A bunch of questions regarding radially magetized ring magnet.

[Magluvin]

Another thing is the difference in coil vs gap.  The first pic shows an overhung coil where only a portion of it is in the densest flux in the gap, and the second, which I prefer, is an underhung coil which all of the windings are in the flux at all times.  In the overhung, parts of the coil that go out of the flux when moving are wasting input power, where the unerhung is not.

So these are things to consider if you are using these rings. Been thinking on it all for some time and Im giving what I know of it to you.  Pushing the levels and limits on these things leads to more output.  If we took a toy motor apart and replaced the magnets with weaker ones, we would have the same or more current flow into the armature and less speed and torque. Now we put in stronger mags, more speed and torque. Then stronger mags, more speed and torque.  So gen or motor, the stronger the mags the better the performance.

Mags

[Magluvin]

Found a good example of how the dual sets of rings would work

Here in the pic JBL is doing it with 3 stacked strontium mags and the top plat has a pole gap and the bottom plate has a pole gap.  So we get rid of the 3 mags, and replace the top and bottom plates with neo ring mags, and replace the inner pole with the 2 smaller ring mags and center them with an inner core rod, and make an outer core tube for the outer dia of the large rings. So the bottom large and small ring have S outer dia and the top rings have N outer dia poles Now the 2 coils, just reverse polarity one coil and connect in series or parallel with each other.

Mags

[Magluvin]

Except my rendition would have taller mags with underhung coils

Mags

[gyulasun]

...
   .... a large ring and a smaller ring that fits the hole of the larger ring with space enough for the coil
to move within, both with same poles in and same poles out, N out S in for example.
...
   .... where the inner ring is attracted to the outer ring, and then the inner rod core and the outer
tube core to close the magnetic circuit. Even if there are loses in the circuit closing cores, the gap
between the magnets will still be denser than having the core carry it to the gap.
...

Hi Mags,

Yes, it is very possible the magnetic flux density would be higher between two radially magnetized
ring magnets when one of them is embedded into the other. I have made a drawing as you described,
and although I did not include the inner rod and the outer tube cores for closing the magnetic circuit,
it is a must to utilize all the flux available in the two rings AND increase magnetic field in the gap manyfold. 
By the way, the magnetic circuit between your mentioned rod and tube could be closed by both a soft steel
ring or by just a third radially magnetized ring magnet, both with the correct ID and OD sizes to fit to the rod
and to the tube.  When you use a third radial magnet between the rod and tube to close the magnetic
circuit of the first two rings, the field in the gap can increase even higher than with a soft iron ring.

...
  If we have 2 mags in attraction with some measuring device to detect the pull force, and then we add
a core to give the outer poles a shorter path to the other, will the pull force between the mags increase? 

Yes it will.  Just consider a magnetic door lock in which a simple ceramic block magnet is used with two
soft iron plates. The plates fully cover the side poles of the magnet (that is magnetized through its thickness)
and the magnetic circuit is closed between an outside third plate (to which the locking happens) and the two
iron plates that outreach the magnet.  The two soft iron plates 'collect' and focus most of the fields coming from
the sides of the magnet and make the attraction force to a third plate stronger, this is the same 'trick'
you can find in pot magnets and of course in speakers.

...
So if we add the outer return cores, and the force between the 2 mags increases substantially, can we only
attribute that to the smaller percentage of flux density possibly added by adding the cores to have even less
flowering at the gap?   I think its more of the shortening of the loop that increases field strength,
along with density, naturally.

I think the first and stronger factor responsible for a stronger field is the less flowering i.e. less possibility
for stray fields and the second factor is the length of the loop. For this latter factor I think the cross section
area and the permeability of the soft iron material used for closing the magnetic circuit do matter.

Gyula

[Magluvin]

Yeah thats the idea you drew there. The second set would be of opposite polarity. And yes the rod and outer tube could be magnets. Id rather they were magnets as I think there is too much leakage with iron core paths.   I said it before, I wonder if the core paths were made of neo but not magnetized, if they would work better than iron?

Mags