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This description of Tesla's "magnifying transmitter", which supposedly used electrical resonance to transmit energy (similar to resonant inductive coupling?) states that the coils (or at least part of them) are submerged in liquid air, which "causes an extraordinary magnification of oscillation in the resonating circuits".

What effect would submerging a coil in liquid air actually have? Does it just decrease resistance of the coils? I know with resonant inductive coupling, efficiency is improved with higher Q factor of the RLC circuit, and Q is improved by reducing the resistive component, so would this make some sense?

Does it have any other effects? Electrical conductivity, magnetic permeability, etc?

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If that's an actual Tesla quote, it sounds like he's talking about the high output voltage during silent operation when no plasma streamers are present.

In that case the max voltage limit of a well-designed Tesla coil is determined by the maximum power limit on the power supply (volts and max current of wall outlet, breaker panel, etc.) Since radio wave losses are insignificant, all of the operating power goes into heating the spark gap, conductors, and the capacitor dielectric. When no spark streamers are present, the voltage rises high, and the entire TC becomes very hot!

Copper resistivity drops by about 10X at LN2 temperature. If input power is adjusted to be constant, then a conductivity drop of 10X gives a resonator current increase of SQRT(10) ~= 3 times. For a simple linear RLC resonator the capacitor voltage is proportional to the inductor current, so liquid nitrogen cooling should increase the main TC output voltage by about 3X. (This is assuming that copper losses were dominant. If spark gap and capacitor dielectric loss was significant, then LN2 would give less than 3X voltage improvement.)

And if spark streamers break out, this math doesn't work. If the majority of the available power was going into the plasma, then reducing the copper losses wouldn't increase the spark length by much.

http://www.copper.org/resources/properties/cryogenic/homepage.html

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Liquid oxygen is paramagnetic. A Tesla coil is basically an air-core, high-frequency transformer. Immersing the coil in liquid air would be like putting a core in the transformer with a permeability higher than non-liquified air.

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That description mentions a "Magnetic Core" already, so it's not an air core. Even if it were, the permeability of liquid air appears to be 1.00287, vs 1.00000037 for gaseous air. Not much of a difference compared to iron. –  endolith May 16 '11 at 21:40
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