This paper reviews magnetic flux signal calculations through pick-up loops using vector spherical harmonic expansion under the quasi-static approximation, and presents a near-analytical method of evaluating the flux through arbitrary parametrizable pick-up loops for each expansion degree. This is done by simplifying the surface flux integral (2D) into a line integral (1D). For the special case of tangential circular sensors, we present a fully analytical recursion calculation. We then compare commonly-used cubature approximations to our (near-)analytical forms, and show that cubature approximations suffer from increasing errors for higher spatial frequency components. This suggests the need for more accurate evaluations for increasingly sensitive sensors that are being developed, and our (near-)analytical forms themselves are a solution to this problem.

本文回顾了在准静态近似下使用矢量球谐展开通过拾取回路计算的磁通量信号，并提出了一种近似解析的方法来评估通过任意参数化拾取回路对每个展开度的磁通量。这是通过将表面通量积分 (2D) 简化为线积分 (1D) 来完成的。对于切向圆形传感器的特殊情况，我们提出了一个完全解析的递归计算。然后，我们将常用的体积近似与我们的（近）分析形式进行比较，并表明体积近似会因较高空间频率分量的误差增加而受到影响。这表明需要对正在开发的越来越敏感的传感器进行更准确的评估，