Significance of hysteresis loop, volume dependence

The hysteresis loop is the closed $B$–$H$ curve traced by a ferromagnetic material when $H$ is taken through a full cycle. It shows how the flux density lags behind the magnetising force, and its shape and area decide where a material can be used.

Significance of the hysteresis loop

      B
      |      ___---- Bs (saturation)
      |   .-'
   Br |--/---.
      | /     \
 -----+-+------+------ H
   -Hc|        Hc
      |

1. Retentivity ($B_r$): the flux density left when $H = 0$. A high $B_r$ is needed for permanent magnets.
2. Coercivity ($H_c$): the reverse field needed to bring $B$ to zero. A small $H_c$ gives a narrow loop (soft iron, silicon steel) for transformer and armature cores. A large $H_c$ gives a wide loop (hard steel, alnico) for permanent magnets.
3. Area of the loop: it equals the energy lost as heat per cycle per unit volume. A narrow loop means low hysteresis loss and high efficiency.
4. Saturation flux density ($B_s$): it fixes the highest flux a core can carry, so it limits the size of the machine.
5. Material selection: the loop shows at a glance whether the material suits an ac machine core (narrow loop) or a permanent magnet (wide loop).

Hysteresis loss depends on volume

Energy spent in one cycle per unit volume is the loop area:

$$w_h = \oint H\,dB\ \ \text{J/m}^3\text{ per cycle}$$

This value depends only on the material and on $B_{max}$, not on the size of the core. A core of volume $V$ has $V$ times as much material taking part in the magnetisation reversal, so

$$\begin{aligned} \text{Energy per cycle} &= w_h V \\ P_h &= w_h V f = \eta B_{max}^{x} f V\ \ \text{W} \end{aligned}$$

where $\eta$ is the Steinmetz coefficient and $x \approx 1.6$ to $2$. Hence $P_h \propto V$: doubling the core volume at the same $B_{max}$ and $f$ doubles the hysteresis loss. This is why machine cores are made as small as the design allows.