Metamaterial-controlled non-hermitian wireless power transfer

Abstract

A wireless power system includes a transmission resonator and a metamaterial including an array of unit cells. The unit cells are individually adjustable to change a resonance frequency of the array. A controller adjusts unit cells in the array of unit cells to set a resonant mode providing coupling coefficients equal to that of the transmitter resonator and a receiver resonator to yield a paritytime symmetric state of the array, the transmitter resonator and the receiver resonator.

Description

METAMATERIAL-CONTROLLED NON-HERMITIAN WIRELESS POWER TRANSFER FIELD

[0001] The application claims priority under 35 U.S.C. §119 and all applicable statutesand treaties from prior United States provisional application serial number 63 / 603,469, which was filed November 28, 2023. FIELD

[0002] A field of the invention is wireless power transfer (WPT).BACKGROUND

[0003] Non-radiative WPT utilizes the magnetic near-field to carry energy and is morecommonly used than radiative WPT due to its advantages of high-power volume and safety. A receiving (Rx) resonator couples with a transmitting (Tx) resonator through magnetic mutual induction. Inductive WPT systems can be described as non-Hermitian systems through coupled-mode theory. Power transfer in such systems is efficient when forming parity-time (PT)-symmetric states, which can be guaranteed by physical symmetry in a strong coupling regime. S.Assawaworrarit, et al, "Robust wireless power transfer using a nonlinear parity– time-symmetric circuit," Nature, vol. 546, no. 7658, pp. 387-390 (2017). However, spontaneous symmetry breaking happens in the weak coupling regime with increasing Tx-Rx separation, where the resonant states become anti-PT- symmetric. In such scenarios, spontaneous symmetry breaking occurs, leading to the formation of resonant states with PT-symmetry broken phase, See, Z. Dong, et al., "Sensitive readout of implantable microsensors using a wireless system locked to an exceptional point," Nature Electronics, vol.2, no.8, pp.335- 342, (2019); R. El-Ganainy, et al., "Non-Hermitian physics and PT symmetry," Nature Physics, vol.14, no.1, pp.11-19 (2018).

[0004] Metamaterials are engineered to have properties that do not typically occur innature. Passive relay resonators have been widely employed in WPT systems to increase the effective coupling between the transmitting (Tx) and receiving (Rx) resonators using metamaterials. Example prior approaches include “Ranaweera, A.L.A.K., Pham, T.S., et all., “An active metasurface for field-localizing wireless power transfer using dynamically reconfigurable cavities,” Scientific reports, 9 (1), p.11735 (2019) and Zhong, W., et al, “General analysis on the use of Tesla's resonators in domino forms for wireless power transfer,” IEEE transactions on industrial electronics, 60 (1), pp.261-270 (2011).

[0005] These approaches can create additional eigenstates with broken PT symmetrywhere the relay resonators consume a significant amount of energy. Specific spatial arrangements of the relay resonators, being dependent on the positions of the Tx and Rx resonators, are required to avoid these states. Such a requirement limits the applications such as free-positioning WPT.

[0006] Another approach is described in Wang, H, et al. “A wearable metasurface forhigh efficiency, free-positioning omnidirectional wireless power transfer,” NewJournal of Physics, 23 (12), p.125003 (2021). This can configure the metasurface according to a targeted current distribution, but it does not provide an ability to optimize the targeted current distribution to form a PT symmetric state. Enhancement to the efficiency is therefore not guaranteed.

[0007] CN115864675 describes a wireless power transmission system that includes S-S type, P-S type, S-P type, and P-P type circuits. According to the size of the load in the application and the quality factor of the transmitting and receiving coils, one of the circuits is selected, and the load is converted to the optimal load by frequency selection and impedance conversion to enhance transmission efficiency. This approach uses a two-body system where the maximum transmission distance is limited by the design of the transmitter and receiver, such as size, number of turns, and geometry.

[0008] WO2010093997 describes a wireless transfer system that includes a sourcemagnetic resonator and a capacitively-loaded conducting loop configured to convert oscillating magnetic fields into electrical energy. The resonator has a keep-out zone around the resonator that surrounds the resonator with a layer of non-lossy material. Metamaterial can be used around high-conductivity surfaces to completely or partially enclose or cover the loss-inducing objects. This approach uses only a limited number of relay resonators without forming a homogenized resonance mode, thus, their coupling coefficient to the transmitter and receiver cannot be controlled.

[0009] Wang et al. US Published Patent Application US20230047663 describes anadvance in wireless power transfer that uses a metasurface. The metasurface includes a plurality of coupled resonators that are configured and arranged to couple within and shape a magnetic near-field distribution from a transmitter into a target distribution toward a target receiver. The plurality of coupledresonators forms a non-uniform impedance distribution pattern to provide the shape of the target distribution. An insulated support structure of the resonators can be thin and flexible, allowing it to be worn by a person, for example, to transfer power to an implanted device. The ‘663 published application does not provide a method to tune the effective coupling coefficient to the Tx and Rx to form the PT-symmetric state, which is an advance provided in the current invention. SUMMARY OF THE INVENTION

[0010] A preferred embodiment provides a wireless power system including atransmission resonator and a metamaterial including an array of unit cells. The unit cells are individually adjustable to change a resonance frequency of the array. A controller adjusts unit cells in the array of unit cells to set a resonant mode providing coupling coefficients equal to that of the transmitter resonator and a receiver resonator to yield a parity-time symmetric state of the array, the transmitter resonator and the receiver resonator. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] FIGs 1A-1C show a preferred metasurface for a wireless power transfer system;

[0012] FIGs. 1D and 1E illustrate the metasurface of FIGs. 1A-1C in a wireless powertransfer system that includes a power transmitter and a receiver;

[0013] FIGs. 2A and 2B respectively show a schematic diagram of a metamaterial-enhanced system of FIGs.1D-1E and the system states controlled by coupling coefficients;

[0014] FIGs. 3A and 3B plot system states at the mismatched coupling condition;

[0015] FIGs.4A-4F illustrate a practical system with the coupling coefficient controlledto achieve the exception point;

[0016] FIG. 5A and FIGs. 5B-5D respectively show an experimental wireless powertransfer system of the invention data concerning testing and simulation of the system;

[0017] FIG. 6 shows that gain of the FIG. 5A experimental system reaches saturationafter four oscillatrions;

[0018] FIG. 7A-7B show a simulation of the unit cell of the FIGs 1A-1C preferredmetasurface;

[0019] FIG. 8 shows the unit cell simulation experimental set up;

[0020] FIG. 9 shows that the configuration of the metamaterial can be uniquely givenby the differential frequency of the center unit cell;

[0021] FIGs. 10A-10B show the metamaterial configuration characterized bydifferential resonance frequencies between the unit cells;

[0022] FIGs. 11A-11C show a preferred automatically configurable metasurface;

[0023] FIGs. 12A and 12B show the efficiency maps with and without the auto-reconfigurable metasurface of FIGs.11A-11C; and

[0024] FIG. 13 shows the experimental results for the measured efficiency at trackingtarget for the FIGs.11A-11C automatically reconfigurable metasurface. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0025] As used in the present application, and the applicable art to which it relates, anexceptional point is a point between strong and weak coupling regimes, where the wireless power transfer is both highly efficient and sees minimal frequency splitting. Frequency splitting increases load power and therefore energy consumption.

[0026] Preferred embodiment systems and methods provide for control of PT-symmetric states of non-Hermitian WPT systems with metamaterials, which minimizes the transmission loss and enables high-efficiency, mid-range, frequency-stable, and free-positioning WPT. Independent control of individual unit cells of the metamaterial provides a unique solution for controlling high- order exceptional points in non-Hermitian WPT systems.

[0027] Preferred embodiments provide a metamaterial-enhanced non-Hermitian WPTsystem that can reach high-order exceptional points without physical rearrangement of the transmitter and receiver. Control of individual unit cells of the metamaterials provides a PT-symmetric state that appears when the absolute coupling coefficients of the metamaterial to the Tx and Rx resonators are the same. As the coupling coefficients are determined by the resonant mode of the metamaterial, the inverse design of the metamaterial configuration with a given targeted mode can achieve the controllable PT symmetry. Preferred systems demonstrate the phase transition between anti-PT-symmetric and PT-symmetric states can be controlled by the metamaterial. High-order exceptional points can be achieved with different spatial arrangements of the Rx resonator, which permits free-positioning WPT.

[0028] The control creates a WPT system that is more efficient and can transfer powerover a long distance to the receivers at arbitrary positions. Near-field is with respect to the operating wavelength, which is in ten to hundreds of meters at kilohertz to megahertz frequency regime, where the WPT systems usually operate. Long distance is determined with respect to the transmitter and receiver's sizes. For example, for phones with centimeter-scale receivers, a transmitting distance over 10 cm would be considered as a long distance.

[0029] The magnetic metamaterial is controlled to enforce a PT-symmetric state of thenon-Hermitian WPT system. By configuring the metamaterial according to the Rx positions, we can always guarantee the PT symmetry, which protects the system from forming additional eigenstates with broken PT symmetry, i.e., higher loss. This makes it possible to create free-positioning WPT systems, which have a wide range of potential commercial applications.

[0030] A preferred WPT system includes a metamaterial with a plurality of individuallyadjustable unit cells. Preferably, every one of the unit cells is adjustable, but tuning freedom can be reduced in view of potential symmetry in the system. For example, if the Tx, Rx, and the metasurface are aligned, the targeted mode would be symmetric to the center. Therefore, the unit cells in the symmetric locations can be linked and adjusted to the same resonant frequency. This is demonstrated experimentally, as demonstrated in FIG.9.

[0031] The unit cells are preferrable arranged in a hexagonal, oblique, or square array,with one or multiple layers. The size, periodicity, and number of unit cells can be designed to match the physical size of the transmitter and receiver. For example, in cases where there is a size mismatch between the Tx and Rx, the unit cell can be designed to have a similar size to the receiver, and the metamaterial to have a similar size to the transmitter, to optimize the effective coupling between the transmitter and the receiver, relayed by the metamaterial. The coupling between the unit cells can be optimized by taking the minimum periodicity with a certain number of layers. Preferably, the overall size of the metamaterial should roughly match with the Tx and the unit cell should match with the Rx. The periodicity should be as small as possible (with the given number of layers) to increase the coupling between the unit cells. A rule of thumb is to optimize the coupling, both the metamaterial's overall coupling tothe transmitting resonator (Tx) and the receiving resonator (Rx), and the unit cells' coupling. The metamaterial is coupled to the Tx and Rx, and periodic adjustment of unit cells based upon measurements conducted by the Tx and Rx, is performed to set a resonant mode providing coupling coefficients equal to the Tx and Rx to yield a parity-time symmetric state. The Tx and Rx both periodically communicate with the metamaterial to provide responses to allow determination of their coupling coefficients. The unit cells are controlled using the metamaterial's phase transition to achieve high-order exceptional points. The control of the unit cells can be mechanical, e.g., compression of a spiral- type unit cell, or electrical, e.g. via an adjustable capacitor.

[0032] A mobile device charging system is provided by a preferred embodiment.Magnetic metamaterials have tunable unit cells that are controlled to enforce a PT-symmetric state. A more efficient and long-distance wireless charging system is provided. A medical implant system is provided by a preferred embodiment, where a medical implant, such as pacemakers and insulin pumps, are receivers charged through a wireless power receiving unit that is small, lightweight, and easy to implant. Another application is free-positioning charging of EVs, as the control to enforce PT-symmetric state of the non- Hermitian WPT system permits vehicles to be charged without the requirement of precise alignment, and can allow movement during charging, i.e., the vehicle need not be stationary. Another application is to charge within an Internet-of- thing (IoT) network, including devices such as robots, sensors, light, and other in-network electronics. For example, it could be used to power robots, sensors, light, and other devices in remote locations. Artisans will appreciate that the present invention uses near-field, but near-field is determined with respect to the wavelength, which is in meters to hundreds of meters depending on the operatingfrequency. But the remote or transmission distance is with respect to the size of the Tx and Rx. For example, a transmission distance of about 8 meters can be used when the operating wavelength is around 130 meters. Eight meters is a substantial separation / remote between the Tx and Rx, while the coupling is still within the near-field regime in view of the wavelength.

[0033] Preferred embodiments of the invention will now be discussed with respect toexperiments and drawings. Broader aspects of the invention will be understood by artisans in view of the general knowledge in the art and the description of the experiments that follows.

[0034] FIGs 1A-1C show a preferred metasurface 100 for a WPT system of theinvention. The metasurface 100 has two insulated support layers 102 and 104, shown bonded together in FIG.1A. Each of the layers 102 and 104 supports and insulates a plurality of coupled resonators 106. Each pair of coupled resonators is a unit cell, and each pair is individually adjustable via mechanical or electrical means to guarantee the PT symmetry. The resonators 106 are configured and arranged to receive a magnetic near field distribution 108 from a transmitter and produced a shaped field 110 toward a target receiver, while allowing relative position changes between the target receiver and the metasurface 100 and transmitter. The resonators 106 in the two layers 102 and 104 preferably are offset from each other but have some overlap when viewed in the direction of the field lines of the field 108. Such partial overlap between the resonators can increase their coupling.

[0035] The field 110 in the example is a more narrowly focused and centered field thanthe received field 108, for example, but can directed based upon measurement and adjustment of each unit cell to provide PT symmetry.

[0036] While the example preferred metasurface 100 includes two layers 102 and 104,more layers can be added and can improve performance as a tradeoff with more complex fabrication. A benefit to additional layers is that one can further reduce the unit cell-to-unit cell separations. Lower separations lead to a stronger near- field coupling between the unit cells and therefore, higher ability to reshape the magnetic field.

[0037] The received magnetic field 108 can be supplied, for example, by an ambientradio-frequency field or a transmitter. The metasurface 100 reshapes the magnetic field 108 into the focused field 110, which contains an enhanced peak power density.

[0038] FIGs. 1D and 1E illustrate the metasurface 100 in a system 130 that includes apower transmitter 132 and a receiver 134. The power transmitter 132 includes a transmitting coil 136 with a power source 138. The receiver 134 includes a receiving coil 140 with a load 142.

[0039] In an example experimental device, each resonator 106 was formed of 10 turnsof AWG 20-Litz wire with a radius of 2 cm and periodicity of 2.2 cm in both the x- and y- directions and included a tunable capacitor 150. A controller 152 can be used to set the compensation capacitor(s) 150 to set the non-uniform impedance distribution pattern according to information about the coupling coefficients between metamaterial and the power transmitter and the receiver, respectively. In the present invention, the controller 152 operates differently than the controller in US20230047663. The present controller 152 calculates an optimal configuration directly based on the coupling coefficient of Tx and Rx to the unit cells, which can be measured by voltage sensors on the unit cells as discussed further below.

[0040] Each unit cell 106 can be treated as an LC resonator, and its impedance can betuned by the value of the capacitor 150. The receiver 134 is located above or was placed on top of the metasurface 100 with a loading resistance of 5 .

[0041] Each unit cell 106 can be made with a spiral coil and a variable chip capacitor150 that is in series with the coil. As an example, the spiral coil has 5 turns, a line width of 0.1 mm, a separation of 0.1 mm, and an outer diameter of 17.5 mm. the unit cell can be modeled as an LC resonator with the imaginary impedance, Im (Z ), given by10L.Im (Z )is tunable by the capacitance. The coupling C between thethe same layer and in the two different layers can be modeled with Neumann’s formula for mutual inductance between two closed circuits.

[0042] Control Theory for Unit Cell Adjustment to Achieve PT Symmetry

[0043] FIGs. 2A and 2B respectively show a circuit diagram of a metamaterial-enhanced WPT system and the system states controlled by coupling coefficients. The example metamaterial 202 is a single layer, i.e., a metasurface, that is between a Tx resonator 204 and a Rx resonator 206. The coupling coefficients are given by1m κ† 1u a t and 2m κ† 2u a t , and can be controlled by the targeted mode, a t

[0044] Thepresented here is general and applicable for metamaterials with anarbitrary number of layers. The metamaterial 202 serves as a controllable relay to bridge the Tx 204 and Rx 206 resonators.

[0045] Effective coupling between the metamaterial and the Tx / Rx resonators isdecided by the targeted resonant mode of the metamaterial, where †1m κ1u a tand 2m κ †2u a t , κ 1u and κ 2u are the coupling coefficients between themetamaterial 202 and204 and Rx 206 resonators, respectively.Specifically,κT 1u 1 u 1 1 uN , 1uj is the coupling coefficient between theTx resonator 204 and the j-th unit cell of the metamaterial 202.0is the coupling between the Tx 204 and Rx 206 resonators. We inversely design the configuration of the metamaterial to shape the resonant mode, and ultimately, change the ratio between1mand2m. As shown in FIG.2B, the system states are dependent on1mand2m.

[0046] A PT-without frequency splitting when the couplingcoefficients are balanced,1m 2 m. The state is stable in the strong coupling regime, and the stabilitythe overall coupling strength,2 21m 2 mand is unstable in the weak coupling regime. In FIG. 2B, the system states controlled by the coupling coefficients1mand2mare shown. PT-symmetric states can form on the dashed line1m 2 m. The real part of the frequency is shown as the 2D plots, and the imaginary part is shown as the 3D plot, where the x and y coordinates represent1mand1mfrom the dashed line. The imaginary part shows that the PT-state is less stable with a lower overall coupling in the strong coupling regime and becomes unstable in the weak coupling regime (only a single point at the perfect matched condition).

[0047] The metamaterial-enhanced WPT system can be modeled through the time-independent coupled-mode theory equation: a1 i 1 g 1 iκ1ui ad 0 1a m i κ1u H mui κ a(1) dt2u ma 2 i 0 iκ 2ui 2 2 a 2

[0048] where a 1 and a 2 are resonance amplitudes of the Tx 204 and Rx 206 resonators,am a u1aT uN represents the metamaterial’s mode. 1 and 2 are theresonance of the Tx and Rx resonators. We1 2 0with0being the operating frequency. g 1 is the net gain of the Tx resonator thattakes into account its intrinsic loss, g 1 g 10 10 , g 10 is the input gain, and 10 isthe intrinsic loss of the Tx resonator. We assume the coupling coefficients to be non-dispersive for simplicity of the solution. The error of this assumption becomes negligible when the eigenfrequency is close to the operating frequency, 0.uand2are the damping coefficients of the unit cell and the Rx resonators,H m is the lossless Hamiltonian of the metamaterial,i u 1 i u 1 Nmetamaterial configuration is given by the distribution of the unit cells’ resonance frequencies,uj.

[0049] We choose a targetedof the metamaterial as aT ta t 1 a tN . To betterevaluate the overall resonance intensity of the metamaterial, the mode is normalized, a Tt a t 1. We control the metamaterial configuration by varying theresonance frequencies of the unit cells to form the targeted mode at the operating frequency without perturbation (coupling from the Tx and Rx resonators), i.e., i0a m H m a m . (2)

[0050] We can solve the configuration asN,i j

[0051] uj0uij a ti / a tj (3)i 1202 serves as a resonator: a1 i 0 g 1 i 1mi 0 ad 1theRx resonators, respectively. In the frequency domain, Eq. (4) becomes

[0055] i a H 3b a (5)i 0 i 1mi 0.i I ) 0 , yields[ 2 i ( g 1 2 m ) ( 2 2 21 m 2 m 0 m g 1 ( 2 m ))]2 2 2, whereequation is rather complicated. However, it can be greatly simplified when the original coupling between the Tx 204 and Rx 206 resonators are weak, 0 0 ,the metamaterial 202 has a low loss,m0, and the net gain of204 resonator matches the loss of the206 resonator, g 1 2 . Under suchconditions, the characteristic equation can be simplified as

[0058] 2 s 22 id 2. (6)d 2 22m 1 m , represents the difference between themetamaterial’s to Rx 206 and Tx 204.

[0060] When the coupling coefficients are matched, 2 22m 1 m , the characteristicequation becomes 2 s 22 0 , which yields a state that

[0061] 0 (7)

[0062] We can see that the eigenfrequency is real, which represents that the state is PT-symmetric. The high efficiency is guaranteed by the PT-symmetry of the state. Please note that there are also two states given with s 22 . But, as themetamaterial 202 only form resonance around0,only accurate around this frequency. These states may not be observable in practice and are less ideal due to the frequency-splitting behavior, thus, will not be discussed.

[0063] As the necessary condition of the state in equation (7), the controller 152 needsto tune the targeted mode of the metamaterial 102, 202 to achieve d 0. Thisstate can theoretically exist regardless of the overall coupling strength. However, it will be highly unstable if it can only exist at the perfectly matched condition where d 0 ; any small mismatch between 22m and 21m will result inannihilating this state. To analyze the stability of this state, we will consider the unbalanced coupling cases when2 22m 1 m.

[0064] In the unbalanced coupling cases, the characteristic equation (equation (6))yields

[0065] ii , (8)

[0066] where i is the imaginary component of the eigenfrequency, which follows

[0067] 2 (s 2i 2 ) d 2 / i (9)

[0068] Although it is difficult to solve equation (9) analytically, we can understand itgraphically, with reference to FIGs.3A and 3B.

[0069] FIGs. 3A and 3B plot system states at the mismatched coupling condition, (A)plot of the two sides of equation (9). (B)iversus d at different overall coupling strength, characterized by s 22 .

[0070] As shown in FIG. 3A, the left side of the equation is a parabola function versusi , and the right side is a hyperbola versus i . The solution is given by theintersection of the two curves.iis non-zero, i.e., the eigenfrequeny is complex, thus, the state is anti-PT-symmetric. To study the phase transition around d 0 ,we look at the condition when d 0. In such a case, the hyperbola terms to theright of Eq. (9) approach to the x and y axis. If s 22 0 , we can get anapproximation that d 2 / i ( s 22 ) ; else, the approximation becomes2 2i ( s 2 ) 0.

[0071] The coupling coefficients, 1m and 2m , are dependent on the metamaterialmode. The controller 152 can shape the targeted mode to be the combination of two modes targeting toward the positions of the Tx 204 and Rx 206 resonators, termed as a tar-Tx and a tar-Rx , respectively. The controller 152 can tune the ratio2 2 between the two modes to control the differences between1mand 2m through a parameter . , with a range of 0 to 90to continuously tune the ratio from a tar-Tx -only to a tar-Rx -only. Note that does notcorrespond to any physical angle but chosen to tune the intensity ratios of the modes, and thus, control the coupling coefficients1mand2m.

[0072] a sin a tar-Tx cos a tar-Rxt sin 2 a †ar-Tx a tar-Tx c 2 † †(10) tos a tar-Rx a tar-Rx sin 2 a tar-Tx a tar-Rxchoose them as the coupling coefficient distribution for reducing the loss of the metamaterial , a tar-Tx κ 1u and a tar-Rx = κ 2u .

[0074] FIGs.4A-4F illustrate a practical system with the coupling coefficient controlledby for exception point. (A)1mand2mas a function of and the distance between the metamateriald Rx . (B) s as a function of and d with theposition of the exceptional point.s at the exceptional point versus dRx.(D- F) Numerically calculated scattering parameter, S21, of the system. The distances between the Tx resonator and the metamaterial are fixed at 10 cm. The distance between the metamaterial and the Rx resonator are varied at (D) 10 cm, (E) 12.5 cm, and (F) 15 cm. The dashed line indicates where2 21m 2 m, which indicates the high-order exceptional points (EP). The load resistance is 10 Ω.

[0075] As shown in FIG. 4A, by scanning , the amplitude of 1m and 2m can becontrolled. Specifically, when is close to 0coupling of the metamaterial is more toward the Tx; when is close to 90 degrees, it is more toward the Rx. A phase transition will happen when couplings are balanced, i.e., d0, as the dashed lines shown in FIG. 4B. As Rx moves away from the metamaterial, i.e., (dRxincreases), the metamaterial require a stronger mode w.r.t. Rx (a tar-Rx ) to balance the coupling, and thus, the phase transition pointshift to a lower . As shown in FIG. 4C, the total coupling strength alsodecreases with dRx. Therefore, the PT-symmetric state becomes more unstable and the phase transition becomes more rapid. We used a Tx coil with a radius of 10 cm and a Rx coil with a radius of 5 cm. The self-inductances of the two coils are 63.83 μH and 31.42 μH, respectively. Both of the two coils have 1 turn to have minimum perturbations to the metamaterial mode. The metamaterial consists of a 3-by-3 square array of unit cells with a periodicity of 10 cm in both the x and y directions. Each unit cell has a radius of 4.55 cm and 5 turns to form resonance at tens of megahertz regime, which ensures a strong near-field coupling to their neighboring unit cells. The unit cells are placed in a planar array without intersection.

[0076] To probe the resonance states, we use an oscillating voltage instead of a negativeresistance to drive the Tx coil and measure the scattering parameter, S21, between the Tx and Rx resonators. As shown in FIG.4D-4F, we simulate the scattering parameter with the varying distances, dRx, of 10 cm, 12.5 cm, and 15 cm between the metamaterial and the Rx resonator, while the distance between Tx and 2 2 metamaterials is fixed at 10 cm. When1mand2mare very unbalanced, the system demonstrates two resonance states corresponding to the anti-PT- symmetric states of a two-body system (formed between Tx and Rx). The metamaterial is away from resonance and does not significantly contribute to 22resonance in such cases. As gets close to the critical condition of1m 2 m, a phase transition to a non-frequency splitting state can beon the theory illustrated previously, the state is PT-symmetric, leading to an increased S21. The phase transition point between the PT-symmetric and anti- PT-symmetric states in this multi-body system is also known as a high-order exceptional point.

[0077] When we change the Rx position, the increase of the separation between themetamaterial and the Rx resonator requires compensation with a higher a Rx toachieve the balanced couplings, which results in the shifting of the exceptional point to a lower . Furthermore, as the decrease of the overall coupling,s, thePT-symmetric state becomes more unstable. As a result, we can see that the increase of the d Rx s leads reduction of the where the frequency splitting ofthe resonance states start to shift, i.e., arapid phase transition (FIG.3F). As the keep increasing of dRx, the PT-symmetric state will become too unstable that it simply cannot exist practically.

[0078] The controller 152 will receive voltage signals as responses to impulse voltageinputs to the Tx and the Rx, respectively, to calculate the coupling coefficients and , and further calculate the optimal impedance.

[0079] Experiment to Confirm Controller 152 Unit Cell Adjustment for PT Symmetry

[0080] FIG.5A shows an experimental WPT system of the invention and FIGs.5B-5Ddata concerning testing and simulation of the system. (A) Experimental setup of the metamaterial enhanced WPT system. (B) Measured and simulated resonance frequency versus height of the resonator. The error bars represent standard deviation of the 9 unit cells. The inset picture shows the unit cell with a tunable height by the top screw. (C) Measured spectrum of the normalized S21at different configurations. (D) 2 for exceptional point, 2 EP , versus thedistance between the metamaterial and the Rx resonator, d Rx . The error barsrepresent standard deviation of three measurements.

[0081] The experiment included a Tx resonator, an arbitrary non-limiting example 9-element metamaterial, and a Rx resonator, as shown in FIG.5A. The unit cells were open-ended spirals with a non-uniform radius. The middle loop with asmaller radius of 2.65 cm and the top and bottom loops with a larger radius of 4.55 cm. This design allows a relatively uniform magnetic field in the vertical direction and also a wide range of tunability when mechanically compressing the structure (FIG.5B). The measured resonance frequencies as a function of the unit cell’s height fits with the simulation. To replicate the simulation, we use a voltage source to drive the Tx resonator and measure the scattering parameter of the system. The resonance frequencies of the unit cells are controlled by the heights of the spiral resonators as shown in FIG.5B. The measured resonance frequencies fit well with the simulation. As shown in FIG. 5C, S21shows a stronger intensity with 2 approximately equal to 35 degrees, which indicate a PT-symmetric state. Phase transition from two anti-PT-symmetric states to a single PT-symmetric state can be observed. Similar to the numerical result shown in FIG. 4, the anti-PT-symmetric states to the left and right of the exceptional point are asymmetric.

[0082] We further demonstrate the control of the system’s state, characterized by ,with the separation between the metamaterial and the Rx resonator. As shown in FIG. 5D, increasing the separation results in the left movement of the exceptional point, which fits with our theoretical prediction. There is a minor mismatch between the experiment and the theory. This is because we overestimated the separation between the metamaterial and the Rx resonator. As different unit cells have different heights, it is difficult to determine the effective z-position of the metamaterial. So, we use the bottom of the metamaterial as a reference to determine d Rx . This will cause an error when we calculate ,especially when the Rxis close to the metamaterial.

[0083] The experiments showed that the controller 152 can control the PT-symmetry innon-Hermitian WPT systems via individual unit cell controllable metamaterials. The metamaterial serves as a controllable relay device that can tune the effective coupling coefficients to the Tx and Rx resonators. Through balancing the coupling,1and2, PT-symmetry can be maintained even when the Tx and Rx resonators aresymmetrically placed and sized. We further demonstrate a metamaterial consisting of a 3-by-3 array of unit cells with tunable resonance frequencies. Both the PT-symmetric state and anti-PT-symmetric states are observed experimentally. By reconfiguring the metamaterial, we can achieve PT-symmetrical state at different Rx positions, showing potential in free- positioning and frequency robust WPT.

[0084] Simplification to the Three-Body System

[0085] Perturbation theory yields

[0086] a 3 body | a 3 body a 3 body H 3 body a 3 body (A1), ,,as K , exist.

[0088] H3-body = K * K (A2)

[0089] As HT 3-body H 3-body , KT K H K * K . We get K K H , K is Hermitian. Replacethe first K as K H to the right side of equation (A1), we get

[0090] H T3-body = K K (A3)

[0091] Take equation (A3) into equation (A1), we get

[0092] a3body|a3body a3body KTK a3body K a3body|K a3body. So,

[0093] a 3 body K a 3 body (A4)

[0094] Multiply by K T to both sides of equation (A4), KT a T3 body K K a 3 body .The leftK Ta 3 body K* a 3 body K a*3body a*3body a 3 body .time-independent CMT equation a3 bodyH3-bodya3 body, as in equation (5).

[0095] PT Symmetry of the Hamiltonian

[0096] PT-symmetric Hamiltonians followPˆT ˆ(H ) T ˆ 1 P ˆ 1 H (A5)

[0097] where P̂ is parity operator, and T̂ is time-reversal operator. For the three-body1 ˆ system in the frequencycan see that forthe three-body Hamiltonian H 3,i k 1 k 0ˆ ˆ ˆm. ,whenm0.

[0100] Transient Analysis

[0101] The temporal coupled-mode theory equation of the three-body system describedby equation (5) is

[0102] it a 3b H 3 b a 3 b (A7)t ,assume a constant voltage,v, feeding to the Tx resonator,gva2. Putting in the Hamiltonian, we get the standard time-dependent first- 1L Txorder ordinary differential equation (ODE) that can be solved numerically. a1 1 k 1 k a v / 2 Ld 0 1Tx

[0104] where t 2 tt0represents normalized time in unit of the number of periods,t01f .set 0.028 , k 0 0 , k 1 0.1 , k 2 0 , and m 0. As shown in, With an a 1 operating frequency of 65 MHz, the time of reaching saturation is around 61.6 ns.

[0105] Full Wave Simulation of the Unit Cell

[0106] FIG. 7A-7B show a simulation of the unit cell. (A) Magnetic field intensitydistribution. (B) Current density and magnetic field distribution. The unit cell has a height of 2 cm. The simulation frequency is 63.6 MHz.

[0107] The resonance frequencies of the unit cells are simulated using COMSOLMultiphysics 6.0, AC / DC Magnetic Fields Module, and Frequency Domain solver. The cubic simulation domain has a diameter of 20 cm. We use constant magnetic field boundary conditions at all the 6 boundaries of the simulation domain. The feeding magnetic field is toward -z direct with an intensity of 1 A / m. The simulated magnetic field is shown in FIG. 7A. As the resonator has open ends, the current density drops to zero at the two ends. The current density is highest at the middle point of the wire constructing the resonator and decrease with the distance to the middle point FIG.7B. The resonance mode the lowest resonance mode of the resonator. As all the current flow to the same direction, the magnetic field distribution is similar to the one of a coil.

[0108] Experiment 2 Set Up.

[0109] FIG. 8 shows the experiment 2 set up. In the experiment, the metamaterial wasused to manipulate the coupling coefficientsk 1andk 2. As discussed above we choose the targeted mode to be a t cos sin kT 1u k 2 u . As the positionof the Rx resonator willmetamaterialconfiguration, given by r distribution of the unitdependent on bothdand . We plot the configurations in a square grid of d , space in FIG.9. Asthe centers of the Tx resonator, thethe Rx resonator are aligned, the configurations are symmetric among the center of the metamaterial. As shown in FIG.9, the configuration can be uniquely given by the differential frequency of the center unit cell (unit cell 5 in FIG. 10A and 10B) with its surrounding unit cells.

[0110] FIGs. 10A-10B show the metamaterial configuration characterized bydifferential,r. (A)rof the center unit cell with its directly adjacent unitcells (marked in blue in the inset). (B)rof the center unit cell with the diagonal unit cells (marked in blue in the inset).

[0111] The sounding unit cells can be separated into two groups: unit cells 2, 4, 6, 8(FIG.10A) and 1, 3, 7, 9 (FIG.10B). The metamaterial configuration is uniquely reflected byrof the sounding unit cells in these two groups. We plotrin d, space. By using this diagram, one can easily acquire the configuration atand d . And we can see that the exceptional point falls in the regionwith a monotonous dependency of r with and d , making the adjustment ofthe configuration to locate the exceptional point straightforward for the controller.

[0112] System Efficiency

[0113] The efficiency of the system is given by2

[0114] 2 L a 2 22 2(A9)a1,a m, anda 2as

[0116] i a i g a i a(A10)ratio betweena 1anda 2as

[0119] a a (A12),we get equation (A13)

[0121] L 22(A13) 1mi 2

[0123] ( 22 s 0 ),2. Therefore, we can see that the d 2coupling coefficients are balanced, 2 210g 1 d0.Otherwise, thedecreases as d increases. To achieve highefficiency, we need the coupling coefficients,1mand2m, to have a similar amplitude.

[0124] Optimization of the metamaterial targeted mode.

[0125] The targeted modes of the metamaterials can theoretically be chosen arbitrarilyif the metamaterial has a zero loss. However, in practice, the metamaterial is not loss-free. We would want to miniaturize the potential loss caused by the resistive loss of the metamaterial resonators. To achieve this, we proceed the following optimization considering the metamaterial’s loss.

[0126] The efficiency is given by2

[0127] 2 L a 2 (A14)targeted mode,a t, asam a m a t. At the operating frequency,0, equation (4) yields that

[0129] a ( iκ†1u a ta 1 iκ†2u a ta 2 )m (A15) m

[0132] ai a ( †2ua † †0 1 mκ 1u κ1κ 2u κ 2ua 2 )a t a t2 (A17) 2 m 0 , which is

[0134] a aa a a a a a a(A18)

[0135] Taking in equations (A15), (A16), and (A18), we can get thata κ † †

[0136] i 1 1u a 2 κ 2u2κ 1u κ 2u a 1 κ 2u κ 2u a 2 at (A19)thea tmust be proportional toκ 1uandκ 2u, as

[0138] a c 1(A20)

[0139] where the coefficients c 1 and c 2 are coefficients to be determined as above.

[0140] AutomaticallyMetasurface for Wireless Power Transfer

[0141] FIGs. 11A-11C show a preferred automatically configurable metasurface 1100.The description here includes example dimensions and impedances, while artisans will appreciate variations of the dimension and impedances are within the scope of the invention. The automatically reconfigurable metasurface 1100 can enhance wireless power transfer (WPT) systems and enable free-positioning capabilities. The example design includes a transmitting (Tx) coil 1136 with an inductance of 5.15 μH, a receiving (Rx) coil 1140 with an inductance of 1.89 μH, and a metasurface comprised of a 5-by-5 periodic array of unit cells with aperiodicity of 3.68 cm, consistent with FIG. 1. Each unit cell in the example metasurface 1100 is implemented as a planar, circular closed-loop coil with an outer radius of 25 mm, a track width of 0.508 mm, and a line gap of 0.305 mm, consistent with FIG.1. The unit cells contain parallel compensation capacitors (with tunable values such as 51 pF, 82 pF, etc.), which can be adjusted in real time using relays or analog switches controlled by a microcontroller as shown schematically in FIG.11C.

[0142] Optimization of the metasurface configuration 1100 is implemented using astochastic gradient descent (SGD algorithm), which is executed by the microcontroller 152. This algorithm continuously adjusts the capacitance values of the unit cells to achieve the configuration that maximizes power transfer efficiency. Starting from an initial configuration, the system iteratively refines the metasurface settings by evaluating the received power at the Rx coil 1140 and adjusting the parameters accordingly. Adjustment can be through the adjustable capacitor to control the unit cells electrically rather than mechanically. The LC (Inductor-capacitor) resonator has the resonating frequency as. The inductance is fixed since it is a pre-defined patter of the planar coilPCB, and the capacitance can be controlled to tune the self- resonating frequency of each unit cell. With the target mode and equation (2) and (3), a controller can calculate the Hamiltonian of the metamaterial, and solve the resonance frequency of each unit cell. The feedback mechanism allows the system to adapt in real time, using gradient information to converge toward an optimal solution. The digital-to-analog converter DAC (152 in FIG. 1E) provides the necessary control signals to fine-tune the capacitance values, ensuring precise adjustments to the electromagnetic field distribution. The useof the SGD as a control algorithm allows for rapid convergence, even in dynamic environments where the Rx coil position may vary, thereby maintaining high- efficiency power transfer across a range of operating conditions.

[0143] The performance of the metasurface-enhanced WPT system of FIG. 11 wasevaluated using both numerical simulations and experimental validation. The numerical simulations modeled a system configuration in which the Tx coil was placed beneath the metasurface and the Rx coil positioned above. The targeted magnetic field mode was configured to maximize coupling between the Tx and Rx coils, resulting in significant improvements in both peak efficiency and spatial coverage. Simulation results demonstrated that the metasurface effectively shaped the magnetic field to enhance power transfer efficiency, even under conditions of misalignment, which is shown in FIGs.12A and 12B, which respectively show Magnetic Field Intensity Map (i.e., the efficiency map) with and without the auto-reconfigurable metasurface 1100.

[0144] In the experimental setup, a function generator supplied a 6.78 MHz RF powersignal to the Tx coil, while power output at the Rx coil was measured using an oscilloscope. FIG.13 shows the experimental results for the measured efficiency at tracking target, indicating that the optimized metasurface enhanced peak efficiency by a factor of 1.91 compared to an untuned configuration and a factor of 2.37 compared to a system without a metasurface. The system maintained high efficiency even under misalignment and with a transmission distance of 10 cm between the Tx and Rx coils. These results indicate that the metasurface- enhanced system effectively overcomes a major challenge in conventional WPT systems: efficiency degradation due to misalignment or increased distance. Furthermore, the experiments validated the dynamic reconfiguration of the unitcells' resonance frequencies, confirming that the metasurface could adapt to varying operating conditions to optimize power transfer.

[0145] The experimental setup also included tests to evaluate the free-positioningcapability of the system, wherein the metasurface tracked a moving Rx coil to maintain high transmission efficiency. Results demonstrated that the metasurface could dynamically reconfigure itself in real-time to optimize power transfer as the Rx coil changed positions. By continuously adjusting the magnetic field distribution based on the Rx coil's movement, the system ensured that the Rx coil received an optimal amount of power, irrespective of its position. FIG.13 illustrates the wide coverage range, from -5 cm to 5 cm offset from the center position, with nearly constant 30% transmission efficiency. This capability is particularly beneficial for applications involving moving targets, such as powering medical implants in dynamic environments or charging portable electronics without requiring precise alignment.

[0146] While preferred embodiments have been described, it should be understood thatother modifications, substitutions and alternatives are apparent to one of ordinary skill in the art. Such modifications, substitutions and alternatives can be made without departing from the spirit and scope of the invention, which should be determined from the appended claims.

[0147] Various features of the invention are set forth in the appended claims.

Claims

CLAIMS 1. A wireless power system, comprising a transmission resonator; a metamaterial comprising an array of unit cells, the unit cells being individually adjustable to change a resonance frequency of the array, and a controller that adjusts unit cells in the array of unit cells to set a resonant mode providing coupling coefficients equal to that of the transmitter resonator and a receiver resonator to yield a parity-time symmetric state of the array, the transmitter resonator and the receiver resonator.

2. The wireless power system of claim 1, wherein the controller controls a phase transition of the metamaterial to create high-order exceptional points.

3. The wireless power system of claim 2, wherein the controller individual adjusts unit cells in the array toward making an absolute coupling coefficient between the metamaterial and the transmitter resonator equal to an absolute coupling coefficient between the metamaterial and the receiver resonator.

4. The wireless power system of claim 3, wherein the unit cells are arranged in one of a hexagonal, oblique, or square array, with one or multiple layers.

5. The wireless power system of claim 4, wherein the metamaterial comprises at least two insulated support layers, and each of the supportlayers supports and insulates a plurality of coupled resonators the define the unit cells.

6. The wireless power system of claim 5, wherein resonators in the at least two insulated support layers are offset from each other by having some overlap when viewed in a direction of field lines.

7. The wireless power system of claim 1, wherein the controller calculates an optimal configuration directly based on the coupling coefficient of the transmitter resonator and the receiver resonator to the unit cells.

8. The wireless power system of claim 7, comprising voltage sensors on the unit cells that measure the coupling coefficient.

9. The wireless power system of any previous claim, wherein the unit cells are electrically adjustable.

10. The wireless power system of any of claims 1-8, wherein the unit cells are mechanically adjustable.

11. The wireless power system of any previous claim, wherein the controller adjusts the unit cells by balancing the real portions of coupling coefficients between the transmission resonator and the receiver resonator.

12. The wireless power system of any previous claim, wherein the controller controls the metamaterial by varying the resonance frequencies of the unitcells to form the targeted mode at an operating frequency without perturbation of coupling from the transmission and receiver resonators.

13. The wireless power system of any previous claim, wherein the controller receives voltage signals as responses to impulse voltage inputs to the transmission and receiver resonators, respectively, and uses the voltage signals to calculate the coupling coefficients.

14. The wireless power system of claim 13, wherein the controller calculates an optimal impedance from the voltage signals. 15 The wireless power system of any previous claim, in a mobile device charging system.

16. The wireless power system of any of claims 1-14, in a medical implant system.

17. The wireless power system of any of claims 1-14, in an EV charging system.

18. The wireless power system of claims 1-14, in an internet of things charging system.

19. The wireless power system of any previous claim, wherein the controller executes a stochastic gradient descent that continuously adjustscapacitance values of the unit cells to achieve s configuration that maximizes power transfer efficiency.

Images