From: "Peter O. Brackett"
Subject: Re: New simple LC filter design program
Date: Thu, 26 Dec 2002 02:12:13 -0500
Organization: MindSpring Enterprises
X-Server-Date: 26 Dec 2002 07:12:11 GMT
X-Newsreader: Microsoft Outlook Express 5.00.2919.6600
Thanks for another great program!
However, as usual we are not privy to your methods of calcualtion. I was
Does your program use the simple, venerable, but approximate, image
parameter design formulae?
If not then perhaps you are aware of, and have been delighted by, the
remarkably simple explicit
formulae for the exact element values of low-pass, all-pole, ladder filters
with Butterworth and
Chebychev characteristic functions. These remarkably simple formulae
involve only the trigonometric
functions powers and square roots, and may be executed easily by hand on a
As a "teaser" the extremely simple formulae, known as Bennett's formulae,
for the "normalized" element
values of the N'th order Butterworth [a.k.a maximally flat] minimum
inductor, low-pass characteristic
function operating between one Ohm source and load resistances [R1 = R2 =
1.0 Ohms] with exact
unity passband ripple bandwidth and exact passband attenuation ripple given
by Ap in dB are:
Ck = 2 eps^(1/N) sin(Pi (2k-1)/2N)): N - odd
Lk = 2 eps^(1/N) sin(Pi (2k-1)/2N)): N - even
Where: eps = sqrt(10^(Ap/10) - 1.0) is the passband ripple factor with Ap
No need to calculate poles and zeros, no need to understand complicated
Chebyshev polynomials, no need tolearn how to read tables and nomographs nor
the effort to carry them around! The results, unlike image parameter
design, are exact and
are not approximations.
Apparently Bennett "discovered" the remarkable Butterworth formulae long
ago, and it was many
years later, and with great difficulty, that Takahashi "proved" the formulae
and introduced several
new ones for the Chebychev case including some predistorted designs.
A table for the element values for the 5th and 6th order filters operating
Ohm terminations with 3dB ripple at unity bandwidth can be generated by hand
on a modern
scientific calculator in moments. Here's a table for Butterworth filters
which I generated by hand
in a couple of minutes.
N C1 L2 C3 L4 C5
5 0.6180 1.6180 2.0000 1.6180 0.6180
6 0.5176 1.4142 1.9319 1.9319 1.4142 0.5176
N L1 C2 L3 C4 L5 C6
Of course the conventional simple element by element reactance
transformations from low-pass to high-pass
and low-pass to band-pass along with frequency and impedance level scaling
may also be easily executed by
hand on a modern scientific calculator.
As with your program, ideal elements are assumed, and consequently the
effects of finite Q of the elements is
not accounted for by these formulae. However, there exists simple
algorithms, derived from sensitivity analysis
by Mayer Blostein Gabor Temes, which I have used for many years, which
estimates the effects of semi-uniform
element value losses and termination mis-matches on the passband response.
Although I use a calculator myself, I have long felt that such exact
formulae could form the basis of a very
elegant and simple "Reg Edward's style" program for all-pole insertion loss
filter design. Such a program
could also readily handle the denormalizations from unity bandwidth and
impedance levels to the practical
final desired targets as well as the conventional reactance transformations
for high-pass and band-pass cases.
If you are interested in more information on these remarkable formulae for
the Butterworth and Chebychev
characteristic functions, there is a nice description of them with an
outline proof along with 13 references to
the original literature in:
Louis Weinberg, "Network Analysis and Synthesis", McGraw-Hill, 1962, [see
Ch. 13, Sec. 13.5]
There is also a write up on how to use them effectively in:
Adel S. Sedra, and Peter O. Brackett, "Filter Theory and Design: Active and
Passive", Matrix, 1978.
Indialantic By-the-Sea, FL
"Reg Edwards" wrote in message
> PROGRAM "LPF_HPF.exe"
> Most filters consist of one or more cascaded, simple, 3-component, T or Pi
> sections and are used in a multitude of applications where pass-band
> frequencies of infinite attenuation in the stop-band, exact impedances
> matches, etc., are not of consequence.
> This easy to use program calculates component values of high-pass,
> T and Pi, basic filter sections. Internal filter loss is neglected.
> Operating inconveniences which may be present in more complicated programs
> can be avoided.
> Filter response vs frequency is in terms of dB insertion loss between Ro
> terminations for 1 to 5 cascaded sections. Frequency range is from audio
> Download in a few seconds program LPF_HPF.exe from website below and run
> Regards from Reg G4FGQ
> For Free Radio Design Software
> Go to http://www.g4fgq.com