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).
**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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