Wireless Power Transfer via StronglyCoupled Magnetic ResonancesAndr Kurs,1* Aristeidis Karalis,2 Robert Moffatt,1 J. D. Joannopoulos,1Peter Fisher,3 Marin Soljacic 1

Using self-resonant coils in a strongly coupled regime, we experimentally demonstrated efficientnonradiative power transfer over distances up to 8 times the radius of the coils. We were ableto transfer 60 watts with ~40% efficiency over distances in excess of 2 meters. We present aquantitative model describing the power transfer, which matches the experimental results to within5%. We discuss the practical applicability of this system and suggest directions for further study.

In the early 20th century, before the electrical-wire grid, Nikola Tesla (1) devoted mucheffort toward schemes to transport powerwire-lessly. However, typical embodiments (e.g., Teslacoils) involved undesirably large electric fields.The past decade has witnessed a surge in the useof autonomous electronic devices (laptops, cellphones, robots, PDAs, etc.). As a consequence,interest in wireless power has reemerged (24).Radiative transfer (5), although perfectly suitablefor transferring information, poses a number ofdifficulties for power transfer applications: Theefficiency of power transfer is very low if theradiation is omnidirectional, and unidirectionalradiation requires an uninterrupted line of sightand sophisticated tracking mechanisms. A recenttheoretical paper (6) presented a detailed analysisof the feasibility of using resonant objects cou-pled through the tails of their nonradiative fieldsfor midrange energy transfer (7). Intuitively, tworesonant objects of the same resonant frequencytend to exchange energy efficiently, while dis-sipating relatively little energy in extraneous off-resonant objects. In systems of coupled resonances(e.g., acoustic, electromagnetic, magnetic, nucle-ar), there is often a general strongly coupledregime of operation (8). If one can operate in thatregime in a given system, the energy transfer isexpected to be very efficient. Midrange powertransfer implemented in this way can be nearlyomnidirectional and efficient, irrespective of thegeometry of the surrounding space, with low in-terference and losses into environmental objects (6).

The above considerations apply irrespectiveof the physical nature of the resonances. Here,we focus on one particular physical embodiment:magnetic resonances (9). Magnetic resonancesare particularly suitable for everyday applica-tions because most of the common materials do

not interact with magnetic fields, so interactionswith environmental objects are suppressed evenfurther. We were able to identify the stronglycoupled regime in the system of two coupledmagnetic resonances by exploring nonradiative(near-field) magnetic resonant induction atmegahertz frequencies. At first glance, suchpower transfer is reminiscent of the usual mag-netic induction (10); however, note that theusual nonresonant induction is very inefficientfor midrange applications.

Overview of the formalism. Efficient mid-range power transfer occurs in particular regionsof the parameter space describing resonantobjects strongly coupled to one another. Usingcoupled-mode theory to describe this physicalsystem (11), we obtain the following set oflinear equations:

a: mt iwm Gmamt nm

ikmnant Fmt 1

where the indices denote the different resonantobjects. The variables am(t) are defined so thatthe energy contained in object m is |am(t)|

2, wmis the resonant angular frequency of that isolatedobject, and Gm is its intrinsic decay rate (e.g.,due to absorption and radiated losses). In thisframework, an uncoupled and undriven oscilla-tor with parameters w0 and G0 would evolve intime as exp(iw0t G0t). The kmn = knm arecoupling coefficients between the resonantobjects indicated by the subscripts, and Fm(t) aredriving terms.

We limit the treatment to the case of twoobjects, denoted by source and device, such thatthe source (identified by the subscript S) is driv-en externally at a constant frequency, and thetwo objects have a coupling coefficient k. Workis extracted from the device (subscript D) bymeans of a load (subscript W) that acts as acircuit resistance connected to the device, andhas the effect of contributing an additional termGW to the unloaded device objects decay rateGD. The overall decay rate at the device is there-fore GD = GD + GW. The work extracted is de-termined by the power dissipated in the load, that

is, 2GW|aD(t)|2. Maximizing the efficiency h of

the transfer with respect to the loading GW, givenEq. 1, is equivalent to solving an impedance-matching problem. One finds that the schemeworks best when the source and the device areresonant, in which case the efficiency is

h GWjaDj2

GSjaSj2 GD GWjaDj2

GWGD

k2GSGD

1 GWGD

k2GSGD

h i 1 GWGD

2h i 2

The efficiency is maximized when GW/GD = [1 +(k2/GSGD)]

1/2. It is easy to show that the key toefficient energy transfer is to have k2/GSGD > 1.This is commonly referred to as the strong coupl-ing regime. Resonance plays an essential role inthis power transfer mechanism, as the efficiency isimproved by approximately w2/GD

2 (~106 fortypical parameters) relative to the case of induc-tively coupled nonresonant objects.

Theoretical model for self-resonant coils.Our experimental realization of the scheme con-sists of two self-resonant coils. One coil (thesource coil) is coupled inductively to an oscillat-ing circuit; the other (the device coil) is coupledinductively to a resistive load (12) (Fig. 1). Self-resonant coils rely on the interplay betweendistributed inductance and distributed capaci-tance to achieve resonance. The coils are madeof an electrically conducting wire of total lengthl and cross-sectional radius a wound into a helixof n turns, radius r, and height h. To the best ofour knowledge, there is no exact solution for afinite helix in the literature, and even in the caseof infinitely long coils, the solutions rely onassumptions that are inadequate for our system(13). We have found, however, that the simplequasi-static model described below is in goodagreement (within ~5%) with experiment.

We start by observing that the current mustbe zero at the ends of the coil, and we make theeducated guess that the resonant modes of thecoil are well approximated by sinusoidal currentprofiles along the length of the conducting wire.We are interested in the lowest mode, so if wedenote by s the parameterization coordinatealong the length of the conductor, such that itruns from l/2 to +l/2, then the time-dependentcurrent profile has the form I0 cos(ps/l) exp(iwt).It follows from the continuity equation forcharge that the linear charge density profile isof the form l0 sin(ps/l) exp(iwt), so that one-halfof the coil (when sliced perpendicularly to itsaxis) contains an oscillating total charge (ofamplitude q0 = l0l/p) that is equal in magnitudebut opposite in sign to the charge in the otherhalf.

As the coil is resonant, the current and chargedensity profiles are p/2 out of phase from each

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1Department of Physics, Massachusetts Institute of Technol-ogy, Cambridge, MA 02139, USA. 2Department of ElectricalEngineering and Computer Science, Massachusetts Institute ofTechnology, Cambridge, MA 02139, USA. 3Department ofPhysics and Laboratory for Nuclear Science, MassachusettsInstitute of Technology, Cambridge, MA 02139, USA.

*To whom correspondence should be addressed. E-mail:akurs@mit.edu

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other, meaning that the real part of one is maxi-mum when the real part of the other is zero.Equivalently, the energy contained in the coilis at certain points in time completely due tothe current, and at other points it is completelydue to the charge. Using electromagnetictheory, we can define an effective inductanceL and an effective capacitance C for each coilas follows:

L m04pjI0j2

ZZdrdr

Jr Jrjr rj 3

1

C 1

4pe0jq0j2ZZ

drdrrrrrjr rj 4

where the spatial current J(r) and charge densityr(r) are obtained respectively from the currentand charge densities along the isolated coil, inconjunction with the geometry of the object. Asdefined, L and C have the property that theenergy U contained in the coil is given by

U 12LjI0j2

12C

jq0j2 5

Given this relation and the equation of con-tinuity, the resulting resonant frequency is f0 =1/[2p(LC)1/2]. We can now treat this coil as astandard oscillator in coupled-mode theory bydefining a(t) = [(L/2)1/2]I0(t).

We can estimate the power dissipated bynoting that the sinusoidal profile of the currentdistribution implies that the spatial average ofthe peak current squared is |I0|

2/2. For a coilwith n turns and made of a material with con-ductivity s, we modify the standard formulasfor ohmic (Ro) and radiation (Rr) resistanceaccordingly:

Ro m0w2s

rl

4pa6

Rr m0e0

rp12

n2wrc

4 23p3

whc

2 7

The first term in Eq. 7 is a magnetic dipoleradiation term (assuming r

light bulb adds a small reactive component toGW, which is compensated for by slightly retun-ing the coil.) We measure the work extracted byadjusting the power going into the Colpittsoscillator until the light bulb at the load glows atits full nominal brightness.

We determine the efficiency of the transfertaking place between the source coil and theload by measuring the current at the midpoint ofeach of the self-resonant coils with a currentprobe (which does not lower the Q of the coilsnoticeably). This gives a measurement of thecurrent parameters IS and ID used in our the-oretical model. We then compute the powerdissipated in each coil from PS,D = GL|IS,D|

2,and obtain the efficiency from h = PW/(PS +PD + PW). To ensure that the experimentalsetup is well described by a two-object cou-pled mode theory model, we position the de-vice coil such that its direct coupling to thecopper loop attached to the Colpitts oscillator iszero. The experimental results are shown inFig. 4, along with the theoretical prediction formaximum efficiency, given by Eq. 2. We wereable to transfer several tens of watts with theuse of this setup, fully lighting up a 60-Wlight bulb from distances more than 2 m away(figs. S3 and S4).

As a cross-check, we also measured the totalpower going from the wall power outlet into thedriving circuit. The efficiency of the wirelesstransfer itself is hard to estimate in this way,however, as the efficiency of the Colpitts os-cillator itself is not precisely known, although itis expected to be far from 100% (15). Still, theratio of power extracted to power entering thedriving circuit gives a lower bound on the ef-ficiency. When transferring 60 W to the loadover a distance of 2 m, for example, the powerflowing into the driving circuit is 400 W. Thisyields an overall wall-to-load efficiency of 15%,which is reasonable given the expected efficien-cy of 40 to 50% for the wireless power trans-

fer at that distance and the low efficiency ofthe Colpitts oscillator.

Concluding remarks. It is essential that thecoils be on resonance for the power transfer to bepractical (6). We find experimentally that thepower transmitted to the load drops sharply aseither one of the coils is detuned from resonance.For a fractional detuning Df/f0 of a few times theinverse loadedQ, the induced current in the devicecoil is indistinguishable from noise.

A detailed and quantitative analysis of theeffect of external objects on our scheme is be-yond the scope of this work, but we note herethat the power transfer is not visibly affected ashumans and various everyday objects, such asmetals, wood, and electronic devices large andsmall, are placed between the two coilsevenin cases where they completely obstruct theline of sight between source and device (figs.S3 to S5). External objects have a noticeableeffect only when they are within a few cen-timeters from either one of the coils. Somematerials (such as aluminum foil, Styrofoam,and humans) mostly just shift the resonant fre-quency, which can in principle be easily cor-rected with a feedback circuit; other materials(cardboard, wood, and polyvinyl chloride) lowerQ when placed closer than a few centimetersfrom the coil, thereby lowering the efficiency ofthe transfer.

When transferring 60 W across 2 m, we cal-culate that at the point halfway between thecoils, the root mean square (RMS) magnitude ofthe electric field is ERMS = 210 V/m, that of themagnetic field is HRMS = 1 A/m, and that of thePoynting vector is SRMS = 3.2 mW/cm

2 (16).These values increase closer to the coils, wherethe fields at source and device are comparable.For example, at distances 20 cm away from the

surface of the device coil, we calculate the maxi-mum values for the fields to be ERMS = 1.4 kV/m,HRMS = 8 A/m, and SRMS = 0.2 W/cm

2. Thepower radiated for these parameters is ~5 W,which is roughly an order of magnitude higherthan cell phones. In the particular geometry thatwe studied, the overwhelming contribution (byone to two orders of magnitude) to the electricnear-field, and hence to the near-field Poyntingvector, comes from the electric dipole momentof the coils. If instead one uses a capacitivelyloaded single-turn loop design (6)which hasthe advantage of confining nearly all of theelectric field inside the capacitorand tailorsthe system to operate at lower frequencies, ourcalculations show (17) that it should be pos-sible to reduce the values cited above for theelectric and magnetic fields, the Poynting vector,and the power radiated so that they fall belowthresholds specified by general safety regulations[e.g., the IEEE safety standards for general publicexposure (18)].

Although the two coils are currently of iden-tical dimensions, it is possible to make the devicecoil small enough to fit into portable deviceswithout decreasing the efficiency. One could, forinstance, maintain the product of the charac-teristic sizes of the source and device coils con-stant, as argued in (6).

We believe that the efficiency of the schemeand the power transfer distances could be appre-ciably improved by silver-plating the coils, whichshould increase theirQ, or by working with moreelaborate geometries for the resonant objects (19).Nonetheless, the performance characteristics of thesystem presented here are already at levels wherethey could be useful in practical applications.

References and Notes1. N. Tesla, U.S. patent 1,119,732 (1914).2. J. M. Fernandez, J. A. Borras, U.S. patent 6,184,651 (2001).3. A. Esser, H.-C. Skudelny, IEEE Trans. Ind. Appl. 27, 872

(1991).4. J. Hirai, T.-W. Kim, A. Kawamura, IEEE Trans. Power

Electron. 15, 21 (2000).5. T. A. Vanderelli, J. G. Shearer, J. R. Shearer, U.S. patent

7,027,311 (2006).6. A. Karalis, J. D. Joannopoulos, M. Soljai, Ann. Phys.,

10.1016/j.aop.2007.04.017 (2007).7. Here, by midrange, we mean that the sizes of the devices

that participate in the power transfer are smaller thanthe distance between devices by a factor of at least 2 to3. For example, if the device being powered is a laptop(size ~50 cm) and the power source (size ~50 cm) is inthe same room as the laptop, the distance of powertransfer could be within a room or a factory pavilion (sizeon the order of a few meters).

8. T. Aoki et al., Nature 443, 671 (2006).9. K. OBrien, G. Scheible, H. Gueldner, in IECON 03, 29th

Annual Conference of the IEEE (http://ieeexplore.ieee.org/xpl/RecentCon.jsp?punumber=9011) (2003).

10. L. Ka-Lai, J. W. Hay, P. G. W. Beart, U.S. patent 7,042,196(2006).

11. H. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

12. The couplings to the driving circuit and the load do nothave to be inductive. They may also be connected by awire, for example. We have chosen inductive coupling inthe present work because of its easier implementation.

13. S. Sensiper, thesis, Massachusetts Institute of Technology(1951).

75 100 125 150 175 200 2250

10

20

30

40

50

60

Distance (cm)

/

TheoryExperiment

Fig. 3. Comparison of experimental and theoret-ical values for the parameter k/G as a function ofthe wireless power transfer distance. The theoryvalues are obtained by using the theoretical k andthe experimentally measured G. The shaded arearepresents the spread in the theoretical k/G due tothe 5% uncertainty in Q.

75 100 125 150 175 200 22500.10.20.30.40.50.60.70.80.9

1

Distance (cm)

(E

fficien

cy)

TheoryFrom experimental Experiment

Fig. 4. Comparison of experimental and the-oretical efficiencies as functions of the wirelesspower transfer distance. The shaded area rep-resents the theoretical prediction for maximumefficiency and is obtained by inserting the the-oretical values from Fig. 3 into Eq. 2, with GW/GD =[1 + (k2/G2)]1/2. The black squares are the max-imum efficiency obtained from Eq. 2 and theexperimental values of k/G from Fig. 3. The reddots present the directly measured efficiency, asdescribed in the text.

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14. We experimented with various power ratings from 5 to75 W.

15. W. A. Edson, Vacuum-Tube Oscillators (Wiley, New York,1953).

16. Note that E cm0H, and that the fields are out of phaseand not necessarily perpendicular because we are not ina radiative regime.

17. See supporting material on Science Online.18. IEEE Std C95.12005 IEEE Standard for Safety Levels

with Respect to Human Exposure to Radio Frequency

Electromagnetic Fields, 3 kHz to 300 GHz (IEEE,Piscataway, NJ, 2006).

19. J. B. Pendry, Science 306, 1353 (2004).20. We thank J. Pendry for suggesting the use of magnetic

resonances and M. Grossman and I. elanovi for technicalassistance. Supported by NSF Materials Research Scienceand Engineering Center grant DMR 02-13282, U.S.Department of Energy grant DE-FG02-99ER45778, and theArmy Research Office through the Institute for SoldierNanotechnologies under contract DAAD-19-02-D0002.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/1143254/DC1SOM TextFigs. S1 to S5

30 March 2007; accepted 21 May 2007Published online 7 June 2007;10.1126/science.1143254Include this information when citing this paper.

Sea Anemone Genome RevealsAncestral Eumetazoan GeneRepertoire and Genomic OrganizationNicholas H. Putnam,1 Mansi Srivastava,2 Uffe Hellsten,1 Bill Dirks,2 Jarrod Chapman,1Asaf Salamov,1 Astrid Terry,1 Harris Shapiro,1 Erika Lindquist,1 Vladimir V. Kapitonov,3Jerzy Jurka,3 Grigory Genikhovich,4 Igor V. Grigoriev,1 Susan M. Lucas,1 Robert E. Steele,5John R. Finnerty,6 Ulrich Technau,4 Mark Q. Martindale,7 Daniel S. Rokhsar1,2*Sea anemones are seemingly primitive animals that, along with corals, jellyfish, and hydras,constitute the oldest eumetazoan phylum, the Cnidaria. Here, we report a comparative analysis ofthe draft genome of an emerging cnidarian model, the starlet sea anemone Nematostellavectensis. The sea anemone genome is complex, with a gene repertoire, exon-intron structure, andlarge-scale gene linkage more similar to vertebrates than to flies or nematodes, implying that thegenome of the eumetazoan ancestor was similarly complex. Nearly one-fifth of the inferred genesof the ancestor are eumetazoan novelties, which are enriched for animal functions like cellsignaling, adhesion, and synaptic transmission. Analysis of diverse pathways suggests that thesegene inventions along the lineage leading to animals were likely already well integrated withpreexisting eukaryotic genes in the eumetazoan progenitor.

All living tissue-grade animals, or eu-metazoans, are descended from the lastcommon ancestor of bilaterians (flies,worms, snails, and humans), cnidarians (anem-ones, jellyfish, and hydra), and ctenophores (combjellies) (1, 2). This eumetazoan ancestor livedperhaps 700 million years ago. Although it is notpreserved in the fossil record (3), we can infermany of its characteristicsflagellated sperm, de-velopment through a process of gastrulation, mul-tiple germ layers, true epithelia lying upon abasementmembrane, a lined gut (enteron), a neuro-muscular system, multiple sensory systems, andfixed body axesbecause these conserved fea-tures are retained by its modern descendants.

Similarly, we can characterize the genome ofthis long-extinct eumetazoan progenitor by com-paring modern DNA and protein sequences and

identifying conserved ancestral features that havean intrinsically slow rate of change and/or arepreserved by selective pressures. Comparisons(46) between fruit fly, nematode, and vertebrategenomes reveal greater genomic complexity inthe vertebrates [and other deuterostomes (7, 8)]as measured by gene content and structure, but atthe same time show that many genes and net-works are shared across bilaterians. Probing theancestral eumetazoan genome requires sequencesfrom even deeper branches of the animal tree,comparing bilaterian and nonbilaterian phyla.

In comparison with bilaterians, cnidarians ap-pear morphologically simple. The phylum is de-fined (2) by a sac-like body plan with a singleoral opening, two epithelial tissue layers, the pres-ence of numerous tentacles, a nerve net, and thecharacteristic stinging cells (cnidocytes, literallytranslated as nettle cells) that give the phylum itsname (fig. S1.1G). The class Anthozoa (floweranimals) includes diverse anemones, corals, andsea pens, all of which lack a medusa stage. Theother cnidarian classes are united by their pelagicmedusae and characteristically linear mitochon-drial genomes (9) into the Medusozoa, includingHydra and related hydroids, jellyfish, and boxjellies. The disparate bilaterian phyla of the earlyCambrian suggest a Precambrian divergence ofthe cnidarian lineage from the bilaterian stem, andindeed some of the oldest animal body and em-bryo fossils are plausibly relics of stem cnidarians[reviewed in (10, 11)].

Among Anthozoan cnidarians, the starlet seaanemone Nematostella vectensis is an emergingmodel system (12, 13). This estuarine burrowinganemone is found on the Atlantic and Pacificcoasts of North America, as well as the coast ofsoutheast England (14). Nematostella culturesare easily maintained in the laboratory; withseparate sexes, inducible spawning, and externalfertilization (12, 15), embryos are availablethroughout the year.

Although cnidarians are often character-ized as simple or primitive, closer study ofNematostella and its relatives has revealedconsiderable molecular (1619) and morpho-logical complexity (13). Based on expressedsequence tag (EST) analyses (17, 18) and thetargeted study of specific gene families[reviewed in (13, 16, 2022)], signaling path-ways and transcription factors involved in theearly patterning and development of bilateriansare present in cnidarian genomes and are ac-tive in development (13, 2328), indicating thatthese pathways and regulatory mechanismspredate the eumetazoan radiation. Perhaps mostnotably, genes that establish the main bodyaxes in bilaterian embryos are also expressedasymmetrically in Nematostella development,even though cnidarians are conventionallyviewed as radial animals [for a critical dis-cussion, see (29)].

Here, we report the draft genome of thestarlet sea anemone and use its gene repertoireand genome organization to reconstruct featuresof the ancestral eumetazoan genome. Analysisof the Nematostella genome in the context ofsequences from other eukaryotes reveals the ge-nomic complexity of this last common cnidarian-bilaterian ancestor. The emerging picture fromthe genome and EST studies (17, 18) is one ofextensive conservation in gene content, structure,and organization between Nematostella and ver-tebrates. We show that even chromosome-scalelinkage has been preserved between Nematostellaand vertebrates. These are the most ancient con-served linkages known outside of prokaryoticoperons. In contrast, the fruit fly and nematodemodel systems have experienced extensive geneloss (18), intron loss (30), and genome rearrange-ment. Thus, from a genomic perspective, theeumetazoan ancestor more closely resembledmodern vertebrates and sea anemones.

Nematostella Genome Assembly and Gene SetThe draft sequence of the Nematostella genomewas produced with the use of a random shotgun

1Department of Energy Joint Genome Institute, Walnut Creek,CA 94598, USA. 2Center for Integrative Genomics andDepartment of Molecular and Cell Biology, University ofCalifornia, Berkeley, CA 94720, USA. 3Genetic InformationResearch Institute, 1925 Landings Drive, Mountain View, CA94043, USA. 4Sars International Centre for Marine MolecularBiology, University of Bergen, Thormhlensgt 55, 5008,Bergen, Norway. 5Department of Biological Chemistry andthe Developmental Biology Center, University of California,Irvine, CA 92697, USA. 6Department of Biology, BostonUniversity, Boston, MA 02215, USA. 7Kewalo Marine Labora-tory, University of Hawaii, Honolulu, HI 96813, USA.

*To whom correspondence should be addressed. E-mail:dsrokhsar@lbl.gov

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DOI: 10.1126/science.1143254, 83 (2007);317 Science

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