Application Report
SBAA237 – July 2017
Compensation Methodology for Error in Sallen-Key Low-
Pass Filter, Caused by Limited Gain-Bandwidth of
Operational Amplifiers
Vito Shen, Thomas Kuehl
ABSTRACT
Analog active filters are essential signal-processing circuits, widely used to modify the frequency spectrum
of an analog signal. The Sallen-Key (SK) filter is a widely applied topology used to realize a low-pass filter.
During the design of the SK filter, the active element is always modeled as an ideal operational amplifier
(op amp), however all real op amps have a limited gain-bandwidth (GBW) product. The limited bandwidth
of the op amp introduces errors into the amplitude and phase responses of the filter.
To diminish these errors, the normal solution is to choose an op amp with a GBW of more than 100x the
–3 dB bandwidth of the filter. Only by choosing an op amp with a high GBW, relative to the bandwidth of
the filter, can it be treated as an ideal op amp. However, the cost of a high-GBW op amp may significantly
increase the cost of the filter. For some applications, the GBW requirement can be so high that it can be
difficult to identify a suitable, cost-effective op amp.
This application report provides a tutorial description of the error analysis for the SK active low-pass filter.
Additionally, the report introduces a new compensation method to reduce the errors caused by the limited
GBW of an op amp. Simulation and test data, which demonstrate the performance achieved with this
compensation technique are presented.
Contents
1
2
3
4
5
Introduction
...................................................................................................................
Ideal Transfer Function and Error Analysis
..............................................................................
Simulation and Test Results
................................................................................................
Conclusions
...................................................................................................................
References
...................................................................................................................
List of Figures
1
2
3
4
5
6
7
Low-Pass Sallen-Key Architecture
........................................................................................
2
2
2
5
8
8
........................................
Schematic for 300-kHz, Sallen-Key, Low-Pass Filter
...................................................................
TINA Spice AC Simulations for Ideal Op Amp and TL081
.............................................................
Modified Schematic for 300-kHz Sallen-Key Low-Pass Filter
.........................................................
Modified Sallen-Key Topology Incorporating Op Amp GBW Compensation
4
5
6
6
TINA Spice AC Simulations of Sallen-Key Filter Using Ideal Op Amp, Standard TL081 Circuit and
Compensated TL081 Circuit
...............................................................................................
7
S21 for Modified Sallen-Key Filter Using TL081
.........................................................................
7
Trademarks
FilterPro is a trademark of Texas Instruments.
WEBENCH is a registered trademark of Texas Instruments.
All other trademarks are the property of their respective owners.
SBAA237 – July 2017
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Compensation Methodology for Error in Sallen-Key Low-Pass Filter, Caused
by Limited Gain-Bandwidth of Operational Amplifiers
Copyright © 2017, Texas Instruments Incorporated
1
Introduction
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1
Introduction
The Sallen-Key (SK) topology is an electronic filter topology used to implement a second-order response.
The topology is particularly valued for its simplicity. The SK is a noninverting filter topology, which can
make it preferable over the Multiple Feed Back (MFB) filter, which is an inverting topology. However, this
is not the only potential advantage. The SK topology may be a better choice if the following apply:
• Gain accuracy is important.
• A unity-gain filter is needed.
• The pole-pair Q is low (for example, Q < 3).
Specific filter performance can be achieved by cascading one or more SK stages. The poles of the SK are
sensitive to the circuit elements, especially to the GBW of the op amp.
This application report develops the ideal SK low-pass transfer function, and then analyzes the frequency
response error contributed by the limited bandwidth of the op amp. The report then introduces a new
compensation methodology to reduce the error caused by the limited gain-bandwidth. Lastly, a real-case
example of an SK low-pass filter is provided. Simulation and testing results confirm the effectiveness of
this new compensation method.
2
Ideal Transfer Function and Error Analysis
Figure 1
shows a circuit diagram for low-pass SK architecture.
a
2
u
C
Input
R
a
1
u
R
OPAMP
C
a
4
u
R
Output
a
3
u
R
Figure 1. Low-Pass Sallen-Key Architecture
Equation 1
provides the Laplace transfer function for the circuit of
Figure 1.
a
3
a
4
a
3
F s
§
a
2
a
4
·
2
s
2
u
RC
u
a
1
a
2
s
u
RC
¨
1 a
1
¸
1
a
3
¹
©
DC gain (where s = 0, see
Equation 2):
a
3
a
4
Av DC
a
3
Characteristic frequency (see
Equation 3):
1
1
Z
o
u
a
1
a
2
RC
(1)
(2)
(3)
2
Compensation Methodology for Error in Sallen-Key Low-Pass Filter, Caused
by Limited Gain-Bandwidth of Operational Amplifiers
Copyright © 2017, Texas Instruments Incorporated
SBAA237 – July 2017
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Ideal Transfer Function and Error Analysis
The quality factor (see
Equation 4):
Q
a
1
a
2
a
2
a
4
1 a
1
a
3
(4)
The ideal Laplace transfer function of the SK can also be expressed as shown in
Equation 5:
a
3
a
4
a
3
F s
2
§
1
·
1
s
2
u ¨
1
¸
s
u
Z
o
¹
Z
o
Q
©
(5)
When the filter performance requirements have been established, the resistance and capacitance values
are determined by calculations or by using filter synthesis software. In the real-filter case, the op amps
used to construct the active filter are not ideal. These op amps have limited GBW which alters the
response of the filter, thereby causing it to deviate from the ideal.
If the limited GBW of the op amp is considered, the frequency response can be modeled as a 1-pole
system.
Equation 6
shows the Laplace transfer function for an actual op amp.
GBW
A s
s
NOTE:
GBW is specified in radians.
(6)
When the transfer function of the op amps has been introduced into the transfer function of the SK, the
equation becomes more complex (see
Equation 7
and
Equation 8).
a
3
a
4
a
3
F s
b
3
s
3
b
2
s
2
b
1
s b
0
(7)
1
§
·
¨
¸
CRa
2
a
4
a
4
1
¨
¸
CRa
1
CR
§
b
0
· ¨
¸
a
3
GBW GBWa
3
¨ ¸
¸
¨
b
1
¸ ¨
2 2
CRa
1
CRa
2
CRa
4
CRa
1
a
4
CRa
2
a
4
¸
CR
¨
b
2
¸ ¨
C R a
1
a
2
GBW GBW GBW GBWa
3
GBWa
3
GBWa
3
¸
¨ ¸ ¨
¨
¸
©
b
3
¹
2 2
2 2
¨
¸
C R a
1
a
2
C R a
1
a
2
a
4
¨
¸
GBW
GBWa
3
©
¹
(8)
This result indicates that when the transfer function of the op amp has been included, the filter order
increases to that of a third-order response. Additionally, the qualify factor (Q) and the characteristic
frequency of the filter have been affected by the GBW of the op amp. These changes alter the response of
the filter.
It has been determined that the response errors caused by the limited-GBW op amp can be effectively
compensated for by adding a resistor, R
comp
, in series with the capacitor, C.
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Compensation Methodology for Error in Sallen-Key Low-Pass Filter, Caused
by Limited Gain-Bandwidth of Operational Amplifiers
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Ideal Transfer Function and Error Analysis
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Figure 2
shows the resulting revised SK topology.
a
2
u
C
Input
R
a
1
u
R
OPAMP
Output
R
comp
a
4
u
R
C
a
3
u
R
Figure 2. Modified Sallen-Key Topology Incorporating Op Amp GBW Compensation
The R
comp
value is selected so that it, combined with capacitor C, results in a corner frequency equal to the
op amp GBW multiplied by the DC feedback factor (see
Equation 9):
1
R
comp
a
3
C
GBW
a
3
a
4
(9)
When R
comp
is added to the standard SK topology, the transfer function can be simplified, as shown in
Equation 10
and
Equation 11.
a
3
a
4
a
3
F s
b
2
s
2
b
1
s b
0
(10)
§
b
0
·
¨ ¸
¨
b
1
¸
¨
b
¸
©
2
¹
§
·
¨
¸
1
¨
¸
¨
a
CRa
2
a
4
¸
1
4
CRa
1
CR
¨
¸
a
3
GBW GBW
a
3
¨
¸
¨
¸
CRa
2
a
4
CRa
2
2 2
¨
¸
C R a
1
a
2
¨
¸
a
3
GBW GBW
©
¹
(11)
Characteristic frequency (see
Equation 12):
1
Z
o
a
2
a
4
a
2
RC
a
1
a
2
a
3
CR
u
GBW CR
u
GBW
The quality factor (see
Equation 13):
(12)
Q
a
2
a
4
a
2
a
1
a
2
a
3
CR
u
GBW CR
u
GBW
a
4
a a
1
a
1
1
2 4
a
3
CR
u
GBW CR
u
GBW
a
3
(13)
R
comp
, in conjunction with capacitor C, adds a zero into the response that cancels the pole associated with
the limited GBW of the op amp.
4
Compensation Methodology for Error in Sallen-Key Low-Pass Filter, Caused
by Limited Gain-Bandwidth of Operational Amplifiers
Copyright © 2017, Texas Instruments Incorporated
SBAA237 – July 2017
Submit Documentation Feedback
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Simulation and Test Results
3
Simulation and Test Results
To verify the response of the revised SK topology, a filter with a second-order, Butterworth low-pass
response, DC gain of 4 V/V (12 dB), and a 300-kHz, –3 dB bandwidth was selected. The Butterworth
response has a Q factor of 0.71 and a characteristic frequency equal to the –3 dB bandwidth of the filter.
The response requirements were entered into the WEBENCH
®
Filter Designer tool from TI, which
generates the resistor and capacitor component values. FilterPro™, from TI, could have been used as
well. Both tools produce a schematic similar to that shown in
Figure 3.
Figure 3
shows the SK filter which realizes a Butterworth, second-order low-pass response, with a –3 dB
bandwidth of 300 kHz, when an ideal op amp is used. However, Filter Designer indicates that to assure
correct filter response with an actual op amp, it should have a GBW of at least 85 MHz. That is a high
GBW requirement and would require the using a high-speed op amp. High-speed op amps tend to cost
more than their lower-GBW counterparts, so the goal here is to use a low-GBW op amp and incorporate
the proposed GBW compensation. The filter responses with and without the compensation can then be
compared.
The TL081 device is a popular JFET input op amp. Its GBW is 3 MHz—much less than the 85 MHz
indicated by the filter program. Despite this, the TL081 is selected to test the new compensation method.
28.5 pF
Input
26.25 kŸ
3.75 kŸ
Output
OPAMP
100 pF
7.5 kŸ
2.5 kŸ
Figure 3. Schematic for 300-kHz, Sallen-Key, Low-Pass Filter
SBAA237 – July 2017
Submit Documentation Feedback
Compensation Methodology for Error in Sallen-Key Low-Pass Filter, Caused
by Limited Gain-Bandwidth of Operational Amplifiers
Copyright © 2017, Texas Instruments Incorporated
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