Determining the cutoff frequency of lowpass filters? Some op-amp questions.

[sfr]

Okay a few questions about filters:

This is a standard active lowpass op-amp-based filter, right? 


If I understand correctly, the cutoff frequency is 1 / (2pi*R₂*C), and the gain in the passband is R₂/R₁?  The stopband drops off at -6dB per octave?

Okay, I've seen something like this in a couple of schematics, however:


I'm assuming the addition of C₂ before the opamp circuit is simply to add another (passive) lowpass filter before the opamp, the filter consisting of R₁ and C₂?  So two lowpass filters in sequence like this makes basically a second-order filter, with a steeper drop off, if the cutoff frequencies are the same?  Or am I mistaken?  If the cutoff frequencys aren't the same, then what, you get a stopband that changes "steepness" during the slope? 

And what is up with R3, that's sort of in the feedback loop and sort of in series with the input?  How does that effect the gain?  The filter?  If C₁ wasn't there and this was just a gain stage, how would I figure out the gain?

Any help would be appreciated, or simply if someone could point me in the correct direction to where I can learn what I need.  I've gotten very interested in active filters lately, and am thinking of picking up "The Active Filter Cookbook" by Lancaster - is this a usefull reference or are there other books I should be looking at?  (Like I'm just going to buy *one* book   )

[GibsonGM]

Awesome questions, SFR!  I'm waiting for some anwers on these things myself 
Maybe for now, take a look thru the op amp basics stuff on here?
http://sound.westhost.com/site-map.htm
Some good info on OA design!

[grapefruit]

They're not exactly the same as the schematic you have, but if you do a web search on VCVS filter you should come up with something.

Stew.

[mountainking]

Picking up a copy of the The Active Filter Cookbook will definately help to answer the questions you have. You should be able to get a copy off ebay for cheap, and while your at it, pickup a copy of The Active Filter "Handbook" by Frank Tedeschi. Until then, search around, theres plenty sites that'll give you info on active filter design.

[Izzy]

I am too waiting for some excellent reply for this question!
Lets wait !

[brett]

Hi
I'm no expert, but it is clear that R1 and C2 form a single-pole low pass filter (6dB/octave above fc at 1/(2.pi.R.C.))
R3 is now in series with the other feedback resistor R2, so the fc changes.

What IS R3 doing ??

What a good question.

[Izzy]

Hmm no one to answer?


I  hope R.G.Keen, Mark Hammer, Aron or some one will answer this very well.

[Sir H C]

I always viewed R3 as a way to keep the cap to ground from killing the opamp stability.  Caps directly on the negative input of an op-amp always kill stability when not in the feedback loop. 

[GibsonGM]

I saw something like this on the ESP site I linked to above...I believe it's a higher-order filter...the resistor in parallel was mentioned.  Could it be that C1 and R3 are doing some secondary filtering?  And also R3 doing something regarding stability? It seems R2, R3 and C1 are doing double duty here, lol.  Going thru the ESP stuff on OA design might help decode this one! I'm only ok up to the simpler 1st order stuff as shown in the 1st diagram...

[R.G.]

The correct advice has already been posted - go get a copy of "The Active Filter Cookbook" by Lancaster. A used copy can be had for $2.18 here:
http://www.abebooks.com/servlet/SearchResults?sts=t&an=lancaster&y=0&tn=active+filter+cookbook&x=0

Look up "multiple feedback filters".

This is an example of the multiple feedback lowpass  filter.

For active filters, setting the gain, the Q (resonance), the frequency, and the amount of ripple in the passband as well as the speed of rolloff at the critical frequency may all interact.

For the mutliple feedback filter shown, the frequency is
F0 = SQRT(1/(R2*R3*C1*C2)/(2*pi)
The expressions for the gain, the Q, the flatness, etc are too complicated to type.

In general, the design of active filters from scratch is not a novice activity. Lancaster's book cases up the mathematics in a usable form and keeps you out of the weeds by telling what is and is not practical for each form of filter.

By the way, one reason the state variable filter is so popular is that the Q, Gain, and flatness are controllable by separate components independently, not by diddling the ratios of capacitors and resistors in complicated ways.