Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 4
,
A
ugu
st
2016
, pp
. 13
95
~
1
405
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
4.8
793
1
395
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Comparison of Voltage Vector
Control Based on Duty Cycl
e
Analysis in Three Phase Four Le
g System
of
Active Filt
er
I
n
d
r
ia
rt
o Yu
nia
n
to
ro
1,2
, Ru
dy
Seti
abud
y
1
,
Ridw
a
n
G
u
n
a
w
a
n
1
1
Department of
Electrical Eng
i
n
eering
,
Faculty
o
f
Engin
eering
,
Universitas Indon
esia, Depok, Ind
onesia
2
Department of
Electrical Eng
i
n
eering
,
Faculty
o
f
Industrial
Tech
nolog
y
,
Univ
ersity
of
Trisak
ti, Jakarta, Indon
esia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Aug 13, 2015
Rev
i
sed
D
ec 10
, 20
15
Accepted Dec 26, 2015
Comparison of
voltag
e
vector
control in variou
s forms of tetrahedron that
result from swit
ching combination on three-phase four-leg s
y
stem of
activ
e
filte
r is present
e
d especi
all
y
a
s
y
m
m
e
tric te
tra
h
edron shape which is a
projection pqr-
c
oordinate in
to
αβ
0-coordinate. Paramete
r
t
e
trah
e
d
rons
s
u
ch
as modulation b
oundar
y
-lin
e, r
e
ferenc
e vector
, switching dur
atio
n time and
duty
cy
cl
e a
r
e desc
ri
be
d. Duty
cyc
l
e
a
n
al
y
s
is con
ducted on
th
e Shen’s model,
the Zhang’s model, th
e Perales’s
model and as
y
mmetric’s model are
pres
ented
.
Th
e
chara
c
t
e
ris
t
i
c
re
s
u
lts
s
howed th
at s
w
itch
i
ng
co
m
b
ination of
each IGBT cond
uctor es
pec
i
al
l
y
its
review on the neutra
l wire.
As
y
m
m
e
tri
c
tetr
ahedron can
be proposed as contro
l technique in thr
ee-ph
ase four-leg
s
y
stem
of ac
tive
filte
r.
Keyword:
Asymmetric Tetrah
ed
ro
n
Duty Cycle
Mo
du
latio
n Bou
n
d
a
ry-lin
e
Refere
nce Vec
t
or
Vol
t
a
ge
Vect
or
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
In
dri
a
rt
o Y
uni
ant
o
ro
,
Depa
rt
m
e
nt
of
El
ect
ri
cal
Engi
neeri
n
g
,
Fac
u
l
t
y
of
En
gi
nee
r
i
n
g
.
Uni
v
ersitas Indone
sia,
Kam
pus
U
I
De
po
k 1
6
4
2
4
, I
n
d
one
si
a.
Em
a
il: in
d
y
_
s
in
gn
et@yah
oo
.co
m
1.
INTRODUCTION
Th
ree-p
h
a
se syste
m
activ
e filter to
po
log
y
has evo
l
v
e
d
from
th
ree-p
h
a
se th
ree-wire
b
e
co
m
e
s
th
ree-
pha
se four-wire and the
n
three-phase four-leg
.
In
three-ph
ase fou
r
-wire syst
e
m
o
f
acti
v
e filter, th
e
fo
urth-
wire
form
s a neutral
wire
which is connecte
d
to m
i
ddle place in c
o
uples of split
capacitors. Meanwhi
l
e, in
three-phase four-leg system
the fourth-wire
as a ne
utral wi
re is conn
ecte
d
to middle plac
e in a pair
of
IGBT
co
ndu
ctor
.
Step
s i
n
cu
rr
en
t co
m
p
en
satio
n co
n
t
r
o
l
on
t
h
r
e
e
-
phase
four-wi
r
e system
re
sem
b
le as on three-phase
t
h
ree-
wi
re sy
st
em
. C
ont
rol
o
f
t
h
e com
p
ensat
i
on cu
rre
nt
i
s
do
ne by
ad
j
u
st
i
ng v
o
l
t
a
ge
-ve
c
t
o
r ei
t
h
er
on t
h
ree
-
p
h
a
se fou
r
-wire syste
m
o
r
th
ree-ph
ase
fo
ur-leg syst
em
a
c
tiv
e filter. Vo
ltag
e
-v
ector i
s
th
ree-d
i
m
e
n
s
io
n
a
l
space-vector
where t
h
e vect
or
resulting of on-off switching
com
b
ination si
gnal from
pa
ir of
IGBT c
o
nductor.
In
sp
ite o
f
d
i
fferen
t to
po
logy,
b
o
t
h
activ
e filter h
a
v
e
same tech
n
i
q
u
e
. Sw
itch
i
ng
com
b
in
atio
n
o
n
th
ree-
dim
e
nsional space-vector m
odulati
on (SVM)
has produce sixtee
n pieces
of voltage-vector [1],
[2]
.
Mo
du
latio
n
p
a
ram
e
ters su
ch
as referen
ce
v
e
cto
r
,
du
ty cycl
e, pu
lse p
a
ttern
, to
tal h
a
rm
o
n
i
cs d
i
stortion
(THD)
and
di
st
ort
i
on
fact
or (
D
F
)
can be
det
e
r
m
i
n
ed by
si
xt
eent
h
of v
o
l
t
a
ge-
v
ect
o
r
i
n
t
h
ree-
di
m
e
nsional
o
f
tetrah
edron
sh
ap
e an
d th
en
v
o
l
tag
e
-v
ector con
t
ro
l on
t
h
ree-p
h
a
se activ
e
filter is regu
lated b
y
all of t
h
em
.
In th
e
Sh
en’s
Mo
d
e
l
[1
], resu
ltin
g of si
x
t
een
th
switch
i
n
g
co
m
b
in
atio
n
s
in
t
h
ree-ph
ase fo
ur-wi
r
e
activ
e filter can
b
e
d
e
scrib
e
d
b
y
th
e inv
a
riant Clark
’
s tran
sfo
r
m
a
tio
n
.
Three d
i
m
e
n
s
io
n
a
l p
o
s
ition
o
f
v
o
l
tag
e
-
vector located in
αβ
0-c
o
or
di
nat
e
s co
nsi
s
t
e
d o
f
t
w
o
poi
nt
of ze
ro
swi
t
c
hi
n
g
vect
or
(Z
SV)
p
o
si
t
i
on a
t
t
h
e
ori
g
i
n
al
co
or
di
nat
e
an
d fo
u
r
t
een p
o
i
n
t
s
o
f
no
n-ze
ro s
w
i
t
c
hi
n
g
vect
ors
(NZ
S
V
)
at
an
ot
he
r. T
h
e fo
u
r
t
eent
h
poi
nt
s o
f
N
Z
S
V
can
be cl
ass
i
fi
ed i
n
t
o
si
x-
p
r
i
s
m
s
cons
ists th
e to
p-b
o
tto
m
o
f
tetrah
ed
ro
ns. I
t
sh
ow
s
f
our
teen
-
pos
si
bl
e p
o
si
t
i
ons
v
o
l
t
a
ge
-ve
c
t
o
r as
refe
re
n
ces i
n
t
h
e
b
o
u
nda
ry
o
f
eac
h
t
e
t
r
ahed
r
o
n
.
Sh
oul
d be
n
o
t
e
d, t
h
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
13
95
–
1
405
1
396
coo
r
di
nat
e
t
r
ansf
orm
a
t
i
on i
s
obt
ai
ne
d fr
o
m
t
h
e i
nvari
an
t
C
l
ark’s t
r
ans
f
o
r
m
a
ti
on w
h
i
c
h pr
o
duce m
a
xi
m
u
m
m
odul
at
i
on b
o
u
n
d
a
r
y
-
l
i
n
e on
αβ
0-c
o
o
r
di
nat
e
are 0.
8
2
Vdc
.
In
t
h
e
Zha
n
g
’
s m
odel
[
2
]
,
co
or
di
nat
e
t
r
ans
f
orm
a
t
i
on base
d
o
n
t
h
e
no
n
-
i
n
vari
ant
C
l
ar
k’s
t
r
ans
f
o
r
m
a
ti
on
i
s
used t
o
pr
od
uce si
xt
ee
nt
h swi
t
c
hi
ng c
o
m
b
i
n
at
i
on p
o
s
i
t
i
ons at
αβγ
-coordinates in three-
pha
se four-leg power converte
r (active filter). It de
scribe
s in one s
ector whic
h has four tetrahedron of each
voltage
-vector occ
upies t
h
re
e-dim
e
ns
ional
space-vector s
o
in
overall t
h
ey ha
ve twe
n
ty-four tetra
h
edrons.
R
e
fere
nce-
vect
ors
of t
e
t
r
a
h
ed
r
ons a
r
e
c
o
m
b
ination of three
pieces of voltage
-vect
or
neares
t neighbors
. It can
be seen
from
the non-inva
r
iant Cl
ark’s transform
a
tion that
m
a
xim
u
m
m
o
dulation boundary-line according to
t
h
e Z
h
an
g’
s m
odel
a
r
e e
q
ual
0.
67
V
d
c.
W
i
t
h
ou
t do
ing Clark
’
s tran
sfo
r
m
a
tio
n
,
in
th
e Perales’s m
o
d
e
l [3
] switch
i
n
g
co
m
b
in
atio
n
o
f
o
n
-off
p
a
ir
o
f
IGBT con
d
u
c
t
o
r i
n
th
ree-
ph
ase fo
ur-wire of
activ
e filter
p
r
o
d
u
ces six
t
een
t
h
vo
ltag
e
-v
ecto
r as
refe
rence
-
v
ect
or l
i
k
e t
h
e
Sh
en’s
m
odel
an
d t
h
e
Zha
n
g’s
m
odel
.
V
o
l
t
a
ge-
v
ect
o
r
s i
n
abc-c
o
or
di
nat
e
s o
f
t
h
e
Peral
e
s’s m
o
d
e
l
whi
c
h i
s
di
r
ect
l
y
m
a
pped i
n
t
h
ree
-
di
m
e
nsi
onal
space
-
v
e
c
t
o
r p
r
od
uce a
do
deca
hed
r
o
n
shape
wi
t
h
a si
de l
e
ngt
h as
m
a
xi
m
u
m
m
odul
at
i
on
bo
u
nda
ry
-l
i
n
e are 1 V
d
c
.
From
si
xt
eent
h
com
b
i
n
at
i
ons
of
vol
t
a
ge
-
v
ect
o
r
s i
t
can be
m
a
de t
w
e
n
t
y
-f
o
u
r
regi
on
p
o
i
n
t
e
r
s
(t
et
rahe
d
r
o
n
s
)
as
refe
rence
-
vect
o
r
s
whi
c
h
m
a
ke
descri
ption three pieces
of volta
ge-vect
or ne
arest nei
g
hbors
.
Thi
s
pa
pe
r ai
m
s
t
o
creat
e a new
m
odel
of
v
o
l
t
a
ge-
v
ec
t
o
r
based
o
n
t
h
e Ki
m
-
Aka
g
i
’
s m
a
ppi
n
g
matrices
m
o
d
e
l (pq
r-p
ower th
eory)
doing
for com
p
ensati
on curre
nt c
o
ntrol s
p
ace
vect
or m
odulation
(SVM
),
i
t
can be
use
d
t
o
harm
oni
cs
el
im
i
n
at
i
on [
4
]
,
[
5
]
.
C
u
rre
nt
C
o
m
p
ensat
i
o
n c
ont
rol
i
n
t
h
ree
-
p
h
ase
f
o
u
r
-
w
i
r
e
syste
m
o
f
active filter is exp
l
ain
e
d
b
y
th
e
pq
r-po
wer t
h
eory. Rein
terpretatio
n of
p
q
r-coo
rd
in
ate
fro
m
m
a
p
p
i
n
g
m
a
t
r
i
ces
m
odel
by
t
h
e Eul
e
r angl
e r
o
t
a
t
i
on
m
e
t
hod i
s
t
u
r
n
ed o
u
t
,
an
d i
t
pro
d
u
ces i
n
a di
ffe
rent
way
w
h
en i
t
i
s
com
p
ared from its
orig
in
al t
h
eory. In
th
e
Kim
-
Ak
ag
i’s map
p
i
ng
m
a
tri
ces
m
o
d
e
l, pq
r-coo
r
d
i
n
a
te is
id
en
tical
to dqn-c
oordinate whe
n
t
h
e
n-axes
coi
n
cide
s on the
0-
a
x
es
, wherea
s in the Euler angle rotation m
e
thod pqr-
co
ord
i
n
a
te is
g
e
n
e
rated from
twice ro
tatio
n [6
]. Proj
ect
io
n
o
f
pqr-co
o
rd
i
n
ate in
to
αβ
0-c
o
or
di
at
e p
r
o
d
u
ce
s
t
w
ent
y
-f
o
u
r
of
asym
m
e
t
r
i
c
vol
t
a
ge
-
v
ect
o
r
i
n
t
h
ree
-
di
m
e
nsi
o
nal
space
-
v
ect
o
r
. T
h
e m
a
xi
m
u
m
m
odul
at
i
on
bounda
ry-line
according t
o
the Euler an
gle
rotation m
e
thod is equal
0.5
Vdc.
2.
MO
DUL
ATI
O
N B
O
U
N
D
A
R
Y
-
LINE
IN
THR
EE-PHASE SYSTEM
ACTIVE FIL
TER
Int
e
r
s
ect
i
o
n
o
f
a re
fere
nce s
i
gnal
fr
om
sources
w
h
i
c
h
i
s
de
ri
ve
d a
n
y
si
gnal
t
h
at
d
o
es
not
ha
v
e
peri
odic si
gnal
but
has a
n
y a
m
plitude there
will be pr
oducing asym
m
e
tric
m
odula
tion.
Square wa
ve pulse
wi
dt
h
m
odul
at
i
on (P
WM
) si
g
n
al
w
h
i
c
h has a
resul
t
of
t
h
i
s
i
n
t
e
rsect
i
o
n
i
n
t
h
e
i
n
t
e
r
v
al
(
0
-
π
/ 2
)
,
(
π
/ 2-
π
), (
π
-
3
π
/ 2), and
(3
π
/ 2
-
2
π
),
it
also has random
and diffe
re
nt from
each
other of duty
cycle.
Any
si
gnal can be
ei
t
h
er t
o
t
a
l
ha
rm
oni
cs or
di
sor
d
e
r
si
g
n
al
.
Asy
m
m
e
t
r
i
c
m
odel
i
n
t
h
i
s
m
odul
at
i
o
n c
a
n be
res
u
l
t
e
d
fr
o
m
pr
o
j
ect
i
on
of t
h
e p
q
r
-
co
or
di
n
a
t
e
of vol
t
a
ge
-
v
ect
o
r
of t
h
e
Eul
e
r an
gl
e ro
t
a
t
i
on
m
e
t
hod
i
n
t
o
αβ
0-c
o
or
d
i
nat
e
.
Fo
uri
e
r a
n
al
y
s
i
s
can n
o
t
be p
e
rf
orm
e
d beca
use o
f
t
h
e res
u
l
t
i
ng sq
uare si
gnal
i
s
n
o
t
per
i
odi
cal
l
y
. To k
eep i
n
separat
ec
fo
rm
ul
at
i
ons
, i
t
i
s
needed t
o
pe
rf
or
m
a dut
y
cycle
analysis. Carri
er-base
d
and hysteresis
m
odulation
i
s
di
f
f
ere
n
t
f
r
o
m
asymm
e
t
r
i
c
m
odul
at
i
on.
The following step of
duty cycle
analysis in space-vect
or m
odul
ation (SVM
) has
done through
measurem
ent such as
followi
ng [7]:
Determ
in
e th
e
switch
i
ng
co
mb
in
ation
,
Ind
e
n
tify
th
e vo
ltag
e
-v
ector po
sitio
n
s
,
Inde
ntify
the refere
nce vector,
Determ
in
e th
e
switch
i
ng
tim
e
du
ration
,
Calculate duty
cycle,
Determ
in
e th
e
p
u
l
se p
a
ttern
,
Th
ree-p
h
a
se t
h
ree-wire system activ
e filter
h
a
s a cap
acit
o
r in
DC-link
voltag
e
th
at is eq
u
a
l 1
Vd
c.
The m
a
xim
u
m
m
odul
at
i
on b
o
u
n
d
a
r
y
-
l
i
m
i
t perm
i
ssi
bl
e
i
s
equal
t
o
DC
-l
i
n
k v
o
l
t
a
ge. R
e
fe
rence ve
ct
or a
n
d d
u
t
y
cycle are determined by dra
w
ing a pe
rpe
n
dicular line as the resultant bet
w
een
two pieces
of voltage vector
i
n
line of the
circ
le. Pythagoras
form
ula can be
det
e
rm
i
n
ed t
h
e refe
re
nce
ve
ct
or l
e
ngt
h i
s
√
3 / 2
= 0,87
V
d
c
.
It
can be co
ncl
u
d
e
d t
h
at
val
u
e o
f
refe
rence
vec
t
or bet
w
een
0.
87
Vdc -
1
V
d
c
i
s
m
odul
at
i
on bo
u
nda
ry
-l
i
n
e
whi
c
h
i
s
a for
b
i
d
de
n
zone
wi
t
h
i
n
t
h
e l
i
m
i
t
s
of t
h
e Qua
n
g’s m
odel
[8]
.
T
h
e P
e
ral
e
s’s m
odul
at
i
on b
o
u
n
d
a
r
y
-
l
i
n
e
d
e
scri
p
tio
n is l
i
k
e
th
e
Qu
ang’s m
o
d
e
l.
Th
ree-p
h
a
se
fo
ur-wire system an
d
fou
r
-leg
syste
m
o
f
activ
e filter is
d
e
v
e
l
o
p
e
d
b
y
Sh
en
[1
] an
d
Zh
an
g
[2
] wh
i
c
h
is ob
tain
ed
v
o
ltag
e
v
ector
th
at h
a
s a v
a
l
u
e s
m
aller th
an
allo
wab
l
e m
o
du
latio
n
b
oun
d
a
ry-lin
e
of
t
h
ree
-
phase
t
h
ree-
wi
re
sy
st
em
t
h
at
i
s
eq
ua
l
0.
82
V
d
c a
n
d
0.
67
V
d
c l
i
k
e
s
h
o
w
n i
n
Fi
g
u
r
e
1.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
C
o
m
p
a
r
i
s
o
n
of
Vol
t
a
ge V
ect
o
r
C
o
nt
rol
B
a
se
d
on
D
u
t
y
C
ycl
e An
al
ysi
s
i
n
T
h
ree
...
. (
I
ndri
a
rt
o Y
uni
a
n
t
o
r
o
)
1
397
Fig
u
re 1
.
Modu
latio
n
bo
und
ary-lin
e
in
three-ph
ase
syste
m
s of activ
e
filter
2.
1.
Co
ordi
n
a
te T
r
ans
f
or
ma
ti
o
n
M
o
del
Using
th
ree-ph
ase fo
ur-leg syste
m
o
f
activ
e filter, Sh
en
[1
] exp
l
ain
e
d
th
e inv
a
rian
t Clark
’
s
t
r
ans
f
o
r
m
a
ti
on
coo
r
di
nat
e
sy
st
em
i
n
w
h
i
c
h
va
ri
abl
e
s a
r
e c
o
n
v
ert
e
d
by
f
o
l
l
o
wi
n
g
e
quat
i
o
n:
(1
)
The si
xt
een
p
o
ssi
bl
e c
o
m
b
i
n
at
i
ons ca
n
be
di
vi
de
d i
n
t
o
t
w
o
vect
or
swi
t
c
hi
ng
w
h
i
c
h
has
a l
e
n
g
t
h
i
s
equal
0.
82
V
d
c. I
n
dat
a
p
r
oces
si
n
g
, co
n
t
rol
vect
or
-spa
ce i
s
necessar
y
for a c
o
n
v
e
nt
i
onal
t
h
ree-
pha
s
e
con
v
e
r
ter
req
u
i
r
es the
re
fere
n
ce vecto
r
.
Usi
n
g t
h
e sa
m
e
form
ul
at
i
on base
d o
n
coo
r
di
nat
e
t
r
a
n
sf
orm
a
t
i
on (t
he n
o
n
-
i
n
vari
ant
C
l
ark’s
tran
sform
a
t
i
o
n
), if th
e ta
bl
e t
r
ansf
o
r
m
a
ti
on i
s
m
a
de i
n
t
o
ab
c-co
or
di
nat
e
t
o
αβ
0
-
coord
i
n
a
te lik
e Sh
en
[1
], th
an
a com
b
ination
of switchi
ng
from
each voltage vector
pr
oduce sixteen poss
ible.
(2)
Zha
n
g
[
2
]
al
so
pr
o
duce
1
6
s
w
i
t
chi
n
g
com
b
i
n
at
i
on
on
-
o
f
f
pa
i
r
o
f
IGB
T
co
n
duct
o
r
-
a,
b
,
c
and
ne
ut
ral
.
2.
2.
Co
ordin
a
te R
o
t
a
ti
on Me
th
o
d
R
o
t
a
t
i
on m
e
t
hod
of E
u
l
e
r a
n
gl
e can be
d
o
n
e
wi
t
h
a cont
i
n
uo
us c
o
n
f
i
g
ura
t
i
on ei
t
h
er t
w
o
or m
o
re t
h
e
C
a
rt
esi
a
n’s
co
or
di
nat
e
.
R
o
t
a
t
i
on R
(
θ
1
,
θ
2
,
θ
3
) is a relation
wh
ich
is
don
e to
m
ove th
e stationa
ry
re
fere
nce
fram
e
to
ward
s
th
e ro
tatin
g referen
c
e fram
e.
In m
a
th
e
m
a
tic
al n
o
t
ation
,
th
e Eu
ler ang
l
es
ro
tatio
n
m
e
th
od can
b
e
d
e
term
i
n
ed as
fo
llows;
R (
θ
1
,
θ
2
,
θ
3
) =
R
x
(
θ
3
) R
y
(
θ
2
) R
z
(
θ
1
)
=
(
3
)
Whe
r
e i
n
0<
θ
1
,
θ
2
<2
π
da
n
0
<
θ
3
<
π
.
Pri
n
ci
pl
e o
f
t
h
e Eul
e
r
a
ngl
e
rot
a
t
i
o
n m
e
t
hod ca
n
be
use
d
t
o
rei
n
t
e
r
p
ret
a
t
i
on
of
t
h
e
K
i
m
-
Aka
g
i
’
s
map
p
i
ng
m
a
trices
m
o
d
e
l in
so
lv
ing
prob
lem
s
wh
ich
is re
lated
to
p
q
r
-coo
rd
in
ate in
[4
],[5
]. Mu
ltip
licatio
n
o
f
twice ro
tation
in
θ
1
,
θ
2
ang
l
e is th
e sam
e
i
n
pr
odu
cing
fo
r
m
u
l
at
i
on as
t
h
e Ki
m
-
Aka
g
i
’
s m
a
ppi
ng m
a
t
r
i
ces
m
o
d
e
l as fo
ll
ows:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
13
95
–
1
405
1
398
=
(4
)
Here
in
X
i, v
(
c
ur
rent
o
r
vol
t
a
ge)
.
E
quat
i
on
4
ca
n be descri
bed
i
n
st
eps su
ch
as
Fi
g
u
re
2
and
Fi
g
u
r
e 3
f
o
r
each of
the Eul
e
r
a
ngle rotation.
Fi
gu
re
2.
C
h
a
n
ge a
fi
rst
of
an
gl
e p
r
od
uci
n
g
r
o
t
a
t
i
on i
n
t
h
e
h
o
ri
z
ont
al
pl
ane
Fi
gu
re
3.
C
h
a
n
ge t
h
e
sec
o
n
d
of
an
gl
e
pr
od
u
c
i
ng
r
o
t
a
t
i
on i
n
t
h
e
vert
i
cal
pl
ane
2.
3.
Asymmetric T
e
trahedr
o
n
Shape
Ro
tatio
n
m
e
th
o
d
of two
or t
h
ree Eu
ler an
gle b
eco
m
e
s more
like a s
p
iral sha
p
e tha
n
a cylindrical
sh
ap
e. A sp
iral sh
ap
e will b
e
easily
an
alyzed
wh
en
it
is c
o
n
s
i
d
ered
as a sk
ewed
v
ector. Three-d
i
m
e
n
s
io
n
a
l
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
C
o
m
p
a
r
i
s
o
n
of
Vol
t
a
ge V
ect
o
r
C
o
nt
rol
B
a
se
d
on
D
u
t
y
C
ycl
e An
al
ysi
s
i
n
T
h
ree
...
. (
I
ndri
a
rt
o Y
uni
a
n
t
o
r
o
)
1
399
space
vector in three
-
phase
four-leg sy
stem
of acti
v
e filter
for this is a
cy
lindrical coordinate
(sha
pe)
with six
prism
such as
Figure
4a,
whi
l
e based
on E
u
ler angle
rotation m
e
thod this
result form
s a
stack of two
pieces
hexa
g
onal
as
skew
ed
vect
o
r
suc
h
as Fi
g
u
r
e 4
b
,
4c. R
o
t
a
t
i
on m
e
t
hod
of t
w
o a
n
gl
es Eul
e
r
pr
o
duc
es p
q
r
-
coo
r
di
nat
e
i
f
i
t
i
s
do
ne t
h
e
pr
o
j
ect
i
o
n
i
n
a
h
o
ri
zo
nt
al
pl
a
n
e (
s
l
i
d
i
n
g si
d
e
) o
b
t
a
i
n
i
ng
n
e
w c
o
o
r
di
nat
e
s d
q
r
-
coo
r
di
nat
e
suc
h
as fi
gu
re-
4
b
and
pr
o
j
ect
i
o
n
i
n
a vert
i
cal
pl
ane (sl
i
d
i
ng
o
b
l
i
que)
o
b
t
a
i
n
i
n
g
αβ
r
-
co
or
di
na
t
e
such
as Figure
-4c.
Fi
gu
re
4.
p
q
r
-
c
o
o
r
di
nat
e
pr
o
j
e
c
t
i
on i
s
pr
o
duc
e;
(a) a
cy
l
i
n
d
r
i
cal
coo
r
di
nat
e
;
(
b
)
d
q
r
-
co
o
r
di
nat
e
;
(c)
αβ
r-
coo
r
di
nat
e
Two hexa
gona
l stacked with a shif
t at a cen
tral point where the he
xa
go
na
l
of t
h
e t
op an
d t
h
e b
o
t
t
o
m
are
not ce
ntral
i
zed, it ca
n
be
conside
r
ed as
a s
k
ewe
d
vect
or
. T
h
e ce
nt
er
part
of
t
h
e
co
m
posi
t
i
on o
f
t
h
e t
w
o
hexa
gonal sha
p
es
el
ongated hexa
gonal
with two-piece ce
nter
point,
na
mely
state-(1,1,1,1) an
d state-(0,0,0,0)
whic
h does
not coincide
with each
othe
r.
Overall, three-di
mensional s
p
a
ce
vector ca
n
be parse
d
int
o
twelve-
p
a
irs of asymmetric te
trah
ed
ron
.
Asymmetric tetrah
ed
ro
n
sh
ape b
e
t
w
een eac
h ot
her
of t
e
t
r
ahe
d
r
o
n i
s
in
d
i
cates h
a
r
m
o
n
i
cs occur
r
e
nce o
n
vo
ltag
e
v
ector
in
w
h
ich
h
a
s go
t 0
.
5
V
d
c. H
e
x
a
gonal p
iles w
h
ich
f
o
r
m
s an
asym
m
e
t
r
i
c
t
e
trahe
d
r
o
n pai
r
whe
n
a
r
e m
a
pped i
n
a t
w
o-
d
i
m
e
nsi
onal
h
o
r
i
zont
al
pl
a
n
e (
v
i
e
we
d f
r
om
the t
o
p)
can
b
e
ob
tain
ed
coo
r
d
i
n
a
te po
in
ts wh
ich
is th
e p
o
s
ition
voltag
e
v
ecto
r
in
αβ
0
-
coor
d
i
n
a
tes su
ch
as Figu
r
e
5a
and
f
r
om
vert
i
cal
pl
ane
(
v
i
e
w
e
d
fr
om
t
h
e si
d
e
) as
sh
ow
n
f
o
l
l
o
wi
n
g
i
n
Fi
gu
re
5b
.
Fig
u
re
5
.
Asy
mme
tric tetrahed
ron
in
αβ
0-co
ord
i
n
a
te; (a)
View
fro
m
th
e t
o
p
(h
orizon
tal); (b
) View from
th
e
side (ve
r
tical)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
13
95
–
1
405
1
400
Each
of t
h
ese
coo
r
di
nat
e
can
be m
a
pped i
n
fo
ur
-
p
ai
r com
b
i
n
at
i
on
of
IGB
T
and
ge
nerat
e
s a vol
t
a
g
e
vect
o
r
i
n
spac
e of t
e
t
r
ahe
d
r
o
n. Si
xt
eent
h
com
b
i
n
at
i
on pai
r
of I
G
B
T
co
n
duct
o
r ca
n be
descri
bed i
n
t
w
ent
y
-
fo
ur c
o
o
r
di
nat
e
s of a pi
l
e
of
reg
u
l
a
rl
y
hex
a
go
nal
,
w
h
i
c
h
m
eans t
h
ey
have t
w
ent
y
-
f
o
u
r
or t
w
el
ve
-
p
ai
rs o
f
asymm
e
tric tetrahe
d
ron. The
twenty
-fourth coordinates c
a
n
be dec
o
m
pose
d
into each of tetrahe
d
ron with
separate
d cent
e
r point,
where
are they
both in
state-(1,1
,1,1) a
nd i
n
state-
(0,0,0,0). Base
d on
fi
gure-5b, each
sid
e
d of
p
r
ism
with
a leng
th
of “
a
”
i
s
e
q
ual
0.
5 Vdc
f
o
r
up
si
de part
of
as
ym
m
e
t
r
i
c
t
e
t
r
ahed
r
o
n
an
d -
0
.
5
Vdc
for
d
o
wn
sid
e
part of asymm
e
t
r
ic tetrah
ed
ron. Tab
l
e-1
will b
e
g
i
v
e
n
p
o
sitio
n
of
tw
en
ty-fo
urth vo
ltag
e
v
ector
as a resu
lt of a sw
itch
i
ng
com
b
in
atio
n
p
a
ir o
f
IG
BT condu
ctor on
th
ree-p
h
a
se syste
m
activ
e filter.
Twenty-
fo
urt
h
v
o
l
t
a
ge
vect
or ca
n
b
e
descri
bed as
an asy
m
m
e
tr
ical tetrah
ed
ron
p
a
ir. A
s
k
e
w
ed vector
ac
tually
resem
b
les th
e Perales’s m
o
del wh
en
d
e
termin
in
g
vo
ltage v
ecto
r
u
s
ed
to
ab
c-coo
r
d
i
nate. In
th
is case, th
e
d
i
fferen
ce of state-(1
,1
,1
,1) an
d
state-(0
,0
,0
,0
)
d
o
e
s no
t co
i
n
cid
e
with
each
o
t
h
e
r at
o
n
e
p
o
s
ition
.
Peral
e
s [3
]
get
t
w
ent
y
-f
o
u
r
vol
t
a
g
e
vect
ors
wi
t
h
o
u
t
C
l
arke t
r
a
n
sf
o
r
m
a
t
i
on. I
n
space
vect
or m
odul
at
i
on (S
VM
)
C
l
ark’s
tran
sform
a
t
i
o
n
is u
s
ed
to ob
tain
v
o
ltag
e
v
e
ctor.
Tabl
e 1. V
o
l
t
a
ge vect
o
r
on
t
h
e
ske
w
e
d
vect
o
r
m
odel
(a=
0
.
5
V
d
c)
Voltage
Vector
Va
(Vd
c
)
V/a
(Vd
c
)
V/a
(Vd
c
)
Voltage
Vector
V/a
(Vd
c
)
V/a
(Vd
c
)
V/a
(Vd
c
)
V
0
0 0
0 V
12
0
-
0
V
1
0
0
-
1
/2
V
13
¼
-
0
V
2
0
0
½
V
14
-1
/4
-
½
V
3
-1
/4
-
-1
/2
V
15
-1
/4
-
0
V
4
¼
-
½ V
16
-1
/2
0
½
V
5
½
0
-1/2
V
17
-3
/4
-
½
V
6
¼
-1
/2
V
18
-1
/2
-
½
V
7
-1
/4
-1
/2
V
19
0
-
½
V8
-1
/2
0
-1
/2
V
20
¼
-
½
V
9
-
1
/2
0
0
V
21
½
0
0
V
10
-3
/4
0 V
22
¼
-
0
V
11
-1
/2
0 V
23
-1
/4
-
0
Refere
nce vect
or is sum
of three pieces
of
voltage
vector nearest neighbor
and its representsation
com
p
ares of
magnitude
(Vdc) from
Va, Vb, a
nd
Vc.
Reference ve
ctor
can be
obtained from
each
of
tetrah
edron
.
After th
e
d
e
term
in
atio
n
o
f
referen
ce
v
ect
or,
nex
t
step
s are
determin
ed
switch
i
ng
tim
e d
u
r
atio
n
and calculated
duty cycle.
In figure-6,
re
fere
nce
vect
o
r
(V
r
e
f)
has
de
scri
b
e
d i
n
a
sp
heri
c
a
l
co
or
di
nat
e
a
n
d
t
h
i
s
redu
ction
aim
s
to
d
e
term
in
e switch
i
ng
tim
e
d
u
ration
an
d du
ty cycle.
Fi
gu
re 6.
R
e
fer
e
nce vect
o
r
(V
ref
)
descri
pt
i
o
n
i
n
a sp
heri
cal
coo
r
di
nat
e
To
g
e
t th
e li
n
e
to
n
e
u
t
ral
v
o
ltag
e
(V
an
, V
bn
, V
cn
, V
nn
)
,
ci
rcui
t
of a
,
b,c
-
co
n
duct
o
r
and
ne
ut
ral
co
ndu
ctor
o
n
th
e activ
e filter
b
a
sed
o
n
a sp
herical coo
r
d
i
n
a
te can
b
e
o
b
t
ai
n
e
d as
fo
llows
[9
]:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
C
o
m
p
a
r
i
s
o
n
of
Vol
t
a
ge V
ect
o
r
C
o
nt
rol
B
a
se
d
on
D
u
t
y
C
ycl
e An
al
ysi
s
i
n
T
h
ree
...
. (
I
ndri
a
rt
o Y
uni
a
n
t
o
r
o
)
1
401
(5a
)
(5
b)
(5c
)
(5
d)
So t
h
e swi
t
c
hi
ng
d
u
rat
i
o
n t
i
m
e
of eac
h a,
b
,
c-c
o
n
d
u
ct
or
a
n
d
ne
ut
ral
c
o
n
duct
o
r a
r
e
obt
a
i
ned
fr
om
t
h
e
fo
llowing
equ
a
tio
n
s
:
=
=
(6a
)
=
=
(6
b)
=
=
(6c
)
=
=
(6
d)
No
ted
th
at T
z
=
,
m
odul
at
i
on i
n
dex
,
M
=
, and t
h
e bo
u
nda
ry
co
n
d
i
t
i
ons eac
h sect
or i
s
0
,
0
. Furt
herm
ore, to
determine the
duty
cycle
of e
ach tetrahe
d
ron ca
n be
use
d
t
o
t
h
e
fo
llowing
equ
a
tio
n
.
(7a
)
(7
b)
(7c
)
(7
d)
Herei
n
Dz
= 1-
(D
1
+D
2
+D
3
).
Value of Vref
and
ф
,
ψ
-a
ngles if incorporated
in
to
t
w
enty-fou
r
t
h
asymmetric tetrah
ed
ron
can
b
e
d
e
term
in
ed
ti
me d
u
ration
and
d
u
t
y cycle, it is requ
ired
f
o
r s
w
i
t
c
hi
n
g
com
b
i
n
at
i
on of b
o
t
h
a,b,c
-
c
o
n
d
u
ct
or
a
n
d
neut
ral
co
n
duc
t
o
r.
3.
DUT
Y
CYCL
E ANAL
YSIS ON
VARIOUS TETRAHE
D
RON SHAPE
Geom
etrical principles are
ne
eded t
o
analys
e the re
fe
re
nce
vector on the t
e
trahe
d
ron
sha
p
e.
A cubic
h
a
s a len
g
t
h
si
d
e
is
2
u
n
it
(cen
ti
m
e
ter) if it is cu
t in cro
s
s sectio
n
t
h
an
resu
ltin
g a tetrah
edro
n
sh
ap
e l
i
k
e
as
Fi
gu
re 7
.
The l
o
n
g
i
t
u
de i
s
co
nnect
e
d
a cent
r
al
poi
nt
t
o
wa
r
d
m
i
ddl
e poi
nt
,
and i
t
can be u
s
ed t
o
det
e
rm
ine t
h
e
val
u
e
of
refe
re
nce vect
or
of t
e
t
r
ahe
d
r
on
(
V
r
e
f). T
h
e
resul
t
i
ng a
ngl
e
bet
w
een t
h
e l
o
n
g
i
t
u
de t
o
t
h
e e
d
ges
pl
ane
defi
ned
Φ
-a
ngl
e w
h
i
l
e
t
h
e res
u
l
t
i
ng a
n
gl
e be
t
w
een t
h
e l
o
n
g
i
t
udes t
o
t
h
e
u
p
ri
ght
pl
anes
d
e
fi
ne
d
ψ
-
a
n
g
l
e s
u
ch
as i
n
equat
i
o
n-
5 an
d eq
uat
i
o
n
-
6
.
W
e
i
g
ht
val
u
e of t
h
e
α
-a
x
e
s,
β
-
a
xes and
γ
-axes are
represente
d duty cycle
(D
1
,D
2
.D
3
).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
13
95
–
1
405
1
402
Fig
u
re
7
.
Geometry an
alyse o
f
tetrah
ed
ron
fro
m
a cu
b
i
c cu
ttin
g
3.
1.
Duty Cycle Analysis
in the Shen’s Model
Th
er
e ar
e six
no
n-
zero v
ect
o
r
sw
itch
i
ng
(N
ZSV
)
an
d
t
w
o z
e
ro
vect
or
s
w
i
t
c
hi
n
g
(ZS
V
)
i
n
o
n
e s
ect
or
(6
0
o
)
whic
h ca
n be
desc
ribe
d as switching
com
b
ination
divided i
n
to
four initial
tetrahedron, whe
r
ein
each
t
e
t
r
ahed
ro
n i
s
defi
ned
by
t
h
r
ee NZS
V
a
n
d
t
w
o Z
S
V
.
F
o
urt
h
o
f
t
e
t
r
a
h
e
d
r
o
n i
n
eac
h s
ect
or ca
n
be p
r
o
d
u
ced
param
e
t
e
r t
e
t
r
ahed
r
on
suc
h
as
refe
rence
vect
or a
n
d
dut
y
cy
cle. In Ta
ble 2, duty cy
cle cal
cu
latio
n
resu
lt in
the
She
n
’s
m
odel
i
s
gi
ven
.
Table
2.
Duty cycle calculation in t
h
e She
n
’s m
odel
No Vref
D
1
D
2
D
3
Dz/2
No
Vref
D
1
D
2
D
3
Dz/2
1
1,
0601
4
0,
80
0,
22
0
0
7
0,
8165
0,
44
0,
56
0
0
2
0,
8165
0,
35
0,
62
0
0
8
1,
0601
4
0,
61
0,
44
0
0
3
0,
8165
0,
44
0,
56
0
0
9
1,
0601
4
0,
80
0,
22
0
0
4
1,
0601
4
0,
61
0,
44
0
0
10
0,
8165
0,
35
0,
62
0
0
5
1,
0601
4
0,
80
0,
22
0
0
11
0,
8165
0,
44
0,
56
0
0
6
0,
8165
0,
35
0,
62
0
0
12
1,
0601
4
0,
61
0,
44
0
0
3.
2.
Dut
y
C
y
cl
e A
n
al
ysi
s
i
n
the Z
h
ang’s
M
o
d
e
l
One state of s
ector (prism
) in the Z
h
ang’s
m
odel
is produced
four piec
es
of tetra
h
edron t
h
ere
f
ore
com
p
rising all
of si
x
prism
in a cylin
drical sha
p
e entirely
has twe
n
ty-four
pieces
of te
trahe
d
ron. Ea
ch of
tetrah
edron
ha
s diffe
rent piec
es, so that anal
ysis each of te
trahe
d
ron s
h
ould be perform
ed differe
n
tly. Using
equat
i
o
n 5,
e
q
uat
i
on 6
a
nd e
quat
i
o
n 7
i
n
p
r
evi
ous di
sc
us
si
on
,
t
h
e para
m
e
t
e
rs
t
e
t
r
ahedr
o
n
suc
h
as v
a
l
u
e
o
f
Vre
f
, swi
t
c
hi
n
g
t
i
m
e
of each t
e
t
r
ahe
d
r
on
and
dut
y
cy
cl
e can be det
e
rm
i
n
ed. In Tabl
e 3, d
u
t
y
cy
cl
e
calcu
latio
n
resu
lt in
th
e Zh
ang
’
s m
o
d
e
l is g
i
v
e
n.
Table
3.
Duty cycle calculation in t
h
e Z
h
a
n
g’s
m
odel
No Vref
D
1
D
2
D
3
Dz/2
No
Vref
D
1
D
2
D
3
Dz/2
1
0.
6301
85
0.
37
0.
48
0.
03
0.
1
13
0.
6700
08
0.
29
0.
51
0.
04
0.
08
2
0.
6292
32
0.
23
0.
56
0.
03
0.
09
14
0.
7374
28
0.
32
0.
56
0.
04
0.
04
3
0.
7700
22
0.
39
0.
53
0.
06
0.
01
15
0.
6316
82
0.
06
0.
59
0.
02
0.
15
4
0.
3853
43
0.
01
0.
39
0.
04
0.
25
16
0.
7700
22
0.
39
0.
53
0.
06
0.
01
5
0.
6311
72
0.
18
0.
55
0.
01
0.
12
17
0.
6311
72
0.
24
0.
55
0.
01
0.
09
6
0.
6311
72
0.
36
0.
4
0.
07
0.
09
18
0.
3866
95
0.
19
0.
24
0.
05
0.
23
7
0.
7706
85
0.
33
0.
57
0.
05
0.
02
19
0.
4997
33
0.
45
0.
14
0.
11
0.
15
8
0.
5105
12
0.
11
0.
48
0.
02
0.
18
20
0.
7706
85
0.
33
0.
57
0.
05
0.
02
9
0.
5984
61
0.
14
0.
55
0.
02
0.
13
21
0.
6292
32
0.
18
0.
69
0.
01
0.
11
10
0.
6301
85
0.
2
0.
56
0.
02
0.
1
22
0.
6301
85
0.
43
0.
4
0.
07
0.
09
11
0.
7700
22
0.
4
0.
53
0.
06
0.
01
23
0.
7700
22
0.
33
0.
57
0.
05
0.
02
12
0.
7700
22
0.
33
0.
58
0.
05
0.
02
24
0.
7700
22
0.
4
0.
53
0.
06
0.
01
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
C
o
m
p
a
r
i
s
o
n
of
Vol
t
a
ge V
ect
o
r
C
o
nt
rol
B
a
se
d
on
D
u
t
y
C
ycl
e An
al
ysi
s
i
n
T
h
ree
...
. (
I
ndri
a
rt
o Y
uni
a
n
t
o
r
o
)
1
403
3.
3.
Duty Cycle Analysis
in
the Perales’s
Model
Tet
r
ahe
d
r
o
n (
r
e
gi
o
n
poi
nt
er)
anal
y
s
i
s
i
n
a
b
c
-
co
or
di
nat
e
o
f
t
h
e Per
a
l
e
s’s
m
odel
i
s
used
qui
t
e
a l
o
t
.
There
are
sixty-four tetra
h
ed
r
ons
t
h
at
ca
n
be
o
b
t
a
i
n
ed
.
Ho
weve
r,
fr
om
t
h
ese si
xt
y
-
f
o
urt
h
t
e
t
r
ah
ed
ro
ns
,
t
a
ki
n
g
twenty-four pi
eces of them
is
enough to dete
rm
ine value
of
duty cycle.
It is interesting i
n
the Pe
rales’s m
odel
whe
n
pa
ram
e
t
e
r of fourt
h
pi
eces of initial
state of tetrah
e
d
ron is known, then this value will be repea
t
ed to
an
o
t
h
e
r.
Using equ
a
tion
5
,
eq
u
ition
6
and
eq
u
a
tion 7 in
p
r
ev
iou
s
d
i
scussio
n
, th
e
p
a
rameters o
f
th
e
fou
r
t
h
in
itial state tet
r
ah
ed
ron
i
n
the Perales’s m
o
d
e
l su
ch as
v
a
lu
e o
f
Vref,
the
switch
i
n
g
d
u
ratio
n
ti
m
e
o
f
each
pha
se a
n
d dut
y
cycle can be
determ
ined.
In Ta
ble
4,
du
ty cycle calcu
latio
n
resu
lt in th
e Perales’s mo
d
e
l is
gi
ve
n.
Table
4.
Duty cycle calculation in t
h
e Pe
ral
e
s’s m
odel
No Vref
D
1
D
2
D
3
Dz/2
No
Vref
D
1
D
2
D
3
Dz/2
1
1.
0601
37
0.
22
0.
88
0
0
13
1.
0601
37
0.
22
0.
88
0
0
2
0.
8165
0.
41
0.
6
0
0
14
0.
8165
0.
41
0.
6
0
0
3
0.
6940
25
0.
16
0.
67
0.
07
0.
09
15
0.
6940
25
0.
16
0.
67
0.
07
0.
09
4
0.
9389
75
0.
58
0.
54
0
0
16
0.
9389
75
0.
58
0.
54
0
0
5
1.
0601
37
0.
22
0.
88
0
0
17
1.
0601
37
0.
22
0.
88
0
0
6
0.
8165
0.
41
0.
6
0
0
18
0.
8165
0.
41
0.
6
0
0
7
0.
6940
25
0.
16
0.
67
0.
07
0.
09
19
0.
6940
25
0.
16
0.
67
0.
07
0.
09
8
0.
9389
75
0.
58
0.
54
0
0
20
0.
9389
75
0.
58
0.
54
0
0
9
1.
0601
37
0.
22
0.
88
0
0
21
1.
0601
37
0.
22
0.
88
0
0
10
0.
8165
0.
41
0.
6
0
0
22
0.
8165
0.
41
0.
6
0
0
11
0.
6940
25
0.
16
0.
67
0.
07
0.
09
23
0.
6940
25
0.
16
0.
67
0.
07
0.
09
12
0.
9389
75
0.
58
0.
54
0
0
24
0.
9389
75
0.
58
0.
54
0
0
3.
4.
Duty Cycle Analysis
In As
ymmetric’s
Model
Based
on
an
alysis p
a
ir
o
f
asy
mme
tric tetr
ahedron sha
p
es
obtained
fr
om
pro
j
ect
i
o
n i
n
αβ
0-c
o
or
di
nat
e
l
i
k
e Fi
gur
e 8,
i
t
can be descri
be
d coo
r
di
na
t
e
posi
t
i
on o
n
t
h
e t
op and t
h
e b
o
t
t
o
m
of tet
r
ahe
d
r
on
pai
r
. Si
x
sect
ors t
h
at
are
on t
h
e t
op a
n
d o
n
t
h
e bottom te
trahedrons
are
produce
d the sa
m
e
value of duty cycle
in one
circle. Th
e center po
in
t of th
e to
p
te
trah
edron
is po
sitio
n
e
d
in
v
ector
V
15
an
d
t
h
e cen
ter po
in
t of th
e
bo
tto
m
o
f
tetrah
edron
is
p
o
s
ition
e
d
in
vecto
r
V
0
. Asy
mme
trical tetr
ahedron a
ppea
r
s as
pieces of prism
form
pairs that
are not balanced bet
w
een t
h
e top and the bottom
.
Re
ference
vector is determ
ined by drawi
n
g a line
co
nn
ecting
th
e cen
ter
po
in
t
with
a cross-sectio
n
o
f
t
h
e
p
r
is
m
.
In
one circle, there
are t
w
enty-four pie
ces
of
refe
rence
vect
or
. The
r
ef
o
r
e,
dut
y
cy
cl
e can
be det
e
rm
i
n
ed. In Fi
gu
re
8, t
e
t
r
ahe
d
r
on
pai
r
i
s
descri
bed
wi
t
h
t
h
e
cen
ter po
in
t b
e
in
g
at
t
h
e V
15
on
th
e top
o
f
tetrah
ed
ron
.
Fi
gu
re
8.
Tet
r
a
h
ed
r
o
n
pai
r
s
o
n
st
at
e-
(1,1,1,1) the
top
of ce
nter point
(V
15
)
Usi
n
g eq
uat
i
o
n 3, e
q
uat
i
on
4 an
d eq
uat
i
o
n 5, t
h
e t
e
t
r
ah
edr
o
n par
a
m
e
ters suc
h
as va
l
u
e of
Vre
f
,
swi
t
c
hi
n
g
t
i
m
e o
f
eac
h t
e
t
r
a
h
edr
o
n a
n
d
dut
y
cy
cl
e can
be
det
e
rm
i
n
ed.
T
a
bl
e
5 i
s
gi
ve
n t
h
e
resul
t
s
o
f
dut
y
cycle calculation in a
s
ymmetric m
odels.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 4
,
Au
gu
st 2
016
:
13
95
–
1
405
1
404
Table 5. Duty
cycle
calculati
o
n
in
t
h
e asymmetric
m
o
d
e
l
No Vref
D
1
D
2
D
3
Dz/2
No
Vref
D
1
D
2
D
3
Dz/2
1
0.
6009
25
0.
12
0.
55
0.
04
0.
15
13
0.
4714
03
0.
1
0.
44
0.
01
0.
21
2
0.
1666
63
0.
06
0.
11
0.
01
0.
4
14
0.
4713
99
0.
1
0.
44
0.
01
0.
21
3
0.
4713
99
0.
26
0.
32
0.
04
0.
19
15
0.
4714
03
0.
26
0.
32
0.
04
0.
19
4
0.
4714
03
0.
26
0.
32
0.
04
0.
19
16
0.
4713
99
0.
26
0.
32
0.
04
0.
19
5
0.
2204
77
0.
1
0.
18
0.
02
0.
35
17
0.
4713
93
0.
1
0.
44
0.
01
0.
21
6
0.
4714
03
0.
16
0.
4
0.
01
0.
21
18
0.
4713
99
0.
1 0.
44
0.
01
0.
21
7
0.
4713
99
0.
26
0.
31
0.
03
0.
19
19
0.
4714
03
0.
26
0.
32
0.
04
0.
19
8
0.
4713
93
0.
28
0.
36
0.
06
0.
15
20
0.
3333
26
0.
36
0.
11
0.
23
0.
15
9
0.
5527
7
0.
12
0.
43
0.
01
0.
21
21
0.
3004
61
0.
14
0.
17
0.
05
0.
32
10
0.
4714
03
0.
1
0.
44
0.
01
0.
21
22
0.
4713
99
0.
1
0.
44
0.
01
0.
21
11
0.
4713
93
0.
16
0.
32
0.
07
0.
22
23
0.
4713
93
0.
24
0.
3
0.
06
0.
19
12
0.
3333
31
0.
12
0.
43
0.
01
0.
21
24
0.
4409
57
0.
09
0.
41
0.
02
0.
23
4.
RESULT AND DIS
C
USSI
ON
The following
com
p
arison
of duty cycle analysis
based
on each tetra
h
e
d
ron m
odel was de
veloped.
In
th
e Sh
en
’s an
d
th
e Perales’s
m
o
d
e
l, v
a
lu
e o
f
du
ty cycle
in
th
e c-cond
ucto
r (D
3
) an
d t
h
e ne
ut
ral
co
n
duct
o
r
(D
z
)
is eq
u
a
l zer
o
, ev
en
t
h
ough
actu
a
lly calcu
latio
n
pr
odu
ces a n
e
g
a
tiv
e valu
e, it ind
i
cates th
at th
e Shen
’
s
and
Perales’s m
o
d
e
l to
cu
rren
t
com
p
en
satio
n
con
t
ro
l is n
o
t
n
e
cessary to
regu
l
a
te switch
i
n
g
co
m
b
in
atio
n
s
fro
m
th
e
c-co
n
duct
o
r a
n
d t
h
e
neut
ral
c
o
n
d
u
ct
o
r
. T
h
e
She
n
’s a
n
d Per
a
l
e
s’s m
odel
even c
o
nd
uct
e
d
on t
h
ree
-
p
h
ase
fo
ur
-
wire system o
f
activ
e filter resu
lted
and
ind
i
cated
th
at
it ca
n
b
e
don
e bu
sim
p
l
i
fyin
g
and
u
s
ing
on
th
ree-p
h
a
se
th
ree-wire syst
e
m
o
f
activ
e filter, sin
ce switch
i
n
g
co
m
b
i
n
atio
n
is on
ly d
o
n
e
in
a,b-co
ndu
ctor (two-ph
a
se
m
odul
at
i
on)
. I
n
t
h
e Z
h
an
g
’
s
m
odel
,
i
t
appears t
h
at
val
u
e of
dut
y
cy
cl
e i
n
ne
ut
ral
wi
re (D
z
) is sm
a
ller th
an
v
a
lu
e
of
d
u
t
y cycle in
asymmetric m
o
d
e
l. It
mean
s, in
t
h
e Zha
n
g
’
s
m
odel
,
t
h
e pe
ri
o
d
of swi
t
c
hi
n
g
i
n
f
o
urt
h
-
leg
of
IGBT mo
du
le co
n
t
ro
ller is sh
orter th
an
asymm
e
tric m
o
d
e
l.
In
Fig
u
re
9
,
re
sult com
p
ariso
n
of
duty cycle betwee
n
She
n
’s,
Zha
n
g’
s, Perales’s, Asymmetric
m
o
d
e
l
is give
n. T
h
e
graph the
She
n
’s and Perales
’
s
lines can
not be
de
scribed because
the
val
u
e of
duty cycle (D
z
)
fo
r
bot
h
of
t
h
e
m
i
s
equal
ze
r
o
.
Fi
gu
re
9.
R
e
sul
t
com
p
ari
s
on
o
f
dut
y
cy
cl
e be
t
w
een
She
n
’s,
Zha
n
g
’
s,
Peral
e
s’s,
Asy
m
m
e
tri
c
m
odel
5.
CO
NCL
USI
O
N
Asy
m
m
e
t
r
i
c
vol
t
a
ge
vect
o
r
m
odel
as a res
u
l
t
of
p
q
r
-
c
o
o
r
di
nat
e
pr
oje
c
t
i
on i
n
t
o
αβ
0-c
o
or
di
nat
e
ca
n
be
u
s
ed
to regu
late cu
rren
t co
m
p
ensatio
n con
t
ro
l. A skew
e
d
vect
or
desc
ri
pt
i
o
n
o
f
a
s
y
m
m
e
t
r
i
c
vol
t
a
ge
vect
or
defi
nes t
h
e
m
odul
at
i
o
n
bo
u
n
d
a
ry
-l
i
n
e as e
q
u
a
l
0.
5 V
d
c
w
h
ere t
h
i
s
val
u
e i
s
l
e
ss t
h
a
n
t
h
e
m
odel
devel
o
p
e
d
by
S
h
en
,
Z
h
ang
an
d P
e
r
a
le
s
.
Th
e p
a
ram
e
ter
calcu
latio
n
o
f
tetr
ahed
r
on a
n
d com
p
are e
x
i
s
t
i
ng m
ode
l ba
sed
on
duty cycle analysis
i
n
ne
ut
ral
wi
re
can be c
o
ncl
u
ded
w
h
ere asy
m
m
e
t
r
i
c
vol
t
a
g
e
vect
o
r
m
odel
have
val
u
e
g
r
eat
er t
h
an a
n
ot
her
.
It
Evaluation Warning : The document was created with Spire.PDF for Python.