Abstract The common approach to create a uniform magnetic field in the μ T range is placing a coil set inside a magnetic shield. One of the main obstacles to achieving the requested uniformity can be a field distortion caused by the shield. We derive an approximate expression for the magnetic field generated by circular coils enclosed in ferromagnetic cylinders, the most used shield geometry. The main advantage of the expression over the known approximation is considering the finite permeability and the finite thickness of the cylindrical shield. To model the cylinder end caps, we propose the finite thickness mirror method, which extends the method of mirror image to the case of finite thickness. The cylinder barrel is modeled as an infinite tube and its effect on the field is analytically solved. The error of the expression and its dependency on the observation point and the shield geometry are discussed. Compared with finite-element simulations our analytic method is consistent within a relative error of less than 0.3% for practical shielding setups. The expression enables the rapid optimization of shield-coupled coil sets with a custom-designed objective function. To show the applicability of our method, we present simulations for the direct selection of the optimum spacing of a coil pair to match the given dimensions of a cylindrical shield.

中文翻译：

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由有限厚度和磁导率的封闭圆柱形屏蔽内的圆形线圈产生的磁场的近似表达式

摘要 在 μ T 范围内产生均匀磁场的常用方法是将线圈组放置在磁屏蔽内。实现所需均匀性的主要障碍之一可能是屏蔽引起的场失真。我们推导出由封闭在铁磁圆柱体中的圆形线圈产生的磁场的近似表达式，铁磁圆柱体是最常用的屏蔽几何结构。与已知近似值相比，该表达式的主要优点是考虑了圆柱屏蔽的有限磁导率和有限厚度。为了模拟圆柱端盖，我们提出了有限厚度镜像法，它将镜像方法扩展到有限厚度的情况。气缸筒被建模为一个无限管，其对场的影响被解析解决。讨论了表达式的误差及其对观测点和屏蔽几何形状的依赖性。与有限元模拟相比，我们的分析方法在实际屏蔽设置的相对误差小于 0.3% 内是一致的。该表达式可以使用定制设计的目标函数快速优化屏蔽耦合线圈组。为了显示我们方法的适用性，我们提供了直接选择线圈对的最佳间距以匹配圆柱形屏蔽的给定尺寸的模拟。该表达式可以使用定制设计的目标函数快速优化屏蔽耦合线圈组。为了显示我们方法的适用性，我们提供了直接选择线圈对的最佳间距以匹配圆柱形屏蔽的给定尺寸的模拟。该表达式可以使用定制设计的目标函数快速优化屏蔽耦合线圈组。为了显示我们方法的适用性，我们提供了直接选择线圈对的最佳间距以匹配圆柱形屏蔽的给定尺寸的模拟。