Quang-Trung Luu, Duc-Hung Tran, Bao-Huy Nguyen, Yem Vu-Van, and Cao-Minh Ta, "Design of Resonators for Coupled Magnetic Resonance-based Wireless Power Transmission System," In Proc. Vietnam Conference on Control and Automation, Da Nang, Nov. 2013.

1. VIETNAM CONFERENCE ON CONTROL AND AUTOMATION (VCCA-2013) 1
Design of Resonators for
Coupled Magnetic Resonance based
Wireless Power Transmission Systems
Quang-Trung Luu, Duc-Hung Tran, Bao-Huy Nguyen, Yem Vu-Van, and Cao-Minh Ta
Abstract—In this paper, we propose the design methodology of
resonators for the wireless power transmission system, based on
solving the optimization problem in order to achieve the required
operation frequency with the highest efficiency and suitable device
structure.
Keywords—Wireless power transfer, coupled magnetic resonance.
Symbols
Symbol Unit Name
η % Transmission efficiency
Γ Decay rate
κ Coupling coefficient
D m Coil diameter
l m Length of coil
h m Height of coil
N Number of turns
a m Cross-sectional radius of coil
H m Distance between two coils
Abbreviations
CMT Coupled Mode Theory
CMR Coupled Magnetic Resonance
FEM Finite Element Method
RFID Radio Frequency Identification
WiTricity Wireless Electricity
I. INTRODUCTION
THE history of wireless power transmission started at
the beginning of the 20th century when Nikola Tesla
implemented a system based on the principle of energy transfer
through electric field [1] . Up to now, there is a lot of efforts for
transmitting have been spent around the world, e.g. LASER,
microwave, magnetic induction, etc. But they have been still
not efficient.
In 2007, the research team WiTricity from the Massachusetts
Institute of Technology (MIT) demonstrated a wireless power
transmission system based on the principle of Couple Magnetic
Resonance (CMR) [2], [3]. This principle is basically different
from the ones previously-mentioned, and allows the high-
efficiency energy transfer in mid-range distance. With that
system, MIT team transferred a 60-watts with 40%-efficiency
Q.T. Luu, D.H. Tran, B.H. Nguyen, and C.M. Ta are with the Center for
Technology Innovation, Hanoi University of Science and Technology, Hanoi,
Vietnam. E-mail: minh.tacao@hust.edu.vn.
Y. Vu-Van is with the School of Electronics and Telecommunications,
Hanoi University of Science and Technology, Hanoi, Vietnam. E-mail:
yem.vuvan@hust.edu.vn.
at the distance of 2 meters. At present, it is recognized as the
best efficient transmission method. CMR is now applied for
various research directions such as:
• Increasing the transmission efficiency by using differ-
ent structures of resonators: Sample et al. [4] (Intel),
Casanova et al. [5], etc.;
• Improving the transmission power to kilowatts-range for
charging electric vehicles (EVs);
• Analyzing the equivalent circuit of the wireless energy
system. This is also a way to solve the system operation
without using coupled-mode theory.
The three directions show us the approach of forward
problem, that is how to calculate the operation parameters of
WPT system (resonant frequency, transmission efficiency, etc.),
by using the theory of coupled modes, FEM simulation, and
experiments.
This paper proposes the method for solving the backward
problem that is how to find out the optimal design of res-
onators, with the constraints of physical dimension, operation
frequency, and transfer distance.
II. COUPLED MAGNETIC RESONANCE BASED WIRELESS
POWER TRANSMISSION
A. Principle of Coupled Magnetic Resonance
In this study, wireless power transmission system is built
on the fundamental principle: improving energy transmission
efficiency based on the coupling of two oscillators. This
principle is called Coupled Magnetic Resonance.
Coupled magnetic resonance based wireless power transmis-
sion system is designed to achieve strongly coupled regime
of resonators. If this optimal regime of operation is satisfied,
the maximum of transferred power will be accomplished.
It is the necessary reason why the best methodology for
transferring power in mid-range (a few meters) is coupled
magnetic resonance.
In the field of electromagnetism, CMR-based wireless power
transmission can be nearly non-radiative and energy flow is
transmitted omnidirectionally, despite of the geometry of the
surrounding space, and with low losses of radiation [3].
Fig. 1 shows two helical coils as resonators. Coupling
coefficient κ is related to the transferred energy between two
resonators, and it indicates the coupling is weak or strong.

2. VIETNAM CONFERENCE ON CONTROL AND AUTOMATION (VCCA-2013) 2
κ
Resonant Coil 1 Resonant Coil 2
Magnetically Resonant
Coupling
Fig. 1. Coupled magnetic resonance between two resonators.
B. Coupled Mode Theory
Coupled Mode Theory was presented at first in the work of
Pierce [6] as a physical approach to analyze coupled resonant
systems. CMT can help us to know clearly about the principles
of energy transfer wirelessly in resonant state [2].
In this model, we have two resonant oscillators (source coil
and device coil), as shown in Fig. 1. From results of [2], [7]
we have the energy equation of WPT system:
˙αS = −j(ωS − jΓS)αS − jκαD + Fe−jωt
˙αD = −j(ωD − jΓD)αD − jκαS
(1)
where:
• α are determined so that the energy stored in object m
is |αm(t)|2
;
• ω = 2πf0: resonant frequency;
• Γ: decay rate;
• Fe−jωt
: driving terms;
• κ coupling coefficient,
and αD, αS, ωD, ωS, ΓD, ΓS are the parameters of source
and device coil, respectively. Equation (1) can be solved by
using eigenfrequencies as shown in (2).
ω1,2 = 1
2 [ωS + ωD − j(ΓS + ΓD)]
±1
2
4κ2
+ (ωS − ωD)2
− (ΓS − ΓD)2
−2j(ΓS − ΓD)(ωS − ωD)
2
.
(2)
In case of two identical resonant coils, with ωS = ωD = ω0
and ΓS = ΓD = Γ, (2) can be simply rewritten as:
ω1,2 = ω0 − jΓ ± κ. (3)
We can see that κ is related to the difference of two
eigenfrequencies in equation: κ = ω1 − ω2. Reform the
differential (1) we have (??).
III. PROPOSED DESIGN METHOD
A. Design Procedure
To solve forward problem, we can use numerical method
or FEM simulation as well as experiment in order to compute
the WPT systems operation parameters which are: resonant
frequency, transfer efficiency, etc. However, with backward
problem, since the constraints of resonator design are very
complicated, we do not have an easy formula to calculate the
characteristics of WPT system.
We propose a procedure which uses mathematical optimiza-
tion method for resonators designing. Generally, optimization
includes finding the optimal values of some objective function
f given a defined set of constraints. For example when we
want to design a pair of resonators consisting of two coils
in order to make the system working at the frequency of
f0 have desired (and possible) efficiency at the distance of
H. Naturally, the approach to solve this problem is to apply
optimization method so that we can find the maximum value of
efficiency function on condition that the system characteristics
(resonant frequency, resonator dimension, transfer distance,
etc.) satisfy the design constraints. Besides, the other way is to
find the minimum value of ∆f = |f − f0| in order to achieve
desired frequency f0 of WPT system with the constraint of
efficiency is η ≥ η0 (η0 is the minimum efficiency), and limit
of dimensions, etc.
Clearly, an optimization problem can be represented in the
following way:
• Given a function f : A → from some set A to the
real numbers;
• Sought an element x0 in A such that f(x0) ≤ f(x) for
all x in A (minimization) or such that f(x0) ≥ f(x) for
all x in A (“maximization”);
• Domain A is a subset of Euclidean space n
, often
defined by a set of constraints, equalities or inequalities
that the members of A have to satisfy. The domain A of
f is called the search space or the choice set, while the
elements of A are called candidate solutions or feasible
solutions. The function f is called objective function
or cost function. A feasible solution that optimizes the
objective function is called an optimal solution.
B. Design Resonators Using Optimization Method
To design the resonator for the WPT system, we have the
supposition and the boundary as follows:
• Desired frequency f0;
• Transfer efficiency η;
• Dimension limit of resonators;
• Transfer distance.
Objective function: f = |f − f0| or η.
Set of constraints: η or f0, constraints of dimension as well as
transfer range.
Thus, we have two approach of optimal problem, as shown in
Fig 2.

3. VIETNAM CONFERENCE ON CONTROL AND AUTOMATION (VCCA-2013) 3
Resonator inductance (L), capacitance
(C), and mutual coefficient (κ)
calculation
Compute the resonant
frequency f0
Define the set of contraints (resonators
dimension: h, D, l, a) and the
transmission distance H
Maximize η
Minimize
Δf = |f-f0|
Compute the transmission
efficiency η
Method 1 Method 2
Fig. 2. Two approaches of optimization-based resonators design.
The calculation process begins with supposing the constraint
of physical dimension of resonators (h, D, l, a) and the air gap
between two resonators (H). From the above suppositions,
the parameters of resonator can be calculated: inductance
(L), capacitance (C) and coupling coefficient (κ), respectively.
Then, we can apply two methods to solve the optimal problem:
• Method 1: Maximize the efficiency η with the desired
value, namely resonant frequency f0, the boundary of
the size and the distance transfer.
• Method 2: Minimize the value of f = |f −f0| based on
efficiency η, resonator dimension and transfer range. It
means that the result will be a resonator and achieved
efficiency η will be higher than the predetermined value
η0 and the resonant frequency is nearly f0.
The maximization and minimization algorithm are per-
formed by using function fmincon in MATLAB software.
C. Case Study
In this section, the design method of resonator will be
presented. The resonator is helical coil (single layer solenoid,
Fig. 3) with number of turns N, length l, height h, cross-
sectional radius a, and coil diameter D = 2r. Besides,
in the system consists of two resonators, we have also the
parameter H which is the distance between two resonators.
All measurement units are defined by the SI. The results are
compared with the other published in [2].
h
D = 2r
2a
N turns
Fig. 3. Physical dimentions of a helical coil.
1) Used Formulas: Used formulas in this section are derived
from the works of Kurs et al. [2], [8], Karalis [9], Grover
[10], Balanis [11], and others in [12], [13]. The characteristics
of coil (resistance R, capacitance C, and inductance L) are
calculated by using the following formulas.
In this model, the energy is lost in two ways: resistive losses
and radiative losses. Their values can be determined as: [8]
Ohmic resistance:
R0 =
µ0ω
20
l
4πa
(Ω). (5)
Radiative resistance:
Rr =
µ0
0
l
4πa
π
12
N2 ωr
c
4
+
2
3π3
ωh
c
2
, (6)
where 0 = 8.854e−12 is the dielectric constant, µ0 = 4πe−7
is the vacuum permeability and c = 3e8 is the speed of light.
Inductance: [9]
L = µ0r ln
8r
a
− 2 N2
(H). (7)
Capacitance [9] Coil capacitance C is defined from the
formula: C = 1/(ω2
L) where L is coil inductance and ω
has the value:
ω∗ =
c4
πζ0
µc/2σ
aNr3
2/7
(rad/s). (8)
where ζ0 = (µ0/ 0)1/2
is the characteristic impedance of free
space, σ is the conductivity and µc is the permeability of the
dielectric between turns of coil (in this case the dielectric is
vacuum, so µc = µ0).
Coupling coefficient: [9]
κ =
π
2
r
H
3 1
log(8r/a) − 2
. (9)

4. VIETNAM CONFERENCE ON CONTROL AND AUTOMATION (VCCA-2013) 4
TABLE I. COMPARE THE OPTIMAL VALUES WITH [2].
Parameter Kurs et al. [2] Optimal Value
N 5.25 5.3749
D 0.6 0.5953
h 0.2 0.1
η 37.46% 37.92%
2) Verify the design method: In this case study, we apply
the first procedure (Method 1) to verify the design method,
with the objective function is η.
From the previous analysis of Coupled Mode Theory in
Section II and [3], [2], we have the fundamental formula of
transfer efficiency as follow:
η =
(ΓW /ΓD)κ2
/(ΓSΓD)
(1 + ΓW /ΓD)κ2/(ΓSΓD) + (1 + ΓW /ΓD)
2 (10)
By using numerical transform, we have:
ηmax =
κ2
(
√
ΓSΓD +
√
ΓSΓD + κ2)
2 , (11)
at
ΓW =
ΓD
1 + κ2/ΓSΓD
. (12)
We use the values given in [2] to verify our proposed
method. The characteristics of coil in [2] are: turns of coil
N = 5.25, coil diameter D = 30cm, coil height h = 20cm,
cross-sectional radius of coil: a = 3mm. Applying these above
formulas, the calculated resonant frequency is 9.48 MHz,
while the theoretical value, the computed value in COMSOL
Multiphysics, and the value achieved by experiment are 10.52
MHz, 9.93 MHz and 9.90 MHz, respectively. The constraints
of resonator dimension and transfer range are given as follow:
• D ∈ [0.1, 1] (m)
• N ∈ [2, 30] (turns)
• h ∈ [0.1, 0.5] (m)
• H = 2m (fixed distance)
Reference point: [N; D; h] = [2; 0.2; 0.1]. Using MATLAB
optimization tool, the optimal values of N, D, and l are given
as:
[N; D; l] = [5.3749; 0.5953; 0.1000]
Comment: Based on the values [N; D; h] = [5.25; 0.6; 0.2] are
given by Kurs et al. in [2], the transfer efficiency is 37.46% at
the distance of 2 meters. With the optimal values calculated,
the proposed method gave the higher efficiency is 37.92% (still
at the distance of 2 meters), but the length of helical coil is
just 0.1m. It means that the size of the wireless power transfer
system can be reduced by using our proposed method. Our
result is compared with [2] in Table I.
Fig. 4. κ/Γ as the function of H at the frequency of 13.56 MHz.
3) Apply for the specific problem: After the proposed
method is verified previous section, we continue this work
with the specific problem as follow:
We know that, at high frequency, it is difficult to transfer
the large amount of power, because of the limitation of power
amplifier. In other hand, if we use high frequency to transmit
high power, it will make noises to other electric devices.
By contrast, if we use low frequency, the size of resonator
will increase. Besides, frequency has to be chosen by following
the ISM bands. From above reasons, we choose the resonant
frequency as 13.56 MHz, which is in range of RFID (Radio
Frequency Identification). The second reason is that RFID
applications also work in mid-range, thus we can apply the
WPT system based on coupled magnetic resonance to build
RFID products.
We define a set of constraints for that problem:
• a = 1.5mm (fixed cross-sectional radius)
• D ∈ [0.1, 1] (m)
• N ∈ [1, 5] (turns)
• h ∈ [0.1, 0.5] (m)
• H = 2m (fixed distance)
• Reference point: [N; D; h] = [1; 0.1; 0.1].
The optimal result is given as:
[N; D; h] = [1.4698; 0.7615; 0.1000].
Based on that result, the efficiency is 47.4% at the distance of
2 meters. The graph of κ/Γ and efficiency as the function of
distance H are shown in Fig. 4 and Fig. ??.
IV. CONCLUSION
In this paper, the complete method of designing the res-
onator is supposed with the desired parameters based on the
optimal method. By applying this method to design resonators,
the transfer efficiency can be reached maximum with prede-
termined constraints of system characteristics.

5. VIETNAM CONFERENCE ON CONTROL AND AUTOMATION (VCCA-2013) 5
REFERENCES
[1] N. Tesla, “Apparatus for Transmitting Electrical Energy,” Patent
1,119,732, 1914.
[2] A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and
M. Soljacic, “Wireless power transfer via strongly coupled magnetic
resonances.” Science, vol. 317, no. 5834, pp. 83–86, Jul. 2007.
[3] A. Karalis, J. Joannopoulos, and M. Soljaˇci´c, “Efficient wireless non-
radiative mid-range energy transfer,” Annals of Physics, vol. 323, no. 1,
pp. 34–48, Jan. 2008.
[4] A. P. Sample, D. A. Meyer, and J. R. Smith, “Analysis , Experimental
Results , and Range Adaptation of Magnetically Coupled Resonators
for Wireless Power Transfer,” IEEE Trans. Ind. Electron., vol. 58, no. 2,
pp. 544–554, 2011.
[5] J. J. Casanova, Z. N. Low, and J. Lin, “A Loosely Coupled Planar
Wireless Power System for Multiple Receivers,” IEEE Trans. Ind.
Electron., vol. 56, no. 8, pp. 3060–3068, 2009.
[6] J. R. Pierce, “Coupling of Modes of Propagation,” Journal of Applied
Physics, vol. 25, no. 2, pp. 179–183, 1954.
[7] H. A. Haus and W. Huang, “Coupled-Mode Theory,” Proc. IEEE,
vol. 79, no. 10, pp. 1505–1518, 1991.
[8] A. Kurs, “Power Transfer Through Strongly Coupled Resonances,” MSc
thesis, MIT, 2007.
[9] A. Karalis, “Mid-range Efficient Insensitive Wireless Energy-Transfer,”
PhD thesis, MIT, 2008.
[10] F. Grover, Inductance Calculation. New York: Dover Publications,
2004.
[11] C. Balanis, Antenna Theory: Analysis and Design. New Jersey: Wiley,
2005.
[12] A. J. Palermo, “Distributed capacity of single-layer,” Proc.s IRE,
vol. 22, no. 7, pp. 897–905, 1934.
[13] R. G. Medhurst, “HF Resistance and Self-Capacitance of Single-Layer
Solenoids,” Wireless Engineer, pp. 35–92, 1947.