Real devices based on this principle exist: current transformers, Rogowski coils, and lights! The latter looks like this:

https://pr-tech.com/product/spanlite-self-illuminated-marker...

And, hilariously, the FAQ informs us that the light contains a “flux capacitor”.

Good point. So for the customary 3-phase power lines the UNC exam is confusing and the given sample solution wrong. In reality you could steal much less than the already ridiculous amount in the exam.

The German railways operate a 1-phase power grid. (After all 3-phase transmission makes only sense if you are able to balance the consumption of the 3 phases at a single place (small area in real life). With relatively few trains using 1 phase at uncoordinated locations that's not the case. And they cannot use the public grid because for historical reasons they operate on a different frequency.

A 1-phase transmission line needs 2 conductors. With my high school physics, I can't tell what that means. Isn't it so that when the max current flows in one direction in one conductor it will flow in the opposite direction in the other one? So what you can steal is determined by d1^2 - d2^2 where dn is your stealing coil's distance from conductor n? Orders of magnitudes worse than in the exam once again.

There are two wires because of Kirchoff's current law -- if the current goes out in one direction, it has to come back somehow. There are "single wire earth return" systems that use the ground as the return wire, for better or for worse.

If you replace d^2 by d^-2 in your equation, you'll be closer. Although, as noted in the OP, the magnetic field around a wire is proportional to distance^-1.

Right, but for many single phase installations a ground/earth return is often employed. This often simplifies things and saves a lot of infrastructure costs.

Whether it's single or 3-phase, calculating the coupling efficiency will, at best, only be a rough guesstimate unless great care is taken to accurately collect and collate all physical and electrical parameters. Not only will interline coupling be relevant so will the fact that three phases are involved and we'd have to take into account standing waves etc. (mind you at 50/60Hz the wavelength is enormous (thousands of kms) so the transmission line effect is likely be trivial at the local level - see note).

I reckon this article deliberately avoided the transmission line/multiphase effect to avoid that controversy or complicatlation.

__

PS: ignoring the transmission line velocity factor and ground effects, the wavelength at 60Hz is 5000km and it's 6000km at 50Hz; therefore the first voltage maxima (lambda/4) will, in both instances, only ocurr at over a thousand km from the source. Standing wave effects due to the presence of three phase will reduce this but it's still a big distance - too big to have much of an effect to vary much over a single farmer's land.

The solution therefore would be to calculate the various vectors for any given point on the line (here, at any area on the farmer's land and take that as constant for calculating the coupling to the secondary (pickup) inductor). It seems to me the most practical way would be to just measure the field strength empirically or to look up the various tables and monographs readily available in power engineering.

Often? Not what I as a complete laymen in power transmission have observed.

The mentioned power grid of the German railways uses 2 conductors. Why they would pay nearly 100% more in infrastructure cost than just using ground as conductor for return current I have no idea. There must be a reason.

The Finnish railways don't have their own power grid, with the frequency being the same as the general grid they can get power from there. However, they have another way of seemingly wasting metal for a return conductor. Normally trains use the rails as return conductor. That seems "free" metal to use, because the train needs more than enough for it for mechanical reasons. However, Finnish railway lines have a second overhead wire next to the railway line and autotransformers or booster transformers (I don't understand the difference) to make sure the current is directed from the rails to the much thinner overhead return wire instead. The overhead wire being insulated from earth in contrast to the rails.

https://en.m.wikipedia.org/wiki/Split-phase_electric_power mentions the same (?) for British railways. As a reason for the extra effort they mention reducing transmission losses and reducing electro-magnetic interference to the environment.

I guess reducing electro-magnetic interference means that the "stealing" scenario from the original posting does really only work for 1-phase single wire power lines where the return current goes via earth. And even there not really well, as the article has shown.

The only place where I have seen single wire power lines is Iceland. 20 kV lines I guessed when I saw them.

> I reckon this article deliberately avoided the transmission line/multiphase effect to avoid that controversy or complicatlation.

I agree. But I would have expected that the assumptions/simplifications had been mentioned as part of the task description.

First, it depends on the country and its rules (some permit it others don't); second, it depends on the application.

The balanced system in your Wiki link is common too and it has advantages over the single system. For starters, it's a balanced power feed from a center-tapped transformer, and thus they are often used in sound studios and laboratories to reduce hum. noise and ground currents. Sometimes in the US you'll see the three-wire 110/220V systems. As they use three-wire feeds, they're more expensive than single-wire ones such as here: https://en.m.wikipedia.org/wiki/Single-wire_earth_return

Earth return is mostly used in rural areas to feed small communities and such. This is to save cost. It is usually only employed on medium voltage feeds (6,600, 11,000 Volt lines). Rather than terminating in a substation, lines generally terminate in a single stepdown transformer. I would imagine it wouldn't be very common in Europe as it's pretty built up/has a large population (when I was living there I never saw any - but then, I was mostly in large cities, so I wouldn't have expected to see them).

I've not been to Iceland but with its smallish population I wouldn't be surprised if single lines with earth return are used there - it would make sense to use them there.

You are also unlikely to see single line feeds if the current is high or the line voltage is very high (intercity transmission lines, etc.) as ground eddies are potentially dangerous. Nor will you see it used where high current without sufficient voltage would lead to excessive I^2R losses.

That said, many railways effectively use ground return. As you mentioned, many electricified lines use the rail tracks as a return and they're effectively earthed. In lines where the rails aren't welded together but only joined by fishplates you'll see multi-strand copper wire connecting each rail to ensure that the return also doesn't suffer excessive I^2R losses.

I've lived in Europe but it's not my usual home so I'm not overly conversant with railway power systems there. However, I do know there are three common European electrification standards for normal railway services (i.e.: standard and Iberian guages - not specialized ones, narrow guage etc.). And roughly by country they are 25kV (Fr & Eastern EU), 15kV (De), and 3kV (Es & It). France, I think, still has some of the older 1.5kV standard. Not sure what Scandinavian countries use but I think it's 15kV and or possibly 25kV (so Finnish railways are likely to be 15kV or 25kV but I'm guessing here).

Therefore, the max/min ratio of voltages in use is 25/1.5 or 16.67 : 1. This is very significant, for what engineering calls for at 1.5kV DC is very different to that at 25kV. AC Both 1.5 and 3kV will nearly always use a single overhead line with earth/rail return. The higher voltages will more likely use two lines (this is likely both an enginering matter and a safety requirement). So this would mostly account for the differences you mention. Note: 1.5 and 3kV are DC and the 15 & 25kV are AC.

(Note: trams and small/light rail often use 750VDC.)

The mixture of different voltages in use across Europe and the fact that they have this high ratio of 16.6 : 1 has called for some innovative solutions. For instance, there are classes of locomotives that now use all three common voltages (3kV, 15kV & 25kV) and they automatically switch between them when moving from one county to another.

However, I do recall a method that was apparently 'much' more efficient (at least from a theoretical perspective). That said, it was still inefficient in absolute terms.

Some decades ago I read a report that someone who lived very nearby to a Navy high power (~1MW) low-frequency RF transmitter used to communicate with submarines stole power by building a suitable antenna to pick up RF energy broadcast from the transmitter (I think he put the antenna coil in his roof but that may just be hearsay).

Note: I used 'low frequency' twice here so I must clarify, the RF TX was indeed very low frequency by radio standards (needed to communicate through seawater) but it was still many times higher than mains frequency. Assuming the transmitter's frequency was somewhere between say 12kHz and 35kHz then it would be between ~200 and 500 times higher than mains frequency, thus coupling the energy would be that much easier. This combined with both an antenna especially built to radiate as well as having a proper ground counterpoise/earthing system would have provided the transmitting station's neighbor with a much better opportunity to steal power.

The report was so long ago I've forgotten most of the pertinent details but it seems that the guy got caught because they discovered an unexpected notch in the antenna's radiation pattern.

If I recall this happened in Florida sometime in the 1960s. Seems the guy powered all his fluorescent lights from the RF.

Would someone with a much better recollection of the facts please clean up the details. Thanks.

We argued about the wisdom of theft, when the electricity company would actually re-connect you to an illegally occupied house anyway.

Why compound your problems with the law?

It made not one iota of difference to the disk's rotation speed as the poles of the meter's magnet are so close to the disk that there was no way to alter the strength of the magnetic field that was at right angles on to the disk.

Electricity utilities have been in the game a long time, they already know all the tricks and design their systems accordingly.

This was a proof of concept experiment after I had had a discussion about a decade ago with a friend of my then boss who in his younger years actually designed electricity meters - he was also president of my local IEEE chapter so he was highly regarded (unfortunately he's now dead).

Anyway, he challenged me to alter the metering rate on one of my meters (I've four 3-phase + off-peak) without actually tampering with the meter or breaking the lead seals. He said I'd have Buckley's chance and he was right.

He said they not only go to inordinate lengths in the design of meters to ensure that it's not possible to stop them but also that they don't change their rate (run slightly slow, etc.) when heated, cooled, exposed to magnetic fields etc., and that before a meter is approved it's tested by certifying authorities for strict compliance.

Mind you, some meter readers don't seem to give a damn. We rented some short-term space off site to the main factory (which itself was an old factory) and some of its meters no longer even have glass covering the dials and others had their lead seals missing altogether. Anyway whenever the meter reader turned up he was never the slightest bit concerned. The boss said for no one to fiddle, for as sure as eggs, someone will forget to remove the 'tweak' before the next reading and the consequences of that would be much more trouble/far outweigh any potential savings gained on the account. :-)

> With 7140 loops of wire required and a length per loop of 20 m, the farmer would need 143 km of wire! At $0.15/ft = $0.50/m, this wire would cost $71,400! This means that even at the maximum energy consumption rate, the payback period would be at least 1000 years!

Also, what if you stick a magnetic metal in the coil to amplify the flux?

Someone needs to do this for today's kids and harvesting 'water' from the air.

Some years ago, while visiting my uncle who still lives on the same farm, I sought and found both the poles and the old threshing motor. The motor was enormous and mounted on a sled though rated for only 9-10 kW. It is not only semiconductors that keep getting smaller!

I think the main way is wires around the meter. How cool were these dudes stealing electricity from Edison, I liked Edison's firewall -

(1886) - "It was hardly worth while to maintain the continuous espionage necessary to detect and punish these pilferers, but the superintendent of the station, Mr. Chamberlain coupled in extra dynamos, and threw as great an increase of current over the system as the safety catches would permit, at various times for about one second while this current was passing; the incandescent lamps would give an unwanted glow, and every induction coil and motor surreptitiously attached to the system would receive an extra current designed to burn it. In this manner the system to occasionally cleared of all trespassers."

https://trove.nla.gov.au/newspaper/article/28360460

See https://en.wikipedia.org/wiki/Time-domain_reflectometer

Now I wonder if you could take advantage of the increase in copper price over the time, you might come out ahead

:-)

Why use such low gauge wire if you are only harvesting 100mA?

And as mentioned, the rogowski coil is neat.