# Talk:Foster's reactance theorem

Odd that for the formulator of this theorem, a function such as x/(1-x^2) is considered to be increasing (for the range x>0). Won't this lead to difficulties in the application of the theorem to situations where values are being compared? 84.227.254.143 (talk) 10:02, 30 March 2014 (UTC)

I don't think you can use Foster's theorem in that way (comparing two discrete measurements) to determine if the network is a Foster network or not.  Clearly, it will fail if there is a pole between the two measurements.  One must have knowledge of the sign of the slope of the function across the whole frequency range. SpinningSpark 13:24, 30 March 2014 (UTC)
There might be another way to formulate it that won't lead to initial confusion. There's a hint of this in the reference to going clockwise in a Smith diagram -- the monotonicity is really a kind of "around the clock" phase monotonicity, like for tan(x). Not being familiar with the field, I wonder if there is a formulation that draws on the specialist literature, but would also satisfy literalists. Perhaps simply "except when crossing poles"? 84.227.254.143 (talk) 15:54, 30 March 2014 (UTC)
The alternative formulation is to say the function always has a positive slope.  A function becoming less negative is still increasing. SpinningSpark 17:07, 30 March 2014 (UTC)

This review is transcluded from Talk:Foster's reactance theorem/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Protonk (talk · contribs) 16:59, 4 October 2014 (UTC)

I'm always happy to review articles like this as I was an electrician in a past life (figuratively), so it's fun to see how little I knew about the underlying circuit theory. :) I've got a few comments below but I think the article is pretty close to GA, with just some work needed to make the explanation and characteristics of the theory a bit more clear.


### style/layout
I think the lede could stand to be a little longer. It covers the basics and isn't required to be longer, but skips past the explanation and much of the history.
Done. SpinningSpark 01:12, 6 October 2014 (UTC)
Is it monotonic or strictly monotonic?
Strictly monotonic. SpinningSpark 01:12, 6 October 2014 (UTC)
"...completely determine the frequency characteristics of a Foster network." should we instead link to Frequency response here?
The frequency response article talks about inputs and outputs, that is, it is considering frequency response as the transfer function of a two-port network.  An impedance function is a one-port.  While one can think of voltage and current as being input and output, all-in-all I think it would be confusing to link to that article. SpinningSpark 01:12, 6 October 2014 (UTC)


### content
"Reactance is the imaginary part of the complex electrical impedance." I think this could benefit from a brief digression explaining that inductance and capacitance are both elements of impedance as well as the fact that the imaginary component is the one which is frequency dependent (as this will be important for the theorem). It may also help explain the importance of the system being lossless/passive.
Done first part.  You are incorrect in thinking that the real part of impedance is not frequency dependent.  It can be frequency dependent even if all the resistors in the network have no frequency dependence.  For instance, the real part of the impedance of a resistor and capacitor in parallel is given by R/(1+(ωCR)2). SpinningSpark 23:05, 6 October 2014 (UTC)
"Foster's theorem applies equally to the admittance of a network..." This is neat (and the following sentence explains why we note it even though the bare application would be trivially true)
Doesn't seem to be anything actionable in that comment. SpinningSpark 23:05, 6 October 2014 (UTC)
Sorry about that. I originally had something else but reading a bit more decided against it. Protonk (talk) 23:34, 6 October 2014 (UTC)
The "explanation" section feels like a bit of throat clearing to get to the immitance form of the theorem. That's ok, so far as it goes, but it's not an explanation, per se. I think the clarity suffers a bit by inserting the application to distributed element capacitance between the assumptions and the link to admittance. We might be better off by taking some time to note the basics (impedance/resistance), making a statement of fosters theory for impedance then showing the dual nature of it as well as the similar outcome for admittance and finally stating the immitance version. The sentence on the discrete/distributed part could be dropped or moved someplace else.
I've moved the distributed element stuff down. SpinningSpark 23:20, 6 October 2014 (UTC)
We're also in the odd position of explaining the theory without specifying why the 'one port' bit is important.
I'm not sure I understand what your difficulty is here.  One-port is merely a term of art.  It has to be a one-port because one cannot speak of the impedance of a multi-port.  One cannot even specify the impedance of one port of a multi-port without specifying what the other ports are doing.  An n-port is described by an nxn impedance matrix, not simply an impedance.  We can, of course, terminate all the other ports in a known impedance, but this merely reduces the network back to a one-port with a 1x1 impedance matrix, in other words just an impedance.  I'm not sure how to work any of that into the article, and I don't believe it would be beneficial.  It would be a bit like stating that a screw is a device for inserting into a threaded hole, and then launching a lengthy discussion on why it is only one hole. SpinningSpark 23:35, 6 October 2014 (UTC)
Ok, let me take another look. Protonk (talk) 02:25, 8 October 2014 (UTC)
"A consequence of Foster's theorem is that the poles and zeroes of any passive immittance function must alternate with increasing frequency." We're kinda stung here by the term "frequency" playing two roles. Do we mean that as frequency increases the poles and zeros must alternate or that they alternate more rapidly as frequency increases? Or both?
Done. SpinningSpark 17:05, 7 October 2014 (UTC)
"Two Foster networks that have identical poles and zeroes will be equivalent circuits in the sense that their immittance functions will be identical." Since these ciruits are by definition passive and lossless is explanation at the end of the sentence necessary? Or (alternately) could it be made more explicit by noting that the assumption proscribes non reactive components?
I'm not understanding the problem here.  The sentence does not mention "passive and lossless", nor does it make an assumption. SpinningSpark 17:05, 7 October 2014 (UTC)
Well they're passive and lossless by virtue of being foster networks, so (if I'm understanding this, which as other comments have borne out may not be the case :) ) wouldn't two foster networks w/ identical poles and zeroes be equivalent circuits in every sense? Protonk (talk) 02:32, 8 October 2014 (UTC)
They are not necessarily topologically equivalent.  For instance, these two networks,

are completely equivalent.  They have the same poles and zeroes and the same impedance function, yet they are completely different topologically.  If that is where the issue lies then we can clarify that.  If it isn't then I have no idea what the problem is.
....unless...this revolves round the scaling factor (stated in the article).  If all the impedances of the first network are scaled by a factor of two for instance, making the component values C1, 1/2C1, and 2L2, then it is still equivalent to the second network in terms of having the same poles and zeroes and the shape of the impedance function remains the same.  However, the total impedance is now double what it was before.  We could say the networks are completely equivalent except in terms of how much they load the source. SpinningSpark 16:37, 8 October 2014 (UTC)
Ok. I think the question arises from the distinction between electrically equivalent and topologically identical. Which is to say if we mean Equivalent circuit, we can just link it without specifying immittance. I think the bit about the scaling factor could do with some clarification but it's relatively straightforward right now. Protonk (talk) 17:08, 8 October 2014 (UTC)
I would rather link to the article those diagrams came from which I wrote some time ago (never put up for GA if you have nothing better to do).  The equivalent circuit article is not very focused and talks a lot about approximations which is not what we mean here at all. SpinningSpark 20:25, 8 October 2014 (UTC)
That works, thanks. Protonk (talk) 21:07, 8 October 2014 (UTC)
"A one-port passive immittance consisting of discrete elements (that is, not a distributed element circuit) can be represented as a rational function of s..." We jump right in here without explaining why impedance is described by a rational function.
This is a well known basic property of analysis.  I have added a brief explanation. SpinningSpark 17:05, 7 October 2014 (UTC)
"Foster in his paper describes..." Probably better as "In his paper, Foster describes..."
Done. SpinningSpark 17:05, 7 October 2014 (UTC)
We go into a lot of detail in the Realization section but leave an explanation of how the different forms reduce to something solvable (for foster) only in the image captions. Would this be better expressed in the body of the article as well? I know we don't want to repeat ourselves too much but I think doing so will make the article a bit more clear.
I think it's fine as it is.  The forms as described in the text are solutions, they do not need reducing.  The elements can take on any value, including zero and infinity in some cases, this does not need stating explicitly.  When explaining how this relates to poles and zeroes at zero and infinity, we need to refer to the relevant diagram.  This is easiest done in the caption of the relevant diagram.  The alternative is to give the diagrams figure numbers so they can be referenced.  Not everyone likes this; I have been criticised at FA by reviewers for this being "too textbook" and try not to do it any more. SpinningSpark 17:05, 7 October 2014 (UTC)
That's fair. Protonk (talk) 02:25, 8 October 2014 (UTC)
"This work was commercially important, large sums of money..." Vonnegut notwithstanding, this should be a semi-colon, no?
Done. SpinningSpark 17:05, 7 October 2014 (UTC)
"Amongst Cauer's many innovations was to extend..." probably "was the extension of"
Done. SpinningSpark 17:05, 7 October 2014 (UTC)
"(the condition of being a Foster network is not a necessary and sufficient condition, for that, see positive-real function)" Is it necessary but not sufficient or neither?
Being a Foster network is by definition sufficient for realisability, but it is a long way from being necessary (a simple resistor is obviously realisable, and, equally obviously, is not Fosterite).  The point being made here, I think, is that the iff condition is what Cauer was looking for.  Rewording to try and make that clear. SpinningSpark 17:05, 7 October 2014 (UTC)
Thanks, Protonk (talk) 16:59, 4 October 2014 (UTC)

Hi Protonk, thanks for reviewing.  It may take me two or three sessions to go through all your comments.  Please be patient. SpinningSpark 01:12, 6 October 2014 (UTC)

It's better to have no short descritption than one that is meaningless
Copied from edit summary without permission.
@Spinningspark: Hiya!  Thanks for reverting my mistake!  The short description I added was "A reactance theory of electrical circuits".  I was trying to distill the first two sentences of the lead to something most people could understand, or at least recognize the field of science.  Can you suggest something simple and accurate? – Reidgreg (talk) 19:25, 1 September 2021 (UTC)

@Spinningspark: Rather than these back-and-forth edits, can we discuss what the short description should be? – Reidgreg (talk) 17:05, 2 September 2021 (UTC)
It's a theorem not a theory, it says so right there in the title.  I'll stop reverting sds that are misleading or just plain wrong when the sd police stop reverting me for writing ones that are too long, .  Sometimes it's just not possible to write an sd that is both accurate and meaningful in less than 40 characters.
Having got that off my chest, let's try to be helpful: "An electrical network theorem" works for me.  Although since reactance is an electrical engineering term the article title pretty well tells you than much.  Almost a case for writing {{Short description|none}} (the template recognises that in case you didn't know). SpinningSpark 18:46, 2 September 2021 (UTC)
Ah, I see.  I wasn't aware of the earlier conflict with the short description.  (The instructions for their purpose and writing them could be clearer.)  Also, I feel a bit dense that I didn't bother to check that there was a difference between a theory and a theorem.  I won't edit it again unless we can agree on something, and hope that this hasn't caused a muddle with the various article nominations.
They don't like A or An, but I'm okay with Electrical network theorem as the short description.  I'd thought electrical circuit would be more accessible than electrical network, if they were roughly equivalent, but if the former is not accurate then we certainly shouldn't use it.  I guess, as with article title policy, we should go with how those familiar with the subject would define it, rather than generalizing it to the point of inaccuracy.  I do think that it's better than no sd.  Having 'electrical' there as the first word will do a lot to help uninformed readers understand the general field of the theorem, even if they don't know what an electrical network is. (Not everyone is going to understand 'reactance' without further context.) – Reidgreg (talk) 23:29, 2 September 2021 (UTC)
My opinion on the situation is to go with the short description without the A/an. I don't have much opinions to state, so this is a short comment. There is a task on the reward board regarding removing these from short descriptions and this just went in my way. Luxtaythe2nd (talk) 18:30, 3 September 2021 (UTC)
@Spinningspark: what do you think, are you okay with Electrical network theorem ? Let me know and I'll set it and fix the wikidata description to match.  BTW, I have a better understanding of your frustrations now, seeing the number of edits over the past 48 hours or so. The script makes it easy for people to edit, if not edit responsibly (myself included). – Reidgreg (talk) 18:34, 3 September 2021 (UTC)
I'm ok with that. Sometimes the article actually is needed to avoid appearing to say something inaccurate.  It's ok in this case, but it really shouldn't be a hard rule. SpinningSpark 19:02, 3 September 2021 (UTC)