Analysis of two-dimensional and, in particular, three-dimensional fringing electric fields in quadrupole mass spectrometers is a challenging task. In this work, we consider an analytical approach for such analysis. We show how to obtain precise amplitudes of multipole components of 2D and 3D electric fields. We describe some mathematical methods for accurate evaluation of realistic three-dimensional electric fields generated by quadrupole mass filters. This technique can be applied to obtain precise amplitudes of multipole components of 3D electric fields as distributions along some axis. All methods described here, are implemented in the PAnalyser application, which we use to analyse electric fields in different quadrupole systems. We provide some practical examples, which include analysis of quadrupole mass filters with hyperbolic and circular electrodes, a 3D Paul ion trap, and a linear ion trap with triangular electrodes. We also investigate 3D fringing electric fields: a DC and Quadrupole fringing fields within quadrupole mass filters with circular electrodes. Another example, interesting both theoretically and practically, is the analysis of the fringing field between two adjacent quadrupole systems – electrodes of a Brubaker pre-filter and an analytical mass filter. We also show how to synthesize an electric field from a given axial expansion of multipole components.

在四极杆质谱仪中分析二维，尤其是三维边缘电场是一项具有挑战性的任务。在这项工作中，我们考虑了这种分析的分析方法。我们展示了如何获得 2D 和 3D 电场的多极分量的精确幅度。我们描述了一些精确评估由四极杆质量过滤器产生的真实三维电场的数学方法。该技术可用于获得 3D 电场多极分量的精确幅度，作为沿某个轴的分布。此处描述的所有方法都在 PAnalyser 应用程序中实现，我们用它来分析不同四极杆系统中的电场。我们提供了一些实际的例子，其中包括对带有双曲线和圆形电极的四极杆质量过滤器、3D Paul 离子阱和带有三角形电极的线性离子阱的分析。我们还研究了 3D 边缘电场：带有圆形电极的四极杆滤质器内的 DC 和四极杆边缘场。另一个在理论和实践上都很有趣的例子是对两个相邻四极杆系统（Brubaker 预过滤器和分析质量过滤器的电极）之间的边缘场的分析。我们还展示了如何从多极组件的给定轴向扩展合成电场。在理论和实践中都有趣的是对两个相邻四极杆系统之间的边缘场分析 - Brubaker 预过滤器和分析质量过滤器的电极。我们还展示了如何从多极组件的给定轴向扩展合成电场。在理论和实践上都有趣的是对两个相邻四极杆系统之间的边缘场分析 - Brubaker 预过滤器和分析质量过滤器的电极。我们还展示了如何从多极组件的给定轴向扩展合成电场。