Complete Set of Partial Differential Equations for Direct Localization of a Magnetic Dipole

Recently, the use of Euler's homogeneity equation has been proposed to obtain a direct algorithm for localizing a magnetic dipole in a free space. This is based on the vector-homogeneity of the magnetic field generated by the source. However, the magnetic field generated is also governed by electromagnetic constraints such as divergence-freeness. In order to obtain the methods that consider the full properties of the dipole-induced magnetic field, we develop an additional set of partial differential equations (PDEs), such that the general solution of the combined equations satisfies a complete or subset of the electromagnetic constraints induced and restricted by the magnetic dipole and field. We apply the weighted integral method considering a suitable form of weight functions (observation functions) for each PDE. In them, the magnetic dipole can be localized from line integrals (measurements) or surface integrals of the magnetic field, while Euler's homogeneity equation requires volume integrals. The methods are verified by numerical simulations and experiments for application in a shorter search and rapid localization of an avalanche beacon.