Magnetic flux and flux linkage

Magnetic flux measures the magnetic field passing through a surface. For a uniform magnetic field perpendicular to a surface, the flux depends on the magnetic flux density and the area of the surface:

Here, $$\Phi$$ is magnetic flux, $$B$$ is magnetic flux density and $$A$$ is the surface area. Magnetic flux is measured in webers, with the symbol $$\mathrm{Wb}$$.

Flux linkage in a coil

A coil has multiple turns, each enclosing a surface. Flux linkage is the total flux linked with all the turns. If every turn links the same flux $$\Phi$$ and the coil has $$N$$ turns, the flux linkage is:

For the same flux through each turn, increasing the number of turns increases the flux linkage.

Flux linkage of a rotating coil

For a rectangular coil in a uniform magnetic field, the flux depends on the coil’s orientation. Let $$\theta$$ be the angle between the field and the normal to the coil: a line perpendicular to its plane. Only the component of the field normal to the coil passes through its area. That component is $$B\cos\theta$$.

For a coil of area $$A$$ with $$N$$ turns, the flux through each turn and the flux linkage are therefore:

When $$\theta = 0^\circ$$, the field is normal to the coil and the flux linkage is greatest: $$N\Phi = BAN$$. When $$\theta = 90^\circ$$, the field is parallel to the plane of the coil, so there is no normal component and the flux linkage is zero.

As the coil continues to rotate, the cosine changes sign when the field passes through the coil in the opposite direction relative to the chosen normal. The flux linkage is then negative; its magnitude is greatest when the field is again normal to the coil.