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How do Sallen key bandpass filters really work?

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Good spot! I was wondering how the gain could be dependent on R1 if both networks made use of Req. At least this explains how they're separable. I should have gone for linear superposition in the first place without source and circuit transformations, then I wouldn't have had my solution as half of one and half of another. I've simulated the new bandpass shown below and it does indeed work out to 0.01v to give an overall gain of as it should ~2. In reference to your point with the geometric mean of the cut off frequencies, this too was something new for me, but it seems to work, so I'm glad you found it useful!
**broken link removed**
So in order to wrap things up somewhat, the original question that launched the thread was how does this filter essentially work. Unfortunately typical analysis of reactances at extreme high and low frequencies is not going to give you enough of a picture of how it works (apart from at those limits). Linear superposition shows the circuit looking as a band pass in the forward direction and a band stop in the reverse. As you'll see from previous postings we can combine the effects of how each of these filters produce the final output in the end by an equation of G = Bandpass Gain/Band Stop Gain. The diagram above for example is how the forward bandpass circuitry would look as part of a 3khz multiple feedback filter. Gain is then attained since the denominator in this fraction is slightly less than unity at the centre or 'resonant' frequency and so the fraction grows and you get your amplification and Q. Without this amplifying element, the maximim Q any passive filter can have is probably about 1/2. I would have to say the reason the circuit is not intuitive to understand in the first instance is because I'm not typically used to looking at circuits being stimulated from both ends (Since Vout is a signal source in its own right when it's being fed back) and passive circuits performing different functions in different directions. Hopefully, if I've made any mistakes here or someone feels the need to add more they should (I learn much more from corrections!). The only thing that I would say that would be perfect, would be to be able to visually inspect the notch 'gain' frequency formula conditions from the bandstop passive circuitry, but the arrangement looks far too complicated at least to me to eyeball it. So in the end deriving the transfer function, like you could have done for the entire filter to start with anyway, is probably the only way you'll get to it, if you didn't know it or its formula already.

Megamox

PS. It'd be nice If I could change the title of the thread to mention Multiple Feedback filters, but I'm sure all of this logic can be applied to pretty much all of those similar types of Sallen Key circuitry now too. I've also seen that sometimes this type of multi-feedback filter goes by the name of deliyannis-friend (at least in my findings through limited internet research).
 
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Hi Megamox - nice summary.
One final comment: The deliyannis-friend topology uses - in addition - positive resistive feedback in order to have one more degree of freedom to boost the quality factor.
However, I am afraid that I must disappoint you: In most cases it is not easy - in many cases: not possible - to evaluate the principle filter characteristic only by visual inspection of the circuit.
For example, the Sallen-Key structure has a positive feedback path as well as negative one (to set the gain). Thus, to derive the resulting Hr you have to consider both feedback pathes, which means: Calculation of the difference Hr,net= Hr+ - Hr-.

W.
 
Hi,

The gain is only independent of R1 if we operate at the center frequency so i dont think it's that important except to calculate the center gain. For the other frequencies it is:

Ampl=(w*C1*Vin*R2*R3)/sqrt((-w^2*C1*C2*R1*R2*R3+R2+R1)^2+(w*C2*R1*R2+w*C1*R1*R2)^2)

The main idea with these kinds of circuits is to be able to find the transfer function from which you can derive everything else. Otherwise you will find that there are many, many different ways to look at the circuit and you might be here forever finding new ways. For example, disconnect C2 from the output, then connect that end of C2 to a source of amplitude -Vin*R3/(2*R1) and that simplifies the circuit a little so you can 'see' how the components work together. When we get a phase shift of -pi/2 at the central node that's when we are at the center frequency. By doing this little topology trick we eliminate the second feedback path and so then we 'almost' have a purely feed forward circuit which makes for simpler 'thinking' about.

We can not see the phase shift of the non inverting terminal (i'll call v3) to ground, but using an imperfect op amp we can, and then we can let the op amp get more and more perfect, and as we do that we see the phase shift of v3 approach zero degrees. With the central node at -90 degrees that means C1 is providing 90 degrees of phase shift to give us zero degrees at v3, and then the inverting amp gives us 180 degrees so we get the output looking like an inverter.

There are many other ways to look at this too though, we could probably be here all day :)
 
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I too was staring at the circuit for so long, I must have seen dozens of equally acceptable topology within it and must have drawn the circuit ten different ways!. It was pretty exhausting trying to narrow it down to one which felt reassuringly right. I like the topological trick, that's some really good insight there MrAl! It'll be tricks like that which will come in handy, especially since as Winterstone mentions with increasingly more complex filters, there are multiple negative AND positive feedback paths to consider.
 
Megamox,

in case you like playing with the topology that was discussed in this thread - here are some further aspects:

* You can cancel the grounded resistor R2 - however, in this case you cannot select Q and the midband gain independently.
* This restriction does not apply in case of additional positive feedback (Deliyannis modification)
* There are two additional component allocations, which also give a bandpass response. Here are the replacements (of course, other values):
R1>>>C
R2>>>R
R3>>>C
C1>>>R
C2>>>R

or

R1>>>C
R2>>>C
R3>>>C
C1>>>R
C2>>>R
 
Yes, I've seen the actual topologies you mention on various websites. One of the initial and still somewhat confusing aspects of active filters seems to be that there is not one specific design that handles 'band pass'. There seems to be a variety of circuits which all have various advantages and weaknesses, even sometimes swapping components within circuits leads to new circuits which perform the same function - but slightly differently. It's probably one of the reasons for my confusion with the multiple feedback filters with sallen key filters which led to the misnaming of the thread. I see I need to be more careful now.

Through learning about this topic, I've got to say, for the beginner there does not seem to be that much entry level theory available on how to understand the inner workings of these designs. I found plenty of resources that state the formula's out right which is fine, but sometimes I prefer a deeper understanding like we have arrived at through this thread. If there are any 'classic' books or resources that anyone feels would provide that type of coverage please feel free to mention them.

One book I have rented from the library which I've found helpful is called the 'Analogue Filter Cookbook', but it's very old circa 1970's and It actually has no rear cover on it! But still if it was still in print, perhaps it might be one I'd think was worth buying. I've also been recommended the art of electronics by horrowitz and hill, which I've already got a copy of. There was supposed to be a really good one from the 1980s if I remember rightly, but I've forgot the title now. In the end though maybe it's not the descriptions that will help, maybe it's the ease you have with identifying transfer function characteristics that comes with experience and practise.

Megamox
 
There seems to be a variety of circuits which all have various advantages and weaknesses, even sometimes swapping components within circuits leads to new circuits which perform the same function - but slightly differently.

I found plenty of resources that state the formula's out right which is fine, but sometimes I prefer a deeper understanding like we have arrived at through this thread. If there are any 'classic' books or resources that anyone feels would provide that type of coverage please feel free to mention them.

One book I have rented from the library which I've found helpful is called the 'Analogue Filter Cookbook', but it's very old circa 1970's and It actually has no rear cover on it! But still if it was still in print, perhaps it might be one I'd think was worth buying. I've also been recommended the art of electronics by horrowitz and hill, which I've already got a copy of.

Hi Megamox, some comments from my side regarding the above three topics:

1) Yes, it is not an easy task to select one of the various available filter structure for a particular application. You have to consider many aspects - like performance, accuracy, adjustment of parameters, cost,
power consumption, technology, frequency ranges, etc. More than that, it is important if real zero`s are required or not (e.g. for elliptical responses).

2.) In principle, there are two basic approaches for filter design: (a) Cascade design (cascade of several 2nd-order stages) and (b) Active simulation of passive reference structures.
Most of the design alternatives following (a) were found by searching a way to create a conjugate-complex pole pair via feedback. In contrast, filter structures following (b) are based on classical passive RLC structures.
In these designs, the inductor is replaced by active circuits - applying one of three methods:
(2.1) Active L-simulation,
(2.2) FDNR-approach using the unique Bruton transformation,
(2.3) state-space realization of internal V-I relations (leading to integrator based topologies)

3.) For a good understanding of filter design I recommend neither Lancaster`s "Cookbook" nor Horowitz/Hill. Both books do not support the understanding at all from the system point of view.
Instead I can recommend (from my own experience) the following authors in alphabetical order:
Antoniou, Bruton, Budak, Deliyannis, Ghausi-Laker, Lam, Martin-Sedra, Rhea, Taylor.

Of course, this list is not complete.
 
Thanks for the information Winterstone, that gives me a nice framework to think about the subject and the books you've recommended do look excellent!

Megamox
 
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