Now to drive deeper in the matter, thicker wires have larger cross-sectional area facilitating better and more efficient flow of electrons (Some say, it is due the larger surface area, but it is not correct as electrons electrons do not travel along the surface of a conductor, it travels through the whole conductor.) resulting in less resistance thus, producing better magnetic field strength in the stator. Also larger surface area helps in heat dissipation.
AFAIK the power generation depends more on the amount of copper in the winding versus the iron in the stator body. In case of a thin wires, the only way to increase the amount of copper (in order to generate more power) is to wind more of it, but a thicker wire does not need to be winded more to increase the amount of copper. While higher amount of copper leads to higher power generation, it is possible to saturate the iron core of the stator, so there is a limit on how much copper a stator body can take. After the limit is crossed winding more would not yield more power, in fact it would lower it due to higher parasitic loss (more weight means more drag).
There are some problems with using thicker wires too, first of all, winding thicker wires require sophisticated machinery and it is not as efficient as winding thinner wires. That's why you see manufacturers using thinner wires, it matters on large scale operations but in case of rewinding stators where one does not need to pump out thousands of stator coils in a day, it does not matter. For the record, the stator coil in the pictures I posted earlier was machine winded. Thicker wires create some unfilled space against flat surfaces, also the winding becomes somewhat oval shaped and it wastes some space. It is also easier to wind thin wires than thicker ones and there are some gaps in the stator core where only thin wires would fit perfectly.
I'm not an electrical engineer and whatever I say should be taken with a grain of salt.
And yes, it is a single piece of wire which has been winded around all the poles.







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