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1、Wireless Power Transfer via Strongly Coupled Magnetic ResonancesAndr Kurs,1* Aristeidis Karalis,2 Robert Moffatt,1 J. D. Joannopoulos,1 Peter Fisher,3 Marin Soljai11Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. 2Department of Electrical Engineering and Compu
2、ter Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. 3Department of Physics and Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.*To whom correspondence should be addressed. E-mail: / www.sciencexpress.org / 7 June 2007 / Page 4
3、 / 10.1126/science.1143254Using self-resonant coils in a strongly coupled regime, we experimentally demonstrate efficient non-radiative power transfer over distances of up to eight times the radius of the coils. We demonstrate the ability to transfer 60W with approximately 40% efficiency over distan
4、ces in excess of two meters. We present a quantitative model describing the power transfer which matches the experimental results to within 5%. We discuss practical applicability and suggest directions for further studies.At first glance, such power transfer is reminiscent of the usual magnetic indu
5、ction (10); however, note that the usual non- resonant induction is very inefficient for mid-range applications.Overview of the formalism. Efficient mid-range power transfer occurs in particular regions of the parameter space describing resonant objects strongly coupled to one another. Using coupled
6、-mode theory to describe this physical system (11), we obtain the following set of linear equationsIn the early 20th century, before the electrical-wire grid, Nikola Tesla (1) devoted much effort towards schemes toa& m(t) = (im - Gm )am (t) + imnan (t) + Fm (t) nm (1)transport power wirelessly. Howe
7、ver, typical embodiments (e.g. Tesla coils) involved undesirably large electric fields. During the past decade, society has witnessed a dramatic surge of use of autonomous electronic devices (laptops, cell- phones, robots, PDAs, etc.) As a consequence, interest in wireless power has re-emerged (24).
8、 Radiative transfer (5), while perfectly suitable for transferring information, poses a number of difficulties for power transfer applications: the efficiency of power transfer is very low if the radiation is omnidirectional, and requires an uninterrupted line of sight and sophisticated tracking mec
9、hanisms if radiation is unidirectional. A recent theoretical paper (6) presented a detailed analysis of the feasibility of using resonant objects coupled through the tails of their non-radiative fields for mid- range energy transfer (7). Intuitively, two resonant objects of the same resonant frequen
10、cy tend to exchange energy efficiently, while interacting weakly with extraneous off- resonant objects. In systems of coupled resonances (e.g. acoustic, electro-magnetic, magnetic, nuclear, etc.), there is often a general “strongly coupled” regime of operation (8). If one can operate in that regime
11、in a given system, the energy transfer is expected to be very efficient. Mid-range power transfer implemented this way can be nearly omnidirectional and efficient, irrespective of the geometry of the surrounding space, and with low interference and losses into environmental objects (6).Consideration
12、s above apply irrespective of the physicalnature of the resonances. In the current work, we focus on one particular physical embodiment: magnetic resonances (9). Magnetic resonances are particularly suitable for everyday applications because most of the common materials do not interact with magnetic
13、 fields, so interactions with environmental objects are suppressed even further. We were able to identify the strongly coupled regime in the system of two coupled magnetic resonances, by exploring non-radiative (near-field) magnetic resonant induction at MHz frequencies.where the indices denote the
14、different resonant objects. The variables am(t) are defined so that the energy contained in object m is |am(t)|2, wm is the resonant frequency of that isolated object, and Gm is its intrinsic decay rate (e.g. due to absorption and radiated losses), so that in this framework an uncoupled and undriven
15、 oscillator with parameters w0 and G0 would evolve in time as exp(iw0t G0t). The kmn = knm are coupling coefficients between the resonant objects indicated by the subscripts, and Fm(t) are driving terms.We limit the treatment to the case of two objects, denoted by source and device, such that the so
16、urce (identified by the subscript S) is driven externally at a constant frequency, and the two objects have a coupling coefficient k. Work is extracted from the device (subscript D) by means of a load (subscript W) which acts as a circuit resistance connected to the device, and has the effect of con
17、tributing an additional term GW to the unloaded device objects decay rate GD. The overall decay rate at the device is therefore GD = GD + GW. The work extracted is determined by the power dissipated in the load, i.e. 2GW|aD(t)|2. Maximizing the efficiency h of the transfer with respect to the loadin
18、g GW, given Eq. 1, is equivalent to solving an impedance matching problem. One finds that the scheme works best when the source and the device are resonant, in which case the efficiency isThe efficiency is maximized when GW/GD = (1 + k2/GSGD)1/2. It is easy to show that the key to efficient energy t
19、ransfer is to have k2/GSGD 1. This is commonly referred to as the strong coupling regime. Resonance plays an essential role in thispower transfer mechanism, as the efficiency is improved by approximately w2/GD2 (106 for typical parameters) compared to the case of inductively coupled non-resonant obj
20、ects.Theoretical model for self-resonant coils. Our experimental realization of the scheme consists of two self- resonant coils, one of which (the source coil) is coupled inductively to an oscillating circuit, while the other (the device coil) is coupled inductively to a resistive load (12) (Fig. 1)
21、. Self-resonant coils rely on the interplay between distributed inductance and distributed capacitance to achieve resonance. The coils are made of an electrically conducting wire of total length l and cross-sectional radius a wound intoGiven this relation and the equation of continuity, one finds th
22、at the resonant frequency is f0 = 1/2p(LC)1/2. We can now treat this coil as a standard oscillator in coupled-mode theory by defining a(t) = (L/2)1/2I0(t).We can estimate the power dissipated by noting that the sinusoidal profile of the current distribution implies that the spatial average of the pe
23、ak current-squared is |I0|2/2. For a coil with n turns and made of a material with conductivity s, we modify the standard formulas for ohmic (Ro) and radiation (Rr) resistance accordingly:0la helix of n turns, radius r, and height h. To the best of our knowledge, there is no exact solution for a fin
24、ite helix in the literature, and even in the case of infinitely long coils, the solutions rely on assumptions that are inadequate for ourRo =2 4a r 42 h 2 (6)system (13). We have found, however, that the simple quasi-R =0 n2+(7)static model described below is in good agreementr 12 c 33 c (approximat
25、ely 5%) with experiment.We start by observing that the current has to be zero at the ends of the coil, and make the educated guess that the resonant modes of the coil are well approximated by sinusoidal current profiles along the length of the conducting wire. We are interested in the lowest mode, s
26、o if we denote by s the parameterization coordinate along the length of the conductor, such that it runs from -l/2 to +l/2, then the time- dependent current profile has the form I0 cos(ps/l) exp(iwt). It follows from the continuity equation for charge that the linear charge density profile is of the
27、 form l0 sin(ps/l) exp(iwt), so the two halves of the coil (when sliced perpendicularly to its axis) contain charges equal in magnitude q0 = l0l/p but opposite in sign.As the coil is resonant, the current and charge density profiles are p/2 out of phase from each other, meaning that the real part of
28、 one is maximum when the real part of the other is zero. Equivalently, the energy contained in the coil is0 The first term in Eq. 7 is a magnetic dipole radiation term (assuming r 2pc/w); the second term is due to the electric dipole of the coil, and is smaller than the first term for our experiment
29、al parameters. The coupled-mode theory decay constant for the coil is therefore G = (Ro + Rr)/2L, and its quality factor is Q = w/2G.We find the coupling coefficient kDS by looking at the power transferred from the source to the device coil, assuming a steady-state solution in which currents and cha
30、rge densities vary in time as exp(iwt).DSSDP= drE (r) J (r)=- dr (A& S(r) + fS (r) J D (r)at certain points in time completely due to the current, and at other points, completely due to the charge. Using electromagnetic theory, we can define an effective inductance L and an effective capacitance C f
31、or each coil as follows:=- 1 drdr J& S(r) + S (r) 4| r - r |0 0 -iMISIDr - r | r - r |3 J D (r )(8)2L =04 | I0 | drdr J(r) J(r)| r - r |where the subscript S indicates that the electric field is due to the source. We then conclude from standard coupled-mode theory arguments that kDS = kSD = k = wM/2
32、(LSLD)1/2. When11(r)(r)the distance D between the centers of the coils is much larger2=C40 | q0 | drdr | r - r |(4)than their characteristic size, k scales with the D-3 dependence characteristic of dipole-dipole coupling. Both k and G are functions of the frequency, and k/G and thewhere the spatial
33、current J(r) and charge density r(r) areobtained respectively from the current and charge densities along the isolated coil, in conjunction with the geometry of the object. As defined, L and C have the property that the energy U contained in the coil is given byefficiency are maximized for a particu
34、lar value of f, which is in the range 1-50MHz for typical parameters of interest.Thus, picking an appropriate frequency for a given coil size, as we do in this experimental demonstration, plays a major role in optimizing the power transfer.12Comparison with experimentally determinedU = 2 L | I0 |par
35、ameters. The parameters for the two identical helical coils built for the experimental validation of the power12transfer scheme are h = 20cm, a = 3mm, r = 30 cm, and n = 2C | q0 |(5)5.25. Both coils are made of copper. The spacing between loops of the helix is not uniform, and we encapsulate the unc
36、ertainty about their uniformity by attributing a 10% (2cm) uncertainty to h. The expected resonant frequency given thesedimensions is f0 = 10.56 0.3MHz, which is about 5% off from the measured resonance at 9.90MHz.The theoretical Q for the loops is estimated to be approximately 2500 (assuming s = 5.
37、9 107 m/W) but the measured value is Q = 95050. 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 (20mm) at this frequency. We therefore use the experimenta
38、lly observed Q and GS = GD = G = w/2Q derived from it in all subsequent computations.We find the coupling coefficient k 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 split
39、ting in the frequencies of the two resonant modes. According to coupled-mode theory, this splitting should be Dw = 2(k2 -G2)1/2. In the present work, we focus on the case where the two coils are aligned coaxially (Fig. 2), although similar results are obtained for other orientations (figs. S1 and S2
40、).Measurement of the efficiency. The maximum theoretical efficiency depends only on the parameter k/(LSLD)1/2 = k/G, which is greater than 1 even for D = 2.4m (eight times the radius of the coils) (Fig. 3), thus we operate in the strongly- coupled regime throughout the entire range of distances prob
41、ed.As our driving circuit, we use a standard Colpitts oscillator whose inductive element consists of a single loop of copper wire 25cm in radius(Fig. 1); this loop of wire couples inductively to the source coil and drives the entire wireless power transfer apparatus. The load consists of a calibrate
42、d light-bulb (14), and is attached to its own loop of insulated wire, which is placed in proximity of the device coil and inductively coupled to it. By varying the distance between the light-bulb and the device coil, we are able to adjust the parameter GW/G so that it matches its optimal value, give
43、n theoretically by (1 + k2/G2)1/2. (The loop connected to the light-bulb adds a small reactive component to GW which is compensated for by slightly retuning the coil.) We measure the work extracted by adjusting the power going into the Colpitts oscillator until the light-bulb at the load glows at it
44、s full nominal brightness.We determine the efficiency of the transfer taking place between the source coil and the load by measuring the current at the mid-point of each of the self-resonant coils with a current-probe (which does not lower the Q of the coils noticeably.) This gives a measurement of
45、the current parameters IS and ID used in our theoretical model. We then compute the power dissipated in each coil from PS,D =GL|IS,D|2, and obtain the efficiency from h = PW/(PS + PD +PW). To ensure that the experimental setup is well described by a two-object coupled-mode theory model, we position
46、the device coil such that its direct coupling to the copper loop attached to the Colpitts oscillator is zero. The experimental results are shown in Fig. 4, along with the theoretical prediction for maximum efficiency, given by Eq. 2. We are able to transfer significant amounts of power using this se
47、tup, fully lighting up a 60W light-bulb from distances more than 2m away (figs. S3 and S4).As a cross-check, we also measure the total power going from the wall power outlet into the driving circuit. The efficiency of the wireless transfer itself is hard to estimate inthis way, however, as the effic
48、iency of the Colpitts oscillator itself is not precisely known, although it is expected to be far from 100% (15). Still, the ratio of power extracted to power entering the driving circuit gives a lower bound on the efficiency. When transferring 60W to the load over a distance of 2m, for example, the power flowing into the driving circuit is 400W. This yields an overall wall-to-load efficiency of 15%, which is reasonable given the expe