Partial-Fraction Expansion?

Someone please refresh my memory...

Isn't there some easy way to do a "partial-fraction expansion" of a transfer function in "S" (Heaviside version of Laplace) and generate an L-R-C ladder network? ...Jim Thompson

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| James E.Thompson                                 |    mens     | 
| Analog Innovations                               |     et      | 
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Jim Thompson
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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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bitrex

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.

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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

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Bob Masta

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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Tim Wescott

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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Phil Hobbs

;-)

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     | 
| Analog Innovations                               |     et      | 
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Jim Thompson
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Must be Pan. I use Giganews and see no such problems. ...Jim Thompson

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| James E.Thompson                                 |    mens     | 
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Jim Thompson

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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Phil Hobbs

So does a gyrator-based low-pass. ...Jim Thompson

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| James E.Thompson                                 |    mens     | 
| Analog Innovations                               |     et      | 
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Jim Thompson

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 usually care about sensitivity factors less than about 3. A unity-gain SK isn't bad at all--see e.g.

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Cheers

Phil Hobbs

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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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Jim Thompson

There's another one, which I think is somewhat better, at

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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

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Phil Hobbs

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