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I am a bit confused on this matter.

Suppose we have an alternating current power source. One can transmit electric power two ways

  1. Through radiation, e.g., via an antenna.
  2. Through a transmission (power) line.

I suppose mechanism 2, if the line is just one single solid line, as opposed to a waveguide or coaxial cable which I understand the Poynting vector is nonzero and points longitudinally within the hollow space, concentrates the power across near surface of the conductor which is essentially of constant area whereas mechanism 1 the power falls off like $1/r^2$. One possibility I thought of is to model mechanism 2 as a wave guide where the electromagnetic wave travels confined within the hollow tube. That answers the question of power concentration. However, that requires the transmission line to be hollow or coaxial.

My questions: Is the power transmission line one isolated solid line or hollow/coaxial? If it is one isolated solid line, how does the power transmit (We know the AC travels along the surface of a conductor within skin depth) ?

Hans
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2 Answers2

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  1. Through a transmission (power) line.

I suppose mechanism 2 concentrates the power across near surface of the conductor ... (We know the AC travels along the surface of a conductor within skin depth)

No. Only current and charges are concentrated near the surface of the conductors. But the electromagnetic energy is located outside of the conductors, because that is where the electric and magnetic fields are.

Consider an open transmission line with two conductors.

enter image description here
(image from electriciantraining.tpub.com - Electromagnetic Fields About A Transmission Line)

The electric field ($\mathbf{E}$) goes from one conductor to the other. And the magnetic field ($\mathbf{H}$) curls around the two conductors. So the electromagnetic energy is in the space between and around the the conductors. The electromagnetic power flow (called the Poynting vector, measured in W/m$^2$) for every point in space can be calculated from $$\mathbf{S}=\mathbf{E}\times\mathbf{H}$$ and its direction turns out to be parallel to the conductors, as expected.

Thomas Fritsch
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  • The premise of your answer is that there are two parallel lines with currents running in opposite directions. I actually thought about that kind of coupling and consider the fields to be akin --- the fields are topologically equivalent --- to those of the coaxial cable in mechanism. Is the pairing necessary to transmit power? When I said about "the power concentrates along the conductor surface", I was thinking about the single solid line rather than a waveguide or coaxial cable (or the coupled parallel lines). – Hans Nov 06 '23 at 22:03
  • I now think one line is alright too. It is not as efficient as two coupled lines which in tern is less efficient than a coaxial cable or a waveguide. Do you agree? – Hans Nov 06 '23 at 22:06
  • I'm not an electric engineering expert. I think for efficient energy transfer at high frequency you need coaxial cables or waveguides. One-line or antenna transmission is inefficient for energy transfer, but may still be sufficient for information transfer (like radio broadcast). – Thomas Fritsch Nov 06 '23 at 22:50
  • I am puzzling over where the radial $\mathbf E$ in, say, a coaxial cable would come from. The distribution of charge should be axially symmetric and the net charge on both the inner and outer shell should be zero. The radial $\mathbf E$ should be zero. What do I miss? – Hans Nov 07 '23 at 05:34
  • @Hans, The E field oscillates, so if the inner conductor is positive at one point along the line it is negative at another point ~1/2 wavelength away, and the net charge remains 0. Same for the outer conductor, although the line is normally treated as unbalanced and the outer conductor taken as ground (0 V) all along its length. – The Photon Nov 07 '23 at 05:56
  • @ThePhoton: To clarify, I am concerned about the radial direction $\mathbf E$. Are you saying the charge density of the shells (inner or outer) is not uniform along the line but is a sinusoidal function of the longitudinal position on the line? Can you derive it either as an answer or provide a reference? – Hans Nov 07 '23 at 06:19
  • @Hans if the current along the line is a sinusoidal function of longitudinal position, then charge density must also be, but 90 degrees out of phase. And current must be sinusoidal because the H field is. This can be a subtle effect in a power line since the wavelength involved is on the order of 5000 or 6000 km. – The Photon Nov 07 '23 at 06:27
  • @ThePhoton: The conclusion does not follow. Current is the product of charge density and velocity. So the functional form of the current is not necessarily inherited by that of the density. – Hans Nov 07 '23 at 06:34
  • @Hans, If current is flowing outward from a region, its charge density must drop. If current is flowing inward to a region, its charge density must fall. This causes the charge density to have a sinusoidal dependence when the current does. See the first image on the Wiki page Transmission Line for a graphic illustration of this effect. – The Photon Nov 07 '23 at 06:39
  • @ThePhoton: You are right. Actually, I just thought of it before seeing your reply. It is the continuity equation $\frac{\partial\rho}{\partial t}+\nabla\cdot \mathbf J=0$. The functional form of both $\rho$ and $\mathbf J$ are consistent with the wave equation of $\mathbf E$ and $\mathbf B$. Thank you for reminding me. – Hans Nov 07 '23 at 07:03
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I suppose mechanism 2, if the line is just one single solid line,

Two conductors are required to form a transmission line. (Three, for three-phase power, AFAIK).

If there is only one line, then it means the earth (the actual dirt and rock) below the line is being used as the second conductor. This is not very efficient but it is done in some rural systems.

... concentrates the power across near surface of the conductor

As pointed out in another answer, the E and H fields associated with the electrical wave are found in the space between the conductors, not within the wires themselves.

We can gain some understanding by modeling the transmission as voltage and current waves on the wires, but if it comes to the question of "where is the power?", it is more physical to say that it is in the fields surrounding the wires. (On the other hand, Feynmann tells us there isn't actually a unique answer to this question)

One possibility I thought of is to model mechanism 2 as a wave guide where the electromagnetic wave travels confined within the hollow tube.

Yes, this is absolutely how transmission lines are treated in electromagnetics. They are a form of waveguide.

However, that requires the transmission line to be hollow or coaxial.

Not true. Just as twisted pair lines can be used to transmit data signals in data communication systems, two and three wire structures without any enclosing conductor can be used to guide power transmissions.

My questions: Is the power transmission line one isolated solid line or hollow/coaxial?

Neither.

enter image description here

(image source: User Kent Murrell on Wikimedia)

The multiple wires connected to the tower form two or more multi-wire transmission lines (likely two 3-phase lines of three conductors each), comparable, as I mentioned above, to the twisted pair lines used in computer networking.

You can see that on each arm of the transmission tower there are two or three wires attached. These are "bundled conductors" whose geometry reduces the incidence of corona discharge around the line.

The Photon
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  • @SolomonSlow, thanks. Edited. – The Photon Nov 07 '23 at 06:42
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    you wrote "Two conductors are required to form a transmission line." but this is not quite true. The Goubau line is one example of an open waveguide transmission medium using surface waves; there are also others using different techniques to slow the phase velocity down, for example via corrugations. These surface waves were a very popular subject of investigation some 60-70 years ago; it is still of general interest just not as a viable long distance link but as antenna launchers, spurious radiating modes of phased arrays, and some such. – hyportnex Nov 07 '23 at 10:06
  • @hyportnex you can also make zero-conductor waveguides, as in optical fiber. But it's not very practical at power transmission frequencies. – The Photon Nov 07 '23 at 14:57
  • On the contrary, long distance VF, LVF, ELF (some are megawatts transmitters) count on the Earth, a dielectric sphere, surrounded by the atmosphere being the waveguide. – hyportnex Nov 07 '23 at 15:26
  • @hyportnex: I did a preliminary browse on the linked article as well the subject of single wire transmission line. This is very interesting. I am particularly interested in the surface wave theory initiated by Arnold Sommerfeld. – Hans Nov 07 '23 at 19:35
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    Great! Both Collin & Zucker: Antenna Theory vol2 and Collin: Field Theory of Guided Waves have longish chapters on surface wave guides and antennas, and you should also check out anything that Arthur Oliner wrote for the nitty-gritty. – hyportnex Nov 07 '23 at 20:09
  • @hyportnex: Just want to add a reference for posterity: F. Stulle, J. Bergoz, Surface Waves for Testing of Beam Instrumentation https://accelconf.web.cern.ch/IPAC2012/papers/moppr006.pdf – Hans Nov 08 '23 at 06:39