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`function of the relative distance DI r between two identical loops with r = 30cm and
`
`a=2cm.
`
`Fig. 22(b) illustrates the strong-coupling factor U and the strong-interference
`
`factor V as a curve in the U -V plane, parametrized with the relative distance DI r
`between the two loops, for the cases with interference and eigenfrequency f-ri (solid),
`with interference and eigenfrequency fu ( dashed), and without interference and
`
`eigenfrequency fu ( dotted).
`
`Fig. 22( c) shows the efficiency enhancement ratio of the solid curve in Fig. 22(b)
`
`relative to the dashed and dotted curves in Fig. 22(b ).
`
`Fig. 23 shows the radiation efficiency as a function of the resonant
`
`eigenfrequency of two identical capacitively-loaded conducting single-tum loops. Results
`
`for two different loop dimensions are shown and for two relative distances between the
`
`identical loops. For each loops dimension and distance, four different cases are examined:
`
`without far-field interference (dotted), with far-field interference but no driving(cid:173)
`
`frequency detuning (dashed) and with driving-frequency detuning to maximize either the
`
`efficiency (solid) or the ratio of efficiency over radiation ( dash-dotted).
`
`Fig. 24 shows CMT results for (a) the coupling factor k and (b) the strong(cid:173)
`
`coupling factor U , for three different m values of subwavelength resonant modes of two
`
`same dielectric disks at distance DI r = 5 ( and also a couple more distances for m = 2 ),
`
`when varying their E in the range 250 ~ E ~ 35. Note that disk-material loss-tangent
`tan 8 = 6-1 o-6
`E - 2·10--4 was used. ( c) Relative U error between CMT and numerical
`FEFD calculations of part (b ).
`
`Fig. 25 shows Antenna Theory (AT) results for (a) the normalized interference
`term 2A/ ,Jov»2 and (b) magnitude of the strong-interference factor IVI, as a function of
`frequency, for the exact same parameters as in Fig.24. ( c) Relative V error between AT
`
`and numerical FEFD calculations of part (b).
`
`Fig. 26 shows results for the overall power transmission as a function of
`
`frequency, for the same set of resonant modes and distances as in Figs.24 and 25, based
`
`on the predictions including interference (solid lines) and without interference, just from
`
`U ( dotted lines).
`
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`Fig. 27 (a) shows the frequencies fu and hi, where the strong-coupling factor U
`
`and the power-transmission efficiency T/ are respectively maximized, as a function of the
`
`transfer distance between the m = 2 disks of Fig. 15. Fig. 27(b) shows the efficiencies
`
`achieved at the frequencies of (a) and, in inset, the enhancement ratio of the optimal (by
`definition) efficiency for hi versus the achievable efficiency at fu. Fig. 27(c) shows the
`
`D -parametrized path of the transmission efficiency for the frequency choices of (a) on
`
`the U - V efficiency map.
`
`Fig. 28 shows results for the radiation efficiency as a function of the transfer
`
`distance at resonant frequency fu, when the operating frequency is detuned (solid line),
`
`when it is not ( dashed line), and when there is no interference whatsoever ( dotted line). In
`
`the inset, we show the corresponding radiation suppression factors.
`
`Figs. 29(a)-(b) show schematics for frequency control mechanisms.
`
`Figs. 30(a)-(c) illustrate a wireless energy transfer scheme using two dielectric
`
`disks in the presence of various extraneous objects.
`
`1. Efficient energy-transfer by 'strongly coupled' resonances
`
`DETAILED DESCRIPTION
`
`Fig. 1 shows a schematic that generally describes one example of the invention, in
`
`which energy is transferred wirelessly between two resonant objects. Referring to Fig. 1,
`
`energy is transferred, over a distance D, between a resonant source object having a
`
`characteristic size 'i and a resonant device object of characteristic size r2 • Both objects
`
`are resonant objects. 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.
`
`The characteristic size of an object can be regarded as being equal to the radius of
`
`the smallest sphere which can fit around the entire object. The characteristic thickness of
`
`an object can be regarded as being, when placed on a flat surface in any arbitrary
`
`configuration, the smallest possible height of the highest point of the object above a flat
`
`surface. The characteristic width of an object can be regarded as being the radius of the
`
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`smallest possible circle that the object can pass through while traveling in a straight line.
`
`For example, the characteristic width of a cylindrical object is the radius of the cylinder.
`
`It is to be understood that while two resonant objects are shown in the example of
`
`Fig. 1, and in many of the examples below, other examples can feature three or more
`
`resonant objects. For example, in some examples, a single source object can transfer
`
`energy to multiple device objects. In some examples, energy can be transferred from a
`
`first resonant object to a second resonant object, and then from the second resonant object
`
`to a third resonant object, and so forth.
`
`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.
`
`Different temporal schemes can be employed, depending on the application, to
`
`transfer energy between two resonant objects. Here we will consider two particularly
`
`simple but important schemes: a one-time finite-amount energy-transfer scheme and a
`
`continuous finite-rate energy-transfer (power) scheme.
`
`1.1 Finite-amount energy-transfer efficiency
`
`Let the source and device objects be 1, 2 respectively and their resonance
`
`eigemodes, which we will use for the energy exchange, have angular frequencies OJ1.2 ,
`frequency-widths due to intrinsic (absorption, radiation etc.) losses r 1 2 and (generally)
`vector fields F1,2 ( r), normalized to unity energy. Once the two resonant objects are
`
`brought in proximity, they can interact and an appropriate analytical framework for
`
`modeling this resonant interaction is that of the well-known coupled-mode theory (CMT).
`
`In this picture, the field of the system of the two resonant objects 1, 2 can be
`approximated by F( r,t) = a 1 (t )F1 ( r )+ a2 (t )F2 ( r), where a 1,2 (t) are the field
`amplitudes, with la1,2 ( t )12 equal to the energy stored inside the object 1, 2 respectively,
`due to the normalization. Then, using e -irot time dependence, the field amplitudes can be
`
`shown to satisfy, to lowest order:
`
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`K
`U = - - = k,JQ1Q2
`,Jr1r2
`
`(18)
`
`that has been set as a figure-of-merit for any system under consideration for wireless
`
`energy-transfer, along with the distance over which this ratio can be achieved (clearly, U
`will be a decreasing function of distance). The desired optimal regime U > 1 is called
`'strong-coupling' regime and it is a necessary and sufficient condition for efficient
`energy-transfer. In particular, for U > 1 we get, from Eq.(15), 1]p* > 17%, large enough
`for practical applications. The figure-of-merit U is called the strong-coupling factor. We
`
`will further show how to design systems with a large strong-coupling factor.
`
`To achieve a large strong-coupling factor U, in some examples, the energy(cid:173)
`transfer application preferably uses resonant modes of high quality factors Q,
`corresponding to low (i.e. slow) intrinsic-loss rates r. This condition can be satisfied by
`designing resonant modes where all loss mechanisms, typically radiation and absorption,
`
`are sufficiently suppressed.
`
`This suggests that the coupling be implemented using, not the lossy radiative far(cid:173)
`
`field, which should rather be suppressed, but the evanescent (non-lossy) stationary near(cid:173)
`
`field. To implement an energy-transfer scheme, usually more appropriate are finite
`
`objects, namely ones that are topologically surrounded everywhere by air, into where the
`
`near field extends to achieve the coupling. Objects of finite extent do not generally
`
`support electromagnetic states that are exponentially decaying in all directions in air
`
`away from the objects, since Maxwell's Equations in free space imply that k 2 =al /c 2
`where k is the wave vector, co the angular frequency, and c the speed oflight, because of
`which one can show that such finite objects cannot support states of infinite Q, rather
`
`,
`
`there always is some amount of radiation. 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 tum 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. One typical way of
`
`achieving a high radiation-Q (Qract) is to design subwavelength resonant objects. When
`
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`the size of an object is much smaller than the wavelength of radiation in free space, its
`
`electromagnetic field couples to radiation very weakly. Since the extent of the near-field
`
`into the area surrounding a finite-sized resonant object is set typically by the wavelength,
`
`in some examples, resonant objects of subwavelength size have significantly longer
`
`evanescent field-tails. In other words, the radiation caustic is pushed far away from the
`
`object, so the electromagnetic mode enters the radiative regime only with a small
`
`amplitude.
`
`Moreover, most realistic materials exhibit some nonzero amount of absorption,
`which can be frequency dependent, and thus cannot support states of infinite Q, rather
`
`there always is some amount of absorption. However, very long-lived ("high-Q") states
`
`can be found, where electromagnetic modal energy is only weakly dissipated. Some
`
`typical ways of achieving a high absorption-Q (Qabs) is to use materials which exhibit
`
`very small absorption at the resonant frequency and/or to shape the field to be localized
`
`more inside the least lossy materials.
`
`Furthermore, to achieve a large strong-coupling factor U, in some examples, the
`
`energy-transfer application preferably uses systems that achieve a high coupling factor k,
`
`corresponding to strong (i.e. fast) coupling rate K, over distances larger than the
`
`characteristic sizes of the objects.
`
`Since finite-sized subwavelength resonant objects can often be accompanied with
`
`a highQ, as was discussed above and will be seen in examples later on, such an object
`
`will typically be the appropriate choice for the possibly-mobile resonant device-object.
`
`In these cases, the electromagnetic field is, in some examples, of quasi-static nature and
`
`the distance, up to which sufficient coupling can be achieved, is dictated by the decay(cid:173)
`
`law of this quasi-static field.
`
`Note, though, that in some examples, the resonant source-object will be immobile
`
`and thus less restricted in its allowed geometry and size. It can be therefore chosen large
`
`enough that the near-field extent is not limited by the wavelength, and can thus have
`
`nearly infinite radiation-Q. Some 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,
`
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`therefore a good coupling can also be achieved over distances quite a few times larger
`
`than a characteristic size of the source- and/or device-object.
`
`2 'Strongly-coupled' resonances at mid-range distances for realistic systems
`
`In the following, examples of systems suitable for energy transfer of the type
`
`described above are described. We will demonstrate how to compute the CMT
`
`parameters w 1 ,2 , Q1 ,2 and k described above and how to choose or design these
`parameters for particular examples in order to produce a desirable figure-of-merit
`U = K/,jr1r2 = k,JQ1 Q2 at a desired distance D. In some examples, this figure-of-merit
`is maximized when w 1 ,2 are tuned close to a particular angular frequency Wu.
`2.1 Self-resonant conducting coils
`
`In some examples, one or more of the resonant objects are self-resonant conducting
`
`coils. Referring to Fig. 3, a conducting wire of length 1 and cross-sectional radius a is
`wound into a helical coil of radius r and height h ( namely with N = .J l
`/ 27rr
`number of turns), surrounded by air. As described below, the wire has distributed
`
`2
`- h
`
`2
`
`inductance and distributed capacitance, and therefore it supports a resonant mode of
`
`angular frequency co . 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
`
`p ( x) across it, to the magnetic field in free space, due to the current distribution j ( x) m
`
`the wire. In particular, the charge conservation equation V · j = iwp implies that: (i) this
`periodic exchange is accompanied by a rr /2 phase-shift between the current and the
`charge density profiles, namely the energy W 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 Pt ( x) and I ( x) are respectively the linear charge and current densities
`in the wire, where x runs along the wire, q O = ½ f dx IPz ( x )I 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 = max {II (x )I}
`
`is the maximum positive value of the linear current distribution, then
`
`0
`
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`radiation quality factors of the resonance are given by Qabs = Z I Rahs and
`
`Qrad = Z I Rrad respectively.
`
`From Eq.(19)-(22) 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 I 2 to +l I 2 , then the current amplitude profile would have the form
`
`/ ( x) = / 0 cos ( ;rx I 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 p 1 ( x) = p 0 sin ( ;rx I l), and thus
`q0 = f0
`
`dxp0 lsin(;rx/ z)I = pJ I ;r. Using these sinusoidal profiles we find the so-called
`
`l/2
`
`"self-inductance" Ls and "self-capacitance" Cs of the coil by computing numerically the
`
`integrals Eq.(19) and (20); the associated frequency and effective impedance are cos and
`
`Zs respectively. The "self-resistances" Rs are given analytically by Eq.(21) and (22)
`
`2 -lfl/
`·
`usmg Irms -
`f
`-l,2
`
`·/ 12-.l 2 11-
`2
`I dx Io cos(;rx, l) - 1 Io, p -qo
`I
`
`~
`
`27
`
`(2.. )2
`h +
`
`Jr
`
`(
`4/v -1 ;r
`
`r2
`
`(4Ncos(nN) J~
`
`')
`
`r
`
`and
`
`)
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`I I
`rn = / 0
`
`2)2
`(2
`-1V 1rr
`Jr
`
`(cos(nN)(l2N 2 -l)-sin(nN)1rN(4N2 -l) J2
`+
`(
`l6N -8N +l Jr
`
`hr
`
`, and therefore the
`
`4
`
`2
`
`)
`
`associated Qs factors can be calculated.
`
`The results for two examples of resonant coils with subwavelength modes of
`
`,\ / 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
`
`(cr=5.998•1W'-7 S/m) was used. It can be seen that expected quality factors at microwave
`
`frequencies are Qs,abs ;;:=:-: 1000 and Qs,rad ;;:=:-: 5000 .
`
`single coil
`
`r=30cm, h=20cm, a=lcm, N=4
`
`Table 1
`
`f(MHz)
`
`13.39
`
`As IT
`I
`
`74.7
`
`Qs,rad
`
`4164
`
`Qs,abs
`
`8170
`
`r=l0cm, h=3cm, a=2mm, N=6
`
`140
`
`21.38
`
`43919
`
`3968
`
`Q
`2758
`
`3639
`
`Referring to Fig. 4, in some examples, energy is transferred between two self(cid:173)
`
`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 p 1,2 ( x) and k2 ( x) , total charges
`and peak currents respectively as q1,2 and 11,2 , and capacitances and inductances
`respectively as C1,2 and L1,2 , which are the analogs of p ( x) , j ( x) , q O
`for the single-coil case and are therefore well defined, we can define their mutual
`
`, C and L
`
`, I O
`
`capacitance and inductance through the total energy:
`
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`medium distances D / r = 10 - 3 the expected coupling-to-loss ratios are in the range
`
`U rv 2-70.
`
`2.1.1 Experimental Results
`
`An experimental realization of an example of the above described system 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. 5. Referring to Fig. 5, A is a single copper loop of radius 25cm that is part of the
`
`driving circuit, which outputs a sine wave with frequency 9.9MHz. 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 K'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.
`
`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 = 3 0 cm and
`
`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~ = 10.56 ± 0.3 MHz, which is
`
`about 5% off from the measured resonance at around 9.90MHz.
`The theoretical Q for the loops is estimated to be ~ 2500 (assuming perfect
`copper of resistivity p = l /CY= 1.7 x 10-sn m) 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 ( ~ 20µm) at this frequency. We have therefore used the experimentally
`observed Q (and r 1 = r 2 = r = cv/(2Q) derived from it) in all subsequent computations.
`The coupling coefficient K can be found experimentally by placing the two self(cid:173)
`
`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
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`resonant modes in the transmission spectrum. According to Eq.( 13) derived by coupled(cid:173)
`- r 2
`
`, when
`
`mode theory, the splitting in the transmission spectrum should be ;sP = 2.J K
`
`2
`
`KA,B are kept very small by keeping A and Bat a relatively large distance. The
`
`comparison between experimental and theoretical results as a function of distance when
`
`the two the coils are aligned coaxially is shown in Fig. 6.
`
`Fig. 7 shows a comparison of experimental and theoretical values for the strong(cid:173)
`coupling factor U =KI r as a function of the separation between the two coils. The
`theory values are obtained by using the theoretically obtained K and the experimentally
`measured r . The shaded area represents the spread in the theoretical U due to the ~ 5%
`uncertainty in Q . As noted above, the maximum theoretical efficiency depends only on
`
`the parameter U, which is plotted as a function of distance in Fig. 7. U is greater than 1
`
`even for D = 2.4m (eight times the radius of the coils), thus the sytem is in the strongly(cid:173)
`
`coupled regime throughout the entire range of distances probed.
`
`The power-generator circuit was a standard Colpitts oscillator coupled inductively
`
`to the source coil by means of a single loop of copper wire 25cm in radius (see Fig. 5).
`
`The load consisted of a previously calibrated light-bulb, and was attached to its own loop
`
`of insulated wire, which was in tum 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 U B = KB Ir was adjusted so that it matched its optimal value,
`
`given theoretically by Eq.(14) as UB* = .J1+u2
`
`• Because of its inductive nature, the loop
`
`connected to the light-bulb added a small reactive component to KB 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.
`
`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(cid:173)
`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 11 and 12 defined above.
`The power dissipated in each coil was then computed from Pi. 2 = rL I 11•2 1
`
`, and the
`
`2
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`efficiency was directly obtained from 77 = P8 I (Pi+~+~). 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. 8, along with the
`
`theoretical prediction for maximum efficiency, given by Eq.(15).
`
`Using this example, we were able to transmit significant amounts of power using
`
`this setup from the source coil to the device coil, fully lighting up a 60W 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 power(cid:173)
`
`transmission 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 transmitting 60W to the load over a distance of 2m, for example, the power
`
`flowing into the driving circuit was 400W. This yields an overall wall-to-load efficiency
`
`of rv 15%, which is reasonable given the expected rv 40% efficiency for the wireless
`
`power transmission at that distance and the low efficiency of the driving circuit.
`
`From the theoretical treatment above, we see that in typical examples it is
`
`important that the coils be on resonance for the power transmission 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 !J..fifo of a few times the
`
`inverse loaded Q, the induced current in the device coil was indistinguishable from
`
`noise.
`
`The power transmission 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 10cm 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
`
`32
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`closer than a few centimeters from the coil, thereby lowering the efficiency of the
`
`transfer.
`
`This method of power transmission is believed safe for humans. When
`
`transmitting 60W (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.
`
`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.
`
`These experiments demonstrated experimentally a system for power transmission
`
`over medium range distances, and found that the experimental results match theory well
`
`in multiple independent and mutually consistent tests.
`
`The efficiency of the scheme and the distances covered can 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.
`
`2.2 Capacitively-loaded conducting loops or coils
`
`In some examples, one or more of the resonant objects are capacitively-loaded
`
`conducting loops or coils . Referring to Fig. 9 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=l and h=O). The plates have a capacitance
`
`C P = E 0 E A Id , 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
`
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`effective inductance L and total effective capacitance C of the coil are different
`
`respectively from Ls and Cs, 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 <Cs, and thus, from the charge conservation equation, the current
`
`distribution j flattens out, so L > Ls . The resonant frequency for this system is
`
`w = 1/ ,JL(C + cp) < ws = 1/ .JLSCS, and I(x) ➔ Io cos(;irxll) ⇒ C ➔ cs ⇒
`
`m ➔ ms , as C P ➔ 0 .
`
`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 cp » cs > C' then OJ~ 1/ .JLc p « OJS and z ~ .JL IC p «Zs' 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.(19). In
`
`the case h = 0 and N integer, the integral in Eq.(19) can actually be computed
`
`analytically, giving the formula L = µ 0 r [ 1n ( 8r I a)- 2] N 2
`
`. Explicit analytical formulas
`, IPI ~ 0 and
`are again available for R from Eq.(21) and (22), since Inns= I 0
`1ml = I 0 N ;irr2 (namely only the magnetic-dipole term is contributing to radiation), so we
`
`can determine also Qabs = OJL I Rahs and Qrad = oJL I Rrad . At the end of the
`
`calculations, the validity of the assumption of constant current profile is confirmed by
`
`checking that indeed the condition CP » Cs <=>OJ« OJs 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 examples, one often prefers coils with very small self(cid:173)
`
`capacitance Cs to begin with, which usually holds, for the types of coils under
`
`consideration, when N = I, so that the self-capacitance comes from the charge
`
`distribution across the single turn, which is almost always very small, or when N > I and
`
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`which again is more accurate for N1 = N 2 =I.
`
`From Eq.(26) it can be seen that the optimal frequency OJu, where the figure-of(cid:173)
`
`merit is maximized to the value U max , is close to the frequency mQ
`
`Q
`1
`2
`
`at which Q1 Q2 is
`
`maximized, since k does not depend much on frequency (at least for the distances D<<}, of
`
`interest for which the quasi-static approximation is still valid). Therefore, the optimal
`
`frequency O.Ju •~ 0JQ
`
`Q
`1
`
`2
`
`is mostly independent of the distance D between the two coils and
`
`lies between the two frequencies mQ
`1
`
`and mQ
`
`2
`
`at which the single-coil Q1 and Q2
`
`respectively peak. For same coils, this optimal frequency is given by Eq.(24) and then
`
`the strong-coupling factor from Eq.(26) becomes
`
`U
`max
`
`= kQ
`max
`
`3
`
`~ ..!_
`D
`]
`[
`
`. ~ 21r2,
`[
`7
`Tio
`
`? ]½
`cm .zv-
`2
`r
`
`(27)
`
`In some examples, one can tune the capacitively-loaded conducting loops or coils,
`so that their angular eigenfrequencies are close to mu within r u , which is half the
`
`angular frequency width for which U > U max I 2 .
`
`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 d