Spherical

Mode A
n
alysis of Wireless Power Transfer
System
Yoon Goo Kim
and
Sangwook Nam
*
School of Electrical Engineering and Computer Science,
Seoul
National University
1 Gwanak

ro Gwanak

gu Seoul Korea
scika@ael.snu.ac.kr
,
snam@snu.ac.kr
*
1. INTRODU
CTION
Recently, wireless power transfer using near

field is receiving much attenti
on and is being studied
wid
ely.
Several analysis models for the
near

field
wireless power transfer have been proposed. These
include an a
nalysis by coupled mode theory
[1]
, a
n analys
is using an equivalent circuit
[2]
and a
n
analysis using filter theory
. Another analysis method involves the use of spherical modes [
3
].
In this
paper we investigate characteristics of wireless power transfer using spherical modes.
In order to use
a wireless power transfer in practice the effect of wireless power transfer on the
human body and electronic devices must be investigated. In addition, such systems
must
adhere to
EMC
regulations. Therefore,
it is needed to calculate
electromagnetic fields
near wireless power
transfer systems.
In this paper, we calculate electromagnetic fields near two coupled antennas
using
spherical modes
.
2.
ANALYSIS OF ANTENNA COUPLING USING SPHERICAL MODES
In order to derive
a
Z

parameter
of
two
coupled
antennas, we f
irst
represent
an antenna as a
scattering matrix
in terms of spherical modes
[
4
].
w v
R
b T S a
(1)
The
radiation,
receiving and scattering properties of an antenna are contained in this matrix equation.
Let
antenna
1
be
on the origin of coo
rdinate 1 (
x
1
,
y
1
,
z
1
axis
) and antenna
2 be
on the origin of
coordinate 3(
x
3
,
y
3
,
z
3
axis
), as shown in Fig. 1.
C
oordinate 2 (
x
2
,
y
2
,
z
2
axis
) is obtained
by translating
coordinate 1
and coordinate 3 is obtained by rotating coordinate 2. It is assumed tha
t two spheres
enclosing each antenna do not overlap.
The coupling of two antennas can be considered as the
cascading of a transmitting antenna network, a space network, and a receiving antenna network, as
shown in Fig. 2.
Here, the antenna networks are exp
ressed as
the
scattering matrix and the space
network is expressed
as
the
impedance matrix
.
In Fig.
2, left side p
ort
s
in the space network
denote
the mode ports for coordinate 1 and
right side ports
are
the mode ports of coordinate 3.
Let the
characterist
ic impedance of antennas be Z
0
and the characteristic impedance of the space network be 1.
For simplicity, we assume that antennas are canonical minimum scattering
(CMS)
antennas that
generate only fundamental modes.
The canonical minimum scattering antenn
a is one which does not
scatter electromagnetic fields when its local port is open

circuited [
5
].
Antennas that are small
compared with wavelength can be modeled as a minimum scattering antenna [
6
].
The impedance
matrix of the space network was derived in
[3] using the addition theorem.
The Z

parameter between
two identical CMS antennas generating only fundamental mode is presented in [
3
]. Once the Z

parameter is
known
, the maximum power transfer efficiency and optimum load impedance can be
determined by th
e formula in [
3
]. According to [
3
], the radiation efficiency of an antenna is
a
key
parameter in wireless power transfer. Fig
.
3 shows maximum power transfer efficiency of the antennas
generating
only
TE
01
(TM
01
) mode for different radiation efficiencies.
It can be seen in Fig.
3
that the
maximum power transfer efficiency increases
with
the
radiation efficiency
.
3.
ELECTROMAGNETIC FIELDS NEAR
WIRELESS POWER TRANSFER SYSTEMS
If we know the voltages and currents of
the
ports in the space network, we can calc
ulate the
electromagnetic fields near wireless power transfer systems and far

field that wireless power transfer
systems radiate.
The region where electromagnetic fields are calculated is divided into two regions
and the method for calculating fields in
ea
ch region is different
(Fig. 4).
This is because
the formulas
of the addition theorem are different in the two regions. In region 1 (blue region), both incoming and
outgoing spherical waves exist. In region 2, only outgoing spherical waves exist. If we kno
w the
voltages and currents of ports
of
the space network, we can determine the spherical mode coefficients
in both regions. From the spherical mode coefficients, we can calculate electromagnetic fields near
wireless power transfer system and far

field tha
t the system radiates.
The
current
s of the space
network
are as follow:
0
1 1 1
1
1
Z
i
I T
0
2 2 2
2
1
Z
i
I T
where
T
j
is
the
modal transmitting pattern (
T
in (1)
) of antenna j
(j=1,2) and
i
j
is
the
current at the
local port of antenna
j (j=1,2).
Here,
I
1
is currents at the mode ports of
coordinate 1 and
I
2
is currents
at the
mode
ports
of
coordinate 2
.
The v
oltages can be determined from the impedance matrix of the
space
network
and the currents.
To verify the theory, we simulated
elect
romagnetic fields near
two coupled helix coils with FEKO.
For
helix
1
, the radius is 30cm, the height is 20cm, and
the
number of turn is
4.25
. For
helix
2
, the
radius is 20cm, the height is 20cm, and
the
number of turn is 6
.5
.
All the antennas are made out
of
copper wire with a diameter of
5
mm.
Two helices are
3m
apart (Fig.
5
).
We attached
the
load of 3

j2
on the port of helix 2 and excite 10V at the port of helix 1.
We calculated electric fields on the sphere
whose
radius is 2m
at 13.56MHz
. Fig.
6
shows
r component of
electric fields against
when
is 0.
R
EFERENCES
[1]
A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, M. Soljacic, “Wireless power
transer via strongly coupled magnetic resonances,”
Science
, vol. 317, no. 5834, pp. 83

86, Ju
l. 2007
.
[2]
A. P. Sample, D. A. Meyer, and J. R. Smith, “Analysis, experimental result, and range adaptation
of magnetically coupled resonators for wireless power transfer,”
IEEE Trans. Industrial Electronics
,
vol. 58, no. 2, pp. 544

554, Feb. 2011
[
3
]
J.
Lee and S. Nam “Fundamental aspects of near

field coupling small antennas for wireless power
transfer,”
IEEE Trans. Antennas Propagat
., vol. 58, no. 11, pp. 3442

3449, Nov. 2010.
[
4
]
J. E. Hansen,
Spherical Near

field Antenna Measurements
, London: Peter P
eregrinus LTd., 1988.
[5
]
W. K. Kahn and H. Kurss, “Minimum

scattering antennas,”
IEEE Trans. Antennas Propagat
., vol.
13, no.5, pp. 671

675, Sep. 1965.
[6
]
P. G. Rogers, “Application of the minimum scattering antenna theory to mismatched antennas,”
IEEE T
rans. Antennas Propagat
., vol. 34, no. 10, pp. 1223

1228, Oct. 1986.
Fig. 1. Coordinate systems and antennas
Fig. 2. Network representation of two coupled antennas
Fig. 3. Maximum power transfer efficiencies of antennas for different radiation efficiencies at
= 0.
Fig. 4.
Two regions where electromagnetic fields are calculated
.
Fig. 5.
Helix a
ntenna conf
iguration and
the sphere where electromagnetic fields are calculated.
Fig. 6.
r
component of electric fields against
. Frequency=13.56MHz. r = 2.
= 0. (a) real part. (b) imaginary part.
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