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Research ArticleMethod for Estimating Optimum Free Resonant Frequencies inOvercoupled WPT System

Dong-Wook Seo and Jae-Ho Lee

Automotive IT Platform Research Section, Electronics and Telecommunications Research Institute (ETRI), 1 Techno Sunhwan-ro 10-gil,Yuga-myeon, Dalseong-gun, Daegu, Republic of Korea

Correspondence should be addressed to Jae-Ho Lee; [email protected]

Received 30 January 2017; Revised 27 March 2017; Accepted 30 March 2017; Published 27 April 2017

Academic Editor: Marta Cavagnaro

Copyright © 2017 Dong-Wook Seo and Jae-Ho Lee. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

In our previous work, we proposed the method to maximize the output power even in the overcoupled state of the wireless powertransfer (WPT) system by controlling free resonant frequencies and derived closed-form expression for optimum free resonantfrequencies of the primary and secondary resonators. In this paper, we propose themutual coupling approach to derive the optimumfree resonant frequencies and show the measured power transfer efficiency (PTE) using the transmission efficiency as well as thesystem energy efficiency. The results of the proposed approach exactly coincide with those of the previous work, and the fabricatedprototype achieves the transmission efficiency of about 80% by tuning the free resonant frequencies to the optimum values in theovercoupled state.

1. Introduction

Recently, the wireless power transfer (WPT) system hasdrawn a great deal of attention as it was applied in vari-ous electric devices including implanted devices [1], mobiledevices [2], and electric vehicles [3]. Furthermore, manyresearchers are now starting to combine the WPT systemwith a communications system, and the combined systemis called the simultaneous wireless information and powertransfer (SWIPT) [4]. In the future, power lines are expectedto be replaced by the WPT system. However, inductive WPTsystems have the following inefficiency; it achieves the maxi-mum output power and power transfer efficiency (PTE) onlyat a specific frequency and at a specific distance (i.e., couplingcoefficient). Figure 1 shows the transmission coefficient of theWPT system of [5] as a function of frequency and as a func-tion of coupling coefficient. For the operating frequency of10MHz, the frequencywith themaximum transmission coef-ficient bifurcates at the critical coupling resulting in the well-known frequency splitting phenomenon. The output powerand the transmission coefficients are generally maximized atthe critical coupling, and they decrease when the couplingcoefficient deviates from the critical coupling, as shown in

Figure 1(b). In the overcoupled state, the excessive couplingbetween the coils results in frequency splitting, producinglow transmission coefficients. On the other hand, low amountof power is transferred from the primary resonator to thesecondary resonator in the undercoupled state, due to the lowcoupling coefficient. It is difficult to operate andmaintain theWPT system at the critical coupling. According to changedoperating circumstances, that is to say, changed couplingcoefficient, some system parameters should be adjusted toachieve the high transmission coefficient.

Figure 2 shows an equivalent circuit model of a two-coil WPT system, where the controllable elements includevoltage source V𝑠, source impedance R𝑠, load impedanceR𝐿, primary coil 𝐿1, secondary coil 𝐿2, and capacitances ofprimary and secondary resonators,𝐶1 and𝐶2. Depending onthe circuit elements, various methods have been introducedto improve the PTE and output power of theWPT system [6–13]. These methods are categorized in two types: a methodthat changes the critical coupling coefficient [6–10] and amethod that mitigates the equivalent coupling coefficient inthe overcoupled state [11–13]. The former methods controlthe resistance or inductance in order to change the criticalcoupling coefficient k𝑐, since k𝑐 is a function of the resistance

HindawiInternational Journal of Antennas and PropagationVolume 2017, Article ID 1830687, 6 pageshttps://doi.org/10.1155/2017/1830687

https://doi.org/10.1155/2017/1830687

2 International Journal of Antennas and Propagation

1

0.8

0.6

0.4

0.2

0

10−1

10−2

10−3 5 67 8

9 1011 12

13 1415

|S21

|

Tran

smiss

ion

coe�

cien

t,

Coupling coe�cient Frequency (

MHz)

Critically coupledOvercoupled Undercoupled

10−1

10−2

7 89 10

11 1213 14

upli H )

pled Underco

(a)

Coupling coe�cient 1.0 0.0

Tran

smiss

ion

coe�

cien

t

1.0

0.0

Critically coupled

Overcoupled Undercoupled

Frequencysplitting Low k

(b)

Figure 1: Transmission coefficient of a WPT system (a) as a function of frequency and coupling coefficient and (b) as a function of couplingcoefficient at the operating frequency of 10MHz.

Zin

RsC1 R1p

VsI1 L1

MR2p C2

RL VLI2L2

R2 = R2p + RL

+

−

R1 = R1p + Rs k = M/√L1L2

Figure 2: Equivalent circuit model of a two-coil resonant WPT system.

and inductance.The lattermethods control the voltage sourceor capacitance to change the operating frequency or freeresonant frequencies only in the overcoupled state.

For the control of resistance, the most commonly usedmethod is the matching networks. This method convertssource and load impedances into optimum values. Our grouprevealed that the WPT system using lumped reactive match-ing networks is equivalent to a two-coil WPT system, andtheWPT system using transformermatching networks couldbe considered as a four-coil WPT system [6]. Additionally,we showed how the source (or load) impedance could bematched to the optimum value not only in the two-coilWPT system but also in the four-coil WPT system. In [7], aswitchable capacitor array circuit was proposed, and it wasshown that several capacitors were enough to implement avariety of matching cases in an 𝐿-section matching network.Our group showed that theWPT system can achieve the highPTE by adjusting only the source impedance [8], and we alsodemonstrated the self-adaptive WPT system achieving thehigh PTE without the feedback of receiver information [9].

Besides the method using matching networks, themethod of selecting the optimum primary and secondarycoils among the coil arrays was suggested in [10].

By adjusting either the amplitude or the operating fre-quency of voltage source, the output power can be improved.Changing the amplitude of the voltage source does notimprove the PTE, while changing the operating frequency inovercoupled state achieves the high PTE [11, 12]. However,this method may result in the violation of frequency reg-ulation for a WPT system in narrow bandwidth. Similarly,our group recently proposed that free resonant frequenciesof resonators are tuned to optimum frequencies in orderto achieve a high PTE in overcoupled state [13]. In [13],the optimum free resonant frequencies were derived in theclosed-form based on a condition having a current ratiobetween the primary and secondary resonators identicalto that in critical coupled state. In addition, the systemenergy efficiency was used to express the PTE instead ofthe more commonly used transmission efficiency. On theother hand, the free resonant frequency tuning method was

International Journal of Antennas and Propagation 3

applied not only to the two-coil WPT system but also to themulticoil WPT system [14]. The authors of [14] determinedthe capacitance and free resonant frequencies by using agenetic algorithm; however, the process does not guaranteea global maximum and takes dramatically increasing timeas the number of resonators increases. Additionally, it isalso cumbersome to apply the principles used in [13] to themulticoil WPT system.

In [15], we have already presented the idea that theeffect of mutual inductance on a reflected inductance can beused to derive the condition of the optimum free resonantfrequencies for a high PTE. In this paper, we form the conceptin greater detail and show that the tuned free resonantfrequencies result in a high PTE using the transmissionefficiency as well as the output power.

2. Formulation

In the circuit model of Figure 2,M is the mutual inductancebetween the primary and secondary coils and is given by

𝑀 = 𝑘√𝐿1𝐿2, 0 ≤ 𝑘 ≤ 1, (1)

where 𝑘 is the coupling coefficient.When no mutual coupling exists, that is, M = k = 0, the

resonance frequency is called the free resonant frequency.The free resonance frequencies of the primary and secondaryresonators are given, respectively,

𝜔1 =1√𝐿1𝐶1,

𝜔2 =1√𝐿2𝐶2.

(2)

In the conventional WPT system, the free resonantfrequencies are tuned to the operating frequency,𝜔0, allowingthe WPT system to function at the operating frequency.However, as the definition of the free resonant frequencymight suggest, the primary and secondary resonators work atthe operating frequency only when there is little or nomutualcoupling. If the mutual coupling becomes stronger, then theresonators affect each other, and the resonant frequencies ofthe resonators are no longer equal to the operating frequency.To see this effect intuitively, we use reflected impedanceconcept.The circuit model of Figure 2 can be described usingthe reflected impedance 𝑍𝑟 as shown in Figure 3, and 𝑍𝑟 isexpressed as

𝑍𝑟 =𝜔2𝑀2

𝑅𝐿 + 𝑅2𝑝 + 𝑗𝜔𝐿2 + 1/𝑗𝜔𝐶2. (3)

The loaded 𝑄 of the primary and secondary series RLCresonators are defined as, respectively,

𝑄1 =𝜔𝐿1𝑅𝑠 + 𝑅1𝑝

,

𝑄2 =𝜔𝐿2𝑅𝐿 + 𝑅2𝑝

.(4)

+

RsC1 R1p

Vs I1

L1

−

Zin

V1Zr

Figure 3: Equivalent circuit model of a two-coil resonant WPTsystem using reflected impedance.

Using (1), (2), and (4), Z𝑟 can be rewritten as

𝑍𝑟 =𝜔𝐿1𝑘2

1/𝑄2 + 𝑗 (1 − 𝜔22/𝜔2), (5)

where the real and imaginary parts are given by

Re {𝑍𝑟} = 𝜔𝐿1 ⋅𝑘2/𝑄2

1/𝑄22 + (1 − 𝜔22/𝜔2)2, (6)

Im {𝑍𝑟} = 𝜔𝐿1 ⋅𝑘2 (𝜔22/𝜔2 − 1)1/𝑄22 + (𝜔22/𝜔2 − 1)

2. (7)

Therefore, due to the mutual coupling, the resistance of(6) and reactance of (7) are added to the primary resonator.Let the right-hand side of (7) be the product of 𝜔L1 anda function of the free resonant frequency of the secondaryresonator, A(𝜔2). Then, the total inductance of the primaryresonator becomes 𝐿1 + 𝐿1 ⋅ 𝐴(𝜔2). In order to work at theoperating frequency, the resonant frequency of the primaryresonator should be

1√𝐿1 [1 + 𝐴 (𝜔2)] ⋅ 𝐶1

= 𝜔1√1 + 𝐴 (𝜔2)

= 𝜔0, (8)

which is satisfied by adjusting the capacitance of the primaryresonator, 𝐶1.

Similarly, in order to operate the secondary resonator atthe operating frequency 𝜔0, the resonant frequency of thesecondary resonator should also be

1√𝐿2 [1 + 𝐵 (𝜔1)] ⋅ 𝐶2

= 𝜔2√1 + 𝐵 (𝜔1)

= 𝜔0, (9)

where B(𝜔1) is given by

𝐵 (𝜔1) =𝑘2 (𝜔21/𝜔2 − 1)1/𝑄21 + (𝜔21/𝜔2 − 1)

2. (10)

Since variables A(𝜔2) and B(𝜔1) are functions of 𝜔2 and𝜔1, respectively, (8) and (9)must be solved simultaneously forthe variables 𝜔2 and 𝜔1.


Start

Set system parameters

Determine initial values

Variation

End

Variation

No

No

Yes

Yes

Compute the variation

of 𝜔1 and 𝜔2

𝜔2

Calculate A(𝜔2)

Update 𝜔1

Calculate B(𝜔1)

Update 𝜔2

rate of 𝜔1 & 𝜔2

Determine �nal 𝜔1 &

Calculate C1 & C2

rate of 𝜔1 < delta

rate of 𝜔2 < delta

Figure 4: Flowchart of the proposed iterative process.

3. Estimation Procedure

It is cumbersome to directly solve the simultaneous equations(8) and (9). Instead, we can find the optimal free resonantfrequencies using an iterative process, as summarized inFigure 4. The detailed steps of the proposed scheme aredescribed as follows.

(1) Set the operating frequency𝜔0 and systemparametersR𝑠, R1𝑝, 𝐿1, R𝐿, R2𝑝, 𝐿2, and 𝑘 of Figure 2. V𝑠, 𝐶1, and𝐶2 are not required.

(2) It is recommended that the initial 𝜔1 and 𝜔2 are setto be two times the operating frequency. If the initial𝜔1 and 𝜔2 are set to be 𝜔0, A(𝜔2) and B(𝜔1) termsbecome zero, which removes the mutual coupling inthe reactance of the reflected impedance.

(3) Calculate A(𝜔2) using (7) and update 𝜔1 using (8).(4) Calculate B(𝜔1) using (10) and update 𝜔2 using (9).(5) Compute the variation of 𝜔1 and 𝜔2. IfΔ𝜔1/𝜔1 and Δ𝜔2/𝜔2 ≤ 0.1%, then go to step(6). Otherwise, go to step (3).

(6) Obtain the final 𝜔1 and 𝜔2.(7) Estimate𝐶1 and𝐶2 by using (2) according to the final𝜔1 and 𝜔2.

4. Simulation and Experiment Results

We simulated and measured the WPT system with theidentical parameter used in [13]. Figure 5 shows schematicsand component values of the experimental setup to validatethe proposed method. The capacitance values of the primary

+DC ++ 10 V

Vs

83.6–90.5 pF

0.072Ω

24 𝜇H

0.019 Ω

6.3 𝜇H 1.0 Ω

−−

−

VL

243.4–344.7 pF10–60mmk = 0.1∼0.2

Figure 5: Schematic of the WPT system prototype.

126

124

122

120

118

116

114

112

110

108

Freq

uenc

y (k

Hz)

0 2 4 6 8 10

Iteration number

125.63kHz

111.73kHz

116.51kHz

109.74kHz

𝜔1 @ k = 0.129𝜔2 @ k = 0.129

𝜔1 @ k = 0.190𝜔2 @ k = 0.190

Figure 6: Free resonant frequencies with respect to iterationnumber for k = 0.129 and 0.190.

and secondary resonators are varied to adjust the free res-onant frequencies to the estimated optimum values in theovercoupled state. The switching frequency of the series-resonant half-bridge inverter is 108 kHz, and IR25603 andIRFP250N are used as the inverter driver and the switchingdevice, respectively. The coils that were used were obtainedfrom ABRACON Corp. Because of high voltage, the filmcapacitors and cement resistors were used as the capacitor ofthe resonators, the source, and load resistors.

Figure 6 shows the estimated free resonant frequencieswith respect to the iteration number. We show two cases ofthe overcoupled state, k = 0.129 and 0.190. In this system, 𝑄1and 𝑄2 are 21.1 and 4.2, respectively, since 𝑄1 ̸= 𝑄2, 𝜔1 ̸= 𝜔2.Additionally, the higher the loaded 𝑄 of the other resonatoris, the more the free resonant frequency deviates from 𝜔0.The optimum free resonant frequencies are reached only afterseveral iterations.

We summarized the results of [13] and the proposediterative process in Table 1. The estimated optimum freeresonant frequencies using the proposed method are verysimilar to those of [13].We highlight the fact that there is onlya maximum difference of 0.04 kHz between the results of twomethods.

International Journal of Antennas and Propagation 5

130

125

120

115

110

Freq

uenc

y (k

Hz)

0.12 0.14 0.16 0.18 0.20

Coupling coe�cient

𝜔1𝜔2

(a)

10

8

6

4

2

00.12 0.14 0.16 0.18 0.20

Term

inat

ed it

erat

ion

num

ber


(b)

Figure 7: Estimated results using the proposedmethod as a function of coupling coefficient: (a) free resonant frequencies and (b) terminatediteration number.

Table 1: The estimated free resonant frequencies using method of[13] and the proposed method with respect to coupling coefficientsto achieve the high PTE.

Couplingcoefficient

𝜔1 & 𝜔2 of[13] [kHz]

𝜔1 & 𝜔2using theproposedmethod[kHz]

Difference[kHz]

Terminatediterationnumber

0.214 112.38 112.39 0.01 5128.54 128.51 0.03

0.190 111.73 111.73 0 4125.63 125.63 0

0.164 110.97 110.97 0 4122.19 122.19 0

0.143 110.28 110.28 0 5119.02 119.02 0

0.129 109.75 109.74 0.01 6116.51 116.51 0

0.116 109.11 109.10 0.01 10113.48 113.44 0.04

Figure 7 shows the estimated optimum free resonantfrequencies of the primary and secondary resonators as afunction of the coupling coefficient, and the iteration numberwhen the proposed process is terminated. As expected, thelarger the coupling coefficient is, the farther away the opti-mum free resonant frequencies deviate from the operatingfrequency (108 kHz). The interesting point is that the smallerthe coupling coefficient is, the more the iterations wereneeded. We hypothesize that it is because the initial valuesof 𝜔1 and 𝜔2 are set to be two times 𝜔0. Nevertheless,convergence is reached within ten iterations.

The efficiency (or power gain) of an electrical system isdefined as the ratio of an output power to an input power.The output power is referred to as a load power, while theinput power is defined as either the total input power fromthe power source, 𝑉𝑠 ⋅ 𝐼1∗, or maximum available power atthe power source, |𝑉𝑠|2/4𝑅𝑠. Efficiency is defined differentlydepending on the definition of the input power. According to[16], the efficiency with the former notation is called “systemenergy efficiency” which is given by 𝜂 = (|𝑉𝐿|2/𝑅𝐿)/(𝑉𝑠 ⋅ 𝐼1∗),and the efficiency with the latter notation is called “transmis-sion efficiency” which is given by 𝜂 = (|𝑉𝐿|2/𝑅𝐿)/(|𝑉𝑠|2/4𝑅𝑠).The difference between them is determined by whether thepower loss by the source resistance is included in the inputpower or not. In general, the system energy efficiency isused in the WPT with negligible source resistance and lowoperation frequency, while the transmission efficiency is usedin the WPT operating at a high frequency and with fixedsource resistance. In this study, we use the transmissionefficiency to represent the performance of the WPT system,becauseV𝑠 andR𝑠 of the generator were fixed as 4.5 Vrms and0.7Ω, respectively.

Figure 8 depicts the measured load power and trans-mission efficiency when the values are optimized for thetuned free resonant frequencies obtained from the proposedprocess. Since the input power in the transmission efficiencyis defined as the maximally delivered power from the powersource, the input power keeps unchanged regardless of statusof the WPT system and load. Then, the transmission effi-ciency has a direct relation to the load power. From Figure 8,we notice that the transmission efficiency is maximized bymaximizing the load power. When the tuned free resonantfrequencies are applied to the overcoupled WPT system, thesystem energy efficiency approaches about 50% in [13] whilethe transmission efficiency approaches about 81%. Obviously,the tuned free resonant frequencies effectively improve the


30

25

20

15

10

5

0

Pow

er (W

)

0.00 0.05 0.10 0.15 0.20


0.8

0.6

0.4

0.2

0.0

E�ci

ency

PL (calculated)PL (measured)Pin (calculated)

Pin (measured)𝜂 (calculated)𝜂 (measured)

Figure 8: Calculated and measured load power and transmissionefficiency.

transmission efficiency and load power in the overcoupledstate.

5. Conclusion

In this paper, we proposed a new approach to derive theoptimum free resonant frequencies in the overcoupled WPTsystem.The optimum values can be obtained through severaliterations and are identical to those obtained in previousworks. Additionally, when the tuned free resonant frequen-cies are adopted in the overcoupled WPT system, the systemachieves the transmission efficiency close to 80% and thesystem energy efficiency close to 50%.

The proposedmethod is very intuitive and simple.There-fore, it can be extended to the sophisticated applications suchas a multicoil WPT system.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

This work was supported by ETRI R&D Program [17ZD1200,Advanced Development of Fusion-Platform and Local PartsMaker Support Project for Context-Aware Smart Vehicle]funded by the Government of Korea.

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[10] J. Kim and J. Jeong, “Range-adaptive wireless power transferusingmultiloop and tunablematching techniques,” IEEE Trans-actions on Industrial Electronics, vol. 62, no. 10, pp. 6233–6241,2015.

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[13] D.-W. Seo, J.-H. Lee, and H.-S. Lee, “Optimal coupling toachieve maximum output power in a WPT system,” IEEETransactions on Power Electronics, vol. 31, no. 6, pp. 3994–3998,2016.

[14] B.-H. Choi and J.-H. Lee, “Design of asymmetrical relayresonators for maximum efficiency of wireless power transfer,”International Journal of Antennas and Propagation, vol. 2016,Article ID 8247476, 8 pages, 2016.

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