Effect of Coupling Between Multiple Transmitters …arazprojects.com/wp-content/uploads/10.1109-TIE.2012...power transfer is also derived under the operation of maximum efﬁciency

2602 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 7, JULY 2013

Effect of Coupling Between Multiple Transmitters orMultiple Receivers on Wireless Power Transfer

Dukju Ahn and Songcheol Hong, Member, IEEE

Abstract—The operation of wireless power transfer systemswith multiple transmitters (TXs) or receivers (RXs) is investi-gated. With multiple TXs or RXs in a limited space, couplingsoccur between TXs or between RXs. The frequency conditionsfor maximum efficiency and power transfer under such couplingsare proposed. Effective resonant frequency of the TXs or the RXsis changed due to such couplings, and driving and/or resonantfrequencies should therefore be adjusted accordingly. The amountand type of the required adjustments are provided. The efficienciesin these conditions are discussed. These concepts are supportedby experiments with couplings between TXs or between RXs.Using the proposed frequency adjustments, 51–65-W power istransferred with 45%–57% efficiency, even with very low couplingcoefficients of 0.025–0.063 from TX to RX. This improvement issignificant compared to the unadjusted cases where less than 4 Wis transferred with only 5%–33% efficiency.

Index Terms—Contactless energy delivery, coupled magneticresonance, frequency tuning, inductive link, inductive powertransfer, wireless energy transmission.

I. INTRODUCTION

W IRELESS POWER TRANSFER (WPT) is now becom-ing more important because of the user-friendly nature

of wireless technology. In contrast, power cables are bulky andlimit the mobility of many electronic appliances. There havebeen attempts to enable a cord-free desk by embedding primarycoils under the desk [1], [2]. In addition, many studies havebeen performed on contactless battery charging for portable cel-lular phones [3]–[6]. Applications to medical implants are alsoconsidered to be important due to the difficulties of insertingpower cables across the skin [7]–[11]. Other applications rangefrom integrated circuit power supply [12], [13] to space vehicleoperation [14]. Recently, mobile laptop charging [15], [16]and roadway-powered electric vehicles [17], [18] are garneringmore interest. Industries are now actively establishing andunifying wireless power standards. For example, the WirelessPower Consortium released the Qi specifications [19]. Theequivalent circuit approach [15], [30], [35]–[39] has meanwhilebecome popular in analyzing the WPT very recently.

One of the advantages and required features of WPT ispower transfer ability from a single transmitter (TX) to multiple

Manuscript received October 20, 2011; revised February 18, 2012 andMarch 29, 2012; accepted April 10, 2012. Date of publication April 30, 2012;date of current version February 28, 2013.

The authors are with the Department of Electrical Engineering and ComputerScience, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail: [email protected]; [email protected]).

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

Digital Object Identifier 10.1109/TIE.2012.2196902

receivers (RXs) or vice versa. With multiple TXs or RXs in alimited space, coupling among TXs or among RXs cannot beavoided. For coupled RXs, consider wirelessly powered dualliquid crystal display monitors on a desk as an example. Sincedual monitors are usually placed side by side, nonnegligiblecoupling between them may exist. A similar situation mayoccur in a mobile phone battery charging pad where multiplephones are wirelessly charged simultaneously. In factory equip-ment or home audio components, vertically stacked secondarycoils can be wirelessly powered by a TX on the floor. In thesecases, coupling between secondary coils could be even higherthan that between primary and secondary coils.

Employing multiple TXs is also beneficial [1], [19], [33], andcoupling between primary coils similarly cannot be avoided.One example can be found in the specifications released bythe Wireless Power Consortium [19]. By activating multiplehexagonal primaries in such a way that each selected primaryoverlaps with every other selected primary, the array of hexag-onal primaries enables free positioning and size adaptation ofsecondary coils [19]. Similar concepts of TX array for sizeadaptation can be found in patent [33, Fig. 63]. Another benefitbrought by multiple primary coils is increased power transfercapability under the given coupling coefficient, as shown in thelater sections of this paper. Apart from the intended use of mul-tiple primary coils as described previously, multiple primarycoils may incidentally overlap each other in various use cases.

However, the effects of coupling between TXs or betweenRXs have not been extensively studied thus far. Previous papers[17] and [20] have suggested that using multiple RXs wouldincrease the overall system efficiency by increasing the effec-tive coupling from TX to RXs. However, the coupling betweenRXs was not considered. References [19] and [33] describeoverlapped multiple primaries powering a single secondary.However, effects and corresponding guidelines under suchoverlapping are not discussed. The paper [21] reports splittingof the voltage transfer function when two RXs are strongly cou-pled. However, either the condition for maximum efficiency orthe required adjustment under such coupling was not discussed.The work in [22] deals with multiple TXs and RXs. However,Casanova et al. [22] did not investigate the impact of couplingbetween TXs or between RXs on system performances.

In this paper, we propose that the resonant and/or the drivingfrequencies should be adjusted according to the couplings be-tween TXs or between RXs. The performance of multiple TXsor RXs with the proposed adjustments is shown to be higherthan that of a single TX and RX system. The prototype systemsdeliver ∼60-W power at 45%–60% efficiency, even with veryloose coupling coefficients from TXs to RXs.

0278-0046/$31.00 © 2012 IEEE

AHN AND HONG: EFFECT OF COUPLING BETWEEN MULTIPLE TXs OR MULTIPLE RXs ON WPT 2603

Fig. 1. Magnetically coupled resonator. k is the coupling coefficient betweenthe TX and RX. RS and RL are source and load resistances, respectively. Rsp

and Rrp are the parasitic resistances of the TX and RX coils, respectively [34].

This paper is arranged as follows. Section II reviews thebasic principles of WPT technology that is based on magneticcoupling and introduces the framework of the discussion usingthe simple case of a TX and an RX. Section III analyzes the caseof a TX with multiple RXs. Section IV discusses multiple TXswith an RX. The experimental setup and results are discussedin Section V. Conclusions are drawn in Section VI.

II. ANALYSIS OF A SINGLE TX AND RX

Starting from the simplest case of a single TX and RX,the frequency conditions of maximum efficiency are derived.Since the conditions of maximum efficiency do not necessarilyguarantee maximum power transfer, a condition of maximumpower transfer is also derived under the operation of maximumefficiency.

Fig. 1 shows magnetically coupled series-resonant circuitswith a driving source of frequency ω, a TX resonant frequencyof ωTX, and an RX resonant frequency of ωRX.

A. Frequency Condition of Maximum Efficiency

The efficiency η is defined as the ratio between the totalpower dissipation in the load and the total power supplied bythe sources [23]

η =RL|I2|2

(RS +Rsp)|I1|2 + (RL +Rrp)|I2|2

=RL|I2/I1|2

(RS +Rsp) + (RL +Rrp)|I2/I1|2(1)

where I1 and I2 are the phasors of rms currents of coils 1 and2, RL|I2|2 is the power consumed in load, RS |I1|2 is the powerconsumed in TX resistance, and Rsp and Rrp are the parasiticresistances of the TX and RX coils, respectively. Equation(1) indicates that the efficiency is maximized when |I2/I1| ismaximized with the given values of RS , Rsp, and Rrp. It isaimed to find the frequency conditions that maximize powertransfer efficiency and capability under the given resistance orparasitic values.

To find |I2/I1|, Kirchhoff’s voltage law (KVL) is employedon the circuit in Fig. 1. The KVL equations on Fig. 1 are

(RTX

jωLTX+ 1− ω2

TX

ω2

)I1 + kI2

√LRX

LTX=

VS

jωLTX(2.a)

kI1

√LTX

LRX+

(RRX

jωLRX+ 1− ω2

RX

ω2

)I2 =0 (2.b)

where VS is the phasor of rms voltage of the ac source, k =M/

√LTXLRX, M is the mutual inductance, RTX = RS +

Rsp, and RRX = RL +Rrp. By solving (2), I2/I1 is foundto be

I2I1

= − k√LTX/LRX

RRX

jωLRX+(1− ω2

RX

ω2

) . (3)

|I2/I1|, and thus the efficiency, is maximized when

ω =

√2LRXω

2RX√

2L2RXω

2RX −R2

RX

∼=ωRX. (4)

The approximation in (4) holds for 2L2RXω

2RX � R2

RX or2Q2

RX � 1, which is valid and even a required condition be-cause the Q factor should be high for efficient power transfer[15], [23]. By employing coupled resonators of high Q’s,the energies which are not coupled to RXs are stored in theTXs without dissipation. Therefore, even with a very lowcoupling between TX and RX, highly efficient transmission ispossible.

Condition (4) is equivalent to maximizing the real part andcanceling the imaginary part of reflected impedance. This canbe seen by inspecting the total output impedance seen by the acsource

Zout=RTX+jωLTX

(1−ω2

TX

ω2

)+ω2k2LTXLRX/RRX

j ωLRX

RRX

(1− ω2

RX

ω2

)+1

(5)

where the reflected impedance is the term that contains k2.High efficiency is achieved at ω = ωRX, because the real partof the reflected impedance becomes large compared to theparasitic RTX.

If the circuits are in parallel resonance, |V2/V1|, rather than|I2/I1|, becomes the parameter to be maximized, and similararguments on series-resonant circuits are applied to the parallel-resonant one owing to the duality. Note that a slight changein the maximum efficiency condition is observed because theparallel-compensated circuit reflects reactance to the primaryside [17], [24]. The maximum efficiency condition for theparallel-resonant counterpart is found as

ω =

√2ωRX

√2(1− k2)− (1−k2)2

Q2RX

2(1− k2)

∼= ωRX√1− k2

(6)

with an approximation of 2Q2RX � 1− k2. The value of k is

around or less than 0.1 in a loosely coupled system. Excludingthis aspect, the same arguments are applicable, and therefore,we will mainly consider series-resonant circuits in this paper.

Note that, regardless of the strength of coupling, the peakefficiency frequency is ω ∼= ωRX. At first glance, this seems toconflict with the well-known fact that overcoupling, which is


Fig. 2. Currents in TX and RX coils in overcoupled regime. The supplyvoltage is 20 V, and the coupling coefficient is 0.15. LTX and LRX are39 and 44 μH, respectively. RTX and RRX are 1 and 4.1 Ω, respectively.The original resonant frequency of TX and RX is 144 kHz.

defined as k > 1/√QTXQRX, where QTX is the Q factor of

the TX, splits the original resonant frequency into two separatefrequencies [15], [25], [35]. However, the power dissipationin the TX should also be taken into account in the efficiencycalculation. Fig. 2 shows this point. Although the peak RXpower consumption appears at two split frequencies, the TXpower consumption also becomes high at those points. How-ever, at the original RX resonant frequency of 144 kHz, theTX power dissipation is minimized. Therefore, the frequencyfor maximum efficiency does not change, even with frequencysplitting.

Under the frequency condition of (4), |I2/I1| becomes

I2I1

= −jkωRXLRX

RRX

√LTX

LRX= −jkQRX

√LTX

LRX. (7)

B. Frequency Condition of Maximum Power Transfer

Under the constraint of the maximum efficiency condition of(4), I2 is calculated as (8), shown at the bottom of the page.

Therefore, the condition to maximize I2, and thus the powertransfer, at a given voltage level is

ωTX = ωRX (9)

i.e., the resonant frequency of the TX should be the sameas that of the RX. This is important because actual circuitcomponents in TXs place limitations on the maximum voltagelevel. The condition of (9) also applies to the parallel-resonantcounterpart.

Under the constraint of (4), condition (9) is equivalent to can-celing the reactive term jωLTX(1− ω2

TX/ω2) at (5), resulting

in minimum reactive power or voltampere reactive.

Fig. 3. Power transfer from a single TX to multiple RXs. RXs couple witheach other with a coupling coefficient of kRX.

III. MULTIPLE COUPLED RXS WITH A TX

A. Solving KVL Equations Directly

Fig. 3 shows the equivalent circuit of power transmissionfrom a TX to two RXs. The RXs are assumed to be coupled witheach other with a coupling coefficient of kRX. Under the mutualcoupling within RXs, the driving frequency for maximumefficiency and the TX resonant frequency for maximum powertransfer are changed. This can be analyzed by KVL equationsfor three coils

(RTX

jωLTX+1−ω2

TX

ω2

)I1+kI2

√LRX

LTX+kI3

√LRX

LTX=

VS

jωLTX

(10.a)

kI1

√LTX

LRX+

(RRX

jωLRX+1−ω2

RX

ω2

)I2+kRXI3=0 (10.b)

kI1

√LTX

LRX+kRXI2+

(RRX

jωLRX+1−ω2

RX

ω2

)I3=0. (10.c)

Denoting IRX as the current in the two identical RXs, IRX/I1is calculated as follows:

IRX

I1= − k

√LTX/LRX

RRX

jωLRX+((1 + kRX)− ω2

RX

ω2

) . (11)

|IRX/I1|, and thus the efficiency, is now maximized when

ω =

√2LRXω

2RX√

(1 + kRX)2L2RXω

2RX −R2

RX

∼= ωRX√1 + kRX

(12)

I2 = − jkLRXω2RX

√LTX/LRXVS

{k2LTXLRXω3RX +RTXRRXωRX + jLTXRRX (ω2

RX − ω2TX)}

(8)


for high-Q RXs. Therefore, to achieve maximum efficiency, thedriving frequency ω should be lowered as the coupling betweentwo RXs becomes high. Similar to the case of single TX andsingle RX, condition (12) maximizes the real part and cancelsthe imaginary part of reflected impedance.

Alternatively, the resonant frequency of RXs ωRX can beshifted upward rather than adapting the driving frequency.However, to adjust the resonant frequency of RXs, the RXshould bear an additional circuitry burden. Generally, the TXis larger and has more space for such additional circuitry.

Substituting (12) into (11) yields

IRX

I1= −jk

QRX√1 + kRX

√LTX

LRX(13)

which indicates that high coupling between RXs would lowerthe peak efficiency value. The transmission efficiency in thecase of two RXs can be defined as

η =RL|I2|2 +RL|I3|2

(RS +Rsp)|I1|2 + (RL +Rrp)|I2|2 + (RL +Rrp)|I3|2

=RL|

√2IRX/I1|2

(RS +Rsp) + (RL +Rrp)|√2IRX/I1|2

. (14)

Substituting (13) into (14) and comparing the result with(1), it can be seen that the effective |IRX/I1| is increased by√2 with two RXs. Therefore, using multiple RXs increases the

overall transfer efficiency. This was discussed in [17] and [20]where the coupling between RXs was neglected.

Under the maximum efficiency condition of (12), IRX is nowcalculated as (15), shown at the bottom of the page.

The IRX, and thus the power transfer, is maximized when

ωTX =ωRX√1 + kRX

. (16)

In other words, the TX resonant frequency should also belowered for maximum power transfer under the maximumefficiency condition. Similar to the case of single TX and singleRX, under the constraint of (12), the condition (16) is equivalentto canceling the reactive output impedance.

The same analysis can be carried out with more than twoRXs. For n coupled RXs, the frequencies in (12) and (16)change to

ω ∼= ωTX =ωRX√

1 + (n− 1)kRX

. (17)

B. Eigenvalue/Eigenvector Approach

More physical insight can be provided through the eigen-value and eigenvector approach to the two-coupled-RX case.For the coupled resonators, it is well known that an original

Fig. 4. Splitting of the resonant frequency of coupled RXs. A TX sees aneffective RX with an even-mode resonant frequency.

resonant frequency splits into even- and odd-mode frequencies[25]–[27]. The frequencies in (12) and (16) are the even-modefrequencies of coupled resonators. For the two lossless coupledresonant RXs, the KVL equation can be written as

(1− ω2

RX

ω2

)I2 + kRXI3 =0 (18.a)

kRXI2 +

(1− ω2

RX

ω2

)I3 =0. (18.b)

Equation (18) can be rearranged as

ω2RX

ω2

(I2I3

)=

(1 kRX

kRX 1

)(I2I3

)(19)

where ω2RX/ω

2 is the eigenvalue which is calculated as

ω =ωRX√1 + kRX

,ωRX√1− kRX

(20)

and the eigenvectors corresponding to each eigenvalue are(I2I3

)=

(1

1

),

(1

−1

). (21)

The first solution of (21) corresponds to the eigenvector ofωRX/

√1 + kRX (even mode), and the second solution corre-

sponds to that of ωRX/√1− kRX (odd mode). For the eigen-

vector of even mode, the two coupled resonators are in phase,while for the eigenvector of odd mode, the coupled resonatorsare out of phase. Since the two RXs are driven identically bya TX, inphase coupled resonance becomes the dominant mode.This explains why the operation in even mode is efficient. Fig. 4summarizes this idea. If the two RXs are driven out of phase forsome reason, odd-mode transmission would be more efficient.Section III-C discusses a related phenomenon.

The same analysis can be applied to more than two coupledRXs. For n coupled RXs, the eigenvalues and eigenvectors are

IRX = − jkLRXω2RX

√1 + kRX

√LTX/LRXVS{

(1 + kRX)RTXRRXωRX + 2LTXLRXk2ω3RX + jLTXRRX

√1 + kRX (ω2

RX − (1 + kRX)ω2TX)

} (15)


Fig. 5. Couplings from a TX to each RX are different. Then, the effectivecoupling can be thought of as a combination of common- and differential-modecouplings.

calculated as

ω=ωRX√

1+(n−1)kRX

,ωRX√1−kRX

,ωRX√1−kRX

, . . . (22)

⎛⎜⎜⎝

I2I3I4...

⎞⎟⎟⎠=

⎛⎜⎜⎝

111...

⎞⎟⎟⎠ ,

⎛⎜⎜⎝

−110...

⎞⎟⎟⎠ ,

⎛⎜⎜⎝

−101...

⎞⎟⎟⎠ , . . . . (23)

C. Different Couplings From a TX to RXs

Section III-B implies the presence of odd-mode resonantfrequency ω = ωRX/

√1− kRX when two RXs are driven out

of phase. Up to this point, the couplings from a TX to RXsare assumed to be the same. If the coupling from a TX differsfor each RX, the effective coupling can be thought of as acombination of common- and differential-mode couplings, asshown in Fig. 5. As the difference between couplings to eachRX increases, the contribution of differential-mode couplingincreases. Then, the efficiencies at the odd-mode resonant fre-quencies of RXs gradually increase. Denoting k2 and k3 as thecoupling coefficients from a TX to RXs #2 and #3, respectively,I2/I1 and I3/I1 are found as

I2I1=

−k2√

LTX/LRX

(RRX

jωLRX+(1−k3

k2kRX

)−ω2

RX

ω2

)(

RRX

jωLRX+(1+kRX)−ω2

RX

ω2

)(RRX

jωLRX+(1−kRX)−ω2

RX

ω2

) (24.a)

I3I1=

−k3√

LTX/LRX

(RRX

jωLRX+(1−k2

k3kRX

)−ω2

RX

ω2

)(

RRX

jωLRX+(1+kRX)−ω2

RX

ω2

)(RRX

jωLRX+(1−kRX)−ω2

RX

ω2

) . (24.b)

At the even-mode frequency ω = ωRX/√1 + kRX, (24) is

reduced to

I2I1

∼= − jk2 + k3

2

QRX√1 + kRX

√LTX

LRX(25.a)

I3I1

∼= − jk2 + k3

2

QRX√1 + kRX

√LTX

LRX. (25.b)

Fig. 6. Power transfer from multiple TXs to an RX. TXs couple with eachother with a coupling coefficient of kTX. The two TXs are simultaneouslydriven in phase by voltage sources.

However, at the odd-mode frequency ω = ωRX/√1− kRX,

(24) is reduced to

I2I1

∼= − jk2 − k3

2

QRX√1− kRX

√LTX

LRX(26.a)

I3I1

∼= + jk2 − k3

2

QRX√1− kRX

√LTX

LRX. (26.b)

From (26), it can be seen that, as the difference between k2and k3 is increased, the efficiency at odd-mode frequency isgradually increased. If k3 ≈ 0, then the efficiencies at even- andodd-mode frequencies would be almost equal.

IV. MULTIPLE TXs WITH AN RX

A similar methodology as that used in Section III couldbe applied to the case of multiple TXs, as shown in Fig. 6.Denoting IRX as the current in an RX and ITX as the current intwo identical TXs 1 and 2, the efficiency is defined as follows:

η =RL|IRX|2

2× (RS +Rsp)|ITX|2 + (RL +Rrp)|IRX|2

=RL|IRX/ITX|2

2× (RS +Rsp) + (RL +Rrp)|IRX/ITX|2. (27)

The efficiency is maximized when |IRX/ITX| is maximized.By solving a system of KVL equations for three coils, IRX/ITX

can be calculated as

IRX

ITX= −

2k√

LTX

LRX

RRX

jωLRX+(1− ω2

RX

ω2

) . (28)

|IRX/ITX| is maximized when

ω ∼= ωRX (29)

for a high-Q RX. The coupling between TXs does not changethe driving frequency required for the maximum efficiency.


This is expected because the optimal driving frequency formaximum efficiency is related to the RX resonant frequency,which does not change in this single-RX case.

Substituting (29) into (28) reduces to

IRX

ITX= −2jkQRX

√LTX

LRX. (30)

Contrary to the coupled RX case of (13), the coupling betweenTXs does not degrade the peak efficiency value. In addition,substituting (30) into (27) and comparing the result with (1),it can be seen that the effective |IRX/ITX| is increased by√2. Therefore, using multiple TXs enhances the efficiency

regardless of the coupling between TXs.While increasing the number of RXs enhances efficiency,

increasing the number of TXs could degrade efficiency in somecases if the coupling coefficient from a newly added TX to anRX is too small. Consider a one-TX one-RX system with acoupling coefficient of k1.

If an additional TX with a weak coupling coefficient ofk2 ∼ 0 with the RX is added, most of the power from theadditional TX would dissipate uselessly in the TX, resultingin efficiency degradation. Therefore, the value of k2 shouldbe higher than a certain level depending on the value ofk1. Simulation revealed that, if the original one-TX one-RXsystem is critically coupled (kc = 1/Q), then a newly addedTX should have a coupling coefficient higher than ∼ 0.6kc toachieve efficiency higher than that of the original one-TX one-RX system. In the undercoupled (ku < kc) range, the requiredvalue decreases to around 0.4ku. If the original one-TX one-RX system is overcoupled (kov > kc), the coupling coefficientof the newly added TX should be around 0.7kov−0.8kov.

To find the condition for maximum power transfer undermaximum efficiency, (29) is substituted into the calculated IRX,resulting in (31), shown at the bottom of the page.

Equation (31), and thus the power transfer to an RX, ismaximized when

ω2TX = (1 + kTX)ω

2RX. (32)

The TX resonant frequency should be shifted to a higher one.This is shown in Fig. 7. With the coupling between TXs, theTX resonant frequency splits into even/odd modes. Since theeven-mode frequency is lower than the original TX resonantfrequency ωTX, ωTX should move to a higher frequency so thatthe new even-mode frequency will match with ωRX.

If there exists phase difference φ between the two separatedvoltage sources in Fig. 6, the magnetic field at the TX coilsis equivalent to that induced by a single voltage source of2Vs cos(ωt+ φ/2) cos(φ/2), which indicates that larger phasedifference lowers transferred power. However, the frequencyconditions of (29) and (32) still apply.

Fig. 7. Splitting of the resonant frequency of coupled TXs. An effective TXwith even-mode resonant frequency sees an RX.

Fig. 8. Diagram of experimental setup for two coupled RXs and a TX.

V. EXPERIMENTAL RESULTS

Two experiments are performed to verify the proposed con-cepts. The first part of the experiment is carried out withinverters and rectifiers. The second part of the experiment isfocused on better illustrating frequency responses and currentflows under the ideal ac source assumption with 50-Ω outputresistance.

A. Experiment With Real Power Circuits

1) Experiment Setup: Fig. 8 shows a diagram of the experi-mental setup for a two-RX system. Fig. 9 shows the schematicsand component values. The capacitance values of the primaryLC tanks in Fig. 9 are varied so as to adjust the resonantfrequencies of the TXs when coupling between TXs or betweenRXs is introduced. The switching frequencies of the invertersare 144 kHz for Fig. 9(a) and 149 kHz for Fig. 9(b). Theswitching frequency of Fig. 9(b) is also adjusted when couplingbetween RXs is introduced.

A series-resonant half-bridge inverter and a parallel-resonantpush–pull inverter are implemented as ac sources for a two-TX/one-RX system and a one-TX/two-RX system, respectively.The utilized switching devices are IRFP250N for the push-pull parallel-resonant inverter and IRFP150N/IRFP9140N for

IRX = − 2jkLRXω2RX

√LTX/LRXVS

{RRXRTXωRX + 2LTXLRXk2ω3RX + jLTXRRX ((1 + kTX)ω2

RX − ω2TX)}

(31)


Fig. 9. Schematic of the designed wireless power systems. (a) With two TXs.(b) With two RXs.

the series-resonant half-bridge inverter. Both devices wereobtained from International Rectifier. The output capacitancesof IRFP150N and IRFP9140N are 450 and 400 pF, respec-tively. The on-resistance of these MOSFETs is in the range of0.035–0.117 Ω. This on-resistance corresponds to the sourceresistance (RS) in Fig. 1. A full-wave rectifier with a capacitivefilter is implemented using FMB-34M diodes from SankenElectric. As a load resistance, 3.8-Ω cement resistors, whichcan withstand up to 120-W power consumption, are used. Thetransferred power is determined by measuring the dc voltageacross the load resistance. Low-loss polypropylene capacitorsare used as compensating capacitors for primary and secondarycoils.

For the one-TX/two-RX system, 2000 strands of 38 AWGlitz wire are used for the TX coil, and 1.2-mm solid copperwire is used for RX coils. The sizes of the TX coil and the RXcoils are 35 cm × 30 cm and 31.5 cm × 22.5 cm, respectively.For the two-TX/one-RX system, 200 strands of 38 AWG litzwire are used for all coils. The dimensions of the TX coilsand RX coil are 31.5 cm × 22.5 cm and 35 cm × 30 cm,respectively. The coil placement configurations are shownin Fig. 10.

2) Component Value Selection: Since the efficiency of wire-less power system is low under the small coupling [29], thecomponent values should be carefully selected. The load resis-tance of 3.8 Ω represents 60-W power under 15-V dc output.For the series-resonant TX of Fig. 9(a), the current ratio (7)should be high to maximize efficiency which is defined as (1).Since the coupling k from TX to RX is only 0.025–0.063,high QRX, and thus the large LRX, is selected to obtain highcurrent ratio (7). The LTX also should be sufficiently large tosuppress the effect of TX parasitic resistance as implied in (1)and (7). From another viewpoint, the effective Q factor of TXcan be defined as ωLTX/(RS +Rsp), where RS represents theMOSFET parasitic [37]. Therefore, high LTX is selected toobtain high-Q TX under the presence of RS .

Fig. 10. Measurement arrangements of TXs and RXs. The configurationsfrom (c) through (f) are used in both systems of two TXs and two RXs.(a) and (b) Negligible coupling between TXs or RXs. (c) kRX or kTX ∼0.033. (d) kRX or kTX ∼0.1. (e) kRX or kTX ∼0.23. (f) kRX or kTX ∼0.31.(g) Single-TX and single-RX configurations to be compared with the two-TXor two-RX system.

An alternative method to enhance the loaded Q of the res-onator is to reduce the loading effect by using an impedancematching network. The well-known small extra feeding loopused in [15] and [23] is one kind of the matching network. Thenet effect of such feeding loop is the “reduced” (by a factorof ω2M2/RL, where M is the mutual inductance betweenthe feeding loop and main resonator) effective source/loadresistances connected in series with the main resonators. How-ever, such impedance matching network is not used in thisexperiment because the source/load resistances are small andinductances are sufficiently large in this experiment, enablinghigh loaded Q without the aid of the matching network. Theadditional impedance matching loop will just reduce transferredpower [38].

For the parallel-resonant TX of Fig. 9(b), however, smallerequivalent resistance (or larger equivalent conductance) in par-allel with L and C is required to increase the power under thegiven supply voltage level. Since the drain voltage of the push-pull parallel-resonant inverter reaches three times the supply


TABLE IRESONATOR CHARACTERISTICS FOR FIG. 9

voltage, the allowable supply voltage is limited. The equivalentparallel conductance of reflected impedance at the secondaryresonant frequency is

Greflected∼= k2

1

ωRXLTXQRX. (33)

Therefore, small TX inductance is chosen in the design ofFig. 9(b). Note that excessively large inductance in Fig. 9(a) orexcessively small TX inductance in Fig. 9(b) might increase theloss of LC tanks. Therefore, there exist an optimum inductancevalue and a corresponding distance limitation of power transfer.Table I provides the characteristics of resonators used in theexperiments.

After the inductances are chosen, the required capacitancesare calculated in order to resonate at the desired frequency. Thedesired resonant frequency is varied depending on the couplingsbetween TXs or between RXs. The proposed frequency condi-tions in Sections III and IV serve as the criterion to determinethe desired resonant frequency.

Although the measurements are performed only for specificsets of dimensions, distance, center frequency, inductance/capacitance, and source/load impedances, the discussions inSections II–IV still apply to other situations in general. The dis-cussions from Sections II–IV do not impose any constraints onimpedance level, nominal operating frequency, or componentvalues provided that 2Q2

RX � 1, which is required for looselycoupled WPT systems. The only quantity that determinesthe required frequency adjustment is the coupling coefficient,which does not depend on the operating frequency or turnnumber of coils.

3) Experiment Result: The magnetic field is measured withan electromagnetic compatibility analyzer (SPECTRAN NF5035) at 52-W output. The measurement points are 20–30 cmapart from the coils. The measured levels are 3.3–6.4 μT, whichare the same with or lower than the International Commissionon Non-Ionizing Radiation Protection regulation [31]. Systemswith shielding and shorter distance from TXs to RXs wouldpresent much lower field strength.

Fig. 11(a) shows that the coupling between RXs lowersthe driving frequency that yields the maximum efficiency, aspredicted in (12). The coupling between TXs, on the otherhand, does not have any meaningful influence on the drivingfrequency, as predicted in (29). Fig. 11(b) shows that, for twocoupled RXs, the required resonant frequency of a TX should

Fig. 11. Frequencies for maximum efficiency and power transfer. (a) Requireddriving frequencies. (b) Required resonant frequency of LC tank of TX.

be lowered, as explained in (16). For two coupled TXs, on theother hand, the resonant frequency of TXs should be upshifted,as explained in (32).

Fig. 12 compares two cases with and without the proposedadjustments of Fig. 11. It is proved that the frequency adjust-ments are necessary to maintain peak performances when thecouplings between TXs or between RXs exist. Without suchadjustments, the performances are significantly degraded as thecoupling between TXs or RXs is increased.

In addition, Fig. 12 also shows that systems with multipleTXs or RXs show higher efficiency and transferred power thansystems with a single TX and RX. In Fig. 12(a), both thetransferred power and efficiency are increased. In Fig. 12(b),whereas the power delivery is significantly improved, a slightefficiency drop is observed when the coupling between TXsis 0–0.033. This is because the coupling coefficients from twoTXs to an RX become slightly low for this specific configu-ration. For the cases of Fig. 10(a) or (c), where the couplingbetween TXs is 0–0.033, the two TXs are placed far away fromthe coaxial line of RX, resulting in lower coupling from twoTXs to an RX. Moreover, the newly introduced second TX(i.e., TX #2) is located 31 cm apart from the RX, which is6 cm more distant from TX #1. The coupling coefficient at31-cm distance is only ∼65% of that at 25-cm distance. If thecoupling of the newly introduced TX coil (TX #2) is small


Fig. 12. Efficiency and transferred power versus couplings between TXsor between RXs. It is shown that the proposed frequency adjustments areessential to maintain peak performances when the couplings between TXs orbetween RXs exist. In addition, systems with multiple TXs or RXs show higherperformance than that with single TX and RX. (a) Two RXs are coupled to eachother. (b) Two TXs are coupled to each other.

compared to the coupling of the original TX (TX #1), the newTX contributes more loss than the original TX. Therefore, theefficiency enhancement is not high for the two-TX system inthis experiment.

Fig. 12(a) shows that the peak efficiency value obtained withthe adjustments is slightly degraded as the coupling betweenRXs is increased, as described in (13). On the other hand,Fig. 12(b) shows that the coupling between TXs does not de-grade the peak efficiency value obtained with the adjustments,as described in (30).

For Fig. 12(b), (29) and (30) seem to imply that no ad-justments are required to solely achieve maximum efficiencyonly, because the optimum driving frequency needed for themaximum efficiency is not changed regardless of couplingbetween TXs. However, a slow efficiency drop in Fig. 12(b) isobserved when the adjustments are not employed. The reasonfor the measured slow efficiency drop is the failure of zero-current switching (ZCS). ZCS fails because the coupling be-tween TXs lowers the effective resonant frequency of TXs,

TABLE IIPERFORMANCE COMPARISON

TABLE IIIRESONATOR CHARACTERISTICS FOR FIGS. 13 AND 14

altering the frequency where ZCS is achieved. Therefore, theeffective resonant frequency, as well as the driving frequency,of TXs should be equal to the RX resonant frequency to obtainmaximum efficiency in soft-switching drivers.

Table II compares the performance with those of other workswhich contain TX inverters and RX rectifiers.

B. Experiment With Ideal AC Source Assumption

The second part of the experiment is performed with threerectangular 19-turn coils of size 31.5 × 22.5 cm. The diameterof the copper wire is 1.2 mm. A series capacitance of 277 pFis added to make the coil resonant at 650–660 kHz. Table IIIsummarizes the characteristics of the resonators used.

The measurements are performed with a three-port networkanalyzer, because the accurate frequency response can be eas-ily measured. The network analyzer measures S-parameters,which can represent the ratios of output voltage to input voltagein frequency domain. In [15], [35], and [36], the S21 of theS-parameter is directly used to illustrate the frequency domaincharacteristics of power transmission.

In this work, however, the S-parameter is converted tothe normalized current that flows through the source/load re-sistances using the relationships in [15] and [32]. This isto illustrate not only the delivered power to RXs but alsothe dissipated power in TXs. The 50-Ω port impedancesof the network analyzer correspond to the source/loadimpedances [15].

Fig. 13 plots the frequency domain currents and efficiencywhen the two RXs are coupled to each other. The couplingcoefficients from a TX to each RX are set to 0.06 and 0.04,respectively. The coupling between RXs is set to 0.33. A similarconfiguration to that in Fig. 10(f) is used except that the TX coilsize is chosen to be the same as the RX coil size.


Fig. 13. Currents and efficiency in frequency domain with two coupled RXs.For (a) and (b), the same supply voltage is assumed. (a) Peak efficiencyfrequency is downshifted due to the RX coupling. The amount of transferredpower becomes small due to the mismatch between the resonant frequency ofcoupled RXs and that of a TX. (b) Transferred power is increased because theresonant frequency of a TX is adjusted to that of coupled RXs.

Although the original resonant frequency of the RXs isaround 650 kHz, the peak efficiency frequency is now shiftedto 565 kHz due to the coupling between RXs. In Fig. 13(a), theamount of power transferred to the RXs is high in the frequencyrange of 560–670 kHz. However, the efficiency peak occursonly at 565 kHz, because, beyond this frequency, the powerconsumed in the TX rises significantly. Since the resonantfrequency of coupled RXs (565 kHz) is different from thatof a TX (660 kHz), the amount of delivered power is low.However, in Fig. 13(b), the resonant frequency of a TX is nowadjusted to be the same as that of coupled RXs. Therefore,the transferred power to RXs is increased. The reference valueto which all values are normalized is the peak TX currentin Fig. 13(a).

Fig. 14(a) shows the normalized efficiency when the twocoupling coefficients from a TX to two RXs are different.Fig. 14(b) shows the arrangements used in the experimentsand corresponding peak efficiency values. In Fig. 14(a), theleft peak corresponds to the even-mode resonant frequency oftwo coupled RXs while the right peak is the odd-mode one.The larger the difference between couplings from a TX to each

Fig. 14. Two RXs with different couplings from a TX. The coupling betweenRXs is fixed to 0.17. (a) Maximum efficiency point still lies in the even-moderesonant frequency of coupled RXs. (b) Measurement arrangements and theirpeak efficiencies.

RX, the higher the efficiency at odd-mode frequency. However,the maximum efficiency still occurs at the even-mode resonantfrequency since the component of the odd mode is not largerthan that of the even mode. The reference values to whichthe efficiencies are normalized are 0.323, 0.597, and 0.821,respectively.

VI. CONCLUSION

This paper has proposed that the resonant and/or the drivingfrequencies should be adjusted according to the couplings be-tween TXs or between RXs. The effectiveness of the proposedadjustments is verified in proof-of-concept circuits.

For maximum efficiency, the driving frequency of TXsshould be set to the resonant frequency of the RXs. If couplingexists between RXs, the effective resonant frequency of thecoupled RXs is changed. To achieve maximum efficiency inthis case, the driving frequency of the TX should be adjustedto this new effective resonant frequency. Unlike the couplingbetween RXs, the coupling between TXs does not change thedriving frequency required for maximum efficiency.


The condition for maximum efficiency does not necessarilyguarantee maximum power transfer. To maximize the amountof power transfer simultaneously, the effective resonant fre-quency of TXs should also be set to the effective resonantfrequency of RXs, which is also equal to the driving frequencyunder the condition of maximum efficiency. The couplingsbetween TXs or between RXs change these effective resonantfrequencies.

In the case of switch-mode drivers, soft switching suchas ZCS or zero-voltage switching is desired. The couplingsbetween TXs change the effective resonant frequency of TXs,altering the frequency where soft switching is achieved. There-fore, not only the driving frequency but also the effectiveresonant frequency of TXs should be set to the RX resonantfrequency in order to achieve soft switching. In this case,the conditions for maximum efficiency and maximum powertransfer are simultaneously satisfied.

These provide fundamental guidelines for efficient opera-tion in WPT systems with coupled multiple TXs or coupledmultiple RXs.

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Dukju Ahn received the B.S. degree in electricalengineering from Seoul National University, Seoul,Korea, in 2007 and the M.S. degree in electrical en-gineering from Korea Advanced Institute of Scienceand Technology, Daejeon, Korea, in 2010, wherehe is currently working toward the Ph.D. degree inelectrical engineering.

His research interests include wireless powertransfer and analog/RF integrated circuit design.

Mr. Ahn was a recipient of the EncouragementPrize in the 17th Human-Tech Thesis Contest from

Samsung Electronics in 2011.

Songcheol Hong (S’87–M’88) received the B.S. andM.S. degrees in electronics from Seoul National Uni-versity, Seoul, Korea, in 1982 and 1984, respectively,and the Ph.D. degree in electrical engineering fromthe University of Michigan, Ann Arbor, in 1989.

Since May 1989, he has been a Faculty Mem-ber with the Department of Electrical Engineeringand Computer Science, Korea Advanced Institute ofScience and Technology, Daejeon, Korea. In 1997,he held short visiting professorships with StanfordUniversity, Palo Alto, CA, and Samsung Microwave

Semiconductor, Suwon, Korea. His research interests are microwave integratedcircuits and systems, including power amplifiers for mobile communications,miniaturized radar, millimeter-wave frequency synthesizers, and novel semi-conductor devices.

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