Hysteresis

H is proportional to I

As I becomes positive, B reaches a peak, as I becomes 0, B maintains some residual flux and keeps some magnetism, then I becomes negative bringing H to 0 and then as I increases in the negative direction, H peaks in the negative direction.

Energy

The area enclosed by the hysteresis loop is the energy lost in one cycle of excitation current. In some cases, we want to have a small hysteresis loop to minimize loss.

Loss

$P_{hysteresis}=K_{h}V{B}_{\max }^{n}f$