There exists a basic type of low-pass filter, called a Butterworth Filter, which is a 2nd-order filter, which therefore has a falloff-rate of -12db /Octave, far above the corner frequency, and this is its general diagram:

Even though it is clear from this diagram that the two capacitors, or the two resistors, are allowed to have different values, the way the design of this filter is mainly taught today, both resistors are made equal, as are both capacitors, thus simplifying the computation of each, once the other has been determined according to what seems practical, applying the same principle as what would be applied for a 1st-order filter.

One basic weakness of this filter, especially in modern applications, is the fact that it will attenuate frequency-components considerably, which are below its corner-frequency. There have historically been two approaches taken to reduce this effect, if any attempt has been made to do so at all:

- C1 can be given twice the value of C2, but R1 and R2 kept equal. This poses the question of whether the corner-frequency will still be correct. And my estimation is that because of the way Electrical Engineers have defined the corner-frequency, the specific frequency-response at that frequency should remain the square root of 1/2 (or, -3db). But, if C1 is larger than C2, then the frequency-response will not be the same at any other point in the curve. I.e., the curve could be flatter, with response-values closer to unity, at frequencies considerably lower than the corner-frequency.
- The operational amplifier stage, which in the basic design is just a voltage-follower, can be transformed into a gain-stage, with a gain slightly higher than one. This is done by placing a voltage-divider from the output of an operational amplifier, to yield the feedback voltage, fed to its inverting input. What needs to be stressed here, is that significantly high gain leads to an unstable circuit.

While either approach can be taken, it is important not to apply both at the same time, as the amount of feedback given by C1 would be exaggerated, and would lead to a hot-spot somewhere in the pass-band of this filter. In general, the trend today would be to use approach (2).

And while it might seem to make sense just to boost the output of the amplifier-stage by the same factor, by which we would want to correct the gain at the corner-frequency, it also needs to be remembered that increasing this open-loop gain not only boosts the output of the active stage once, but additionally boosts the amount of feedback leading back in to the positive input of the operational-amplifier, by way of C1. This tends to resemble an infinite series – the power-series – which needs to be made convergent through the use of an alpha of much less than two.

And so, if we wanted our stage to have as Q the square root of 2, then it would seem to make most sense to use an alpha of 1.29 .

Now, one important context in which these low-pass filters can be used, is in a sequence of up to 3 chained together, so that within 2 octaves of separation, a higher frequency can be suppressed to -70db with respect to a lower frequency.

(Edit 05/03/2017 : )

When chaining them, the fact that each stage would normally attenuate its own declared pass-band becomes more of a problem, and then approach (2) definitely seems better-suited to dealing with that. But still, significant coloration of the frequency-response curve, within the pass-band, would ensue. And so an additional trick which may be used, is that each stage is given a slightly different corner-frequency. Let us say that the corner-frequencies are to be offset progressively by 25%, and that the first starts at 20 kHz. This would mean that three corner-frequencies of 20, 25 and 31.25 kHz would need to be applied in practice. And then the fact that the third of those has already reached 31 kHz, explains why it becomes impractical to chain more than 3 stages in this way.

The fact that each pass-band is not completely flat, accounts for ripple in the response-curve which can be measured just below the lowest corner-frequency. And, placing them closer together yields lower-quality results, with potentially more ripple.

One observation which deserves mention, is that when a lower-order filter is given a higher value of Q (than 1), it will seem to achieve a sharper falloff-curve, than its order would suggest by itself.

This will only take place close to the corner-frequency, and if the stages do not have the same corner-frequency, then the effect will not compound strongly between them. Further, this effect will also produce a local peak in response, just below the corner-frequency, for which reason higher values of Q should be used with caution.

I suppose that a meaningful question to ask would be, ‘Why was approach (1) above ever used?’ And the reason only has to do with the fact that historically, circuits were designed using discrete components, and that the cost of designing a circuit could therefore depend linearly on the number of components needed – including resistors and capacitors. Approach (1) requires no higher number of components, than the standard Butterworth Filter would have required, and therefore offers an improvement at no additional cost.

The first use of operational amplifiers implied that Small-Scale Integration already existed, in which one physical chip might have delivered 1 or 2 operational amplifiers, but took up space on the circuit-board as if it was itself one discrete component. And if a circuit-designer wanted to use transistors with special properties, in the 1970s one such transistor might have cost him $ 50 !

Today we live in a world where Large-Scale Integration is often available, and for custom circuits, Medium-Scale is as available. And so to put the extra components can take place on one chip. In fact, if a chip is to be mass-produced with one design, then indeed 3 Butterworth Filters in a chain can also be made monolithic, at no apparent increase in cost…

Dirk