Effect of Wireless Power Link Load Resistance onthe Efficiency of the Energy Transfer

Mariusz Bojarski, Erdem Asa, Dariusz CzarkowskiNYU Polytechnic School of Engineering

Brooklyn, New York, USA

mb4496@nyu.edu, ea1145@nyu.edu, dc1677@nyu.edu

Abstract—With the proper impedance matching system, a highefficiency can be acquired in wireless power transfer applications.Variations of the coupling coefficient factor could deviate, how-ever, the impedance matching system from the designed consider-ations. In this paper, the transmitter reflected impedance from thereceiver side is analyzed to avoid divergence of the impedancematching network considering a wide air gap range betweentransmitter and receiver sides. A 2 kW contactless system isdesigned to investigate an optimum impedance requirement bytesting several displacement gaps between coils. Experimental re-sults are demonstrated to reveal a relation between the efficiencyand the reflected impedance.

I. INTRODUCTION

Contactless energy transfer has experienced recently a grow-

ing attention for its numerous potential applications such as

smartphone charging platforms [1]-[2], medical implant de-

vices [3]-[4], and an electric vehicle charging [5]-[6]. Starting

from the transmitter side and ending at the receiver side,

the contactless energy transfer system must be well designed

and organized in order to acquire high efficiency [7]-[8]. An

impedance matching network is important to improve the

system performance and is usually used at both transmitter

and receiver sides. However, due to the coupling coefficient

variation, the contactless system overall impedance diverges

from the designed characteristic, which means that the de-

signed impedance matching may be not working effectively

causing a decrease in the system efficiency as compared to

the designed performance values.

Fig. 1. A general wireless power transfer system block diagram.

A conventional wireless energy transfer system is demon-

strated in Fig. 1. As seen in the figure, the system consists

of two main stages: the transmitter and receiver platforms.

The role of the first stage with impedance matching network

is to deliver energy to the second stage. The dc output

voltage is provided to the load by the second stage with an

impedance matching network, a high-frequency rectifier, and

a non-isolated dc/dc converter.

Power management, system controller synthesis, and cir-

cuit topologies are explored in the literature [9]-[13]. Self-

tuning power regulator [14], implicit adaptive controller [15],

directional tuning control [16], and more flexible solution,

active tuning of parallel compensated receiver with tri-state

boost converter topology [17] are submitted for wireless power

transfer applications. Using a buck converter in the receiver

side, GaN based transmitter with adaptive receiver is designed

in [18]. A boost converter topology with low switching fre-

quency is demonstrated in [19]. In [20]-[22], researchers have

investigated reflected power to the transmitter side for low

power inductive power transfer applications by using cascaded

boost and buck, buck-boost, and integrated buck and boost

converter, respectively. Optimal resonant load transformation

for the biomedical implants is proposed in [23]. However, there

is no paper considering the high power applications.

In this paper, wireless power link load influences on the

impedance which are seen by the inverter are investigated by

variable load in the receiver side. In order to reveal the system

efficiency improvement, the system is tested with several

distance variation between coils. Especially for high power

applications such as EV chargers, the reflected impedance

consideration brings a solution for EMI problems that can

be minimized providing transmitter side control at constant

frequency and regulating the output with phase shift or dc

link control. The converter model controllability is also tested

deriving the circuit behavior and transfer function of the

converter extracted for the reflected impedance approach. The

system performance is confirmed with experimental results

designing a wireless power transfer system at 2 kW in the

laboratory conditions. A related theoretical analysis with ver-

ification of the converter is discussed more detailed in the

following chapters.

II. ANALYSIS OF THE WIRELESS POWER LINK

A. Electrical Model

In order to perform an theoretical analysis a proper electrical

model of the circuit is required. The wireless power link can

be represented as two coupled inductors and two resonant

capacitors connected in series. The schematic of the circuit

model is shown in Fig. 2.

In this model VI is an ac voltage source, RL is a load

resistance, LP and LS are two coupled inductors with series

resistances RS and RP . K is a coupling factor between the9781-4799-6075-0/14/$31.00 c©2014 IEEE

Fig. 2. Schematic of the wireless power link connected to the AC source andresistive load.

two coils. CP and CS are resonant capacitors. The two coupled

inductors can be equivalently modeled as a transformer with

proper leakage and magnetizing inductances. To simplify

analysis both coils LP and LS are assumed to be identical and

equal to L. Then the model can be equivalently represented

by the circuit shown in Fig. 3.

Fig. 3. Equivalent circuit of the wireless power link with two identical coilsconnected to the AC source and resistive load.

In this model VI is an ac voltage source, RL is a load

resistance, LL and LM are leakage and magnetizing in-

ductances related with coupled inductors, RS and RP are

series resistances of the coils, and CP and CS are resonant

capacitors. This model is related to the model from Fig. 2 with

the following equations.

LM = K√

LPLS = KL,

LL = L− LM = (1−K)L, (1)

For the analysis purpose model can be simplified and

generalized as shown in Fig. 4.

Fig. 4. Generalized circuit of the wireless power link with two identical coilsconnected to the AC source and resistive load.

The generalized circuit model is more compact and it

simplifies derivation of the equations. Impedances in the model

are

ZP =1

jωCP

+RP + jωLL,

ZS =1

jωCS

+RS + jωLL,

ZM = jωLM . (2)

B. Circuit Analysis

The wireless power link analysis is performed based on the

circuit shown in Fig. 4. It is assumed that the circuit is loaded

with a pure resistance RL. As the first step of the analysis,

the voltage across the impedance ZM is calculated.

VZM =V[ZM||(ZS +RL)]

ZM||(ZS +RL) + ZP

=

=V(ZMZS + ZSRL)

ZMZS + ZMRL + ZMZP + ZSZP + ZSRL

(3)

Then, the voltage across the load resistance RL is calculated

as

VRL =VZMRL

ZS +RL

=

=VZMRL

ZMZS + ZMRL + ZMZP + ZSZP + ZSRL

(4)

The secondary side current, which is same as the load

current is equal to

IS =VRL

RL

=

=VZM

ZMZS + ZMRL + ZMZP + ZSZP + ZSRL

(5)

In order to calculate the primary side current, which is same

as the voltage source current, the impedance ZT seen by the

source is calculated.

ZT = ZP +ZMZS + ZMRL

ZM + ZS +RL

=

=ZMZS + ZMRL + ZMZP + ZSZP + ZPRL

ZM + ZS +RL

(6)

Then, the primary side current can be calculated as

IP =V

ZT

=

=V(ZM + ZS +RL)

ZMZS + ZMRL + ZMZP + ZSZP + ZPRL

(7)

The primary and the secondary currents ratio is

IP

IS=

ZM + ZS +RL

ZM

(8)

Losses in the circuit can be calculated based on the currents.

The losses are related to the real parts of the impedances

ZP and ZS , which are RP and RS . It can be seen that

these impedances can also include additional components, for

instance lead cables, which makes this analysis more general.

The circuit losses are calculated using the following equation.

Ploss = |IP|2RP + |IS|

2RS (9)

In order to calculate the efficiency of the circuit, equations

for the losses and the output power are needed. The output

power can be calculated based on the secondary side current

as

Po = |IS|2RL (10)

Then, the efficiency can be calculated using the following

equation.

η =Po

Po + Ploss=

1

1 + RS

RL+ RP |ZM+ZS+RL|2

RL|ZM|2

(11)

From the above equation it can be concluded that the effi-

ciency does not depend on an imaginary part of the impedance

ZP , which represents the leakage inductance of the primary

side. Secondly, the equation shows that low values of the

magnetizing inductance impedance ZM and large values of

the secondary side leakage impedance ZS may lead to a poor

efficiency.

It can be derived, that maximum efficiency is obtained at

the resonant frequency of the secondary side of the wireless

power link. Then ZM + ZS = RS , and equation (11) can be

rewritten as

η =Po

Po + Ploss=

1

1 + RS

RL+ RP |RS+RL|2

RL|ZM|2

(12)

When RL >> RS , which is usually the case, efficiency can

be approximated as

η =Po

Po + Ploss=

1

1 + RS

RL+ RPRL

|ZM|2

(13)

The maximum of above equation is obtained when function

f(RL) =RS

RL

+RPRL

|ZM|2(14)

is minimized. Derivative of above-mentioned function f(RL)is

df

dRL

=−RS

R2L

+RP

|ZM|2(15)

In can be derived that the maximum is obtained when

RS

R2L

=RP

|ZM|2

RL =

√

RS |ZM|2

RP

(16)

If RS = RP it can be reduced to

RL = |ZM| = ωKL (17)

The obtained optimum load resistance point is related to

the wireless power link primary and secondary current ratio

through the equation (8). As mentioned, at the resonance

frequency ZM + ZS = RS . As RS << RL the current ratio

for optimum load resistance can be simplified to

IP

IS=

RL

ZM

(18)

The above equation introduces the straightforward method

of verifying optimum load conditions of the wireless power

link.

III. SIMULATION RESULTS

Analytical results are verified with AC simulations. Circuit

model shown in Fig. 2 is used for simulations with parameters

shown in Table I.

TABLE ISIMULATION PARAMETERS

Parameter Value Unit Parameter Value Unit

f 145 kHz LP 30 µH

VI 200 V LS 30 µH

CP 40 nF RS 100 mΩ

CS 40 nF RP 100 mΩ

Simulation results are shown in Fig. 5 through Fig. 11.

Fig. 5 shows the efficiency of the wireless power link as

a function of the load resistance RL. It clearly shows that

operating at high or low load resistance results with poor

efficiency, especially at low values of coupling factor. The

optimum load resistance points were calculated using equation

(17) and marked on the plot. It can be observed that theoretical

predictions are in agreement with the simulation results.

Fig. 5. Wireless power link efficiency vs load resistance for various values ofcoupling factor. Optimum load resistance points are marked with ’X’ symbols.

On the next plot, shown in Fig. 6, the wireless power

link primary and secondary current ratio is presented. The

current ratios for optimum load resistances were calculated

using equation (18) and market on the plot. It can be seen

that it is consistent with the previous plot shown in Fig. 5.

Fig. 7 shows the optimum load resistance of the wireless

power link as a function of the coupling factor. It can be seen

that results are in perfect agreement with an analytical equation

(17).

The next two plots are shown for two cases, the optimum

adjustable resistance, and fixed value to optimum for a lowest

coupling factor. Fig. 8 shows the efficiency of the wireless

Fig. 6. Wireless power link primary and secondary current ratio vs loadresistance for various values of coupling factor. Optimum load resistancepoints are marked with ’X’ symbols.

Fig. 7. Wireless power link optimum load resistance vs coupling factor.

power link as a function of the coupling factor. Fig. 9 shows

the output power of the wireless power link as a function of the

coupling factor. It can be seen that adjusting load resistance

is increasing efficiency and decreasing power variations over

the changes of the coupling factor.

Fig. 8. Wireless power link efficiency vs coupling factor for optimum loadresistances.

In the next step, the influence of the series resistances

of wireless power link model (RP and RS) is investigated.

For that purpose the primary side series resistance RP is

reduced to 25 mΩ. Then, the optimum load resistance point

is calculated using equation (11) and related current ratio is

Fig. 9. Wireless power link output power vs coupling factor for optimumload resistances.

calculated using equation (18). The theoretical prediction is

then verified with simulation and the results are presented in

Fig. 10 and Fig. 11.

Fig. 10. Wireless power link efficiency vs load resistance for various values ofcoupling factor. Optimum load resistance points are marked with ’X’ symbols.For RP = 25 mΩ.

Fig. 11. Wireless power link primary and secondary current ratio vs loadresistance for various values of coupling factor. Optimum load resistancepoints are marked with ’X’ symbols. For RP = 25 mΩ.

IV. EXPERIMENTS

To confirm the performance of the system and correctness

of theoretical analysis, experimental results are provided. The

setup used for experiments is shown in Fig. 12.

Fig. 12. Experimental setup block diagram.

Fig. 13. Wireless power link used for experiment setup for 12 inches distancebetween coils.

Both of the used coils has dimensions 2.5 by 2.5 feet and

4 turns. In the experiments, setups with various distances

were used. The coupling factor was measured as described in

[24]. The summary of distances and related coupling factors

between the coils are presented in Table II. The wireless power

link setup is presented in Fig. 13.

TABLE IICOUPLING COEFFICIENTS

d [inches] VP [V] VS [V] K

4 6.12 1.39 0.227

6 6.08 1.05 0.173

8 6.16 0.832 0.135

10 6.08 0.616 0.101

12 6.08 0.500 0.082

The VP and VS values presented in the table are peak-to-peak

voltage measurements performed on primary and secondary

side coils in order to obtain coupling factor values.

The wireless power link consist of the two above-mentioned

coils and two resonant capacitors. The summary of the wireless

power link parameters is presented in Table III.

A full-bridge current driven Class D circuit is used as the

rectifier. Thus, the load resistance RL seen by the wireless

power link is

TABLE IIIWIRELESS POWER LINK PARAMETERS

Parameter Value Unit Parameter Value Unit

LP 24.9 µH rLP 45 mΩ

CP 41.1 nF rCP 4.5 mΩ

LS 25.6 µH rLS 47 mΩ

CS 41.3 nF rCS 3.5 mΩ

RL = 0.81RLDC (19)

The rectifier is built using four DSEI2X101-06A diodes. The

series resistance Rd introduced by the rectifier is calculated as

follows

Rd =1.4 V

Idc+ 9.4 mΩ (20)

where Idc is the output dc current through the load resistance

RLDC . The values used in the equation are based on the used

diode datasheet and are representing two diodes connected in

series.

As an inverter, a 2 kW two phase resonant inverter with a

common resonant circuit is used [25]. The operating frequency

is chosen to be 151.5 kHz, which is slightly above the resonant

frequency.

Fig. 14 and Fig. 15 show the efficiency and output power

characteristics with variable load and various distances be-

tween coils. On the plots the calculated optimum load resis-

tance points are marked. Those values were calculated using

equation (16), where RS = 60 mΩ, and RS = Rd + 100 mΩto include series resistances of the rectifier, inverter, and

connecting cables.

Fig. 14. Wireless power link efficiency vs load resistance for various distancesbetween coils. The actual measurement point are marked with ”x” symbols.The calculated optimum load resistance points are marked with ”o” symbols.

From the plot in Fig. 14, it can be seen that efficiency

decreases when the distance D between the wireless power

link coils increases. The plot also shows, that theoretical

prediction of the optimum load resistance is close to the

measured one, which verify correctness of the theoretical

analysis. It can be also concluded, that proper selection of load

resistance is especially important when the coupling factor has

a low value, because then the efficiency is dropping fast with

a load variation.

Fig. 15. Wireless power output power vs load resistance for various distancesbetween coils. The actual measurement point are marked with ”x” symbols.The calculated optimum load resistance points are marked with ”o” symbols.

From the plot in Fig. 15, it can be seen that the system

output power is rising with the load increment linearly.

The extracted optimum load resistance is plotted in Fig. 16.

As seen in the figure, the optimum resistance does not change

much between 6” and 8” inches. It is related to the variable

resistance step in the experiment, which was around 3 Ω.

Fig. 16. Wireless power link optimum load resistance vs distances betweencoils. The actual measurement point are marked with ”x” symbols.

The comparison of the optimum load and the constant

load at 4.9 Ω is given for efficiency and power analysis in

Fig. 17 and Fig. 18. As predicted from theoretical analysis,

the optimum load gives higher efficiency and lower power

variation for wide range of distances between the coils.

V. CONCLUSIONS

In this study, the load resistance effects on the efficiency of

the wireless power transfer are examined. The experimental

results are established with a variable load for various air

gap distances. The possible system efficiency improvement is

revealed in these circumstances. Considering the importance

of the efficiency in high power applications, the impedance

matching disturbance can be preventable with the proper load

and a careful design of the wireless power link. The converter

model is presented and analyzed. The theoretical results are

Fig. 17. Wireless power link efficiency vs distances between coils foroptimum load resistances. The actual measurement point are marked with”x” symbols.

Fig. 18. Wireless power link output power vs distances between coils foroptimum load resistances. The actual measurement point are marked with”x” symbols.

in good agreement with the simulation and the experimental

results. Laboratory prototype of a 2 kW wireless power

transfer system shows that the total efficiency can be improved

by up to 8% compared to the constant load application.

REFERENCES

[1] Y. Jang and M. M. Jovanovic, ”A Contactless Electrical Energy Transmis-sion System for Portable Telephone Battery Chargers,” IEEE Transactions

on Industrial Electronics, vol. 50, no. 3, pp. 520–527, Jun. 2003.

[2] K. C. Wan, Q. Xue, X. Liu, and S. Y. Hui, ”Passive Radio FrequencyRepeater for Enhancing Signal Reception and Transmission in a WirelessCharging Platform,” IEEE Transactions on Industrial Electronics, vol. 61,no. 4, pp. 1750–1757, Apr. 2014.

[3] A. K. RamRakhyani and G. Lazzi, ”On the Design of Efficient MultiCoil Telemetry System for Biomedical Implants,” IEEE Transactions on

Biomedical Circuits and Systems, vol. 7, no. 1, pp. 11–23, Feb. 2013.

[4] M. Q. Nguyen, Z. Hughes, P. Woods, Y. S. Seo; S. Rao, and J. C. Chiao,”Field Distribution Models of Spiral Coil for Misalignment Analysis inWireless Power Transfer Systems,” IEEE Transactions on Microwave

Theory and Techniques, vol. 62, no. 4, pp. 920–930, Apr. 2014.

[5] U. K. Madawala, M. Neath, and D. J. Thrimawithana, ”A Power Fre-quency Controller for Bidirectional Inductive Power Transfer Systems,”IEEE Transactions on Industrial Electronics, vol. 60, no. 1, pp. 310–317,Jan. 2013.

[6] F. Musavi and W. Eberle, ”Overview of Wireless Power Transfer Tech-nologies for Electric Vehicle Battery Charging,” IET Power Electronics,vol. 7, no. 1, pp. 60–66, Jan. 2014.

[7] W. Zhang, S. C. Wong, C. K. Tse, and Q. Chen, ”Design for EfficiencyOptimization and Voltage Controllability of Series Series CompensatedInductive Power Transfer Systems,” IEEE Transactions on Power Elec-

tronics, vol. 29, no. 1, pp. 191–200, Jan. 2014.[8] C. Florian, F. Mastri, R. P. Paganelli, D. Masotti, and A. Costanzo, ”The-

oretical and Numerical Design of a Wireless Power Transmission LinkWith GaN Based Transmitter and Adaptive Receiver,” IEEE Transactions

on Microwave Theory and Techniques, vol.62, no.4, pp. 931–946, Apr.2014.

[9] L. Chen, J. Boys, and G. Covic, ”Power Management for Multiple-pickup IPT Systems in Materials Handling Applications,” IEEE Journal

of Emerging and Selected Topics in Power Electronics, in press.[10] A. K. Swain, S. Devarakonda, and U. K. Madawala, ”Modeling, Sen-

sitivity Analysis, and Controller Synthesis of Multipickup BidirectionalInductive Power Transfer Systems,” IEEE Transactions on Industrial

Informatics, vol. 10, no. 2, pp. 1372–1380, May 2014.[11] C. Park, S. Lee, G. H. Cho, S. Y. Choi, and C. T. Rim, ”Two-

Dimensional Inductive Power Transfer System for Mobile Robots UsingEvenly Displaced Multiple Pickups,” IEEE Transactions on Industry

Applications, vol. 50, no. 1, pp. 558–565, Feb. 2014.[12] C. Liu, A. P. Hu, B. Wang, and N. C. Nair, ”A Capacitively Coupled

Contactless Matrix Charging Platform With Soft Switched TransformerControl,” IEEE Transactions on Industrial Electronics, vol. 60, no. 1, pp.249–260, Jan. 2013.

[13] J. E. James, D. R. Robertson, and G. A. Covic, ”Improved AC Pickupsfor IPT Systems,” IEEE Transactions on Power Electronics, vol. 29, no.12, pp. 6361–6374, Dec. 2014.

[14] G. A. Covic, J. T. Boys, A. M. W. Tam, and J. C. H. Peng, ”SelfTuning Pick-ups for Inductive Power Transfer,” IEEE Power Electronics

Specialists Conference (PESC), pp. 3489–3494, Jun. 2008.[15] J. U. W. Hsu, A. Swain, and A. P. Hu, ”Implicit adaptive controller for

wireless power pickups,” IEEE Industrial Electronics and Applications

(ICIEA), pp. 514–519, Jun. 2011.[16] J. U. W. Hsu, A. P. Hu, and A. Swain, ”A Wireless Power Pickup Based

on Directional Tuning Control of Magnetic Amplifier,” IEEE Transactions

on Industrial Electronics, vol. 56, no. 7, pp. 2771–2781, Jul. 2009.

[17] Z. Pantic and S. M. Lukic, ”Framework and Topology for Active Tuningof Parallel Compensated Receivers in Power Transfer Systems,” IEEE

Transactions on Power Electronics, vol. 27, no. 11, pp. 4503–4513, Nov.2012.

[18] C. Florian, F. Mastri, R. P. Paganelli, D. Masotti, and A. Costanzo, ”The-oretical and Numerical Design of a Wireless Power Transmission LinkWith GaN-Based Transmitter and Adaptive Receiver,” IEEE Transactions

on Microwave Theory and Techniques, vol. 62, no. 4, pp. 931–946, Apr.2014.

[19] H. Jiang, B. Lariviere, D. Lan, J. Zhang, J. Wang, R. Fechter, M.Harrison, and S. Roy, ”A Low Switching Frequency AC-DC BoostConverter for Wireless Powered Miniaturized Implants,” IEEE Topical

Conference on Biomedical Wireless Technologies, Networks, and Sensing

Systems, pp. 40–42, Jan. 2014.[20] M. Fu, C. Ma, and X. Zhu, ”A Cascaded Boost-Buck Converter for

High Efficiency Wireless Power Transfer Systems,” IEEE Transactions

on Industrial Informatics, vol. 10, no. 3, pp. 1972–1980, Aug. 2014.[21] Y. Huang, N. Shinohara, and T. Mitani, ”A Constant Efficiency of

Rectifying Circuit in an Extremely Wide Load Range,” IEEE Transactions

on Microwave Theory and Techniques, vol. 62, no. 4, pp. 986–993, Apr.2014.

[22] D. Ahn and S. Hong, ”Wireless Power Transfer Resonance CouplingAmplification by Load-Modulation Switching Controller,” IEEE Trans-

actions on Industrial Electronics, in press.[23] R. F. Xue, K. W. Cheng, and M. Je, ”High Efficiency Wireless Power

Transfer for Biomedical Implants by Optimal Resonant Load Transfor-mation,” IEEE Transactions on Circuits and Systems I: Regular Papers,vol. 60, no. 4, pp. 867–874, Apr. 2013.

[24] Lukas Heinzle, ”Measuring the Mutual Interaction between CoaxialCylindrical Coils with the Bode 100,” Smart Measurement Solution,

OMICRON LAB, 2013.[25] M. Bojarski, D. Czarkowski, F. de Leon, Q. Deng, M. K. Kazimierczuk,

and H. Sekiya, ”Multiphase resonant inverters with common resonantcircuit,” in Proc. IEEE International Symposium on Circuits and Systems

(ISCAS), 2014, pp. 2445–2448.