Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
5, N
o
. 4
,
A
ugu
st
2015
, pp
. 64
4
~
65
5
I
S
SN
: 208
8-8
7
0
8
6
44
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
Modeling of Balanced and Unba
lanced Three-Phase Induction
Motor under Balanced and Unbalanced Supply Based on
Winding Function Method
Mohammad Jann
ati
1
, To
le
Sutikno
2
,
Nik
Rumz
i Nik I
d
ris
3
, Mo
hd Juna
idi
Abdul Azi
z
4
1,3,4
UTM-PROT
ON Future Driv
e L
a
bora
t
or
y,
Fa
cult
y
of
Ele
c
tri
c
al
Engine
ering
,
Universiti
Tekno
logi Ma
la
ysia
81310 Skudai, Johor Bahru, Malay
s
ia
2
Department of Electrical
Eng
i
n
eering
,
Faculty
o
f
Indus
trial
Tech
nolog
y
,
Univ
ersitas Ahmad Dahlan
UAD 3
rd
Campu
s
, Janturan
5516
4, Yog
y
ak
arta, I
ndonesia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ma
r 8, 2015
Rev
i
sed
May 12
, 20
15
Accepted
May 30, 2015
An accura
te m
odel of bal
a
nced
and unbalan
ced
three-ph
ase Indu
ction Motor
(IM) under balanced
and unbal
a
nced
suppl
y
co
nditions based
on Winding
Function Metho
d
(WFM) is pr
esented
in
this
work. In this
paper
,
th
e
unbalan
ced con
d
ition in three-p
h
ase IM
is lim
it
ed to stator winding open-
phase fault. The
analy
s
is of presented
m
odels is shown in details which allow
predicting the
perform
ance of
3-phase IM
under differen
t
conditions.
Computer simulations were obtaine
d using
the MATLAB software for
a
three-ph
as
e s
qui
rrel
cag
e IM
.
M
A
TLAB s
i
m
u
lation
res
u
l
t
s
s
how that
th
e
oscilla
tion of
the speed and
elec
trom
agnet
i
c torque h
a
s incre
a
se
d
considerab
ly
du
e to
the open-ph
ase fault
in stator
windings.
Keyword:
Balan
ced
I
nductio
n
M
o
to
r
Balan
ced
Supply
M
odel
i
n
g
Un
bal
a
nce
d
I
n
duct
i
o
n
M
o
t
o
r
Un
bal
a
nce
d
Su
ppl
y
W
i
n
d
i
n
g F
u
nct
i
on M
e
t
h
o
d
Copyright ©
201
5 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
s:
M
oham
m
ad Jannat
i
a
n
d
Ni
k
R
u
m
z
i
Ni
k Id
ri
s
UTM
-
PR
OT
O
N
F
u
tu
re
Dri
v
e
Lab
o
rat
o
ry
,
Faculty of Electrical
En
gi
nee
r
i
n
g
,
Un
i
v
ersiti Tekn
o
l
o
g
i
Malaysia
8
131
0 Sk
ud
ai,
Jo
hor
Bah
r
u
,
Malaysia
Em
a
il: j
a
n
n
a
ti
m
9
4
@
yah
o
o
.
co
m
,
n
i
k
r
u
m
zi
@fk
e
.u
tm
.
m
y
1.
INTRODUCTION
Three
-
phase
In
duct
i
o
n M
o
t
o
r
s
(IM
s) are c
o
m
m
onl
y
em
pl
o
y
ed i
n
m
a
ny
i
ndust
r
i
a
l
appl
i
cat
i
ons d
u
e t
o
th
eir reliab
ility
, robu
stn
e
ss, lo
w co
st, goo
d p
e
rform
a
n
ce
an
d
n
eed
little
m
a
in
ten
a
n
ce co
m
p
ared
with o
t
her
types of electri
cal
m
achines
[1].
The
d-
q m
odel
i
s
o
n
e
of
t
h
e
m
o
st
general
l
y
m
odel
s
fo
r t
h
r
ee-p
h
ase
IM
s
whi
c
h
has
bee
n
prese
n
t
e
d
b
y
Park
.
D
e
tailed
d
-
q
m
o
d
e
ling
i
s
u
s
ed
t
o
r
e
pr
esen
t
h
ealth
y IMs and
m
o
to
r
s
un
d
e
r
f
a
u
lt con
d
ition
s
[
2
]-[5]. Th
is
m
odel decreas
es the
num
b
er of equatio
n
s
need
ed fo
r sim
u
latio
n
.
However, it requ
ires
so
m
e
m
o
d
i
ficatio
n
i
n
m
o
d
e
l stru
cture for each
fau
lt con
d
ition
i
n
3
-
p
h
a
se IM
[6
]. M
o
reov
er th
e
d-q
m
o
d
e
l is b
a
sed
o
n
th
e
su
ppo
sitio
n th
at th
e stato
r
wi
nd
ing
s
are si
n
u
s
o
i
d
a
l
d
i
stri
bu
ted
.
Th
is assu
m
p
tio
n
is cau
s
ed
t
h
e
h
a
rm
o
n
i
cs
of th
e
wi
n
d
i
n
gs di
st
r
i
but
i
o
n are
re
m
oved i
n
t
h
e
m
o
t
o
r anal
y
s
i
s
. Det
a
i
l
e
d m
odel
i
n
g
of
3-
p
h
ase IM
un
de
r fa
ul
t
condition assis
t
s unde
rstandi
n
g m
o
tor
dyna
m
i
c beha
vior
for choosing a
p
propriate m
e
thods
to detect
faults
and
c
h
o
o
si
n
g
sui
t
a
bl
e c
ont
ro
l
st
rat
e
gi
es.
A
t
ech
ni
q
u
e
bas
e
d
on
t
h
e
real
di
st
ri
but
i
o
n
o
f
st
at
o
r
wi
n
d
i
n
gs
f
o
r
m
odel
i
ng o
f
t
h
ree-
pha
se IM
has
been
pr
o
p
o
se
d by
T
o
l
i
y
at
et al.
[7
],
[8
]. In
th
is techn
i
qu
e wh
ich
is called
W
i
n
d
i
n
g Fu
nc
t
i
on
M
e
t
h
o
d
(
W
FM
)
ha
s be
en use
d
t
o
st
udy he
althy electrical
m
achines and m
a
ny fa
m
i
liar
faults in electrical
m
achines suc
h
as
crac
ke
d r
o
t
o
r en
d ri
n
g
s,
br
o
k
en
rot
o
r
bars
, sh
o
r
t
ci
rcui
t
an
d ab
no
rm
al
co
nd
itio
ns
o
f
t
h
e stator
wind
in
g
s
[9
]-[16
]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
JECE Vo
l. 5
,
N
o
. 4
,
Au
gu
st 2
015
:
64
4
–
65
5
6
45
Whe
n
t
h
e
3-phase IM is c
o
nnected di
rectly to
powe
r s
u
ppl
y
or
an inverte
r
in the case
of electrical
dri
v
es, t
h
e
ope
rat
i
on
of t
h
e m
achi
n
e ca
nn
ot
be d
o
n
e u
nde
r
ope
rat
i
o
n o
f
b
a
l
a
nced
po
wer
sup
p
l
y
. Fr
om
m
a
ny
researc
h
es,
t
h
e
un
bal
a
nced
p
o
we
r s
u
p
p
l
y
h
a
s dam
a
gi
ng
resu
lt on
th
e
IM p
e
rf
or
m
a
nce. It induces l
o
sses,
vi
b
r
at
i
o
n
,
heat
i
ng a
n
d
n
o
i
s
e [
17]
-
[
2
2
]
.
C
ons
eq
ue
nt
l
y
, un
bal
a
nci
n
g
d
e
t
ect
i
on i
n
t
h
e v
o
l
t
a
ge a
p
p
l
i
e
d i
s
m
a
ndat
o
ry
.
In t
h
i
s
w
o
rk
, we p
r
esent
m
odel
of
heal
t
h
y
and
faul
t
y
t
h
re
e-p
h
ase IM
(t
h
r
ee-
pha
se IM
un
de
r st
at
or
wi
n
d
i
n
g
ope
n-
ci
rcui
t
fa
ul
t
)
u
nde
r
bal
a
nce
d
and
u
n
b
al
ance
d
po
wer
su
p
p
l
y
com
b
i
n
ed t
o
t
h
e wi
ndi
ng
f
unct
i
o
n
th
eory. Th
is pap
e
r is o
r
g
a
n
i
zed
as fo
llows: After in
tro
ductio
n
in
section
1
,
in
section 2
,
W
F
M m
o
d
e
l o
f
healthy and fa
ulty three-phas
e IM u
nde
r b
a
l
a
nced a
nd
u
n
b
al
ance
d s
u
p
p
l
y
i
s
di
scusse
d.
The
per
f
o
r
m
a
nce o
f
t
h
e prese
n
t
e
d
m
e
t
hods i
s
ana
l
y
zed and chec
ked
usi
n
g
M
a
t
l
a
b soft
ware i
n
sect
i
on 3 an
d sect
i
on 4 co
nc
l
udes
t
h
e pa
pe
r.
2.
WFM MOD
E
L OF HEA
L
THY
AN
D FAU
LTY
THR
E
E-
PHA
S
E IM UN
D
E
R
BA
LA
NC
ED AND
U
N
B
A
L
ANCED
SU
PPLY
The sq
ui
r
r
el
cage r
o
t
o
r
of
3-
p
h
ase IM
an
d e
qui
val
e
nt
ci
rcu
i
t
of squi
rrel
ca
ge r
o
t
o
r i
n
W
F
M
i
s
show
n
in
Figur
e
1
an
d Figu
r
e
2 r
e
sp
ectiv
ely.
Fi
gu
re
1.
S
qui
r
r
el
cage
r
o
t
o
r
Fi
gu
re
2.
Eq
ui
val
e
nt
ci
rc
ui
t
o
f
s
qui
rrel
ca
ge
rot
o
r i
n
WFM
Mo
reo
v
e
r, t
h
e
eq
u
a
tion
s
of health
y 3
-
p
h
a
se IM with
“m
” ro
t
o
r
b
a
rs can
b
e
written
as eq
u
a
tion
s
(1
) an
d
(2
)
[
7
],
[8
].
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Mod
e
lin
g o
f
Ba
lan
ced and
Un
ba
lan
c
ed
Th
ree-
Pha
s
e In
ductio
n
Mo
to
r und
er …
(Moha
mm
ad
Jan
n
a
ti)
64
6
dt
d
I
L
I
T
I
L
I
L
dt
d
I
R
V
I
L
I
L
dt
d
I
R
V
rm
rm
r
rm
sr
T
s
e
s
T
sr
r
rr
r
r
r
r
r
r
sr
s
ss
s
s
s
s
s
(1
)
whe
r
e:
e
b
mr
r
r
r
r
b
r
r
r
r
e
b
mr
b
r
r
r
r
r
r
b
r
r
e
b
mr
b
r
r
b
r
r
r
r
b
r
r
e
b
mr
rr
e
b
b
b
b
e
b
b
e
b
b
b
b
e
b
r
crm
cr
cr
brm
br
br
arm
ar
ar
sr
cc
cb
ca
bc
bb
ba
ac
ab
aa
ss
s
s
s
s
T
T
rm
r
r
r
T
rm
r
r
r
T
rm
r
r
r
T
c
b
a
s
T
c
b
a
s
T
c
b
a
s
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
r
r
r
R
v
v
v
V
i
i
i
I
i
i
i
I
v
v
v
V
m
m
m
m
m
m
2
2
2
2
2
0
0
2
0
0
0
0
0
2
0
0
2
,
,
0
0
0
0
0
0
0
0
0
,
,
,
,
3
2
1
3
2
3
1
3
2
3
2
1
2
1
3
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
(2
)
In (2
),
[R
s
] is 3×3 c
o
nsists of resistance
of
each coil, [R
r
] is
m
×
m
m
a
trix
wh
ere, R
e
is th
e end
ri
ng
segm
ent resistance and R
b
is th
e ro
tor b
a
r resistan
ce. Th
e
matrix
[L
ss
] is
3×3 m
a
trix. The m
u
tual inductance
matrix
[L
sr
]
i
s
3×m
m
a
t
r
i
x
i
n
cl
ude
d
of t
h
e
m
u
t
u
al
i
nduct
a
nces
bet
w
ee
n t
h
e st
at
o
r
c
o
i
l
s
and
t
h
e
rot
o
r
b
a
rs. L
mr
i
s
t
h
e m
a
gnet
i
z
i
ng i
n
d
u
ct
an
ce
of
eac
h r
o
t
o
r
bar
.
L
b
and
L
e
are rotor
bar leakage
inducta
nce and
rot
o
r e
n
d ri
ng
leakage i
n
duct
a
nce. M
o
re
ove
r
, L
rirj
is
the m
u
tual inductanc
e
betwee
n t
w
o
rot
o
r
ba
rs.
The
m
o
t
o
r
t
h
at
i
s
st
u
d
i
e
d
i
n
t
h
i
s
pape
r has
28 r
o
t
o
r ba
rs and
36 st
at
or sl
ot
s. Fi
g
u
re 3 a
nd Fi
gu
re 4
sh
ow th
e t
u
rn
fun
c
tion
of th
e stato
r
ph
ases
an
d
t
h
e turn
fu
n
c
tion
o
f
first ro
tor b
a
r for th
e h
ealth
y m
a
ch
ine
resp
ectiv
ely (fo
r th
e turn
fu
nctio
n
of second
ro
to
r
b
a
r, the wav
e
form
o
f
Fig
u
re 4
is shifted
to
th
e righ
t b
y
2
π
/28
=
π
/1
4).
In
WFM
,
wi
n
d
i
ng f
unct
i
o
n
i
s
defi
ned
as fol
l
owi
n
g
eq
uat
i
o
n [7]
,
[
8
]
:
n
n
N
(3
)
whe
r
e,
n
(
φ
)
i
s
t
h
e t
u
r
n
fu
nct
i
on
an
d
˂
n(
φ
)
˃
i
s
t
h
e
ave
r
a
g
e
val
u
e
o
f
t
u
rn
f
unct
i
on
. B
a
sed
o
n
e
quat
i
on
(
3
)
,
Fi
gu
re 3 an
d F
i
gu
re 4, t
h
e wi
ndi
ng f
u
nct
i
o
n of t
h
e st
at
or
ph
ases and t
h
e
wi
ndi
ng f
u
nct
i
o
n of fi
r
s
t
rot
o
r ba
r ar
e
sho
w
n i
n
Fi
gu
r
e
5 a
n
d Fi
gu
re
6
respect
i
v
el
y
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
JECE Vo
l. 5
,
N
o
. 4
,
Au
gu
st 2
015
:
64
4
–
65
5
6
47
Fi
gu
re 3.
Tu
r
n
fu
nct
i
o
n of st
at
or
p
h
ases
Figu
re 4.
Tu
r
n
fu
nctio
n of firs
t
rot
o
r ba
r
Fi
gu
re 5.
W
i
nd
i
ng f
unct
i
o
n of
st
at
or p
h
ases
Fi
gu
re 6.
W
i
nd
i
ng f
unct
i
o
n of
fi
rst
rot
o
r
ba
r
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Mod
e
lin
g o
f
Ba
lan
ced and
Un
ba
lan
c
ed
Th
ree-
Pha
s
e In
ductio
n
Mo
to
r und
er …
(Moha
mm
ad
Jan
n
a
ti)
64
8
The
m
u
tual
inducta
nce between windings
B
and A (L
BA
) in term
s o
f
tu
rn
fun
c
tion
an
d wi
n
d
i
ng
fun
c
tion
is calcu
lated
b
y
[7
]:
d
n
N
g
rl
L
B
A
o
BA
2
0
(4
)
whe
r
e “
r
” is rotor
radi
us, “l” i
s
stack
length,
“g” is effective
air ga
p,
n
B
(
φ
)
i
s
t
u
r
n
f
u
nct
i
o
n
of
wi
n
d
i
n
g B
an
d
N
A
(
φ
) i
s
wi
n
d
i
ng
fu
nct
i
o
n o
f
wi
n
d
i
n
g A.
M
o
re
ove
r,
μ
o
=4
π
E-
7. F
r
om
Fi
gu
res 3
-
6 a
n
d
equat
i
o
n (
4
),
L
ss
, L
sr
and L
rr
ca
n be calculated
as:
L
aa
:
9
127
2
2
4
2
5
3
6
2
5
4
2
2
2
225
215
235
225
205
195
195
185
55
45
185
55
45
35
15
5
25
15
N
g
rl
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
g
rl
L
o
o
aa
(5
)
L
ab
:
6
3
3
2
3
3
3
4
3
5
3
6
2
6
6
6
2
6
3
6
3
5
3
4
3
3
3
2
3
2
235
225
225
215
215
205
205
195
195
185
185
175
175
165
165
155
145
135
135
125
125
55
45
35
55
45
35
25
15
5
25
15
N
g
rl
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
g
rl
L
o
o
ab
(6
)
L
ac
:
6
3
3
2
3
3
3
4
3
5
3
6
2
6
6
6
2
6
3
6
3
5
3
4
3
3
3
2
3
2
235
225
225
215
215
205
205
195
195
185
185
115
115
105
105
95
85
75
65
55
75
65
45
35
55
45
35
25
15
5
25
15
N
g
rl
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
d
N
N
g
rl
L
o
o
ac
(7
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 5
,
N
o
. 4
,
Au
gu
st 2
015
:
64
4
–
65
5
6
49
There
f
ore L
ss
is ob
tain
ed
as equ
a
tio
n (8
).
9
127
6
6
6
9
127
6
6
6
9
127
2
N
g
rl
L
o
ss
(8
)
L
sr
:
d
n
N
g
rl
L
r
s
o
sr
2
0
whi
c
h gi
ves
,
)
2
)(
(
2
1
:
0
N
g
rl
d
N
g
rl
L
o
o
sr
rm
rm
rm
(9
)
rm
o
o
sr
rm
N
g
rl
d
N
d
N
g
rl
L
rm
rm
2
2
)
2
)(
(
2
2
:
(1
0)
)
2
)(
(
2
:
2
N
g
rl
d
N
g
rl
L
o
o
sr
rm
rm
rm
(1
1)
rm
o
o
sr
rm
N
g
rl
d
N
d
N
g
rl
L
rm
rm
2
4
)
2
)(
(
2
2
:
2
2
2
2
(1
2)
There
f
ore L
sr
is ob
tain
ed
as e
q
uat
i
o
n
(
1
3).
2
2
2
2
2
2
2
0
2
rm
rm
rm
rm
rm
rm
o
sr
g
rlN
L
(1
3)
As m
e
ntioned bef
o
re
, the m
o
tor that is st
udi
ed in this pape
r has 2
8
r
o
to
r b
a
rs (
α
=2
π
/28=
π
/14) a
nd
3
6
stator sl
ots. Therefore equation
(1
3) can be
written
as equation
(14).
2
14
27
28
55
14
27
28
14
13
28
27
14
13
0
28
1
1
rm
rm
rm
rm
rm
rm
o
r
a
g
rlN
L
(1
4)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Mod
e
lin
g o
f
Ba
lan
ced and
Un
ba
lan
c
ed
Th
ree-
Pha
s
e In
ductio
n
Mo
to
r und
er …
(Moha
mmad
Jan
n
a
ti)
65
0
There
f
ore L
ar1
(inductance
between the
phase “a” of the
st
ator wi
ndi
ng and fi
rst rotor ba
r) is
obtaine
d a
s
f
o
l
l
ows:
1
6
1
5
1
4
1
3
1
2
2
1
1
r
a
r
a
r
a
r
a
r
a
r
a
ar
L
L
L
L
L
L
L
(1
5)
whic
h gives
,
180
360
180
14
.
357
180
86
.
742
2
180
14
.
357
180
14
.
247
180
72
.
385
180
14
.
247
180
230
180
58
.
38
180
230
180
220
180
42
.
191
180
220
180
14
.
217
180
42
.
411
2
180
14
.
217
180
210
180
28
.
194
180
210
180
14
.
207
180
28
.
404
2
180
14
.
207
180
200
180
14
.
197
180
200
180
14
.
197
180
14
.
397
2
180
14
.
197
180
190
180
200
180
190
180
14
.
187
180
390
2
180
14
.
187
180
180
180
86
.
202
180
180
180
14
.
177
180
86
.
382
2
180
14
.
177
180
14
.
167
180
72
.
205
180
14
.
167
180
50
180
58
.
38
180
50
180
40
180
42
.
11
180
40
180
14
.
37
180
42
.
51
2
180
14
.
37
180
30
180
28
.
14
180
30
180
14
.
27
180
28
.
44
2
180
14
.
27
180
20
180
14
.
17
180
20
180
14
.
17
180
32
.
37
2
180
14
.
17
180
10
180
18
.
20
180
10
180
14
.
7
180
36
.
30
2
180
14
.
7
0
180
04
.
23
1
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
rm
o
ar
g
rlN
L
(1
6)
Th
e sam
e
p
r
o
c
ess can b
e
done fo
r
“L
ar2
, L
ar3
, …,
L
br1
, L
br2
, …
and
L
cr1
, L
cr
2
, …”.
L
rr
:
2
2
2
0
1
1
1
1
g
rl
d
n
N
g
rl
L
o
r
r
o
r
r
(1
7)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
ECE Vo
l.
5
,
No.
4
,
Au
gu
st 2
015
:
64
4
–
65
5
6
51
As the rotor bars are the sa
m
e
, therefore
the ge
n
e
ral f
o
rm
of roto
r in
ducta
nces are
obtaine
d as
following equation:
392
g
rl
L
o
r
r
j
i
(1
8)
Equations (1) and (2) can be
written as (19)
and (20).
c
b
b
a
c
b
a
s
s
s
s
c
b
b
a
dt
d
i
i
i
r
r
r
r
v
v
v
v
0
0
(1
9)
28
2
1
28
28
2
2
1
1
28
28
2
2
1
1
0
0
0
1
1
1
0
r
r
r
cr
br
cr
br
cr
br
br
ar
br
ar
br
ar
c
b
a
cc
bc
cb
bb
ca
ba
bc
ac
bb
ab
ba
aa
c
b
b
a
i
i
i
L
L
L
L
L
L
L
L
L
L
L
L
i
i
i
L
L
L
L
L
L
L
L
L
L
L
L
(2
0)
Eq
uations of 3
-
p
h
ase
IM
wh
en
o
n
e of the
stator phases
ope
ne
d ha
ve t
h
e sim
i
lar structure to the
healthy
3-
pha
s
e
m
achine eq
u
a
tions.
T
h
e
onl
y
diffe
re
nt is t
h
at, i
n
the
fa
ulty
m
ode,
the
r
o
w a
n
d
c
o
lum
n
fo
r the
faulted
p
h
ase is rem
oved. T
h
eref
ore
,
d
u
ri
ng
stator wi
ndi
ng
ope
n-
p
h
ase fa
ult, (
1
9
)
an
d (
2
0) c
h
a
nge t
o
(
2
1)
a
n
d
(22) (i
n this
pa
per it is assum
e
d that a
phase
cut-off
fa
ult is occ
u
rred in
phase “c”
of t
h
e stator
windings).
c
b
b
a
b
a
s
s
s
c
b
b
a
dt
d
i
i
r
r
r
v
v
v
v
0
(2
1)
28
2
1
28
28
2
2
1
1
28
28
2
2
1
1
...
...
r
r
r
cr
br
cr
br
cr
br
br
ar
br
ar
br
ar
b
a
cb
bb
ca
ba
bb
ab
ba
aa
c
b
b
a
i
i
i
L
L
L
L
L
L
L
L
L
L
L
L
i
i
L
L
L
L
L
L
L
L
(2
2)
3.
SIMULATION RESULTS
The
WF m
o
d
e
l prese
n
ted i
n
the sectio
n
2 has
bee
n
im
plem
ented in the M
a
tlab (M
-File)
envi
ro
nm
ent. The
3-
pha
se I
M
used i
n
thi
s
pa
per
is
7
.
5Hp,
400
V,
60Hz,
2
P
o
l
es.
Th
eir
d
e
tailed
m
o
to
r
param
e
ters are
give
n as
f
o
llo
ws:
Effective
air
g
a
p: g=
0.
9
8
7
4
E
-
3m
Stack
leng
th
: l=1
0
2
.
41
28
E-
3m
Ro
to
r r
a
d
i
u
s
:
r=6
3
.
296
8E-3
m
Stator resistanc
e
:
r
s
=1.76
Ω
Rotor ba
r
resistance: R
b
=68.34E-6
Ω
Rotor end
ring
segm
ent resistance: R
e
=1.56E
-6
Ω
Rotor ba
r lea
k
age inductance
: L
b
=0.28E-6H
Rotor end
ring
leakage i
n
duct
a
nce: L
e
=0
.0
3E
-
6
H
Inertia: J=0.03kg.m
2
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN:
208
8-8
7
0
8
Mod
e
lin
g o
f
Ba
lan
ced and
Un
ba
lan
c
ed
Th
ree-
Pha
s
e In
ductio
n
Mo
to
r und
er …
(Moha
mmad
Jan
n
a
ti)
65
2
The
3-
p
h
ase
m
o
tor studie
d
u
nde
r tw
o
diffe
rent s
o
urce
co
nditio
n
s: A
:
sinus
oidal
3-
pha
se p
o
w
e
r
su
pp
ly (Figu
r
e 7(
a) and
Figur
e
8
(
a)
) and
B
:
u
n
b
a
lan
ced no
n-
sinu
so
id
al
3
-
ph
ase
p
o
wer supp
ly (
F
ig
ure 7(
b)
and Fig
u
re 8(
b
)). Fig
u
re 7(a
)
and
Fig
u
re
7(
b) s
h
ow t
h
e sim
u
lation res
u
lts of t
h
e 3
-
p
h
a
s
e IM
u
nde
r
h
ealthy
con
d
ition a
n
d Figu
re 8
(
a) a
n
d Fig
u
re
8(
b
)
sho
w
the
sim
u
lation res
u
lts o
f
the 3
-
p
h
ase
I
M
un
der
ope
n
-
pha
se
fault. I
n
Fig
u
r
e
7 and Fig
u
r
e
8 a step load torq
ue eq
ual to 5
N
.m
at third seco
nd is ap
plied. M
o
reo
v
e
r, in
Figure
8 a
pha
s
e cut-off
fault
is ha
ppe
ne
d a
t
starting a
n
d i
n
phase
“c”.
T
h
e supply voltage
values
use
d
in
Fig
u
r
e
7
(
a)
, Fig
u
r
e
7
(
b)
,
Figur
e
8
(
a)
an
d Figu
r
e
8(
b)
ar
e:
Figu
re 7(a
)
:
V
a
=4
0
0*c
os
(1
20
*
p
i*t)
V
b
=4
00
*c
os
(1
20
*
p
i*t-
2
*pi/3
)
V
c
=400
*
c
o
s
(
1
2
0*p
i*
t+2*p
i/3
)
Figu
re 7(
b
)
:
V
a
=4
0
0*c
os
(1
20
*
p
i*t)
V
b
=3
50
*c
os
(1
20
*
p
i*t-
2
*pi/3
)+2
0
*
cos
(
3*
(1
20
*
p
i*t-
2
*pi/3
))
V
c
=3
0
0*c
os
(1
20
*
p
i*t+2
*
p
i/3)+
3
0*c
os
(5
*(
12
0
*pi
*t+2
*pi
/3))
Figu
re 8(a
)
:
V
a
=4
0
0*c
os
(1
20
*
p
i*t)
V
b
=4
00
*c
os
(1
20
*
p
i*t-
2
*pi/3
)
Figu
re 8(
b
)
:
V
a
=4
0
0*c
os
(1
20
*
p
i*t)
V
b
=3
00
*c
os
(1
20
*
p
i*t-
2
*pi/3
)+2
0
*
cos
(
4*
(1
20
*
p
i*t-
2
*pi/3
))
Figure 7
and Figure 8
illu
strate the waveform
of
stator a-axis current,
first rotor
bar current,
electrom
a
gnetic torque
and
m
achine spe
e
d. It is
obse
r
v
e
d
fro
m
th
e stat
o
r
and
ro
tor
cu
rr
en
t
wav
e
fo
rm
s
th
at
machine currents are
balanced
and si
nusoidal but with
different am
plitu
des in
healthy, faulty and bal
a
nced
and u
nbala
nce sinus
oidal
a
nd no
n
-
sin
u
s
o
idal
sou
r
ce
co
n
d
iti
ons
. B
a
sed
o
n
sim
u
lation res
u
lts of Fi
gu
re
8, it is
concl
uded t
h
at, the
oscillations of th
e speed
and electrom
a
gnetic t
o
rque has
increased consider
ably due to
the
ope
n
-
p
h
ase
fa
ult in stator
win
d
in
gs. M
o
r
e
ove
r,
base
d
on t
h
is Fig
u
r
e
,
the stato
r
an
d r
o
to
r c
u
r
r
en
ts hav
e
increased at
open-phase co
ndition
com
p
ared with norm
al condition.
Moreover, in
Figure 7(a)
the
m
o
tor
spee
d reac
h t
o
steady-state after
̴
0.2s, in
Figure
7(b)
the m
o
tor spee
d reach to stea
dy-state after
̴
0.
3s
in
Figure
8(a
)
the
m
o
tor spee
d
reach to steady
-
state after
̴
1.
2
s
in Fi
gu
re
8(
b
)
the m
o
tor
sp
eed rea
c
h t
o
st
eady
-
state after
̴
1.4
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
JECE Vo
l. 5
,
N
o
. 4
,
Au
gu
st 2
015
:
64
4
–
65
5
6
53
(a)
(b
)
Fig
u
re
7
.
Sim
u
latio
n
resu
lts
of
h
ealth
y 3-ph
ase IM; (a):
b
a
lan
ced su
pp
ly, (b): unb
alan
ced
su
pp
ly
Evaluation Warning : The document was created with Spire.PDF for Python.