Hysteresis with AC excitation

Hysteresis is the lagging of flux density B behind the magnetising force H in a ferromagnetic material. Under ac excitation H goes through a full cycle every period, so the material traces a closed hysteresis loop each cycle and loses energy as heat.

Mechanism

• A ferromagnetic material is made of small domains, each fully magnetised. With no field they point randomly, so the net B is zero.
• As H rises, domains in line with H grow and others turn, so B rises steeply and then saturates when almost all are aligned.
• When H is reduced, many domains do not turn back; B stays higher than on the rising curve. Turning them back requires work, which is lost as heat. This is the cause of hysteresis.

Hysteresis loop with ac excitation

              B
              |   ___ a  (+Bmax)
          b  _|-'   /
            / |    /
     c     /  |   /    
  --*-----/---+--/---*--- H
         /    | /    f
        /  ___|'
   d  _/-'    |e
   (-Bmax)    |

With an ac current $i = I_m \sin\omega t$, H varies sinusoidally between $+H_m$ and $-H_m$:

1. H rises 0 → $+H_m$: B rises to $+B_m$ (point a).
2. H falls to 0: B falls only to residual flux density $B_r$ (point b), called retentivity.
3. H reversed to $-H_c$: B becomes zero (point c); $H_c$ is the coercive force.
4. H to $-H_m$: B reaches $-B_m$ (point d).
5. H back through 0 ($-B_r$, point e) and $+H_c$ (point f) to $+H_m$: the loop closes.

The loop is traced f times per second. For each cycle the energy lost per unit volume is $\oint H\,dB$, the area of the loop.

Hysteresis loss

$$P_h = \eta\, B_m^{1.6} f\, V \quad \text{W}$$

where $\eta$ is the Steinmetz coefficient (J/m³), $B_m$ the peak flux density, $f$ the frequency and $V$ the core volume. The index 1.6 varies from 1.5 to 2.5 for different materials.

Effects in machines

• Heats the core and lowers efficiency of transformers, motors and generators.
• Makes the exciting current non-sinusoidal and slightly leading the flux (core-loss component).
• Reduced by using soft magnetic materials (silicon steel, CRGO steel) with a narrow loop, and by keeping $B_m$ moderate.