Multipole analysis of an elongated magnetic source by a cylindrical sensor array

This paper covers basics of the magnetic characterization of an elongated source. The core element of the work is to obtain a magnetic multipole image (MI) of the magnetic source in terms of the prolate spheroidal harmonic expansion, where the MI encapsulates all the main features of the external field of the source. A method for the prolate spheroidal multipole analysis of a magnetic source enclosed within a cylindrical measurement surface is developed. The spheroidal harmonic coefficients of the MI are restored from measurements of the x component of the magnetic field (B_x) over a finite cylindrical array. The general form of B_x is expressed as a prolate spheroidal harmonic expansion. The harmonies (field patterns) of the expansion are classified by their symmetries about the axis of rotation and about the yz plane. The sets of linear independent patterns of unit intensity are calculated for all multipoles with a given type of symmetry. The sets of filtering K functions are developed, where the K functions are orthonormal to the field patterns of same symmetry. The expansion coefficients, which represent the multipole image of the source, are determined by matrix multiplication of the measured B_x and the K functions.