Operation of Bandpass active filter

[CataM]

The input capacitor (C1) is not part of the filter function and its value is chosen very large in order not to influence the desired characteristics.

What is the purpose of the bypass capacitor C1 then? just to block the DC component?
The C6 capacitor is not used either(in the transfer function and in the circuit analysis), what is it purpose?

 

[Audioguru]

What is the purpose of the bypass capacitor C1 then? just to block the DC component?

Yes, and it can also form another highpass filter if you want.

The C6 capacitor is not used either (in the transfer function and in the circuit analysis), what is it purpose?

There are a few schematics in this thread, maybe you are talking about C6 in post #5?
Since all the other circuits assume that you are powering the circuit with a positive and negative supply then its DC reference voltage is ground (0V). But you are powering the circuit with only a positive supply so it uses R5 and R7 to make a half-the-supply reference voltage and C6 filters any noise out of it.

 

[wherati]

It's interesting, after staring at the schematics for a bit, you realize that the structure is surprisingly simple. It just sums a weighted input with the the output of the adder, its first integral and its second integral (pretty obvious in hindsight). The response is:

H(s)=sum( g*vin + a*x + b*x/s + c*x/s^2), where x is the summer output and a,b,c and g are set by the component values and feedback.

This of course simplifies to a plain old biquad, and we can tap at different points in the loop to pick different zeros to get HP (first order zero), BP (second order zeros) or LP response (no zeros). I've look at 3 different structures and they all work the same way.

The insight I'm still lacking is a clear mental picture of advantages and disadvantages of the various structures, particularly when taking non-ideal opamp behavior into account.

For example, FvM suggested the structure here (https://en.wikipedia.org/wiki/State_variable_filter) as a better implementation. I have not been able to figure out how it's better, except that the tuning equations seem to be more decoupled and easier to handle.

 

[FvM]

I have not been able to figure out how it's better, except that the tuning equations seem to be more decoupled and easier to handle.

Isn't that reason enough? Particularly if you are designing a filter with tunable Q?

 

[LvW]

This of course simplifies to a plain old biquad, and we can tap at different points in the loop to pick different zeros to get HP (first order zero), BP (second order zeros) or LP response (no zeros). I've look at 3 different structures and they all work the same way.
The insight I'm still lacking is a clear mental picture of advantages and disadvantages of the various structures, particularly when taking non-ideal opamp behavior into account.

wherati, as you probably know, there are many different filter topologies - and all show the same behaviour, provided the active devices are assumed to be ideal.
And it is really an involved task to select a particular structure for a specific application because differences between the variuos alternatives can be observed only if realistic properties are considered for the active devices (opamp, OTA, CC).
And - of course, there are other (technical, economic) criteria for making a decision:
Tunability, indpendent control of pole parameters, frequency range, number of elements active/passive, sensitivity to tolerances active/passive, ...