Hard magnets have low permeability over almost the entire B-H curve; you'll do little better than air cored, with the exception of excessive eddy current losses if a fairly conductive material is used (e.g., SmCo, NdFeB).
After a whole lot of magnetization (NdFeB requires >10^5 A/m, which is >1T in free space, to change state!), you'll change the magnet's field from axial to tangential. Also, as the magnetization slips, a large amount of flux will suddenly flow, which is to say, if a constant current is applied, the voltage will shoot up suddenly; or if a constant voltage is applied, rate of current rise dI/dt will slow down considerably as the magnetic domains flip, one by one.
The nonlinearity of the core will look like this:
- Small signals: very low permeability (around 1), little hysteresis loss. The winding *might* make a good inductor (in the sense of, the impedance is mostly inductive reactance), but it's little better than air.
- Medium signals, enough to exceed the coercive force: average permeability is greater than unity, most of which is hysteresis loss. In other words, the tan delta (loss angle) of the impedance is large, maybe, I don't know,
50% (electrolytic capacitors are much better!).
- Large signals, up to and past saturation: losses remain fairly constant (you can't push more hysteresis than the biggest loop on the B-H curve), inductance goes back down (because in saturation, it has about the same permeability as in any other state).
This may be somewhat familiar from plain old laminated iron cores, which exhibit low initial permeability (maybe ~800 or so), excellent operating permeability (5k, up to 20-40k for annealed, grain-oriented toroids), and less and less going into saturation. The difference is, a hard magnet's variation is severely exaggerated, making it useless for inductor or transformer duty.
Tim