Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
6,
No.
1,
February
2016,
pp.
431
–
438
ISSN:
2088-8708,
DOI:
10.11591/ijece.v6i1.9039
431
Using
Squar
e
W
a
v
e
Input
f
or
W
ir
eless
P
o
wer
T
ransfer
Kazuya
Y
amaguchi
*
,
T
akuya
Hirata
*
,
and
Ichijo
Hodaka
**
*
Interdisciplinary
Graduate
School
of
Agriculture
and
Engineering,
Uni
v
ersity
of
Miyazaki,
Japan
**
Department
of
En
vironmental
Robotics,
F
aculty
of
Engineering,
Uni
v
ersity
of
Miyazaki,
Japan
Article
Inf
o
Article
history:
Recei
v
ed
Sep
16,
2015
Re
vised
Dec
19,
2015
Accepted
Jan
2,
2016
K
eyw
ord:
Resonant
phenomenon
Square
w
a
v
e
State
space
W
ireless
po
wer
transfer
ABSTRA
CT
A
wireless
po
wer
transfer
(WPT)
circuit
is
composed
of
a
transmitting
circuit
with
an
A
C
po
wer
supply
and
a
recei
ving
circuit
with
a
load,
and
the
circuits
are
wirelessly
connected
each
other
.
Then
a
designer
chooses
the
w
a
v
e
form
of
the
A
C
po
wer
supply
.
Man
y
papers
about
WPT
adopt
a
sinusoidal
w
a
v
e
as
the
input.
The
frequenc
y
of
the
sinusoidal
w
a
v
e
is
generally
determined
to
the
resonant
frequenc
y
of
the
circuit
for
high
po
wer
transfer
.
Since
the
number
of
circuit
elements
in
the
po
wer
supply
to
generate
a
square
w
a
v
e
is
much
less
than
that
of
a
sinusoidal
w
a
v
e,
WPT
with
a
square
w
a
v
e
input
should
be
treated.
In
f
act,
some
papers
adopt
a
square
w
a
v
e
as
the
input,
and
adjust
the
frequenc
y
of
the
square
w
a
v
e
to
the
resonant
frequenc
y
of
the
circuit.
In
this
paper
,
we
e
xamine
ho
w
the
frequenc
y
of
a
square
w
a
v
e
input
af
fects
po
wer
and
ef
ficienc
y
of
WPT
circuits,
and
propose
a
procedure
ho
w
to
determine
the
frequenc
y
of
the
input
to
impro
v
e
po
wer
and
ef
ficienc
y
.
Finally
we
discuss
which
w
a
v
e
should
be
adopted
as
an
input
and
ho
w
the
frequenc
y
of
the
input
should
be
determined,
re
g
ardless
of
whether
resonant
phenomena
occur
or
not.
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
In
these
years,
technology
of
po
wer
transfer
without
electric
wires
between
a
po
wer
supply
and
a
de
vice
to
be
po
wered,
which
is
called
wir
eless
po
wer
transfer
(WPT),
has
been
acti
v
ely
studied.
WPT
is
e
xpected
to
mak
e
our
life
con
v
enient
at
man
y
situations.
One
is
that
the
electric
v
ehicles
will
be
char
ged
just
by
parking
them
at
a
specified
place.
Especially
,
man
y
systems
to
char
ge
the
electric
v
ehicles
at
the
distance
of
dozens
of
centimeters
with
more
than
80
percent
ef
fici
enc
y
are
already
reported
by
man
y
de
v
elopers
of
WPT
.
On
the
other
hand,
there
is
a
problem
that
a
po
wer
transfer
is
not
suited
for
long
distance
since
it
is
dif
ficult
to
a
v
oid
the
leakage
inductances
between
a
po
wer
supply
and
the
de
vices.
Since
an
ef
ficient
po
wer
t
ransfer
w
as
reported
where
the
g
ap
of
transmission
w
as
0.6
meters
by
[1],
man
y
studies
about
WPT
ha
v
e
been
reported.
[2]
presented
a
mathematical
e
xpression
of
ef
ficienc
y
with
the
quality
f
actor
Q
instead
of
the
circuit
constants,
and
in
v
estig
ated
ho
w
ef
fi
cienc
y
depended
on
Q.
A
mathematical
e
xpression
of
ef
ficienc
y
with
Q
for
another
WPT
circuit
is
deri
v
ed
for
series
and
parallel
circuits
in
[3].
The
e
xpressions
of
ef
ficienc
y
and
the
optimal
resisti
v
e
load
with
a
relay
circuit
betwe
en
a
transmitting
and
a
recei
ving
circuits
are
deri
v
ed
in
terms
of
Q
in
[4],
and
it
is
sho
wn
that
Q
increases
if
a
relay
circuit
is
inserted
betwee
n
a
transmitting
circui
and
recei
ving
circuit
[5].
An
equi
v
alent
circuit
to
WPT
is
used
in
order
to
analyze
WPT
circuits
in
[6].
The
electromagnetic
field
with
resonant
phenomena
is
analyzed
by
[7].
It
is
pro
v
ed
that
a
DC
po
wer
supply
can
be
utilized
as
an
input
of
WPT
with
an
oscillator
in
[8].
Man
y
papers
about
WPT
adopt
a
sinusoidal
w
a
v
e
as
an
input
of
a
WPT
circuit.
[9]
and
[10]
ar
gue
that
a
square
w
a
v
e
is
approximated
as
a
sinusoidal
w
a
v
e
whi
ch
has
the
same
frequenc
y
as
the
frequenc
y
of
the
square
w
a
v
e
with
resonant
phenomena.
This
is
attrib
uted
to
the
f
act
that
if
the
fundamental
frequenc
y
of
the
square
w
a
v
e
which
has
man
y
frequencies
is
adjusted
to
the
resonant
frequenc
y
,
the
influence
of
frequencies
apart
from
the
fundamental
frequenc
y
will
be
suppressed.
In
f
act,
the
range
of
frequenc
y
by
a
po
wer
supply
does
not
al
w
ays
cont
ain
the
resonant
frequenc
y
.
This
situation
should
be
in
v
estig
ated
as
well
as
dri
ving
the
po
wer
supply
at
the
resonant
frequenc
y
.
Evaluation Warning : The document was created with Spire.PDF for Python.
432
ISSN:
2088-8708
In
this
paper
,
we
clarify
ho
w
the
dri
ving
frequenc
y
of
a
square
w
a
v
e
input
af
fects
output
po
wer
and
ef
ficienc
y
of
WPT
,
while
man
y
papers
adopt
a
sinusoidal
w
a
v
e
as
an
input.
W
e
calculate
the
v
alues
of
output
po
wer
and
ef
ficienc
y
without
approximating
the
square
w
a
v
e
as
a
sinusoidal
w
a
v
e.
The
calculation
is
e
x
ecuted
for
three
types
of
the
circuits,
and
the
dif
ference
between
output
po
wers
with
the
sinusoidal
and
the
square
w
a
v
es
is
e
xamined
in
vie
w
of
the
characteristics
of
these
w
a
v
es
and
resonant
phenomena.
Finally
,
we
discuss
which
w
a
v
es
should
be
adopted
as
an
input
on
each
situations,
and
decide
the
frequenc
y
of
the
periodic
w
a
v
es
to
realize
a
high
po
wer
WPT
.
2.
THE
AMPLITUDES
OF
INPUT
V
OL
T
A
GES
A
circuit
which
is
commonly
utiliz
ed
for
WPT
is
sho
wn
in
Figure
1,
and
we
consider
the
circuit
for
a
simple
WPT
.
L
1
;
L
2
are
the
self
inductances
to
transfer
ener
gy
electromagnetically
,
and
M
is
the
mutual
inductance
which
is
determined
by
the
radius,
the
number
of
turns,
and
the
distance
of
the
coils.
The
load
R
L
is
resisti
v
e
such
as
a
battery
.
R
1
;
R
2
;
C
1
,
and
C
2
are
the
parasitic
f
actors
of
the
transmitting
circuit
and
recei
ving
circuit.
Figure
1.
A
simple
WPT
circuit
Here
we
assume
that
the
input
v
oltage
u
is
either
a
sinusoidal
w
a
v
e
or
a
square
w
a
v
e,
and
we
will
in
v
estig
ate
in
the
sequel
ho
w
the
choice
of
w
a
v
e
form
impro
v
es
or
contaminates
the
system
of
WPT
.
In
order
to
figure
out
results
from
the
choice
of
input
w
a
v
e
form,
we
should
determine
the
amplitude
of
w
a
v
e
forms
for
the
inputs
to
ha
v
e
the
same
output
po
wer
.
Then,
we
consider
the
bare
circuit
in
Figure
2
instead
of
Figure
1.
In
the
circuit
in
Figure
2,
the
po
wer
supply
and
the
load
(put
R
L
=
100[]
)
are
borro
wed
from
Figure
1.
If
we
consider
a
sinusoidal
w
a
v
e
and
a
square
w
a
v
e
with
amplitudes
described
in
the
table
in
Figure
2,
we
ha
v
e
the
same
output
a
v
erage
po
wer
5mW
in
the
table.
Thus
we
adopt
the
amplitudes
of
v
oltages
in
the
sequel
of
this
paper
.
Figure
2.
A
bare
circuit
(left)
and
a
table
of
v
oltage
and
input
po
wer
(right)
3.
INPUT
AND
OUTPUT
PO
WER,
AND
EFFICIENCY
The
beha
vior
of
WPT
circuit
in
Figure
1
is
go
v
erned
by
Kirchhof
f
’
s
la
w
,
Ohm’
s
la
w
,
and
other
la
ws
of
electric
elements.
Those
la
ws
can
be
described
by
algebraic
and
dif
ferential
equations
with
time-v
ariables
of
v
oltages
and
currents
in
the
circuit.
After
eliminating
unnecessary
v
ariables
from
the
equations,
we
ha
v
e
dif
ferential
equations
dri
v
en
by
the
input
v
oltage
as
in
the
equation
(1)
that
is
called
the
state
space
equation,
since
the
state
v
ariables
v
1
,
v
2
,
IJECE
V
ol.
6,
No.
1,
February
2016:
431
–
438
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
433
i
1
and
i
2
represent
the
whole
beha
vior
of
the
circuit.
If
we
choose
the
current
i
2
as
an
output
of
the
system,
we
can
e
xpress
the
input
to
output
beha
vior
as
a
transfer
function
written
by
(2).
_
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
G
(
s
)
=
M
C
1
C
2
s
3
C
1
C
2
s
4
+
(
R
1
L
2
+
R
3
L
1
)
C
1
C
2
s
3
+
(
L
1
C
1
+
L
2
C
2
+
R
1
R
3
C
1
C
2
)
s
2
+
(
R
1
C
1
+
R
3
C
2
)
s
+
1
(2)
In
order
to
visualize
the
input
to
output
relation,
we
substitute
concrete
v
alues
into
v
alues
of
circuit
elements
as
in
the
table
belo
w
.
Then
we
ha
v
e
three
log
-
log
plots
respecti
v
ely
for
the
sinusoidal
input
(left
column)
and
the
square
w
a
v
e
input
(right
column)
in
Figure
3.
The
first
ro
w
of
plots
is
the
g
ain
plots
of
transfer
function
G
(
s
)
in
(2).
The
second
ro
w
depicts
the
a
v
erage
po
wer
P
1
at
the
po
wer
supply
and
P
2
at
the
load.
The
third
ro
w
is
the
ef
ficienc
y
which
is
defined
by
the
ratio
of
the
output
po
wer
P
2
to
the
input
po
wer
P
1
.
The
horizontal
ax
es
are
all
the
angular
frequenc
y
of
input
v
oltages.
Needless
to
say
,
the
ouput
po
wer
P
2
and
ef
ficienc
y
are
desired
to
be
higher
for
an
y
WPT
circuit.
A
designer
or
user
of
WPT
circuits
choose
the
best
dri
ving
frequenc
y
in
vie
w
of
the
last
tw
o
ro
ws
of
plots.
Using
Squar
e
W
ave
Input
for
W
ir
eless
P
ower
T
r
ansfer
(Kazuya
Y
ama
guc
hi)
Evaluation Warning : The document was created with Spire.PDF for Python.
434
ISSN:
2088-8708
Figure
3.
Gain,
po
wer
,
and
ef
ficienc
y
v
ersus
frequenc
y
Notice
that
the
resonant
angula
r
frequenc
y
is
equal
to
10
6
rad/sec,
as
seen
in
the
first
ro
w
(tw
o
plots
in
the
first
are
identical
since
the
transfer
function
is
uniquely
determined
by
t
he
circuit;
it
does
not
depend
on
what
input
w
a
v
e
is
used).
In
this
case,
the
peak
frequenc
y
in
po
wer
is
equal
to
that
in
g
ain,
although
that
is
not
al
w
ays
true
in
general.
Moreo
v
er
,
the
ef
ficienc
y
reaches
maximum
at
the
peak
frequenc
y
.
This
is
the
case
in
both
of
sinusoidal
w
a
v
e
and
square
w
a
v
e
inputs.
The
dif
ference
of
left
and
right
plots
is
t
he
number
of
local
peaks
in
po
wer
and
ef
ficienc
y
.
This
comes,
of
course,
from
that
square
w
a
v
es
ha
v
e
man
y
dif
ferent
frequencies,
and
that
v
oltages
and
current
s
are
gi
v
en
by
superposition
of
ones
by
separate
sinusoidal
inputs
with
dif
ferent
frequencies.
In
the
follo
wing
section,
we
will
discuss
the
situation
abo
v
e
in
a
mathematical
w
ay
.
4.
CALCULA
TION
OF
PO
WER
BY
SQ
U
ARE
W
A
VE
INPUT
The
partial
sum
of
F
ourier
series
of
a
square
w
a
v
e
with
the
amplitude
1
is
written
by
the
follo
wing.
f
u
sq
=
1
2
+
2
n
X
k
=1
1
2
k
1
sin(2
k
1)
!
t
(3)
where
!
=
2
=T
and
T
is
the
period
of
the
square
w
a
v
e.
If
we
put
n
!
1
,
the
abo
v
e
con
v
er
ges
the
square
w
a
v
e
pointwizely
e
xcept
for
the
uncontinuous
points.
Since
the
WP
T
circuit
consists
of
linear
and
time-in
v
ariant
elements,
the
state
space
representation
is
also
linear
and
time-in
v
ariant
as
in
(1).
Then
we
ha
v
e
transfer
functions
from
the
IJECE
V
ol.
6,
No.
1,
February
2016:
431
–
438
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
435
input
u
to
an
y
of
state
v
ariables
v
1
,
v
2
,
i
1
and
i
2
.
Therefore,
it
is
straightforw
ard
t
o
obtain
the
periodic
and
stationary
solution
to
the
equation
(1)
with
the
periodic
input
(3).
If
we
put
the
number
n
in
(3)
as
lar
ge
enough,
we
ha
v
e
approximated
solutions
to
the
e
xact
solutions
with
square
w
a
v
e
inputs.
Here
note
that
we
should
focus
on
po
wer
which
is
product
of
v
oltage
and
current,
when
we
refer
to
the
performance
of
WPT
system.
Since
the
relation
between
i
n
put
v
oltage
and
po
wer
at
the
load
is
not
linear
,
it
is
not
straightforw
ard
to
obtain
the
mathematical
e
xpression
of
po
wer
.
Ne
v
ertheless,
we
can
deduce
a
simple
e
xpression
of
po
wer
.
F
or
this
end,
we
write
P
(
u
)
as
the
a
v
erage
po
wer
at
the
load
by
a
periodic
input
u
.
It
can
be
v
erified
that
P
(
u
1
+
u
2
)
=
P
(
u
1
)
+
P
(
u
2
)
holds
if
u
1
and
u
2
are
orthogonal,
that
is,
Z
T
0
u
1
(
t
)
u
2
(
t
)
dt
=
0
,
where
T
is
the
period
of
u
1
and
u
2
.
Therefore
P
(
f
u
sq
)
is
written
as
P
(
f
u
sq
)
=
P
1
2
+
P
2
sin
!
t
+
P
2
3
sin
3
!
t
+
+
P
2
(2
n
1)
sin(2
n
1)
!
t
:
(4)
Another
appar
ant
b
ut
important
identity
is
P
(
u
)
=
2
P
(
u
)
(
is
an
arbitrary
real
number).
Then
P
(
f
u
sq
)
in
(4)
is
refined
as
P
(
f
u
sq
)
=
2
2
n
X
k
=1
1
2
k
1
2
P
(sin(2
k
1)
!
t
)
:
(5)
The
equation
(5)
gi
v
es
ho
w
to
calculate
the
po
wer
with
square
w
a
v
e
inputs
based
on
po
wer
with
sinusoidal
w
a
v
e
inputs.
A
rough
estimation
for
(5)
is
gi
v
en
by
P
(
f
u
sq
)
=
0
:
41
P
(sin
!
t
)
+
0
:
045
P
(sin
3
!
t
)
+
0
:
016
P
(sin
5
!
t
)
:
(6)
The
coef
ficients
0
:
41
,
0
:
045
and
0
:
016
respecti
v
ely
reflect
ho
w
the
po
wers
P
(sin
!
t
)
,
P
(sin
3
!
t
)
and
P
(sin
5
!
t
)
with
sinusoidal
inputs
respecti
v
ely
contrib
ute
to
the
po
wer
P
()
with
the
square
input.
W
ith
this
observ
ation,
it
is
reasonable
that
we
should
adjust
the
dri
ving
frequenc
y
of
square
w
a
v
e
input
to
the
resonant
frequenc
y
of
WPT
circuit,
just
lik
e
sinusoidal
w
a
v
e
input,
when
we
intend
to
obtain
high
output
po
wer
.
This
could
be
true
if
the
po
wer
plot
v
ersus
frequenc
y
with
sinusoidal
inputs
has
a
single
steep
peak.
In
other
w
ords,
we
may
ne
glect
higher
frequencies
and
concentrate
on
the
fundamental
frequenc
y
,
also
stated
in
[9,
10].
F
or
some
WPT
circuits,
po
wer
plot
v
ersus
frequenc
y
with
sinusoidal
inputs
ine
vitably
ha
v
e
plural
or
gentle
peaks,
where
we
should
tak
e
higher
frequencies
as
well
as
the
fundam
ental
frequenc
y
into
account.
One
important
observ
ation
is
to
see
a
local
peak
at
one
third
frequenc
y
of
the
resonant
frequenc
y
as
the
po
wer
plot
with
square
w
a
v
e
inputs
in
Figure
3,
because
square
w
a
v
e
has
three
times
frequenc
y
of
the
fundamental
frequenc
y
.
Such
situations
are,
in
general,
dif
ficult
to
cope
with,
and
little
is
kno
wn,
or
e
v
en
not
referred
to
in
the
literature.
Plot
of
ef
ficienc
y
is
not
al
w
ays
monotone
in
general,
e
v
en
if
g
ain
plot
and
po
wer
plot
are
monotone
as
in
Figure
3.
In
f
act,
maximizing
output
po
wer
does
not
al
w
ays
mean
maximizing
ef
ficienc
y
as
discussed
in
[11,
12,
13].
5.
PO
WER
AND
EFFICIENCY
WITH
THE
V
ARIOUS
CIRCUITS
Po
wer
and
ef
ficienc
y
about
the
circuits
of
Figure
4
and
Figure
5
are
sho
wn
belo
w
.
A
WPT
circuit
with
a
relay
circuit
in
Figure
5
can
be
ef
ficient
in
order
to
increase
a
distance
between
a
po
wer
supply
and
a
resisti
v
e
load
to
transfer
po
wer[14].
Here
we
omit
to
write
the
state
space
representation
and
transfer
function,
and
readers
can
refer
to
[15]
about
the
calculation
and
e
xpression
of
them.
These
results
sho
w
that
the
a
v
erage
po
wers
with
square
w
a
v
e
inputs
are
locally
high
at
the
neighborhoods
of
the
resonant
frequenc
y
and
one
third
of
the
res
onant
frequenc
y
as
in
Figure
3.
As
we
can
see
thes
e
results,
if
the
frequenc
y
of
input
is
restricte
d
to
v
alues
lo
wer
than
the
resonant
frequenc
y
,
we
should
use
a
square
w
a
v
e
rather
than
a
sinusoidal
w
a
v
e
as
a
v
oltage
input
of
WPT
circuit.
Using
Squar
e
W
ave
Input
for
W
ir
eless
P
ower
T
r
ansfer
(Kazuya
Y
ama
guc
hi)
Evaluation Warning : The document was created with Spire.PDF for Python.
436
ISSN:
2088-8708
Figure
4.
Bode
diagram
and
po
wer
and
ef
ficienc
y
with
a
parallel
circuit
IJECE
V
ol.
6,
No.
1,
February
2016:
431
–
438
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
437
Figure
5.
Bode
diagram
and
po
wer
and
ef
ficienc
y
with
a
relay
circuit
Using
Squar
e
W
ave
Input
for
W
ir
eless
P
ower
T
r
ansfer
(Kazuya
Y
ama
guc
hi)
Evaluation Warning : The document was created with Spire.PDF for Python.
438
ISSN:
2088-8708
6.
CONCLUSION
In
this
paper
,
we
ha
v
e
e
xamined
ho
w
po
wer
and
ef
ficienc
y
of
wireless
po
wer
transfer
are
af
fected
by
choosing
type
of
w
a
v
e
form
of
A
C
po
wer
supply
v
oltage,
sinusoidal
w
a
v
es
and
square
w
a
v
es,
and
then,
we
ha
v
e
suggested
which
type
of
w
a
v
e
form
should
be
chosen
for
a
better
wireless
po
wer
transfer
.
W
e
ha
v
e
confirmed
that
the
frequenc
y
of
A
C
po
wer
supply
should
be
adjusted
to
the
resonant
frequenc
y
of
the
circuit
if
the
sinusoidal
input
is
chosen,
as
man
y
other
papers
also
concluded.
Ho
we
v
er
,
it
is
not
al
w
ays
possible
to
use
the
resonant
frequenc
y
,
especially
if
the
frequenc
y
is
too
high
to
implement
the
synchronous
input.
F
or
such
a
dif
ficult
situation,
we
ha
v
e
proposed
to
utilize
the
characteristics
that
square
w
a
v
es
contain
(infinitely)
man
y
frequencies.
Then
we
ha
v
e
sho
wn
that
the
higher
po
wer
can
be
obt
ained
e
v
en
if
the
fr
equenc
y
of
input
is
res
tricted
to
a
lo
wer
frequenc
y
than
the
resonant
frequenc
y
.
This
gi
v
es
a
ne
w
insight
for
the
de
v
elopment
of
wireless
po
wer
transfer
system
dri
v
en
by
square
w
a
v
e
inputs.
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Evaluation Warning : The document was created with Spire.PDF for Python.