TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 10, Octobe
r 20
14, pp. 7167
~ 717
5
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.652
3
7167
Re
cei
v
ed
Jul
y
27, 201
4; Revi
sed Aug
u
st 13, 2014; Accepted Aug
u
s
t 25, 2014
Indirect Flux-Oriented Control of Faulty Single-Phase
Induction Motors
M. Jannati*,
N. R. N. Idris, M. J. A. Az
iz
Univers
i
ti T
e
knolo
g
i Mal
a
ysia
UT
M-PROT
O
N F
u
ture Drive
Lab
orator
y, F
a
cult
y
of Elec
tric
al Eng
i
ne
eri
ng,
Universiti T
e
knol
ogi Ma
la
ysi
a
,
813
10 Sku
dai,
Johor Ba
hru, MALAYSIA
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: jann
atim94
@
y
a
h
o
o
.com
A
b
st
r
a
ct
Most types
of el
ectrical
ma
chin
es suc
h
a
s
Inducti
on
M
o
tors (IMs) ca
n b
e
mo
de
led
by
a
n
equ
ival
ent 2-p
hase
mac
h
in
e
(d-q mo
de
l). F
o
r instanc
e a
singl
e-ph
ase
IM w
i
th
tw
o main a
nd a
u
xil
i
a
r
y
w
i
ndin
g
s can
be
mod
e
l
ed a
s
an eq
uiva
le
n
t
2-phas
e IM.
Also a fau
l
ty singl
e-p
has
e IM (singl
e-ph
ase
IM
und
er op
en-p
h
a
se fault) can
be mod
e
le
d as
an unb
al
ance
d
2-ph
ase IM. T
h
is study con
f
irms this conc
ep
t
by
mo
del
ing
a
sin
g
le-
phas
e I
M
w
i
th on
e of
its stator
p
has
es, op
en. In
a
dditi
on th
e stu
d
y sh
ow
s that
th
e
vector control
of this faulty IM c
an be perf
o
rmed by so
me mo
dific
a
tions
in
the vector control of he
alth
y IM
(the prop
ose
d
vector control
i
n
th
is pa
per is
based
on Ind
i
rect F
l
ux-O
riented Contr
o
l (IFOC)). MATLA
B
simulati
on res
u
lts show
the go
od perfo
r
m
anc
e of the prop
os
ed techn
i
q
ue.
Ke
y
w
ords
: mo
deli
ng, sin
g
l
e
-p
hase IM, ope
n-
phas
e fault, IFOC, simul
a
tio
n
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
One of the m
o
st kn
own failure
s in stato
r
wi
ndin
g
s of the Indu
ction
Motors (IMs)
is ope
n
circuit. Some
cau
s
e
s
such as me
ch
an
ical sti
rrin
g
machi
ne, blo
w
n fuses, o
p
en coil
s a
nd
etc.
make thi
s
fail
ure. Recently, various te
ch
nique
s have
been d
e
velop
ed to indi
cate
stator
windin
g
failure
s in
IMs [1-3]. In [1] ba
sed
on
qu
ery table
s
an
d ne
ural
net
works a
nd i
n
[2], a tech
ni
que
based o
n
unf
amiliar in
put
observe
r an
d
Extended Ka
lman Filter
(E
KF) have b
e
e
n
pro
p
o
s
ed f
o
r
detectio
n
of fault in IM sta
t
or win
d
ing
s
.
In [3], a bala
n
ce
d 3-pha
se small
ampli
t
ude si
gnal
with
high f
r
equ
en
cy was u
s
ed
to dete
c
t o
p
e
n
-ph
a
se faul
t
in
stator wi
n
d
ing
s
. Thi
s
method
provides
almost imme
diate ope
n st
ator win
d
ing
detectio
n
and
is assume
d in this pap
er.
One of the m
o
st gen
eral t
e
ch
niqu
es for controlling th
e spe
ed an
d torque in IM
s is Flux-
Oriente
d
Co
ntrol (FO
C
). In the last deca
d
e
s
, various
control method
s we
re introdu
ce
d
for
controlling
si
ngle-pha
se I
M
s (
or
2-p
h
a
se IM
s)
whi
c
h a
r
e li
sted
as follo
ws: In [4-6], Roto
r
Flux-
Oriente
d
Co
n
t
rol (RFO
C)
of 2-pha
se I
M
with
hysteresi
s
cu
rrent cont
rolle
r ha
s
been p
r
e
s
ent
ed.
In [7], Stator Flux-Orie
n
te
d Control (S
FOC) of
sing
le-ph
a
se IM with cu
rrent doubl
e se
qu
ence
controlle
r wa
s pre
s
e
n
ted. A novel deco
upling vect
o
r
control of sin
g
le-p
ha
se IM was sugg
est
e
d
in [8]. Comm
on p
r
o
b
lem
s
encounte
r
ed
in
the
conv
e
n
tional vecto
r
control of si
ngle-pha
se I
M
s
has bee
n di
scu
s
sed in
pa
per [9]. In thi
s
p
ape
r, a m
e
thod b
a
sed
on FO
C fo
r a
symmetri
c
al
2-
pha
se IM as repla
c
eme
n
t has be
en prop
ose
d
. In
[10-1
2
], several ex
act model
s fo
r vector
control
of 2-ph
ase IMs ba
sed
on F
O
C have
bee
n peop
osed.
High p
e
rfo
r
m
ance FO
C of 2-ph
ase IM with
rotor spe
ed estimation using
Mod
e
l
Referen
c
e
Ad
a
p
tive Sys
t
em (MRAS) in [
13], EKF in Ref,
[14-16] an
d motor mod
e
l in [6, 17] has been p
r
e
s
ent
ed.
In spite of excellent performa
nce of
conventional vector
co
ntrol
for IMs, its ability in
controlling faulty motors is not good [19, 20]. Th
is study concerns
with
the
probl
em of modeli
ng
and ve
cto
r
control
of fault
y
IMs. Thi
s
p
aper ex
plai
ns a m
odeli
ng
method
for a
sin
g
le
-pha
se
IM
unde
r o
pen
-p
hase fault. B
a
se
d o
n
thi
s
modelin
g,
a n
e
w
ve
ctor co
ntrol
m
e
thod
based on
F
O
C
i
s
prop
osed. Th
is cont
rol techniqu
e can b
e
used to co
ntrol a singl
e
-
pha
se IM un
der he
althy and
faulty con
d
itions (the
faul
t con
d
ition in
this
p
ape
r i
s
limited
to
open
-ph
a
se f
ault). Thi
s
st
udy
sho
w
s that th
e vecto
r
cont
rol of
healthy
and
faul
ty si
ngle-pha
se
IM ca
n b
e
p
e
rformed
by u
s
ing
transfo
rmatio
n matrices a
n
d
some m
odif
i
cation
s in the
conventio
nal
FOC for 3
-
ph
ase IM.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 716
7
– 7175
7168
2. Modeling of a Single-p
h
ase IM
w
i
th
One Open
e
d
Phase
Suppo
se th
at a p
h
a
s
e
cut
off fault ha
s bee
n o
c
curred in
the
au
xiliary win
d
in
g of a
singl
e-p
h
a
s
e
IM (b axis).
Assumi
ng sinusoidal
wa
veform for the s
p
atial dis
t
ribution of the
windi
ng
s, stator and
rotor fl
ux axes ca
n be sh
own as
follows:
Figure 1. Stator and
Roto
r Windi
ng'
s Flu
x
Axes
d
and
q
comp
onent
s of the stator fluxe
s
can b
e
writte
n as follo
wing
equation
s
:
o
as
s
o
as
s
q
d
sin
,
co
s
(1)
In (1),
o
i
s
the an
gle bet
wee
n
a
s
a
nd
d
s
axes. Th
e tran
sform
a
tion vecto
r
s
must b
e
perp
endi
cul
a
r to each othe
r, therefore fro
m
Equation (1):
(2)
If we suppo
se that
o
=0, the followin
g
norm
a
lized transfo
rmatio
n
matrix for the stator
v
a
riable
s
([
T
s
]
)
is obtain
ed.
0
1
s
T
(3)
Because the
rotor vari
ables are
still in the bal
anced condition, the
decom
position matrix
for roto
r v
a
ria
b
les ([
T
r
])
re
mains u
n
cha
nged a
s
follo
ws [21]:
2
1
2
1
2
1
3
2
co
s
3
2
cos
cos
3
2
sin
3
2
sin
sin
3
2
r
T
(4)
In (4
),
γ
i
s
th
e angl
e b
e
tween
a
r
and
d
s
axes. By ap
plying (3)
an
d (4
), the
d-q
model
of
singl
e-p
h
a
s
e
IM under o
p
e
n
-ph
a
se fault is obtain
ed a
s
followi
ng eq
uation
s
:
2
or
0
0
cos
sin
o
o
o
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Indire
ct Flux-Oriente
d
Co
n
t
rol of Faulty
Single-Ph
ase
Induction Mo
tors (M. Jan
n
a
ti)
7169
Stator and rot
o
r volta
ge eq
uation
s
:
s
qr
s
dr
s
ds
r
r
r
r
ds
r
r
r
r
r
ds
ds
ds
ds
s
ds
i
i
i
dt
d
L
R
L
M
L
dt
d
L
R
dt
d
M
dt
d
M
dt
d
L
R
v
0
0
0
0
0
0
0
0
0
0
0
0
(5)
Stator and rot
o
r flux eq
uati
ons:
s
qr
s
dr
s
ds
r
r
ds
ds
ds
s
qr
s
dr
s
ds
i
i
i
L
L
M
M
L
0
0
0
0
0
0
0
0
0
0
0
0
0
(6)
Electrom
agn
etic torqu
e
eq
uation
s
:
s
qr
s
ds
ds
e
i
i
M
Pole
2
(7)
As can b
e
seen from Eq
uqation
(5
)-(7),
the
singl
e
-
pha
se
IM e
quation
s
u
n
d
e
r o
pen-
phase fault is the
sam
e
as the
healthy singl
e-phase IM
equations
with t
w
o
mai
n
and auxillary
windi
ng
s (the
healthy singl
e-ph
ase IM equation
s
with
two main an
d auxillary wi
nding
s are as
Equation
(8)-(10)). The o
n
l
y
difference b
e
twee
n eq
uat
ions
of faulty and h
ealthy IM is that, in the
faulty mode:
L
qs
=M
qs
=R
qs
=0 but in the healthy mode
we have different value
s
of
L
ds
, L
qs
,
M
ds
,
M
qs
,
R
ds
and
R
qs
(
L
ds
≠
L
qs
,
M
ds
≠
M
qs
and
R
ds
≠
R
qs
).
Stator and rot
o
r volta
ge eq
uation
s
:
s
qr
s
dr
s
qs
s
ds
r
r
r
r
qs
ds
r
r
r
r
r
qs
r
ds
qs
qs
qs
ds
ds
ds
s
qs
s
ds
i
i
i
i
dt
d
L
R
L
dt
d
M
M
L
dt
d
L
R
M
dt
d
M
dt
d
M
dt
d
L
R
dt
d
M
dt
d
L
R
v
v
0
0
0
0
0
0
(8)
Stator and rot
o
r flux eq
uati
ons:
s
qr
s
dr
s
qs
s
ds
r
qs
r
ds
qs
qs
ds
ds
s
qr
s
dr
s
qs
s
ds
i
i
i
i
L
M
L
M
M
L
M
L
0
0
0
0
0
0
0
0
(9)
Electrom
agn
etic torqu
e
eq
uation
s
:
s
qr
s
ds
ds
s
dr
s
qs
qs
e
i
i
M
i
i
M
Pole
2
(10
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 10, Octobe
r 2014: 716
7
– 7175
7170
In (8)
-(1
0),
v
s
ds
, v
s
qs
, i
s
ds
, i
s
qs
, i
s
dr
, i
s
qr
,
λ
s
ds
,
λ
s
qs
,
λ
s
dr
and
λ
s
qr
are the
d-q axe
s
voltages,
curre
n
ts, an
d
fluxes of the
stator a
nd rotor
in the
stationary
referen
c
e fra
m
e (su
perscript
s
).
R
ds
,
R
qs
and
R
r
d
enote the sta
t
or and roto
r resi
stan
ce
s.
L
ds
, L
qs
, L
r
, M
ds
and
M
qs
denote the stator,
and the roto
r self and mutual indu
ctan
ces.
r
and
τ
e
are machine
speed a
nd electroma
gne
tic
torque.
3. Vector Co
ntrol of a Sin
g
le-pha
se IM
w
i
th
One O
p
ened Pha
s
e Bas
e
d on IFOC
In the FOC te
chni
que fo
r 3-pha
se IM and
in
the health
y mode, co
nventional o
r
ba
lance
d
rotational
tra
n
sformation
([
T
s
e
]) which
is
appli
ed
to the m
a
ch
ine e
quatio
n
s
i
s
a
s
foll
o
w
s
(co
n
vention
a
l
or balan
ce
d rotational
transfo
rmat
io
n is a transformation m
a
trix to transfer
equatio
ns fro
m
stationa
ry referen
c
e fra
m
e to rotating
refere
nce fra
m
e) [22]:
e
e
e
e
e
s
T
cos
sin
sin
cos
(11
)
In this
equ
ation,
θ
e
i
s
th
e
angl
e b
e
twe
en the
statio
nary
referen
c
e fram
e a
nd
rotating
referen
c
e fra
m
e (i
n this p
aper, th
e
su
perscript
e
i
ndicates the
variable
s
are
in the
rotati
ng
referen
c
e fra
m
e). Beca
use of unequal
inductan
c
e
s
and re
sista
n
ce
s in the singl
e-p
h
a
s
e
IM
model, thi
s
matrix can
n
o
t be
used f
o
r
singl
e-pha
se IM
s ve
cto
r
control. In
this
study, th
e
followin
g
un
b
a
lan
c
ed
rotat
i
onal tra
n
sfo
r
mation for
bo
th stator volt
age a
nd
cu
rrent varia
b
les is
prop
osed
(in
this p
ape
r, a
vector control
tech
niqu
e fo
r
singl
e-p
h
a
s
e IM
with diff
erent
value
s
of
L
ds
, L
qs
, M
ds
, M
qs
, R
ds
and R
qs
is pro
p
o
s
ed. It is obvious by sub
s
tations
L
qs
=M
qs
=R
qs
=0 in the
equatio
ns, thi
s
method
can
be use
d
for faulty single
-
p
hase IM).
Propo
se
d un
balan
ce
d rota
tional tran
sform
a
tion for stator voltag
e an
d curre
n
t vari
able
s
:
e
ds
qs
e
e
ds
qs
e
e
vs
e
is
M
M
M
M
T
T
cos
sin
sin
cos
(12
)
Whe
r
e,
s
qs
s
ds
e
vs
e
qs
e
ds
s
qs
s
ds
e
is
e
qs
e
ds
v
v
T
v
v
i
i
T
i
i
(13
)
Usi
ng (11)-(1
3
) an
d after
simplifying, the
equatio
ns
of single
-
p
h
a
s
e IM are obt
ained a
s
followin
g
equ
ations:
Rotor flux
equations
:
s
qr
s
dr
e
s
e
s
r
r
e
s
s
qs
s
ds
e
is
e
is
qs
ds
e
s
e
qr
e
dr
e
s
i
i
T
T
L
L
T
i
i
T
T
M
M
T
T
0
0
0
0
1
1
(14
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Indire
ct Flux-Oriente
d
Co
n
t
rol of Faulty
Single-Ph
ase
Induction Mo
tors (M. Jan
n
a
ti)
7171
After simplifying, Equation
(14
)
ca
n be written as Equ
a
tion (15
)
.
e
qr
e
dr
r
r
e
qs
e
ds
ds
ds
e
qr
e
dr
i
i
L
L
i
i
M
M
0
0
0
0
(15
)
Rotor voltag
e
equation
s
:
s
qr
s
dr
e
s
e
s
r
r
r
r
r
r
r
r
e
s
s
qs
s
ds
e
is
e
is
qs
ds
r
qs
r
ds
e
s
e
s
i
i
T
T
dt
d
L
R
L
L
dt
d
L
R
T
i
i
T
T
dt
d
M
M
M
dt
d
M
T
T
0
0
1
1
(16
)
After simplifying, Equation
(16
)
ca
n be written as:
e
qr
e
dr
r
r
r
e
r
r
e
r
r
r
e
qs
e
ds
ds
ds
e
r
ds
e
r
ds
i
i
dt
d
L
R
L
L
dt
d
L
R
i
i
dt
d
M
M
M
dt
d
M
0
0
(17
)
Electrom
agn
etic torqu
e
eq
uation:
s
qs
s
ds
e
is
e
is
ds
qs
T
e
s
T
e
s
s
qr
s
dr
s
qs
s
ds
ds
qs
s
qr
s
dr
s
qr
s
ds
ds
s
dr
s
qs
qs
e
i
i
T
T
M
M
T
T
i
i
Pole
i
i
M
M
i
i
Pole
i
i
M
i
i
M
Pole
0
0
2
0
0
2
2
1
1
(18
)
After simplifying Equation
(18) can be
written as:
)
(
2
e
qr
e
ds
e
dr
e
qs
ds
e
i
i
i
i
M
Pole
(19
)
Based
o
n
E
quation
(1
5),
(1
7) and
(19),
RFO
C
equatio
ns of
sin
g
le
-pha
se IM a
r
e
obtaine
d a
s
followin
g
e
q
u
a
tions
(in
RF
OC m
e
thod,
the roto
r flux
vector i
s
ali
gned
with d
-
axis;
λ
dr
e
=
ǀ
λ
r
ǀ
and
λ
qr
e
=0):
dt
d
T
i
M
r
e
ds
ds
r
1
(20
)
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Vol. 12, No. 10, Octobe
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7
– 7175
7172
r
r
e
qs
ds
r
e
T
i
M
(21
)
e
qs
r
ds
r
e
i
L
M
Po
l
e
2
(22
)
In Equation (20),
T
r
is rot
o
r time co
nst
ant (
T
r
=L
r
/
R
r
).
As sho
w
n b
y
using the p
r
opo
se
d
rotational tra
n
sformation, RFO
C
equ
ations of si
n
g
le-pha
se IM cha
nge into bala
nce
d
equatio
ns.
Based
on E
q
uation
(20
)
-(2
2
), Figu
re
2
can
be
pro
p
o
s
ed fo
r IRFO
C of b
o
th he
althy and fa
u
l
ty
singl
e-p
h
a
s
e
IM. In this block di
agram, the arro
ws
illu
strate the p
a
rts that requi
re to be modifi
ed
for co
ntrollin
g
faulty single-pha
se IM
(a
s mentione
d b
e
fore by sub
s
tations
L
qs
=M
qs
=R
qs
=0 i
n
t
h
e
equatio
ns, thi
s
meth
od
can
be u
s
ed
for f
aulty singl
e-p
hase IM). In
summ
ery, Ta
ble 1
sho
w
s t
h
e
comp
ari
s
o
n
b
e
twee
n two vector
cont
rol tech
niqu
es.
Figure 2. Block
Diag
ram o
f
the Propose
d
IFOC
for Controllin
g He
althy and Fau
l
ty Single-pha
s
e
IM
Table 1. Co
m
pari
s
on b
e
tween Two Vect
or Co
ntrol Te
chni
que
s
Transform
ation
matrix fo
r stator
variables
Proposed unbala
n
ced rotational
transformation for
stator voltage an
d current variabl
es based on
equation (12
)
:
PI Controllers
Health
y
single-
phase IM
1
0
0
1
s
T
e
ds
qs
e
e
ds
qs
e
e
vs
e
is
M
M
M
M
T
T
cos
sin
sin
cos
---
---
--
---
--
---
--
--
Fault
y
single-
phase IM
0
1
s
T
0
cos
sin
sin
cos
qs
e
ds
qs
e
e
ds
qs
e
e
vs
e
is
M
M
M
M
M
T
T
Regulation of PI
controller coefficients
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TELKOM
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ISSN:
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Indire
ct Flux-Oriente
d
Co
n
t
rol of Faulty
Single-Ph
ase
Induction Mo
tors (M. Jan
n
a
ti)
7173
4. Simulation Resul
t
s
To verify the effectiveness of
the prop
ose
d
drive
system for bo
th healthy an
d faulty
singl
e-p
h
a
s
e
IM, a vecto
r
control d
r
ive sy
stem
i
s
simulate
d u
s
i
ng MATLAB
software.
T
he
controlle
r, which
wa
s used for the speed
control
of the balanced a
nd unb
a
lan
c
ed mot
o
r, is
based o
n
Fi
g
u
re
2. Rung
e
–
Kutta algo
rithm is
u
s
ed f
o
r
solving
th
e healthy
an
d faulty sin
g
l
e
-
pha
se dyn
a
m
ic e
quatio
n
s
. A si
ngle
-
p
hase IM i
s
fed fro
m
a S
i
ne Pul
s
e
Wi
dth Mod
u
lati
on
(SPWM
)
, 2-l
e
g Voltage
So
urce Inve
rter
(VSI) a
s
us
ed in [4]. In
s
i
mulation
s
,
th
e
re
fe
r
e
nc
e sp
ee
d
is 500
rpm.
Figure 3
sh
o
w
s the
simul
a
tion re
sult
s of
the
c
onventi
onal ve
ctor controlle
r. At time t=2
s
,
a pha
se
cut
out fault occu
rs
and the
si
ngle-pha
se
I
M
become
s
unbal
an
ced. Simulation sh
ows
that the co
n
v
entional ve
ctor co
ntroll
er can
not pro
perly cont
rol
the unb
a
lan
c
ed m
o
tor
(see
Figure 3
(
b
)).
A co
nsi
derab
le o
scill
ation
is al
so
seen
in the
ele
c
tro
m
agneti
c
to
rque
after fa
u
l
t
con
d
ition (a
b
out 0.8N.m around t
he ave
r
age a
m
ou
nt of 0N.m). In
Figure 4, the same
pro
c
e
s
s is
repe
ated
but
this tim
e
af
ter the
fault
occurre
n
ce t
h
e p
r
op
osed
modification
s in
the ve
ctor
controlle
r are appli
ed. S
i
mulation
re
sults of
the
Figure 4
sh
ow that the
pro
p
o
s
ed v
e
ctor
controller reduces
th
e
torque
oscillation consi
d
erably
(this ti
me the torque osci
llation is about
0.4N.m aroun
d the averag
e
amount of 0
N
.m).
(a) Stator
cu
rrent
(b) Spe
e
d
(c
) Tor
que
Figure 3. Simulation Results of the Conv
entional IFO
Controlle
r
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TELKOM
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KA
Vol. 12, No. 10, Octobe
r 2014: 716
7
– 7175
7174
(a) Stator
cu
rre
nt
(b) Spe
e
d
(
c
)
T
or
q
ue
Figure 4. Simulation Results of the Prop
ose
d
IFO Co
ntrolle
r
5. Conclusio
n
In this pa
pe
r, a mod
e
ling
method
and
a ne
w
sch
e
m
e
for ve
ctor
control
of both
singl
e-
pha
se IM a
n
d
sin
g
le-pha
se IM und
er
o
pen p
h
a
s
e fa
ult (faulty sin
g
le-p
ha
se IM
) ba
se
d on
u
s
ing
transfo
rmatio
n matri
c
e
s
h
a
s b
een
pre
s
ented. The
p
e
rform
a
n
c
e o
f
the pre
s
ent
ed IFO
C
sch
e
me
is highly satisfactory for contro
lling faul
ty single-phase IM espe
ci
ally in decre
asing the speed
and torque
ri
pple
s
. In this
way, this tech
nique
see
m
s
to be suita
b
le
method for
critical in
du
stri
a
l
appli
c
ation
s
whe
r
e we ne
ed a fault-tole
rant co
ntrol
system.
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ces
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r, R
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y
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ork
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TELKOM
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ISSN:
2302-4
046
Indire
ct Flux-Oriente
d
Co
n
t
rol of Faulty
Single-Ph
ase
Induction Mo
tors (M. Jan
n
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