磁共振无线充电鼻祖级论文-通过强耦合磁共振进行无线电力传输

Wireless Power Transfer via Strongly Coupled Magnetic Resonances

André Kurs,1* Aristeidis Karalis,2 Robert Moffatt,1 J. D. Joannopoulos,1 Peter Fisher,3 Marin Soljačić1

12

Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. Department of Electrical Engineering and Computer 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: akurs@mit.edu

通过强耦合磁共振进行无线电力传输 AndréKurs，1 * Aristeidis Karalis，2 Robert Moffatt，1 J. D. Joannopoulos，1 Peter Fisher，3 MarinSoljačić1

1麻省理工学院物理系，剑桥，MA 02139，USA。

2马萨诸塞理工学院电气工程与计算机科学系，剑桥，MA 02139，USA。 马萨诸塞理工学院核科学物理与实验室3，剑桥，MA 02139，USA。 *应向谁发信。 电子邮件：akurs@mit.edu

Using 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 distances 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.

在强耦合状态下使用自谐振线圈，我们实验证明高达线圈半径的八倍的有效的非辐射功率传输。 我们展示了在距离超过两米的情况下以大约40％的效率传输60W的能力。 我们提出一个描述电力传输的定量模型，将实验结果与5％以内相匹配。 讨论实际适用性，并提出进一步研究的方向。

1

1. 30 March 2007; accepted 21 May 2007

2. Published online 7 June 2007; 10.1126/science.11432 Include this information 3. when citing this paper.

In the early 20th century, before the electrical-wire grid, Nikola Tesla (1) devoted much effort towards schemes to transport power wirelessly. However, 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 (2–4). 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 mechanisms 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 frequency 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 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). 20世纪初，电线网格之前，尼古拉·特斯拉（Nikola

Tesla）（1）非常努力地进行无线传输的方案。然而，典型的实施例（例如特斯拉线圈）涉及不期望的大电场。在过去十年中，社会出现了自主电子设备（笔记本电脑，手机，机器人，PDA等）的大量使用。因此，对无线电力的兴趣重新出现（2-4）。辐射传输（5）虽然完美适用于传输信息，但对电力传输应用造成了许多困难：如果辐射是全向的，则电力传输的效率非常低，如果辐射是辐射，则需要不间断的视线和复杂的跟踪机制是单向的最近的理论论文（6）详细分析了通过其非辐射场的尾部耦合的共振物体在中等能量转移中的可行性（7）。直观地，相同共振频率的两个共振物体倾向于有效地交换能量，同时与外来的非共振物体相互作用弱。在耦合谐振（例如声学，电磁，磁性，核等）的系统中，通常存在一般的“强耦合”的操作（8）。如果在给定的系统中可以在该制度下运作，预计能量转移将非常有效。不论周围空间的几何形状如何，以及对环境物体的干扰和损失都很小，实现这种方式的中距离功率传输几乎是全方位和有效的（6）。

Considerations above apply irrespective of the physical nature 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 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.At first glance, such power transfer is reminiscent of the usual magnetic induction (10); however, note that the usual non- resonant induction is very inefficient for mid-range applications. 上述注意事项不考虑谐振的物理性质。

在目前的工作中，我们专注于一个特定的物理实施例：磁共振（9）。

磁共振特别适用于日常应用，因为大多数普通材料不与磁场相互作用，因此与环境物体的相互作用被进一步抑制。

我们能够通过探索MHz频率下的非辐射（近场）磁共振感应来识别两个耦合磁共振系统中的强耦合状态。乍一看，这种功率传输让人联想到通常的磁感应（10）;

然而，请注意，对于中端应用，通常的非谐振感应非常低效。

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-mode theory to describe this physical system (11), we obtain the following set of linear equations 公式概述。

有效的中距离功率传递发生在描述谐振物体的参数空间的特定区域中，该谐振物体彼此强耦合。 使用耦合模式理论来描述这个物理系统（11），我们得到以下的线性方程组

where the indices denote the different resonant objects. The variables am(t) are defined so that the energy contained in object m is |am(t)|2, m is the resonant frequency of that isolated object, and m is its intrinsic decay rate (e.g. due to absorption and radiated losses), so that in this framework an uncoupled and undriven oscillator with parameters 0 and 0 would evolve in time as exp(i0t – 0t). The mn = nm are coupling coefficients between the resonant objects indicated by the subscripts, and Fm(t) are driving terms.

其中指数表示不同的共振物体。 变量am（t）被定义为使得物体m中包含的能量为| am（t）| 2，m是该孤立物体的共振频率，m是其本征衰减速率（例如由于吸收和

辐射损耗），因此在这个框架中，具有参数0 and 0的非耦合和未驱动振荡器将随时间演化为 exp(i0t – 0t). 。 mn= nm是由下标表示的谐振物体之间的耦合系数，Fm(t) 是驱动项。 We limit the treatment to the case of two objects, denoted by source and device, such that the source (identified by the subscript S) is driven externally at a constant frequency, and the two objects have a coupling coefficient . 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 contributing an additional term W to the unloaded device object's decay rateD. The overall decay rate at the device is therefore D= D+ W.. The work extracted is determined by the power dissipated in the load, i.e.2W|aD(t)|2.. Maximizing the efficiency  of the transfer with respect to the loading W, 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 is

我们将处理在由源和设备表示的两个对象的情况下，使得源（由下标S标识）以恒定频率从外部驱动，并且两个对象具有耦合系数。

通过负载（下标W）从设备（下标D）提取工作，该负载用作连接到设备的电路电阻，并且具有对未加载设备对象的衰减速率D的附加项W的作用。 因此，器件的整体衰减率为D=D+W。

提取的工作由负载中消耗的功率确定，即2W| aD（t）| 2。

最大化转移效率相对于负载W，给定方程式 1，相当于解决阻抗匹配问题。 人们发现，当源和设备共振时，该方案效果最好，在这种情况下效率就是这样

The efficiency is maximized when W/D= 1 + 2/SD)1/2.. It is easy to show that the key to efficient energy transfer is to have 2/SD> 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 2/D2 (~106 for typical parameters) compared to the case of inductively coupled non-resonant objects.

当W/D= 1 + 2/SD)1/2..时，效率最大化。 很容易表明，有效能量转移的关键是具有 2/SD> 1.。这通常被称为强耦合状态。

与电感耦合非谐振对象相比，谐振在这种功率传递机制中起着至关重要的作用，因为效率提高了约 2/D2（典型参数为~106）

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). 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 intoa helix of n turns, radius r, and height h. To the best of our knowledge, there is no exact solution for a finite helix in the literature, and even in the case of infinitely long coils, the solutions rely on assumptions that are inadequate for our static model described below is in good agreement (approximately 5%) with experiment. 自谐振线圈的理论模型。

我们的实验方案包括两个自谐振线圈，其中一个（源极线圈）感应耦合到一个振荡电路，而另一个（器件线圈）感应耦合到一个电阻负载（12） 1）。

自谐振线圈依赖于分布电感和分布电容之间的相互作用来实现谐振。 线圈由总长为l的导电线制成，横截面半径a为n匝的螺旋线，半径r和高度h。

据我们所知，在文献中没有一个有限螺旋的确切解决方案，即使在无限长的线圈的情况下，解决方案依赖于我们以下描述的静态模型不足的假设是很好的协议.

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, so 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(s/l) exp(it). It follows from the continuity equation for charge that the linear charge density profile is of the form 0 sin(s/l) exp(it), so the two halves of the coil (when sliced perpendicularly to its axis) contain charges equal in magnitudeq0 = 0l/but opposite in sign.

我们首先观察到线圈末端的电流必须为零，并且使得有根据的猜测线圈的谐振模式通过沿着导线长度的正弦电流分布很好地近似。

我们对最低模式感兴趣，所以如果我们用s表示沿着导体长度的参数化坐标，使得它从-L/ 2到+ L/2运行，则时间依赖的电流分布具有形式 I0 cos(s/l) exp(it).。 从电荷的连续性方程式可以看出，线性电荷密度分布形式为0 sin(s/l)

exp(it)，所以线圈的两半（垂直于其轴线切片）包含费用在q0 = 0l/相等但符号相反。 As the coil is resonant, the current and charge density profiles are /2 out of phase from each other, meaning that the real part of one is maximum when the real part of the other is zero. Equivalently, the energy contained in the coil is 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 for each coil as follows:

当线圈共振时，电流和电荷密度分布彼此相位为/2 ，这意味着当另一个的实部为零时，一个的实部为最大。

等效地，包含在线圈中的能量完全是由于电流而在某些时间点，而在其他点，完全是由于电荷。 使用电磁理论，我们可以定义每个线圈的有效电感L和有效电容C如下：

where the spatial current J(r) and charge density (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 by 其中空间电流J（r）和电荷密度

(r)分别从沿着隔离线圈的电流密度和电荷密度结合物体的几何形状获得。 如所定义，L和C具有由线圈中包含的能量U给出的性质

Given this relation and the equation of continuity, one finds that the resonant frequency is f0 = 1/2[(LC)1/2]. We can now treat this coil as a standard oscillator in coupled-mode theory by defining a(t) = [(L/2)1/2]I0(t).

给定这种关系和连续性方程，发现谐振频率为f0 = 1/2[(LC)1/2]。 我们现在可以通过定义a(t) = [(L/2)1/2]I0(t).，将该线圈视为耦合模式理论中的标准振荡器。

We can estimate the power dissipated by noting that the sinusoidal profile of the current distribution implies that the spatial average of the peak current-squared is |I0|2/2. For a coil with n turns and made of a material with conductivity , we modify the standard formulas for ohmic (Ro) and radiation (Rr) resistance accordingly:

我们可以通过注意到电流分布的正弦曲线意味着峰值电流平方的空间平均值为|I0|2/2来估计功耗。

对于具有n匝并且由导电a材料制成的线圈，我们相应修改欧姆（Ro）和辐射（Rr）电阻的标准公式：

The first term in Eq. 7 is a magnetic dipole radiation term (assuming r << 2c/); the second term is due to the electric dipole of the coil, and is smaller than the first term for our experimental parameters. The coupled-mode theory decay constant for the coil is therefore  = (Ro+ Rr)/2L, and its quality factor is Q = /2.

方程式的第一个术语 7是磁偶极辐射项（假设<< 2c/）;

第二项是由于线圈的电偶极子，并且小于我们的实验参数的第一项。

因此线圈的耦合模式理论衰减常数为  = (Ro+ Rr)/2L,，其品质因子为Q = /2。

We find the coupling coefficient DSby looking at the power transferred from the source to the device coil, assuming a steady-state solution in which currents and charge densities vary in time as exp(it) 我们通过观察从源到器件线圈的功率来找到耦合系数DS，假设电流和电荷密度随时间变化的稳态解决方案为exp(it)。

where the subscript S indicates that the electric field is due to the source. We then conclude from standard coupled-mode theory arguments thatDS= SD=  = M/2[(LSLD)1/2]. When the distance D between the centers of the coils is much larger than their characteristic size, scales with

theD3dependence characteristic of dipole-dipole coupling. Both  and  are functions of the frequency, and /and the efficiency are maximized for a particular 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.

其中下标S表示电场是源于源。 然后，我们从标准耦合模式理论论证得出结论：DS= SD=  = M/2[(LSLD)1/2].。 当线圈中心之间的距离D远大于其特征尺寸时，以偶极偶极耦合的D -

3

依赖特性为依据。

和是频率的函数，对于特定的f值，/和效率最大化，对于典型的感兴趣参数，其值在1〜50MHz的范围内。因此，为给定线圈选择合适的频率

尺寸，正如我们在这个实验演示中所做的那样，在优化功率传递方面起着重要的作用。 Comparison with experimentallydeterminedparameters.The parameters for the two identical helical coils built for the experimental validation of the powertransfer scheme are h = 20cm, a = 3mm, r = 30 cm, and n=5.25. Both coils are made of copper. The spacing between loops of the helix is not uniform, and we encapsulate the uncertainty about their uniformity byattributing a 10% (2cm) uncertainty to h. The expected resonant frequency given these dimensions is f0 = 10.56 ± 0.3MHz, which is about 5% off from the measured resonance at 9.90MHz.

与实验确定的参数进行比较。 用于实验验证功率传输方案的两个相同螺旋线圈的参数为h = 20cm，a = 3mm，r = 30cm，n = 5.25。 两个线圈均由铜制成。

螺旋环之间的间距不均匀，我们通过将10％（2cm）不确定度归因于h来封装其均匀性的不确定性。 给定这些尺寸的预期谐振频率为f0 =

10.56±0.3MHz，与9.90MHz的测量谐振相比，约为5％。

The theoretical Q for the loops is estimated to be approximately 2500 (assuming  = 5.9 × 107 m/) but the measured value is Q = 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m) at this frequency. We therefore use the experimentally observed Q and S= D=  = /2Q derived from it in all subsequent computations. 估计环路的理论值约为2500（假设

=

5.9

×

107

m/），但测量值为Q

=

950±50。我们认为这种差异主要是由于在铜线表面上导电不良的氧化铜层的影响，在这个频率下电流受到短的皮肤深度（〜20微米）的。

因此，我们在所有后续计算中使用实验观察到的Q和S= D=  = /2Q 。

We find the coupling coefficient  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. According to coupled-mode theory, this splitting should be  = 222)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).

我们通过将两个自谐振线圈（微调，通过稍微调整h，隔离时相同的谐振频率）相隔一段距离D并测量两个谐振模式的频率中的来实验地找到耦合系数。 根据耦合模式理论，这种应该是 的结果（图S1和S2）。

Measurement of the efficiency. The maximum theoretical efficiency depends only on the parameter /[(LSLD)1/2] = /, 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 probed. 测量效率。 合状态下运行。

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 calibrated 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 W/ so that it matches its optimal value, given theoretically by (1 + 2/2)1/2. (The loop connected to the light-bulb adds a small reactive component to Wwhich 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 its full nominal brightness. 作为我们的驱动电路，我们使用标准的Colpitts振荡器，其感应元件由半径为25厘米的单线铜线组成（图1）;

该电路环感应地耦合到源极线圈并驱动整个无线电力传输装置。

负载由校准的灯泡（14）组成，并且连接到其自身的绝缘电线回路，绝缘电线被放置在设备线圈附近并感应耦合到其上。

通过改变灯泡和器件线圈之间的距离，我们可以调整参数W/，使其与理论上由 2/2)1/2给出的最优值相匹配。

（连接到灯泡的回路为W添加一个小的无功分量，通过轻轻地重新调整线圈来补偿）。我们通过调整进入Colpitts振荡器的功率来测量所提取的功率，直到负载上的灯泡

(1

+

最大理论效率仅取决于参数/[(LSLD)1/2]

=

/，即使对于D

=

2.4m（八倍于线圈的半径），它也大于1（图3），因此，我们在所有探测距离范围内的强耦

=

222)1/2。

在本工作中，我们集中在两个线圈同轴对准的情况（图2），尽管对于其他方向获得了类似

以其全名义亮度发光。

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 the current parameters ISand IDused in our theoretical model. We then compute the power dissipated in each coil from PS,D=

L|IS,D|2, and obtain the efficiency from  = PW/(PS+ PD+PW). To ensure that the experimental setup is well described by a two-object coupled-mode theory model, we position 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 setup, fully lighting up a 60W light-bulb from distances more than 2m away (figs. S3 and S4).

我们通过用电流探针（不会明显降低线圈的Q）来测量每个自谐振线圈的中点处的电流来确定在源极线圈和负载之间发生的转换的效率。 ）这给出了我们理论模型中使用的当前参数IS和ID的测量。 我们然后计算从PS，D =消耗在每个线圈的功率PS,D=L|IS,D|2，并从= PW /（PS + PD + PW）获得效率。 为了确保通过双目标耦合模式理论模型很好地描述了实验装置，我们定位器件线圈，使其与连接到Colpitts振荡器的铜环的直接耦合为零。

实验结果如图4所示，与最大效率的理论预测一致，由式2我们能够使用这种设置转移大量的电力，从距离超过2m的距离（图S3和S4）完全照亮60W的灯泡。

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 efficiency 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 expected efficiency of roughly 40% for the wireless power transfer at that distance and the low efficiency of the Colpitts oscillator.

作为交叉检查，我们还测量从墙壁电源插座进入驱动电路的总功率。

然而，由于Colpitts振荡器本身的效率并不准确，因此无线传输本身的效率很难估计，尽管预计距离100％（15）远。

然而，提取的功率与进入驱动电路的功率的比率给出了效率的下限。 例如，当传送到负载超过2m的负载时，流入驱动电路的功率为400W。

这导致15％的总体墙对负载效率，这是合理的，因为在该距离处的无线功率传输的预期效率大约为40％，并且Colpitts振荡器的低效率。

Concluding remarks. It is essential that the coils be on resonance for the power transfer to be practical (6). We find experimentally that the power transmitted to the load drops sharply as either one of the coils is detuned from resonance. For a fractional detuning f/f0 of a few times the inverse loaded Q, the induced current in the device coil is indistinguishable from noise. 结语

为了实现电力传输，线圈必须进行谐振（6）。

我们实验发现，传输到负载的功率会随着任一个线圈与谐振失谐而急剧下降。 对于反向负载Q的几倍的分数失谐f/ f0，器件线圈中的感应电流与噪声无法区分。 A detailed and quantitative analysis of the effect of external objects on our scheme is beyond the scope of the current work, but we would like to note here that the power transfer is not visibly affected as humans and various everyday objects, such as metals, wood, and electronic devices large and small, are placed between the two coils, even in cases where they completely obstruct the line of sight between source and device (figs. S3 to S5). External objects have a noticeable effect only when they are within a few centimeters from either one of the coils. While some materials (such as aluminum foil, styrofoam and humans) mostly just shift the resonant frequency, which can in

principle be easily corrected with a feedback circuit, others (cardboard, wood, and PVC) lower Q when placed closer than a few centimeters from the coil, thereby lowering the efficiency of the transfer.

对外部物体对我们的方案的影响的详细和定量分析超出了目前的工作范围，但我们要在这里注意到，电力传递并不会受到人类和各种日常用品如金属，木材

和大型和小型的电子设备放置在两个线圈之间，即使在完全阻挡源和设备之间的视线（图S3至S5）的情况下也是如此。

外部物体只有当距任何一个线圈都在几厘米以内时才具有明显的效果。

虽然一些材料（如铝箔，泡沫聚苯乙烯和人类）大多只是移动谐振频率，这原则上可以用反馈电路轻松校正，但其他材料（纸板，木材和PVC）放置得比几厘米更低 从而降低了转换的效率。

When transferring 60W across 2m, we calculate that at the point halfway between the coils the RMS magnitude of theelectric field is Erms= 210V/m, that of the magnetic field is Hrms= 1A/m, and that of the Poynting vector is Srms= 3.2mW/cm2 (16). These values increase closer to the coils, where the fields at source and device are comparable. For example, at distances 20cm away from the surface of the device coil, we calculate the maximum values for the fields to be Erms= 1.4kV/m, Hrms= 8A/m, and Srms= 0.2W/cm2. The power radiated for these parameters is approximately 5W, which is roughly an order of magnitude higher than cell phones. In the particular geometry studied in this article, the overwhelming contribution (by one to two orders of magnitude) to the electric near-field, and hence to the near- field Poynting vector, comes from the electric dipole moment of the coils. If instead one uses capacitively-loaded single- turn loop design (6) - which has the advantage of confining nearly all of the electric field inside the capacitor - and tailors the system to operate at lower frequencies, our calculations show (17) that it should be possible to reduce the values cited above for the electric field, the Poynting vector, and the power radiated to below general safety regulations (e.g. the IEEE safety standards for general public exposure(18).)

当在2m处传输60W时，我们计算出，在线圈中间点的电场的RMS幅度为Erms = 210V / m，磁场的RMS幅度为Hrms = 1A / m，而坡印亭矢量的RMS值为Srms = 3.2mW / cm2（16）。这些值越来越接近线圈，源和器件上的场可比。例如，在距离器件线圈表面20cm的距离处，我们计算出场的最大值为Erms = 1.4kV / m，Hrms = 8A / m，Srms = 0.2W / cm2。为这些参数辐射的功率约为5W，这比手机高出一个数量级。在本文研究的特殊几何中，对电近场，从而对近场坡印廷矢量的压倒性贡献（一到两个数量级）来自线圈的电偶极矩。如果相反，一个使用容性加载的单回路环路设计（6） 其优点在于将几乎所有的电场在电容器内

并且使系统在较低频率下工作，我们的计算表明（17）应该可以将上述电场，坡印亭矢量和辐射功率降低到一般安全规定以下（例如，IEEE公开曝光安全标准（18））。

Although the two coils are currently of identicaldimensions, 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, as argued in (6).

尽管两个线圈当前具有相同的尺寸，但是可以使器件线圈足够小以适应便携式器件而不降低效率。 例如，如（6）所述，可以保持源极和器件线圈的特征尺寸的乘积恒定。 We believe that the efficiency of the scheme and the power transfer distances 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 (19). Nevertheless, the performance

- -

characteristics of the system presented here are already at levels where they could be useful in practical applications.

我们认为，通过镀银线圈可以显着提高该方案的效率和功率传递距离，这些线圈应该增加Q值，或者通过为谐振物体（19）处理更精细的几何形状。 然而，这里提出的系统的性能特征已经在可用于实际应用的水平上。 References and Notes

1. 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. Indust. 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. Soljačić, Ann. Phys., 10.1016/j.aop.2007.04.017(2007). 7. Here, by mid-range, we mean that the sizes of the devices which participate in the power transfer are at least a few times smaller than the distance between the devices. For example, if the device being powered is a laptop (size ~ 50cm), while the power source (size ~ 50cm) is in the same room as the laptop, the distance of power transfer could be within a room or a factory pavilion (size of the order of a fewmeters).

8. T. Aoki, et al., Nature 443, 671 (2006).

9. K. O’Brien, G. Scheible, H. Gueldner, 29th Annual Conference of the IEEE 1, 367(2003). 10. L. Ka-Lai, J. W. Hay, P. G. W., U.S. patent7,042,196 (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 donothave to be inductive. They may also be

connected by a wire, for example. We have chosen inductive coupling in the present work because of its easier implementation.

13.S. Sensiper, thesis, Massachusetts Institute of Technology (1951). 14.We experimented with various power ratings from 5W to 75W. 15.W. A. Edson, Vacuum-Tube Oscillators (Wiley, New York,1953).

16.Note that E c0H, and that the fields are out of phase and not necessarily perpendicular because we are not in a radiativeregime.

17.See supporting material on ScienceOnline.

18.IEEE Std C95.1—2005 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. The authors would like to thank John Pendry for suggesting the use of magnetic

resonances, and Michael Grossman and Ivan Čelanović for technical assistance. This work was supported in part by the Materials Research Science and Engineering Center program of the National Science Foundation under Grant No. DMR 02-13282, by the U.S. Department of Energy under Grant No. DE- FG02-99ER45778, and by the Army Research Office through the Institute for Soldier Nanotechnologies under Contract No. DAAD-19-02-D0002.

Fig. 1. Schematic of the experimental setup. 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 ’s represent direct couplings between the objects indicated by the arrows. The angle

between coil D and the loop A is adjusted to ensure that their direct coupling is zero, while coils S and D are aligned coaxially. The direct couplings between B and A and between B and S are negligible.

图1.实验设置示意图

A是半径为25cm的单个铜环路，它是驱动电路的一部分，其输出频率为9.9MHz的正弦波。 S和D分别是文中引用的源和器件线圈。 B是连接到负载（“灯泡”）的线圈。 各种表示由箭头指示的物体之间的直接耦合。

调节线圈D和环路A之间的角度，以确保它们的直接耦合为零，同时线圈S和D同轴对齐。 B和A之间以及B和S之间的直接耦合可以忽略不计。

Fig. 2. Comparison of experimental and theoretical values for as a function of the separation between coaxially aligned source and device coils (the wireless power transfer distance.)

图2.作为同轴对准的源极和器件线圈之间的间隔的函数的的实验和理论值的比较（无线功率传输距离）

Fig. 3. Comparison of experimental and theoretical values for the parameter / as a function of the wireless power transfer distance. The theory values are obtained by using the theoretical  and the experimentally measured . The shaded area represents the spread in the theoretical / due to the 5% uncertainty in Q.

图3。该参数的/实验值和理论值作为一种无线电力传输的距离函数的比较。采用理论与实验得到的理论值。阴影区域代表由于在问：5%不确定性理论/蔓延

Fig. 4. Comparison of experimental and theoretical efficiencies as functions of the wireless power transfer distance. The shaded area represents the theoretical prediction for maximum efficiency, and is obtained by inserting theheoretical values from Fig. 3 into Eq. 2 [with

The black dots are the maximum efficiency obtained from Eq. 2

and the experimental values of / from Fig. 3. The red dots present the directly measured efficiency, as described in thetext.