Effect of Wireless Power Link Load Resistance on
the Efficiency of the Energy Transfer
Mariusz Bojarski, Erdem Asa, Dariusz Czarkowski
NYU 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 high
efficiency can be acquired in wireless power transfer applications.
Variations of the coupling coefficient factor could deviate, however, the impedance matching system from the designed considerations. In this paper, the transmitter reflected impedance from the
receiver side is analyzed to avoid divergence of the impedance
matching network considering a wide air gap range between
transmitter and receiver sides. A 2 kW contactless system is
designed to investigate an optimum impedance requirement by
testing several displacement gaps between coils. Experimental results are demonstrated to reveal a relation between the efficiency
and the reflected impedance.
I. I NTRODUCTION
Contactless energy transfer has experienced recently a growing attention for its numerous potential applications such as
smartphone charging platforms [1]-[2], medical implant devices [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 designed 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 demonstrated 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.
c
9781-4799-6075-0/14/$31.00 2014
IEEE
Power management, system controller synthesis, and circuit topologies are explored in the literature [9]-[13]. Selftuning 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 frequency 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 verification of the converter is discussed more detailed in the
following chapters.
II. A NALYSIS 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 the
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 =
Fig. 2. Schematic of the wireless power link connected to the AC source and
resistive 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.
=
V[ZM ||(ZS + RL )]
=
ZM ||(ZS + RL ) + ZP
V(ZM ZS + ZS RL )
Z M Z S + Z M RL + Z M Z P + Z S Z P + Z S R L
(3)
Then, the voltage across the load resistance RL is calculated
as
VRL =
=
VZM RL
=
Z S + RL
VZM RL
Z M Z S + Z M RL + Z M Z P + Z S Z P + Z S R L
(4)
The secondary side current, which is same as the load
current is equal to
IS =
Fig. 3. Equivalent circuit of the wireless power link with two identical coils
connected 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 inductances 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.
p
LM = K LP LS = KL,
LL = L − LM = (1 − K)L,
=
VZM
Z M Z S + Z M RL + Z M Z P + Z S Z P + Z S R L
(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 +
=
(1)
For the analysis purpose model can be simplified and
generalized as shown in Fig. 4.
VRL
=
RL
Z M Z S + Z M RL
=
Z M + Z S + RL
Z M Z S + Z M RL + Z M Z P + Z S Z P + Z P RL
Z M + Z S + RL
(6)
Then, the primary side current can be calculated as
IP =
=
V
=
ZT
V(ZM + ZS + RL )
Z M Z S + Z M RL + Z M Z P + Z S Z P + Z P RL
(7)
The primary and the secondary currents ratio is
Fig. 4. Generalized circuit of the wireless power link with two identical coils
connected 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
1
+ RP + jωLL ,
jωCP
1
ZS =
+ RS + jωLL ,
jωCS
ZM = jωLM .
ZP =
IP
Z M + Z S + RL
=
IS
ZM
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 |2 RP + |IS |2 RS
(2)
(8)
(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 |2 RL
(10)
Then, the efficiency can be calculated using the following
equation.
IP
RL
=
IS
ZM
(18)
The above equation introduces the straightforward method
of verifying optimum load conditions of the wireless power
link.
III. S IMULATION R ESULTS
Po
=
η=
Po + Pl oss
1+
1
RS
RL
+
RP |ZM +ZS +RL |2
RL |ZM |2
(11)
From the above equation it can be concluded that the efficiency 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 + Pl oss
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 + Pl oss
1+
RS
RL
1
+
RP RL
|ZM |2
Analytical results are verified with AC simulations. Circuit
model shown in Fig. 2 is used for simulations with parameters
shown in Table I.
TABLE I
S IMULATION 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.
(13)
The maximum of above equation is obtained when function
f (RL ) =
RS
RP RL
+
RL
|ZM |2
(14)
is minimized. Derivative of above-mentioned function f (RL )
is
−RS
RP
df
=
+
2
dRL
RL
|ZM |2
(15)
In can be derived that the maximum is obtained when
RP
RS
2 = |Z |2
RL
M
s
RS |ZM |2
RL =
RP
Fig. 5. Wireless power link efficiency vs load resistance for various values of
coupling factor. Optimum load resistance points are marked with ’X’ symbols.
(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
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 load
resistance for various values of coupling factor. Optimum load resistance
points are marked with ’X’ symbols.
Fig. 9. Wireless power link output power vs coupling factor for optimum
load 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. 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. 10. Wireless power link efficiency vs load resistance for various values of
coupling factor. Optimum load resistance points are marked with ’X’ symbols.
For RP = 25 mΩ.
Fig. 8. Wireless power link efficiency vs coupling factor for optimum load
resistances.
Fig. 11. Wireless power link primary and secondary current ratio vs load
resistance for various values of coupling factor. Optimum load resistance
points are marked with ’X’ symbols. For RP = 25 mΩ.
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
IV. E XPERIMENTS
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.
TABLE III
W IRELESS P OWER L INK PARAMETERS
Fig. 12. Experimental setup block diagram.
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
1.4 V
+ 9.4 mΩ
(20)
Idc
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 between coils. On the plots the calculated optimum load resistance points are marked. Those values were calculated using
equation (16), where RS = 60 mΩ, and RS = Rd + 100 mΩ
to include series resistances of the rectifier, inverter, and
connecting cables.
Rd =
Fig. 13. Wireless power link used for experiment setup for 12 inches distance
between 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 II
C OUPLING C OEFFICIENTS
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
Fig. 14. Wireless power link efficiency vs load resistance for various distances
between 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. 17. Wireless power link efficiency vs distances between coils for
optimum load resistances. The actual measurement point are marked with
”x” symbols.
Fig. 15. Wireless power output power vs load resistance for various distances
between 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. 18. Wireless power link output power vs distances between coils for
optimum 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.
Fig. 16. Wireless power link optimum load resistance vs distances between
coils. 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. C ONCLUSIONS
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
R EFERENCES
[1] Y. Jang and M. M. Jovanovic, ”A Contactless Electrical Energy Transmission 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 Frequency
Repeater for Enhancing Signal Reception and Transmission in a Wireless
Charging 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 Multi
Coil 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 in
Wireless 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 Frequency 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 Technologies 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 Efficiency
Optimization and Voltage Controllability of Series Series Compensated
Inductive Power Transfer Systems,” IEEE Transactions on Power Electronics, vol. 29, no. 1, pp. 191–200, Jan. 2014.
[8] C. Florian, F. Mastri, R. P. Paganelli, D. Masotti, and A. Costanzo, ”Theoretical and Numerical Design of a Wireless Power Transmission Link
With 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 Multiplepickup 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, Sensitivity Analysis, and Controller Synthesis of Multipickup Bidirectional
Inductive 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, ”TwoDimensional Inductive Power Transfer System for Mobile Robots Using
Evenly 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 Transformer
Control,” 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 Pickups
for 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, ”Self
Tuning 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 Tuning
of 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, ”Theoretical and Numerical Design of a Wireless Power Transmission Link
With 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 Boost
Converter 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 Coupling
Amplification by Load-Modulation Switching Controller,” IEEE Transactions 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 Transformation,” 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 Coaxial
Cylindrical 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 resonant
circuit,” in Proc. IEEE International Symposium on Circuits and Systems
(ISCAS), 2014, pp. 2445–2448.