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EfficiencyAnalysisofMagneticResonanceWirelessPowerTransferWithIntermediateResonantCoil

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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 10, 2011 389

Efficiency Analysis of Magnetic Resonance WirelessPower Transfer With Intermediate Resonant Coil

JinWook Kim, Hyeon-Chang Son, Kwan-Ho Kim, and Young-Jin Park, Member, IEEE

Abstract—This letter presents an efficiency analysis of a mag-netic resonance wireless power transfer (WPT) system with anintermediate resonant coil. A helical coil and a spiral coil withan additional capacitor are considered as resonant coils for theWPT system. The intermediate resonant coil is set up coaxiallyand perpendicular to both the Tx and Rx resonant coils in orderto observe the efficiency change according to the directions.The power efficiency is calculated using the temporal coupledmode theory (CMT). Impedance matching conditions are alsoshown by using the CMT. Analysis results show that using anintermediate coil properly improves efficiency and extends thedistance between the transmitter and receiver. Both calculatedand measured efficiencies are in good agreement. It is also shownthat the intermediate resonant system has a good efficiency and issuperior to nonintermediate systems.

Index Terms—Coupled mode theory (CMT), intermediate coil,magnetic resonant coupling, wireless power transfer (WPT).

I. INTRODUCTION

W IRELESS power transfer (WPT) using magnetic fieldresonance in a near-field region has attracted much at-

tention since WPT via strongly coupled magnetic resonanceswas reported [1]–[4]. In [2] and [4], a resonance-based WPTsystem is analyzed using a circuit-based model, and design andoptimization procedures were reported.

Recently, an intermediate resonant coil between transmit-ting (Tx) and receiving (Rx) resonant coils was used [5].Compared to the two resonant coil systems in [2] and [4],an intermediate resonant coil system with the same resonantfrequency between Tx and Rx resonant coils can be appliedeffectively to extend the distance of power delivery or increasepower transfer efficiency. However, to date, detailed analysesof power transfer efficiency depending on Tx, Rx, and interme-diate resonant coils are not available.

In this letter, the power efficiency of a magnetic resonancewireless power transfer system with an intermediate resonantcoil is analyzed. The intermediate resonant system (intermediatesystem) is very useful in extending transfer distance and can beapplied to household electric appliances such as a wall-mountedTV. In particular, the intermediate resonant coil is geometricallyperpendicular to a Tx resonant coil and a Rx resonant coil. Aspiral coil is used to reduce the volume of the intermediate res-onant coil, while helical coils are used for Tx and Rx resonant

Manuscript received March 08, 2011; accepted April 21, 2011. Date of pub-lication May 02, 2011; date of current version May 16, 2011.

The authors are with the Department of Power Electric Equipment Informa-tion and Communication, University of Science and Technology (UST) andKorea Electrotechnology Research Institute (KERI), Ansan 426-170, Korea(e-mail: yjpark@keri.re.kr).

Color versions of one or more of the figures in this letter are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LAWP.2011.2150192

Fig. 1. Configuration of a magnetic resonance WPT system with an interme-diate resonant coil.

coils. High- capacitors are added to the coils in order to ad-just the resonant frequency of the resonant coils and reduce theworse effects on power transfer. Otherwise, variation in the res-onant frequency caused by external objects is increased due tothe small amount of self-capacitance in the helical and spiralcoils.

In the following sections, efficiencies derived in the inter-mediate system are shown using the CMT and the modifiedCMT. The efficiencies are compared to an arrangement in whichthe intermediate resonant coil is changed coaxially and perpen-dicularly to the normal direction of both the Tx and Rx res-onant coils. The frequency shifting and change of impedancematching conditions are also shown. The measured data arecompared to theoretical calculations.

II. DERIVATION OF THE POWER TRANSFER EFFICIENCY

Fig. 1 shows the configuration of a magnetic resonance WPTwith an intermediate resonant coil. The variables of , ,

, , and are denoted as coupling coefficientsbetween coils. The intermediate resonant coil is placed be-tween the Tx and Rx resonant coils, and the central axis is notaligned with that of Tx and Rx resonant coils. and arecenter-to-center spacing distances between the Tx resonant coiland the intermediate resonant coil, between Tx and Rx resonantcoils, respectively.

By applying CMT, the intermediate system is presented asfollows:

(1)

1536-1225/$26.00 © 2011 IEEE

390 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 10, 2011

where

1 (Tx), 2 (Rx), and (intermediate resonant coil);

mode amplitude of each resonant coil;

resonant angular frequency of each resonant coil;

intrinsic decay rate of each resonant coil;

coupling coefficient between resonant coils (or a coiland a resonant coil);relative decay rate in the Rx resonant coil due to thepower transfer to the load coil [6].

Power transfer efficiency for the WPT system using an inter-mediate resonant coil is derived as shown in (2) at bottom of thepage. For maximum power efficiency, the parameter is derivedas follows:

(3)

Now, by using the power efficiency formula in (2), the powertransfer efficiency is analyzed in three different practical cases.

A. Case 1 ( , , )

In this case, the Tx and Rx resonant coils are identical. How-ever, an intermediate resonant coil is different from the Tx andRx resonant coils. The coupling coefficient between the Tx andintermediate resonant coils is not the same as the coupling co-efficient between the Rx and intermediate resonant coils. islarge enough to neglect the direct coupling coefficient betweenthe Tx and Rx resonant coils.

By substituting for and for at (2)and (3), respectively, the power efficiency can be simplified asfollows:

(4)To maximize the efficiency, the parameter,

should be satisfied with (5) when [1]

(5)By the relation among the arithmetic, geometric, and har-

monic averages, the maximum efficiency can be achieved forthe case , that is, because is included

Fig. 2. Efficiencies of nonintermediate and intermediate systems with varying���� � � � � ���� � .

in common. Thus, maximum efficiency is obtained with the in-termediate resonant coil in the center between the Tx and Rxresonant coils.

B. Case 2 ( , , )

The condition for maximum efficiency is obtained asfrom Case 1. The system is satisfied with

from Case 2 in general cases. On the basis of the prior case, thepower transfer efficiency is derived as follows:

(6)The efficiency is expressed as the function of variable

. The case is compared to a noninterme-diate case—that is, the system of only Tx and Rx resonant coilswithout the intermediate resonant coil.

Fig. 2 shows the efficiencies for the two cases with varying. The circular-marked line and the square-marked line indi-

cate the efficiencies of a nonintermediate system and an interme-diate system, respectively. The results show that the efficiencyof the intermediate system is 5% points less than that of the non-intermediate system for the value of lower than 100, whilethe efficiencies of both cases for higher than 100 are almostthe same. In addition, it should be noted that an efficiency ofbetter than 90% can be achieved even in a perpendicularly ar-ranged intermediate system with higher than 27.

C. Case 3 (Case 2 Including Source and Load Coils)

In this case, the source and load coils are added in the con-dition of Case 2. For the analysis, a modified CMT formula is

(2)

KIM et al.: EFFICIENCY ANALYSIS OF MAGNETIC RESONANCE WIRELESS POWER TRANSFER WITH INTERMEDIATE RESONANT COIL 391

Fig. 3. Efficiencies versus normalized frequency for three different matchingconditions �� � ���.

obtained by referring to [7] and is shown in (7) at the bottom ofthe page.

Here, are field amplitudes of incident field and re-flect field at the source, and are field amplitudes at theload. The rates of field amplitudes are scat-tering parameters. By applying the conditions obtained in theprior cases ( , ), the field amplitudetransmitted to the load from source is obtained as

(8)

where , ,, . Power transfer efficiency

is also obtained as follows:

(9)

The impedance matching condition is obtained by finding themaximum power transfer efficiency of (9). The matching condi-tion is when . Thederived efficiency formula is identical with (6) for the matchingcondition.

Fig. 3 shows efficiencies versus normalized frequency forthree different matching conditions with and .In the case of the undercoupling condition , threepeak frequencies are observed. Maximum power transferefficiency is obtained at the center frequency. In the case ofthe overcoupling condition , peak frequencyis observed at the center frequency. In the critical coupling

, the system has the best efficiency at the centerfrequency compared to other conditions. In the under- and

Fig. 4. Schematic drawing of the Tx/Rx and intermediate resonant coils.

TABLE ISPECIFIC PARAMETERS OF RESONANT COILS AND COUPLING COEFFICIENTS

overcoupling conditions, maximum power transfer efficiencycannot be achieved. Thus, satisfying the impedance matchingcondition is very important for higher efficiency.

III. EFFICIENCY OF THE INTERMEDIATE SYSTEM

Tx and Rx resonant coils are designed using a helical coil( mm, mm, turns, mm)as shown in Fig. 4 (left). The intermediate resonant coil is asingle-layer spiral coil ( mm, mm,

turns, mm) as shown in Fig. 4 (right). Two coilsare made of copper pipe. To adjust the resonant frequency ofthe resonant coils, a lumped high- capacitor is connected inparallel. Single-loop coils are made to form a source coil anda load coil. They are then placed for coupling and impedancematching as shown in Fig. 1.

A vector network analyzer (Agilent 4395A) is used to findthe resonant frequency and measure transmission behavior be-tween Tx and Rx. The measured intrinsic decay rate, -factor,connected lumped capacitance, and resonant frequency of theresonant coils are illustrated in Table I. The results show that theresonant frequencies of the fabricated resonant coils are almostthe same because the resonant frequencies are adjusted using

(7)

392 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 10, 2011

Fig. 5. Fabricated intermediate system (a) coaxially arranged intermediatesystem and (b) perpendicularly arranged intermediate system.

high- capacitors. For measuring the coupling coefficient ,splitting frequencies between the Tx (or Rx) coil and the in-termediate resonant coil are checked according to distance asshown in [8].

Fig. 5 shows two fabricated intermediate systems: One iswith the intermediate resonant coil coaxially arranged; theother is with the intermediate resonant coil perpendicularlyarranged. The intermediate resonant coil is placed in the centerbetween the two coils. For the perpendicularly arranged inter-mediate system, the spiral intermediate resonant coil is placedat ( 230 mm, 0, ). As shown in Table I, are about fourtimes and five times bigger than in the cases of perpen-dicularly and coaxially arranged systems, respectively, when

cm, cm. Thus, the condition issatisfied in the distance over 80 cm. The matching conditionsare changed according to movement of the source and loadcoils. Therefore, the source and load coils are set up properly toget the critical coupling in the Case 3 for the maximum powertransfer efficiency.

Fig. 6 shows the comparison between theoretical and mea-sured results for three different systems. The measured intrinsicdecay rates of coils and coupling coefficients are used in theoret-ical calculation. First, the calculated and measured results of thenonintermediate system are the circular-marked solid line andthe gray circles, respectively. Second, the square-marked andtriangular-marked solid lines are the theoretical results of thecases in which the intermediate systems were arranged coaxiallyand perpendicularly, respectively, while the gray squares and tri-angles are the measured results. Finally, the measurements andthe calculations for the three different systems are shown. Thecomparison results for three cases show that the calculations arein excellent agreement with the measurements. Moreover, theefficiency of the coaxially arranged intermediate system is thebest at the same distance. The reason is that the coupling coef-ficient is the highest because the strongest magnetic field existsat the normal direction of the helical coil.

IV. CONCLUSION

The efficiency formulas of magnetic resonance wirelesspower transfer using an intermediate resonant coil were derived

Fig. 6. Efficiencies in cases of the coaxially and perpendicular arranged inter-mediate systems and nonintermediate system. �� � � ���.

and analyzed by using temporal CMT in the intermediatesystem. The theoretical calculations have a good agreementwith the measured results. It is shown that, compared to thenonintermediate system, efficiency is improved considerably inthe cases of not only a coaxially arranged intermediate system,but also a perpendicularly arranged intermediate system. It isalso shown that the power efficiency of the coaxially arrangedintermediate system is the best. However, from a practical pointof view, the perpendicularly arranged intermediate systemcan be widely used to extend the range of wireless powertransfer and enhance efficiency since the intermediate resonantcoil can be implemented adaptively in the space. Examplesare wall-mounted TVs and furniture embedded systems. Thederived formulas can be also used for optimization of theefficiency in the intermediate system.

REFERENCES

[1] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, andM. Soljacic, “Wireless power transfer via strongly coupled magneticresonances,” Sci. Exp., vol. 317, pp. 83–86, Jul. 2007.

[2] A. P. Sample, D. T. Meyer, and J. R. Smith, “Analysis, experimentalresults, and range adaptation of magnetically coupled resonators forwireless power transfer,” IEEE Trans. Ind. Electron., vol. 58, no. 2, pp.544–554, Feb. 2011.

[3] Y. D. Tak, J. M. Park, and S. W. Nam, “Mode-based analysis of res-onant characteristics for near-field coupled small antennas,” IEEE An-tennas Wireless Propag. Lett., vol. 8, pp. 1238–1241, 2009.

[4] A. K. Ramrakhyani, S. Mirabbasi, and M. Chiao, “Design and opti-mization of resonance based efficient wireless power delivery systemfor biomedical implants,” IEEE Trans. Biomed. Circuits Syst., vol. 5,no. 1, pp. 48–63, Feb. 2011.

[5] R. E. Hamam, A. Karalis, J. D. Joannopoulos, and M. Soljacic, “Effi-cient weakly-radiative wireless energy transfer: An EIT-like approach,”Ann. Phys., vol. 324, pp. 1783–1795, 2009.

[6] A. Kurs, R. Moffatt, and M. Soljacic, “Simultaneous mid-range powertransfer to multiple devices,” Appl. Phys. Lett., vol. 96, pp. 044102-1–044102-3, 2010.

[7] H. A. Haus, Waves and Fields in Optoelectronics. Upper SaddleRiver, NJ: Prentice Hall, 1984, pp. 197–234.

[8] A. Karalis, “Novel photonic phenomena in nanostructured materialsystems with applications and mid-range efficient insensitive wirelessenergy-transfer,” Ph.D. dissertation, Dept. Elect. Eng. Comput. Sci.,MIT, Cambridge, MA, 2008.