How to find the component values of a Butterworth low pass filter

Using the term "unit circle" will get you in trouble with DSP types -- it has a specific meaning in sampled-time filtering that doesn't have much to do with Butterworth filters.

There are a number of different circuits that can realize a Butterworth low-pass filter, and low-pass filters aren't the only ones that can be called "Butterworth". Each circuit topology has specific requirements for component values, and in each case you have to choose to account for various non-ideal component effects or not.

Yes, but it goes by the individual circuit.

A Butterworth filter is "maximally flat" in the passband, meaning that it's amplitude response changes as little as possible, and is monotonically decreasing toward cutoff. A Chebychev filter has an amplitude response that rises and falls in the passband, but always between the same two points -- it is "equiripple". In return for tolerating the ripple in the passband the Chebychev gives you a steeper falloff from the passband to the stopband.

Entire books are written about designing and implementing filters, and they can be highly technical. Your best bet is to go buy a copy of the ARRL Handbook, which does a pretty good job of explaining this stuff, and doesn't assume that you have a BS in electrical engineering to start with.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

You sit down and build up the equation for the impedance of the circuit and its derivatives and equate them to that dictated by the "magic equation." This is considered a homework problem :-) for, say, 2nd and 3rd order circuits, but for anything much higher than that either you sit down and write a program to do it... or just use tables or software like most people have done for decades now.

There are many, many books that have these tables, and I'm sure they're on the Internet as well. "Analog and Digital Filter Design" by Steve Winder is a book I like that strikes a good balance between "cookbook" and "theoretical" (leaning towards the cookbook side, however). For software, for free you can get AADE's Filter Designer, or try Elsie. Fancier software tends to support more topologies -- NuHertz Solutions has their Filter Solutions, but it's not cheap (hundreds of bucks to thousands of dollars); it's probably about the cheapest package that will let you specify arbitrary poles and zeroes and it "does the rest" (completely synthesizes a circuit for you).

For Butterworth and Chebyshev I believe you can find closed form expressions for component values. For fancier filters (elliptical being the prime example), there are no closed form expressions for the component values in terms of common functions.

This is the game you're playing:

-- Higher order filters have faster (steeper) dropoff between the passband and stopband

-- Accepting ripple in the passband or stopband response lets you obtain more steepness for a given order filter.

-- The more ripples, the more you gain... although this is a relatively 'slow' function

-- Accepting finite (rather than infinite) loss in the stopband also buys you steepness at a given order

Chebyshev has ripple in the passband but still infinite loss.

BTW, for low order filters, in some cases the finite Q (loss) of the inductors involved tends to smooth out the ripples and make a Chebyshev look a lot like a Butterworth!

Yes, for Chebyshev. However, as soon as you go for the 'accepting finite loss' option, you need to add components to add zeroes to the transfer function (this occurs in Elliptic filters, so-called Inverse Chebyshev, etc.).

The equations always change. Butterworth comes from making the passband as flat as possible, Chebyshev comes from constraining the ripple to a specified amount in the passband, Bessel comes from making group delay as falt as possible, etc.

Nope, the concept behind them is the same. I do think it's useful to keep in mind that you (yes, you!) can sit down and desgin a filter just by throwing a few zeroes and poles around the S plane, writing down the transfer function, and equating it to a bunch of components... All the filters with Big Names attached to them are that way because they were designed very systematically and, by now, most filters people actually want designed can be easily accomdated by these well-known designs. Still, there's a lot of room for "tweaking" standard filters by tossing in, e.g., another pole to built various hybrid-type filters.

---Joel Kolstad

assembly

recently.

filter are

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equation?

Butterworth

Your best bet is to find a good filter design text-book. I like "Electronic Filter Design" by Arthur B. Williams and Fred J. Taylor. I've got the second edition - ISBN 0-07-070434-1. Since then there has been a third edition

- ISBN 0-07-070430-9 - which was also out of print the last time I looked, though Amazon have three (from $130).

Don Lancaster's "Active Filter Cookbook" ISBN: 075062986X has its fans, and it is still in print, and it is a lot cheaper at $33. If you are only interested in Butterworth filters, it will probably provide what you need.

The structural difference between Butterworth and Chebyshev flters is purely a question of component values (which happen to be reflected in the coefficients in the magic equation, as is spelled out in text-books of filter design).

The concept behind the difference lies in the frequency response of the filters and their real-time response.

A Chebyshev low-pass filter gives better rejection of high frequencies than a Butterworth filter, but if you put a step change in voltage into a Chebyshev filter, the output will show worse ringing than you'd see on the output of a Butterworth filter. A Bessel (or phase-linear) low pass filter gives even poorer high frequency rejection than the Butterworth filter, but drive it with a step function and you don't see any ringing on the output.

Incidentally, your magic equation corresponds to a 2-pole Butterworth filter - one built with just two reactive components (inductor and capacitor or two capacitors). You can make higher order filters -

4-pole and 6-pole filters are popular - which you find yourself using you need to roll off the frequency response more sharply. This is also covered in any decent text-book.

Bill Sloman, Nijmegen

Theoretically, once you've derived your butterworth polynomial, you physically realize your filter by partial fraction expansion or continued fraction expansion. But in practice, no one does that, they simply work from the normalized tables that are widely published in many texts, and scale to impedance and frequency.

Zverev's text probably was most comprehensive for published tables. But for cheaper and likely sufficient, you could simply get Winder's book.

If you want to get down to the derivations (and synthesis via expansion), then Aram Budek, Kendall Su, and Earnst A. Guillemin (sp?) books can get you there.

The Budak book is probably the best for the money. The Guillemin book is particularly gnarly. I can't find a used one...

too fast

I was lucky... the 6B students (EE honors), in 1959, didn't have to take passive circuit theory from Guillemin, we had H.B. Lee instead... marvelous instructor... and we got real values, not just 1H, 1 ohm and

1F ;-)

...Jim Thompson

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|  James E.Thompson, P.E.                           |    mens     |
|  Analog Innovations, Inc.                         |     et      |
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Or a gratuitous plug for where I got my copy

A recent book which has Butterworth and Chebyshev tables is R W Rhea "HF filter design and computer simulation" McGraw Hill

Best Regards

Jens

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Thanks for you answer! =)

"Jens T> >

esis&x=39

filter

I buy thru abe too. $22.50 is great for the Guillemin text.

Look on

- there are five copies, starting at 1/10th the price.

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