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From: "Peter O. Brackett"
Subject: Re: Asymmetrical frequency response of lumped element bandpass filters
Date: Wed, 16 Oct 2002 10:57:38 -0400
Organization: MindSpring Enterprises
References: <3DAAF2B0.FA01AC9F@niobiumfive.co.uk> <firstname.lastname@example.org> <3DAD4EE6.E32D1AC0@niobiumfive.co.uk>
X-Server-Date: 16 Oct 2002 14:57:31 GMT
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Lumped element band pass filters designed using the standard low pass to
band pass transformation [a.k.a. reactance transformation] will always
have equal numbers of loss poles [a.k.a. transmission zeros, the zeros of
the polynomial P(s)] in the upper and lower stopbands. Hence sucn
transformed low pass filters will always have an asymmetrical frequency
response with higher rates of cutoff on the lower stop band side than on the
upper stop band side.
To eliminate this and to achieve arithmetic symmetry, or symmetry with
steeper upper side cutoffs, or for that matter any prescribed symmetry
you will need to perform/solve an approximation problem which does not use
the restrictions inherent in the classical low pass to band pass reactance
Instead to achieve arbitrary symmetry, one must place un-equal numbers of
loss poles in the upper and lower stop bands. Usually to achieve say
arithmetic symmetry in a band pass design it is necessary to place more loss
poles in the upper stop band and/or at infinite frequency than in the lower
stop band and/or at zero frequency. Of course one should "place" these loss
poles [zeros of P(s)] optimally to find a characteristic function K(s) of
the filter to satisfy the symmetry requirements using an iterative
approximation routine [Remez method] and then finally synthesize the filter
from the resulting loss poles P(s) and reflection zeros F(s).
Often when adjusting only the position and number of imaginary axis loss
poles a good fit may be found to a given arbitrary symmetry requirement,
but... to obtain even finer control of the symmetry it is possible to
introduce additional "parameters" [The so called parametric band pass
filters.] into the approximation and these additional parameters [either
real axis reflection zeros or real axis loss poles or some combination
thereof] can then be used by the [Remez] routines to fine tune the
approximation as closely as you wish to match arbitrary symmetry
requirments. The result of the approximation is then an optimally matched
characteristic function [K(s) = F(s)/P(s)] with the required symmetry. The
lumped element filter can then be synthesized from the open and short
circuit reactance functions resulting from those reflection zero F(s) and
natural mode E(s) polynomials.
All of this is explained in various papers and fully detailed along with
practical examples in the book:
A. S. Sedra and P. O. Brackett
"Filter Theory and Design: Active and Passive"
Matrix Publishers, Portland, OR 1978.
ISBN 0-916-46914-2, LCCN: 76-39745
You should find this book shelved in your local technical library at:
LC Shelf Call: TK7872.F5S42 -or- Dewey: 621.3815'33.
Most advanced computer programs for filter design can be used to design
band pass filters with arbitrary symmetry in this way. For example, the
program Filtor developed by myself and several others at the University
of Toronto and its' use illustrated by some designs in the above book is
one example of such a program.
Unfortunately it is not possible to obtain such band pass designs with the
simple methods of reactance transformations from low pass prototypes. To
achieve arbitrary symmetry in the band pass requires the use of computer
programs which perform the design [approximation and synthesis] of arbitrary
general parameter lumped element filters.
The techniques for design and synthesis these advanced band pass filter
designs were developed several decades ago [circa 1960-1970's] and are no
longer taught in engineering curricula or any modern texts having been
squeezed out of the curricula by more modern job related skills training.
Indialantic By-the-Sea, FL.
"The Technical Manager" wrote in message
> Max Foo wrote:
> > was this design using a narrow band approximation algoritm?
> It is designed using the usual lowpass to bandpass transform found in
> filter textbooks.
> > sounds like your describing a capacitive coupled bpf, that is each LC
> > section couples to the next LC section via a cap. An inductive coupled
> > will have the response flipped.
> It is a standard bandpass filter configuration. Not a capacitive or an
> inductive coupled filter.
> > On Mon, 14 Oct 2002 17:37:04 +0100, The Technical Manager
> > wrote:
> > >Why is it that lumped element LC bandpass filters have an asymmetrical
> > >frequency response where the slope between the passband and stopband on
> > >the high frequency side is less steep than the slope on the low
> > >frequency side ? Filters constructed from distributed elements normally
> > >have perfectly symmetrical frequency responses.
> > >
> > >Is it something to do with the admittance vs frequency response of
> > >distributed elements being periodic with frequency whereas the
> > >admittance vs frequency response of an LC resonator isn't periodic with
> > >frequency and exhibits an admittance of either zero or infinite ohms -
> > >depending on whether it is parallel or series LC circuit - only at
> > >DC and infinity Hz ?
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