"Interleaving" of LC filters

[Joel Kolstad]

I think that most people are aware of the standard transforms used to
convert lowpass filter prototypes into highpass filters, bandpass filters,
etc. If you look at the result, it's clear that the bandpass filter, for
instance, is an 'interleaved' lowpass and high pass filter design. That
struck me as more coincidental than anything profound, but I've recently
learned that this interleaving is a more general result and apparently can
be done with many different LC filters. That is, if I have two, say, 3
section filters, I can match up the series and shunt sections in each filter
and interleave the two (shunt sections placed in parallel, series sections
placed in series).

So now I'm curious... how does this actually work? Is the idea just that
each starting filter must be operating at frequencies far enough apart from
one another that at any given frequency the components from one filter are
near-opens for shunt components and near-shorts for series components, so
that the two designs don't interact much? Or is there a more general
mathematical basis for why the interleaving works, and you could
successfully interleave something like a notch filter in the middle of a
bandpass filter?

Thanks,
---Joel Kolstad

(Who'll shortly need to be getting new bookshelves after acquiring copies of
Zverev, Matthei/Young/Jones, etc...)

 

[Jens Tingleff]

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I think that most people are aware of the standard transforms used to
convert lowpass filter prototypes into highpass filters, bandpass filters,
etc. If you look at the result, it's clear that the bandpass filter, for
instance, is an 'interleaved' lowpass and high pass filter design.

Hmm. What about the transform of a lowpass filter which has a transmission
zero?

That
struck me as more coincidental than anything profound, but I've recently
learned that this interleaving is a more general result and apparently can
be done with many different LC filters. That is, if I have two, say, 3
section filters, I can match up the series and shunt sections in each
filter and interleave the two (shunt sections placed in parallel, series
sections placed in series).

At best only in some cases, I'll bet. Trivially, a lowpass filter (shunt C,
series L, shunt C, etc) will have lower order if you bundle all the caps
together in parallel...

Also - in general - the component values have to change, even in cases where
you can actually move sections around.

What can happen is that one characteristic of the moved section can still be
found in a re-arranged filter, but the rest of the characteristic is
changed. If you have a parallel resonant circuit in series which breaks the
connection from input to output at a certain frequency, it'll still break
the connection at that frequency if you move another section in front of
it. However, the response of the filter at other frequencies can very well
change if you move the section around without changing its component
values.

[...]

Or is there a more general
mathematical basis for why the interleaving works, and you could
successfully interleave something like a notch filter in the middle of a
bandpass filter?

There's loads of mathematical background. Try to read parts of "synthesis of
passive networks" by Ernst A Guillemin (good university libraries would
have a copy, alternatively, it's available second hand from, say,
www.abebooks.com) - recommended!

You can put a notch filter in a bandpass filter, but only the notch will
come though, the combined response is nothing like the product of the two
responses. If you put a buffer between two complete filters, then the
response will be the product, if you put two filters together without a
buffer (i.e. allow them to load each other), the combined response will
have some characteristics from both, but will not be a lot like the product
of the responses.

Hope this helps a little

Best Regards

Jens


- --
Key ID 0x09723C12, [email protected]
Analogue filtering / 5GHz RLAN / Mdk Linux / odds and ends
http://www.tingleff.org/jensting/ +44 1223 211 585
"YOU ARE WITNESSING A THREE-QUARTER VIEW OF TWO ADULTS SHARING A TENDER
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[Phil Hobbs]

I think that most people are aware of the standard transforms used to
convert lowpass filter prototypes into highpass filters, bandpass filters,
etc. If you look at the result, it's clear that the bandpass filter, for
instance, is an 'interleaved' lowpass and high pass filter design. That
struck me as more coincidental than anything profound, but I've recently
learned that this interleaving is a more general result and apparently can
be done with many different LC filters. That is, if I have two, say, 3
section filters, I can match up the series and shunt sections in each filter
and interleave the two (shunt sections placed in parallel, series sections
placed in series).

So now I'm curious... how does this actually work? Is the idea just that
each starting filter must be operating at frequencies far enough apart from
one another that at any given frequency the components from one filter are
near-opens for shunt components and near-shorts for series components, so
that the two designs don't interact much? Or is there a more general
mathematical basis for why the interleaving works, and you could
successfully interleave something like a notch filter in the middle of a
bandpass filter?

Thanks,
---Joel Kolstad

(Who'll shortly need to be getting new bookshelves after acquiring copies of
Zverev, Matthei/Young/Jones, etc...)

The lowpass to highpass transformation is a conformal map in complex
frequency. If you transform the transfer function from f into
g=fc**2/f, where fc is the 3 dB frequency, all the factors of f/fc
become factors of gc/g, which is the same as transforming Ls into Cs
and Cs into Ls.

The lowpass to bandpass transformation involves a somewhat more complex
mapping, something like g=f/f_0 - f_0/f, where f_0 is the centre
frequency. The centre of the passband is mapped to dc, and both dc and
infinity are mapped to infinity. Thus a capacitor, whose dc impedance
is infinite, changes into a parallel LC, and an inductor, whose dc
impedance is ideally zero, changes into a series LC. In each section,
you use *the same values as the original lowpass*, and chose the new
element to resonate at f_0.

The reactance of each section now changes twice as fast with frequency,
but you get one image of the passband on each side of f_0, so the full
bandwidth is the same as the lowpass prototype's.

Because the transformation is nonlinear, you can't usefully apply this
transformation to linear phase filters unless they are very narrow-band.

Cheers,

Phil Hobbs