# Partial-Fraction Expansion?

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Isn't there some easy way to do a "partial-fraction expansion" of a
transfer function in "S" (Heaviside version of Laplace) and generate

...Jim Thompson
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| James E.Thompson                                 |    mens     |
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Re:Partial-Fraction Expansion?
Wrote in message:

I don't know that there's any easy way to do it by hand, no.
That's why for the "standard" filter types, someone went to the
trouble of computing "normalized" filter tables for ladder
networks.

With something like SciPy or Matlab, if you have a candidate
filter topology and the transfer function of your filter it
should be possible to write a program that can find the transfer
function of the network and solve for the component
values

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Re: Partial-Fraction Expansion?
On 01/27/2015 07:15 PM, bitrex wrote:

A partial-fraction expansion will give you a sum of quadratic terms,
each of which will look like a parallel LRC section.  That leads
directly to a realization: buffer the tank voltage and sum the results.
Getting the ladder realization is tougher, as others have noted.

Cheers

Phil Hobbs

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Re: Partial-Fraction Expansion?
On Tue, 27 Jan 2015 16:38:34 -0700, Jim Thompson wrote:

There's something about alt.binaries.schematics.electronic that either
Pan or Giganews doesn't like -- if I reply without snipping that from the
"Newsgroups" line, things quietly fail.

You don't want partial fraction expansion.  Partial fraction expansion
turns a big transfer function into a sum of a bunch of smaller transfer
functions.

You would like to be able to factor your transfer function so that you
could have the product of a bunch of second-order transfer functions,
each specifying a filter in cascade.  You'd like that, but it only works
with active filters: with passive filters the stages interact.

Worse, the handbook filters usually don't deal with finite-Q coils, and
if this is for your Q>100 filter, they'll be useless.

If this were 1950, the answer would be to make pi-wound coils on forms
inside of cans, with built-in capacitors, with amplifiers between
stages.  Fancy circuits would use two coils in each can, to get two
resonators per amplifier.  But even in 1950, no one would try for a
bandwidth/carrier ratio of 1:100.

--
www.wescottdesign.com

Re: Partial-Fraction Expansion?
On Wed, 28 Jan 2015 20:28:29 -0600, Tim Wescott

Even with active filters this would be a bear. The stages
may not interact, but the component values and tolerances
get really fiddly.  I liked the "equal-component" Sallen-Key
topology for 2nd-order stages, since it was easier to find
matching values rather than specific quirky ratios to set
the Q.  Then the Q is set by the stage gain, which you can
always trim manually.  Nevertheless, it was way too much
work!

Best regards,

Bob Masta

DAQARTA  v7.60
Data AcQuisition And Real-Time Analysis
www.daqarta.com
Scope, Spectrum, Spectrogram, Sound Level Meter
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Re: Partial-Fraction Expansion?
On Thu, 29 Jan 2015 13:37:59 +0000, Bob Masta wrote:

Jim is pretty consistently ignoring suggestions that a 100:1
carrier:bandwidth ratio may cause problems.  It's his money.  (Well, it's
his client's money).

--
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Re: Partial-Fraction Expansion?
wrote:

;-)

But I'm not ignoring.  I'm pulling (:-) the client toward an
injection-locked oscillator.

BTW, Gyrator type filters have non-interacting Q and Fo setting
components.

Sallen-Key should be renamed Sucky-Krap >:-}

...Jim Thompson
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| James E.Thompson                                 |    mens     |
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Re: Partial-Fraction Expansion?
On 01/29/2015 12:17 PM, Jim Thompson wrote:

If you're trying to do anything fancy, sure, but gentle filters such as
low-order Bessel and Gaussian work fine--the component value sensitivity
isn't serious.  I use gentle SKs all the time, because of one shining
virtue: they have exactly unity DC gain, which sometimes matters a lot.

Cheers

Phil Hobbs

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Dr Philip C D Hobbs
Principal Consultant
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Re: Partial-Fraction Expansion?
On Thu, 29 Jan 2015 12:23:29 -0500, Phil Hobbs

So does a gyrator-based low-pass.

...Jim Thompson
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Re: Partial-Fraction Expansion?
On 01/29/2015 01:13 PM, Jim Thompson wrote:

But it needs twice as many op amps, no?  Those cost money in my world,
whereas good resistors and decent capacitors are cheap, so I don't
SK isn't bad at all--see e.g.
http://www.maximintegrated.com/en/app-notes/index.mvp/id/738 .

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
We've slightly trimmed the long signature. Click to see the full one.
Re: Partial-Fraction Expansion?
On Thu, 29 Jan 2015 13:26:29 -0500, Phil Hobbs

Cheap is as cheap does ;-)

OpAmps are free in my world.

Nice tutorial though.  I'll have to commit that to a macro to spit out
quick filter requirements.

...Jim Thompson
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| James E.Thompson                                 |    mens     |
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Re: Partial-Fraction Expansion?
On 01/29/2015 01:43 PM, Jim Thompson wrote:

There's another one, which I think is somewhat better, at
http://www.ti.com/lit/ml/sprp524/sprp524.pdf .

They show that the 'key' is to use unity gain and equal resistors, which
gets the sensitivity down to about the level of a biquad.  The main
issue remaining is that the ratio of the capacitors becomes large if you
need a Q >~ 5. The TI guys recommend using a biquad for the high-Q
section (there's usually only one) and low sensitivity SKs for the rest.

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
We've slightly trimmed the long signature. Click to see the full one.
Re: Partial-Fraction Expansion?
wrote:

[snip]

[snip]

Must be Pan.  I use Giganews and see no such problems.

...Jim Thompson
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| James E.Thompson                                 |    mens     |