Is magnetic hysteresis present even with a DC bias?

Question

Consider the following circuit:

With Vin(t) = A sin(ωt) for some given frequency ω.

The relationship between Vin and Vout is certainly non-linear, because of the effect of hysteresis in L's magnetic core.

My question: is hysteresis non-linearity present even if Vin(t) = A sin(ωt) + B, where B is greater (possibly much greater) than A?

My intuition is that since the polarity of the current through L does not change, then there is no hysteresis. An alternative intuition tells me that maybe it is the sign of the derivative of the current what counts; i.e., the current rises through one curve, and lowers through a different curve (different sections of the magnetization curve), thus causing non-linearity. However, I'm not sure that the math checks out with this alternative intuition, in that the output (the value of the y-axis) would diverge if you repeatedly go down at a low slope and then go back up at a higher slope.

Answer 1

By adding a DC bias, you are just shifting the hysteresis loop to another operating point. You are not eliminating it. The smaller the AC component of the current, the smaller the hysteresis loop and the related non-linear effects of the core.

The linear relationship of \$B = \mu H\$ is only an approximation in this case.

_{Source: Magnetic Hysteresis}