wireless power transfer

536 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 57, NO. 7, JULY 2010
A Study of Loosely Coupled Coils
for Wireless Power Transfer
Chih-Jung Chen, Tah-Hsiung Chu, Member, IEEE, Chih-Lung Lin, and Zeui-Chown Jou
Abstract—Nonradiative wireless power transfer using magnet-
ically coupled coils is studied in order to transfer a predeter-
mined amount of power at the maximum efficiency. Accordingly,
a conceptual wireless power transfer system and a tuning method
are presented. Such a study is essential for effectively exploiting
the inherent ability of a given pair of coupled coils. With the
equations for inductance and resistance calculations, the system
performance is evaluated and verified with well-known experi-
mental results and circuit simulations.
Index Terms—Contactless power, inductive power transfer,
loosely coupled coils, wireless power.
I. INTRODUCTION
MAGNETICALLY coupled coils have been widely used
for a variety of applications requiring contactless or
wireless power transfer (WPT), such as transcutaneous power
transmission for biomedical devices [1]–[3], radio-frequency
identification [4], and so on [5]–[11]. Unlike the coils of a
transformer that are wound around a magnetic core to attain
tight coupling, the coils for WPT are usually loosely coupled
due to the absence of a common magnetic core to confine and
guide most of the magnetic flux.
Fig. 1 illustrates a pair of coupled coils L1 and L2 with
quality factors Q1 and Q2 and coupling coefficient K12. The
coupled coils are driven by a sinusoidal voltage source with
an rms amplitude of Vs and an angular frequency ω while
terminated in a load with ZL = RL + jXL. For the lossless
case, the quality factors are assumed to be infinite, and the real
power delivered to the load ZL is given by
Re{POUT} =
K2
12L2RL
L1
1
[ωL2 (1 −K2
12)+XL]
2
+R2
L
|VS|2
.
(1)
As the coupling is tight, i.e., K12 ≈ 1, (1) reduces to
Re{POUT}|K12≈1 ≈ (L2/L1) RL/ X2
L + R2
L |VS|2
. (2)
Manuscript received September 22, 2009; revised January 20, 2010;
accepted March 2, 2010. Date of publication June 1, 2010; date of current
version July 16, 2010. This work was supported in part by the National
Science Council of Taiwan under Grant NSC 97-2221-E-002-057-MY2 and
in part by Darfon Electronics Corporation. This paper was recommended by
Associate Editor D. Heo.
C.-J. Chen is with the Department of Communications Navigation and Con-
trol Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
(e-mail: cjchen@mail.ntou.edu.tw).
T.-H. Chu is with the Department of Electrical Engineering, National Taiwan
University, Taipei 106, Taiwan (e-mail: thc@ntu.edu.tw).
C.-L. Lin is with the Department of Electrical Engineering, National Cheng
Kung University, Tainan 701, Taiwan.
Z.-C. Jou is with the Department of Advanced Technology, Darfon Electron-
ics Corporation, Taoyuan 33347, Taiwan.
Digital Object Identifier 10.1109/TCSII.2010.2048403
Fig. 1. Schematic of a pair of coupled coils.
According to (2), the tightly coupled coils, also known as a
transformer, can, therefore, be used for voltage level conversion
and a galvanic separation.
By contrast, for the loosely coupled coils, i.e.,K12 1, (1)
can be simplified as
Re{POUT}|K12 1 ≈
K2
12L2
L1
RL
(ωL2 + XL)2 + R2
L
|VS|2
. (3)
It shows that the power transfer capability of loosely coupled
coils is proportional to the second power of the coupling
coefficient.
To enhance the power transfer capability, the loosely coupled
coils generally need to be compensated capacitively to obtain
the current magnification resulting from the resonance effect
[12]. The power transfer capability enhancement via current
magnification becomes evident by rewriting (1) as
Re{POUT} = K2
12ω2
L1L2/RL(ωL2 + XL)2
|I1|2
. (4)
Capacitive compensation is crucial to the implementations of
those loosely coupled applications [1]–[11].
In addition to the power transfer capability, the operating
efficiency of the coupled coils is of concern to many applica-
tions. As long as the lossless coils are considered, the maximum
power transfer efficiency of the coupled coils is expected to be
100%. However, coils have electrical losses.
Without being compensated appropriately for minimizing the
power dissipation, the operating distance between a given pair
of coupled coils can hardly be increased. Nonetheless, there
appears to be relatively little systematic research that tackles
the above two issues as an integral part of the design of a WPT
system.
In this brief, the power transfer capability and efficiency of a
given pair of coupled coils are simultaneously studied. Based
on the capacitive compensation scheme presented in [8], a
conceptual WPT system is presented. Explicit design equations
for tuning the WPT system are derived for one to transfer a
predetermined amount of power via the coupled coils at the
maximum efficiency. To verify the feasibility of the conceptual
1549-7747/$26.00 © 2010 IEEE

CHEN et al.: STUDY OF LOOSELY COUPLED COILS FOR WIRELESS POWER TRANSFER 537
WPT system, the characteristics of the system evaluated with
inductance and resistance calculation equations are verified
with the experimental results in [8] and circuit simulations.
II. FORMULATION
A. Power Transfer Efficiency
By taking into account the quality factors in Fig. 1, the power
transfer efficiency η or Re{POUT}/Re{PIN} is acquired as (5),
shown at the bottom of the page.
Judging by (5), one can realize that the power transfer
efficiency is dependent on the load ZL. As a result, the optimum
load for maximum power transfer efficiency can be found by
letting
∂η/∂RL = 0
∂η/∂XL = 0.
(6)
The optimum load and the maximum power transfer efficiency
are then given by
Zopt
L = (ωL2
√
1 + Δ)/Q2 − jωL2 (7)
and [13]
ηmax
= Δ/(1 +
√
1 + Δ)2
(8)
where Δ = K2
12Q1Q2. Equations (7) and (8) indicate that the
coupled coils need to be terminated in the optimum load to
achieve the maximum efficiency, which can be higher than
17% as Δ > 1. The coupled coils are said to be operated in a
strongly coupled regime when Δ > 1 [8], [9]. According to the
definitions, it should be noted that the loosely coupled coils, i.e.,
K12 1, can still be operated in a strongly coupled regime,
i.e., Δ > 1, if the quality factors, i.e., Q1 and Q2, are high
enough.
To illustrate the maximum efficiency at which a given pair
of coupled coils can be operated, we consider two identical
circular coils having a radius of r and a cross-sectional radius
of a. The self-inductance is known as
L = rμ0 [ln(8r/a) − 2] . (9)
When these two coils are aligned coaxially and separated
by a distance d, the coupling coefficient of the coils can be
formulated as [14]
K12 =
2r
[ln(8r/a) − 2]
√
d2 + 4r2
×
π/2
0
(2 sin2
φ − 1)dφ
1 − 4r2 sin2
φ (d2 + 4r2)
. (10)
For an exemplary case that r = 100a and Q1 = Q2 = Q,
the maximum efficiency for different normalized distances, i.e.,
Fig. 2. Calculated maximum power transfer efficiency of a pair of circular
coils with quality factor Q for different normalized distances.
d/r, can be computed with (8) and (10). As shown in Fig. 2,
depending on the quality factor, the coupled coils can be oper-
ated in a strongly coupled regime, i.e., Δ > 1, over a distance
up to certain times the coil radius. For instance, the boundaries
of the strongly coupled regime are about d = 3.4r and 5.7r for
Q = 150 and 600, respectively.
In practice, an output matching network is required to trans-
form the load ZL to the optimum load given by (7) to achieve
the maximum transmission efficiency given by (8).
B. Power Transfer Capability
The input impedance ZIN of a pair of coupled coils termi-
nated in the optimum load can be derived as
ZIN|ZL=Zopt
L
= (ωL1/Q1)(1 + ηmax
+ jQ1). (11)
In other words, when terminated in the optimum load for a
specific coupling coefficient K12, the input power is determined
and so is the output power.
For practical applications, the input matching network can be
used to transform ZIN to an arbitrary impedance ZIN. Conse-
quently, the output power can be adjusted to the predetermined
value according to
Re{POUT}|ZL=Zopt
L
=(ηmax
√
1+Δ)(ωL1/Q1)|VS/ZIN|
2
.
(12)
III. CONCEPTUAL WPT SYSTEM
Fig. 3 depicts a generic system for contactless power transfer
or WPT. While the output matching network transforms the
load RL to Zopt
L to operate the system at the maximum effi-
ciency, the input matching network plays a role to set the input
impedance ZIN, which determines the power transfer capability
or the amount of power extracted from the driving circuit [15].
Accordingly, one can transfer a predetermined amount of power
at the maximum efficiency. The flexibility and the practicability
of the matching networks are then the key consideration of the
WPT system design.
η =
ωK2
12L2RL
ωK2
12L2RL + ω2K2
12L2
2 Q2 + ω2L2
2 (1/Q1Q2
2 + 1/Q1) + R2
L/Q1 + X2
L/Q1 + 2ωL2RL/Q1Q2 + 2XLωL2/Q1
(5)

Fig. 3. Generic system for contactless or wireless power transfer.
Fig. 4. Schematic of the conceptual WPT system.
A. Matching Network Topology
Recently, a WPT scheme has shown promising results for
midrange applications [8]. Instead of using lumped inductors
and capacitors, the system consists of two self-resonant helical
coils, i.e., distributed resonators.
Since distributed resonators behave much like lumped ones
in the vicinity of the resonant frequencies [16], [17], one can
model and implement the power transfer scheme in [8] with
lumped components, as shown in Fig. 4. For loosely coupled
applications, i.e., K12 1, the parasitic couplings, including
K23, K14, and K34, are negligible. The approximation of ignor-
ing these parasitic couplings is justified by circuit simulations
in Section IV-C.
It is important to understand that the lump-element circuit
represents a good approximation of the distributed one over a
limited frequency range. Such a circuit theory approximation
is essential for the analysis and the synthesis of a large class
of microwave filters that are typically composed of distributed
resonators [17].
Comparing Fig. 3 with Fig. 4, one can see that, in the con-
ceptual WPT system, the matching networks are formed by C1,
L3, C2, and L4 and the couplings between L1, L3 and L2, L4.
With the introduction of the lump-element model in Fig. 4,
the power transfer scheme in [8] can be analyzed and explained
by circuit theory, which leads to a considerable mathematical
simplification compared to the coupled-mode theory in [8] or
the method of moment in [18]. In addition, by implementing
the scheme with lumped elements, including coils and high-Q
variable capacitors, it becomes more tunable and compact.
In Fig. 4, Q1 and Q2 encapsulate the intrinsic losses
of both the capacitor and the inductor so that Q1,2 =
QL1,2
QC1,2
/(QL1,2
+ QC1,2
). Moreover, due to the intrinsic
losses of L3 and L4, the maximum attainable efficiency of the
WPT system is somewhat lower than that calculated from (8).
B. Tuning Method
As stated in [8], the maximum efficiency can be achieved
by varying K24 and slightly retuning the resonator consisting
of L2 and C2. However, the explicit equations for tuning input
and output matching networks are not addressed in [8].
Fig. 5. Equivalent circuit of the conceptual WPT system.
Fig. 6. Comparison of the scheme in [8] and the conceptual WPT system.
In order to acquire the equations for tuning the coupling
coefficients K13 and K24 and the natural angular frequencies
ω1 and ω2, i.e., 1/
√
L1C2 and 1/
√
L2C2, the circuitry in
Fig. 4 is rearranged as shown in Fig. 5. By equating ZL to the
optimum load given by (7), one can readily deduce that
K24 = (1 + Δ)1/4
(ω2L2
4 + R 2
L) Q2ωL4RL (13)
ω2 = ω 1 − ωL4 Q2RL
√
1 + Δ (14)
where RL = RL + ωL4/Q4. Next, let ZIN be a pure resistance
RIN under the circumstances of (13) and (14). One then finds
K13 =
ω2L2
3 + R2
IN
Q1RINωL3
×
(1 + Δ) Δ2 + 8Δ + 8 + 4(Δ + 2)
√
1 + Δ
4Δ2 + 12Δ + 8 + (Δ2 + 8Δ + 8)
√
1 + Δ
(15)
ω1 = ω 1 − (ωL3/Q1RIN)
√
1 + Δ (16)
where RIN = RIN − ωL3/Q3.
IV. VERIFICATION
In Fig. 6, the scheme in [8] is depicted along with the
conceptual WPT system consisting of four identical single-loop
coils L1–L4 and two capacitors C1 and C2. Let the operating
frequency f0, coil radius r, and cross-sectional radius of copper
wire a be 10 MHz, 30 cm, and 3 mm, respectively, which are
the same values as those in [8]. The characteristics of the coils

CHEN et al.: STUDY OF LOOSELY COUPLED COILS FOR WIRELESS POWER TRANSFER 539
Fig. 7. Comparison of the coupling coefficients.
Fig. 8. Comparison of the power transfer efficiency.
can, therefore, be evaluated and verified with the experimental
results in [8].
The coil inductance calculated with (9) is 1.77 μH. As the
coil is electrically small, i.e., 2πr/λ 1, the radiation loss is
negligible compared with the ohmic loss. The ohmic skin effect
resistance is evaluated as 0.041 Ω with
RS = (r/2a) μ0ω/2σ (17)
where σ = 5.7 × 107
(m∗
Ω)−1
. Consequently, the calculated
quality factor of the coils QL is 2712 at 10 MHz.
A. Coupling Coefficients and Power Transfer Efficiency
We first check the coupling coefficients between the coils
with the experimental results in [8]. The coupling coefficients of
the coils are computed with (10). As shown in Fig. 6, the helical
resonators have a length of 20 cm. The distance between D and
D is, therefore, 40 cm. To interpret the couplings at a single
distance, an effective distance between D and D is defined as
D = D − 30. As depicted in Fig. 7, the calculated coupling
coefficients are in line with the experimental results in [8], when
interpreted with the effective distance. Note that the coupling
of modes κ in [8] is related to the coupling coefficient K12 by
κ = ωK12/2 [8]. The unit of κ is misprinted in [8]. It should be
107
/s, as can be found in the supporting online material of [8].
Suppose C1 and C2 lower the quality factors Q1 and Q2 to
1000 for comparison with the measured value of Q = 950 ± 50
in [8]. The maximum efficiency of the conceptual WPT system
for different K12, i.e., D , is calculated with (8). As shown
in Fig. 8, the calculated results are also in agreement with the
experimental results in [8].
Fig. 9. Calculated distances between L1, L3 and L2, L4 for tuning the
conceptual WPT system.
Fig. 10. Calculated capacitances for tuning the conceptual WPT system.
B. Tuning Method
With the design equations, the WPT system can be tuned
accordingly to transfer a predetermined amount of power at the
maximum efficiency.
For example, suppose that one wants to transfer 60 W to a
200-Ω load with the WPT system driven by a Class E power
amplifier (PA). The power extracted from the Class E PA can
be estimated as [15]
PIN = 8V 2
DD/ (π2
+ 4)RIN (18)
where VDD is the direct-current voltage at which the PA oper-
ates. For VDD = 100 V, K13, K24, ω1, and ω2 are calculated
for various K12 or D and then changed into d13, d24, C1, and
C2 as presented in Figs. 9 and 10.
C. Circuit Simulations
With the Advanced Design System (ADS) of Agilent
Technologies to take into account the parasitic couplings,
simulated results for D = 75 and 150 cm are presented to
verify the conceptual WPT system. In the circuit simulations,
the WPT system is driven by a sinusoidal voltage source with
an rms amplitude of 75.95 V, which is calculated from (18)
with VDD = 100.
When D = 75 and 150 cm, K12 are 0.0145 and 0.0024,
respectively. The evaluated tuning parameters, i.e., K13, K24,
C1, and C2, and the parasitic couplings, i.e., K23, K14, and
K34, are tabulated in Table I. The design goals tabulated in
Table II are calculated with the above-derived equations that
ignore the parasitic couplings.

TABLE I
EVALUATED TUNING PARAMETERS AND PARASITIC COUPLINGS
TABLE II
DESIGN GOALS
Fig. 11. Simulated frequency responses for (a) power transfer efficiency and
output power, and (b) input impedance.
TABLE III
ADS SIMULATION RESULTS AT 10 MHz
Fig. 11 depicts the simulated frequency responses of the
WPT system. The simulated results at 10 MHz are tabulated
in Table III for comparison with the design goals in Table II.
Furthermore, for D = 150 cm, the simulated results at
10 MHz show that the currents flowing in the WPT system are
|I3| = 1.9 A, |I1| = 23.1 A, |I2| = 14.6 A, and |I4| = 0.552 A.
In comparison with the current flowing in L3, the current
magnification in L1 and L2 is apparent.
The conceptual WPT system is verified by observing Fig. 11
and comparing the simulated results in Table III with the design
goals in Table II.
V. CONCLUSION
A conceptual WPT system and the tuning method have been
presented for one to transfer a predetermined amount of power
via the loosely coupled coils at the maximum efficiency. The
tuning method can substantially increase the operating distance,
as well as the utility, of a WPT system.
The conceptual WPT system is implemented with two cou-
pled resonators consisting of lumped coils and high-Q vari-
able capacitors. Compared to the distributed resonators in
[8], the conceptual WPT system is more compact. Further-
more, it might be dynamically tuned for different operating
distances.
According to the analysis, the natural frequencies of the res-
onators are not exactly identical. The coupled resonators with
identical natural frequency or the double-tuned transformer is
known to have two resonant frequencies, i.e., even and odd
modes [17]. The natural frequencies need to be slightly dis-
placed from each other to avoid the power transfer bifurcation
phenomenon [11].
REFERENCES
[1] G. Wang, W. Liu, M. Sivaprakasam, and G. A. Kendir, “Design and
analysis of an adaptive transcutaneous power telemetry for biomedical
implants,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 10,
pp. 2109–2117, Oct. 2005.
[2] P. Li and R. Bashirullah, “A wireless power interface for rechargeable
battery operated medical implants,” IEEE Trans. Circuits Syst. II, Exp.
Briefs, vol. 54, no. 10, pp. 912–916, Oct. 2007.
[3] M. Akay, Wiley Encyclopedia of Biomedical Engineering. Hoboken, NJ:
Wiley, 2006.
[4] K. Finkenzeller, RFID Handbook: Fundaments and Applications in
Contactless Smart Cards and Identification. Hoboken, NJ: Wiley, 2003.
[5] K. W. Klontz, D. M. Divan, D. W. Novotny, and R. D. Lorenz,
“Contactless power delivery system for mining applications,” IEEE
Trans. Ind. Appl., vol. 31, no. 1, pp. 27–35, Jan. 1995.
[6] J. de Boeij, E. Lomonova, J. L. Duarte, and A. J. A. Vandenput,
“Contactless power supply for moving sensors and actuators in high-
precision mechatronic systems with long-stroke power transfer capability
in x-y plane,” Sens. Actuators A, Phys., vol. 148, no. 1, pp. 319–328,
Nov. 2008.
[7] J. L. Villa, J. Sallan, A. Llombart, and J. F. Sanz, “Design of a high
frequency inductively coupled power transfer system for electric vehicle
battery charge,” Appl. Energy, vol. 86, no. 3, pp. 355–363, Mar. 2009.
[8] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and
M. Soljacic, “Wireless power transfer via strongly coupled magnetic res-
onances,” Sci. Express, vol. 317, no. 5834, pp. 83–86, Jul. 2007.
[9] F. Segura-Quijano, J. García-Cantón, J. Sacristán, T. Osés, and A. Baldi,
“Wireless powering of single-chip systems with integrated coil and exter-
nal wire-loop resonator,” Appl. Phys. Lett., vol. 92, no. 7, pp. 074 102-1–
074 102-3, Feb. 2008.
[10] X. Liu and S. Y. Hui, “Optimal design of a hybrid winding structure
for planar contactless battery charging platform,” IEEE Trans. Power
Electron., vol. 23, no. 1, pp. 455–463, Jan. 2008.
[11] C. S. Wang and G. A. Govic, “Power transfer capability and bifurcation
phenomena of loosely coupled inductive power transfer systems,” IEEE
Trans. Ind. Appl., vol. 51, no. 1, pp. 148–157, Feb. 2004.
[12] J. Bird, Electrical Circuit Theory and Technology. Burlington, MA:
Newnes, 2003.
[13] W. H. Ko, S. P. Liang, and C. D. F. Fung, “Design of radio-frequency
powered coils for implant instruments,” Med. Biol. Eng. Comput., vol. 15,
no. 6, pp. 634–640, Nov. 1977.
[14] S. Ramo, J. R. Whinnery, and T. V. Duzer, Field and Waves in Communi-
cation Electronics. Hoboken, NJ: Wiley, 1994.
[15] M. K. Kazimierczuk and D. Czarkowski, Resonant Power Converters.
New York: Wiley-Interscience, 1995.
[16] D. Kajfez and E. J. Hwan, “Q-factor measurement with network
analyzer,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 7,
pp. 666–670, Jul. 1984.
[17] R. J. Cameron, R. Mansour, and C. M. Kudsia, Microwave Filters
for Communication Systems Fundamentals, Design and Applications.
New York: Wiley-Interscience, 2007.
[18] X. C. Wei and E. P. Li, “Simulation and Experimental comparison
of different coupling mechanisms for the wireless electricity transfer,”
J. Electromagn. Waves Appl., vol. 23, no. 7, pp. 925–934, 2009.

wireless power transfer

More Related Content

What's hot

Similar to wireless power transfer

More from Mimar Sinan Saraç

Recently uploaded

wireless power transfer