I don't think I quite got the right understanding of what  you were trying to say with this part:
"Notice: This is a single conductive wire, not a coil. The length of that conductor in the above equation is what is of prime importance. This means that the core (and rotor) must be "wide" to optimize the length of the conductor and thus the output. Also the equation states the "active length" of the conductor. I interpreted this to mean that the magnets must be long or as long as the conductor in order for the total conductive surface area of the wire to be active.

Interpretation; A wide core is a requirement. The magnets are as long as the wire, the wider the core the longer the magnet."

By "active length", it seems to me we are referring to the part of the coil that the magnet passes over at 90 degrees, thus the tendency to make oval shaped coils to mimize the "wasted" copper at the short ends of the coil where the flux is not cutting at 90 degrees.  With such an elongated coil, I would think it prudent to have the dimension of the magnet that is parallel to the long side of the coils be approx the same length as the coils, minus the end parts of the coil.  Perhaps this is what you mean by the "core".  I take this all to mean that the radial length of the coil (i.e. the dimension measured parallel to the radius of the stator) should be as large as practical, but may be limited by the length of the magnets available.  I think it also means that a round coil is the "least" efficient shape for a coil.

As for the diameter of the stator, yes, I think it is interesting to note that even without increasing the number of magnets, you will get more voltage out of a larger diameter genny.  This property could be very useful for those working on VAWT gennys.  Probably a lot cheaper to double the diameter of the genny than to go to a dual rotor design with twice the magnets.

Thanks again or bringing science back into the picture.  I think we all benefit from doing that now and then.

Regarding "round is bad", I meant truly round bad, elongated good.

Ed, Thanks for posting, as usual awesome work!! I remember a long time ago we where taught that bending a wire at 90 deg ceases emf oscillations for interference applications (transmission lines)? But if this statement was true then how could a square transformer produce maximum power output work? I guess I was trying to state that, I have tested square elongated coils without any issues, so not sure what your saying about angles on coils, care to elaborate?

On the other hand, diameter is important. for two reasons, first you require less wind speed for the same rpm's. If the magnets are moving faster across the surface of the coil due to more magnets and larger diameter then output power will increase.

A friend of mine just built a 5 kw unit using square (flat wire) coils and 18 inch disks. The power of this unit is hideous, you can arc weld with it. Short the phases and there is no way in hell that it moves. The second reason was as mentioned, more space to play.

The way I work it out is to start with the fact that the average voltage induced in each coil is determined by the average rate of flux change through the coil.  If the coil can encompass all the flux from each magnet as it passes and you know the total area of the magnets and the flux density then you know all that you need to know.

Average voltage will be 2 times the total flux times the revs per second.  2 times, because the coils get the flux in both directions.  Or another way to explain that 2 is to say that a coil has 2 sides.

There are also some zones of lesser flux between the one pole and the next as you move around the circle.  the outer turns of the coils can pick up some of this extra flux in some cases - in others it tends to cancel because the turns are too large or too small to benefit from them.

The highest voltage will be induced in turns that are exactly as wide as the distance between magnet centres and as long as possible beyond the magnets so as to pick up stray flux at the ends.  The ideal coil would be one in which every turn was this shape, but in reality we are also governed by the resistance of the wire.  We could put a huge amont of very slender wires in and get any voltage we like - and you can do this just as well in an axial flux machine as in a converted motor, by the way.  I have produced hundreds of volts (where I wanted to) with axial flux machines.

So all you really need to know is the size, and number of the magnets, their flux density in this configuration, and the speed of rotation and you can work out the average voltage produced in one turn of wire.  The peak is usually about 1.56 times the mean.  REctifiers start to conduct and the machine cuts in when the peak voltage is far enough above battery voltage to overcome the doide drop.  So you can choose the cut in speed based on the number of turns in each coil and the number of coils in series and star etc..

The shape of the coil is something you can agonise over but it will not make a big difference.  The ideal coil would have straight radial legs (for an axial machine) or axial legs (for a radial machine).  However this is not the optimum path for getting the most voltage for the lowest resistance in the real world.  There are a lot of compromises to be made to get that optimum, but none of them make a lot of difference compared to the strength, number and size of the magnets in the rotor.  In radial machines with corse the optimum shape is often very noisy and so the coil legs are skewed to reduce cogging vibration.  This will result in some loss of performance but you need to do it for practical reasons.