Analysis of Mutual Inductance Characteristics of Rectangular Coils Based on Double-Sided Electromagnetic Shielding Technology and Study of the Effects of Positional Misalignment

Abstract

In wireless power transfer systems, the relative positional misalignment between transmitting and receiving coils significantly impacts the system’s mutual inductance characteristics, thereby constraining the system’s output power stability and transmission efficiency optimization potential. Hence, accurate formulas for calculating mutual inductance are crucial for optimizing coil structures and achieving mutual inductance stability. This study focuses on the mutual inductance characteristics of rectangular coils under positional misalignment conditions in a dual-sided electromagnetic shielding environment. Initially, the research deduces the incident magnetic flux density induced by the current in rectangular coils through the dual Fourier transform and magnetic vector potential method. Subsequently, Maxwell’s equations and boundary conditions are employed to analytically examine the induced eddy currents within the shielding layer, allowing for the calculation of reflected magnetic flux density. Based on these analyses, the study derives a formula for mutual inductance using the magnetic flux density method. A prototype was built for experimental verification. The experiment results show that the maximum error between the measured mutual inductance and the calculated result is less than 3.8%, which verifies the feasibility and the accuracy of the proposed calculation method. Simulations and empirical validation demonstrate the superior accuracy and practicality of the proposed formula. This research not only offers an innovative technological pathway for enhancing the stability and efficiency of wireless power transfer systems but also provides a solid theoretical foundation and guiding framework for coil design and optimization.

1. Introduction

Dynamic wireless charging (DWC) technology can realize the nonstop charging for electric vehicles, effectively increase the cruising range of vehicles, and reduce the waiting time of parking charging [1,2]. With the rapid development of technologies such as radio frequency circuits, sensor networks, neural networks, and wireless communication technology, as an important component, DWC has been widely applied in various fields [3,4,5,6,7]. Wireless power transfer (WPT) technology can solve endurance problems for electric vehicles [8], and it is widely used in medical devices, mobile electronic terminals, underwater mechanical equipment, refs. [9,10,11], etc. The WPT system mainly consists of transmitting and receiving coils. As the most important factor that directly determines the system’s output power, mutual inductance is related to many parameters such as the position and structure of the coils [12,13,14], etc.

Finite element analysis (FEA) is one approach to calculate the mutual inductance of different coil structures. Taking into account the skin effect and mutual couplings, the system was modeled using a matrix equation and appropriate formulas [15]. However, the fine mesh density of an FEA simulation requires intensive computational resources as the dimensions of the coil system increases [16]. In extreme cases, the FEA calculation may not converge because of the low quality of meshing. In addition, the FEA may have problems such as nonconvergence and a long simulation process.

Analytical models are approach to calculated the mutual inductance of a coil topology. By using such models, a parameterized sweep analysis can be conducted without considering the model meshing configuration [17,18]. Analytical models also provide better insight into the underlying physics of the magnetic field distribution of a specific coil topology [19]. So far, there are several methods for calculating the mutual inductance of circular and rectangular coils: for instance, Heuman’s lambda function [20], Bessel and Struve functions [21], and Biot–Savart law [22,23].

Electromagnetic shielding is an effective method to change the magnetic field distribution. Unlike the traditional coils, the magnetic field of rectangular coils with electromagnetic shielding not only contains the incident magnetic field generated by the current-carrying coil but also includes the reflected magnetic field generated by induced eddy currents in the shielding [24]. The reasonable design can enhance the working magnetic field, and weaken the non-working magnetic field, enhancing the coupling coefficient of the transmitting and receiving coils and reducing the electromagnetic pollution to the environment [25]. An optimization method for mutual inductance was proposed, and the structural parameters that met the design requirements were obtained using the proposed optimization method [26]. It is significant to study the mutual inductance of rectangular coils with electromagnetic shielding. As for analytical models including the magnetic shield, the transfer efficiency of the nonlinear system is stable and high [27,28]. In [29], the mutual inductance between the transmitting coil and the receiving coil of any relative position and shape is expressed by the Kalantarov–Zeitlin method. In [30], the mutual inductance between two rectangular coils at different relative spatial positions is calculated by analytical models including the magnetic shield about different positional misalignment. In [31], the mutual inductance calculation formula between circular coils at any relative position is derived by using Maxwell’s equations, boundary conditions, and a parameter vector. The literature [32] took filamentous circular coils and cylindrical spiral coils as research objects and derived the non-coaxial mutual inductance formula of the two coils based on the Neumann formula. In [33], the mutual inductance of mutually perpendicular coils is calculated based on the second-order vector model. In [34], the magnetic induction intensity is derived in order to calculate the magnetic flux passing through the small blocks and then the mutual inductance. In [35], the mutual inductance calculation between rectangular filament coils at any relative position is given through Biot–Savart law and the geometric structure method. In [36], the mutual inductance between two single-turn rectangular coils at any relative position is given by the spatial rectangular coordinate transformation and magnetic flux density method alongside the mutual inductance calculation formula of rectangular spiral coils. Kushwaha et al. considered a magnetic shield in the transmitter pad only [37]. Kushwaha et al. [38] proposed an analytical model of a rectangular coil with finite magnetic shields. An analytical calculation of the mutual inductance is then carried out with respect to the conductivity and the permeability of the shield, accounting for ferrite and the aluminum shields [39].

However, in most wireless power supply devices, the position of the receiving coil is not fixed, and there is an inevitable positional misalignment between the transmitting and receiving coils, which causes the fluctuations of mutual inductance, degrades the performance, and affects the system security [9,40]. If an accurate calculation formula for mutual inductance of the coils structure can be given, it is significant to optimize the coil structure [41,42]. The completely analytical calculation of inductance for circular coils with bilateral finite cores in case of vertical misalignment (VM), horizontal misalignment (HM), and angular misalignment (AM) is realized [43]. The angular misalignment has a significant impact on WPT [44]. A harmonic model is developed for calculation using a series–series compensated system in tight coupling [45]. The proposed method is a more generalized and simpler one that can be used to calculate the mutual inductance of any size of coils, either spiral or circular, with any lateral and angular misalignment [46]; it is a single band.

In [47], only the nutation angle is considered in the angular misalignment case, and a general misalignment, which includes both the lateral and angular misalignment, is not studied. Mutual inductance between circular spiral coils is derived using the magnetic vector potential approach [48]. This method can calculate the mutual inductance of magnetic couplers with different receiver structures and receiver positons [49]. The formula for coil mutual inductance is derived using Maxwell’s equations and boundary conditions and a tactic for adapting the mutual inductance with the aid of an intelligent optimization algorithm, which is utilized for steadying the mutual inductance of the transmitting coil [50]. Luo and Wei [51] proposed a mutual inductance calculation method for a horizontally misaligned coil by Maxwell’s equations, in which only the magnetic shielding on one side of the coil is considered, but the thickness and width of the magnetic shielding is not. Luo et al. [17] derived the mutual inductance of the rectangular coil with bilateral magnetic shields by the separation of variables method. These studies failed to calculated the mutual inductance of rectangular coils with bilateral finite magnetic shields at arbitrary positions. The mutual inductance formulation of the rectangular coil at vertical misalignment is first obtained by the subdomain partition method (SPM) and the mutual inductance calculation method for rectangular coils with bilateral finite magnetic shielding at an arbitrary position [18].

From the parallel and coaxial circular coils to those in arbitrary relative positions, and from air-core coils to those with electromagnetic shielding, the research on the calculation methods of their mutual inductance has been studied. However, research on the calculation method of the mutual inductance of rectangular coils is relatively limited. The mutual inductance fluctuation rate of rectangular coils is smaller than that of circular coils when they are offset.

The calculation method of rectangular coil mutual inductance in complex, and misaligned environments is one of the important reasons for a WPT system based on rectangular coils. In this paper, a feasible mutual inductance calculation of a double-sided electromagnetic shielding rectangular coils is proposed. A prototype was built for experimental verification. Then, an optimization method of the mutual inductance is presented, and the parameters of the proposed structure that meets the design requirements are obtained by using the proposed optimization method. The mutual inductance calculation formula is verified by simulation and experiment. The experiment results show that the max error between the measured mutual inductance and the calculated result is less than 3.8%. This research not only offers an innovative technological pathway for enhancing the stability and efficiency of wireless power systems but also provides a solid theoretical foundation and guiding framework for coil design and optimization.

2. Magnetic Distribution of Rectangular Coils with Double-Sided Shielding

The structure of rectangular coils with double-sided shielding is shown in Figure 1. The model is divided into eight sections in the vertical direction. $t_2$ and $t_7$ represent the height of Region 2 and Region 7, which is aluminum. $t_3$ and $t_6$ represent the height of Region 3 and Region 6, which is ferrite. The upper surface of Region 3 is $z=0$. $a_1$, $a_2$ and $s_1$ represent the half-length, half-width and height of the transmitting coil, $b_1$, $b_2$ and $s_2$ represent the half-length, half-width and height of the receiving coil. There is the magnetic flux density generated by the current of the coil ($B_i$) and the reflected magnetic field model of the double-sided electromagnetic shielding ($B_r$). To simplify the calculation, it is generally believed that the incident magnetic field cannot pass through the electromagnetic shielding layer; that is, the incident magnetic flux density only exists in Regions 3 and 4.

2.1. Incident Magnetic Flux Density

The magnetic vector potential produced by a current flowing through a conductor in the air at any point $P(x,y,z)$ is as follows [52]:

$$A(x,y,z)={\displaystyle \frac{{\mu }_0}{4\pi }}{\int }_V{\displaystyle \frac{J(x^{\prime },y^{\prime },z^{\prime })\mathrm{d}{v}^{\prime }}{R}},$$

(1)

$$R=\sqrt{{(x-x^{\prime })}^2+{(y-y^{\prime })}^2+{(z-z^{\prime })}^2},$$

(2)

where ${\mu }_0$ is the magnetic conductivity of the vacuum, J is the current density, V is the current distribution of the conductor, and R is the distance from the source point $(x^{\prime },y^{\prime },z^{\prime })$ to $P(x,y,z)$. The double Fourier transform is used to solve the equation [53],

$$b(\xi ,\eta ,z)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }B(x,y,z)e^{-\mathrm{j}(x\xi +y\eta )}\mathrm{d}x\mathrm{d}y,$$

(3)

$$B(x,y,z)={\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }b(\xi ,\eta ,z)e^{\mathrm{j}(x\xi +y\eta )}\mathrm{d}\xi \mathrm{d}\eta ,$$

(4)

where $\xi$, $\eta$ is the Fourier integral variable.

Substituting (1) into (3), we can obtain

$$a(\xi ,\eta ,z)={\displaystyle \frac{{\mu }_0}{2}}{\int }_v{e}^{-\mathrm{j}(x^{\prime }\xi +y^{\prime }\eta )}J(x^{\prime },y^{\prime },x^{\prime }){\displaystyle \frac{e^{-k\mid z-s_1\mid }}{k}}\mathrm{d}{v}^{\prime },$$

(5)

$$k=\sqrt{{\xi }^2+{\eta }^2}.$$

(6)

The relationship between the incident magnetic flux density $B_i$ and the magnetic vector potential $A_i$ is as follows:

$$B_i=▽\times A_i.$$

(7)

Substituting the Fourier transform into (7):

$$\left\{\begin{array}{c}\begin{array}{cc}{b}_{ix}={\widehat{B}}_{ix}={({\displaystyle \frac{\partial A_{iz}}{\partial y}}-{\displaystyle \frac{\partial A_{iy}}{\partial z}})}^{\wedge }=-\mathrm{j}\eta a_{iz}-{\displaystyle \frac{\partial a_{iy}}{\partial z}}\hfill & \\ b_{iy}={\widehat{B}}_{iy}={({\displaystyle \frac{\partial A_{ix}}{\partial z}}-{\displaystyle \frac{\partial A_{iz}}{\partial y}})}^{\wedge }={\displaystyle \frac{\partial a_{ix}}{\partial z}}+\mathrm{j}\xi a_{iz}.\hfill & \\ b_{iz}={\widehat{B}}_{iz}={({\displaystyle \frac{\partial A_{iy}}{\partial x}}-{\displaystyle \frac{\partial A_{ix}}{\partial y}})}^{\wedge }=-\mathrm{j}\xi a_{iy}+\mathrm{j}\eta a_{ix}\hfill \end{array}\hfill \end{array}\right.$$

(8)

The component $a_x$ of the magnetic vector potential in the X-axis direction is generated only by the wires $l_1$ and $l_3$ which are parallel to the X-axis. The expression of $a_x$ is as follows:

$$\begin{array}{cc}\hfill a_x=& a_{x1}-a_{x3}={\displaystyle \frac{{\mu }_{0}I{e}^{\mathrm{j}{a}_2\eta }{e}^{-k\mid z-z_0\mid }}{2k}}({\int }_{-a_1}^{a_1}{e}^{-\mathrm{j}{x}^{\prime }\xi }\mathrm{d}{x}^{\prime }-{\int }_{-a_3}^{a_3}{e}^{-\mathrm{j}{x}^{\prime }\xi }\mathrm{d}{x}^{\prime })\hfill \\ \hfill & ={\displaystyle \frac{\mathrm{j}2{\mu }_{0}Isin(\xi a_1)sin(\eta a_2)e^{-\mid z-z_0\mid k}}{\xi k}}.\hfill \end{array}$$

(9)

Similarly, the component of the magnetic vector potential $a_y$ in the Y-axis direction is

$$a_y={\displaystyle \frac{-\mathrm{j}2{\mu }_{0}Isin(\eta a_2)sin(\xi a_1)e^{-\mid z-s_1\mid k}}{\eta k}}.$$

(10)

Substituting (9) and (10) into (8), the magnetic flux density in Region 4 $(0⩽z⩽s_1)$ is

$$b_{ix}={\displaystyle \frac{\mathrm{j}2{\mu }_{0}Isin(\xi a_1)sin(\eta a_2)}{\eta }}{e}^{-s_{1}k}{e}^{kz}=C_{ix}{e}^{kz},$$

(11)

$$b_{iy}={\displaystyle \frac{\mathrm{j}2{\mu }_{0}Isin(\xi a_1)sin(\eta a_2)}{\xi }}{e}^{-s_{1}k}{e}^{kz}=C_{iy}{e}^{kz},$$

(12)

$$b_{iz}={\displaystyle \frac{-2{\mu }_{0}Iksin(\xi a_1)sin(\eta a_2)}{\xi \eta }}{e}^{-s_{1}k}{e}^{kz}=C_{iz}{e}^{kz}.$$

(13)

The magnetic flux density in Region 5 $(s_1<z<d)$ is

$$b_{ix}^{\prime }={\displaystyle \frac{-\mathrm{j}2{\mu }_{0}Isin(\xi a_1)sin(\eta a_2)}{\eta }}{e}^{s_{1}k}{e}^{-kz}=C_{ix}^{\prime }{e}^{-kz},$$

(14)

$$b_{iy}^{\prime }={\displaystyle \frac{-\mathrm{j}2{\mu }_{0}Isin(\xi a_1)sin(\eta a_2)}{\xi }}{e}^{s_{1}k}{e}^{-kz}=C_{iy}^{\prime }{e}^{-kz},$$

(15)

$$b_{iz}^{\prime }={\displaystyle \frac{-2{\mu }_{0}Iksin(\xi a_1)sin(\eta a_2)}{\xi \eta }}{e}^{s_{1}k}{e}^{-kz}=C_{iz}^{\prime }{e}^{-kz}$$

(16)

2.2. Reflected Magnetic Flux Density

2.2.1. Electromagnetic Shielding Region

In these regions $(-t_2-t_3<z<0)$ or $(d<z<d+t_7+t_8)$, that is, Region 2, Region 3, Region 6 and Region 7, there are induced eddy currents. According to Maxwell’s equations, the reflection magnetic flux density is

$${▽}^2{B}_n-\mathrm{j}\omega {\mu }_0{\mu }_n{\sigma }_n{B}_n=0,$$

(17)

$$▽·B_n=0,$$

(18)

where $\omega$ is the source angular frequency, ${\mu }_n$ is the magnetic conductivity in region n, and ${\sigma }_n$ is the specific conductance in region n.

Substituting (17) and (18) into (3),

$${\displaystyle \frac{{\partial }^2{b}_n}{\partial z^2}}-({\xi }^2+{\eta }^2+\mathrm{j}\omega {\mu }_0{\mu }_n{\sigma }_n)b_n=0,$$

(19)

$$-\mathrm{j}\xi b_{nx}-\mathrm{j}\eta b_{ny}+{\displaystyle \frac{\partial b_{nz}}{\partial z}}=0.$$

(20)

According to Equation (19), the vertical component of the reflection magnetic flux density in region n can be expressed as

$$b_{nz}=C_n{e}^{{\lambda }_{n}z}+C_n^{\prime }{e}^{-{\lambda }_{n}z},$$

(21)

$${\lambda }_n=\sqrt{k^2+\mathrm{j}\omega {\mu }_0{\mu }_n{\sigma }_n}.$$

(22)

There is no current density about the Z-direction in these regions,

$$\xi b_{ny}=\eta b_{nx}.$$

(23)

By Equations (20) and (23), we can obtain

$$\left\{\begin{array}{c}\begin{array}{cc}{b}_{nx}={\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial b_{nz}}{\partial z}}\hfill & \\ b_{ny}={\displaystyle \frac{\eta }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial b_{nz}}{\partial z}}.\hfill \end{array}\hfill \end{array}\right.$$

(24)

2.2.2. Region 4 and Region 5

For Region 4 and Region 5, $(0<z<d)$. According to Maxwell’s equations, the reflection magnetic flux density is

$${▽}^2{B}_r=0,$$

(25)

$$▽·B_r=0.$$

(26)

Substituting (25) and (26) into (3), we can obtain

$${\displaystyle \frac{{\partial }^2{b}_r}{\partial z^2}}-({\xi }^2+{\eta }^2)b_r=0,$$

(27)

$$-\mathrm{j}\xi b_{rx}-\mathrm{j}\eta b_{ry}+{\displaystyle \frac{\partial b_{rz}}{\partial z}}=0.$$

(28)

The expression for the reflection magnetic flux is,

$$b_{rz}=C_{rz}{e}^{kz}+C_{rz}^{\prime }{e}^{-kz},$$

(29)

there is no eddy current in the following regions:

$$\left\{\begin{array}{c}\begin{array}{cc}{b}_{rx}={\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial b_{rz}}{\partial z}}\hfill & \\ b_{ry}={\displaystyle \frac{\eta }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial b_{rz}}{\partial z}}.\hfill \end{array}\hfill \end{array}\right.$$

(30)

2.2.3. Region 1 and Region 8

In these regions, $(z<-t_2-t_3)$ or $(z>t_7+t_8)$ for Region 1 and Region 8, respectively. The reflected magnetic flux density goes to zero, z tends to both positive and negative infinity, and the reflected magnetic flux densities can be, respectively, expressed as

$$b_{1z}=C_{1z}·e^{kz},$$

(31)

$$b_{8z}=C_{8z}·e^{-kz}.$$

(32)

There is no eddy current in the following regions:

$$\left\{\begin{array}{c}\begin{array}{cc}{b}_{nx}={\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial b_{nz}}{\partial z}}\hfill & \\ b_{ny}={\displaystyle \frac{\eta }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial b_{nz}}{\partial z}}.\hfill \end{array}\hfill \end{array}\right.$$

(33)

2.3. Boundary Conditions

According to the boundary conditions (the tangential component of magnetic field intensity and the normal component of magnetic flux density are continuous on both sides of the interface), $H_x$ and $B_z$ at the plane $(z=-t_2-t_3,z=-t_3,z=0,z=d+t_7,z=d+t_7+t_8)$ are continuous.

$$\left\{\begin{array}{c}\begin{array}{cc}{\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{1z}{e}^{kz})={\displaystyle \frac{1}{{\mu }_2}}{\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{2z}{e}^{{\lambda }_{2}z}+C_{2z}^{\prime }{e}^{-{\lambda }_{2}z})z=-t_2-t_3\hfill & \\ {\displaystyle \frac{1}{{\mu }_2}}{\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{2z}{e}^{{\lambda }_{2}z}+C_{2z}^{\prime }{e}^{-{\lambda }_{2}z})={\displaystyle \frac{1}{{\mu }_3}}{\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{3z}{e}^{{\lambda }_{3}z}+C_{3z}^{\prime }{e}^{-{\lambda }_{3}z})z=-t_3\hfill & \\ {\displaystyle \frac{1}{{\mu }_3}}{\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{3z}{e}^{{\lambda }_{3}z}+C_{3z}^{\prime }{e}^{-{\lambda }_{3}z})={\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{rz}{e}^{kz}+C_{rz}^{\prime }{e}^{-kz})+{\displaystyle \frac{\xi }{\mathrm{j}k}}{C}_{iz}{e}^{kz}z=0\hfill & \\ {\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{rz}{e}^{kz}+C_{rz}^{\prime }{e}^{-kz})-{\displaystyle \frac{\xi }{\mathrm{j}k}}{C}_{iz}{e}^{-kz}={\displaystyle \frac{1}{{\mu }_6}}{\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{6z}{e}^{{\lambda }_{6}z}+C_{6z}^{\prime }{e}^{-{\lambda }_{6}z})z=d\hfill & \\ {\displaystyle \frac{1}{{\mu }_6}}{\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{6z}{e}^{{\lambda }_{6}z}+C_{6z}^{\prime }{e}^{-{\lambda }_{6}z})={\displaystyle \frac{1}{{\mu }_7}}{\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{7z}{e}^{{\lambda }_{7}z}+C_{7z}^{\prime }{e}^{-{\lambda }_{7}z})z=d+t_6\hfill & \\ {\displaystyle \frac{1}{{\mu }_7}}{\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{7z}{e}^{{\lambda }_{7}z}+C_{7z}^{\prime }{e}^{-{\lambda }_{7}z})={\displaystyle \frac{\xi }{\mathrm{j}{k}^2}}{\displaystyle \frac{\partial }{\partial z}}(C_{8z}{e}^{-kz})z=d+t_6+t_7\hfill \end{array}\hfill \end{array}\right.$$

(34)

$$\left\{\begin{array}{c}\begin{array}{cc}{C}_{1z}{e}^{kz}=C_{2z}{e}^{{\lambda }_{2}z}+C_{2z}^{\prime }{e}^{-{\lambda }_{2}z}z=-t_2-t_3\hfill & \\ C_{2z}{e}^{{\lambda }_{2}z}+C_{2z}^{\prime }{e}^{-{\lambda }_{2}z}=C_{3z}{e}^{{\lambda }_{3}z}+C_{3z}^{\prime }{e}^{-{\lambda }_{3}z}z=-t_3\hfill & \\ C_{3z}{e}^{{\lambda }_{3}z}+C_{3z}^{\prime }{e}^{-{\lambda }_{3}z}=C_{rz}{e}^{kz}+C_{rz}^{\prime }{e}^{-kz}+C_{iz}^{\prime }{e}^{kz}z=0\hfill & \\ C_{rz}{e}^{kz}+C_{rz}^{\prime }{e}^{-kz}+C_{iz}^{\prime }{e}^{kz}=C_{6z}{e}^{{\lambda }_{6}z}+C_{6z}^{\prime }{e}^{-{\lambda }_{6}z}z=d\hfill & \\ C_{6z}{e}^{{\lambda }_{6}z}+C_{6z}^{\prime }{e}^{-{\lambda }_{6}z}=C_{7z}{e}^{{\lambda }_{7}z}+C_{7z}^{\prime }{e}^{-{\lambda }_{7}z}z=d+t_6\hfill & \\ C_{7z}{e}^{{\lambda }_{7}z}+C_{7z}^{\prime }{e}^{-{\lambda }_{7}z}=C_{8z}{e}^{-kz}z=d+t_6+t_7\hfill \end{array}\hfill \end{array}\right.$$

(35)

The magnetic flux density in the region where the receiving coil is located can be obtained,

$$b_{5z}=C_{rz}{e}^{kz}+C_{rz}^{\prime }{e}^{-kz}+C_{iz}^{\prime }{e}^{kz},$$

(36)

here,

$$C_{rz}={\displaystyle \frac{1}{4}}V{C}_{7z},$$

(37)

$$C_{rz}^{\prime }={\displaystyle \frac{1}{4}}W{C}_{7z}-C_{iz}^{\prime },$$

(38)

$$C_{iz}^{\prime }={\displaystyle \frac{-2{\mu }_{0}Iksin(\xi a_1)sin(\eta a_2)}{\xi \eta }}{e}^{s_{1}k},$$

(39)

$$C_{7z}={\displaystyle \frac{(m_2+1)F-(m_2-1)e^{2{\lambda }_2(-t_2-t_3)}B}{(m_2-1)e^{2{\lambda }_2(-t_2-t_3)}A-(m_2+1)E}},$$

(40)

$$V=(1+m_6)H{e}^{({\lambda }_6-k)d}+(1-m_6)N{e}^{(-{\lambda }_6-k)d},$$

(41)

$$W=(1-m_6)H{e}^{({\lambda }_6+k)d}+(1+m_6)N{e}^{(-{\lambda }_6+k)d},$$

(42)

$$\begin{array}{cc}\hfill A=& {\displaystyle \frac{1}{16m_2{m}_3}}\{(m_2+m_3)[(m_3+1)V+(m_3-1)W]e^{({\lambda }_2-{\lambda }_3)}{t}_3\hfill \\ \hfill & +(m_2-m_3)[(m_3-1)V+(m_3+1)W]e^{({\lambda }_2+{\lambda }_3)}{t}_3\},\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill B=& {\displaystyle \frac{1}{4m_2{m}_3}}[(m_2+m_3)(1-m_3)e^{({\lambda }_2-{\lambda }_3)t_3}\hfill \\ \hfill & +(m_2-m_3)(-m_3-1)e^{({\lambda }_2+{\lambda }_3)t_3}]C_{iz}^{\prime }\hfill \\ \hfill +& {\displaystyle \frac{1}{4m_2{m}_3}}[(m_2+m_3)(1+m_3)e^{({\lambda }_2-{\lambda }_3)t_3}\hfill \\ \hfill & +(m_2-m_3)(m_3-1)e^{({\lambda }_2+{\lambda }_3)t_3}]C_{iz},\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill E=& {\displaystyle \frac{1}{16m_2{m}_3}}\{(m_2-m_3)[(m_3+1)V+(m_3-1)W]e^{(-{\lambda }_2-{\lambda }_3)}{t}_3\hfill \\ \hfill & +(m_2+m_3)[(m_3-1)V+(m_3+1)W]e^{(-{\lambda }_2+{\lambda }_3)}{t}_3\},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill F=& {\displaystyle \frac{1}{4m_2{m}_3}}[(m_2+m_3)(m_3-1)e^{(-{\lambda }_2+{\lambda }_3)t_3}+\hfill \\ \hfill & (m_2-m_3)(m_3+1)e^{(-{\lambda }_2-{\lambda }_3)t_3}]C_{iz}\hfill \\ \hfill +& {\displaystyle \frac{1}{4m_2{m}_3}}[(m_2-m_3)(1-m_3)e^{(-{\lambda }_2-{\lambda }_3)t_3}+\hfill \\ \hfill & (m_2+m_3)(-m_3-1)e^{(-{\lambda }_2+{\lambda }_3)t_3}]C_{iz}^{\prime },\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill N=& (1-{\displaystyle \frac{m_7}{m_6}})e^{({\lambda }_6+{\lambda }_7)(d+t_6)}\hfill \\ \hfill & +(1+{\displaystyle \frac{m_7}{m_6}}){\displaystyle \frac{(m_7+1)}{(m_7-1)}}{e}^{({\lambda }_6-{\lambda }_7)(d+t_6)}{e}^{2{\lambda }_7(d+t_6+t_7)},\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill H=& (1+{\displaystyle \frac{m_7}{m_6}})e^{(-{\lambda }_6+{\lambda }_7)(d+t_6)}\hfill \\ \hfill & +(1-{\displaystyle \frac{m_7}{m_6}}){\displaystyle \frac{(m_7+1)}{(m_7-1)}}{e}^{(-{\lambda }_6-{\lambda }_7)(d+t_6)}{e}^{2{\lambda }_7(d+t_6+t_7)},\hfill \end{array}$$

(48)

$$m_i={\displaystyle \frac{{\lambda }_i}{{\mu }_{i}k}}.$$

(49)

Substituting (36) into (4),

$$B_{5z}(x,y,z)={\displaystyle \frac{1}{4{\pi }^2}}{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }(C_{rz}{e}^{kz}+C_{rz}^{\prime }{e}^{-kz}+C_{iz}^{\prime }{e}^{kz})e^{\mathrm{j}(x\xi +y\eta )}\mathrm{d}\xi \mathrm{d}\eta$$

(50)

3. Mutual Inductance Calculating for a Rectangular Coil with Misalignment in a Double-Sided Electromagnetic Shield

Using mutual inductance calculating for a rectangular coil with misalignment in a double-sided electromagnetic shield [54],

$$M={\displaystyle \frac{\oint B\mathrm{d}s}{I}},$$

(51)

where s is the region of the receiving coil, and I is the current of the transmitting coil.

Substituting (50) into (51), the mutual inductance for a rectangular coil in a double-sided electromagnetic shield can be obtained based on the magnetic flux density,

$$\begin{array}{cc}\hfill & M={\displaystyle \frac{1}{4{\pi }^{2}I}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-b_1}^{b_1}{\int }_{-b_2}^{b_2}(C_{rz}{k}^{s_2}+C_{rz}^{\prime }{k}^{-s_2}+C_{iz}{k}^{s_2})e^{\mathrm{j}(x\xi +y\eta )}\mathrm{d}x\mathrm{d}y\mathrm{d}\xi \mathrm{d}\eta \hfill \\ \hfill & ={\displaystyle \frac{1}{4{\pi }^{2}I}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }[(C_{rz}+C_{iz}^{\prime })e^{k{s}_2}+C_{rz}^{\prime }{e}^{-k{s}_2}]{\displaystyle \frac{-1}{\xi \eta }}(e^{\mathrm{j}\xi b_1}-e^{-\mathrm{j}\xi b_1})(e^{\mathrm{j}\eta b_2}-e^{-\mathrm{j}\eta b_2})\mathrm{d}\xi \mathrm{d}\eta .\hfill \end{array}$$

(52)

The mutual inductance between multi-turn coils can be calculated by (53).

$$M_{total}=\sum _{m=1}^{N_T}\sum _{n=1}^{N_R}{M}_{mn},$$

(53)

where $N_T$ is the number of turns of transmitting coils, $N_R$ is the number of turns of receiving coils, and $M_{mn}$ is the the mutual inductance between the m-th turn of the transmitting coil and the n-th turn of the receiving coil.

4. Verification

The calculation was carried out with the model that is divided into regions as shown in Figure 1. The magnetic vector potential and induced eddy currents for each region are obtained. According to the boundary conditions, the magnetic is obtained (50); finally, the mutual inductance is calculated using (52), and the calculation time is not more than 15 s.

The simulation was carried out using Ansys Maxwell 15.0, and the model is shown in Figure 2. The transmitting coil and the electromagnetic shielding layer below it were fixed, while the receiving coil and the shielding layer above it were adjusted to change the relative positions. The condition of vertical and horizontal misalignment was discussed.

The experimental device is shown in Figure 3. The 3D movable platform was developed independently by the group, and the non-magnetic acrylic material was used to minimize the influence of the environment. The mutual inductance was measured measured by an IM3536 Impedance Analyzer, and the frequency was set at 85.0 kHz. In addition, winding the long terminal blocks connected to the impedance analyzer can also reduce the impact of the current on the terminal blocks on the electromagnetic field.

The mutual inductance is calculated by measuring the difference in inductance when the transmitting coil and the receiving coil are connected in the same direction or in the opposite direction. The two spiral coils are connected in the same direction, the self-inductance is (54), and when in the opposite direction, it is (55). Here, $L_T$ and $L_R$ are the self-inductance of the transmitting coil and the receiving coil, respectively. The mutual inductance of the coils is calculated in (56).

$$L_0=L_T+L_R+2M,$$

(54)

$$L_1=L_T+L_R-2M,$$

(55)

$$M={\displaystyle \frac{\mid L_0-L_1\mid }{4}},$$

(56)

4.1. Optimize Parameters

The parameters of the proposed structure that meet the design requirements are obtained by using the optimization method. The mutual inductance is determined by the length and width of the coils, the number of turns and the relative position between the coils. The mutual inductance of different parameters is calculated by the formula in order to obtain the optimal parameters.

The simulation results of vertical misalignment of the receiving coil migrated from a height of 40 $\mathrm{mm}$ to 75 $\mathrm{mm}$. ${\epsilon }_1$ is the simulation error, ${\epsilon }_2$ is the experimental error, $M_c$ is the calculated value of the proposed formula for mutual inductance in this paper, $M_m$ is the simulated value and $M_s$ is the experimental value. We set the conditional judgment as ${\epsilon }_1$, ${\epsilon }_2$ < 5%.

$${\epsilon }_1={\displaystyle \frac{\mid M_c-M_m\mid }{M_m}},$$

(57)

$${\epsilon }_2={\displaystyle \frac{\mid M_c-M_s\mid }{M_s}}.$$

(58)

The optimization method is shown in Figure 4.

The result that meets the requirements that is obtained in the simulation is given in Figure 5, Figure 6 and Figure 7. The step is the increment of independent variables in the numerical calculation process. As shown in Figure 5, $N_1$ represents the turn of transmitting coils, and $N_2$ represents the turn of receiving coils.

When the receiving coils migrated along the X-axis, the mutual inductance is shown in Figure 6.

When the receiving coils migrated along the Y-axis, the mutual inductance is shown in Figure 7.

From the Matlab simulation diagram, we can obtain parameters of the coils, and they are shown in

Table 1. Parameters of coils and magnetic shielding.

Parameters	Value	Parameters	Value
inner length	120 $\mathrm{mm}$	length of aluminum shielding	600 $\mathrm{mm}$
inner width	100 $\mathrm{mm}$	height of aluminum shielding	5 $\mathrm{mm}$
turn of transmitting coil	12	magnetic conductivity of aluminum	1
turn of receiving coil	10	specific conductance of aluminum of aluminum	3.8 × 107 $\mathrm{S}/\mathrm{m}$
height of transmitting coil	4 $\mathrm{mm}$	length of ferrite shielding	550 $\mathrm{mm}$
height of receiving coil	40 to 75 $\mathrm{mm}$	height of ferrite shielding	15 $\mathrm{mm}$
diameter of coil	2 $\mathrm{mm}$	magnetic conductivity of ferrite	1000
variation of coil	4.8 $\mathrm{mm}$	specific conductance of aluminum of ferrite	0.01 $\mathrm{S}/\mathrm{m}$

.

4.2. Vertical Misalignment

The simulation results of vertical misalignment of the receiving coil show that it migrated from a height of 40 $\mathrm{mm}$ to 75 $\mathrm{mm}$. During the process, the relative position between the shielding layer above the receiving coil and the receiving coil is unchanged. The calculated, simulated, and experimental values of mutual inductance are shown in

Table 2. Mutual inductance and error during vertical misalignment.

Height/mm	Calculated/ μH	Simulated/ μH	Experimental/μH	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$
40	42.6490	42.6130	42.5172	0.08%	0.31%
45	36.9970	36.9390	36.8606	0.16%	0.37%
50	32.3810	32.3220	32.2424	0.18%	0.43%
55	28.5640	28.4960	28.3880	0.24%	0.62%
60	25.3330	25.2890	25.1469	0.17%	0.74%
65	22.6110	22.5580	22.4315	0.23%	0.80%
70	20.2750	20.2160	20.1041	0.29%	0.85%
75	18.2560	18.1950	18.0914	0.34%	0.91%

.

The simulation error and experimental error for the vertical misalignment are less than $1\%$. The calculated, simulated, and experimental results have consistency. Figure 8 shows the calculated, simulated, and experimental values of mutual inductance. The mutual inductance is maximum when the receiving coil is the lowest. As the height of the receiving coil increases, the transmission distance (the vertical distance between the receiving coil and the transmitting coil) also increases. Consequently, the magnetic flux between decreases, leading to a gradual decrease in mutual inductance.

4.3. Horizontal Misalignment

When the receiving coil and the transmitting coil are in parallel planes but not coaxial, the misalignment distance of the receiving coil along the X-axis is $\Delta x$. The calculated, simulated, and experimental values of mutual inductance are shown in

Table 3. Mutual inductance and error during horizontal misalignment.

Δx/mm	Calculated/ μH	Simulated/ μH	Experimental/ μH	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$
0	18.2560	18.1950	18.0914	0.34%	0.91%
10	18.0620	17.9770	17.9746	0.36%	0.49%
20	17.5000	17.4250	17.4119	0.43%	0.51%
30	16.6050	16.5040	16.5003	0.61%	0.63%
40	15.4260	15.3200	15.3116	0.69%	0.75%
50	14.0420	13.9100	13.9322	0.95%	0.79%
60	12.5210	12.3720	12.4064	1.20%	0.92%
70	10.9260	10.7560	10.8079	1.58%	1.09%
80	9.3153	9.1282	9.1736	2.05%	1.54%
90	7.7446	7.5327	7.5778	2.81%	2.20%
100	6.2633	6.0342	6.1103	3.80%	2.50%

, the receiving coil at height is 75 $\mathrm{mm}$, and $\Delta x$ is from 0 $\mathrm{mm}$ to 100 $\mathrm{mm}$ with a step size of 10 $\mathrm{mm}$.

Table 3 shows that ${\epsilon }_1$ does not exceed 3.80% and ${\epsilon }_2$ does not exceed 2.5% when the receiving coil is along the X-axis. Figure 9 shows the calculated, simulated, and experimental values of mutual inductance. The mutual inductance is maximum when $\Delta x$ is the lowest. As $\Delta x$ increases, the magnetic flux between decreases, and then the mutual inductance decreases.

4.4. Error Analysis

There is a method equivalent error when calculating mutual inductance because in the theoretical calculation, the multi-turn planar spiral rectangular coil is equivalent to several standard rectangular coils, but in the actual experiment, there is an angle between the turns of the multi-turn planar spiral rectangular coil, which is not the standard rectangular coil. And there is a measuring equipment error, because the wire of the probe and coil of the impedance analyzer of the measuring equipment are connected through the clamp, leading to a contact error between them. In the theoretical calculation, the influence of the external magnetic field is ignored in the vacuum environment, but in the test process, the coil parameters will be affected by the surrounding magnetic field, resulting in measurement errors. According to the data in Table 2 and Table 3, the error is less than 3.80%. The results show that the calculated, simulated, and experimental measurements match. The proposed method of this article is compared with other methods in the literature mentioned in

Table 4. Comparison of the methods.

Literature	Method	$\mathit{\epsilon }$
[22]	Biot–Savar	5.84%
[20]	Bessel	8.57%
[18]	vector coordinate	4.7%
[39]	3-D Analytical Model	6%
our work		5.84%

.

In conclusion, the calculated, simulated, and experimental values of mutual inductance under the condition of vertical and horizontal misalignment are consistent. It demonstrates the accuracy of the mutual inductance formula for the double-sided shielded rectangular coil in this paper.

5. Conclusions

In this paper, the mutual inductance calculation formula of the double-sided electro-magnetic shielding rectangular coil is derived using the magnetic flux density method. A model for the reflected magnetic field of the double-sided shielded rectangular coil is given, and the reflected magnetic flux density generated by the induced eddy currents in the electromagnetic shielding layer is calculated. The mutual inductance calculation formula for the double-sided shielded rectangular coil is deduced using the magnetic flux density method, providing a feasible method for the mutual inductance calculation for such coils. The experimental and simulated results convincingly demonstrate the accuracy of the mutual inductance formula of rectangular coils with misalignment with magnetic shielding in this paper.

This analytical method can accurately numerically solve the mutual inductance between coils. Compared with finite element simulation, the proposed method is capable of model parameterization, and the parameters can be flexibly modified according to the requirements. A prototype is built for experimental verification, and the parameters are optimized by the mutual inductance calculation formula. The proposed method not only provides a theoretical basis for the fast optimization of mutual inductance between coils in wireless charging systems for electric vehicles but also provides a theoretical basis for the next study of mutual inductance calculation methods between rectangular coils with double-sided magnetic shields.