Hysteresis loop energy, hard and soft magnetic materials

Magnetic hysteresis is the lagging of flux density B behind the magnetising force H when a ferromagnetic material is taken through a cycle of magnetisation. As a result the B–H curve forms a closed loop, and energy equal to its area is lost per cycle.

Hysteresis loop

              B
              |    __ a
          b  _|.-'  /
            / |    /
     c     /  |   /
  --*-----/---+--/---*--- H
         /    | /    f
        / ___.|'
     d -'     |e

ob = residual flux density (retentivity), oc = coercive force, a and d = positive and negative saturation.

Proof: area ∝ energy loss per unit volume

Ring: mean length $l$, area $A$, $N$ turns, current $i$. By Ampere's law $Hl = Ni$, so $i = Hl/N$. When B changes by $dB$ the induced emf is $e = NA\,dB/dt$. Energy supplied by the source in time $dt$:

$$\begin{aligned} dW &= e\,i\,dt = NA\frac{dB}{dt}\cdot\frac{Hl}{N}\,dt = (Al)\,H\,dB \end{aligned}$$

Per unit volume, $dw = H\,dB$. For one complete cycle

$$w = \oint H\,dB = \text{area of B–H loop} \quad (\text{J/m}^3)$$

The energy stored while B rises is only partly returned while B falls; the net amount, equal to the loop area, is converted into heat. Total hysteresis loss:

$$P_h = (\text{loop area}) \times V \times f \ \text{W} \approx \eta B_m^{1.6} f V$$

Hard and soft magnetic materials

Soft materials are chosen for ac machines because their small loop area gives low core loss; hard materials are chosen where the magnetism must be kept.