Inductors and Inductance — Question 10

Question 10

An air gap can dominate a linear-core inductor even when the core permeability is large

A magnetic circuit has N turns, common cross-section A, core path length ellc, relative permeability mur and air-gap length g. Neglect leakage and fringing. Use $NI=H_c\ell_c+H_g g$, $B=\mu_0\mu_rH_c=\mu_0H_g$ and $\Lambda=NBA$.

1. Derive the inductance.

2. Find its gap-dominated limit and the approximate fractional change when g increases by a small delta g.

Question 10 - Solution

An air gap can dominate a linear-core inductor even when the core permeability is large

(a) Method and calculation

$$NI=\frac B{\mu_0}\left(\ell_c/\mu_r+g\right)\Rightarrow B=\frac{\mu_0NI}{g+\ell_c/\mu_r}.$$$$\boxed{L=\frac{\mu_0N^2A}{g+\ell_c/\mu_r}}.$$

(b) Interpretation and further calculation

$$g\gg\ell_c/\mu_r:\quad\boxed{L\simeq\mu_0N^2A/g}.$$$$\boxed{\Delta L/L\simeq-\Delta g/g}\quad\text{in the gap-dominated limit}.$$

Physical check

The supplied linear-core model excludes saturation and hysteresis. A large permeability alone cannot overcome a dominant air gap.