Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
6,
No.
6,
December
2016,
pp.
3217
–
3221
ISSN:
2088-8708
3217
A
General
Method
to
P
arameter
Optimization
f
or
Highly
Efficient
W
ir
eless
P
o
wer
T
ransfer
Kazuya
Y
amaguchi
1
,
T
akuya
Hirata
2
,
and
Ichijo
Hodaka
3
1
Department
of
Control
Engineering,
National
Institute
of
T
echnology
,
Nara
Colle
ge,
Japan
2
Interdisciplinary
Graduate
School
of
Agriculture
and
Engineering,
Uni
v
ersity
of
Miyazaki,
Japan
3
Department
of
En
vironmental
Robotics,
F
aculty
of
Engineering,
Uni
v
ersity
of
Miyazaki,
Japan
Article
Inf
o
Article
history:
Recei
v
ed
Mar
26,
2016
Re
vised
No
v
8,
2016
Accepted
No
v
21,
2016
K
eyw
ord:
wireless
po
wer
transfer
resonant
phenomenon
impedance
matching
state
space
representation
ABSTRA
CT
This
paper
proposes
a
ne
w
and
general
method
to
optimize
a
w
orking
frequenc
y
and
a
load
resistance
in
order
to
realize
highly
ef
ficient
wireless
po
wer
transfer
.
It
should
be
noticed
that
neither
resonant
frequenc
y
nor
matched
impedance
maximizes
ef
ficienc
y
of
wireless
po
wer
transfer
circuit,
in
general.
This
paper
establishes
a
mathematical
model
of
a
c
ommonly
used
wireless
po
wer
transfer
circuit,
and
deri
v
es
a
mathematical
e
xpression
of
circuit
ef
ficienc
y
which
in
v
olv
es
a
w
orking
frequenc
y
,
a
load
resistance
and
the
other
parameters
as
symbols.
This
enables
us
to
find
the
optimal
w
orking
frequenc
y
a
nd
load
resista
nce.
The
result
of
this
paper
is
compared
with
results
by
a
method
based
on
resonance
and
impedance
matching,
and
then
clarified
by
a
numerical
e
xample.
Copyright
c
2016
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Ichijo
Hodaka
Department
of
En
vironmental
Robotics,
F
aculty
of
Engineering,
Uni
v
ersity
of
Miyazaki
1-1,
Gakuen
Kibanadai
Nishi,
Miyazaki,
889-2192,
Japan
Email:
hijhodaka@cc.miyazaki-u.ac.jp
1.
INTR
ODUCTION
Supplying
electric
po
wer
to
electric
de
vices
generally
needs
electric
wires.
T
ransferring
electric
po
wer
without
electric
wires
is
called
wireless
po
wer
trans
fer
(WPT),
where
we
can
a
v
oid
messy
cable
connections
and
reduce
possibility
of
gett
ing
an
electric
shock.
The
method
to
transfer
electric
po
wer
from
a
v
oltage
source
to
a
load
is
based
on
electromagnetic
phenomena
between
transmitting
and
recei
ving
sides,
which
originates
from
the
w
ork[1].
Stimulated
by
the
w
ork[2],
numerous
number
of
researches
about
WPT
ha
v
e
been
reported.
WPT
with
high
ef
ficienc
y
e
xpressed
in
terms
of
coupling
coef
ficient
and
quality
f
actor
is
used
for
a
medical
application[3].
A
load
resistance
which
realizes
high
ef
ficienc
y
is
applied
for
a
DC-DC
con
v
erter
in
an
implanted
syst
em[4].
A
relay
circuit
between
a
transmitting
circuit
and
a
recei
ving
circuit
is
proposed
to
impro
v
e
ef
ficienc
y
of
WPT[5].
An
oscillation
circuit
is
utilized
as
a
transmitting
circuit
for
a
transmission
with
a
DC
po
wer
supply
tar
geted
to
implants[6].
Man
y
papers
use
the
idea
of
impedance
matching
to
maximize
po
wer
at
a
load,
which
is
re
vie
wed
in
[7].
It
is
realized
by
matchi
ng
impedance
of
a
l
oad
to
output
impedance
of
a
po
wer
source.
Since
matched
impedance
includes
a
w
orking
frequenc
y
,
man
y
papers
choose
a
resonant
frequenc
y
as
the
w
orking
frequenc
y
to
generate
a
sinusoidal
v
oltage
input
at
the
po
wer
source.
This
choice
enables
to
mak
e
reactance
of
matched
impedance
zero,
and
match
the
real
parts
of
output
impedance
of
the
po
wer
source
to
a
resisti
v
e
load.
On
the
other
h
a
nd
,
it
is
pointed
out
in
[8]
that
ef
ficienc
y
which
is
defined
as
the
ratio
of
the
load
po
wer
and
the
input
po
wer
is
not
al
w
ays
maximized
with
resonance.
There
are
a
superior
load
and
frequenc
y
of
input
than
a
load
and
frequenc
y
with
impedance
matching
and
resonance
to
obta
in
high
ef
ficienc
y
.
Therefore
we
should
adjust
the
w
orking
frequenc
y
of
po
wer
source
to
a
frequenc
y
re
g
ardless
of
resonant
frequenc
y
.
In
this
paper
,
the
optimal
load
resistance
based
on
a
mathematical
e
xpression
of
ef
ficienc
y
to
maximi
ze
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJECE
,
DOI:
10.11591/ijece.v6i6.10508
Evaluation Warning : The document was created with Spire.PDF for Python.
3218
ISSN:
2088-8708
ef
ficienc
y
of
a
WPT
circuit
is
ne
wly
obtained
by
using
the
state
space
approach
which
represents
the
beha
vior
of
circuit
as
a
set
of
dif
ferential
equations.
This
e
xplains
that
methods
by
impedance
matching
and
resonance
in
the
other
literature
are
not
optimal
in
a
sense
of
parameter
optimization.
Finally
,
a
numerical
calculation
of
ef
ficienc
y
is
sho
wn
to
compare
our
method
with
the
other
method
for
ef
ficient
WPT
.
2.
ANGULAR
FREQ
UENCY
AND
LO
AD
RESIST
ANCE
FOR
MAXIMAL
EFFICIENCY
In
this
paper
,
the
circuit
on
Figure
1
is
analyzed.
Figure
1.
A
simple
WPT
circuit
On
the
abo
v
e
circuit,
the
left
side
is
called
transmitting
side,
and
the
right
side
is
called
recei
ving
side.
u
is
a
sinusoidal
input
for
WPT
.
R
1
;
R
2
;
C
1
;
C
2
are
parasitic
f
actors
of
the
circuit.
L
1
and
L
2
are
self
inductances,
and
M
is
the
mutual
inductance
between
L
1
and
L
2
.
R
L
is
a
load
resistance
which
consumes
ener
gy
.
v
1
and
v
2
are
the
v
oltage
of
C
1
and
C
2
,
and
i
1
and
i
2
are
the
current
of
L
1
and
L
2
.
In
this
section,
we
attempt
to
maximize
ef
ficienc
y
which
is
defined
by
the
ratio
of
the
load
po
wer
and
the
input
po
wer
.
T
o
maximize
ef
ficienc
y
,
a
mathematical
e
xpression
of
ef
ficienc
y
is
e
xpressed
by
composing
a
mathematical
model
as
follo
ws.
_
x
=
Ax
+
B
u;
x
=
v
1
v
2
i
1
i
2
T
(1)
A
=
1
2
6
6
4
0
0
C
1
0
0
0
0
C
2
L
2
M
R
1
L
2
R
3
M
M
L
1
R
1
M
R
3
L
1
3
7
7
5
;
B
=
1
2
6
6
4
0
0
L
2
M
3
7
7
5
=
L
1
L
2
M
2
;
R
3
=
R
2
+
R
L
:
Such
model
is
called
the
state
space
equation,
and
v
1
;
v
2
;
i
1
;
and
i
2
are
state
v
ariables
which
represent
the
beha
vior
of
the
circuit
in
the
model.
The
mathematical
e
xpression
of
ef
ficienc
y
is
described
by
finding
the
state
solution
from
the
model
as
belo
w
.
=
R
L
M
2
C
2
2
!
4
(
R
1
L
2
2
+
R
3
M
2
)
C
2
2
!
4
+
R
1
(
2
L
2
+
R
2
3
C
2
)
C
2
!
2
+
R
1
(2)
where
!
is
the
angular
frequenc
y
of
u
.
Then
we
assume
!
and
L
1
;
L
2
;
C
1
;
C
2
as
!
=
!
0
=
r
1
L
1
C
1
=
r
1
L
2
C
2
:
(3)
If
the
condition
(3)
is
satisfied,
the
ef
ficienc
y
0
is
determined
as
0
=
R
L
M
2
R
3
(
M
2
+
R
1
R
3
L
2
C
2
)
:
(4)
From
e
xpression
(4),
the
load
resistance
R
L0
which
maximizes
e
xpression
(4)
is
deri
v
ed
as
belo
w[9].
R
L0
=
s
R
2
R
2
+
(
!
0
M
)
2
R
1
:
(5)
IJECE
V
ol.
6,
No.
6,
December
2016:
3217
–
3221
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
3219
The
maximal
ef
ficienc
y
under
the
condition
(3)
is
found
by
adjusting
R
L
to
R
L0
in
(5).
Ho
we
v
er
the
condition
(3)
is
not
al
w
ays
fulfiled,
and
therefore
the
standard
condition
which
maximizes
ef
ficienc
y
should
be
deri
v
ed.
is
seen
to
be
maximized
at
!
=
!
opt
=
1
p
L
2
C
2
s
2
L
2
2
L
2
R
2
3
C
2
:
(6)
by
solving
@
=@
!
=
0
and
assuming
2
L
2
R
2
3
C
2
>
0
[10].
If
2
L
2
R
2
3
C
2
0
,
ef
ficienc
y
increases
monotonically
as
!
increases,
and
has
no
maximum.
If
R
3
approaches
to
zero,
!
opt
in
(6)
approches
to
!
0
in
(3).
Although
a
load
is
normally
attached
to
man
y
WPT
circuits,
and
hence
these
e
xpressions
imply
dif
ferent
v
alues
each
other
.
Similarly
the
load
resistance
R
Lopt
which
maximizes
is
found
by
solving
@
=@
R
L
=
0
in
the
follo
wing.
R
Lopt
=
s
R
2
R
2
+
(
!
M
)
2
R
1
+
!
L
2
1
!
C
2
2
:
(7)
If
!
=
1
=
p
L
2
C
2
,
R
Lopt
in
(7)
becomes
same
as
R
L0
in
(5).
It
has
been
re
v
ealed
that
the
e
xpressions
!
0
and
R
L0
which
are
adopted
as
an
ideal
angular
frequenc
y
and
load
resistance
are
the
limited
conditions
in
!
opt
and
R
Lopt
which
are
found
in
this
paper
.
Then
we
should
clarify
which
angular
frequenc
y
and
load
resistance
are
appropriate
for
ef
ficient
WPT
.
3.
OPTIMAL
LO
AD
RESISIT
ANCE
T
O
REALIZE
HIGH
PO
WER
W
e
discuss
a
condition
for
high
po
wer
of
WPT
.
As
a
method
to
impro
v
e
the
po
wer
of
a
load,
impedance
matching
which
is
used
by
fitting
the
load
impedance
to
the
comple
x
conjug
ate
of
output
impedance
of
input
is
well
kno
wn.
If
impedance
matching
is
realized,
the
maximal
po
wer
of
load
is
obtained.
F
or
the
circuit
at
Figure
1,
the
condition
for
impedance
matching
is
e
xamined.
The
equi
v
alent
cir
cuit
of
Figure
1
is
described
as
belo
w[11,
12].
Figure
2.
The
equi
v
alent
circuit
of
Figure
1
The
synthetic
impedance
Z
seen
from
R
L
is
deri
v
ed
from
Figure
2.
Z
=
!
2
C
1
C
2
(
R
2
k
1
+
R
1
k
2
)
j
f
!
4
M
2
C
1
C
2
+
!
2
C
1
C
2
(
R
1
R
2
k
1
k
2
)
+
!
(
C
1
k
1
+
C
2
k
2
)
k
1
k
2
g
!
2
C
1
C
2
(
k
1
j
R
1
)
:
(8)
k
1
=
!
L
1
1
!
C
1
;
k
2
=
!
L
2
1
!
C
2
Then
we
assume
!
and
L
1
;
L
2
;
C
1
;
C
2
as
(3),
and
Z
becomes
Z
=
R
2
+
M
2
R
1
L
1
C
1
:
(9)
In
terms
of
impedance
matching,
the
ideal
R
Lmat
which
maximizes
the
a
v
erage
po
wer
of
R
L
is
found
in
the
follo
wing[7].
R
L
=
R
Lmat
=
R
2
+
M
2
R
1
L
1
C
1
:
(10)
R
Lmat
has
been
obtained
by
applying
impedance
matching
under
the
condition
(3).
Ho
we
v
er
it
maximizes
the
po
wer
of
R
L
on
the
condition
(3),
and
therefore
R
Lmat
is
not
al
w
ays
satisfied.
Moreo
v
er
!
0
in
(3)
has
no
rele
v
ance
with
the
objecti
v
e
to
maximize
the
po
wer
of
load
and
ef
ficienc
y
.
A
Gener
al
Method
to
P
ar
ameter
Optimization
for
Highly
Ef
ficient
W
ir
eless
...
(Kazuya
Y
ama
guc
hi)
Evaluation Warning : The document was created with Spire.PDF for Python.
3220
ISSN:
2088-8708
4.
NUMERICAL
CALCULA
TION
OF
EFFICIENCY
The
tw
o
types
of
angular
frequencies
!
0
and
!
opt
,
and
the
tw
o
types
of
load
resistances
R
L0
and
R
Lopt
ha
v
e
been
deri
v
ed
in
the
pre
vious
section.
Then
it
is
e
xamined
which
angular
frequenc
y
and
load
resistance
are
appropriate
to
impro
v
e
e
f
ficienc
y
by
a
numerical
calculation.
F
or
the
numerical
calculation,
we
set
the
v
alues
of
elements
as
belo
w
.
T
able
1.
v
alues
of
elements
R
1
1
R
2
4
L
1
;
L
2
5
:
20
10
5
H
M
1
:
00
10
5
H
C
1
;
C
2
9
:
14
10
8
F
Then
!
0
=
4
:
59
10
5
[rad
=
sec]
;
!
opt
=
5
:
48
10
5
[rad
=
sec]
;
R
L0
=
10
:
0[]
;
R
Lopt
=
14
:
4[]
.
On
this
situation,
ef
ficienc
y
is
calculated
as
Figure
3.
Figure
3.
Ef
ficienc
y
by
our
method
and
the
others
While
ef
ficienc
y
at
!
=
!
0
and
R
L
=
R
L0
is
0
:
429
,
ef
ficie
nc
y
at
!
=
!
opt
and
R
L
=
R
Lopt
is
0
:
449
.
Man
y
papers
use
!
=
1
=
p
LC
as
an
angular
frequenc
y
of
input.
Our
result
sho
ws
that
we
should
not
use
!
=
1
=
p
LC
b
ut
!
opt
,
and
also,
we
should
not
use
the
load
resistance
R
L0
b
ut
R
Lopt
in
order
to
obtain
highest
ef
ficienc
y
.
5.
CONCLUSION
In
this
paper
,
we
proposed
a
method
to
accomplish
the
highest
ef
ficienc
y
of
wireless
po
wer
transfer
.
W
e
ha
v
e
sho
wn
ef
ficienc
y
by
our
method
is
indeed
higher
than
ef
ficienc
y
by
con
v
entional
m
ethods
using
reso-
nant
phenomena
and
impedance
matching.
The
k
e
y
to
accomplish
the
highest
ef
ficienc
y
has
been
mathematical
e
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erage
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ef
ficienc
y
of
wireless
po
wer
transfer
circuits.
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