TELK
OMNIKA
Indonesian
Journal
of
Electrical
Engineering
V
ol.
12,
No
.
5,
Ma
yl
2014,
pp
.
3697
3704
DOI:
http://dx.doi.org/10.11591/telk
omnika.v12.i5.4525
3697
Modelling
of
a
Witricity
System
Using
GSSA
Method
Lan
Jian
yu
*
,
T
ang
Houjun
,
a
nd
Geng
Xin
Shanghai
Jiao
T
ong
Univ
ersity/Depar
tment
of
Electr
ical
Engineer
ing
NO
.
800,
RD
.
Dong
Chuan,
Shanghai,
200240,
China,
008613524956457
*
e-mail:
jian
yu
lan@163.com
Abstract
With
the
r
apid
de
v
elopment
of
mobile
app
liances
,
wireless
po
w
er
tr
ansf
er
technique
has
been
a
hot
issue
f
or
researchers
.
A
resonant
coupled
po
w
er
system
called
witr
icity
with
high
efficiency
and
the
middle
r
ange
tr
ansf
er
distance
is
presented
b
y
MIT
.
The
main
circuit
of
the
witr
icity
system
acts
as
a
resonant
con
v
er
ter
oper
ating
in
high
frequency
.
The
con
v
er
ter
is
a
comple
x
time
v
ar
iant
and
non-linear
systems
.
So
,
it
is
difficult
to
obtain
its
accur
ate
mathematical
m
odels
.
In
this
paper
,
the
gene
r
aliz
ed
state-space
a
v
er
aging
method
is
applied
to
model
this
con
v
er
ter
.
With
appropr
iate
v
alues
f
or
the
circuit
par
ameters
,
n
umer
ical
results
are
compared
with
which
in
time
domain
model.
The
results
sho
w
that
theoretical
analysis
are
w
ell
ag
ree
with
the
sim
ulation,
and
b
y
proposed
method
the
computational
time
is
remar
ka
b
ly
reduced.
K
e
yw
or
ds:
wireless
po
w
er
,
witr
icity
,
gener
aliz
ed-state-space-a
v
er
aging
Cop
yright
c
2014
Institute
of
Ad
v
anced
Engineering
and
Science
.
All
rights
reser
v
ed.
1.
Intr
oduction
Without
cords
and
plugs
,
electronic
and
electr
ical
products
has
been
more
fr
iendly
and
saf
er
than
tr
aditi
onal
method
in
special
o
ccasion
such
as
coal
mine
,
under
w
ater
and
implanted
de
vices
[1,
2,
3,
4,
5].
The
wireless
po
w
er
tr
ansf
er
system
based
on
inductiv
ely
coupled
po
w
er
tr
ansf
er
method
has
the
bottlenec
k
of
shor
t
tr
ansf
er
distance
.
If
the
distance
betw
een
energy
tr
ansmitter
and
receiv
er
is
more
than
tens
millimetres
,
the
system
efficiencies
will
drop
shar
ply
.
Ho
w
e
v
er
,
t
he
theor
y
of
magnetic
resonance
coupling
pro
v
es
a
method
of
wireless
po
w
er
tr
ans-
f
er(WPT)
in
longer
distances
.
Andre
[6]
presented
his
w
or
k
about
this
method
which
called
witr
icity
(wireless
electr
icity).
A
100
W
b
ulb
w
as
lit
1m
apar
t
from
the
source
b
y
magnetic
resonance
cou-
pling
method.
Currently
,
consider
ab
le
eff
or
ts
ha
v
e
been
made
to
the
de
v
elopment
of
this
po
w
er
tr
ansf
er
method.
It
ha
s
been
sho
wn
that
the
distance
of
wireless
po
w
er
tr
ansf
er
is
significantly
increased
b
y
placing
inter
mediate
resonators
betw
een
tr
ansmitter
and
recei
v
e
r
.
Though
abo
v
e
researches
ha
v
e
g
reat
contr
ib
utes
to
the
tr
ansf
er
distance
and
efficiency
[7,
8,
9,
10,
11,
12],
the
studies
about
mathematical
model
of
the
witr
icity
system
are
seldom.
The
circuit
of
witr
icity
requires
oper
ating
with
a
high
frequency
,
nor
mally
more
than
1
MHz,
to
obtain
high
tr
ansf
er
effi-
ciency
in
long
distance
.
Usually
,
a
resonant
con
v
er
ter
with
PWM
s
witches
is
emplo
y
ed
to
dr
iv
e
the
witr
icity
system.
The
po
w
er
con
v
er
ter
model
is
f
ast
changing
with
the
time
in
nature
because
of
the
s
witching
beha
viour
,
so
this
resonant
con
v
er
ter
is
a
time
v
ar
ying,
nonlinear
and
comple
x
system.
Thus
,
it
is
difficult
to
acquire
the
e
xact
mathematical
model
of
this
con
v
er
ter
.
There
are
tw
o
methods
a
v
ailab
le
to
model
a
circuit.
One
is
based
on
the
circuit
topology
,
which
can
be
sim
ulated
b
y
some
soft
pac
kages
lik
e
SIMULIK
or
PSIM.
The
shor
tcoming
of
this
method
is
that
it
requires
v
ast
resources
of
computer
and
a
long
sim
ulation
time
.
The
other
method
is
to
der
iv
e
its
mathematical
model
directly
through
mathematical
method.
The
adv
a
ntage
is
that
an
y
par
ameter
can
be
change
to
obser
v
e
char
acter
istics
of
proposed
system
in
diff
erent
condition.
Based
on
this
idea,
the
state
space
a
v
er
aging
method
is
de
v
eloped
f
or
the
analysis
and
design
of
PWM
con
v
er
ters
[13].
Because
this
me
thod
considers
only
the
DC
components
of
v
ar
iab
les
,
it
is
not
v
alid
f
or
modelling
of
resonant
con
v
er
ters
in
which
the
A
C
comp
onents
are
the
main
con-
tr
ib
utions
.
In
addition,
this
method
is
based
on
an
assumption
that
the
v
ar
iab
les
should
be
m
uch
slo
w
in
time
domain
than
s
witching
frequencies
.
Ho
w
e
v
er
,
in
a
resonant
con
v
er
ter
,
the
s
witching
Receiv
ed
September
26,
2013;
Re
vised
December
26,
2013;
Accepted
J
an
uar
y
16,
2014
Evaluation Warning : The document was created with Spire.PDF for Python.
3698
ISSN:
2302-4046
frequencies
are
near
to
its
nature
resonant
frequencies
.
After
that,
a
modified
state
space
a
v
er-
aging
approach
is
proposed,
which
is
applied
to
model
the
quasi
resonant
con
v
er
ters
with
high
accur
acy
.
But
it
till
canno
t
descr
ibe
the
resonant
con
v
er
ter
in
detail.
Sanders
[14]
proposed
a
gener
aliz
ed
state
space
a
v
er
aging
(GSSA
)
method
to
model
resonant
con
v
er
ters
.
Compared
to
other
resonant
con
v
er
ter
modelling
techniques
,
this
GSSA
method
represents
v
ar
iab
les
both
in
a
slo
w
time-v
ar
ying
DC
v
ar
iab
les
and
f
ast
oscillator
y
A
C
v
ar
iab
les
.
This
method
is
par
ticular
ly
suitab
le
to
f
ull
resonant
con
v
er
ters
.
Car
los[15]
has
successfully
emplo
y
ed
this
method
in
mod-
elling
a
full-b
r
idge
rectifier
.
Besides
,
GSSA
method
w
as
applied
to
the
har
monics
estimation
of
PWM
con
v
er
ters
[16,
17,
18,
19].
In
the
application
of
resonant
con
v
er
ters
,
a
contactless
po
w
er
supply
system
is
modelled
using
GSSA
method
with
a
completer
9th
order
system
[20].
And
then,
Xin[21]
emplo
y
ed
this
method
to
der
iv
e
the
mathematical
model
of
an
inductiv
ely
coupled
po
w
er
tr
ansf
er
system,
and
a
rob
ust
controller
has
designed
based
on
this
model.
In
addition,
a
mathematical
model
to
sim
ulate
tr
anscutaneous
energy
tr
ansmission
systems
b
y
GSSA
method
is
presented[22],
and
through
selectiv
e
modal
analysis
method,
the
system
w
as
k
eeping
up
to
the
first-order
.
Ho
w
e
v
er
,
the
mathematical
model
of
the
witr
icity
system
based
on
GSSA
or
other
modelling
method
is
seldom
to
be
f
ound
up
to
date
.
In
this
paper
,
the
GSSA
method
is
emplo
y
ed
to
the
modelling
of
a
witr
icity
system
and
the
steps
are
presented
in
detail.
The
compar
ison
of
the
GSSA
method
and
circuit
model
sim
ulation
in
time
do
main
b
y
PSIM
is
analysis
,
which
sho
ws
that
GSSA
has
a
sufficient
precision
to
present
the
dynamic
system
of
a
resonant
con
v
er
ter
.
This
paper
is
organiz
ed
as
f
ollo
ws
.
In
section
2
the
GSSA
method
is
re
vie
w
ed.
In
section
3,
the
mathematical
model
is
der
iv
ed
b
y
GSSA
method
and
n
umer
ical
analysis
is
presented
at
last.
The
conclusions
are
giv
en
in
section
4.
2.
Over
vie
w
of
GSSA
Method
The
GSSA
method
is
based
on
the
concept
that
an
y
w
a
v
ef
or
m
x
(
t
)
on
a
time
inter
v
al
[
t;
t
+
T
]
can
be
e
xpressed
in
F
our
ier
ser
ies
as
,
x
(
t
)
=
1
2
A
0
+
1
X
n
=1
(
a
n
cos
(
!
nt
)
+
b
n
sin
(
!
nt
))
,
(1)
where
t
2
[0
;
T
]
,
!
=
2
=T
,
and
T
is
the
per
iod
of
a
sliding
windo
w
.
It
is
mention
to
note
that
the
v
ar
iab
le
x
(
t
)
is
not
necessar
ily
per
iodic
and
can
be
imagined
so
b
y
summing
the
windo
w
repeats
o
v
er
all
time
domain.
Thus
,
a
F
our
ier
ser
ies
e
xists
and
its
coefficients
can
be
deter
mined
from,
a
n
(
t
)
=
2
T
Z
t
+
T
t
x
(
t
)
cos
(
n!
t
)
dt
,
(2)
and
b
n
(
t
)
=
2
T
Z
t
+
T
t
x
(
t
)
sin
(
n!
t
)
dt
.
(3)
It
is
clear
that
the
a
v
er
age
v
alue
A
0
=
2
and
the
amplitude
p
a
2
n
+
b
2
n
of
each
component
change
with
time
.
In
other
w
ord,
the
coefficient
a
n
and
b
n
can
reflect
the
en
v
elope
of
the
or
iginal
signal
x
(
t
)
.
F
or
resonant
con
v
er
ters
,
v
ar
iab
les
usually
ha
v
e
both
quasi-sin
usoidal
and
direct
com-
ponents
,
and
if
an
ap
propr
iate
sliding
windo
w
is
selected,
the
fundamental
ter
m
of
the
v
ar
iab
le
in
each
sliding
windo
w
will
represented
of
the
or
iginal
signal
x
(
t
)
w
ell.
In
gener
al,
the
per
iod
T
is
chosen
near
the
nature
resonant
per
iod
of
the
con
v
er
ter
under
studied.
The
v
alue
of
n
represents
the
accur
acy
le
v
el
of
the
signals
and
if
n
approaches
infinity
,
the
appro
ximation
error
theoretically
approaches
z
eros
.
3.
Mathematical
Modelling
of
the
Witricity
System
The
witr
icity
system
emplo
ys
a
high
frequency
resonant
con
v
er
ter
to
gener
at
e
the
e
xcita-
tion
of
the
po
w
er
tr
ansf
er
system.
In
this
par
t,
the
oper
ational
pr
inciple
of
witr
icity
will
be
re
vie
w
first,
and
then
the
circuit
model
is
discussed.
At
last
the
mathematical
model
is
der
iv
ed
b
y
GSSA.
TELK
OMNIKA
V
ol.
12,
No
.
5,
Ma
yl
2014
:
3697
3704
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
3699
3.1.
Operational
Principles
of
Witricity
System
Witr
icity
is
based
on
the
near-field,
strongly
coupled
magnetic
resonance
,
and
the
funda-
mental
pr
inciple
is
that
resonant
objects
e
xchange
energy
efficiently
,
while
non-resonant
objects
inter
act
w
eekly
.
Figure1
sho
ws
the
schematic
of
a
witr
icity
system
using
tw
o
magnetically
coupled
resonators
,
which
includes
the
source
coil,
the
de
vice
coil,
the
energy
source
,
and
the
load
[].
S
o
u
r
c
e
L
o
a
d
S
o
u
r
c
e
c
oi
l
E
n
e
r
g
y
f
l
o
w
D
e
v
i
c
e
c
o
i
l
1
2
2
1
k
k
Figure
1.
Basic
components
of
witr
icity
system
Obser
v
ed
from
figure
1,
energy
resonates
betw
een
the
source
coils
and
the
de
vice
coils
through
the
electromagnetic
fields
,
though
there
is
a
big
gap
betw
een
the
source
coils
and
the
de
vice
coils
which
gener
ates
a
lo
w
coupling
coefficient
betw
een
this
tw
o
coils
.
This
ph
ysical
phenomenon
can
be
e
xplained
using
the
coupled
mode
the
or
y
[].
As
seen
from
Figure
1,
it
can
assume
that
this
tw
o
coils
are
tw
o
resonators
,
and
a
s
(
t
)
and
a
d
(
t
)
represent
the
amplitude
of
t
hese
resonators
.
In
which,
a
s
(
t
)
is
the
source
resonator
and
a
d
(
t
)
is
the
de
vice
resonator
respectiv
ely
.
The
tw
o
resonators
obe
y
the
f
ollo
wing
equations
[].
da
s
(
t
)
dt
=
(
i!
s
s
)
a
s
(
t
)
+
ik
sd
a
d
(
t
)
(4)
da
d
(
t
)
dt
=
(
i!
d
d
)
a
d
(
t
)
+
ik
ds
a
s
(
t
)
(5)
Where
,
!
s
and
!
d
are
the
individual
angular
frequency
of
source
coils
and
de
vice
coils
,
respec-
tiv
ely;
k
=
k
12
=
k
21
is
the
coupling
coefficient
betw
een
the
source
and
de
vice;
s
and
d
are
the
individual
intr
insic
deca
y
r
ates
f
or
the
source
and
de
vice
,
respectiv
ely
.
Define
the
coupling
f
actor
as
C
F
=
!
k
2
p
s
d
.
(6)
When
this
meets
the
f
ollo
wing
conditions
,
C
F
1
and
!
s
=
!
d
,
the
witr
icity
system
can
tr
ansf
er
po
w
er
efficiently
from
source
coils
and
de
vice
coils[].
3.2.
Cir
cuit
Model
In
this
paper
,
the
source
de
vice
emplo
ys
a
half-br
idge
to
e
xcite
the
resonant
con
v
er
ter
,
and
the
ser
ial-par
allel
compensation
net
is
used
which
is
sho
wn
in
figure
2.
D
r
i
v
e
r
M
1
C
2
C
L
R
1
L
2
L
dc
U
Figure
2.
Schematic
of
proposed
half-br
idge
con
v
er
ter
In
figure2,
L
1
and
L
2
denote
the
pr
imar
y
coil
self-inductances
and
the
secondar
y
coil
self-inductances
,
respectiv
ely;
C
1
and
C
2
represent
compensated
capacitors
on
both
sides;
M
is
Title
of
man
uscr
ipt
is
shor
t
and
clear
,
implies
research
results
(First
A
uthor)
Evaluation Warning : The document was created with Spire.PDF for Python.
3700
ISSN:
2302-4046
the
m
utual
inductance
betw
een
the
pr
imar
y
coil
and
secondar
y
coil
and
R
L
is
the
equiv
alent
A
C
resistor
of
loads
.
By
means
of
m
utual
inductance
theor
y
,
the
equiv
alent
circuit
of
this
system
is
deduced
and
sho
wn
in
Figure3.
M
1
C
2
C
L
R
1
L
2
L
r
Z
(
)
c
u
t
d
c
+
U
0
1
(
)
i
t
2
(
)
i
t
!
Figure
3.
Mutual
inductance
equiv
alent
circuit
of
proposed
con
v
er
ter
In
figure3,
Z
r
is
the
reflected
impedance
from
the
secon
dar
y
side
and
u
c
(
t
)
is
the
induced
v
oltage
of
secondar
y
side
.
Z
r
can
be
e
xpressed
b
y
equation
(7).
Z
r
=
!
2
M
2
Z
2
(7)
In
which,
Z
2
is
the
par
alleled
impedance
of
secondar
y
side
,
which
is
e
xpressed
in
equation
(8).
Z
2
=
j
!
L
s
+
R
L
1
+
j
!
C
2
R
L
(8)
Substituting
(8)
into
(7)
the
reflected
resistance
and
reactance
from
the
secondar
y
coil
to
the
pr
imar
y
is
,
respectiv
ely
,
Re
Z
r
=
!
2
M
2
R
L
R
2
L
(
!
2
C
2
L
2
1)
2
+
!
2
L
2
2
(9)
and
Im
Z
r
=
!
3
M
2
[
C
2
R
2
L
(
!
2
C
2
L
2
1)
+
L
2
]
R
2
L
(
!
2
C
2
L
2
1)
2
+
!
2
L
2
2
(10)
Then,
the
equiv
alent
impedance
looking
from
the
input
side
of
the
half-br
idge
in
v
er
ter
is
,
Z
eq
=
j
!
L
1
+
1
j
!
C
1
+
Z
r
(11)
3.3.
Mathematical
Models
Using
GSSA
According
to
Kirchhoffs
circuit
la
ws
,
the
equiv
alent
circuit
equations
are
as
f
ollo
ws
,
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
s
(
t
)
U
dc
=
L
1
di
1
(
t
)
dt
+
u
c
1
(
t
)
+
i
1
Z
r
i
1
(
t
)
=
C
1
du
c
1
(
t
)
dt
j
!
M
i
1
(
t
)
=
L
2
di
2
(
t
)
dt
+
u
2
(
t
)
i
2
(
t
)
=
C
2
du
2
(
t
)
dt
u
c
2
(
t
)
R
,
(12)
where
i
1
(
t
)
and
i
2
(
t
)
represent
the
resonant
current
of
pr
imar
y
side
and
secondar
y
side
respec-
tiv
ely
,
and
u
1
(
t
)
denotes
the
v
oltag
e
of
pr
imar
y
capacitor
while
u
2
(
t
)
is
the
v
olta
ge
of
the
sec-
ondar
y
capacitor
.
Besides
,
s
(
t
)
is
the
s
witch
function,
which
can
be
e
xpressed
as
,
s
(
t
)
=
(
1
nT
t
<
(2
n
+
1)
T
=
2
0(2
n
+
1)
T
=
2
t
<
(
n
+
1)
T
.
(1
3)
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3701
Because
the
circuit
model
of
the
witr
icity
system
is
a
resonant
con
v
er
ter
which
can
filter
out
har
monic
components
,
the
dynamic
system
associated
with
a
z
ero
and
first
order
F
our
ier
coefficient
e
xpressions
can
pro
vide
the
enou
gh
accur
acy
.
So
,
b
y
using
GSSA
method,
the
state
v
ar
iab
les
can
be
e
xpressed
b
y
the
z
ero
and
first
order
components
and
the
function
is
as
f
ollo
ws
,
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
h
i
1
i
1
=
x
1
+
j
x
2
h
u
c
1
i
1
=
x
3
+
j
x
4
h
i
2
i
1
=
x
5
+
j
x
6
h
u
c
2
i
1
=
x
7
+
j
x
8
h
i
1
i
0
=
x
9
h
u
c
1
i
0
=
x
10
h
i
2
i
0
=
x
11
h
u
c
2
i
0
=
x
12
.
(14)
Then,
from
equation
11,
the
diff
erential
equations
in
state-space
ha
v
e
the
f
ollo
wing
matr
ix
f
or
m,
d
~
x
dt
=
A
~
x
+
U
dc
.
(15)
Where
,
~
x
representing
the
z
ero-order
and
first-order
state
v
ar
iab
les
,
is
defined
as
a
column
v
ector
,
~
x
=
x
1
x
2
x
3
x
4
:::
x
9
x
10
x
11
x
12
T
.
(16)
In
addition,
the
matr
ix
A
can
be
divided
into
f
our
which
is
e
xpressed
as
,
A
=
A
1
0
0
A
2
,
(17)
where
A
1
and
A
2
are
sho
wn
as
f
ollo
ws
,
A
1
=
2
6
6
6
6
6
6
6
6
6
6
6
4
Z
r
L
1
1
L
1
0
0
1
C
1
0
0
0
j
!
M
L
2
0
0
1
L
2
0
0
1
C
2
1
R
L
C
2
3
7
7
7
7
7
7
7
7
7
7
7
5
(18)
Title
of
man
uscr
ipt
is
shor
t
and
clear
,
implies
research
results
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and
A
2
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
Re
(
Z
r
)
L
1
!
+
Im
Z
r
L
1
1
L
1
0
0
0
0
0
!
Im
Z
r
L
1
Re
Z
r
L
1
0
1
L
1
0
0
0
0
1
C
1
0
0
!
0
0
0
0
0
1
C
1
!
0
0
0
0
0
0
!
M
L
2
0
0
0
!
1
L
2
0
!
M
L
2
0
0
0
!
0
0
1
L
2
0
0
0
0
1
C
2
0
1
R
L
C
2
!
0
0
0
0
0
1
C
2
!
1
R
L
C
2
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(19)
In
addition,
the
coefficient
b
is
e
xpressed
as
,
b
=
0
0
0
0
0
2
L
1
:::
0
0
.
(20)
T
o
obtain
the
n
umer
ical
solution,
t
he
contin
uous
linear
model
should
be
tr
ansf
er
to
the
discrete
mode
first.
Through
the
equation
(15)
and
(16),
the
model
is
coded
in
MA
TLAB
,
and
the
results
are
compared
with
the
sim
ulation
results
in
time
domain
b
y
PSIM
softw
are
pac
kage
.
x
(
k
+
1)
=
x
(
k
)
+
u
(
k
)
(21)
[
;
]
=
f
(
A;
B
;
T
s
)
(22)
where
,
k
=
0
;
1
;
2
:::n:
The
par
ameter
v
alues
is
sho
wn
in
tab
le
1,
and
the
sim
ulat
ion
results
are
sho
wn
in
figure
4
and
figure
5.
Figure
4
sho
ws
the
w
a
v
ef
or
ms
of
resonant
current
of
pr
imar
y
side
,
in
which
the
red
line
represents
the
solution
from
GSSA
while
the
b
lue
lines
is
the
results
from
the
PISM
pac
kage
.
T
ab
le
1.
The
P
erf
or
mance
of
the
witr
icity
system
V
ar
iab
le
Result
Unit
L
1
18
H
C
1
800
pF
L
2
18
H
C
2
800
pF
M
5
H
R
L
12
Compar
ing
cur
v
es
in
figure
4,
the
resonant
current
cur
v
es
based
on
GSSA
method
is
v
er
y
similar
to
the
sim
ulation
result
from
PSIM
pac
kage
.
And
figure
5
sho
ws
the
errors
of
these
tw
o
methods
.
F
rom
which
w
e
can
see
that
the
errors
is
alw
a
ys
under
0.2
A.
thus
,
it
is
clear
ly
that
the
first
appro
ximation
is
sufficiently
precise
.
TELK
OMNIKA
V
ol.
12,
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.
5,
Ma
yl
2014
:
3697
3704
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TELK
OMNIKA
ISSN:
2302-4046
3703
0
2
2
1
1
0
4
2
6
T
i
m
e
(
s
)
C
u
r
r
e
n
t
(
A
)
Figure
4.
Resonant
current
of
pr
imar
y
side
using
GSSA
and
sim
ulation
method
0
0
.
5
0
.
5
0
.
2
0
.
2
0
1
0
5
1
5
T
i
m
e
(
s
)
C
u
r
r
e
n
t
(
A
)
Figure
5.
Errors
betw
een
the
GSSA
method
and
the
sim
ulation
method
4.
Conc
lusion
In
this
paper
,
a
tutor
ial
mater
ial
of
a
method
f
or
modelling
the
witr
icity
system
is
presented.
The
GSSA
me
thod
has
been
re
vie
w
ed
first.
Then
the
circuit
model
of
the
witr
icity
is
introduced.
Based
on
which,
the
mathematical
model
is
der
iv
ed
using
GSSA
method,
and
the
steps
are
pre-
sented
in
detail.
The
results
of
GSSA
are
compared
with
the
sim
ulation
from
PSIM
pac
kage
in
time
domain,
which
sho
ws
that
there
is
little
error
betw
een
each
other
.
The
n
umer
ical
analyses
sho
w
that
a
first
order
appro
ximation
of
GSSA
is
enough
to
present
the
precision
of
a
witr
icity
sys-
tem,
and
there
is
no
need
to
include
higher-order
ter
ms
in
the
analysis
.
The
models
der
iv
ed
from
the
GSSA
method
are
suitab
le
f
or
dynamic
analysis
of
the
witr
icity
system,
and
its
computational
time
has
been
consider
ab
ly
reduced
compared
to
other
time
domain
sim
ulation
method.
Ac
kno
wledg
ement
This
w
or
k
is
suppor
ted
b
y
the
National
Natur
al
Science
Fund
of
China
(51277120)
and
the
ITER
project(2011GB113005).
The
authors
also
thanks
to
the
help
b
y
State
Energy
Smar
t
Gr
id
R&D
Center(Shanghai).
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3704
ISSN:
2302-4046
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