Firstly I would like to bury this idea that coil sides at right angles to the motion generate the voltage and the end connections contribute nothing but resistance.

The basic equation for an emf induced in a wire moving at constant speed though a uniform field is E=Blv. Where B is the field strength, l is the length of the wire and v is the velocity. This a very special and almost useless case, you have to complete the circuit and in most real cases the return wire has an equal and opposite voltage induced in it so it doesn't work.

Although I have never thought of this before, it must be the basis of the homopolar generator. The simplest form is the Faraday disc machine and it needs a sliding contact to overcome the return connector problem. the same problem dogs all homopolar dc machines.

For the alternators that we think of it is much safer to consider the voltage to be proportional to the rate of change of flux. You are looking at flux linkage through a coil and all the conductor is actually necessary to complete the circuit and carry the current.

Looking at things at a very basic level the shape of the coil doesn't matter as long as all the flux is linked, in other words the hole is bigger than the magnet.

Sorting this lot out into a practical machine does cause a lot of confusion as there are so many variables. Let us forget any form of slotted iron core as that changes things far too much. If we stick to the air gap alternator we run into trouble as soon as we add turns side by side as the flux linkage then becomes progressive and we have a distributed winding which changes the waveform and alters the ratios between peak, rms and mean voltage. We also run into more trouble unless the coil leg width is equal to the space between magnets or less than it. If more then there is cancellation mixed up with the distribution and the emf is reduced.

Now add the complication that most of the methods we use doesn't put both coil legs in the magnet gap at the same time and you can see that changing any one factor will mess up the answers.

For a given shape and number of magnets there will be a geometry that gives the best compromise between voltage and circuit resistance and that will give most power at a given speed. If we fix one constraint such as disc diameter then we may benefit by changing the number, size or shape of the magnets while still maintaining the same magnet area.

If you take Hugh's machine with a given disc size with the magnets one way round you may reduce the output by turning the magnets round. The actual result will depend a lot on the type of winding. The 4/3 single layer winding is a totally odd thing and has no real commercial counterpart, it just works well , is easy to wind and suits the axial construction where overlapped coils are tricky except for very thin stators. It works best when the gap between magnets is about equal to the magnet width.

The conventional full pitched fully wound three phase winding works best when the gap is about half magnet width so that the third ( inactive) phase falls within the gap.

I am sure someone can do a computer programme to analyse all these variables but it is beyond me to do it. Working on voltage alone is completely useless, it has to also include coil resistance to find the optimum power out for each case.

I hope this helps, but it is not simple.

I have just come inside for a coffee after doing some tests myself on a new axial I'm playing with 24mags 50mmx10mm round mags. N45 17mm airgap 14" disk.
 I wound 1 coil as a wedge at 105turns.... 6.8v 100rpm center diam about 52mm Coil weight was 193 grams    15.4 turns per volt or 28grams of copper/volt ..1mm wire

 I wound 1 coil as a circle 52mm diameter  105 turns 7.0v 100rpm weight 157 grams 15turns/volt 24grams/volt ..1mm wire

 I wound 1 coil as a circle 44mm diam inner (smaller than the mags) 105 turns 5.9v 100rpm weight 130grams 17.8 turns/volt but 22grams per volt.  1mm wire

All with 1mm wire as a test coil.

It seems from this that although the straight wire of the wedge cuts the flux at right angles,compared to the circular coil, it's output was on a par with the circular one of the same inner diam.

 The smaller holed circle was lower in output as far as turns per volt were concerned, but was streets ahead of the wedge and slightly better than the correct (50mm) sized circle in terms of resistance per volt (grams of copper per volt) as the circle length was obviously shorter.

So at this stage of testing, a slightly undersized circular coil gives the best bounce per ounce of copper and so gives a lower resistance stator for the same volts out..

The real coils will be 1.8mm wire... so I have wound one of the smaller diam (44mm inner hole) with 110T  1.8mm wire.... voltage is 7.1 at 100rpm. 542 grams. I shall now uncoil it and wind the same piece of wire into a 50mm circle hole coil, test that, and  then a wedge to suit the 50mm magnets.

In other words whats the best shape and size for max Volts for 542 Grams of 1.8mm copper wire

What has all this got to do with dead copper??

I feel it is volts per gram of a given size wire that counts towards the lowest resistance stator. So between 22grams/volt and 28grams/volt there must be dead copper wire of differing amounts.

Doesn't look too flash for the wedge at this stage.

Perhaps it would be better not to use the term cancellation at all once we progress to coils with multiple turns. The effect of cancellation is much the same as not linking all the turns.

If you make the hole smaller than the magnet you can't link all the turns at the same time. This doesn't really matter as you will progressively manage to link each turn at some point and the effect is the same as distribution of a coil by not having all its turns at one point. It will almost certainly still continue to increase volts even if you wind it full with no hole at all. it again will change the waveform and most likely up to a point it will make things more of a sine wave, which at least makes the other equations apply more accurately.

Also the centre of the coil has short turns and contributes far less resistance so there is no harm in winding smaller. I see no reason to carry this to the radial dimension ( for round magnets it would probably be better to make coils elliptical)  but then you still have magnet shape and spacing to mess things up.

Not only do you have to worry about the hole in relation to magnet size, the position of the mean line of the coil in relation to the pole pitch also has an effect. All these 4/3 type windings are short pitched compared with a conventional winding with coil mean line spaced on the pole pitch.Flux leaves on magnet and returns via another, the total flux linking the coil is what matters.

If what you find experimentally works best then that is the way to go. Too much maths here sometimes. Fine when it works but I don't like it when it doesn't agree with experiment.