Full-Wave Semi-Analytical Modeling of Planar Spiral Inductors in Layered Media

In this paper, we present a full-wave semi-analytical solution to calculate the self and mutual impedances of two coupled spiral inductors with rectangular cross sections.In low-frequency electromagnetism, the self and mutual impedance of planar spiral inductors can be obtained based on the eddy current approximation, where the displacement current is disregarded.As the frequency increases, the size of the system can be designed to be smaller.However, the displacement current becomes more important in inductively-coupled systems.By directly deriving the Maxwell's equations without the eddy current assumption, the obtained full-wave model could be applied to both homogeneous and planarly layered media for wireless power transfer systems.Compared to the traditional methods, the newly derived impedances show a considerable discrepancy at GHz frequencies for millimeter-sized inductors, indicating the significance of the displacement current if the operating frequency of wireless power transmission reaches the GHz-range.Nomenclature E, H: Electric and magnetic fields δ(x): Dirac Delta function ρ, ϕ, z: Coordinates of the cylindrical coordinate system ρ n , z n : Radius and position of loop/coil n ρ ni , ρ no : Inner and outer radii of coil n, see Fig. 3 σ: Conductivity ε 0 , μ 0 : Permittivity and permeability of free space ε r , μ r : Relative permittivity and permeability B m , D m : The amplitudes of upgoing and downgoing waves in layer m d m : Location of layer m, see Fig. 3 F (z, z 1 ): Propagation function in layered media f, ω: Operation frequency and angular frequency H (1) 0 (x) , H (2) 0 (x): Hankel functions I, J: Electric current and electric current density in a loop/coil k, k ρ , k z : Wave number and its ρ and z components k m , k mz : Wave number and its z component in layer m t m : Thickness of layer m, see Fig. 3