A Closed-Form Formula for Magnetic Dipole Localization by Measurement of Its Magnetic Field and Spatial Gradients

This paper presents a simple algorithm and a compact sensor for localization of a magnetic dipole. First, we derive an explicit localization formula, where the dipole position, irrespective of its posture, is directly reconstructed from the magnetic field and its spatial gradient tensor. Then, we develop a sensor unit consisting of three orthogonal loop coils, three orthogonal planar gradiometers, and three orthogonal coaxial gradiometers, for measuring the magnetic field, nondiagonal, and diagonal components of the gradient tensor, respectively. Localization experiments are conducted, where the maximum error is about 7 mm when the source-sensor distance is from 80 to 140 mm