Retentivity, coercivity, energy per cycle

Retentivity and coercivity

              B
              |      .------- +Bs
              |    /
          Br -+---/---.
              |  /    |
  ------------+-/-----+-------- H
            -Hc /     +Hc
     -Bs ------'      |
              |

• Retentivity (Br): the flux density left in a magnetic material when the magnetising force H is brought back to zero after saturating it. It is the B-axis intercept of the hysteresis loop.
• Coercivity (Hc): the reverse magnetising force (A/m) needed to bring the residual flux density to zero. It is the H-axis intercept of the loop.
• Core materials for machines should have low Br and low Hc, so that the loop is narrow. Permanent magnets need both high.

Energy spent per cycle per unit volume is proportional to loop area

Consider a core of length l, cross-section a, wound with N turns carrying a current i that varies slowly.

Field strength     H = N i / l          ->  i = H l / N
Flux               φ = B a
Induced emf        e = N dφ/dt = N a dB/dt

The source must supply power e·i against this emf. In a time dt the energy taken from the source is

dW = e i dt
   = (N a dB/dt)(H l / N) dt
   = a l · H dB

With a·l = V, the volume of the core, the energy per unit volume in this step is

dW/V = H dB

For one complete cycle of magnetisation the energy per unit volume is

W = ∮ H dB = area of the B–H loop (J/m³ per cycle)

The scales of the loop are H in A/m and B in tesla, so the area has the unit (A/m)(T) = J/m³. Hence the energy lost per cycle per unit volume is directly proportional to (in fact equal to) the area of the hysteresis loop. At frequency f the hysteresis loss is Pₕ = f × V × (loop area) watts.