Quick and dirty
filter design
By Dave Erickson
Intro: Butterworth,
the filter, not the pancake syrup.
Butterworth filters are
very handy. Also known as maximally flat, they have a nice smooth
frequency response. They are the compromise between frequency and
step response. The do have about 6% or so overshoot, so be careful
when processing square waves or steps. A 2nd order Butterworth is
pretty simple. Here's the low-math approach. When I design a
filter, I typically do the design, do the component scaling in
Excel, then simulate it with LT Spice, and compare the frequency
and step response to the expected. Then do a bit of tolerance
variation to see if 5%, 2% or 1% capacitors are needed. 1%, 0.5%
and 0.1% resistors are pretty cheap, so the capacitors are the
limiting component for accuracy. The op-amp selection also can
affect the response, particularly the high-frequency roll-off.
Simulation is your friend in filter design.
For a 2nd order butterworth, there are any number of
implementations to get a specific response. Virtually unlimited.
You can vary any or all of the resistors and capacitors, the
amplifier gain, use an inverting or a non-inverting amp, use an IC
opamp or transistors, you name it. And they will all provide the
same response. So where to start?
I like to minimize the total number of components, and to minimize
number of different components. You do need to make a few
decisions. And need to use just a bit of math.
Let's start with a simple example. R1, R2 = 20K, C1 = 10nF, C2 =
5nF. Gain = 1. Sallen-Key.
The cutoff frequency of a Sallen-Key filter is: Fc = 1 /
(2*pi*(sqrt(R1*R2*C1*C2))
The capacitor ratio affects
the damping ratio. This provides a Butterworth response
at a cutoff (Fc) of 2.25 KHz. The two resistors are the same
value, and the capacitors are in a 2:1 ratio. The math to prove
this can be daunting, so here is a bit of hand-waving. The
polynomial of a Butterworth is 
.
The capacitors with equal-value resistors are in the ratio of
SQRT(2) and 1/SQRT(2). so 2:1.
This seems nice and simple, but actually getting 2:1
capacitors can be tricky. Caps often come in the E12 (10, 12, 15, 18, 22...) or E24 series: (10,
11, 12, 13, 15, 16, 18...) E12 doesn't provide accurate 2:1. E24
series however has many 2:1 values. Or you can parallel (or
series) 2 caps to provide the 2x (or 1/2) value. This has the
advantage of minimizing the number of values used. Typically you
will select the closest capacitors, then tweak the resistors to
get the closest cutoff you need. Also 10:5.1 (=1.96), or even 10:
4.7 (=2.13), 22:10: (=2.2), are all pretty close to 2:1.
Regarding capacitor selection, best to use film capacitors for
audio range and for accuracy, and NPO ceramic's (or Mica's) for
higher frequency filters. Be very careful if you try to use
electrolytics or other ceramics.
Here is an interesting simulation. These 4 filters all have the
same 1.1KHz Butterworth response, but the differences in the
stop-bands are interesting. The rises are due to the output
impedance of the opamps at high frequencies, being fed by the RC
values. Note that the low impedance
(2K) filters are worse, since the low impedance resistors cause
feed-forward current to fight the op-amp output impedance.
Everything is a compromise. If you don't care much
about stop-band response above 30KHz or so, then they will all
work. But if you are filtering fast pulses with content above 1Mhz
or so, then be careful.
BTW the op-amps I picked are LT specific. The LT1057 is BiFet,
similar in speed to a TL072 or LF351, about 5MHz. The LT1468 is my
go-to medium-fast (80Mhz) opamp. I used it for 4th-order 80KHz
Butterworh filters and fast DAC buffers.
Here is a nice, simple 2nd order Butterworth
Calculator
Here is the nice Analog Devices
filter tool. It does many different
Filters
4th order Butterworth
The first time I needed a
low frequency 4th order Butterworth, I simply cascaded two 2nd
order Butterworth filters. This does NOT provide a true 4th order
Butterworth response, and the calculated Fc for the net filter is
actually -6dB, not 3dB. But it was close enough for the
application I had.
Later I worked on a proper 4th order, 80KHz filter. I suspect the
original designer used a web filter design tool. For a proper 4th
order, made from two cascaded 2nd order sections, the polynomial
equation is:
Each section has its
own Fc and damping ratio. In this design case, there
were two channels, and the frequency responses and time responses
had to match quite closely. So we used 1% COG capacitors and 0.5%
resistors. Here is the schematic for one filter. It uses LT1801, a
single supply, 80MHz, +5V,
op-amp.
And a simulation:
Passive LC
filter design
I used to design video
digitizers for Datacube. Low-pass filters for anti-aliasing. I
typically used 75 ohm 3 or 5 pole Pi filters: CLC or CLCLC
butterworth. Cutoff was typically 5-6MHz. My boss had an
original Zverev "Handbook
of Filter Synthesis" from 1967. This excellent hardcover
book has tables and graphs for almost all LC filters. It is out of
print now but available on line and used. I would enter the
coefficients into a spreadsheet, along with the normalized
impedance and desired frequency, and calculate the L and C
values.
For active filter design, I use these books:
Don Lancaster's "Active
Filter Cookbook" Signed copy!
Howard M. Berlin "Design of Active Filters, with Experiments"
Howard W Sams & Co.
and the ancient Burr-Brown "Handbook
of Operational Amplifier Active RC Networks"
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Last updated 6/14/2025