I. INTRODUCTION

The concept of wireless power transfer (WPT) based on magnetic coupling theory was proposed by Nikola Tesla more than a century ago [1]. WPT uses magneto-quasistatic fields to transfer energy over short to medium distances and has been increasingly applied in electric vehicles, consumer electronics, and implantable medical devices due to its superiority and usability in humid and lousy environments [2]-[4].

Magnetically coupled resonant (MCR) WPT technology can deliver power efficiently over medium distances [5]. The basic MCR WPT system usually consists of two coils, namely, transmitting and receiving coils, which operate at the same resonant frequency. Numerous studies have focused on the analysis, design, and optimization of the basic two-coil MCR WPT system [6]-[8], whose output power and transmission efficiency noticeably drop with increased transfer distance [9] or the variation of position in space [10], [11].

For further enhancement of transmission characteristics, intermediate coils were introduced between the transmitting and receiving coils of the MCR WPT system in previous works. Three- and four-coil MCR WPT systems have been extensively proposed for increasing output power and transmission efficiency, particularly in long-distance applications [12], [13]. Similar to the position of coils in a two-coil WPT system, that in a three-coil WPT system considerably influences the transmission characteristics. Studies have been conducted to reveal the effect of intermediate coil on the performance of a system. [14] showed that three-coil inductive links can significantly improve transmission efficiency, particularly at long coupling distances, by transforming any arbitrary load impedance to the optimal impedance required at the input of the inductive link. The transmission efficiency of WPT with an intermediate coil was analyzed in [15], and the results showed that the intermediate resonant system had good transmission efficiency and was superior to nonintermediate systems. [16] discovered that placing an intermediate coil slightly near the transmission coil achieved better transmission efficiency than placing it slightly near the receiving coil. In [17], a three-coil WPT system structure was proposed and proven capable of significantly reducing the electromagnetic field emission caused by coil misalignment. In [18], the influence of spatial scales on the output power of three-coil system was investigated, but only one variable spatial scale was analyzed, meanwhile, the transmission efficiency was not considered.

In this paper, the transmission characteristics of a three-coil MCR WPT system versus the spatial scales, including axial, lateral, and angular misalignments, are discussed for stabilizing the output power and transmission efficiency at varying positions of the intermediate and receiving coils. Section II describes the basic theories and the key parameters of the three-coil MCR WPT system. According to equivalent circuit theory, the circuit model of the three-coil MCR WPT system is built, and the expressions of the output power and transmission efficiency are derived by the fundamental harmonic analysis (FHA) method. In Section III, the transmission characteristics of the system are analyzed and illustrated when the mutual inductances between the three coils are proportional. We find that the robustness of the output power and the transmission efficiency can be improved by an adjustment in the position of the intermediate coil according to the receiving coil. A prototype of the three-coil MCR WPT system is built and tested to validate the theoretical analysis in Section IV, which is followed by the conclusion in Section V.

II. MODELING AND ANALYSIS OF TRANSMISSION CHARACTERISTICS

Fig. 1 shows the circuit diagram of the three-coil MCR WPT system, which includes a half-bridge inverter; a voltage doubler rectifier; and three resonant coils, namely, the transmitting, intermediate, and receiving coils. The mutual inductances between two adjacent coils are defined as M₁₂ and M₂₃, and the mutual inductance between transmitting and receiving coils is ignored for simplification. The input voltage is inverted by the half-bridge inverter and employed to drive the resonant coils. Q₁ and Q₂ are the power switches driven in a complementary manner with the same duty cycle.

Fig. 1. Configuration of the three-coil MCR WPT system.

Fig. 2 shows the equivalent circuit of the three-coil MCR WPT system, where U_s is the equivalent alternating voltage source, L₁–L₃ are the self-inductances of the resonant coils, with parasitic resistances R₁–R₃. C₁–C₃ are the series resonant capacitors designed to resonate with L₁–L₃. R_S is the impedance of the AC voltage source. R_W is the equivalent load resistance.

Fig. 2. Equivalent circuit of the system.

For the half-bridge configuration, the norm of \(\dot{U}_s\) can be calculated as

\(U_{S}=\left\|\dot{U}_{S}\right\|=\frac{\sqrt{2}}{\pi} V_{i n}.\)       (1)

The equivalent load resistance (R_W) can be expressed by the load resistance (R_L) as

\(R_{W}=\frac{2}{\pi^{2}} R_{L}.\)       (2)

The resonant frequency is defined as

\(f_{r}=1 /(2 \pi \sqrt{L_{r} \cdot C_{r}}),\)       (3)

where L_r and C_r are the resonant inductor and capacitor, respectively. If the three-coil MCR WPT system operates in a resonant state, then L₁ = L₂ = L₃ = L_r, and C₁ = C₂ = C₃ = C_r.

According to Fig. 2 and Kirchhoff’s voltage law, the following can be obtained:

\(\left[\begin{array}{ccc} R_{S}+R_{1} & j \omega M_{12} & 0 \\ j \omega M_{12} & Z_{2} & j \omega M_{23} \\ 0 & j \omega M_{23} & R_{3}+R_{W} \end{array}\right]\left[\begin{array}{c} \dot{I}_{1} \\ \dot{I}_{2} \\ \dot{I}_{3} \end{array}\right]=\left[\begin{array}{c} \dot{U}_{s} \\ 0 \\ 0 \end{array}\right],\)       (4)

where ω = 2πf_r, and \(\dot{I}_1\)–\(\dot{I}_3\) are the fundamental harmonics of the resonant currents.

If R_1S = R₁ + R_S and R_3W = R₃ + R_W, then \(\dot{I}_1\)–\(\dot{I}_3\) can be solved from (4).

\(\left\{\begin{array}{l} \dot{I}_{1}=\frac{U_{S}\left(\omega^{2} M_{23}^{2}+R_{2} R_{3 W}\right)}{\omega^{2} M_{12}^{2} R_{3 W}+\omega^{2} M_{23}^{2} R_{1 S}+R_{1 S} R_{2} R_{3 W}} \\ \dot{I}_{2}=\frac{-j U_{S} \omega M_{12} R_{3 W}}{\omega^{2} M_{12}^{2} R_{3 W}+\omega^{2} M_{23}^{2} R_{1 S}+R_{1 S} R_{2} R_{3 W}} \\ \dot{I}_{3}=\frac{-U_{S} \omega^{2} M_{12} M_{23}}{\omega^{2} M_{12}^{2} R_{3 W}+\omega^{2} M_{23}^{2} R_{1 S}+R_{1 S} R_{2} R_{3 W}} \end{array}\right.\)       (5)

Therefore, the output power of the system P_o can be expressed as

\(P_{o}=\frac{U_{s}^{2} \omega^{4} M_{12}^{2} M_{23}^{2} R_{W}}{\left(\omega^{2} M_{12}^{2} R_{3 W}+\omega^{2} M_{23}^{2} R_{1 S}+R_{1 S} R_{2} R_{3 W}\right)^{2}}.\)       (6)

Radiation and power losses are ignored for simplification. Thus, the transmission efficiency η can be derived as

\(\eta=\frac{\omega^{4} M_{12}^{2} M_{23}^{2} R_{W}}{\left(\omega^{2} M_{23}^{2}+R_{2} R_{3 W}\right)\left(\omega^{2} M_{12}^{2} R_{3 W}+\omega^{2} M_{23}^{2} R_{1 S}+R_{1 S} R_{2} R_{3 W}\right)}.\)       (7)

From (6) and (7), the 3D graphs of the output power and the transmission efficiency versus M₁₂ and M₂₃ are illustrated in Fig. 3, and the specifications are as follows: V_in = 24 V, f_s = f_r = 200 kHz, R_S = 0.1 Ω, R_L = 5 Ω, R₁ = 0.093 Ω, R₂ = 0.21 Ω, and R₃ = 0.093 Ω. The values of R₁–R₃ are measured by the impedance network analyzer and adjusted according to the experimental results. Fig. 3 indicates that P_o and η are sensitive to variations in M₁₂ and M₂₃ and P_o and η generally decrease with M₁₂ and M₂₃.

Fig. 3. 3D graphs of P_o and η versus M₁₂ and M₂₃: (a) P_o; (b) η.

In realistic applications, R_L always changes. Fig. 4 plots how R_L influences P_o and η, from which we can conclude that P_o decreases with an increase in R_L in a specific variation range of M₁₂ (or M₂₃). However, the variation of R_L has less influence on η compared with that of P_o. For simplification, the following analysis is based on a constant R_L (5 Ω).

Fig. 4. Transfer characteristics under different loads: (a) P_o (R_L = 2 Ω); (b) P_o (R_L = 50 Ω); (c) η (R_L = 2 Ω); (d) η (R_L = 50 Ω).

III. ANALYSIS OF SPATIAL SCALES FOR STABLE OUTPUT POWER AND TRANSMISSION EFIFFIENCY

As transmission characteristics are subject to the variation of mutual inductances in a three-coil MCR WPT system, the possibility of achieving stable transmission characteristics by the adjustment of mutual inductances should be determined.

Fig. 5 shows the contour lines extracted from Fig. 3 by the insertion of horizontal planes into Fig. 3 with different magnitudes that represent diverse transfer characteristics, from which we can conclude that P_o and η can remain stable within specific ranges if M₁₂ is proportional to M₂₃.

Fig. 5. Contour with various mutual inductances: (a) P_o; (b) η.

If M₂₃ = k ·M₁₂ and k is the proportional coefficient, then (6) can be simplified as

\(P_{o}=\frac{1}{\left(a_{1}+\frac{a_{2}}{M_{12}^{2}}\right)^{2}}\)       (8)

\(\left\{\begin{array}{l} a_{1}=\frac{R_{1 S}}{U_{S} \sqrt{R_{W}}} \cdot k+\frac{R_{3 W}}{U_{S} \sqrt{R_{W}}} \cdot \frac{1}{k} \\ a_{2}=\frac{R_{1 S} R_{2} R_{3 W}}{U_{S} \omega^{2} \sqrt{R_{W}}} \cdot \frac{1}{k} \end{array}\right.\)       (9)

and (7) can be simplified as

\(\eta=\frac{1}{\frac{b_{1} b_{3}}{M_{12}^{4}}+\frac{b_{1} b_{4}+b_{2} b_{3}}{M_{12}^{6}}+\frac{b_{2} b_{4}}{M_{12}^{8}}}\)       (10)

\(\left\{\begin{array}{l} b_{1}=\frac{1}{\omega^{2} R_{W}} \\ b_{2}=\frac{R_{2} R_{3 W}}{\omega^{4} R_{W}} \cdot \frac{1}{k^{2}} \\ b_{3}=\frac{R_{1 S}}{\omega^{2} R_{W}} \cdot k^{2}+\frac{R_{3 W}}{\omega^{2} R_{W}} \cdot \frac{1}{k^{2}} \\ b_{4}=\frac{R_{1 S} R_{2} R_{3 W}}{\omega^{4} R_{W}} \cdot \frac{1}{k^{2}} \end{array}\right.\)       (11)

where a₁, a₂, b₁, b₂, b₃, and b₄ are constants related to the system parameters and k.

Based on (8)-(11) and M₂₃ = k ·M₁₂, two clusters of curves of P_o and η versus M₁₂ are illustrated in Fig. 6, which infers that coefficient k considerably influences P_o and η. If k is designed appropriately (for example, k≥1), then P_o and η will be nearly stable within a specific range of M₁₂. Theoretically, k can be random in practice, as shown in Fig. 6. To simplify the analysis, we select k = 1.

Fig. 6. Curves of P_o and η versus M₁₂: (a) P_o; (b) η.

To further investigate the requirements on spatial scales to achieve stable transmission characteristics, the relationships between mutual inductances and spatial scales are first obtained [19]. Fig. 7 illustrates the behavior of mutual inductance under lateral and angular misalignments and indicates that misalignments substantially affect mutual inductance. With the substitution of the mutual inductance into (8) and (10), the tendency of P_o and η versus the spatial scales can be determined.

Fig. 7. 3D graphs of mutual inductance versus lateral and angular misalignments.

Case 1: One variable spatial scale

In this study, three typical spatial scales are considered in the following analysis, namely, the axial misalignment (d), the lateral misalignment (Δ), and the angular misalignment (α), as shown in Fig. 8. Meanwhile, three different cases are discussed, namely, one-, two-, and three-variable spatial scales. The specifications are given as follows: r₁ = r₂ = r₃= 0.11 m, N₁ = N₃ = 6, and N₂ = 14.

Fig. 8. Coil layout with axial, lateral, and angular misalignments.

1) Axial Misalignment d (Δ = 0, α = 0)

With the assumption that d₂₃ = k_d ·d₁₂, Fig. 9 shows the clusters of curves of P_o and η versus d₁₂ under different k_d values. A remarkable discrepancy exists between the curves with different k_d values, and most curves fluctuate on a large scale. When k_d = 1 (M₂₃ = M₁₂), the variation of P_o and η is relatively minor within the specific range of d₁₂ (0 ≤ d₁₂ ≤ 0.2 m), which means that the transmission characteristics of the system are approximately constant. However, if k_d ≠ 1 or d₁₂ > 0.2 m, then P_o and η will fluctuate severely, which is detrimental to providing a stable power for loads.

Fig. 9. Curves of P_o and η versus d₁₂: (a) P_o; (b) η.

2) Lateral Misalignment Δ (d = 0, α = 0)

With the assumption that Δ₂ = k_Δ·Δ₃, Fig. 10 illustrates how k_Δ affects the tendency of P_o and η under different Δ₃ values. We can infer that P_o is more sensitive to k_Δ than η to k_Δ. Moreover, when k_Δ = 0.5 (M₂₃ = M₁₂), P_o and η decrease slightly, which means that k_Δ=0.5 is also the best when only lateral misalignment is considered.

Fig. 10. Curves of P_o and η versus Δ₃: (a) P_o; (b) η.

3) Angular Misalignment α (d = 0, Δ = 0)

With the assumption that α₂ = k_α·α₃, Fig. 11 illustrates the correlation between P_o, η, and α₃, where the maximal α₃ is supposed to be 90°. Compared with other curves, the ones when k_α equals 0.5 (M₂₃ = M₁₂) are nearly constant over the full range of α₃, which means that if α₂ is strictly the half of α₃ (0 ≤ α₃ ≤ 90°), then the system can always realize stable transmission characteristics.

Fig. 11. Curves of P_o and η versus α₃: (a) P_o; (b) η.

From the above analysis, we can conclude that if only one spatial scale is considered, then a specific spatial range exists to keep the transmission characteristics of the three-coil MCR WPT system stable under the following conditions: d₂₃ = d₁₂, Δ₂ = 0.5·Δ₃, or α₂ = 0.5·α₃.

Case 2: Two-variable spatial scales

Under the constraints of d₂₃ = d₁₂, Δ₂ = 0.5·Δ₃, and α₂ = 0.5·α₃, the transmission characteristics of the three-coil MCR WPT system when any two variable spatial scales are considered are analyzed as follows.

Figs. 12–14 demonstrate the curves of P_o and η versus d₁₂, Δ₃, and α₃. As shown in Fig. 12, P_o and η decrease with the increase of d₁₂ and Δ₃. However, the transmission characteristics present less sensitivity to α than to d and Δ, as shown in Figs. 13 and 14. The spatial scale ranges for stable P_o and η are d₁₂ ≤ 0.2 m, Δ₃ ≤ 0.25 m, and α₃ ≤ 90° under the given specifications, which are consistent with those of the one-variable spatial scale.

Fig. 12. P_o and η versus d₁₂ and Δ₃: (a) P_o; (b) η.

Fig. 13. P_o and η versus Δ₃ and α₃: (a) P_o; (b) η.

Fig. 14. P_o and η versus Δ₃ and α₃: (a) P_o; (b) η.

Case 3: Three-variable spatial scales

Fig. 15 shows the curve clusters of P_o and η versus three spatial scales under the conditions of d₂₃ = d₁₂, Δ₂ = 0.5·Δ₃, and α₂=0.5·α₃. The transmission characteristics decay with the increasing axial misalignment. In a specific range (d₁₂ ≤ 0.1 m), lateral and angular misalignments only slightly impact P_o and η; thus, the curves (red) are noticeably near each other and change smoothly. However, when d₁₂ > 0.1 m, P_o and η are considerably influenced by lateral and angular misalignments, as shown by the blue and green curves. Meanwhile, axial and lateral misalignments have a greater influence on the transmission characteristics than does the angular misalignment.

Fig. 15. P_o and η with axial, lateral, and angular misalignments:(a) P_o; (b) η.

From the analysis above, we can deduce that the three-coil MCR WPT system can realize a stable power transfer within a specific spatial scale range if the positions of the three coils are appropriately arranged to guarantee equal or proportional mutual inductances between two adjacent coils.

The misalignment range for stable P_o and η is not fixed because the transfer characteristics of the system depend on multiple parameters. Therefore, in practice, the misalignment range for stable P_o and η is determined by the specification of the system, especially by the proportional coefficient, k.

IV. EXPERIMENTAL VERIFICATION

To validate the theoretical analysis, a prototype of the three-coil MCR WPT system is built in the laboratory, as shown in Fig. 16, and the specifications are listed in Table I.

Fig. 16. Photo of the prototype.

TABLE I SPECIFICATIONS OF THE SYSTEM

Figs. 17–19 show the theoretical and experimental results of the output power and transmission efficiency versus the one-variable spatial scales, respectively, where the solid curves represent the theoretical results and the dashed ones are the experimental data.

Fig. 17. Theoretical and experimental data with different axial misalignments (d₁₂ = d₂₃, Δ₂ = Δ₃ = 0 m, α₂ = α₃ = 0°): (a) P_o; (b) η.

Fig. 18. Theoretical and experimental data with different lateral misalignments (d₁₂ = d₂₃ = 0.25 m, Δ₂ = 0.5 Δ₃, α₂ = α₃ = 0°): (a)P_o; (b) η.

Fig. 19. Theoretical and experimental data with different angular misalignments (d₁₂ = d₂₃ = 0.25 m, Δ₂ = Δ₃ = 0 m, α₂ = 0.5 α₃): (a) P_o; (b) η.

Figs. 20-22 illustrate the curves of the output power and transmission efficiency versus the two-variable spatial scales, respectively. The experimental results are also generally consistent with the theoretical results.

Fig. 20. Theoretical and experimental data with different axial and lateral misalignments (d₁₂ = d₂₃, Δ₂ = 0.5 Δ₃, α₂ = α₃ = 0°): (a) P_o (Δ₃ = 0 m); (b) η (Δ₃ = 0 m); (c) P_o (Δ₃ = 0.1 m); (d) η (Δ₃ = 0.1 m).

Fig. 21. Theoretical and experimental value with variant lateral and angular misalignments (d₁₂ = d₂₃ = 0.25 m, Δ₂ = 0.5 Δ₃, α₂ = 0.5 α₃):(a) P_o (Δ₃ = 0 m); (b) η (Δ₃ = 0 m); (c) P_o (Δ₃ = 0.1 m); (d) η (Δ₃ = 0.1 m).

Fig. 22. Theoretical and experimental value with variant axial and angular misalignments: (a) P_o (d₁₂ = 0 m); (b) η (d₁₂ = 0 m); (c) P_o (d₁₂ = 0.22 m); (d) η (d₁₂ = 0.22 m).

Fig. 23 demonstrates the output power and transmission efficiency at three spatial scales. As depicted, the angular misalignment has less effect on the transmission characteristics than do the axial and lateral misalignments; and this finding is consistent with the analysis in Section III.

Fig. 23. Theoretical and experimental data with different axial, lateral and angular misalignments: (a) P_o (d₁₂=0.1m, α₃=0°); (b) η (d₁₂ = 0.1 m, α₃ = 0°); (c) P_o (d₁₂ = 0.1 m, α₃ = 90°); (d) η (d₁₂ = 0.1 m, α₃ = 90°); (e) P_o (d₁₂ = 0.25 m, α₃ = 0°); (f) η (d₁₂ = 0.25 m, α₃ = 0°); (g) P_o (d₁₂ = 0.25 m, α₃ = 90°); (h) η (d₁₂ = 0.25 m, α₃ = 90°).

Fig. 24 shows the key experimental waveforms under different spatial scales, where i₁ and i₃ are the currents of the transmitting and receiving coils, respectively. As shown in Fig. 24, the phase difference between i₁ and i₃ is 180°. Before the switch turns on, the drain current is negative, which indicates that the drain-to-source voltage (v_ds) is clamped to zero and the switch can achieve zero-voltage-switching.

Fig. 24. Experimental waveforms: (a) Lateral misalignment (Δ₂ = 0.5Δ₃ = 5 cm, α₂ = α₃ = 0°); (b) Angular misalignment (Δ₂ = Δ₃ = 0 cm, α₂ = 0.5α₃ = 10°); (c) General misalignment (Δ₂ = 0.5Δ₃ = 5 cm, α₂ = 0.5α₃ = 10°).

V. CONCLUSIONS

This research investigates a three-coil MCR WPT system with transmitting, intermediate, and receiving coils. Via the equivalent circuit and FHA method, the output power and the transmission efficiency of the system with axial, lateral, and angular misalignments are analyzed in detail. We find that different spatial scales have varied influences on the transmission characteristics. Compared with axial and lateral misalignments, the angular misalignment has a minor impact on the system. By adjusting the position of the intermediate and receiving coils to ensure equal or proportional mutual inductances between two adjacent coils, the WPT system can achieve relatively stable transmission characteristics within a specific range of spatial scales. The conclusions are validated by experiments, whose results are consistent with those in the theoretical analysis.

ACKNOWLEDGMENT

This work was financially supported by the National Natural Science Foundation of China (51505223, 51877103), the Fundamental Research Funds for the Central Universities of China (NS2018020), the Natural Science Foundation of Jiangsu Province, China (BK20151471).