Magnetic reluctance ($\mathcal{R}=\int dl/({\mu }_0{\mu }_{r}A)$)

The magnetic reluctance of a linear section of magnetic circuit is the ratio of the winding's integrated magnetic field intensity to the magnetic flux, with units $\mathrm{A}\mathrm{W}{\mathrm{b}}^{-1}$. For uniform length $\ell$, cross-sectional area $A$, and relative magnetic permeability ${\mu }_r$, $\mathcal{R}=\ell /({\mu }_0{\mu }_{r}A)$. Series sections carrying the same magnetic flux have additive magnetic reluctances, and $NI=\Phi {\sum }_i{\mathcal{R}}_i$.

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, ${\mu }_{\mathrm{core}}=B_m/H_m$.

For a linear-radius taper from $R$ to $r$ over length $s$, constant absolute permeability $\mu$, and a common approximately axial flux, integrate $dx/[\mu \pi \{R-(R-r)x/s{\}}^2]$ to obtain the displayed reluctance. Leakage, fringing, nonuniform transverse fields and saturation limit this one-dimensional magnetic-circuit approximation.