Thank you for reading this. the circuit is simply a SinWave voltage source (ideal) connected to a single coil with an closed iron core inside. It would like the picture below while ignore the labels: Only consider saturation and hysteresis of the core and assuming no resistance,no eddie current, no core loss or leakage, nothing else. the relation between the voltage of the source: e, the current in the coils I, and the flux ø, will look like this from ALL textbooks: SO many text books just say:" Farady said v = -dø/dt, so if v is sin(something) then ø is cos(something)". I totally disagree, that's the induced voltage from a driving flux. Flux has nothing to do with voltage. flux is the result of current: N*i. However I am also aware of the fact that i won't be sinusoidal, as the inductance of the coil will be none constant, but I can not explain why ø is not affected in shape at all.. Now, when I look at the Maxwell's 4th equation the E inside Maxwell's 4th equation in this case means the E inside of the wires. and H it generated around the winding. that does imply a relationship of V_magnetizing = d( flux_result ) / dt... I thought 4th equation does not apply here, but apparently it does apply inside the winding wires. but... what about the current side of the story, the current flowing in the wires, current will generate flux as well. did I apply maxwell's 4th equation right?, if I did, what about the flux generated by the currrent? Thank you.