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 Post subject: Tuning state-variable active filters
PostPosted: Dec Sat 21, 2019 3:26 pm 
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I'm building a 60 Hz active filter to provide a clean external 60 Hz reference signal to my HP 6920B meter calibrator: viewtopic.php?p=3123574#p3123574

The filter is a fourth-order Butterworth bandpass with a Q of 10. I'm implementing it with two second-order state-variable bandpass filter sections. Don Lancaster's Active Filter Cookbook covers all this quite nicely.

The piece that I'm missing is information on how to tune the filter once it's been constructed. Lancaster's book 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 do have a GR 1650-A bridge, so I could measure the capacitance to 1% and just set the resistance to the calculated value, but I imagine there's a better way to do this. Does anyone have any experience with this? Any pointers to reference material?

Since each section is a pole-pair, I don't think I can just adjust each of the two RC time constants individually to peak up the filter response. They must interact.

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 Post subject: Re: Tuning state-variable active filters
PostPosted: Dec Sat 21, 2019 8:02 pm 
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Use multi turn pots for the resistance , use 5% resistors for half the calculated resistance and a multi turn pot in series for the calculated value which should give you a wide latitude for fine tuning .

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 Post subject: Re: Tuning state-variable active filters
PostPosted: Dec Sat 21, 2019 8:06 pm 
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Do you have an electronic CAD package which will permit you to simulate the operation of your circuit?

-EB

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 Post subject: Re: Tuning state-variable active filters
PostPosted: Dec Sat 21, 2019 10:13 pm 
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Mark, I do plan to use multi-turn trimmers. My question is about the process of actually adjusting them. My intuition says the two adjustments in each second-order state variable filter interact, so I don't think just tweaking for max at the desired frequency will work.

EB, I suppose I could generate a spice netlist and simulate it. Why?

I guess I could use Spice to check my intuition about the adjustments interacting. It seems pretty clear to me though that each integrator affects the position of both of the poles in the pole-pair. In the s-plane, the poles are the roots of the quadratic equation formed by multiplying the transfer function of each of the integrators. That implies that tweaking the time constant of one integrator affects the position of both poles.

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 Post subject: Re: Tuning state-variable active filters
PostPosted: Dec Sun 22, 2019 12:05 am 
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stevebyan wrote:
Mark, I do plan to use multi-turn trimmers. My question is about the process of actually adjusting them. My intuition says the two adjustments in each second-order state variable filter interact, so I don't think just tweaking for max at the desired frequency will work.

EB, I suppose I could generate a spice netlist and simulate it. Why?

I guess I could use Spice to check my intuition about the adjustments interacting. It seems pretty clear to me though that each integrator affects the position of both of the poles in the pole-pair. In the s-plane, the poles are the roots of the quadratic equation formed by multiplying the transfer function of each of the integrators. That implies that tweaking the time constant of one integrator affects the position of both poles.
If I am understanding your design concept accurately, you plan to cascade two second order state-variable circuits to get a total of four poles. Is that correct? If so there should be little or no interaction between adjusting each pole.

In a traditional second-order state-variable circuit where each pole is formed by using an opamp (as integrator), the opamps isolate each pole from affecting the other one.

The place where it gets complicated is when the filter is completely passive. Without the opamps buffering everything, each stage of a passive filter loads the stage preceding it. This can get touchy when striving for a perfect amplitude and phase response.

The “active filter cookbook” is still my favorite reference on this subject.

-EB

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 Post subject: Re: Tuning state-variable active filters
PostPosted: Dec Sun 22, 2019 12:50 am 
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electricboyo wrote:
If I am understanding your design concept accurately, you plan to cascade two second order state-variable circuits to get a total of four poles. Is that correct? If so there should be little or no interaction between adjusting each pole.

In a traditional second-order state-variable circuit where each pole is formed by using an opamp (as integrator), the opamps isolate each pole from affecting the other one.

Yep, that's the architecture I'm using. Yeah, that makes sense, each integrator forms one root of the transfer function, so they should be independent. So I can just tune each one for max response at the correct frequency for the pole constellation.

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 Post subject: Re: Tuning state-variable active filters
PostPosted: Jan Wed 08, 2020 8:30 am 
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Greetings to Steve and the Forum:

If I understand what you are trying to do correctly, you are making a narrow bandpass filter for 60 Hz to supply a reference to your meter calibrator. In that case, phase change across the filter does not seem to me to be relevant. I would simply peak the filter for best response at 60 Hz, then look at the output with a distortion analyzer and tune for minimum distortion. This will ensure best gain at 60 Hz while any noise outside of the desired passband will show up as distortion. You will probably have to monitor the filter output with an audio voltmeter as well as a distortion analyzer simultaneously to be sure that you maintain the correct level for which the distortion analyzer was set. This can be done by varying the amplitude of the input to the filter. You will be busier than a one-armed paper-hanger because of control interactions, but you should be able to get it all together with some patience.

Good Luck,

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 Post subject: Re: Tuning state-variable active filters
PostPosted: Apr Fri 03, 2020 11:01 pm 
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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 results are pretty interesting.

Image

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.

Image

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

Image

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.

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