In
te
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n
ation
a
l Jou
rn
al
o
f Po
we
r
Elec
tron
ic
s an
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D
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ive S
y
stem
(IJ
PED
S
)
V
o
l.
10, N
o.
1, Mar
ch 20
19,
p
p.
538~
5
4
7
IS
S
N
: 2088-
86
94,
D
O
I
:
10.11
59
1
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ped
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v10
.
i
1.pp
5
38-
54
7
538
Jou
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score
.
com
/
j
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p
hp/IJ
PED
S
Dynamic analysis of the hi
gh-p
ower factor three-phase AC to
DC converter u
s
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id resonant technique
R
a
h
i
mi B
ah
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om,
Moh
a
m
m
ad
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awa
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i S
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Facul
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E
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r
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Engi
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,
U
niversiti T
e
knol
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,
M
alay
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a
Art
i
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fo
ABSTRACT
A
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R
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Jun
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2
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c
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K
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:
AC-DC converter
Cur
r
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n
jec
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Dy
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a
mi
c
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alysi
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H
ybri
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on
ve
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Co
pyri
gh
t © 2
019 In
stit
u
t
e
of Advanced
En
gi
neeri
n
g
an
d
S
c
ien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
R
a
h
i
mi
B
ah
arom,
Fa
cult
y
o
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l
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c
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rica
l
En
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erin
g,
Un
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r
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Tekn
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A
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A,
4
045
0
Sh
ah
Ala
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S
el
a
n
go
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M
al
a
y
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a
.
Em
ail:
rahim
i
6
5
7
9
@
gm
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TR
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odul
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o
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o
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i
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perfor
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term
s
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pow
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f
actor,
eff
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c
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cy,
r
e
l
i
ab
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i
mp
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p
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s
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three
-
p
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ase
A
C
-D
C
co
n
v
erte
r
s
[
1],
[2]
.
I
n
a
d
d
i
t
i
o
n
to
t
ha
t,
t
he
c
urr
e
nt
i
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t
i
o
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t
ec
h
n
i
q
ue
w
as
mor
e
p
ra
ctica
l
due
t
o
i
t
s
sim
p
li
c
i
ty
[
3].
B
y
u
s
i
ng
the
hybr
id
r
eso
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an
t
co
nfi
gura
t
i
o
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t
he
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c
urrent
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y
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rob
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em
o
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t
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se
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c
irc
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c
onf
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at
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c
a
n
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a
ll
ow
i
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the
c
o
n
t
ro
l
of
t
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p
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t
vo
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t n
o
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oad
or sm
a
ll l
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d
it
io
ns.
M
o
r
e
over,
t
he
u
se
o
f
hi
g
h sw
i
t
ch
i
ng fr
eq
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n
c
y w
i
ll
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t
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to
s
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a
ll
ele
m
e
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t
val
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o
f
t
h
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re
so
n
a
nt
t
a
n
k
an
d
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nce
impr
ov
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n
g
the
po
w
e
r
de
nsit
y
of
t
h
e
c
o
nv
e
r
t
e
r t
opolo
g
y
.
I
n
t
h
i
s
rese
arc
h
p
a
p
er,
dyna
m
i
c
anal
ys
is
o
f
the
thre
e-p
h
a
s
e
A
C
-
D
C
C
IH
RC
ha
s
bee
n
d
eve
l
o
p
e
d
.
Thi
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n
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el
a
n
a
l
y
si
s
h
a
v
e
b
een
d
eriv
e
d
b
as
e
d
o
n
t
h
e
st
ea
dy
-st
a
t
e
c
irc
u
it
a
n
al
ys
i
s
a
s
de
vel
o
ped
in
[
4
]
,
[5]
t
o
impro
v
e
t
h
e u
n
d
ersta
n
d
i
ng of the be
h
av
i
our of the
c
i
rc
u
it top
o
l
ogy
i
n
t
e
rms
o
f
i
t
s
c
ont
ro
ll
a
b
i
lit
y
,
s
t
a
b
i
l
i
t
y
a
nd
susta
i
na
bil
i
t
y.
F
urthe
r
mor
e
,
the
de
ve
l
o
p
e
d
dyna
mic
a
n
a
l
y
s
i
s
c
a
n
p
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Int J
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k
e
t
h
i
s
c
i
r
cu
it t
o
pol
og
y o
p
e
r
a
te
a
s
hy
br
id
r
eso
n
a
n
t
co
nver
t
e
r
.
F
i
gure
1.
The
t
hree
-phase
A
C-
D
C
c
urr
e
nt in
j
ecti
o
n hy
bri
d
r
eso
n
a
nt
con
v
e
rt
e
r
t
op
ol
o
g
y
3.
DYNA
M
I
C A
NALYS
I
S
OF
T
H
REE
-
PHAS
E
A
C
TO
DC CONVE
RTER US
I
NG
CURREN
T
INJ
E
C
T
ION HYBRI
D
RESONANT TE
C
H
N
IQUE
Th
is
s
ec
ti
on
d
e
scr
i
be
s
the
d
e
riva
ti
o
n
o
f
a
dy
nam
i
c
a
n
a
l
ysis
f
or
t
he
t
hr
e
e
-pha
se
A
C
t
o
D
C
us
in
g
curr
ent
i
n
j
e
c
t
i
on
hy
bri
d
r
e
s
o
n
a
n
t
t
e
ch
n
i
q
u
e
.
T
he
d
y
n
am
ic
a
nal
y
s
i
s
f
or
t
he
C
IH
RC
w
a
s
done
b
y
c
o
n
s
i
d
er
i
n
g
tha
t
t
he
c
o
nver
t
e
r
w
oul
d i
n
v
o
l
ve tw
o
s
t
a
ges
:
a.
The
line
-
fre
q
u
e
ncy
rec
tifie
r
b.
H
i
g
h
-fre
q
u
enc
y
r
esona
nt
c
irc
u
i
t
The
a
n
a
l
ys
is
o
f
l
i
n
e-
freq
ue
nc
y
of
t
he
t
hree
-phase
P
WM
A
C
t
o
D
C
c
o
nv
e
r
ter
w
a
s
b
a
se
d
on
t
he
st
a
ndard
m
eth
od
a
s
d
isc
u
sse
d
i
n
[
6].
The
r
e
su
lt
ing
c
i
rcu
i
t
equ
a
t
i
ons
t
ha
t
w
e
re
e
xpre
s
se
d
i
n
s
ta
t
e
-spac
e
for
m
w
e
r
e
t
hen
a
v
er
aged
t
o
rem
o
v
e
t
he
r
ip
p
l
e.
A
fter
t
ha
t,
t
he
d
irec
t
a
n
d
q
u
a
dra
t
u
r
e
(d
-q
)
t
r
a
n
s
f
o
r
mat
i
o
n
met
hod
w
a
s
a
dop
t
e
d
t
o
e
l
i
mina
t
e
t
he
time
var
i
a
n
ce
i
n
the
eq
ua
t
i
o
n
s.
I
n
o
rd
er
t
o
mo
d
e
l
th
e
high
-f
re
qu
e
n
c
y
r
e
s
o
n
a
n
t
st
a
g
e,
t
he
f
und
a
m
e
n
t
a
l
freq
u
e
n
c
y
m
e
t
ho
ds
a
s
disc
usse
d
i
n
[
7]
w
e
re
a
dopte
d
.
To
m
atc
h
t
he
l
i
n
e
fre
q
uenc
y
equa
t
i
o
n
s
of
t
h
e
t
hr
ee-
ph
ase
P
W
M
A
C
-D
C
c
o
n
v
e
r
ter
w
ith
t
he
h
ig
h-
fre
q
uenc
y
r
e
so
nan
t
s
tage
e
qua
t
i
o
n
s,
t
h
e
pow
er
bala
n
c
e
d
r
elat
io
ns
f
or
t
he D
C
l
i
n
k
m
etho
ds w
e
r
e
e
m
plo
y
e
d
.
3
.
1
.
L
i
ne
f
requency
rec
t
ifier
F
i
gure
2
shows
the
s
i
mplified
d
i
a
gram
o
f
the
line-frequency
r
ec
tifier
.
T
he
t
ota
l
D
C-
l
i
n
k
vol
ta
ge
V
DC
w
a
s
ob
tai
n
e
d
f
rom
a
se
ries
c
on
nec
t
ed
o
f
the
tw
o
sp
lit
D
C
l
ink
c
a
p
a
c
it
ors
repr
esen
ted
b
y
t
he
D
C
l
i
n
k
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
53
8 –
54
7
54
0
ca
paci
t
o
r,
C
DC
.
The
l
i
ne
c
urre
nts
a
nd
s
u
p
p
l
y
v
o
lta
ge
s
w
e
r
e
r
epre
sente
d
by
i
i
a
n
d
V
i
r
esp
e
ct
iv
el
y
,
w
h
e
re
i
repr
esents
t
he
1
,
2
a
n
d
3
pha
ses.
T
he
l
i
n
e
i
nduc
tance
an
d
its
s
m
a
l
l
r
e
s
i
s
t
a
n
c
e
w
e
r
e
r
e
p
r
e
s
e
n
t
e
d
b
y
L
s
a
nd
R
s
re
sp
ec
ti
v
e
ly
.
Th
e
t
h
ree
-
ph
a
s
e
di
ode
b
ridge
rectif
ier
were
d
enot
ed
b
y
D
j
,
wher
e
j
=1…
6
.
T
h
e
s
w
it
c
h
in
g
fu
nc
ti
o
n
o
f
eac
h
d
i
odes
le
g
w
e
r
e
r
epr
e
sent
ed
b
y
S
i
w
h
e
r
e
i
i
s
one
‘
1’
w
hen
t
h
e
up
pe
r
dio
d
e
o
f
t
he
l
eg
i
s
c
o
ndu
c
tin
g
whi
l
e
i
i
s
ze
ro ‘0’
w
he
n t
h
e
l
o
w
e
r dio
d
e
is
c
on
d
u
ct
i
ng.
F
i
gure
2.
A
si
m
plifie
d o
f
t
he
line freq
u
e
n
c
y
A
C-D
C
c
onve
rter
Ba
se
d
o
n
t
h
e
c
irc
u
it
c
o
n
f
ig
ura
tio
n
sh
ow
n
in
F
igure
1,
t
he
p
ha
se
v
ol
t
a
ge
e
qua
tio
n
ca
n
be
d
erive
d
b
y
app
l
yi
ng
th
e Kir
c
h
h
o
f
f's v
o
lta
ge
l
aw
,
re
sult
in
g in
:
v
i
R
L
v
v
(
1
)
wher
e
R
s
i
s
the
seri
e
s
r
es
i
s
t
a
n
c
e
of
t
h
e
l
i
n
e
i
ndu
ct
o
r
.
v
DN
i
s
the
in
pu
t
vo
lt
age
o
f
t
he
f
irst
l
eg
a
t
p
o
in
t
D
,
w
i
t
h
respe
c
ts
t
o
t
h
e
nega
tive
po
i
n
t,
N
o
f
the
D
C
-
l
i
n
k.
v
NO
i
s
t
h
e
vo
l
t
a
g
e
betw
ee
n
poi
nt
N
a
nd
t
he
s
t
a
r
p
o
i
nt
o
f
the
in
put
s
u
p
p
l
y
v
o
l
t
a
g
e.
B
y
c
o
n
s
i
d
er
i
n
g
the
s
w
itc
hi
n
g
f
un
ct
i
on
n
o
t
a
t
io
n
of
v
DN
=
V
DC
S
1
,
w
h
e
r
e
S
1
i
s
a
func
t
i
o
n
of
c
o
n
duc
t
i
ng
of
d
i
ode
D
1
,
t
h
u
s
,
(
1
)
c
a
n
b
e
s
i
mp
l
i
f
i
ed
a
s
:
v
i
R
L
V
S
v
(
2
)
The me
tho
d
fo
r
der
i
v
e
t
h
e fir
s
t
p
h
a
s
e
e
q
ua
tion ca
n
be ap
p
l
i
e
d t
o so
l
v
e
pha
se
s 2 an
d
3.
T
h
e
refore,
th
e
pha
se
2
a
n
d
p
h
a
se
3
e
q
u
a
tio
ns
c
an
b
e
de
rive
d
a
s
;
v
i
R
L
V
S
v
(
3
)
v
i
R
L
V
S
v
(
4
)
F
o
r
a
three
-
pha
se
s
yste
m
w
ith
ou
t a
neu
t
ra
l
l
i
ne
,
i
i
i
0
(
5
)
By
a
ssum
i
n
g
t
ha
t
the
t
h
re
e-pha
se
A
C
su
pp
l
y
v
o
lta
ge
s
w
e
r
e
b
a
l
anc
e
d
,
t
h
e
su
mmat
ion
of
t
h
e
in
sta
n
ta
ne
o
u
s vo
l
t
a
g
e
s
w
il
l b
e
e
qua
l
to
z
e
r
o.
v
v
v
0
(
6
)
Re
arr
a
ng
i
ng (2
)
,
(
3) a
nd (4)
wil
l
r
e
s
ul
t i
n
;
i
L
di
dt
V
S
v
v
R
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
D
y
nam
ic
a
n
a
l
y
si
s
o
f
the h
i
gh-
po
wer f
a
c
t
o
r
t
h
ree
-
p
h
a
s
e
AC
to D
C
conve
r
t
e
r
using
…
(Rahim
i
Ba
har
om
)
54
1
i
L
di
dt
V
S
v
v
R
i
L
di
dt
V
S
v
v
R
S
ubst
i
t
ut
i
ng t
h
e a
bove
i
n
t
o (
5
),
h
ence
;
v
V
∑
S
(
7
)
Th
e
in
put
s
i
d
e
o
f
t
h
e
c
on
v
e
rte
r
c
a
n
b
e
obt
a
i
n
e
d
by
s
ub
st
itu
t
in
g
(
7)
i
n
t
o
(2)
,
(
3)
a
nd
(
4
)
repr
esente
d
by t
h
e
fol
l
ow
i
n
g:
L
ı
R
i
S
∑
S
V
v
(
8
)
L
ı
R
i
S
∑
S
V
v
(
9
)
L
ı
R
i
S
∑
S
V
v
(
1
0
)
or in
the
m
a
trix form
as:
(
11)
Her
e
:
i
i
i
R
00
0
R
0
00
R
L
00
0L
0
00
L
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎡
S
1
3
S
V
v
S
1
3
S
V
v
S
1
3
S
V
v
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎤
The
circu
i
t (
11)
ca
n
be
simpli
fie
d
by
ave
r
ag
i
ng t
h
e
sw
itc
hi
n
g
f
u
n
c
t
i
ons,
henc
e
rem
ovi
n
g
t
he hi
g
h
fre
que
nc
y
inf
o
rma
t
i
o
n
[8].
F
u
r
therm
o
re,
by
e
m
pl
o
y
i
n
g
t
he
t
ransf
or
ma
t
i
o
n
t
o
a r
o
t
a
tin
g
refe
renc
e
fra
m
e,
t
he
sup
p
l
y
f
re
qu
e
n
cy tim
e va
ria
n
ce
c
an be
rem
o
v
e
d.
3.1.
1. Averagin
g
p
roced
u
re
F
o
r
a nat
u
ra
l
s
a
m
p
lin
g sin
u
s
o
ida
l
pu
l
se-w
i
d
t
h
m
od
u
l
a
tio
n,
t
he s
wi
t
c
h
i
ng
p
o
i
nt
s
with
in
o
ne
s
wi
t
c
hi
ng
peri
od
a
r
e
n
o
t
s
ym
me
trical.
H
o
w
e
ver
,
w
hen
the
sw
i
t
c
h
ing
frequ
en
c
y
i
s
m
u
ch
h
i
g
he
r
th
an
t
h
e
l
i
n
e
fre
que
nc
y,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
53
8 –
54
7
54
2
the
m
o
d
u
la
t
i
n
g
w
a
ve
c
a
n
b
e
rega
rded
a
s
a
c
onsta
n
t
w
ith
i
n
e
ach
s
w
i
tc
h
i
ng
pe
ri
od.
T
h
e
refore,
the
sw
itc
h
i
ng
pat
t
ern
is
c
los
e
t
o
be
s
ymm
e
tric
a
l
[
9]
.
In
t
hi
s
w
a
y,
t
he
l
oca
l
ave
r
age
v
a
l
u
e
or
dut
y
rat
i
o
S
can
b
e
used
t
o
repla
ce
t
h
e
sw
itc
hi
ng fu
nc
ti
o
n
S
i
i
n
a
ny
s
p
ec
ifi
c
p
e
r
i
od.
T
he
u
se
o
f
c
u
rrent i
n
j
e
c
t
i
o
n
sin
u
s
o
i
d
a
l
P
W
M
sc
hem
e
resul
t
i
ng
i
n
t
he
s
up
p
l
y
c
u
rre
n
t
t
ha
t
is
i
n
pha
s
e
w
ith
t
he
f
u
n
d
a
m
e
nta
l
o
f
t
h
e
P
W
M
v
o
l
tage,
V
PW
M
1
.
T
h
e
V
PW
M
1
i
s
l
a
ggi
ng
t
h
e
su
ppl
y
vo
lt
ag
e
,
V
s
b
y
an
a
ngl
e
ϕ
as show
n
in F
igure
3,
w
it
h the
a
ssump
t
i
on
tha
t t
h
e p
h
ase
ang
l
e
w
a
s
unaffe
c
te
d
w
i
t
h
t
he
sm
a
ll
re
s
istanc
e R
s
.
F
i
gure
3.
F
un
d
a
me
nta
l
fr
e
que
nc
y p
h
as
or
d
i
a
gram
By
r
efe
rrin
g
t
o
t
h
e
p
h
a
s
or
d
ia
gram
s
how
n
i
n
F
ig
ure
2
an
d
by
u
si
ng
t
h
e
r
e
l
at
io
nshi
p
i
n
(
2
)
,
i
t
ca
n
b
e
seen tha
t t
h
e fu
ndam
e
n
t
a
l
of t
h
e P
W
M
v
o
lta
ge o
f pha
se
i
=
R
,
Y
,B
w
h
e
re
R
=
1
,
Y
=
2
a
nd
B=3
ca
n
b
e
e
xp
re
sse
d
as;
v
f
M
sin
ωt
∅
i
1
(
12)
And for phas
e i,
L
ı
i
R
V
v
(
1
3
)
There
f
ore
,
t
he
l
oca
l
a
ve
rage
s
w
itc
hi
n
g
f
unc
ti
on
S
in m
a
tr
i
x
e
c
oul
d be ob
t
a
i
ne
d by
c
o
mpa
r
in
g
(1
3)
w
it
h
(
1
1)
.
Th
us, the
re
s
ul
ti
n
g
equa
ti
on c
a
n
be e
xpresse
d
as:
S
∑
S
sin
ωt
∅
i1
(
14)
Fi
n
a
lly
, t
h
e ma
t
r
ix
e
in (
11) c
an be
simp
li
fie
d
a
s foll
ow
s:
e
⎣
⎢
⎢
⎢
⎡
V
sin
ωt
∅
v
V
si
n
ωt
∅
v
V
si
n
ωt
∅
v
⎦
⎥
⎥
⎥
⎤
(
1
5
)
3.1.
2.
El
im
in
a
t
i
o
n
of
t
h
e
t
ime-var
i
a
n
c
e
The
trans
f
or
ma
t
i
o
n
t
o
a
ro
t
a
ti
n
g
r
efe
r
enc
e
f
ra
me
t
ec
hn
i
q
ue
w
a
s
a
d
opte
d
t
o
e
l
i
m
in
ate
the
tim
e
-
vary
in
g
sw
itc
hin
g
f
u
n
ct
i
on
in
(
11)
a
nd
use
d
t
o
furt
her
si
m
p
li
fy
t
he
m
ode
l.
I
n
th
is
w
a
y
,
t
h
r
ee-
phase
q
uan
t
i
tie
s
w
e
r
e
t
ransfor
m
e
d
t
o
d
i
rec
t
a
n
d
qua
drat
ure
axi
s
(
d-q)
c
om
p
one
n
t
s
u
s
i
n
g
t
he
t
r
a
nsform
ati
on
ma
t
r
i
x
,
T
-1
.
Th
en
,
the
d-q
q
u
an
t
i
t
i
e
s
were
t
rans
for
m
e
d
b
ack
t
o
t
h
e
i
r
t
h
ree
-
p
h
ase
q
ua
n
tit
ies
us
in
g
the
in
v
e
rse
m
a
trix
T
.
Th
i
s
proce
s
s
i
s
k
n
o
w
n
as
t
he
n
on-
pow
e
r
i
n
v
a
r
ia
n
t
P
ark
transfo
r
ma
t
i
o
n
[
8].
The
r
efore,
t
he
t
ransfor
m
a
t
i
o
n
ma
t
r
ix
T
-
1
c
an
b
e e
xpressed
as;
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
D
y
nam
ic
a
n
a
l
y
si
s
o
f
the h
i
gh-
po
wer f
a
c
t
o
r
t
h
ree
-
p
h
a
s
e
AC
to D
C
conve
r
t
e
r
using
…
(Rahim
i
Ba
har
om
)
54
3
⎣
⎢
⎢
⎢
⎡
cos
ω
t
c
os
ωt
c
o
s
ω
t
s
i
n
ω
t
s
i
n
ω
t
s
i
n
ω
t
⎦
⎥
⎥
⎥
⎤
(
16)
Mea
n
w
h
ile,
th
e
i
nve
rse
ma
t
r
ix
T
i
s
gi
v
e
n
by:
co
s
ω
t
si
n
ω
t
1
co
s
ω
t
s
i
n
ω
t
1
co
s
ω
t
s
i
n
ω
t
1
(
1
7
)
The
re
la
tio
ns
h
i
p
be
tw
ee
n
the
ve
ct
or
x
i
n
the
st
a
t
i
onar
y
f
ram
e
a
nd
the
new
sta
t
e
ve
c
t
or
x
r
i
n
t
h
e
rota
ti
n
g
fr
a
m
e
c
an
b
e e
xpress
e
d
as foll
o
ws:
i
i
i
(
18)
The
ori
g
ina
l
s
ta
te ve
c
t
or a
lt
e
r
nat
i
vel
y
c
an be
expr
essed a
s
:
(
19)
w
h
er
e
the
d,
q
,
and
zer
o
s
e
que
nc
e
c
o
mp
one
nts
of
t
he
l
i
n
e
c
u
rre
n
t
s
a
re
r
epr
e
sent
ed
b
y
i
D
,
i
Q
a
nd
i
o
respe
c
t
i
ve
l
y
. D
i
f
fer
e
n
t
i
a
tin
g
(19) g
ives
:
(
20)
By
s
ubst
i
t
u
tin
g
(
1
9)
a
n
d
(
20)
i
nto
(11),
yiel
d
i
ng
:
(
21)
Mu
lt
ip
lyi
n
g ou
t
and r
ear
rangi
ng (2
1)
re
s
u
l
t
i
ng i
n
:
(
22)
By
u
sing
ZT
=
TZ
a
s
follow
s
,
w
h
ich
g
i
ve
s
:
(
23)
Th
e
r
efo
r
e,
(
24)
Whe
r
e,
i
s
;
R
ωL
0
ωL
R
0
00
R
(
2
5
)
and
from
(25)
,
T
-1
e
can
b
e
se
para
te
d in
t
o
:
⎣
⎢
⎢
⎢
⎢
⎡
V
si
n
ωt
∅
V
si
n
ωt
∅
V
si
n
ωt
∅
⎦
⎥
⎥
⎥
⎥
⎤
v
v
v
(
26)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
53
8 –
54
7
54
4
Th
e
r
efo
r
e;
⎣
⎢
⎢
⎢
⎢
⎡
V
sin
ωt
∅
V
sin
ωt
∅
V
sin
ωt
∅
⎦
⎥
⎥
⎥
⎥
⎤
V
si
n
∅
V
cos
∅
0
(
27)
A
ssumin
g
t
hat
t
h
e
d-
q
t
r
ansfo
r
m
a
tio
n of
t
he thre
e
-p
ha
se su
p
p
l
y
vo
l
t
a
g
e
yie
l
d
s
:
v
v
v
V
V
0
(
2
8
)
wher
e
V
D
a
nd
V
Q
a
re
t
he
d
a
nd
q
c
om
p
one
n
t
s
of
t
he
t
hr
ee-
p
h
ase
s
u
p
p
l
y
v
o
l
t
a
g
es
v
1
,
v
2
,
and
v
3
.
B
y
sub
s
t
i
t
u
tin
g
(2
5),
(26),
(
27)
a
nd
(
2
8
)
i
n
to
(
2
4
),
t
he
t
hre
e
-p
ha
se
l
i
n
e
f
r
e
quen
c
y
rec
t
i
f
i
e
r
can
b
e
rep
r
e
s
ent
e
d
by
the f
o
l
l
ow
i
n
g set
of t
ime
-
in
va
rian
t,
state-spa
ce
equat
i
o
n
s in
a
r
otating fram
e
a
s
the
reference:
L
00
0L
0
00
L
ı
ı
ı
R
ωL
0
ω
L
R
0
00
R
i
i
i
f
M
si
n
∅
V
f
M
co
s
∅
V
0
(
29)
By
n
e
g
l
e
c
t
ing
the
ne
u
t
r
a
l
l
i
ne
i
n
t
h
e
sym
m
e
t
r
i
cal
t
hree
-pha
se
s
ystem
,
t
he
z
ero
seque
nce
c
o
mpo
n
e
n
t
i
0
c
a
n
b
e
ig
no
re
d
.
B
y
u
s
in
g
t
h
e
t
r
a
n
sf
o
r
ma
tio
n
t
o
a
rot
a
ti
ng
r
e
f
e
re
nce
fra
m
e
m
etho
d,
t
h
e
d
-
a
x
i
s
is
9
0
°
a
he
a
d
of the
pha
sor
o
f
the su
p
p
l
y
vo
lta
ge
, result
ing
i
n
V
D
= 0
a
n
d
V
Q
√
2
V
s
.
A
s
a
r
esul
t
of t
his, the tim
e varia
n
t
equa
t
i
o
n
s c
a
n
be
expr
esse
d b
y
m
ult
i
pl
y
i
ng
ou
t (29)
to be
.
L
ı
R
i
ω
L
i
f
M
sin
∅
(
3
0
)
L
ı
ω
L
i
R
i
f
M
co
s
∅
√
2
V
(31)
sin
ϕ c
a
n
be ob
t
ai
ne
d
from
the
f
un
dam
e
n
t
a
l
fre
que
ncy
p
h
a
s
or dia
gr
am
in F
i
g
u
r
e
3.
sin
Φ
(
3
2
)
The
d-q tra
n
sf
orm
a
tion
o
f
t
h
e
l
i
n
e
c
u
rren
t
g
i
v
e
s
:
I
√
i
i
(
3
3
)
i
i
tan
∅
Th
e
r
efo
r
e,
√
2
I
i
i
tan
∅
√
2
I
i
1
1
tan
∅
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
D
y
nam
ic
a
n
a
l
y
si
s
o
f
the h
i
gh-
po
wer f
a
c
t
o
r
t
h
ree
-
p
h
a
s
e
AC
to D
C
conve
r
t
e
r
using
…
(Rahim
i
Ba
har
om
)
54
5
As a resul
t
of;
i
i
2t
a
n
∅
1
t
a
n
∅
F
i
nal
l
y
,
t
h
e
re
sona
n
t
cur
rent
c
an
b
e de
ri
ved a
s
:
I
(
3
4
)
3.2.
High
-
f
req
u
e
n
c
y r
e
son
a
n
t
st
a
g
e
Base
d
o
n
t
he
f
un
dam
e
n
t
a
l
freque
nc
y
a
p
pro
x
i
ma
t
i
o
n
a
s
disc
usse
d
i
n
[4],
t
h
e
h
igh-
fre
q
u
e
n
c
y
resona
n
t
st
a
g
e
w
a
s
ob
ta
ine
d
b
y
a
ssum
i
ng
tha
t
t
he
r
e
s
ona
n
t
c
i
r
cu
i
t
r
e
s
po
n
de
d
im
m
e
dia
t
e
l
y
t
o
t
h
e
s
ma
l
l
-
s
i
gna
l
c
h
a
n
ge
s.
Under
trans
i
e
n
t
c
o
n
d
i
t
io
n
s
,
t
h
e
dif
f
ere
n
ce
b
etwe
e
n
t
he
t
o
t
al
a
c
t
i
ve
pow
e
r
f
l
o
w
i
n
g
i
n
t
o
the
D
C
-
l
i
n
k
t
h
ro
ug
h
the
t
h
re
e-pha
se
r
ecti
f
ier
a
nd t
h
e p
o
w
e
r draw
n thro
u
g
h
t
h
e tr
ans
ist
o
r sw
it
c
h
in
g leg
m
u
st
f
low
int
o
or
o
u
t
o
f
t
h
e
D
C
-lin
k
capa
c
i
t
o
r.
T
hi
s
i
s
t
he
d
y
n
am
i
c
v
e
r
si
o
n
o
f
t
h
e
st
e
a
dy-s
t
ate
p
o
wer
bala
nce
in
(
15)
,
w
h
e
r
e
it
w
as
assum
e
d t
h
at
t
he
di
ffe
re
nc
e i
n
p
o
w
er
w
as zero
[8].
The
refore;
3V
I
V
I
√
I
cos
θ
V
C
(
35)
B
y
u
s
i
ng
θ
DC-D
C
in [5]
to el
im
inate the
θ
DC
-
D
C
term
s
in (
35),
i
t
w
ill
resul
t
i
n:
3V
I
V
I
√
√
V
C
3V
I
V
C
(
3
6
)
wher
e;
V
V
2
√
2
f
M
By
su
b
s
tit
ut
i
ng V
PW
M
1
int
o (
36)
,
y
i
e
l
ds;
C
√
(
3
7
)
In or
d
er
t
o
eli
m
ina
t
e
the
cos
θ
term
f
r
om [5] yielding;
√
√
√
I
Z
√
2
0
(
38)
In
o
rder
t
o
c
o
m
p
l
e
te
t
he
m
ode
l
of
t
he
c
on
verter
,
t
h
e
dy
n
a
m
i
c
r
e
la
t
i
onsh
i
p
be
tw
ee
n
t
h
e
o
u
t
put
vo
lta
ge,
V
o
a
nd
the
ou
tpu
t
c
urre
n
t
,
I
o
w
a
s
r
e
q
u
i
r
e
d
.
I
o
i
s
the
loc
a
l
a
ve
ra
ge
o
f
the
r
ect
ified
br
i
d
ge
c
ur
rent,
I
B
tha
t
w
a
s
f
e
d
i
n
t
o t
h
e para
l
l
e
l
com
bina
tio
n of
t
he l
oa
d
r
e
sis
t
or
an
d
fi
l
t
e
r ca
pa
ci
tor
a
s
i
llus
t
rated in
F
ig
ure
4 an
d
F
i
gure
5.
I
Z
i
s
the
sm
al
l
c
u
rr
ent
di
sturba
nce
pl
a
c
e
d
i
n
para
llel
w
i
t
h
t
he
l
o
a
d
resis
t
or
R
L
a
nd
is
a
n
a
n
a
l
yt
ica
l
to
ol
t
o
e
x
a
m
in
e
t
h
e
l
o
ad
c
ur
rent
c
ha
nge
s.
T
he
d
i
ffe
ren
t
ial
equa
ti
on
f
o
r
t
he
o
ut
p
u
t
f
i
lt
e
r
c
a
p
ac
it
o
r
,
C
o
c
an
b
e
expre
s
se
d a
s
:
I
I
C
(
3
9
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SSN: 2088-
8694
I
nt
J
P
ow
Elec
& Dr
i
S
y
st, Vol. 10,
N
o.
1, Mar
c
h 2
0
1
9
:
53
8 –
54
7
54
6
wher
e,
C
dV
dt
I
I
V
R
I
I
I
F
i
gure
4.
O
utput
r
ec
tifie
r un
it
under
d
ynam
i
c c
ond
iti
o
n
s
F
i
gure
5.
C
ur
rent
f
low
thro
ug
h ou
t
p
u
t
rec
t
ifie
r un
i
t
u
nder
trans
i
e
n
t
c
o
n
d
i
t
i
on
s
So
,
I
2
π
√
2
I
I
I
I
I
C
dV
dt
Hen
c
e,
I
√
2
I
C
(
4
0
)
Whe
r
e,
V
I
R
(
4
1
)
S
ubst
i
t
ute
(41)
i
nt
o (
39),
result
ing
i
n
;
√
√
(
4
2
)
and
fu
nct
i
o
n
v
r
epr
e
sents t
h
e
rig
h
t
h
an
d si
de
o
f (
42).
4.
CONCL
U
S
ION
I
n
t
h
i
s
pa
per
,
a
d
yn
a
m
ic
a
na
ly
sis
o
f
t
he
t
h
r
ee
-phase
A
C
to
D
C
c
o
n
v
ert
e
r
usi
n
g
c
u
rr
ent
i
n
jec
tio
n
hy
bri
d
r
e
s
o
n
a
n
t
t
e
c
h
n
i
q
u
e
w
a
s
derive
d.
I
n
ge
ner
a
l,
it
t
h
e
de
si
g
n
o
f
d
y
n
a
m
ic
a
na
ly
sis
w
a
s
to
p
r
ovi
de
a
n
ac
cura
t
e
p
red
i
cti
o
n
of
t
he
o
ut
p
u
t
v
o
lta
ge
,
l
i
n
e
c
urr
e
n
t
,
D
C
-lin
k
vo
l
t
a
g
e
a
n
d
r
e
so
na
nt
c
urre
n
t
t
ra
nsie
n
t
s
respo
nd.
T
h
i
s
m
a
the
m
a
tica
l
m
ode
l
ca
n
be
u
se
d
t
o
d
e
v
el
o
p
a
s
ma
l
l
-si
g
n
a
l
m
ode
l
t
o
d
esi
gn
a
c
l
ose
d
l
o
op
con
t
ro
l
l
er
t
o
r
e
gu
l
a
te
t
he
D
C
ou
tpu
t
vol
tage
,
w
h
ilst
a
t
t
he
s
am
e
ti
m
e
a
l
s
o
bei
n
g
c
a
pa
b
l
e
t
o
p
e
rfor
m
t
he
pow
e
r
fa
ct
or c
orre
cto
r
ope
rati
on.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
El
e
c
&
D
ri S
yst
I
S
S
N
:
2088-
86
94
D
y
nam
ic
a
n
a
l
y
si
s
o
f
the h
i
gh-
po
wer f
a
c
t
o
r
t
h
ree
-
p
h
a
s
e
AC
to D
C
conve
r
t
e
r
using
…
(Rahim
i
Ba
har
om
)
54
7
ACKNOW
LEDG
E
MEN
T
S
A
u
t
hors
gra
t
ef
ull
y
a
ck
now
le
dge
t
he
f
ina
n
c
i
al
s
u
p
p
o
rt
fro
m
Inst
i
tut
e
o
f
R
e
sea
r
ch
M
a
n
a
g
emen
t
and
In
no
v
a
ti
on
(I
R
MI)
Un
i
v
ersiti
T
ekn
o
lo
gi
M
A
R
A
Gran
t
No
:
6
0
0
-IRMI/
M
y
R
A
5
/
3
/
B
ESTARI (0
2
9
/
201
7).
REFE
RENCES
[1]
A.
V
.
T.Ch
and
r
asekar,
et al
.
,
"A
S
tud
y
a
nd
R
evi
e
w
o
f
C
urren
t
I
nject
io
n
Techniq
u
es,
"
I
n
te
r
n
a
t
i
o
na
l J
o
u
r
na
l of
T
echn
o
lo
gy a
n
d
En
gi
neeri
ng Science
[I
JT
ES
]
,
vo
l
1
,
pp.
1
008
-1001
3,
S
e
p
t
2
01
3.
[2]
M
.
Y
.
Tarnin
i,
"
H
a
rm
oni
c
F
i
ltratio
n
b
y
C
u
r
rent
I
nj
ecti
on
a
n
d
S
h
u
n
t
Capacit
o
rs
T
ech
ni
que,"
in
20
13
3rd
Int
e
rna
t
i
o
n
a
l
Co
n
f
er
e
n
ce on
Electric P
o
wer an
d
E
n
er
gy Con
vers
i
o
n
S
y
st
e
m
s (
EPE
CS
)
,
pp.
1
-
5
,
Oct
2
013
.
[3]
M
.
T
aleb
a
nd
T.
H
.
O
r
tm
eyer,
"E
xami
nati
on
o
f
th
e
curren
t
i
nject
ion
t
echnique
(dis
t
r
ib
u
tio
n
n
e
tw
orks
),
"
in
I
EEE
Tran
sac
t
io
ns
on
Po
we
r De
liv
e
r
y
,
vo
l.
7
,
p
p
.
4
4
2
-448,
J
an
.
19
92
.
[4]
R
.
B
a
h
a
r
o
m
,
M
.
N
.
S
e
r
o
j
i
,
M
.
K
.
M
.
S
a
l
l
e
h
a
n
d
K
.
S
.
M
u
h
a
m
m
a
d
,
"
A
hig
h
p
o
w
er
f
a
c
t
o
r
th
ree-ph
a
s
e
AC-DC
curren
t
i
nj
ecti
o
n
hy
bri
d
r
es
on
ant
con
v
ert
e
r,"
IECON 2
0
1
6
-
42
nd
An
nu
al
C
o
n
f
e
r
e
n
c
e
o
f
th
e
I
E
EE
In
du
st
ri
al
El
ectro
n
i
cs
S
o
ciety
,
Floren
ce,
pp.
3
12
3-31
28
,
2
0
1
6
.
[5]
R.
B
ah
arom
et
al
.,
"
A
n
al
ysis
o
f
Three-P
h
ase
A
C
-DC
Cu
rrent
I
nj
ec
t
i
o
n
H
yb
rid
Reso
na
nt
C
on
ve
rte
r
"
,
Jo
ur
nal
o
f
T
e
leco
mmu
ni
cati
on
, Electr
onic and
Com
puter Engin
eeri
n
g
(
J
TEC)
,
Vo
l
.
9,
No
.2
-7
, 20
1
6
.
[6]
R.
W
u
,
S
.
B.
D
ewan
an
d
G
. R.
S
l
emon
, "
A
P
W
M
AC
t
o
DC
co
n
v
erter
wi
th
f
ix
e
d
s
wi
tc
h
i
ng
f
r
e
qu
e
n
c
y
,
"
Co
nfer
ence
Reco
rd of
t
h
e
198
8 IEEE
In
d
u
st
ry
A
ppli
c
ati
o
n
s
So
ciety An
n
u
a
l
M
eeting
,
P
i
ttsb
u
rg
h,
P
A,
U
S
A
,
pp.
706
-71
1
vo
l.
1,
1
98
8
.
[7]
R.
L
.
S
t
e
i
gerw
ald
,
"
Hi
gh
-F
requen
c
y
Reso
nan
t
T
ran
s
isto
r
D
C
-DC
Co
nv
e
r
te
rs,"
i
n
IE
EE
T
r
an
sa
c
tions
on In
dustrial
El
ectro
n
i
cs
,
v
o
l
.
IE
-3
1
, pp
. 18
1
-
1
91
, May 1
98
4
.
[8]
M
.
N
.
S
e
r
o
j
i
,
"
A
n
a
l
y
s
i
s
a
n
d
C
o
n
t
r
o
l
D
e
s
i
g
n
o
f
a
H
i
g
h
-
P
o
w
e
r
F
a
c
t
or,
Three-P
h
ase
AC/
D
C
Con
v
ert
e
r
With
H
ig
h
-
F
r
equ
e
ncy
R
e
son
a
nt
C
urren
t
I
n
j
ect
io
n,
"
Ph.
D
.
di
ssertati
on,
Dep
a
r
t
m
e
n
t
of
Electr
o
n
i
c,
Elect
ri
c
a
l
an
d
Comp
ut
er
En
gi
neeri
ng,
U
n
iver
sity
of B
i
r
m
ing
ham
, D
ecemb
e
r 20
07
.
[9]
R.
W
u
,
S
.
B
.
D
e
w
an
an
d
G
. R.
S
l
emon
,
"An
a
lys
i
s
of
a
n
AC-
t
o-D
C
v
ol
tage
s
ource
converter
using P
W
M
with
p
has
e
and amplitude
c
ont
rol," in
IEE
E
T
r
an
sactio
ns
on
Industry App
l
ica
t
ion
s
,
vo
l
.
2
7
, pp
. 35
5
-3
64
,
Ma
rc
h-Ap
ril 19
91
.
Evaluation Warning : The document was created with Spire.PDF for Python.