An Eight-Coil Wireless Power Transfer Method for Improving the Coupling Tolerance Based on Uniform Magnetic Field

Abstract

In wireless power transfers (WPTs), it is challenging to obtain a constant output of power (COP) and constant transmission efficiency (CTE) in large coupling variation ranges. In this study, the eight-coil WPT system achieves a uniform magnetic field (UMF) in the transmitter and receiver sides using two transmitting (Tx) coils and two receiving (Rx) coils, respectively. COP and CTE are then achieved with large coupling variation ranges. The circuit model and equations of the transmission characteristics are first obtained based on the structure and working principle of the Helmholtz coil. The model of the mutual inductance and equation of the impedance coupled factor are then developed. The laws of the transmission characteristic are also determined by adopting a simulation tool and equations of the transmission characteristics. Finally, the eight-coil WPT experimental system is designed. In a fixed-frequency mode, the COP and CTE are achieved when the coupling and misalignment distances are changed within a quarter or one-fifth of the relay coil diameter, respectively. This topology provides an efficient solution for problems faced in practical applications, such as wireless chargers of kitchen appliances and automatic mobile robots of small size.

1. Introduction

In the early 20th century, Nikola Tesla pioneered wireless power transfer (WPT) technology [1]. Since then, great progress has been made [2]. The magnetic coupling resonant WPT technology [3] has been widely applied in many application domains, such as consumer electronics, electric vehicles, and household appliances [4].

In a two-coil WPT system, Figure 1 shows a three-dimensional graph of the normalized output power (NOP), where frequency splitting occurs in the over-coupled region [5], and the NOP rapidly decreases in the under-coupled region. In the critical-coupled point, NOP reaches its maximum value, while in the over-coupled region, it reaches its maximum value at low and high frequencies with distance and alignment changes among the coils. In addition, in a three-coil WPT system, the receiver coil (or relay coil) undergoes the vibration phenomenon [6,7]. Thus, the WPT system is very sensitive to the changes in the coupling and alignment distances among the coils. It is then challenging to reach a constant output power (COP) and constant transmission efficiency (CTE) with a long-range distance and misalignment [8].

In order to address the sensitivity of the output power (OP) and transmission efficiency (TE) to the changes in the coupling and alignment distances among coils, two types of methods are usually adopted: frequency tracking and impedance matching. In the over-coupled region, when the coupling and alignment distances change, the frequency tracking method is used to adjust the operating frequency in order to track the maximum OP. However, this method requires measuring the parameters of the WPT system and providing the frequency tracking function with a digital signal processor. In [9], a dual-frequency-detuning method is proposed to obtain a COP in large coupled distances. However, this method involves complex system parameter calculations and a system control strategy. In [10], a reflection coefficient of the frequency tuning method is proposed to enhance the OP and TE. However, this method is complex because it requires measuring the reflection coefficient to adjust the operating frequency. In the over-coupled region, the impedance matching method modifies the circuit parameters of the WPT system to achieve the maximum OP and TE. However, this method is unable to adapt to long-range coupling and misalignment distance changes. In [11], a tracking method for optimizing TE is proposed with impedance matching using full current mode. However, this method requires a specific mechanism for detection tracking. It has been shown that the two-side impedance-angle design is efficient in optimizing the CE [12]. However, this method involves a complex system design. In general, it is still challenging to achieve COP and CTE with long-range coupling and changes in alignment distances.

The aforementioned types of methods are both traditional approaches. Figure 2 shows a schematic diagram of the distribution of the magnetic field intensity, presenting Helmholtz coils that are a pair of identical circular coils parallel to each other, separated by a radius distance, and wrapped together [13,14]. The current flows through the two coils in the same direction, which generates a uniform magnetic field (UMF) between them.

Based on the aforementioned results, using the structure and working principle of the Helmholtz coil, a topology of the eight-coil WPT system with two transmitting (Tx) coils and two receiving (Rx) coils is proposed, allowing the achievement of the UMFs on the Tx and Rx sides. The UMFs on the Tx and Rx sides are obtained using the two Tx coils and two Rx coils. In addition, COP and CTE are achieved in a fixed-frequency mode. The circuit model is first developed, and the transfer characteristics are studied. The OP and TE are then simulated. Afterward, the eight-coil WPT experimental system is developed. The constant transfer power and efficiency are finally obtained with large coupling variation ranges in the fixed-frequency mode and open space. This topology provides a solution for problems faced in practical applications, such as the wireless charging of kitchen appliances and automatic mobile robots of small sizes.

Note that in Figure 1, Ψ is the normalized OP, ξ is the frequency detuning factor, and τ is the impedance coupled factor, while in Figure 2, R is the radius of the Helmholtz coils, and B, B₁, and B₂ are the intensities of the magnetic field.

2. The Proposed Method

The circuit model and equations of the transmission characteristics are first established based on the structure and working principle of the Helmholtz coil. This allows us to analyze the transmission characteristics of the eight-coil WPT system. The model of the mutual inductance and the equation of the impedance coupled factor are then obtained. The laws of the transmission characteristic are finally determined by applying a simulation tool and equations of the transmission characteristics.

2.1. Model of the Eight-Coil WPT System and Equations of the Transmission Characteristics

Based on the aforementioned results, this paper proposes a topology consisting of two-Tx-coil and two-Rx coil, allowing us to obtain the UMF. The relay1 loop obtains the UMF using one coil inserted into the Tx loop. The relay2 loop also obtains the UMF using one coil inserted into the Rx loop. COP and CTE are then reached. Note that the distances of the two-Tx-coil and two-Rx coils are given constant values, and the distance of the relay1 and relay2 coils is given a variable.

A sketch of the eight-coil WPT system comprising two Tx coils, relay1 coils, relay2 coils, and two Rx coils is shown in Figure 3. Figure 4 shows an equivalent circuit of the eight-coil WPT system with two-Tx-coil (M-Tx L₁ and S-Tx L₃), relay1 coils (Ry₁ L₂ and Ry₁′ L₂′), relay2 coils (Ry₂ L₅ and Ry₂′ L₅′), and two-Rx-coil (M-Rx L₄ and S-Rx L₆). The two Tx coils are linked into the Tx loop. The M-Tx and S-Tx loops are then obtained using a circuit similar to that of the transmitter, as shown in Figure 4. Similarly, the M-Rx and S-Rx loops are obtained using a circuit similar to that of the receiver, as shown in Figure 4.

The parameters of this eight-coil WPT system are shown in

Table 1. Parameters of the eight-coil WPT system.

Parameter	Value
Input power	${\dot{U}}_S$
Load voltage	${\dot{U}}_{L1}$, ${\dot{U}}_{L2}$
Winding ohmic resistance of the coil	R1, R2, R3, R4, R5, R6
Load resistance	RL1, RL2
Inductance of the coil	L1, L2, L2′, L3, L4, L5, L5′, L6
Capacitance	CS, C2, C5, CL
Coupled distance	d1, d2, d3, d4, d5, ds = d1 + d2, dL = d4 + d5
Mutual inductance	M1, M2, M3, M4, M5

. Note that, for the convenience of analysis, the mutual inductance of the M-Tx and S-Tx coils is ignored because M₁, M₂, M₃, M₄, M₅, and M₆ are much larger. The mutual inductance of the M-Rx and S-Rx coils is also ignored.

A detailed analysis of the transmission characteristics of this eight-coil WPT system is provided by the equivalent circuit model.

Table 2. Adopted values for the M-Tx, Ry₁, S-Tx, Ry₁′, Ry₂, M-Rx, Ry₂′, and S-Rx.

Parameter	Value
Capacitances	CS = C2 = C5 = CL = C
Inductances	L1 = 2L2 = 2L2′ = L3 =L4 = 2L5 = 2L5′ = L6 = L
Resistance	R4 + RL1 = R6 + RL2 = R
Ratio	RL1 = RL2 = βR, R1 = R2 = R3 = R5 = σR (σ = 1)
Operating angular frequency	ω
Resonance frequency	f0 = ω0/2π
Frequency detuning factor	ξ = Q0(ω/ω0–ω0/ω)
Resonant angular frequency	ω0 = 1/(LC)0.5, ω1 = ω2 = ω3 = ω4 = ω5 = ω6 = ω0
Quality factor	Q0 = ω0L/R = 1/(ω0CR), Q1 = Q2 Q3 = Q5 = ω0L/R1 = 1/(ω0CR1) = Q0/σ, Q4 = Q6 = ω0L/(R4 + RL1) = 1/(ω0C(R6 + RL2)) = Q0

shows the adopted values for the M-Tx, Ry₁, S-Tx, Ry₁′, Ry₂, M-Rx, Ry₂′, and S-Rx [7].

According to Figure 4, Table 1 and Table 2, and Kirchhoff’s voltage law, the eight-coil WPT system comprising two-Tx-coil and two-Rx-coil can be expressed as

$$\left\{\begin{array}{l}{Z}_1{\dot{I}}_1-j\omega M_1{\dot{I}}_2={\dot{U}}_S\\ Z_2{\dot{I}}_2-j\omega M_1{\dot{I}}_1-j\omega M_2{\dot{I}}_3-j\omega M_3{\dot{I}}_5=0\\ Z_3{\dot{I}}_3-j\omega M_2{\dot{I}}_2={\dot{U}}_S\\ Z_5{\dot{I}}_5-j\omega M_3{\dot{I}}_2-j\omega M_4{\dot{I}}_4-j\omega M_5{\dot{I}}_6=0\\ Z_4{\dot{I}}_4-j\omega M_4{\dot{I}}_5=0\\ Z_6{\dot{I}}_6-j\omega M_5{\dot{I}}_5=0\end{array}\right.$$

(1)

The self-impedance values of Z₁, Z₂, Z₃, Z₄, Z₅, and Z₆ are given by [7]

$$Z_1=Z_2=Z_3=Z_4=Z_5=Z_6=(1+j\xi )R$$

(2)

The impedance coupled factors τ₁, τ₂, τ₃, τ₄, and τ₅, which denote the impedance coupled ability, are expressed as [7]

$${\tau }_n={\displaystyle \frac{\omega M_n}{R}},{\tau }_n>0,(n=1,2,3,4,5)$$

(3)

Based on Equations (1)–(3) and the parameters presented in Table 2, the currents can be expressed as

$$\left\{\begin{array}{l}{\dot{I}}_1={\displaystyle \frac{{(1+j\xi )}^4+{(1+j\xi )}^2({{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2-{\tau }_1{\tau }_2)+({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)-{\tau }_1({\tau }_1+{\tau }_2)({\tau }_4+{\tau }_5)}{{(1+j\xi )}^5+{(1+j\xi )}^3({{\tau }_1}^2+{{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2)+(1+j\xi )({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)}}{\displaystyle \frac{{\dot{U}}_S}{R}}\\ {\dot{I}}_2=j{\displaystyle \frac{({\tau }_1+{\tau }_2)[{(1+j\xi )}^2+{\tau }_4+{\tau }_5]}{{(1+j\xi )}^4+{(1+j\xi )}^2({{\tau }_1}^2+{{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2)+({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)}}{\displaystyle \frac{{\dot{U}}_S}{R}}\\ {\dot{I}}_3={\displaystyle \frac{{(1+j\xi )}^4+{(1+j\xi )}^2({{\tau }_1}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2-{\tau }_1{\tau }_2)+({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)-{\tau }_2({\tau }_1+{\tau }_2)({\tau }_4+{\tau }_5)}{{(1+j\xi )}^5+{(1+j\xi )}^3({{\tau }_1}^2+{{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2)+(1+j\xi )({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)}}{\displaystyle \frac{{\dot{U}}_S}{R}}\\ {\dot{I}}_4=-j{\displaystyle \frac{({\tau }_1+{\tau }_2){\tau }_3{\tau }_4}{{(1+j\xi )}^4+{(1+j\xi )}^2({{\tau }_1}^2+{{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2)+({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)}}{\displaystyle \frac{{\dot{U}}_S}{R}}\\ {\dot{I}}_5=-{\displaystyle \frac{({\tau }_1+{\tau }_2){\tau }_3(1+j\xi )}{{(1+j\xi )}^4+{(1+j\xi )}^2({{\tau }_1}^2+{{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2)+({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)}}{\displaystyle \frac{{\dot{U}}_S}{R}}\\ {\dot{I}}_6=-j{\displaystyle \frac{({\tau }_1+{\tau }_2){\tau }_3{\tau }_5}{{(1+j\xi )}^4+{(1+j\xi )}^2({{\tau }_1}^2+{{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2)+({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)}}{\displaystyle \frac{{\dot{U}}_S}{R}}\end{array}\right.$$

(4)

By applying Equation (4), the OP P_out can be written as

$$\begin{array}{l}{P}_{out}=|{\dot{I}}_4{|}^2{R}_{L1}+|{\dot{I}}_6{|}^2{R}_{L2}\\ ={\displaystyle \frac{\beta {({\tau }_1+{\tau }_2)}^2{{\tau }_3}^2({{\tau }_4}^2+{{\tau }_5}^2)}{\begin{array}{l}\{{[1-6{\xi }^2+{\xi }^4+(1-{\xi }^2)({{\tau }_1}^2+{{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2)+({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)]}^2\\ +{\xi }^2{[4-{\xi }^2+2({{\tau }_1}^2+{{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2)]}^2\}\end{array}}}{\displaystyle \frac{{{\dot{U}}_S}^2}{R}}\end{array}$$

(5)

By applying Equation (4), the TE η can be expressed as

$$\begin{array}{l}\eta ={\displaystyle \frac{P_{out}}{P_{in}}}={\displaystyle \frac{|{\dot{I}}_4{|}^2{R}_{L1}+|{\dot{I}}_6{|}^2{R}_{L2}}{|{\dot{I}}_1{|}^2{R}_1+|{\dot{I}}_2{|}^2{R}_2+|{\dot{I}}_3{|}^2{R}_3+|{\dot{I}}_4{|}^2(R_4+R_{L1})+|{\dot{I}}_5{|}^2{R}_5+|{\dot{I}}_6{|}^2(R_6+R_{L2})}}\\ ={\displaystyle \frac{\beta (1+{\xi }^2){({\tau }_1+{\tau }_2)}^2{{\tau }_3}^2({{\tau }_4}^2+{{\tau }_5}^2)}{\begin{array}{l}\{[1-6{\xi }^2+{\xi }^4+(1-{\xi }^2)({{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2-{\tau }_1{\tau }_2)+({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)\\ -{\tau }_1({\tau }_1+{\tau }_2)({\tau }_4+{\tau }_5){]}^2+{[4-4{\xi }^3+2\xi ({{\tau }_1}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2-{\tau }_1{\tau }_2)]}^2\\ +[1-6{\xi }^2+{\xi }^4+(1-{\xi }^2)({{\tau }_1}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2-{\tau }_1{\tau }_2)+({{\tau }_1}^2+{{\tau }_2}^2)({{\tau }_4}^2+{{\tau }_5}^2)\\ -{\tau }_2({\tau }_1+{\tau }_2)({\tau }_4+{\tau }_5){]}^2+{[4-4{\xi }^3+2\xi ({{\tau }_2}^2+{{\tau }_3}^2+{{\tau }_4}^2+{{\tau }_5}^2-{\tau }_1{\tau }_2)]}^2\\ +(1+{\xi }^2)\{{[({\tau }_1+{\tau }_2)(1-{\xi }^2)+{\tau }_4+{\tau }_5]}^2+4{\xi }^2{({\tau }_1+{\tau }_2)}^2\}+(1+{\xi }^2){({\tau }_1+{\tau }_2)}^2{{\tau }_3}^2({{\tau }_4}^2+{{\tau }_5}^2)\\ +[{(1-{\xi }^2)}^2+4{\xi }^2]{({\tau }_1+{\tau }_2)}^2{{\tau }_3}^2\}\end{array}}}\end{array}$$

(6)

Equations (5) and (6), respectively, present the OP and TE equations (i.e., the transmission characteristic equations). In these equations, complex relationships exist between the underlying parameters. Therefore, it is difficult to reveal the laws of the OP and TE. In the sequel, the laws of the transmission characteristics are revealed by applying a simulation tool as well as Equations (5) and (6).

2.2. Model of the Mutual Inductance and Equation of the Impedance Coupled Factor

Figure 5 shows a sketch of the mutual inductance of the eight-coil system that applies two-Tx-coil and two-Rx-coil to achieve the UMF on the two sides [14] and to further achieve the COP and CTE. The underlying parameters are determined using the Biot–Savart law [15]. The mutual inductances are shown in Figure 5.

Table 3. Parameters of the magnetic induction intensity, mutual inductance, and geometric structure.

Parameter	Value
Permeability of vacuum (H/m)	μ0 = 4π × 10−7
Magnetic induction intensity of L1 at the position of L2	B1 = μ0(n1n2)0.5r12i1/(2(r12 + d12)3/2)
Magnetic induction intensity of L3 at the position of L2	B2 = μ0(n2n3)0.5r32i3/(2(r32 + d22)3/2)
Magnetic induction intensity of L2′ at the position of L5	B3 = μ0(n2′n5)0.5r2′2i2/(2(r2′2 + d32)3/2)
Magnetic induction intensity of L5′ at the position of L4	B4 = μ0(n4n5′)0.5r5′2i5/(2(r5′2 + d42)3/2)
Magnetic induction intensity of L5′ at the position of L6	B5 = μ0(n5′n6)0.5r5′2i5′/(2(r5′2 + d52)3/2)
Radius of the M-Tx, Ry1, S-Tx, Ry1′, Ry2, M-Rx, Ry2′, and S-Rx	r1, r2, r3, r2′, r5, r4, r5′, r6
Turn number of the M-Tx, Ry1, S-Tx, Ry1′, Ry2, M-Rx, Ry2′, and S-Rx	n1, n2, n3, n2′, n5, n4, n5′, n6
Geometric center of the M-Tx, Ry1, S-Tx, Ry1′, Ry2, M-Rx, Ry2′, and S-Rx	O1, O2, O3, O2′, O5, O4, O5′, O6
Mutual inductance between L1 and L2	M1 = πμ0(n1n2)0.5(r1r2)2/(2(r12 + d12)3/2)
Mutual inductance between L3 and L2	M2 = πμ0(n2n3)0.5(r2r3)2/(2(r32 + d22)3/2)
Mutual inductance between L2′ and L5	M3 = πμ0(n2′n5)0.5(r2′r5)2/(2(r2′2 + d32)3/2)
Mutual inductance between L5′ and L4	M4 = πμ0(n4n5′)0.5(r4r5′)2/(2(r5′2 + d42)3/2)
Mutual inductance between L5′ and L6	M5 = πμ0(n5′n6)0.5(r5′r6)2/(2(r5′2 + d52)3/2)

presents the parameters of the magnetic induction intensity, mutual inductance, and geometric structure that are used to derive Equation (7).

Besides Equation (3) and Table 3, the τ₁, τ₂, τ₃, τ₄, and τ₅ used to draw the curves of the transmission characteristics are given by

$$\left\{\begin{array}{l}{\tau }_1={\displaystyle \frac{\pi \omega {\mu }_0{(n_1{n}_2)}^{0.5}{(r_1{r}_2)}^2}{2R{({r_1}^2+{d_1}^2)}^{3/2}}}={\displaystyle \frac{\pi \omega {\mu }_0{(n_1{n}_2)}^{0.5}{(r_1{r}_2)}^2}{2R{({r_1}^2+{(\frac{d_1+d_2}{2}+x)}^2)}^{3/2}}}\\ {\tau }_2={\displaystyle \frac{\pi \omega {\mu }_0{(n_2{n}_3)}^{0.5}{(r_2{r}_3)}^2}{2R{({r_3}^2+{d_2}^2)}^{3/2}}}={\displaystyle \frac{\pi \omega {\mu }_0{(n_2{n}_3)}^{0.5}{(r_2{r}_3)}^2}{2R{({r_3}^2+{(\frac{d_1+d_2}{2}-x)}^2)}^{3/2}}}\\ {\tau }_3={\displaystyle \frac{\pi \omega {\mu }_0{({n_2}^{\prime }{n}_5)}^{0.5}{({r_2}^{\prime }{r}_5)}^2}{2R{(r_2{\prime }^2+{d_3}^2)}^{3/2}}}={\displaystyle \frac{\pi \omega {\mu }_0{({n_2}^{\prime }{n}_5)}^{0.5}{({r_2}^{\prime }{r}_5)}^2}{2R{(r_2{\prime }^2+z^2)}^{3/2}}}\\ {\tau }_4={\displaystyle \frac{\pi \omega {\mu }_0{({n_4{n}_5}^{\prime })}^{0.5}{({r_4{r}_5}^{\prime })}^2}{2R{(r_5{\prime }^2+{d_4}^2)}^{3/2}}}={\displaystyle \frac{\pi \omega {\mu }_0{({n_4{n}_5}^{\prime })}^{0.5}{({r_4{r}_5}^{\prime })}^2}{2R{(r_5{\prime }^2+{(\frac{d_4+d_5}{2}+x)}^2)}^{3/2}}}\\ {\tau }_5={\displaystyle \frac{\pi \omega {\mu }_0{({n_5}^{\prime }{n}_6)}^{0.5}{({r_5}^{\prime }{r}_6)}^2}{2R{(r_5{\prime }^2+{d_5}^2)}^{3/2}}}={\displaystyle \frac{\pi \omega {\mu }_0{({n_5}^{\prime }{n}_6)}^{0.5}{({r_5}^{\prime }{r}_6)}^2}{2R{(r_5{\prime }^2+{(\frac{d_4+d_5}{2}-y)}^2)}^{3/2}}}\end{array}\right.$$

(7)

2.3. Law of the Transmission Characteristics of the Eight-Coil WPT System

2.3.1. Law of the Transmission Characteristics of the Frequency with Coupling Distance

According to Equations (5) and (6), the normalized OP and TE are shown in Figure 6. The normalized OP and TE have three states: the under-coupled region, the critical-coupled point, and the over-coupled region. In the under-coupled region (τ₃ < 1), the normalized OP and TE rapidly decrease. In the critical-coupled point (τ₃ = 1), the normalized OP and TE reach their maximum values, while in the over-coupled region (τ₃ > 1), they reach their maximum values at low and high frequencies with distance and alignment changes among coils. Thus, the law of the transmission characteristics of the frequency with a coupling distance for the eight-coil WPT system should be studied in detail.

2.3.2. Law of the Transmission Characteristics of the Two-Tx-Coil System

The radius of the experimental Tx coil is 25 mm, and the Helmholtz coils are needed to separate the radius distances. According to Figure 4 and Figure 5, it is assumed that the coupled distance of the M-Tx and S-Tx coils (d₁ + d₂) has a constant value of 24 mm and M₃ = 0. In particular, the eight-coil WPT system is converted to a two-Tx-coil system comprising the M-Tx, Ry₁, S-Tx, and Ry₁′ coils. The obtained results are shown in Figure 7. The OP and TE curves of the two-Tx-coil system are then drawn using Matlab R2019b and by substituting Equation (7) into Equations (5) and (6), as shown in Figure 8.

The M-Tx and S-Tx coils are placed in parallel, and the two-Tx-coil generates the UMF [14]. Thus, the OP and TE of point O₂ are obtained by superposition. The obtained results are shown in Figure 8. The WPT system reaches the COP and CTE between points O₁ and O₃. Outside these points, OP and TE are significantly decreased. That is, at the Tx side and between the two-Tx-coil (L₁ and L₃), the two-Tx-coil system reaches COP and CTE using the structure and working principle of the Helmholtz coil.

2.3.3. Law of the Transmission Characteristics of the Two-Rx-Coil System

Because the radius of the experimental coil is 25 mm, according to Figure 4 and Figure 5, in the eight-coil WPT system, it is assumed that the d₁ + d₂, d₃, and d₄ + d₅ coupled distances have constant values of 24, 8, and 24 mm, respectively. Using Matlab R2019b and substituting Equation (7) into Equations (5) and (6) yields the OP and TE curves presented in Figure 9.

The M-Rx and S-Rx coils are placed in parallel, and the two-Rx-coil generates the UMF [14]. Thus, the OP and TE of point O₅′ between points O₄ and O₆ are superimposed with a parallel mode. The obtained results are shown in Figure 9. It can be clearly seen that for OP, the curve of τ₄ + τ₅ is identical to the sum of the curves of τ₄ and τ₅. Similarly, for TE, the curve of τ₄ + τ₅ is identical to the sum of the curves of τ₄ and τ₅. The two-Rx-coil system reaches the COP and CTE between points O₄ and O₆. Outside these points, OP and TE are significantly decreased. Thus, at the Rx side, and between the L₄ and L₆ Rx coils, the two-Rx-coil system reaches the COP and CTE using the structure and working principle of the Helmholtz coil.

2.3.4. Law of the Transmission Characteristics of the Relay-Coil System

According to Figure 4 and Figure 5, it is assumed that the coupled distance of d₁ + d₂ has a constant value of 24 mm. In addition, M₃ = 0 and M₅ = 0. In particular, the eight-coil WPT system is converted to the relay-coil system comprising the M-Tx, Ry₁, S-Tx, Ry₁′, Ry₂, and Ry₂′ coils. The obtained results are shown in Figure 10. Afterward, using Matlab and substituting Equation (7) into Equations (5) and (6) yields the OP and TE curves presented in Figure 11.

It can be seen from Figure 3 and Figure 10 that since the relay1 loop is placed in a UMF, OP, and TE reach a constant value when z (d₃) is in the range of 0–10 mm. In other words, the energy is transferred from the Tx side to the Rx side with constant values.

In summary, by adopting the structure and working principle of the Helmholtz coil, on the two sides, the two-Tx-coil and two-Rx-coil systems can provide a UMF. Furthermore, they reach the COP and CTE. Those of the eight-coil WPT system are finally obtained with a suitable coupling distance d₃. In the sequel, experiments are first conducted, and the simulation and experimental results are then compared.

3. Experimental Results

An eight-coil WPT experimental block diagram is shown in Figure 12, where AC220V is the input power resource, C_S, C₂, and C_L are the capacitors, L₁, L₂, L₂′, L₃, L₄, L₅, L₅′, and L₆ are the inductors, D₁-D₄ are the rectifier diodes, C_K is the filter capacitor, R_L is the load resistor, and U_L is the load voltage.

The eight-coil WPT experimental equipment comprises a power amplifier, wave generator, oscilloscope, capacitors, transmitter, relay1, relay2, receiver, load, M-Tx coil, S-Tx coil, relay1 coils, relay2 coils, M-Rx coil, and S-Rx coil, as shown in Figure 13. The parameters of the eight-coil WPT experimental system are shown in

Table 4. Parameters of the eight-coil WPT experimental system.

Parameter	M-Tx, S-Tx, M-Rx, S-Rx	Ry1, Ry1′, Ry2, Ry2′
Inside diameter of the coil φ/mm	30	32
Outside diameter of the coil Φ/mm	50	40
Layers of the coil	2	3
Number of turns	30	20
Frequency f0/kHz	135	135
Inductance L/μH	50.0	29.0
Capacitance C/nF	27.83	47.98
Distance d1 + d2 or d4 + d5 mm	24
Distance d3 mm	2, 8, or 12
Impedance scaling factor σ	1
Frequency detuning factor ξ	0
Load RL Ω	0.5
Input Voltage US/V	16
Input power W	40

. The experimental video of the eight-coil WPT system is shown in Video S1 (Supplementary Materials).

3.1. Law of the Transmission Characteristics with Frequency for the Eight-Coil WPT System

According to Table 4, in the conducted experiments, d₁ + d₂ and d₄ + d₅ are both set to 24 mm, while f₀ and U_S are set to 135 kHz and 16 V, respectively. The UMFs of the two sides of the system are obtained using these values. The OP and TE of the system are then studied at many frequencies and values of the coupled distance (d₃).

In addition, in the conducted experiments, d₃ is set to 2, 8, and 12 mm. The input and output voltages and currents are measured at many frequencies. The results of the experimental data are shown in Figure 14. It can be seen that when d3 is set to 8 mm and 12 mm, the high and flat OP and TE peaks reach almost 135 kHz. When d₃ is set to 2 mm, the peaks of the OP and TE occur at 115 kHz and 155 kHz, respectively. According to Figure 6 and Figure 14, the frequency splitting of the OP and TE occurs. Thus, when d₃ is set to 8 mm and 12 mm, the high and flat OP and TE peaks are obtained at the resonance frequency 135 kHz. This verifies the necessity of using the structure and working principle of the Helmholtz coil to achieve the UMF on the two sides and further achieve high and flat OP and TE peaks.

3.2. COP and CTE with Long-Range Coupling and Misalignment Distances

3.2.1. COP and CTE with Long-Range Coupling Distance

Based on the aforementioned results, the OP and TE reach maximum values at 115, 135, and 155 kHz. Thus, the law of the transmission characteristics with a frequency for the eight-coil WPT system should be studied in detail. When the system functions, the aforementioned frequencies remain constant. The experimental data are obtained by measuring the input and output voltage function of the z (d₃) value. The obtained results are shown in Figure 15.

It can be clearly seen that with 135 kHz, COP and CTE are obtained when d₃ is in the range of 0–10 mm. However, with 115 kHz and 155 kHz, their values significantly decrease with the increase in d₃. Thus, at 135 kHz, the system achieves the COP and CTE with a large coupling distance (0–10 mm), which is a quarter of the relay coil diameter (40 mm).

3.2.2. COP and CTE with Long-Range Misalignment Distance

It can be clearly observed from Figure 15 that the maximum values of the OP and TE are obtained at 135 kHz. Therefore, when ω is tuned and retains a constant value of 135 kHz, the experimental data are obtained by measuring the input and output voltages at the different misalignment distances between the Ry₁′ and Ry₂ coils. The results of the experimental data are shown in Figure 16. It can be clearly seen that, for 135 kHz and d₃ = 8 mm, the COP and CTE are reached with a large misalignment distance (0–8 mm), which is one-fifth of the relay coil diameter (40 mm).

In general, by inserting the relay loops into the UMF, the COP and CTE are achieved when the coupling and misalignment distances are changed within a quarter and one-fifth of the relay coil diameter, respectively.

4. Discussion

In the eight-coil WPT system, the UMFs on the Tx and Rx sides are obtained using the two-Tx-coil and two-Rx-coil, and the COP and CTE are then achieved with large coupling variation ranges. It can be clearly seen that the method to obtain the UMF is a key factor in addressing the sensitivity of the OP and TE to the changes in the coupling and alignment distances among coils. Thus, proposing another novel structure of the coil to achieve the UMF is a future research study. For example, using a symmetrical structure coil to constrain the magnetic field to obtain the UMF is an effective approach, which reduces the occupied space of the multi-coil.

5. Conclusions

This paper used the structure and working principle of the Helmholtz coil to propose a novel eight-coil WPT method for improving the coupling tolerance based on UMF. The UMFs on the Tx and Rx sides were obtained using the two-Tx-coil and two-Rx-coil. In addition, in a fixed-frequency mode, the COP and CTE were achieved when the coupling and misalignment distances were changed within a quarter or one-fifth of the relay coil diameter, respectively. This topology provides an efficient solution for problems faced in practical applications, such as wireless chargers of kitchen appliances and automatic mobile robots of small size.