Experts have expended plenty of ink analyzing the venerable Sallen Key low pass filter (SKF) topology since its description by R.P. Sallen and E.L. Key of MIT Lincoln Labs in 1955.
1 Depending on what assumptions you bring into the analysis, results can be very different.
Almost all efforts immediately constrain implementations by unity gain in the op amp, equal R or equal C, or both. When multi-stage filters are implemented with gain, most of the gain is placed in the earliest stages. These assumptions don’t always lead to particularly good results. But they are entrenched in the literature and most online design tools as a sort of design mythology.
How do these “myths” arise? For a single second-order stage, the designer is really only after three performance elements: dc gain, ω0 (the characteristic frequency of a second-order pole pair), and Q (an indication of the complexness of the poles). However, the circuit gives you five elements to resolve: the two R’s, the two C’s, and the amplifier gain.
Constraining your design flow to just unity gain and equal C (a very typical approach in academia) removes two of those pesky degrees of freedom, leaving you a unique set of R’s that will hit the ω0 and Q once you have chosen some desirable equal C values.
Many authors also deem equal C desirable to get better matching in the C’s, particularly when the effort is headed toward IC implementations. Also, in the earliest days, it was no small task to come up with amplifiers fast enough to implement these filters reliably without considerable production variation, and unity gain certainly helps with that issue.
The last 25 years have brought major advances in op amps and passives. Low-power/low-cost voltage feedback (VFA) op amps with unity gain stability and bandwidths in excess of 1 GHz are available. Also useful are current feedback op amps (CFA) that hold relatively constant bandwidth over gain, so they are very uesful in SKF stages that require gain. These are available at low power and low cost with bandwidths exceeding 500 MHz over gains from 1 to 10.
Some sources imply the CFA topology cannot be used in an active filter circuit. Much like the ancient fear that viewing the Medusa would turn you into stone, this myth spreads unnecessary trepidation where in fact these devices are very useful in SKF low pass stages. Finally, C0G dielectric, 1% multilayer ceramic capacitors (MLCCs) with low temperature drift are also available at low cost.
From Myth To Reality
With these improvements in the SKF building blocks, it seems time to move beyond these legacy constraints and deliver better solutions. Since the absolute accuracy of the R’s and C’s have improved, and amplifier bandwidths over a wide range of gains is far better today, it’s time to leave behind these myths of low or unity gain, equal C, or equal R and focus instead on more interesting issues inside the SKF.
Given the under-constrained nature of the SKF, it is still important to realize that an infinite range of R and C combinations will give you the same target ω0 and Q (if you have a particular gain in mind at low frequencies setting the amplifier gain). The next most interesting thing is to bias the R and C selections to improve the dynamic range in the passband.
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