Yes, I have studied homopolar "dynamos" for a while.
Ampere built one, Faraday built one, I think most "pioneers"
in the old days built versions of it. They never noticed any
over unity effects, but they were intrigued by the fact that
such generators also work when you attach the magnet to
the metal disc and spin the entire thing, and that they did
not work when the disc was held stationary and the magnets
were spun. They thought it might be an asymmetrical effect.
Turned out not to be of course, but that was one of the things
that fascinated quite a number of the old guys. Even Einstein
in his earlier years thought that is was an anomalous dynamo.
Tesla indeed did a little work on them although as far as I know
not much at all, and the coolest thing he came up with was to
carve special "conductive paths" in the disc surface which he claimed
allowed the electrical output to flow more easily, thus producing
a slightly higher output.
You may want to look up the Indian mr Tewari and his homopolar
generator which, it is claimed, he demostrated as slightly over unity
at a conference (in Germany?) about a century ago.
And then of course you may want to look up the late Bruce dePalma
who studied homopolar generators at MIT and designed and tested
several what he called "N-machines", of which he was convinced
they can produce over unity. As far as I know he was never able to
succesfully prove that before he passed away.
Some have speculated that perhaps if Bruce had been able to build
a superconductive N-machine, he might have been able to prove it.
Most of the variations are only minor variations on the theme.
Tewari for example used electromagnets to generate the magnetic
field, and an iron rotor. DePalma at some point was looking at a
version, which I believe Faraday had also designed a version of,
that used a mercury pool instead of brush contacts at the discs rim,
so there would be minimal physical friction and good conduction.
Some have designed and built versions consisting of multiple zones
of alternating magnetic field and multiple seperated "spokes" on the wheel,
so that the different "spokes" generated alternating currents while
the wheel spun. Some have claimed these no longer qualify as homopolar
dynamos because the magnetic field is not unidirectional, but I disagree.
The basic function is still the same, it still depends on movement of a
conductor through a magnetic field that is perpendicular to the direction
of motion, and that still generates a current perpendicular to both.
It is in fact a variation of the "Hall effect", and also works if the conductor
is stationary but the electron "gas" inside the conductor is moving.
A variation that I seem to recall Bruce dePalma did come up with,
but one that I have not seen in the work of most others, is one where
we don't use a disc but rather a pipe. Imagine we take a copper pipe.
We also take two pipes of the same length but one slightly wider
and the other slightly less wide, so that the copper pipe fits neatly inside
the one and over the other. These second pipes must be made of magnetic
material, so ideally we'd have two permanent magnets shaped like a cylinder.
Crucial here is that these magnetic cylinders must be magnetised radially,
in the sense that the North pole is located on the outer surface of the cylinders
and the South pole on the inner surface.
Now we attach two copper end caps to the copper pipe, with an axle fixed to them.
If we now spin the entire cylindrical setup around its central axis,
a DC current will run through the pipe from the top to the bottom or vice versa
depending on the magnetic field polarisation and the direction of rotation.
And obviously the end caps can now be connected to a wire, and DC will run through
the wire.
That is a cylindrical version of the "homopolar disc generator", and it works.
But as far as I know nobody has been able to actually get OU from such a dynamo.
And if anyone did manage, they certainly are keeping it a secret.
