (2)
vout(t) is a square waveform with:- vout max 2,068 V- vout min -2,068 V- frequency 500 kHz
R = 22 kΩC = 47 pFvin(t) is a triangular waveform with:
- vin max 2 V- vin min 0 V- frequency 500 kHz
vin(t)
vout(t)
C
R
vinvout
Cs Cs
Stefano Gregori Basics of OP AMP-RC Circuits 12
Inverting lossy integrator
R1
R2
vinvout
C
in
21
out1
1 V
RsCR
V
Stefano Gregori Basics of OP AMP-RC Circuits 13
Inverting weighted summing integrator
R1
v1voutR2
v2
Rn
vn
C
n
k k
kout R
VsC
V1
1
Stefano Gregori Basics of OP AMP-RC Circuits 14
Subtractor
2
321
1031
1
0 VRRRRRR
VRR
Vout
R1
R0
v1vout
R2
v2
R3
Stefano Gregori Basics of OP AMP-RC Circuits 15
Integrator and differentiator
integrator
differentiator
frequency behavior
R
C
vinvout
C
R
vinvout
integrator differentiator
R = 1 kΩC = 1 nF
2
1V fRCπ
A
fRCπA 2V 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)
Stefano Gregori Basics of OP AMP-RC Circuits 16
Low-pass and high-pass circuits
low-passvoutlpvin
R
C
C
vin vouthp
R
low-pass circuit
high-pass circuit
frequency behavior
high-pass
R = 1 kΩC = 1 nF
Stefano Gregori Basics of OP AMP-RC Circuits 17
Inverting first-order section
v1v2
R1
C1
R2
C2
Z1
Z2
v1v2
2
1
1
2
1
2 YY
ZZ
VV
11
22
11
1
2
1
2
CsRCsR
RR
VV
inverting lossing integrator
Stefano Gregori Basics of OP AMP-RC Circuits 18
Noninverting first-order section
2
1
1
2
1
2 1 1YY
ZZ
VV
11
122
11
1
2
1
2
CsRCsR
RR
VV
noninverting lossing integrator
Z1
Z2
v1
v2
v1
v2
R1
C1
R2
C2
Stefano Gregori Basics of OP AMP-RC Circuits 19
Finite-gain single op-amp configuration
V1
RCthreeport V2
V3
i2
i3i1
Many second-order or biquadratic filter circuits use a combination of a grounded RC threeport and an op-amp
3332321313
3232221212
3132121111
VyVyVyIVyVyVyI
VyVyVyI
μEEI
/0
23
3
μyy
yEE
3332
31
1
2
Stefano Gregori Basics of OP AMP-RC Circuits 20
Infinite-gain single op-amp configuration
32
31
1
2
yy
EE
V1
RCthreeport V2
V3
Stefano Gregori Basics of OP AMP-RC Circuits 21
Gain reduction
V1
NV2
Z
V1
NV2
Z1
Z2
V1'
To reduce the gain to α times its original value (α < 1) we make
21
2
1
1
ZZZ
αVV
Z
ZZZZ
21
21and
solving for Z1 and Z2, we get
αZZ 1 Z
αZ
11
2and
Stefano Gregori Basics of OP AMP-RC Circuits 22
Gain enhancement
A simple scheme is to increase the amplifier gain and decrease the feedback of the same amount
V1
RCthreeport V2
V3 K
1/KV2/K
μyy
KyEE
3332
31
1
2
Stefano Gregori Basics of OP AMP-RC Circuits 23
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
sZ
ssZ 11)(
the corresponding network functions with the dimension of the admittance must satisfy
ssYsY 1)(
the corresponding network functions that are dimensionless must satisfy
sHsH 1)(
Stefano Gregori Basics of OP AMP-RC Circuits 24
RC-CR transformation (2)
67212)( 2
1
2
ssVV
sH
1 F
v1' v2'2
1/2 F
3
2
v1 v2
12
2
1/3 F
1/2 F
)12(672)(
2
1
111
ssss
IV
sZ
27612)( 2
2
1
2
ss
sVV
sH
)2(276)(
2
1
111
ss
ssIV
sZ
N N’
Stefano Gregori Basics of OP AMP-RC Circuits 25
Sallen-Key filters
lowpass filter
highpass filter
frequency behavior
C
vin
vout
C
R
R
C
vin
vout
C
RR
lowpass highpass
bandpass
R = 1 kΩC = 1 nF
Stefano Gregori Basics of OP AMP-RC Circuits 26
2 4 6 8 10
0.2
0.4
0.6
0.8
1
2 4 6 8 10
0.2
0.4
0.6
0.8
1
2 4 6 8 10
0.2
0.4
0.6
0.8
1
2 4 6 8 10
0.2
0.4
0.6
0.8
1
2 4 6 8 10
0.2
0.4
0.6
0.8
1
Types of biquadratic filters
012
0
bsbsGb
012
2
bsbsGs
012
1
bsbssGb
01
20
22
bsbsasa
012
012
bsbsbsbs
G
lowpass highpass bandpass bandreject allpass