Talk:Skin effect/Archive 2

Archive 1

Archive 2

Eddy Currents

The association of eddy currents with skin effect, is self contradictory and incorrect. In the figure showing circular currents, all of the horizontal components of current, are cancelled by opposite horizontal components, induced a little further down the wire. Skin effect is simply a difference in inductance, of different paths within the wire. In a straight wire, the inductance of a path increases, as it approaches the center of the wire, where there are more short paths for the H field, within the wire. User:Newtonez (talk) 13:02 15 August 2017 Newtonez (talk) 20:06, 15 August 2017 (UTC)

The eddy current explanation has a solid reliable reference. You would need some equally solid references to refute it. You would also need reliable sources to assert that the skin effect is due to differences in path inductance. I don't think that it is because I don't think that it would reproduce the frequency dependence. The alternate explanation that I am aware of and which does have reliable sources is that the E-field inside the wire is propagated in from the outside and good conductors rapidly attenuate propagating EM fields. In fact, the formula : $J=J_{\mathrm {S} }\,e^{-{(1+j)d/\delta }}$ is easily derived from that analysis. Constant314 (talk) 20:43, 15 August 2017 (UTC)

I am also critical of the claim that Eddy currents are the underlying reason for skin effect, and that the resulting current density variation with depth in a wire is the superposition of the homogeneous "original" current and opposing Eddy currents depending on radial position. It's true that there is a reference (a book) cited by the article, however I would challenge its claimed solidity and reliability, since no mathematical proof is given there for the claim that "these emf's are greater at the center than at the circumference, so the potential difference tends to establish currents that oppose the current at the center and assist it at the circumference", as put by the author's own words. In the case of homogeneous current density distribution in a cylindrical wire (as in th case of DC), the H-field magnitude is zero in the centre and increases linearly with radial position up to the outer surface. Why should then the change in H-field be maximum in the centre, as frequency goes gradually up from zero? I would need to see a mathematical proof to believe.

On the other hand, for an explanation as well as a mathematical derivation of why and how skin effect comes to be, without even mentioning the notion of Eddy currents at all, one can see any one of D. K. Cheng's books on electromagnetics. All it takes is to start with electromagnetic fields assumed to propagate as a plane wave in a good conductor and then simplify the attenutation constant using the fact that the conductivity is much larger than the frequency-permittivity product. In my view, both skin effect and Eddy currents are observed manifestations of electromagnetism that is (for the scope here) sufficiently well-described by Maxwell's equations alone. I don't think they are in a cause/consequence relationship with one another. — Preceding unsigned comment added by 185.34.132.4 (talk) 13:28, 9 February 2018 (UTC)

I'm sorry, but these objections are both mistaken. "Skin effect is simply a difference in inductance, of different paths within the wire" is clearly inaccurate if you'd work it out. A clear counterexample is where the wire's conductance is high or infinite in which case all the current is very close to the surface; that cannot be explained by looking at the inductances of different wire radii which are not (much) affected by the metal's conductivity. The second explanation in terms of EM waves impinging on a metal conductor is perfectly correct but is NOT inconsistent with a more detailed examination of the metal shielding electric fields or magnetic fields alone. There are always multiple ways of looking at these problems and one cannot generally separate "cause and effect" unless you want to go to the time domain (where this problem would become much more difficult).

With an external E field only, the field is canceled inside the metal due to polarization charges near/at the surface which is perfect at DC or high conductivity. Now consider an external changing (AC) H field crossing the wire which implies a nonzero curl E in the wire's direction. Let the wire have infinite conductivity. A net E field inside the wire would cause an infinite current and is impossible. The only place you can have a large curl E that doesn't produce a current is outside the wire or especially right at its surface. Currents at that surface, opposite between the edges where the external H enters and exits (thus creating a current loop) MUST cancel that H inside the wire (this is irrespective of normal currents along the wire which have no curl). Or an external changing (AC) H field in the direction of the wire will create currents curling along the edge of the wire. Again the longitudinal H field it creates must cancel the external H field in order that there is no infinite current further inside the wire. If this current were not right at the edge then you would still have a noncanceled H field outside it creating a current there, so the current winds up at the edge. So it's easy to place the eddy currents when you have infinite conductivity. The important thing to note is that both the E and H cancellation can occur on a very small spatial scale relative to the wavelength so you don't need to consider an EM wave. But the fact that there is a dissimilar explanation for EM waves impinging on a conductor (thus at scales greater than a wavelength) shouldn't bother anyone unless the predicted results are inconsistent. If someone wants to add text looking at the problem differently that's fine as long as you can make it understandable (the difficult part!). Interferometrist (talk) 15:14, 9 February 2018 (UTC)

I do not have Cheng’s book, but I presume that his approach is the same as Hayt, which is you just assume a locally plane wave propagating normally to the surface, write down the propagation constant, and voila, you are done. It works well when the skin depth is small compared to the thickness and I would support having it in the article as another explanation. But gets into trouble when the skin depth is on the order of the diameter. The wave is not a plane wave (it is a cylindrical wave) and when it reaches the other side it interferes with the wave coming in on that side, and it is reflected at the conductor/air interface. In this case, the eddy current explanation is still accessible. The locally plane wave assumption also gets into trouble when the conductor is not round. When the conductor is a high aspect ratio rectangle, as in a stripline, Cheng’s approach does not reproduce the current crowding that first sends the current to the short sides and then into the corners as frequency rises. But the approach of mutual inductance between current filaments does and is commonly used in CAD tools for layout and analysis of advanced circuit boards. This is discussed in Clayton Paul’s, Analysis of Multiconductor Transmission Lines, Wiley, 2008. I believe you can even find that bit of the book on Amazon or elsewhere on the internet. Constant314 (talk) 00:45, 10 February 2018 (UTC)

- "The second explanation in terms of EM waves impinging on a metal..."

There is no assumption of waves impinging onto the conductor from outside. The analysis assumes propagation entirely within a good conductor (note, not a perfect conductor, which can contain no E-field), and not across a boundary.

- "Let the wire have infinite conductivity. A net E field inside the wire would cause an infinite current and is impossible."

Correct statement given the assumption. However, this is really where we must carefully differentiate between perfect and imperfect conductors. A non-zero skin depth with an exponentially decaying current density profile is observed only in imperfect conductors - a perfect conductor (speaking for AC only) will only have surface current density, and zero currents in its volume (limit case with zero skin depth). For imperfect conductors, the argument of impossibility of infinite current holds no more, for we may well have a large but finite amount of current in such a material, and a very small E-field that is sufficient to drive that current.

- "The only place you can have a large curl E that doesn't produce a current is outside the wire or especially right at its surface."

In space, a point of consideration either lies inside the conductor, or outside it. If it is inside, the E-field has to be either zero (in a perfect conductor) or very small (in a good but imperfect conductor). We cannot treat the surface of an imperfect conductor as a special place that acts like a transition between the conductor and the (presumed) vacuum surrounding it, where $\mathbf {J} =\sigma \mathbf {E}$ is allowed to not hold. It simply holds everywhere and it's only the spatial variation of $\sigma$ that produces different current magnitudes at different positions for the same E-field magnitude, that is, whether we are in one material or in the other, meaning whether we have one $\sigma$ value or another. The conclusion of skin effect and exponential decay of current density with increasing depth beneath the surface cannot be reached by attributing that special status to the surface.

- "The important thing to note is that both the E and H cancellation can occur on a very small spatial scale relative to the wavelength so you don't need to consider an EM wave."

Not quite true. A skin depth is, yes, very small compared to the free-space wavelength. With the extremely small phase velocity inside a good conductor, it can be shown that a skin depth corresponds to a radian of electrical length, i.e., $\delta ={\lambda \over {2\pi }}$ where $\lambda$ is the wavelength inside the good conductor. I've just realized this is actually discussed in the article.

- "... you just assume a locally plane wave propagating normally to the surface..."

No, there is no assumption regarding any surfaces or boundaries. The wave is assumed to propagate completely within the conducting medium, and the only conclusion reached from there is that the wave will attenuate very fast in the direction of propagation. Of course to talk about skin effect, we need a boundary, and for the currents (as well as fields) to decay exponentially with increasing depth, we need the propagation to be perpendicular to the surface. This, however, is not dependent on any of the attenutation constant, the boundary being planar, the wave being a plane wave, or so on. It is guaranteed rather by the extremely small phase velocity of the wave in a good conductor, ${\sqrt {{2\omega } \over {\mu \sigma }}}$, which, combined with Snell's law of refraction, requires that any penetration into a good conductor, even at grazing angles of incidence, is followed by near-zero angles of refraction, resulting in all propagation in a good conductor being almost normal to the surface. This produces a more generic explanation to skin effect, which, after all, is not a phenomenon that is observed exclusively in wires. In fact, skin depth is defined as the depth below a planar boundary of an otherwise unbounded medium of the considered material, where the current density or the fields go down to $1/e$ of their surface value.

- "When the conductor is a high aspect ratio rectangle, as in a stripline, Cheng’s approach does not reproduce the current crowding that first sends the current to the short sides and then into the corners as frequency rises."

This is only natural. What is referred to here as Cheng's approach is a set of inferences off of Maxwell's equations regarding propagation inside a good conductor, derived to explain skin effect. That current crowding mentioned here, however, does not originate from the propagation characteristics inside an imperfect conductor being the way they are. In that sense it is unrelated to skin effect. In fact, concentration of the surface currents around the edges of a stripline can be observed when the structure is perfectly conducting too, in spite of the zero penetration depth. And needless to say, the distribution can perfectly well be predicted by solving Maxwell's equations for the given geometry.

- It would be greatly satisfying to the mind to be able to arrive at the same conclusion of exponentially decaying current density with increasing depth into a good conductor, using one or the other explanation. The Eddy current explanation presented in the article, however, appears mathematically less complete to me. There is after all an open question: Why would the Eddy currents counteracting the original current distribution (that supposedly tries to spread evenly across the wire) be stronger in the centre of the wire than near its surface, while the H-field is at its weakest in the centre, and strongest on the surface? Could someone present a methematical derivation for the spatial distribution of Eddy currents, such that when we superpose this distribution of counteracting currents onto a homogeneously distributed "original" current, we get the exact same current density profile we obtain by Maxwell's equations?

185.34.132.4 (talk) 13:47, 13 February 2018 (UTC)

Listen, I'll answer a couple things, but this page isn't for general discussion of the content, but for editors to coordinate their contributions. First, you objected to "The important thing to note is that both the E and H cancellation can occur on a very small spatial scale relative to the wavelength so you don't need to consider an EM wave" by mentioning the wavelength INSIDE the conductor whereas I was obviously talking about the external wavelength which shows you didn't even TRY to understand my point. OUTSIDE the conductor due to IMPINGING fields, the E and H fields can have rather arbitrary magnitudes over a small scale which is small compared to the wavelength. THEREFORE it must be that these fields are screened well inside the conductor by different mechanisms so as to accommodate any combination of E and H. For E it is the conduction currents -> polarization charge density; for H it is the effect of eddy currents. For current along the wire, the latter is the important one.

"Why would the Eddy currents counteracting the original current distribution (that supposedly tries to spread evenly across the wire) be stronger in the centre of the wire than near its surface, while the H-field is at its weakest in the centre" - because the curl of H is greatest at the center which is what drives E which drives J. I think you could have figured that out yourself! And then you ask "Could someone present a methematical derivation for the spatial distribution of Eddy currents". Well I think that could be found but looking at the result with complex bessel functions I doubt it's simple. What would be simpler is showing that the expression given for J(r) satisfies Maxwells equations, but I still doubt it's easy but if you'd like to find that, then go ahead and include the math. I don't think either needs to be in this article but would expect that at least the latter is included in one of the refs. That should be sufficient. Interferometrist (talk) 21:30, 13 February 2018 (UTC)

- "... you didn't even TRY to understand my point."

I'd like to invite you to not get personal by making assumptions on my effort or willingness to understand you. Truthfully, I have all the intention of understanding and addressing all arguments here, regardless of whom they come from, as objectively as I can. This is a good debate on a rather technical matter, and I am happy to be part of it, let's keep to its standards of high quality.

- "... the E and H fields can have rather arbitrary magnitudes over a small scale which is small compared to the wavelength. THEREFORE it must be that these fields are screened well inside the conductor..."

I must note perhaps that I am not objecting to that the fields are screened inside the conductor. Albeit of small scale with reference to the free-space wavelength, this distance where the fields decay, that is, where the skin effect phenomenon takes place, is contained inside the conductor, in its entirety. I fail to see how we could use the wavelength from another medium (free space, in this case) to call this distance electrically small, overlooking that it is not the local wavelength inside the subject medium. Anyway, this is a minor detail I think, as the claim in the article that I am not sure I agree with (i.e., that Eddy currents are the underlying cause of skin effect) is not based on this argument. I will therefore not comment on this one any more.

- "... because the curl of H is greatest at the center which is what drives E which drives J. I think you could have figured that out yourself!"

Provided that the initial assumption for the current distribution across the wire (before it is altered by the induced Eddy currents) is that it is uniform, this is simply not true. Let's do the math: Assume a uniformly distributed current density along a $z$-directed cylindrical wire, $\mathbf {J} =J_{0}{\hat {\mathbf {z} }}$. The magnetic field is then $\mathbf {H} =({J_{0}}/2)\rho {\hat {\boldsymbol {\rho }}}$ inside the wire. Taking the curl, we simply get back to the initially assumed current density: $\nabla \times \mathbf {H} =J_{0}{\hat {\mathbf {z} }}$, which is of constant magnitude inside the wire, so no, it does not peak at the centre. This is no surprise, as the curl-of-H equation must return the current density judging by its right-hand side. So we basically go forth and back between the current distribution and the magnetic field caused by it using Ampere's circuital law. We cannot account for Eddy currents without referring to Faraday's law of induction ($\nabla \times \mathbf {E} =-{\partial \mathbf {B} }/{\partial t}$). And even when we do refer to the law of induction under the initially-uniform current density assumption, we get $\nabla \times \mathbf {E} =0$ along the centre of the wire, since here we simply have $\mathbf {H} =0$, and consequently so is its time derivative. In fact, irrespective of the spatial distribution of the current density, as long as we have only time-harmonic fields, the time derivative is a linear operator, meaning the time derivative of any quantity will be proportional to its own value across the entire space, meaning ${\partial \mathbf {B} }/{\partial t}$ will peak where the B-field itself peaks.

- "... I think that could be found but looking at the result with complex bessel functions I doubt it's simple. What would be simpler is showing that the expression given for J(r) satisfies Maxwells equations..."

Again, let's not restrain the generic notion of skin effect to the special case of current along cylindrical wires. I am not asking for a derivation of the current distributions involving Bessel functions, that outcome is something that is specific to one certain geometry. I am rather expressing dissatisfaction by the (lack of) mathematical completeness of the argument that Eddy currents are the cause of skin effect. I would be equally content if I saw a derivation of the current distribution in, for example, a conducting medium that is unbounded except one planar face. It is a fact that skin effect would be observed in this case too, and fields and current density would decay exponentially with increasing depth into the conductor. This result is well captured by the wave analysis approach, whereas I have yet to see some similar piece of math predicting the same result building from Eddy currents, without involving propagating waves and attenuation coefficient. I am talking about this geometry because it is a rather simple one where Bessel functions do not appear, it is of course not a must to use that one.

- "I don't think either needs to be in this article but would expect that at least the latter is included in one of the refs. That should be sufficient."

And I agree. I am not discussing the math because I think it needs to be included in the article, but because I am not assured math supports us when we try to explain skin effect by Eddy currents. To keep all this still connected to how we should edit the article, here go a few questions: Do we consider the given explanation verifiable? Should it remain in the article? If it should, should we not add references to a source that mathematically proves it, rather than saying "opposing Eddy currents are strongest in the centre of the cylindrical wire" only verbally, even after seeing that $\nabla \times \mathbf {E} =0$ exactly there?

185.34.132.4 (talk) 15:07, 20 February 2018 (UTC)

I think that your comments have two salients and it would be useful to separate them. I think, on one hand, that you are arguing for an explanation based on an EM wave propagating into the conductor. In response to that, yes, it is a useful explanation. If you read the whole article you will find bits and pieces of that explanation. If you want to improve those pieces, then I encourage you to do so. If you are ambitious and want to add a sub-section that brings all that together, I would support that and would try to help by providing alternate references and wording suggestions. If you are suggesting that someone should do that, then we are done: someone may do that when they want to do that.

The other salient seems to be that the eddy current explanation is wrong. If you put in the article that it is wrong or delete it, you will probably encounter strong resistance because it has reliable references. Wikipedia is not the arbitrator of what is right and wrong (see WP:RIGHTGREATWRONGS). It is the amalgamator of reliable sources, even if the sources disagree!

If you have two electric fields that exactly cancel each other inside a conductor, there will be no net current density there. You are quite justified in saying that the E fields cancel and there is no current. You are also justified in saying that each E field creates a current density and the currents cancel. That is my understanding of the eddy current explanation. This is the same explanation as dividing the conductor up into many strands of current and calculating their mutual inductances and then solving for the current in each strand and finding that the current decreases as you get further into the conductor. This is in fact a method used to compute losses of high frequency circuit board traces; it applies at both low frequency and high frequency and arbitrary cross-sections. If you want to improve the eddy current explanation, then you are welcome to try.

So, in summary, if you want to improve the explanation based on wave propagation, then I encourage you. if you want to improve the eddy current explanation. I encourage that too. It you want to delete the eddy current explanation or say that it is wrong, you will probably encounter frustrating resistance which may end up in an edit war. Constant314 (talk) 00:37, 21 February 2018 (UTC)

I'm sorry for saying that you didn't "try" to understand. I should have said you didn't understand that point and you still don't but never mind. When you are in the frequency domain you cannot generally talk about cause and effect because cause and effect are generally identified due to their order in time and we've taken time out of the picture. So you are very mistaken when you posit: "Provided that the initial assumption for the current distribution across the wire (before it is altered by the induced Eddy currents) is that it is uniform...." No, it doesn't have a chance to become uniform because the magnetic field is generated as soon as the current starts and the skin effect takes place immediately. And it is only after some time that it finally DOES become uniform (after the skin effect has died down). But this is why it's so hard and confusing to do it in the time domain. But because it is the frequency domain, perhaps there shouldn't be language saying that the eddy currents "cause" the skin effect but rather are "associated" with the skin effect, and when the so-computed eddy currents are added to the drift current (caused by a small voltage applied across the wire) you find the net current smallest at the center of the wire. Go ahead and edit in wording such as that if you wish. Interferometrist (talk) 20:14, 22 February 2018 (UTC)

Interferometrist, I agree with this comment about "order" of things and the difference between their meanings in time and frequency domains, and that the fields/currents never become uniform, even momentarily. I meant or implied no such thing as fields first distribute evenly and only later in time come the cancelling fields/currents. I do realize that frequency domain is an idealisation, that even if we try to create a setup with all quantities containing only one pure sinusoid, we do not get to avoid the intial transients, and that AC fields and currents will never appear deep inside the conductor (and disappear only afterwards). The word "initial" (used for the assumed current density) as well as all its posterior implications (like the induced E-field and the Eddy currents driven by that), refer only to an order of steps of calculation, and not to an order in physical time. I thought this awareness was obvious but maybe I had to phrase it more clearly while expressing my point anyway. All in all that does not seem to me as the source of the disagreement here.

Constant314, constructive advice regarding how to proceed with the article. I would like to point out few things though. First off, I have no interest in an edit war, I am aware that it will not help the article, nor Wikipedia's purpose in general. Another thing is, there is one reference cited in the article that supports the claim that the opposing Eddy currents are the strongest in the centre of a cylindrical wire - not multiple. Yet another is, I offer to discuss its reliability as opposed to labeling it as reliable without further discussion. I have checked the reference and seen no mathematical proof to the claim in it, this is my reason to question its reliability. If others that also have read the reference have reasons to think why it is reliable, I would like to hear. If the community thinks this is not the place to discuss the reliability of a reference cited by the article, well, then that's it, we stop here. But if there is motivation towards making use of technical knowledge made available here, then I'd be happy to contribute. Without anything that resembles an agreement here on the talk page about how to proceed, I will not edit the article.

As a generic comment I feel the need to mention that in linear electromagnetics (as is the case in hand), the moment we say "cancel out" we need to remember that in mathematical terms we mean superposition (likewise for when we say "add up"). When we phrase an explanation to a phenomenon saying two things cancel out, the default mathematical proof of this explanation goes by writing the expressions for the two quantities and mathematically adding them to show that the sum indeed is an expression representing the verbally described one. This is clearly missing in the Eddy current explanation. The propagating wave explanation, on the other hand, does not superpose anything with the wave that has penetrated into the conductor. It only describes (mathematically) how that same wave is attenuated by absorption along its path in the medium, due to the high conductivity. That is one substantial difference between the two explanations. 185.34.132.4 (talk) 13:18, 26 February 2018 (UTC)

A source does not have to have a mathematical proof to be reliable. We prefer reliable secondary sources, which frequently, simply use the results of proofs published in primary sources. I have added another reliable reference for the eddy current explanation.Constant314 (talk) 21:15, 27 February 2018 (UTC)

Reversion of 23 Feb. 2018

I found that the user who reverted my changes to the lede had been too busy to have carefully considered the reversion (and more importantly, what the best text would have been to replace it rather than just the previous wording). In particular, he/she had no more than 2 minutes and 25 seconds since having edited (reverted) a different page before deciding to revert my edit and write an edit summary (and less than a minute before reverting another page!):

User:Wtshymanski

2018-02-23

(diff | hist) 03:52:33 Solder spam; IP address appears to resolve to same company, COI Undid revision 827119436 by 70.60.53.26 (talk)

(diff | hist) 03:51:35 Skin effect longer, but was no clearer. Undid revision 827112431 by Interferometrist (talk)

(diff | hist) 03:49:10 Citizens band radio no in-line URLs Undid revision 827106980 by 41.182.3.128 (talk)

So I am restoring my changes (which are partly linked to the above discussion, which I doubt the reverting editor read through) and will ask for any disagreements with my edit to be addressed specifically and with at least an attempt at a better edit. Interferometrist (talk) 12:25, 23 February 2018 (UTC)

The phrasing "The bulk of the electric current flow is then along the outer layers of the conductor, greatly falling off at depths much in excess of the skin depth parameter δ (see figure). " is very wordy and hard to read - the second clause "greatly falling off..." seems to be very loosely jointed to the rest of the sentence. If it takes longer than 2 minutes 25 seconds to comprehend a minor change to an article, clearly the change is a failure. --Wtshymanski (talk) 21:08, 23 February 2018 (UTC)

Without implying that it is better, I do prefer the original version. But that does not mean that it is impossible to improve the lede. Constant314 (talk) 21:19, 27 February 2018 (UTC)

Is the recent citation to Kevan Hashemi, Brandeis University a relaible source?

I have examined the link. It is hand written and somewhat difficult to follow, but I did not see any errors. Kevan Hashemi is an employee of Brandeis University, but he is not a professor and he does not have a long list of publications in academic journals. In fact, his resume does not list any publications. Although interesting, I think that this source must be considered no better than anyone's personal blog and as such should not be cited. Constant314 (talk) 20:30, 2 October 2018 (UTC)