Analytical approximations for magnetic coupling coefficients between adjacent coils

This paper presents a simple yet novel two-dimensional modelling approach for approximating the coupling coefficient between neighbouring inductors as a function of co-planar separation and relative angular displacement. The approach employs simple geometric arguments to predict the effective magnetic flux between inductors. Two extreme coil geometry regimes are considered; planar coils (i.e. on printed circuit board), and solenoid coils, each with asymmetric ferrite cores about the central magnetic plane of the inductor. The proposed geometric approximation is used to predict the coupling coefficient between sensors as a function of separation distance and angular displacement and the results are validated against two-dimensional finite element modelling results. The analytical approximations show excellent agreement with the FE analysis, predicting comparable trends with changing separation and angular displacement, enabling best fitting to 2D FE and 3D numerical data with a residual standard deviation of less than 0.5% for the planar coil approximation. The work demonstrates the validity of the analytical approximation for predicting coupling behaviour between neighbouring coils. This has practical uses for the automated estimation of the physical separation between coils, or the curvature of surfaces they are rested or adhered to.

本文提出了一种简洁新颖的二维建模方法，可根据共面间距（co-planar separation）与相对角位移（relative angular displacement），近似求解相邻电感器（inductor）之间的耦合系数（coupling coefficient）。该方法借助简单的几何推导（geometric arguments），预测电感器间的有效磁通量（effective magnetic flux）。本研究涵盖两类极端的线圈几何构型（coil geometry regimes）：印制电路板（printed circuit board, PCB）上的平面线圈（planar coils），以及围绕电感器中心磁平面配置非对称铁氧体磁芯（ferrite cores）的螺线管线圈（solenoid coils）。所提出的几何近似方法被用于根据间距与角位移，预测传感器间的耦合系数，并将所得结果与二维有限元建模（two-dimensional finite element modelling）结果进行了验证对比。该解析近似（analytical approximations）结果与有限元分析（finite element analysis）结果吻合极佳，在间距与角位移变化时可呈现一致的变化趋势，可实现与二维有限元与三维数值数据（numerical data）的最优拟合，平面线圈近似场景下的剩余标准偏差（residual standard deviation）小于0.5%。本研究证实了该解析近似方法用于预测相邻线圈耦合特性（coupling behaviour）的有效性，其实际应用包括自动估算线圈间的物理间距，或是计算线圈放置或附着的表面曲率。