Magnetic Materials — Question 9

Question 9

A nonlinear reversible law has different secant and differential permeabilities

A hypothetical nonhysteretic material obeys $M(H)=M_s\tanh(H/H_0)$ for positive H. Define secant permeability as B/H and differential permeability as dB/dH.

1. Derive both permeabilities from B=mu0(H+M).

2. Find their limits at H approaching zero and at H much greater than H0, explaining the meaning of saturation.

Question 9 - Solution

A nonlinear reversible law has different secant and differential permeabilities

(a) Method and calculation

$$\boxed{\mu_{\mathrm{sec}}=\mu_0\left[1+\frac{M_s}{H}\tanh(H/H_0)\right]}.$$$$\boxed{\mu_{\mathrm{diff}}=\mu_0\left[1+\frac{M_s}{H_0}\operatorname{sech}^2(H/H_0)\right]}.$$

(b) Interpretation and further calculation

$$H\to0:\quad\mu_{\mathrm{sec}},\mu_{\mathrm{diff}}\to\mu_0(1+M_s/H_0).$$$$H\gg H_0:\quad M\to M_s,\quad\boxed{\mu_{\mathrm{sec}},\mu_{\mathrm{diff}}\to\mu_0}.$$

Physical check

M saturates, but B continues to increase through the vacuum term mu0H; a nonlinear curve alone does not establish hysteresis.