### Author Topic: 7th Order Butterworth Filter Help  (Read 5776 times)

0 Members and 1 Guest are viewing this topic.

#### mawyatt

• Super Contributor
• Posts: 3248
• Country:
##### Re: 7th Order Butterworth Filter Help
« Reply #25 on: July 09, 2021, 02:40:05 pm »
An interesting active filter we did ~45 years ago and got published by EDN or Electronics Design (can't remember), then later Walter Jung of Analog Devices called to discuss this and use it in an ap note or one of his books (also can't remember, getting old sucks   This filter is a unique implementation of the 3rd Order Butterworth polynomial, which is S^3 + 2*S^2+2*S+1. This can be simply factored into S+1 times S^2 + S +1 which offers a implementation as a 1st Order followed by a 2nd Order, or the reverse. If the 2nd order is a Sallen-Key Low Pass type except with the junction of the feedback capacitor and input resistor is followed by a unity gain buffer to the second resistor. With this implementation the 3rd Order Butterworth can be created with all equal value resistors and all equal value capacitors, and the amplifiers are unity gain, all of which is a big advantage in implementation since as mentioned precision capacitors are difficult to find and expensive, but if all the same value then a little better

Another interesting property is if the resistors and capacitors are exchanged the result is a 3rd Order Butterworth High Pass filter at the same corner frequency of 1/RC (radians/s).

Once this filter implementation was discovered long ago we utilized this active filter in numerious applications over the years and it performed admirably

If interested we can provide more details, but think this was posted here at sometime.

Best,
« Last Edit: July 09, 2021, 02:43:56 pm by mawyatt »
Curiosity killed the cat, also depleted my wallet!
~Wyatt Labs by Mike~

#### T3sl4co1l

• Super Contributor
• Posts: 21667
• Country:
• Expert, Analog Electronics, PCB Layout, EMC
##### Re: 7th Order Butterworth Filter Help
« Reply #26 on: July 09, 2021, 03:33:08 pm »
Huh, wouldn't think that would be all that useful back in the day, opamps being much more expensive than resistors or capacitors?  But yeah, other than that, buffering to isolate RC sections is nice, just as helpful as isolating LC sections but you may get to save a few BOM items.

I'm fond of putting the real pole out in front, which works for Sallen-Key, MFB or other types -- see this for example:
http://sim.okawa-denshi.jp/en/MultipleFB3Lowkeisan.htm
As usual, the added impedance in front [going from 2nd to 3rd order], affects all the other values, so you're again using a diversity of resistors, and you need a low impedance source (typically from a proceeding op-amp).  Seems I can usually get reasonable results with common values (decades say) that are already in use, so it's not too big a deal on BOM size.

Can also make an even higher order SK (or other type too, I suppose), by extending the RC from the feedback node to +in, with additional RC stages.  Downside is, because there's an even higher order network inside the loop (more phase shift), it's even less stable (more sensitive to component values).  For this reason, it's generally preferred to stick to 2nd and 3rd order sections, and cascade them.

(Digital filters are the same way, you can do a higher order IIR filter but it's that much more sensitive to errors, requiring extra bits (in both the coefficients and accumulators) to preserve numerical stability.  Better to cascade biquad sections for the same overall response.)

Worthy of note, MFB is generally better than SK for a number of reasons:
- Both are limited by amp output impedance and GBW; MFB's inverting op-amp performs slightly better.
- Finite GBW (or less than unity gain for the follower, or less than infinite DC gain for op-amps in general) places a zero in the stopband.  Which is to say, at high frequencies, the op-amp is no longer "able" to control the output, and feed-forward terms from the direct input-to-output path take over.
- The 3rd order forms, both have a "hard" real pole (the input RC to GND).  This ensures a decreasing stopband, even above GBW.
- However, MFB has another: the node between R2, R3 and R4 is hard bypassed to ground by C2.  This gives better asymptotic response.
- MFB is less sensitive to component variation than SK.  That is, the response is nearly 1:1 proportional to any given component value.  Whereas ,SK has higher coefficients depending on filter type, meaning values need to be more precise.

OTOH, nice thing about SK is it can be done with plain old emitter followers, or even FET or tube followers if you don't mind the slightly low gain.  Here's one I did for an all-tube receiver:
https://www.seventransistorlabs.com/Images/Rx_AF_Bandpass.png
https://www.seventransistorlabs.com/Images/Rx_AF_Bandpass_Response.png
(6V6 being a pretty close SPICE model to the 5702 I actually used; or if you prefer, 2N3819 biased to around 3mS would do pretty much the same thing.)
Note the pairs of RC feedback components; it's a bandpass.  You can combine HP and LP filters in one circuit, each still being 2nd (or even 3rd?) order.  At least if they're not too close together that their poles start overlapping (in which case you'll have something more of a tuned-resonator architecture, and a different topology may be preferable?).  Beware, it may not be quite as easy as just superimpose both circuits and go; I recommend setting up in a simulator and adjusting values for best AC response.  Good opportunity to select common values as well.

Tim
Seven Transistor Labs, LLC
Electronic design, from concept to prototype.
Bringing a project to life?  Send me a message!

#### mawyatt

• Super Contributor
• Posts: 3248
• Country:
##### Re: 7th Order Butterworth Filter Help
« Reply #27 on: July 09, 2021, 04:22:20 pm »
Quad package Op Amps were cheap even back then, so amp cost wasn't an issue. 1% caps were the overall cost driver since 1% resistors were also cheap. Agree that using the 1st order in front is best because this attenuates the higher frequencies passively before it "sees" the op amp. One of the weak issues with the Sallen-Key is the input signal can bypass the op-amp right to the output thru the feedback cap and totally depends on the op-amp output impedance to attenuate the signal. This is the limiting factor in stop band attenuation with SK configurations and why the methods of reducing the Op-Amp output impedance were mentioned earlier. We utilized both methods successfully and got text book filter responses even with general purpose Op-Amps.

BTW some of the best passive filter designs I have seen were in the original Sony CD players before oversampling was employed, and the original Stanford Research function generators.

Best,
Curiosity killed the cat, also depleted my wallet!
~Wyatt Labs by Mike~

#### TimFox

• Super Contributor
• Posts: 7942
• Country:
• Retired, now restoring antique test equipment
##### Re: 7th Order Butterworth Filter Help
« Reply #28 on: July 09, 2021, 04:25:02 pm »
I also recommend the passive network before the active-filter, for reasons you state.
For hobbyist purposes, measuring the capacitors with a DE-5000 or similar LCR meter can easily sort capacitors to better than 1%.  I do this often when matching capacitors.

#### mbless

• Regular Contributor
• Posts: 227
• Country:
##### Re: 7th Order Butterworth Filter Help
« Reply #29 on: July 09, 2021, 06:58:17 pm »
I ordered parts to test the passive filter with lower impedance. I just realized the caps are only 5% tolerance, so I'll measure their capacitance before soldering them.

I'm going to go ahead and design a board with an active filter while waiting for the passive parts. I found AD and TI both have active filter calculators, which have been helpful to ensure the selected opamp meets bandwidth requirements. It looks like they also tune the resistor and capacitors to the specific opamp, so perhaps they're using fairly accurate models... throwing caution into the wind.

1/ I am not convinced that the problem is defined, as the measurement of the interference may be picking up noise. The OP should use a better probing technique to verify.
3/ Doubt if 7 th order is needed. Filter spec should be reexamined.

7th order Butterworth is needed based on the values I gave earlier: 3dB at 250kHz and >34dB at 500kHz.
The device is a black box, but I figured out how to feed in a voltage to test its functionality. Using a AAA alkaline battery, which from my understanding are essentially noise free, I checked the output voltage. The DC voltage was spot on, but it had an AC component which is the ~200mVpp, ~500kHz noise I'm trying to filter out. And to put the question of my probing to rest, I measured the battery voltage in-situ and got an AC component of 0.7mVrms and 6mVpp. So that's higher noise than I'd expect from a battery (likely scope noise floor and not perfect probing), but no, my probing is not causing the 200mVpp noise.

#### jonpaul

• Super Contributor
• Posts: 3366
• Country:
##### Re: 7th Order Butterworth Filter Help
« Reply #30 on: July 10, 2021, 06:07:33 am »
Hello we never used active filters above audio band.

We used Sallen-Key three pole extensively.

Also elliptic /Cauer passive.

For a 500 kHz noise filter   passive is the simplest.

If you have electronics driving and receiving the filter signal, you are free to choose whatever Zo are convenient.

Finally passive filters can have unequal Zin and Zout.

Enjoy,

Jon

Jean-Paul  the Internet Dinosaur

#### Kleinstein

• Super Contributor
• Posts: 14192
• Country:
##### Re: 7th Order Butterworth Filter Help
« Reply #31 on: July 10, 2021, 07:08:54 am »
With passive fitlers the tolerance of the capacitors is usually less critical than with active fitlers. The more critical parts are the inductors: they can couple magnetically to other inductors and with a ferromagnetic core the inductance in temperature dependent and can change from mechanical stress. Core materials also are not perfectly linear, which can be a problem at higher amplitudes.

Slight deviations in the capacitors would change the filter characteristics a little, but it would still be a good low pass filter. The relatively low Q Butterworth filter characteristics is not very sensitive. So 5% capacitiors should be nothing to worry about. The larger deviation is likely from the inductors, including the parasitic capacitance that causes the self resonance. One could at least include those parasitics in the simulation and if needed adjust the capacitors.

There is no absolute need to use a Butterworth filter characteristics.  A Chebychev type filter would give a sharper drop and may get away with lower order. Especially if this is about suppressing a known 500 kHz interference, a notch componenet (like in a Cauer filter) can be a good solution too. As a passive filter the Butterwirth type may still be good, as it has relativey low sensitivity to the component accuracy and higher order is not that much effort.

#### mawyatt

• Super Contributor
• Posts: 3248
• Country:
##### Re: 7th Order Butterworth Filter Help
« Reply #32 on: July 10, 2021, 01:21:52 pm »
Hello we never used active filters above audio band.

We used Sallen-Key three pole extensively.

Also elliptic /Cauer passive.

For a 500 kHz noise filter   passive is the simplest.

If you have electronics driving and receiving the filter signal, you are free to choose whatever Zo are convenient.

Finally passive filters can have unequal Zin and Zout.

Enjoy,

Jon

We were able to use active filters into the 100s of KHz region back then, employing some of the techniques mentioned. Later when higher ft transistors became available we moved into the MHz region and more recently around ~2010 DARPA supported development of a ~100GHz Op-Amp which produced GHz region active filters. This Op-Amp was based around Cherry-Hooper circuit, a circuit we utilized often back then.

Interesting you mention the unequal Zin and Out for passive filters, I remember way back in grad school Microwave Filters course we had a take home exam which required proof of Bartlett's Bi-Section Theorem which is used to modify the passive filter network components for unequal source and load impedances.

Best,
Curiosity killed the cat, also depleted my wallet!
~Wyatt Labs by Mike~

Smf