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Correct - however, does this justify the dimensioning with equal components?
I don`t think so - because this has severe disadvantages.
Selection of equal component values requires a fixed gain rater close to the stability limit (which is "3") and the filter parameter sensitivity upon tolerances is rather large.
A better design is based on unity gain or gain of "2" - with the consequence of unequal resistor values.
Correct - however, does this justify the dimensioning with equal components?
I don`t think so - because this has severe disadvantages.
Selection of equal component values requires a fixed gain rater close to the stability limit (which is "3") and the filter parameter sensitivity upon tolerances is rather large.
A better design is based on unity gain or gain of "2" - with the consequence of unequal resistor values.
I think the original question is about a Sallen Key low-pass with equal resistors. You are apparently referring to the "equal caps gain" setting. The resulting gain will however depend on the filter Q. In so far I don't exactly understand the statement about gain "near the stability limit".
As opposite design method, you can chose an arbitrary gain for the Sallen Key filter and calculate R and C values for the intended filter prototype. The stability margin will be always reached with filter Q -> infinity.
Ok - I agree. The wording "near the stability limit" may sound a bit too "dramatic".
More than that, my answer was not related to lowpass filters only (the questioner did not speak of lowpass filters only).
What I mean is the following:
1.) For S&K lowpass filters, the "equal component" design (two equal R, two equal C) results in a pole Q of Qp=1/(3-Acl) with Acl=closed-loop gain. Example: Chebyshev (1 dB ripple) with Qp=0.9565 and Acl=3-1/Qp=1.9545.
Hence, both gain determining resistors must have rather tight tolerances to meet this specific gain requirement.
2.) The situation is even more critical for bandpass filters. In this case, the equal component design results in Qp=SQRT(2)/4-Acl). Example: Q=Qp=5 with Acl=3.72. (Note that for Acl=4 the circuit oscillates already).
3.) Sensitivity calculations show that the pole Q is much more susceptible to the two gain determining resistors (active sensitivity) than to the two resistors in the passive network (which define the pole frequency).
Therefore, with regard to component tolerances it seems to be advantageous to use the unity gain concept Acl=1 (opamp with 100% feedback) for both lowpass and bandpass filters - even if we have to live with unequal components.
I am not sure when it was invented but the S-K filter was in common use in 1945, an era of slide rules, nomographs and lined pads. Ease of computation was a major feature of the S-K filter. Also in the age of 10% and 20% standard component tolerances only having to stock one or two special tight tolerance parts was a boon. Electrical performance is not always the biggest factor in filter design.
Nowadays, with more computing power than the Apollo space program available at your fingertips, there are many options that used to be impractical.
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