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Battery Charging Profile-Based ParameterDesign of A 6.78-MHz Class E2 Wireless

Charging System

Ming Liu, Student Member, IEEE, Chen Zhao, Jibin Song, Student Member, IEEE,and Chengbin Ma, Member, IEEE

Abstract—Wireless power transfer (WPT) working at sev-eral megahertz (MHz), 6.78 or 13.56 MHz, is widely consid-ered to be a promising candidate for charging electronic de-vices. The so-called Class E2 converter combining the soft-switching based Class E power amplifier (PA) and Class Erectifier are known to be suitable for high-frequency appli-cations with improved efficiency. However, the charging ofbatteries usually need to follow a specific profile, in whichbattery voltage and charging current vary over time. Theinput reactance of the Class E rectifier also becomes ob-vious at MHz. This non-neglectable and varying reactancesignificantly lowers system efficiency and complicates pa-rameter design. In this paper, a systematic design approachis developed that minimizes the energy loss of a 6.78-MHzClass E2 wireless charging system during the entire batterycharging cycle. A LC matching network is added to improvethe loading conditions of the Class E PA and couplingcoils, and provide new degrees of freedom in the parameterdesign. Average power loss is defined based on analyti-cally derived system efficiency and a discretized batterycharging profile. It serves as an objective function that isminimized through the proposed battery charging profile-based parameter design. In final experiments, the proposeddesign achieves a 24.5% reduction of the average powerloss when comparing with that through the conventionaldesign.

Index Terms—Battery charging system, charging profile,efficiency, lithium-ion battery, megahertz wireless powertransfer.

I. INTRODUCTION

IN recent years, wireless power transfer (WPT) throughinductive resonance coupling became popular to charge

various electronic devices (e.g., cellphones, laptop computers,

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Manuscript received Sep. 01, 2016; revised Dec. 30, 2016; acceptedJan. 29, 2017. This work was supported by the Shanghai NaturalScience Foundation under Grant 16ZR1416300.

M. Liu, J. Song, and C. Ma are with the University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong Uni-versity, Minhang, Shanghai 200240, China (e-mail: mikeliu@sjtu.edu.cn;jibinsong@sjtu.edu.cn; chbma@sjtu.edu.cn). C. Zhao is with UnitedAutomotive Electronic Systems Co., Ltd, Pudong, Shanghai 201206,China (e-mail: chen.zhao3@uaes.com).

wearable devices, medical implant devices) and even electricvehicles [1], [2]. For high-power applications, WPT work-ing at kilohertz (kHz) is making rapid progress in termsof coil design, compensation topologies, control strategies,etc. [3]–[6]. At the same time, in order to reduce size andweight of WPT systems, it is preferred to further increasethe operating frequency to several megahertz (MHz) suchas 6.78 and 13.56 MHz. A higher operating frequency alsohelps to improve spatial freedom, namely a longer transferdistance and higher tolerance to coupling coil misalignment,which is particularly beneficial for charging mobile devices.However, the power capability of the present switching devicesis insufficient when operating at MHz. At present the MHzWPT is usually considered to be suitable for mid-range andlow-power applications [2], [7]–[9].

A practical challenge for the MHz WPT is the highswitching loss when using conventional hard-switching-basedpower amplifiers (PAs) and rectifiers. Class E PA and rectifierare promising candidates to build high-efficiency MHz WPTsystems because of their soft-switching properties. The ClassE PA was first introduced for high-frequency applicationsin [10]. It has been applied in MHz WPT systems and shownimprovements thanks to its high efficiency and simple topol-ogy [11]–[14]. Similarly, the Class E rectifier has also beenproposed for high-frequency rectification [15]. Its applicationin WPT was first investigated in [16]. A high efficiency ofthe rectifier, 94.43%, was reported at an 800 kHz operatingfrequency. Thus the combination of both Class E PA andrectifier, i.e., a Class E2 converter, is expected to enable ahigh-efficiency wireless charging system working at MHz.

Lithium-ion batteries are now widely used in consumerelectronic devices mostly due to its high energy density [17],[18]. A typical charging profile of the lithium-ion batteriesusually consists of two modes, constant current (CC) modeand constant voltage (CV) mode. The battery is first chargedin CC mode. When its voltage reaches a nominal value, thecharging system enters the CV mode, in which the chargingcurrent rapidly drops. In order to prolong the battery cycle-life,a wireless charging system must supply current and voltageaccurately following a specific battery charging profile [19]. Inreal applications, the charging profiles can be tracked througheither the input voltage control of the charging system or theregulation circuit between the charger and batteries.

In the conventional Class E2 converter for WPT applica-

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tions, the system parameters are optimized targeting on onlya single specific operating condition, i.e., fixed coil relativeposition and final load [20]. However, different with theconventional constant resistive loads, the battery voltage andcurrent vary when following a charging profile. In addition,the input reactance of the Class E rectifier is non-neglectableat MHz and changes with the varying battery voltage andcurrent. This obvious and varying input reactance of the ClassE rectifier increases the power loss and complicates the designof a high-efficiency MHz Class E2 wireless battery chargingsystem. In KHz wireless charging systems, compensationnetworks (such as a LLC network) and designs have beendiscussed to achieve a load-variation-independent resonantfrequency [5]. However, these existing approaches are notvalid anymore for the MHz Class E2 wireless battery chargingsystems due to the non-neglectable and varying input reactanceof the Class E rectifier.

This paper proposes a systematic and general parameterdesign approach that achieves a high-efficiency Class E2 MHzwireless battery charging system. The influence of the non-neglectable input reactance of the Class E rectifier is firstanalyzed when working at 6.78 MHz and following a batterycharging profile. This aspect is quite different with the KHzWPT systems, in which the input impedance of the rectifieris pure resistive. A LC matching network is added that workswith the series compensation capacitor of the receiving coilto mitigate the influence of the non-neglectable and varyinginput reactance of the rectifier. The LC matching network alsoprovides new degrees of freedom in the following parameterdesign. In order to minimize the energy loss of the MHz ClassE2 wireless battery charging system during the entire chargingcycle, average power loss is defined and calculated based on adiscretized battery charging profile and analytically derivedsystem efficiency. This average power loss is used as theobjective function in the following optimization of the systemdesign parameters. Finally, both the analytical derivations anddesign optimization are validated in experiments.

II. PRELIMINARY ANALYSIS

The configuration of a classical 6.78-MHz Class E2 wire-less battery charging system is shown in Fig. 1. It consistsof the Class E PA, coupling coils, Class E rectifier, andlithium-ion batteries. Ltx and Lrx represent the transmittingand receiving coils, respectively; rtx and rrx are equivalentseries resistance (ESR) of Ltx and Lrx; Ctx and Crx arecompensation capacitors; Zin is the input impedance of thecoupling coils, and Zout is the input impedance seen fromthe receiving coil; Vbat and Ibat are the battery voltage andcharging current, respectively.

The typical charging profile of a lithium-ion battery usuallyconsists of two modes, constant current (CC) mode andconstant voltage (CV) mode. The wireless battery chargingsystem is expected to provide accurate charging current andvoltage that well follow the desired charging profile. Fig. 2shows an example charging profile of a lithium-ion batterypack with four SANYO UR18650A batteries included (4S1P,4 series 1 parallel). This battery pack is used in the final

k

Class E Rectifier

Lrx

Crx

rrx

Ibat

VbatLtx

Ctx

rtx

Class E PA

ZoutZin

Series Resonant Coils Batteries

Vin

Fig. 1. The configuration of a classical 6.78-MHz Class E2 wirelessbattery charging system.

experiments, section V. In this charging profile, the constantcharging current is 1 A and the constant battery voltage is 16.8V, i.e., the nominal charging current and voltage, Inombat andV nombat . The varying charging voltage and current will influence

the values of Zout and Zin, and thus negatively affect theefficiencies of the coupling coil and Class E PA, as discussedbelow. Their influences become obvious in the MHz Class E2

wireless charging systems due to the highly nonlinear behaviorof the Class E rectifier when working at MHz.

CC Mode CV Mode

Vb

at (V

) Ib

at (A

)

Time (min)

Fig. 2. The charging profile of the SANYO UR18650A lithium-ion batterypack.

TABLE IPARAMETERS OF COUPLING COILS.

k Ltx Lrx Ctx Crx rtx rrx0.2 3.34 µH 3.34 µH 165 pF 165 pF 0.7 Ω 0.7 Ω

The performance of the above 6.78-MHz Class E2 wirelessbattery charging system in Fig. 1 is investigated using awell-known nonlinear radio-frequency (RF) circuit simulationsoftware, advanced design system (ADS) from Keysight. Theparameters of the coupling coils are given in Table I. In thesimulation, a half-wave Class E rectifier is employed. Thesame Class E rectifier is used in the following analysis andexperiments. Fig. 3(a) shows the simulation results of theresistance and reactance of Zout, Rout and Xout, under thecharging profile given in Fig. 2. The capacitive, i.e., negative,reactance Xout dramatically increases in the CV mode. Thislarge reactance significantly affects the efficiency and thuspower transfer capability of the coupling coils, as shown inFig. 3(b). Fig. 4 gives the simulation results of the inputimpedance of the coupling coils. Due to the increasing Xout,the resistance and reactance of Zin, Rin and Xin, decrease

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rapidly in the CV mode. The small resistance of Zin will leadto the high power loss on the self-resistance of Ltx, rtx, andparasitic resistances of the components in the Class E PA. Inaddition, the varying Rin and Xin, i.e., the load of the PA,may negatively affect the ZVS operation of the highly load-sensitive Class E PA.

Zout

()

Ω

Time (min)

CC Mode CV Mode

(a)

CC Mode CV Mode

Time (min)

coil

η

(b)

Fig. 3. Simulation results of the input impedance seen from the receivingcoil and efficiency of the coupling coils. (a) Zout. (b) ηcoil.

Zin

()

Ω

Time (min)

CC Mode CV Mode

Fig. 4. Simulation results of the input impedance of the coupling coilsZin.

Crk

Crx

Vdc

Q Cs

Dr

C0

CoLf

Ldc

Ltx Lrx

rtx rrxVG

+Ls

Cp

Zin

Pin

Zout

Pout

Zrec

Prec PO

Class E rectifierCoupling coilsClass E PA

Pdc

Matching

Network

Vbat

Ibat

Fig. 5. Circuit topology of the proposed 6.78-MHz Class E2 wirelessbattery charging system.

As discussed above, following a required battery chargingprofile leads to a changing impedance characteristics, andthus significantly affects the efficiencies of the coupling coiland Class E PA when operating at MHz. It complicates thedesign and implementation of high-efficiency MHz Class E2

wireless charging systems. In order to mitigate the influencesof the changing battery charing current and voltage, a LCmatching network (MN) can be added into the classical ClassE2 wireless charging system. As shown in Fig. 5, the matching

network locates between the coupling coils and Class Erectifier. It improves the loading conditions of the couplingcoils and Class E PA, and introduces new degrees of designfreedom that optimize the overall system efficiency under thebattery charging profile. In Fig. 5, the compensation capacitorof the transmitting coil, Ctx, is absorbed into the seriescapacitor of the Class E PA, C0. Ldc and Lf are the dc filterinductors of the Class E PA and rectifier, respectively. Cs isthe parallel capacitors of the Class E PA. Cr and Co are theshunt capacitor and dc output capacitor of the Class E rectifier.Ls and Cp are the series inductor and parallel capacitor of theLC network, respectively.

III. EFFICIENCY DERIVATION

The equivalent circuit model of the proposed 6.78-MHzClass E2 wireless battery charging system is given in Fig. 6.This circuit model facilitates the following derivation of thesystem efficiency. In the circuit model, the two dc filterinductors, Ldc and Lf , are represented by their equivalentseries resistances (ESRs) rLdc

and rLf, respectively. The ac

inductor of the LC matching network is modeled as a seriescombination of its inductance Ls and ESR rLs

. Table IIsummarizes the definitions of all the circuit parameters. Zout isthe input impedance seen by the receiving coil, and Zin is theinput impedance of the coupling coils. Rdc is the equivalentresistance of the entire charging system seen by the dc powersupply. Idc and Ibat are the dc input and output currents ofthe charging system. iin and iout are the sinusoidal input andoutput currents of the coupling coils. irec is the sinusoidalinput current of the Class E rectifier. iQ is the switch currentwhen Q is on. Similarly iDr is the current that flows throughthe diode when Dr is on. Pin and Pout are the input andoutput powers of the coupling coil. Prec is the input power ofthe rectifier.

TABLE IICIRCUIT PARAMETERS

Parameter DefinitionrLdc

ESR of dc filter Ldc

rQ on-resistance of MOSFET QCS shunt capacitor of PAC0 series capacitor of PALtx inductance of transmitting coilrtx ESR of transmitting coilLrx inductance of receiving coilrrx ESR of receiving coilLm mutual inductanceCp parallel capacitor of MNLs series inductor of MNrLs ESR of ac filter Ls

rLfESR of dc filter Lf

Cr parallel capacitor of rectifierVf diode forward voltage droprDr on-resistance of diodeCo dc output filter

According to [21], the equations for the Class E rectifierthat involve in the following efficiency derivation can be

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rQ Cs

C0

Vdc

rLdcIdc

iin

iQ

Lm

Lrx-LmLtx-Lm

rtx rrx

Crx Cr

rDrVf+

VbatrLf

iout

Ibat

Co

irec

iCp

Ls

Cp

Ibat

Prec

Zrec

Rdc

Pout

Zout

Pin

Zin iDr

rLs

Fig. 6. Equivalent circuit model of the proposed 6.78-MHz Class E2 wireless battery charging system.

obtained,

Im,rec = − Ibatsinϕ1

, (1)

tanϕ1 =1− cos 2πD

sin 2πD + 2π(1−D), (2)

2πωVbatCr

Ibat= 1 +

[sin 2πD + 2π(1−D)]2

1− cos 2πD

− 2π2(1−D)2 − cos 2πD, (3)

and

Rrec =2Vbatsin

2ϕ1

Ibat, (4)

Xrec = − 1

π

[a+ b

ωCr

], (5)

a = π(1−D) + 2π(1−D) sinϕ1 sin(ϕ1 − 2πD), (6)

b = sin 2πD +1

4sin(2ϕ1 − 4πD)− 1

4sin 2ϕ1, (7)

where ϕ1 is the initial phase of irec; D is the duty cycle ofthe rectifying diode; ω is the operating frequency, 6.78-MHzhere; Rrec and Xrec are the resistance and reactance of Zrec;Im,rec is the amplitude of irec; a and b are the intermediatevariables.

As shown in Fig. 6, the below relationship exists,

Pout = Prec + PLs,loss, (8)

where PLs,loss is the power loss on Ls. Because iout and irecare sinusoidal, the above equation can be rewritten as

I2m,outRout = I2m,recRrec + I2m,recrLs , (9)

where Im,out is the amplitude of iout and Rout is the resistanceof the input impedance seen by the receiving coil, Zout. Fromthe circuit model in Fig. 6, Zout(= Rout + jXout) can bederived as

Rout =Rrec+rLs

(ωCp)2[(Rrec+rLs )

2+( 1ωCp

+ωLs+Xrec)2] , (10)

Xout =(ωCp)(Rrec+rLs )

2+(Xrec+ωLs)

(ωCp)2[(Rrec+rLs )

2+( 1ωCp

+ωLs+Xrec)2] . (11)

Substituting (1) into (9) gives

Im,out =Ibat

|sinϕ1|

√Rrec + rLs

Rout. (12)

Similarly,

I2m,inRin = I2m,inrtx + I2m,outrrx + I2m,outRout, (13)

where Im,in is the amplitude of iin and Rin is the resistanceof the input impedance of the coupling coils, Zin. Thus

Im,in =Ibat

|sinϕ1|

√Rrec + rLs

Rout

√Rout + rrxRin − rtx

. (14)

Again, from the circuit model, Zin(= Rin + jXin) can bederived as follows,

Rin = rtx +ω2Lm

2(Rout+rrx)

(Rout+rrx)2+ (Xout + ωLrx − 1

ωCrx)2 , (15)

Xin = ωLtx −ω2Lm

2(Xout + ωLrx − 1ωCrx

)

(Rout+rrx)2+ (Xout + ωLrx − 1

ωCrx)2 .

(16)In the above equations, Lm is the mutual inductance,

Lm = k√

LtxLrx, (17)

where k is the mutual inductance coefficient. The dc inputcurrent and equivalent resistance of the wireless chargingsystem can be obtained as follows [22],

Idc =Im,in

g, (18)

Rdc =π2 − g(2π cosϕ2 + 4 sinϕ2)

4πωCS, (19)

where

g =2π sin(φ+ ϕ2) + 4cos(φ+ ϕ2)

4 cosϕ2 sin(φ+ ϕ2) + π cosφ, (20)

φ = arctanXin − 1

ωC0

Rin, (21)

ϕ2 = arctanπ2

2 −4−πωCS

[2Rin+π(Xin− 1

ωC0)]

π+π2ωCSRin−2πωCS(Xin− 1ωC0

). (22)

ϕ2 is initial phase of iin; g and φ are the intermediatevariables. Substituting (14) into (18) gives

Idc =Ibat

g |sinϕ1|

√Rrec + rLs

Rout

√Rout + rrxRin − rtx

. (23)

There are also power losses from the two dc filter inductors,Ldc and Lf , switch Q and diode Dr. Thus the overall systemefficiency is expressed as below,

ηsys =IbatVbat

I2dcRdc + I2dcrLdc+ I2batrLf

+ PQ,loss + PDr,loss,

(24)

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ηsys =4πVbat

Ibat[(Rrec+rLs)(Rout+rrx)][4π(Rdc+rLdc)+(2π+8 cosϕ2g+πg2)rQ]

Rout(Rin−rtx)g2sin2ϕ1+ 4π

[Ibat(rLf

+ erDr) + fVf

] . (29)

where

PQ,loss =1

2π

∫ π

0

i2QrQdωt =

(1

2+ 2 cosϕ2

g

π+

g2

4

)I2dcrQ,

(25)

PDr,loss =1

2π

∫ 2πD

0

(i2DrrDr + iDrVf )dωt

= eI2batrDr+ fIbatVf . (26)

The coefficients are

f = 2π + cotϕ1 −cos(ϕ1 − 2πD)

sinϕ1, (27)

e = D +D

2sin2ϕ1

− cos(ϕ1 − 2πD)

π sinϕ1

+3 sin 2ϕ1 + sin(2ϕ1 − 4πD)

8πsin2ϕ1

. (28)

Substituting (23)(25)(26) into (24) gives (29), the overallefficiency of the wireless charging system, ηsys. It can beseen that ηsys depends on the battery voltage Vbat, chargingcurrent Ibat, parasitic parameters of the components, andsystem parameters. Note that the influences of the systemparameters are represented by Rdc, Rin, Rout, and Rrec in(29).

As discussed in section II, when operating at MHz, fol-lowing the battery charging profile leads to a varying batteryvoltage (Vbat) and charging current (Ibat), and thus a largeand varying input reactance of the Class E rectifier (Xrec). Itwill not only negatively affect the efficiency of the wirelesscharging system but also present challenges in the optimizeddesign of the system. A systematic parameter design approachis developed below that minimizes the energy loss of the MHzClass E2 wireless battery charging system over the entirecharging cycle.

IV. DESIGN PARAMETERS OPTIMIZATION

At given time t in the battery charging profile, the outputpower of the charging system, Po, is

Po(t) = Ibat(t)Vbat(t). (30)

Then the total power loss, Ploss, at time t is

Ploss(t) =Po(t)

ηsys(t)− Po(t) =

Ibat(t)Vbat(t)

ηsys(t)[1− ηsys(t)] .

(31)Here ηsys(t) can be calculated by substituting Ibat(t) andVbat(t) into (29). For the following optimization, the chargingprofile is evenly divided into N pieces. Thus the average powerloss based on the discretized battery charging profile, namelythe objective function of the optimization problem, is definedas

P avgloss =

N∑i=1

Ibat(ti)Vbat(ti)ηsys(ti)

[1− ηsys(ti)]

N. (32)

As shown in Fig. 5, the purpose of the LC matching net-work is to improve the loading condition of the coupling coils.Thus the inductor Ls should be designed to partly compensatethe non-neglectable reactance of the input impedance of theClass E rectifier operating at MHz. From (5),

Ls = −Xrec

ω

∣∣∣∣V nombat ,Inom

bat

=1

π

[a+ b

ω2Cr

]∣∣∣∣V nombat ,Inom

bat

, (33)

which targets the nominal operating condition, i.e., the nom-inal battery voltage and charging current, V nom

bat and Inombat .Note that Xrec is determined by Cr, Vbat, and Ibat [refer to(3)(5)]. Then the ESR of Ls, rLs , can be calculated,

rLs =ωLs

QLs

=1

πQLs

[a+ b

ωCr

]∣∣∣∣V nombat ,Inom

bat

, (34)

where QLs is the quality factor of Ls. Note that a candidateCr is given for calculating Ls during the below designoptimization.

The design parameters, the capacitors, in the below opti-mization are finalized as follows,

x = [CS , C0, Crx, Cp, Cr], (35)

that minimizes the average power loss, P avgloss over the entire

charging profile. Here x is a vector. CS and C0 are theparallel and series capacitors of the Class E PA; Crx is thecompensation capacitor of the receiving coil; Cp is the parallelcapacitor of the LC matching network; and Cr is the shuntcapacitor of the Class E rectifier. Again, due to the non-neglectable Xrec, Crx is also taken as a design parameter thatfurther lowers the influence of Xrec during the entire charingcycle. Thus, unlike in the conventional design, the resonanceof the receiving coil is not pre-assumed here. The constantparameters in the proposed optimized design of the wirelesscharging system are

pcon = [Ltx, Lrx, Vf , rDr, rtx, rrx, rLdc

, rLf, rQ, ω, k].

(36)

Candidate x from

(xlower, x

upper)

Calculate Ls,rLs

using candidate x

and equ. (33)(34)

Battery

charging profile

Calculate

using equ. (32)

Constant

parameters

Pcon

Ibat, VbatP lossavg

I batnom, V bat

nom

Fig. 7. Relationship between the constant and design parameters, andbattery charging profile in the optimized design.

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The design optimization problem is formulated as follows,

minx

P avgloss (x,pcon, Ibat,Vbat) (37)

s.t. x ∈ (xlower,xupper), (38)

where xlower and xupper are the lower and upper feasiblebounds of x, respectively. Ibat and Vbat represent the dis-credited battery charging profile,

Ibat = [Ibat(t1), Ibat(t2), . . . , Ibat(tN )], (39)Vbat = [Vbat(t1), Vbat(t2), . . . , Vbat(tN )]. (40)

The average power loss, P avgloss, is determined by the design

and constant parameters of the MHz wireless charging system,and the specific battery charging profile [refer to (32) andFig. 7]. The purpose of the optimization problem is to findan optimal set of the design parameters, xopt, that achievesthe lowest average power loss over a specific charging profile.The proposed design optimization is a general solution thatminimizes the energy loss when charging batteries with dif-ferent profiles, i.e., different Ibat and Vbat. Given the natureof the above optimization problem, it is appropriate to applygenetic algorithm (GA), a popular population-based heuristicapproach, to locate a global or at least near-to-global optimalsolution [23], [24].

Fig. 8. Experimental setup of the proposed 6.78-MHz Class E2 wirelessbattery charging system.

V. EXPERIMENTAL RESULTS

As shown in Fig. 8, an experimental Class E2 wirelesscharging system working at 6.78 MHz is built up to validatethe above proposed configuration and parameter design. Thelithium-ion battery pack uses four SANYO UR18650A bat-teries connected in 4S1P. In experiments, the battery chargingprofile in Fig. 2 is implemented through the control of thedc input voltage of the wireless charging system, i.e., theoutput voltage of the power supply. A high-accuracy sam-pling resistor is connected in series with the battery pack

for charging current measurement. The National Instruments(NI) CompactRio system is employed to measure/sample thebattery voltage and charging current, Vbat(t) and Ibat(t), andcontrol the dc output voltage of the power supply based onthe voltage and current feedback signals. In the CC mode, thepower supply is controlled by the NI compactRIO system toincrease its dc output voltage in order to achieve the constantbattery charging current. Similarly, after the battery voltagereaches the nominal value, the power supply dc output voltageis decreased to keep the constant battery voltage, i.e., theCV mode. When the battery charging current is lower thanthe termination current (here 0.05 A recommended by thedatasheet of the battery cell), the charging cycle stops. Thewireless battery charging system has the same configuration asthat in Fig. 5. Its constant parameters, pcon, are summarizedin Table III. The distance of the coupling coils is 30 mmcorresponding to a mutual inductance coefficient, k=0.2. AMOSFET SUD06N10 and a silicon carbide diode STPSC406are used as the switch Q and rectifying diode Dr of the ClassE PA and rectifier, respectively. The parasitic capacitors ofSUD06N10 and STPSC406 are assumed to be constant, 50pF and 35 pF, according to their datasheets. Note that thesetwo parasitic capacitances are included in the calculated finalCr and CS , respectively.

TABLE IIICONSTANT PARAMETERS

ω Ltx Lrx Vf rDr rQ6.78 MHz 3.34 µ H 3.34 µ H 1.2 V 0.3 Ω 0.225 Ω

k rtx rrx rLdcrLf

-0.2 0.7 Ω 0.7 Ω 0.2 Ω 0.2 Ω -

In the experiments, the feasible range of the design param-eters, x = [CS , C0, Crx, Cp, Cr], are given as

xlower = [100 pF, 100 pF, 100 pF, 100 pF, 100 pF],

xupper = [2000 pF, 2000 pF, 2000 pF, 2000 pF, 2000 pF] (41)

Here the feasible ranges are estimated based on the induc-tances of the coupling coils and equivalent loads under thebattery charging profile. By applying the design proceduresdeveloped in section IV and charging profile in Fig. 2, theoptimal design parameters, xopt, and thus Ls, the inductor ofthe LC matching network, are calculated as follows,

xopt = [105 pF, 250 pF, 195 pF, 192 pF, 120 pF], (42)Ls = 323 nH. (43)

The inductance of Ls is finalized as 330 nH, which iscommercially available. In the above optimization, the totalnumber of the sampling instants of the charging profile, N , ischosen as 50. There are 28 sampling instants in the CC mode,and 22 instants in the CV mode. Thus the system efficiencyduring the CC mode is particularly emphasized.

In the experiments, the battery charging profile is accuratelyfollowed by controlling the dc output voltage of the powersupply Vdc (see Fig. 5). Fig. 9(a) and (b) show Vdc and Po, theinput voltage and output power of the experimental wirelesscharging system under the single charging profile in Fig. 2.

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In the CC mode, the current is controlled as 1 A and thebattery voltage increases until it reaches the nominal value16.8 V at charge time t=115 min. After entering the CV mode,Vdc decreases rapidly in order to maintain the nominal batteryvoltage. The output power of the wireless charging systemshows a similar trend.

CC Mode CV Mode

Vdc (V

)

Time (min)

28 points 22 points

(a)

CC Mode CV Mode

PO

(W

)

Time (min)

(b)

Fig. 9. The dc input voltage and output power of the experimentalwireless charging system under a single charging profile. (a) Vdc. (b)Po.

Fig. 10–12 compare the system and component-level effi-ciencies and overall power loss when applying the conven-tional and proposed designs. In the conventional design, thedesign parameters are optimized targeting on only a singleoperation condition, i.e., the nominal battery voltage andcharging current, V nom

bat and Inombat [20]. The goals of theconvectional design are 1) resonance of the receiving coilexactly at the target frequency, 6.78 MHz here; 2) ZVSoperation of the Class E PA when its load, i.e., the inputimpedance of the coupling coil Zin, is as equal as the oneunder V nom

bat (= 16.8 V) and Inombat (=1 A); 3) a duty cycleD (=0.5) of the rectifying diode, Dr, in the Class E rectifier,again, under V nom

bat and Inombat [21]. Its circuit model can beobtained by removing the LC matching network in Fig. 5. Theresults of the conventional design are calculated and listed inTable IV based on the constant parameters given in Table IIIand the nominal operating condition.

As shown in Fig. 10 and 11, the proposed design demon-strates obvious improvements in both system efficiency andpower loss during the entire charging cycle. The experimental

TABLE IVRESULTS OF CONVENTIONAL DESIGN

CS C0 Crx Cr

130 pF 168 pF 165 pF 420 pF

CC Mode CV Mode

Time (min)

sys

η

Proposed Design (exp.)Proposed Design (cal.)

Conventional Design (exp.)

Fig. 10. Experimental results of the system efficiency.

Proposed Design (exp.)Proposed Design (cal.)

Conventional Design (exp.)

CC Mode CV Mode

Plo

ss (W

)

Time (min)

Fig. 11. Experimental results of the power loss.

results using the proposed design well match the calculatedresults, which verifies the correctness of the analytical deriva-tions in section III. The conventional design achieves opti-mized performance only when the nominal operating condi-tion, its design target, is met, i.e., V nom

bat =16.8 V and Inombat =1 Aat t=115 min. When the operation deviates from this nominalone, its performance starts to deteriorate. The proposed designoutpaces the conventional design within the most chargingcycle because it is optimized targeting on the battery chargingprofile rather than a specific single operating condition.

Note that mostly due to improved coupling coil efficiencyin the proposed parameter design, the final system efficiencywhen using the conventional design is not obviously higher(1.0% percentage improvement) even under the nominal op-erating condition [see Figs. 10 and 12]. In the conventionaldesign, the design parameters are optimized with the givencoupling coils and targeting on the nominal battery voltage andcharging current, i.e., a specific fixed nominal load. However,this nominal load does not necessarily equal to the final loadthat corresponds to the highest achievable efficiency of thegiven coupling coils [25]. It is because practically coupling

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coils are usually designed to first meet other requirements suchas targeted size, transfer distance, and power level. Meanwhile,thanks to the LC matching network and proposed parameterdesign, the coupling coils work under a better loading condi-tion (namely a load different with the one seen by the receivingcoil in the conventional design), and thus with obviouslyimproved efficiency. This aspect can be particularly explainedby Fig. 12(a) and (b), the component-level efficiencies inthe experiments. Since the efficiency of the Class E rectifieris largely determined by the battery voltage and chargingcurrent, its efficiencies in the two designs are close. Thusthe efficiencies of the coupling coils (ηcoil) and Class E PA(ηpa) are shown in the two subfigures. Besides the improvedcoupling coil efficiency, using the proposed design the Class EPA also demonstrates higher efficiency during the most high-power period (e.g., charging time from 0 to 105 min) [referto Fig. 9(b)]. Note that the PA efficiency in the conventionaldesign is higher when the operating condition is close to thenominal one, i.e., its design target, while the target of theproposed design is to minimize the energy loss over the entirecharging cycle.

Proposed Design Conventional Design

CC Mode CV Mode

Time (min)

coil

η

(a)

Proposed Design Conventional Design

CC Mode CV Mode

Time (min)

pa

η

(b)

Fig. 12. Experimental results of the efficiencies of the coupling coils andClass E PA. (a) ηcoil. (b) ηpa.

Table V lists the average power loss, P avgloss, and the total

energy loss, Eloss, within the single battery charging profile.Here Eloss equals the production of P avg

loss and total chargingtime, 221 minutes. With the proposed parameter design, theaverage power loss is reduced by 24.5% when comparingwith that in the conventional design. The saved total energyis 11536 J. From the above experimental results, the averageefficiencies of the conventional design and proposed designare 74.5% and 80.9%, respectively, namely 8.6% improvementpercentage of the system efficiency over the entire chargingcycle. Because the objective function in the optimizationproblem is the average power loss, (32), naturally the proposeddesign assigns higher priority to achieving higher efficiencyduring the high-power period. This leads to the observedhigher improvement in the average power loss than that inthe average system efficiency, again 24.5% versus 8.6%.

TABLE VAVERAGE POWER LOSS AND TOTAL ENERGY LOSS.

Con. design Proposed designPavgloss 3.55 W 2.68 W

Eloss 47073 J 35537 J

For reference purposes, an additional charging profile (solidline) and experimental/calculated results are shown in Fig. 13and 14, respectively, when using a different type of batterycell (4S1P), TNL-ITR18650. The total charging time of theTNL-ITR18650 battery pack is 208 min and the pack voltagereaches its nominal value at 95 min. Due to the differ-ent charging profile, the design parameters are recalculatedthrough the proposed design, x

′

opt=[126 pF, 240 pF, 180 pF,262 pF, 110 pF] and L

′

s=328 nH. Similar improvementsin the system efficiency and power loss are observed inFig. 14(a) and (b). Table VI summarizes the average powerloss in the conventional design and proposed design, P avg

loss,when charging the two different battery packs. Applying theproposed design methodology, the average power loss duringthe entire charging cycle is significantly reduced by about 25%for the both battery packs.

Vb

at (V

) Ib

at (A

)

Time (min)

Fig. 13. Charging profiles of TNL-ITR18650 battery pack and SANYOUR18650 battery pack.

Proposed Design (exp.)

Proposed Design (cal.)

Conventional Design (exp.)

CC Mode CV Mode

Time (min)

sys

η

(a)

CC Mode CV Mode

Plo

ss (W

)

Time (min)

Proposed Design (exp.)

Proposed Design (cal.)

Conventional Design (exp.)

(b)

Fig. 14. Experimental results when charging TNL-ITR18650 batterypack. (a) system efficiency. (b) power loss.

TABLE VIAVERAGE POWER LOSS WHEN CHARGING TWO BATTERY PACKS.

UR18650 ITR18650Con. design 3.55 W 3.36 W

Proposed design 2.68 W 2.46 WReduction 24.5% 26.8%

VI. CONCLUSIONS

This paper proposes a systematic design methodology for ahigh-efficiency 6.78-MHz Class E2 wireless battery chargingsystem. Through the preliminary analysis, it shows that the

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non-neglectable and varying input reactance of the Class Erectifier adversely affects the efficiency of the system whenfollowing the battery charging profile. The average powerloss is then defined based on the analytically derived systemefficiency and discretized charging file. The design parameters,the five capacitors and one inductor, are optimized later thatminimizes the average power loss during the entire batterycharging cycle. The experiments show that the proposed designachieves a significant reduction of the average power loss,24.5%, when comparing with that through the conventionaldesign. The design methodology proposed in this paper isa general solution that enables high-efficiency MHz wirelesscharging under any specific battery charging profile.

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[5] S. Li, W. Li, J. Deng, T. D. Nguyen, and C. C. Mi, “A double-sidedLCC compensation network and its tuning method for wireless powertransfer,” IEEE Trans. Veh. Technol., vol. 64, no. 6, pp. 2261–2273, Jun.2015.

[6] W. Zhong and S. Hui, “Maximum energy efficiency tracking for wirelesspower transfer systems,” IEEE Trans. on Power Electron., vol. 30, no. 7,pp. 4025–4034, Jul. 2015.

[7] M. Pinuela, D. C. Yates, S. Lucyszyn, and P. D. Mitcheson, “MaximizingDC-to-load efficiency for inductive power transfer,” IEEE Trans. PowerElectron., vol. 28, no. 5, pp. 2437–2447, May. 2013.

[8] W. Zhong, C. Zhang, X. Liu, and S. Hui, “A methodology for makinga three-coil wireless power transfer system more energy efficient than atwo-coil counterpart for extended transfer distance,” IEEE Trans. PowerElectron., vol. 30, no. 2, pp. 933–942, Feb. 2015.

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[12] S. Aldhaher, P.-K. Luk, A. Bati, and J. Whidborne, “Wireless powertransfer using Class E inverter with saturable DC-feed inductor,” IEEETrans. Ind. Appl., vol. 50, no. 4, pp. 2710–2718, Jul. 2014.

[13] S. Aldhaher, P.-K. Luk, and J. F. Whidborne, “Electronic tuning ofmisaligned coils in wireless power transfer systems,” IEEE Trans. PowerElectron., vol. 29, no. 11, pp. 5975–5982, Nov. 2014.

[14] S. Liu, M. Liu, M. Fu, C. Ma, and X. Zhu, “A high-efficiency Class-Epower amplifier with wide-range load in wpt systems,” in Proc. IEEEWireless Power Transfer Conference (WPTC), Boulder, CO, USA, May.2015.

[15] W. Nitz, W. Bowman, F. Dickens, F. Magalhaes, W. Strauss, W. Suiter,and N. Ziesse, “A new family of resonant rectifier circuits for high fre-quency DC-DC converter applications,” in Proc. Appl., Power Electron.Conf., pp. 12–22, New Orleans, LA, USA, Feb. 1988.

[16] S. Aldhaher, P.-K. Luk, K. El Khamlichi Drissi, and J. Whidborne,“High-input-voltage high-frequency Class E rectifiers for resonant in-ductive links,” IEEE Trans. Power Electron., vol. 30, no. 3, pp. 1328–1335, Mar. 2015.

[17] H. V. Venkatasetty and Y. U. Jeong, “Recent advances in lithium-ionand lithium-polymer batteries,” in Proc. The Seventeenth Annual BatteryConference on Applications and Advances, pp. 173–178, Long Beach,CA, USA, Jan. 2002.

[18] A. Khaligh and Z. Li, “Battery, ultracapacitor, fuel cell, and hybridenergy storage systems for electric, hybrid electric, fuel cell, and plug-in hybrid electric vehicles: State of the art,” IEEE Trans. Veh. Technol,vol. 59, no. 6, pp. 2806–2814, Jul. 2010.

[19] X. Qu, H. Han, S. C. Wong, C. K. Tse, and W. Chen, “Hybrid IPTtopologies with constant current or constant voltage output for batterycharging applications,” IEEE Trans. Power Electron., vol. 30, no. 11,pp. 6329–6337, Nov. 2015.

[20] P.-K. Luk, S. Aldhaher, W. Fei, and J. Whidborne, “State-space modelingof a Class E2 converter for inductive links,” IEEE Trans. PowerElectron., vol. 30, no. 6, pp. 3242–3251, Jun. 2015.

[21] M. Kazimierczuk, “Class E low dvd/dt rectifier,” Electric Power Appli-cations, IEE Proceedings B, vol. 136, no. 6, pp. 257–262, Nov. 1989.

[22] M. Albulet, RF power amplifiers. SciTech Publishing, 2001.[23] M. Li, S. Azarm, and A. Boyars, “A new deterministic approach using

sensitivity region measures for multi-objective robust and feasibilityrobust design optimization,” Journal of mechanical design, vol. 128,no. 4, pp. 874–883, Dec. 2006.

[24] M. J. Neath, A. K. Swain, U. K. Madawala, and D. J. Thrimawithana,“An optimal PID controller for a bidirectional inductive power transfersystem using multiobjective genetic algorithm,” IEEE Trans. PowerElectron., vol. 29, no. 3, pp. 1523–1531, Mar. 2014.

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Ming Liu (S’15) received the B.S. degree fromSiChuan University, Sichuan, China, in 2007,and the M.S. degree from the University ofScience and Technology Beijing, Beijing, China,in 2011, both in mechatronic engineering. Heis currently working toward the Ph.D. degreein electrical and computer engineering fromthe University of Michigan-Shanghai Jiao TongUniversity Joint Institute, Shanghai Jiao TongUniversity, Shanghai, China.

His research interests include high frequencypower electronic circuits such as high frequency resonant convertersand megahertz wireless power transfer systems, general powerelectronics and applications, circuit and system-level optimization.

Chen Zhao received the B.S. degree inelectrical engineering and automation from EastChina University of Science and Technology,Shanghai, China, in 2011, and Ph.D. degreein electrical and computer engineering fromUniversity of Michigan-Shanghai Jiao TongUniversity Joint Institute, Shanghai Jiao TongUniversity, China, in 2016. He is currentlya systems engineer with United AutomotiveElectronic Systems Co., Ltd, Shanghai, China.

His research interests include modeling andtesting of lithium-ion batteries and control of battery-ultracapacitorhybrid energy storage systems.

Jibin Song (S’16) received the B.S. degreein measurement & control technology andinstrumentation from Jilin University, Jilin,China, in 2016. He is currently working towardthe Ph.D. degree in electrical and computerengineering, University of Michigan-ShanghaiJiao Tong University Joint Institute, ShanghaiJiao Tong University, Shanghai, China.

His research interests include powerelectronics, design and optimization of multiple-receiver megahertz wireless power transfer

systems.

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Chengbin Ma (M’05) received the B.S. (Hons.)degree in industrial automation from EastChina University of Science and Technology,Shanghai, China, in 1997, and the M.S. andPh.D. degrees both in electrical engineeringfrom University of Tokyo, Japan, in 2001 and2004, respectively. He is currently an associateprofessor of electrical and computer engineeringwith the University of Michigan-Shanghai JiaoTong University Joint Institute, Shanghai JiaoTong University, China. He is also with a joint

faculty appointment in School of Mechanical Engineering, ShanghaiJiao Tong University. Between 2006 and 2008, he held a post-doctoral position with the Department of Mechanical and AeronauticalEngineering, University of California Davis, USA. From 2004 to 2006, hewas a R&D researcher with Servo Laboratory, Fanuc Limited, Japan.

His research interests include wireless power transfer, networkedhybrid energy systems such as microgrids and smart grids, andmechatronic control.