TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5110 ~ 51
2
0
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.592
3
5110
Re
cei
v
ed Fe
brua
ry 15, 20
14; Re
vised
Ma
rch 16, 20
14; Accepted
April 2, 2014
An Exact Model for Rotor Field-Oriented Control of
Single-Phase Induction Motors
M. Jannati*,
A. Monadi, S. A. Anbaran
,
N. R. N. Idri
s, M. J. A. Az
iz
Univers
i
ti T
e
knolo
g
i Mal
a
ysia,
UT
M-PROTON Future Driv
e
Lab
orator
y, F
a
cult
y
of Electric
al Eng
i
ne
eri
ng,
Univers
i
ti T
e
knolo
g
i Mal
a
y
s
ia,
8131
0 Skud
ai,
Johor Bahr
u, MALAYSIA
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
Ja
nnatim9
4
@
yaho
o.co
m
A
b
st
r
a
ct
T
h
is w
o
rk presents a new
Rotor F
i
eld-Or
iente
d
Contro
l
(RF
O
C) tech
niq
ue for sing
le-p
has
e
Inductio
n
M
o
to
rs (IMs). T
he p
r
opos
ed
metho
d
us
es tw
o
rot
a
tion
al tra
n
sfor
mati
ons, w
h
ic
h
extract fro
m
t
h
e
steady-state
equ
ival
ent c
i
r
c
uit of s
i
ng
le
-phas
e IM
. It is
prove
d
by us
ing
pr
o
pose
d
rot
a
tio
n
a
l
transformatio
n
s
, the sing
le-p
hase IM asy
m
metric
al e
q
u
a
ti
ons ch
ang
e i
n
to symmetrical
equ
atio
ns. In th
e
prop
osed tec
h
niq
ue, the ass
u
mptio
n
of (M
q
/M
d
)
2
=L
qs
/L
ds
=a
2
w
h
ich is usually us
ed in ot
her F
O
C of single-
phas
e IMs, i
s
not consid
ered.
Perfor
mance of the
propos
ed techn
i
qu
e is assesse
d usi
n
g
MAT
L
AB/SIMULINK. Extensiv
e si
mu
lati
on r
e
sults sh
ow
the
perfor
m
a
n
ce
a
nd c
o
rrectness
of the
pro
pos
e
d
meth
od.
Ke
y
w
ords
: ne
w
RF
OC
techni
que, rotatio
n
a
l
transformatio
n
s
, single-
phas
e
inducti
on
moto
r
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
Single-pha
se
Induction
Motors (IMs) hav
e been
broadly em
ployed in lo
w po
wer
appli
c
ation
s
. In the
s
e a
ppli
c
ation
s
th
e m
o
tor o
p
e
r
ate
at low
efficien
cy, fixed spee
d an
d con
s
u
m
e
about 10% o
f
electrical e
nergy. Du
rin
g
the la
st de
cad
e
s, si
ngle
-
pha
se IM drives have be
en
thoroughly di
scussed by
researchers. Single-phase
IMs with t
w
o main and
auxiliary wi
ndings
can
be
con
s
i
dere
d
a
s
two
-
pha
se
IMs,
since th
ese st
ator
windin
g
s are
displa
ce
d 90
ο
fr
om ea
c
h
other. A Four-Switch Invert
er (FSI) topol
ogy is us
ed i
n
this paper
as illust
rated in Figure 1, has
been p
r
op
ose
d
as a lo
w co
st solutio
n
for single
-
ph
ase
IM drives [1, 2].
Variabl
e spe
ed d
r
ives
ca
n provide reli
able
dyn
a
mic system
s a
n
d
importa
nt sa
vings in
energy cu
sto
m
and
co
sts of the elect
r
ical
ma
chin
es [1-11], [13-16]. Recen
t
ly, some hi
gh
perfo
rman
ce singl
e-p
h
a
s
e IM
drive
syst
ems we
re
p
r
opo
sed. In th
ese
schem
es, rotor an
d st
ator
Field-Orie
nte
d
Control (F
OC) prin
ciple
s
are a
dapt
e
d
to the singl
e-ph
ase IM model [1-3], [6-10].
Windi
ng
asy
mmetry d
ue t
o
the
differe
n
t
indu
ctan
ce
s and
resi
stan
ce
s of th
e m
a
in a
nd
auxili
ary
stator win
d
in
gs i
n
singl
e-pha
se IM
ca
use
s
extra
couplin
g bet
ween
win
d
ing
s
. This a
s
ymmetry
effects o
n
th
e motor o
p
e
r
ation and
produ
ce
s torq
u
e
and
cu
rre
n
t
pulsatio
n
s [
11]. By using
a
trans
formation matrix, the c
o
mpens
a
tion of t
he si
ngl
e-ph
ase IM a
s
ymmetry
wa
s p
r
esented i
n
[2]. In the pre
v
ious meth
od
s for ve
ctor
control
of
singl
e-ph
ase o
r
u
nbala
n
ced two-ph
ase IMs,
the
a
s
s
u
mp
tion
of (
M
q
/M
d
)
2
= L
qs
/L
ds
is normally applied [2-3], [6-10]. The differen
c
e
s
in the d and q
stator
with consi
deri
ng of
(M
q
/M
d
)
2
=
L
qs
/L
ds
and without con
s
id
ering
of(M
q
/M
d
)
2
= L
qs
/L
ds
, are
reflecte
d in th
e spe
ed a
nd
hen
ce the to
rque respon
se
s of the drive
system. In thi
s
pap
er, a n
e
w
RFO
C
st
rate
gy for singl
e-pha
se IM driv
e is in
vestig
ated. In the pro
posed sch
e
m
e
, the backward
comp
one
nt i
n
the
stator
voltages
du
e
to this a
s
sumption
((M
q
/M
d
)
2
=
L
qs
/L
ds
)as previou
s
ly
ignored in [2
-3], [6-10] will
be taken int
o
acco
un
t. The re
maind
e
r of this work
is organi
ze
d
as
follows. The
modelin
g of the sin
g
le
-pha
se IM is
p
r
e
s
ented in
se
ction 2. In se
cti
on 3, the mai
n
idea of pro
p
o
s
ed ve
ctor co
ntrol for sin
g
l
e
-ph
a
se
IM discusse
d and
sub
s
equ
entl
y
a new cont
rol
strategy b
a
sed on
RFO
C
is pr
esented
. The effectiv
ene
ss
of the
prop
osed m
e
thod i
s
verif
i
ed
usin
g MATL
AB/SIMULINK and p
r
e
s
e
n
ted in
se
cti
on 4. Fin
a
lly, con
c
lu
sion
is p
r
e
s
ente
d
i
n
se
ction 5.
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TELKOM
NIKA
ISSN:
2302-4
046
An Exact Mo
del for Roto
r
Field-Orie
nte
d
Contro
l of Single-P
h
a
s
e Indu
ction Mot
o
rs
(M. Ja
nna
ti)
5111
2. Modeling of the Single
-
Phase IM
The ba
sic d
r
i
v
e system wh
ich is
studie
d
in this pap
er i
s
sh
own in Figure 1.
Figure 1. Single-p
h
a
s
e IM Drive System
Negl
ectin
g
the co
re
satura
tion, the electric
al a
nd me
cha
n
ical equ
a
tions
of the single
-
pha
se IM in
the stato
r
referen
c
e
fra
m
e (sup
erscript “
s
”) a
r
e
given a
s
foll
ows (All of t
h
e
para
m
eters in
Equation (1
) have bee
n de
fined in [2]):
(
1
)
3. Equations
of Proposed
RFO
C
for Single-Pha
se IM
Steady-state
equivalent
circuit of the si
n
g
le-p
ha
se IM
can b
e
sho
w
n as Fi
gure 2
[12]. In
this
Figure,
V
m
,
V
a
,
I
m
and
I
a
are the m
a
in and auxiliary
voltages
and
currents,
"
a
" is
the turn
ratio (
α
=
N
a
/
N
m
) and "
j
" is
the sq
uare root of "-1".
E
fm
,
E
fa
,
E
bm
and
E
ba
are
the forward
and
backward vol
t
age of ma
gn
etizing
bra
n
ch
of the mai
n
and a
u
xiliary
windi
ng
s.
R
f
,
R
b
,
X
f
and
X
b
are th
e fo
rwa
r
d a
nd
ba
ckward
stator resi
st
an
ce
an
d ind
u
cta
n
ce
in mai
n
win
d
ing.
R
lm
,
R
la
,
X
lm
and
X
la
are the lea
k
a
ge resi
stan
ce a
n
d
indu
ctan
ce
of the main
and a
u
xiliary windin
g
. With
followin
g
cha
nge of variabl
es:
(
2
)
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TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5110 – 51
20
5112
Figure 2. Steady-state Eq
uivalent
Circu
i
t of the Single-ph
ase IM
Figure 3 can
be sim
p
lified
as two
bala
n
c
ed fo
rwar
d
a
n
d ba
ckwa
rd
circuit a
s
follo
ws [13,
14]:
Figure 3. Simplified Equiva
lent Circuit of Single-pha
s
e
IM
Whe
r
e:
(
3
)
As you can
see, the equiv
a
lent
ci
rcuit o
f
single
-
ph
ase IM splits to
two ci
rcuits,
each of
them indicates a bala
n
ced motor wh
ich rotate
s
i
n
the forwa
r
d and ba
ckward di
re
ction.
By
con
s
id
erin
g F
i
gure 3 a
nd E
quation
s
(3
)
we have:
(
4
)
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TELKOM
NIKA
ISSN:
2302-4
046
An Exact Mo
del for Roto
r
Field-Orie
nte
d
Contro
l of Single-P
h
a
s
e Indu
ction Mot
o
rs
(M. Ja
nna
ti)
5113
Equation
(4
) i
s
the t
r
an
sformation mat
r
ixes fo
r
chan
gi
ng the va
riabl
es from u
n
ba
lanced
form to
bala
n
c
ed
form
(e.g
., Figure
2
to
Figure 3
)
.
With sub
s
tation
of Equatio
ns
(5) in
equ
a
tio
n
s
(4), we ca
n o
b
tain Equatio
n (6) a
nd (7
).
(
5
)
Rotation
al tra
n
sformation f
o
r voltage variable
s
:
(
6
)
Rotation
al tra
n
sformation f
o
r cu
rrent variable
s
:
(
7
)
In this eq
uati
on,
θ
e i
s
the
angle
betwe
en the
statio
nary refere
nce frame
and
rotating
referen
c
e fra
m
e. It is exp
e
cted
by usi
ng (6
) a
nd (7), unb
alan
ced si
ngle
-ph
a
se IM
equa
t
ion
s
cha
nge into
balan
ce
d eq
uation
s
form.
By applying
of these rotational tra
n
sf
ormatio
n
s to
the
equatio
ns of
singl
e-p
h
a
s
e
IM and with
o
u
t the assum
p
tion of(
M
q
/
M
d
)
2
=
L
qs
/
L
ds
, we have (in t
h
e
previou
s
p
r
e
s
ented meth
o
d
s for F
O
C of
single
-
ph
ase
IM the assu
mption of (
M
q
/
M
d
)
2
=
L
qs
/
L
ds
is
c
o
ns
idered [2-3], [6-10]):
Rotor voltag
e
equation
s
:
(
8
)
After simplifying the equ
ations of
rotor voltage
s ca
n b
e
written a
s
:
Rotor voltag
e
equation:
(
9
)
Rotor flux eq
uation
s
:
(
1
0
)
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ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5110 – 51
20
5114
After simplifying the equ
ations of ro
tor flux can be
wri
tten as follows:
Rotor flux eq
uation:
(
1
1
)
Electrom
agn
etic torqu
e
eq
uation:
(
1
2
)
Therefore, we have:
(
1
3
)
Stator voltage equatio
ns:
(
1
4
)
Equation
s
of stator voltage
can be
simpli
fied as follo
wi
ng equ
ation:
(15
)
Gene
rally, E
quation
(1
5)
are
in
clude
d
two te
rms;
fo
rwa
r
d
term
s
(su
p
e
r
scri
pt “
e
”) an
d
backward terms (su
p
e
r
scri
pt “-
e
”). It is
becaus
e
of unequal ma
in and
auxiliary res
i
s
t
anc
e
s
and
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Exact Mo
del for Roto
r
Field-Orie
nte
d
Contro
l of Single-P
h
a
s
e Indu
ction Mot
o
rs
(M. Ja
nna
ti)
5115
indu
ctan
ce
s in the si
ngle
-
p
hase IM equ
ations.
Equ
a
tion bet
ween f
o
rward and
b
a
ckward te
rm
is
as
follows
:
(
1
6
)
As expe
cted,
usin
g propo
sed tr
an
sform
a
tion matrixe
s
(Eq
u
ation
s
(6) a
nd
(7)) e
quation
s
of the rotor voltage, flux, torqu
e
and
stator vo
ltage
are obtai
ned
like bal
an
ced
motor. In RF
OC
method, the
rotor flux vector i
s
alig
ned with d-a
x
is; (
λ
dr
e
=
׀
λ
r
׀
,
λ
qr
e
= 0), based on
this
assumptio
n
, Equation (15) can be
cla
ssi
fied as follo
ws:
(
1
7
)
Whe
r
e:
(
1
8
)
In Equation
(18),
v
ds
d-f
,
v
ds
d-
b
,
v
qs
d-f
and
v
qs
d-b
can
be g
enerated
by De
cou
p
ling Circuit
a
n
d
v
ds
ref-f
,
v
ds
ref-b
,
v
qs
ref-f
and
v
qs
re
f
-
b
can be ge
nerate
d
by current PI cont
rolle
r as it is
sho
w
n in Fi
g
u
re
4.
Acco
rdi
ng to (9), (1
1) an
d (13), the equa
tions of RFO
C
can b
e
formulated an
d sho
w
n a
s
Equation (19) and Figu
re 5
resp
ectively (In Equation (19),
T
r
is
rotor time c
o
ns
tant).
(19
)
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ISSN: 23
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046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5110 – 51
20
5116
Figure 4. Vector
Control o
f
Single-
ph
ase IM Acco
rdin
g to Equation
(18)
Figure 5. Vector Cont
rol of Single-pha
se
IM Acco
rding
to Equation (19)
4. Results a
nd Comparis
ons
4.1. Perform
ance Ev
aluation
The p
r
opo
se
d co
ntrolle
r b
a
se
d on Fi
gure 4 and
Figu
re 5 is a
pplie
d
to a comm
ercial 0.25
hp sin
g
le-ph
a
se IM with
the nominal
values an
d p
a
ram
e
ters as in Table 1. The develo
p
e
d
scheme
pe
rfo
r
man
c
e i
s
the
n
sim
u
lated
with different
v
a
lue
s
of rotor spe
ed. Exten
s
ive si
mulatio
n
results a
r
e prese
n
ted to ev
aluate the pro
posed si
ngle
-
pha
se
IM driv
e perfo
r
man
c
e.
Figure 6, illustrates the ref
e
re
nce and the real rotor
speed
signal
s towards t
w
o different
steady-state
rotor speed
values
(the
referen
c
e
re
al
sp
eed va
rie
s
fro
m
zero to the
rated
a
nd
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Exact Mo
del for Roto
r
Field-Orie
nte
d
Contro
l of Single-P
h
a
s
e Indu
ction Mot
o
rs
(M. Ja
nna
ti)
5117
rated to ze
ro
value. A load torque eq
ual
to 1 N.m is introdu
ce
d at t = 9 s and re
moved at t = 11
s). It is seen
that the real rotor sp
eed
si
gnal
s are
so
accurate that hardly ca
n b
e
distingui
sh
ed
from the
corresp
ondi
ng
ref
e
ren
c
e
spee
d
sig
nal
s eve
n
after
applyin
g
loa
d
to
rqu
e
(the
o
scill
ation
of spe
ed i
s
about
0.2
rpm in
ste
a
d
y-state
and
after ap
plying loa
d
torq
ue). T
he m
o
tor
electroma
gne
tic torq
ue i
s
also
sh
own i
n
Figu
re
6(b
)
. It can b
e
seen that th
e
electroma
gne
tic
torque h
a
s a
quick re
sp
on
se with n
o
pul
sation
s.
(a)
(b)
Figure 6. Simulation Results of RFO
C
at Zero a
nd No
minal Co
mm
and Spee
d; (a) Spee
d, (b)
Torq
ue
(a)
(b)
(c
)
Figure 7. Simulation Results of RFO
C
fo
r a Tr
a
p
e
z
oid
a
l Comm
and
Speed; (a
) Stator cu
rrents,
(b) Spe
ed, (
c
)
Torqu
e
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ISSN: 23
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046
TELKOM
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KA
Vol. 12, No. 7, July 201
4: 5110 – 51
20
5118
Figure
7
sh
ows
si
mulation
re
su
lts
of
the
com
m
and
and
a
c
tual
roto
r
sp
e
ed
a
c
cording
to
prop
osed me
thod for a tra
pezoidal
com
m
and spee
d fr
om 500
rpm
to -500
rpm. It is evident fro
m
Figure 7(b) t
hat the re
al
spe
ed follo
ws the
comm
and
spee
d. The auxilia
ry and mai
n
st
ator
curre
n
ts
and
electroma
gne
tic torq
ue fo
r
trape
zoid
al
comman
d
spe
ed a
r
e
sho
w
n in Fi
gure 7
(
a)
and Fig
u
re
7
(
c) re
sp
ectiv
e
ly. In this case, a
s
can
be seen in
F
i
gure
7(b
)
, b
y
using p
r
o
p
o
sed
controlle
r, the spee
d oscilla
tion at steady
-state
i
s
~ 0.0
7
rpm at rotor
spe
e
d of 500
rpm.
Figure 8
an
d
Figu
re
9
sh
ows the
go
o
d
pe
rfor
m
a
n
c
e of the
prop
ose
d
d
r
ive
system fo
r
controlling
si
ngle-pha
se I
M
in the dif
f
eren
ce
valu
es of
spee
d
(±5
00rpm,
±100
0rpm a
n
d
±170
0rpm) a
nd at
very lo
w
spe
ed
ope
ration
re
spe
c
ti
vely. It can
b
e
seen
from
Figure 6
-
9
th
at the
dynamic pe
rforma
nce of t
he p
r
opo
se
d
driv
e
sy
st
e
m
f
o
r v
e
ct
or
c
ont
rol of
si
ngle-pha
se I
M
is
extremely accepta
b
le.
(a)
(b)
Figure 8. Simulation Results of RFO
C
in
the Differen
c
e Values of Comman
d
Spe
ed; (a) Spe
e
d
,
(b) T
o
rq
ue
(a)
(b)
Figure 9. Simulation Results of RFO
C
at Low
Spe
ed O
peratio
n; (a)
Speed, (b
) Speed Erro
r
4.1. Compari
s
ons
Based o
n
Equation (15),
the difference in
the d and q stator voltages bet
wee
n
the
con
d
ition
s
in
whi
c
h th
e
sup
positio
n of
(
M
q
/
M
d
)
2
=
L
qs
/
L
ds
is
c
o
ns
idered (e.g.,
[2-3], [6-10],
[14-16])
and othe
rwi
s
e is as follo
wi
ng equ
ation:
(
2
0
)
An evaluatio
n between t
he ste
ady-st
a
te ro
to
r sp
eed respon
se of the RF
OC
with
considering of (
M
q
/
M
d
)
2
=
L
qs
/
L
ds
and
with no
con
s
ide
r
ing of
(
M
q
/
M
d
)
2
=
L
qs
/
L
ds
is
demon
strated
in
Figure 10. S
m
all magnitude of
osc
illati
ons at the rated
reference
rotor speed
can be observ
ed in
the speed
re
spo
n
ses
wh
e
n
the
sup
p
o
s
ition (
M
q
/
M
d
)
2
=
L
qs
/
L
ds
is employed.
A
s
can be se
e
n
in
Figure 1
0
, by
usi
ng
co
nve
n
tional
co
ntro
ller
(supp
ositi
o
n (
M
q
/
M
d
)
2
=
L
qs
/
L
ds
),
the speed oscillat
i
on
at steady
-sta
te is
~ 0.2rp
m
at roto
r
speed
of
180
0rpm
but by
usin
g p
r
opo
sed controller the
spe
ed o
scillat
i
on red
u
ced ~ 0.08rpm at rotor sp
eed of
1800
rpm.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
An Exact Mo
del for Roto
r
Field-Orie
nte
d
Contro
l of Single-P
h
a
s
e Indu
ction Mot
o
rs
(M. Ja
nna
ti)
5119
It is con
c
lud
e
d
in
com
p
a
r
i
s
on
with
the
previou
s
pro
posed
schem
es fo
r F
O
C o
f
singl
e-
phas
e
or t
w
o-phas
e
IMs
(e.g., [2-3
], [6-10], [14
-
16]),the pro
p
o
s
e
d
controll
er i
n
this
re
se
arch
prod
uces fe
wer rip
p
le
s in the torqu
e
an
d spe
ed.
(a)
(b)
Figure 10. Simulation Results of Comp
arison bet
we
en Speed
Re
spo
n
se in Single-p
ha
se IM (a)
not assumin
g
(
M
q
/
M
d
)
2
=
L
qs
/
L
d
s
(b) assu
ming (
M
q
/
M
d
)
2
=
L
qs
/
L
d
s
5. Conclusio
n
An accurate t
e
ch
niqu
e for spe
ed co
ntrol
of single-p
h
a
s
e IMs ba
se
d
on RFO
C
ha
s been
pre
s
ente
d
. The pro
p
o
s
ed
method employs rotati
onal tran
sformations that
transfo
rm the
unbal
an
ced single-pha
se I
M
equation
s
i
n
to equation
s
of RFOC tha
t
has the sa
me stru
cture as
the balan
ce
d
motor. Unli
ke other RFO
C
met
hod im
plemente
d
for the singl
e-pha
s
e IMs, the
proposed technique
does
not utilize th
e supposition (
M
q
/
M
d
)
2
=
L
qs
/
L
ds
. Simulation re
sults
prov
ed
the techni
que
validity.
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ea,
C
B
Jaco
bin
a
, E
RCD S
ilva,
AM
N Lim
a
. Vect
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g
i
es for
sin
g
l
e
-p
hase
in
ducti
on
motor drive s
y
s
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ems.
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T
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80.
[2]
MBR Correa,
CB Jacob
i
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AMN Lima, ER
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l
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r
ient
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ntrol of a
Singl
e Phase
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n
Moto
r Drive.
IEEE Transacti
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Industria
l Elect
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M
Jemli,
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zza, M Boussak, M Gossa. S
ensor
le
ss I
ndir
e
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i
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O
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ie
ntatio
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pee
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ontro
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h
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i
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hi
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Spee
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E
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O
M
NIKA Indo
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our
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l
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he Prelim
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arch fo
r Impleme
n
tati
on of Improv
ed DT
C Sch
e
me of H
i
gh
Performanc
e P
M
SM Drives.
T
E
LKO
M
NIKA Indo
nesi
an J
o
u
r
nal of El
ectric
al En
gin
eeri
n
g
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[6]
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a
lla
h.
A New
Method for Sp
eed Se
nsorl
e
s
s
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a
se Inducti
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ilter
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ICEE. 2011.
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a
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p
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a
se Inducti
on
Motor Drives
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sig
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a
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ss torqu
e
o
p
t
i
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i
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e p
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orless s
pee
d
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M
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w
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as
e
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Moto
r Drive Usin
g ISF
O
C Strategy.
IJST
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ansactions of Ele
c
trical Eng
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B
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C da
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ource In
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w
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o
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Evaluation Warning : The document was created with Spire.PDF for Python.