unity gain Sallen-Key low pass filter designs

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emrr
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unity gain Sallen-Key low pass filter designs

Post by emrr »

I was recently off micro-managing some unity gain Sallen-Key low pass filter designs using calculators at this site, shooting for Butterworth (0.707 QF) response. Why Butterworth? I don't honestly know, sounds like the right approach:
http://sim.okawa-denshi.jp/en/OPseikiLowkeisan.htm

Design from the second calc gives two different resistor values for either damping ratio or quality factor selection.  

As Wayne said to me privately:
If this is a unity gain SK filter why aren't the LPF resistors of equal value?
For a unity-gain SK LPF R=R. The FB cap is 2/d, the shunt cap, d/2.
Huh. What does it mean? I promised to bring the observations here for dissection by smarter people.

So, backtracking and using the first calc, entering all values using one R value from this calc:
http://www.calculatoredge.com/electroni ... 20pass.htm

Gives a different cutoff and QF.

I then ran the numbers a couple of ways; identical R value, QF (0.707), and damping (1). I got the following results using C values of 0.1 and 0.047.

R1/2 (2)281R
8261hz cutoff
QF 0.729
Damping ratio 0.685
Osc freq 6014
Overshoot 1.05

R1/2 374R/226R
7985hz cutoff
QF 0.7067 (closest to Butterworth)
Damping ratio 0.7074
Osc freq 5643
Overshoot 1.04

R1/2 732R/115R
8001hz cutoff
QF 0.49 (closest to critically damped)
Damping ratio 1
Osc freq none
Overshoot none 

I would guess for audio I am overreacting to the differences here, but am curious about their practical meaning, if any. I can only guess the uncredited source Z of the calculators plays a large part in the results. I also (shame, shame) haven't built up all three and had a close look and listen yet.

I started exploring my vast ignorance with books, google, and wikipedia, and found the following:

http://en.m.wikipedia.org/wiki/Sallen–K ... #section_5
A second-order Butterworth filter, which has maximally flat passband frequency response, has a Q of 1/?2 (0.707) Because there are two parameters and four unknowns, the design procedure typically fixes one resistor as a ratio of the other resistor and one capacitor as a ratio of the other capacitor.


http://en.m.wikipedia.org/wiki/Q_factor#section_4
Quality factors of common systems
? A unity gain Sallen–Key filter topology with equivalent capacitors and equivalent resistors is critically damped (i.e., Q=1/2).
? A second order Butterworth filter (i.e., continuous-time filter with the flattest passband frequency response) has an underdamped Q=1/?2.

Q factor and damping
? A system with low quality factor (Q < ½) is said to be overdamped. Such a system doesn't oscillate at all, but when displaced from its equilibrium steady-state output it returns to it by exponential decay, approaching the steady state value asymptotically. A second-order low-pass filter with a very low quality factor has a nearly first-order step response; the system's output responds to a step input by slowly rising toward an asymptote.
? A system with high quality factor (Q > ½) is said to be underdamped. Underdamped systems combine oscillation at a specific frequency with a decay of the amplitude of the signal. Underdamped systems with a low quality factor (a little above Q = ½) may oscillate only once or a few times before dying out. As the quality factor increases, the relative amount of damping decreases. A high-quality bell rings with a single pure tone for a very long time after being struck. A purely oscillatory system, such as a bell that rings forever, has an infinite quality factor. More generally, the output of a second-order low-pass filter with a very high quality factor responds to a step input by quickly rising above, oscillating around, and eventually converging to a steady-state value.
? A system with an intermediate quality factor (Q = ½) is said to be critically damped. Like an overdamped system, the output does not oscillate, and does not overshoot its steady-state output (i.e., it approaches a steady-state asymptote). Like an underdamped response, the output of such a system responds quickly to a unit step input. Critical damping results in the fastest response (approach to the final value) possible without overshoot.
In negative feedback systems, the dominant closed-loop response is often well-modeled by a second-order system. The phase margin of the open-loop system sets the quality factor Q of the closed-loop system; as the phase margin decreases, the approximate second-order closed-loop system is made more oscillatory (i.e., has a higher quality factor).
http://en.m.wikipedia.org/wiki/Butterwo ... #section_6
The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband. It is also referred to as a maximally flat magnitude filter.
I imagine someone can easily tell me why I am spending too much time on these differences, or that the calculator is wrong in it's assumptions, etc etc. Any insight appreciated.
Best,

Doug Williams
Electromagnetic Radiation Recorders
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JR.
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Re: unity gain Sallen-Key low pass filter designs

Post by JR. »

It is much too early in the morning for this...

It has been a while since I messed with lots of filters but it was an excuse to dust off my old Burr-Brown opamp text, that I cut my teeth on learning filter design.

They call the Sallen and Key a VCVS topology or voltage controlled voltage source (sorry TMI).

Now what exactly is the question?

The Q (it's called alpha or 1/alpha in my old book) is a function of the cap ratios and resistor ratios. You can probably scale both independently up and down to realize the same tuning frequency product and quality factor ratio(s).

for unity gain alpha (1/Q) equation simplifies to = SQRT(R2C2/R1C1)+SQRT(R1C2/R2C1) So you can make the Rs smaller (while in the same ratio) and Cs larger in same ratio, to realize the same cut off frequency and same Q.

This makes it easier to dial in values...

JR

PS: Back in the '70s when I was doing a lot of filter design associated with BBD delay lines I didn't like the Sallen and Key topology to filter out the very HF clock pulses from BBD outputs. The S & K topology depends on the opamp output impedance being low at very high frequency for the input RC to eat the clock edges. I preferred topologies where the input RC was a real pole to ground. An old preference that has stayed with me, even though opamps have gotten better.
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emrr
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Re: unity gain Sallen-Key low pass filter designs

Post by emrr »

Initially the question concerns the fact that this one particular calculator spits out two resistor values rather than two identical as is typically defined. Why is this is the only place I find results with two R values?

That it gives multiple ways to calculate and see analysis, allowed me to blindly explore the differences in approach. Ultimately, I am curious about the practical meaning of those differences; do they matter in practical application, or is this best ignored in favor of the R=R approach? Is the R=R approach inherently a practical simplification? Overshoot is the biggest difference I see in the analysis, and it is so small as to seem practically ignorable. I would guess the shape around cutoff differs slightly, but the slope becomes the same eventually. I imagine source Z may have to be well defined for any calculation to give data so tightly applicable, thus rendering this calculators results as ever so slightly misleading. I am pretty sure I am deep in minutia, I suppose looking for confirmation of that thought, or explanation of when and where a choice of approach might actually matter.
Best,

Doug Williams
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JR.
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Re: unity gain Sallen-Key low pass filter designs

Post by JR. »

emrr wrote:Initially the question concerns the fact that this one particular calculator spits out two resistor values rather than two identical as is typically defined. Why is this is the only place I find results with two R values.
It's some free website, who knows?

If I was writing the application I would look at common values of capacitors since they are generally available in less values than resistors, then tweak the resistors than can be sourced in even 1% increments to fine tune the filter.

Back in the '70s I actually wrote my own computer program to help me design complex multi-stage cascaded filters. It was crude using a tab command on a dot matrix printer to plot out filter results. But it was a huge luxury to be able to stick in real capacitor values and tweak away. I was designing multi-pole Chebyschev alignments even with HF pre/de-emphsis integrated into the Chebyschev's real pole on de-emphasis side. Hairy stuff.
That it gives multiple ways to calculate and see analysis, allowed me to blindly explore the differences in approach. Ultimately, I am curious about the practical meaning of those differences; do they matter in practical application, or is this best ignored in favor of the R=R approach?
In practice we design filters to use parts we already have in our system.It is always cheaper to bring in a new resistor than capacitor. Scaling the filter values too high impedance has noise and stray capacitance issues, too low impedance raises drive impedance issues. It is also easier to source small cap values in higher quality dielectric than larger values.
Is the R=R approach inherently a practical simplification? Overshoot is the biggest difference I see in the analysis, and it is so small as to seem practically ignorable. I would guess the shape around cutoff differs slightly, but the slope becomes the same eventually. I imagine source Z may have to be well defined for any calculation to give data so tightly applicable,
If I understand what you are asking, source impedance is added to the first resistor value, while a source impedance that changes with frequency will have a somewhat different impact.
thus rendering this calculators results as ever so slightly misleading. I am pretty sure I am deep in minutia, I suppose looking for confirmation of that thought, or explanation of when and where a choice of approach might actually matter.
While not a math geek the raw equations will tell you what you need to know... of course non-ideal op amp behavior matters... more back in the '70s than now.

Don't over think this... it's a filter.

JR
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emrr
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Re: unity gain Sallen-Key low pass filter designs

Post by emrr »

JR. wrote: Don't over think this... it's a filter.
Yup, that's really what I'm after. I don't know enough here to know what I know or don't know. I think the rest of the varied results I gathered are theoretically microscopic, might apply in very tight tolerance work at Los Alamos, and nowhere else.
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Doug Williams
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mediatechnology
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Re: unity gain Sallen-Key low pass filter designs

Post by mediatechnology »

If this is a unity gain SK filter why aren't the LPF resistors of equal value?
For a unity-gain SK LPF R=R. The FB cap is 2/d, the shunt cap, d/2.
Huh. What does it mean? I promised to bring the observations here for dissection by smarter people.
My question related to their being two types of Sallen-Key filter calculations.

Image
From http://sim.okawa-denshi.jp/en/OPseikiLowkeisan.htm


The top one is the "unity gain" topology.
The bottom example is sometimes called the "equal component value" topology.

In the top unity gain example R1=R2 and the ratio of C1/C2 set the damping.

In the "equal component" circuit, R1=R2 and C1=C2 with the ratio of R4/R3 setting damping.

FWIW I just checked a HPF filter I did recently using the cool site Doug found. http://sim.okawa-denshi.jp/en/OPseikiLowkeisan.htm

The sim matched reality so I suppose for that set of values, 20 Hz, 470 nF, 24K, 12K, Q=0.707 it works.


http://sim.okawa-denshi.jp/en/OPseikiLowkeisan.htm
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Re: unity gain Sallen-Key low pass filter designs

Post by emrr »

Yes, that's what I see almost everywhere. The unity gain calculator at that one site, which has the most extensive selection of calculators I've found, delivers results with R1 and R2 having different values.
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Doug Williams
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Re: unity gain Sallen-Key low pass filter designs

Post by mediatechnology »

I forced them equal.
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Re: unity gain Sallen-Key low pass filter designs

Post by mediatechnology »

I did 1.6Hz, 1F and it returned 0.1R for both R1 and R2 for a Q=0.707.

Edit: Q was 0.5.
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Re: unity gain Sallen-Key low pass filter designs

Post by emrr »

Another 1st-2nd grade tangent. I had an 'ahah!' moment when I made a wiring mistake in a derived crossover. First we see how it should look, and recombine.

Image

Then we see what happens when the derived LPF is wired opposite polarity by mistake. The phase change at the crossover point affects response, with a nice big bump where it should be flat.

Image
Best,

Doug Williams
Electromagnetic Radiation Recorders
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