Abstract
The ability to wirelessly power electrical devices is becoming of greater urgency as a component of energy conservation and sustainability efforts. Due to health and safety concerns, most wireless power transfer (WPT) schemes utilize very low frequency, quasistatic, magnetic fields; power transfer occurs via magnetoinductive (MI) coupling between conducting loops serving as transmitter and receiver. At the “long range” regime – referring to distances larger than the diameter of the largest loop – WPT efficiency in free space falls off as (1/d)^{6}; power loss quickly approaches 100% and limits practical implementations of WPT to relatively tight distances between power source and device. A “superlens”, however, can concentrate the magnetic near fields of a source. Here, we demonstrate the impact of a magnetic metamaterial (MM) superlens on longrange nearfield WPT, quantitatively confirming in simulation and measurement at 13–16 MHz the conditions under which the superlens can enhance power transfer efficiency compared to the lensless freespace system.
Introduction
The superlens, which can refocus not only propagating farfield waves but also nonpropagating, nearfield waves, has been one of the more provocative concepts to emerge from the field of metamaterials^{1}. A superlens comprises a medium whose electric permittivity ε and magnetic permeability μ both take on the value of −1. The superlens structure offers a means of controlling and manipulating the nearfields that would otherwise decay rapidly away from a source. Initially, the superlens was proposed in the context of optics, where its use was suggested as a means of forming an image with resolution greater than that implied by the diffraction limit^{2,3}. The superlens functions via the excitation of magnetic and electric surface modes that couple to the near fields of an object placed on one side of the slab, subsequently bringing them to a focus on the opposite side. Since electricity and magnetism are nearly decoupled in the near field, it was realized early on that a superlens with either ε = −1 or μ = −1 could focus the near field of electric or magnetic sources, respectively. Imaging with a superlens has been demonstrated at visible and infrared wavelengths using thin layers of materials such as silver or silicon carbide whose dielectric functions take the value of ε = −1 at particular wavelengths^{4}.
At low frequencies, where magnetism is much more prevalent, superlenses based on structured metamaterials characterized as artificial magnetic permeability media have been pursued for a variety of applications, including as flux guides to enhance resolution in magnetic resonance imaging^{5,6}. More recently, as interest in WPT schemes has risen, the use of superlenses and other metamaterialbased components to enhance transfer efficiency has been suggested^{7}. At the low frequencies typical of inductive WPT schemes, the excitation wavelength exceeds 10 m, whereas the dimension of the coils and loops is on the decimeter or centimeter scale. Thus, the loops can be initially approximated as magnetic dipoles as a route to gaining an intuitive understanding of the limits and behavior of the WPT system.
A general treatment of a magnetoinductive WPT system was carried out by Kurs et al., who made use of coupledmode theory to calculate the expected efficiency of two selfresonant coils separated by a distance d in free space^{8}. In that work and the subsequent analysis of Urzhumov et al.^{9}, power transfer efficiency is defined as the power dissipated in a load placed on the receiver coil (Rx) divided by the total power dissipated in the transmitter (Tx) and receiver circuits, as well as in any intermediate “relay systems” (such as a repeater coil, lens or metamaterial layer):
Once the Rx and Tx circuits and an optional relay system are specified, power dissipation rates can be calculated using the coupled mode theory for arbitrary Rx and Tx configurations^{8}, or the simplified coupleddipole formalism for small resonant coils^{9}. Coupled mode theory treats the Rx and Tx coils as resonators whose interaction is indicated by a mode coupling coefficient, κ, whereas the coupleddipole theory uses the conventional notion of mutual inductance, L_{21}. Both theories predict the same longrange behavior for the WPT efficiency (1), namely, η ~ d^{−6}. The physical origin of this power law is due to the 1/d^{3} dependence of magnetic field in the near field of a magnetic dipole source. The overall system efficiency in magnetoinductive schemes is therefore inherently limited by the divergence of magnetic flux in free space, because power transfer is ultimately related to the amount of flux from the first coil that can be captured in the “aperture” of the second coil. Given that any coil or loop will behave roughly as a magnetic dipole, it is unlikely that any redesign or engineering of the coils can possibly enhance WPT efficiency of any system that is already optimally impedancematched.
If the efficiency of a resonant WPT system is to be improved, a means must be found to recapture and refocus the otherwise divergent magnetic flux. Traditional lenses based on conventional materials only focus the farfields and therefore are irrelevant to this application. Nearfield lenses of several configurations^{10,11,12} have been proposed; however those configurations provide partial field focusing with efficiency significantly lower than the perfect focusing promised by the superlens^{13}.
The use of a nearfield superlens in conjunction with resonant power transfer was first considered by Wang et al^{14}. Earlier, similar metamaterial lenses were proposed for magnetic resonance imaging applications^{15,16}. Examining both isotropic^{14} as well as anisotropic^{17} versions of the negativepermeability medium, these works provided significant numerical evidence and some experimental evidence, of enhanced efficiency. Here, we address the question raised by the studies in Ref. 14,17, as to whether metamaterial superlenses can improve WPT efficiency (and conversely, reduce power loss) in the longrange nearfield transfer regime.
Theoretically, a positive answer to this important question was given in Ref. 9. Figure 1 is based on the analytical solutions from Ref. 9 and it shows that a metamaterial with finite and realistic resistive/magnetic loss can help deliver magnetic flux and generate alternating current in a resonant Rx circuit, leading to overall improvement in WPT efficiency. This performance boost was predicted in the longrange, highload transfer regime, where both the transfer distance d and the resistive load in the receive circuit exceed certain thresholds. This Report presents experimental evidence that longrange WPT efficiency using a MM slab can exceed the maximum efficiency obtainable, ceteris paribus, in free space.
Results
Negativepermeability superlens design
Geometrically, the superlens is perhaps the simplest possible configuration for nearfield focusing: it consists of a uniform layer of isotropic, negative permeability. This property simplifies the superlens design, which could be as simple as a double or tripleperiodic arrangement of identical unit cells. In addition, the superlens is translationally invariant, at least in the large aperture limit, which reduces the need for fine mechanical alignment of the WPT system relative to the lens. Figure 2 illustrates the geometry of our superlens implementation; the unit cell geometry, design, homogenization and effective permeability are detailed in Methods. The major challenge for designing negativepermeability metamaterials at relatively low frequencies (MHz regime and lower) is the design of sufficiently low loss, strong magnetic dipole resonators of deeply subwavelength dimensions. Here, we have opted for unit cells of size ~ 2 cm and achieved sufficiently large inductance by using multiturn planar coils^{18}.
Enhancement of Magnetoinductive coupling with finite aperture superlens
Once a metamaterial layer with desired effective permeability is designed, we can characterize its effect on the coupling between two magnetic dipoles. We approximate a finiteaperture slab as a disk of the same diameter, which enables highly efficient axisymmetric description along the lines of Ref. 9,17 in a 2D rotationallysymmetric geometry modeled in COMSOL Multiphysics. We use the retrieved permeability components of the metamaterial slab, each fitted to a Lorentzian resonance shape, as the components of the slab diagonal permeability tensor μ = [μ_{T} μ_{T} μ_{N}] (assuming all offdiagonal elements are zero and relative permittivity is ε_{r} = +1). The transmitting magnetic dipole is simulated as a current loop of radius r = 1 cm carrying a fixed current I = 1A. We define the magnetic field transmission coefficient as T = H_{Rx}/H_{Tx}^{2}, where H_{Tx} = I/2r is used to approximate the total field radiating from the Tx loop and H_{Rx} is the field measured at a point on the axisofrevolution a distance d away from the loop. It is convenient to define transmission enhancement factor due to the slab, using the ratio of transmission coefficients with and without the slab G = T^{slab}/T^{vac}.
Since the simulated coil is small relative to the slab and can be approximated as a dipole, we use the simulation results to validate the analytical solutions for the inductance between two dipoles with and without the MM slab ( and , respectively) as expressed in Ref. 9. In Figure 3 we plot the simulated and analytical enhancement factors (the latter, defined as , we evaluate numerically with Mathematica). Results indicate that the slab enhances the field transmission coefficient T by roughly a factor of five over an appreciable frequency region. In addition, we observe a striking agreement between simulation and analytical results as a function of frequency.
Measurements of enhancement with nonresonant coils
To experimentally verify the effect of the metamaterial “superlens” on WPT between two coils, we begin by constructing a slab from the fabricated MM. To assemble a 1layer slab, we first designed the x and yoriented MM elements in rows with oppositefacing slits, which can be assembled into a winecrate pattern, as shown in Figure 2A. The zoriented resonators were fabricated across a single sheet, which was then placed perpendicular to the winecrate and held together by additional slits cut around the winecrate. In a similar fashion, we assembled a 3layer MM slab (Figure 2B). To demonstrate a dipoletodipole WPT system, we used two small copper coils with crosssectional wire diameter of 1.6 mm wound into a circular loop of radius 1 cm, aligned the coils coaxially and separated by distance d along their shared axis and connected them to ports 1 and 2 of an Agilent Vector Network Analyzer (VNA). Since our coils are nonresonant, to compute the experimental WPT efficiency from the measured Sparameters we must first remove the losses resulting from the mismatch between the network analyzer and the coils. Calculations needed to account for these losses are detailed below in Methods.
Figure 4 shows the simulated and measured transmission coefficients for free space and in the presence of a 1 and 3layer slab, across several coiltocoil distances. In addition, for the 1layer slab, we report the transmission coefficient of the two indefinite MM layers: one composed of only the zfacing MMs while the other consists of the xyintersecting MMs.
Discussion
Our central experimental results as shown in Fig. 4 are generally in very good agreement with numerical simulations, with the exception of the curves corresponding to power transfer distance d = 8 cm. The latter case needs additional discussion, considering that the distance from Tx and Rx coils to the surface of the metamaterial slab (of thickness L = 6 cm) is only (d − L)/2 = 1 cm, or roughly one half of the lateral unit cell dimension (array periodicity). In our numerical simulations (Fig. 4A,C), we first simulate a single period of the 2D array and retrieve all three principal values of the effective permeability tensor. Then, we replace the slab with a layer of homogeneous magnetic medium having the same complex permeability components as retrieved. In doing so, near field effects not describable in terms of effective magnetic permeability are lost. The physical fields on the surface of an array can be Fourier transformed and all waves with transverse wavenumbers k > k_{0} = ω/c are necessarily evanescent. However, out of this infinite spectrum, the effective medium description is only adequate for components with , where a is the metamaterial lattice constant. The higherorder Fourier components with k ≥ k_{Bloch} decay rapidly away from the interface – as exp(−k_{Bloch}z) or faster. They can be picked up by a near field probe placed at z = 1/k_{Bloch} (or closer), which corresponds to our measurements taken with d = 8 cm. However, these harmonics are negligibly small at distances d > 15 cm, which explains the excellent agreement between effective medium models and measurements in that regime (Fig. 4).
To summarize, we have demonstrated that a resonant array acting as an effective medium with negativedefinite magnetic permeability enhances nearfield transmission of quasistatic magnetic fields between nonresonant magnetic loop antennas. Significant enhancement is seen only at frequencies where at least one component of effective magnetic permeability has a negative real part. The enhancements due to different components of permeability tensor are investigated (Fig. 3) and it is shown that the strongest effect is obtained when all three components of Re(μ) are negative (see Fig. 3, red and blue curves). We attribute the latter effect to the excitation of magnetoinductive surface waves existing in triplenegative permeability layers^{19}. Enhancements in power transmission coefficient in the range of +15 to +30 dB are observed for all transfer distances from 8–24 cm; those transfer distances are 4–12 times greater than the diameter of both transmitter and receiver coils. Much higher enhancement is anticipated with largeraperture superlenses and multiturn, selfresonant Tx/Rx coils.
Methods
Magnetic metamaterial design and fabrication
Our aim is to design and fabricate an isotropic metamaterial (MM) exhibiting Re{μ} < 0 at 13.56 MHz with minimal losses. Assuming operational frequency close to 13.56 MHz, the freespace wavelength is λ_{o} ≈ 22 m; a conventional metamaterial whose elements are roughly λ_{o}/10 in size would be far too large for practical implementations. Instead, we must demonstrate the desired behavior with elements whose size is only several centimeters, on the order of λ_{o}/1000. To achieve this, we significantly increase the metamaterial unit cell inductance by utilizing the doublesided rotated coil design shown in Figure 5, which sandwiches a substrate between two viaconnected multiturn coils. The coils are rotated with respect to one another to form a composite circuit in which the inductances of two individual coils are added in series, resulting in the total inductance improved by a factor of four relative to the configuration with inductance in parallel.
We form the complete metamaterial unit cell by positioning three identical resonators perpendicular to each other, as shown in Figure 5C and iteratively tweak the design in CST microwave studio using the standard Sparameters retrievals^{20,21} to obtain Re{μ} = −1 in the desired ISM frequency band.
The figure of merit (FOM) for our design is the inverse losstangent ratio at the frequency where Re{μ} = −1:
To minimize the loss tangent we choose a lowloss Rogers 4350 substrate and construct the metamaterial using a 1ounce (34 μm thick) copper clad; the skin depth in Cu at 10 MHz is about 20 μm. The Sparameter retrieval method lets us compute the transverse permeability components μ_{x} and μ_{y} by setting periodic boundary conditions along the x and y directions and enforcing a normally (z) incident, transversely polarized plane wave. The final design, whose retrieved permeability is shown in Figure 6, utilizes coils with 17 turns on each side of the substrate. Each turn is 200 μm wide and the gap between turns is set to 200 μm as well. To reduce the design's sensitivity to fabrication errors, we insert three vias 200 μm in diameter into the outermost leg in each coil and increase the width of that leg to 500 μm such that there are 150 um between each via and the metal's edge. The total unitcell size, including a 1 mm gap between adjacent unitcells, is 1.894 cm.
Although all three orthogonal coils in the unitcell are identical, this does not mean the MM's permeability is isotropic because the normal (z) and transverse (x,y) boundary conditions observed by fields propagating through the slab are significantly different from each other.
Fieldaveraging homogenization method for finitethickness, anisotropic metamaterial layers
While the standard Sparameter retrieval techniques^{20,21} allow one to compute the components of effective permeability and permittivity tangential to the surface of a MM layer, the normal components are difficult to retrieve since they are not excited by a normally incident, transversely polarized plane wave. Here we present a quasimagnetostatic field averaging retrieval method and outline how we used it to compute both the transverse and normal components of our MM. Quasielectrostatic fieldaveraging homogenization was described in detail in Ref. 22., where effective ε was expressed through the capacitance of a unit cell submerged into a curlfree electric field. Here, we extend this method to quasistatic permeability retrieval using the electricmagnetic duality theorem. For brevity we assume the medium to be uniaxial with permeabilities μ_{T} and μ_{N}; the method is applicable to a general orthotropic medium with an orthorhombic lattice.
We simulate a unit cell of dimensions a_{x} × a_{y} × a_{z} in COMSOL Multiphysics's RF module. Air surrounded the cell along z while periodic boundary conditions (BCs) are along x and y such that unitcell is part of a slab. Across the faces normal to the zaxis we assign an Electric Field which varies with z, . By using , we can be sure that the electric field has a linear variation in z and thus its curl is virtually uniform in the entire domain. From Faraday's law, this Efield leads to , uniform magnetic field H_{y}. By using the duality theorem together with the definition of capacitance, C = εA/d = Q/V, we can replace ε with μ, electric charge Q with magnetic charge Q_{m} and electric voltage V with magnetic potential V_{m} and extract effective permeability according to
With H polarized along , we obtain μ_{y}, one of the permeability principal values, by substituting d = a_{y}, A = a_{x} × a_{z} and B = B_{y} into (3).
To compute μ_{x}, the remaining transverse component, one replaces the incident field E_{x}(z) with E_{y}(z) and utilizes the appropriate fields and dimensions in (3). To compute the normal component μ_{z}, however, an additional subtle change has to be made. We begin by exciting an incident field E_{y}(x) of the form , which gives rise to an almost uniform Hfield H_{z}. However, such a field still violates the periodic condition along x slightly; therefore on the x = const faces we use Floquet (phaseshifted periodic) boundary condition with the phase shift given by k_{x}a_{x}. We then compute μ_{z} from (3), substituting d = a_{z}, A = a_{x} × a_{y}.
We perform fieldaveraging retrievals across the 8 MHz–16 MHz frequency range and compute the transverse and normal components of μ for both the 1 and 3layers configurations. We then fit each retrieved permeability to a Lorentzian curve defined as
where F is a constant representing the oscillator's strength, ω_{0} = 2πf is the angular resonance frequency and y = ω_{0}/2Q. The resulting fitted parameters of the 1layer MM are
and the parameters fitted from the 3Layer MM retrieval were calculated to be
Here we have used the superscripts 1 and 3 to distinguish between the 1 and 3layer slabs, respectively and the subscripts N and T to distinguish between the normal and transverse permeability components, respectively. Not surprisingly, comparing the fitted parameters for the 1 and 3layer MM suggests that as more layers are added the metamaterial behaves in a more isotropic fashion.
We note that the Lorentzian parameters provide a quality fit for the complex permeability curve only in the vicinity of the fundamental magnetic resonance studied. Good quality of fit is maintained at least through the frequency where Re(μ) crosses zero (roughly 16 MHz), that is, in the entire frequency band of interest.
Maximum transducer power gain calculations
Before we conduct WPT measurements with the nonresonant coils setup, we perform a calibration that moves reference planes of a VNA to the end of the cables that are connected to the coils (see Figure 7). This enables us to retrieve the direct coiltocoil transmission efficiency. In Ref. 23. Pozar describes a suitable metric called the Maximum Transducer Power Gain, , which is the gain that would be achieved if a lossless matching network was inserted between the NA's reference planes and the nonresonant loops. Pozar defines in terms of only S parameters; here we summarize the calculations outlined in Ref. 23.
Transducer Power Gain G_{T} is the ratio of power delivered to the load, P_{L}, to the power available from the source, p_{s}:
where Γ_{L} = (Z_{L} − Z_{0})/(Z_{L} + Z_{0}) is the reflection coefficient seen looking toward the load, Γ_{S} = (Z_{S} − Z_{0})/(Z_{S} + Z_{0}) is the reflection coefficient seen looking toward the source and Γ_{in} is the reflection coefficient seen looking toward the input of the two port network
where Z_{in} is the impedance seen looking into port 1 of the terminated network. Similarly, Γ_{out} is the reflection coefficient seen looking into port 2 of the network when port 1 is terminated by Z_{S}:
The Maximum Transducer Power Gain, , occurs when and . In the general case with a bilateral two port network, Γ_{in} is affected by Γ_{out} and vice versa, so that the input and output must be matched simultaneously. Equating and with the RHS of (8) and (9), respectively, yields
where Δ = S_{11}S_{22} − S_{12}S_{21}. Substituting (10) into (9) and rearranging the terms results in the quadratic equation
yielding the solutions
where B_{1} = 1 + S_{11}^{2} − S_{22}^{2} − Δ^{2}, , B_{2} = 1 + S_{22}^{2} − S_{11}^{2} − Δ^{2} and . To convert the recorded Sparameters to , then, one must first calculate (in order) Δ, C_{1}, C_{2}, B_{1}, B_{2}, Γ_{L}, Γ_{S} and Γ_{in} and use these parameters in (12).
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Acknowledgements
This work was financially supported by the Toyota Research Institute of North America.
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D.R.S. and Y.U. developed quantitative models for superlensassisted power transfer and assisted in the analysis of simulations and experiments, Y.U., D.H. and K.S. developed the simulation algorithms, G.L. and K.S. performed numerical calculations, G.L. and P.S. designed PCB layouts, P.S., M.R. and J.E. constructed the metamaterial lens, M.R., J.E., P.S., T.N. and J.S.L. designed and conducted the experiments and J.S.L. developed the project. All authors participated in discussing the results and writing the manuscript.
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Lipworth, G., Ensworth, J., Seetharam, K. et al. Magnetic Metamaterial Superlens for Increased Range Wireless Power Transfer. Sci Rep 4, 3642 (2014). https://doi.org/10.1038/srep03642
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