A Novel Analytical Formulation of the Magnetic Field Generated by Halbach Permanent Magnet Arrays

Abstract

This paper presents an analytical study of the air-gap magnetic field of a surface permanent magnet (SPM) linear, slot-less machine with a Halbach PM configuration, under the no-load condition. While other analytical formulations of the magnetic field generated by PMs are available, they exhibit some drawbacks, such as only providing a Fourier series, or being suitable to determine magnetic field average values, but not local magnetic field distributions. On the contrary, the proposed approach allows the determination of a unique, closed-form formulation for the slot-less machine air-gap field. This is obtained starting from the complex expression of the magnetic field of a conductor, inside the air gap, between two parallel smooth iron surfaces, obtained by means of the method of images. The magnetic field due to an infinitesimal conductor belonging to a current sheet is then integrated along a segment, providing the expression of the magnetic field due to the corresponding linear current density distribution, for current sheets perpendicular or parallel to the iron surfaces. Any Halbach PM segment disposition can, hence, be obtained via a suitable combination of field distributions generated by couples of current sheets with perpendicular and parallel orientation. Lastly, the no-load magnetic field expression with a Halbach array of PMs is retrieved. The proposed analytical model provides an accurate representation of the magnetic field distribution produced by any Halbach array, with an arbitrary number of segments and orientations. Additionally, the results obtained from the proposed analytical expressions are compared with FEM simulations realized by commercial software, and show an excellent agreement.

How the Magnetization Angle of a Linear Halbach Array Influences Particle Steering in Magnetic Drug Targeting—A Systematic Evaluation and Optimization

“…For this reason, despite the high relative errors, the calculations in this paper continue with the fitted values for the consideration of the forces. For a more accurate fitting result, as in [51], fitting functions with more degrees of freedom or analytical approaches, such as in [56,57,101], must be used.…”

“…Also, in most of the literature, 90 • is used for steering or attracting magnetic nanoparticles [47,49,[52][53][54]. In other applications of Halbach arrays, various magnetization angles are investigated and discussed, e.g., 45 • by Zhang et al [55], 30 • by Di Gerlando et al [56], 18 • and 22.5 • by Shen et al [57], and 90 • by Li et al [58].…”

Noise and vibration modelling of Permanent Magnet Synchronous Motors - a review

“…Local differences occur when the edge of the magnet is in front of a slot. The Halbach arrangement for slotless structures results in a perfect curve overlap with 2D FEM simulations [46]. In some areas, errors of 10% on the airgap field distribution due to armature reaction were found [48].…”

“…The slotted condition is then considered through a complex airgap permeance function. Then, authors extend the work by analysing the airgap magnetic field distribution due to armature reaction and due to the permanent magnets arranged in an Halbach array, in [46], [48], respectively. Results show that the analytical approach matches almost everywhere the FEM results.…”

“…For this reason often it is combined with other methods [42]. In particular it gives results in terms of magnetic field as average quantities, not in as a local distribution [46]. Han et al [45] propose an increase of the complexity defining a MEC model for an IPM synchronous motor equipped with three-phase flux barriers in the rotor.…”