Analytical Model of Coils with Magnetic Shields of Finite Size and Permeability and Analysis of Effective Coupling Regions

Based on Maxwell's system of equations, this paper derives the magnetic flux density under the condition of magnetic shielding with finite permeability and finite size by using the magnetic vector potential and two-dimensional Fourier transform. On this basis, the mutual inductance calculation equation of two rectangular coils and the analytical equation of the effective coupling region boundary are derived. A super-rounded rectangular sub-side coil design is proposed to enhance the coil coupling based on the effective coupling region boundary. Unlike the traditional approximate calculation method that employs a large number of assumptions, the proposed method is an analytical method for calculating the circuit parameters, which allows the mutual inductance between the coils and the effective coupling region to be solved accurately. Finally, several sets of rectangular helical coils are used for model validation, and the theoretically calculated values of mutual inductance are in good agreement with the finite element simulation results and the experimental results, which verifies the accuracy of the analytical expression for the boundary of the effective coupling region. The proposed analytical model can be used for the initial design and optimization of coils used in WPT systems.