Shim Coils Design for Halbach Magnet Based on Target Field Method

Abstract

A modified target field method is presented to the construction of shim and gradient coils for Halbach-type magnet featuring cylindrical magnet and transversal magnetic field direction. With Biot–Savart law, magnetic field is deduced from current density distribution, and two solutions corresponding to relations between spherical harmonic functions degree m and current density degree k, namely $m=k\pm 1$ are derived. Then combined with magnetic field distribution of target points, current density coefficient $U_{\rm q}$ is adopted to calculate stream function, based on which the coils' structure can be obtained. An appropriate objective function integrating dissipated energy and relative error is utilized to further improve the design. Examples of X2–Y2 and Y coils indicate that solution corresponding to $m=k-1$ exhibits good agreement with theory in terms of obtaining lower dissipated energy and smaller error rate, suggesting that the algorithm provides an effective, preferable approach for the design of Halbach shim coils.