Small-signal response of a gapped magnetic core ($\kappa ={\mu }_{0}n/[d+{\mu }_0\ell /{\mu }_{\mathrm{core}}]$)

For a uniform core of path length $\ell$, a narrow air gap $d$, and $n$ winding turns, Ampère's circuital law gives $nI=\ell H_{\mathrm{core}}+dB/{\mu }_0$ when leakage and fringing are negligible. If a reversible branch has differential magnetic permeability ${\mu }_{\mathrm{core}}=dB/dH$, linearization gives $dB/dI=\kappa$. An offset in the operating branch changes the bias field, not this differential formula. For $B=B_m\tanh (H/H_m)$ at the origin, ${\mu }_{\mathrm{core}}=B_m/H_m$.