 Inductance of a rectangular-section toroid

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.