Analysis and Comparison of Secondary Series- and Parallel-Compensated Inductive Power Transfer...

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IEEE TRANSACTION ON POWER ELECTRONICS, VOL.X, NO. X, MONTH 2013 1

Analysis and Comparison of Secondary Series andParallel Compensated Inductive Power TransferSystems Operating for Optimal Efficiency and

Load-independent Voltage-Transfer RatioWei Zhang,Student Member, IEEE,Siu-Chung Wong,Senior Member, IEEE,Chi K. Tse,Fellow, IEEE,and

Qianhong Chen,Member, IEEE

Abstract—Secondary series and parallel compensations arewidely used in inductive power transfer (IPT) systems forvarious applications. These compensations are often studiedunder some isolated constraints of maximum power transfer,optimal efficiency at a particular loading condition, etc. Theseconstraints constitute an insufficient set of requirements forengineers to select appropriate compensation techniques to beused as a voltage converter with optimal efficiency and loadingconditions. This paper studies the characteristics of the IPTsystem at various frequencies of operation utilizing the twocompensation techniques to work as a voltage converter. Thefrequencies that can provide maximum efficiency of operationand load-independent voltage-transfer ratio are analyzed. Theoptimal frequencies corresponding to the two compensationtechniques are found and compared to facilitate the design ofvoltage converters with efficient power conversion and load-independent frequency of operation. The analysis is supportedby experimental measurements.

Index Terms—Inductive power transfer, series-series compen-sation, series-parallel compensation, resonance power converter,loosely-coupled transformer, voltage transfer function.

NOMENCLATURE

ω Operating angular frequency of the systemRL Equivalent loading resistance of the systemRP Winding resistance of the transformer primaryRS Winding resistance of the transformer secondaryk Coupling coefficientLP Transformer primary inductanceLS Transformer secondary inductanceM Transformer mutual inductanceGV Output-to-input voltage-transfer functionGV id GV with ideal (lossless) componentsGV−S GV with series-compensation, likewiseXy−S stands

for Xy with series-compensationGV−P GV with parallel-compensation, likewiseXy−P stands

for Xy with parallel-compensation

Manuscript received March 6, 2013; revised July 1, 2013; accepted July 8,2013. This work is supported by Hong Kong Polytechnic University underCentral Research Grant G-YJ90.

W. Zhang, S.C. Wong and C. K. Tse are with the Department of Elec-tronic and Information Engineering, The Hong Kong Polytechnic University,Kowloon, Hong Kong (Email: [email protected])

Q. Chen is with the Nanjing University of Aeronautics and Astronautics,Nanjing 210016, China

CS Capacitance for secondary series or parallel-compensation

ZS Impedance of the secondary network including the load-ing resistanceRL

ZP Impedance of the primary networkωS = 1

LSCS, for brevity LS represents bothLS−S and

LS−P . The same applies toCS .ωP = 1

LPCP

ωQ Operating frequencies at whichGV−P is RL-independent

ωH , ωL Operating frequencies at whichGV−S is RL-independent whereωH > ωL

Zr Equivalent secondary-side impedance reflected to theprimary

RO = RS +RL

QL−S Quality factor of secondary-side resonant tank withseries compensation and output loadRL

QO−S Quality factor of secondary-side resonant tank withseries compensation and output loadRO

ωM The operating frequency to achieve the theoretical max-imum efficiency for secondary series-compensation

λ = k2QP

QS

QL−P Quality factor of secondary-side resonant tank withparallel compensation and output loadRL

ωN Operating frequency to achieve the maximum efficiencyfor secondary parallel-compensation

QLp Value of QL−P operating atωN to achieve the maxi-mum efficiency for secondary parallel-compensation

ωnp = ωωP

ωns = ωωS

I. I NTRODUCTION

I N the application of magnetic induction for the transfer ofelectric power, voltage to voltage conversion is often desir-

able for effective operation. Development in modern powerelectronics has enabled many new loosely-coupled power-transfer applications. The inductive power transfer (IPT)sys-tem is designed to deliver power to a load over a relativelylarge air gap via magnetic coupling. With an extra degree offreedom on electrical isolation and the absence of mechanicalcontact between the power supply side and the load, the IPT

0000–0000/00$00.00c© 2013 IEEE



2 IEEE TRANSACTION ON POWER ELECTRONICS, VOL.X, NO. X, MONTH 2013

Primary

compensation

Secondary

compensation

Loosely coupled transformer

LP LS

RP RS

AC input Loading

RL

M

Fig. 1. Circuit model of IPT System.

system has the advantages of low maintenance cost, highreliability and the ability to operate in ultra clean or ultradirty environment. This technology has found applicationsin many high power fields such as continuously poweredelectric vehicles (the roadway powered EVs) [1]–[5], and incontactless EV battery charging when the vehicle is parkingin a power station [6]–[8]. IPT can also be found in the areaof relatively low power applications, such as wireless powersupplies to home appliances [9]–[11], and wireless powersupplies to implantable devices [12]–[15] in medical use.

Since IPT systems extend the fundamental principle of mag-netic induction of the widely used electromechanical deviceswith good coupling such as transformers and induction motorsto the use of loosely coupled transformers, additional develop-ment of working principles for the effective transfer of poweris needed [16]–[21]. The loosely coupled transformer can bemodeled and shown in Fig. 1, with primary and secondaryresonant tanks to transfer power via the mutual coupling ofthe conducting primary and secondary winding conductors.This is almost identical to the traditional transformer modelexcept that the value of mutual inductance cannot be assumedto be much larger than the leakage inductances of the primaryand secondary windings. In Fig. 1, an equivalent AC voltagesource or current source can be adopted as the power supply,RL is the equivalent loading resistor of the system,RP andRS are the winding resistances of the transformer primary andsecondary respectively, the coupling coefficientk = M√

LPLS

of the loosely coupled transformer is much lower than thatof a traditional transformer, and finallyLP , LS and M arethe transformer primary, secondary, and mutual inductances,respectively.

Research has been carried out in order to select the mostappropriate compensation techniques for specific applications[6]–[8], [22]–[26]. Research has also been carried out lookingfor coupling-insensitive IPT [27], [28]. However, there israrely a general study of the IPT system to work as a voltageconverter providing related information for the design ofoptimal conversion efficiency, voltage transfer characteristicsversus operating frequencies and loading conditions.

In this paper, a general analysis is proposed for the IPT con-verter using secondary series- and parallel-compensations toachieve an optimal operating efficiency and a load-independentvoltage-transfer ratio. The paper is structured as followsforthe study of the converter using secondary series- and parallel-compensations. The voltage-transfer ratio of the converter isstudied in Section II. The converter efficiency is studied inSection III. The input impedance of the compensated trans-former is analyzed to guarantee soft switching in Section IV.Two IPT prototypes providing an output power of 24 W and

LPRP

Voltage AC source

MiS

iP

Primary loop

RS

Loading

RL

LS

MiP

iS

Series compensation

Parallel compensation

RS

Loading

RL

LS

MiP

iS

CS

CS

Secondary loop

CP

Fig. 2. Subcircuits for the illustration of series-compensation and parallel-compensation of the transformer primary and secondary.

a conversion efficiency of higher than 90% at an air-gap of35 mm is developed to validate the analytical findings inSection V. Finally, Section VI concludes the paper.

II. V OLTAGE TRANSFERRATIO

Fig. 2 gives an equivalent circuit of Fig. 1 for the analysis ofsteady-state transfer functions. The primary of the transformeris driven by a modulated AC voltage source which is an equiv-alent voltage readily generated from a pulse-width-modulatedDC voltage source using either a simple full- or half-bridgeswitching circuit. For a parallel resonant compensated primary,an equivalent current source is needed. Due to the difficultyof energy storage in the form of a simple current source,extra components will be needed to transfer energy on demandfrom a voltage source, incurring extra loss. Therefore, seriesresonant primary compensation will be used in this paper. Thesecondary can be compensated either by a series resonantcircuit or a parallel resonant circuit as shown on the right-hand-side of Fig. 2.

In the subsequent analysis, a frequency-domain equivalentcircuit is adopted and only the fundamental component isconsidered here for simplicity [1], [6]–[8], [29]. The fun-damental component approximation is sufficiently accuratefor a high quality-factor resonant circuit that works nearresonance, where the tank-inductor-current and tank-capacitor-voltage waveforms are sinusoidal.

From Fig. 2, the output voltage acrossRL of the series- orparallel-compensated secondary is calculated with supportingparameters as shown in Table I. The output-to-input voltage-transfer functionsGV−S andGV−P of secondary series andparallel compensation are expressed in (1) and (2), respec-tively. The tailing subscript−S or −P of GV indicates series-or parallel-compensation respectively. This labeling methodwill be used for other variables throughout this paper.

GV−S =voutvin

=ωM

ZPZS−S + ω2M2RL. (1)

GV−P =voutvin

=ωM

ZPZS−P + ω2M2(

1

jωCS

||RL). (2)



ZHANG et al.: ANALYSIS AND COMPARISON OF SECONDARY SERIES AND PARALLEL COMPENSATED IPT SYSTEMS 3

TABLE IINTERMEDIATE PARAMETERS NEEDED FOR CALCULATINGGV .

Series compensation Parallel compensationZS jωLS + 1

jωCS+RS +RL jωLS +RS + 1

jωCS||RL

vout jωMRL

ZS−S+RLip ωM

1

jωCS||RL

ZS−Pip

ioutωM

ZS−Sip ωM

1

jωCS||RL

ZS−PRLip

ZP jωLP + 1jωCP

+RP

ip vin

(

ZP + ω2M2

ZS

)−1

In (1) and (2),ZS (ZS−S or ZS−P ) is the impedance of thesecondary network including the loading resistanceRL, andZP is the impedance of the primary network.

A desirable feature of|GV−S | [30], [31] is that it isRL-independent when the converter is operated at the frequencyof ωH or ωL. The frequencies can be found by solving theequation∂GV −S

∂RL= 0 for ω using (1) and assumingRP = 0

andRS = 0, i.e.,

GV id−S(ωL) = −j

√

LS

LP

ω2P + ω2

S +∆

(1− 2k2)ω2P + ω2

S +∆, (3)

GV id−S(ωH) = −j

√

LS

LP

ω2P + ω2

S −∆

(1− 2k2)ω2P − ω2

S +∆, (4)

ωL =

√

ω2P + ω2

S −∆

2(1− k2), and (5)

ωH =

√

ω2P + ω2

S +∆

2(1− k2), where (6)

∆ =√

(ω2P + ω2

S)2 − 4(1− k2)ω2

Pω2S , (7)

ωP =1√

LPCP

, and (8)

ωS =1√

LSCS

. (9)

To account for the loss due toRP andRS and the associatedoutput voltage drop, (5) and (6) are substituted separatelyinto(1) to obtain

GV−S(ωL) =GV id−S(ωL)

β(ωL)GV id−S(ωL) + 1, and (10)

GV−S(ωH) =GV id−S(ωH)

β(ωH)GV id−S(ωH) + 1, where (11)

β(ω) = β1(ω) + β2(ω) + β3(ω), (12)

β1(ω) = RP

RL + j(

ωLS − 1ωCS

)

kRLω√LPLS

, (13)

β2(ω) = RS

j(

ωLP − 1ωCP

)

kRLω√LPLS

, and (14)

β3(ω) =RPRS

kRLω√LPLS

. (15)

Similarly, we explore an operating frequency of thesecondary-parallel-compensated converter at which|GV−P |

ROeq

RL

iout

vin·Gvid

vin·Gv

Fig. 3. Output Thevenin’s equivalent circuit of the converter, where(GV id, ROeq, GV ) can be (GV id−S(ωL), ROLeq−S , GV −S(ωL)),(GV id−S(ωH), ROHeq−S , GV −S(ωH)) or(GV id−P (ωQ), ROeq−P , GV −P (ωQ)).

is also load-independent. The ideal load-independent voltage-transfer function is calculated by solvingωQ in the equation∂GV −P

∂RL= 0 using (2) and assumingRP = 0 and RS = 0,

i.e.,

GV id−P (ωQ) = −j1

k

√

LS

LP

, and (16)

ωQ =ωP√1− k2

. (17)

As expected, bothGV id−P (ωQ) andωQ are load-independent.To account for the loss due toRP andRS and the associated

output voltage drop, (17) is substituted into (2) to obtain

GV−P (ωQ) = αGV id−P (ωQ), where (18)

α−1 = α1 + α2 + α3, (19)

α1 = jωQCS

(

LS

k2LP

RP +RS

)

, (20)

α2 =

(

1 +RS

RL

)(

1− jRP

k2ωQLP

)

, and (21)

α3 =RP

k2LP

(

LS

RL

+RSCS

)

. (22)

A Thevenin’s equivalent circuit of the steady-state-voltage-transfer function can thus be modeled as in Fig. 3, where

ROLeq−S = RL

(

GV id−S(ωL)

GV−S(ωL)− 1

)

= RLβ(ωL)GV id−S(ωL), (23)

ROHeq−S = RL

(

GV id−S(ωH)

GV−P (ωH)− 1

)

= RLβ(ωH)GV id−S(ωH), and (24)

ROeq−P = RL

(

GV id−P (ωQ)

GV−P (ωQ)− 1

)

= RL(α1 + α2 + α3 − 1), (25)

are equivalent output resistances of the converters due tonon-zeroRP andRS . Their impacts on the voltage transferfunction will be illustrated in Section V.

Fig. 4 depicts the SPICE simulation results of the voltagetransfer ratio at various loading conditions for both series andparallel compensated secondary circuits with zeroRP andRS .In the simulation,k is set as 0.2, andRP = RS = 0. InFig. 4(a),CP is selected to resonate withLP = 30 µH atωS/(2π) = 200 kHz, andωP = ωS , whereas in Fig. 4(b),CP

is selected to resonate withLP (1−k2) atωQ/(2π) = 200 kHz,andωQ = ωS .




150 175 200 225 250

Frequency (kHz)

0

1.25

3.75

5

2.5|Gv-s|

H/2L/2

RL=31.8

RL=7.55

RL=1.79

(a)

0

6.25

12.5

18.75

25

150 175 200 225 250

Frequency (kHz)

Q/2

|Gv-p|

RL=44.1

RL=195

RL=796

(b)

Fig. 4. The voltage transfer ratio of the converter using (a)secondary seriescompensation and (b) secondary parallel compensation.

For the secondary series compensation, the converter hastwo operating frequencies to realize the load-independentvoltage transfer ratio. For secondary parallel compensation,it has one operating frequency to realize the load-independentvoltage transfer ratio. However, when the converter transferefficiency versus operating frequency is taken into considera-tion, as will be studied in the next section, the relative meritsof the two secondary compensation techniques will becomemore apparent.

III. C ONVERSIONEFFICIENCY

The efficiency versus operating frequency of secondary-side series and parallel compensations will be studied in thissection. The efficiencies at the frequency points to achieveaload-independent voltage-transfer ratio atωH andωQ are ourfocuses. As the capacitive input impedance of the converteroperating atωL cannot achieve soft switching using H-bridgeswitches [30], [31], the analysis on this frequency point willbe omitted in the following analysis for its lower efficiency.

To start the analysis, the current-dependent-voltage sourceωMiS in the primary loop of Fig. 2 can be replaced by anequivalent impedanceZr which represents eitherZr−S orZr−P . The impedanceZr is calculated by dividingωMiSwith iP . In this way, the primary loop is decoupled from thesecondary loop. From Fig. 2 we have

Zr =ω2M2

ZS

(26)

An expression for the power transfer efficiency can beobtained by solely considering active power, and thus the

efficiency ηP in the primary loop and the efficiencyηS inthe secondary loop are calculated separately as

ηP =ℜ(Zr)

RP + ℜ(Zr), and (27)

ηS =ℜ(ZS)−RS

ℜ(ZS), (28)

where the operator “ℜ” represents the real component of thecorresponding variable. Therefore,

η = ηP ηS . (29)

A. Efficiency Analysis of Secondary Series Compensation

Specifically, using secondary series compensation, we have

ℜ(Zr−S) =ω2M2RO

R2O +XS

2

=ω2k2LPLSRO

R2O +XS

2 , (30)

whereRO = RS + RL, XS = ωLS − 1ωCS

. Therefore, theefficiency expression using normalized quantities is

η(ω) =1− a(ω)

1 + b(ω), (31)

where

a(ω) =QL−S

QL−S +QS(ω),and (32)

b(ω) =1 +Q2

O(ω)(ωωS

− ωS

ω)2

k2 ωωS

ωωP

QO(ω)QP (ω)(33)

where the quality factors are varying with frequency. Theapproximate Dowell’s equation [32] which is valid for fre-quencies much lower than the self-resonant frequency of theinductor can be used in this paper. The quality factors can bewritten as follows.

QS(ω) =2QS,max

ωQS

ω+ ω

ωQS

(34)

QP (ω) =2QP,max

ωQP

ω+ ω

ωQP

(35)

QO(ω) =QS(ω)QL−S

QS(ω) +QL−S

(36)

QL−S =

√

LS

CS

1

RL

(37)

The quality factorQS maximizes atQS,max whenω = ωQS,

and likewise forQP . In practice, we have enough freedom tochoose the location ofωQS

or ωQPby selecting appropriate

conductor wire material and configuration. The design cantherefore be divided into two separate procedures, iterativelywhenever necessary. The first procedure is to calculate in cir-cuit topology for the optimal operation frequency by assuminga givenQS,max or QP,max, i.e., by using constantQS,max andQP,max in the calculation without involving the complexitiesof the approximate Dowell’s equations (34) and (35). Oncethe optimal operating frequency is found, a second procedure




will be followed by constructing the appropriate transformerwith ωQS

or ωQPpeaking at the optimal frequency for the

application. This procedure is valid for constant frequencyoperation of the converter, provided that the converter isdesigned to operate at the angular frequencyωQS

= ωQP.

Using the afore-mentioned procedures, theQP (ω) andQS(ω) in (34) and (35) are considered as constant and re-placed byQP andQS in the subsequent analysis for simplicity.

From the efficiency expression (31) of secondary seriescompensation, given aQO, which is adjustable usingQL−S ,there may exist a local maximum ofη at ω = ωM which iscalculated fromdη

dω= 0 and expressed as

ωM =ωS

√

1− 12Q2

O

. (38)

From (38),ωM is load-dependent. The theoretical maximumefficiency can be achievable in a narrow range of operatingfrequencies near the peaking ofQS,max and QP,max usingvariable frequency control which may add control complexity.To achieve the theoretical maximum efficiency for a muchwider range of loading conditions, corresponding to a muchwider range of operating frequencies, variable transformersor constant switching of physical transformers peaking atQS,max andQP,max for the range of frequencies is/are needed.However, such design is impractical. Besides, from the anal-ysis in Section II,ωH of load-independentGV−S is alsoload-independent. Hence, there does not exist an operatingfrequency to achieve both maximum efficiency and load-independent voltage-transfer ratio.

To achieve load-independent voltage-transfer ratio and op-timal efficiency is thus of interest. From (36) and (38),ωM

has a theoretical range of[ωS ,∞) and it increases withdecreasingQO ∈ (QO,critical, QS ], whereQO,critical = 1√

2is the minimumQO equal and below which the converterefficiency cannot be maximized by anyωM , andQO = QS asQL−S → ∞. The power transfer efficiency atωM is calculatedas

η∣

∣

∣

QS>QO>QO,criticalω=ωM

=1− QO

QS

1 +4Q2

O−1

4k2Q3

OQP

. (39)

Given the operation frequencyωM , (39) can be used forfinding local extrema ofη by varyingQO. The local extremaare the roots of the polynomial given by settingdη

dQOof (39)

to zero and the resulting polynomial is given as

Q4O +

2QO −QS

k2QP

Q2O +

−4QO + 3QS

4k2QP

= 0. (40)

It is normally true thatQS ≫ QO. Thus, (40) can beapproximated as

Q4O +

−QS

k2QP

Q2O +

3QS

4k2QP

= 0. (41)

The approximation allows the merging of two closely lo-cated roots into a single double root. Two extrema correspond-ing to a local maximum and a local minimum can be identified

ωH

0.6

0.7

0.8

0.9

1

ωS

0.5ωS

1.5ωS

2ωS

2.5ωS

Efficiency

Frequency

Eff

icie

ncy

Op

timal

frequ

ency

QO,critical 0.9 1.2 1.9 3.7 QS

QO

(a)

0.6

0.7

0.8

0.9

1

S

0.5 S

1.5 S

2 S

2.5 S

QO,critical 0.9 1.2 1.9 3.7 QS

QO

H

Efficiency

FrequencyE

ffic

iency

Optim

al frequen

cy

QO1

QO2

QOH

(b)

Fig. 5. The simulated efficiency and optimal frequency (ωM ) versusQL

when (a)λ > 13

and (b)λ < 13

.

from (41) as

QO1 =

√

1

2λ

(

1 +√1− 3λ

)

, and (42)

QO2 =

√

1

2λ

(

1−√1− 3λ

)

(43)

respectively, whereλ = k2QP

QS< 1

3 . Whenλ ≥ 13 , the local

extrema disappear andη in (39) is monotonically increasingwith decreasingQO. A typical plot of (39) is shown in Fig. 5.In this figure, the parameters are set asQP = QS = 100,ωS = 2π× 200 kHz, and the coupling coefficient in Fig. 5(a)is k = 0.65 to satisfy λ > 1

3 , whereas in Fig. 5(b),k isassigned as 0.18 to guaranteeλ < 1

3 .Therefore, two types of efficiency curves are identified

according to the value ofλ as shown in Fig. 5:

1) For λ ≥ 13 , the optimal efficiency increases monoton-

ically with decreasingQO by increasingωM from ωS

to ∞. Since the optimal efficiency operating atωH andclose toQOH is higher than the one operating atωS

and close toQS , the efficiency performance working atωH can be better than that working atωS .

2) For λ ≤ 13 , the optimal efficiency peaks atQO1.

For loading range close toQO1, ωM is just slightlyhigher thanωS which is therefore an ideal constantfrequency for best efficiency operation of the converter.The efficiency performance operating atωH peaks at aloading ofQOH which can be far fromQO1. Therefore,the efficiency performance working atωH will normallybe lower than that operating atωS .

A comparison of efficiency performance working at constant




0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1 10 100 1000

k=0.1

k=0.4

k=0.8

@ωS @ωH

QL

η

Fig. 6. An efficiency comparison of secondary series compensation converterworking atωH andωS versus load quality factor.

frequenciesωS andωH is shown in Fig. 6, which confirms theprevious observations from Fig. 5.

If the primary and secondary coils are constructed with thesame material and physical structure (QP = QS), λ ≥ 1

3requiresk > 0.577. This is challenging for the majority ofIPT applications, where the air gap is normally large. Withan aim of looking for load-independent voltage-transfer ratioand higher efficiency, the efficiency characteristic of secondaryparallel compensation will be studied in Section III-B.

B. Efficiency Analysis of Secondary Parallel Compensation

We apply similar analysis as in Section III-A for thesecondary parallel compensation in this subsection. We have

ℜ(Zr−P ) =

ω2k2LPLS(RS +RL + ω2C2SR

2LRS)

(RS +RL − ω2LSCSRL)2 + (ωLS + ωCSRLRS)2,

(44)

ℜ(ZS−P ) = RS +RL

1 + ω2C2SR

2L

. (45)

Then the efficiency using normalized quantities is given as

η(ω) =1

1 + c(ω)

1

1 + d(ω), (46)

where

c(ω) =1

QLQS

+ω2QL−P

ω2SQS

(47)

d(ω) =

1QP

[

( 1QS

+QL − ω2

ω2

S

QL)2 + (1 + QL

QS)2]

k2ω2

ωPωS

[

1QS

+QL + ω2

ω2

S

Q2

L

QS

] , and

(48)

QL−P = RL

√

CS

LS

. (49)

The frequency that achieves maximum efficiency can befound by solving the root ofdη−P

dω= 0, giving

ωN =ωS

Γ(1 +

1

QSQL

)1

2 , (50)

k=0.1

k=0.2

k=0.8

@ωM @ωN

0.01 0.1 1 10 100 10000

0.2

0.4

0.6

0.8

1

η

QO,critical

QL

Fig. 7. A maximum achievable efficiency comparison of secondaryseriescompensation and parallel compensation versus load quality factor.

where

Γ = (1 +QP

QS

k2)1

4 . (51)

Substituting (50) into (46),η(ωN ) has a maximum atQL−P =QLp, which is given by

QLp =

√

k2QPQS

Γ2 +Q2S(Γ

2 − 1)2 + 1

k2QPQS +Q2S

. (52)

SinceQSQL ≫ 1 in a practical converter design, (50) canbe simplified as

ωN =ωS

Γ. (53)

The simplifiedωN will be used for our subsequent analysis.The expression of the maximum efficiency of secondaryparallel compensation can be readily obtained and is omittedhere for brevity.

Fig. 7 gives a comparison of the maximum achievableefficiencies of secondary series and parallel compensationsversus the loading conditionQL. As shown in Fig. 7, near-identical theoretical maximum efficiency can be achievedexcept whenλ ≥ 1

3 . For instance, atk = 0.8 in Fig. 7,the maximum efficiency of the converter with secondaryseries compensation operating at a much higher operatingfrequency is slightly better than that of the secondary parallelcompensation converter.

Fig. 8 gives a comparison of the efficiencies of secondaryseries and parallel compensations operating at their corre-sponding frequenciesωH and ωN (compensated to haveωN = ωQ) to achieve load-independent voltage transfer ratios.The secondary parallel compensation operating atωN hasefficiency advantage over the secondary series compensationoperating atωH at weak coupling (λ ≤ 1

3 ), and the situationreverses at strong coupling (λ > 1

3 ).

IV. I NPUT PHASE ANGLE AND SOFT SWITCHING

A half-bridge or full-bridge converter is commonly usedfor the modulation of a DC voltage to drive a resonantcircuit. Fig. 9 shows the same primary circuit and two typesof secondary-compensated circuits used in this paper. Eachswitch of the bridge circuit can be considered as a parallel




k=0.1

k=0.3

k=0.7

@ωN @ωH

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100 1000QL

η

Fig. 8. An efficiency comparison of secondary series compensation operatingat ωH and parallel compensation atωN = ωQ with constant|Gv |.

connection of an active MOSFET switch, an anti-parallel-bodydiode and a parasitic capacitor.

The converter achieves soft switching property by operatingat above resonance mode [33], where the resonant-tank currentis lagging the driven voltage modulated by the active switches.In this operating mode, the active switch (Q1 or Q4) enjoysturn-on zero voltage switching (ZVS) due to the early turn-onbody diode associated with the active switch. The body diodeis turned on by the residue current of the resonant tank whichlags behind the turn-off action of the other switch (Q3 orQ2) of the half-bridge circuit. At the same time the energyof the parasitic capacitor associated with the active switch(Q1 or Q4) is recycled back to the resonant tank. When thediode is turned on, the capacitor is fully discharged and theresonant tank current has not been fully depressed, the voltageacross the active switch is kept zero. During this time interval,the active switch can turn on at zero-voltage, i.e. ZVS. Therequirement of high turn-off speed of the body diode can beextended to the time when the parallel active switch turns off.The benefits of converter operating above resonance is thatthere is no switching loss apart from the turn-off switchingloss of the active switch. The turn-off loss of active switchescan, however, be reduced by paralleling the switch with anexternal capacitor as a lossless snubber.

In contrast, when the converter operates at below resonantmode, the resonant-tank current is leading the driven voltagemodulated by the active switches. The active switches can turnoff near the instant that the resonant tank current crosses zero,and therefore they have the benefit of zero current switching(ZCS) at turn-off. However, all the other switching losses areretained, i.e., the switches suffer from turn-on switchinglosses,diode switching losses, parasistic capacitance charge storagelosses through internal discharge of the active switch duringturn on due to high-voltage switching. These analyses in [33]have been shown to be valid experimentally in [34] using aspecific loss model.

The soft switching property of the IPT converter is thusreduced to the study of the input phase angleθin between theinput voltage and current in this section. When the equivalentAC voltage source in Fig. 1 or Fig. 2 is generated from a poweramplifier, it is desirable to haveθin = 0 to maximize the power

vin

Q1

Q3

Q2

Q4

CP CS

LP LS

M

D1

D3

D2

D4

Cf

RL

(a) Secondary series compensation

vin

Q1

Q3

Q2

Q4

CP

CSLP LS

M

D1

D3

D2

D4

Cf

RL

(b) Secondary parallel compensation

Fig. 9. Circuit diagrams for experiment.

transfer capability and minimize the VA rating of the poweramplifier. However, in this paper, the equivalent AC voltagesource is generated from a H-bridge converter which is muchsimpler and more efficient by using zero-voltage switching atan inductive phase angle ofθin > 0. The input phase anglesat ωH of secondary series compensation and atωQ (andωN )of secondary parallel compensation will be studied.

The input phase angle is calculated as

θin =180◦

πtan−1 ℜ(Zin)

ℑ(Zin)(54)

where “ℑ” is the imaginary component of the correspondingvariable.

A. Secondary Series Compensation

For secondary series compensation, we have

ℜ(Zin−S) = ωLP

ωnsk2 1QL−S

(ωns − 1ωns

)2 + 1Q2

L−S

, and (55)

ℑ(Zin−S) = ωLP

1− 1

ω2np

+k2(1− ω2

ns)

(ωns − 1ωns

)2 + 1Q2

L−S

,

(56)

where

ωnp =ω

ωP

, and (57)

ωns =ω

ωS

. (58)

From (55),ℜ(Zin−S) is always positive. The sign ofZin−S

is thus determined by the sign ofℑ(Zin−S) in (56). SinceRP andRS are small and insignificant for the determinationof the input phase angle, they will be neglected. We take thederivative ofℑ(Zin−S) to obtain

dℑ(Zin−S)

dQL−S

= −2ωLP k2QL−Sω

4ns(ω

2ns − 1)

(Q2L−S(ω

2ns − 1)2 + ω2

ns)2, (59)




TABLE IICOMPONENTS ANDPARAMETERS USED IN THE CONVERTER.

Circuit components ValuePower MOSFETQ1-Q4 IRF540ND1-D4 for series compensation STPS20H100CGD1-D4 for parallel compensation STPSC606GLoosely-coupled transformer NP = 30 turns of Litz wire(Fig. 11 gives the dimension) NS = 29 turns of Litz wireCapacitanceCf 220µFk = 0.182 (air gap = 35mm)

LP = 31.45µH @200 kHzLS = 32.55µH @200 kHz

Capacitances for series compensation CP = 19.45nF / 630 VCS = 20.13nF / 630 V

Capacitances for parallel compensationCP = 20.13nF / 630 VCS = 20.13nF / 630 V

k = 0.616 (air gap = 10mm)LP = 32.72µH @200 kHzLP = 31.66µH @200 kHz

Capacitances for series compensation CP = 19.35nF / 630VCS = 20.00nF / 630V

Capacitances for parallel compensationCP = 31.20nF / 630VCS = 17.03nF / 630V

which is always less than zero for allω > ωS . Therefore, thetheoretical minimum ofℑ(Zin−S) is atQL → ∞.

Putting (6) into (56), we obtain

limQL−S→∞

ℑ(Zin−S)(ωH) = 0 (60)

As ωH > ωS , Zin−S is always inductive when it operates atωH . As a result, operating atωH can provide soft switchingwhenZin−S is driven by a half-bridge or full-bridge switchingcircuit.

B. Secondary Parallel Compensation

Similar analytical procedure can be conducted for secondaryparallel compensation. The real and imaginary parts ofZin−P

are calculated as

ℜ(Zin−P ) = ωLP

1ωns

1QL−P

(ωns − 1ωns

)2 + 1Q2

L−P

, and (61)

ℑ(Zin−P ) = ωLP [1−1

ω2np

+ k2(1− ω2

ns)− 1Q2

L−P

(ωns − 1ωns

)2 + 1Q2

L−P

].

(62)

To have load-independent voltage-transfer ratio and optimalefficiency, the working frequencyω needs to beωQ andωN , i.e., ω = ωP√

1−k2= ωS

(1+k2)1

4

, ωnp = 1√1−k2

and

ωns =1

(1+k2)1

4

. Substitutingωnp andωns into (62),ℑ(Zin−P )

is always greater than zero. Therefore, when the compensatedcircuit has load-independent voltage-transfer ratio and optimalefficiency, soft switching is achieved automatically.

Fig. 10 plotsθin versus loading condition for the two typesof secondary compensations.

V. EXPERIMENTAL VALIDATION

The full-bridge circuits shown in Figs. 9(a) and (b) arebuilt to verify the results from the analyses. The prototypesare constructed with the components and parameters given

0.1 1 10 1000

30

60

90

QL

k=0.1

k=0.3

k=0.5

(a)

0.1 1 10 1000

30

60

90

QL

k=0.1

k=0.3

k=0.5

(b)

Fig. 10. The input phase angle of (a) secondary series compensation atωH

and (b) secondary parallel compensation atωN = ωQ.

d=49.5

r=40.5

d1=

90

d2=

9

g=35

(a)

(b)

Fig. 11. Physical dimension of the loosely-coupled transformer: (a) dimensionin mm and (b) photograph.

in Table II. The physical dimension of the loosely-coupledtransformer is shown in Fig. 11.

For the secondary series compensation,CS is selected toresonate withLS at 200 kHz, andLPCP = LSCS , whilefor parallel compensation,CS is selected to resonate with√1 + k2LS at 200 kHz andCP to resonate with(1− k2)LP

at 200 kHz.




Frequency (kHz)

Vo

ltag

e tr

ansf

er r

atio

QL=21.3

QL=5.31

QL=1.06

(a)

Frequency (kHz)

Volt

age

tran

sfer

rat

io

QL=20.1

QL=6.28

QL=1.54

(b)

Frequency (kHz)

Vo

ltag

e tr

ansf

er r

atio

QL=8.33

QL=3.49

QL=0.91

(c)

Frequency (kHz)

Vo

ltag

e tr

ansf

er r

atio

QL=0.33

QL=2.63

QL=12.6

(d)

Fig. 12. Voltage transfer ratio atk = 0.182 of (a) secondary seriescompensation and (b) secondary parallel compensation, and atk = 0.616 of(c) secondary series compensation and (d) secondary parallel compensation.

Fig. 12 gives the experimental voltage-transfer ratios of thesecondary series and parallel converters versus the operatingfrequency and working at three output loading conditions. Thetrend of the voltage-transfer ratios are similar to that shown inFig. 4. A comparison of the measured and calculated voltage-transfer ratios at the load-independent operating frequenciesωH and ωQ is shown in Fig. 13. It can be observed thatdiscrepancy between calculated and measured results is moresignificant with lighter loads, i.e., higherQL values forthe secondary parallel compensation and lowerQL values

0.1 1 10 1000.7

0.8

1

QL

1.1

|Gv/G

vid

|

0.9

Measurement results

Calculation results

Series Parallel

Fig. 13. Comparison of calculated and experimental voltage-transfer ratios.Solid lines are calculatedGV

GV idusing equations (4) and (11) for secondary

series compensation and equations (16) and (18) for secondary parallelcompensation. Data points are experimental data transformed from Figs. 12(c)at ωH

2π≈ 325 kHz and (d) at

ωQ

2π≈ 200 kHz.

vi

(a)

v

i

(b)

Fig. 14. Experimental waveforms of full-bridge-modulated voltage v andtransformer primary currenti at (a)QL = 0.25 and (b)QL = 7.93 of thesecondary parallel compensated converter.

for the secondary series compensation. The discrepancy canbe explained by observing the inductor waveforms shownin Fig. 14(a) at heavy load and Fig. 14(b) at light loadconditions. The fundamental component approximation usedin the calculations can generally give accurate results foroperation with sinusoidal inductor-current waveform as shownin Fig. 14(a) and less accurate results for operation withimperfect sinusoidal inductor-current waveform as shown inFig. 14(b) [35] due to more significant contribution of higherharmonic components.

The efficiencies versusQL (adjusted usingRL) at a constantoutput power of 24 W (adjusted usingVin) are measured forthe two prototype converters. A constant power operation can




N of parallel compensation

S of series compensation

H of series compensation

QL

Eff

icie

ncy

(a)

QL

Eff

icie

ncy




(b)

Fig. 15. Experimental efficiency versus loading condition at(a) k = 0.182and (b)k = 0.616.

maintain a smaller converter temperature variation and thus asmaller variation of converter component parameters, leadingto a more consistent efficiency measurement. The measuredefficiencies of the secondary series compensated converterop-erating atωH andωS , and the secondary parallel compensatedconverter operating atωN are compared as shown in Fig. 15.

It is noted that the efficiency atωS is measured by anintentional 2% increase ofCP than that shown in Table IIto guarantee soft switching. The soft switching condition isautomatically satisfied when the converter is operating atωH

and ωN , as analysed in Section III-A. Therefore, all theefficiency curves are measured under soft switching.

Besides, the output voltage of the converter using secondaryparallel compensation is much higher than that of the seriescompensation, which can be seen from theGv curves shownin Fig. 12. To account for the large output voltage difference ofthe two types of secondary compensation, the output voltagesare measured including the rectifier diode drops for betterfairness in efficiency comparison.

From Fig. 15(a), wherek is small, the maximum efficiencyof the secondary series compensated converter operating atωH is generally lower than the converter operating atωS .The efficiency difference is increasing with increasingQL.The maximum efficiency of secondary parallel compensationis similar to the secondary series compensation operating atωS , regardless of the value ofk. At a higher k, as shownin Fig. 15(b), the maximum efficiency of the secondary seriescompensated converter operating atωH can be higher than theconverter operating atωS and the secondary parallel converteroperating atωN with decreasingQL. The experimental results

0 5 10 15 20 250.7

0.8

0.9

QL

1

Eff

icie

ncy




(a)

0 1 2 3 4 5 6 7

QL

0.85

0.9

0.95

1

Eff

icie

ncy




(b)

Fig. 16. Calculated efficiency versus loading condition at (a) k = 0.182 and(b) k = 0.616.

confirm the analytical results shown in Fig. 16. Comparingthe results from Figs. 15 and 16, the calculated efficienciesfitwell with the experimental measurements. The good matchof efficiencies operating atωS with 2% increase ofCP

confirms thatCP does not affect efficiency for this specificcompensation and soft-switching is verified experimentally.However, whenQL and k are small and the converter isoperating atωN of the secondary parallel compensation, sig-nificant discrepancy between the calculated and experimentalmeasured efficiencies is identified. The discrepancy is dueto the nonlinear operation of the converter as discontinuousoutput current is observed experimentally, and the fundamentalcomponent approximation used in the calculation becomes lessaccurate.

VI. CONCLUSION

The operating frequencies of an inductive power transfervoltage converter are studied with primary series compensationand secondary series or parallel compensation. The frequenciesOperating for Maximum Efficiency (OME) and Operatingfor Load-Independent Voltage-Transfer Ratio (OLIVTR) areidentified and compared with the two secondary compen-sation techniques. For the secondary series compensation,the frequenciesωH OLIVTR and ωM OME exist. However,ωM is not OLIVTR and the variable-frequency control canbe complicated. Fortunately, operating at the series resonantfrequencyωS is found to have an efficiency close to thatoperating atωM . However, bothωM andωS are not OLIVTR.For the secondary parallel compensation, the frequenciesωQ

OLIVTR and ωN OME exist. Moreover, the frequenciesωQ




andωN can be designed to be identical. Converters operatingat the frequencies identified are compared. Two experimentalprototypes are built. The experimental result confirms theanalytical result.

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Wei Zhang (S’11) received the BSc degree in elec-trical engineering from HeFei University of Technol-ogy, Hefei, China in 2007, and the MSc degree fromNanjing University of Aeronautics and Astronautics,Nanjing, China in 2010. He is currently workingtoward the Ph.D. degree in power electronics at theHong Kong Polytechnic University, Kowloon, HongKong.

His current research interests include inductivepower transfer system and resonant converters.




Siu-Chung Wong (M’01–SM’09) received the BScdegree in Physics from the University of Hong Kong,Hong Kong, in 1986, the MPhil degree in electron-ics from the Chinese University of Hong Kong in1989, and the PhD degree from the University ofSouthampton, U.K., in 1997.

Dr. Wong joined the Hong Kong Polytechnic in1988 as an Assistant Lecturer. He is currently anAssociate Professor of the Department of Electronicand Information Engineering, The Hong Kong Poly-technic University, where he conducts research in

power electronics. Dr. Wong is a senior member of the IEEE and a memberof the Electrical College, The Institution of Engineers, Australia. He is aneditor of theEnergy and Power Engineering Journaland a member of theEditorial Board of theJournal of Electrical and Control Engineering.

In 2012, Dr Wong was appointed as a Chutian Scholar Chair Professor bythe Hubei Provincial Department of Education, China and the appointmentwas hosted by Wuhan University of Science and Technology, Wuhan, China.In 2013, Dr Wong was appointed as Guest Professor by the School ofElectrical Engineering, Southeast University, Nanjing, China. He was avisiting scholar at the Center for Power Electronics Systems, Virginia Tech,VA, USA on November 2008, Aero-Power Sci-tech Center, Nanjing Universityof Aeronautics and Astronautics, Nanjing, China on January2009, and Schoolof Electrical Engineering, Southeast University, Nanjing, China on March2012.

Chi K. Tse (M’90–SM’97–F’06) received the BEng(Hons) degree with first class honors in electricalengineering and the PhD degree from the Universityof Melbourne, Australia, in 1987 and 1991, respec-tively.

He is presently Chair Professor of Electronic En-gineering at the Hong Kong Polytechnic University,Hong Kong. From 2005 to 2012, he was the Headof Department of Electronic and Information Engi-neering at the same university. His research interestsinclude complex network applications, power elec-

tronics and chaos-based communications. He is the author of the booksLinearCircuit Analysis(London: Addison-Wesley, 1998) andComplex Behavior ofSwitching Power Converters(Boca Raton: CRC Press, 2003), co-author ofChaos-Based Digital Communication Systems(Heidelberg: Springer-Verlag,2003), Digital Communications with Chaos(London: Elsevier, 2006),Re-construction of Chaotic Signals with Applications to Chaos-Based Commu-nications (Singapore: World Scientific, 2007) andSliding Mode Control ofSwitching Power Converters: Techniques and Implementation (Roca Raton:CRC Press, 2012) and co-holder of 4 US patents and 2 other pending patents.

Currently Dr. Tse serves as Editor-in-Chief for theIEEE Circuits andSystems Magazineand Editor-in-Chief ofIEEE Circuits and Systems SocietyNewsletter. He was/is an Associate Editor for the IEEE TRANSACTIONS ON

CIRCUITS AND SYSTEMS PART I—FUNDAMENTAL THEORY AND APPLI-CATIONS from 1999 to 2001 and again from 2007 to 2009. He has also beenan Associate Editor for the IEEE TRANSACTIONSON POWERELECTRONICS

since 1999. He is an Associate Editor of theInternational Journal of SystemsScience,and also on the Editorial Board of theInternational Journal ofCircuit Theory and Applicationsand International Journal and Bifurcationand Chaos.He also served as Guest Editor and Guest Associate Editor foranumber of special issues in various journals.

Dr. Tse received the L.R. East Prize from the Institution of Engineers,Australia, in 1987, the Best Paper Award from IEEE TRANSACTIONS ON

POWER ELECTRONICS in 2001 and the Best Paper Award fromInternationalJournal of Circuit Theory and Applicationsin 2003. In 2005 and 2011,he was selected and appointed as IEEE Distinguished Lecturer. In 2007,he was awarded the Distinguished International Research Fellowship by theUniversity of Calgary, Canada. In 2009, he and his co-inventors won theGold Medal with Jury’s Commendation at the International Exhibition ofInventions of Geneva, Switzerland, for a novel driving technique for LEDs.In 2010, he was appointed the Chang Jiang Scholars Chair Professorshipby the Ministry of Education of China and the appointment was hosted byHuazhong University of Science and Technology, Wuhan, China. In 2011, hewas appointed Honorary Professor by RMIT University, Melbourne, Australia.

Qianhong Chen (M’06) was born in HubeiProvince, China, in 1974. She received the B.S.,M.S., and Ph.D. degrees in electrical engineeringfrom Nanjing University of Aeronautics and Astro-nautics, Nanjing, China, in 1995, 1998 and 2001,respectively.

In 2001, she joined the Teaching and ResearchDivision of the Faculty of Electrical Engineering atNanjing University of Aeronautics and Astronautics,China, and is currently a professor with the Aero-Power Sci-Tech Center in the College of Automation

Engineering. From April 2007 to January 2008, she was a research associatein the Department of Electronic and Information Engineering,Hong KongPolytechnic University, Hong Kong, China. Her research interests includeapplication of integrated-magnetics, inductive power transfer converters, soft-switching dc/dc converters, power factor correction, and converter modeling.She has published over 30 papers in international journals and conferences,and is the holder of 7 patents.

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