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Design of Edge-Enhanced Coil Structure to Obtain Constant Mutual Inductance With Horizontal Misalignment in Wireless Power Transfer Systems of Electric Vehicles
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Zhongqi LI1, Zhongbang CHEN2, Minsheng YANG3, Yu CHENG2, Xinbo XIONG2, Shoudao HUANG1
CPSS Transactions on Power Electronics and Applications | 2024, 9(2) : 141 - 151
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CPSS Transactions on Power Electronics and Applications | 2024, 9(2): 141-151
• Papers •
Design of Edge-Enhanced Coil Structure to Obtain Constant Mutual Inductance With Horizontal Misalignment in Wireless Power Transfer Systems of Electric Vehicles
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Zhongqi LI1, Zhongbang CHEN2, Minsheng YANG3, Yu CHENG2, Xinbo XIONG2, Shoudao HUANG1
Affiliations
  • 1 Hunan University College of Electrical and Information Engineering Changsha 410082 China
  • 2 Hunan University of Technology College of Traffic Engineering Zhuzhou 412007 China
  • 3 Hunan University of Arts and Sciences College of Electrical Engineering Changde 415000 China

    • Zhongqi Li was born in China in 1985. He received the M.Sc. degree from Hunan University of Technology, China in 2012, and the Ph.D. degree from the Hunan University in 2016. From 2016, he is working Assistant Professor in Hunan University of Technology, China. From 2020, he is now working post-doctoral fellow in Hunan University. His research interests include wireless power transfer systems and soft-switching power converters.

    Zhongbang Chen was born in China in 1999. He received a Bachelor's degree in Automation from Software Engineering Institute of GuangZhou, GuangZhou, China, in 2017. He is currently pursuing the Master's degree in Control Theory and Control Engineering at Hunan University of Technology. His current research interests include wireless power transfer systems.

    Minsheng Yang graduated from Hunan University with a Ph.D. in 2012, and is currently the Vice Dean of the School of Computer and Electrical Engineering at Hunan University of Arts and Sciences, where he has long been engaged in research and teaching related to the fields of electrical energy conversion and control, electrical control technology, electrical equipment intelligence and intelligent control.

    Yu Cheng was born in China in 1998. He received a Bachelor's degree in Mechanical Design-Manufacture and Automation from College of Science and Technology Ningbo University, NingBo, China, in 2021. He is currently pursuing the Master's degree in Electronic Information at Hunan University of Technology. His current research interests include wireless power transfer systems.

    Xinbo Xiong was born in China in 1998. He received a Bachelor's degree in Electrical Engineering and Automation from Hohai University, NanJing, China, in 2020. He is currently pursuing the Master's degree in Energy Power at Hunan University of Technology. His current research interests include wireless power transfer systems.

    Shoudao Huang received the B.Eng. and Ph.D. degrees in electrical engineering from the College of Electrical and Information Engineering, Hunan University, Changsha, China, in 1983, and 2005, respectively. He is currently a Full-Time Professor with the College of Electrical and Information Engineering, Hunan University. His research interests include motor design and control, power electronic system and control, and wind energy conversion systems.

Published: 2024-06-10 doi: 10.24295/CPSSTPEA.2024.00006
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Horizontal misalignment to the Y and Xaxes can be as much as half the side length of a transmitting resonant coil or 10 cm for both dynamic and static wireless power transfer (DSWPT) systems. Misalignment to the Yaxis and Xaxis may cause DSWPT systems to malfunction due to fluctuations in mutual inductance. In this paper, a structure of edgeenhanced coil (EEC) is proposed. The mutual inductance expression of the EEC structure is then established. Moreover, the variation of the mutual inductance of the EEC structure is obtained based on the mutual inductance expression. The study demonstrates that the mutual inductance of the EEC structure can be increased while reducing its fluctuation. The problem that quasiconstant mutual inductance is obtained at the expense of mutual inductance value is solved. Therefore, the high transmission efficiency of DSWPT systems can be obtained, and the transmission efficiency and output power can be maintained almost constant with the misalignment to the Yaxis or Xaxis. The calculated, simulated, and measured results validating the effectiveness of the EEC structure are shown.

Coil structure  /  mutual inductance optimization  /  quasi-constant mutual inductance  /  wireless power transfer
Zhongqi LI, Zhongbang CHEN, Minsheng YANG, Yu CHENG, Xinbo XIONG, Shoudao HUANG. Design of Edge-Enhanced Coil Structure to Obtain Constant Mutual Inductance With Horizontal Misalignment in Wireless Power Transfer Systems of Electric Vehicles[J]. CPSS Transactions on Power Electronics and Applications, 2024 , 9 (2) : 141 -151 . DOI: 10.24295/CPSSTPEA.2024.00006
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I. Introduction
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WIRELESS power transfer (WPT) technology is receiv- ing increasing attention because of the convenience and safety of this technology [1]-[3]. This technology is widely used in electric vehicles (EVs)[4], automatic guided vehicles (AGVs)[5],[6], underwater vehicles (UVs)[7],[8], and drones [9],[10]. The advantage of WPT technology is that EVs can be charged while on the road using a dynamic WPT system, eliminating the need for batteries. At the same time, the static WPT technology provides smart charging services for EVs at home or office. Therefore, a set of coil structure should be used to simultaneously satisfy the dynamic and static WPT system requirements of EVs.
However, the unavoidable horizontal misalignment along the Y-axis and X-axis causes a change of the mutual inductance (MI) between the transmitting and receiving coils in the WPT system. The horizontal misalignment along the Y-axis may reach half the side length of a transmitting resonance coil in a dynamic WPT system for EVs, as shown in Fig. 1(a). According to SAE J2954, in a static wireless charging system for electric vehicles, the horizontal misalignment along the Y-axis and the X-axis can reach a least 7.5cm and 10cm, respectively, as shown in Fig. 1(b). Large horizontal misalignment may result in the change of MI in the dynamic WPT system. The change of MI may lead to fluctuation in system performance, including the efficiency and output power.
At present, the main methods for solving the effect of horizontal misalignment on output power and efficiency fluctuation are impedance matching methods and coil structure design methods. Impedance matching methods including LC, T, LCC-LCC [11],[12], LCL and CL [13], LCC-C [14], LCC-S [15], and DC-DC converters [16] are proposed by scholars. These impedance-matching networks can improve the system performance. However, it is difficult to further improve the performance of WPT systems with large misalignment because the change of MI in the dynamic WPT systems is too fast and large.
The method of designing coil structure may be an effective method to deal with the rapid and large variation in MI. Two types of coil structures are commonly used. The first type is the long-track coil structures; the second type is the short-individual coil structures. Several long-track coil structures, such as E-core structure, U-core structure, W-core structure, S-core structure [17], and I-core structure [18], are proposed successively to keep the MI constant. These structures have good misalignment tolerance. However, the MI of the long-track coil structures is very small. Thus, the short-individual coil structures are proposed to increase the MI. A structure of DD coil is proposed to enhance the MI. This structure has good misalignment in the Y-axis. However, the MI in the X-axis is highly variable [19]. Therefore, it is recommended to overlap a single-pole (Q) coil and a double-pole (DD) coil to form a DDQ coil to improve the misalignment capability in the X-axis [20]. Then, an asymmetric coil structure is proposed. Experiments show that the MI is constant when the misalignment along Y-axis or X-axis direction are within 10cm [21]. However, the efficiency of this structure is only 65.6% because of the small MI. An asymmetrical three-coil structure is proposed to improve efficiency and keep the MI constant. However, the maximum misalignment along the Y-axis or X-axis direction is only 10cm with this structure [22]. To improve the misalignment distance, a structure with one transmit coil and four cascaded receive coils is proposed. The maximum misalignment along the Y-axis direction can reach 24cm and the fluctuation in MI is 8.4%. However, the MI value of this structure is reduced by 42.79% compared with the traditional structure [23]. To further reduce the reduction rate of MI and the ratio of fluctuation in MI, a reverse series coil is added to the transmitting coil [24]. The fluctuation in MI is 6.4% when the misalignment reaches 50% of the diameter of the transmitting coil. Moreover, a structure of unsymmetrical and the opposite series coil is proposed. The ratio of fluctuation in MI is 4.2% when the misalignment is within 50% of the side length of the transmitting coil [25]. However, compared with the traditional structure, the MI value of this structure is reduced by 32.81%. In summary, the problem of obtaining constant MI at the expense of MI values has not been solved.
In this paper, an edge-enhanced coil (EEC) structure is proposed. By using the EEC structure, not only the MI fluctuation is reduced, but also the MI value is increased. Therefore, higher efficiency and smaller output power fluctuation are also achieved by the proposed structure compared to the traditional structure. Experimental results verify the effectiveness of the EEC structure.
II. Edge-Enhanced Coil Structure
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A. EEC Structure
An edge-enhanced coil (EEC) structure is proposed, as shown in Fig. 2. The EEC structure consists of a transmitting coil and a receiving coil. The transmitting coil is labeled ${T}_{x}$, the receiving coil is labeled ${R}_{x}$. The receiving coil is composed of ${R}_{x1},{R}_{x2}$, and ${R}_{x3}$. The MI at the receiving resonance coil edge can be increased by adding ${R}_{x1}$ and ${R}_{x3}$ into ${R}_{x2}$ when the receiving resonance coil is moved in the Y-axis. Parameters ${\Delta }_{x}$ and ${\Delta }_{y}$ are the misalignment along the X-axis and Y-axis, respectively. Parameters ${I}_{1}$ and ${I}_{2}$ are the currents of ${T}_{x}$ and ${R}_{x}$, respectively. $D$ is the transmitting distance between ${T}_{x}$ and ${R}_{x}$.
Fig. 3 shows the proposed ${R}_{x}$ and ${T}_{x}$. Parameters ${l}_{1 -\text{inner }}$, ${l}_{2 -\text{inner }},{l}_{3 -\text{inner }}$ and ${l}_{4 -\text{inner }}$ are the inner length of ${R}_{x1},{R}_{x2},{R}_{x3}$, and ${T}_{x}$, respectively. Parameters ${l}_{1 -\text{outer }},{l}_{2 -\text{outer }},{l}_{3 -\text{outer }}$ and ${l}_{4 -\text{outer }}$ are the outer length of ${R}_{x1},{R}_{x2},{R}_{x3}$ and ${T}_{x}$, respectively. Parameters ${h}_{1 -\text{inner }},{h}_{2 -\text{inner }},{h}_{3 -\text{inner }}$ and ${h}_{4 -\text{inner }}$ are the inner width of ${R}_{x1},{R}_{x2}$, ${R}_{x3}$, and ${T}_{x}$, respectively. Parameters ${h}_{1 -\text{ outer }},{h}_{2 -\text{ outer }},{h}_{3 -\text{ outer }}$ and ${h}_{4\text{-outer }}$ are the outer width of ${R}_{x1},{R}_{x2},{R}_{x3}$ and ${T}_{x}$, respectively.
There are two outstanding characteristics of the EEC structure. First, the size of ${T}_{x}$ is smaller than that of ${R}_{x}$. The purpose of this design is to reduce MI fluctuation when ${R}_{x}$ is moved in the Y-axis and X-axis directions. Second, ${R}_{x}$ is composed of ${R}_{x1}$, ${R}_{x2}$ and ${R}_{x3}$. The size of ${R}_{x2}$ is larger than that of ${R}_{x1}$ and ${R}_{x3},{R}_{x1}$ is the same as ${R}_{x3}$ in size. ${R}_{x1}$ and ${R}_{x3}$ are on the left and right of ${R}_{x2}$, respectively. ${R}_{x1}$ and ${R}_{x3}$ are connected to ${R}_{x2}$ in series in the same direction. The purpose of this design is to increase the magnetic flux density on both sides of ${R}_{x2}$ in the Y-axis direction with the misalignment along the Y-axis. Therefore, not only can the MI be increased, but the MI volatility can also be reduced.
When ${R}_{x}$ is moved in the Y-axis direction, the MI between ${T}_{x}$ and ${R}_{x2}$ may be decreased with the increase of misalignment distance. This phenomenon is consistent with the law of MI of the traditional coil structure. However, in the EEC structure, ${R}_{x1}$ and ${R}_{x3}$ are added into ${R}_{x}$. The MI between ${T}_{x}$ and ${R}_{x1}$ may be sharply increased and the MI between ${T}_{x}$ and ${R}_{x3}$ may be slightly decreased. The decrease in MI between ${T}_{x}$ and ${R}_{x2}$ and MI between ${T}_{x}$ and ${R}_{x3}$ are nearly the same as the increase in the MI between ${T}_{x}$ and ${R}_{x1}$. Therefore, the MI between ${T}_{x}$ and ${R}_{x}$ can be maintained at a constant. Meanwhile, the MI of the EEC structure is larger than that of the traditional coil structure. This is because the total MI with the EEC structure is equal to the sum of the MI between ${T}_{x}$ and ${R}_{x1}$, the MI between ${T}_{x}$ and ${R}_{x2}$, and the MI between ${T}_{x}$ and ${R}_{x3}$ ; whereas the total MI with the traditional structure is equal to the MI between ${T}_{x}$ and ${R}_{x2}$ only.
B. Law of MI of EEC Structure
The MI between ${T}_{x}$ and ${R}_{x}$ of the EEC structure can be obtained as follows:
${M}_{{Tx}- {Rx}}= {M}_{{Tx}- {Rx1}}+ {M}_{{Tx}- {Rx2}}+ {M}_{{Tx}- {Rx3}}$
where ${M}_{{Tx}- {Rx}}$ is the MI between ${T}_{x}$ and ${R}_{x},{M}_{{Tx}- {Rx1}}$ is the MI between ${T}_{x}$ and ${R}_{x1},{M}_{{Tx}- {Rx2}}$ is the MI between ${T}_{x}$ and ${R}_{x2}$, ${M}_{{Tx}- {Rx3}}$ is the MI between ${T}_{x}$ and ${R}_{x3}$.
${M}_{{Tx}- {Rx}}$ may be maintained constant when the change value of ${M}_{{Tx}- {Rx1}}$ equals the change value of the sum of ${M}_{{Tx}- {Rx2}}$ and ${M}_{{Tx}- {Rx3}}$. The value of ${M}_{{Tx}- {Rx1}}$ may be increased and the value of ${M}_{{Tx}- {Rx2}}$ and ${M}_{{Tx}- {Rx3}}$ may be decreased with different misalignment along the Y-axis direction owing to the two outstanding characteristics in the EEC structure.
Fig. 4(a) shows that ${M}_{{Tx}- {Rx}}$ is not decreased monotonically as ${\Delta }_{y}$ is increased. Fig. 4(a) shows that ${M}_{{Tx}- {Rx}}$ may be increased when ${\Delta }_{y}$ is increased. This is because the increase in ${M}_{{Tx}- {Rx1}}$ is larger than that of the decrease in the sum of ${M}_{{Tx}- {Rx2}}$ and ${M}_{{Tx}- {Rx3}}$. However, ${M}_{{Tx}- {Rx}}$ in the traditional structure is reduced as the misalignment is increased, as shown in Fig. 4(b).
C. S-S Topology in EEC Structure
The equivalent circuit model is obtained from the EEC coil structure as shown in Fig. 5. Parameters ${L}_{1},{L}_{2},{R}_{1}$, and ${R}_{2}$ are the self-inductance and internal resistance of the transmitting and receiving coils, respectively. ${C}_{\mathrm{{Tx}}}$ and ${C}_{\mathrm{{Rx}}}$ are defined as the resonant capacitance of the transmitting and receiving coils. M is the mutual inductance between the coils and ${R}_{\mathrm{s}}$ is the internal resistance of the power supply ${V}_{\mathrm{s}}$. According to Fig. 5 the matrix of Kirchhoff’s voltage equation is obtained as
$\left\lbrack \begin{matrix}{Z}_{1}& \mathrm{j}{\omega M}\\\mathrm{j}{\omega M}& {Z}_{2}\end{matrix}\right\rbrack \left\lbrack \begin{matrix}{I}_{\mathrm{{Tx}}}\\{I}_{\mathrm{{Rx}}}\end{matrix}\right\rbrack =\left\lbrack \begin{matrix}{V}_{\mathrm{S}}\\ 0 \end{matrix}\right\rbrack $
The currents in the transmitting and receiving coils are given
$\left\{\begin{array}{l}{I}_{\mathrm{{Tx}}}= \frac{{Z}_{2}{V}_{\mathrm{S}}}{{Z}_{1}{Z}_{2}+ {\left(\omega M\right)}^{2}}\\{I}_{\mathrm{{Rx}}}= -\frac{\mathrm{j}{\omega M}{V}_{\mathrm{S}}}{{Z}_{1}{Z}_{2}+ {\left(\omega M\right)}^{2}}\end{array}\right.$
The transmitting coil impedance ${Z}_{1}$ and receiving coil impedance ${Z}_{2}$ are expressed as follows:
$\left\{\begin{array}{l}{Z}_{1}= {R}_{\mathrm{S}}+ {R}_{1}+ \mathrm{j}\omega {L}_{1}- \mathrm{j}\frac{1}{\omega {C}_{\mathrm{{Tx}}}}\\{Z}_{2}= {R}_{2}+ {R}_{\mathrm{L}}+ \mathrm{j}\omega {L}_{2}- \mathrm{j}\frac{1}{\omega {C}_{\mathrm{{Rx}}}}\end{array}\right.$
According to (3) and (4) the input power ${P}_{\text{in }}$, output power ${P}_{\text{out }}$ and transmission efficiency $\eta$ of the system can be obtained
${P}_{\text{in }}= {V}_{\mathrm{S}}{I}_{\mathrm{{Tx}}}= \left|\frac{{Z}_{2}{V}_{\mathrm{S}}{}^{2}}{{Z}_{1}{Z}_{2}+ {\omega }^{2}{M}^{2}}\right|$
${P}_{\text{out }}= {I}_{\mathrm{{Rx}}}^{2}{R}_{\mathrm{L}}= \left|{-\frac{{V}_{\mathrm{S}}{}^{2}{\left(\omega M\right)}^{2}{R}_{\mathrm{L}}}{{\left({Z}_{1}{Z}_{2}+ {\omega }^{2}{M}^{2}\right)}^{2}}}\right|$
$\eta =\left|\frac{{P}_{\text{out }}}{{P}_{\text{in }}}\right|= \left|{-\frac{{\left(\omega M\right)}^{2}{R}_{\mathrm{L}}}{{Z}_{1}{Z}_{2}^{2}+ {\left(\omega M\right)}^{2}{Z}_{2}}}\right|$
III. Calculation and Optimization Method of COUPLING COEFFICIENT
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In this section, a calculation method for the mutual inductance and self-inductance of the rectangular coil is presented to calculate the coupling coefficient of the proposed structure fast. The coupling coefficient is defined as follows:
${k}_{12}= \frac{{M}_{12}}{\sqrt{{L}_{1}{L}_{2}}}$
where ${k}_{12}$ is the mutual inductance between two coils, ${M}_{12}$ is the mutual inductance between two coils, ${L}_{1}$ and ${L}_{2}$ are the self-inductance of two coils.
A. Self-Inductance Calculation of the Rectangular Coil
A schematic diagram of a single-turn rectangular coil is shown in Fig. 6. The half-length and half-width of the rectangular coil are ${a}_{1}$ and ${a}_{2}$, respectively. ${z}_{0}$ is the height of the rectangular coil. ${\operatorname{Coil}}_{1}$ is separated into four parts $\left({{l}_{1},{l}_{2},{l}_{3}}\right.$, and ${l}_{4}$). The parameter I is the current of the ${\operatorname{Coil}}_{1}$.
For a current-carrying conductor with a current density $J$ in air, the magnetic vector of any point $P\left({x, y, z}\right)$ is as follows:
$ A\left({\mathrm{x},\mathrm{y},\mathrm{z}}\right)= \frac{{\mu }_{0}}{4\pi }{\int }_{\mathrm{v}}\frac{J\left({{x}^{\prime },{\mathrm{y}}^{\prime },{\mathrm{z}}^{\prime }}\right)\mathrm{d}{v}^{\prime }}{R}$
where $J$ is the current density and $v$ is the current distribution of ${\text{Coil}}_{1}.R$ is the distance from $P\left({x, y, z}\right)$ to the source point $\left({{x}^{\prime }{y}^{\prime },{z}^{\prime }}\right)$.
$ R =\sqrt{{\left( x -{x}^{\prime }\right)}^{2}+ {\left( y -{y}^{\prime }\right)}^{2}+ {\left( z -{z}^{\prime }\right)}^{2}}$
The dual Fourier transformation and its inverse transformation are used to solve (9)[26].
$ b\left({\xi,\eta, z}\right)= {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }B\left({x, y, z}\right){\mathrm{e}}^{-\mathrm{j}\left({{x\xi }+ {y\eta }}\right)}\mathrm{d}x\mathrm{\;d}y $
$ B\left({x, y, z}\right)= \frac{1}{4{\pi }^{2}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }b\left({\xi,\eta, z}\right){\mathrm{e}}^{\mathrm{j}\left({{x\xi }+ {y\eta }}\right)}\mathrm{d}\xi \mathrm{d}\eta $
where $\xi$ and $\eta$ are the double Fourier integral variables.
Substituting (9) into (11),(13) can be obtained as follows:
$ a\left({\xi,\eta, z}\right)= \frac{{\mu }_{0}}{2}{\int }_{v}\frac{1}{k}{\mathrm{e}}^{-\mathrm{j}\left({{x}^{\prime }\xi +{y}^{\prime }\eta }\right)}J\left({{x}^{\prime },{y}^{\prime },{z}^{\prime }}\right){\mathrm{e}}^{-k\left|{z -{z}^{\prime }}\right|}\mathrm{d}{v}^{\prime }$
The relationship between the incident magnetic flux density and the magnetic vector potential is as follows:
$ B =\nabla \times A $
Substituting the Fourier transform into (14),(15) is obtained as follows:
$\left\{\begin{array}{l}{b}_{x}= -\mathrm{j}\eta {a}_{\mathrm{z}}- \frac{\partial {a}_{y}}{\partial z}\\{b}_{y}= \frac{\partial {a}_{x}}{\partial z}+ \mathrm{j}\xi {a}_{z}\\{b}_{z}= -\mathrm{j}\xi {a}_{y}+ \mathrm{j}\eta {a}_{x}\end{array}\right.$
where
$ k =\sqrt{{\xi }^{2}+ {\eta }^{2}}$
The wires ${l}_{1}$ and ${l}_{3}$ are parallel to the X-axis, and the wires ${l}_{2}$ and ${l}_{4}$ are parallel to the Y-axis. Therefore, the component ${a}_{x}$ of the magnetic vector potential in the X-axis direction is generated only by the wires ${l}_{1}$ and ${l}_{3}$. According to (13), the expression of ${a}_{x}$ is as follows:
$\begin{array}{l} {a}_{{x}}= {a}_{{x}_{1}}- {a}_{{x}_{3}}\\= \frac{{\mu }_{0}I{\mathrm{e}}^{\mathrm{j}{a}_{2}\eta }{\mathrm{e}}^{-\left|{z -{z}_{0}}\right| k}}{2k}\left({{\int }_{-{a}_{1}}^{{a}_{1}}{\mathrm{e}}^{-\mathrm{j}{x}^{\prime }\xi }\mathrm{d}{x}^{\prime }- {\int }_{-{a}_{1}}^{{a}_{1}}{\mathrm{e}}^{-\mathrm{j}{x}^{\prime }\xi }\mathrm{d}{x}^{\prime }}\right)\end{array}\\= \frac{\mathrm{j}2{\mu }_{0}I\sin \left({\xi {a}_{1}}\right)\sin \left({\eta {a}_{2}}\right){\mathrm{e}}^{-\left|{z -{z}_{0}}\right| k}}{\xi k}$
where ${a}_{x1}$ and ${a}_{x3}$ represent the magnetic vector potential components, which are generated by wires ${l}_{1}$ and ${l}_{3}$, respectively.
Similarly, the component of the magnetic vector potential ${a}_{y}$ in the Y-axis direction is obtained as follows:
${a}_{y}= \frac{-\mathrm{j}2{\mu }_{0}I\sin \left({\eta {a}_{2}}\right)\sin \left({\xi {a}_{1}}\right){\mathrm{e}}^{-\left|{z -{z}_{0}}\right| k}}{\eta k}$
Substituting (17) and (18) into (16),(19)-(21) can be obtained as follows:
${b}_{x}= \frac{\mathrm{j}2{\mu }_{0}I\sin \left({\xi {a}_{1}}\right)\sin \left({\eta {a}_{2}}\right)}{\eta }$
${b}_{y}= \frac{\mathrm{j}2{\mu }_{0}I\sin \left({\xi {a}_{2}}\right)\sin \left({\eta {a}_{1}}\right)}{\xi }$
${b}_{z}= \frac{-2{\mu }_{0}{Ik}\sin \left({\xi {a}_{1}}\right)\sin \left({\eta {a}_{2}}\right)}{\xi \eta }$
Substituting (19),(20), and (21) into (12), the magnetic flux density in the $z ={z}_{0}$ plane is obtained.
${B}_{x}= {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{\mathrm{j}2{\mu }_{0}I\sin \left({\xi {a}_{1}}\right)\sin \left({\eta {a}_{2}}\right)}{4{\pi }^{2}\eta }{\mathrm{e}}^{\mathrm{j}\left({{x\xi }+ {y\eta }}\right)}\mathrm{d}\xi \mathrm{d}\eta $
${B}_{y}= {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{\mathrm{j}2{\mu }_{0}I\sin \left({\xi {a}_{2}}\right)\sin \left({\eta {a}_{1}}\right)}{4{\pi }^{2}\xi }{\mathrm{e}}^{\mathrm{j}\left({{x\xi }+ {y\eta }}\right)}\mathrm{d}\xi \mathrm{d}\eta $
${B}_{z}= {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{-2{\mu }_{0}{Ik}\sin \left({\xi {a}_{1}}\right)\sin \left({\eta {a}_{2}}\right)}{4{\pi }^{2}{\xi \eta }}{\mathrm{e}}^{\mathrm{j}\left({{x\xi }+ {y\eta }}\right)}\mathrm{d}\xi \mathrm{d}\eta $
For a single rectangular coil, only the magnetic flux density in the $Z$-axis direction needs to be considered when the self-inductance is calculated. According to (24), the self-inductance calculation formula of a single rectangular coil is obtained as follows:
B. Mutual Inductance Calculation Method of the Rectangular Coil
A method for calculating the mutual inductance of rectangular coils based on the vector potential is proposed. A schematic diagram of two rectangular coils, ${\mathrm{{Coil}}}_{1}$ and ${\mathrm{{Coil}}}_{2}$, is shown in Fig. 7. ${a}_{1}$ and ${a}_{2}$ are the half-length and half-width of ${\text{Coll}}_{1}$. ${b}_{1}$ and ${b}_{2}$ are the half-length and half-width of ${\text{Coil}}_{2}$. ${\text{Coil}}_{1}$ is separated into four parts $\left({{l}_{1},{l}_{2},{l}_{3}}\right.$, and $\left.{l}_{4}\right).{z}_{0}$ is the vertical distance between $o$ and ${o}_{1}$, and ${z}_{1}$ is the vertical distance between $o$ and ${o}_{2}$. The parameter I is the current of ${\operatorname{Coil}}_{1}$.
It is noted that $z \neq {z}_{0}$ when the mutual inductance between ${T}_{x}$ and ${R}_{x}$ is calculated. By substituting (17) and (18) into (15), the following is obtained:
${b}_{ix}= {C}_{ix}\cdot {\mathrm{e}}^{-{kz}}$
${b}_{iy}= {C}_{iy}\cdot {\mathrm{e}}^{-{kz}}$
${b}_{iz}= {C}_{iz}\cdot {\mathrm{e}}^{-{kz}}$
where
${C}_{ix}= \frac{-\mathrm{j}2{\mu }_{0}I\sin \left({\xi {a}_{1}}\right)\sin \left({\eta {a}_{2}}\right)}{\eta }{\mathrm{e}}^{{z}_{0}k}$
${C}_{iy}= \frac{-\mathrm{j}2{\mu }_{0}I\sin \left({\xi {a}_{1}}\right)\sin \left({\eta {a}_{2}}\right)}{\xi }{\mathrm{e}}^{{z}_{0}k}$
${C}_{iz}= \frac{-2{\mu }_{0}{Ik}\sin \left({\xi {a}_{1}}\right)\sin \left({\eta {a}_{2}}\right)}{\xi \eta }{\mathrm{e}}^{{z}_{0}k}$
By substituting (26),(27), and (28) into (12), the magnetic flux density ${B}_{i}$ is obtained as follows:
${B}_{ix}\left({x, y, z}\right)= \frac{1}{4{\pi }^{2}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{C}_{ix}\cdot {\mathrm{e}}^{-{kz}}\cdot {\mathrm{e}}^{\mathrm{j}\left({{x\xi }+ {y\eta }}\right)}\mathrm{d}\xi \mathrm{d}\eta $
${B}_{iy}\left({x, y, z}\right)= \frac{1}{4{\pi }^{2}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{C}_{iy}\cdot {\mathrm{e}}^{-{kz}}\cdot {\mathrm{e}}^{\mathrm{j}\left({{x\xi }+ {y\eta }}\right)}\mathrm{d}\xi \mathrm{d}\eta $
${B}_{iz}\left({x, y, z}\right)= \frac{1}{4{\pi }^{2}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{C}_{iz}\cdot {\mathrm{e}}^{-{kz}}\cdot {\mathrm{e}}^{\mathrm{j}\left({{x\xi }+ {y\eta }}\right)}\mathrm{d}\xi \mathrm{d}\eta $
According to (32),(33), and (34), the mutual inductance between ${\text{ Coil }}_{1}$ and ${\text{ Coil }}_{2}$ can be obtained
where ${b}_{1d}$ and ${b}_{2d}$ are the misalignment of the X-axis and Y-axis, respectively.
The mutual inductance between multi-turn coils is calculated as follows:
$ M =\mathop{\sum }\limits_{{m = 1}}^{{N}_{1}}\mathop{\sum }\limits_{{n = 1}}^{{N}_{2}}{M}_{\mathrm{{mn}}}$
where ${N}_{1}$ is the number of turns of ${\operatorname{Coil}}_{1},{N}_{2}$ is the number of turns of ${\operatorname{Coil}}_{2}, m$ is the M-th turn of ${\operatorname{Coil}}_{1}$, and $n$ is the $n$-th turn of ${\text{Coil}}_{2}$.
The calculation method of the coupling coefficient provides the theoretical basis for analyzing and optimizing the coupling coefficient of the proposed structure in the next subsection.
C. Optimization Method of Parameters of EEC Structure
In the previous section, it is theoretically analyzed that the MI fluctuation rate may be reduced and the MI value may be increased by using the EEC structure at the same time. In this section, an optimization method of parameters of the EEC structure is proposed to obtain the quasi-constant MI when ${\Delta }_{y}$ and ${\Delta }_{x}$ are within 10% of the side length of ${T}_{x}$ and 10cm, respectively. The optimization process is as follows:
1) Parameters initialization: $D$ is set to 15cm. The copper wire diameter is set to 0.25cm. Parameters ${\varepsilon }_{1\;Y}$ and ${\varepsilon }_{2\;Y}$ are the fluctuations in MI when ${\Delta }_{y}$ is within 50% of the side length of the transmitting coil. ${\varepsilon }_{1\_ X}$ and ${\varepsilon }_{2\_ X}$ are the fluctuations in MI when ${\Delta }_{x}$ is within ${10}\mathrm{\;{cm}}.{\varepsilon }_{1\_ {Y}^{* }},{\varepsilon }_{2\_ {Y}^{* }},{\varepsilon }_{1\_ {X}^{* }}$, and ${\varepsilon }_{2\_ {X}^{* }}$ are set to $5\%.{M}_{{Tx}- {Rx}- {0}^{* }}$ is set to ${25\mu }\mathrm{H}$. The coupling coefficient between ${T}_{x}$ and ${R}_{x}\left({k}_{{Tx}- {Rx}- {0}^{* }}\right)$ is set to 0.11 . All rates of fluctuation in MI along the Y-axis and X-axis are defined as
${\varepsilon }_{1\_ Y}= \left({{M}_{{Tx}- {Rx}- \max \_ Y}- {M}_{{Tx}- {Rx}- 0}}\right)/{M}_{{Tx}- {Rx}- 0}$
$\varepsilon_{2 \_Y}=\left|\left(\begin{array}{lllll}M_{T x} & R x & \min \_Y-M_{T x} & R x & 0\end{array}\right) / M_{T x} \quad R x \quad 0\right|$
$\varepsilon_{1_{-} X}=\left(M_{T x-R x-\max \_X}-M_{T x-R x-0}\right) / M_{T x-R x-0}$
$\varepsilon_{2 \_X}=\left|\left(M_{T x-R x-\min \_X}-M_{T x-R x-0}\right) / M_{T x-R x-0}\right|$
where ${M}_{{Tx}- {Rx}- 0}$ is the MI between ${T}_{x}$ and ${R}_{x}$ with ${\Delta }_{x}= 0\mathrm{\;{cm}}$ and ${\Delta }_{y}= 0\mathrm{\;{cm}}.{M}_{{Tx}- {Rx}- \max \_ Y}$ and ${M}_{{Tx}- {Rx}- \min \_ Y}$ are the maximum and minimum MI between ${T}_{x}$ and ${R}_{x}$ with misalignment along the Y-axis, respectively. ${M}_{{Tx}- {Rx}- \max X}$ and ${M}_{{Tx}- {Rx}- \min X}$ are the maximum and minimum MI between ${T}_{x}$ and ${R}_{x}$ with misalignment along the X-axis, respectively.
2) Constraint conditions setting: Constraint conditions of each coil parameter are as follows: ${21}\mathrm{\;{cm}}\leq {l}_{1 -\text{ inner }}\leq {23}\mathrm{\;{cm}}$, ${54}\mathrm{\;{cm}}\leq {l}_{2 -\text{ inner }}\leq {56}\mathrm{\;{cm}},{21}\mathrm{\;{cm}}\leq {l}_{3 -\text{ inner }}\leq {23}\mathrm{\;{cm}},{26}\mathrm{\;{cm}}\leq$ ${l}_{4 -\text{inner }}\leq {27}\mathrm{\;{cm}},{57}\mathrm{\;{cm}}\leq {h}_{1 -\text{inner }}\leq {58}\mathrm{\;{cm}},{54}\mathrm{\;{cm}}\leq {h}_{2 -\text{inner }}\leq$ ${56}\mathrm{\;{cm}},{57}\mathrm{\;{cm}}\leq {h}_{3 -\text{ inner }}\leq {58}\mathrm{\;{cm}},{26}\mathrm{\;{cm}}\leq {h}_{4 -\text{ inner }}\leq {27}\mathrm{\;{cm}}$. The parameters of ${R}_{x1}$ and ${R}_{x3}$ are set to be the same. ${N}_{1}$ and ${N}_{2}$ are the numbers of turns for ${T}_{x}$ and ${R}_{x2}$, respectively. ${N}_{3}$ is the number of turns for ${R}_{x1}$ and ${R}_{x3}.{N}_{1}$ is changed from 16 turns to 20 turns. ${N}_{2}$ is changed from 9 turns to 13 turns. ${N}_{3}$ is changed from 3 turns to 7 turns. The step of the number of turns is 1 turn. The variation of both the length and width is $1\mathrm{\;{cm}}$.
3) Calculation: The MI with the EEC structure can be calculated according to (35) and (36) with different misalignment along the Y-axis, and X-axis. ${\varepsilon }_{1, Y}$ and ${\varepsilon }_{2, Y}$ can be calculated by (37) and (38). ${\varepsilon }_{1, X}$ and ${\varepsilon }_{2, X}$ can be calculated by (39) and (40).
4) Conditional judgment and output solutions: Two conditions should be met at the same time. ① ${\varepsilon }_{1\_ Y}< {\varepsilon }_{1\_ Y *},{\varepsilon }_{2\_ Y}< {\varepsilon }_{2\_ Y *}$, ${\varepsilon }_{1\_ X}< {\varepsilon }_{1\_ {X}^{* }}$, and ${\varepsilon }_{2\_ X}< {\varepsilon }_{2\_ {X}^{* }}$. ② ${M}_{{Tx}- {Rx}- 0}> {M}_{{Tx}- {Rx}- {0}^{* }}$ and ${k}_{{Tx}- {Rx}- 0}>$ ${k}_{{Tx}- {Rx}- {0}^{* }}$. Solutions that meet the above two conditions are output. The flowchart of the proposed method is shown in Fig. 8.
IV. Verification
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The experimental setup and simulation model are built based on the EEC structure. The measurement of the MI is obtained. Measurement results agree with calculation results and simulation results. The experimental results show that the MI value of the EEC coil structure increases and then decreases when ${\Delta }_{y}$ is within ${50}\%$ of the side length of the transmitting coil, and the fluctuation rate is almost constant.
A. Experimental and Simulation Setup
The simulation model of the EEC structure is established by using Ansys Maxwell, as shown in Fig. 9. Simulation results of the MI can be obtained.
According to the optimization results in the previous section, the dimensions of each coil can be obtained as shown in Table I. The diameter of the Litz wire is 0.25mm. Measurement parameters of ${T}_{x}$ and ${R}_{x}$ are shown in Table II.
In addition, an experimental setup is established, including a power source, ${T}_{x},{R}_{x}$, and load, as shown in Fig. 10(a). The transmitting coil is shown in Fig. 10(b) and the receiving coil is shown in Fig. 10(c).
B. Mutual Inductance Results
As described in reference [26],[27], the MI calculation results can be obtained by using MATLAB. The simulation results of MI are obtained by using Ansys Maxwell model. The measurement results of the MI are obtained by using an IM3536 precision LCR meter.
The MI of the EEC structure versus ${\Delta }_{v}$ is shown in Fig. 11(a). The MI is almost constant at ${35000\mu }\mathrm{H}$. For different ${\Delta }_{v}$, the MI and the rates of variation of the MI are shown in Table III. As shown in Table III, the maximum MI is ${35.625\mu }\mathrm{H}$ at ${\Delta }_{v}=$ ${8.5}\mathrm{\;{cm}}$, the minimum $\mathrm{{MI}}$ is ${33.260\mu }\mathrm{H}$ at ${\Delta }_{v}= {17.0}\mathrm{\;{cm}}$, and the $\mathrm{{MI}}$ is ${34.925\mu }\mathrm{H}$ at ${\Delta }_{v}= 0\mathrm{\;{cm}}$. According to (37) and (38), the rates of variation in the MI ${\varepsilon }_{1\;Y}$ and ${\varepsilon }_{2\;Y}$ are 2.00% and 4.77%, respectively, with different ${\Delta }_{v}$.
The MI of the EEC structure versus ${\Delta }_{x}$ is shown in Fig. 11(b). The MI is also nearly constant at ${35.000\mu }\mathrm{H}$. The MI and the rates of fluctuation in MI are shown in Table IV with ${\Delta }_{x}$. As shown in Table IV, the maximal MI is ${34.925\mu }\mathrm{H}$ with ${\Delta }_{x}=$ ${0.0}\mathrm{\;{cm}}$ and the minimal MI is ${33.250\mu }\mathrm{H}$ with ${\Delta }_{x}= {10.0}\mathrm{\;{cm}}$. According to (39) and (40), the rates of fluctuation in MI ${\varepsilon }_{1\mathrm{X}}$ and ${\varepsilon }_{2\mathrm{X}}$ are equal to ${0.00}\%$ and ${4.80}\%$ with the misalignment along the X-axis, respectively.
The research results show that the fluctuation of MI is less than 5.0% for both the misalignment along the Y-axis and X-axis directions, which meets the design requirements.
C. Efficiency and Output Power
Parameter ${\Delta }_{v}$ is changed from 0cm to 17cm with a step of 1.7cm. And Parameter ${\Delta }_{x}$ is changed from 0cm to 10cm with a step of 2cm. Parameter $D$ is set at 15cm. The model of the power amplifier is Aigtek ATA-3090, The gain of the power amplifier is set to 15 . The load resistor $\left({R}_{L}\right)$ is set at ${42.0\Omega }$. The input voltage of ${T}_{x}\left({U}_{1}\right)$, the input current of ${T}_{x}\left({I}_{1}\right)$, the output voltage of ${R}_{x}\left({U}_{2}\right)$, and the output current of ${R}_{x}\left({I}_{2}\right)$ of the proposed structure are measured by using a Yokogawa WT5000 Power analyzer. The measuring schematic diagram of the output power $\left({P}_{\text{out }}\right)$ and transmission efficiency $\left(\eta \right)$ is shown in Fig. 12.
When the misalignment along the Y-axis $\left({\Delta }_{v}\right)$ is within 17.0 $\mathrm{{cm}}$, the measured $\eta$ and ${P}_{\text{out }}$ versus different ${\Delta }_{v}$ are shown in Fig. 13(a). As can be seen in Fig. 13(a), $\eta$ is nearly constant with different ${\Delta }_{y}$, the maximum efficiency is 95.35% with ${\Delta }_{y}= {8.5}\mathrm{\;{cm}}$, and the minimum efficiency is 94.03% with ${\Delta }_{v}= {17.0}\mathrm{\;{cm}}$. The rate of change in $\eta$ is only ${1.38}\%.{P}_{\text{out }}$ is varied from 115.36 W to 119.35W with ${\Delta }_{v}$ varying from 0cm to 6.8cm. And ${P}_{\text{out }}$ is varied from 118.97 W to 126.74W with ${\Delta }_{v}$ varying from 8.5cm to 17.0cm. It can be concluded that the variation in ${P}_{\text{out }}$ is 9.86% with the different ${\Delta }_{y}$.
When ${\Delta }_{x}$ is within 10.0cm, the measured $\eta$ and ${P}_{\text{out }}$ versus different ${\Delta }_{x}$ are shown in Fig. 13(b). As can be seen in Fig. 13(b), $\eta$ is also nearly constant for different ${\Delta }_{x}$, the maximum efficiency is 95.35% with ${\Delta }_{x}= {0.0}\mathrm{\;{cm}}$, and the minimum efficiency is 94.13% with ${\Delta }_{x}= {10.0}\mathrm{\;{cm}}$. The rate of change in efficiency is ${1.27}\%.{P}_{\text{out }}$ is varied from 114.32 W to ${125.32}\mathrm{\;W}$ when misalignment along the X-axis is within ${10.0}\mathrm{\;{cm}}$. The rate of change in ${P}_{\text{out }}$ is ${9.62}\%$ with misalignment along the X-axis. The research results show that the fluctuations of the transfer efficiency and ${P}_{\text{out }}$ are very small. This is due to the fluctuation of MI is small with different misalignments in the proposed structure.
D. Comparison
The conventional structure is composed of ${T}_{x}$ and ${R}_{x2}$ only, whereas the EEC structure is composed of ${T}_{x},{R}_{x1},{R}_{x2}$, and ${R}_{x3}$. The performance comparison between the conventional structure and the EEC structure is presented. It is found that the MI of the conventional structure is smaller than that of the EEC structure and the fluctuation in MI of the conventional structure is larger than that of the EEC structure. Therefore, the fluctuation in $\eta$ and ${P}_{\text{out }}$ of the conventional structure are larger than those of the EEC structure. Experimental results further prove the progressiveness of the EEC structure.
The MI of the conventional structure versus different ${\Delta }_{v}$ is shown in Fig. 14. As shown in Fig. 14, the maximum MI is ${27.274\mu }\mathrm{H}$ with ${\Delta }_{v}= {0.0}\mathrm{\;{cm}}$ and the minimum $\mathrm{{MI}}$ is ${21.923\mu }\mathrm{H}$ with ${\Delta }_{y}= {17.0}\mathrm{\;{cm}}$. In addition, ${\varepsilon }_{1\mathrm{\;Y}}$ and ${\varepsilon }_{2\mathrm{\;Y}}$ are 0.00% and 19.60%, respectively, when ${\Delta }_{v}$ is within 17.0cm. The MI and the rates of fluctuation of the MI with the misalignment along the X axis are the same as those with the misalignment along the Y axis. This is because ${T}_{x}$ and ${R}_{x2}$ are square coil structures.
When ${\Delta }_{v}$ is within 17.0cm, the rates of fluctuation in the MI with the conventional structure are much greater than those with the EEC structure. The maximum rate of fluctuation in the MI with the conventional structure is 19.60%, while the maximum rate of fluctuation in the MI with the proposed structure is only 4.80%. Fig. 15 shows $\eta$ and ${P}_{\text{out }}$ versus misalignment along the Y-axis with the traditional structure. $\eta$ is varied from 94.99% to ${92.55}\%$ and ${P}_{\text{out }}$ is varied from 168.23 W to 244.71 W. The rates of fluctuation in ${P}_{\text{out }}$ and $\eta$ are ${45.40}\%$ and ${2.57}\%$, respectively. The $\eta$ and ${P}_{\text{out }}$ with the misalignment along the X-axis are the same as those with the misalignment along the Y-axis. This is because ${T}_{x}$ and ${R}_{x2}$ are square coil structures. The ${P}_{\text{out }}$ of the conventional structure is larger than that of the EEC structure because too large MI can lead to power splitting in the proposed structure [25].
The performance comparison between other references and our work is shown in Table V. Compared to the literature [17],[18],[23], and [25], only our proposed structure can reduce the fluctuation in MI and increase the MI value at the same time. Therefore, $\eta$ is the highest in our work. In addition, the MI can be maintained almost constant in our work when the receiving coil is moved in the Y-axis or X-axis directions simultaneously. Therefore, the proposed structure in our work has two directions of offset capability. The proposed structure can be used to simultaneously satisfy the dynamic and static wireless charging requirements of EVs.
V. Conclusion
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In this paper, an edge-enhanced coil (EEC) structure is proposed. Moreover, an optimization method of parameters of the EEC structure is proposed to obtain the optimal coil parameters. It is found that the MI of the EEC structure can not only be kept quasi-constant, but also be increased with misalignment along the Y-axis. The problem that quasi-constant MI is obtained at the expense of MI value is solved. The experimental results show that the MI reaches ${35.625\mu }\mathrm{H}$ with ${\Delta }_{v}= {8.5}\mathrm{\;{cm}}$ and the MI equals ${34.925\mu }\mathrm{H}$ with ${\Delta }_{v}= {0.0}\mathrm{\;{cm}}$ in the EEC structure. The fluctuation in MI is within 4.80%. The transmission efficiency of the EEC structure is also higher due to the increase of MI. And the maximum transmission efficiency is 95.35% when ${\Delta }_{v}$ and ${\Delta }_{x}$ are within 50% of the side length of ${T}_{x}$ and 10.0cm, respectively.
It is worth noting that the transfer efficiency and output power can be maintained nearly constant without additional auxiliary control devices, although the maximal ${\Delta }_{y}$ reaches 50% of the side length of ${T}_{x}$. This will result in significant cost savings and reduce the difficulty of controlling the DSWPT systems. Therefore, the EEC structure can be used to simultaneously satisfy the dynamic and static wireless charging requirements of EVs. In future research, we will study on adding a magnetic medium to the EEC structure to obtain a greater MI and a smaller fluctuation in MI.
Funding
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  • National Program on Key Research Project(2022YFB3403200)
  • Natural Science Foundation of Hunan Province(2022JJ30226)
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Appendix
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Year 2024 volume 9 Issue 2
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Cite this Article
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Article Info
doi: 10.24295/CPSSTPEA.2024.00006
  • Receive Date：2023-09-01
  • Online Date：2025-07-05
  • Published：2024-06-10
Article Data
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History
  • Received：2023-09-01
  • Revised：2023-12-20
  • Accepted：2024-05-07
Funding
National Program on Key Research Project(2022YFB3403200)
Natural Science Foundation of Hunan Province(2022JJ30226)
Affiliations
    1 Hunan University College of Electrical and Information Engineering Changsha 410082 China
    2 Hunan University of Technology College of Traffic Engineering Zhuzhou 412007 China
    3 Hunan University of Arts and Sciences College of Electrical Engineering Changde 415000 China

Corresponding:

Minsheng Yang.
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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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