design for efficiency optimization and voltage controllability of series–series compensated...

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

Design for Efficiency Optimization and VoltageControllability of Series-series Compensated

Inductive Power Transfer SystemsWei Zhang, Student Member, IEEE, Siu-Chung Wong, Senior Member, IEEE, Chi K. Tse, Fellow, IEEE, and

Qianhong Chen, Member, IEEE

Abstract—Inductive power transfer (IPT) is an emergingtechnology that may create new possibilities for wireless powercharging and transfer applications. However, the rather complexcontrol method and low efficiency remain the key obstructingfactors for general deployment. In a regularly compensated IPTcircuit, high efficiency and controllability of the voltage transferfunction are always conflicting requirements under varying loadconditions. In this paper, the relationships among compensationparameters, circuit efficiency, voltage transfer function andconduction angle of the input current relative to the input voltageare studied. A design and optimization method is proposed toachieve a better overall efficiency as well as good output voltagecontrollability. An IPT system design procedure is illustratedwith design curves to achieve a desirable voltage transfer ratio,optimizing between efficiency enhancement and current rating ofthe switches. The analysis is supported with experimental results.

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

I. INTRODUCTION

IN the application of magnetic induction to transfer electric

power wirelessly, good coupling is the fundamental factor

for effective operation. Development in modern power elec-

tronics, however, enables many new loosely coupled energy

transmission applications. An inductive power transfer (IPT)

system is designed to deliver power to a load over rela-

tively large air gap via magnetic coupling. The fundamental

principles of such systems are identical to the widely used

electromechanical devices with good coupling such as trans-

formers and induction motors, where the leakage inductance is

much lower than the mutual inductance. Because of electrical

isolation and the absence of mechanical contact between the

power supply side and the load, the IPT system has advantages

of low maintenance cost, high reliability and the ability to

operate in ultra clean or ultra dirty environment. Common

applications of this technology include wireless power sup-

ply to home appliances such as electric toothbrush, wireless

Manuscript received September 14, 2012; revised December 24, 2012; ac-cepted February 12, 2013. This work is supported by Hong Kong PolytechnicUniversity under Central Research Grant GU-809.

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

charging of mobile phones using a charging platform [1]–

[3], and in medical use such as wireless power supply to

implantable devices [4], [5]. Medium to high power applica-

tions of this technology include continuous power transfer to

people movers [6], and contactless battery charging for moving

actuator [7] or electric vehicles [8]–[10].

It is common that IPT involves a large separation between

the primary and secondary winding [11], [12]. Therefore, the

electric characteristics of the transformers used are very differ-

ent from those of conventional transformers which have good

coupling between the windings. Due to large winding separa-

tion, IPT has a relatively large leakage inductance, as well as

increased proximity-effect and winding resistances. Further-

more, for an IPT system where the primary and secondary

winding are separated by an air gap, the magnetizing flux is

significantly reduced. Thus, a higher magnetizing current and

circuit compensation for the larger leakage inductances are

needed.

Compensation is often required for both primary and sec-

ondary of the loosely-coupled transformer to enhance the

power transfer capability, to minimize the VA rating of the

power supply, and to regulate the value of current in the

supply loop and the voltage of the receiving loop with a higher

efficiency [2], [4], [13]–[15]. Several studies have been carried

out in order to select the most appropriate compensation topol-

ogy for the IPT systems, depending on the application and

taking into account the stability of the configuration [15], [17].

However, these studies focused on either system efficiency

or system controllability but rarely considered both. Usually,

the compensation technique is selected to pursue maximum

power transfer capability, whereas the voltage controllability

is mostly neglected, leading to significant fluctuation in the

output voltage under load change. An additional secondary

power regulator is therefore needed for applications requiring

a stable output voltage.

In this paper, a compensation technique considering both

good output voltage controllability and high system efficiency

is proposed. A systematic design procedure for the circuit and

loosely coupled transformer parameters are elaborated.

0000–0000/00$00.00 c©2013 IEEE



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

LP LS

MCP CSRP RS

vin RL

Transformer model

Fig. 1. S-S compensation topology.

jωM iS

iP

CP CSRP RS

vin

RL

Primary loop Secondary loop

LP

jωM iP

LS

iS

Fig. 2. Equivalent circuit model of Fig. 1.

II. CIRCUIT CHARACTERISTICS

A. Circuit Model and Efficiency

A commonly-used loosely-coupled transformer model, as

shown in Fig. 1, with series-series (S-S) external capacitors

compensation is analyzed in this paper. Transformer induc-

tances LP in the primary, LS in the secondary and mutual

inductance M are components of the transformer model shown

in Fig. 1. RP and RS are the winding resistances (which

are varying with frequency [16]) of the transformer primary

and secondary, respectively. CP and CS are the primary and

secondary external compensation capacitors, for enhancing

energy transfer from an AC source vin to an output loading

resistance RL. The AC source is generally an equivalent

voltage generated from a half- or full-bridge switching circuit

operating at an angular frequency ω.

In the subsequent analysis, a frequency-domain equivalent

circuit is adopted and only the fundamental component is

considered here for simplicity [9], [14], [15]. The fundamental

component approximation is sufficiently accurate for a high

quality factor resonant circuit that works near resonance. Fig. 2

gives an equivalent circuit of Fig. 1 for the analysis of steady

state solutions. The current-dependent source ωMiS in Fig. 2

can be replaced by an equivalent impedance Zr which is

calculated by dividing ωMiS with iP . In this way, the primary

loop is decoupled from the secondary loop. We have

Zr =ω2M2

Zs +RL

, (1)

where Zs is the impedance of the secondary network given by

Zs = jωLS +1

jωCS

+RS . (2)

An expression for power transfer efficiency can be obtained

by solely considering active power and thus the efficiency ηPin the primary loop and the efficiency ηS in the secondary

loop are calculated separately as

ηP =ℜ(Zr)

RP + ℜ(Zr), and (3)

ηS =RL

RL +RS

, (4)

where the operator “ℜ” represents the real component of the

corresponding variable, and

ℜ(Zr) =ω2M2RO

R2O +XS

2

=ω2k2LPLSRO

R2O +XS

2 (5)

where RO = RS +RL, XS = ωLS − 1ωCS

, and the coupling

coefficient is given by

k =M√LPLS

. (6)

The transfer efficiency ηT is therefore given by

ηT = ηP ηS . (7)

Using normalized quantities, we have

ηT (ω) =1− QO

QS

1 +1+Q2

O

(

ωωS

−ωSω

)

2

k2 ωωS

QOω

ωPQP

(8)

where

QS =

√

LS

CS

1

RS

, (9)

QO =

√

LS

CS

1

RO

, (10)

QP =

√

LP

CP

1

RP

, (11)

ωP =1√

LPCP

, (12)

ωS =1√

LSCS

. (13)

In practice, the quality factors in equations (9) to (11) are vary-

ing with frequency. The approximate Dowell’s equation [16]

which is valid for frequencies much lower than the self-

resonant frequency of the inductor is used in this paper. The

quality factors can be written as follows.

QS(ω) =2QS,max

ωQS

ω+ ω

ωQS

(14)

QP (ω) =2QP,max

ωQP

ω+ ω

ωQP

(15)

QO(ω) =QS(ω)QL

QS(ω) +QL

(16)

QL =

√

LS

CS

1

RL

(17)

The quality factor QS maximizes at QS,max when ω = ωQS,

likewise for QP . Substituting (14) to (16) into (8), we have

ηT (ω) =1− a(ω)

1 + b(ω), (18)



ZHANG et al.: DESIGN FOR EFFICIENCY OPTIMIZATION AND VOLTAGE CONTROLLABILITY OF S-S COMPENSATED IPT SYSTEMS 3

where

a(ω) =QL

QL +QS(ω), and (19)

b(ω) =1 +Q2

O(ω)(

ωωS

− ωS

ω

)2

k2 ωωS

QO(ω)ωωP

QP (ω). (20)

B. Maximum Efficiency and Operating Frequency

The power loss caused by winding resistances of the loosely

coupled transformer has been considered by (18). The power

loss imposed on the switching components should be con-

sidered in order to achieve an accurate calculation for the

overall efficiency. Switching loss is composed of conduction

loss and transient loss. We will show in Section IV-A that

conduction and switching losses of the switching components

can be modeled as equivalent resistances of RDS(on) and

Rf,eqv respectively, of which RDS(on) does not depend on

frequency and Rf,eqv increases linearly with the operating

frequency. In practice, it can be design such that switching

devices satisfy RP ≫ Rf,eqv. Therefore RDS(on) + Rf,eqv

can be readily absorbed into RP without significantly affect-

ing the frequency property of the QP (ω). From (18), the

power transfer efficiency ηT can be maximized at a particular

operating frequency ω = ωM given a load quality factor

QL. Qualitatively, ηT is maximized for a minimum a and a

minimum b as a and b are both non-negative. It can be readily

observed that a is minimized when ω = ωS . The numerator

of b is minimized as 1 when ω = ωS and denominator of b

is maximized when ω is a bit higher than ωQSor ωQP

. So, it

is common to design ωQSclose to ωQP

and compensate the

circuit by choosing ωM = ωS ≈ ωQS≈ ωQP

.

When the fixed operating frequency ωS is used, the corre-

sponding efficiency can be obtained by substituting ω = ωS

into (18), i.e.,

ηT (QL)∣

∣

∣

QS>QO>QO,min

ω=ωS

=

1− 1

1+QS,max

QL

1 +1+

QS,max

QL

k2QS,max

ωSωP

QP,max

, (21)

where ηT has a maximum value when QL = QLp, i.e.,

ηT,max =c

(

1 +√1 + c

)2 (22)

QLp =QS,max√1 + c

(23)

where c = k2 ωS

ωPQS,maxQP,max. It is readily observed that

ηT,max increases with increasing c.

Essentially, a power converter should be designed for a

suitable range of loading for best efficiency. It can be achieved

by introducing a design parameter 0 < γ < 1, such that

the converter efficiency ηT is always larger than γηT,max for

the desired range of loading within [QL,min, QL,max], where

QL,min < QLp < QL,max. The boundaries of QL,min and

QL,max are found by equating the RHS of (21) to γηT,max

0.9 1.6 2.4 ∞

QL1.2 4.7

0.6

0.7

0.8

0.9

1

Transferefficiency

QL,min=1.17 QL,max=17.4

ηT,max

0.9ηT,max

Fig. 3. Simulation results of efficiency versus QL.

and solving for the roots, which are given as follows:

QL,min =QS,max

A+B, and (24)

QL,max =QS,max

A−B, (25)

where

A =2 + 2c1(1− γ) + c[2 + c1(1− γ)]

2γc1(26)

B =

√

(1− γ)(1 + c)c22γc1

(27)

c1 =√1 + c (28)

c2 = 8(1 + c1) + 4c(2 + c1) + c2(1− γ) (29)

For illustration and verification, SPICE simulations using

the transformer model in Fig. 1 are conducted. In the simu-

lation, we use QP = QS = 100 and k = 0.2. The operating

frequency is ωS

2π = 150 kHz. For this given set of parameters,

ηT,max is obtained as 0.90 at QL = 4.99 using (23). Using

γ = 0.9, QL,min and QL,max are calculated as 1.17 and 17.39,

respectively, using equations (23) to (25). The simulation

results shown in Fig. 3 confirm the theoretical findings in this

subsection.

C. Output to Input Voltage Transfer Ratio

For converters with conventional transformers, the ratio

of the output voltage compared with the input voltage is

determined by the turns ratio of the transformer. For IPT

applications using loosely coupled transformer which has

compensation in both primary and secondary sides, the com-

pensation capacitors are involved in the voltage transfer func-

tion. Therefore, the voltage transfer function will be studied

subsequently for optimal operation and circuit design.

From Fig. 2, the output voltage across RL can be calculated

using KVL as

vout = jωMRL

Zs +RL

iP , (30)

where iP can be calculated from the decoupled primary loop

of Fig. 2 using (1) as

iP =vin

Zp +ω2M2

Zs+RL

(31)




0

0.8

1.6

2.4

3.2

120 130 140 150 160

Frequency (kHz)

|Gv|

180170

ωS/2π

ωH/2πωL/2π

QLp

3QLp

0.3QLp

Fig. 4. Output to input voltage transfer ratio versus different operatingfrequencies at various loads.

where

Zp = jωLP +1

jωCP

+RP . (32)

The output-to-input voltage transfer function Gv is deter-

mined by substituting (31) into (30) as

Gv =vout

vin=

jωMRL

Zp(Zs +RL) + ω2M2, (33)

From Section II-B, the converter can operate at ω = ωS .

It will be interesting to know the voltage transfer function

operating at ωS when the load changes. We have plotted Gv

using (33) for the set of parameters used in Section II-B and

compared with a simplified Gv with zero RP and RS , and

found that there is no significant difference. Therefore, RP and

RS can be assumed zero for simplicity for the subsequence

analysis of Gv in this subsection.

The curves shown in Fig. 4 are approximated values of |Gv|with zero RP and RS at various loads. Usually ωP is assigned

as ωS [14], [15] by selecting a suitable CP , such that Zp is

zero according to (32). Therefore, for maximum efficiency and

operating at ω = ωS ,

|Gv(ωS)| =RL

ωSM. (34)

The |Gv| at ωS is directly proportional to RL, making it

difficult to have good voltage regulation without an additional

power converter. In Fig. 4, however, there are two frequency

points where the voltage transfer ratio is constant when

load changes. To calculate the two frequency points, further

manipulation of (33) [19] is conducted to give

Gv =1

Zp

jωM+ δ

jω3MCSCPRL

, (35)

where

δ =ω4

ω2Pω

2S

(k2 − 1) + ω2(1

ω2P

+1

ω2S

)− 1 (36)

It can be readily seen from (36) that if δ = 0, then Gv is

independent of RL. Solving for roots ωL and ωH of (36), the

normalized frequencies at which Gv is independent of RL and

can be obtained as

ωl =ωL

ωS

=

√

ω2n + 1− λ5

2(1− k2)(37)

ωh =ωH

ωS

=

√

ω2n + 1 + λ5

2(1− k2). (38)

where λ5 =√

(ω2n − 1)2 + 4k2ω2

n and ωn = ωP

ωS. Operating

at frequency ωh or ωl (shown in Fig. 4), the converter has

|Gv(ωl)| = k

√

LS

LP

ω2n + 1− λ5

(1− 2k2)ω2n − 1 + λ5

(39)

|Gv(ωh)| = k

√

LS

LP

ω2n + 1 + λ5

(1− 2k2)ω2n + 1 + λ5

. (40)

The value of |Gv| at ωl or ωh has no relationship with the

loading resistance. Therefore, if the converter is adjusted to

operate at frequency ωh or ωl, the voltage controllability is

significantly improved when load changes.

D. Input Phase Angle

The input phase angle between the input voltage and current

(θin) when working at ωH and ωL is studied to guarantee an

inductive input impedance. For H-bridge converters, which is

adopted in our experiment, an inductive input impedance is

necessary to realize soft switching for better efficiency.

The inductive or capacitive characteristic of Zin is decided

by the sign of its imaginary part, which is given by

ℑ(Zin) = ZP

(

ωnP −k2ω2

ωPωSωnS

ω2nS + 1

Q2

O

)

, (41)

where ωnS =(

ωωS

− ωS

ω

)

and ωnP =(

ωωP

− ωP

ω

)

. We take

the derivative of ℑ(Zin) to obtain

dℑ(Zin)

dQO

= −2 k2ω2

ωPωSωnS

(

ω2nS + 1

Q2

O

)2

Q3O

, (42)

which is always smaller (greater) than zero for all ω > ωS

(ω < ωS). Therefore the theoretical minimum (maximum) of

ℑ(Zin) is at QO → ∞. Substituting either (37) or (38) into

(41), we obtain limQO→∞ ℑ(Zin) = 0. Hence, Zin is always

inductive (capacitive) when it operates at ωH (ωL). As a result,

operating at ωH can provide soft switching when Zin is driven

by a half-bridge or full-bridge switching circuit.

The value of θin is calculated as

θin =180

πtan−1 ℜ(Zin)

ℑ(Zin)(43)

where

ℜ(Zin)

ℑ(Zin)=ω4h(1− k2) + ω2

h(k2 − 2) +

ω2

h

Q2

O

+ 1

k2ω3

h

QO

−(ω2

h − 1)2 +ω2

h

Q2

O

k2ω5

h

ω2nQO

. (44)




45

90

-45

0

-90

Inputphaseangle

∞ 2.175 1.087 0.707QO

θin@ ωL

θin@ ωH

Fig. 5. Simulated results of input phase angle when the equivalent circuit isoperated at ωL or ωH .

Fig. 5 shows the simulation results which confirm the input

phase angle analysis. ωH is suitable for H-bridge converters

for its inductive input phase angle. In the next section, we will

study the tuning of CS and CP for voltage controllability and

best efficiency.

III. PARAMETER OPTIMIZATION AND CIRCUIT DESIGN

A. Compensation Parameter Optimization

In Section II, we have studied three operational angular

frequencies ωS , ωL and ωH for the S-S tunned loosely coupled

transformer circuit. It has been proven in Section II-B that the

converter can give optimal efficiency by operating at the fixed

frequency ωS for a load range of [QO,min, QO,max]. On the

contrary, operating at ωS as described in Section II-C, the

converter gives the worse load voltage regulation. Meanwhile,

operating at ωH , the converter achieves the best output voltage

controllability with soft switching. It will be desirable to have

both good efficiency and good load voltage regulation.

Essentially, operation frequency cannot deviate from ωH for

good load voltage regulation. It will be desirable to design ωH

as close to ωS as possible. From (38), we have

∂ωh

∂ωn

=ωn

(1− k2)ωh

(

λ6

λ5

)

(45)

where

λ6 = λ5 + ω2n + 2k2 − 1. (46)

We are now going to show that ∂ωh

∂ωn> 0 for all ωn ≥ 0. With

1 ≥ k ≥ 0, ωh will be decreasing and has a minimum of1√

1−k2at ωn = 0.

The sign of (45) is determined by λ6 whose slope is given

as

∂λ6

∂ωn

= ωn

λ7

λ5. (47)

where λ7 = 2λ5 + 4k2 + 2(ω2n − 1). The sign of (47) is

determined by λ7 whose slope is given as

∂λ7

∂ωn

=16k2(1− k2)ωn

λ35

. (48)

It can be readily seen from (48) that ∂λ7

∂ωn> 0. Therefore λ7

has a minimum of 4k2 at ωn = 0. Therefore ∂λ6

∂ωn> 4k2 > 0

0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

1.2

k = 0.1

k = 0.2

k = 0.3

k = 0.4

k = 0.5

ωh

ωn

Fig. 6. Variation of ωh versus ωn for various k.

and λ6 has a minimum of 2k2 at ωn = 0. Hence, ∂ωh

∂ωn> 0

and

ωh,min =1√

1− k2at ωn = 0. (49)

From (49), ωh cannot be designed for maximum efficiency

at ωh,min as ωn cannot be zero. As a compromise, ωh can be

chosen as ωh,γ1= γ1ωh,min where γ1 ≥ 1. The corresponding

ωn,γ1can be solved using (38) as

ωn,γ1=

√

γ1(γ1 − 1)

γ1 + k2 − 1(50)

A plot of ωh versus ωn using (38) at various coupling

coefficients is shown in Fig. 6 which illustrates that ωh

decreases toward 1√1−k2

as ωn decreases. The data points in

Fig. 6 indicate the position of ωh when ωn = ωn,γ1. We will

therefore expect that the converter operating at ωH for better

voltage controllability can have a better efficiency by choosing

the compensation of ωn = ωn,γ1rather than the conventional

compensation of ωn = 1. The efficiency equation (8) can be

rewritten as

ηT =1− QO

QS

1 +1+Q2

O

(

ωh− 1

ωh

)

2

k2ω2

hωn

QOQP

. (51)

Therefore, a transfer efficiency enhancement ηTen can be

calculated using the difference of efficiency designed with

ωn = ωn,γ1and ωn = 1, where 0 < ωn,γ1

≤ 1. The actual

enhancement operating at ωh is given as

ηTen = ηT (ωn = ωn,γ1)− ηT (ωn = 1). (52)

Fig. 7 shows the efficiency enhancement versus k and QO

with various values of ωn,γ1. Fig. 7 gives us the information

that

1) the efficiency enhancement is increasingly obvious with

larger QO and smaller ωn,γ1,

2) the enhancement peaks at smaller k and vanishes when

k = 0 or 1.




6 12 15 18

15%

12%

9%

3%

0%3

ηTen

QO

ωn,γ1=0.95

ωn,γ1=0.9

ωn,γ1=0.8ωn,γ1=0

6%

9

(a)

0 0.2 0.4 0.6 0.80

3%

6%

9%

k1

ωn,γ1=0

ωn,γ1=0.95

ωn,γ1=0.9

ωn,γ1=0.8ηTen

(b)

Fig. 7. Efficiency enhancement with various ωn,γ1(a) versus QO (k is

fixed at 0.2) and (b) versus k (QO is fixed at 10).

d=49.5

r=40.5

d1=90

d2=9

g=35

Fig. 8. Sample transformer geometry and dimensions in mm.

B. Circuit design

From the analysis in Section III-A, the design with ωn,γ1

can improve transfer efficiency and operating at ωH can keep

the voltage transfer ratio constant. In this section, we focus

on choosing the value of ωn,γ1and give design procedures

for the system considering higher efficiency and voltage con-

trollability.

A design example is illustrated in this subsection. The

specifications are given as follows. The converter has a rated

power (Pout) of 18 W, an output voltage of 12 V and an input

voltage of 48 V.

The size of the transformer is application specific. In this

design example, the air gap between the two coils (g) is 35

mm. Spiral circular geometry coils are adopted here for its

better coupling [11]. It has outer diameter d1 = 90 mm and

inner diameter d2 = 9 mm. More detailed dimensions are

shown in Fig. 8.

The coupling coefficient can be calculated according to the

dimensions shown in Fig. 8 by using Neumann’s formula. The

self and mutual inductances for circular windings with N , NP

and NS turns can be approximately calculated as

L =µ0

4πN2

∮ ∮

dl · dl′r

(53)

M =µ0

4πNPNS

∮ ∮

dl · dl′r

,

which are applied to the sample transformer in Fig. 8, giv-

ing [22]

L =µ0

8πN2d ·Ψ (54)

M =µ0

4πNPNSd · Φ

where

d =d1 + d2

2, (55)

Ψ =(1 + ρ)3

ρ2(1.7424 + 3.29κ3 lnκ− 2.27κ3

+ 0.3702κ5 + 0.0826κ7 + 0.0312κ9), (56)

ρ =r

d, (57)

κ =d2

d1, (58)

and the value of Φ can be determined by the table look-

up scheme in [22] to be 1.4 for the example transformer.

Substituting (54) into (6), we obtain

k =2Ψ

Φ= 0.18. (59)

Quality factors QP and QS of the transformer windings can

be estimated according to the wire material and manufacturer

data sheets. For accurate efficiency estimation, quality factors

can be measured after the completion of the design of the

transformer physical dimensions. In our design, QP,max and

QS,max are both estimated as 100 when working at 200 kHz

for the litz wires used.

Using the estimated quality factors, the converter transfer

efficiency and the input phase angle at various QO and ωn,γ1

are plotted using (51) and (43). The sample design curves are

as shown in Fig. 9.

The selection of ωn,γ1should take into account the effi-

ciency enhancement and the amount of inactive power which

affects the current rating of switches. We select ωn,γ1as 0.7.

When ωn,γ1is chosen, optimal QO is selected according to

Fig. 9(a).

The inductance of the secondary coil can be calculated using

(60). In the example design, QO is approximately 5, and LS =33.51 µH. The number of turns NS is calculated to be around

29 turns based on (55). Using the current in the secondary

loop, which can be calculated by is = vout

RLto choose the

secondary total wire sectional area.

LS =RL

ωS(1

QO− 1

QS). (60)

The compensation capacitor in the secondary is calculated

using (13) as CS = 18.9 nF.

The value of LP is determined by substituting the selected

ωn,γ1and LS into (40), according to the desired voltage




5 10 150.72

0.76

0.84

0.88

0.92

0.80

1QO

ηT

ωn,γ1=1

ωn,γ1=0.9

ωn,γ1=0.8

ωn,γ1=0.7

ωn,γ1=0

(a)

0

30

60

90

5 10 151QO

θin

ωn,γ1=1

ωn,γ1=0

ωn,γ1=0.9

ωn,γ1=0.8

ωn,γ1=0.7

(b)

Fig. 9. Design curves at various ωn,γ1(a) efficiency versus QO and (b)

input phase angle versus QO .

transfer ratio. In our example, |Gv| = 0.25 and LP = 38.21µH. As (40) is calculated without considering RP and RS ,

which may result in a lower voltage transfer ratio. Therefore,

in transformer fabrication, LP should be designed slightly

smaller to satisfy the requirement of Gv . Based on (55), the

number of turns of LP is calculated as 30. Using the output

power and the approximate efficiency, the input active power

can be calculated. The input phase angle in Fig. 9 indicates the

proportion of input active power to inactive power. Therefore,

the apparent power as well as input current can be estimated

for the determination of sectional area of the litz wire used in

the primary. In our design, we use ready-made litz wire in our

laboratory that can surely satisfy the current requirement.

Finally, the compensation capacitor in the primary can be

calculated according to (12), (13) at the selected ωn,γ1as

CP = 33.15 nF.

IV. EVALUATION

A. Overall efficiency calculation

A full-bridge circuit shown in Fig. 10 with four MOSFETs

is used for driving the converter circuit. Besides the power

loss caused by winding resistances of the loosely coupled

transformer, the power loss imposed on the switching com-

ponents cannot be neglected in order to have an accurate

overall efficiency calculation. Switching loss is composed

of conduction loss and transient loss, which are calculated

separately in the following.

Switching transient loss calculation is conducted first. Due

to the impedance experienced by the full-bridge circuit is

inductive, zero-voltage-on of the four MOSFET switches can

be realized. The turn-on loss is assumed zero. When turned

off, however, the voltage and current stresses on the four

vin

Q1

Q3

Q2

Q4

CP CS

LP LS

M

D1

D3

D2

D4

Cf RO

Fig. 10. Circuit diagram for calculation and experimentation

ttf

vT=vin

iT=iin

0

Fig. 11. MOSFETs turn off transient

MOSFETs overlap, as shown in Fig. 11. The figure assumes

that the current through MOSFET reaches zero before its

voltage rises up to vin when it turn off. It is easy to realize

by increasing the capacitance between drain and source by

external capacitor.

The switching loss of each MOSFET in unit time can be

expressed as

PLs = fI2in

16CDS

t2f , (61)

where CDS is the drain-to-source capacitance and tf is the

fall time upon switching of the MOSFETs selected. Iin is

the MOSFET drain current just before the switching transient,

which is given as

Iin =√2iin sin θin. (62)

Hence,

PLs = i2inf (tf sin θin)

2

8CDS

= i2inRf,eqv, (63)

where Rf,eqv is the equivalent switching loss resistance of the

MOSFET.

For conduction loss of the MOSFETs, it can be calculated

as i2inRDS(on), where RDS(on) is the conduction resistance of

the MOSFET selected. The power loss of the rectifier bridge

diode in the secondary loop is calculated by multiplying its

voltage drop and output current.

According to the design parameters in Section III-B, the

transformer is built and shown in Fig. 12. In addition, the

quality factors of the windings are measured under 1 A current,

which is shown in Fig. 13. Table I gives the measured

parameters of the prototype and the components used in

experiment.

The calculated overall efficiency using MOSFET IRF640N

with RDS(on) = 0.18 Ω, CDS = 1.96 nF, tf = 35 ns, as well

as the voltage transfer ratio at ωn,γ1= 0.7 are shown in Fig 14.

The efficiency at ωn,γ1= 1 is also shown in Fig 14(b) for




Fig. 12. Transformer prototype builded as the sample.

QP

QS

Qualityfactor

Frequency (kHz)

Fig. 13. QP and QS of the prototype transformer.

comparison. With RP + 2RDS(on) = 0.794 Ω and 2Rf,eqv =0.032 Ω, conduction loss is dominating when operating at

around 200 kHz. The switching loss may be higher if higher

working frequency is selected.

B. Experimental Evaluation

A prototype is constructed using the components in Table I

to verify the analytical results. The measured ωH and Gv for

ωn,γ1= 0.7 is depicted in Fig. 15 (a), showing the good

voltage regulation at ωH with respect to load change. This

agrees well with the calculation result in Fig. 14(a). Efficiency

versus QO has been measured and compared with calculation

results for the two ωn,γ1values of 0.7 and 1, as Fig. 15 (b)

shows. By decreasing the ωn,γ1value, there is an obvious

improvement in efficiency especially at heavy load.

In addition, more experiments are conducted by decreasing

the air gap. Fig. 16 shows the measured efficiency results of

g = 27 mm (k = 0.264) and g = 20 mm (k = 0.355) with

ωn,γ1= 1 and ωn,γ1

= 0.7 respectively.

V. CONCLUSION

The operating frequency of an inductive power transfer

system that achieves an optimal efficiency, a load independent

voltage transfer ratio and an inductive input phase angle is

studied in this paper. A compensation technique to optimize

the system efficiency and output voltage controllability of

an inductive power transfer system is proposed. The value

of ωP compared to ωS is selected as a design parameter

for the efficiency enhancement. A circuit design procedure

is explicated with an example experimental prototype. Exper-

imental measurements confirm the analytical results and the

0

0.2

0.4

0.6

0.8

145 165 185 205 225 245 265

RL=4

RL=8

RL=20

Frequency (kHz)

|Gv|

(a)

0.68

0.72

0.76

0.8

0.84

1 5 9 13QO

Overallefficiency

ωn,γ1=1

ωn,γ1=0.7

(b)

Fig. 14. Calculated results of (a) voltage transfer ratio and (b) overallefficiency.

|Gv|

Frequency (kHz)

RL=4

RL=8

RL=20

(a)

QO

Efficiency

ωn,γ1=1

ωn,γ1=0.7

(b)

Fig. 15. Measured results of (a) voltage transfer ratio and (b) overallefficiency.




TABLE IPOWER COMPONENTS USED IN THE CONVERTER.

Circuit components ValuePower MOSFET Q1-Q4 IRF640NCapacitance CP 33.12 nF / 630VCapacitance CS 18.9 nF / 630VSchottky diode D1-D4 STPS20H100CGCapacitance Cf 220 µFLoosely-coupled transformer NP =30 turns of Litz wire

LP =32.78µH QP =94.9@200 kHzNS=29 turns of Litz wireLP =31.46µH QP =89.0@200 kHzair gap = 35 mmk =0.182

QO

Efficiency

ωn,γ1=1

ωn,γ1=0.7

(a)

QO

Efficiency

ωn,γ1=1

ωn,γ1=0.7

(b)

Fig. 16. Measured results of (a) k = 0.264 and (b) k = 0.355.

design procedures in achieving both efficiency optimization

and voltage controllability.

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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 the Energy and Power Engineering Journal and a member of theEditorial Board of the Journal 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.He was a visiting scholar at the Center for Power Electronics Systems, VirginiaTech, VA, USA on November 2008, Aero-Power Sci-tech Center, NanjingUniversity of Aeronautics and Astronautics, Nanjing, China on January 2009,and School of Electrical Engineering, Southeast University, Nanjing, Chinaon March 2012.

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 books Linear

Circuit Analysis (London: Addison-Wesley, 1998) and Complex Behavior of

Switching 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) and Sliding Mode Control of

Switching 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 the IEEE Circuits and

Systems Magazine and Editor-in-Chief of IEEE Circuits and Systems Society

Newsletter. 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 TRANSACTIONS ON POWER ELECTRONICS

since 1999. He is an Associate Editor of the International Journal of Systems

Science, and also on the Editorial Board of the International Journal of

Circuit Theory and Applications and International Journal and Bifurcation

and Chaos. He also served as Guest Editor and Guest Associate Editor for anumber 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 from International

Journal of Circuit Theory and Applications in 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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