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Computation of magnetic fields from field components on a plane grid

机译：从平面网格上的场分量计算磁场的计算

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An algorithm is presented to calculate the field components of the magnetic field [B-x(x, y, z), B-y(x, y, z), B-z(x, y, z)] at a point (x, y, z) in space, from the knowledge of the components [B-x(x, y = 0, z), B-y(x, y = 0, z), B-z(x, y = 0, z)] on a "reference plane", which is normal to the y-axis at y = 0. The algorithm, which is a general one and is not restricted to fields with mid-plane symmetry is based on the Maclaurin series expansion of the magnetic field components at any point in space in terms of the distance (y) of the point from the reference plane. The coefficients of the Maclaurin series expansion are expressed in terms of the on-plane field components and their partial derivatives with respect to spatial coordinates (x, z). The field components are usually generated from magnetic field measurements on a rectangular grid on the plane. This algorithm was employed in 1986 in the RAYTRACE computer code to help calculate the optical properties of magnets and of the Alternating Gradient Synchrotron (AGS) at the Brookhaven National Laboratory (BNL). A general mathematical formulation of this algorithm based on the differential algebraic method was presented by Makino in 2011. This paper presents the step by step derivation of the algorithm and provides the necessary formulas to be introduced by the reader in any computer code which requires the field components generated by magnetic devices. In addition provides an example of the use of the algorithm and its limitations as applied to a Halbach type magnet with or without median plane symmetry. (C) 2019 Elsevier Inc. All rights reserved.
机译：本文提出了一种算法，根据在y=0时垂直于y轴的“参考平面”上的分量[B-x（x，y=0，z）、B-y（x，y=0，z）、B-z（x，y=0，z）]的知识，计算空间中某一点（x，y，z）处的磁场[B-x（x，y，z）、B-y（x，y=0，z）]的场分量。该算法是一种通用算法，不局限于具有中平面对称性的场，它基于空间中任意点的磁场分量的麦克劳林级数展开，即该点与参考平面的距离（y）。麦克劳林级数展开的系数用平面场分量及其相对于空间坐标（x，z）的偏导数表示。磁场分量通常由平面上矩形网格上的磁场测量产生。1986年，在布鲁克海文国家实验室（BNL）的光线追踪计算机代码中，该算法被用于帮助计算磁铁和交替梯度同步加速器（AGS）的光学特性。2011年，Makino提出了基于微分代数方法的该算法的通用数学公式。本文介绍了算法的逐步推导，并提供了读者在任何需要磁性器件产生的磁场分量的计算机代码中引入的必要公式。此外，还提供了一个示例，说明了该算法的使用及其适用于具有或不具有中间平面对称性的Halbach型磁铁的局限性。（C） 2019爱思唯尔公司版权所有。

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