Great ! Good for you!
The maths is excitingly complex but what it means is that for 50Hz AC power, the current flows in the edge of the wire rather than the middle. For Copper at 50Hz the thickness of the skin is about 8mm. (For iron it's about 1mm) Therefore once you exceed a Copper wire 16mm in diameter then you just cannot get more current down it by making the wire fatter. This limits the amount of current you can deliver and it means that you can't have lots of high current draw (ie acceleration) in an AC system.
Hah! You're right that the maths are interesting - but you won't produce a black-and-white result of "you just cannot get more current". You do. But the conductivity is progressively degraded, it doesn't hit a brick (copper?) wall. In power distribution systems the solution is to run parallel lines a few cms apart. In rail systems, we can take advantage of some prior knowledge about how the load is distributed across the electrical sections (eg max 6 trains or 20MW or whatever per section) and provide separate feeders and transformer to each. This is exactly what we do. With additional power cables running along the track to the next "section".
DC on the other hand doesn't have the same issues.
Er, Doesn't have the same ISSUE singular. Most of the problems of leakage, losses, flashover and conductor impedances are very very comparable. Switching has to be different, and corrosion is VERY different (and very significant - but you didn't raise that factor).
The current capacity of a wire/rail is depandant on it's size. The current will flow more or less evenly throughout the entire rail. There are also far fewer issues to do with the metal used in the 3rd rail - here you can use iron. This is important as it's much harder and can take more frequent wear than Copper wire can.
Now I'm really enganged! I think we've just begun to confuse Overhead vs 3rd rail with AC vs DC. Third rail and its power delivery connections are going to corrode in a completely different way to overhead DC of the same voltage. Just as copper and iron corrode differently. And sure, you can put up with a few mms loss on a big fat rail more than an o/h conductor, and change the pickup on either rail quite quickly and cheaply, BUT when you compare AC vs DC you are introducing 5 factors: Direct vs Alternating, conductor material, friction with vehicle pick-ups, voltage, and where the power is "transformed" (quote marks 'cos some traction converts on way and some the other and I think some transforms then chops it back again). But . . .
The BIG reason for switching from DC to AC in "the old days" was voltage. You can transmit the same power as a DC network over MUCH smaller and lighter cable with AC and then transform it down to the required voltage where its needed. And this is exactly what the rail network covering an entire coutry needs - long distance power distribution.
You just CANNOT distribute DC over large distances.
Providing the power supply (ie substations) can handle it, the DC 3rd rail can deliver more peak acceleration. That's why in the past metro systems have used 3rd rail systems rather than OHLE.
But only if the power reaches them, as high voltage AC.
In fact I can't see that this conclusion follows from your arguments!
The Metro systems are local - the need to distribute power is minimal.
The "peak accelleration" is simply the required current, and the designed voltage, with minimal losses introduced to the traction system. If you manage to get the designed current to the vehicle, it should perform correctly. If you can't, then its because the local network has too many local losses, either in the infrastructure as you've suggested, or in user demand (other trains and water on the conductors etc), all of which can be, AND SHOULD BE, designed out of the supply system.
Of course, modern control and electronics can overcome many issues (but not the skin effect!) and now other issues such as what's already there and nearby predominate.
Agreed. But I don't agree that the distribution "problems" should be passed on to the designers of train electrics. I maintain tht the most effective power distribution system is AC (not necc. 50Hz but it works ok) and high voltage. I also maintain that the delivery to the rail network should be very localised (eg independant drops to the local voltage at less than 10kms in dense urban networks and much more in high speed/intercity sections) but the final drop from distribution network to train? It shouldn't matter - its got to be safe, reliable and maintainable, losses must be accepted if they're a requirement of a safe network and a maintainable network.
Both AC and DC seem to have reasonably comparable records in safety (though I am not adequately informed or qualified to assert this). Maintenance of DC includes the corrosion issue. Maintenence of either includes a transformer somewhere or other. On a stanchion or on a train? Is it THAT a big factor?
In conclusion - the "skin effect" is a challenge easily surmountable and one which international electrical power distribution networks have dealt with quite adequately.
(Now, shall we argue next about nuclear-powered trains, or bio-gas turbines or transporting coal by electric trains powered by coal? Tee Hee)[
) Its when you put one foot down and ground yourself thats going to cause you problems.
