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I am thinking about using a Sallen-Key bandpass filter for an audio project I am working on.

While trying to understand the circuit, I came across two similar, but slightly different circuits.

The first circuit is from Art of Electronics and the second is from Wikipedia. The circuit from Wikipedia is pretty intuitive since it's just a high pass filter and a low pass filter with bootstrapping.

  1. Are these two circuits equivalent? (Meaning, they have the same transfer function, neglecting the gain in Fig.1 for the moment.)
  2. Can someone explain intuitively how the first circuit (AoE) realizes a bandpass filter?

From Art of Electronics:

From Art of Electronics

From the Wikipedia article on the Sallen-Key topology:

From Wikipedia on Sallen-Key topology

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    \$\begingroup\$ Well, if all you want is an intuition about the first one then just note an HF shorting cap has been moved but still performs a similar function. I'd expect the gains and damping to be different, so they are not directly equivalent in that sense. The frequency should be the same, though. \$\endgroup\$
    – jonk
    Aug 21, 2021 at 3:54

3 Answers 3

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Can someone explain \$\boxed{\color{red}{\text{intuitively}}}\$ how the first circuit (AoE) realizes a bandpass filter?

enter image description here

You then need to ask yourself what does C2 do in this circuit and, you might correctly conclude that it serves as high frequency attenuator (in either position) to currents flowing from R1 or R2. In other words currents from R1 and R2 are shunted to ground. This of course, is part of the low-pass functionality in both circuits.

And, at high frequencies, the next question might be about the placement of C1 relative to C2? If we recognize that C1 passes high frequencies much more easily than low frequencies, the position of C1 is not much of a big deal in either scenario.

But, the devil is in the detail and, if we focussed on minutia we would see some differences in the response but, like I implied when I said "minutia" it's a trivial difference.

Are these two circuits equivalent?

If you ignored trivial differences then I'd say yes but, if you were interested in the minutia then I have to say no. It all depends on exactly what you are wanting from a Sallen Key bandpass filter.

If you were really interested, then you'd simulate/model both circuits and adjust circuit values so that they produced an almost identical transfer function. Then when you have them with almost identical TFs you'd be hard pressed to justify one over the other in most applications.

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  • \$\begingroup\$ Quote Andy aka: "I said "minutia" it's a trivial difference.". I think, the difference between both circuits is not trivial. Simple example: If we design the most right circuit using two equal capacitors and three equal resistors and a gain of "2", we get a Q of 0.7071 (a pair of two complex poles) and a midband gain of unity. However, using the same components for the most left circuit, the midband gain is app 3.5dB lower and the bandwidth ist remarkably larger. That means the Q-value is much smaller (below 0.5) and we now have two real poles . \$\endgroup\$
    – LvW
    Aug 22, 2021 at 9:28
  • \$\begingroup\$ @lvw - if you use the same values in both circuits then of course, the TF differences are not trivial because the circuits are different. Given that no component values were given in the original circuit, it's a fair-game (as I said in my answer) to manipulate the values in one circuit to (very nearly) match the performance of the other. I am not claiming that the two circuits perform identically with the same values used in each. \$\endgroup\$
    – Andy aka
    Aug 22, 2021 at 9:33
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    \$\begingroup\$ Yes - I agree to your comment. Of course, both circuits can be designed for any desired bandpass characteristics - however, with a different set of formulas. \$\endgroup\$
    – LvW
    Aug 22, 2021 at 11:26
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Answer to 1): No, both circuits have not the same transfer function. A different topology gives another function. But both circuits can be used for any desired values for Qp (bandwidth, selectivity) and midfrequency wp=wo.

More than that, there is even a third alternative for which the following replacement is valid (ref. to the second figure): Replace: R1 with C1, Rf with C2, C2 with R3, R2 with C4, C1 with R5.

Answer to 2.): A bandpass contains always a series capacitor (to realize a stop for DC) and a grounded parallel capacitor (to realize a stop for infinite frequencies). This requirement is fulfilled in the first shown circuit.

However, without the feedback resistor R2 we would have one of the classical passive bandpass filters with an additional gain stage at its output. This passive filter would allow Q-values (Q=fo/bandwith) below 0.5 only (very bad selectivity). It is the purpose of R2 to provide a certain amount of positive feedback (thereby enhancing the Q-value) and, thus, to allow each desired selectivity.

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After you've convinced yourself that the transfer function (damping, resonance, and gain) of the two topologies can be identical, but attained with different values for the passive components, the next aspect to look as is other effects of the topology.

I haven't analyzed the two relative to each other, but a few aspects do come to mind considering effects other than the transfer function.

The salient difference is that one has a high-pass before low-pass, and the other a low-pass before high-pass. The op-amp feedback in the wikipedia topology is applied after the passive input low-pass structure, meaning it will have reduced high-pass energy, and therefore possibly better input noise rejection for a given gain-bandwidth product.

  1. Internal voltages and currents, and what they demand of the OP-amp: I'd consider the two to be identical in terms of internal gain, feedback amplitude and the ability to handle bias, but different in terms of the required gain-bandwidth product required of the op-amp, to deal with out-of-band rejection of strong signals.

To simulate and analyze this effect you would have to use non-ideal op-amp models and specifically look at signals in the bandpass's rejection frequency range.

  1. Internal noise sources: the two differ also differ around the resistor that shunts the positive op-amp input. Thermal noise generated by R2 in the "art of electronics" version has a high pass path via C2 to ground (aside other paths), whereas the noise in R3 of the wikipedia version has path via C1 + C2. Arguably this could make a difference and it would depend strongly on the specs of the op-amp used.

To simulate and analyze this effect you could place a voltage noise source in series with the resistor in question and inspect the noise power and noise bandwidth at the filter's output. You could repeat this with other nodes in the circuit as well.

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