Why isnt the flux in phase with the current in an AC analysis and/or why is the inductance complex-valued?
Eddy currents and hysteresis make the flux lag the current and make the inductance complex-valued. The imaginary part of the resulting inductance is associated with eddy current and hysteresis losses. Consider an inductance, L, that is complex-valued. The inductance could be decomposed as: L = Lr – j Li where Lr and Li are the real and imaginary components of L respectively. Now, consider the impedance that is implied: Z = j ω L, where ω represents the frequency in radians per second. Expanding this out, we get: Z= j ω Lr + ω Li The imaginary part of the inductance contributes a real part to the impedance, associated with losses. You could interpret the (ω Li) term as the frequency-dependent part of the circuit’s resistance. The real part of the inductance the forms the imaginary part of the impedance, associated with inductive energy storage.
