= Small-signal response of a gapped magnetic core
{title2=$\kappa=\mu_0n/[d+\mu_0\ell/\mu_{\rm core}]$}
For a uniform core of path length $\ell$, a narrow air gap $d$, and $n$ winding turns, <Ampère's circuital law> gives $nI=\ell H_{\rm core}+dB/\mu_0$ when leakage and fringing are negligible. If a reversible branch has differential <magnetic permeability> $\mu_{\rm core}=dB/dH$, 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, $\mu_{\rm core}=B_m/H_m$.
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