Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
Vol.
4, No. 4, Decem
ber
2014, pp. 430~
438
I
S
SN
: 208
8-8
6
9
4
4
30
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
Speed Sensorless Direct Rotor Fi
eld-Oriented Control of Single-
Phase In
duction Mot
o
r Usin
g Extended Kalman Filter
Mohammad
Jann
ati,
Se
yed
Hesam
As
gari
, Nik
Ru
mzi Nik Idris,
Mo
hd Juna
idi Abdul Aziz
UTM-PROTON
Future Drive Laborator
y
,
Facu
lty
of Electri
cal Engineer
ing, Univ
ersiti
Tekno
logi
Malay
s
ia, Johor
Bahru, MALAYSIA
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Apr 12, 2014
Rev
i
sed
Jun
4
,
2
014
Accepte
d J
u
l
2, 2014
Nowaday
s
, Field-Oriented C
ontr
o
l (FOC) strateg
i
es broadly
used
as a v
ecto
r
based con
t
roller
for Single-Phase Induc
tion
Motors (SPIMs). This paper
is
focused on Dir
e
ct Rotor
FOC (DRFOC) of SPIM. In
the proposed
technique,
transform
a
tion
m
a
trices ar
e a
pplied in ord
e
r
to control th
e
m
o
tor by
converting the u
nbalan
ced SPIM equations
to the balan
ced
equations (in this
paper th
e SPIM with two differ
e
nt stator
wind
ings is considered). Besides
this contro
l
technique, a meth
od for speed
es
timation o
f
SPIM based on
Extend
ed Kalm
a
n
Filter (
E
KF) to achi
e
ve th
e hi
gher perform
anc
e
of SPIM
drive s
y
s
t
em is
presented
.
Simulation r
e
sults
are
provided to
dem
onstrate the
high performan
ce of
the presen
ted techniques
.
Keyword:
DRFOC
EKF
Spee
d se
ns
orl
e
ss
SPIM
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
:
Sey
e
d Hesam
Asga
ri,
Facu
lty of Electri
cal Engineering,
Un
i
v
ersiti Tekn
o
l
o
g
i
Malaysia,
U
T
M Sku
d
a
i,
8
131
0 Joho
r,
Malaysia.
Em
a
il: seyed
h
e
sam
a
sg
ari@gmail.co
m
1.
INTRODUCTION
Single
-
Phase Induction Motors (SPIMs) a
r
e
use
d
in
bo
th
do
m
e
stic
an
d
ind
u
s
t
r
ial p
u
rpo
s
es. Th
ey can
be em
pl
oy
ed
i
n
ai
r c
o
ndi
t
i
one
rs,
fa
ns,
r
e
fri
ge
rat
o
rs,
c
o
m
p
resso
rs,
d
r
y
e
rs,
was
h
i
n
g
m
achi
n
es an
d
ot
he
r
applications.
Gene
rally, the
stator
of SPIMs
has
two windings whic
h
ar
e
orthogonal in
s
p
ace. They
are
d
i
fferen
t
in
term
s o
f
re
sistan
ces and
im
p
e
d
a
n
c
es.
Sin
c
e th
e m
a
in
an
d aux
iliary stato
r
wind
ing
s
are
u
n
b
a
lan
c
ed
, therefo
r
e SPIM will b
e
en
coun
tered
th
e
to
rq
u
e
p
u
l
sation
s
[1
]. Co
n
s
equen
tly, stu
d
y
ing ab
ou
t
SPIM
s
has
bee
n
i
n
c
r
ease
d
dra
m
at
i
call
y
. It
ha
s bee
n
recom
m
e
nde
d
by
t
h
e
r
e
searche
r
s,
va
r
i
ous
st
u
d
i
e
s,
w
h
i
c
h
foc
u
se
d o
n
im
pr
o
v
i
n
g t
h
e pe
rf
orm
a
n
ce and efficiency of t
h
e SPIMs
. The
s
e researc
h
es,
prese
n
ted
desi
gn a
n
d
opt
i
m
i
zati
on,
st
udy
o
n
t
h
e
p
o
we
r
fact
o
r
, rese
arc
h
on
i
m
prove
d m
odel
i
n
g
an
d a
n
al
y
s
i
s
, p
r
og
r
e
ss
o
n
i
m
p
r
ov
em
en
t o
f
to
rqu
e
p
e
rfo
r
m
an
ce and
resear
ch
in
g on
in
f
l
u
e
n
ce
o
f
har
m
o
n
i
c [
2
]-[4].
V
a
r
i
ab
le
Fr
eq
u
e
n
c
y
C
ont
r
o
l
(
V
FC
)
t
echni
q
u
es
wh
i
c
h are
used i
n
t
h
e Va
ri
abl
e
Fr
eque
ncy
D
r
i
v
e
s
(V
FDs
)
ha
ve
adva
nt
age
s
i
n
t
e
rm
s
of sa
vi
n
g
of e
n
er
gy
an
d
hi
g
h
pe
rf
o
r
m
a
nce appl
i
cat
i
o
ns o
f
IM
s [
5
]
-
[
7]
.
VFC
m
e
t
hods
are cat
eg
ori
z
e
d
i
n
t
o
scalar
and vec
t
or base
d
cont
rol. Th
e
scalar m
e
thods
will not
be a
b
le to
fulfill the
requirem
e
nt of dy
nam
i
c
dri
v
es an
d has
a sl
ow react
i
on t
o
t
r
a
n
si
ent
but
, vect
or c
ont
rol
i
s
an excel
l
e
nt
cont
r
o
l
m
e
t
hod t
o
han
d
l
e
t
r
ansi
ent
a
nd s
a
t
i
s
fy
t
h
e req
u
i
rem
e
nt
of dy
nam
i
c dri
v
es.
Gen
e
rally v
ecto
r
co
n
t
ro
l is classified
in
to
Direct
Tor
q
ue C
o
nt
r
o
l
(DTC
) a
nd
Fi
el
d-
Ori
e
nt
e
d
C
ont
r
o
l
(F
O
C
). F
O
C
m
e
t
hod i
s
pr
op
ose
d
b
r
oa
dl
y
as a vect
o
r
base
d co
nt
r
o
l
l
e
r fo
r IM
s a
n
d i
s
cl
assi
fi
ed
i
n
t
o
St
at
or F
O
C
(SF
O
C
)
a
nd R
o
t
o
r
FO
C
(R
FOC
)
. A
not
her
cl
assi
fi
cat
i
on
o
f
t
h
i
s
m
e
t
hod
i
s
al
so
pe
rf
orm
e
d
base
d
on
t
h
e cal
cul
a
t
i
o
n
o
f
r
o
t
o
r
fl
u
x
po
si
t
i
on
whi
c
h i
n
cl
ude
s
Direct FOC
(DFOC) and
Indi
r
ect FO
C (I
FOC)
[8
].
Fro
m
a rev
i
ew o
f
literatu
re, t
h
ere are m
a
n
y
p
a
p
e
rs wh
ich
h
a
v
e
b
e
en
sugg
ested
for v
ect
o
r
co
n
t
ro
l of
SPIM
s
base
d on
FOC
.
In
2
0
0
0
, C
o
r
r
ea et
al
. i
nvest
i
g
at
ed IR
F
O
C
t
echni
que
f
o
r S
P
I
M
.
I
n
t
h
e
pr
op
os
e
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Spee
d
Se
ns
orl
e
ss Di
rect
R
o
t
o
r
Fi
el
d-
Ori
e
nt
e
d
C
ont
r
o
l
of
Si
ngl
e
-
Ph
ase
I
n
d
u
ct
i
o
n
...
(
M
o
h
a
m
m
a
d
Ja
n
nat
i)
43
1
technique, t
h
e
y
eliminated the asymm
e
try ter
m
s
of
t
h
e SPIM
equat
i
o
ns
by
sel
ect
i
ng ap
pr
o
p
ri
at
e
t
r
ans
f
o
r
m
a
ti
on
f
o
r
t
h
e
st
at
or
v
a
ri
abl
e
s
[9]
.
It
was s
u
gge
st
ed
i
n
[
9
]
,
t
o
use
h
y
s
t
e
resi
s cu
rre
nt
c
ont
r
o
l
l
e
r.
I
n
[1
0]
,
C
o
r
r
ea et
al
.
p
r
o
p
o
sed
vect
or
co
nt
r
o
l
o
f
SPI
M
base
d
on
I
S
FOC
.
I
n
t
h
e
pr
op
ose
d
t
e
c
hni
q
u
e,
f
o
r
re
duct
i
on
o
f
electro
m
a
g
n
e
tic to
rq
u
e
o
s
cillatio
n
s
in
SPIM, th
ey d
e
si
gn
ed
a dou
b
l
e-seq
u
e
n
ce cu
rren
t con
t
ro
ller to
con
t
ro
l
st
at
or c
u
r
r
ent
[
10]
.
I
n
[
1
1]
,
de
cou
p
l
i
n
g
vect
o
r
c
ont
r
o
l
of
SP
IM
has
be
en
p
r
op
ose
d
by
V
a
e
z
-Zade
h
a
n
d R
e
i
c
y
.
In
t
h
e
p
r
o
p
o
se
d
vect
o
r
c
ont
r
o
l
m
e
t
hod i
n
p
a
per
[
12]
,
t
h
e
m
a
xim
u
m
pot
e
n
t
i
a
l
o
p
erat
i
o
n
o
f
S
P
IM
i
s
o
b
t
ai
ned
according to maxim
u
m
torque per am
pere. In [13], [
14], t
h
e aut
h
ors propos
ed a
nd im
ple
m
ented ISFOC and
IRFOC f
o
r S
P
I
M
respectivel
y
.
In fact
they used the same variable cha
ngi
ng
base
d o
n
[9]
,
t
o
el
i
m
inat
e t
h
e
asym
m
e
t
r
i
cal
term
s i
n
SPIM
.
To ha
ve hi
gh
perf
o
r
m
a
nce vect
o
r
co
nt
r
o
l
t
echni
q
u
es s
u
ch as [
9
]
-
[
14]
,
i
t
i
s
necessa
ry
t
o
e
m
pl
oy
feedbac
k
s
p
eed c
o
nt
r
o
l
.
Fo
r t
h
i
s
pu
rp
o
s
e, an
en
coder is no
rm
ally
u
s
ed
to
p
r
ov
ide th
is
i
n
f
o
rm
at
i
on. U
s
i
ng se
ns
or ca
uses m
o
re i
n
st
rum
e
nt
at
i
on, i
n
creasi
ng c
o
st
and si
ze,
dec
r
ease r
o
bust
n
e
ss an
d
reliab
ility o
f
t
h
e driv
e system
.
Th
erefo
r
e, in
stead
of
i
m
p
l
e
m
en
tatio
n
o
f
sen
s
or, it is b
e
tter to
ap
p
l
y sp
eed
estim
a
tion techniques.
Gene
ra
lly, speed
estimatio
n
is categ
o
r
ized
in
to
two
m
a
in
p
a
rts, speed
esti
m
a
tio
n
b
a
sed
on
m
o
t
o
r m
ode
l
and
spee
d e
s
t
i
m
a
ti
on t
h
r
o
u
g
h
si
g
n
al
i
n
ject
i
o
n
[
8
]
.
It
i
s
p
r
o
pos
ed
spee
d es
t
i
m
a
t
i
on m
e
t
hods i
n
IM
s by
di
ffe
re
nt
aut
h
ors
.
In
[1
5]
, a st
udy
has bee
n
p
r
o
p
o
se
d fo
r sens
o
r
l
e
ss IR
FOC
o
f
SPIM
.
T
h
e a
ppl
i
e
d
m
e
t
hod f
o
r est
i
m
a
t
i
on of spee
d i
n
[1
5]
i
s
bas
e
d o
n
SPIM
m
odel
.
In p
a
pe
r [1
6]
, a
m
e
t
hod
for I
S
F
O
C
of
SPIM
w
ith
esti
m
a
tio
n
o
f
r
o
t
o
r
speed
b
a
sed
on
th
e m
o
to
r
cu
r
r
e
n
t
s an
d
r
e
f
e
r
e
n
ce
q
-
ax
is cur
r
e
n
t
h
a
s b
e
en
pr
oposed
. In
[1
7]
,
M
odel
R
e
fere
nce A
d
ap
t
i
v
e
Sy
st
em
(M
R
A
S)
st
rat
e
gy
has bee
n
u
s
ed f
o
r
spee
d
sens
orl
e
ss IR
F
O
C
of
SPIM. Th
e MRAS sp
eed
sen
s
orless v
ect
o
r
co
n
t
ro
l o
f
IM
s is sen
s
itiv
e t
o
v
a
riatio
n
s
of resistan
ce [18]. For
th
is, in
[19
]
,
MRAS strategy b
y
an
o
n
lin
e stato
r
resi
stance estim
a
tor and i
n
[20], a
Recursi
v
e Lea
s
t Square
(RLS) al
g
o
rithm is e
m
p
l
o
y
ed
to
calcu
late th
e SPIM p
a
ra
meters in
sen
s
o
r
less
v
ector co
n
t
ro
l of th
is
m
o
to
r.
Using
Ex
tended
Kalm
an
Filter (EKF) is an
o
t
h
e
r t
echn
i
qu
e to
estimate th
e ro
tor sp
eed
.
Si
n
c
e th
e
n
o
n
lin
earities an
d un
certain
ties of IM are
well-su
ited
to
the EKF, th
erefore it wo
u
l
d
b
e
ab
le to
estim
at
e th
e
param
e
t
e
rs sim
u
l
t
a
neo
u
sl
y
at
t
h
e sho
r
t
i
n
t
e
rv
al
of t
i
m
e
[21]
,
[22]
. M
o
reo
v
e
r
, i
n
t
h
i
s
m
e
t
hod t
h
e m
easurem
ent
and
sy
st
em
noi
ses w
h
i
c
h
are
not
n
o
rm
al
l
y
consi
d
ere
d
i
n
t
h
e previ
ously
presented technique
s for SPIM
s suc
h
as [
15]
-
[
20]
a
r
e re
gar
d
e
d
.
In
t
h
i
s
pa
pe
r, a
n
ovel
t
e
c
h
ni
q
u
e
f
o
r
DR
F
O
C
o
f
S
P
IM
(u
n
b
al
anced
t
w
o-
p
h
a
s
e IM
)
wi
t
h
est
i
m
at
i
o
n
of
m
echani
cal
spee
d
usi
n
g
EKF
i
s
di
scuss
e
d a
n
d
veri
fi
e
d
usi
n
g
M
A
T
L
A
B
/
SIM
U
LI
N
K
.
The
prese
n
t
e
d E
K
F i
n
t
h
i
s
pa
pe
r i
s
t
h
e co
nve
nt
i
onal
E
K
F
whi
c
h has
bee
n
de
vel
o
pe
d o
f
SPIM
.
B
e
si
d
e
s t
h
e
rem
oving of mechanical
speed
sensor
suc
h
as t
ach
oge
n
e
rat
o
r a
n
d enc
ode
r, t
h
e
pr
o
p
o
se
d DR
F
O
C
i
n
t
h
i
s
work, elim
in
at
es th
e pu
re in
t
e
g
r
ation
wh
ich is used in
I
F
OC. U
s
i
n
g in
tegr
atio
n op
er
ator in
t
h
e
v
ector
co
n
t
r
o
l
o
f
IM su
ffers
th
e well-kno
wn
d
i
fficu
lties of in
tegratio
n
effect esp
eciall
y
at th
e lo
w frequ
en
cies [23]. Th
e
resul
t
s
o
f
t
h
i
s
r
e
search s
h
ow t
h
at
t
h
e p
r
o
p
o
s
e
d spee
d se
ns
o
r
l
e
ss co
nt
r
o
l
fo
r SPIM
has rea
s
on
abl
y
g
o
o
d
t
o
r
q
ue
an
d sp
eed respo
n
s
e
d
y
n
a
m
i
cs
and
satisfact
o
r
y track
ing
cap
a
b
ility.
2.
SPIM MODE
L
The m
a
the
m
atical
m
odel of
squi
rrel ca
ge
SPIM ca
n
be
sho
w
n in
a sta
tionary
refe
re
n
ce fram
e
as
f
o
llow
s
[9
]:
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
(1
)
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
r
r
l
e
s
qr
s
ds
ds
s
dr
s
qs
qs
e
F
dt
d
J
Pole
i
i
M
i
i
M
Pole
2
)
(
2
(2
)
(3
)
(4
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
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-86
94
I
J
PED
S
Vo
l.
4
,
No
.
4
,
D
ecem
b
er
2
014
:
43
0 – 438
43
2
Whe
r
e,
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 a
nd
q
axes voltages
,
currents a
n
d
fl
u
x
es
of
t
h
e
st
at
or a
n
d
rot
o
r
,
R
ds
,
R
qs
,
R
r
,
L
ds
, L
qs
, L
r
, M
ds
and
M
qs
denote t
h
e d and
q axes
resistances, sel
f
a
nd
m
u
tual
inductances of the st
at
or a
nd r
o
t
o
r.
M
o
re
ove
r,
r
,
τ
e
,
τ
l
,
J
and
F
are the m
o
tor speed, electromagneti
c
t
o
r
que
, l
o
a
d
t
o
r
q
ue, i
n
e
r
t
i
a
and
vi
sc
ou
s f
r
i
ct
i
on coe
ffi
ci
en
t, resp
ectiv
ely. Based
on
Equ
a
tio
n
(1
)-(4
) it is
assu
m
e
d
th
at t
h
e m
a
in
an
d
au
x
iliary stat
o
r
wind
ing
s
h
a
v
e
d
i
fferen
t
v
a
lu
es (
R
ds
≠
R
qs
,
L
ds
≠
L
qs
and
M
ds
≠
M
qs
). To
com
p
ensat
e
t
h
e asym
m
e
t
r
y
in u
nbal
a
nce
d
IM
s i
n
[2
4]
-
[
2
9
]
,
Jan
n
at
i
et
al
. pr
op
ose
d
t
h
e use
of
un
b
a
l
a
nce
d
trans
f
orm
a
tion m
a
trices for
stator
vo
ltag
e
an
d cu
rren
t
v
a
riab
les. In
t
h
is
wo
rk
, sim
i
lar
to
[
2
5
]-[
29
],
th
ese
tran
sform
a
t
i
o
n
m
a
trices are em
p
l
o
y
ed
to
com
p
en
sate
th
e
asy
m
m
e
try b
e
tween th
e m
a
in
and
aux
iliary stator
wi
n
d
i
n
gs i
n
S
P
IM
(i
n [
2
7]
-[
29]
, t
h
e t
r
a
n
sf
orm
a
t
i
on
m
a
t
r
ices have b
een
use
d
fo
r IR
F
O
C
of t
h
ree
-
p
h
a
s
e IM
unde
r
ope
n-phase fa
ult). T
h
es
e m
a
trices are
as follows:
Tran
sf
orm
a
t
i
on m
a
t
r
i
x
fo
r st
a
t
or
v
o
l
t
a
ge
vari
abl
e
s:
s
qs
s
ds
e
e
ds
qs
e
e
ds
qs
s
qs
s
ds
e
vs
e
qs
e
ds
v
v
M
M
M
M
v
v
T
v
v
cos
sin
sin
cos
(5
)
Transfo
r
m
a
tio
n
m
a
trix
for stato
r
cu
rren
t v
a
ri
ab
les:
s
qs
s
ds
e
e
qs
ds
e
e
qs
ds
s
qs
s
ds
e
is
e
qs
e
ds
i
i
M
M
M
M
i
i
T
i
i
cos
sin
sin
cos
(6
)
Whe
r
e,
θ
e
i
s
t
h
e a
ngl
e
bet
w
e
e
n t
h
e
st
at
i
ona
ry
re
fere
n
ce fr
am
e
and
t
h
e ro
tor flu
x
-
o
rie
n
te
d
r
e
fe
renc
e
fram
e
(in this pape
r su
pe
rs
cript “
e
” in
d
i
cates th
at th
e v
a
riab
les are in
th
e ro
tatin
g referen
ce
frame and
su
perscrip
t “
s
”
indicates that
the va
riables a
r
e in the stationa
ry
re
fere
nce
fram
e
. Ii is sh
ow
n
by
usin
g
these
tran
sform
a
t
i
o
n
m
a
trices, th
e stato
r
and
ro
t
o
r v
a
riab
le
s
o
f
th
e m
a
in
an
d
au
x
iliary wi
n
d
in
g
s
are tran
sfo
r
m
e
d
i
n
t
o
eq
uat
i
o
ns t
h
at
have si
m
i
lar st
ruct
ure t
o
bal
a
nce
d
IM
e
quat
i
o
ns
. The
st
at
or an
d r
o
t
o
r v
o
l
t
a
ge eq
uat
i
ons
,
rot
o
r
fl
u
x
e
q
u
a
t
i
ons a
n
d el
e
c
t
r
om
agnet
i
c
t
o
r
q
ue e
quat
i
o
n
aft
e
r
ap
pl
y
i
ng
Eq
uat
i
o
n
(5
)
and
E
quat
i
o
n
(
6
)
are
gi
ve
n by
(
7
)
-
(
1
1)
.
St
at
or v
o
l
t
a
ge equat
i
o
ns:
e
qr
e
dr
qs
qs
e
qs
e
qs
e
qs
e
ds
qs
qs
qs
e
qs
e
qs
ds
e
qs
e
ds
i
i
dt
d
M
M
M
dt
d
M
i
i
dt
d
L
R
L
L
dt
d
L
R
v
v
e
qs
e
ds
qs
ds
ds
qs
qs
ds
ds
qs
qs
ds
ds
qs
e
qs
ds
ds
qs
e
qs
ds
ds
qs
qs
ds
ds
qs
i
i
dt
d
L
L
M
M
R
R
M
M
L
L
M
M
L
L
M
M
dt
d
L
L
M
M
R
R
M
M
)
(
)
(
)
(
)
(
)
(
)
(
2
2
2
2
2
2
2
2
2
2
2
2
(7
)
Whe
r
e,
e
qs
e
ds
e
e
e
e
e
e
e
qs
e
ds
i
i
i
i
2
2
sin
cos
sin
cos
sin
cos
(8
)
Ro
to
r vo
ltag
e
eq
u
a
tion
s
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Spee
d
Se
ns
orl
e
ss Di
rect
R
o
t
o
r
Fi
el
d-
Ori
e
nt
e
d
C
ont
r
o
l
of
Si
ngl
e
-
Ph
ase
I
n
d
u
ct
i
o
n
...
(
M
o
h
a
m
m
a
d
Ja
n
nat
i)
43
3
e
qs
e
ds
qs
qs
r
e
qs
r
e
qs
i
i
dt
d
M
M
M
dt
d
M
)
(
)
(
0
0
e
qr
e
dr
r
r
r
r
e
r
r
e
r
r
i
i
dt
d
L
R
L
L
dt
d
L
R
)
(
)
(
(9
)
R
o
t
o
r
fl
ux
eq
u
a
t
i
ons:
e
qr
e
dr
r
r
e
qs
e
ds
qs
qs
e
qr
e
dr
i
i
L
L
i
i
M
M
0
0
0
0
(1
0)
El
ect
rom
a
gnet
i
c
t
o
r
q
ue e
quat
i
on:
)
(
2
e
qr
e
ds
e
dr
e
qs
qs
e
i
i
i
i
M
Pole
(1
1)
In (
7
),
ω
e
i
s
t
h
e an
gl
e bet
w
ee
n t
h
e
st
at
i
ona
r
y
refe
renc
e
f
r
a
m
e and t
h
e
rot
o
r
flu
x
refe
re
n
ce fram
e
. As
can
be see
n
fr
om
Equat
i
o
n
(
7
)
-
(
1
1),
u
s
i
n
g
Eq
uat
i
o
n
(
5
)
a
n
d
E
quat
i
o
n
(
6
)
,
t
h
e
asy
m
met
r
i
cal
equat
i
o
n
s
of
SPIM
c
h
a
nge
d
i
n
t
o
sy
m
m
e
t
r
ical
equat
i
ons
.
Th
us, t
h
e F
O
C
p
r
i
n
ci
pl
es c
a
n
be a
ppl
i
e
d
.
3.
DRF
OC OF
S
P
IM
In t
h
i
s
st
u
d
y
,
t
h
e DR
FOC
t
e
chni
que
f
o
r
ve
ct
or c
ont
rol
of
SPIM
was
us
ed. B
a
se
d
on
(
7
)
-
(
1
1) a
n
d
after sim
p
lifyi
n
g
of eq
u
a
tions, th
e equ
a
tio
ns of th
e R
F
OC
techni
que
for
a SPIM a
r
e
obtained as
following
equat
i
o
ns
(i
n
t
h
i
s
m
e
t
hod t
h
e
rot
o
r
fl
u
x
vect
or
i
s
al
i
gne
d
w
i
t
h
d
-
axi
s
):
r
r
e
qs
qs
r
e
T
i
M
(1
2)
e
qs
r
r
qs
e
i
L
M
Pole
2
(1
3)
dt
d
T
i
M
r
e
ds
qs
r
1
e
ds
ref
ds
d
ds
e
ds
v
v
v
v
,
e
qs
ref
qs
d
qs
e
qs
v
v
v
v
(1
4)
(1
5)
Whe
r
e,
)
(
)
(
2
r
r
e
ds
qs
r
qs
r
qs
qs
e
qs
e
d
ds
T
i
M
L
M
L
M
L
i
v
r
r
qs
e
r
qs
qs
e
qs
e
d
qs
L
M
L
M
L
i
v
)
(
2
dt
di
L
M
L
i
M
M
R
M
R
v
e
ds
r
qs
qs
e
ds
ds
ds
qs
qs
ds
ref
ds
)
(
)
2
(
2
2
2
2
dt
di
L
M
L
i
M
M
R
M
R
v
e
qs
r
qs
qs
e
qs
ds
ds
qs
qs
ds
ref
qs
)
(
)
2
(
2
2
2
2
e
qs
e
ds
e
e
e
e
ds
ds
qs
qs
ds
e
qs
e
ds
i
i
M
M
R
M
R
v
v
2
cos
2
sin
2
sin
2
cos
2
2
2
2
(1
6)
(1
7)
(1
8)
(1
9)
(2
0)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l.
4
,
No
.
4
,
D
ecem
b
er
2
014
:
43
0 – 438
43
4
Whe
r
e,
T
r
=L
r
/
R
r
i
s
t
h
e rot
o
r
t
i
m
e
const
a
nt
. In (
1
4) an
d
(1
5)
,
v
ds
d
and
v
qs
d
are gener
a
t
e
d usi
n
g
Deco
u
p
l
i
ng C
i
rcui
t
an
d
v
ds
ref
and
v
qs
ref
are g
e
nerat
e
d usi
n
g
curre
nt
PI co
n
t
rol
l
e
rs i
n
t
h
e
R
F
OC
bl
oc
k d
i
agram
of t
h
e S
P
IM
(s
ee Fi
gu
re 2
)
.
M
o
re
ove
r, t
h
e
val
u
es
of
ω
r
,
λ
r
and
θ
e
in (12)-(20) are calcu
lated
u
s
ing
estimated
val
u
es
of r
o
t
o
r
d and
q axi
s
fl
uxe
s as fol
l
o
w
s
(i
n t
h
i
s
pa
per
,
t
h
e
m
o
t
o
r spe
e
d an
d r
o
t
o
r
d and
q axi
s
fl
u
x
e
s are
esti
m
a
ted
u
s
ing
EKF):
2
2
ˆ
ˆ
ˆ
qr
dr
r
qr
dr
r
ˆ
ˆ
tan
ˆ
1
(2
1)
(2
2)
4.
EKF FOR
ROTOR SPEE
D ESTIMATION IN
SPIM
In
t
h
is p
a
p
e
r, an
Ex
tend
ed
Kal
m
an
Filter is u
s
ed
to
estim
at
e th
e m
ech
an
ical sp
eed
and
ro
tor flux
es.
The st
at
e s
p
ace
m
odel
of
SP
I
M
i
s
sh
ow
n
by
Eq
uat
i
o
n (
2
3):
t
w
Bu
Ax
x
,
t
v
Cx
y
(2
3)
Whe
r
e,
A
n
,
B
n
and
C
n
are th
e
in
pu
t an
d ou
tpu
t
m
a
trix
es of
syste
m
an
d
x
,
y
and
u
a
r
e t
h
e
system
state
matrix
, syste
m
o
u
t
pu
t m
a
trix
an
d
system
in
pu
t
m
a
trix
resp
ectiv
ely. Th
e
covaria
n
ce m
a
trices of
w
(
t
) a
n
d
v
(
t
)
are defi
ne
d as
f
o
l
l
o
w
s
(
w
(
t
): s
y
ste
m
noise;
v
(
t
): m
easure
m
e
n
t noise):
cov
t
ww
E
w
Q
,
t
vv
E
v
R
cov
(2
4)
In th
is
filter, t
h
e state m
a
trix
(
x
n
) is t
h
e stat
o
r
d an
d q ax
is cur
r
e
n
t
s, ro
t
o
r d an
d q ax
is f
l
ux
es an
d
ro
t
o
r
sp
eed
,
the in
pu
t m
a
trix
(
u
n
) i
s
st
at
o
r
d
an
d
q a
x
i
s
vo
l
t
a
ges an
d t
h
e
out
put
m
a
t
r
i
x
(
y
n
)
i
s
st
at
or d and
q
axis c
u
rrents.
T
n
l
n
r
n
qs
n
ds
n
qs
n
ds
n
i
i
x
)
(
)
(
)
(
)
(
)
(
)
(
(2
5)
T
n
qs
n
ds
n
v
v
u
(2
6)
T
n
qs
n
ds
n
i
i
y
(2
7)
B
a
sed
on
d
-
q
m
odel
of
SP
I
M
(E
quat
i
o
n
(
1
)
-
(
4
)
)
a
n
d E
q
uat
i
o
n
(
2
5)
-(
2
7
)
, t
h
e m
a
t
r
i
x
es
A
n
,
B
n
and
C
n
are
obt
ai
ne
d a
s
Eq
uat
i
o
n (
2
8)
.
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
1
1
0
0
0
0
0
1
1
0
0
2
5
.
1
2
5
.
1
0
0
1
0
0
0
1
0
0
0
1
1
0
0
0
0
1
1
2
1
2
2
2
2
2
2
2
2
1
2
1
2
2
1
n
n
r
s
dr
qs
r
s
qr
ds
r
r
r
r
r
qs
r
r
r
r
r
ds
r
r
qs
r
r
qs
r
r
qs
qs
r
r
ds
r
r
ds
r
r
ds
ds
n
C
dt
k
dt
k
B
J
dt
JL
M
Pole
dt
JL
M
Pole
dt
L
R
dt
R
dt
L
R
M
dt
R
dt
L
R
dt
L
R
M
dt
L
k
R
M
dt
L
k
R
M
dt
L
R
M
R
k
dt
L
k
R
M
dt
L
k
R
M
dt
L
R
M
R
k
A
(2
8)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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S
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:
208
8-8
6
9
4
Spee
d
Se
ns
orl
e
ss Di
rect
R
o
t
o
r
Fi
el
d-
Ori
e
nt
e
d
C
ont
r
o
l
of
Si
ngl
e
-
Ph
ase
I
n
d
u
ct
i
o
n
...
(
M
o
h
a
m
m
a
d
Ja
n
nat
i)
43
5
Whe
r
e:
r
ds
ds
L
M
L
k
2
1
,
r
qs
qs
L
M
L
k
2
2
(2
9)
Usi
n
g (
2
8),
(2
9) a
nd E
K
F al
go
ri
t
h
m
(Equa
t
i
ons (
3
0)
-(
36
)
)
, t
h
e r
o
t
o
r s
p
e
e
d an
d r
o
t
o
r fl
uxe
s can
be
esti
m
a
ted
(
i
n
(3
0)
-(3
6)
,
H
is th
e m
a
trix
o
f
ou
tpu
t
pred
iction
,
P
n
is e
r
ror c
ova
riance m
a
trix and
Φ
is th
e
m
a
trix
of state predict
i
on).
E
K
F Al
gori
t
h
m
:
Pre
d
i
c
t
i
on of
S
t
at
e:
n
n
n
n
n
u
x
n
n
x
,
,
,
1
1
1
(3
0)
Whe
r
e,
n
n
n
n
n
n
n
n
n
n
n
n
u
x
B
x
x
A
u
x
n
n
,
,
,
1
1
(3
1)
Esti
m
atio
n
o
f
Erro
r Cov
a
riance Matrix
:
Q
dx
d
P
dx
d
P
n
n
n
n
x
x
T
n
n
x
x
n
n
1
(3
2)
Co
m
p
u
t
atio
n
of Kalm
an
Filte
r Gain
:
1
1
1
1
1
1
R
x
H
P
x
H
x
H
P
K
n
n
n
n
n
n
x
x
T
n
n
x
x
x
x
T
n
n
n
(3
3)
Whe
r
e,
1
1
1
,
n
n
n
n
n
n
n
x
x
C
n
x
H
(3
4)
State Estim
atio
n:
n
x
H
y
K
x
x
n
n
n
n
n
n
n
n
,
1
1
(3
5)
Upd
a
te
o
f
th
e
Erro
r Cov
a
riance Matrix
:
1
1
1
n
n
x
x
n
n
n
n
n
P
x
H
K
P
P
n
n
(3
6)
B
a
sed
on
E
qua
t
i
on
(3
0)
-(
3
6
)
,
t
h
e bl
ock
di
a
g
r
a
m
of E
K
F
can
be
sh
ow
n
as F
i
gu
re
1.
Fi
gu
re
1.
B
l
oc
k
di
ag
ram
of E
K
F
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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:
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l.
4
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.
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ecem
b
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2
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:
43
0 – 438
43
6
5.
SIMULATIONS
AN
D R
E
SU
LTS
In
th
is section, MATLAB si
m
u
latio
n
results
were o
b
tained
fo
r a SP
I
M
. The sim
u
lated SPI
M
param
e
ters are:
Vo
ltag
e
:
1
10V,
f
=60Hz
,
N
o
. o
f
p
o
l
e
s=
4,
R
ds
=7.14
,
R
qs
=2.02
,
R
r
=4.12
,
L
ds
=0
.1
885H,
L
qs
=0
.1
844
H,
L
r
=0.
1
82
6H
,
M
qs
=0
.1
772
H,
J
=0
.01
46k
g.m
2
The si
m
u
l
a
t
e
d dri
v
e sy
st
em
i
s
prese
n
t
e
d i
n
F
i
gu
re 2
.
As s
h
ow
n i
n
t
h
i
s
fi
g
u
re
, t
h
e SP
IM
was fe
d
by
two-leg
Vo
ltage So
urce Inv
e
rt
er (VSI). In
the si
m
u
latio
n
te
st, th
e
m
o
t
o
r speed
vari
ed
fr
om
zero t
o
±400
r
p
m
(a trapez
oidal refe
rence s
p
ee
d) as s
h
own in Figure 3(
a), a
nd t
h
e control
dri
v
e system
was fee
d
back
with the
EKF.
Fi
gu
re
2.
B
l
oc
k
di
ag
ram
of t
h
e p
r
o
p
o
sed
s
p
e
e
d se
ns
orl
e
ss
DR
FOC
f
o
r
SP
IM
(a)
(b
)
(c)
(d
)
(e)
Fig
u
re
3
.
Sim
u
latio
n
resu
lts
of th
e sp
eed
sens
orless
DRFOC for a
trapez
oidal refe
re
nce s
p
eed
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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PED
S
I
S
SN
:
208
8-8
6
9
4
Spee
d
Se
ns
orl
e
ss Di
rect
R
o
t
o
r
Fi
el
d-
Ori
e
nt
e
d
C
ont
r
o
l
of
Si
ngl
e
-
Ph
ase
I
n
d
u
ct
i
o
n
...
(
M
o
h
a
m
m
a
d
Ja
n
nat
i)
43
7
Figure
3(a
)
presents the
refe
rence s
p
ee
d a
nd es
tim
ated speed, while Figure
3(
b) s
h
o
w
s t
h
e e
r
r
o
r
betwee
n re
fere
nce s
p
eed
and estim
a
ted speed. Fi
gures
3(
c) an
d
3(
d)
sh
ow t
h
e re
fer
e
n
ce spee
d a
nd
m
o
t
o
r
spee
d and the error
betwee
n refe
rence s
p
ee
d and actual
speed
respecti
v
ely. Figure
3(e
)
shows t
h
e sim
u
la
ted
el
ect
rom
a
gnet
i
c
t
o
r
que
of
S
P
IM
. T
h
e si
m
u
l
a
t
i
on
resul
t
s
have
bee
n
s
h
o
w
n t
h
e
go
o
d
spe
e
d a
n
d t
o
r
q
ue
i
d
ent
i
f
i
cat
i
on
per
f
o
r
m
a
nce o
f
t
h
e
pr
op
ose
d
dri
v
e sy
st
em
(e.
g
., t
h
e osci
l
l
at
i
ons o
f
el
ect
rom
a
gnet
i
c
t
o
r
que i
s
abo
u
t
0.
2N
.m
).
Fig
u
r
e
4
sho
w
s th
e
go
od
p
e
rfo
r
m
an
ce of
the p
r
opo
sed dr
ive syste
m
f
o
r speed
sen
s
o
r
less
D
R
FO
C
of
SPIM at zero a
nd l
o
w spee
d
operation
(
ω
ref
=
0
a
nd
ω
ref
=5
0r
pm
). It
can
be
seen
fr
om
Fi
gu
re 4 t
h
at
t
h
e
dy
nam
i
c
perform
a
nce of the propos
ed drive system
f
o
r s
p
eed se
ns
orless of SPIM
at zero and low spee
d is extrem
ely
acceptable.
Fi
gu
re
5 s
h
ows
t
h
e si
m
u
l
a
t
i
on res
u
l
t
s
o
f
t
h
e
pr
o
pose
d
c
o
nt
r
o
l
l
e
r
un
der
l
o
a
d
(
s
t
e
p l
o
ad
).
F
r
om
t
=
0s t
o
t=1
.
2s, t
h
e
v
a
l
u
e
o
f
th
e lo
ad
is 0N.m
an
d fro
m
t=1
.
2
s
to
t
=
1
.
5
s
, th
e v
a
l
u
e of t
h
e lo
ad is 1N.m
. Resu
lts show
th
at th
e p
r
op
osed
con
t
ro
ller
for v
ect
o
r
co
ntro
l of SPIM is also
robu
st to
th
e lo
ad
torqu
e
v
a
riatio
ns and
pr
o
duce
d
g
o
o
d
resul
t
s
(i
n t
h
i
s
case, as can be seen i
n
Fi
g
u
re 5
(
b),
by
us
i
ng p
r
o
p
o
se
d cont
rol
l
e
r, t
h
e t
o
r
q
ue
o
s
cillatio
n
after app
l
yin
g
l
o
ad
to
rqu
e
an
d
p
h
a
se cu
t-off
a
n
d
at stead
y
state is ~ 0
.
1N.m
at lo
ad
to
rqu
e
of
1N
.m
).
(a)
(b
)
Fi
gu
re
4.
Si
m
u
l
a
t
i
on res
u
l
t
s
o
f
t
h
e
spee
d
sen
s
orl
e
ss
DR
F
O
C
at
zero
an
d l
o
w
spe
e
d;
(a)
Spee
d,
(
b
)
To
r
que
(a)
(b
)
Fi
gu
re
5.
Si
m
u
l
a
t
i
on res
u
l
t
s
o
f
t
h
e
spee
d
sen
s
orl
e
ss
DR
F
O
C
u
nde
r l
o
a
d
;
(
a
) S
p
ee
d,
(b
) T
o
r
q
ue
6.
CO
NCL
USI
O
N
Thi
s
pa
pe
r m
a
de a co
nt
ri
b
u
t
i
on t
o
t
h
e s
p
eed se
ns
orl
e
s
s
DR
FOC
of
SPIM
.
Fi
rst
,
by
ap
pl
y
i
ng
t
r
ans
f
o
r
m
a
ti
on
m
a
t
r
i
ces
t
o
t
h
e SPIM
equat
i
ons
, a no
vel
D
R
FOC
fo
r SP
I
M
i
s
present
e
d
.
Seco
ndl
y
,
i
n
or
der t
o
get
hi
g
h
e
r
pe
rf
orm
a
nce of
SP
IM
dri
v
e, a s
p
e
e
d est
i
m
at
i
on m
e
t
hod
base
d
on
EKF i
s
pr
o
pos
ed
. The si
m
u
l
a
t
i
o
n
resul
t
s
i
n
t
h
i
s
pape
r dem
onst
r
at
e t
h
e go
o
d
per
f
o
r
m
a
nce of t
h
e su
gge
st
ed m
e
t
hod
s, i
n
bot
h co
nt
r
o
l
l
i
ng a
n
d
spee
d estim
ation strategies
.
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ES
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HW Beatt
y
, JL
Kirtley
.
Electri
c
m
o
tor handbook
. New York
, McGraw Hill. 1998
.
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S
SN
:
2
088
-86
94
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J
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S
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l.
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,
No
.
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ase
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