Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
8,
No.
3,
June
2018,
pp.
1331
–
1335
ISSN:
2088-8708
1331
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
The
Interaction
between
Load
Cir
cuits
and
Decision
of
Fr
equency
f
or
Efficient
W
ir
eless
P
o
wer
T
ransfer
Kazuya
Y
amaguchi
Department
of
Control
Engineering,
National
Institute
of
T
echnology
,
Nara
Colle
ge,
Japan
Article
Inf
o
Article
history:
Recei
v
ed:
Dec
14,
2017
Re
vised:
Feb
6,
2018
Accepted:
Feb
22,
2018
K
eyw
ord:
wireless
po
wer
transfer
coupling
coef
ficient
state
space
representation
ABSTRA
CT
This
paper
deri
v
es
an
e
xpression
of
ef
ficienc
y
of
wireless
po
wer
transfer
on
a
situa-
tion
that
there
are
tw
o
de
vices
to
w
ards
one
A
C
po
wer
supply
.
The
interaction
between
a
po
wer
supply
and
load
is
paid
attention
on
a
con
v
entional
wireless
po
wer
transfer
system,
in
contrast,
the
interaction
between
loads
must
be
tak
en
account
of
on
the
sit-
uation
too.
This
is
attrib
uted
to
a
possibility
that
a
load
disturbs
the
ener
gy
transmitted
from
a
po
wer
supply
to
another
load.
Moreo
v
er
each
load
needs
dif
ferent
frequenc
y
of
po
wer
supply
for
the
ideal
transfer
since
the
y
ha
v
e
dif
ferent
natural
frequencies
on
man
y
situations.
This
paper
models
a
circumstance
that
there
are
a
po
wer
supply
and
tw
o
loads
with
a
state
space
equation,
and
proposes
ho
w
to
decide
a
frequenc
y
of
po
wer
supply
to
realize
ef
ficient
transfer
for
each
load.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Kazuya
Y
amaguchi
Department
of
Control
Engineering,
National
Institute
of
T
echnology
,
Nara
Colle
ge
22
Y
ata-cho,
Y
amatok
oriyama,
Nara,
J
AP
AN
Email:
k-yamaguchi@ctrl.nara-k.ac.jp
1.
INTR
ODUCTION
W
ireless
Po
wer
T
ransfer
(WPT)
system
is
focuse
d
on
the
areas
of
ener
gy
,
information,
com
munica-
tion,
and
control,
etc.
In
WPT
,
electric
po
wer
is
transferred
on
electromagnetic
field
by
coils
based
on
some
electromagnetic
theory
.
The
basic
theory
of
transfer
electromagneticall
y
w
as
establ
ished
by
Nik
ola
T
esla[1],
and
WPT
is
researched
acti
v
ely
from
the
study
of
WPT
on
strongly
coupled
magnetic
resonance[2].
The
v
arious
methods
are
attempted
according
to
some
papers
to
impro
v
e
ef
ficienc
y
and
po
wer
,
and
to
reduce
ener
gy
loss
of
WPT
.
[3]
focuses
on
circuit
topology
,
and
it
mentions
that
the
combination
of
series
or
parallel
of
circuit
af
fects
the
intensity
of
po
wer
.
A
phase
dif
ference
between
the
tw
o
transmission
coils
is
controlled
to
increase
ef
ficienc
y[4].
The
se
v
eral
coils
are
utilized
for
transmission
of
po
wer
to
enlar
ge
po
wer
,
ef
ficienc
y
,
and
the
distance
between
a
po
wer
supply
and
loads[5][6].
Moreo
v
er
WPT
is
applied
for
the
v
arious
de
vices,
places,
and
situations.
In
the
medical
area,
a
capsule
endoscop
y
is
char
ged
from
the
outside
of
human
body[7],
and
an
ef
fect
to
human
body
is
in
v
estig
ated
because
the
electromagnetic
w
a
v
e
is
generated
on
transmission
wirelessly[8].
In
the
area
of
semiconductor
,
an
application
of
WPT
for
the
semiconductors
is
e
xamined
on
lo
w
po
wer
and
high
frequenc
y[9].
In
terms
of
the
long
distance
transmission,
char
ge
from
a
po
wer
supply
on
the
ground
to
a
balloon
at
high
altitude
is
re
vie
wed
with
electric
propulsion[10].
As
a
merit
of
WPT
,
char
ging
se
v
eral
de
vices
simultaneously
by
closing
the
m
near
a
po
wer
supply
is
listed
since
WPT
system
does
not
need
an
y
connection
of
wire.
Ho
we
v
er
,
when
the
de
vice
A
is
char
ged
by
a
po
wer
supply
,
the
other
de
vice
B
which
is
put
near
A
may
disturb
e
xchanging
ener
gy
between
the
po
wer
supply
and
A.
Therefore
the
interaction
between
de
vices
must
be
in
v
estig
ated
in
addition
to
the
transmission
of
ener
gy
from
po
wer
supply
to
loads[11].
In
this
paper
,
the
e
xchange
of
ener
gy
between
a
po
wer
supply
and
tw
o
loads
is
analyzed
with
a
m
athe-
matical
model
bas
ed
on
modern
control
theory
.
Especially
the
coupling
coef
ficients
between
a
transmitting
coil
and
de
vice
coil,
and
between
de
vice
coils
are
focused
for
finding
transportation
ef
ficienc
y
.
Finally
,
the
ef
fect
of
J
ournal
Homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v8i3.pp1331-1335
Evaluation Warning : The document was created with Spire.PDF for Python.
1332
ISSN:
2088-8708
a
load
circuit
which
is
not
aimed
to
transfer
ener
gy
is
re
v
ealed,
and
the
optimal
angular
frequenc
y
to
maximize
ef
ficienc
y
is
obtained.
2.
DESIGN
OF
WPT
CIRCUIT
AND
DERIV
A
TION
OF
MA
THEMA
TICAL
MODEL
At
first,
a
circuit
treated
in
this
paper
is
sho
wn
as
belo
w
.
Figure
1.
A
WPT
circuit
which
has
tw
o
loads
In
Figure
1,
the
left
side
is
a
transmitting
circuit,
and
the
circuits
on
the
right
side
are
recei
ving
ci
rcuits,
also
tw
o
recei
ving
circuits
are
distinguished
as
recei
ving
circuit
1
and
recei
ving
circuit
2
each
other
.
These
three
circuits
are
connected
electromagnetically
with
three
coils
L
0
,
L
1
,
and
L
2
.
R
0
is
a
parasitic
resistance
in
the
transmitting
circuit,
and
R
1
and
R
2
are
the
load
resistances
on
each
recei
ving
circuit,
and
C
0
,
C
1
,
and
C
2
are
parasitic
capacitances
on
each
circuit.
k
01
,
k
02
,
and
k
12
are
the
coupling
coef
ficients
between
each
coil.
In
this
section,
the
optimal
angular
frequenc
y
of
po
wer
supply
which
maximizes
ef
ficienc
y
is
deri
v
ed.
Then
a
mathematical
model
is
found
to
deri
v
e
the
e
xpressions
of
po
wer
and
ef
ficienc
y
in
the
follo
wing
[12].
_
x
=
Ax
+
B
u;
x
=
v
0
v
1
v
2
i
0
i
1
i
2
T
(1)
A
=
1
2
6
6
6
6
6
6
4
0
0
0
C
0
0
0
0
0
0
0
C
1
0
0
0
0
0
0
C
2
12
2
1
12
R
0
2
R
1
1
R
2
2
02
0
2
R
0
02
R
1
0
R
2
1
0
01
1
R
0
0
R
1
01
R
2
3
7
7
7
7
7
7
5
;
B
=
1
2
6
6
6
6
6
6
4
0
0
0
12
2
1
3
7
7
7
7
7
7
5
=
L
0
L
1
L
2
(
k
2
01
+
k
2
02
+
k
2
12
2
k
01
k
02
k
12
1)
01
=
L
0
L
1
(1
k
2
01
)
02
=
L
0
L
2
(1
k
2
02
)
12
=
L
1
L
2
(1
k
2
12
)
0
=
L
0
p
L
1
L
2
(
k
01
k
02
k
12
)
1
=
L
1
p
L
2
L
0
(
k
12
k
01
k
02
)
2
=
L
2
p
L
0
L
1
(
k
02
k
12
k
01
)
:
This
model
is
a
state
space
equation
to
obtain
the
state
solutions,
and
it
has
the
state
v
ariables
v
0
;
v
1
;
v
2
;
i
0
;
i
1
;
i
2
,
which
are
v
oltages
and
currents
at
each
capacitor
and
coil.
The
abo
v
e
equation
is
solv
ed
with
re
g
ard
to
these
v
ariables
to
find
the
state
solutions,
po
wers,
and
ef
ficiencies.
IJECE
V
ol.
8,
No.
3,
June
2018:
1331
–
1335
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1333
3.
FINDING
OF
EFFICIENCY
FOR
OBT
AING
THE
OPTIMAL
ANGULAR
FREQ
UENCY
3.1.
Deri
v
ation
of
the
expr
essions
of
efficiencies
By
the
mathematical
m
odel,
a
mathematical
e
xpression
of
ef
ficienc
y
is
obtained.
Ef
ficienc
y
is
defined
as
the
ratio
of
the
load
po
wer
and
the
input
po
wer
,
and
tw
o
ef
ficiencies
must
be
distinguished
because
there
are
tw
o
loads
in
Figure
1.
In
this
paper
,
the
tw
o
ef
ficiencies
1
and
2
are
written
as
belo
w
.
1
=
P
1
P
0
2
=
P
2
P
0
:
(2)
Where
P
0
is
the
po
wer
of
po
wer
supply
,
and
P
1
and
P
2
are
the
load
po
wers
of
R
1
and
R
2
each
other
.
From
the
equation
(1),
1
and
2
are
found
theoretically
,
ho
we
v
er
it
is
dif
ficult
to
describe
all
of
the
e
xpression
of
1
and
2
,
and
therefore
the
second
order
approximation
as
re
g
ards
k
01
,
k
02
,
and
k
12
is
applied
because
these
v
alues
are
less
than
1
.
The
e
xpressions
of
1
and
2
with
second
order
approximation
are
deri
v
ed
as
follo
ws.
1
=
P
1
P
0
=
k
2
01
R
1
L
0
L
1
C
2
1
!
4
R
0
L
2
1
C
2
1
!
4
+
R
0
(
R
2
1
C
1
2
L
1
)
C
1
!
2
+
R
0
2
=
P
2
P
0
=
k
2
02
R
2
L
0
L
2
C
2
2
!
4
R
0
L
2
2
C
2
2
!
4
+
R
0
(
R
2
2
C
2
2
L
2
)
C
2
!
2
+
R
0
:
(3)
k
12
is
not
contained
in
the
e
xpressions
(3),
hence
it
is
ob
vious
that
its
ef
fect
to
ef
ficienc
y
is
little
compared
to
k
01
and
k
02
in
terms
of
ef
ficienc
y
.
Moreo
v
er
the
optimal
angular
frequencies
!
opt1
and
!
opt2
which
maximize
1
or
2
each
other
are
obtained
from
e
xpressions
(3)
as
belo
w
.
!
optn
=
s
2
(2
L
n
R
2
n
C
n
)
C
n
:
(4)
n
=
1
or
2
The
desirable
angular
frequenc
y
for
ef
ficient
WPT
has
been
re
v
ealed
in
this
section.
Furthermore
a
numerical
calculation
is
sho
wn
by
the
e
xpression
(4)
in
the
follo
wing
section.
3.2.
Numerical
calculation
of
efficiencies
In
the
former
section,
we
find
the
optimal
frequenc
y
by
the
mathematical
model
(1).
In
this
s
ection,
the
v
ariation
of
ef
ficienc
y
with
respect
to
angular
frequenc
y
is
in
v
estig
ated
based
on
the
e
xpression
(4).
From
the
equation
(3),
a
numerical
calculation
of
1
and
2
with
practical
v
alues
of
circuit
eleme
nts
(T
able
1)
is
sho
wn
in
Figure.
2.
T
able
1.
v
alues
of
elements
R
0
1
C
0
1nF
R
1
10
C
1
100nF
R
2
100
C
2
10nF
L
0
1
H
k
01
0
:
2
L
1
10
H
k
02
0
:
2
L
2
70
H
k
12
0
:
2
The
Inter
action
between
Load
Cir
cuits
and
Decision
of
F
r
equency
for
...
(Kazuya
Y
ama
guc
hi)
Evaluation Warning : The document was created with Spire.PDF for Python.
1334
ISSN:
2088-8708
angular freque
ncy[rad/sec]
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1x10
6
1x10
7
ef
ficiency
Figure
2.
A
numerical
calculation
of
ef
ficienc
y
By
the
e
xpression
(4),
we
find
that
!
opt1
=
1
:
41
10
6
[rad
=
sec]
;
!
opt2
=
2
:
24
10
6
[rad
=
sec]
,
and
1
and
2
are
maximized
with
these
angular
frequencies
each
other
.
In
man
y
cases,
the
load
circuits
ha
v
e
its
unique
optimal
frequenc
y
which
maximizes
ef
ficienc
y
,
and
therefore
it
is
rare
to
maximize
ef
ficiencies
from
a
po
wer
supply
to
the
se
v
eral
loads
simultaneously
as
sho
wn
in
Figure
2.
And
it
is
seen
that
1
and
2
con
v
er
ge
to
0
at
lo
w
frequenc
y
,
and
constant
v
alue
at
high
frequenc
y
.
The
correctness
is
pro
v
ed
by
taking
the
limit
of
the
e
xpression
(3)
with
re
g
ard
to
!
.
4.
CONCLUSION
In
this
paper
,
we
supposed
the
situation
that
there
are
tw
o
loads
to
w
ards
one
po
wer
supply
,
and
found
the
optimal
angular
frequenc
y
for
ef
ficient
WPT
.
From
the
e
xpression
(4),
it
is
ob
vious
that
the
optimal
angular
frequenc
y
is
not
af
fected
by
the
elements
on
transmitting
ci
rcuit.
Thus
the
elements
in
recei
ving
circuits
are
significant
when
ef
ficient
WPT
is
desired,
and
the
ideal
frequenc
y
is
dif
ferent
with
each
recei
ving
circuit.
More-
o
v
er
the
interaction
between
the
coils
on
recei
ving
circuits
is
little
compared
wi
th
the
interaction
between
the
coils
on
transmitting
and
recei
ving
circuit.
Hence
the
coupling
between
a
transmitting
coil
and
each
recei
ving
coil
is
more
significant
than
the
coupling
between
recei
ving
coils.
REFERENCES
[1]
N.
T
esla,
U.
S.
patent
1,
119,
732,
1914.
[2]
A.
K
urs,
A.
Karalis,
R.
Mof
f
att,
J.
D.
Joannopoulos,
P
.
Fisher
,
and
M.
Solja
˘
ci
´
c,
“W
ireless
Po
wer
T
ransfer
via
Strongly
Coupled
Magnetic
Resonances”,
Science
,
v
ol.
317,
pp.
83-86,
2007.
[3]
Z.
Bi,
T
.
Kan,
C.
C.
Mi,
Y
.
Zhang,
Z.
Zhao,
and
G.
A.
K
eoleian,
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(Kazuya
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