Higher-order anti-PT symmetry for efficient wireless power transfer in multiple-transmitter system

Abstract

Wireless power transfer (WPT) offers significant advantages, particularly due to its flexibility, enabling diverse applications. However, conventional single-transmitter, single-receiver systems are limited by their sensitivity to lateral disturbances, frequency instability, and strict distance constraints. Recently, multiple-transmitter, single-receiver (MTSR) systems have gained attention for their potential to enhance system flexibility and reliability. In this work, we propose an efficient second-order anti-parity‒time (anti-PT) symmetry by introducing two transmitters that simultaneously exchange energy with the external channel. This concept is further extended to third-order anti-PT symmetry for efficient WPT in MTSR systems. By leveraging interference between shared sources, we construct virtual coupling instead of relying on traditional resistive losses. Remarkably, our system maintains frequency stability, broad bandwidth, and robust high-efficiency power transfer even when the resonant frequencies of the transmitter and receiver coils are mismatched. This innovation challenges conventional understanding and opens new directions for WPT technology.

Graphical abstract

Keywords

non-Hermitian physics / anti-PT symmetry / wireless power transfer

1 Introduction

Wireless power transfer (WPT) eliminates the need for conventional wired connections, minimizing the reliance on large batteries and providing key benefits such as enhanced flexibility, convenience, and improved safety [1-3]. Magnetic resonance-based WPT has advanced rapidly in recent years [4-8], driven by improvements in system design and resonator materials. Fine-tuning system parameters allows for efficient energy transfer, optimizing performance under varying conditions. However, these optimal parameters are highly sensitive to the coupling strength between the source and receiver resonators, complicating stable energy transfer to a moving receiver while maintaining high efficiency [9-12]. The traditional single-transmitter, single receiver (STSR) architecture suffers from sensitivity to lateral disturbances, dynamic frequency adjustment requirements, and strict distance constraints, which limit efficiency [2, 4]. The multiple transmitter, single receiver (MTSR) configuration, where multiple transmitters supply energy to a single receiver, enhances system flexibility and reliability [13-17]. In environments like smart homes and offices, where devices are constantly moving, a single transmitter struggles to meet all energy demands [18]. A multi-transmitter system offers spatial coverage, ensuring efficient energy delivery to devices at any location [13]. In fields such as healthcare, industrial automation, and intelligent transportation, where device locations and statuses are unpredictable, multi-transmitter systems provide a more reliable power supply, reducing downtime and maintenance costs [16].

Non-Hermitian parity−time (PT) symmetry [19-22], a mechanism rooted in quantum physics, has revealed a range of fascinating effects that can significantly enhance the performance of WPT systems [23-27]. Studies suggest that by carefully balancing gain and loss, PT-symmetric systems can achieve robust dynamic WPT, demonstrating phenomena such as single-mode lasing, loss-induced transmission, and unidirectional reflection [28-30]. However, most PT-symmetric WPT systems require frequency tracking to maintain efficient energy transfer, which increases both complexity and cost. Higher-order PT-symmetric WPT systems offer frequency stability independent of transmission distance, but their stringent symmetry places significant constraints on practical applications [31, 32]. On the other hand, anti-PT symmetry, as the counterpart to PT symmetry, defines another fascinating category of non-Hermitian systems , have gained attention due to unique properties like topological singularities [33-35] and edge states [36-38]. Anti-PT-symmetric systems have been realized on various platforms, including coupled waveguides [39], micro-wave magnonics [40], optical fibers [41], artificial meta-atoms [42], coupled resonance rings [43], circuit systems [44], and flying atoms [45]. More recently, the level-locking effect in anti-PT-symmetric systems has been used to achieve stable and efficient WPT [46]. Nevertheless, these systems often depend on complex circuitry, such as nonlinear gain sources or adaptive power amplifiers, which significantly limits their wider adoption.

To address these challenges, we propose a novel anti-PT multi-transmitter WPT system. By leveraging interference between shared sources to replace traditional resistive losses [39], we construct virtual coupling and achieve second-order anti-PT symmetry. This is then extended to third-order anti-PT symmetry, resulting in the development of an MTSR system. Notably, even when the resonant frequencies of the transmitter and receiver coils do not match, our system maintains frequency stability, wide bandwidth, and robust, efficient power transfer. This innovation challenges conventional understanding, providing valuable insights into optimal WPT operating conditions and introducing a new approach for efficient dynamic MTSR WPT. Furthermore, the anti-PT-symmetric platform we propose is highly adaptable and can be extended to more advanced high-order systems. This approach holds significant promise for both improving WPT efficiency and advancing the study of anti-PT-symmetric phenomena.

2 Results

Second-rder anti-PT-symmetric systems. In the Hamiltonian of a physical system, the distinction between anti-PT symmetry and PT symmetry lies in the difference of sign, as indicated by H_{\text{anti-PT}}=\pm \mathrm{i}{H}_{\mathrm{P}\mathrm{T}}. This difference is reflected in the PT transformation, where PTH(PT {)}^{-1}=\pm H. A positive sign corresponds to PT symmetry, while a negative sign denotes anti-PT symmetry. Due to their inherent similarities, both symmetries share common properties, behaviors, and applications across various contexts. In this work, we construct a second-order PT-symmetric system, as shown within the red dashed box in Fig.1(a), where two resonators (blue) are driven simultaneously by an external source that generates two input signals S_{1+}. The corresponding simplified energy level diagram is shown in the red dashed box in Fig.1(b). Additionally, a symmetric frequency detuning 2\mathrm{\Delta } is introduced between the two resonators for {\omega }_1={\omega }_0+ \mathrm{\Delta } and {\omega }_2={\omega }_0-\mathrm{\Delta }, thus creating a frequency mismatch. The dynamic equation of the coupled resonators in the red box can described by temporal coupled mode theory as [47]

(1)\begin{array}{rl}& \frac{\mathrm{d}{a}_1}{\mathrm{d}t}=\left[-\mathrm{i}\left(\omega -{\omega }_1\right)-g_1-{\mathrm{\Gamma }}_1\right]a_1+\sqrt{2g_1}{S}_{1+}-\sqrt{g_1{g}_2}{a}_2{\mathrm{e}}^{\mathrm{i}\phi },\\ & \frac{\mathrm{d}{a}_2}{\mathrm{d}t}=\left[-\mathrm{i}\left(\omega -{\omega }_2\right)-g_2-{\mathrm{\Gamma }}_2\right]a_2+\sqrt{2g_2}{S}_{1+}-\sqrt{g_1{g}_2}{a}_1{\mathrm{e}}^{\mathrm{i}\phi },\end{array}

where g_i and {\mathrm{\Gamma }}_i (i=1, 2) denote the radiative loss and dissipative loss of the harmonic modes a_j=A_j{\mathrm{e}}^{-\mathrm{i}\omega t} in two resonators, respectively. The phase impactor \phi correspons to zero because two resonators are simultaneously coupled to an external source without phase difference [48]. The associated dynamic equation can be represented as H_2\psi =\omega \psi, where \psi ={\left(a_1,a_2\right)}^{\mathrm{T}}. Considering the same radiative loss g_1=g_2=g, same dissipative loss {\mathrm{\Gamma }}_1= {\mathrm{\Gamma }}_2= \mathrm{\Gamma }, and zero reflected waves S_{1-}=S_{1+} -\sqrt{2g_1} a_1- \sqrt{2g_2}{a}_2=0, the effective Hamiltonian of the second-order system can be written as

(2)H_2=\left(\begin{array}{cc}{\omega }_0+\mathrm{\Delta }-\mathrm{i}{\mathrm{\Gamma }}_1+\mathrm{i}g& \mathrm{i}{\kappa }_0\\ \mathrm{i}{\kappa }_0& {\omega }_0-\mathrm{\Delta }-\mathrm{i}{\mathrm{\Gamma }}_2+\mathrm{i}g\end{array}\right),

where the virtual coupling \mathrm{i}{\kappa }_0=\mathrm{i}\sqrt{g_1{g}_2}= \mathrm{i}g. According to Eq. (2) , we can find that \left(PT\right)H_2(PT{)}^{-1}=P{H}_2^{\ast }P=-H_2, confirming that the system exhibits anti-PT symmetry. By solving the characteristic equation \left|H_2-\omega I\right|=0 (where I is the identity matrix), the evolution of the real and imaginary parts of the eigenfrequencies is calculated, as shown in Fig.1(c). Under the influence of virtual coupling, the system’s eigenvalues approach and eventually coalesce at exceptional point (EP) as the coupling strength increases. The characteristic frequency evolution of the second-order anti-PT-symmetric system with virtual coupling can be found in the Supplementary Note A.

Third-order anti-PT-symmetric systems. The second-order system is extended by incorporating an additional resonator with {\omega }_3={\omega }_0 (yellow), as depicted in Fig.1(a), forming a composite third-order system. In this configuration, the previously detuned two resonators (blue) are coupled to the newly added resonator through real couplings {\kappa }_1 and {\kappa }_2, respectively. The intrinsic loss of resonator 3 is also assume as {\mathrm{\Gamma }}_3=\mathrm{\Gamma }, the Hamiltonian of the third-order system is then expressed as

(3)H_3=\left(\begin{array}{ccc}{\omega }_0+\mathrm{\Delta }-\mathrm{i}\mathrm{\Gamma }+\mathrm{i}g& {\kappa }_1& \mathrm{i}{\kappa }_0\\ {\kappa }_1& {\omega }_0-\mathrm{i}\mathrm{\Gamma }-\mathrm{i}\gamma & -{\kappa }_2\\ \mathrm{i}{\kappa }_0& -{\kappa }_2& {\omega }_0-\mathrm{\Delta }-\mathrm{i}\mathrm{\Gamma }+\mathrm{i}g\end{array}\right).

Similarly, the system satisfies the relation \left(PT\right)H_3(PT{)}^{-1}=P{H}_3^{\ast }P=-H_3, which indicates that the system also exhibits anti-PT symmetry.

The presence of real coupling causes the eigenenergy levels to split progressively as the coupling strength increases. In this system, both dissipative real coupling and coherent virtual coupling coexist, and their interplay results in a unique phenomenon known as level pinning [41], as illustrated in the energy level structure shown in Fig.1(b). This phenomenon proves beneficial for achieving stable and robust WPT and for advancing the study of anti-PT-symmetric platforms. Similarly, the eigenfrequencies of the third-order anti-PT-symmetric system were calculated, with the results depicted in Fig.1(d). Notably, one real eigenvalue remains constant regardless of changes in the coupling strength, indicating that WPT systems based on this model no longer require dynamic frequency tracking, thereby significantly reducing both system complexity and operational costs.

Comparison of two configurations for a third-order anti-PT-symmetric system. We next construct the real WPT system using effect circuits, focusing on two practical configurations based on the proposed physical model. Configuration I of the third-order anti-PT-symmetric system is shown in Fig.2(a), where the receiver resonator is placed between two coaxial transmitter resonators along the x-axis, with a distance d_1 between them. This configuration is designed for long-range power transfer, as it accommodates greater separation between components while maintaining stability. Fig.2(b) depicts configuration II, where the transmitter resonators are spaced by a along the x-axis (the influence of neighboring coupling between transmitting coils can be found in Supplementary Note B), and the receiver resonator is positioned at the center of their plane, with a separation d_2 along the y-axis. This configuration offers better integration, making it suitable for compact or embedded systems, though it may have a shorter transmission range compared to configuration I. Depending on the specific application, either configuration can be selected to optimize for transmission distance or system integration.

The equivalent circuit diagrams for both configurations are presented in Fig.2(c). In these diagrams, the inductances and capacitances of the two transmitter coils are denoted as L_1 (L_2) and C_1 (C_2), respectively. The receiver coil is characterized by inductance L_3 and capacitance C_3, with a load resistance R_L connected to the receiver. It is important to note that in both configurations, the transmitting coils are wound in opposite directions (one clockwise and one counterclockwise) to generate the magnetic field shown in Fig.2(a) and (b). In Fig.2(c), the small circle on L_2 indicates counterclockwise winding, while L_1 and L_3 are wound clockwise.

The Kirchhoff’s equation can be written as

(4)\begin{array}{rl}& U_S=\mathrm{j}\left(\omega L_1-\frac{1}{\omega C_1}\right)I_1+I_1{r}_1+\mathrm{j}\omega M_1{I}_3,\\ & U_S=\mathrm{j}\left(\omega L_2-\frac{1}{\omega C_2}\right)I_2+I_2{r}_2-\mathrm{j}\omega M_2{I}_3,\\ & \mathrm{j}\omega M_1{I}_1-\mathrm{j}\omega M_2{I}_2=\mathrm{j}\left(\omega L_3-\frac{1}{\omega C_3}\right)I_3+(r_3+R_L)I_3,\\ & U_S=(I_1+I_2)Z_S,\end{array}

where U_S represents the output voltage of the power supply, r_{1,2} is the internal resistance loss of the system, M_{1,2} is mutual inductance between the resonant coils and Z_S denotes the equivalent input impedance of the system.

When I_3=0, it corresponds to the second-order case shown in the red dashed box of Fig.2(c) [which corresponds to the red dashed box in the Fig.1(a) and (b) models]. By utilizing the relationship between the circuit equations and the coupled mode dynamics equations: a_n= -\mathrm{i}{L}_{n}d{I}_n/d_{\tau }, \tau = \omega \mathrm{t} and {\omega }_0=1/\sqrt{L{C}_0}, {\omega }_1={\omega }_0+ \mathrm{\Delta }=1/\sqrt{L{C}_1}, {\omega }_2={\omega }_0-\mathrm{\Delta }=1 /\sqrt{L{C}_2}, Eq. (4) can be rewritten as

(5)\begin{array}{rl}& \left({{\omega }_1}^2-{\omega }^2-\mathrm{i}\omega \frac{Z_S}{L_1}+\frac{\omega r_1}{L_1}\right)a_1-\mathrm{i}\omega \frac{Z_S}{L_2}{a}_2+\mathrm{i}\frac{M_1\omega }{2L}{a}_3=0,\\ & \left({{\omega }_2}^2-{\omega }^2-\mathrm{i}\omega \frac{Z_S}{L_2}+\frac{\omega r_2}{L_2}\right)a_2-\mathrm{i}\omega \frac{Z_S}{L_1}{a}_1-\mathrm{i}\frac{M_2\omega }{2L_2}{a}_3=0,\\ & \left({{\omega }_0}^2-{\omega }^2-\mathrm{i}\omega \frac{Z_L}{L_3}+\frac{\omega r_3}{L_3}\right)a_3+\mathrm{i}\frac{M_1\omega }{2L_1}{a}_1-\mathrm{i}\frac{M_2\omega }{2L_2}{a}_2=0.\end{array}

We define the following: g_j= Z_S/(2L_j)(j=1,2), {\mathrm{\Gamma }}_j= r_j/(2L) (j=1,2,3), \mathrm{\Delta }={\omega }_{1,2}+{\omega }_0. Using the approximation relationship {\omega }^2-{\omega }_0^2\approx 2\omega (\omega -{\omega }_0), under the condition of zero reflected wave S_{1-}=-S_{1+} +\sqrt{2g_1} a_1+ \sqrt{2g_2}{a}_2=0 the corresponding dynamical equation for the open system is

(6)\begin{array}{rl}& \frac{\mathrm{d}{a}_1}{\mathrm{d}t}=\left[-\mathrm{i}({\omega }_0+\mathrm{\Delta })-g_1-{\mathrm{\Gamma }}_1\right]a_1-\mathrm{i}{\kappa }_1{a}_3+\sqrt{2g_1}{S}_{1+}-\sqrt{g_1{g}_2}{a}_2,\\ & \frac{\mathrm{d}{a}_2}{\mathrm{d}t}=\left[-\mathrm{i}({\omega }_0-\mathrm{\Delta })-g_2-{\mathrm{\Gamma }}_2\right]a_2+\mathrm{i}{\kappa }_2{a}_3+\sqrt{2g_2}{S}_{1+}-\sqrt{g_1{g}_2}{a}_1,\\ & \frac{\mathrm{d}{a}_3}{\mathrm{d}t}=(-\mathrm{i}{\omega }_0-\gamma -{\mathrm{\Gamma }}_3)a_3-\mathrm{i}{\kappa }_1{a}_1+\mathrm{i}{\kappa }_2{a}_2.\end{array}

The dynamical equation can be expressed as

(7)H_3\left(\begin{array}{c}{a}_{{}_1}\\ a_{{}_3}\\ a_{{}_2}\end{array}\right)=\omega \left(\begin{array}{c}{a}_{{}_1}\\ a_{{}_3}\\ a_{{}_2}\end{array}\right).

By setting L_1= L_2= L_3= L, it follows that g_1=g_2=g , and the system’s equivalent Hamiltonian is as shown in Eq. (3) of the model. By setting a_3 in Eq. (7), the Hamiltonian for the second-order system is obtained, as expressed in Eq. (2). The coupling strength of the two configurations as a function of distance is shown in Fig.2(d). Both configurations exhibit exponential decay of coupling strength with distance [49]. At the same distance, the coupling strength of configuration I is higher than that of Configuration II; however, as the distance increases, the coupling strengths of both configurations converge. The data points represent experimental measurements, and the curves are fitted exponential functions. In addition, we also compared the efficiency of the third-order anti-PT-symmetric MTSR WPT system with the traditional MTSR WPT system (see more details in Supplementary Note C).

Superior transmission efficiency of third-order anti-PT-symmetric system. To streamline the calculation and minimize the complexity, we assume that {\kappa }_1={\kappa }_2=\kappa, {\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_2={\mathrm{\Gamma }}_3=\mathrm{\Gamma }, thereby simplifying the system’s parameters. With this assumption, the transmission efficiency of the system can be expressed as follows:

(8)\eta ={\left|S_{21}\right|}^2={\displaystyle \frac{16\gamma g{\mathrm{\Delta }}^2{\kappa }^2}{{\left(B+AC+2Dg\right)}^2}},

where A=\mathrm{\Gamma }-\mathrm{i}\delta \omega,B=\gamma \left({\mathrm{\Gamma }}^2+{\mathrm{\Delta }}^2-2\mathrm{i}\mathrm{\Gamma }\delta \omega -\delta {\omega }^2\right), C={\mathrm{\Gamma }}^2+{\mathrm{\Delta }}^2+ 2{\kappa }^2-{\omega }^2-2\mathrm{i}\mathrm{\Gamma }\delta \omega +2\omega {\omega }_0-{\omega }_0^2, D={\mathrm{\Gamma }}^2+2{\kappa }^2- {\omega }^2+\gamma A-2\mathrm{i}\mathrm{\Gamma }\delta \omega +2{\omega }_0\delta \omega, and \delta \omega =\omega -{\omega }_0.

The WPT efficiency of this system is compared with that of a traditional second-order WPT system, with the latter’s schematic shown in the blue shaded region of Fig.2(a). The theoretical results in Fig.2(e) and (f) reveal that, under identical conditions, the third-order anti-PT-symmetric system consistently outperforms the traditional second-order WPT system in both ideal lossless [\mathrm{\Gamma }/\left(2\pi \right) =0] and realistic loss-including scenarios [\mathrm{\Gamma }/\left(2\pi \right) =100\mathrm{H} \mathrm{z}]. The anti-PT symmetry eliminates frequency tracking requirements, ensuring stable operation even under detuning, a key advantage over PT-symmetric systems.

Detuning effects on system efficiency and impedance matching. Further analysis reveals that variations in the detuning (\mathrm{\Delta }) degree of freedom significantly affect both the eigenvalues and the efficiency of the system. Building upon the results presented in Fig.1(d), we conduct an in-depth investigation of this effect. As shown in Fig.3(a), we calculate and analyze the system’s response for different values of \mathrm{\Delta }=i \kappa (i=0 ,0.5,1, \sqrt{2},1.8,2) using Eq. (3). Notably, when \mathrm{\Delta }=\sqrt{2}\kappa, the system’s Hamiltonian eigenvalues exhibit a globally zero imaginary branch (orange branch), indicating that the system enters a distinct state.

To further explore the relationship between detuning and system efficiency, we analyze the efficiency-detuning dependence, with results shown in Fig.3(b). We observe that the efficiency reaches its maximum when \mathrm{\Delta }=\sqrt{2}\kappa, which coincides with the condition where the imaginary part of the eigenvalues is globally zero, marking this point as the system’s optimal operating condition. The pentagrams in Fig.3(b) represent calculation results, with different colors corresponding to the points in Fig.3(a).

Furthermore, we analyze the system efficiency from the perspective of circuit impedance by rewriting the circuit Eq. (4) as

(9)\begin{array}{rl}& U_S=i_1{Z}_1+\mathrm{i}\omega M{i}_3,\\ & U_S=i_2{Z}_2-\mathrm{i}\omega M{i}_3,\\ & \mathrm{i}\omega M(i_1-i_2)=i_3{Z}_r,\\ & U_S=(i_1+i_2)R_S,\end{array}

where Z_1= \mathrm{i}\left(\omega L_1-\frac{1}{\omega C_1}\right) +r_1, Z_2= \mathrm{i}\left(\omega L_2-\frac{1}{\omega C_2}\right) +r_2, Z_r= \mathrm{i}\left(\omega L_3-\frac{1}{\omega C_3}\right) +\left(r_1+R_L\right). Solving for the current is

(10)\begin{array}{rl}& I_1=-\frac{-2M^2{\omega }^2{U}_S-U_S{Z}_2{Z}_r}{M^2{\omega }^2{Z}_1+M^2{\omega }^2{Z}_2+Z_1{Z}_2{Z}_r},\\ & I_2=-\frac{-2M^2{\omega }^2{U}_S-U_S{Z}_1{Z}_r}{M^2{\omega }^2{Z}_1+M^2{\omega }^2{Z}_2+Z_1{Z}_2{Z}_r},\\ & I_3=\frac{\mathrm{i}M\omega \left(U_S{Z}_1-U_S{Z}_2\right)}{M^2{\omega }^2{Z}_1+M^2{\omega }^2{Z}_2+Z_1{Z}_2{Z}_r}.\end{array}

The input impedances of the two transmitting coils (Txs) are given by

(11)\begin{array}{rl}& Z_{\mathrm{T}\mathrm{x}1}=\frac{M^2{\omega }^2{Z}_1+M^2{\omega }^2{Z}_2+Z_1{Z}_2{Z}_r}{2M^2{\omega }^2-Z_2{Z}_r},\\ & Z_{\mathrm{T}\mathrm{x}2}=\frac{-M^2{\omega }^2\left(Z_1+Z_2\right)+Z_1{Z}_2{Z}_r}{-2M^2{\omega }^2+Z_1{Z}_r}.\end{array}

The input impedance Z_{\mathrm{i}\mathrm{n}} of the system is

(12)Z_{\mathrm{i}\mathrm{n}}=Z_{\mathrm{T}\mathrm{x}1}{Z}_{\mathrm{T}\mathrm{x}2}/\phantom{Z_{\mathrm{T}\mathrm{x}1}{Z}_{\mathrm{T}\mathrm{x}2}(Z_{\mathrm{T}\mathrm{x}1}+Z_{\mathrm{T}\mathrm{x}2}}(Z_{\mathrm{T}\mathrm{x}1}+Z_{\mathrm{T}\mathrm{x}2}).

The three-dimensional plot of the system’s input impedance Z_{\mathrm{i}\mathrm{n}} as a function of detuning and coupling is presented in Fig.3(c). At \mathrm{\Delta }=\sqrt{2}\kappa, the impedance matches the source impedance of 50\mathrm{\Omega }, in agreement with the results shown in Fig.3(a) and (b). In summary, the analysis of the eigenvalue imaginary part, efficiency, and impedance as a function of detuning reveals that \mathrm{\Delta }= \sqrt{2}\kappa represents the optimal operating point for the system. At this condition, the system not only maximizes efficiency but also achieves ideal impedance matching, highlighting the significant role of detuning in system performance. These findings provide valuable insights for further optimization and enhanced transmission efficiency in system design.

Experimental validation and performance of the third-order anti-PT-symmetric WPT System. Based on the theoretical model outlined earlier, we fabricated an experimental prototype and performed experimental validation, as depicted in Fig.4(a). The magnetically coupled resonant coils were constructed using 2 mm diameter litz wire, wound in two layers around an acrylic ring with a radius of 6 cm and a total of 20 turns. To ensure consistency, the inductance of all three coils — two transmitters (Tx1 and Tx2) and the receiver (Rx)—were precisely set to 230 μH. The receiver resonant coil was equipped with an 11 nF lumped capacitor to fine-tune the system’s resonance angular frequency, which was set to approximately 100 kHz, with a load resistance of 100 Ω. The experimental setup was configured with four interfaces: interfaces I and II were connected to the network analyzer (Keysight E5071C), while interfaces III and IV were connected to voltage probes on an oscilloscope (Keysight DSOS054A). Specifically, interfaces 1 and 2 were linked to Tx1, and interfaces 3 and 4 to Tx2. It is important to emphasize that in this system, multiple transmitters are configured in a parallel connection, simultaneously injecting energy to create an interference with zero phase difference. This interference generates a common source, and the gain produces an interference effect, establishing a virtual coupling. The virtual coupling mechanism enables efficient and robust energy transfer through the interference effect, not only enhancing the system’s power transmission efficiency but also improving its flexibility and stability in practical applications.

In the experiment, it is important to note that to achieve opposite coupling coefficients at the receiver coil, the two transmitter coils were wound in reverse directions. On the one hand, it is a necessary condition for constructing positive and negative coupling in the physical model of the anti-PT system. On the other hand, it can significantly suppress unnecessary next nearest neighbor coupling between the transmitting ends. Through this optimized coupling management mechanism, the system achieves energy transfer efficiency improvement while ensuring operational stability, thereby minimizing parasitic energy dissipation and potential instability factors in traditional designs. Additionally, the system’s detuning was dynamically adjusted by varying the lumped capacitor values in the transmitter coils. This adjustment was achieved using dip switches, as shown in the zoomed-in view in the lower-right corner of Fig.4(a). During the experiment, as the distance between the coils d_1(d_2) was varied, the capacitances of C_1 and C_2 were adjusted accordingly via the switches. Specifically, in configuration I, the capacitance of C_1 increased from 6.03 nF to 10.13 nF, while C_2 decreased from 26.03 nF to 11.98 nF. Although configurations I and II differ in their setup, the experimental results are the same for both configurations with the same coupling strength \kappa. Due to the narrower adjustable range of \kappa in configuration II, only the experimental results for configuration I are presented. Fig.4(b) illustrates the relationship between system efficiency, coupling strength, and frequency. The colored surface represents the theoretical results, while the green spheres correspond to the experimental measurements. The experimental data shows good agreement with the theoretical calculations.

Due to the difficulty in adjusting the output of the gain source, we investigated the trend of detuning by varying the values of \mathrm{\Delta }=10, 21, 30, 42, and 50 kHz. The results for the second-order anti-PT-symmetric system are shown in Fig.4(c), where we measured the system’s reflection coefficient and observed the frequency splitting. As \mathrm{\Delta } increased, the degree of splitting became more pronounced. The spheres in the figure represent the projections of the reflection coefficient trough minima onto the plane, while the magenta curve corresponds to the theoretical calculations. The experimental data closely align with the theoretical predictions.

Asymmetric perturbations were applied to compare the efficiency degradation between the PT-symmetric and anti-PT-symmetric systems, as shown in Fig.4(d). For the anti-PT-symmetric system, the system was excited at a fixed locking frequency {\omega }_0, with the position of one transmitter coil (TX1) and the receiver coil (RX) fixed, while a random perturbation was applied to the position of the other transmitter coil (TX2), disrupting the symmetry of the coupling. For the PT-symmetric system, the same perturbation was applied to TX2, with all other conditions unchanged. The perturbation strength \xi was related to the micro perturbation (\text{δ}\kappa) by \xi =\text{δ}\kappa /50. Perturbation strengths of \xi =15%, 20%, 23%, 25%, and 30% were tested, with experimental results shown by circles and stars. The third-order anti-PT-symmetric system (orange curve) exhibited superior robustness to perturbations.

The third-order anti-PT-symmetric WPT system was evaluated by measuring the transmission coefficient. Fig.4(e), (f), and (g) show the transmission coefficient as a function of frequency (f) and coupling strength (\kappa) from calculation, simulation, and experiment. In the strong coupling regime, the system maintains high transmission across a wide frequency range, as highlighted by the pink regions. To highlight the system’s frequency bandwidth advantage, Fig.4(h) shows the frequency bandwidth corresponding to different \kappa values when the transmission coefficient exceeds 0.9. Simulations were conducted using Advanced Design System (ADS) software. As the coupling strength \kappa increases, the frequency bandwidth also expands, with \mathrm{\Delta } f surpassing 10 kHz when \kappa > 10 kHz. The calculated (green), simulated (orange), and experimental (pink) results are in good agreement. In conclusion, the third-order anti-PT WPT system demonstrates a significant enhancement in frequency bandwidth, showcasing its potential for high-efficiency WPT across a broad frequency spectrum. The broadband performance of this system can be further applied to magnetic actuation capabilities [50]. Furthermore, we can expand it to a multi transmitter and multi receiver WPT system to meet the demand for multi-target power supply in the Internet of things era [51]. For detailed discussion and expansion, please refer to the Supplementary Note D.

3 Conclusion

In summary, we present a novel third-order anti-PT symmetric multi-transmitter WPT system that seamlessly integrates non-Hermitian physics with WPT, embedding anti-PT symmetry into a simple MTSR-based circuit architecture. By exploiting interference between shared sources to replace traditional resistive losses, we establish virtual coupling and achieve second-order anti-PT-symmetry system, which is subsequently extended to third-order anti-PT symmetry, resulting in the MTSR WPT system. Notably, even when the resonant frequencies of the transmitter and receiver coils are mismatched, our system maintains frequency stability, wide bandwidth, and robust, efficient power transfer. This work not only provides valuable insights for optimizing WPT system operation but also opens new research avenues for the development of efficient, dynamic MTSR WPT technologies. Further exploration of the anti-PT-symmetric platform and higher-order system structures will not only deepen our understanding of non-Hermitian phenomena but also enhance the frequency stability, energy transfer efficiency, and wideband robustness of WPT systems, offering promising implications for applications in smart homes, medical devices, and industrial automation.

References

[1] M. Song, P. Jayathurathnage, E. Zanganeh, M. Krasikova, P. Smirnov, P. Belov, P. Kapitanova, C. Simovski, S. Tretyakov, and A. Krasnok, Wireless power transfer based on novel physical concepts, Nat. Electron. 4(10), 707 (2021)

[2] A. P. Sample, D. T. Meyer, and J. R. Smith, Analysis, experimental results, and range adaptation of magnetically coupled resonators for wireless power transfer, IEEE Trans. Ind. Electron. 58(2), 544 (2011)

[3] C. Zeng, Z. Guo, K. Zhu, C. Fan, G. Li, J. Jiang, Y. Li, H. Jiang, Y. Yang, Y. Sun, and H. Chen, Efficient and stable wireless power transfer based on the non-Hermitian physics, Chin. Phys. B 31(1), 010307 (2022)

[4] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljacic, Wireless power transfer via strongly coupled magnetic resonances, Science 317(5834), 83 (2007)

[5] Z. Miao, D. Liu, and C. Gong, Efficiency enhancement for an inductive wireless power transfer system by optimizing the impedance matching networks, IEEE Trans. Biomed. Circuits Syst. 11(5), 1160 (2017)

[6] F. Wen and X. Huang, Optimal magnetic field shielding method by metallic sheets in wireless power transfer system, Energies 9(9), 733 (2016)

[7] T. Imura and Y. Hori, Maximizing air gap and efficiency of magnetic resonant coupling for wireless power transfer using equivalent circuit and neumann formula, IEEE Trans. Ind. Electron. 58(10), 4746 (2011)

[8] K. Sun, R. Fan, X. Zhang, Z. Zhang, Z. Shi, N. Wang, P. Xie, Z. Wang, G. Fan, H. Liu, C. Liu, T. Li, C. Yan, and Z. Guo, An overview of metamaterials and their achievements in wireless power transfer, J. Mater. Chem. C 6(12), 2925 (2018)

[9] X. Y. Zhang, C. D. Xue, and J. K. Lin, Distance-insensitive wireless power transfer using mixed electric and magnetic coupling for frequency splitting suppression, IEEE Trans. Microw. Theory Tech. 65(11), 4307 (2017)

[10] R. D. Fernandes, J. N. Matos, and N. B. Carvalho, Resonant electrical coupling: Circuit model and first experimental results, IEEE Trans. Microw. Theory Tech. 63(9), 2983 (2015)

[11] J. Zhou, B. Zhang, W. Xiao, D. Qiu, and Y. Chen, Nonlinear parity–time-symmetric model for constant efficiency wireless power transfer: Application to a drone-in-flight wireless charging platform, IEEE Trans. Ind. Electron. 66(5), 4097 (2019)

[12] A. Krasnok, D. G. Baranov, A. Generalov, S. Li, and A. Alù, Coherently enhanced wireless power transfer, Phys. Rev. Lett. 120(14), 143901 (2018)

[13] I. J. Yoon and H. Ling, Investigation of near-field wireless power transfer under multiple transmitters, IEEE Antennas Wirel. Propag. Lett. 10, 662 (2011)

[14] Y. Luo,Z. Dai,Y. Yang, A single-transmitter multi-receiver wireless power transfer system with high coil misalignment tolerance and variable power allocation ratios, Electronics (Basel) 13(19), 3838 (2024)

[15] J. Liu, Y. Min, J. Gao, A. Yang, and J. Zhou, Design and optimization of a modular wireless power system based on multiple transmitters and multiple receivers architecture, Front. Energy Res. 10, 896575 (2022)

[16] D. H. Kim and D. Ahn, Maximum efficiency point tracking for multiple-transmitter wireless power transfer, IEEE Trans. Power Electron. 35(11), 11391 (2020)

[17] B. L. Cannon, J. F. Hoburg, D. D. Stancil, and S. C. Goldstein, Magnetic resonant coupling as a potential means for wireless power transfer to multiple small receivers, IEEE Trans. Power Electron. 24(7), 1819 (2009)

[18] V. K. Awaar,M. N. S. Rani,P. Jugge, S. V. Bellal,B. N. Rao,H. M. Al-Jawahry, Multi-transmitter based wireless power transmission for electrical vehicles, in: 2024 IEEE 4th International Conference on Sustainable Energy and Future Electric Transportation (SEFET) 1, 2024

[19] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14(1), 11 (2018)

[20] C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80(24), 5243 (1998)

[21] G. Lévai and M. Znojil, Systematic search for PT-symmetric potentials with real energy spectra, J. Phys. Math. Gen. 33(40), 7165 (2000)

[22] Z. Guo, J. Jiang, Y. Wang, J. Alvarez-Cuervo, A. T. Martin-Luengo, S. Hu, J. Jiang, P. A. Gonzalez, J. Duan, and H. Chen, Exceptional point empowered nearfield routing of hyperbolic polaritons, Sci. Bull. (Beijing) 69(22), 3491 (2024)

[23] S. Assawaworrarit, X. Yu, and S. Fan, Robust wireless power transfer using a nonlinear parity–time-symmetric circuit, Nature 546(7658), 387 (2017)

[24] J. Song, F. Yang, Z. Guo, X. Wu, K. Zhu, J. Jiang, Y. Sun, Y. Li, H. Jiang, and H. Chen, Wireless power transfer via topological modes in dimer chains, Phys. Rev. Appl. 15(1), 014009 (2021)

[25] H. Zhang,Z. Guo,Y. Li,Y. Yang,Y. Chen, H. Chen, A universal non-Hermitian platform for bound state in the continuum enhanced wireless power transfer, Front. Phys. (Beijing) 19(4), 43209 (2024)

[26] Y. H. Wu, L. Kang, and D. H. Werner, Generalized symmetry in non-Hermitian wireless power transfer systems, Phys. Rev. Lett. 129(20), 200201 (2022)

[27] X. L. Hao, K. Yin, J. L. Zou, R. B. Wang, Y. G. Huang, X. K. Ma, and T. Y. Dong, Frequency-stable robust wireless power transfer based on high-order pseudo-Hermitian physics, Phys. Rev. Lett. 130(7), 077202 (2023)

[28] J. Li,B. Zhang, A wireless power transfer system based on quasi‐parity–time symmetry with gain–loss ratio modulation, Int. J. Circuit Theory Appl. 51(3), 1039 (2023)

[29] S. Longhi, PT-symmetric laser absorber, Phys. Rev. A 82(3), 031801 (2010)

[30] Y. D. Chong,L. Ge,A. D. Stone, PT-symmetry breaking and laser-absorber modes in optical scattering systems, Phys. Rev. Lett. 106(9), 093902 (2011)

[31] M. Sakhdari, M. Hajizadegan, and P. Y. Chen, Robust extended-range wireless power transfer using a higher-order PT-symmetric platform, Phys. Rev. Res. 2(1), 013152 (2020)

[32] Z. W. Guo,J. Jiang,X. Wu,H. Y. Zhang,S. Y. Hu, Y. Q. Wang,Y. H. Li,Y. P. Yang,H. Chen, Rotation manipulation of high-order PT-symmetry for robust wireless power transfer, Fundamental Research, doi: 10.1016/j.fmre.2023.11.010 (2023)

[33] X. L. Zhang, T. Jiang, and C. T. Chan, Dynamically encircling an exceptional point in anti-parity–time symmetric systems: asymmetric mode switching for symmetry-broken modes, Light Sci. Appl. 8(1), 88 (2019)

[34] Z. Feng and X. K. Sun, Harnessing dynamical encircling of an exceptional point in anti-PT-symmetric integrated photonic systems, Phys. Rev. Lett. 129(27), 273601 (2022)

[35] L. Wang, N. Liu, C. Wu, and G. Chen, Dynamical encircling of multiple exceptional points in anti-PT symmetry system, Opt. Express 32(12), 21616 (2024)

[36] S. M. Zhang, H. S. Xu, and L. Jin, Tunable aharonov-bohm cages through anti-PT-symmetric imaginary couplings, Phys. Rev. A 108(2), 023518 (2023)

[37] H. C. Wu, L. Jin, and Z. Song, Topology of an anti-parity–time symmetric non-Hermitian Su–Schrieffer–Heeger model, Phys. Rev. B 103(23), 235110 (2021)

[38] S. L. Ke, D. Zhao, J. X. Liu, Q. J. Liu, Q. Liao, B. Wang, and P. X. Lu, Topological bound modes in anti-PT-symmetric optical waveguide arrays, Opt. Express 27(10), 13858 (2019)

[39] F. Yang, Y. C. Liu, and L. You, Anti-PT symmetry in dissipatively coupled optical systems, Phys. Rev. A 96(5), 053845 (2017)

[40] Y. Yang, Y. P. Wang, J. W. Rao, Y. S. Gui, B. M. Yao, W. Lu, and C. M. Hu, Unconventional singularity in anti-parity–time symmetric cavity magnonics, Phys. Rev. Lett. 125(14), 147202 (2020)

[41] A. Bergman, R. Duggan, K. Sharma, M. Tur, A. Zadok, and A. Alù, Observation of anti-parity–time-symmetry, phase transitions and exceptional points in an optical fibre, Nat. Commun. 12(1), 486 (2021)

[42] Y. Yang, X. Xie, Y. Li, Z. Zhang, Y. Peng, C. Wang, E. Li, Y. Li, H. S. Chen, and F. Gao, Radiative anti-parity–time plasmonics, Nat. Commun. 13(1), 7678 (2022)

[43] C. Qin, B. Wang, S. Fan, and P. Lu, Gauge-flux-induced anti-PT phase transitions for extreme control of channel-drop tunneling, Laser Photonics Rev. 18(4), 2300458 (2024)

[44] Y. Choi, C. Hahn, J. W. Yoon, and S. H. Song, Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators, Nat. Commun. 9(1), 2182 (2018)

[45] P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, Anti-parity–time symmetry with flying atoms, Nat. Phys. 12(12), 1139 (2016)

[46] Z. Guo, F. Yang, H. Zhang, X. Wu, Q. Wu, K. Zhu, J. Jiang, H. Jiang, Y. Yang, Y. Li, and H. Chen, Level pinning of anti-PT symmetric circuits for efficient wireless power transfer, Natl. Sci. Rev. 11(1), nwad172 (2023)

[47] S. Fan, W. Suh, and J. D. Joannopoulos, Temporal coupled-mode theory for the Fano resonance in optical resonators, J. Opt. Soc. Am. A 20(3), 569 (2003)

[48] R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, Phase-coupled plasmon-induced transparency, Phys. Rev. Lett. 104(24), 243902 (2010)

[49] Z. Guo, H. Jiang, Y. Li, H. Chen, and G. S. Agarwal, Enhancement of electromagnetically induced transparency in metamaterials using long range coupling mediated by a hyperbolic material, Opt. Express 26(2), 627 (2018)

[50] K. Sun, C. Wang, J. Tian, Z. Zhang, N. Zeng, R. Yin, W. Duan, Q. Hou, Y. Zhao, H. Wu, and R. Fan, Magnetic‐driven broadband epsilon‐near‐zero materials at radio frequency, Adv. Funct. Mater. 34(2), 2306747 (2024)

[51] J. Jiang, S. Hu, Z. Guo, S. Ke, X. Wu, Y. Wu, Y. Wang, H. Jiang, Y. Yang, and H. Chen, Topological corner states for wireless power transfer, Rev. Phys 13(1), 100112 (2025)

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