AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Kraków, Poland.
AGH University of Science and Technology, al. A. Mickiewicza 30, 30-059 Kraków, Poland; AMAG Ltd., ul. Zaścianek 14, 30-209 Kraków, Poland.
J Magn Reson. 2014 Jan;238:52-62. doi: 10.1016/j.jmr.2013.10.026. Epub 2013 Nov 14.
Halbach hollow cylinder dipole magnets of a low or relatively low aspect ratio attract considerable attention due to their applications, among others, in compact NMR and MRI systems for investigating small objects. However, a complete mathematical framework for the analysis of magnetic fields in these magnets has been developed only for their infinitely long precursors. In such a case the analysis is reduced to two-dimensions (2D). The paper details the analysis of the 3D magnetic field in the Halbach dipole cylinders of a finite length. The analysis is based on three equations in which the components of the magnetic flux density Bx, By and Bz are expanded to infinite power series of the radial coordinate r. The zeroth term in the series corresponds to a homogeneous magnetic field Bc, which is perturbed by the higher order terms due to a finite magnet length. This set of equations is supplemented with an equation for the field profile B(z) along the magnet axis, presented for the first time. It is demonstrated that the geometrical factors in the coefficients of particular powers of r, defined by intricate integrals are the coefficients of the Taylor expansion of the homogeneity profile (B(z)-Bc)/Bc. As a consequence, the components of B can be easily calculated with an arbitrary accuracy. In order to describe perturbations of the field due to segmentation, two additional equations are borrowed from the 2D theory. It is shown that the 2D approach to the perturbations generated by the segmentation can be applied to the 3D Halbach structures unless r is not too close to the inner radius of the cylinder ri. The mathematical framework presented in the paper was verified with great precision by computations of B by a highly accurate integration of the magnetostatic Coulomb law and utilized to analyze the inhomogeneity of the magnetic field in the magnet with the accuracy better than 1 ppm.
Halbach 空心圆柱型磁体的低或相对低纵横比由于其应用而引起了相当大的关注，特别是在用于研究小物体的紧凑型 NMR 和 MRI 系统中。然而，对于这些磁体的无限长前体，已经开发出了用于分析磁场的完整数学框架。在这种情况下，分析简化为二维（2D）。本文详细分析了有限长 Halbach 偶极圆柱磁体中的 3D 磁场。分析基于三个方程，其中磁通密度 Bx、By 和 Bz 的分量被扩展为径向坐标 r 的无穷幂级数。级数中的零项对应于均匀磁场 Bc，它由于磁体长度有限而被高阶项所扰动。该方程组补充了一个关于磁体轴上磁场分布 B(z)的方程，这是首次提出的。结果表明，定义为复杂积分的特定 r 次幂系数中的几何因子是均匀性分布（B(z)-Bc）/Bc 的泰勒展开式的系数。因此，可以以任意精度轻松计算 B 的分量。为了描述由于分段引起的场的扰动，从二维理论中借用了另外两个方程。结果表明，除非 r 不接近圆柱内半径 ri，否则可以将二维方法应用于由分段引起的 3D Halbach 结构的扰动。本文提出的数学框架通过对磁静库仑定律的高精度积分来计算 B，并利用其来分析磁体中磁场的不均匀性，精度优于 1 ppm，从而得到了很好的验证。
