Wireless energy transfer

Abstract

Disclosed is an apparatus for use in wireless energy transfer, which includes a first resonator structure configured to transfer energy non-radiatively with a second resonator structure over a distance greater than a characteristic size of the second resonator structure. The non-radiative energy transfer is mediated by a coupling of a resonant field evanescent tail of the first resonator structure and a resonant field evanescent tail of the second resonator structure.

Claims

1. A source assembly for use with a wireless power system including a device resonator and a load coupled to the device resonator to receive power from the device resonator, the device resonator having a resonant frequency ω.sub.2, an intrinsic loss rate Γ.sub.2, and an intrinsic quality factor Q.sub.2=ω.sub.2/(2Γ.sub.2)>100, the source assembly comprising: a source resonator having a resonant frequency ω.sub.1, an intrinsic loss rate Γ.sub.1, and an intrinsic quality factor Q.sub.1=ω.sub.1/(2Γ.sub.1)>100; and a current probe configured to measure a current in the source resonator, wherein the source resonator and the device resonator are configured to resonantly and wirelessly couple electromagnetic power from the source resonator to the device resonator over a range of distances D between the source resonator and the device resonator using non-radiative electromagnetic induction having a coupling coefficient K.

2. The source assembly of claim 1, further comprising a power supply coupled to the source resonator to provide power to the source resonator.

3. The source assembly of claim 1, wherein the intrinsic loss rates satisfy κ/√{square root over (Γ.sub.1Γ.sub.2)}>2 over the range of distances D between the source resonator and the device resonator.

4. The source assembly of claim 1, further comprising a monitor configured to measure an efficiency of power coupled to the load from the source resonator using information from the current measurement.

5. The source assembly of claim 4, further comprising a frequency adjuster configured to adjust a frequency of the wireless energy transfer based on the measurement by the monitor.

6. The source assembly of claim 5, wherein the frequency adjuster is configured to adjust the resonant frequency of the source resonator.

7. The source assembly of claim 4, wherein the resonators wirelessly exchange information based on the measurement by the monitor.

8. The source assembly of claim 1, wherein f.sub.1=ω.sub.1/(2π) and f.sub.1 is between 1 MHz and 10 MHz.

9. The source assembly of claim 1, wherein Q.sub.1>200.

10. The source assembly of claim 9, wherein f.sub.1=ω.sub.1/(2π) and f.sub.1 is between 10 KHz and 1 MHz.

11. A device assembly for use with a wireless power system including a source resonator and a power supply coupled to the source resonator to provide power to the source resonator, the source resonator having a resonant frequency ω.sub.1, an intrinsic loss rate Γ.sub.1, and an intrinsic quality factor Q.sub.1=ω.sub.1/(2Γ.sub.1)>100, the device assembly comprising: a device resonator having a resonant frequency ω.sub.2, an intrinsic loss rate Γ.sub.2, and an intrinsic quality factor Q.sub.2=ω.sub.2/(2Γ.sub.2)>100; and a current probe configured to measure a current in the device resonator, wherein the source resonator and the device resonator are configured to resonantly and wirelessly couple electromagnetic power from the source resonator to the device resonator over a range of distances D between the source resonator and the device resonator using non-radiative electromagnetic induction having a coupling coefficient κ.

12. The device assembly of claim 11, further comprising a load coupled to the device resonator to receive power from the device resonator.

13. The device assembly of claim 11, wherein the intrinsic loss rates satisfy κ/√{square root over (Γ.sub.1Γ.sub.2)}>2 over the range of distances D between the source resonator and the device resonator.

14. The device assembly of claim 11, further comprising a monitor configured to measure an efficiency of power coupled to the load from the source resonator using information from the current measurement.

15. The device assembly of claim 14, further comprising a frequency adjuster configured to adjust a frequency of the wireless energy transfer based on the measurement by the monitor.

16. The device assembly of claim 15, wherein the frequency adjuster is configured to adjust the resonant frequency of the device resonator.

17. The device assembly of claim 14, wherein the resonators wirelessly exchange information based on the measurement by the monitor.

18. The device assembly of claim 11, wherein f.sub.2=ω.sub.2/(2π) and f.sub.2 is between 1 MHz and 10 MHz.

19. The device assembly of claim 11, wherein Q.sub.2>200.

20. The device assembly of claim 19, wherein f.sub.2=ω.sub.2/(2π) and f.sub.2 is between 10 KHz and 1 MHz.

21. A vehicle configured for use with a wireless power system including a source resonator and a power supply coupled to the source resonator to provide power to the source resonator, the source resonator having a resonant frequency ω.sub.1, an intrinsic loss rate Γ.sub.1, and an intrinsic quality factor Q.sub.1=ω.sub.1/(2Γ.sub.1)>100, the vehicle comprising: a device resonator having a resonant frequency ω.sub.2, an intrinsic loss rate Γ.sub.2, and an intrinsic quality factor Q.sub.2=ω.sub.2/(2Γ.sub.2)>100; a load coupled to the device resonator to receive power from the device resonator; and a current probe configured to measure a current in the device resonator, wherein the source resonator and the device resonator are configured to resonantly and wirelessly couple electromagnetic power from the source resonator to the device resonator over a range of distances D between the source resonator and the device resonator using non-radiative electromagnetic induction having a coupling coefficient κ.

22. The vehicle of claim 21, further comprising a monitor configured to measure an efficiency of power coupled to the load from the source resonator using information from the current measurement.

23. The vehicle of claim 22, further comprising a frequency adjuster configured to adjust a frequency of the wireless energy transfer based on the measurement by the monitor.

24. The vehicle of claim 23, wherein the frequency adjuster is configured to adjust the resonant frequency of the device resonator.

25. The vehicle of claim 22, wherein the resonators wirelessly exchange information based on the measurement by the monitor.

26. The vehicle of claim 21, wherein Q.sub.1>200 and Q.sub.2>200, wherein the load is configured to provide power to a vehicle, wherein the intrinsic loss rates satisfy κ/√{square root over (Γ.sub.1Γ.sub.2)} >5 over the range of distances D, and wherein the power provided to the load from the device resonator is greater than about 10 Watt.

27. The vehicle of claim 26, wherein f.sub.1=ω.sub.1/(2π) and f.sub.2=ω.sub.2/(2π), and f.sub.1 and f.sub.6 are between 10 kHz and 1 MHz.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows a schematic of a wireless energy transfer scheme.

(2) FIG. 2 shows an example of a self-resonant conducting-wire coil.

(3) FIG. 3 shows a wireless energy transfer scheme featuring two self-resonant conducting-wire coils

(4) FIG. 4 shows an example of a capacitively loaded conducting-wire coil, and illustrates the surrounding field.

(5) FIG. 5 shows a wireless energy transfer scheme featuring two capacitively loaded conducting-wire coils, and illustrates the surrounding field.

(6) FIG. 6 shows an example of a resonant dielectric disk, and illustrates the surrounding field.

(7) FIG. 7 shows a wireless energy transfer scheme featuring two resonant dielectric disks, and illustrates the surrounding field.

(8) FIGS. 8a and 8b show schematics for frequency control mechanisms.

(9) FIGS. 9a through 9c illustrate a wireless energy transfer scheme in the presence of various extraneous objects.

(10) FIG. 10 illustrates a circuit model for wireless energy transfer.

(11) FIG. 11 illustrates the efficiency of a wireless energy transfer scheme.

(12) FIG. 12 illustrates parametric dependences of a wireless energy transfer scheme.

(13) FIG. 13 plots the parametric dependences of a wireless energy transfer scheme.

(14) FIG. 14 is a schematic of an experimental system demonstrating wireless energy transfer.

(15) FIGS. 15-17. Plot experiment results for the system shown schematically in FIG. 14.

DETAILED DESCRIPTION

(16) FIG. 1 shows a schematic that generally describes one embodiment of the invention, in which energy is transferred wirelessly between two resonant objects.

(17) Referring to FIG. 1, energy is transferred, over a distance D, between a resonant source object having a characteristic size L.sub.1 and a resonant device object of characteristic size L.sub.2. Both objects are resonant objects. The source object is connected to a power supply (not shown), and the device object is connected to a power consuming device (e.g. a load resistor, not shown). Energy is provided by the power supply to the source object, transferred wirelessly and non-radiatively from the source object to the device object, and consumed by the power consuming device. The wireless non-radiative energy transfer is performed using the field (e.g. the electromagnetic field or acoustic field) of the system of two resonant objects. For simplicity, in the following we will assume that field is the electromagnetic field.

(18) It is to be understood that while two resonant objects are shown in the embodiment of FIG. 1, and in many of the examples below, other embodiments may feature 3 or more resonant objects. For example, in some embodiments a single source object can transfer energy to multiple device objects. Similarly, in some embodiments multiple sources can transfer energy to one or more device objects. For example, as explained at in the paragraph bridging pages 4-5 of U.S. Provisional Application No. 60/698,442 to which the present application claims benefit and which is incorporated by reference above, for certain applications having uneven power transfer to the device object as the distance between the device and the source changes, multiple sources can be strategically placed to alleviate the problem, and/or the peak amplitude of the source can be dynamically adjusted. Furthermore, in some embodiments energy may be transferred from a first device to a second, and then from the second device to the third, and so forth.

(19) Initially, we present a theoretical framework for understanding non-radiative wireless energy transfer. Note however that it is to be understood that the scope of the invention is not bound by theory.

(20) Coupled Mode Theory

(21) An appropriate analytical framework for modeling the resonant energy-exchange between two resonant objects 1 and 2 is that of “coupled-mode theory” (CMT). The field of the system of two resonant objects 1 and 2 is approximated by F(r,t)≈a.sub.1(t)F.sub.1(r)+a.sub.2(t)F.sub.2(r), where Γ.sub.1,2(r) are the eigenmodes of 1 and 2 alone, normalized to unity energy, and the field amplitudes a.sub.1,2(t) are defined so that |a.sub.1,2(t)|.sup.2 is equal to the energy stored inside the objects 1 and 2 respectively. Then, the field amplitudes can be shown to satisfy, to lowest order:

(22) $\begin{array}{cc}\frac{{\mathrm{da}}_1}{dt}=-i\left({\omega }_1-{\mathrm{iT}}_1\right)a_1+i\kappa a_2,\frac{{\mathrm{da}}_2}{dt}=-i\left({\omega }_2-i{\Gamma }_2\right)a_2+i\kappa a_1& \left(1\right)\end{array}$
where ω.sub.1,2 are the individual angular eigenfrequencies of the eigenmodes, Γ.sub.1,2 are the resonance widths due to the objects' intrinsic (absorption, radiation etc.) losses, and κ is the coupling coefficient. Eqs. (1) show that at exact resonance (ω.sub.1=ω.sub.2 and Γ.sub.1=Γ.sub.2), the eigenmodes of the combined system are split by 2κ; the energy exchange between the two objects takes place in time˜π/2κ and is nearly perfect, apart for losses, which are minimal when the coupling rate is much faster than all loss rates (κ»Γ.sub.1,2). The coupling to loss ratio κ/√{square root over (Γ.sub.1Γ.sub.2)} serves as a figure-of-merit in evaluating a system used for wireless energy-transfer, along with the distance over which this ratio can be achieved. The regime κ/√{square root over (Γ.sub.1Γ.sub.2)}»1 is called “strong-coupling” regime.

(23) In some embodiments, the energy-transfer application preferably uses resonant modes of high Q=ω/2 Γ, corresponding to low (i.e. slow) intrinsic-loss rates F This condition may be satisfied where the coupling is implemented using, not the lossy radiative far-field, but the evanescent (non-lossy) stationary near-field.

(24) To implement an energy-transfer scheme, usually finite objects, namely ones that are topologically surrounded everywhere by air, are more appropriate. Unfortunately, objects of finite extent cannot support electromagnetic states that are exponentially decaying in all directions in air, since, from Maxwell's Equations in free space: {right arrow over (k)}.sup.2=ω.sup.2/c.sup.2 where {right arrow over (k)} is the wave vector, co the angular frequency, and c the speed of light.

(25) Because of this, one can show that they cannot support states of infinite Q. However, very long-lived (so-called “high-Q”) states can be found, whose tails display the needed exponential or exponential-like decay away from the resonant object over long enough distances before they turn oscillatory (radiative). The limiting surface, where this change in the field behavior happens, is called the “radiation caustic”, and, for the wireless energy-transfer scheme to be based on the near field rather than the far/radiation field, the distance between the coupled objects must be such that one lies within the radiation caustic of the other.

(26) Furthermore, in some embodiments, small Q.sub.κ=ω/2κ corresponding to strong (i.e. fast) coupling rate κ is preferred over distances larger than the characteristic sizes of the objects. Therefore, since the extent of the near-field into the area surrounding a finite-sized resonant object is set typically by the wavelength, in some embodiments, this mid-range non-radiative coupling can be achieved using resonant objects of subwavelength size, and thus significantly longer evanescent field-tails. As will be seen in examples later on, such subwavelength resonances can often be accompanied with a high Q, so this will typically be the appropriate choice for the possibly-mobile resonant device-object. Note, though, that in some embodiments, the resonant source-object will be immobile and thus less restricted in its allowed geometry and size, which can be therefore chosen large enough that the near-field extent is not limited by the wavelength. Objects of nearly infinite extent, such as dielectric waveguides, can support guided modes whose evanescent tails are decaying exponentially in the direction away from the object, slowly if tuned close to cutoff, and can have nearly infinite Q.

(27) In the following, we describe several examples of systems suitable for energy transfer of the type described above. We will demonstrate how to compute the CMT parameters ω.sub.1,2, Q.sub.1,2 and Q.sub.κ described above and how to choose these parameters for particular embodiments in order to produce a desirable figure-of-merit κ/√{square root over (Γ.sub.1Γ.sub.2)}=√{square root over (Q.sub.1Q.sub.2)}/Q.sub.κ. In particular, this figure of merit is typically maximized when ω.sub.1,2 are tuned to a particular angular frequency {tilde over (ω)}, thus, if {tilde over (Γ)} is half the angular-frequency width for which √{square root over (Q.sub.1Q.sub.2)}/Q.sub.κ is above half its maximum value at {tilde over (ω)}, the angular eigenfrequencies ω.sub.1,2 should typically be tuned to be close to 6 to within the width {tilde over (Γ)}.

(28) In addition, as described below, Q.sub.1,2 can sometimes be limited not from intrinsic loss mechanisms but from external perturbations. In those cases, producing a desirable figure-of-merit translates to reducing Q.sub.κ (i.e. increasing the coupling). Accordingly we will demonstrate how, for particular embodiments, to reduce Q.sub.κ.

(29) Self-Resonant Conducting Coils

(30) In some embodiments, one or more of the resonant objects are self-resonant conducting coils. Referring to FIG. 2, a conducting wire of length l and cross-sectional radius a is wound into a helical coil of radius r and height h (namely with N=√{square root over (l.sup.2−h.sup.2)}/2πr number of turns), surrounded by air. As described below, the wire has distributed inductance and distributed capacitance, and therefore it supports a resonant mode of angular frequency ω. The nature of the resonance lies in the periodic exchange of energy from the electric field within the capacitance of the coil, due to the charge distribution ρ(x) across it, to the magnetic field in free space, due to the current distribution j(x) in the wire. In particular, the charge conservation equation ∇.Math.j=iωρ implies that: (i) this periodic exchange is accompanied by a π/2 phase-shift between the current and the charge density profiles, namely the energy U contained in the coil is at certain points in time completely due to the current and at other points in time completely due to the charge, and (ii) if ρ.sub.l(x) and I(x) are respectively the linear charge and current densities in the wire, where x runs along the wire, q.sub.o=½∫dx|ρ.sub.l(x)| is the maximum amount of positive charge accumulated in one side of the coil (where an equal amount of negative charge always also accumulates in the other side to make the system neutral) and I.sub.o=max {|I(x)|} is the maximum positive value of the linear current distribution, then I.sub.o=ωq.sub.o. Then, one can define an effective total inductance L and an effective total capacitance C of the coil through the amount of energy U inside its resonant mode:

(31) $\begin{array}{cc}U\equiv \frac{1}{2}{I}_o^{2}L.Math.L=\frac{{\mu }_0}{4\pi I_o^2}\int \int {\mathrm{dxdx}}^{\prime }\frac{j\left(x\right).Math.j\left(x^{\prime }\right)}{.Math.x-x^{\prime }.Math.},& \left(2\right)\\ U\equiv \frac{1}{2}{q}_o^2\frac{1}{C}.Math.\frac{1}{C}=\frac{1}{4{\mathrm{\pi .Math.}}_o{q}_o^2}\int \int {\mathrm{dxdx}}^{\prime }\frac{\rho \left(x\right).Math.\rho \left(x^{\prime }\right)}{.Math.x-x^{\prime }.Math.}& \left(3\right)\end{array}$
where μ.sub.o and ε.sub.o are the magnetic permeability and electric permittivity of free space. With these definitions, the resonant angular frequency and the effective impedance are given by the common formulas ω=1/√{square root over (LC)} and Z=√{square root over (L/C)} respectively.

(32) Losses in this resonant system consist of ohmic (material absorption) loss inside the wire and radiative loss into free space. One can again define a total absorption resistance R.sub.abs from the amount of power absorbed inside the wire and a total radiation resistance R.sub.rad from the amount of power radiated due to electric- and magnetic-dipole radiation:

(33) $\begin{array}{cc}{P}_{abs}\equiv \frac{1}{2}{I}_o^2{R}_{abs}.Math.R_{abs}\approx {\zeta }_c\frac{l}{2\pi a}.Math.\frac{I_{rms}^2}{I_o^2}& \left(4\right)\\ P_{rad}\equiv \frac{1}{2}{I}_o^2{R}_{rad}.Math.R_{rad}\approx \frac{{\zeta }_o}{6_{iT}}\left[{\left(\frac{\omega .Math.p.Math.}{c}\right)}^2+{\left(\frac{\omega \sqrt{.Math.m.Math.}}{c}\right)}^4\right],& \left(5\right)\end{array}$
where c=1/√{square root over (μ.sub.oε.sub.o)} and ξ.sub.o=√{square root over (μ.sub.o/ε.sub.o)} are the light velocity and light impedance in free space, the impedance ζ.sub.c is ζ.sub.c=1/σδ=√{square root over (μ.sub.oω/2σ)} with σ the conductivity of the conductor and δ the skin depth at the frequency ω,

(34) $I_{rms}^2=\frac{1}{l}\int \mathrm{dx}{.Math.l\left(x\right).Math.}^2,$
p=∫dx r ρ.sub.l(x) is the electric-dipole moment of the coil and m=½∫dx r×j(x) is the magnetic-dipole moment of the coil. For the radiation resistance formula Eq. (5), the assumption of operation in the quasi-static regime (h, r«λ=2πc/ω) has been used, which is the desired regime of a subwavelength resonance. With these definitions, the absorption and radiation quality factors of the resonance are given by Q.sup.abs=Z/R.sub.abs and Q.sup.rad=Z/R.sub.rad respectively.

(35) From Eq. (2)-(5) it follows that to determine the resonance parameters one simply needs to know the current distribution j in the resonant coil. Solving Maxwell's equations to rigorously find the current distribution of the resonant electromagnetic eigenmode of a conducting-wire coil is more involved than, for example, of a standard LC circuit, and we can find no exact solutions in the literature for coils of finite length, making an exact solution difficult. One could in principle write down an elaborate transmission-line-like model, and solve it by brute force. We instead present a model that is (as described below) in good agreement (˜5%) with experiment. Observing that the finite extent of the conductor forming each coil imposes the boundary condition that the current has to be zero at the ends of the coil, since no current can leave the wire, we assume that the resonant mode of each coil is well approximated by a sinusoidal current profile along the length of the conducting wire. We shall be interested in the lowest mode, so if we denote by x the coordinate along the conductor, such that it runs from −l/2 to +l/2, then the current amplitude profile would have the form I(x)=I.sub.o cos (πx/l), where we have assumed that the current does not vary significantly along the wire circumference for a particular x, a valid assumption provided a«r. It immediately follows from the continuity equation for charge that the linear charge density profile should be of the form ρ.sub.l(x)=ρ.sub.o sin (πx/l), and thus q.sub.o=∫.sub.0.sup.1/2dx ρ.sub.o|sin(πx/l)|=ρ.sub.ol/π. Using these sinusoidal profiles we find the so-called “self-inductance” L.sub.s and “self-capacitance” C.sub.s of the coil by computing numerically the integrals Eq. (2) and (3); the associated frequency and effective impedance are ω.sub.s and Z.sub.s respectively. The “self-resistances” R.sub.s are given analytically by Eq. (4) and (5) using

(36) $I_{rms}^2=\frac{1}{l}{\int }_{-l/2}^{l/2}\mathrm{dx}{.Math.I_o\cos \left(\pi x/l\right).Math.}^2={\left(\frac{1}{2}I\right)}_o^2,.Math.p.Math.=q_o\sqrt{{\left(\frac{2}{\pi }h\right)}^2+{\left(\frac{4N\cos \left(\pi N\right)}{\left(4N^2-1\right)\pi }r\right)}^2}\mathrm{and}$$.Math.m.Math.=I_o\sqrt{{\left(\frac{2}{\pi }N\pi r^2\right)}^2+{\left(\frac{\cos \left(\pi N\right)\left(12N^2-1\right)-\sin \left(\pi N\right)\pi N\left(4N^2-1\right)}{\left(16N^4-8N^2+1\right)\prime \tau }hr\right)}^2},$
and therefore the associated Q.sub.s factors may be calculated.

(37) The results for two particular embodiments of resonant coils with subwavelength modes of λ.sub.s/r≥70 (i.e. those highly suitable for near-field coupling and well within the quasi-static limit) are presented in Table 1. Numerical results are shown for the wavelength and absorption, radiation and total loss rates, for the two different cases of subwavelength-coil resonant modes. Note that, for conducting material, copper (σ=5.998.Math.10{circumflex over ( )}−7 S/m) was used. It can be seen that expected quality factors at microwave frequencies are Q.sub.s.sup.abs≥1000 and Q.sub.s.sup.rad≥5000.

(38) TABLE-US-00001 TABLE 1 single coil λ.sub.s/r f (MHz) Q.sub.s.sup.rad Q.sub.s.sup.abs Q.sub.s = ω.sub.s/2Γ.sub.s r = 30 cm, h = 20 cm, 74.7 13.39 4164 8170 2758 a = 1 cm, N = 4 r = 10 cm, h = 3 cm, 140 21.38 43919 3968 3639 a = 2 mm, N = 6

(39) Referring to FIG. 3, in some embodiments, energy is transferred between two self-resonant conducting-wire coils. The electric and magnetic fields are used to couple the different resonant conducting-wire coils at a distance D between their centers. Usually, the electric coupling highly dominates over the magnetic coupling in the system under consideration for coils with h»2r and, oppositely, the magnetic coupling highly dominates over the electric coupling for coils with h«2r. Defining the charge and current distributions of two coils 1,2 respectively as ρ.sub.1,2(x) and j.sub.1,2 (x), total charges and peak currents respectively as q.sub.1,2 and I.sub.1,2, and capacitances and inductances respectively as C.sub.1,2 and L.sub.1,2, which are the analogs of ρ(x), j(x), q.sub.o, I.sub.o, C and L for the single-coil case and are therefore well defined, we can define their mutual capacitance and inductance through the total energy:

(40) 0$\begin{array}{cc}U\equiv U_1+U_2+\frac{1}{2}\left(q_1*q_2+q_2*q_1\right)/M_C+\frac{1}{2}\left(q_1^{*}{q}_2+q_2^{*}{q}_1\right)M_L.Math.1/M_C=\frac{1}{4{\mathrm{\pi .Math.}}_o{q}_1{q}_2}\int \int {\mathrm{dxdx}}^{\prime }\frac{{\rho }_1\left(x\right).Math.{\rho }_2\left(x^{\prime }\right)}{.Math.x-x^{\prime }.Math.},M_L=\frac{{\mu }_o}{4\pi I_1{I}_2}\int \int {\mathrm{dxdx}}^{\prime }\frac{j_1\left(x\right).Math.j_2\left(x^{\prime }\right)}{.Math.x-x^{\prime }.Math.}u,& \left(6\right)\end{array}$
where U.sub.1=½q.sub.1.sup.2/C.sub.1=½I.sub.1.sup.2L.sub.1, U.sub.2=½q.sub.2.sup.2/C.sub.2=½I.sub.2.sup.2L.sub.2 and the retardation factor of u=exp (iω|x−x′|/c) inside the integral can been ignored in the quasi-static regime D«λ of interest, where each coil is within the near field of the other. With this definition, the coupling coefficient is given by κ=ω√{square root over (C.sub.1C.sub.2)}/2M.sub.C+ωM.sub.L/2√{square root over (L.sub.1L.sub.2)} ⇔Q.sub.κ.sup.−1=(M.sub.C/√{square root over (C.sub.1C.sub.2)}).sup.−1+(√{square root over (L.sub.1L.sub.2)}/M.sub.L).sup.−1.

(41) Therefore, to calculate the coupling rate between two self-resonant coils, again the current profiles are needed and, by using again the assumed sinusoidal current profiles, we compute numerically from Eq. (6) the mutual capacitance M.sub.C,s and inductance M.sub.L,s between two self-resonant coils at a distance D between their centers, and thus Q.sub.κ,s is also determined.

(42) TABLE-US-00002 TABLE 2 pair of coils D/r Q = ω/2Γ Q.sub.κ = ω/2κ κ/Γ r = 30 cm, h = 20 cm, 3 2758 38.9 70.9 a = 1 cm, N = 4 5 2758 139.4 19.8 λ/r ≈ 75 7 2758 333.0 8.3 Q.sub.s.sup.abs ≈ 8170, Q.sub.s.sup.rad ≈ 4164 10 2758 818.9 3.4 r = 10 cm, h = 3 cm, 3 3639 61.4 59.3 a = 2 mm, N = 6 5 3639 232.5 15.7 λ/r ≈ 140 7 3639 587.5 6.2 Q.sub.s.sup.abs ≈ 3968, Q.sub.s.sup.rad ≈ 43919 10 3639 1580 2.3

(43) Referring to Table 2, relevant parameters are shown for exemplary embodiments featuring pairs or identical self resonant coils. Numerical results are presented for the average wavelength and loss rates of the two normal modes (individual values not shown), and also the coupling rate and figure-of-merit as a function of the coupling distance D, for the two cases of modes presented in Tablet. It can be seen that for medium distances D/r=10−3 the expected coupling-to-loss ratios are in the range κ/Γ˜2−70.

(44) Capacitively-Loaded Conducting Loops or Coils

(45) In some embodiments, one or more of the resonant objects are capacitively-loaded conducting loops or coils. Referring to FIG. 4 a helical coil with N turns of conducting wire, as described above, is connected to a pair of conducting parallel plates of area A spaced by distance d via a dielectric material of relative permittivity E, and everything is surrounded by air (as shown, N=1 and h=0). The plates have a capacitance C.sub.p=ε.sub.oεA/d, which is added to the distributed capacitance of the coil and thus modifies its resonance. Note however, that the presence of the loading capacitor modifies significantly the current distribution inside the wire and therefore the total effective inductance L and total effective capacitance C of the coil are different respectively from L.sub.s and C.sub.s, which are calculated for a self-resonant coil of the same geometry using a sinusoidal current profile. Since some charge is accumulated at the plates of the external loading capacitor, the charge distribution p inside the wire is reduced, so C<C.sub.s, and thus, from the charge conservation equation, the current distribution j flattens out, so L>L.sub.s. The resonant frequency for this system is

(46) ω=1/√{square root over (L(C+C.sub.p))}<ω.sub.s=1/√{square root over (L.sub.sC.sub.s)}, and I(x).fwdarw.I.sub.o cos (πx/l).Math.C.fwdarw.C.sub.s.Math.ω.fwdarw.ω.sub.s, as C.sub.p.fwdarw.0.

(47) In general, the desired CMT parameters can be found for this system, but again a very complicated solution of Maxwell's Equations is required. Instead, we will analyze only a special case, where a reasonable guess for the current distribution can be made. When C.sub.p»C.sub.s>C, then ω≈1/√{square root over (LC.sub.p)}«ω.sub.s and Z≈√{square root over (L/C.sub.p)}«Z.sub.s, while all the charge is on the plates of the loading capacitor and thus the current distribution is constant along the wire. This allows us now to compute numerically L from Eq. (2). In the case h=0 and N integer, the integral in Eq. (2) can actually be computed analytically, giving the formula L=μ.sub.or[ln (8r/a)−2]N.sup.2. Explicit analytical formulas are again available for R from Eq. (4) and (5), since I.sub.rms=I.sub.o, |p|≈0 and |m|=I.sub.oNπr.sup.2 (namely only the magnetic-dipole term is contributing to radiation), so we can determine also Q.sup.abs=ωL/R.sub.abs and Q.sup.rad=ωL/R.sub.rad. At the end of the calculations, the validity of the assumption of constant current profile is confirmed by checking that indeed the condition C.sub.p»C.sub.s⇔ω«ω.sub.s is satisfied. To satisfy this condition, one could use a large external capacitance, however, this would usually shift the operational frequency lower than the optimal frequency, which we will determine shortly; instead, in typical embodiments, one often prefers coils with very small self-capacitance C.sub.s to begin with, which usually holds, for the types of coils under consideration, when N=1, so that the self-capacitance comes from the charge distribution across the single turn, which is almost always very small, or when N>1 and h»2Na, so that the dominant self-capacitance comes from the charge distribution across adjacent turns, which is small if the separation between adjacent turns is large.

(48) The external loading capacitance C.sub.p provides the freedom to tune the resonant frequency (for example by tuning A or d). Then, for the particular simple case h=0, for which we have analytical formulas, the total Q=ωL/(R.sub.abs+R.sub.rad) becomes highest at the optimal frequency

(49) $\begin{array}{cc}\tilde{\omega }={\left[\frac{c^4}{\pi }\sqrt{\frac{{\mathrm{.Math.}}_o}{2\sigma }}.Math.\frac{1}{aN{r}^3}\right]}^{2/7},& \left(7\right)\end{array}$
reaching the value

(50) $\begin{array}{cc}\tilde{Q}=\frac{6}{7\pi }{\left(2{\pi }^2{\eta }_o\frac{\sigma a^2{N}^2}{r}\right)}^{3/7}.Math.\left[\ln \left(\frac{8r}{a}\right)-2\right].& \left(8\right)\end{array}$
At lower frequencies it is dominated by ohmic loss and at higher frequencies by radiation. Note, however, that the formulas above are accurate as long as {tilde over (ω)}«ω.sub.s and, as explained above, this holds almost always when N=1, and is usually less accurate when N>1, since h=0 usually implies a large self-capacitance. A coil with large h can be used, if the self-capacitance needs to be reduced compared to the external capacitance, but then the formulas for L and {tilde over (ω)}, {tilde over (Q)} are again less accurate. Similar qualitative behavior is expected, but a more complicated theoretical model is needed for making quantitative predictions in that case.

(51) The results of the above analysis for two embodiments of subwavelength modes of λ/r≥70 (namely highly suitable for near-field coupling and well within the quasi-static limit) of coils with N=1 and h=0 at the optimal frequency Eq. (7) are presented in Table 3. To confirm the validity of constant-current assumption and the resulting analytical formulas, mode-solving calculations were also performed using another completely independent method: computational 3D finite-element frequency-domain (FEFD) simulations (which solve Maxwell's Equations in frequency domain exactly apart for spatial discretization) were conducted, in which the boundaries of the conductor were modeled using a complex impedance ζ.sub.c=√{square root over (μ.sub.oω/2σ)} boundary condition, valid as long as ζ.sub.c/ζ.sub.o«1 (<10.sup.−5 for copper in the microwave). Table 3 shows Numerical FEFD (and in parentheses analytical) results for the wavelength and absorption, radiation and total loss rates, for two different cases of subwavelength-loop resonant modes. Note that for conducting material copper (σ=5.998.Math.10.sup.7 S/m) was used. (The specific parameters of the plot in FIG. 4 are highlighted with bold in the table.) The two methods (analytical and computational) are in very good agreement and show that, in some embodiments, the optimal frequency is in the low-MHz microwave range and the expected quality factors are Q.sup.abs≥1000 and Q.sup.rad≥10000.

(52) TABLE-US-00003 TABLE 3 single coil λ/r f (MHz) Q.sup.rad Q.sup.abs Q = ω/2Γ r = 30 cm, a = 2 cm 111.4 (112.4) 8.976 (8.897) 29546 (30512) 4886 (5117) 4193 (4381) ϵ = 10, A = 138 cm2, d = 4 mm r = 10 cm, a = 2 mm 69.7 (70.4) 43.04 (42.61) 10702 (10727) 1545 (1604) 1350 (1395) ϵ = 10, A = 3.14 cm.sup.2, d = 1 mm

(53) Referring to FIG. 5, in some embodiments, energy is transferred between two capacitively-loaded coils. For the rate of energy transfer between two capacitively-loaded coils 1 and 2 at distance D between their centers, the mutual inductance ML can be evaluated numerically from Eq. (6) by using constant current distributions in the case ω«ω.sub.s. In the case h=0, the coupling is only magnetic and again we have an analytical formula, which, in the quasi-static limit r«D«λ and for the relative orientation shown in FIG. 4, is M.sub.L≈πμ.sub.o/2.Math.(r.sub.1r.sub.2).sup.2N.sub.1N.sub.2/D.sup.3, which means that Q.sub.κ∝(D/√{square root over (r.sub.1r.sub.2)}).sup.3 is independent of the frequency co and the number of turns N.sub.1, N.sub.2. Consequently, the resultant coupling figure-of-merit of interest is

(54) $\begin{array}{cc}\frac{\kappa }{\sqrt{{\Gamma }_1{\Gamma }_2}}=\frac{\sqrt{Q_1{Q}_2}}{Q_{\kappa }}\approx {\left(\frac{\sqrt{r_1{r}_2}}{D}\right)}^3.Math.\frac{{\pi }^2{\eta }_o\frac{\sqrt{r_1{r}_2}}{\lambda }.Math.N_1{N}_2}{\underset{J=1,2}{.Math.}{\left(\sqrt{\frac{{\mathrm{\pi \eta }}_o}{\lambda \sigma }}.Math.\frac{r_j}{a_j}{N}_j+\frac{8}{3}{\pi }^5{{\eta }_o\left(\frac{r_j}{\lambda }\right)}^4{N}_j^2\right)}^{1/2}},& \left(9\right)\end{array}$
which again is more accurate for N.sub.1=N.sub.2=1.

(55) From Eq. (9) it can be seen that the optimal frequency ω, where the figure-of-merit is maximized to the value , is that where √{square root over (Q.sub.1Q.sub.2)} is maximized, since Q.sub.κ does not depend on frequency (at least for the distances D«λ of interest for which the quasi-static approximation is still valid). Therefore, the optimal frequency is independent of the distance D between the two coils and lies between the two frequencies where the single-coil Q.sub.1 and Q.sub.2 peak. For same coils, it is given by Eq. (7) and then the figure-of-merit Eq. (9) becomes

(56) $\begin{array}{cc}\widetilde{\left(\frac{\kappa }{\Gamma }\right)}=\frac{\tilde{Q}}{Q_{\kappa }}\approx {\left(\frac{r}{D}\right)}^3.Math.\frac{3}{7}{\left(2{\pi }^2{\eta }_o\frac{\sigma a^2{N}^2}{r}\right)}^{3/7}.& \left(10\right)\end{array}$
Typically, one should tune the capacitively-loaded conducting loops or coils, so that their angular eigenfrequencies are close to {tilde over (ω)} within {tilde over (Γ)}, which is half the angular frequency width for which √{square root over (Q.sub.1Q.sub.2)}/Q.sub.κ>/2.

(57) Referring to Table 4, numerical FEFD and, in parentheses, analytical results based on the above are shown for two systems each composed of a matched pair of the loaded coils described in Table 3. The average wavelength and loss rates are shown along with the coupling rate and coupling to loss ratio figure-of-merit κ/Γ as a function of the coupling distance D, for the two cases. Note that the average numerical Γ.sup.rad shown are again slightly different from the single-loop value of FIG. 3, analytical results for Γ.sup.rad are not shown but the single-loop value is used. (The specific parameters corresponding to the plot in FIG. 5 are highlighted with bold in the table.) Again we chose N=1 to make the constant-current assumption a good one and computed M.sub.L numerically from Eq. (6). Indeed the accuracy can be confirmed by their agreement with the computational FEFD mode-solver simulations, which give i through the frequency splitting (=2κ) of the two normal modes of the combined system. The results show that for medium distances D/r=10−3 the expected coupling-to-loss ratios are in the range κ/Γ˜0.5−50.

(58) TABLE-US-00004 TABLE 4 pair of coils D/r Q.sup.rad Q = ω/2Γ Q.sub.κ = ω/2κ κ/Γ r = 30 cm, a = 3 30779 4216 62.6 (63.7) 67.4 (68.7) 2 cm ϵ = 10, 5 29577 4194 235 (248) 17.8 (17.6) A = 138 cm2, 7 29128 4185 589 (646) 7.1 (6.8) d = 4 mm 10 28833 4177 1539 (1828) 2.7 (2.4) λ/r ≈ 112 Q.sup.abs ≈ 4886 r = 10 cm, a = 3 10955 1355 85.4 (91.3) 15.9 (15.3) 2 mm ϵ = 10, 5 10740 1351 313 (356) 4.32 (3.92) A = 3.14 cm.sup.2, 7 10759 1351 754 (925) 1.79 (1.51) d = 1 mm 10 10756 1351 1895 (2617) 0.71 (0.53) λ/r ≈ 70 Q.sup.abs ≈ 1546

(59) Optimization of √{square root over (Q.sub.1Q.sub.2)}/Q.sub.κ

(60) In some embodiments, the results above can be used to increase or optimize the performance of a wireless energy transfer system which employs capacitively-loaded coils. For example, the scaling of Eq. (10) with the different system parameters one sees that to maximize the system figure-of-merit κ/Γ one can, for example: Decrease the resistivity of the conducting material. This can be achieved, for example, by using good conductors (such as copper or silver) and/or lowering the temperature. At very low temperatures one could use also superconducting materials to achieve extremely good performance. Increase the wire radius a. In typical embodiments, this action is limited by physical size considerations. The purpose of this action is mainly to reduce the resistive losses in the wire by increasing the cross-sectional area through which the electric current is flowing, so one could alternatively use also a Litz wire or a ribbon instead of a circular wire. For fixed desired distance D of energy transfer, increase the radius of the loop r. In typical embodiments, this action is limited by physical size considerations. For fixed desired distance vs. loop-size ratio D/r, decrease the radius of the loop r. In typical embodiments, this action is limited by physical size considerations. Increase the number of turns N. (Even though Eq. (10) is expected to be less accurate for N>1, qualitatively it still provides a good indication that we expect an improvement in the coupling-to-loss ratio with increased N.) In typical embodiments, this action is limited by physical size and possible voltage considerations, as will be discussed in following sections. Adjust the alignment and orientation between the two coils. The figure-of-merit is optimized when both cylindrical coils have exactly the same axis of cylindrical symmetry (namely they are “facing” each other). In some embodiments, particular mutual coil angles and orientations that lead to zero mutual inductance (such as the orientation where the axes of the two coils are perpendicular) should be avoided. Finally, note that the height of the coil h is another available design parameter, which has an impact to the performance similar to that of its radius r, and thus the design rules are similar.

(61) The above analysis technique can be used to design systems with desired parameters. For example, as listed below, the above described techniques can be used to determine the cross sectional radius a of the wire which one should use when designing as system two same single-turn loops with a given radius in order to achieve a specific performance in terms of κ/Γ at a given D/r between them, when the material is copper (σ=5.998.Math.10.sup.7S/m):
D/r=5, κ/Γ≥10, r=30 cm.Math.a≥9 mm
D/r=5, κ/Γ≥10, r=5 cm.Math.a≥3.7 mm
D/r=5, κ/Γ≥20, r=30 cm.Math.a≥20 mm
D/r=5, κ/Γ≥20, r=5 cm.Math.a≥8.3 mm
D/r=10, κ/Γ≥1, r=30 cm.Math.a≥7 mm
D/r=10, κ/Γ≥1, r=5 cm.Math.a≥2.8 mm
D/r=10, κ/Γ≥3, r=30 cm.Math.a≥25 mm
D/r=10, κ/Γ≥3, r=5 cm.Math.a≥10 mm

(62) Similar analysis can be done for the case of two dissimilar loops. For example, in some embodiments, the device under consideration is very specific (e.g. a laptop or a cell phone), so the dimensions of the device object (r.sub.d, h.sub.d, a.sub.d, N.sub.d) are very restricted. However, in some such embodiments, the restrictions on the source object (r.sub.s, h.sub.s, a.sub.s, N.sub.s) are much less, since the source can, for example, be placed under the floor or on the ceiling. In such cases, the desired distance is often well defined, based on the application (e.g. D˜1 m for charging a laptop on a table wirelessly from the floor). Listed below are examples (simplified to the case N.sub.s=N.sub.d=1 and h.sub.s=h.sub.d=0) of how one can vary the dimensions of the source object to achieve the desired system performance in terms of κ/√{square root over (Γ.sub.sΓ.sub.d)}, when the material is again copper (σ=5.998.Math.10.sup.7S/m):
D=1.5 m, κ/√{square root over (Γ.sub.sΓ.sub.d)}≥15, r.sub.d=30 cm, a.sub.d=6 mm.Math.r.sub.s=1.158 m, a.sub.s≥5 mm
D=1.5 m, κ/√{square root over (Γ.sub.sΓ.sub.d)}≥30, r.sub.d=30 cm, a.sub.d=6 mm.Math.r.sub.s=1.15 m, a.sub.s≥33 mm
D=1.5 m, κ/√{square root over (Γ.sub.sΓ.sub.d)}≥1, r.sub.d=5 cm, a.sub.d=4 mm.Math.r, =1.119 m, a.sub.s≥7 mm
D=1.5 m, κ/√{square root over (Γ.sub.sΓ.sub.d)}≥2, r.sub.d=5 cm, a.sub.d=4 mm.Math.r, =1.119 m, a.sub.s≥52 mm
D=2 m, κ/√{square root over (Γ.sub.sΓ.sub.d)}≥10, r.sub.d=30 cm, a.sub.d=6 mm.Math.r.sub.s=1.518 m, a.sub.s≥7 mm
D=2 m, κ/√{square root over (Γ.sub.sΓ.sub.d)}≥20, r.sub.d=30 cm, a.sub.d=6 mm.Math.r, =1.514 m, a.sub.s≥50 mm
D=2 m, κ/√{square root over (Γ.sub.sΓ.sub.d)}≥0.5, r.sub.d=5 cm, a.sub.d=4 mm.Math.r.sub.s=1.491 m, a.sub.s≥5 mm
D=2 m, κ/√{square root over (Γ.sub.sΓ.sub.d)}≥1, r.sub.d=5 cm, a.sub.d=4 mm.Math.r.sub.s=1.491 m, a.sub.s≥36 mm

(63) Optimization of Q.sub.κ

(64) As will be described below, in some embodiments the quality factor Q of the resonant objects is limited from external perturbations and thus varying the coil parameters cannot lead to improvement in Q. In such cases, one may opt to increase the coupling to loss ratio figure-of-merit by decreasing Q.sub.κ (i.e. increasing the coupling). The coupling does not depend on the frequency and the number of turns. Therefore, the remaining degrees of freedom are: Increase the wire radii a.sub.1 and a.sub.2. In typical embodiments, this action is limited by physical size considerations. For fixed desired distance D of energy transfer, increase the radii of the coils r.sub.1 and r.sub.2. In typical embodiments, this action is limited by physical size considerations. For fixed desired distance vs. coil-sizes ratio D/r.sub.1r.sub.2, only the weak (logarithmic) dependence of the inductance remains, which suggests that one should decrease the radii of the coils r.sub.1 and r.sub.2. In typical embodiments, this action is limited by physical size considerations. Adjust the alignment and orientation between the two coils. In typical embodiments, the coupling is optimized when both cylindrical coils have exactly the same axis of cylindrical symmetry (namely they are “facing” each other). Particular mutual coil angles and orientations that lead to zero mutual inductance (such as the orientation where the axes of the two coils are perpendicular) should obviously be avoided. Finally, note that the heights of the coils h.sub.1 and h.sub.2 are other available design parameters, which have an impact to the coupling similar to that of their radii r.sub.1 and r.sub.2, and thus the design rules are similar.

(65) Further practical considerations apart from efficiency, e.g. physical size limitations, will be discussed in detail below.

(66) It is also important to appreciate the difference between the above described resonant-coupling inductive scheme and the well-known non-resonant inductive scheme for energy transfer. Using CMT it is easy to show that, keeping the geometry and the energy stored at the source fixed, the resonant inductive mechanism allows for ˜Q.sup.2 (˜10.sup.6) times more power delivered for work at the device than the traditional non-resonant mechanism. This is why only close-range contact-less medium-power (W) transfer is possible with the latter, while with resonance either close-range but large-power (˜kW) transfer is allowed or, as currently proposed, if one also ensures operation in the strongly-coupled regime, medium-range and medium-power transfer is possible. Capacitively-loaded conducting loops are currently used as resonant antennas (for example in cell phones), but those operate in the far-field regime with D/r»1, r/λ˜1, and the radiation Q's are intentionally designed to be small to make the antenna efficient, so they are not appropriate for energy transfer.

(67) Inductively-Loaded Conducting Rods

(68) A straight conducting rod of length 2 h and cross-sectional radius a has distributed capacitance and distributed inductance, and therefore it supports a resonant mode of angular frequency ω. Using the same procedure as in the case of self-resonant coils, one can define an effective total inductance L and an effective total capacitance C of the rod through formulas (2) and (3). With these definitions, the resonant angular frequency and the effective impedance are given again by the common formulas ω=1/√{square root over (LC)} and Z=√{square root over (L/C)} respectively. To calculate the total inductance and capacitance, one can assume again a sinusoidal current profile along the length of the conducting wire. When interested in the lowest mode, if we denote by x the coordinate along the conductor, such that it runs from −h to +h, then the current amplitude profile would have the form I(x)=I.sub.o cos (πx/2h), since it has to be zero at the open ends of the rod. This is the well-known half-wavelength electric dipole resonant mode.

(69) In some embodiments, one or more of the resonant objects are inductively-loaded conducting rods. A straight conducting rod of length 2h and cross-sectional radius a, as in the previous paragraph, is cut into two equal pieces of length h, which are connected via a coil wrapped around a magnetic material of relative permeability μ, and everything is surrounded by air. The coil has an inductance L.sub.C, which is added to the distributed inductance of the rod and thus modifies its resonance. Note however, that the presence of the center-loading inductor modifies significantly the current distribution inside the wire and therefore the total effective inductance L and total effective capacitance C of the rod are different respectively from L.sub.s and C.sub.s, which are calculated for a self-resonant rod of the same total length using a sinusoidal current profile, as in the previous paragraph. Since some current is running inside the coil of the external loading inductor, the current distribution j inside the rod is reduced, so L<L.sub.s, and thus, from the charge conservation equation, the linear charge distribution ρ.sub.l flattens out towards the center (being positive in one side of the rod and negative in the other side of the rod, changing abruptly through the inductor), so C>C.sub.s. The resonant frequency for this system is ω=1/√{square root over ((L±L.sub.c)C)}<ω.sub.s=1/√{square root over (L.sub.sC.sub.s)}, and I(x).fwdarw.I.sub.o cos (πx/2h).Math.L.fwdarw.L.sub.s .Math.ω.fwdarw.ω.sub.s, as L.sub.c.fwdarw.0.

(70) In general, the desired CMT parameters can be found for this system, but again a very complicated solution of Maxwell's Equations is required. Instead, we will analyze only a special case, where a reasonable guess for the current distribution can be made. When L.sub.c»L.sub.s>L, then ω≈1/L.sub.cC«ω.sub.s and Z≈√{square root over (L.sub.c/C)}»Z.sub.s, while the current distribution is triangular along the rod (with maximum at the center-loading inductor and zero at the ends) and thus the charge distribution is positive constant on one half of the rod and equally negative constant on the other side of the rod. This allows us now to compute numerically C from Eq. (3). In this case, the integral in Eq. (3) can actually be computed analytically, giving the formula 1/C=1/(πε.sub.oh)[ln(h/a)−1]. Explicit analytical formulas are again available for R from Eq. (4) and (5), since I.sub.rms=I.sub.o, |p|=q.sub.oh and |m|=0 (namely only the electric-dipole term is contributing to radiation), so we can determine also Q.sup.abs=1/ωCR.sub.abs and Q.sup.rad=1/ωCR.sub.rad. At the end of the calculations, the validity of the assumption of triangular current profile is confirmed by checking that indeed the condition L.sub.c »L.sub.s⇔ω»ω.sub.s is satisfied. This condition is relatively easily satisfied, since typically a conducting rod has very small self-inductance L.sub.s to begin with.

(71) Another important loss factor in this case is the resistive loss inside the coil of the external loading inductor L.sub.c and it depends on the particular design of the inductor. In some embodiments, the inductor is made of a Brooks coil, which is the coil geometry which, for fixed wire length, demonstrates the highest inductance and thus quality factor. The Brooks coil geometry has N.sub.Bc turns of conducting wire of cross-sectional radius a.sub.Bc wrapped around a cylindrically symmetric coil former, which forms a coil with a square cross-section of side r.sub.Bc, where the inner side of the square is also at radius r.sub.Bc (and thus the outer side of the square is at radius 2r.sub.Bc), therefore N.sub.Bc≈(r.sub.Bc/2a.sub.Bc).sup.2 The inductance of the coil is then L.sub.c=2.0285μ.sub.or.sub.BcN.sub.Bc.sup.2≈2.0285, μ.sub.or.sub.Bc.sup.5/8a.sub.Bc.sup.4 and its resistance

(72) $R_c\approx \frac{1}{\sigma }\frac{l_{\mathrm{Bc}}}{\pi a_{\mathrm{Bc}}^2}\sqrt{1+\frac{\mu \omega \sigma }{2}{\left(\frac{a_B}{2}\right)}^2},$
where the total wire length is l.sub.Bc≈2π(3r.sub.Bc/2)N.sub.Bc≈3πr.sub.Bc.sup.3/4a.sub.Bc.sup.2 and we have used an approximate square-root law for the transition of the resistance from the dc to the ac limit as the skin depth varies with frequency.

(73) The external loading inductance L.sub.c provides the freedom to tune the resonant frequency. (For example, for a Brooks coil with a fixed size r.sub.Bc, the resonant frequency can be reduced by increasing the number of turns N.sub.Bc by decreasing the wire cross-sectional radius a.sub.Bc. Then the desired resonant angular frequency ω=1/√{square root over (L.sub.cC)} is achieved for a.sub.BC≈(2.0285μ.sub.or.sub.Bc.sup.5ω.sup.2C).sup.1/4 and the resulting coil quality factor is Q.sub.c≈0.169μ.sub.oσr.sub.Bc.sup.2ω/√{square root over (1+ω.sup.2μ.sub.oσ√{square root over (2.0285μ.sub.o(r.sub.Bc/4).sup.5C)})}). Then, for the particular simple case L.sub.c»L.sub.s, for which we have analytical formulas, the total Q=1/ωC(R.sub.c+R.sub.abs+R.sub.rad) becomes highest at some optimal frequency CD, reaching the value Q, both determined by the loading-inductor specific design. (For example, for the Brooks-coil procedure described above, at the optimal frequency {tilde over (Q)}≈Q.sub.c≈0.8 (μ.sub.oσ.sup.2r.sub.Bc.sup.3/C).sup.1/4) At lower frequencies it is dominated by ohmic loss inside the inductor coil and at higher frequencies by radiation. Note, again, that the above formulas are accurate as long as {tilde over (ω)}«ω.sub.s and, as explained above, this is easy to design for by using a large inductance.

(74) The results of the above analysis for two embodiments, using Brooks coils, of subwavelength modes of λ/h≥200 (namely highly suitable for near-field coupling and well within the quasi-static limit) at the optimal frequency {tilde over (ω)} are presented in Table 5. Table 5 shows in parentheses (for similarity to previous tables) analytical results for the wavelength and absorption, radiation and total loss rates, for two different cases of subwavelength-loop resonant modes. Note that for conducting material copper (σ=5.998.Math.10.sup.7S/m) was used. The results show that, in some embodiments, the optimal frequency is in the low-MHz microwave range and the expected quality factors are Q.sup.abs≥1000 and Q.sup.rad≥100000.

(75) TABLE-US-00005 TABLE 5 Q = single rod λ/h f (MHz) Q.sup.rad Q.sup.abs ω/2Γ h = 30 cm, a = 2 cm (403.8)  (2.477) (2.72 * (7400) (7380) μ = 1, r.sub.Bc = 2 cm, 10.sup.6) a.sub.Bc = 0.88 mm, N.sub.Bc = 129 h = 10 cm, a = 2 mm (214.2) (14.010) (6.92 * (3908) (3886) μ = 1, r.sub.Bc = 5 mm, 10.sup.5) a.sub.Bc = 0.25 mm,

(76) In some embodiments, energy is transferred between two inductively-loaded rods. For the rate of energy transfer between two inductively-loaded rods 1 and 2 at distance D between their centers, the mutual capacitance Mc can be evaluated numerically from Eq. (6) by using triangular current distributions in the case ω«ω.sub.s. In this case, the coupling is only electric and again we have an analytical formula, which, in the quasi-static limit t«D«λ and for the relative orientation such that the two rods are aligned on the same axis, is 1/M.sub.C≈½πε.sub.o.Math.(h.sub.1h.sub.2).sup.2/D.sup.3, which means that Q.sub.κ∝(D/√{square root over (h.sub.1h.sub.2)}) is independent of the frequency ω. Consequently, one can get the resultant coupling figure-of-merit of interest

(77) $t\frac{\kappa }{\sqrt{{\Gamma }_1{\Gamma }_2}}=\frac{\sqrt{Q_1{Q}_2}}{Q_{\kappa }}.$
It can be seen that the optimal frequency {tilde over (ω)}, where the figure-of-merit is maximized to the value , is that where √{square root over (Q.sub.1Q.sub.2)} is maximized, since Q.sub.κ does not depend on frequency (at least for the distances D«λ of interest for which the quasi-static approximation is still valid). Therefore, the optimal frequency is independent of the distance D between the two rods and lies between the two frequencies where the single-rod Q.sub.1 and Q.sub.2 peak. Typically, one should tune the inductively-loaded conducting rods, so that their angular eigenfrequencies are close to 6 within {tilde over (Γ)}, which is half the angular frequency width for which Q.sub.1Q.sub.2/Q.sub.κ>/2

(78) Referring to Table 6, in parentheses (for similarity to previous tables) analytical results based on the above are shown for two systems each composed of a matched pair of the loaded rods described in Table 5. The average wavelength and loss rates are shown along with the coupling rate and coupling to loss ratio figure-of-merit κ/Γ as a function of the coupling distance D, for the two cases. Note that for Γ.sup.rad the single-rod value is used. Again we chose L.sub.c»L.sub.s to make the triangular-current assumption a good one and computed M.sub.c numerically from Eq. (6). The results show that for medium distances D/h=10−3 the expected coupling-to-loss ratios are in the range κ/Γ˜0.5−100.

(79) TABLE-US-00006 TABLE 6 pair of rods D/h Q.sub.κ = ω/2κ κ/Γ h = 30 cm, a = 2 cm 3    (70.3) (105.0) μ = 1, r.sub.Bc = 2 cm, 5  (389) (19.0) a.sub.Bc = 0.88 mm, N.sub.Bc = 129 7 (1115) (6.62) λ/h ≈ 404 10 (3321) (2.22) Q ≈ 7380 h = 10 cm, a = 2 mm 3  (120) (32.4) μ = 1, r.sub.Bc = 5 mm, 5  (664) (5.85) a.sub.Bc = 0.25 mm, N.sub.Bc = 103 7 (1900) (2.05) λ/h ≈ 214 10 (5656) (0.69) Q ≈ 3886

(80) Dielectric Disks

(81) In some embodiments, one or more of the resonant objects are dielectric objects, such as disks. Consider a two dimensional dielectric disk object, as shown in FIG. 6, of radius r and relative permittivity E surrounded by air that supports high-Q “whispering-gallery” resonant modes. The loss mechanisms for the energy stored inside such a resonant system are radiation into free space and absorption inside the disk material. High-Q.sub.rad and long-tailed subwavelength resonances can be achieved when the dielectric permittivity ε is large and the azimuthal field variations are slow (namely of small principal number m). Material absorption is related to the material loss tangent: Q.sub.abs˜Re{ε}/Im{ε}. Mode-solving calculations for this type of disk resonances were performed using two independent methods: numerically, 2D finite-difference frequency-domain (FDFD) simulations (which solve Maxwell's Equations in frequency domain exactly apart for spatial discretization) were conducted with a resolution of 30 pts/r; analytically, standard separation of variables (SV) in polar coordinates was used.

(82) TABLE-US-00007 TABLE 7 single disk λ/r Q.sup.abs Q.sup.rad Q Re{ϵ} = 20.01 (20.00) 10103 (10075) 1988 (1992) 1661 (1663) 147.7, m = 2 Re{ϵ} = 9.952 (9.950) 10098 (10087) 9078 (9168) 4780 (4802) 65.6, m = 3
The results for two TE-polarized dielectric-disk subwavelength modes of λ/r≥10 are presented in Table 7. Table 7 shows numerical FDFD (and in parentheses analytical SV) results for the wavelength and absorption, radiation and total loss rates, for two different cases of subwavelength-disk resonant modes. Note that disk-material loss-tangent Im{ε}/Re{ε}=10.sup.−4 was used. (The specific parameters corresponding to the plot in FIG. 6. are highlighted with bold in the table.) The two methods have excellent agreement and imply that for a properly designed resonant low-loss-dielectric object values of Q.sub.rad≥2000 and Q.sub.abs˜10000 are achievable. Note that for the 3D case the computational complexity would be immensely increased, while the physics would not be significantly different. For example, a spherical object of ε=147.7 has a whispering gallery mode with m=2, Qrad=13962, and λ/r=17.

(83) The required values of ε, shown in Table 7, might at first seem unrealistically large. However, not only are there in the microwave regime (appropriate for approximately meter-range coupling applications) many materials that have both reasonably high enough dielectric constants and low losses (e.g. Titania, Barium tetratitanate, Lithium tantalite etc.), but also E could signify instead the effective index of other known subwavelength surface-wave systems, such as surface modes on surfaces of metallic materials or plasmonic (metal-like, negative-ε) materials or metallo-dielectric photonic crystals or plasmono-dielectric photonic crystals.

(84) To calculate now the achievable rate of energy transfer between two disks 1 and 2, as shown in FIG. 7 we place them at distance D between their centers. Numerically, the FDFD mode-solver simulations give ic through the frequency splitting (=2κ) of the normal modes of the combined system, which are even and odd superpositions of the initial single-disk modes; analytically, using the expressions for the separation-of-variables eigenfields E.sub.1,2(r) CMT gives κ through κ=ω.sub.1/2.Math.∫d.sup.3rε.sub.2(r)E*.sub.2(r)E.sub.1(r)/∫d.sup.3rε(r)|E.sub.1(r)|.sup.2 where ε.sub.j (r) and ε(r) are the dielectric functions that describe only the disk j (minus the constant ε.sub.o background) and the whole space respectively. Then, for medium distances D/r=10−3 and for non-radiative coupling such that D<2r.sub.C, where r.sub.C mλ/2π is the radius of the radiation caustic, the two methods agree very well, and we finally find, as shown in Table 8, coupling-to-loss ratios in the range κ/Γ □1-50. Thus, for the analyzed embodiments, the achieved figure-of-merit values are large enough to be useful for typical applications, as discussed below.

(85) TABLE-US-00008 TABLE 8 two disks D/r Q.sup.rad Q = ω/2Γ ω/2κ κ/Γ Re{ϵ} = 147.7, 3 2478 1989 46.9 (47.5) 42.4 (35.0) m = 2 λ/r ≈ 20 5 2411 1946 298.0 (298.0) 6.5 (5.6) Q.sup.abs ≈ 10093 7 2196 1804 769.7 (770.2) 2.3 (2.2) 10 2017 1681 1714 (1601) 0.98 (1.04) Re{ϵ} = 65.6, 3 7972 4455 144 (140) 30.9 (34.3) m = 3 λ/r = 10 5 9240 4824 2242 (2083) 2.2 (2.3) Q.sup.abs ≈ 10096 7 9187 4810 7485 (7417) 0.64 (0.65)

(86) Note that even though particular embodiments are presented and analyzed above as examples of systems that use resonant electromagnetic coupling for wireless energy transfer, those of self-resonant conducting coils, capacitively-loaded resonant conducting coils and resonant dielectric disks, any system that supports an electromagnetic mode with its electromagnetic energy extending much further than its size can be used for transferring energy. For example, there can be many abstract geometries with distributed capacitances and inductances that support the desired kind of resonances. In any one of these geometries, one can choose certain parameters to increase and/or optimize √{square root over (Q.sub.1Q.sub.2)}/Q.sub.κ or, if the Q's are limited by external factors, to increase and/or optimize for Q.sub.κ.

(87) System Sensitivity to Extraneous Objects

(88) In general, the overall performance of particular embodiment of the resonance-based wireless energy-transfer scheme depends strongly on the robustness of the resonant objects' resonances. Therefore, it is desirable to analyze the resonant objects' sensitivity to the near presence of random non-resonant extraneous objects. One appropriate analytical model is that of “perturbation theory” (PT), which suggests that in the presence of an extraneous object e the field amplitude a.sub.1(t) inside the resonant object 1 satisfies, to first order:

(89) $\begin{array}{cc}\frac{{\mathrm{da}}_1}{dt}=-i\left({\omega }_1-i{\Gamma }_1\right)a_1+i\left({\kappa }_{11-e}+i{\Gamma }_{1-e}\right)a_1& \left(11\right)\end{array}$
where again ω.sub.1 is the frequency and Γ.sub.1 the intrinsic (absorption, radiation etc.) loss rate, while is the frequency shift induced onto 1 due to the presence of e and Γ.sub.1-e is the extrinsic due to e (absorption inside e, scattering from e etc.) loss rate. The first-order PT model is valid only for small perturbations. Nevertheless, the parameters κ.sub.11-e, Γ.sub.1-e are well defined, even outside that regime, if a.sub.1 is taken to be the amplitude of the exact perturbed mode. Note also that interference effects between the radiation field of the initial resonant-object mode and the field scattered off the extraneous object can for strong scattering (e.g. off metallic objects) result in total radiation-Γ.sub.1-e's that are smaller than the initial radiation-Γ.sub.1 (namely Γ.sub.1-e is negative).

(90) The frequency shift is a problem that can be “fixed” by applying to one or more resonant objects a feedback mechanism that corrects its frequency. For example, referring to FIG. 8a, in some embodiments each resonant object is provided with an oscillator at fixed frequency and a monitor which determines the frequency of the object. Both the oscillator and the monitor are coupled to a frequency adjuster which can adjust the frequency of the resonant object by, for example, adjusting the geometric properties of the object (e.g. the height of a self-resonant coil, the capacitor plate spacing of a capacitively-loaded loop or coil, the dimensions of the inductor of an inductively-loaded rod, the shape of a dielectric disc, etc.) or changing the position of a non-resonant object in the vicinity of the resonant object. The frequency adjuster determines the difference between the fixed frequency and the object frequency and acts to bring the object frequency into alignment with the fixed frequency. This technique assures that all resonant objects operate at the same fixed frequency, even in the presence of extraneous objects.

(91) As another example, referring to FIG. 8b, in some embodiments, during energy transfer from a source object to a device object, the device object provides energy to a load, and an efficiency monitor measures the efficiency of the transfer. A frequency adjuster coupled to the load and the efficiency monitor acts to adjust the frequency of the object to maximize the transfer efficiency.

(92) In various embodiments, other frequency adjusting schemes may be used which rely on information exchange between the resonant objects. For example, the frequency of a source object can be monitored and transmitted to a device object, which is in turn synched to this frequency using frequency adjusters as described above. In other embodiments the frequency of a single clock may be transmitted to multiple devices, and each device then synched to that frequency.

(93) Unlike the frequency shift, the extrinsic loss can be detrimental to the functionality of the energy-transfer scheme, because it is difficult to remedy, so the total loss rate Γ.sub.1[e]=Γ.sub.1+Γ.sub.1-e (and the corresponding figure-of-merit κ.sub.[e]/√{square root over (Γ.sub.1[e]Γ.sub.z[e])}, where κ.sub.[e] the perturbed coupling rate) should be quantified. In embodiments using primarily magnetic resonances, the influence of extraneous objects on the resonances is nearly absent. The reason is that, in the quasi-static regime of operation (r«λ) that we are considering, the near field in the air region surrounding the resonator is predominantly magnetic (e.g. for coils with h«2r most of the electric field is localized within the self-capacitance of the coil or the externally loading capacitor), therefore extraneous non-conducting objects e that could interact with this field and act as a perturbation to the resonance are those having significant magnetic properties (magnetic permeability Re{μ}>1 or magnetic loss Im{μ}>0). Since almost all every-day non-conducting materials are non-magnetic but just dielectric, they respond to magnetic fields in the same way as free space, and thus will not disturb the resonance of the resonator. Extraneous conducting materials can however lead to some extrinsic losses due to the eddy currents induced on their surface.

(94) As noted above, an extremely important implication of this fact relates to safety considerations for human beings. Humans are also non-magnetic and can sustain strong magnetic fields without undergoing any risk. A typical example, where magnetic fields B˜1T are safely used on humans, is the Magnetic Resonance Imaging (MRI) technique for medical testing. In contrast, the magnetic near-field required in typical embodiments in order to provide a few Watts of power to devices is only B˜10.sup.−4T, which is actually comparable to the magnitude of the Earth's magnetic field. Since, as explained above, a strong electric near-field is also not present and the radiation produced from this non-radiative scheme is minimal, it is reasonable to expect that our proposed energy-transfer method should be safe for living organisms.

(95) One can, for example, estimate the degree to which the resonant system of a capacitively-loaded conducting-wire coil has mostly magnetic energy stored in the space surrounding it. If one ignores the fringing electric field from the capacitor, the electric and magnetic energy densities in the space surrounding the coil come just from the electric and magnetic field produced by the current in the wire; note that in the far field, these two energy densities must be equal, as is always the case for radiative fields. By using the results for the fields produced by a subwavelength (r«λ) current loop (magnetic dipole) with h=0, we can calculate the ratio of electric to magnetic energy densities, as a function of distance D.sub.p from the center of the loop (in the limit r««D.sub.p) and the angle θ with respect to the loop axis:

(96) $\begin{array}{cc}\frac{u_e\left(x\right)}{u_m\left(x\right)}=\frac{{\mathrm{.Math.}}_o{.Math.E\left(x\right).Math.}^2}{{\mu }_o{.Math.H\left(x\right).Math.}^2}=\frac{\left(1+\frac{1}{x^2}\right){\sin }^2\theta }{\left(\frac{1}{x^2}+\frac{1}{x^4}\right)4{\cos }^2\theta +\left(1-\frac{1}{x^2}+\frac{1}{x^4}\right){\sin }^2\theta };x=2\pi \frac{D_p}{\lambda }.Math.\frac{\underset{S_p}{∯}{u}_e\left(x\right)\mathrm{dS}}{\underset{S_p}{∯}{u}_m\left(x\right)\mathrm{dS}}=\frac{1+\frac{1}{x^2}}{1+\frac{1}{x^2}+\frac{3}{x^4}};x=2\pi \frac{D_p}{\lambda },& \left(12\right)\end{array}$
where the second line is the ratio of averages over all angles by integrating the electric and magnetic energy densities over the surface of a sphere of radius D.sub.p. From Eq. (12) it is obvious that indeed for all angles in the near field (x«1) the magnetic energy density is dominant, while in the far field (x»1) they are equal as they should be. Also, the preferred positioning of the loop is such that objects which may interfere with its resonance lie close to its axis (θ=0), where there is no electric field. For example, using the systems described in Table 4, we can estimate from Eq. (12) that for the loop of r=30 cm at a distance D.sub.p=10r=3m the ratio of average electric to average magnetic energy density would be ˜12% and at D.sub.p=3r=90 cm it would be ˜1%, and for the loop of r=10 cm at a distance D.sub.p=10r=1 m the ratio would be ˜33% and at D=3r=30 cm it would be ˜2.5%. At closer distances this ratio is even smaller and thus the energy is predominantly magnetic in the near field, while in the radiative far field, where they are necessarily of the same order (ratio.fwdarw.1), both are very small, because the fields have significantly decayed, as capacitively-loaded coil systems are designed to radiate very little. Therefore, this is the criterion that qualifies this class of resonant system as a magnetic resonant system.

(97) To provide an estimate of the effect of extraneous objects on the resonance of a capacitively-loaded loop including the capacitor fringing electric field, we use the perturbation theory formula, stated earlier, Γ.sub.1-e.sup.abs=ω.sub.1/4.Math.∫d.sup.3r Im {ε.sub.e (r)}|E.sub.1(r)|.sup.2/U with the computational FEFD results for the field of an example like the one shown in the plot of FIG. 5 and with a rectangular object of dimensions 30 cm×30 cm×1.5m and permittivity ε=49+16i (consistent with human muscles) residing between the loops and almost standing on top of one capacitor (˜3 cm away from it) and find Q.sub.c-h.sup.abs˜10.sup.5 and for ˜10 cm away Q.sub.c-h.sup.abs˜5.Math.10.sup.5. Thus, for ordinary distances (˜1m) and placements (not immediately on top of the capacitor) or for most ordinary extraneous objects e of much smaller loss-tangent, we conclude that it is indeed fair to say that Q.sub.c-e.sup.abs.fwdarw.∞. The only perturbation that is expected to affect these resonances is a close proximity of large metallic structures.

(98) Self-resonant coils are more sensitive than capacitively-loaded coils, since for the former the electric field extends over a much larger region in space (the entire coil) rather than for the latter (just inside the capacitor). On the other hand, self-resonant coils are simple to make and can withstand much larger voltages than most lumped capacitors.

(99) In general, different embodiments of resonant systems have different degree of sensitivity to external perturbations, and the resonant system of choice depends on the particular application at hand, and how important matters of sensitivity or safety are for that application. For example, for a medical implantable device (such as a wirelessly powered artificial heart) the electric field extent must be minimized to the highest degree possible to protect the tissue surrounding the device. In such cases where sensitivity to external objects or safety is important, one should design the resonant systems so that the ratio of electric to magnetic energy density u.sub.e/u.sub.m is reduced or minimized at most of the desired (according to the application) points in the surrounding space.

(100) In embodiments using resonances that are not primarily magnetic, the influence of extraneous objects may be of concern. For example, for dielectric disks, small, low-index, low-material-loss or far-away stray objects will induce small scattering and absorption. In such cases of small perturbations these extrinsic loss mechanisms can be quantified using respectively the analytical first-order perturbation theory formulas
Γ.sub.1-e.sup.rad=ω.sub.1∫d.sup.3r Re{ε.sub.e(r)}|E.sub.1(r).sup.2/U
and
Γ.sub.1-e.sup.abs=ω.sub.1/4.Math.∫d.sup.3r Im{ε.sub.e(r)}|E.sub.1(r)|.sup.2/U
where U=½∫d.sup.3rε(r)|E.sub.1(r)|.sup.2 is the total resonant electromagnetic energy of the unperturbed mode. As one can see, both of these losses depend on the square of the resonant electric field tails E.sub.1 at the site of the extraneous object. In contrast, the coupling rate from object 1 to another resonant object 2 is, as stated earlier,
κ=ω.sub.1/2.Math.∫d.sup.3rε.sub.2(r)E*.sub.2(r)E.sub.1(r)/∫d.sup.3rε(r)|E.sub.1(r).sup.2
and depends linearly on the field tails E.sub.1 of 1 inside 2. This difference in scaling gives us confidence that, for, for example, exponentially small field tails, coupling to other resonant objects should be much faster than all extrinsic loss rates (κ»Γ.sub.1-e), at least for small perturbations, and thus the energy-transfer scheme is expected to be sturdy for this class of resonant dielectric disks. However, we also want to examine certain possible situations where extraneous objects cause perturbations too strong to analyze using the above first-order perturbation theory approach. For example, we place a dielectric disk c close to another off-resonance object of large Re{ε}, Im{ε} and of same size but different shape (such as a human being h), as shown in FIG. 9a, and a roughened surface of large extent but of small Re{ε}, Im{ε} (such as a wall w), as shown in FIG. 9b. For distances D.sub.h/w/r=10.sup.−3 between the disk-center and the “human”-center or “wall”, the numerical FDFD simulation results presented in FIGS. 9a and 9b suggest that, the disk resonance seems to be fairly robust, since it is not detrimentally disturbed by the presence of extraneous objects, with the exception of the very close proximity of high-loss objects. To examine the influence of large perturbations on an entire energy-transfer system we consider two resonant disks in the close presence of both a “human” and a “wall”. Comparing Table 8 to the table in FIG. 9c, the numerical FDFD simulations show that the system performance deteriorates from κ/Γ.sub.c˜1−50 to κ[hw]/Γ.sub.c[hw] □0.5-10 i.e. only by acceptably small amounts.

(101) Inductively-loaded conducting rods may also be more sensitive than capacitively-loaded coils, since they rely on the electric field to achieve the coupling.

(102) System Efficiency

(103) In general, another important factor for any energy transfer scheme is the transfer efficiency. Consider again the combined system of a resonant source s and device d in the presence of a set of extraneous objects e. The efficiency of this resonance-based energy-transfer scheme may be determined, when energy is being drained from the device at rate Γ.sub.work for use into operational work. The coupled-mode-theory equation for the device field-amplitude is

(104) $\begin{array}{cc}\frac{d{a}_d}{dt}=-i\left(\omega -i{\Gamma }_{d\left[e\right]}\right)a_d+i{\kappa }_{\left[e\right]}{a}_s-{\Gamma }_{work}{a}_d,& \left(13\right)\end{array}$
where Γ.sub.d[e], Γ=.sub.d[e].sup.rad+Γ.sub.d[e].sup.abs=Γ.sub.d[e].sup.rad+(Γ.sub.d.sup.abs+Γ.sub.d-e.sup.abs) is the net perturbed-device loss rate, and similarly we define Γ.sub.s[e] for the perturbed-source. Different temporal schemes can be used to extract power from the device (e.g. steady-state continuous-wave drainage, instantaneous drainage at periodic times and so on) and their efficiencies exhibit different dependence on the combined system parameters. For simplicity, we assume steady state, such that the field amplitude inside the source is maintained constant, namely a.sub.s(t)=A.sub.se.sup.−iωt, so then the field amplitude inside the device is a.sub.d (t)=A.sub.de.sup.−iωt with A.sub.d/A.sub.s=iκ.sub.(e)/(Γ.sub.d[e]+Γ.sub.work). The various time-averaged powers of interest are then: the useful extracted power is P.sub.work=2Γ.sub.work|A.sub.d|.sup.2, the radiated (including scattered) power is P.sub.rad=2Γ.sub.s(e).sup.rad|A.sub.s|.sup.2+2Γ.sub.d(e).sup.rad|A.sub.d|.sup.2 the power absorbed at the source/device is P.sub.s/d=2Γ.sub.s/d.sup.abs|A.sub.s/d|.sup.2, and at the extraneous objects P.sub.e=2Γ.sub.s-e.sup.abs|A.sub.s|.sup.2+2Γ.sub.d-e.sup.abs|A.sub.d|.sup.2. From energy conservation, the total time-averaged power entering the system is P.sub.total=P.sub.work+P.sub.rad+P.sub.s+P.sub.d+P.sub.e. Note that the reactive powers, which are usually present in a system and circulate stored energy around it, cancel at resonance (which can be proven for example in electromagnetism from Poynting's Theorem) and do not influence the power-balance calculations. The working efficiency is then:

(105) 0$\begin{array}{cc}{\eta }_{work}\equiv \frac{P_{work}}{P_{total}}=\frac{1}{1+\frac{{\Gamma }_{d\left[e\right]}}{{\Gamma }_{work}}\left[1+\frac{1}{fo{m}_{\left[e\right]}^2}{\left(1+\frac{{\Gamma }_{work}}{{\Gamma }_{d\left[e\right]}}\right]}^2\right]},& \left(14\right)\end{array}$
where fom.sub.[e]=κ.sub.[e]/√{square root over (Γ.sub.s(e)Γ.sub.d(e))} is the distance-dependent figure-of-merit of the perturbed resonant energy-exchange system. To derive Eq. (14), we have assumed that the rate Γ.sub.supply, at which the power supply is feeding energy to the resonant source, is Γ.sub.supply=Γ.sub.s[e]+κ.sup.2(Γ.sub.d[e]+Γ.sub.work) such that there are zero reflections of the fed power P.sub.total back into the power supply.

(106) [Example: Capacitively-Loaded Conducting Loops]

(107) Referring to FIG. 10, to rederive and express this formula (14) in terms of the parameters which are more directly accessible from particular resonant objects, e.g. the capacitively-loaded conducting loops, one can consider the following circuit-model of the system, where the inductances L.sub.s, L.sub.d represent the source and device loops respectively, R.sub.s, R.sub.d their respective losses, and C.sub.d, C.sub.d are the required corresponding capacitances to achieve for both resonance at frequency ω. A voltage generator V.sub.g is considered to be connected to the source and a work (load) resistance R.sub.w to the device. The mutual inductance is denoted by M.

(108) Then from the source circuit at resonance (ωL.sub.s=1/ωC.sub.s):
V.sub.g=I.sub.sR.sub.s−jωMI.sub.d.Math.½V*.sub.gI.sub.s=½|I.sub.s|.sup.2R.sub.s+½jωMI*.sub.dI.sub.s,
and from the device circuit at resonance (ωL.sub.d=1/ωC.sub.d):
0=I.sub.d(R.sub.d+R.sub.w)−jωMI.sub.s.Math.jωMI.sub.s=I.sub.d(R.sub.d+R.sub.w)
So by substituting the second to the first:
½V*.sub.gI.sub.s½|I.sub.s|.sup.2R.sub.s+½|I.sub.d|.sup.2(R.sub.d+R.sub.w).
Now we take the real part (time-averaged powers) to find the efficiency:

(109) $P_g\equiv \mathrm{Re}\left\{\frac{1}{2}{V}_g^{*}{I}_s\right\}=P_s+P_d+P_w.Math.{\eta }_{work}\equiv \frac{P_w}{P_{tot}}=\frac{R_w}{{.Math.\frac{I_s}{I_d}.Math.}^2{R}_s+R_d+R_w}.\mathrm{Namely},{\eta }_{work}=\frac{R_w}{\frac{{\left(R_d+R_w\right)}^2}{{\left(\omega M\right)}^2}{R}_s+R_d+R_w},$
which with Γ.sub.work=R.sub.w/2L.sub.d, Γ.sub.d=R.sub.d/2L.sub.d, Γ.sub.s=R.sub.s/2 L.sub.s, and κ=ωM/2√{square root over (L.sub.sL.sub.d)}, becomes the general Eq. (14). [End of Example]

(110) From Eq. (14) one can find that the efficiency is optimized in terms of the chosen work-drainage rate, when this is chosen to be Γ.sub.work/Γ.sub.d[e]=Γ.sub.supply Γ.sub.s[e]=√{square root over (1+fom.sub.[e].sup.2)}1. Then, η.sub.work is a function of the fom.sub.[e] parameter only as shown in FIG. 11 with a solid black line. One can see that the efficiency of the system is η>17% for fom.sub.[e]>1, large enough for practical applications. Thus, the efficiency can be further increased towards 100% by optimizing fom.sub.[e] as described above. The ratio of conversion into radiation loss depends also on the other system parameters, and is plotted in FIG. 11 for the conducting loops with values for their parameters within the ranges determined earlier.

(111) For example, consider the capacitively-loaded coil embodiments described in Table 4, with coupling distance D/r=7,a “human” extraneous object at distance D.sub.h from the source, and that P.sub.work=10 W must be delivered to the load. Then, we have (based on FIG. 11) Q.sub.s[h].sup.rad=Q.sub.d[h].sup.rad˜10.sup.4, Q.sub.s.sup.abs=Q.sub.d.sup.abs˜10.sup.3, Q.sub.κ˜500, and Q.sub.d-h.sup.abs.fwdarw.∞, Q.sub.s-h.sup.abs˜10.sup.5 at D.sub.h˜3 cm and Q.sub.s-h.sup.abs˜5.Math.10.sup.5 at D.sub.h˜10 cm. Therefore fom.sub.[h]˜2, so we find η≈38%, P.sub.rad≈1.5 W, P.sub.s≈11 W, P.sub.d≈4 W, and most importantly η.sub.h≈0.4%, P.sub.h=0.1 W at D.sub.h˜3 cm and η.sub.h≈0.1%, P.sub.h=0.02 W at D.sub.h˜10 cm

(112) Overall System Performance

(113) In many cases, the dimensions of the resonant objects will be set by the particular application at hand. For example, when this application is powering a laptop or a cell-phone, the device resonant object cannot have dimensions larger that those of the laptop or cell-phone respectively. In particular, for a system of two loops of specified dimensions, in terms of loop radii r.sub.s,d and wire radii a.sub.s,d the independent parameters left to adjust for the system optimization are: the number of turns N.sub.s,d, the frequency f, the work-extraction rate (load resistance) Γ.sub.work and the power-supply feeding rate Γ.sub.supply.

(114) In general, in various embodiments, the primary dependent variable that one wants to increase or optimize is the overall efficiency η. However, other important variables need to be taken into consideration upon system design. For example, in embodiments featuring capacitively-loaded coils, the design may be constrained by, for example, the currents flowing inside the wires I.sub.s,d and the voltages across the capacitors V.sub.s,d. These limitations can be important because for ˜Watt power applications the values for these parameters can be too large for the wires or the capacitors respectively to handle. Furthermore, the total loaded Q.sub.tot=ωL.sub.d/(R.sub.d+R.sub.w) of the device is a quantity that should be preferably small, because to match the source and device resonant frequencies to within their Q's, when those are very large, can be challenging experimentally and more sensitive to slight variations. Lastly, the radiated powers P.sub.rad,s,d should be minimized for safety concerns, even though, in general, for a magnetic, non-radiative scheme they are already typically small.

(115) In the following, we examine then the effects of each one of the independent variables on the dependent ones. We define a new variable wp to express the work-drainage rate for some particular value of fom.sub.[e] through Γ.sub.work/Γ.sub.d[e]=√{square root over (1+wp.Math.fom.sub.[e].sup.2)}. Then, in some embodiments, values which impact the choice of this rate are: Γ.sub.work/Γ.sub.d[e]=1⇔wp=0 to minimize the required energy stored in the source (and therefore I.sub.s and V.sub.s), Γ.sub.work/Γ.sub.d[e]=√{square root over (1+fom.sub.[e].sup.2)}>1⇔wp=1 to increase the efficiency, as seen earlier, or Γ.sub.work/Γ.sub.d[e]»1⇔wp»1 to decrease the required energy stored in the device (and therefore I.sub.d and V.sub.d) and to decrease or minimize Q.sub.tot=ωL.sub.d/(R.sub.d+R.sub.w)=ω/[2(Γ.sub.d+Γ.sub.work)]. Similar is the impact of the choice of the power supply feeding rate Γ.sub.supply, with the roles of the source and the device reversed.

(116) Increasing N.sub.s and N.sub.d increases κ/√{square root over (Γ.sub.sΓ.sub.d)} and thus efficiency significantly, as seen before, and also decreases the currents I.sub.s and I.sub.d, because the inductance of the loops increases, and thus the energy U.sub.s,d=½L.sub.s,d|I.sub.s,d|.sup.2 required for given output power P.sub.work can be achieved with smaller currents. However, increasing N.sub.d increases Q.sub.tot, P.sub.rad,d and the voltage across the device capacitance V.sub.d, which unfortunately ends up being, in typical embodiments one of the greatest limiting factors of the system. To explain this, note that it is the electric field that really induces breakdown of the capacitor material (e.g. 3 kV/mm for air) and not the voltage, and that for the desired (close to the optimal) operational frequency, the increased inductance L.sub.d implies reduced required capacitance C.sub.d, which could be achieved in principle, for a capacitively-loaded device coil by increasing the spacing of the device capacitor plates d.sub.d and for a self-resonant coil by increasing through h.sub.d the spacing of adjacent turns, resulting in an electric field (≈V.sub.d/d.sub.d for the former case) that actually decreases with N.sub.d; however, one cannot in reality increase d.sub.d or h.sub.d too much, because then the undesired capacitance fringing electric fields would become very large and/or the size of the coil might become too large; and, in any case, for certain applications extremely high voltages are not desired. A similar increasing behavior is observed for the source p.sub.rad,s and V.sub.s upon increasing N.sub.s. As a conclusion, the number of turns N.sub.s and N.sub.d have to be chosen the largest possible (for efficiency) that allow for reasonable voltages, fringing electric fields and physical sizes.

(117) With respect to frequency, again, there is an optimal one for efficiency, and Q.sub.tot is approximately maximum, close to that optimal frequency. For lower frequencies the currents get worse (larger) but the voltages and radiated powers get better (smaller). Usually, one should pick either the optimal frequency or somewhat lower.

(118) One way to decide on an operating regime for the system is based on a graphical method. In FIG. 12, for two loops of r.sub.s=25 cm, r.sub.d=15 cm, h.sub.s=h.sub.d=0, a.sub.s=a.sub.d=3 mm and distance D=2m between them, we plot all the above dependent variables (currents, voltages and radiated powers normalized to 1 Watt of output power) in terms of frequency and N.sub.d, given some choice for wp and N.sub.s. The Figure depicts all of the dependencies explained above. We can also make a contour plot of the dependent variables as functions of both frequency and wp but for both N.sub.s and N.sub.d fixed. The results are shown in FIG. 13 for the same loop dimensions and distance. For example, a reasonable choice of parameters for the system of two loops with the dimensions given above are: N.sub.s=2, N.sub.d=6, f=10 MHz and wp=10, which gives the following performance characteristics: η.sub.work=20.6%, Q.sub.tot=1264, I.sub.s=7.2 A, I.sub.d=1.4 A, V.sub.s=2.55 kV, V.sub.d=2.30 kV, P.sub.rad,s=0.15 W, P.sub.rad,d=0.006 W. Note that the results in FIGS. 12 and 13, and the just above calculated performance characteristics are made using the analytical formulas provided above, so they are expected to be less accurate for large values of N.sub.s, N.sub.d, still they give a good estimate of the scalings and the orders of magnitude.

(119) Finally, one could additionally optimize for the source dimensions, since usually only the device dimensions are limited, as discussed earlier. Namely, one can add r.sub.s and a, in the set of independent variables and optimize with respect to these too for all the dependent variables of the problem (we saw how to do this only for efficiency earlier). Such an optimization would lead to improved results.

Experimental Results

(120) An experimental realization of an embodiment of the above described scheme for wireless energy transfer consists of two self-resonant coils of the type described above, one of which (the source coil) is coupled inductively to an oscillating circuit, and the second (the device coil) is coupled inductively to a resistive load, as shown schematically in FIG. 14. Referring to FIG. 14, A is a single copper loop of radius 25 cm that is part of the driving circuit, which outputs a sine wave with frequency 9.9 MHz. s and d are respectively the source and device coils referred to in the text. B is a loop of wire attached to the load (“light-bulb”). The various κ's represent direct couplings between the objects. The angle between coil d and the loop A is adjusted so that their direct coupling is zero, while coils s and d are aligned coaxially. The direct coupling between B and A and between B and s is negligible.

(121) The parameters for the two identical helical coils built for the experimental validation of the power transfer scheme were h=20 cm, a=3 mm, r=30 cm, N=5.25. Both coils are made of copper. Due to imperfections in the construction, the spacing between loops of the helix is not uniform, and we have encapsulated the uncertainty about their uniformity by attributing a 10% (2 cm) uncertainty to h. The expected resonant frequency given these dimensions is f.sub.o=10.56±0.3 MHz, which is about 5% off from the measured resonance at around 9.90 MHz.

(122) The theoretical Q for the loops is estimated to be ˜2500 (assuming perfect copper of resistivity ρ=1/σ=1.7×10.sup.−8 Ωm) but the measured value is 950±50. We believe the discrepancy is mostly due to the effect of the layer of poorly conducting copper oxide on the surface of the copper wire, to which the current is confined by the short skin depth (˜20 μm) at this frequency. We have therefore used the experimentally observed Q (and Γ.sub.1=Γ.sub.2=Γ=ω/(2Q) derived from it) in all subsequent computations.

(123) The coupling coefficient κ can be found experimentally by placing the two self-resonant coils (fine-tuned, by slightly adjusting h, to the same resonant frequency when isolated) a distance D apart and measuring the splitting in the frequencies of the two resonant modes in the transmission spectrum. According to coupled-mode theory, the splitting in the transmission spectrum should be Δω=2√{square root over (κ.sup.2−Γ.sup.2)}. The comparison between experimental and theoretical results as a function of distance when the two the coils are aligned coaxially is shown in FIG. 15.

(124) FIG. 16 shows a comparison of experimental and theoretical values for the parameter κ/Γ as a function of the separation between the two coils. The theory values are obtained by using the theoretically obtained κ and the experimentally measured Γ. The shaded area represents the spread in the theoretical κ/Γ due to the ˜5% uncertainty in Q.

(125) As noted above, the maximum theoretical efficiency depends only on the parameter κ/√{square root over (Γ.sub.1Γ.sub.2)}=κ/Γ, plotted as a function of distance in FIG. 17. The coupling to loss ratio κ/Γ is greater than 1 even for D=2.4 m (eight times the radius of the coils), thus the system is in the strongly-coupled regime throughout the entire range of distances probed.

(126) The power supply circuit was a standard Colpitts oscillator coupled inductively to the source coil by means of a single loop of copper wire 25 cm in radius (see FIG. 14). The load consisted of a previously calibrated light-bulb, and was attached to its own loop of insulated wire, which was in turn placed in proximity of the device coil and inductively coupled to it. Thus, by varying the distance between the light-bulb and the device coil, the parameter Γ.sub.work/Γ was adjusted so that it matched its optimal value, given theoretically by √{square root over (1+κ.sup.2/(Γ.sub.1Γ.sub.2))}. Because of its inductive nature, the loop connected to the light-bulb added a small reactive component to Γ.sub.work which was compensated for by slightly retuning the coil. The work extracted was determined by adjusting the power going into the Colpitts oscillator until the light-bulb at the load was at its full nominal brightness.

(127) In order to isolate the efficiency of the transfer taking place specifically between the source coil and the load, we measured the current at the mid-point of each of the self-resonant coils with a current-probe (which was not found to lower the Q of the coils noticeably.) This gave a measurement of the current parameters I.sub.1 and I.sub.2 defined above. The power dissipated in each coil was then computed from P.sub.1,2=ΓL|I.sub.1,2|.sup.2, and the efficiency was directly obtained from η=P.sub.wor/(P.sub.1+P.sub.2±P.sub.work). To ensure that the experimental setup was well described by a two-object coupled-mode theory model, we positioned the device coil such that its direct coupling to the copper loop attached to the Colpitts oscillator was zero. The experimental results are shown in FIG. 17, along with the theoretical prediction for maximum efficiency, given by Eq. (14).

(128) Using this embodiment, we were able to transfer significant amounts of power using this setup, fully lighting up a 60 W light-bulb from distances more than 2m away, for example. As an additional test, we also measured the total power going into the driving circuit. The efficiency of the wireless transfer itself was hard to estimate in this way, however, as the efficiency of the Colpitts oscillator itself is not precisely known, although it is expected to be far from 100%. Nevertheless, this gave an overly conservative lower bound on the efficiency. When transferring 60 W to the load over a distance of 2m, for example, the power flowing into the driving circuit was 400 W. This yields an overall wall-to-load efficiency of ˜15%, which is reasonable given the expected ˜40% efficiency for the wireless power transfer at that distance and the low efficiency of the driving circuit.

(129) From the theoretical treatment above, we see that in typical embodiments it is important that the coils be on resonance for the power transfer to be practical. We found experimentally that the power transmitted to the load dropped sharply as one of the coils was detuned from resonance. For a fractional detuning Δf/f.sub.0 of a few times the inverse loaded Q, the induced current in the device coil was indistinguishable from noise.

(130) The power transfer was not found to be visibly affected as humans and various everyday objects, such as metallic and wooden furniture, as well as electronic devices large and small, were placed between the two coils, even when they drastically obstructed the line of sight between source and device. External objects were found to have an effect only when they were closer than 10 cm from either one of the coils. While some materials (such as aluminum foil, styrofoam and humans) mostly just shifted the resonant frequency, which could in principle be easily corrected with a feedback circuit of the type described earlier, others (cardboard, wood, and PVC) lowered Q when placed closer than a few centimeters from the coil, thereby lowering the efficiency of the transfer.

(131) We believe that this method of power transfer should be safe for humans. When transferring 60 W (more than enough to power a laptop computer) across 2m, we estimated that the magnitude of the magnetic field generated is much weaker than the Earth's magnetic field for all distances except for less than about 1 cm away from the wires in the coil, an indication of the safety of the scheme even after long-term use. The power radiated for these parameters was ˜5 W, which is roughly an order of magnitude higher than cell phones but could be drastically reduced, as discussed below.

(132) Although the two coils are currently of identical dimensions, it is possible to make the device coil small enough to fit into portable devices without decreasing the efficiency. One could, for instance, maintain the product of the characteristic sizes of the source and device coils constant.

(133) These experiments demonstrated experimentally a system for power transfer over medium range distances, and found that the experimental results match theory well in multiple independent and mutually consistent tests.

(134) We believe that the efficiency of the scheme and the distances covered could be appreciably improved by silver-plating the coils, which should increase their Q, or by working with more elaborate geometries for the resonant objects. Nevertheless, the performance characteristics of the system presented here are already at levels where they could be useful in practical applications.

(135) Applications

(136) In conclusion, we have described several embodiments of a resonance-based scheme for wireless non-radiative energy transfer. Although our consideration has been for a static geometry (namely κ and Γ.sub.e were independent of time), all the results can be applied directly for the dynamic geometries of mobile objects, since the energy-transfer time κ.sup.−1 (˜1 μs−1 ms for microwave applications) is much shorter than any timescale associated with motions of macroscopic objects. Analyses of very simple implementation geometries provide encouraging performance characteristics and further improvement is expected with serious design optimization. Thus the proposed mechanism is promising for many modern applications.

(137) For example, in the macroscopic world, this scheme could potentially be used to deliver power to for example, robots and/or computers in a factory room, or electric buses on a highway. In some embodiments source-object could be an elongated “pipe” running above the highway, or along the ceiling.

(138) Some embodiments of the wireless transfer scheme can provide energy to power or charge devices that are difficult or impossible to reach using wires or other techniques. For example some embodiments may provide power to implanted medical devices (e.g. artificial hearts, pacemakers, medicine delivery pumps, etc.) or buried underground sensors.

(139) In the microscopic world, where much smaller wavelengths would be used and smaller powers are needed, one could use it to implement optical inter-connects for CMOS electronics, or to transfer energy to autonomous nano-objects (e.g. MEMS or nano-robots) without worrying much about the relative alignment between the sources and the devices. Furthermore, the range of applicability could be extended to acoustic systems, where the source and device are connected via a common condensed-matter object.

(140) In some embodiments, the techniques described above can provide non-radiative wireless transfer of information using the localized near fields of resonant object. Such schemes provide increased security because no information is radiated into the far-field, and are well suited for mid-range communication of highly sensitive information.

(141) A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention.