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