Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
V
o
l. 5,
N
o
.
1
,
Ju
ly 20
14
, pp
. 1~1
4
I
S
SN
: 208
8-8
6
9
4
1
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
/
IJPEDS
Modeling and Simulation of
a Carrier-based PWM Voltage
Source Inverter for a Nine Ph
ase Induction Machine Drive
Omo
n
ow
o D
a
vi
d
Mo
mo
h
Department o
f
C
o
mputer,
Electrical
and
Information Technolog
y
,
Indiana
University
-Purdue University
(IPFW)
F
o
rt W
a
y
n
e
,
Ind
i
ana
,
Uni
t
ed S
t
at
es
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mar 29, 2014
Rev
i
sed
May 22
, 20
14
Accepte
d
J
u
n 1, 2014
The ana
l
y
s
is
of
a carri
er-bas
ed
P
W
M two level voltag
e
s
ource i
nverter for a
nine ph
ase
ind
u
ction
m
achin
e
drive
s
y
st
em
is present
e
d
in
this p
a
per
.
Methods
for generating zero
-
sequenc
e sign
als during b
a
lanced
and
unbalan
ced con
d
ition ar
e estab
l
ished. Sim
u
latio
n results for the anal
y
s
is are
presented
.
Two f
a
ult
conditions involving
th
e voltage source inv
e
rter and
the
nine-phas
e
s
quir
r
el
cag
e indu
ctio
n m
achine
lo
ad
are
inves
tig
at
ed.
F
o
r the
two
fault s
c
enarios
c
ons
idered,
the e
ffects
on th
e per
f
orm
a
nce ch
arac
teris
t
i
c
s
o
f
the induction machin
e load
are
highli
ghted.
The simulation resu
lts obtain
e
d
show that the two imbalance
conditions
consi
d
ered result
in
substantial
oscilla
tions on
the e
l
e
c
trom
agn
e
ti
c torqu
e
of t
h
e m
achin
e wit
h
att
e
ndan
t
reduction in
th
e torque rating
.
Th
ere
is
a
l
so la
rge
slip in
the
rotor
speed.
Keyword:
Ex
isten
ce fu
n
c
tio
n
Harm
o
n
i
c inj
e
ctio
n
Mu
ltip
h
a
se m
ach
in
es
Ope
n
-p
hase fa
ult
Vol
t
a
ge
-s
ou
rce
i
nve
rt
er
Copyright ©
201
4 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
:
Om
ono
wo
Da
v
i
d M
o
m
o
h
Depa
rt
m
e
nt
of
C
o
m
put
er, El
e
c
t
r
i
cal
and
I
n
f
o
rm
ati
on Tec
h
n
o
l
o
gy
Ind
i
an
a Un
iv
ersity-Pu
r
du
e
Univ
ersity (IPFW)
Fort
W
a
y
n
, Indiana, United
St
ates
Em
ail: m
o
m
ohd@i
p
fw.e
du
1.
INTRODUCTION
Multiphase m
achines
are
electrical gene
rat
o
rs
or m
o
to
rs
with
th
e nu
m
b
er
o
f
phases greater
t
h
an
three (n > 3). When com
p
ared to three pha
se
m
achine
s
,
m
u
ltiphase
m
achi
n
es ha
ve
great
er de
gre
e
s of
free
dom
whi
c
h
has e
n
abl
e
d
som
e
im
prov
em
ent
s
i
n
t
h
e
sy
st
em
’s per
f
o
r
m
a
nce [1]
-
[
5]
. Th
e a
dva
nt
a
g
es
of
m
u
l
tip
h
a
se m
a
ch
in
e ov
er th
eir th
r
ee phase
counterparts are well doc
um
ent
e
d i
n
[6]
-
[
12]
. T
h
ey
i
n
cl
ude:
h
i
gh
er
reliab
ility an
d
i
n
creased
p
o
wer d
e
nsity, en
h
a
n
c
ed fau
lt to
leran
t
cap
ab
ility, ex
ten
d
e
d
sp
eed
/to
rqu
e
cap
ab
ility, redu
ced
am
p
litu
de/in
creased
freq
u
e
n
c
y of
p
u
l
satin
g
torqu
e
, red
u
c
ed
ro
t
o
r harm
o
n
i
c
cu
rren
ts,
an
d
red
u
ce
d cu
rre
nt
per
pha
se wi
t
h
o
u
t
i
n
crea
si
ng t
h
e
vol
t
a
ge pe
r p
h
ase.
Al
so, i
n
m
u
l
t
i
pha
se m
achi
n
e, ext
r
a
-
t
o
r
que ca
n
be
pr
o
duce
d
fr
o
m
t
h
e i
n
t
e
ract
i
ons
o
f
cu
rre
n
t
and
spat
i
a
l
harm
oni
cs
of
t
h
e sam
e
orde
r. F
o
r
i
n
st
ance, i
n
ni
n
e
phase m
achi
n
e, t
h
e t
h
i
r
d, fi
ft
h, an
d se
vent
h harm
oni
cs ca
n be ha
rne
ssed
t
o
generat
e
av
erage
t
o
r
que
w
h
i
c
h
a
dds
u
p
t
o
t
h
e t
o
r
q
ue
pr
od
uce
d
by
t
h
e
f
u
n
d
a
m
ent
a
l
cur
r
ent
com
pone
nt
.
At the core of a
m
u
ltiphase mach
ine dri
v
e is the powe
r
electronics
technology. The advancem
ent in
po
we
r el
ect
ro
n
i
cs t
echnol
ogy
has m
a
de i
t
possi
bl
e t
o
pr
od
uce any
n
u
m
b
er of
p
h
ases us
i
ng a DC
/
A
C
vol
t
a
g
e
sou
r
ce i
n
vert
e
r
(VS
I
)
.
C
a
rri
e
r
-
b
ase
d
si
n
u
so
i
d
al
pul
se
-wi
d
t
h
m
odul
at
i
on
(SP
W
M
)
i
s
t
h
e
m
o
st
po
pul
a
r
a
n
d
wi
del
y
use
d
P
W
M
t
echni
q
u
e
. Thi
s
i
s
bec
a
use o
f
t
h
e si
m
p
l
e
im
pl
em
ent
a
t
i
on i
n
b
o
t
h
anal
og a
n
d di
gi
t
a
l
realizations when
com
p
are
d
to
th
e s
p
ace
-v
ect
or P
W
M
(S
VP
W
M
),
w
h
i
c
h i
s
f
o
un
d t
o
be m
o
re i
n
t
e
n
s
e fr
o
m
co
m
p
u
t
atio
n
a
l an
d
co
m
p
lex
ity v
i
ew
po
in
ts [1
3
]
,
[
1
5
]
. Carr
ier
-
b
a
sed
SPW
M
also
know
n
as t
h
e co
mp
ar
ison
p
u
l
se-
w
i
d
th
mo
du
lator
co
m
p
ar
es a h
i
gh
-f
r
e
q
u
e
n
c
y tr
iangul
ar
(
dou
b
l
e edge)
or
saw
-
too
t
h (
s
ing
l
e-
ed
g
e
)
car
r
i
er
wi
t
h
re
fere
nce
si
gnal
s
(m
od
ul
at
i
on si
gnal
s
) t
h
e
r
eby
c
r
ea
t
i
ng
gat
i
ng
p
u
l
ses fo
r t
h
e
s
w
i
t
c
hes i
n
t
h
e
p
o
we
r
ci
rcui
t
[
15]
,
[
1
6]
.
The
ne
utral point
of m
o
st ac m
o
tor drive and
utilit
y interface a
pplicatio
ns is isolated
a
nd
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 5
,
No
. 1
,
Ju
ly 20
14
:
1
–
14
2
con
s
eq
ue
nt
l
y
, t
h
ere
i
s
no
neut
ral
cu
rre
nt
pat
h
.
The
a
b
sence
o
f
t
h
e
neut
ral
cur
r
ent
pat
h
i
n
t
h
e l
o
ad
pr
ovi
des
a
deg
r
ee o
f
free
dom
i
n
det
e
rm
i
n
i
ng t
h
e
dut
y
cy
cl
e of t
h
e inve
rter s
w
itches [17]. The differe
nce in potential
bet
w
ee
n t
h
e l
o
ad ne
ut
ral
poi
n
t
(‘
n
’) and the
center
poi
nt of the dc
-link ca
pacitor
(‘
o
’)
o
f
th
e VSI is called
th
e
zero
-
se
que
nce
v
o
l
t
a
ge,
no
v
. T
h
e
zer
o-se
q
u
enc
e
v
o
l
t
a
ge ca
n
t
a
ke a
n
y
val
u
e
whi
c
h ca
n
be s
u
bse
que
n
t
ly
i
n
ject
e
d
i
n
t
o
t
h
e m
odul
at
i
o
n
si
g
n
al
s t
o
ac
h
i
eve any
o
f
t
h
e
f
o
l
l
o
wi
ng
des
i
rabl
e
pr
op
ert
i
e
s:
red
u
ce
d s
w
i
t
c
hi
n
g
lo
sses, im
p
r
o
v
ed
wav
e
fo
rm
q
u
a
lity, an
d
in
creased
lin
ear
mo
du
latio
n
range [3
]. If th
e injected
zero-sequ
en
ce
si
gnal
i
s
co
nt
i
n
u
o
u
s, i
t
pr
o
d
u
ces a co
nt
i
n
u
ous
P
W
M
(C
P
W
M
)
sc
hem
e
; ho
we
ver,
w
h
e
n
i
t
i
s
di
scont
i
n
u
o
u
s
with
th
e po
ten
t
ial fo
r th
e m
o
du
lato
r to
h
a
v
e
p
h
a
se seg
m
en
t
s
clam
p
e
d
to
t
h
e
p
o
s
itiv
e or
n
e
g
a
tiv
e d
c
rai
l
s, th
e
m
odulation sc
hem
e
is calle
d discontinuous P
W
M
(DP
W
M). In DP
WM, there is
no switchi
ng (and
con
s
eq
ue
nt
l
y
n
o
s
w
i
t
c
hi
n
g
l
o
s
s
es) i
n
t
h
ose
i
n
t
e
rval
s
w
h
en
t
h
ere i
s
di
sco
n
t
i
n
uo
us
m
odul
at
i
o
n
.
A
car
ri
er
b
a
sed
P
W
M
m
e
t
hod com
p
ri
si
ng o
f
al
l
DP
W
M
sch
e
m
e
s i
s
call
e
d t
h
e gene
ral
i
zed di
sc
ont
i
n
u
o
u
s
P
W
M
(G
DP
WM
)
[1
3]
, [1
8]
.
M
u
ch g
r
ou
n
d
has bee
n
co
ver
e
d o
n
P
W
M
s
c
hem
e
s for a m
u
lt
i
phase VS
I usi
n
g ei
t
h
er
carri
er
-ba
s
ed
P
W
M
o
r
S
V
P
W
M
t
echni
q
u
e
.
The
goal
of t
h
i
s
pa
per i
s
t
o
im
pl
em
ent
a t
h
or
o
u
g
h
si
m
u
l
a
ti
on a
nd a
n
al
y
s
i
s
of a
carrier-ba
s
ed
SPW
M voltage
source inve
rter for a ni
ne-pha
se in
du
ction
mach
in
e
d
r
i
v
e. Th
is is do
n
e
t
h
ro
ugh
m
odel
i
ng a
n
d
sim
u
l
a
t
i
on o
f
t
h
e
dri
v
e sy
st
e
m
. In s
o
d
o
i
n
g
,
t
h
e
w
o
rk
p
r
o
pos
ed
i
n
[
3
]
i
s
ext
e
nde
d t
o
a
ni
ne-
pha
se
sy
st
em
.
Al
so,
bey
o
nd t
h
e w
o
r
k
d
one f
o
r fi
ve-
pha
se VSI
dri
v
e sy
st
em
i
n
[3]
and [15]
, pl
ot
s sh
o
w
i
n
g t
h
e
perform
a
nce characteristics
of t
h
e
nine
-phase in
du
ction
mach
in
e are
presente
d in thi
s
pa
per. Figure 1 is a
si
m
p
lified
sche
m
a
tic d
i
ag
ram
o
f
a two-level co
nv
erter
(
V
SI
) s
u
ppl
y
i
n
g
a
ni
ne
-p
hase
sq
ui
rrel
ca
ge i
n
d
u
ct
i
o
n
machine.
2
dc
v
2
dc
v
Fi
gu
re
1.
Tw
o
-
l
e
vel
ni
n
e
-
pha
s
e
VS
I s
u
ppl
y
i
n
g
a
ni
ne
-
phase
squi
rrel
ca
ge i
n
d
u
ct
i
o
n m
achi
n
e
2.
MO
DELIN
G
OF CO
NTIN
UO
US C
A
R
R
I
ER-B
ASED
PWM
FO
R N
I
NE
P
H
ASE
The
princi
ples
of ca
rrier-base
d
PW
M
for a t
h
ree
-
ph
ase
VSI are als
o
applicable to a m
u
ltiphase
VSI
[1
3]
. C
o
nse
q
u
e
nt
l
y
, t
h
e l
o
a
d
vol
t
a
ge eq
uat
i
ons
fo
r t
h
e
ni
ne-
p
hase V
S
I
sup
p
l
y
i
n
g ni
ne
-p
hase s
q
ui
rrel
cage
i
n
d
u
ct
i
o
n
m
a
chi
n
e a
s
depi
ct
e
d
i
n
Fi
g
u
r
e
1 a
r
e
gi
ve
n as
f
o
l
l
o
ws:
js
js
js
js
jn
i
r
pi
L
v
(1
)
jn
v
are t
h
e phase t
o
ne
ut
ral
out
pu
t
vol
t
a
ge fr
om
t
h
e VSI
,
js
i
are the phase curre
nts fr
o
m
th
e VSI to
th
e lo
ad,
js
L
are the induction m
achin
e stator inducta
n
c
e
s,
js
r
are t
h
e i
nduct
i
o
n m
achine st
at
or resi
st
ances,
p
is a
diffe
re
ntial ope
rator
(
d/
dt
),
j
= a, b,
c, d,
e,
f
,
g
,
h,
i
.
The t
u
r
n
-
o
n a
n
d
t
u
rn
-o
ff
of
t
h
e s
w
i
t
c
hi
n
g
devi
ces
f
o
r
t
h
e ni
n
e
-
pha
se
VSI
s
h
o
w
n i
n
Fi
g
u
re
1
ar
e
rep
r
ese
n
t
e
d by
an exi
s
t
e
nce
fu
nct
i
o
n
.
Th
e exi
s
t
e
nce f
u
nct
i
on
has a val
u
e o
f
u
n
i
t
y
and zer
o w
h
en t
h
e
switch
i
ng
d
e
v
i
ce is tu
rn
ed
on
an
d
turn
ed
o
f
f
resp
ectiv
ely. In
its si
m
p
lest f
o
rm
, an
ex
isten
ce fu
n
c
tion
o
f
a two
-
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Mo
del
i
n
g
a
n
d
Si
m
u
l
a
t
i
o
n
of
a
C
a
rri
er-
b
ase
d
PWM V
o
l
t
a
ge
So
urce
I
n
vert
e
r f
o
r…
(
O
m
o
n
o
w
o
D
a
vi
d
Mo
mo
h)
3
level conve
rter can
be re
prese
n
ted as
i
h
g
f
e
d
c
b
a
j
S
jk
,
,
,
,
,
,
,
,
,
and
n
p
k
,
.
j
d
e
pi
ct
s t
h
e l
o
a
d
pha
se
to
wh
ich th
e switch
i
ng
d
e
v
i
ce is con
n
ected
an
d
k
represents th
e top
(
p
)
an
d bo
tto
m
(
n
) devi
ce o
f
an
i
n
vert
e
r
l
e
g. C
o
nse
que
nt
l
y
, fr
om
Fi
gure
1,
S
ap
and
S
an
tak
e
a v
a
lu
e o
f
zero
or u
n
ity and
th
e two
con
s
titu
t
e
th
e
exi
s
t
e
nce
fu
nct
i
on
of
t
h
e t
op
devi
ce a
n
d
bot
t
o
m
devi
ce of
t
h
e i
n
vert
er
l
e
g c
o
n
n
ect
ed
t
o
p
h
ase ‘a
’
of t
h
e
ni
ne
p
h
a
se indu
ction
m
ach
in
e lo
ad [3
], [13
]
.
There
are
2
9
= 5
1
2
switch
i
ng
p
o
s
sib
ilities
(arran
g
e
m
e
n
t
s) du
ri
n
g
t
h
e o
p
e
ratio
n
o
f
a n
i
n
e
-p
h
a
se
VSI.
The
ope
rat
i
o
n
of a
car
ri
er-
b
as
ed P
W
M
ca
n
b
e
di
vi
ded
i
n
t
o
t
w
o
m
odes-l
i
n
e
a
r m
odul
at
i
o
n
m
ode and
n
o
n
l
i
n
ear
m
odul
at
i
on m
ode. U
n
der t
h
e l
i
near m
odul
at
i
on m
ode, t
h
e
p
eak o
f
a
m
odul
at
i
on si
g
n
al
i
s
l
e
ss t
h
an o
r
eq
ual
t
o
the peak
of the
carrier signal and c
o
nsequ
e
ntly
th
e g
a
in
is ap
pro
x
i
m
a
tel
y
u
n
ity. Ho
wev
e
r, wh
en
the p
e
ak
of a
m
odul
at
i
on si
gnal
i
s
great
e
r
t
h
an t
h
e
pea
k
o
f
t
h
e
car
ri
er si
g
n
al
,
ove
r
-
m
odul
at
i
on
o
ccurs
an
d t
h
e
gai
n
i
s
gene
ral
l
y
l
e
ss t
h
an
uni
t
y
. F
o
r
ope
rat
i
o
n i
n
t
h
e l
i
n
ear
m
odul
at
i
on regi
on
, m
odul
at
i
on
i
s
defi
ned
as
t
h
e rat
i
o
of
the fundam
ental com
ponent
am
plitude of
the line-t
o
-neut
r
al (phase
) inverter
output voltage to one-ha
lf of t
h
e
D
C
bu
s vo
ltage [3
], [19
]
. Th
is is g
i
v
e
n as:
dc
j
j
V
V
M
5
.
0
(2
)
Whe
r
e
M
j
i
s
the m
odul
at
i
o
n
i
ndex (m
agni
t
ude
of t
h
e m
odul
at
i
o
n)
,
V
j
is th
e
m
a
g
n
itu
de o
f
th
e fu
nd
amen
tal
i
nve
rt
er
out
p
u
t
v
o
l
t
a
ge, a
n
d
V
dc
i
s
t
h
e m
a
gni
t
ude
o
f
t
h
e
dc
b
u
s
vol
t
a
ge.
Th
e vo
ltag
e
between
th
e
j
th
i
nvert
e
r
p
h
as
e and t
h
e cen
t
e
r poi
nt
o
f
t
h
e dc
-l
i
nk ca
paci
t
o
r
(‘
o
’)
ot
he
rwi
s
e
k
n
o
w
n a
s
p
o
l
e
(
s
w
i
t
c
hed) i
nve
rt
e
r
p
h
ase
v
o
l
t
a
g
e
,
jo
v
, is related to th
e lo
ad
p
h
a
se v
o
ltag
e
,
jn
v
, as
fo
llows:
no
jn
jo
v
v
v
(3
)
Whe
r
e
no
v
i
s
t
h
e
c
o
m
m
on-n
ode
z
e
ro
-se
que
nce
v
o
l
t
a
ge as
de
fi
n
e
d i
n
sect
i
o
n
I
abo
v
e.
T
h
e c
o
n
s
t
r
ai
nt
i
m
posed
by
Ki
rch
h
o
f
f’s
V
o
ltage La
w
(
KVL
)
on
th
e two
switch
e
s i
n
an
inv
e
rter leg
i
s
su
ch
t
h
at th
e
ex
isten
ce
fun
c
t
i
o
n
s
fo
r t
h
e
t
o
p a
n
d
bot
t
o
m
devi
ces
m
u
st
be com
p
l
i
m
e
nt
ary
of
ea
ch
ot
he
r. T
h
i
s
i
s
ex
pre
ssed
i
n
(4
).
1
jn
jp
S
S
(4
)
Vio
l
atin
g
t
h
is
co
nstrain
t
will
cau
se th
e sho
r
t
circu
itin
g
of
th
e d
c
bu
s
v
o
ltag
e
.
Howev
e
r, fo
r ab
so
lu
te contro
l of
currents a
n
d
output voltages
,
one
de
vice in each le
g m
u
st be turne
d
on at all ope
rating tines. T
h
e
r
e is a
relatio
n
s
h
i
p b
e
tween
t
h
e switch
e
d vo
ltag
e
, t
h
e ex
isten
ce
fun
c
tio
ns, and
the d
c
bu
s
vo
ltage g
i
v
e
n
as fo
llows:
jn
jp
dc
jo
S
S
v
v
2
1
(5
)
From
(4
),
jp
jn
S
S
1
(6
)
The
c
o
m
p
l
i
m
e
nt
ary
pr
ope
rt
y
of
jp
S
and
jn
S
can be f
u
rt
her
e
x
pres
se
d
as:
jp
jn
jn
jp
S
S
S
S
(7
)
By su
b
s
titu
ting (6
) i
n
to
(5
), we h
a
v
e
:
1
2
2
1
jp
dc
jo
S
v
v
(8
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 5
,
No
. 1
,
Ju
ly 20
14
:
1
–
14
4
The
ni
ne
swi
t
c
hed
v
o
l
t
a
ges
e
x
p
r
esse
d i
n
(
3
)
can
be
sum
m
ed
up
as
fol
l
o
ws
:
no
i
a
j
i
a
j
jn
jo
v
v
v
9
(9
)
A symmetrical
in
du
ction
m
a
c
h
in
e i
n
a
n
o
rmal o
p
e
ration
con
d
ition
represen
ts a
b
a
lan
c
ed
star-co
n
n
ected
lo
ad.
C
onse
q
uent
l
y
,
t
h
e s
u
m
of t
h
e
pha
se c
u
r
r
ent
s
equal
t
o
ze
ro
.
Th
us:
i
a
j
js
i
0
(1
0)
Sub
s
titu
tin
g
(10
)
in
t
o
(1) im
p
lies th
at th
e su
m o
f
th
e in
v
e
rter p
h
a
se-to-n
e
u
t
ral ou
tpu
t
v
o
ltag
e
(lo
a
d
v
o
l
tag
e
)
eq
u
a
ls to
zero
as shown in
(11
)
.
i
a
j
jn
v
0
(1
1)
B
y
t
a
ki
ng i
n
t
o
co
nsi
d
erat
i
o
n E
quat
i
o
n
(1
0) a
n
d (
1
1), t
h
e com
m
on-
n
ode ze
r
o
-se
q
u
e
nce v
o
l
t
a
ge
can b
e
ex
pressed
i
n
term
s o
f
th
e swit
ch
ed vo
ltag
e
as fo
llo
ws:
i
a
j
jo
no
v
v
9
1
(1
2)
Also
,
b
y
sub
s
titu
tin
g
(8
) in
t
o
(12), th
e zero
-
sequ
en
ce
v
o
ltag
e
can
b
e
ex
pressed
in
term
s o
f
th
e ex
isten
ce
fu
nct
i
o
n a
n
d
t
h
e dc
b
u
s
v
o
l
t
a
g
e
as
gi
ve
n i
n
(
1
3)
.
i
a
j
jp
dc
no
S
v
v
2
9
9
(1
3)
By su
b
s
titu
ting
(1
3) and
(8) in
to
(3
), th
e in
v
e
rter
ph
ase-to
-n
eu
tral voltag
e
is ex
p
r
essed
in
term
s
o
f
th
e
exi
s
t
e
nce
fu
nct
i
ons
an
d
dc
b
u
s
v
o
l
t
a
ge as
f
o
l
l
ows:
i
h
g
f
e
d
c
b
a
j
q
S
S
v
v
i
j
q
a
q
qp
jp
dc
jn
,
,
,
,
,
,
,
,
,
,
8
9
(1
4)
2.
1. Ni
ne
th-
H
arm
o
ni
c
In
jec
t
i
o
n
It h
a
s b
e
en
prop
o
s
ed
th
at th
e
in
j
ection
of the
n
th
-ha
r
m
oni
c i
n
t
o
t
h
e refe
re
nce m
odul
at
i
o
n si
g
n
al
s of
an
n-
p
hase
sys
t
e
m
effectively increases
th
e lin
ear m
o
du
latio
n
rang
e
with
ou
t m
o
v
i
ng
i
n
to
ov
er-m
o
d
u
latio
n
regi
on [
2
0]
, [2
1]
. Thi
s
t
ech
ni
que
has l
e
d t
o
hi
g
h
er
out
put
f
u
n
d
am
ent
a
l
vol
t
a
ge t
h
an usi
n
g si
m
p
l
e
si
nus
oi
dal
carri
er
base
d
P
W
M
[
3
]
,
[1
5]
.
The
o
p
t
i
m
a
l
l
e
vel
o
f
t
h
e
n
th
-
h
arm
oni
c com
pone
nt
i
s
fo
u
n
d
t
o
be:
n
n
v
v
j
n
2
sin
(1
5)
Equ
a
tio
n
(15
)
t
a
k
e
s a po
sitiv
e sig
n
fo
r
n
= 3,
7, 1
1
,
15
,….
,
and
neg
a
t
i
v
e si
gn
fo
r
n
= 5
,
9,
13,
1
7
, ……
. Th
us
,
fo
r a ni
ne-
p
h
a
se sy
st
em
, t
h
e i
n
ject
ed
ni
net
h
ha
rm
oni
c i
s
negat
i
v
e a
n
d t
h
e res
u
l
t
i
ng ze
ro
-se
que
nce si
gnal
i
s
gi
ve
n as:
)
9
sin(
9
18
sin
)
(
9
t
v
t
v
j
no
(1
6)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Mo
del
i
n
g
a
n
d
Si
m
u
l
a
t
i
o
n
of
a
C
a
rri
er-
b
ase
d
PWM V
o
l
t
a
ge
So
urce
I
n
vert
e
r f
o
r…
(
O
m
o
n
o
w
o
D
a
vi
d
Mo
mo
h)
5
The m
a
xim
u
m
possi
bl
e
m
o
d
u
l
a
t
i
on i
n
dex i
n
th
e lin
ear reg
i
on
in
the cas
e of the
n
th
ha
rm
oni
c i
n
ject
i
on
has
been
de
ri
ve
d t
o
be:
)
2
cos(
1
n
M
(1
7)
It has been
found that inj
ecting th
e
third
harm
onic leads to an i
n
crease
of 15
.4
7
%
in
th
e f
und
am
en
tal o
u
t
pu
t
voltage
while there is an
in
cr
ease of
5.
12% b
y
in
j
ecting th
e f
i
f
t
h
h
a
r
m
o
n
i
c [15
],
[2
0
]
.
Fo
llowing
the sa
m
e
procedure, it can
be derived t
h
at the
inj
ection of t
h
e ni
neth harm
onic
will l
ead to an increase
of 1.54%
in the
fundam
ental output voltage.
2.2. De
terminati
on of
the Comm
on-node Z
ero-sequenc
e Voltage
with Balanced L
o
ad
Existence
fu
n
c
tions are m
odulatio
n p
u
lse
s
havi
n
g
val
u
es of ze
ro
o
r
unity
. T
h
e F
o
u
r
ier se
ries
app
r
oxim
a
tion of
the e
x
istenc
e f
unctio
ns a
r
e
rep
r
ese
n
ted as
follo
ws
[
3
]
,
[
1
3
]
:
ip
ip
hp
hp
gp
gp
fp
fp
ep
ep
dp
dp
cp
cp
bp
bp
ap
ap
M
S
M
S
M
S
M
S
M
S
M
S
M
S
M
S
M
S
1
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
1
5
.
0
(1
8)
Whe
r
e
M
ap
,
M
bp
,
M
cp
,
M
dp
,
M
ep
,
M
fp
,
M
gp
,
M
hp
,
M
ip
are
the car
rier-
b
as
ed m
odulatio
n
signals.
Thei
r
values
range bet
w
een -1 and 1. For
im
ple
m
entatio
n, the e
x
is
tenc
e fu
nctions a
r
e
gene
rated by
com
p
aring the
high
-
frequency carrier signal (t
ri
angle si
gnal
)
havi
ng positive peak an
d negative peak values of
1
and -1
respectively. T
h
e freque
ncy of the ca
rrier si
gnal adopted for t
h
is work is
10kHz. After som
e
si
m
p
lific
ations
that
involve
substituting (18) and (8)
int
o
(3), the m
o
dulation si
gnals
for
the top
nine switching devices can
be e
x
p
r
esse
d a
s
:
i
h
g
f
e
d
c
b
a
j
M
M
v
v
v
v
v
v
v
M
o
jp
dc
no
dc
jn
dc
no
jn
jp
,
,
,
,
,
,
,
,
2
2
2
*
*
(1
9)
Whe
r
e
*
jp
M
are the
refe
re
nce m
o
d
u
lation
sig
n
a
ls fo
r t
h
e
pha
ses,
jp
M
are t
h
e correspondi
ng actual
m
odulation sig
n
als res
p
o
n
sibl
e for
gene
ratin
g the switchi
n
g
f
unctio
ns,
*
o
M
is resp
o
n
sible f
o
r
pr
od
ucin
g the
zer
o-
seq
u
e
n
c
e sig
n
a
l (
c
o
mmo
n-n
od
e vo
ltag
e
)
inj
ection
in
th
e car
r
i
er-based
PW
M VSI
.
Th
e
m
o
d
u
latio
n
s
signals sy
nt
hes
i
zed in (
1
9
)
are
com
p
ared wit
h
the carrier si
gnal such t
h
at the respective sw
itch
turns on when
the corres
ponding m
odulation signal is gr
eat
er than t
h
e triangle carrier si
gnal and vice ve
rsa. F
u
rt
herm
ore, as
the de
vice turns
on, the
com
p
lim
entary
switch on
the particula
r
leg tur
n
s of
f owi
n
g
to
the Kirc
hh
o
f
f
’
s
Voltage
La
w c
onst
r
aint im
po
sed
o
n
the
ex
istence function
as m
e
ntioned earlier.
A properly selected zer
o-se
quence si
gnal c
a
n extend the l
i
near
ity region of the carrier-base
d
PW
M
VSI
.
F
r
om
the earlier
wo
r
k
s
on
this s
u
bject
[3]
,
[
13]
,
the
exp
r
essi
on
f
o
r
gene
ratin
g the
zero
-
se
que
nce
signal
(com
m
on-n
o
d
e
v
o
ltage)
is:
max
min
)
1
(
)
2
1
(
5
.
0
v
v
v
v
dc
no
(
2
0
)
Whe
r
e
v
min
and
v
max
repre
s
en
t the instantan
e
ou
s m
i
nim
u
m
and m
a
xim
u
m
m
a
gnitu
des of the
nine re
f
e
rence
m
odulating
vo
ltages as sh
o
w
n in
(2
1
)
.
can
assum
e
any
value
betwee
n
zero a
n
d
unit
y
but
5
.
0
is
considered t
h
e best
overall in every
linear condition and it is the value ch
osen
for t
h
is work.
The value,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-86
94
I
JPEDS
Vo
l.
5
,
No
.
1
,
Ju
ly 20
14
:
1
–
14
6
5
.
0
, ha
s also
bee
n
fo
u
n
d
to
give
s
t
ate vector
P
W
M
(S
VP
WM
) s
c
hem
e
.
*
*
*
*
*
*
*
*
*
max
*
*
*
*
*
*
*
*
*
min
,
,
,
,
,
,
,
,
max
,
,
,
,
,
,
,
,
min
in
hn
gn
fn
en
dn
cn
bn
an
in
hn
gn
fn
en
dn
cn
bn
an
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
(2
1)
By d
i
v
i
d
i
n
g
both
sid
e
s of
(
20) b
y
0.
5v
dc
,
t
h
e expressi
on
for the
ave
r
a
g
e value of
*
o
M
is given
a
s
:
*
max
*
min
*
max
min
)
1
(
)
2
1
(
5
.
0
)
1
(
5
.
0
)
2
1
(
5
.
0
M
M
M
v
v
v
v
v
v
o
dc
dc
dc
no
(
2
2
)
Whe
r
e
*
min
M
and
*
max
M
are the insta
n
taneo
u
s m
i
nim
u
m
and m
a
xim
u
m
m
a
gnitudes
of t
h
e
refe
rence
m
odulation si
g
n
als f
o
r
the
ni
n
e
p
h
ases.
Figu
re 2 is the
schem
a
tic diagram
represe
n
t
i
ng th
e
nine
-p
hase car
rier-
b
a
s
ed P
W
M
v
o
ltage so
urc
e
inve
rter inc
o
r
p
o
ratin
g
bot
h th
e ninth
ha
rm
onic zer
o-se
q
u
e
n
ce an
d the
co
m
m
on-n
o
d
e ze
ro
-se
que
nce in
jection
techniq
u
e.
A
m
odel c
o
rres
p
on
din
g
to
this sche
m
a
tic diagram
was i
m
pl
em
ented usi
n
g
M
A
TL
AB
/Sim
ulink.
2.3. De
terminati
on of
the Comm
on-node Z
ero
-sequenc
e Voltage
with
Unbal
a
nced
Load
The comm
on-node zero-sequence
vo
ltage si
gnal
for the
ni
ne-phase syst
em
will not be zero when the
load is unbalanced
si
nce there will be
a resul
t
ant current
js
i
in
(1) as a result
of th
e
im
balance. C
o
n
se
que
nt
ly
,
the procedure
adopted t
o
det
e
rm
in
e the common-node zero sequence
signa
l will not be applicable here.
Following the procedure adopted for fi
ve-phase system in [3], the system
voltage equation shown in (3)
will
be re-a
rra
nge
d as
f
o
llow
s
:
*
*
no
jn
jo
v
v
v
(2
3)
Figu
re
2.
Ni
ne-
pha
se car
rier
-b
ased
P
W
M
tec
hni
que
In
(2
3)
, the p
o
l
e
(switche
d) i
n
verter
v
o
ltage is exp
r
esse
d in
term
s of the refere
nce inverter output voltage
*
jn
v
,
an
d th
e av
er
age co
mm
o
n
-
node n
e
u
t
r
a
l
v
o
ltag
e
*
no
v
.
Eq
uation
(
1
8
)
can be gene
ric
a
lly
expres
sed
as:
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I
JPEDS
I
S
SN:
208
8-8
69
4
Mod
e
lin
g and
S
i
mu
la
tion
o
f
a Ca
rrier-ba
sed PWM Vo
ltag
e
S
o
u
r
ce In
verter fo
r… (Omo
no
wo Da
vid Momoh
)
7
i
h
g
f
e
d
c
b
a
j
M
S
jp
jp
,
,
,
,
,
,
,
,
,
1
5
.
0
(
2
4
)
Substituting (24) into (8) gi
ves:
jp
dc
jo
M
v
v
5
.
0
(2
5)
Also, substituti
ng (25) i
n
to
(23) gi
ves:
*
*
5
.
0
no
jn
jp
dc
v
v
M
v
(2
6)
Equ
a
tio
n
(2
6)
can
b
e
r
e
-
a
r
r
a
ng
ed
to
g
i
v
e
:
*
*
*
*
5
.
0
5
.
0
o
jp
jp
dc
no
dc
jn
jp
M
M
M
v
v
v
v
M
(2
7)
Whe
r
e,
dc
no
o
dc
jn
jp
v
v
M
v
v
M
5
.
0
5
.
0
*
*
*
*
(2
8)
Fr
o
m
(
27)
,
*
*
jp
o
jp
M
M
M
(2
9)
Equ
a
tio
n
(2
9)
can
b
e
ex
pr
essed
in m
a
trix form
as follows:
*
*
*
*
*
*
*
*
*
1
1
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
ip
hp
gp
fp
ep
dp
cp
bp
ap
ip
hp
gp
fp
ep
dp
cp
bp
ap
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
(3
0)
Ju
st lik
e in
[
3
], Equ
a
tion
(30
)
can be represented
as
follows:
y
Ax
(3
1)
Whe
r
e,
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I
S
SN:
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088
-86
94
I
JPEDS
Vo
l.
5
,
No
.
1
,
Ju
ly 20
14
:
1
–
14
8
1
1
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
1
A
(3
2)
On cl
oser l
o
ok at (32), the
r
e are ten col
u
m
n
s a
nd nine
ro
ws. T
h
is i
m
plies that there are
nine
eq
u
a
tion
s
with ten
unk
nown in
(27
)
.
The
solution to (31) is i
ndeterm
inate and
by invoking the
Moore-
Pen
r
ose
ps
eu
d
o
-i
nve
rse m
e
thod
. T
h
e solution is
obtained as:
y
AA
A
x
T
T
1
(3
3)
C
onse
q
uently
,
the m
odulatio
n
sig
n
als are
o
b
t
a
ined a
s
:
i
j
k
a
k
kp
jp
jp
M
M
M
*
*
9
10
1
(3
4)
By r
e
-
a
rr
an
g
i
ng
(
3
4
)
,
th
e co
m
m
o
n
-
no
d
e
zero-
sequ
en
ce sig
n
al as a re
sult
of load im
balance is obtained as:
i
a
k
kp
o
M
M
*
*
10
1
(3
5)
3.
R
E
SU
LTS AN
D ANA
LY
SIS
The m
odel of
a carrier-based PW
M two le
vel voltage
source
inve
rter for 9-phase
induction
m
achine
dri
v
e was
de
v
e
lope
d an
d si
m
u
lated usin
g
M
A
TLAB
/
Si
m
u
link.
Detailed analy
s
is en
com
p
assing th
e nint
h
harm
onic zer
o
-
seq
u
e
n
ce in
je
ction an
d c
o
m
m
on-n
ode ze
ro
-seque
nce signal injection wa
s im
ple
m
ented. The
analy
s
is of the
sy
stem
under
b
a
lanced a
nd
un
balance
d
lo
a
d
was also car
rie
d
o
u
t. F
o
r this
wo
rk
, the f
r
e
q
uency
of the
carrier signal
(
V
tri
)
use
d
is 1
0
k
H
z
an
d
t
h
e value
o
f
the dc bu
s volt
a
ge (
V
dc
)
utilized was 150 V.
The
param
e
ters fo
r
the ni
ne-
p
h
ase
squi
rrel ca
ge i
n
d
u
ctio
n m
ach
ine used as the
syste
m
load are listed in
Tabl
e
1.
Th
e pro
to
typ
e
mach
in
e u
s
ed
was a th
r
e
e-p
h
ase in
du
ction
mach
in
e (
w
ith
a r
a
ted
p
e
ak
vo
ltag
e
of
180
V) that was re
wo
u
nd to a nin
e
phase m
achine. C
o
n
se
que
ntly, the peak voltage for
the
nine phase m
a
chine is
60
V.
Table
1.
Param
e
ters o
f
t
h
e
Nine-
P
hase
S
qui
r
r
el C
a
ge
I
n
d
u
c
tion M
a
c
h
ine
Para
m
e
ter value
Rated Power
3 Hor
s
epower
(
hp)
Stator Resistance
(
R
s
) 0.
99
Ω
Referred Rotor Re
sistance (
r
R
)
0.
66
Ω
M
a
gnetizing induc
tance (
L
ms
)
0.
0404
H
Stator
L
e
akage I
nductance (
ls
L
)
0.
0034 H
Refer
r
e
d Rotor
L
e
akage I
nductance (
lr
L
)
0.
0034 H
Nu
m
b
er
of Poles
4
M
o
m
e
nt of I
n
er
tia
(
J
)
0.
089 kg.m
2
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I
JPEDS
I
S
SN:
208
8-8
69
4
Mod
e
lin
g and
S
i
mu
la
tion
o
f
a Ca
rrier-ba
sed PWM Vo
ltag
e
S
o
u
r
ce In
verter fo
r… (Omo
no
wo Da
vid Momoh
)
9
3.
1.
PW
M Al
gori
t
hm f
o
r
B
a
l
a
nce
d
Mo
du
l
a
ti
o
n
Si
gn
al
s and
B
a
l
a
nced
L
oad
Sim
u
lation res
u
lts fo
r the c
a
rrier
-ba
s
ed
P
W
M
v
o
ltage s
o
u
r
ce in
ve
rter
in w
h
ich t
h
e
m
odulatio
n
signals
f
o
r
all the
nine
phas
e
s are
bala
nce
d
a
n
d the
loa
d
is also
bala
nc
ed a
r
e
pre
s
ent
e
d
here
. T
h
e
value,
5
.
0
, was chose
n
for t
h
e c
o
m
m
on-node ze
ro-
sequ
en
ce vo
ltag
e
in
j
ection
.
T
h
e effect of
this value
of
on
the m
o
d
u
lation si
gn
al can
be see
n
in
Fig
u
r
e 3
pa
rt (
b
)
. Figu
re 4 part (a)
sho
w
s
the
i
nve
rter out
put p
h
a
s
e
‘a
’
voltage while part
(b) shows
the load
cur
r
e
nt wh
en
a lo
ad
to
rqu
e of
9
Nm
was applied to
the induction
m
o
tor.
Figure 5 shows the inve
rter
out
put line-t
o
-line voltage
for adjacent
pha
ses (phase ‘a
’
and
phase ‘b’), and
nona
djace
nt phases
(phase ‘a’ and
phase
‘c
’).
Figu
re
3.
P
h
as
e ‘a’
m
odulatio
n si
gnal
(a)
Gi
ven
re
fe
re
nce
m
odulation si
g
n
al (
b
)
actual
m
odulation si
g
n
al
with
5
.
0
In Figure
6, t
h
e electrom
a
gne
tic torque
and
rot
o
r
spee
d characteristics of th
e i
n
duction
m
achine are
prese
n
ted
.
It ca
n
be see
n
f
r
om
the fi
gu
res t
h
a
t
the ra
ted
electrom
a
gnetic tor
que
o
f
the
ni
ne
-p
hase s
q
uirrel
cage
in
du
ctio
n m
a
c
h
in
e is abo
u
t
12
Nm
.
Par
t
s
(
a
) and
(b)
r
e
sp
ectiv
ely sh
ow th
e electro
m
a
g
n
e
tic to
rqu
e and
ro
tor
spee
d of the
m
achine during free accel
era
tion pe
riod while parts (c) and
(d) show sa
m
e
after a 9Nm
load
tor
que is a
p
plied to t
h
e m
achine at 3 sec
o
n
d
s. T
h
e
nine
-
p
h
a
se stator
curr
en
ts
o
f
t
h
e squirrel cage induction
m
achine are
s
h
ow
n i
n
the
pl
ot o
f
Fi
gu
re
7.
Figu
re
4.
(a
) P
h
ase
‘a’
loa
d
v
o
ltage
(b
)
pha
s
e
‘a’
stator
cu
rr
ent
5.
9
7
5.
9
7
5
5.
9
8
5.
9
8
5
5.
9
9
5.
9
9
5
6
-1
-0
.
5
0
0.
5
1
Ti
m
e
[
s
]
ma
re
f
5.
9
7
5.
9
7
5
5.
9
8
5.
9
8
5
5.
9
9
5.
9
9
5
6
-1
-0
.
5
0
0.
5
1
Ti
m
e
[
s
]
ma
(a
)
(b
)
5.
9
4
5.
9
5
5.
96
5.
9
7
5.
9
8
5.
9
9
6
-
200
-
100
0
100
200
Ti
m
e
[
s
]
v
an
[V]
5.
9
4
5.
9
5
5.
96
5.
9
7
5.
9
8
5.
9
9
6
-1
0
-5
0
5
10
Ti
m
e
[
s
]
i
as
[A]
(a
)
(b
)
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I
S
SN:
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088
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94
I
JPEDS
Vo
l.
5
,
No
.
1
,
Ju
ly 20
14
:
1
–
14
10
Figu
re
5.
(a
) Li
ne-t
o-line
v
o
ltage
fo
r
pha
se ‘a
’ a
n
d
p
h
ase
‘
b
’
(
b
) line
-
to
-line
v
o
ltage
fo
r
ph
ase ‘a
’ a
n
d
p
h
a
s
e
‘c’
3.
2.
PW
M Al
gori
t
hm f
o
r U
nbal
a
nced
M
o
dul
ati
o
n
Si
gn
al
s and
B
a
l
a
n
ced
L
o
ad
In this segm
ent of sim
u
lation results analysis,
the carrie
r
-base
d
PW
M
voltage source inve
rter
has
un
balance
d
m
o
d
u
lation si
gn
als. Ho
weve
r,
it is
m
a
de
to supply a balanced load
(sy
m
m
e
trical nine-p
hase
induction m
a
c
h
ine
)
. T
h
e im
balance
was
occasione
d
by
interc
hanging t
h
e
phase
angl
es of the m
o
dulation
signals (
r
efe
r
e
n
ce v
o
ltages
)
f
o
r
pha
se ‘a’ a
n
d p
h
ase ‘
b
’
; also, the m
a
gnitude
fo
r p
h
ase
‘
d’ m
odulatio
n
signal
was cha
n
ged
fr
om
0.8 to 0.
7
8
.
This tran
slated to the
pea
k
v
a
lue of
ph
ase‘
d’ re
fe
rence
vo
ltage chan
gin
g
from
60
V t
o
47
V.
The c
o
m
m
on-no
de
ne
utral v
o
ltage is
gene
r
a
ted th
ro
u
gh t
h
e
pr
oced
u
r
e l
a
id d
o
w
n i
n
se
ction
2
su
bsectio
n
2.
3 ab
ov
e.
Figu
re
6.
Plots
of
electrom
a
gn
etic torq
ue a
n
d
r
o
tor
s
p
eed
5.
9
7
5.
9
7
5
5.
98
5.
9
8
5
5.
9
9
5.
9
9
5
6
-200
-100
0
100
200
Ti
m
e
[
s
]
v
ab
[V
]
5.
9
7
5.
9
7
5
5.
98
5.
9
8
5
5.
9
9
5.
9
9
5
6
-200
-100
0
100
200
Ti
m
e
[
s
]
v
ac
[V
]
(a
)
(b
)
0
1
2
3
-1
0
0
10
20
Ti
m
e
[
s
]
T
e
[N
m
]
3
4
5
6
-5
0
5
10
Ti
m
e
[
s
]
T
e
[N
m
]
0
1
2
3
-
200
0
20
0
40
0
Ti
m
e
[
s
]
w
r
[r
a
d
/
s]
3
4
5
6
320
340
360
380
Ti
m
e
[
s
]
w
r
[r
a
d
/
s]
(a
)
(c
)
(d
)
(
b
)
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