Computation of the flux linkage of windings from magnetic scalar potential finite element solutions

Formulas are derived for computing the flux linkage of windings modelled in finite element computations by regions with given current density. It is shown that, if there is a single winding only in the finite element model, its flux linkage is the ratio of the sum of the magnetic energy and co-energy to the winding current. Alternatively, if there are other current-carrying conductors present in the problem region, the sum of the energy and co-energy has to be supplemented by a surface integral of the product of the magnetic scalar potential and the flux density over a surface enclosing the winding. Finally, these formulas valid for current-carrying coils not embedded in one another are generalised to coils carrying no current and/or surrounding other coils with non-zero current. The results do not involve the magnetic vector potential and are hence useful if the finite element solution is in terms of a magnetic scalar potential. The relationships obtained are valid even if the problem is non-linear.