Retentivity, coercivity and hysteresis energy

Retentivity and coercivity

• Retentivity (residual flux density, $B_r$): the flux density that remains in a magnetic material when the magnetising force H is reduced to zero after saturating it. It measures how much magnetism the material keeps.
• Coercivity (coercive force, $H_c$): the reverse magnetising force needed to reduce the residual flux density to zero. It measures how hard it is to demagnetise the material.

             B
             |   __ +Bsat
        Br  _|-'  /
           / |   /
   -Hc    /  |  /
  ---*---/---+-/----- H
        /    |/

Permanent magnets need high $B_r$ and high $H_c$; transformer cores need low $H_c$.

Proof that $W_h = \oint H\,dB$

Consider a ring of magnetic material with mean length $l$ (m), cross-sectional area $A$ (m²), uniformly wound with $N$ turns carrying current $i$.

1. By Ampere's circuital law the field intensity is

$$H l = N i \;\Rightarrow\; i = \frac{H l}{N}$$

2. If the flux density changes by $dB$ in time $dt$, the flux changes by $A\,dB$, and the emf induced in the coil is

$$e = N\frac{d\phi}{dt} = N A \frac{dB}{dt}$$

3. To keep the current flowing against this emf, the source supplies power $p = e\,i$. The energy supplied in time $dt$ is

$$\begin{aligned} dW &= e\, i\, dt \\ &= N A \frac{dB}{dt} \cdot \frac{Hl}{N}\, dt \\ &= A\,l\,H\,dB \end{aligned}$$

4. $Al$ is the volume of the core, so the energy per unit volume is $dw = H\,dB$.

5. Over a complete cycle of magnetisation (B going from $+B_m$ to $-B_m$ and back):

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

During the parts of the cycle where H and dB have the same sign energy is taken from the source; where they have opposite signs energy is returned. The integral round the closed loop is the net energy absorbed, which equals the area of the hysteresis loop and is dissipated as heat.

Hence hysteresis power loss for a core of volume $V$ at frequency $f$:

$$P_h = f\,V \oint H\,dB \quad \text{W}$$