Effective resistance of inductor In a lab experiment, we connected a simple circuit: an AC voltage source, connected (in series) to a variable resistor and an inductor. We measured the current in the circuit, and the voltage that falls over the inductor.
We calculated the phase difference between the voltages and used it to calculate $V_L$, and used it to calculate $R_L$, the effective resistance of the inductor.
We got that $R_L(I)$ rises up to a maxima, and then decreases, but we couldn't understand why - as we understand, $R_L=2\pi fL$, so it should be constant...
What did we not understand?
 A: There are a couple of nonlinear magnetic material effects that might be at play here, although this answer must be described as speculation without more detail.  Both effects are more pronounced if your inductor is ungapped.
1) At very low current levels (corresponding to very low levels of magnetic field H), the inductance can be lower than nominal.  (The B-H characteristic of the magnetic core material has a lower slope right at the origin.)  As current increases from these low levels, the calculated inductor impedance $Z_L =2 \pi f L$ would increase and then stabilize at the nominal value.
2) As current continues to increase, eventually the inductor starts to saturate. (The B-H characteristic flattens at high fields.)  Inductance then becomes a decreasing function of current, so the calculated inductor impedance would decrease.
You can see some B-H curves illustrating these effects in the wikipedia article on "Saturation (magnetic)".
Introducing a gap in the magnetic core reduces the component's inductance but stabilizes it against these effects.  
