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IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 5, MAY 2011
Relay Effect of Wireless Power Transfer Using Strongly Coupled Magnetic Resonances Fei Zhang1 , Steven A. Hackworth1 , Weinong Fu2 , Chengliu Li1 , Zhihong Mao1 , and Mingui Sun1 1
Departments of Neurosurgery and Electrical Engineering, University of Pittsburgh, Pittsburgh, PA 15260 USA 2 Department of Electrical Engineering, Hong Kong Polytechnic University, Hong Kong
Wireless power transfer using strongly coupled electromagnetic resonators is a recently explored technology. Although this technology is able to transmit electrical energy over a much longer distance than traditional near field methods, in some applications, its effective distance is still insufficient. In this paper, we investigate a relay effect to extend the energy transfer distance. Theoretical analysis is performed based on a set of coupled-mode equations. Experiments are conducted to confirm the theoretical results and demonstrate the effectiveness of the relay approach. Our results show that the efficiency of power transfer can be improved significantly using one or more relay resonators. This approach significantly improves the performance of the present two-resonator system and allows a curved path in space to be defined for wireless power transfer using smaller resonators. Index Terms—Relay effect, strongly coupled magnetic resonance, wireless electricity, wireless power transfer.
I. INTRODUCTION
T
RANSMITTING power without wires or cables [1]–[4] has many applications, such as wirelessly recharging mobile devices and powering an array of wireless sensors. The currently available wireless power transfer techniques can be roughly classified into the radiative and nonradiative modes. The first class includes radio-frequency transmission using highly directional antennas, while the second class includes widely utilized inductively coupled transformers. However, high-power radiation poses safety concerns and requires a complicated tracking system, while nonradiative modes only work in close-range across a small gap. To overcome these problems, a new wireless power transfer technology, commonly called wireless electricity, has been reported [5], [6]. This technique was based on nonradiative strongly coupled magnetic resonance in the midrange, defined as several times the resonator size. With two identical coils utilized as wireless electric resonators, a 60 watt bulb was fully illuminated wirelessly seven feet away from the power source with a power transfer efficiency of 40% [5]. This pioneer system worked well even when the line-of-sight between the two resonators was blocked by a nonresonant object, yet was almost non-respondent to nonresonant objects. The wireless electricity has been studied for practical application to a variety of fields, such as transportation [7], biomedicine [8], [9] and consumer electronics [10]. Recently, new wireless electric system designs were reported, such as powering multiple devices with a single source [11], [12] and transferring power to two devices on both sides of a source with a higher efficiency [13]. Despite these significant developments [7]–[13], the current approaches are still based on the original scheme of “source-device” or “source-devices”. As a result, the effective transmission distance is still limited to the midrange. In this work, we investigate a relay effect in magnetic resonance and present a new ‘source-relay(s)-device’ scheme. We will demonstrate, by both Manuscript received May 29, 2010; revised July 29, 2010; accepted October 05, 2010. Date of current version April 22, 2011. Corresponding author: M. Sun (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2087010
theoretical treatments and experimental studies, that this scheme extends beyond the limit of midrange, allowing much longer and more flexible power transmission without sacrificing efficiency. II. INTERACTION IN LOSSLESS PHYSICAL SYSTEM Coupled mode theory based on a set of integrable coupled mode equations accounts well for the mechanisms of the wireless electric system, where power swapping follows a periodic manner and can be expressed analytically. Considering the modal signals (or state variables in terms of linear dynamic systems) a1 (t) and a2 (t) of two lossless objects with natural frequencies !1 and !2 , we have [14] da1 (t) dt da2 (t) dt
= i!1 a1 (t) + k12 a2 (t)
(1)
= i!2 a2 (t) + k21 a1 (t)
(2)
Where k12 and k21 are the coupling coefficients between two modes. According to conservation of energy, the time rate of energy change must vanish and then can be expressed as d 2 2 ja1 j + ja2 j dt
da31 (t) da1 (t) 3 + a1 (t) dt dt da23 (t) da2 (t) 3 + a2 (t) + a2 (t) dt dt
= a1 (t)
=0
(3)
Substituting (1) and (2) into (3), we have 3 3 a1 (t)k12 a2 (t) + a31 (t)k12 a2 (t)+ 3 3 a2 (t)k21 a1 (t) + a32 (t)k21 a1 (t) = 0
(4)
Since a1 (t) and a2 (t) can have arbitrary initial amplitudes and phases, the coupling coefficients should be related by 3 k12 + k21
= 0:
(5)
We can obtain two homogeneous equations in a1 (t) and a2 (t) from (1) and (2). If a1 (t) = A1 1 ej!t , we will have ! 2 0 (!1 + !2 ) 1 ! + (!1 !2 + k12 k21 ) = 0
(6)
which yields the roots !=
!1 + !2 2
0018-9464/$26.00 © 2011 IEEE
6
!1 0 !2 2
2
+ jk12 j2
!1 + !2 2
6
0
(7)
ZHANG et al.: RELAY EFFECT OF WIRELESS POWER TRANSFER
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Fig. 2. Basic components of present wireless electric system. Fig. 1. Energy exchanges in two-object lossless system. (a) Symmetric resonant case. (b) Nonresonant case.
It indicates that the two frequencies of the coupled system are separated by 0 . In particular, when !1 = !2 , the difference between the two natural frequencies of the coupled modes is 2 0 (or 2jk12 j). Suppose, initially, that at t = 0, a1 (0) and a2 (0) are specified, then the two solutions of (1) and (2) are expressed by
a1 (t) = a1 (0) cos 0 t 0 j !2 0 !1 sin 0 t
2 0
+ k12 a2 (0) sin 0 t 1 ej[(!
0 a2 (t) = k21 a1 (0) sin 0 t
0
(8) Fig. 3. Energy exchanges of a two-object lossy system. (a) Resonant strong coupling with identical resonators. (b) Resonant strong coupling for resonators with different decays. (c)Resonant weak coupling. (d) Nonresonant case.
+a2 (0) cos 0 t j !12 !2 sin 0 t 0 0
1
+! )=2]t
ej [(!
0
+! )=2]t
(9)
Let us consider the case where a1 (0) = 1, a2 (0) = 0 and !1 = !2 = ! , we have a1 (t) = cos 0 t 1 ej!t and a2 (t) = sin 0 t 1 ej!t . Mode 1 is fully excited at t = 0, but at 0 t = =2, all the excitation appears in mode 2. At 0 t = , the excitation returns to mode 1
and mode 2 is unexcited. The process repeats periodically. Hence, the excitation is transferred back and forth with frequency 2 0 (or 2jk12 j). Fig. 1 shows energy exchange between two modes. Here, Fig. 1(a) indicates that resonant energy swapping (k = 500; 000 f1 = f2 = 1 MHz) can be efficient and complete. If f1 6= f2 (k = 500; 000, f1 = 1:2 MHz, f2 = 1 MHz), the energy exchange (Fig. 1(b)) is not complete, but partial and inefficient. III. INTERACTION IN REAL PHYSICAL SYSTEM A. Two-Object Lossy System To easily link coupled mode theory to the present wireless electric system, we have the following coupled differential equations:
daS (t) = (i! 0 0 )a (t) + ik a (t) (10) S S S SD D dt daD (t) = (i! 0 0 )a (t) + ik a (t) (11) D D D DS S dt where aS (t) and aD (t) denote, respectively, the modal signals at the source and device objects, !S;D = 2fS;D are the individual angular frequencies, jkSD j = jkDS j are the coupling coefficients,
and 0S;D are the individual intrinsic decay rates. Applying the Laplace transform method to (10) and (11), we can obtain aS (t)=e (0(b 0 a)) sin dt =d + cos dt (12)
aD (t)=e
2ik sin where we define a = i!S d = sqrt(4k2 a2 + 2ab
dt
2
2
=d (13) 0 0S , b = i!D 0 0D , c = a + b and 2 0 0 b ), and assume the source possesses the full amount of energy (normalized) at t = 0.
2
1) Resonant Case: If we assume that the source and device are identical, we have ! = !S = !D , 0 = 0S = 0D and k = jkSD j = jkDS j. Under these conditions, (12) and (13) yield aS (t) = e(i!00)T cos(kt) and aD (t) = ie(i!00)T sin(kt).
Then, the total energy of this system, decreasing at a exp(020t) rate, is expressed by P (t) =pjaS (t)j2 + jaD (t)j2 = e020t . a) Strong coupling k=p 0S 0D 1: As a key distance-dependent figure-of-merit, k= 0S 0D is known as the relative coupling parameter. It represents, intuitively, the ratio of “how fast energy is transferred between the source and device” to “how fast it is dissipated due to intrinsic losses in these resonators”. Our desired p strong coupling regime for wireless electricity is provided by k= 0S 0D 1. When this inequality is satisfied, the coupling rate is much higher than the loss rate. However, to realize a wireless electric system, “strong coupling” and “resonance” must be guaranteed simultaneously. The fact that the coupling, in theory, is inversely proportional to the cube of the separation distance, and in practice, may decay even more steeply, leading to the ‘mid-range’ limitation of the present wireless electric systems. As shown in Fig. 2, the current wireless electric system consists of two resonators (Source and Device), a driving loop, and an output loop. The source resonator is coupled inductively with the driving loop linked to an oscillator to obtain energy for the system. Similarly, the device coil is coupled inductively with the output loop to supply power to an external load. The characteristics of the coupled system described in Section III-A-1-a provide insights into the working of wireless electricity. The energy exchange for fS = fD = 1 MHz, k = 500; 000 and 0 = 1; 000 is shown in Fig. 3(a). It can be seen that the source and device energies are continually and completely exchanged via a strong energy channel. Since the range of the near-field surrounding a finite-sized resonant object is proportional to the wavelength, this midrange nonradiative resonant coupling can only be achieved using sub-wavelength resonators and thus significantly longer evanescent field-tails [6]. This condition is also true for source and device with different decay rates (0S = 100 and 0D = 10; 000) as shown in Fig. 3(b). The total energy decay at a rate of exp(0(0S + 0D )t) should be compensated by an externalpsource to maintainpoperation. b) Weak coupling k= 0S 0D 1 or k= 0S 0D < 1: If k=0 1 is not satisfied, the system will not resonate since system energy will be lost before an avenue for wireless energy transfer is formed in space. This case is shown in Fig. 3(c) with k = 500; 000 and 0 = 250; 000. It is clear that, in order to transmit power at a
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 5, MAY 2011
high efficiency, the coupling rate should be much higher or faster than the decay rate. 2) General Nonresonant Case: It is known that two non-resonant objects (e.g., Fig. 3(d) with fS = 1 MHz, fD = 2 MHz, k = 500; 000 and 0 = 1; 000) interact weakly and exchange energy ineffectively. This is also true even under strong coupling, as is the case in Fig. 3(d). It can be observed that the energy absorbed by the device is always very small and the total system energy is also decaying with a rate of exp(0(0S +0D )t). Hence, no efficient energy exchange and transfer are achieved in this non-resonant case.
Fig. 4. Main components of relayed wireless electric system.
B. Multi-Object Resonant System The previously described results tell us that strongly coupled resonance can increase efficiency by effectively ‘tunneling’ the magnetic field from the source to the device. Specifically, when two resonators are in the midrange (namely a few times the resonator size), their near fields (evanescent waves) will strongly couple with each other, which will allow the total energy focused in a specific resonant frequency to tunnel/transfer from one resonator to the other resonator within times much shorter than the times of losses in the system. However, the current system configuration with the “Source-Device” scheme must operate in the midrange, which is often too short in practical applications, especially when the resonator size must be made small. We attack this fundamental limitation by providing a relay solution with additional relay resonator(s). For the relayed system, energy transfer operation is modeled in the matrix shown
. . .
. . .
. . .
. . .
. . .
..
. . .
.
. . .
(14)
where subscript n represents an n-object physical system. As shown in Fig. 4, the relayed wireless electric scheme consists of at least three resonators (source, relay(s) and device), a driving loop and an output loop. The resonant evanescent strong coupling mechanism between source and device can be mediated by the presence of the relay(s). Through the overlaps of the non-radiative lossless near-fields both between source and relay and between relay and device, the distance of efficient energy transfer is extended despite k not being large enough to provide strong coupling at the distance between the source and device. In this way, “far-range” wireless power transfer can be achieved. In order to clearly understand how this system works, for the ease of analysis, we will focus on the resonant system having three identical resonators. With the initial conditions aS (0) = 1, aR (0) = 0 and aD (0) = 0, from (14) we can obtain
p = e i!0 t cos( p2kt)=2p+ e i!0 = ie i!0 t sin(p 2kt)= 2 aD = e i!0 t cos( 2kt)=2 0 e i!0 aS aR
(
0)
(
(
(
0)t
=2
(
0)t
=2
0)
0)
(15) (16) (17)
In Fig. 5(a), the total system energy decays at a rate of
exp(030t) (fS;R;D = 1 MHz, k = 500; 000, 0 = 1000). It can
be observed that the energy of the relay resonator is kept at a lower level compared to source and device. It is interesting that
Fig. 5. Energy exchange for one-relay and conventional systems. (a) One-relay system; (b) conventional system.
Fig. 6. Fabrication and measurement of designed resonator. (a) Resonator design; (b) measured smith chart.
the energy leaves the relay (to device) as soon as it reaches relay (from source), and that the frequency of energy oscillation of the relay is twice the device or the source. With the presence of the relay resonator, the device and source can interact more strongly, exchange energy more quickly and deliver power more efficiently when compared with the system without the relay resonator (Fig. 5(b)). Obviously, if we increase the time axis in Fig. 5(b), a sinusoidal energy exchange can still take the maximum at a certain time point. However, at this time, the total system energy will be reduced to a much lower level due to the losses of the system. IV. EXPERIMENTAL RESULTS We constructed several identical resonators with a thin-film design (Fig. 6(a), details are provided in [8]) resonating at 7 MHz. The measured Q value was 50 based on impedance reflection using the Agilent 8753ES vector network analyzer. The actual value could be higher because of the practical difficulty in precise Q-measurement. The radius and height of the resonator were 8.1 cm and 5 cm, respectively. Due to the unique structure of multiple conductor strips, three main resonant frequencies were obtained from the smith chart (Fig. 6(b)) at 7.02 MHz, 24.64 MHz and 47.85 MHz, which match well with our simulation (6.91 MHz, 25.88 MHz, and 51.05 MHz, respectively). This enables simultaneous power transfer and data communication through the use of different resonant frequencies. We tuned all resonators individually to the same value using the Agilent 8753ES vector network analyzer by slightly adjusting the width of the vertical strip conductors in our resonator (Fig. 6(a)). This adjustment varied the capacitance of the LC resonant tank circuit resulting in the desired frequency tuning. To facilitate both quantitative and qualitative evaluations of the system, we used either a resistor (for quantitative measurement) or an LED (for qualitative demonstration) as the system load. A seven-turn coil is used as the output loop to connect the system load. Fig. 7(a) shows a functional 7 MHz wireless electric system without relay over a 50 cm separation. Due to the long distance, the LED could not be lit. However, when we placed a relay resonator in the system, the LED was illuminated fully, as shown in Fig. 7(b).
ZHANG et al.: RELAY EFFECT OF WIRELESS POWER TRANSFER
Fig. 7. Conventional and relayed power transfer in the vertical direction. (a) Conventional system; (b) one-relay system; (c) double-relay system.
Fig. 8. Measurements of conventional and relayed wireless electric systems. (a) Efficiencies versus distance. (b) Location of optimal relay position.
Fig. 9. Directional guidance via the relay resonator.
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V. DISCUSSIONS As for the practical issues raised in [15], there is still a long way for wireless electricity to be widely and safely adopted in the human society, replacing some of the presently wired systems. In this early stage of development, theoretical advancements are important and necessary. This paper thus focuses on theoretical aspects and proof of concepts rather than particular applications of wireless electricity.Nevertheless,weenvisionmanypotentialapplications,such as transportation, industry, medicine, and consumer electronics in which prototypes of wirelessly powered laptop, TV and cell phone have been demonstrated using the two-resonator designs [10]. We believe that the relay effect studied in this paper will alleviate the short-range problem in some of the present designs and accelerate practical adaptations of the wireless electricity. VI. CONCLUSION In this paper, an extension of the present ‘source-device’ wireless electric scheme has been presented, where one or more relay resonators were added to extend the transmission distance, increase power transfer efficiency, and allow a curved wireless transmission path in space. The proposed ‘source-relay(s)-device’ scheme overcomes the midrange limitation of the present wireless electric systems and paves the way for new applications which require long range, arbitrarily directed wireless energy transfer, and/or small resonators. ACKNOWLEDGMENT The authors acknowledge the help with RF instruments and measurements from the group of Professor Marlin H. Mickle. This work was supported in part by US Army contract No. W81XWH-050C-0047 and NIH grant No. U01 HL91736.
REFERENCES Fig. 7(c) shows that the distance between the source and device can beincreasedandmorepowercanbetransferredwithonemorerelay. After replacing the LED with a resistor, we measured RF power values at both the input terminals of the driving loop and the load terminals of the output loop. The power transfer efficiencies are shown in Fig. 8(a). It can be seen that the conventional wireless electric system can only achieve about 10% efficiency over a 30 cm distance, whereas the efficiency of the relayed system can reach up to 46%. Fig. 8(b) shows the power transfer efficiency as the relay resonator moved between the source and device resonators which were separated by a distance of 50 cm. It was observed that the midpoint between source and device is the best position for the relay to achieve the maximum efficiency in the “source-relay-device” scheme. In a separate experiment, we found that we could utilize the nature of magnetic resonances to guide the direction of magnetic flux and hence control the transmission direction using relays. We believe that this was due to the attraction of the magnetic flux by relay. Thus, with the help of relay(s), a wireless energy connection can turn a corner, just like a wire connection. As shown in Fig. 9(a), if there was no relay, the LED, which was connected to the output coils mounted on the device resonator, did not glow. After we placed a relay resonator as shown in Fig. 9(b), the LED was illuminated. This indicated that the relay system guided magnetic flux around the 90 corner. After realizing this ‘L’ shaped wireless connection, in Fig. 9(c), we added two additional relays to form a ‘Z’ shape. Removing any of the two relays turned off the LED, as Fig. 9(d), where the first relay was removed. These observed phenomena strongly support the promise of future applications of “source-relay(s)-device” scheme.
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