3rd order Sallen-Key LPF and HPF design?

[Fancy Lime]

Hey all,

I was wondering if there is good design guideline for third order Sallen-Key filters with and without gain. I am trying to get close to Butterworth response without using crazy component values and with some restrictions concerning the impedance. If I am correctly informed, there is no analytical solution for figuring this out on third order filters (unlike for second order), so it has to be done numerically, which is just trial and error for lazy people with access to a computer. But maybe there are a few general pointers as to where to start optimizing? One method I have used in the past was to tack a passive one pole filter in front of a two pole with the same corner frequency. If you make the impedance of the single pole one or two orders of magnitude lower than that of the two pole and give the two pole a slight pass band resonance, you end up close enough to Butterworth. But in this case, I am trying to run a third order low and third order high pass from the same opamp to make a third order bandpass as part of a very primitive cab sim. This works quite well with second order because the LPF corner is about two decades above the HPF corner, but only if the impedances are chosen correctly. The HPF, which comes first, needs to have a much lower impedance than the HPF. That means, if I want to transfer this concept to third order, I would like to have all three series resistors of the LPF and the three series caps of the HPF to be the same. I know it can be done and I have seen a few examples that did that with just an HPF but not an LPF. Also, I would not mind to have the option to give the whole thing a gain of 6db and all examples I know are for 0db gain.

Thanks,
Andy

[ElectricDruid]

This is the best (only?) tool I know for this job:

http://sim.okawa-denshi.jp/en/Fkeisan.htm

Generally, it's a case of starting with a first shot, and then refining it several times. I try to eliminate awkward values or minimise the number of values, and often, with a few experiments, this is possible. I usually start by putting my required cutoff value in and seeing what it gives me, then putting tweaked values back in to its component boxes and seeing if I can adjust it and still get the response close enough to what I need.

If you need 6dB gain, start with the 2nd order stage using the 2-pole SK tool and add the 6dB gain in, and then use those values for the 3rd order tool later on. It adds another stage of tweaking, but it's not impossible.

I don't know of a better way to do it than trial-and-error, sorry.

HTH,
Tom

[GGBB]

Texas Instruments also has an online filter design tool: https://webench.ti.com/filter-design-tool/

[PRR]

> good design guideline for third order Sallen-Key filters....

IMHO, THE boot-camp is Active Filter Cookbook by Don Lancaster
"This Don Lancaster classic is by far the best-selling active filter book of all time. It gives you everything you need to know to build active lowpass, bandpass, and highpass filters."
https://www.google.com/books/edition/Active_Filter_Cookbook/abAkCPDMTBQC
Don has posted it as a free download: https://www.tinaja.com/ebooks/afcb.pdf
If you like paper, ABE Amazon etc have plenty around $20 pre-owned.

> ....with and without gain.

There is a gain of ~~1.3 in some implementations; unity-gain is a complication (you can just pad or tap). If you *also* need GAIN, consider throwing the 19 cents for another opamp. Many of the cookbook forms want an input buffer, a great place to take gain. (Unless your out-of-band crap is overloading the system.)

[Rob Strand]

Better than books.

Assumes: E24 resistors, E12 caps.
Do not use for E12 resistors.
Butterworth response shape preserved quite well.
Resistor resolution determines accuracy of -3dB frequency f3.

Gives reasonable success without tweaking parts and stuffing up the response shape.





Scaled resistors up by 10 and caps down by 10 in sim:

[Rob Strand]

Here's the response of all possible choices of E12 series caps and corresponding E24 series resistors (stepped in 2xE24 increments).

The response  at 1kHz, the nominal -3dB point, is +0.23dB, -0.56dB.    The range represents the variation due the *nominal* E12 cap values.   Not bad considering no tweaking of parts is required (which is the whole idea of this filter design).

[Fancy Lime]

Thanks guys, very useful info here!

@Tom: Yepp, Okawa is what I usually use for simple filters. Great little tool, just lacks the third order SK with gain

@Gord: ooh, nice, did not know that one! Thanks!

@Paul: thanks for the link, I did not know that there was a PDF version freely available. I think this is the book whence I got the method of tacking on another single pole with a tenth of the impedance.

@Rob: now that is precisely what I had in mind! Thanks for sharing! I'll have to give that a thorough seeing to with spice.

Cheers,
Andy