I have some really powerful neodymium magnets lying around, as well as some 30 AWG copper magnet wire. I needed some magnets for cabinets and mounting things, so I decided to get a few extra.

Obviously you could wind a wire many times in the same direction around a ferrite core (a bolt or something) to get a decent inductance. But what if you used a neodymium magnet as the core? What would it's inductance look like? How would it behave in pure DC, PWMed, and AC circuits? And what different ways could you wind it to get the maximum inductance or achieve other effects? I understand the basics of inductors in circuits (inductive reactance, series/parralel, phase shift, LC/resonance etc.) but do not know too much about how different cores affect them.

I even have a few of these. I imaging with such strong magnetic fields you could get very large inductances.
https://www.kjmagnetics.com/proddetail.asp?prod=BY0X08-N52

 

The strong neodymium magnet will behave much like a air-core for the coil if it's already saturated. The permanent magnet flux will be there but you won't be able to induce a large amount of additional flux into the magnet (NdFeB material resists changing its magnetic state) from the coil. The total field you get is the sum of that from the permanent magnet and the slightly better than air-core coil.

A simple iron core would likely saturate at a much higher flux.

 

The video was helpful, but did not really talk about how it would behave in various circuits. So what inductances could I achieve, and would it be polarized in any way?

 

Can you find some iron/steel that has same diameter as magnet. Cut iron/steel to same weight/mass as magnet.

Frequency Sweep same coil with the two cores and measure coil voltage and current.

Which one has the largest phase difference?

 

The video was helpful, but did not really talk about how it would behave in various circuits. So what inductances could I achieve, and would it be polarized in any way?

The inductance would be about the same as the inductor with air for a core. Inductance is a measure of how much energy is stored in the induced magnetic field when flux is changed. With a iron core the flux change goes from zero to some number as current changes. With a neodymium magnet core you have the almost opposite property, it resists changes in its flux so you don't get the additional induced magnetic field and little additional inductance from the magnetic core. Magnets don't amplify magnets.

 

Obviously you could wind a wire many times in the same direction around a ferrite core (a bolt or something) to get a decent inductance. But what if you used a neodymium magnet as the core?

I can tell you what would happen: disappointment.

What would it's inductance look like? How would it behave in pure DC, PWMed, and AC circuits?

Read what @nsaspook wrote.

And what different ways could you wind it to get the maximum inductance or achieve other effects?

Winding pattern will have the same effect it has in any other inductor.

I even have a few of these. I imaging with such strong magnetic fields you could get very large inductances.
https://www.kjmagnetics.com/proddetail.asp?prod=BY0X08-N52

WRONG.

 

Can you find some iron/steel that has same diameter as magnet. Cut iron/steel to same weight/mass as magnet.

Frequency Sweep same coil with the two cores and measure coil voltage and current.

Which one has the largest phase difference?

I wish I could, but I do not have the equipment. Simulators can be notoriously inaccurate in these kinds of situations with so many variables and unwanted characteristics.

So basically it would be a complete waste and I should just use them for something else like a solenoid or motor?

 

What is the phase shift of a saturated core?

 

What is the phase shift of a saturated core?

The phase shift of the induced current won't change but the complex impedance of the device will as it become more resistive as self inductance drops.
Look up how to use a https://en.wikipedia.org/wiki/Transductor
https://ieeexplore.ieee.org/document/5239834/