T
E
L
KO
M
N
I
KA
T
e
lec
om
m
u
n
icat
ion
,
Com
p
u
t
i
n
g,
E
lec
t
r
on
ics
an
d
Cont
r
ol
Vol.
18
,
No.
1
,
F
e
br
ua
r
y
2020
,
pp.
519
~
529
I
S
S
N:
1693
-
6930,
a
c
c
r
e
dit
e
d
F
ir
s
t
G
r
a
de
by
Ke
me
nr
is
tekdikti
,
De
c
r
e
e
No:
21/E
/KP
T
/2018
DO
I
:
10.
12928/
T
E
L
KO
M
NI
KA
.
v18i1.
12440
519
Jou
r
n
al
h
omepage
:
ht
tp:
//
jour
nal.
uad
.
ac
.
id/
index
.
php/T
E
L
K
OM
N
I
K
A
PWM
c
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ol
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ase
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k
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A.
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u
m
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ar
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men
t
o
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ect
r
i
cal
E
n
g
i
n
eeri
n
g
,
U
n
i
v
ers
i
t
y
o
f
K
u
f
a
,
Iraq
Ar
t
icle
I
n
f
o
AB
S
T
RA
CT
A
r
ti
c
le
h
is
tor
y
:
R
e
c
e
ived
J
a
n
26
,
2019
R
e
vis
e
d
J
un
2
,
20
19
Ac
c
e
pted
Ju
l
1
8
,
20
19
In
t
h
i
s
p
a
p
er
t
w
o
v
er
y
effi
ci
e
n
t
p
u
l
s
e
w
i
d
t
h
m
o
d
u
l
a
t
i
o
n
t
ec
h
n
i
q
u
es
w
ere
d
i
s
cu
s
s
e
d
n
ame
d
Si
n
p
u
l
s
e
w
i
d
t
h
m
o
d
u
l
a
t
i
o
n
an
d
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ect
o
r
p
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w
i
d
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mo
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l
a
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i
o
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h
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as
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c
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u
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re
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h
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l
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t
ra
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p
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cl
amp
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ro
d
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ce
d
an
d
t
h
e
b
as
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c
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d
ea
ab
o
u
t
s
p
ac
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v
ect
o
r
p
u
l
s
e
w
i
d
t
h
mo
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l
a
t
i
o
n
f
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r
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h
ree
-
l
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el
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a
g
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rce
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ert
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h
a
s
b
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n
d
i
s
c
u
s
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ed
i
n
d
et
a
i
l
.
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eare
s
t
t
h
ree
v
ec
t
o
r
s
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p
ace
v
ect
o
r
p
u
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w
i
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t
h
mo
d
u
l
at
i
o
n
co
n
t
r
o
l
al
g
o
ri
t
h
m
i
s
ad
o
p
t
ed
as
t
h
e
co
n
t
r
o
l
s
t
rat
e
g
y
fo
r
t
h
e
t
h
ree
p
h
as
e
t
h
ree
l
e
v
el
N
PC
i
n
v
ert
er
i
n
o
r
d
er
t
o
c
o
mp
e
n
s
a
t
e
t
h
e
n
eu
t
ral
-
p
o
i
n
t
s
h
i
f
t
i
n
g
.
Mat
h
emat
i
cal
fo
rmu
l
at
i
o
n
fo
r
ca
l
cu
l
at
i
n
g
s
w
i
t
ch
i
n
g
s
e
q
u
e
n
ce
h
as
d
et
e
rmi
n
e
d
.
Co
mp
ara
t
i
v
e
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al
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i
s
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p
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ri
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y
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f
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o
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l
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a
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n
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ert
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n
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rm
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l
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d
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d
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l
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ra
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eg
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er
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n
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d
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h
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d
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l
a
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erms
o
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d
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t
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l
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zat
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f
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s
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a
g
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l
o
w
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rre
n
t
ri
p
p
l
e
an
d
red
u
ced
s
w
i
t
c
h
i
n
g
freq
u
en
c
y
.
Sp
ace
v
ect
o
r
p
u
l
s
e
w
i
d
t
h
mo
d
u
l
a
t
i
o
n
p
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o
v
i
d
e
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ad
v
a
n
t
a
g
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e
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t
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fu
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d
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en
t
al
o
u
t
p
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t
v
o
l
t
a
g
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an
d
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s
ef
u
l
i
n
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mp
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armo
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c
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ce
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c
i
n
g
t
o
t
al
h
armo
n
i
c
d
i
s
t
o
rt
i
o
n
.
K
e
y
w
o
r
d
s
:
M
ult
i
-
leve
l
c
onve
r
ter
s
N
e
utr
a
l
-
point
-
c
lampe
d
inver
ter
S
pa
c
e
ve
c
tor
puls
e
width
modul
a
ti
on
T
hr
e
e
-
leve
l
inver
ter
Th
i
s
i
s
a
n
o
p
en
a
c
ces
s
a
r
t
i
c
l
e
u
n
d
e
r
t
h
e
CC
B
Y
-
SA
l
i
ce
n
s
e
.
C
or
r
e
s
pon
din
g
A
u
th
or
:
Ar
ka
n
A.
Ka
dum
,
De
pa
r
tm
e
nt
of
E
lec
tr
ica
l
E
nginee
r
ing
,
Unive
r
s
it
y
of
Ku
f
a
,
I
r
a
q.
E
mail:
a
r
ka
na
.
doa
be
l@uof
ku
f
a
.
e
du.
iq
1.
I
NT
RODU
C
T
I
ON
M
ult
il
e
ve
l
inver
ter
topol
ogies
[
1
,
2]
ha
ve
got
s
pe
c
ial
a
tt
e
nti
on
du
r
ing
the
e
a
r
li
e
r
two
de
c
a
de
s
due
to
their
s
igni
f
ica
nt
a
dva
ntage
s
c
ompar
e
d
to
the
c
l
a
s
s
ica
l
two
leve
l
inver
ter
s
.
As
c
ompar
e
d
with
t
wo
-
leve
l
inver
ter
s
,
mul
ti
leve
l
inver
ter
s
ha
ve
mul
ti
ple
a
dva
ntage
s
,
f
or
e
xa
mpl
e
,
low
ha
r
moni
c
s
in
output
volt
a
ge
s
a
nd
c
ur
r
e
nt,
les
s
dv/dt
,
lowe
r
powe
r
los
s
e
s
a
c
r
os
s
s
w
it
c
hing
de
vice
s
,
les
s
c
omm
on
mode
vol
tage
s
,
a
n
d
higher
qua
li
ty
output
wa
ve
f
or
m
[
3
-
7]
.
T
he
r
e
f
or
e
a
ne
w
f
a
mi
ly
of
mul
ti
leve
l
inver
te
r
s
ha
s
e
mer
ge
d
a
s
the
s
olut
ion
f
or
dif
f
e
r
e
nt
a
ppli
c
a
ti
ons
;
s
uc
h
a
s
,
AC
powe
r
s
uppli
e
s
,
lar
ge
powe
r
f
ul
e
ngine
dr
iver
s
,
t
r
a
ns
mi
s
s
ion,
dis
tr
ibut
ion
s
ys
tems
a
nd
medium
volt
a
ge
gr
id
[
8
-
12]
.
Dif
f
e
r
e
nt
types
of
topol
ogies
of
powe
r
c
onve
r
s
io
n
ha
ve
be
e
n
pr
opos
e
d
with
the
a
im
of
im
pr
oving
the
tot
a
l
ha
r
moni
c
dis
tor
ti
on
(
T
HD
)
a
nd
e
f
f
icie
nc
y,
a
nd
r
e
duc
ing
the
c
ompl
e
xit
y
of
c
ont
r
ol.
T
he
mos
t
known
a
r
e
:
c
a
s
c
a
de
d
H
-
br
idge
(
C
HB
)
inver
ter
,
f
ly
ing
-
c
a
pa
c
it
or
s
(
F
C
)
c
onve
r
ter
,
pa
c
ke
d
U
c
e
ll
s
,
[
13
-
16]
a
nd
ne
utr
a
l
point
c
lampe
d
(
NPC
)
inver
ter
[
1
]
.
T
his
pa
pe
r
us
e
s
the
NPC
topol
ogy
be
c
a
us
e
it
ha
s
the
a
dva
ntage
s
s
uc
h
a
s
:
low
s
witching
f
r
e
que
nc
y,
DC
-
li
nk
c
a
pa
c
it
or
s
a
r
e
c
omm
on
to
thr
e
e
pha
s
e
s
a
nd
r
e
a
c
ti
ve
c
ur
r
e
nt
c
a
n
be
c
ontr
oll
e
d
[
14]
.
T
he
diode
-
c
lampe
d
thr
e
e
leve
l
ne
utr
a
l
-
point
c
lampe
d
(
NPC
)
topol
ogy
ha
s
be
e
n
the
mos
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
1
,
F
e
br
ua
r
y
2020
:
519
-
529
520
wide
ly
us
e
d
one
a
m
ong
a
ll
mul
ti
leve
l
inver
ter
top
ologi
e
s
due
to
their
us
e
of
a
unique
DC
s
our
c
e
of
volt
a
ge
c
ompar
e
d
with
the
C
HB
inver
te
r
s
a
nd
higher
pe
r
f
o
r
manc
e
whe
n
c
ompar
e
d
to
the
F
L
inver
ter
s
[
17
,
18
]
.
On
the
other
ha
nd,
s
e
ve
r
a
l
methods
of
modul
a
ti
o
n
tec
hniques
ha
ve
be
e
n
pr
opos
e
d
f
or
thi
s
type
of
c
onve
r
ter
.
T
he
mos
t
known
tec
hniques
a
r
e
:
s
e
lec
ti
ve
ha
r
moni
c
e
li
mi
na
ti
on
puls
e
width
modul
a
ti
on
(
P
W
M
)
,
s
inus
oidal
P
W
M
with
a
nd
without
ha
r
moni
c
in
j
e
c
ti
on
,
s
pa
c
e
ve
c
tor
modul
a
ti
on
,
s
igm
a
de
lt
a
P
W
M
,
a
nd
c
los
e
d
loop
modul
a
ti
on
tec
hniques
e
xis
t
to
c
ont
r
ol
the
inver
te
r
[
19
-
22]
.
Among
the
pos
s
ibl
e
mul
ti
leve
l
topol
ogies
,
the
s
ine
tr
iangle
P
W
M
(
S
P
W
M
)
a
nd
s
p
a
c
e
ve
c
tor
P
W
M
(
S
VPWM
)
a
r
e
pr
oba
bly
the
mos
t
popular
modes
a
nd
the
mos
t
c
omm
on
P
W
M
ge
ne
r
a
ti
on
t
e
c
hniques
f
or
thr
e
e
-
leve
l
thr
e
e
-
pha
s
e
mul
ti
leve
l
inver
ter
s
.
T
he
S
VPWM
is
ba
s
ica
ll
y
divi
de
d
int
o
two
c
las
s
if
ica
ti
ons
:
the
ne
a
r
e
s
t
thr
e
e
ve
c
tor
s
(
NT
V)
met
hod
a
nd
the
He
xa
gons
method.
T
he
S
VPWM
include
s
th
e
r
e
f
e
r
e
nc
e
volt
a
g
e
s
pa
c
e
ve
c
tor
s
ynthes
is
f
or
t
he
NT
V
a
ppr
oa
c
h,
by
s
witching
be
twe
e
n
the
ne
a
r
e
s
t
thr
e
e
volt
a
ge
s
pa
c
e
ve
c
tor
s
[
23
-
26]
.
T
he
output
volt
a
ge
ve
c
tor
is
s
ynthes
ize
d
f
or
the
he
xa
gon's
a
ppr
oa
c
h
us
ing
a
me
thod
s
im
il
a
r
to
the
t
r
a
dit
ional
two
-
leve
l
c
onve
r
ter
.
T
his
pa
pe
r
modeling
a
nd
s
im
ulation
of
a
thr
e
e
-
leve
l
NPC
inve
r
ter
ha
ve
be
e
n
pe
r
f
or
med
with
S
P
W
M
a
nd
S
VPWM
tec
hniques
us
ing
M
AT
L
AB
/S
im
uli
nk
s
of
twa
r
e
.
T
he
r
e
s
ult
s
of
the
c
ompr
e
he
ns
ive
c
ompar
is
on
a
r
e
pr
e
s
e
nted
in
two
wa
ys
.
T
his
pa
pe
r
will
be
divi
de
d
int
o
s
ix
s
e
c
ti
ons
.
A
b
r
ief
r
e
view
of
the
thr
e
e
leve
l
NPC
inver
ter
is
p
r
ovided
in
s
e
c
ti
on
2.
S
e
c
ti
ons
3
de
s
c
r
ibes
the
thr
e
e
leve
l
P
W
M
tec
hniques
.
T
he
p
r
opos
e
d
thr
e
e
-
leve
l
s
pa
c
e
volt
a
ge
ve
c
tor
a
lg
or
it
hm
ha
s
given
in
s
e
c
ti
on
4.
S
e
c
ti
on
5
wil
l
pe
r
f
o
r
m
the
s
im
ulation
s
uc
c
e
s
s
iv
e
ly.
F
inally,
a
t
the
e
nd
of
th
is
pa
pe
r
,
a
ge
ne
r
a
l
c
onc
lus
ion
is
given.
2.
T
HRE
E
-
L
E
VE
L
DI
OD
E
-
CL
AM
P
E
D
I
NV
E
RT
E
R
T
OP
OL
OG
Y
F
igur
e
1
s
hows
the
thr
e
e
-
pha
s
e
thr
e
e
-
leve
l
diode
-
c
lampe
d
inver
ter
(
NPC
)
topol
ogy.
F
r
o
m
F
igur
e
1
,
e
a
c
h
pha
s
e
of
the
inver
ter
s
ha
r
e
d
the
DC
-
li
nk
s
up
ply.
T
he
c
e
nter
of
e
a
c
h
pha
s
e
is
c
onne
c
ted
to
the
c
omm
on
point
of
the
s
e
r
ies
c
a
pa
c
it
or
s
.
T
he
inve
r
ter
is
f
e
e
ding
a
n
AC
a
thr
e
e
-
pha
s
e
load.
T
h
r
e
e
-
leve
l
output
c
ons
is
ti
ng
of
leve
ls
-
Vdc
,
0,
+
Vdc
de
pe
nding
on
the
DC
-
bu
s
volt
a
ge
.
T
a
ble
1
s
hows
the
f
unc
ti
oning
pr
inciple
.
I
n
or
de
r
to
obtain
the
de
s
ir
e
d
thr
e
e
-
leve
l
volt
a
ge
s
,
the
c
onve
r
ter
mus
t
e
ns
ur
e
c
ompl
e
menta
r
it
ies
be
twe
e
n
the
pa
ir
s
of
s
witche
s
:
(
S
i1
,S
i4
)
a
nd
(
S
i2
,S
i3
)
whe
r
e
'i'
is
the
pha
s
e
indi
c
a
tor
(
i
=
a
,
b,
c
)
,
Vio
is
the
pha
s
e
-
to
-
f
ictive
mi
ddle
point
volt
a
ge
.
T
a
ble
1
s
hows
s
witching
of
the
i
-
pha
s
e
in
F
igur
e
1
with
s
witching
s
tate
s
a
nd
c
or
r
e
s
ponding
output
volt
a
ge
leve
ls
.
F
igur
e
1
.
T
h
r
e
e
-
pha
s
e
thr
e
e
-
leve
l
NPC
inver
ter
c
ir
c
uit
T
a
ble
1
.
R
e
lations
hip
be
twe
e
n
s
witching
de
vice
s
a
nd
output
leve
l
S
w
it
c
hi
ng
S
it
ua
ti
ons
i
-
pha
s
e
of
N
P
C
i
nve
r
te
r
T
e
r
mi
na
l
V
ol
ta
ge
S
i1
S
i2
S
i3
S
i4
P
1
1
0
0
+
V
dc
/
2
O
0
1
1
0
0
N
0
0
1
1
-
V
dc
/
2
3.
T
HRE
E
-
L
E
VE
L
P
WM
T
E
CHNI
QU
E
S
I
n
many
a
ppli
c
a
ti
ons
,
the
output
volt
a
ge
o
f
t
he
inver
ter
is
of
ten
r
e
qui
r
e
d
to
va
r
y
due
to
the
f
oll
owing
r
e
a
s
ons
:
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
P
W
M
c
ontr
ol
tec
hniques
for
thr
e
e
phas
e
thr
e
e
lev
e
l
inve
r
ter
dr
ive
s
(
A
r
k
an
A
.
K
adum)
521
−
T
o
c
ompens
a
te
the
input
volt
a
ge
va
r
iations
.
−
T
o
c
ompens
a
te
f
or
the
r
e
gulation
of
the
inver
te
r
s
.
−
F
or
the
p
r
ovis
ion
of
s
pe
c
ial
loads
that
will
ne
e
d
va
r
iation
of
volt
a
ge
with
f
r
e
que
nc
y.
3.
1.
S
in
u
s
oid
al
p
u
ls
e
wid
t
h
m
od
u
lat
ion
t
e
c
h
n
iq
u
e
T
he
p
r
inciple
o
f
the
s
inus
oidal
c
a
r
r
ier
-
ba
s
e
d
puls
e
width
modul
a
ti
on
(
S
P
W
M
)
tec
hnique;
a
high
-
f
r
e
que
nc
y
tr
iangula
r
c
a
r
r
ier
wa
ve
V
r
is
c
ompar
e
d
with
a
s
inus
oidal
c
ontr
ol
s
ignal
V
c
a
t
the
de
s
ir
e
d
f
r
e
que
nc
y.
T
he
int
e
r
s
e
c
ti
on
of
V
c
a
nd
V
r
wa
ve
s
de
ter
mi
ne
s
the
s
witching
ins
tant
s
a
nd
c
omm
uta
ti
on
of
the
m
odulate
d
puls
e
.
A
tr
a
ns
it
ion
in
P
W
M
wa
ve
f
or
m
is
ge
ne
r
a
ted
a
t
e
a
c
h
c
ompar
e
matc
h
point
.
W
he
n
s
inus
oidal
wa
ve
ha
s
magnitude
higher
than
the
tr
i
a
ngular
wa
ve
the
P
W
M
output
is
pos
it
ive
a
nd
w
h
e
n
V
c
is
s
maller
than
V
r
,
the
output
is
ne
ga
ti
ve
.
T
he
inver
te
r
’
s
s
witching
f
r
e
que
nc
y
f
s
e
s
tablis
he
s
by
the
f
r
e
que
nc
y
of
tr
iangle
wa
ve
f
or
m
V
c
.
T
he
f
unda
menta
l
f
r
e
que
nc
y
c
omponent
in
the
inver
te
r
outpu
t
vo
lt
a
ge
c
a
n
be
c
ontr
oll
e
d
by
a
mpl
i
tude
modul
a
ti
on
index;
we
de
f
i
ne
the
modul
a
ti
on
index
m
i
a
s
f
oll
ows
:
m
i
=
V
c
/
V
r
(
1)
whe
r
e
V
c
a
nd
V
r
a
r
e
the
pe
a
k
va
lues
of
the
m
odulating
a
nd
c
a
r
r
ier
wa
ve
s
,
r
e
s
pe
c
ti
ve
ly.
T
he
a
mpl
it
ude
modul
a
ti
on
index
m
i
is
us
ua
ll
y
a
djus
ted
by
va
r
yi
ng
V
c
while
ke
e
ping
V
r
f
ixed.
T
he
f
r
e
que
nc
y
mo
dulation
index
is
de
f
ined
by
:
m
f
=
f
r
/
f
1
(
2)
whe
r
e
,
f
1
a
nd
f
r
a
r
e
the
f
r
e
que
nc
ies
of
the
mod
ulating
a
nd
c
a
r
r
ier
wa
ve
s
,
r
e
s
pe
c
ti
ve
ly.
T
he
f
un
da
menta
l
c
omponent
V
o
u
t
1
of
the
ou
tput
volt
a
ge
ha
s
th
e
pr
ope
r
ty
a
s
de
picte
d
in
e
qua
ti
o
n
be
low
in
a
li
ne
a
r
modul
a
ti
on
r
e
gion:
V
o
u
t
1
=
m
i
*
V
d
m
i
≤
1.
0
(
3)
i
n
(
1
)
s
hows
that
the
modul
a
ti
on
index
a
nd
a
mpl
it
ude
of
the
f
unda
menta
l
c
omp
one
nt
of
the
output
volt
a
ge
va
r
ies
li
ne
a
r
ly.
T
he
m
i
va
lue
v
a
r
ied
f
r
om
z
e
r
o
to
one
;
i
t
is
de
f
ined
a
s
the
li
ne
a
r
c
ontr
ol
r
a
nge
of
s
inus
oidal
c
a
r
r
ier
P
W
M
.
T
h
r
e
e
leve
l
pul
s
e
width
modul
a
ted
wa
ve
f
or
ms
c
a
n
be
ge
ne
r
a
ted
by
s
ine
c
a
r
r
ier
P
W
M
.
S
ine
c
a
r
r
ier
P
W
M
is
ge
ne
r
a
ted
by
c
ompar
ing
the
thr
e
e
r
e
f
e
r
e
nc
e
c
ontr
ol
s
i
gna
ls
with
two
tr
iangula
r
c
a
r
r
ier
wa
ve
s
[
6
,
12]
.
T
he
th
r
e
e
-
leve
l
s
ine
-
P
W
M
inver
ter
is
im
p
leme
ntation
a
s
a
two
-
leve
l
inver
ter
us
ing
the
s
a
me
pr
inciple.
He
r
e
s
ine
c
a
r
r
ier
P
W
M
is
ge
ne
r
a
ted
by
c
ompar
ing
the
thr
e
e
r
e
f
e
r
e
nc
e
c
ontr
ol
s
ignals
with
two
tr
iangula
r
c
a
r
r
ier
wa
ve
s
.
T
he
c
or
r
e
s
ponding
puls
e
s
a
r
e
ge
ne
r
a
ted
whic
h
a
r
e
to
be
s
uppli
e
d
to
the
inve
r
ter
ga
te
de
vice
s
.
T
he
thr
e
e
r
e
f
e
r
e
nc
e
c
ontr
ol
s
ignals
a
r
e
p
ha
s
e
s
hif
t
by
a
n
a
ngle
2π
/3
a
nd
4π
/3
with
s
a
me
a
mpl
it
ude
.
T
wo
c
a
r
r
ier
wa
ve
s
a
r
e
in
pha
s
e
e
a
c
h
o
ther
with
dc
volt
a
ge
of
f
s
e
t.
F
or
thr
e
e
-
pha
s
e
S
P
W
M
;
=
2
(
4)
whe
r
e
:
V
p
e
a
k
is
the
pe
a
k
va
lue
of
the
f
unda
menta
l
c
omponent
of
the
pha
s
e
-
to
-
n
e
utr
a
l
volt
a
ge
,
v
dc
is
the
DC
l
ink
volt
a
ge
.
F
or
thr
e
e
-
pha
s
e
s
pa
c
e
-
ve
c
tor
S
VPWM
,
[
13]
.
=
√
3
(
5)
F
or
nor
mal
s
tea
dy
-
s
tate
ope
r
a
ti
on,
0
<
m
≤
1
.
3.
2.
S
p
ac
e
ve
c
t
or
p
u
ls
e
wid
t
h
m
od
u
la
t
ion
T
he
ba
s
ic
pr
inciple
o
f
S
VPWM
de
pe
nds
on
s
ynt
he
s
izing
the
ve
c
tor
of
r
e
f
e
r
e
nc
e
volt
a
ge
by
ti
me
a
ve
r
a
ging
of
the
two
ve
c
to
r
s
pr
oduc
e
d
by
the
inve
r
ter
.
T
he
r
e
f
e
r
e
nc
e
volt
a
ge
ve
c
tor
is
the
r
e
quir
e
d
c
omm
a
nd
volt
a
ge
whic
h
that
s
hould
be
given
a
s
r
e
qui
r
e
d
to
the
a
ppli
c
a
ti
on.
S
pa
c
e
ve
c
tor
P
W
M
tec
hnique
is
b
a
s
e
d
on
r
otating
r
e
f
e
r
e
nc
e
volt
a
ge
s
pa
c
e
ve
c
tor
a
ppr
oxim
a
ti
on.
T
he
r
otating
r
e
f
e
r
e
nc
e
volt
a
ge
ve
c
tor
r
e
pr
e
s
e
nts
the
s
pa
ti
a
l
ve
c
tor
s
um
of
the
th
r
e
e
-
pha
s
e
volt
a
ge
in
the
α
-
β
s
pa
c
e
.
T
he
a
mpl
it
ude
o
f
the
ve
c
tor
a
nd
t
he
pha
s
e
a
ngle
of
the
thr
e
e
-
pha
s
e
c
a
n
be
de
ter
mi
ne
d
by
the
ins
tanta
ne
ous
va
lue
s
of
the
volt
a
ge
s
.
I
f
the
magnit
ude
s
a
r
e
s
inus
oidal
a
nd
ba
lanc
e
d,
the
ve
c
tor
it
will
r
otate
r
a
pidl
y
in
a
f
ixed
a
ngular
a
nd
ha
ve
a
c
ons
tant
a
mpl
it
ude
.
W
he
n
c
ons
ider
ing
the
thr
e
e
pha
s
e
of
the
inver
ter
,
ther
e
a
r
e
27
s
witching
s
tate
.
E
a
c
h
of
thes
e
s
witchi
ng
s
tate
s
c
a
n
be
r
e
pr
e
s
e
nted
a
s
a
ve
c
tor
us
ing
(
6
)
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
1
,
F
e
br
ua
r
y
2020
:
519
-
529
522
→
=
+
=
2
3
(
0
+
2
3
+
4
3
)
(
6)
=
−
(
)
(
7)
whe
r
e
V
a
, V
b
,
V
c
a
r
e
the
r
e
f
e
r
e
nc
e
thr
e
e
-
pha
s
e
volt
a
ge
s
a
nd
V
α
a
nd
V
β
a
r
e
the
c
omponents
of
r
e
f
e
r
e
nc
e
ve
c
tor
in
α
-
β
c
oor
dinate
s
ys
tem.
T
hr
ough
the
us
e
C
lar
k
tr
a
ns
f
or
mation,
the
a
-
b
-
c
thr
e
e
-
pha
s
e
c
oor
dinate
s
ys
tem
is
tr
a
ns
f
or
med
to
2
-
dim
e
nti
ona
l
α
-
β
f
r
a
me
that
is
he
l
pf
ul
i
n
s
e
c
tor
identif
ica
ti
on
by
f
o
ll
owing
e
xpr
e
s
s
ion:
[
]
=
2
3
[
1
−
1
2
−
1
2
0
√
3
2
−
√
3
2
]
[
]
(
8)
T
he
27
output
vo
lt
a
ge
ve
c
tor
s
in
the
th
r
e
e
-
leve
l
VSI
a
s
s
hown
in
F
igur
e
2.
T
he
s
pa
c
e
-
ve
c
tor
diagr
a
m
s
hown
in
F
igur
e
2
c
ons
is
ts
of
6
majo
r
tr
iangula
r
s
e
c
tor
s
.
E
a
c
h
majo
r
s
e
c
ti
on
r
e
pr
e
s
e
nts
π
/3
of
the
f
unda
menta
l
c
yc
le.
T
he
r
e
a
r
e
4
mi
nor
tr
iangul
a
r
s
e
c
tor
s
withi
n
e
a
c
h
major
s
e
c
tor
.
T
he
r
e
f
or
e
,
th
e
tot
a
l
is
24
mi
nor
s
e
c
tor
s
in
the
plane
.
T
he
volt
a
ge
ve
c
tor
is
r
e
pr
e
s
e
nted
by
the
ve
r
ti
c
e
s
of
thes
e
s
e
c
tor
s
.
B
a
s
e
d
on
the
a
mpl
it
ude
the
th
r
e
e
-
leve
l
inver
ter
ve
c
tor
s
a
r
e
d
ivi
de
d
int
o
lar
ge
,
medium
,
s
mall,
a
nd
z
e
r
o
ve
c
tor
s
,
they
a
r
e
li
s
ted
a
s
s
hown
in
T
a
ble
2
.
As
c
a
n
be
s
e
e
n
in
F
igur
e
2
,
the
vol
tage
ve
c
tor
s
a
r
e
loca
ted
a
t
the
va
r
ious
poin
ts
of
the
two
he
xa
gons
int
e
r
lea
ve
d
a
c
c
or
ding
to
their
s
witching.
T
he
volt
a
ge
ve
c
tor
s
of
gr
oup
lar
ge
volt
a
ge
ve
c
tor
s
,
with
a
mp
li
tudes
of
2V
dc
/3
a
nd
loca
ted
a
t
the
c
o
r
ne
r
s
of
the
oute
r
he
xa
gon.
T
he
medium
vo
lt
a
ge
ve
c
tor
is
t
he
vol
tage
v
e
c
tor
of
a
mpl
it
ude
V
dc
/√
3
a
nd
is
loca
ted
a
t
the
mi
dpoint
of
the
outer
he
xa
gon.
S
mall
g
r
oup
volt
a
ge
ve
c
tor
s
with
a
mpl
it
ude
s
of
V
dc
/3
a
nd
a
r
e
loca
ted
a
t
the
c
or
ne
r
s
of
the
inner
he
xa
gon.
W
he
n
the
r
otating
vol
tage
ve
c
tor
f
a
ll
s
int
o
a
c
e
r
tain
s
e
c
tor
in
a
thr
e
e
-
pha
s
e
thr
e
e
-
leve
l
i
nve
r
ter
,
a
djac
e
nt
volt
a
ge
ve
c
tor
s
a
r
e
s
e
lec
ted
to
s
ynthes
ize
the
de
s
ir
e
d
r
otating
volt
a
ge
ve
c
tor
ba
s
e
d
on
the
pr
i
nc
ipl
e
of
ve
c
t
or
s
ynthes
is
,
r
e
s
ult
ing
in
P
W
M
wa
ve
f
or
ms
in
thr
e
e
-
pha
s
e
.
T
he
s
e
c
tor
in
whic
h
Vo*
r
e
s
ides
c
a
n
be
e
xa
mi
ne
d
by
e
xa
mi
ning
the
pha
s
e
a
ngle
a
nd
the
magnitude
of
a
r
otating
r
e
f
e
r
e
nc
e
volt
a
ge
ve
c
to
r
V*.
F
igur
e
2
.
S
pa
c
e
volt
a
ge
ve
c
tor
s
in
th
r
e
e
-
leve
l
inver
ter
T
a
ble
2
.
Voltage
ve
c
tor
s
a
nd
s
witching
s
tate
s
V
e
c
to
r
na
me
ve
c
to
r
c
la
s
s
if
ic
a
ti
on
Z
e
r
o ve
c
to
r
[
O
O
O
]
[
P
P
P
]
[
N
N
N
]
S
ma
ll
ve
c
to
r
P
-
ty
pe
N
-
ty
pe
[
P
P
O
]
[
P
O
O
]
[
O
N
N
]
[
O
N
N
]
[
O
P
P
]
[
O
P
O
]
[
N
O
O
]
[
N
N
O
]
[
P
O
P
]
[
O
O
P
]
[
O
N
O
]
[
N
O
N
M
e
di
um ve
c
to
r
[
N
P
O
]
[
P
O
N
]
[
O
P
N
]
[
P
N
O
]
[
O
N
P
]
[
N
O
P
]
L
a
r
ge
ve
c
to
r
[
P
P
N
]
[
P
N
N
]
[
N
P
N
]
[
P
N
P
]
[
N
N
P
]
[
N
P
P
]
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
P
W
M
c
ontr
ol
tec
hniques
for
thr
e
e
phas
e
thr
e
e
lev
e
l
inve
r
ter
dr
ive
s
(
A
r
k
an
A
.
K
adum)
523
4.
T
HRE
E
-
L
E
VE
L
I
NV
E
RT
E
RS
S
P
AC
E
VOL
T
AGE
VE
CT
OR
AL
GO
RI
T
H
M
As
s
hown
in
F
igur
e
3,
the
pr
im
a
r
y
tas
k
of
S
VP
W
M
is
to
de
ter
mi
ne
,
whic
h
s
e
c
tor
a
nd
whic
h
of
the
r
e
gion
,
then
de
ter
mi
ne
the
c
or
r
e
s
ponding
out
put
s
pa
c
e
volt
a
ge
ve
c
tor
s
.
T
he
r
e
a
li
z
a
ti
on
of
the
volt
a
ge
ve
c
tor
S
VPWM
is
s
ubs
umed
int
o
the
p
r
opos
e
d
S
V
P
W
M
tec
hnique
f
oll
owing
s
teps
:
−
T
r
a
ns
f
or
mation
o
f
3
-
pha
s
e
to
2
-
pha
s
e
.
−
De
ter
mi
ning
the
s
e
c
tor
c
ontaining
the
ti
p
of
the
r
e
f
e
r
e
nc
e
ve
c
tor
.
−
De
ter
mi
ne
the
r
e
gion
in
the
c
or
r
e
s
ponding
s
e
c
tor
.
−
De
duc
ti
on
of
opti
mu
m
s
witching
s
e
que
nc
e
.
−
Ge
ne
r
a
ti
on
of
ga
ti
ng
s
ignals
to
the
inve
r
ter
de
vice
s
.
−
C
a
lcula
ti
ng
the
dur
a
ti
on
o
f
s
witching
ve
c
tor
s
.
S
tep
1:
Us
ing
pa
r
k
’
s
tr
a
ns
f
or
mation
to
c
a
lcula
te
V
α
, V
β
, V
r
e
f
a
nd
the
a
ngle
(
θ)
(
8
).
S
tep
2:
Af
ter
c
a
lcula
ti
ng
θ
,
de
te
r
mi
ne
the
s
e
c
tor
a
c
c
or
ding
to
V
r
e
f
pos
it
ion
of
a
s
s
hown
in
T
a
ble
3
S
tep
3
:
De
ter
m
ine
the
r
e
gion
in
the
c
or
r
e
s
ponding
s
e
c
tor
.
T
he
r
e
f
e
r
e
nc
e
volt
a
ge
ve
c
tor
c
a
n
be
f
ound
in
s
pa
c
e
ve
c
tor
diagr
a
m
in
a
ny
r
e
gion
(
1
-
4)
o
f
a
ny
s
e
c
to
r
(
1
-
6)
a
s
s
hown
in
F
ig
ur
e
2.
T
o
e
s
ti
mate
the
r
e
gion
in
s
e
c
tor
1,
s
pa
c
e
ve
c
tor
diagr
a
m
f
o
r
m
1
a
nd
m
2
is
s
hown
in
F
ig
ur
e
3,
a
nd
c
or
r
e
s
pondi
ng
s
witching
logi
c
is
given
in
T
a
ble
4.
(
a)
(
b)
F
igur
e
3.
S
pa
c
e
ve
c
tor
diagr
a
m
f
or
m
1
a
nd
m
2
in
s
e
c
tor
1
T
a
ble
3.
S
e
c
tor
c
a
lcula
ti
on
ba
s
e
d
on
the
loca
ti
on
o
f
V
re
f
R
a
nge
of
(
de
gr
e
e
)
0
≤
<
60
60
≤
<
120
120
≤
<
180
180
≤
<
240
240
≤
<
300
300
≤
<
360
L
oc
a
ti
on of
V
r
e
f
S
e
c
to
r
I
S
e
c
to
r
I
I
S
e
c
to
r
I
I
I
S
e
c
to
r
I
V
S
e
c
to
r
V
S
e
c
to
r
V
I
1
=
−
=
−
(
2
√
3
)
co
s
(
3
)
=
(
−
√
3
)
=
2
√
3
(
3
−
)
(
9)
2
=
=
s
i
n
(
/
3
)
=
2
√
3
(
10)
=
2
/
3
(
11)
T
a
ble
4.
L
ogic
us
e
d
to
f
ind
the
r
e
gion
in
whic
h
V
r
e
f
is
loca
ted
X
1
a
nd X
2
P
os
it
io
n of
V
r
e
f
X
1
, X
2
a
nd (
X
1
+
X
2
)
<
0.5
R
e
gi
on 1
X
2
>
0.5
R
e
gi
on 2
X
1
>
0.5
R
e
gi
on 3
X
1
an
d
X
2
<
0.5 a
nd
(
X
1
+
X
2
)
>
0.5
R
e
gi
on 4
S
tep
4
:
C
a
lcula
ti
on
o
f
s
witching
t
im
e
.
De
ter
mi
ne
t
he
s
witching
ti
me
of
e
a
c
h
s
witch
a
c
r
os
s
a
ll
the
r
e
gions
.
T
he
S
VPWM
tec
hnique
f
or
thr
e
e
-
leve
l
inver
ter
s
is
c
a
ll
e
d
"
volt
a
ge
-
time
e
qua
li
z
a
ti
on
”
pr
inciple
[
10
]
.
F
o
r
e
xa
mpl
e
,
c
ons
ider
ing
that
the
r
e
f
e
r
e
nc
e
ve
c
tor
Vr
e
f
is
loca
ti
ng
in
s
e
c
tor
I
a
nd
in
r
e
gion
2
,
f
or
th
is
r
e
gion,
the
ne
a
r
e
s
t
thr
e
e
volt
a
ge
s
pa
c
e
ve
c
tor
s
a
r
e
V1
,
V7
a
nd
V2
a
r
e
s
hown
in
F
i
g
ur
e
4.
Onc
e
the
ne
a
r
e
s
t
thr
e
e
ve
c
tor
s
ha
ve
be
e
n
iden
ti
f
ied
,
by
us
ing
the
volt
-
s
e
c
ond
ba
lanc
e
method,
the
on
-
ti
me
c
a
lcula
ti
ons
of
the
c
or
r
e
s
ponding
ve
c
tor
s
c
a
n
be
d
e
ve
loped
a
s
the
f
oll
owing
e
xpr
e
s
s
ion
s
.
W
he
n
the
volt
a
ge
in
the
s
e
c
ond
s
e
c
tor
is
e
qua
l
to
the
ti
me
s
ync
hr
oniza
ti
on
e
qua
ti
on;
c
a
n
be
w
r
it
ten
a
s
in
(1
2
).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
1
,
F
e
br
ua
r
y
2020
:
519
-
529
524
=
1
+
7
+
2
(
12)
=
+
+
whe
r
e
T
s
is
the
s
a
mpl
ing
ti
me,
t
a
, t
b
a
nd
t
c
a
r
e
s
witching
on
-
ti
mes
f
or
the
V
1
,
V
7
a
nd
V
2
ve
c
tor
s
r
e
s
pe
c
ti
ve
ly.
1
=
3
0
2
=
3
/
3
=
2
3
=
1
2
+
√
3
2
√
3
6
+
1
2
√
3
3
7
=
√
3
/
6
B
y
s
olvi
ng
the
3
e
qua
ti
ons
a
bove
,
the
on
-
ti
me
o
f
t
he
c
onc
e
r
ne
d
ve
c
tor
s
c
a
n
be
c
omput
e
d
by
[
14]
:
=
−
2
=
2
(
3
+
)
−
=
−
2
(
3
−
)
=
2
√
3
whe
r
e
,
T
s
is
the
s
ys
tem
s
a
mpl
ing
c
ontr
ol
c
yc
le,
a
nd
V
re
f
,
θ
is
the
a
mpl
it
ude
a
nd
a
ngle
o
f
the
r
e
f
e
r
e
nc
e
volt
a
ge
ve
c
tor
.
F
igur
e
4
.
S
ynthes
ize
d
r
e
f
e
r
e
nc
e
ve
c
tor
in
the
f
i
r
s
t
s
e
c
tor
T
he
on
-
ti
mes
f
o
r
other
r
e
gions
in
the
f
ir
s
t
s
e
c
tor
e
quil
ibr
iu
m
ti
me
e
qua
ti
ons
c
a
n
be
c
omput
e
d
by
us
ing
s
im
il
a
r
pr
oc
e
dur
e
a
s
e
xpr
e
s
s
e
d
in
T
a
ble
5.
T
he
s
e
e
qua
ti
ons
c
a
n
be
made
va
li
d
f
or
other
s
pa
c
e
ve
c
tor
diagr
a
m
withi
n
the
s
e
c
tor
s
[
18]
.
T
a
ble
5
.
E
qua
ti
on
s
o
f
s
witching
ti
mes
f
or
s
e
c
tor
-
1
r
e
gi
on
O
n t
im
e
t
a
t
b
t
c
1
2
(
3
−
)
−
2
(
3
+
)
2
2
−
2
2
(
3
+
)
−
−
2
(
3
−
)
3
2
−
2
(
3
−
)
2
−
2
(
3
+
)
4
2
−
2
(
3
+
)
2
2
(
3
−
)
−
S
tep
5
:
I
nve
r
ter
's
ga
ti
ng
s
ignals
ge
ne
r
a
ti
on
T
he
ne
xt
s
tep
in
im
pleme
ntation
of
S
VPWM
i
s
the
s
e
lec
ti
on
of
the
r
e
dunda
nt
s
tate
s
opti
mal
s
witching
s
e
que
nc
e
of
whic
h
is
us
e
f
ul
in
ba
lanc
ing
of
DC
li
nk
volt
a
ge
s
,
f
a
ult
to
ler
a
nc
e
a
nd
s
witching
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
P
W
M
c
ontr
ol
tec
hniques
for
thr
e
e
phas
e
thr
e
e
lev
e
l
inve
r
ter
dr
ive
s
(
A
r
k
an
A
.
K
adum)
525
f
r
e
que
nc
y
r
e
duc
ti
on
e
tc
.
T
a
ble
6
s
hows
a
ll
pos
s
ibl
e
s
witching
s
e
que
nc
e
f
or
a
ll
s
e
c
tor
1
r
e
gions
.
I
n
th
e
s
e
c
ond
r
e
gion
of
the
f
ir
s
t
s
e
c
tor
o
f
the
r
e
f
e
r
e
nc
e
ve
c
tor
,
the
or
de
r
of
s
witching
a
nd
pha
s
e
thr
e
e
-
pha
s
e
s
pir
a
l
of
the
thr
e
e
-
leve
l
inver
ter
a
r
e
s
hown
in
F
ig
u
r
e
5.
Af
ter
the
s
witching
ti
mes
ha
ve
e
laps
e
d,
V
r
e
f
i
n
whic
h
a
ppli
c
a
ti
on
of
volt
a
ge
s
pa
c
e
ve
c
tor
s
a
c
c
or
ding
to
s
e
c
tor
a
nd
r
e
gion
P
W
M
s
ignals
s
hould
be
ge
ne
r
a
ted.
T
a
ble
6
.
P
os
s
ibl
e
s
witching
s
e
que
nc
e
f
or
s
e
c
tor
I
R
e
gi
on
S
w
it
c
hi
ng s
e
que
nc
e
R
e
gi
on 1
PPO
-
P
O
O
-
OOO
-
OON
-
O
N
N
a
nd s
ymm
e
tr
y
R
e
gi
on 2
PPO
-
P
O
O
-
P
O
N
-
OON
-
O
N
N
a
nd s
ymm
e
tr
y
R
e
gi
on 3
P
O
O
-
P
O
N
-
P
N
N
-
O
N
N
a
nd s
ymm
e
tr
y
R
e
gi
on 4
PPO
-
PPN
-
P
O
N
-
O
O
N
a
nd s
ymm
e
tr
y
F
igur
e
5
.
S
witching
s
ignals
f
or
s
e
c
tor
-
1
s
e
c
ond
r
e
g
ion
5.
S
I
M
UL
AT
I
ON
RE
S
UL
T
S
M
AT
L
AB
/S
im
uli
nk
s
of
twa
r
e
is
us
e
d
to
s
im
ulate
t
he
NPC
S
VPWM
th
r
e
e
pha
s
e
th
r
e
e
leve
l
inve
r
ter
s
c
he
me
s
hown
in
F
ig
ur
e
6.
I
n
pa
r
ti
c
ular
,
we
e
mpl
oye
d
S
-
f
unc
ti
ons
to
e
na
ble
li
ne
a
r
pr
ogr
a
ms
to
be
im
pleme
nted
in
S
im
uli
nk
models
.
I
n
the
s
im
uli
nk
model,
the
s
im
ulation
is
pe
r
f
or
med
unde
r
the
pa
r
a
mete
r
s
s
hown
in
T
a
ble
7
.
T
wo
P
W
M
tec
hniques
f
o
r
c
on
tr
oll
ing
the
inve
r
ter
thr
e
e
-
pha
s
e
thr
e
e
-
leve
l
NPC
s
tr
uc
tur
e
ha
ve
be
e
n
s
im
ulate
d
in
thi
s
pa
pe
r
a
s
a
c
ompar
a
ti
ve
s
tudy.
F
ig
ur
e
6
.
T
he
s
im
uli
nk
model
of
the
pr
opos
e
d
th
r
e
e
-
pha
s
e
thr
e
e
-
leve
l
diode
c
lampe
d
inver
ter
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
1
,
F
e
br
ua
r
y
2020
:
519
-
529
526
T
a
ble
7
.
T
hr
e
e
-
pha
s
e
boos
t
inver
ter
pa
r
a
mete
r
s
(
M
AT
L
AB
/m
ode
l)
I
nput
D
C
vol
ta
ge
400V
A
C
l
oa
d pe
r
pha
s
e
R
e
s
is
ta
n
c
e
10Ω
I
nduc
ta
nc
e
1 m H
S
w
it
c
hi
ng f
r
e
que
nc
y
5x10
6
Hz
C
a
r
r
ie
r
S
ig
na
l
F
r
e
que
nc
y
(2
-
10)
K
H
z
F
unda
me
nt
a
l
F
r
e
que
nc
y
50 H
z
5.
1.
S
im
u
lat
io
n
s
r
e
s
u
lt
s
u
s
in
g
S
P
WM
t
e
c
h
n
i
q
u
e
F
r
om
the
s
im
ulation
model
in
F
igur
e
6
,
the
outpu
t
volt
a
ge
s
a
nd
pha
s
e
c
ur
r
e
nt
f
r
om
the
pr
opos
e
d
inver
ter
a
r
e
s
hown
in
F
igur
e
7.
F
igu
r
e
7
(
a
)
s
ho
ws
the
s
im
ulate
d
wa
ve
f
or
ms
of
the
thr
e
e
-
ph
a
s
e
s
our
c
e
volt
a
ge
s
inver
ter
output
li
ne
to
li
ne
volt
a
ge
,
pha
s
e
to
ne
ut
r
a
l
volt
a
ge
,
a
nd
pha
s
e
c
ur
r
e
nt
wa
ve
f
or
ms
f
or
R
L
load.
T
he
c
or
r
e
s
ponding
ha
r
mon
ic
s
pe
c
tr
um
is
s
ho
wn
in
F
ig
u
r
e
7
(
b)
.
(
a
)
(
b)
F
igur
e
7
.
(
a
)
Output
li
ne
-
to
-
li
ne
volt
a
ge
,
li
ne
-
ne
utr
a
l
volt
a
ge
a
nd
pha
s
e
c
ur
r
e
nt
f
or
thr
e
e
pha
s
e
th
r
e
e
l
e
ve
l
inver
ter
by
S
P
W
M
a
lgor
it
hm
.
(
b)
L
ine
-
to
-
li
ne
volt
a
ge
,
li
ne
-
ne
utr
a
l
volt
a
ge
a
nd
pha
s
e
c
ur
r
e
nt
F
F
T
T
a
ble
8
.
%
T
HD
output
vo
lt
a
ge
s
a
nd
pha
s
e
c
ur
r
e
nt
of
F
ig
ur
e
7
L
L
V
ol
ta
ge
N
L
V
ol
ta
ge
P
ha
s
e
c
ur
r
e
nt
F
unda
me
nt
a
l
(
50H
z
)
V
a
lu
e
329.3V
190.6 V
19.4A
T
H
D
35.60%
35.60%
12.93
5.
2.
S
im
u
lat
io
n
s
r
e
s
u
lt
s
u
s
in
g
S
V
P
WM
t
e
c
h
n
i
q
u
e
T
he
S
VPWM
s
im
uli
nk
s
ys
tem
model
is
the
s
a
me
a
s
that
of
the
S
P
W
M
s
ys
tem
e
xc
e
pt
f
or
the
modul
a
ti
ng
wa
ve
f
o
r
m
volt
a
ge
s
,
whic
h
a
r
e
ge
ne
r
a
ted
by
pr
opos
e
d
a
lgo
r
it
hm
e
xplaine
d
in
s
e
c
ti
on
3.
2
.
o
f
the
pr
opos
e
d
inver
ter
.
F
igur
e
8
(
a
)
s
hows
the
volt
a
ge
a
nd
c
ur
r
e
nt
wa
ve
f
or
ms
obtaine
d
by
c
ontr
ol
li
ng
the
thr
e
e
leve
l
S
VPWM
inver
ter
,
the
ha
r
mon
ic
s
pe
c
tr
um
of
the
s
our
c
e
c
ur
r
e
nt
a
f
ter
c
ompens
a
ti
on
is
s
hown
in
F
ig
ur
e
8
(
b
)
.
F
r
om
the
s
im
ulation
r
e
s
ult
,
it
c
a
n
obs
e
r
ve
d
that
th
e
S
P
W
M
tec
hnique
ha
s
tot
a
l
ha
r
moni
c
dis
tor
ti
on
of
li
ne
volt
a
ge
(
T
HD
)
is
35
.
60%
a
nd
the
a
mpl
it
u
de
of
the
f
i
r
s
t
ha
r
mony
a
s
329
.
3.
I
n
a
ddit
ion,
the
T
HD
of
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
P
W
M
c
ontr
ol
tec
hniques
for
thr
e
e
phas
e
thr
e
e
lev
e
l
inve
r
ter
dr
ive
s
(
A
r
k
an
A
.
K
adum)
527
the
pha
s
e
c
ur
r
e
nt
is
12.
93
%
,
the
a
mpl
it
ude
of
th
e
f
i
r
s
t
ha
r
moni
c
19
.
4
A.
W
he
r
e
the
ou
tput
li
ne
v
olt
a
ge
of
the
inver
ter
s
witche
d
by
S
VPWM
,
the
T
HD
is
28.
25%
a
nd
the
a
mpl
it
ude
of
the
f
ir
s
t
ha
r
moni
c
is
5
67.
7
V;
the
T
HD
o
f
the
pha
s
e
c
ur
r
e
nt
is
3
.
66%
,
a
nd
the
a
m
pli
tude
of
the
f
i
r
s
t
ha
r
moni
c
is
178
A.
(
a
)
(
b)
F
igur
e
8
.
(
a
)
Output
li
ne
-
to
-
li
ne
volt
a
ge
,
li
ne
-
ne
utr
a
l
volt
a
ge
a
nd
pha
s
e
c
ur
r
e
nt
f
or
thr
e
e
pha
s
e
th
r
e
e
l
e
ve
l
inver
ter
by
S
P
W
M
a
lgor
it
hm
.
(
b)
li
ne
-
to
-
li
ne
volt
a
ge
,
li
ne
-
ne
utr
a
l
volt
a
ge
a
nd
pha
s
e
c
ur
r
e
nt
F
F
T
T
a
ble
9
.
%
T
HD
output
vo
lt
a
ge
s
a
nd
pha
s
e
c
ur
r
e
nt
of
F
ig
ur
e
8
L
L
V
ol
ta
ge
N
L
V
ol
ta
ge
P
ha
s
e
c
ur
r
e
nt
F
unda
me
nt
a
l
(
50H
z
)
V
a
lu
e
382.5
V
221.8 V
22.19
A
T
H
D
28.77%
28.80%
11.8
5.
3.
Com
p
ar
is
on
of
S
P
WM
&
S
V
P
WM
I
n
ve
r
t
e
r
s
T
he
pe
r
f
or
manc
e
of
S
pa
c
e
Ve
c
tor
P
uls
e
W
idt
h
M
odulation
is
c
ompar
e
d
a
ga
ins
t
P
uls
e
W
idt
h
M
odulation
method
f
or
thr
e
e
-
leve
l
inver
ter
.
T
he
c
ompar
is
on
is
done
on
the
ba
s
is
of
s
im
ulation
r
e
s
ul
ts
a
nd
it
is
with
r
e
s
pe
c
t
to
ha
r
mon
ic
c
ontent
with
wi
de
r
a
nge
o
f
c
a
r
r
ier
f
r
e
que
nc
y
a
nd
modul
a
ti
o
n
index.
T
he
pe
r
f
o
r
manc
e
of
S
P
W
M
a
nd
S
VPWM
th
r
e
e
-
leve
l
inver
ter
s
f
or
s
im
ulate
d
r
e
s
ult
s
with
r
e
s
pe
c
t
to
pe
r
c
e
ntage
tot
a
l
ha
r
moni
c
dis
tor
ti
on
of
li
ne
to
li
ne
volt
a
ge
ove
r
a
wide
r
a
nge
of
c
ur
r
ier
f
r
e
que
nc
y
is
s
hown
in
F
igur
e
9.
I
t
is
obs
e
r
ve
d
the
pe
r
c
e
ntage
tot
a
l
ha
r
mo
nic
dis
tor
ti
on
of
li
ne
to
li
ne
volt
a
ge
im
p
r
ove
s
f
o
r
S
VPWM
inver
ter
a
s
c
ompar
e
d
to
S
P
W
M
inver
ter
.
T
he
tw
o
P
W
M
tec
hniques
with
thr
e
e
-
leve
l
inver
ter
a
r
e
s
im
ulate
d
f
or
dif
f
e
r
e
nt
modul
a
ti
on
indi
c
e
s
a
t
a
c
ons
tant
s
witching
f
r
e
que
nc
y
of
3000Hz
.
F
igu
r
e
10
r
e
s
ult
s
s
how
the
pe
r
f
or
manc
e
c
ur
ve
f
or
modul
a
ti
on
index
vs
T
otal
Ha
r
moni
c
Dis
tor
ti
on
.
Ac
c
or
ding
to
F
igu
r
e
10
,
it
s
note
that
the
incr
e
a
s
ing
o
f
the
modul
a
ti
on
index
f
or
t
hr
e
e
-
leve
l
S
VPWM
c
a
n
a
c
hieve
les
s
ha
r
moni
c
d
is
tor
ti
on
c
ompar
e
d
to
S
P
W
M
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
S
N
:
1693
-
6930
T
E
L
KO
M
NI
KA
T
e
lec
omm
un
C
omput
E
l
C
ontr
o
l
,
Vol.
18
,
No
.
1
,
F
e
br
ua
r
y
2020
:
519
-
529
528
(
a
)
(
b)
F
igur
e
9.
(
a
)
P
e
r
c
e
nt
T
HD
o
f
V
LL
Vs
F
t
ri
f
or
S
VP
W
M
a
nd
S
P
W
M
i
nve
r
ter
,
(
b
)
P
e
r
c
e
nt
T
HD
of
pha
s
e
C
ur
r
e
nt
Vs
F
t
ri
f
o
r
S
V
P
W
M
a
nd
S
P
W
M
i
nve
r
ter
(
a
)
(
b)
F
igur
e
10.
(
a
)
P
e
r
c
e
nt
T
HD
o
f
V
LL
Vs
modul
a
ti
on
index
f
or
S
VPWM
a
nd
S
P
W
M
I
nve
r
ter
,
(
b
)
P
e
r
c
e
nt
T
HD
of
pha
s
e
C
ur
r
e
nt
Vs
modul
a
ti
on
i
nde
x
f
or
S
VPWM
a
nd
S
P
W
M
I
nve
r
ter
6.
CONC
L
USI
ON
I
n
th
is
pa
pe
r
,
a
s
pa
c
e
ve
c
tor
ba
s
e
d
S
V
P
W
M
s
c
he
me
is
c
ompa
r
e
d
with
s
ine
-
tr
iangle
S
P
W
M
s
c
he
me
f
or
a
NPC
th
r
e
e
-
leve
l
inver
ter
.
T
he
modeling
a
n
d
s
im
ulation
o
f
the
thr
e
e
pha
s
e
thr
e
e
leve
l
diode
-
c
lampe
d
inver
ter
is
done
us
ing
the
M
AT
L
AB
/S
im
uli
nk
pr
ogr
a
m.
F
r
o
m
t
he
r
e
s
ult
s
it
is
obs
e
r
ve
d
that
the
s
pa
c
e
ve
c
tor
puls
e
width
modul
a
ti
on
tec
hnique
is
be
tt
e
r
pe
r
f
or
manc
e
than
c
las
s
ica
l
c
onve
nti
ona
l
method
(
S
P
W
M
)
.
T
he
s
pa
c
e
ve
c
tor
puls
e
width
modul
a
ti
on
t
e
c
hnique
pr
oduc
e
s
f
e
we
r
ha
r
moni
c
s
a
s
c
om
pa
r
e
d
to
the
c
onve
nti
ona
l
P
W
M
method.
T
he
s
inus
oidal
pul
s
e
width
modul
a
ti
on
(
S
P
W
M
)
gives
a
va
lue
of
0.
6
12
*Vdc
howe
ve
r
the
s
pa
c
e
ve
c
tor
method
s
hows
that
th
e
maximum
output
obtaine
d
is
0
.
707
Vdc
whic
h
is
15
%
higher
than
S
P
W
M
.
T
he
tot
a
l
ha
r
moni
c
dis
tor
ti
o
n
(
T
HD
)
of
the
output
wa
ve
f
or
m
is
r
e
duc
e
d
by
47%
than
the
s
inus
oidal
P
W
M
.
F
r
om
the
r
e
s
ult
s
it
is
c
lea
r
t
ha
t,
S
VPWM
ga
ve
les
s
c
ur
r
e
nt
r
ippl
e
,
a
nd
les
s
T
HD
than
the
S
P
W
M
modul
a
ti
on
s
c
he
me.
S
pa
c
e
ve
c
tor
pu
l
s
e
width
modul
a
ti
on
(
S
VPWM
)
tec
hnique
s
howe
d
be
tt
e
r
pe
r
f
or
manc
e
with
r
e
duc
e
d
T
HD
be
twe
e
n
0.
55
a
nd
0.
75
of
modul
a
ti
on
index
.
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