Resonant cavity mode enabled wireless power transfer

1 Resonant cavity mode enabled wireless power transferMatthew J. Chabalko and Alanson P. Samplea)Disney Research, Pittsburgh, 4720 Forbes Avenue, Lower Level, Suite 110, Pittsburgh, Pennsylvania 15213,USA(Received 21 August 2014; accepted 3 December 2014; published online 15 December 2014)This letter proposes using the electromagnetic Resonant modes of a hollow metallic structure toprovide wireless power to small receivers contained anywhere inside. The coupling between alarge chamber (used as a cavity resonator) and a small wire loop (used as a receiver) is studied. Ananalytic expression for the coupling coefficient between the fields in the cavity and a loop receiveris derived. This model is validated against simulation and experimental results. Finally, wirelesspower transfer is demonstrated at an efficiency of over 60% for large volumes in the structure, eventhough the receiver is a small cm square shaped loop and the distance to the source probe isgreater than 1 m.

2 This technique for wireless power transfer has thus far been unexplored, and theresults here serve as a starting point for Resonant cavity mode wireless power systems with manyreceivers having arbitrary locations and AIP Publishing LLC.[ ]The challenge for wireless power transfer (WPT) sys-tems is to provide highly efficient power transfer at long dis-tances and over large volumes of space, thus enabling devicecharging in an unencumbered and seamless fashion. ExistingWPT technologies like near-field magnetoquasistatic (MQS)WPT are well suited for two dimensional charging pads, where the efficiency drops off rapidly as the source and re-ceiver are separated by more than a coil 3 Broadcast far-field WPT methods can transfer power atgreater distances but it is not possible to maintain high effi-ciency while powering many devices over a large , point-to-point far-field WPT systems canmaintain high end-to-end efficiency but requires sophisti-cated control and tracking mechanisms to maintain high effi-ciency, which limits many potential ,5 This work aims to provide wireless power anywhere in aconfined three dimensional volume of space.

3 This is accom-plished by stimulating the natural electromagnetic resonantmodes of a metallic structure with low level electromagneticfields, so that energy can be efficiently coupled to a small re-ceiver placed within the structure. This technique can beextended to arbitrary shaped volumes, but for the purposesof this letter we confine our initial setup to a rectangular box,as shown in Figure1(a)depicts the outline of a metal-lic structure with a receiver inside and (b)shows thefield distribution of one of the electromagnetic modes. Byusing one or more Resonant modes of the cavity the receiverscan be powered in nearly any orientation and/or position aslong as strong coupling between the cavity mode fields andreceivers is achieved. When sets of receivers are placed inregions of uniform magnetic fields, power is divided evenlyamong them allowing for simultaneous recharging. Oneapplication in the industrial sector is the wireless rechargingof 10 s 100 s of tools and devices when placed in a securemetal storage cabinet parameters central to determining WPT efficiencybetween two resonators are the coupling coefficient (j, units:rad/s) between the resonators, and the quality factors (Q-fac-tors) of each ,6 Analysis of theQ-factors of cav-ities and coils has been described individually in ,7In this work, coupled mode theory (CMT) is used to derivean analytical expression for the coupling coefficient betweena rectangular cavity resonator and a small square shaped,single-turn, coil receiver.

4 This expression forjis then usedto calculate the efficiencies of WPT that can be expectedbased on the setup s geometry. The analytic model is vali-dated against finite element method (FEM) simulations andexperimental analysis begins with coupled mode theory, wherewe posit the coupling of two generic lossless resonators as afunction of time. Later, it will be shown that this generalanalysis can be conformed to the specific coupling between acavity resonator and a coil receiver. First, each resonator isdefined to have a Resonant frequency and amplitude,x1,a1andx2,a2(withx1,2 2pf1,2), respectively, and that theyhave the time dependence exp(jx1,2t). Using these defini-tions, standard CMT is used to write the differentialFIG. 1. (a) Diagram of the cavity resonator with a square receiver placedinside. The receiver has lengthsand unit normal~n, and is located at position(xo,yo,zo). The dotted red and green lines indicate coil measurement planeand line, respectively, for experiments presented in this letter.

5 (b) Shows theTM110field distribution. The color map is the magnitude of the magneticfield,j~Hj: Red, large; Blue, small. The white arrows are the~H-field )Electronic mail: AIP Publishing LLC105, 243902-1 APPLIED PHYSICS LETTERS105, 243902 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to On: Wed, 24 Dec 2014 19:23:49equations that describe the coupled resonators amplitudeevolution over time8,9ddta1 jx1a1 jj12a2;ddta2 jx2a2 jj21a1;(1)wherej12 j#21 jis the coupling coefficient between thetwo resonators and*indicates the complex , as is standard in CMT,a1,2are defined such thattheir stored energy is given by Energy ja1;2j2. Since thecoupling coefficient is instrumental in determining the effi-ciency of energy transfer between the two resonators, we useenergy conservation arguments to derive an approximateexpression for the coupling coefficient between the that the power fed from resonator one into resonatortwo (P21) must be equal to the time rate of change of energyin resonator two, Eq.

6 (1)enables this to be written as9P21 ddtja2j2 jja1a#2$jj#a#1a2:(2)Equation(2)is an explicit expression relatingP21toa1,2andj, valid for two coupled resonators. IfP21anda1,2areknown, thenjcan be found uniquely. Thus, we turn ourattention to our specific cavity -to-coil coupled mode systemand re-derive this expression, but using physical quantitiesunique to our setup. First, we deriveP21 the power flowingfrom resonator one (the cavity mode) into resonator two (thesquare loop receiver with capacitor to form anLCresonator).Neglecting any coupling via the electric field, and usinganalysis similar to that in , the power flowing from thechamber to the coil can be written in terms of the magneticfluxes crossing the surface of the receiver loopP21 i2d/1$/2 dt /2L2d/1$/2 dt;(3)where/1is the instantaneous total normal flux due to thecavity mode fields crossing the coil s surface, simi-larly the instantaneous time dependent flux crossing the coilsurfaceAdue to the fields generated by a current,i2, circulat-ing in the coil.

7 In the rightmost expression in Eq.(3),wehave used the usual relation/2 L2i2, whereL2is the in-ductance of the receiver coil. We will be more explicit withevaluating/1later, as it is related to field distributions of aparticular cavity ,/1;2is reformulated in terms ofU1;2 the time de-pendent complex envelope functions of the fluxes/1;2t U1;2ejx1;2t U#1;2e$jx1;2t2:(4)Then, substituting Eq.(4)into Eq.(3), and assumingthat the resultingddtU1;2terms are small compared to thejxU1,2terms such that we can neglectddtU1;2terms, we getP21 14L2jx1U1ejx1tU#2e$jx2t$jx1U#1e$jx1tU2ej x2t!":(5)The form of(5)is similar to that of(2). This means, the cou-pling coefficient can be retrieved by inspection if we writeour generic resonator amplitudes,a1,2from(2), in terms ofU1,2. Using that substitution,U1,2!a1,2, in Eq.(5), alongwith some algebraic manipulation allows us to re-write(5)tomatch the from of(2)and retrieve the coupling make this substitution, we require that the total energystored in resonators one and two isja1;2j2, as was defined inEqs.

8 (1)and(2). This mathematical manipulation is accom-plished by introducing three parameters:ais the total mag-netic energy stored in the chamber,bis the total fluxcrossing the receiver due to the chamber s modal magneticfields (~H), andfis a constant relating to the energy stored inthe coil resonator. These parameters are evaluated usinga V loj~Hj2dV;(6)b Alo~H'~n dA;(7)f 1ffiffiffiffiffiffiffi2L2p:(8)Here,Vis the volume of the chamber,~nis the unit nor-mal vector of the coil [as in (a)], andlois the perme-ability of free space (assuming an air filled chamber). Usingthese parameters and some algebraic manipulation, we cannormalizea1,2such thatja1j2is the total magnetic energystored in resonator one (the cavity mode) andja2j2is thetotal energy stored in resonator two (theLCtank formed bythe coil and capacitor). The explicit expressions fora1,2area1 U1a1=2bejx1t;a2 U2fejx2t:(9)After the above normalization,a1,2fits the framework ofCMT, and Eq.(9)can be substituted into Eq.

9 (5)resulting inP21 jx14L2ba1=21fa1a#2$jx14L2ba1=21fa#1a2:(1 0)Now that(10)is in the same form as(2), the couplingcoefficient between the cavity mode and a loop receiver (j)can be determined by inspectionj 14x1bL2a1=2f ffiffiffi2p4x1bffiffiffiffiffiffiffiL2ap :(11)For this initial investigation the lowest order TM modeof the cavity is chosen as it allows for the unambiguous studyof the coupling mechanism of the system. It should be notedthat the following technique can be used for any arbitrarymode structure. A plot of the TM110mode s field distributionis shown in (b), as simulated using COMSOL the TM110mode, and with reference to the coordinatesystem in (and for a cavity with dimensionsa(b(d),the components of the magnetic field within the chamber canbe written as10Hx Eojpbxlosinpxa% (12)243902-2 M. J. Chabalko and A. P. SampleAppl. Phys. , 243902 (2014) This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to On: Wed, 24 Dec 2014 19:23:49Hy $Eojpaxlocospxa% (13)whereEois the peak value of theelectricfield.))

10 Given these~H-fields,(6)and(7)can next be used to computea, andb. Inthis case, for the magnetic energy stored in the chamber, weevaluate(6)to finda E2o8p2a2 d2 dlox21ab;(14)wherex1is the Resonant frequency of the the purposes of comparing the theoretical model tomeasured results, we will restrict the receiver coil to lie flat inthex-zplane. In this orientation, the receiver will only coupleto they-component of the cavity s magnetic field, which willbe reflected in the plots ofj. Thus, evaluating Eq.(7)for asquare coil with side lengths, unit normal~n ~ay( ,y-directed), and inductanceL2, centered at position (xo,yo,zo),the magnetic flux coupled from the chamber to the coil isb jEo2sx1cospxoa%&sinpyob%&sinps2a%&:(15)F inally,jcan be determined for the TM110mode bysubstituting Eqs.(14)and(15)in Eq.(11).The utility in derivingjanalytically is that it is one ofthe three key parameters needed to determine the maximumefficiencies of the cavity -to-coil WPT system.