ELG4139: Op Amp-based Active Filters • Advantages: – Reduced size and weight, and therefore parasitics. – Increased reliability and improved performance. – Simpler design than for passive filters and can realize a wider range of functions as well as providing voltage gain. – In large quantities, the cost of an IC is less than its passive counterpart. • Disadvantages: – Limited bandwidth of active devices limits the highest attainable pole frequency and therefore applications above 100 kHz (passive RLC filters can be used up to 500 MHz). – The achievable quality factor is also limited. – Require power supplies (unlike passive filters). – Increased sensitivity to variations in circuit parameters caused by environmental changes compared to passive filters. • For applications, particularly in voice and data communications, the economic and performance advantages of active RC filters far outweigh their disadvantages.
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ELG4139: Op Amp-based Active Filters
• Advantages:
– Reduced size and weight, and therefore parasitics.
– Increased reliability and improved performance.
– Simpler design than for passive filters and can realize a wider range of
functions as well as providing voltage gain.
– In large quantities, the cost of an IC is less than its passive counterpart.
• Disadvantages:
– Limited bandwidth of active devices limits the highest attainable pole
frequency and therefore applications above 100 kHz (passive RLC
filters can be used up to 500 MHz).
– The achievable quality factor is also limited.
– Require power supplies (unlike passive filters).
– Increased sensitivity to variations in circuit parameters caused by
environmental changes compared to passive filters.
• For applications, particularly in voice and data communications, the
economic and performance advantages of active RC filters far outweigh
their disadvantages.
First-Order Low-Pass Filter
ffB
Bi
f
ffi
f
i
f
ff
f
f
f
ff
f
f
ff
i
f
i
o
CRf
ffjR
R
CfRjR
R
Z
ZfH
CfRj
RZ
R
CfRj
R
fCj
RZ
Z
Z
V
VfH
2
1
)/(1
1
21
1)(
21
21
2
1
111
)(
A low-pass filter with a dc gain of –Rf /Ri
dttvRC
tv
t
o in
0
1
First-Order High-Pass Filter
iiB
B
B
i
f
ii
if
i
i
ii
if
ii
f
i
f
ffi
ii
i
f
i
o
CRf
ffj
ffj
R
R
CRfj
CRfj
R
R
CRfj
CRfj
CfjR
R
Z
ZfH
RZCfj
RZ
Z
Z
v
vfH
2
1
)/(1
)/(
21
2
21
2
2
1)(
2
1
)(
A high-pass filter with a high frequency gain of –Rf /Ri
Higher Order Filters
n
B
n
i
fn
n
ffjR
R
fHfHfHfH
)/(1
1)1(
)()()()( 21
Single-Pole Active Filter Designs
High Pass Low Pass
)/1()/1(
1
1
11
1
RCs
s
RCsRC
sRC
sCR
sRC
sCR
v
v
in
out
RCs
RC
v
v
in
out
/1
/1
7
Two-Pole (Sallen-Key) Filters
-
+
+V
-V
R1
Rf1
Rf2
C1
vin
vout
C2
R2
-
+
+V
-V
R1
Rf1
Rf2
C2
vin
vout
R2
C1
Low Pass Filter High Pass Filter
8
Three-Pole Low-Pass Filter
-
+
+V
-V
R1
Rf1
Rf2
C1
vin
C2
R2
-
+
+V
-V
R3
Rf3
Rf4
C3
vout
Stage 1 Stage 2
9
Two-Stage Band-Pass Filter
R2
R1
vin
C1
C2
Rf1
Rf2
C4
C3
R3
R4
+V
-V
vout
Rf3
Rf4
+
-
+
-
+V
-V
Stage 1
Two-pole low-pass
Stage 2
Two-pole high-pass
BW
f1
f2
f
Av
Stage 2
response
Stage 1
response
fo
BW = f2 – f1
Q = f0 / BW
10
Multiple-Feedback Band-Pass Filter
R1
R2
C1
C2
vin
Rf
+V
-V
-
+v
out
11
Transfer function H(j)
Transfer
Function
)( jHV
oV
i
)(
)()(
jV
jVjH
i
o
)Im()Re( HjHH
22 )Im()Re( HHH
)Re(
)Im(tan 1
H
HH 0)Re( H
)Re(
)Im(tan180 1
H
HH o
0)Re( H
12
Frequency Transfer Function of Filters
H(j)
HL
HL
o
o
o
o
ffffjH
fffjH
ffjH
ffjH
ffjH
ffjH
and 0)(
1)(
Filter Pass-Band (III)
1)(
0)(
Filter Pass-High (II)
0)(
1)(
Filter Pass-Low (I)
response phase specific a has
allfor 1)(
Filter shift)-phase(or Pass-All (V)
and 1)(
0)(
Filter (Notch) Stop-Band (IV)
fjH
ffffjH
fffjH
HL
HL
Bode Plot
To understand Bode plots, you need to use Laplace transforms!
The transfer function of the circuit is:
1
1
/1
/1
)(
)(
sRCsCR
sC
sV
sVA
in
o
v
R
Vin(
s)
b
v
f
fj
RCfjRCjfA
1
1
21
1
1
1)(
where fc is called the break frequency, or corner
frequency, and is given by: RCf
c
2
1
14
Bode Plot (Single Pole)
o
jCRj
jH
1
1
1
1)(
2
1
1)(
o
jH
2
101011log20)(log20)(
o
dBjHjH
o
dBjH
10log20)(
For >>o
R
C VoV
i
Single pole low-pass filter
15
dBjH )(
(log)x
x
2x
10
6d
B2
0d
B slope
-6dB/octave
-20dB/decade
o
jH
10log20)(
For octave apart,
1
2
o
dBjH 6)(
For decade apart, 1
10
o
dBjH 20)(
16
Bode Plot (Two-Pole)
R1
R2
C1
C2
vi v
o
21 oo
2
2
2
1
10111log20)(
oo
jH
Corner Frequency
• The significance of the break frequency is that it represents the frequency where
Av(f) = 070.7-45
• This is where the output of the transfer function has an amplitude 3-dB below the input amplitude, and the output phase is shifted by -45 relative to the input.
• Therefore, fc is also known as the 3-dB frequency or the corner frequency.
Bode plots use a logarithmic scale for frequency, where a decade is
defined as a range of frequencies where the highest and lowest
frequencies differ by a factor of 10.
Magnitude of the Transfer Function in dB
2
/1
1)(
b
v
fffA
b
bb
bdBv
ff
ffff
fffA
/log20
/1log10/1log20
/1log201log20)(
22
2
• See how the above expression changes with frequency:
– at low frequencies f << fb, |Av|dB = 0 dB
• low frequency asymptote
– at high frequencies f >>fb,
|Av(f)|dB = -20 log f/ fb
• high frequency asymptote
Real Filters
• Butterworth Filters – Flat Pass-band.
– 20n dB per decade roll-off.
• Chebyshev Filters – Pass-band ripple.
– Sharper cut-off than Butterworth.
• Elliptic Filters – Pass-band and stop-band ripple.
– Even sharper cut-off.
• Bessel Filters – Linear phase response – i.e. no signal distortion in pass-band.
Filter Response Characteristics
The magnitude response of a Butterworth filter.
Butterworth Filters
Magnitude response for Butterworth filters of various
order with = 1. Note that as the order increases, the
response approaches the ideal brickwall type
transmission.
Sketches of the transmission characteristics of a representative even- and odd-
order Chebyshev filters.
Chebyshev Filters
First-Order Filter Functions
First-Order Filter Functions
Second-Order Filter Functions
Second-Order Filter Functions
Second-Order Filter Functions
Second-Order LCR Resonator
The Antoniou inductance-simulation circuit. (b) Analysis of the circuit assuming ideal op
amps. The order of the analysis steps is indicated by the circled numbers.
Second-Order Active Filter: Inductor Replacement
The Antoniou inductance-simulation circuit. Analysis of the circuit assuming ideal op amps.
The order of the analysis steps is indicated by the circled numbers.
Second-Order Active Filter: Inductor Replacement
Realizations for the various second-order filter functions using the op amp-RC resonator of