Tweaking the tuning of second-order state-variable bandpass filters

[stevebyan]

I'm trying to clean up the harmonics from our AC power line to feed a pure 60 Hz sine wave to the external reference input on my HP 6920B meter calibrator. I've designed a fourth-order Butterworth bandpass filter implemented using two second-order state-variable bandpass filter sections. So far so good.

The piece that I'm missing is information on how to tune each section of the filter once it's been constructed. There are two adjustments per second-order state-variable section. Lancaster's Active Filter Cookbook is silent on this subject, as is my copy of Williams and Taylor's Electronic Filter Design Handbook. I'll be using 5% polystyrene capacitors I have on hand. I plan to make the frequency-controlling resistances tweakable, to compensate for variations in the capacitance. But how do I know what direction and how much to tweak?

I wrote some code in the Racket programming language to generate root-locus plots of the second-order state variable filter as the values of resistors R1 (the resistor in the first integrator) and R2 (the resistor in the second integrator) are varied around their ideal value.


The red lines are the R1 root-locus and the blue lines are the R2 root-locus. Let’s zoom in on the positive pole.


You can see that R2 mostly affects the frequency of the filter, while R1 mostly affects the damping or Q of the filter.


Zooming in on the R1 root-locus, we can see that it also has a small effect on frequency of the filter, but in practice it should be negligible.

So, what have I learned? To tune the second-order bandpass state variable filter, one should first tune R2 to the desired center frequency of the filter. Then, one should tune R1 to set the 3 dB bandwidth for the desired Q.

Is my analysis correct? Any other tips on tuning state variable filters?

[Etesla]

It might not be a bad idea to throw the ideal version of your circuit into a spice simulator like LTspice and run an AC analysis on it. From there, you can easily adjust resistor values and see what real impact they have on the filters frequency response. This would let you know with certainty and pretty much no math what changing each of your resistor values up and down actually impacts.