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The authors of Power Systems Analysis calculate the inductance per unit length (henrys/meter) of a transmission line attributed only to the flux inside the conductor as "flux linkages per ampere." The flux is drawn in the x-y plane (i.e., in the page). Is the flux linkage also in the x-y plane?

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"In the tubular element of thickness $dx$" the flux per meter of length is $d\phi$. "The flux linkages $d\lambda$ per meter of length, which are caused by the flux in the tubular element, are the product of the flux per meter of length and the fraction of the current linked" (p149): \begin{align} \mathrm{d\lambda}=\frac{πx^2}{πr^2}{d\phi} \end{align} What is the direction of $d\lambda$? Given that the flux is directed concentrically (as shown in the figure), is the corresponding flux linkage also directed concentrically in the cross section of the conductor?

Note: The figure above assumes uniform distribution of current throughout the cross section (p145).

artist_and_not_EE_by_training
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  • Why are you looking for "flux linkage" inside a conductor? That is a dubious idea. Flux linkage is just a term referring to magnetic flux through a closed path that is not a simple circle but has many turns. – Ján Lalinský Feb 24 '23 at 22:54
  • Flux linkage seems important to understanding inductance in a transmission line (https://www.electrical4u.com/inductance-in-power-transmission-line/) – artist_and_not_EE_by_training Feb 24 '23 at 23:07
  • Calculation in section "Calculation of Inductance of Single Conductor" on that website is bizarre - in addition to using the term "flux linkage" in a perplexing way, it lacks explicit description of the closed path for which the magnetic flux is being calculated, and the resulting formula for self-inductance is achieved only after introducing an arbitrary limit of integration at $x=D$. Thus the resulting formula is useless - inductance depends on arbitrary choice of $D$ and diverges to infinity as $D$ goes to infinity. – Ján Lalinský Feb 25 '23 at 01:21
  • Author of that text assumes magnetic field is that of straight current-carrying wire of infinite length. However, infinite wire has infinite self-inductance per unit length, which means realistic calculation has to take into account real length of the wire $\ell$, and the resulting $\lambda$ depends on $\ell$. – Ján Lalinský Feb 25 '23 at 01:23
  • They did get finite result for the first contribution to self-inductance (due to region of the wire), but there is no way to check that a part of self-inductance is right. Experiments with self-induction only manifest total self-inductance, nature does not care how we split calculation on paper. – Ján Lalinský Feb 25 '23 at 01:39
  • Rosa gives an argument that his weighing method is right because the "internal" self-inductance it assigns to the region of the wire predicts the correct magnetic energy in the wire. This is true, but I don't see how it is relevant to total self-inductance. He also calculates total self-inductance of finite wire, and he does take into account contribution from the outside. – Ján Lalinský Feb 25 '23 at 02:33
  • I don't have much confidence in Rosa's method and argument for it. Also, self-induction is due to induced electric field, and while it can be analyzed using magnetic flux in simple cases where the relevant magnetic flux is well-defined, we are not obliged to use magnetic flux. Instead, we may work directly with induced electric field and derive net self-induced EMF. In case this EMF comes out as proportional to $\frac{dI}{dt}$ (typically it does), then we define self-inductance as magnitude of that constant (thus always a positive number). – Ján Lalinský Feb 25 '23 at 02:51
  • I found a probable origin of this bizarre idea that "flux weighing" plays role in calculation of self-inductance in straight conductors, an article by Edward Rosa in Bulletin of the Bureau of Standards, see e.g. https://g3ynh.info/zdocs/refs/NBS/Rosa1908.pdf – Ján Lalinský Feb 25 '23 at 03:05
  • See also this very good answer by Edgar Bonet: https://physics.stackexchange.com/a/11805/31895 – Ján Lalinský Feb 25 '23 at 05:10

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My initial thought on this seems to be consistent with Wikipedia.

In circuit theory, flux linkage is a property of a two-terminal element.

Basically, the term "flux linkage" is an idea in circuits and more of an engineering aid. The actual field lines and flux for a particular surface (which must be defined before one can integrate) are the more physical ideas.

Full disclosure, the second sentence says [citation needed] in that article, so it may be a bit of a point of controversy. The article definitely needs improving, but my understanding is consistent with it. I checked a few more places and everyone refers to linkage in terms of wires, so this seems to be the norm.

The interior magnetic field in a conducting wire does lead to observed effects of course, and those will be predicted at the very least by Maxwell's equations and the Lorentz force law. There may be a way to define "equivalent flux linkage" for a wire and simplify the analysis, I'm not sure. I have never seen this and am skeptical of it.

There is a bit of technical point here. Since the $\vec{E}$ field is no longer irrotational when there is a time varying $\vec{B}$ field, the "total voltage" is no longer path independent. This is why you get a concept like flux linkage. It is the path enforced by the wires, with a highly conductive path and highly resistive boundaries, that lead to a particular "total voltage" or "EMF" as it can be called:

$$ EMF = \oint \vec{E} \cdot d\vec{r} $$

I personally don't like the whole EMF thing and just think of it as voltage. Anyway, in the interior of a conductor, the path of the contour integral is not defined, so I don't know how a "flux linkage" would be defined. The behavior of the fields will be predicted by solving Maxwell's equations in a material.

Poisson Aerohead
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  • "I personally don't like the whole EMF thing and just think of it as voltage." This will cause you trouble when analyzing perfect inductor in AC circuits, where induced EMF acts against potential gradient, so that net electric field in the inductor coils can be zero. It is better to reserve the use of "voltage" to differences of potential (which are often of interest and thus the thing we want to measure), and not use it for induced EMF (which is a general concept for any closed loop and it is not common to measure its value).

     – Ján Lalinský Feb 24 '23 at 22:40
  • Setting the V / EMF philosophical discussion aside, do you agree with my actual answer? I have never heard the term "flux linkage" used any other way. – Poisson Aerohead Feb 24 '23 at 22:42
  • I don't think it's a good answer. The question is about a dubious word group "flux linkage inside a conductor", I don't think we know what is meant. – Ján Lalinský Feb 24 '23 at 22:51
  • Well, my answer is that it isn't a defined idea as far as I can see. Flux linkage is defined for a defined path (space curve) which is enforced usually by a wire coiled into a shape. A conductor is a 3D continuum, there is no space curve. Maybe I should say it that way? – Poisson Aerohead Feb 24 '23 at 22:54
  • That's better, and close to how I understand it as well. Flux linkage is just a term for magnetic flux through a closed path that has more than one turn. – Ján Lalinský Feb 24 '23 at 22:57
  • @JánLalinský my textbook derives an expression for the inductance of a transmission line by "integrating from the center of the conductor to its outside edge to find... the total flux linkages inside the conductor" – artist_and_not_EE_by_training Feb 24 '23 at 23:13
  • @PoissonAerohead are you saying that a conductor is not a "two-terminal element", hence flux linkage is not defined for a conductor? – artist_and_not_EE_by_training Feb 24 '23 at 23:15
  • The way you have asked the question, it sounds like you want to know about the internal flows within the space of the conductor itself. That is a 3 dimensional medium, not a "two terminal element" in circuit theory. Is that what you are asking about? Regarding your textbook, that is just ordinary flux, or "linkage" with one turn. – Poisson Aerohead Feb 24 '23 at 23:21
  • @artist_and_not_EE_by_training the textbook probably talks about self-inductance per unit length associated with magnetic energy in the region of the wire (sometimes called "internal self-inductance"). This introdution of separate concept of "internal self-inductance" is a dubious and the concept is misleading. It only works if current is uniform - for surface current, this "internal" self-inductance is zero. Total self-inductance (the one you can measure in lab) is different, and its calculation based on magnetic flux has to take into account also the magnetic flux outside the wire. – Ján Lalinský Feb 25 '23 at 02:16