ETC AB-017

®
SALLEN-KEY LOW-PASS FILTER DESIGN PROGRAM
By Bruce Trump and R. Mark Stitt (602) 746-7445
Although low-pass filters are vital in modern electronics,
their design and verification can be tedious and time
consuming. The Burr-Brown FilterPro™ program makes
it easy to design unity-gain low-pass active filters. The
program supports the most commonly used all-pole filters: Butterworth, Chebyshev, and Bessel.
Butterworth—maximally flat magnitude. This filter has
the flattest possible pass-band magnitude response. Attenuation is –3dB at the design cutoff frequency. Attenuation above the cutoff frequency is a moderately steep
–20dB/decade/pole. The pulse response of the Butterworth
filter has moderate overshoot and ringing.
Chebyshev—equal ripple magnitude. (Sometimes translated Tschebyscheff or Tchevysheff). This filter response
has steeper attenuation above the cutoff frequency than
Butterworth. This advantage comes at the penalty of
amplitude variation (ripple) in the pass-band. Unlike
Butterworth and Bessel responses, which have 3dB attenuation at the cutoff frequency, Chebyshev cutoff frequency is defined as the frequency at which the response
falls below the ripple band. For even-order filters, all
ripple is above the 0dB DC response, so cutoff is at 0dB—
see Figure 1a. For odd-order filters, all ripple is below the
0dB DC response, so cutoff is at –(ripple) dB—see Figure
1b. For a given number of poles, a steeper cutoff can be
achieved by allowing more pass-band ripple. The
Chebyshev has even more ringing in its pulse response
than the Butterworth.
Bessel—maximally flat delay, (also called Thomson).
Due to its linear phase response, this filter has excellent
pulse response (minimal overshoot and ringing). For a
given number of poles, its magnitude response is not as
flat, nor is its attenuation beyond the –3dB cutoff frequency as steep as the Butterworth. It takes a higher-order
Bessel filter to give a magnitude response similar to a
given Butterworth filter, but the pulse response fidelity of
the Bessel filter may make the added complexity worthwhile.
SUMMARY
Butterworth
Advantages—Maximally flat magnitude response in the
pass-band.
Disadvantages—Overshoot and ringing in step response.
Chebyshev
Advantages—Better attenuation beyond the pass-band than
Butterworth.
Disadvantages—Ripple in pass-band. Even more ringing
in step response than Butterworth.
Bessel
Advantages—Excellent step response.
Disadvantages—Even poorer attenuation beyond the passband than Butterworth.
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5
6
7
FILTER RESPONSE vs FREQUENCY
0
0
Ripple
–10
4-Pole Chebyshev
3dB Ripple
–30
–40
10
–20
11
Ripple
–10
5-Pole Chebyshev
3dB Ripple
12
–30
13
–40
fC /10
fC
–50
fC /100
10f C
Normalized Frequency
1990 Burr-Brown Corporation
fC /10
fC
10f C
Normalized Frequency
FIGURE 1a. Response vs Frequency of Even-Order
(4-pole), 3dB-Ripple Chebyshev Filter
Showing Cutoff at 0dB.
AB-017C
1
Application Bulletin Number 17
+10
Filter Response (dB)
Filter Response (dB)
FILTER RESPONSE vs FREQUENCY
©
3
9
+10
–50
fC /100
2
8
FilterPro™, Burr-Brown Corp.
–20
1
FIGURE 1b. Response vs Frequency of Odd-Order
(5-pole), 3dB-Ripple Chebyshev Filter
Showing Cutoff at –3dB.
Printed in U.S.A. August, 1991
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Even-order filters designed with this program consist of
cascaded sections of Sallen-Key complex pole-pairs.
calculates the closest standard 1% resistor values. To
select standard 1% resistor values, use the arrow keys to
move the cursor to the Display menu selection. Then press
<ENTER>. Because the program selects the closest 1%
resistor for one resistor in each pole-pair, and then calculates the exact value for the second resistor before selecting the closest 1% value for the second resistor, it produces the most accurate filter design that can be implemented with 1% resistors.
Odd-order filters contain an additional real-pole section.
Figures 2 to 5 show the recommended cascading arrangement. Lower Q stages are placed ahead of high Q stages
to prevent op amp output saturation due to gain peaking.
The program can be used to design filters up to 7th order.
USING THE FilterPro™ PROGRAM
With each data entry, the program automatically calculates values for filter components. This allows you to use
a “what if” spreadsheet-type design approach. For example, you can quickly determine, by trial-and-error, how
many poles are needed for a given roll-off.
Using the “Scale Resistors” menu option allows you to
scale the computer-selected resistor value to match the
application. The default value of 10kΩ is suggested for
most applications.
Higher resistor values, e.g. 100kΩ, can be used with FETinput op amps. At temperatures below about 70°C, DC
errors and excess noise due to op amp input bias current
will be small. However, noise due to the resistors will be
increased by the square-root of resistor increase.
RESISTOR VALUES
The program automatically selects standard capacitor values and calculates exact resistor values for the filter you
have selected. In the “1% display” option, the program
Optional
Buffer
Real Pole Section
Complex Pole-Pair Section
Optional
Buffer
C2A
R1
A2
VO
R1A
A1
R2A
A3
VO
A1
VIN
C1
VIN
FIGURE 2. Real Pole Section (unity-gain, first-order
Butterworth) f–3dB = 1/(2π R1C1)
C1A
FIGURE 3. Second-Order, Unity-Gain, Low-Pass Filter
Using Sallen-Key Configuration for Complex Pole-Pair.
Complex Pole-Pair Section A
Optional
Buffer
Complex Pole-Pair Section N
C2A
R1A
R1X
A3
R2A
C2X
AN
R2X
VO
A1
VIN
C1A
C1X
FIGURE 4. Even-Order, Unity-Gain, Low-Pass Active Filter Using Cascaded Sallen-Key Complex Pole-Pairs.
Optional
Buffer
Real Pole Section
Complex Pole-Pair Section A
R1A
R1
A2
C2A
Complex Pole-Pair Section N
R1X
A3
R2A
C2X
AN
R2X
VO
A1
VIN
C1
C1A
C1X
FIGURE 5. Odd-Order, Unity-Gain, Low-Pass Active Filter Using One Real Pole Followed by Cascaded Sallen-Key
Complex Pole-Pairs.
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Lower resistor values, e.g. 500Ω, are a better match for
high-frequency filters using the OPA620 op amp.
amp common-mode input capacitance to the actual value of
C1. The program then automatically recalculates the exact
or closest 1% resistor values for accurate filter response.
Capacitor Values
Compared to resistors, capacitors with tight tolerances are
more difficult to obtain and can be much more expensive.
Using the “capacitor menu” option allows you to enter
actual measured capacitor values. The program will then
select exact or closest standard 1% resistor values as before.
In this way, an accurate filter response can be assured with
relatively inexpensive components.
Op Amp Selection
It is important to choose an op amp that can provide the
necessary DC precision, noise, distortion, and bandwidth.
In a low-pass filter section, maximum gain peaking at fn
(the section’s natural frequency) is very nearly equal to Q.
As a rule of thumb, for a unity-gain Sallen-Key section,
the op amp bandwidth should be at least 100 • Q3 • fn. For
a real-pole section, op amp bandwidth should be at least
50 • fn. For example, a 20kHz 5-pole Butterworth filter
needs a 8.5MHz op amp in the Q = 1.62 section.
If the common-mode input capacitance of the op amp used
in a filter section is more than approximately 0.25% of C1,
it must be considered for accurate filter response. A capacitor menu option allows you to change the values of program-selected capacitors as explained earlier. To compensate for op amp capacitance, simply add the value of the op
To aid in selection of the op amp, a program option can
display fn and Q for each section. Press <ENTER> in the
Display option of the menu. Although Q is formally
Attach Disk Sleeve Here.
Call (602) 741-3978 to down-load a DOS-compatible executable file. Down-load the
FILTER1 file from the components, analog circuit functions area. File transfers are supported
by XMODEM, Kermit, ASCII and Sealink protocols. Communications settings are 300/
1200/2400 baud, 8-N-1.
Or,
Call John Conlon, Applications Engineer
(800) 548-6132 for a DOS compatible 5-1/4" disk.
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defined only for complex poles, it is convenient to use a
Q of 0.5 for calculating the op amp gain required in a realpole section.
output requires an op amp slew-rate
6.3V/µs. Burr-Brown offers an excellent
op amps which can be used for high
active filters. The guide below lists
choices.
The slew rate of the op amp must be greater than
π • VOp-p • FILTER BANDWIDTH for adequate full-power
response. For example, a 100kHz filter with 20Vp-p
of at least
selection of
performance
some good
OP AMP SELECTION GUIDE, (IN ORDER OF INCREASING SLEW RATE.)
TA = 25°C, VS=±15V, specifications typ, unless otherwise noted, min/max specifications are for high-grade model.
OP AMP
MODEL
BW
typ
(MHz)
FPR (1)
typ
(kHz)
SR
typ
(V/µs)
VOS
max
(µV)
VOS/dT
max
(µV/°C)
NOISE
at 10kHz
(nV/√Hz)
CCM (3)
(pF)
OPA177
0.6
3
0.2
10
±0.1
8
1
8
30
1.9
25
±0.6
2.7
1
(2)
4.5
280
18
500
±5
8
4
OPA2604(2)
dual
10
400
25
2000
±5 typ
10
10
OPA602(2)
6
500
35
250
±2
12
3
(2)
6
500
35
1000
±3 typ
12
3
OPA27
OPA2107
dual
OPA404
quad
OPA627(2)
16
875
55
100
±0.8
4.5
7
OPA620
(VS = ±5V)
300
16MHz
(5Vp-p)
250
500
±8 typ
2.3
(at 1MHz)
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NOTES: (1) Unless otherwise noted, FPR is full power response at 20Vp-p as calculated from slew rate. (2) These op amps have FET inputs. (3) Commonmode input capacitance.
CAPACITOR SELECTION
Capacitor selection is very important for a high-performance filter. Capacitor behavior can vary significantly
from ideal, introducing series resistance and inductance
which limit Q. Also, nonlinearity of capacitance vs voltage causes distortion.
Simple design procedures for the UAF42 allow implementation of Butterworth, Chebyshev, Bessel, and other types
of filters. An extra FET-input op amp in the UAF42 can be
used to form additional stages or special filter types such as
band-reject and elliptic. The UAF42 is available in a standard 14-pin DIP. For more information about the UAF42
request Burr-Brown Product Data Sheet PDS-1070.
Common ceramic capacitors with high dielectric constants, such as “high-K” types can cause errors in filter
circuits. Recommended capacitor types are: NPO ceramic,
silver mica, metallized polycarbonate; and, for temperatures up to 85°C, polypropylene or polystyrene.
EXAMPLES OF FILTER RESPONSE
Figures 6a and 6b show actual measured magnitude response plots for 5th-order 20kHz Butterworth, 3dB
Chebyshev and Bessel filters designed with the program.
The op amp used in all filters was the OPA627. As can be
seen in Figure 5, the initial roll-off of the Chebyshev filter
is fastest and the roll-off of the Bessel filter is the slowest.
However, each of the 5th-order filters ultimately rolls off
at –N • 20dB/decade, where N is the filter order (–100dB/
decade for a 5-pole filter).
THE UAF42 UNIVERSAL ACTIVE FILTER
For other filter designs, consider the Burr-Brown UAF42
Universal Active Filter. It can easily be configured for a
wide variety of low-pass, high-pass, or band-pass filters. It
uses the classical state-variable architecture with an inverting amplifier and two integrators to form a pole-pair. The
integrators include on-chip 1000pF, 0.5% capacitors. This
solves one of the most difficult problems in active filter
implementation—obtaining tight tolerance, low-loss capacitors at reasonable cost.
The oscilloscope photographs show the step response for
each filter. As expected, the Chebyshev filter has the most
ringing, while the Bessel has the least.
4
10
+3
0
Bessel
–10
Filter Response (dB)
Filter Response (dB)
0
–20
–30
Butterworth
–40
–50
3dB Chebyshev
–60
–3
–6
–12
–18
–21
–80
–24
1k
10k
Frequency (Hz)
Butterworth
–15
–70
–90
200
Bessel
–9
3dB Chebyshev
–27
400
200k
1k
10k
20k
40k
Frequency (Hz)
FIGURE 6a. Gain vs Frequency for 5th-Order 20kHz
Butterworth, 3dB Chebyshev, and Bessel
Unity-Gain Low-Pass Filters Showing Overall Filter Response.
FIGURE 6b. Gain vs Frequency for 5th-Order 20kHz
Butterworth, 3dB Chebyshev, and Bessel
Unity-Gain Low-Pass Filters Showing Transition Band Detail.
FIGURE 7. Step Response of 5th-Order 20kHz
Butterworth Low-Pass Filter.
FIGURE 8. Step Response of 5th-Order 20kHz 3dB Ripple
Chebyshev Low-Pass Filter.
FILTER THD, 20kHz, 5 POLE
THD + Noise (%)
1
Chebyshev
Butterworth
Bessel
0.1
0.01
0.001
10
10 2
103
10 4
105
Frequency (Hz)
FIGURE 9. Step Response of 5th-Order 20kHz Bessel
Low-Pass Filter.
FIGURE 10. Measured Distortion for the Three 20kHz
Low-Pass Filters.
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The information provided herein is believed to be reliable; however, BURR-BROWN assumes no responsibility for inaccuracies or omissions. BURR-BROWN
assumes no responsibility for the use of this information, and all use of such information shall be entirely at the user’s own risk. Prices and specifications are subject
to change without notice. No patent rights or licenses to any of the circuits described herein are implied or granted to any third party. BURR-BROWN does not
authorize or warrant any BURR-BROWN product for use in life support devices and/or systems.
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