Code is here https://github.com/mohsen1/axial-flux-motor-explainer

Feel free to steal! This was one shot with Claude Code. You can take it and adopt it to your need

Yeah so the relationship between speed, power, frequency, size (both in the direction of primary flux excitation and in the direction orthogonal to both that and the movement), and torque at nominal values of current density (for a given conductor losses are proportional to the square or this value and to the total mass of that conductor in the machine; that's independent of any of the other scaling parameters; note this is absolute power not percentage) and peak flux limitations (core saturation, permanent magnet demagnetization), are sadly not trivial if you express them in a way that is even just _valid_ for the modern days where we can support electrical frequencies up to around a megahertz at scales up to around 100 kW, and even harder when you remember that core material has severe frequency dependence of it's limits.

E.g. for example for a given electrical frequency and decent radial flux synchronous machine, power density is quite static and torque density can actually be dialed quite freely from 2-pole machine (turboset in gas turbine running on the grid at 3600 rpm (or 3000 rpm outside NA and some Pacific Islands) to 40(+) (example deployed at Hoover dam, 180 rpm). At those higher pole counts, the center of the rotor is no longer electromagnetically active, because the magnetic field lines keep to a narrow ring only about as thick as each pole is wide. Unfortunately it's mechanically not that trivial to handle a cylindrical shell with a small air gap (this needs to be significantly smaller (about at least 10x) than the pole width) when using substantial torque and speed.

Circumferential velocity is practically limited by hoop strength of whatever the outer region of the rotor is made of, even if it's all very nicely balanced, because eventually the magnetic armature flux source (wires or magnets) will fly out.

Higher electrical frequencies limit the field winding core's magnetic permeability (magnetic field/force strength amplification relative to vacuum, for same electrical current) which hurts efficiency by dropping the useful mechanical power component of field voltage while the voltage resulting from the current (that needs to happen to cause the magnetic field in the direction of movement that causes the mechanical force) due to wiring resistance stays. (I think the permeability gives the ratio between voltage and current for otherwise identical mechanical load conditions and winding shape?)

Thinner wires have less fill factor because the insulation has to stay the same thickness as per-winding voltage stays, but magnetically inactive terminations are less wasteful (for losses and mass) when a decent number of effective turns (>>1, think >10~50 for most of the benefits) are used.

Note while the armature necessarily has an even number of poles in it's construction (north/south), the field is not forced to that.

Indeed, the iirc most smooth torque (under practical mechanical feasibility limitations and without undue sacrifice of efficiency) results from having a prime number (of field windings, in WYE-style connection) exactly one off from the armature pole count. Note that for low losses all these torque-smoothing techniques _require_ only a single electrically directly driven winding in each slot (per mechanical field pole) and with that only GCD(field_slots, (armature_poles / 2)) windings get to share an electrical half-bridge (one single wire going to a single voltage-output terminal on the electronics board; note mainstream BLDCs have 3 of these, classic fridge compressors have 2, and modern stepper motors (e.g. 3D printer) have 4).

Any time you have multiple windings driven by different electrical source voltages you're wasting heat in the winding because the lowest-loss would require all conductor in the slot to to perfectly evenly share current.

There's just one problem with that: you need a nearby slot with exactly opposite phase to even possibly use more than a single (half) turn of "winding" in the slot.

If the voltage is still enough to not loose too much in the connections, you can use transistors developed for efficiently powering modern computer chips from comfortable voltages like 12V, but even then a "winding" has to be much longer than an armature pole to mitigate the losses of spreading the return current sideways to where a slot carries the current in the reverse direction. Once the voltage at the transistor is over around 10V the benefits of more precise control of the field magnetization to the armature position (and how the shapes distort the field lines from anything that would look like a sine wave) could be useful. In theory that'd also provide direct access to electronically control the air gap (well, net force normal to the air gap "surface") which _could_ be an alternative to mechanical bearings for very thin-shell constructions. See maglev trains for a pretty practical application of using an electric motor to also levitate the "rotor" in a place where a mechanical bearing ("train wheels + bogies") performs poorly.

https://archive.org/details/TWTK_WIN