ETC AB-034

APPLICATION BULLETIN
®
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MFB 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, FILTER2,
makes it easy to design low-pass active filters. The program
is intended to aid in the design of low-pass filters implemented with the Multiple Feedback (MFB) topology. Because there are instances where the Sallen-Key filter topology is a better choice, the program also supports Sallen-Key
low-pass filter design.
An ideal low-pass filter would completely eliminate signals
above the cutoff frequency, and perfectly pass signals below
cutoff (in the pass-band). In real filters, various trade-offs
are made in an attempt to approximate the ideal. Some filter
types are optimized for gain flatness in the pass-band, some
trade-off gain variation (ripple) in the pass-band for steeper
roll-off, still others trade-off both flatness and rate of roll-off
in favor of pulse-response fidelity. FILTER2 supports the
three most commonly used all-pole filter types: 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). (Also transliterated
Tschebychev, Tschebyscheff or Tchevysheff.) This filter
type 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-gain
DC response, so cutoff is at 0dB—see Figure 1A. For oddorder filters, all ripple is below the 0dB-gain 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 time 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 which approaches that of a given
Butterworth filter, but the pulse response fidelity of the
Bessel filter may make the added complexity worthwhile.
FILTER RESPONSE vs FREQUENCY
FILTER RESPONSE vs FREQUENCY
+10
+10
0
Ripple
Filter Response (dB)
Filter Response (dB)
0
–10
–20
4-Pole Chebyshev
3dB Ripple
–30
–40
–50
fC/100
f C/10
fC
–30
–50
fC/100
10fC
f C/10
fC
10fC
Normalized Frequency
FIGURE 1A. Response vs Frequency of Even-Order
(4-pole), 3dB Ripple Chebyshev Filter Showing Cutoff at 0dB.
1991 Burr-Brown Corporation
–20
5-Pole Chebyshev
3dB Ripple
–40
Normalized Frequency
©
Ripple
–10
FIGURE 1B. Response vs Frequency of Odd-Order (5pole), 3dB Ripple Chebyshev Filter Showing
Cutoff at –3dB.
AB-034B
Printed in U.S.A. July, 1993
A2
R1
VO
Complex
Pole-Pair
Section A
A1
A1
VIN
VIN
VO
C1
Optional Buffer
Optional Buffer
Real Pole Section
FIGURE 2. Real Pole Section (Unity-Gain, First-Order
Butterworth; f–3dB = 1/2 • π • R1 • C1).
FIGURE 3. Second-Order Low-Pass Filter.
Complex
Pole-Pair
Section A
A2
R1
A1
VIN
VO
C1
Optional Buffer
Real Pole Section
FIGURE 4. Third-Order Low-Pass Filter.
Complex
Pole-Pair
Section A
A1
VIN
•••
Complex
Pole-Pair
Section N
VO
Optional Buffer
FIGURE 5. Even-Order Low-Pass Filter Using Cascaded Complex Pole-Pair Sections.
Complex
Pole-Pair
Section A
A2
R1
A1
VIN
•••
Complex
Pole-Pair
Section N
VO
C1
Optional Buffer
Real Pole Section
FIGURE 6. Odd-Order Low-Pass Filter Using Cascaded Complex Pole-Pair Sections Plus One Real-Pole Section.
2
SUMMARY
Butterworth
Advantages
Disadvantages
R2
C1
Maximally flat magnitude response in the
pass-band. Good all-around performance.
Pulse response better than Chebyshev. Rate
of attenuation better than Bessel.
R1
R3
VIN
A2
C2
Some overshoot and ringing in step response.
VO
Chebyshev
Advantages
Better attenuation beyond the pass-band
than Butterworth.
Disadvantages
Ripple in pass-band. Considerable ringing
in step response.
FIGURE 7. MFB Complex Pole-Pair Section.
(Gain = –R2/R1)
Bessel
C2
Advantages
Best step response—very little overshoot
or ringing.
Disadvantages
Slower rate of attenuation beyond the passband than Butterworth.
R1
A2
R2
VO
VIN
C1
CIRCUIT IMPLEMENTATION
Even-order filters designed with this program consist of
cascaded sections of complex pole-pairs. Odd-order filters
contain an additional real-pole section. Figures 2 through 6
show the recommended cascading arrangement. The program automatically places lower Q stages ahead of higher Q
stages to prevent op amp output saturation due to gain
peaking. The program can be used to design filters up to 8th
order.
FIGURE 8. Unity-Gain Sallen-Key Complex Pole-Pair
Section. (Gain = 1)
C2
R1
R2
VIN
FILTER ORDER
FIGURE
1 pole
2 poles
3 poles
4 or more poles (even order)
5 or more poles (odd order)
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
A2
C1
R3
VO
R4
TABLE I. Filter Circuit vs Filter Order.
FIGURE 9. Sallen-Key Complex Pole-Pair Section.
(Gain = 1 + R4/R3)
COMPLEX POLE-PAIR CIRCUIT
The choice of a complex pole-pair circuit depends on performance requirements. FILTER2 supports the two most commonly used op amp pole-pair circuit topologies:
At unity-gain, the Sallen-Key topology inherently has excellent gain accuracy. This is because the op amp is used as a
unity-gain buffer. With the MFB topology, gain is determined by the R2/R1 resistor ratio. The unity-gain Sallen-Key
topology also requires fewer components—two resistors vs
three for MFB.
• Multiple Feedback (MFB)—shown in Figure 7.
• Sallen-Key—shown in Figures 8 and 9.
The MFB topology (sometimes called Infinite Gain or Rauch)
is often preferred due to assured low sensitivity to component variations—see sensitivity section. There are instances,
however, where the Sallen-Key topology is a better choice.
The Sallen-Key topology may also be preferable for high-Q
high frequency filter sections. In these sections the value
required for C1 in a MFB design can be quite low for
reasonable resistor values. Low capacitor values can result
in significant errors due to parasitic capacitances.
As a rule of thumb, the Sallen-Key topology is better if:
1) Gain accuracy is important, AND
2) A unity-gain filter is used, AND
3) Pole-pair Q is low (e.g. Q < 3)
3
11) To change the display screen press <ENTER> on the
Display line. Available display screens are: Component
values; fn, Q, Response; Sensitivities.
The best filter design may be a combination of MFB and
Sallen-Key sections.
Use the appropriate circuits along with component values
generated by the FILTER2 program to build your filter
design.
TO RETURN TO DOS
To exit the program and return to DOS, press <F1>.
USING THE FILTERPRO™ PROGRAM
With each data entry, the program automatically calculates
filter performance and values for all 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 PLOT FEATURE
A Plot feature allows you to view graphical results of filter
gain and phase vs frequency. This feature is useful for
comparing filter types.
To view a plot of the current filter design, press <F2>.
GETTING STARTED
The first time you use the program, you may want to follow
these steps.
GRAPHIC DISPLAY COMMANDS
While viewing the graphic display, several commands can
be used to compare filter responses:
Type FILTER2 <ENTER> to start the program.
S—Saves the plot of the current design for future recall.
At this point, you have already designed a 3-pole, 1kHz
Butterworth filter. Component values are shown on the
display screen. For a different filter design, use the arrow
keys to move the cursor to the menu selection desired.
R—Recalls the Saved plot and plots it along with the current
design.
P—Recalls the Previous plot of the last design plotted from
the main program (by pressing <F2>) and plots it along with
the current design. You can recall the Previous design and
the Saved design to plot all three together.
Start at the top of the menu and work your way down. Onscreen prompts, to the left of the menu selections, will guide
you in program use. Refer to this bulletin for more detail, if
needed.
C—Clears the display and replots only the current design.
1) Choose pole-pair Circuit: MFB or Sallen-Key
2) Select Filter type: Butterworth
Chebyshev
Bessel
GRAPHIC DISPLAY CURSOR CONTROL
While viewing the graphics display you can also use the left/
right arrow keys to move a cursor and view gain and phase
for plotted filter responses. The gain/phase of the current
design is always displayed. In addition, the gain/phase of the
Recalled or Previous design can be viewed by pressing R or
P.
3) For Chebyshev filter type, enter Ripple: >0dB to 4dB
4) Enter Number of poles: 1 to 8
5) Enter filter Cutoff frequency: (Hz)
The following steps are optional:
6) If you want to view the gain/phase response of the
current filter design at a particular frequency, enter the
frequency of interest on the Response @ fx line. The
gain/phase information can be viewed on the fn, Q,
Response display window—see step 11.
TO PRINT RESULTS
To print results press <F3>. All three display screens will
automatically be printed.
7) If you want to change the resistor scaling, enter a value
on the Scale Resistors line.
SENSITIVITY
Sensitivity is the measure of the vulnerability of a filter’s
performance to changes in component values. The important
filter parameters to consider are natural frequency (fn) and
Q.
8) If you want to change the gain of a section, press
<ENTER> on the Gain Entry line. Default value for
gain is 1.0V/V in each section.
9) If you want to enter your own capacitor values, press
<ENTER> on the Capacitor Menu line.
fn SENSITIVITY FOR BOTH
MFB AND SALLEN-KEY
Sensitivity of fn to resistor, capacitor, and amplifier gain
variations is always low for both the Sallen-Key and MFB
filter topologies.
10) If you want to design with standard 1% resistors instead
of exact resistors, press <ENTER> on the Resistors
line.
4
f
USING THE SENSITIVITY DISPLAY FEATURE
To use the Sensitivity display option, move the cursor to the
Display menu, press <ENTER>, move the cursor to the
Sensitivity selection, and press <ENTER> again. The display shows sensitivity of fn and Q to each component for
each filter section. The format is Sf; SQ.
f
SR = SC = ±0.5%/%
SfK = 0
Where:
SfR, SfC, SfK = Sensitivity of fn to resistor,
capacitor, and gain variations
(%/%)
Rather than displaying the derivative with respect to component variations, the program calculates fn and Q change for
a 1% change in component values. This gives a more
realistic sensitivity value for real-world variations.
Q SENSITIVITY
For the MFB topology, sensitivities to Q are also always
low, but sensitivities for the Sallen-Key topology can be
quite high—exceeding 2 • K • Q2. At unity gain, the SallenKey Q sensitivity to resistor and capacitor variations will
always be low. Unfortunately, however, the sensitivity of
the unity-gain Sallen-Key pole-pair to op amp gain can be
high.
USING THE SCALE RESISTORS MENU OPTION
The Scale Resistors option allows you to scale the computer-selected resistor values to match the application. Move
the cursor to the Scale Resistors menu selection and enter
your seed resistor value. The default value of 10kΩ is
suggested for most applications.
Q Sensitivity for MFB Pole-Pair
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. Remember, however, that noise due to the resistors
will be increased by √n where n is the resistor increase
multiplier.
Q
SC = ±0.5%/%
Q
SR = ±
SQ
K =
R2 – R3 – K • R3
2 ( R 2 + R 3 + K • R 3)
K • R3
R2 + R3 + K • R3
(MFB complex
pole-pair)
(MFB complex
pole-pair)
Lower resistor values, e.g. 500Ω, are a better match for highfrequency filters using the OPA620 or OPA621 op amps.
Q
Notice, by inspection: SR is always less than ±0.5%/%,
Q
and SK is always less than 1.0%/%.
CAPACITOR VALUES
Compared to resistors, capacitors with tight tolerances are
more difficult to obtain and can be much more expensive.
The Capacitor menu option allows you to enter actual
measured capacitor values. In this way, an accurate filter
response can be achieved with relatively inexpensive components.
Q Sensitivity for Gain = 1 Sallen-Key Pole-Pair
SQ
C = ±0.5%/%
SQ
R = ±
R1 – R2
2 ( R 1 + R 2)
(Sallen-Key
complex pole-pair)
So, SQ
R is always less than 0.5%/%.
2
Q2 < SQ
K < 2 • Q (Sallen-Key complex pole-pair)
USING THE CAPACITOR MENU OPTION
To use the Capacitor menu option, move the cursor to the
Capacitor menu selection and press <ENTER>. Move the
cursor to any capacitor and enter your value. Prompts on the
left of the screen advise min/max capacitor entry limits.
With each capacitor entry, the program will select exact or
closest standard 1% resistor values as before.
Where:
SQR , SQC , SQK = Sensitivity of f and Q to resistor,
capacitor, and gain variations
(%/%)
K = Op amp gain (V/V)
Figure 7 circuit, K = R2/R1
Figure 8 circuit, K = 1.0
Figure 9 circuit, K = 1 + R4/R3
COMPENSATE FOR OP AMP
INPUT CAPACITANCE—SALLEN-KEY ONLY
If the common-mode input capacitance of the op amp used
in a Sallen-Key filter section is more than approximately
C1/400 (0.25% of C1), it must be considered for accurate
filter response. You can use the Capacitor menu option to
compensate for op amp input capacitance by simply adding
the value of the op amp common-mode input capacitance to
the actual value of C1. The program then automatically
NOTE: FilterPro™ always selects component values so
unity-gain Sallen-Key SQK will be closer to Q2 than to 2 • Q2.
However, FILTER2 will allow you to design Sallen-Key
pole-pairs with high sensitivities (high Qs and GAIN >> 1).
You must make sure that sensitivities to component variations do not make these designs impractical. A feature in the
Display menu allows you to view the fn and Q sensitivity of
filter sections to resistor and capacitor variations.
5
recalculates the exact or closest 1% resistor values for
accurate filter response. No compensation for op amp input
capacitance is required with MFB designs.
press <ENTER>, move the cursor to the fn & Q selection,
and press <ENTER> again. The fn and Q information is also
useful when trouble-shooting filters by comparing expected
to actual response of individual filter sections.
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.
OP AMP SLEW RATE
For adequate full-power response, the slew rate of the op
amp must be greater than π • VOp-p • FILTER BANDWIDTH. For
example, a 100kHz filter with 20Vp-p output requires an op
amp slew rate of at least 6.3V/µs. Burr-Brown offers an
excel-lent selection of op amps which can be used for high
performance active filters. The guide on P-7 lists some good
choices.
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.
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, band-pass, or bandreject (notch) 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.
OP AMP SELECTION
It is important to choose an op amp that can provide the
necessary DC precision, noise, distortion, and speed.
OP AMP BANDWIDTH
In a low-pass filter section, maximum gain peaking is very
nearly equal to Q at fn (the section’s natural frequency). So,
as a rule of thumb:
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
Inverse Chebyshev. The UAF42 is available in a standard
14-pin DIP. For more information, request the Burr-Brown
Product Data Sheet PDS-1070 and Application Bulletin
AB-035.
For an MFB section: Op amp bandwidth should be at least
100 • GAIN • fn.
High-Q Sallen-Key sections require higher op amp bandwidth.
For a Sallen-Key section: For Q > 1, op amp gainbandwidth should be at least
100 • GAIN • Q3 • fn.
For Q ≤ 1, op amp gainbandwidth should be at least
100 • GAIN • fn.
For a real-pole section: Op amp bandwidth should be at
least 50 • fn.
Although Q is formally defined only for complex poles, it is
convenient to use a Q of 0.5 for calculating the op amp gain
required in a real-pole section.
EXAMPLES OF MEASURED
MFB FILTER RESPONSE
Figures 10 and 11 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 10, 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).
For example, a unity-gain 20kHz 5-pole, 3dB ripple
Chebyshev MFB filter with a 2nd pole-pair fn of 19.35kHz
and a Q of 8.82 needs an op amp with unity gain bandwidth
of at least 17MHz. On the other hand, a 5-pole Butterworth
MFB filter, with a worst case Q of 1.62 needs only a 3.2MHz
op amp. The same 5-pole Butterworth filter implemented
with a Sallen-Key topology would require a 8.5MHz op amp
in the high-Q section.
The oscilloscope photographs (Figures 12-14) show the step
response for each filter. As expected, the Chebyshev filter
has the most ringing, while the Bessel has the least. Figure
15 shows distortion plots vs frequency for the three filters.
See Application Bulletin AB-017 for measured Sallen-Key
filter performance of the same three designs.
USING THE fn AND Q DISPLAY OPTION
To aid in selection of the op amp, a feature in the Display
menu section allows you to view pole-pair section fn and Q.
To use this feature move the cursor to the Display menu,
6
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.
FPR (1)
typ
(kHz)
OP AMP
MODEL
BW
typ
(MHz)
OPA177
0.6
3
0.2
10
OPA27
8
30
1.9
25
SR
typ
(V/µs)
VOS
max
(µV)
VOS/dT
max
(µV/°C)
NOISE
at 10kHz
(nV/√Hz)
CCM (3)
(pF)
±0.1
8
1
±0.6
2.7
1
4.5
280
18
500
±5
8
4
OPA602 (2)
6
500
35
250
±2
12
3
OPA404 quad (2)
6
500
35
1000
±3 typ
12
3
OPA2107 dual
(2)
(2)
16
875
55
100
±0.8
4.5
7
OPA620 (VS = ±5V)
300
16MHz(5Vp-p)
250
500
±8 typ
2.3 @ 1MHz
1
OPA627
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) Common-mode input
capacitance.
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)
–27
400
200k
Butterworth
–15
–70
–90
200
Bessel
–9
3dB Chebyshev
1k
10k
20k
40k
Frequency (Hz)
FIGURE 10. Gain vs Frequency for Fifth-Order 20kHz
Butterworth, Chebyshev, and Bessel UnityGain MFB Low-Pass Filters, Showing Overall Filter Response.
FIGURE 11. Gain vs Frequency for Fifth-Order 20kHz
Butterworth, Chebyshev, and Bessel UnityGain MFB Low-Pass Filters, Showing Transition-Band Detail.
7
FIGURE 12. Step Response of Fifth-Order 20kHz
Butterworth Low-Pass MFB Filter.
FIGURE 13. Step Response of Fifth-Order 20kHz
Chebyshev Low-Pass MFB Filter.
FILTER THD, 20kHz, 5 POLE
100
THD + Noise (%)
10
1
Chebyshev
Butterworth
Bessel
0.1
0.01
0.001
10
100
1k
10k
100k
Frequency (Hz)
FIGURE 15. Measured Distortion for the Three 20kHz MFB
Low-Pass Filters.
FIGURE 14. Step Response of Fifth-Order 20kHz Bessel
Low-Pass MFB Filter.
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.
8