For low values of current before the onset of saturation, the inductance is a function of the gap and is:

L_{max}= \frac{N^{2}}{R_{g}} = \frac{\mu _{0}N^{2}A_{g}}{g} = \frac{(4\pi\times10^{-7})(365)^{2}(2.072\times10^{-4})}{0.5\times10^{-3}} \times10^{3}= 69.4  mH

L is found from Equation 10.7:

L= \frac{N^{2}}{R_{g}}\frac{1}{\frac{\mu_{\mathrm{eff}}}{\mu_{r}} }   (10.7)

\mu _{\mathrm{eff}}= \frac{l_{c}}{g} = \frac{84}{0.5}= 168

And for i = 1.5 A:

a = H_{0} = 100 A/m

b = \frac{Ni}{g}+ H_{0}\mu _{\mathrm{eff}} -H_{m}= \frac{(365)(1.5)}{0.5\times10^{-3}} + (100)(168)-1.3\times10^{6}= -18.82\times10^{4 }  A/m

c = -H_{m}\mu _{\mathrm{eff} }= -(1.3\times10^{6})(168)= -2.184\times10^{8}  A/m

μ_{r}=\frac{-b+ \sqrt{b^{2}-4ac} }{2a} = \frac{(18.82\times10^{4})+\sqrt{(-18.82\times10^{4})^{2}-(4)(100)(-2.184\times10^{8})} }{(2)(100)} = 2693

L  = \frac{N^{2}}{R_{g}} \frac{1}{1+ \frac{\mu_{\mathrm{eff} }}{\mu_{r}} } = (69.4\times10^{-3})\frac{1}{1+ \frac{168}{2693} } = 65.3  mH

Leff is found from Equation 10.15:

L_{\mathrm{eff} }= L(i)+ i\frac{dL(i)}{di} = L(i)+ i\frac{dL}{d\mu_{r}}\frac{d\mu_{r}}{di}              (10.15)

\frac{dL}{d\mu_{r}}= L\frac{\mu_{\mathrm{eff} }/\mu^{2}_{r}}{1+ \mu_{\mathrm{eff} }/\mu_{r}} = (0.0653)\frac{168/(2693)^{2}}{1+ 168/2693} = 1.424\times10^{-6}H

 

\frac{d\mu_r}{di} = \left[-\frac{1}{2a}+ \frac{b}{2a\sqrt{b^{2}-4ac} } \right] \frac{N}{g}

= \left[ -\frac{1}{(2)(100)}+ \frac{-18.82\times10^{4}}{(2)(100)\sqrt{(18.82\times10^{4})^{2}-(4)(100)(-2.184\times10^{8})} } \right]\times \frac{365}{0.5\times 10^{-3}}

= -5.61 \times10^{3}  A^{-1}

 

L_{\mathrm{eff} }= L(i)+ i\frac{dL}{du_{r}}\frac{d\mu_{r}}{di} = 0.0653+ (1.5)(1.424\times10^{-6})(-5.61\times10^{3})= 53.3  mH

This process is repeated for the range of current and the plots of L and L_{\mathrm{eff} } are shown in Figure 10.4.