Hysteresis loop

A hysteresis loop is the graphical fingerprint of a ferromagnetic material. Start with an unmagnetised sample and slowly increase the applied H-field: the magnetization climbs along the initial-magnetisation curve, at first slowly (reversible domain-wall bowing), then rapidly as walls unpin and sweep through the material, and finally saturates as every domain aligns with the field. Reverse the field's direction and the magnetization does not retrace the same curve — it stays large all the way down past H = 0, leaving a remanent magnetization M_r, and only drops to zero when the reversed field reaches the coercive field −H_c. Continue reversing to saturation, cycle back, and the system traces a closed loop: the hysteresis loop.

The loop's shape encodes the material's character. A hard ferromagnet (good for permanent magnets) has a large remanence, a large coercive field, and a wide, square loop: neodymium-iron-boron magnets hold about M_r ≈ 1.1 T and require H_c ≈ 900 kA/m to demagnetise. A soft ferromagnet (good for transformer cores and electromagnets) has the opposite: small coercive field, narrow and tall loop, so that the magnetization can be cycled quickly with minimal energy loss. Pure iron with silicon addition, for instance, has H_c ≈ 40 A/m — forty times smaller than the external H-field of a hand-held magnet.

The area enclosed by the loop is the energy dissipated per unit volume per cycle. As the field drives M around the loop, microscopic domain walls jump past pinning defects irreversibly, and the work done by the external circuit goes into heat. For a transformer cycling 50 times a second, this hysteresis loss can reach tens of watts per kilogram of core. Every transformer designer's first optimisation is to choose a core material whose loop encloses the smallest possible area at the operating frequency — pushing M_r and H_c as small as metallurgy allows while keeping the permeability high.