A Novel Multipolarity Decoupled Magnetic Coupler Applied to Multiple-Receiver Wireless Charging System with Load-Independent CV and CC Outputs

Abstract

Simultaneously enabling wireless charging for multiple electronic devices is a distinctive advantage of wireless power transfer (WPT). Nevertheless, the development of dual-receiver WPT systems is constrained by several challenges, including undesired cross-coupling effects, suboptimal spatial utilization, complex control strategies, and insufficient system stability. To overcome the limitations, this article develops a multipolarity decoupled four-coil WPT system with constant voltage (CV) and constant current (CC). The proposed system suppresses undesired cross-coupling to negligible levels, thereby reducing the system complexity. In addition, the compensation network can be designed in a straightforward manner, providing improved design flexibility. A detailed mathematical derivation is presented to rigorously demonstrate the load-independent CV and CC output characteristics. Meanwhile, the inverter can achieve zero phase angle (ZPA), thereby improving the power factor of the WPT system. In addition, the multipolarity decoupled mechanism of the four-coil magnetic coupler is analyzed in detail theoretically. Finally, an experimental prototype is built and tested. The experimental results demonstrate a strong agreement with the theoretical analysis, ensuring load-independent CV and CC outputs of 68 V and 3.5 A, respectively. The system achieves a measured peak efficiency of 85.97%.

1. Introduction

The proliferation of electronic devices has made intelligent and diversified charging methods a prevailing trend in future technology development. Wireless power transfer (WPT) eliminates the need for direct electrical connections between power sources and devices, thereby addressing the safety concerns associated with the conventional wired charging method in harsh environments [1,2]. Due to its superior reliability, safety, flexibility, and user-friendliness, WPT has demonstrated significant application potential in a wide range of scenarios, including implantable medical devices [3,4], industrial robots [5,6], underwater equipment [7], wearable devices [8,9], autonomous aerial vehicles [10], and electric vehicles [11].

Compared to the wireless charging of a single electronic device, WPT systems offer even greater advantages when applied to simultaneously charging multiple devices [12]. On one hand, extensive research has been conducted on single-receiver WPT systems, focusing on aspects such as modeling, analysis, parameter identification, optimization, and control at the magnetic-coupler and system-topology levels [13]. For instance, transmitter array configurations with star-shaped coils have been explored in order to achieve orientation-tolerant power transfer for free-moving objects in IoT and biomedical applications [14]. On the other hand, dual-receiver WPT systems featuring different output characteristics have also garnered research attention, particularly those capable of delivering constant voltage (CV) and constant current (CC) outputs simultaneously [15,16]. However, one of the key limitations in the current research lies in the difficulty of sufficiently suppressing unnecessary cross-coupling, especially the coupling interference of the receivers. Such undesired coupling can interfere with the independence of multiple output ports and degrade the overall efficiency of the system [17,18]. Accordingly, how to provide different types of output according to the load requirements without unnecessary cross-coupling becomes a significant design bottleneck for developing a dual-receiver WPT system.

In the existing research, the cross-coupling among receivers is negligible when the receiver coils are relatively small in size and sufficiently spaced apart [19]. Nevertheless, this approach lacks theoretical decoupling and becomes invalid as the system power level increases, where the influence of cross-coupling becomes non-negligible. In [20], a WPT system with multiple load-independent CV outputs is proposed. Although paralleling multiple double-T resonant circuits at the common AC terminal of the receivers can mitigate coupling interference, maintaining CV outputs require wireless communication modules and PI controllers, which leads to increased system complexity. In addition, PT-symmetric WPT mechanisms have been investigated for improving positional robustness and achieving a flexible current–ratio adjustment [21]. However, such PT-symmetric schemes mainly focus on robustness enhancement or current–ratio regulation, rather than receiver-to-receiver decoupling and simultaneous load-independent CV/CC outputs. Mai et al. [22] propose a decoupling circuit topology that suppresses cross-coupling by employing additional inductors or capacitors. However, these passive components must be precisely configured between the receivers, and the extra components require additional physical space, thereby limiting the system’s mobility. In [23], a single-transmitter WPT system with dual-channel CC outputs is proposed, where dual CC outputs are achieved by modulating specific control parameters. However, the two receivers cannot be operated at the same resonant frequency. An auxiliary circuit for power flow control is proposed in [24] to mitigate cross-coupling interference. However, the necessity for operating the receivers at multiple frequencies imposes inherent constraints on system design and control. The frequency bifurcation approach and the power division control method are proposed for the multi-receiver WPT system [25,26]. However, when the operating frequencies are closely spaced, unwanted power transfer between receivers may occur, thereby compromising the independence and stability of each receiver’s CV and CC output.

The objective of suppressing cross-coupling can also be addressed through the design of a magnetic coupler. In [27], a magnetic shielding plate is employed in the WPT system, and, thus, only the coupling coefficients between adjacent coils are considered in the analysis, with the cross-coupling effects between other coils being neglected. However, the incorporation of DC/DC converters in each repeater unit for output current regulation introduces increased costs and additional power losses, and poses significant challenges in achieving a broad range of parameter adjustments. Notably, decoupled-double D (DDD) coils with dual-resonating-frequency (DRF) compensation [28] and concentric bucking planar coils (BPCs) [29] have been investigated to suppress the cross-coupling in dual-receiver WPT systems. The DDD-based method requires frequency overlay modulation (FOM) to enable dual-channel power transfer at distinct operating frequencies, which may increase the excitation and parameter-design complexity. In contrast, the BPC-based method achieves receiver decoupling through the flux cancellation produced by the bucking-coil layout, but it requires the careful configuration of the winding structure and geometric parameters, which may constrain the independent optimization of the coupling coefficient and coil quality factor. In addition, load-independent CV and CC outputs and zero phase angle (ZPA) are seldom addressed in these approaches. In [15], a dual-receiver WPT system realizes CC and CV outputs through magnetic-coupler and mutual-inductance design. However, its output characteristics and decoupling performance depend on specific coil structures and mutual-inductance configurations. In [16], two mutually perpendicular solenoid coils are adopted as receivers to achieve natural decoupling and load-independent CC/CV outputs. However, this structure does not include the relay coil adopted in this work, and the transmitter mainly transfers power to the two receivers through direct coupling paths. This leads to a difference in spatial arrangement flexibility compared with the proposed scheme.

To address the existing limitations in dual-receiver WPT systems, such as the insufficient suppression of coupling interference [19,25], circuit and control complexity [20,22], robustness-oriented PT operation without simultaneous receiver decoupling [21], interference caused by non-uniform operating frequencies [23,24], and limited design flexibility in magnetic couplers [27,28,29], a WPT system with load-independent CV and CC outputs is proposed in this article. The Q-shaped structure is adopted as the transmitter coil. Two spatially orthogonal helical coils are employed as the two receivers. A Q-shaped coil connected in series with the two orthogonal helical coils forms the relay coil. The proposed magnetic coupler exhibits natural-decoupling characteristics and suppresses the undesired cross-coupling within the system to negligible levels. Simultaneously, the system enables load-independent CV and CC outputs along with the ZPA characteristic. The main contributions of this work are summarized as follows:

• A multipolarity-decoupling mechanism based on the spatial coil orientation is proposed. This mechanism suppresses undesired cross-couplings to negligible levels without additional passive decoupling components or active control circuits. Compared with turn-count-based or inner-radius-based decoupling approaches, the proposed mechanism provides greater flexibility in magnetic-coupler design.
• A four-coil topology is developed to achieve load-independent CV and CC outputs without wireless communication or feedback control. Different from existing passive-decoupling CC/CV schemes, the proposed topology introduces a relay coil to form two independent coupling paths, thereby suppressing undesired direct coupling from the transmitter to the receiver coils and improving the spatial arrangement flexibility of the dual-receiver system.

The manuscript is organized as follows. In Section 2, a comprehensive mathematical framework is developed for the proposed WPT system. Theoretical derivations are carried out to establish the conditions for realizing load-independent CV and CC outputs, as well as the ZPA characteristic. Section 3 introduces the four-coil magnetic coupler, and a multipolarity-decoupling mechanism is analyzed in detail. Section 4 presents simulation results that verify the CV and CC output characteristics as well as the ZPA behavior of the WPT system. In Section 5, to further verify the feasibility and accuracy of the dual-receiver WPT system, a miniaturized experimental prototype is built and tested. Finally, a conclusion is drawn in Section 6.

2. Circuit Analysis

The entire circuit topology diagram of the proposed dual-output-port WPT system based on integrated multipolarity-decoupling four coils is shown in Figure 1. E is the system input DC voltage of the single-phase full-bridge inverter composed of four MOSFETs (S₁S₄). Such an inverter is equipped at the source side to generate AC power for feeding the transmitter coil. The fundamental harmonics approximation method is used to analyze the proposed WPT system. The fundamental component of the output voltage of the inverter can be expressed as follows:

$$u_1\left(t\right)={\displaystyle \frac{4}{\mathsf{\pi }}}E\sin \left(\omega t\right)$$

(1)

where ω = 2πf₀. The operating frequency f₀ is configured to be 85 kHz according to SAE J2954.

Therefore, the root mean square (RMS) value of u₁(t) can be expressed as follows:

$$U_1={\displaystyle \frac{2\sqrt{2}}{\mathsf{\pi }}}E\angle 0°$$

(2)

Inductances L₁, L₂, L_RA, and L_RB are the self-inductances of the transmitter coil, relay coil, first-stage receiver coil, and second-stage receiver coil. L_srB is the second-stage receiver compensation inductance. R₁, R₂, R_rA, R_rB, and R_srB are the parasitic resistances of each coil. R_A and R_B are the equivalent resistances of the two loads, respectively. I₁, I₂, I_rA, I_rB, and I_srB are the current phasors flowing through each circuit loop, respectively. C₁, C₂, C_A, C_B, and C_rB are the resonant capacitors of the circuit. C_fA and C_fB are the filter capacitors of two receivers, respectively. M₁₂, M_A1, M_B1, M_2A, M_2B, and M_AB represent the mutual inductances between the four coils. Owing to the multipolarity-decoupling mechanism, the undesired mutual inductances M_A1, M_B1, and M_AB can theoretically be minimized toward zero under ideal geometric symmetry, while only small residual values remain in the practical prototype. Therefore, for the dominant output-characteristic analysis, only the three main mutual inductances, namely, M₁₂, M_2A, and M_2B, are considered in the following circuit analysis. This will be further explained in the magnetic coupler section, where the design methodology of the decoupling mechanism is discussed.

The two receivers are connected correspondingly by full-bridge rectifiers composed of diodes (D₁–D₄) and (D₅–D₈), in order to convert the AC current I_rA and I_srB into DC charging currents I_A and I_B to power the receiver loads. To ensure that resonance conditions are provided on each side of the entire circuit architecture, all inductors and capacitors should satisfy the following equation:

$$\left\{\begin{array}{l}{L}_1={\left({\omega }^2{C}_1\right)}^{-1} L_2={\left({\omega }^2{C}_2\right)}^{-1}\\ L_{\mathrm{RA}}={\left({\omega }^2{C}_{\mathrm{A}}\right)}^{-1} L_{\mathrm{srB}}={\left({\omega }^2{C}_{\mathrm{rB}}\right)}^{-1}\\ L_{\mathrm{RB}}={\left({\omega }^2{C}_{\mathrm{B}}\right)}^{-1}+{\left({\omega }^2{C}_{\mathrm{rB}}\right)}^{-1}\end{array}\right.$$

(3)

To facilitate the subsequent analysis of the output characteristics of the WPT system, the minor parasitic resistances of the coils are neglected. To justify this approximation, the parasitic resistance in each coil branch is compared with the dominant effective impedance in the corresponding loop. In the worst case, the combined parasitic resistance on the CC-side receiver, R_rB + R_srB = 0.15 Ω, is approximately 3.7% of the minimum equivalent load resistance R_EB,min. Therefore, its influence on the dominant output-characteristic analysis is limited. Moreover, the experimental prototype still achieves the expected CV output of 68 V and CC output of 3.5 A, which further validates this approximation. The equivalent circuit diagram of the proposed WPT system is shown in Figure 2.

In Figure 2, R_EA and R_EB are equivalent output load resistances reflected to the input sides of two rectifiers, respectively. Their mathematical relationship can be calculated as follows:

$$\left\{\begin{array}{l}{R}_{\mathrm{EA}}={\displaystyle \frac{8}{{\mathsf{\pi }}^2}}{R}_{\mathrm{A}}\hfill \\ R_{\mathrm{EB}}={\displaystyle \frac{8}{{\mathsf{\pi }}^2}}{R}_{\mathrm{B}}\hfill \end{array}\right.$$

(4)

According to Kirchhoff’s voltage law (KVL), the voltage relationship of each circuit loop in the WPT system can be expressed as follows:

$$\left\{\begin{array}{l}{U}_1=(j\omega L_1+{\displaystyle \frac{1}{j\omega C_1}})I_1+j\omega M_{12}{I}_2\\ 0=j\omega M_{12}{I}_1+(j\omega L_2+{\displaystyle \frac{1}{j\omega C_2}})I_2-j\omega M_{2\mathrm{A}}{I}_{\mathrm{rA}}-j\omega M_{2\mathrm{B}}{I}_{\mathrm{rB}}\\ 0=-j\omega M_{2\mathrm{A}}{I}_2+\left(j\omega L_{\mathrm{RA}}+{\displaystyle \frac{1}{j\omega C_{\mathrm{A}}}}+R_{\mathrm{EA}}\right)I_{\mathrm{rA}}\\ 0=-j\omega M_{2\mathrm{B}}{I}_2+\left(j\omega L_{\mathrm{RB}}+{\displaystyle \frac{1}{j\omega C_{\mathrm{B}}}}+{\displaystyle \frac{1}{j\omega C_{\mathrm{rB}}}}\right)I_{\mathrm{rB}}-{\displaystyle \frac{1}{j\omega C_{\mathrm{rB}}}}{I}_{\mathrm{srB}}\\ 0=\left({\displaystyle \frac{1}{j\omega C_{\mathrm{rB}}}}+j\omega L_{\mathrm{srB}}+R_{\mathrm{EB}}\right)I_{\mathrm{srB}}-{\displaystyle \frac{1}{j\omega C_{\mathrm{rB}}}}{I}_{\mathrm{rB}}\end{array}\right.$$

(5)

where the relationships between ac inputs (U_A, U_B, I_rA, and I_srB) and dc outputs (U_rA, U_rB, I_A, and I_B) can be obtained from the following equation:

$$\left\{\begin{array}{l}{U}_{\mathrm{A}}={\displaystyle \frac{2\sqrt{2}}{\mathsf{\pi }}}{U}_{\mathrm{rA}}\\ U_{\mathrm{B}}={\displaystyle \frac{2\sqrt{2}}{\mathsf{\pi }}}{U}_{\mathrm{rB}}\\ I_{\mathrm{rA}}={\displaystyle \frac{\sqrt{2}\mathsf{\pi }}{4}}{I}_{\mathrm{A}}\\ I_{\mathrm{srB}}={\displaystyle \frac{\sqrt{2}\mathsf{\pi }}{4}}{I}_{\mathrm{B}}\end{array}\right.$$

(6)

By solving Equations (2)–(6), the input current phasor I₁, the first-stage receiver charging voltage U_rA, and second-stage receiver charging current I_B can be calculated as follows:

$$\left\{\begin{array}{l}{I}_1={\displaystyle \frac{{\omega }^2{C_{\mathrm{rB}}}^2{M_{2\mathrm{B}}}^2{R}_{\mathrm{EA}}{R}_{\mathrm{EB}}+{M_{2\mathrm{A}}}^2}{R_{\mathrm{EA}}{M}_{12}^2}}{U}_1\\ U_{\mathrm{rA}}={\displaystyle \frac{M_{2\mathrm{A}}E}{M_{12}}}\\ I_{\mathrm{B}}={\displaystyle \frac{8M_{2\mathrm{B}}E}{{\mathsf{\pi }}^2\omega L_{\mathrm{srB}}{M}_{12}}}\end{array}\right.$$

(7)

Accordingly, the following conclusion can be directly inferred from Equation (7): the first-stage receiver is capable of achieving load-independent CV output, whereas the second-stage receiver enables load-independent CC output. Furthermore, Equation (7) indicates that U_rA is directly proportional to M_2A and inversely proportional to M₁₂, while I_B is directly proportional to M_2B and inversely proportional to M₁₂. Therefore, under small perturbations with the other parameters fixed, U_rA exhibits a positive normalized sensitivity of +1 to M_2A and a negative normalized sensitivity of −1 to M₁₂. Similarly, I_B exhibits a positive normalized sensitivity of +1 to M_2B and a negative normalized sensitivity of −1 to M₁₂. When multiple mutual inductances vary simultaneously, the output deviation is determined by the combined perturbations of these parameters. Thus, the allowable deviations of the mutual inductances should be determined according to the required output regulation accuracy in practical applications, rather than by a fixed universal tolerance. The allowable deviations of the compensation capacitors cannot be directly derived from Equation (7), because capacitor tolerances disturb the resonance conditions in Equation (3); their quantitative tolerance analysis will be investigated in future work.

Moreover, the input impedance of the proposed WPT system is defined as Z_in = U₁/I₁. Based on Equation (7), the input impedance Z_in of the system is expressed as follows:

$$Z_{\mathrm{in}}={\displaystyle \frac{R_{\mathrm{EA}}{M_{12}}^2}{{\omega }^2{C_{\mathrm{rB}}}^2{M_{2\mathrm{B}}}^2{R}_{\mathrm{EA}}{R}_{\mathrm{EB}}+{M_{2\mathrm{A}}}^2}}$$

(8)

From Equation (8), Z_in seen from the inverter is purely resistive, which implies that the proposed circuit structure can realize ZPA operation at the designated operating frequency f₀ by incorporating the appropriate resonant compensation elements. As a result, the inverter is responsible solely for delivering active power to the loads, which means that it can enhance the system efficiency and reduce the power losses.

3. Design of Magnetic Coupler

The diagram of the proposed magnetic coupler with multipolarity-decoupling characteristics is illustrated in Figure 3. The first-stage receiver coil L_RA and second-stage receiver coil L_RB are wound perpendicularly onto the receiver-side magnetic core, where the wires are arranged in horizontal and vertical orientations, respectively. The Q-shaped transmitter coil L₁ is selected to establish the main power-transfer path while suppressing the undesired direct couplings from L₁ to the receiver coils through geometric flux cancellation. As analyzed below, the Y–Z-plane flux component generated by L₁ does not intersect L_RA, whereas the X–Z-plane flux components pass through L_RA from opposite directions and cancel each other, thereby suppressing M_A1. The same spatial-symmetry mechanism suppresses M_B1 for L_RB. The relay coil L₂ consists of a Q-shaped coil connected in series with two spatially orthogonal helical coils. The Q-shaped part maintains the main coupling with L₁, while the two helical coils independently establish the coupling paths M_2A and M_2B with L_RA and L_RB, respectively.

To elucidate the multipolarity-decoupling characteristics of the proposed magnetic coupler structure, this paper conducts a detailed theoretical analysis based on electromagnetic field theory. As the proposed dual-receiver WPT system necessitates magnetic decoupling between coils L_RA and L_RB, the mutual inductance M_AB is expected to approach zero under ideal geometric symmetry. Upon the application of excitation currents I_rA and I_rB to the coils L_RA and L_RB, respectively, the magnetic flux generated by coil L_RA is oriented parallel to the X–Z plane, while the magnetic flux generated by coil L_RB is oriented parallel to the Y–Z plane. For further analysis, the mutual magnetic flux linkage ψ_AB, representing the portion of magnetic flux generated by coil L_RB that links with coil L_RA, can be mathematically expressed as follows:

$${\psi }_{\mathrm{AB}}={\displaystyle \iint {\overrightarrow{B}}_{\mathrm{RB}}}\cdot \mathrm{d}{\overrightarrow{S}}_{\mathrm{A}}$$

(9)

where B_RB represents the magnetic flux density vector generated by coil L_RB, and S_A represents the effective surface area of the coil L_RA that is oriented perpendicular to the magnetic field vector. The magnetic flux distribution indicates that the magnetic flux produced by the coil L_RB does not intersect with the coil L_RA. Therefore, S_A evaluates to zero, and ψ_AB also theoretically approaches zero. The mutual inductance M_AB between coils L_RA and L_RB is determined by the following equation:

$$M_{\mathrm{AB}}={\displaystyle \frac{{\psi }_{\mathrm{AB}}}{I_{\mathrm{rB}}}}$$

(10)

The calculated result shows that M_AB theoretically approaches zero under ideal geometric symmetry, thereby confirming that the proposed structure is capable of achieving the decoupling characteristic between coils L_RA and L_RB.

Furthermore, the proposed WPT system necessitates a decoupling mechanism between the transmitter coil L₁ and the receiver coils (L_RA and L_RB), such that the mutual inductances M_A1 and M_B1 should be suppressed to near-zero levels under ideal geometric symmetry. When excitation currents I₁, I_rA, and I_rB are added to coils L₁, L_RA, and L_RB, respectively, the magnetic flux generated by coil L₁ is oriented parallel to the X–Z plane and Y–Z plane, while the magnetic flux generated by coil L_RA is oriented parallel to the X–Z plane. For further analysis, the mutual magnetic flux linkage ψ_A1, representing the portion of magnetic flux generated by coil L₁ that links with coil L_RA, can be expressed as follows:

$${\psi }_{\mathrm{A}1}={\displaystyle \iint {\overrightarrow{B}}_1}\cdot \mathrm{d}{\overrightarrow{S}}_{\mathrm{A}}$$

(11)

where B₁ represents the magnetic flux density vector generated by coil L₁. The magnetic flux generated by L₁ that is oriented parallel to the Y–Z plane does not intersect with the coil L_RA. Meanwhile, the magnetic flux components parallel to the X–Z plane intersect coil L_RA from opposite directions, resulting in positive and negative mutual magnetic flux linkages that cancel each other out. As a result of the above analysis, it can be concluded that ψ_A1 theoretically approaches zero. The mutual inductance M_A1 between coils L₁ and L_RA is determined by the following equation.

$$M_{\mathrm{A}1}={\displaystyle \frac{{\psi }_{\mathrm{A}1}}{I_1}}$$

(12)

Under ideal geometric symmetry, the calculated result indicates that MA1 theoretically approaches zero, demonstrating the decoupling tendency between coils L₁ and L_RA. By utilizing the analogous analytical method, it can be concluded that the mutual inductance M_B1 between coils L₁ and L_RB also theoretically approaches zero. Therefore, the proposed structure successfully realizes the decoupling characteristic between coils L₁ and L_RB. It is worth emphasizing that variations in the height of the receiver coils have no impact on the analytical results regarding the mutual magnetic flux linkages. The dimensional parameters of the magnetic coupler are listed in

Table 1. The dimensional parameters of the proposed magnetic coupler.

Parameters	Value	Parameters	Value
dq	265 mm	ds	159 mm
da	272 mm	db	163 mm
w	65 mm	h1	60 mm
h2	60 mm	--	--

.

To suppress the adverse impact of the skin effect, high-frequency Litz wire consisting of 400 strands with an overall bundle diameter of 2.8 mm is selected for fabricating the coils. Ferrite plates PC95 with a thickness of 2.5 mm are selected as the magnetic core to improve magnetic coupling. h₁ and h₂ are the separation distance between the magnetic cores.

4. Simulation Verification

4.1. Multipolarity Decoupling Characteristics Validation

In order to validate the design concept of the proposed WPT system, this paper designs a set of circuit parameters, as shown in

Table 2. Theoretical circuit parameters.

Parameters	Value	Parameters	Value
L1	125.97 μH	L2	585.91 μH
LRA	233.93 μH	LRB	203.78 μH
LsrB	28.04 μH	C1	27.74 nF
C2	5.91 nF	CA	15.08 nF
CB	19.87 nF	CrB	125.32 nF
CfA	300 μF	CfB	300 μF
M12	46.39 μH	M2A	56.47 μH
M2B	54.28 μH	MA1	0.008 μH
MB1	0.009 μH	MAB	0.012 μH
f0	85 kHz	E	60 V

.

In Table 2, the mutual inductances M_A1, M_B1, and M_AB are 0.008 μH, 0.009 μH, and 0.012 μH, respectively. This result provides a compelling validation of the multipolarity-decoupling characteristics inherent to the proposed magnetic coupler.

4.2. Load-Independent CC and CV Validation

According to the designed parameters, the relationship curves of the first-stage receiver output voltage U_rA and the second-stage receiver output current I_B under different loads and the frequency are drawn, as shown in Figure 4.

As illustrated in Figure 4, at the operating frequency of 85 kHz, U_rA and I_B are maintained at constant values of 68 V and 3.5 A, respectively. Therefore, the proposed WPT system can enable load-independent CV and CC outputs.

4.3. ZPA Validation

The frequency sweep curves of the input impedance angle of the system under different load conditions are obtained, as shown in Figure 5. The simulation result is consistent with the theoretical analysis, thereby validating the system’s ability to operate under the condition of achieving ZPA operation.

5. Experimental Verification

5.1. Experimental Setup

As depicted in Figure 6, a miniaturized experimental prototype of the proposed WPT system is built. This laboratory prototype is mainly utilized to verify the correctness of the theoretical analysis. Although the proposed topology is scalable in principle, a practical extension to higher power levels requires the careful consideration of engineering limitations, including magnetic-core saturation, thermal management, coil losses, the voltage/current stress of resonant components, electromagnetic compatibility, safety constraints, and misalignment sensitivity. These issues are beyond the scope of the present proof-of-concept prototype and will be investigated in future work targeting higher-power implementations.

The experimental setup consists of the following components: a DC power supply (PA20010-TD, PINTECH, Guangzhou, China); a full-bridge inverter composed of four power MOSFETs (C2M0045170D, Wolfspeed, Inc., Durham, NC, USA); a function/arbitrary waveform generator (DG1032, RIGOL Technologies Co., Ltd., Suzhou, China) for generating drive signals; a magnetic coupler; two rectifiers comprising SiC Schottky-barrier diodes (C3D16060D, Wolfspeed, Inc., Durham, NC, USA); two DC load resistors R_A and R_B; and an oscilloscope (DSOX3024G, Keysight Technologies, Santa Rosa, CA, USA). The inductance and capacitance values were measured using a ZX8590-10M LCR meter (ZXP, Changzhou, China). The reported output powers and efficiencies are calculated directly from the measured voltages and currents, without further instrument-uncertainty propagation analysis. The detailed experimental prototype parameters are listed in

Table 3. Experimental circuit parameters.

Parameters	Value	Parameters	Value
L1	125.68 μH	L2	585.67 μH
LRA	233.71 μH	LRB	204.12 μH
LsrB	28.04 μH	C1	27.81 nF
C2	5.89 nF	CA	14.99 nF
CB	19.83 nF	CrB	125.29 nF
CfA	920 μF	CfB	920 μF
R1	0.1 Ω	R2	0.43 Ω
RrA	0.13 Ω	RrB	0.13 Ω
RsrB	0.02 Ω	M12	46.11 μH
M2A	56.29 μH	M2B	54.28 μH
MA1	0.086 μH	MB1	0.093 μH
MAB	0.382 μH	f0	85 kHz
E	60 V	--	--

. The measured residual mutual inductances M_A1, M_B1, and M_AB are 0.086 μH, 0.093 μH, and 0.382 μH, respectively, accounting for only 0.15%, 0.17%, and 0.68% of the dominant coupling inductance M_2A. These residual mutual inductances are much smaller than the main power-transfer mutual inductances M₁₂, M_2A, and M_2B. Moreover, the simulation results based on near-zero cross-couplings and the experimental results with the measured residual mutual inductances both confirm the expected CV output of 68 V and CC output of 3.5 A. This confirms that the residual mutual inductances have a negligible influence on the main output performance of the proposed system.

5.2. Experimental Results of ZPA

The experimental waveforms of the first-stage receiver output voltage U_rA, the second-stage receiver output current I_B, and the output voltage U₁ and output current I₁ of the inverter are illustrated in Figure 7. In Figure 7a, the loads R_A and R_B are set to 60 Ω and 10 Ω, respectively. In Figure 7b, the loads R_A and R_B are set to 45 Ω and 5 Ω. As demonstrated in the waveform results, the system maintains the constant output voltage of 68 V and the constant output current of 3.5 A, which confirm the load-independent CC and CV characteristics. Furthermore, the coincidence of the zero-crossing points of U₁ and I₁ indicates that the system can operate under the ZPA conditions.

5.3. Misalignment Analysis at Specific Symmetric Positions

When the magnetic coupler is misaligned, the mutual inductance between coils also varies accordingly. The magnetic coupler misalignment along X-axis and Y-axis direction is shown in Figure 8.

During the displacement, the relative positions among the relay coil and the two receiver coils remain fixed, and the assembly is shifted as a whole. In Figure 8, α and β represent the axial misalignment distances along the X-axis and Y-axis, respectively. With the increase in α and β, the variations in mutual inductances are illustrated in Figure 9.

In Figure 9, M₁₂ initially increases and then decreases, reaching a value equal to that under the aligned condition at a misalignment of 22 mm and 40 mm, respectively. In contrast, M_A1 and M_B1 exhibit a monotonically increasing trend. M_2A, M_2B, and M_AB remain approximately constant during the variation of α and β. The impact of mutual inductance variations on the system output voltage and current is illustrated in Figure 10. The system output voltage U_rA and current I_B initially decrease and then increase with misalignment. Based on the original experimental data in Figure 10, the percentage deviations of U_rA and I_B from their nominal values of 68 V and 3.5 A are further calculated. Within the tested X-axis displacement range of α = 0–100 mm, the percentage deviation ranges from −10.41% to +68.82% for U_rA and from −9.71% to +66.29% for I_B. Within the tested Y-axis displacement range of β = 0–100 mm, the percentage deviation ranges from −8.16% to +20.85% for U_rA and from −6.86% to +25.71% for I_B. These results indicate that the nominal outputs are recovered only near specific symmetric misalignment positions, rather than over the entire lateral displacement range.

The variation trends in Figure 10 are consistent with the sensitivity analysis derived from Equation (7). Since M_2A and M_2B remain approximately constant during misalignment, whereas M₁₂ varies with displacement, U_rA and I_B mainly follow the variation of M₁₂ and deviate from or recover to their nominal values accordingly. As shown in Figure 9, when α exceeds 70 mm or β exceeds 110 mm, M_A1 or M_B1 exceeds 1 μH, indicating that the residual direct coupling becomes noticeably larger than that under the aligned condition. In this case, the theoretical prerequisite of the proposed decoupling mechanism is no longer strictly satisfied. Therefore, α ≤ 70 mm and β ≤ 110 mm are defined as the misalignment ranges within which the decoupling condition remains approximately valid for the experimental prototype, rather than the regulation ranges for maintaining the nominal CV and CC outputs. Within these ranges, the output values still vary with M₁₂, and the nominal outputs of 68 V and 3.5 A are recovered only near the specific symmetric positions shown in Figure 11.

At the specific symmetric positions 2–5, M₁₂, M_2A, M_2B, and M_AB remain close to their values under the aligned condition, while M_A1 and M_B1 increase by less than 0.5 μH. Therefore, the multipolarity-decoupling characteristics are still approximately maintained at these specific positions, and the measured U_rA and I_B recover to 68 V and 3.5 A, respectively. It should be emphasized that this output recovery occurs only at the specific symmetric positions, namely, positions 2 and 4 under an X-axis misalignment of α = ±22 mm and positions 3 and 5 under a Y-axis misalignment of β = ±40 mm. This behavior is caused by the geometric symmetry of the proposed magnetic coupler, rather than indicating general lateral misalignment tolerance. At other non-symmetric misalignment positions, U_rA and I_B vary with M₁₂ and deviate from their nominal values.

5.4. Dynamic Experimental Response

To assess the robustness and stability of the CV and CC outputs under varying load conditions, the dynamic experimental response waveforms of U₁, I₁, U_rA, and I_B are shown in Figure 12. In the first load variation, with R_A held constant at 60 Ω and R_B varying abruptly from 10 Ω to 5 Ω, the output current I_B deviation is as low as 1.3% and the output voltage U_rA remains unchanged. In the second load variation, with R_B held constant at 5 Ω and R_A varying suddenly from 60 Ω to 45 Ω, the output voltage U_rA deviation is as low as 0.6%. Accordingly, throughout the dynamic response, U_rA and I_B fluctuate little and remain nearly stable at 68 V and 3.5 A, respectively. The dynamic experimental result demonstrates that the WPT system can achieve independent CV and CC outputs across the dual output ports without mutual interference. For simultaneous load variations at the two receivers, Equation (7) indicates that, when the residual cross-couplings are negligible, U_rA and I_B do not directly depend on the load resistance of the other receiver. Therefore, the two output ports are expected to respond independently to their respective load disturbances. The present dynamic tests verify this port independence through two representative single-port load-step cases. Simultaneous load-step transients at both receivers have not been experimentally tested in this work and will be further investigated in future studies. In addition, the open-circuit tests in Section 5.5 show that one receiver can remain functional when the other receiver is disconnected. However, the transient process of fast receiver disconnection and reconnection has not been separately tested and will be considered in future dynamic reliability experiments. The wider load ranges of R_A = 30–70 Ω and R_B = 5–21 Ω are evaluated through the static tests in Section 5.5, while wider-range dynamic load-step tests remain part of future work.

5.5. Efficiency Analysis

According to the experimental results under different load-resistance conditions, the system efficiency and output power are depicted in Figure 13. P_outA and P_outB are the output power from R_A and R_B, respectively. The system efficiency is defined as the DC-to-DC efficiency from the DC input of the full-bridge inverter to the DC outputs of the two rectifiers, where P_in = E × I_in, P_outA = U_rA²/R_A, and P_outB = I_B² × R_B. Here, Iin is the measured DC input current. This efficiency includes the losses along the measured power-conversion path, including inverter switching and conduction losses, coil copper losses, ferrite-core losses, compensation-capacitor ESR losses, and rectifier-diode conduction losses. The CV-port efficiency, CC-port efficiency, and total efficiency are defined as ECV = P_outA/P_in × 100%, ECC = P_outB/P_in × 100%, and EFT = (P_outA + P_outB)/P_in × 100%, respectively. The reported efficiencies are calculated directly from the measured voltages and currents, without including the instrument uncertainty propagation. Within the measured load range, the system achieves a peak efficiency of 85.97% at a total output power of 219.01 W.

Owing to the absence of direct electrical interconnection and mutual inductive coupling between the two receivers, the system ensures the independent operation of each receiver. Consequently, the disconnection of one receiver does not impair the functionality of the other. Figure 14 shows the variation curves of the output power and the system efficiency at different R_A and R_B values. Under the condition where the second-stage receiver is an open circuit, P_outA and the system efficiency gradually degrade with the rising R_A. Under the condition where the first-stage receiver is an open circuit, P_outB increases progressively with the rise in R_B, while the system efficiency initially improves due to better impedance matching and subsequently declines as the load deviates further from the optimal condition. To further benchmark the proposed system against representative multi-receiver WPT systems,

Table 4. Quantitative comparison with representative multi-output and multi-receiver WPT systems.

Ref.	Freq.	Coils	Output Type	Decoupling Method	Extra Control	Peak η (%)
[15]	200 kHz	3	CC + CV	Magnetic-coupler/mutual-inductance design	No	84.5
[16]	100 kHz	4	CC + CV	Natural decoupling of perpendicular solenoid coils	No	92.83
[20]	85 kHz	2	Multi-CV	Cascaded double-T resonant circuits	Yes	91.12
[22]	85 kHz	3	Dual independent voltage outputs	Passive L/C decoupling circuit	No	93
[23]	100/300 kHz	3	Multi-CC	MFRC and band-pass filtering	Yes	N/R
[28]	100/300 kHz	4	Dual-frequency dual-channel transfer	Orthogonal DD coils with DRF compensation	FOM required	83
[29]	85 kHz	3	N/R	Concentric bucking coil/BPC	No	N/R
This work	85 kHz	4	CC + CV	Spatial multipolarity geometry	No	85.97

N/R = not reported in the referenced work.

summarizes the operating frequency, coil number, output type, decoupling method, extra control requirement, and peak efficiency of existing methods and this work.

6. Conclusions

This paper proposes a WPT system based on integrated multipolarity-decoupling four coils. The measured residual mutual inductances M_A1, M_B1, and M_AB account for only 0.15%, 0.17%, and 0.68% of the dominant coupling inductance M_2A, respectively, indicating that the proposed magnetic coupler exhibits effective natural-decoupling characteristics. With the undesired cross-couplings suppressed to negligible levels, the system achieves load-independent CV and CC outputs without observable mutual interference. The proposed multipolarity-decoupling method depends on the geometric winding configuration of the coils rather than the precise design of the turns or the inner radius, thereby offering greater design flexibility for the magnetic coupler. Furthermore, the integration of the coils onto the magnetic cores not only improves the space utilization but also contributes to the compactness and practicality of the proposed magnetic coupler. Under varying load conditions, the system maintains CV and CC outputs at two receivers, respectively, indicating strong robustness and stability. Within the designed load range, the experimental results show that the system can obtain a maximum efficiency of 85.97%.