passive network with transcendental poles or zeros

Is it possible to build a passive network (R,L,C) for which the voltage tranfer function Vo(s)/Vi(s) has transcendental poles or zeros?

nukey

Reply to
hugocoolens
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Ask your professor?

I'm fairly certain the answer is "no." To build an arbitrary network, your only choices are to stick R's, L's, and C's in series or parallel with one another (as complex as you like), and via induction you can attempt a back-of-the-envelope proof that, starting with any of R, L, and C you have an algebraic function for an impedance (or, more generally, an immitance), and sticking any of these in series or parallel with one another still levels you with a plain old rational function. Once you're convinced that all immitances can be expressed as rational functions, it's just a bit of, say, citing Thevenin or Norton to demonstrate that a transfer function has to be rational as well.

These sorts of problems come from the study of network theory that was developed back in the early- to mid-20th century; a lot of this sort of material is no longer taught today as most people don't end up having to make much use of it in practice. (...I'd almost bet you a nickel that the IEEE receives papers from folks having "newly discovered," say, Foster's reactance theorm on a fairly regular basis...)

---Joel

Reply to
Joel Koltner

A lossless 1 uH/1uF tank resonates at 1/sqrt(2*pi) MHz. That's a couple of transcendental poles right there. I suspect that the OP was asking the question you answered, however.

Cheers

Phil Hobbs

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Dr Philip C D Hobbs
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Reply to
Phil Hobbs

Ah, good point; I'll remember that for future reference --> rational polynomial with irrational coefficients is not an algebraic function.

Reply to
Joel Koltner

What are we calling 'transcendental' here? I was taking 'transcendental zero' as being H(s) = (1 + e^(s*a))/(some other stuff) -- you appear to be taking a plain old complex number as "transcendental".

If you want to know "can a passive network have _complex_ zeros", the answer is "sure, and it can be just an RC network, to boot".

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Tim Wescott
Wescott Design Services
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Reply to
Tim Wescott

The general interpretation of the OP's question seems to be "Can a transfer function contain a transcendental function in s." I interpreted the question a little differently - if you define a "zero" to be "some number, real or complex, that when substituted for s sets the numerator of the transfer function equal to zero" and for a pole the same but substituting "denominator" for numerator. So the question would be could that number be transcendental.

I think this interpretation is probably not what the OP intended, since obviously a transcendental number can't be the root of a polynomial in s, by definition. Of course for mathematical completeness following along the same lines as Mr. Koltner's argument you would have to prove that a transfer function made up only of RLCs always has numerators and denominators that are polynomials in s, and that you couldn't,for example, somehow construct from that a function similar to the one you mentioned, like 1 + e^s. That would have a zero or pole at s = i*pi. My intuition tells me that at least for that example it's impossible to get a transfer function in the s domain in such a form, as outside the locations of the poles and zeros the transfer function would be oscillating around, and if the input function to the Laplace transform is of an order where the Laplace transform converges the -st term in the transform prevents that from happening.

Reply to
Bitrex

If you consider distributed elements then you can get transfer functions with exp(s) terms.

Pere

Reply to
o pere o

They look only irrational to me, not transcendental.

Sylvia.

Reply to
Sylvia Else

Pi is a transcendental irrational, as opposed to an algebraic irrational like sqrt(2). But as I said, I think the OP was actually mis-stating a homework problem--I expect that the prof was trying to get the class to think about the implications of lumped-element transfer functions always being ratios of polynomials, as opposed to transcendental functions like exponentials and sines and cosines. That's the question Joel answered.

Cheers

Phil Hobbs

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Dr Philip C D Hobbs
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Reply to
Phil Hobbs

Yep. In the early '60's I fabbed some distributed R/C networks. But, IIRC, they roll off as sqrt(s). ...Jim Thompson

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| James E.Thompson, CTO                            |    mens     |
| Analog Innovations, Inc.                         |     et      |
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Reply to
Jim Thompson

The original poster said R's, L's, and C's, however... which I took to be ideal elements and finite in extent.

Interestingly point though, in that an infinite number of R/L/C's -- as one can model, e.g. a lossy transmission line -- has a limit that can become transcendental. (Appropriate XKCD comic:

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I think most people are exposed to or perform a lab with building artificial transmissoin lines out of lumped elements. When I did it back in school I was kinda surprised at how many extra elements you needed to move our the corner frequency (of the artificial line) by, say, another decade.

---Joel

Reply to
Joel Koltner

Speaking of artificial transmission lines... is anyone here working with so-called metamaterials, which are often modeled like a traditional transmission line but by swapping the positions of the inductors and capacitors (...so that the artificial version has a finite *low frequency* corner whereas traditional artificial transmission lines have finite *high frequency* corners)?

Reply to
Joel Koltner

Oops, sorry - I just saw sqrt(2), and missed the pi.

Sylvia.

Reply to
Sylvia Else

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I'm having the same problem. I google transcendental pole and zero just to make sure it isn't something I learned then forgot. Complex and real works for me, but transcendental is for functions, right?

Reply to
miso

No worries--I got the formula wrong anyway! It's 1/(2*pi*sqrt(LC)), which is still transcendental.

Cheers

Phil Hobbs

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Dr Philip C D Hobbs
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Reply to
Phil Hobbs

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If the numerator and denominator polynomials of the driving point immitance function have rational coefficients, then the roots won't be transcendental. If the coefficients are "already" transcendental, then the roots will be too.

Am I missing something here?

Reply to
Simon S Aysdie

That's right. Because there's a factor of pi in all the formulas for the positions of the poles and zeros, if the component values are rational numbers, all the pole and zero positions have transcendental values. It was not the most imaginative homework question I ever heard of.

As I've said a couple of times, I think that the homework question should have been "With lumped elements only, is it possible to make a transfer function that is a transcendental function of frequency?" (As Joel pointed out in the very first reply, all you can make is some ratio of polynomials, which are algebraic functions, not transcendental ones.)

Cheers

Phil Hobbs

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Dr Philip C D Hobbs
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Reply to
Phil Hobbs

Is it interesting that the realization of a filter such as a Bessel, is an approximation (truncated series) of

H(s) =3D e^(-s) ?

Of course, the polynomials resulting from the truncation are the ordinary one's we're talking aboout.

Guassian also uses a truncated series too, IIRC.

Reply to
Simon S Aysdie

How could you get to a Bessel filter from truncated series of exp(-s) ?

BTW, Bessel filter results from the requirement of maximum flat phase at zero frequency.

Vladimir Vassilevsky DSP and Mixed Signal Design Consultant

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Reply to
Vladimir Vassilevsky

The Storch method, although I don't have the paper. It is mentioned in many filter synthesis texts. Su gives a concise description of Storch in Su's book Analog Filters.

Synthesis of Constant Time-Delay Ladder Networks Using Bessel Polynomials, L. Storch Proceedings of IRE, Vol. 7, 1954, pp. 536=96541

Reply to
Simon S Aysdie

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