Filters can be lossy or, in theory, lossless. An example of a lossy
filter is a first order lowpass, consisting of a series R and shunt C.
But let's talk about the passive lossless variety, made solely of
inductors and capacitors, since I think that's what you're asking about.
In practice, inductors in particular can have appreciable loss, and this
complicates analysis a great deal. But for many applications, for
example HF filters that aren't too sharp, loss can be negligible for
practical purposes. So I'll further simplify things by talking about
only theoretically lossless LC filters.

A passive lossless filter can't achieve any frequency selection by means
of loss, so it depends entirely on mismatch. Among other things, that
means that a passive filter works properly only when both the source and
load impedances are the ones it was designed for. A lossless lowpass
filter has zero loss only at DC. At DC, or very low frequencies, then,
the input is matched to the output. If the filter is designed for 50
ohms in and out, for example, you'd see 50 ohms resistive at the filter
input when the output is terminated in 50 ohms. It can also be designed
for other transformation ratios -- imagine the same filter with a
broadband 10:1 impedance transformer at one end. There are other ways to
effect the transformation, but the end result is the same.

But as you go up in frequency, the attenuation of the filter increases.
In the case of an LC filter, that means -- it has to mean -- that a
mismatch is occurring. The attenuation typically rises slowly and not
too much until you approach the cutoff frequency, but there are an
infinite number of possible filter shapes, and some can vary pretty
wildly in the pass band (the frequency range from DC to cutoff).
Butterworth, Chebyshev, and a number of other canonical types have a
substantial amount of attenuation, and therefore mismatch, at
frequencies quite a bit below cutoff.

An interesting passive LC filter type is a "quarter wave" lowpass
filter. It's so called because it mimics a quarter wave transmission
line over a moderate range of frequencies. This is a pi section filter
(although like any other pi, it can also be realized as a tee)
consisting of a series inductor and shunt capacitors. Each has a
reactance at the operating frequency equal to the source and load
resistance, which for the simple form I'm describing, are equal. This
filter is unusual(*) in that it *is* perfectly matched at the operating
frequency, which is just below the cutoff frequency. The cutoff isn't
particularly sharp, but sections can be cascaded for better high
frequency attenuation without changing the impedance match at the
operating frequency. It's a really handy tool for homebrew transmitters,
where additional harmonic attenuation is needed, since sections can be
added without necessitating output circuit redesign.

(*) It's unusual in my experience with modern filter design, but I
suspect this might be a common characteristic in "image parameter"
designed filters -- a technique I never learned.

Roy Lewallen, W7EL

Paul Burridge wrote:

Hi all,

On page 57 of RF circuit Design, Chris Bowick sets out a filter design
example. I've posted this to a.b.s.e under the same subject header. He
claims that the filter in question - a low pass Butterwoth - matches
50 ohms source to 500 ohms load. However, having checked this out with
the aid of a Smith Chart, it appears there is some capacitive
reactance present that would require the addition of a shunt inductor
to neutralize. However, this would of course totally screw up the
filter's characteristics. Upon closer examination, it appears
impossible that this type of arrangement could ever be designed
without introducing some reactance into the signal path. Or am I nuts?
I'd always thought of these kind of filters as being purely resistive
overall at Fo but is that really the case? It don't look like it...

Design criteria:

Centre frequency: 35Mhz
Response -60dB at 105Mhz
zero ripple(!)
Rs 50 ohms
Rl 500 ohms

Correction:

Roy Lewallen wrote:
. . .
A passive lossless filter can't achieve any frequency selection by means
of loss, so it depends entirely on mismatch. Among other things, that
means that a passive filter works properly only when both the source and
load impedances are the ones it was designed for. A lossless lowpass
filter has zero loss only at DC. . .

The last sentence should read:

A lossless lowpass filter has zero attenuation only at DC.

The attenuation is often called "mismatch loss", but in the remainder of
what I wrote, I use the term loss only to mean dissipative loss -- which
"mismatch loss" isn't.

Roy Lewallen, W7EL

Roy:
[snip]
The last sentence should read:

A lossless lowpass filter has zero attenuation only at DC.
:
:
Roy Lewallen, W7EL

Ummmm... no that statement is only true for one type of approximation
polynomial.

A lossless low pass filter has zero attenuation at its' reflection
coefficient zeros.

If it is a maximally flat low pass. a.k.a. Butterworth. then all of the
reflection zeros
are located at DC, but for any other type, e.g. Chebychev, Cauer/Darlington,
General Parameter,
etc, etc... this is not true.

Such a filter will have zero loss at the designed reflection zeros which are
distributed at various
appropriate frequencies across the passband according to the dictates of the
approximation
polynomials.

Aside: Reflection zeros are also known as Return Loss [Echo Loss] poles.
These are the
pass band frequencies of zero loss for lossless LC filters designed
according to modern
insertion loss methods. No one really knows where the reflection zeros of
an image
parameter LC filter are, one has to find them by analysis after the design.
Whereas
with insertion loss design the frequencies of zero loss [the reflection
zeros] are specified
by the approximation polynomials, specifically the reflection zero
polynomial usually
designated by F(s). In fact modern insertion loss design begins with a
specification
of attenuation ripple between zero loss and the maximum loss in the pass
band. The
frequencies of zero loss then become the zeros of the reflection zero
polynomial F(s).
The attenuation in the stop band results in the specification of the loss
pole polynomial
P(s) whose zeros are the so called loss poles or attenuation poles. The
natural mode
polynomial of the filter E(s) whose zeros are known as the natural modes or
sometimes
just "the filter poles" is formed from the loss poles and reflection zeros
using Feldtkeller's
Equation.

E(s)E(-s) = P(s)P(-s) +k^2F(s)F(-s)

In the approximation process the stopband attenuation is set first by
"placing" the loss poles
in the stopband, i.e. determining the polynomial P(s). Then from the
desired passband
attenuation and type of approximation desired; maximally flat, equiripple,
etc... the
reflection zeros F(s) are determined and finally from Feldtkeller's Equation
and the ripple
factor k, the natural modes or E(s) is determined.

Then the LC filter is synthesized from either or both of the short circuit
or open circuit
reactance functions which are formed from even and odd parts of E and F, for
example.

X = (Eev - Fev)/(Eod + Fod), etc...

You can review all of this in the very practical and professionally oriented
textbook:

Adel S. Sedra and Peter O. Brackett, "Filter Theory and Design: Active and
Passive",
Matrix Publishers, Champaign, IL 1978.

Another good practical and professionally oriented textbook is:

Louis Weinberg, "Network Analysis and Synthesis", McGraw-Hill, New York,
1962.

If you can get a copy of:

R. Saal and E. Ulbrich, "On the design of filters by synthesis", IRE Trans.
Vol. CT-5,
No. 4, pp.284-327, Dec. 1958.

Bind it firmly and keep it in your library forever... you will have the
whole story in a nutshell.

Saal and Ulbrich is "the bible" on LC filter design.

--
Peter
Freelance Professional Consultant
Signal Processing and Analog Electronics
Indialantic By-the-Sea, FL

You're correct, and I apologize. In fact, the example I gave of a
"quarter wave" filter contradicts the statement about the attenuation. I
was thinking of a Butterworth when I wrote it, but as you point out and
as my own example shows, there are many other types for which the
statement is wrong.

I apologize for the error. Thanks for the correction.

Roy Lewallen, W7EL

Peter O. Brackett wrote:

Roy:
[snip]

The last sentence should read:

A lossless lowpass filter has zero attenuation only at DC.

:
:

Roy Lewallen, W7EL


Ummmm... no that statement is only true for one type of approximation
polynomial.

. . .

You're correct, and I apologize. In fact, the example I gave of a
"quarter wave" filter contradicts the statement about the attenuation. I
was thinking of a Butterworth when I wrote it, but as you point out and
as my own example shows, there are many other types for which the
statement is wrong.

I apologize for the error. Thanks for the correction.

Roy Lewallen, W7EL

Peter O. Brackett wrote:

Roy:
[snip]

The last sentence should read:

A lossless lowpass filter has zero attenuation only at DC.

:
:

Roy Lewallen, W7EL


Ummmm... no that statement is only true for one type of approximation
polynomial.

. . .

Roy:
[snip]
The last sentence should read:

A lossless lowpass filter has zero attenuation only at DC.
:
:
Roy Lewallen, W7EL

Ummmm... no that statement is only true for one type of approximation
polynomial.

A lossless low pass filter has zero attenuation at its' reflection
coefficient zeros.

If it is a maximally flat low pass. a.k.a. Butterworth. then all of the
reflection zeros
are located at DC, but for any other type, e.g. Chebychev, Cauer/Darlington,
General Parameter,
etc, etc... this is not true.

Such a filter will have zero loss at the designed reflection zeros which are
distributed at various
appropriate frequencies across the passband according to the dictates of the
approximation
polynomials.

Aside: Reflection zeros are also known as Return Loss [Echo Loss] poles.
These are the
pass band frequencies of zero loss for lossless LC filters designed
according to modern
insertion loss methods. No one really knows where the reflection zeros of
an image
parameter LC filter are, one has to find them by analysis after the design.
Whereas
with insertion loss design the frequencies of zero loss [the reflection
zeros] are specified
by the approximation polynomials, specifically the reflection zero
polynomial usually
designated by F(s). In fact modern insertion loss design begins with a
specification
of attenuation ripple between zero loss and the maximum loss in the pass
band. The
frequencies of zero loss then become the zeros of the reflection zero
polynomial F(s).
The attenuation in the stop band results in the specification of the loss
pole polynomial
P(s) whose zeros are the so called loss poles or attenuation poles. The
natural mode
polynomial of the filter E(s) whose zeros are known as the natural modes or
sometimes
just "the filter poles" is formed from the loss poles and reflection zeros
using Feldtkeller's
Equation.

E(s)E(-s) = P(s)P(-s) +k^2F(s)F(-s)

In the approximation process the stopband attenuation is set first by
"placing" the loss poles
in the stopband, i.e. determining the polynomial P(s). Then from the
desired passband
attenuation and type of approximation desired; maximally flat, equiripple,
etc... the
reflection zeros F(s) are determined and finally from Feldtkeller's Equation
and the ripple
factor k, the natural modes or E(s) is determined.

Then the LC filter is synthesized from either or both of the short circuit
or open circuit
reactance functions which are formed from even and odd parts of E and F, for
example.

X = (Eev - Fev)/(Eod + Fod), etc...

You can review all of this in the very practical and professionally oriented
textbook:

Adel S. Sedra and Peter O. Brackett, "Filter Theory and Design: Active and
Passive",
Matrix Publishers, Champaign, IL 1978.

Another good practical and professionally oriented textbook is:

Louis Weinberg, "Network Analysis and Synthesis", McGraw-Hill, New York,
1962.

If you can get a copy of:

R. Saal and E. Ulbrich, "On the design of filters by synthesis", IRE Trans.
Vol. CT-5,
No. 4, pp.284-327, Dec. 1958.

Bind it firmly and keep it in your library forever... you will have the
whole story in a nutshell.

Saal and Ulbrich is "the bible" on LC filter design.

--
Peter
Freelance Professional Consultant
Signal Processing and Analog Electronics
Indialantic By-the-Sea, FL

Correction:

Roy Lewallen wrote:
. . .
A passive lossless filter can't achieve any frequency selection by means
of loss, so it depends entirely on mismatch. Among other things, that
means that a passive filter works properly only when both the source and
load impedances are the ones it was designed for. A lossless lowpass
filter has zero loss only at DC. . .

The last sentence should read:

A lossless lowpass filter has zero attenuation only at DC.

The attenuation is often called "mismatch loss", but in the remainder of
what I wrote, I use the term loss only to mean dissipative loss -- which
"mismatch loss" isn't.

Roy Lewallen, W7EL