John in CR said:
At what electrical rpm do we need to start worrying about skin effect on our phase wires?
Math to the rescue!
Let's say we're pretty conservative and say you want your wire radius r to equal skin depth δ, so that current density in the very center of your wire is still ~.37 (1/e) what it is on the outside of the wire. It's more or less an approximation for at what point skin effect starts to manifest itself; below that point the difference between any AC and DC resistance due to skin effect would be immeasurably small.
In my case I use 10AWG stranded THHN from the hardware store for my phases, which we can approximate as a solid wire since the strands aren't insulated from each other. Diameter D of that conductor is ~2.6mm, so let r=δ=1.3e-3m
δ is approximated by sqrt(2*ro/(omega*µ)), where ro is the resistivity of the conductor, omega the angular frequency, and µ the magnetic permeability. For copper, ro=1.68e-8Ω*m and µ=1.25e-6H/m. Solving for frequency, omega=2*ro/(µδ^2). Omega is just 2π*f, so f=ro/(πµδ^2)
Plugging in numbers, f=(1.68e-8)/(pi*1.25e-6*(1.3e-3)^2), f=2.57e-3=2.57KHz, or about 150,000 electrical RPM - 150k! You would see a 10% increase in resistance at about 5.4khz (~325k ERPM). For a point of reference, a 5300 (12 poles) in a 20" wheel at 50mph will run at 10k ERPM. If you put a BMC in the same wheel and the same speed, you're looking at ~67k ERPM. Hell, you have to pay extra for a special firmware to get a Kelly controller to handle that.
So at full speed, there is no reasonable hub motor setup that could possibly make us worry about skin effect in our phase wires, at least from commutation from one phase to the next at cruising speed.
One possibility, though, might be at low speeds under high-frequency PWM. I don't know how most other controllers switch, but Kellys' FET choice and driver circuits seem to be optimized for fast switching (as opposed to low Rds), and they claim a 16KHz operating frequency (and imply it has something to do with higher efficiency). Maybe at that frequency it makes a difference - in 1 meter of 10AWG at 100 amps and 16.6KHz, the increase in effective resistance from skin effect would cost you ~19w over the length of that wire, which if you think about it is a lot of heat in a wire, but not even worth thinking about when you're throwing 100a into a hub for a few seconds at a time. But I don't know enough about controllers and switching and PWM to know if "operating frequency" necessarily has anything to do with the frequency of current in the phase wires.
