Inductance of a toroidal winding ($L\simeq \mu N^{2}A/\ell$)

For a slender winding around a toroid with $N$ turns, core area $A$, mean magnetic path length $\ell$ and approximately constant magnetic permeability $\mu$, Ampère's law gives $H\ell =NI$. The linked magnetic flux is $N\Phi =N\mu HA$, so $L=N\Phi /I=\mu N^{2}A/\ell$. This approximation neglects flux leakage and variation across the core and assumes operation below magnetic saturation. With a nonlinear magnetic core, an incremental inductance instead depends on differential permeability at the operating point.

For a uniformly permeable toroid with inner radius $a$, outer radius $b$ and axial height $h$, an ideal poloidal winding gives $H_{\varphi }=NI/(2\pi r)$. Integrating the magnetic flux density across its section gives $\Phi =\mu NIh\ln (b/a)/(2\pi )$ and hence the displayed self-inductance. It retains radial field variation when $b-a$ is not small. For two fully linked windings, replace $N^2$ by $N_1{N}_2$ to obtain their mutual inductance. Discrete turns, leakage and nonlinear core response limit the ideal formula.

In this magnetic circuit approximation, each circular path has air-gap length $d$, core length $2\pi r-d$, and the same normal magnetic flux density in air and core. Ampère's law then gives $B(r)\simeq {\mu }_0{\mu }_{r}NI/[2\pi r+({\mu }_r-1)d]$; integrating across the section gives the displayed mutual inductance. Even $d\ll a$ can strongly reduce inductance when ${\mu }_{r}d/a$ is appreciable. This is a path estimate, not an exact local field solution in the slit geometry. It neglects magnetic fringing and redistribution between paths. In particular, a literal field proportional to $\hat{\mathbf{\varphi }}/(r+q)$ everywhere in a current-free core is not curl-free; transverse corrections are required. Narrow-gap circuit approximations and their limitations are discussed in ^[ref].

Let a slender toroid have core cross-section $A$, mean length $\ell$ and large relative magnetic permeability $\mu$. Cut a narrow gap of perpendicular width $t$ whose faces make angle $\theta$ with the local core direction. Continuity of normal magnetic flux density gives $B_{g,n}=B\sin \theta$; continuity of tangential magnetic field intensity gives $H_{g,t}=B\cos \theta /(\mu {\mu }_0)$. The gap field is nearly normal to the faces for large $\mu$. Its effective flux area is $A/\sin \theta$, so its leading magnetic reluctance is $t\sin \theta /({\mu }_{0}A)$. Adding core reluctance gives the displayed self-inductance. At $45^{\circ }$, $L\simeq {\mu }_0{N}^{2}A/(\ell /\mu +t/\sqrt{2})$ and $H_g\simeq NI/(t+\sqrt{2}\ell /\mu )$. This retains $\mu t/\ell$ while neglecting fringing and corrections of relative order $t/\ell$ or $1/\mu$.