I have found this circuit:


To me, this just looks like a two-pole passive filter going into a non inverting amplifier, the only difference is that that one capacitor is connected to the output. What exactly is the point of connecting the capacitor that way, does that somehow create resonance? And how does changing the resistance of the "Q control" resistor alter the resonance?

Also, this is a two-pole filter. If I wanted a four-pole filter, would I be able to just add two more RC circuits in front of the input? If so, would either of the two additional capacitors need to be connected to the output?

 

It is more complicated than that. Read about Sallen-Key filters

https://en.wikipedia.org/wiki/Sallen–Key_topology
http://www.ti.com/lit/an/sloa024b/sloa024b.pdf

 

A couple of additional sources:
https://www.academia.edu/15449434/Design_of_Analog_Filters_Rolf_Schaumann_Mac_E_Van_Valkenburg
http://www.sophphx.caltech.edu/Physics_5/Experiment_5.pdf

I particularly like the first.

OOPs, that first link seems to have changed. That book is probably available elsewhere or is very cheap as second hand (I got a pristine copy for about $10).

 

Here's a calculator to do a 3-pole active filter with one opamp.
That's as many poles as can be practically done while still achieving a rapid rolloff at the corner frequency.
Cascaded RC filters have a low Q and soft rolloff characteristics.

 

What exactly is the point of connecting the capacitor that way

It reduces the damping factor (higher Q) to get a sharper rolloff.

Below is the simulation of a 2-pole Butterworth active filter and a 2-pole passive filter.
Note how much sharper the rolloff is for the active filter at the corner even though they eventually rolloff at the same frequency and slope.

Edit: I realized my passive filter design was not optimum so below is the improved version.
The difference between the passive and active rolloffs is now less.

 

It is more complicated than that. Read about Sallen-Key filters

https://en.wikipedia.org/wiki/Sallen–Key_topology
http://www.ti.com/lit/an/sloa024b/sloa024b.pdf

Wow thanks, that helped a lot!!! Ok, I now know that making the two Rs equal and two Cs equal allows the gain of the amplifier to control the resonance. The cutoff = 1/(2πRC) and Q = 1/(3-K) where K is the gain. Ok, that makes sense. The only problem is that the curve is nonlinear, but I can compensate for that by using a digipot controlled by a microcontroller.

 

Here's a calculator to do a 3-pole active filter with one opamp.
That's as many poles as can be practically done while still achieving a rapid rolloff at the corner frequency.
Cascaded RC filters have a low Q and soft rolloff characteristics.

Ok, well I can put two of these two pole filters together to get four poles, but will I be able to get the same results as far as variable Q isolated from R and C values? Would I have to make one of these stages unity gain?

 

Get a free copy of the FilterPro Webench software form Texas Instruments.
It helps you design active filters with any number of poles.

 

Analog Devices also has a filter "wizard."

https://www.analog.com/designtools/en/filterwizard/

TI and Analog designs will be slightly different. Analog's are based on VanValkenburg's studies (link above, personal communication with an Analog engineer) and may include Q enhancement, which is describe well in VanValkenburg. I have not seen that in the TI FilterPro, but I have not used that wizard as much as Analog's. NB: With Q enhancement, oscillation can become a problem.

 

According to this article (https://www.electronics-tutorials.ws/filter/filter_5.html),
"When cascading together filter circuits to form higher-order filters, the overall gain of the filter is equal to the product of each stage. For example, the gain of one stage may be 10 and the gain of the second stage may be 32 and the gain of a third stage may be 100. Then the overall gain will be 32,000."

So if I understand correctly, I can simply connect two of these filters and make the first unity gain and the other variable gain. That way, the variable gain of the second can control the Q and I can have 4 poles. Does this make sense/sound correct?

 

I can simply connect two of these filters and make the first unity gain and the other variable gain. That way, the variable gain of the second can control the Q and I can have 4 poles. Does this make sense/sound correct?

That would work, but I'm not sure that the filter response would be optimum.

 

That would work, but I'm not sure that the filter response would be optimum.

What do you mean?

 

What do you mean?

I mean flat response in the passband with an optimally fast rolloff at the corner.

 

What do you mean?

What it means is that you cannot cascade two filters and get the response you might expect. This is because the probable mismatch between the output impedance of the first filter and the input impedance of the second filter will change the overall response of the cascaded filter.

 

What it means is that you cannot cascade two filters and get the response you might expect. This is because the probable mismatch between the output impedance of the first filter and the input impedance of the second filter will change the overall response of the cascaded filter.

Well, what if I just add two RC filters on the input? Would that work? And if not, what should I do? I want to have a 4 pole filter where I can control the resonance. This filter topology is almost perfect except that it's only two poles. Is there really no way to add two more poles and have the resonance be controlled with the gain of the amplifier the way it is in this Sallen Key circuit? Also, since I'm using a single supply, I'm gonna have to DC bias the input and put a buffer (so that the DC biasing with resistors doesn't effect the filter) I just thought I should mention that.

 

I want to have a 4 pole filter where I can control the resonance.

You can put two of your 2-pole active filters in cascade and adjust the Q of each to your liking.

 

Well, what if I just add two RC filters on the input? Would that work? And if not, what should I do? I want to have a 4 pole filter where I can control the resonance. This filter topology is almost perfect except that it's only two poles. Is there really no way to add two more poles and have the resonance be controlled with the gain of the amplifier the way it is in this Sallen Key circuit? Also, since I'm using a single supply, I'm gonna have to DC bias the input and put a buffer (so that the DC biasing with resistors doesn't effect the filter) I just thought I should mention that.

You can do anything you like. What you get may not be what you expect. You can figure out what that is by analysis or simulation. You can play with values by trial and error or you can understand filter mechanics well enough to design what you want.