Analytical Expression of the Mirror Coefficient by Joint Analytical Calculation Method

To address the universal challenge that the mirror coefficient (MC) cannot be analytically expressed, the joint analytical calculation (JAC) method is initially proposed in this article. Selecting the magnetic vector potential (MVP) as a uniform analytical target, the JAC method constructs a bridge, linking the truncated region eigenfunction expansion (TREE) method and the mirror method. Furthermore, an analytical expression of MC is derived based on the uniqueness principle of the magnetic field. The JAC method comprehensively reflects the effect of magnetic medium parameters, coil parameters, and field point coordinates on MC. Furthermore, the double TREE method is proposed to further improve the analytical accuracy of MVP. The experiment indicates that the analytical error of MVP is shrunk by a maximum of 3.16%. Then, magnetic flux density (MFD) is solved according to the analytical expressions of MC. The experiment shows that the JAC method and the mirror method have a maximum error of 4.08% and 27.99% for calculating MFD, respectively.

I. Introduction

Analytical calculation of magnetic field has been widely applied in the design and optimization of electromagnetic systems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], which is more flexible and efficient than the finite element analysis (FEA) tools. For analytical calculation of magnetic field, magnetic flux density (MFD) and magnetic vector potential (MVP) are the most critical due to almost entire magnetic field parameters can be derived from them. At present, the analytical calculation of MFD and MVP has been applied to a large number of industry processes, including analysis of transformer winding losses [1], the design of electromagnetic sensors [2], analysis of magnetic flux leakage to evaluate the defects depth of the magnetic medium [3], transmitting coils position detection for electric vehicle in wireless power transfer (WPT) systems [4], design of eddy-current sensors [5], current field measurement [6], [7], [8], eddy current nondestructive test (NDT) [9], atomic sensors [10], and ac resistance evaluation for Litz coils [11], [12]. These applications almost involve finite-size magnetic mediums. Traditionally, the width of the magnetic medium width is assumed infinite, and then, the MFD and MVP are derived from the mirror method [12] or separation of variables method [13].