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
. 3
,
Febr
u
a
r
y
201
5,
pp
. 35
5
~
36
5
I
S
SN
: 208
8-8
6
9
4
3
55
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
The Self Excited Induction
Generator with Observation
Magneti
z
ing Charact
eri
st
ic in the Air Gap
Ridwan
Gu
nawan*, Feri
Yu
si
var
*,
B
udi
y
a
nt
o
Y
a
n
*
*
* Department of
Electrical Eng
i
n
eering
,
Univ
ersity
of Indon
esia,
Depok 16424, In
donesia
** Departmen
t
o
f
Electr
i
cal
Engineering
,
Univ
ersity
of
Muhammadiy
a
h, Jakar
t
a 10
510
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Sep 9, 2014
Rev
i
sed
Jan
4, 2
015
Accepte
d
Ja
n 15, 2015
This paper discusses The Self Excita
ted Indu
ction Gener
a
tor
(SEIG) by
approach
ing th
e induction machine, ph
y
s
ically and math
ematically
which
then tr
ans
f
orm
e
d from
three-ph
as
e fram
e
ab
c t
o
two-axis
fram
e
, dir
ect
-axis
and quadratur-
a
xis. Based on the reactive pow
er demand of the induction
machine, cap
acitor
mounted on the stator of
the induction mach
in
e then do
es
the ph
y
s
ic
al and
m
a
them
atic
al
a
pproach of th
e s
y
s
t
em
to ob
tain
a s
p
ace
s
t
at
e
m
odel. Under known relationship
s
, m
a
gnetiza
tion
react
anc
e
and m
a
gnetiz
ing
current
is
not
l
i
near
, s
o
do m
a
them
at
ica
l
app
r
oach
to th
e m
a
gnet
i
za
tion
reac
tan
ce and m
a
gnetiz
ation c
u
rrent
ch
ara
c
t
e
r
i
s
tic curve to obtain the
m
a
gnetiz
ation r
eac
tanc
e equa
ti
on us
ed in the calcul
a
t
i
on. Obtain
ed s
t
ate
s
p
ace m
odel and the m
a
gnetic
react
ance equ
a
tion is
s
i
m
u
lated b
y
us
ing
Runge Kutta m
e
thod of fourth or
der. The equ
a
tio
ns of reactanc
e
, i
s
si
m
u
late
d
b
y
first using the poly
nomial
equa
tion and second using th
e exponent
equation,
and then to compar
e thos
e result between
th
e polynomial and
exponent equations.
The load voltag
e
at d axis and q ax
is using th
e
poly
nomial lags
640µs to the exponent
equatio
n. The poly
nomial voltage
magnitude
is less than 0
.
6068Vo
lt from th
e
expo
nent vo
ltag
e
magnitude.
Keyword:
dq
0 t
r
ans
f
orm
a
t
i
o
n
I
ndu
ctio
n Mach
in
e
Mag
n
e
tizatio
n
SEIG
State Space
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
:
R
i
dwa
n
Gu
na
wan
,
Depa
rt
m
e
nt
of
El
ect
ri
cal
Engi
neeri
n
g
,
U
n
i
v
er
sity of
In
don
esia,
D
e
pok
1
642
4, I
ndo
n
e
sia.
Em
a
il: rid
w
an@eng
.u
i.ac.i
d
1.
INTRODUCTION
The length of the air-gap
has
a si
gnificant influe
nce on the characte
r
is
tics of a
n
electric
machine, the
air-g
ap
leng
th
h
a
s to
b
e
in
creased
con
s
id
erab
ly fro
m
the value obtaine
d for a standa
rd electric
m
o
tor. The
effi
ci
ency
o
f
m
o
t
o
r i
s
hi
ghl
y
depe
nde
nt
o
n
t
h
e r
o
t
o
r ed
dy
cur
r
e
n
t
l
o
ss
es. Ai
r
gap
fl
u
x
o
f
i
n
duct
i
o
n
m
o
t
o
rs
cont
ai
n
s
ri
ch
harm
oni
cs.
A
fl
ux m
oni
t
o
ri
ng sc
hem
e
can gi
v
e
rel
i
a
bl
e and acc
urat
e i
n
fo
rm
at
i
on abo
u
t
electrical
m
a
c
h
ine c
o
nditions. Any cha
n
ge
in air ga
p,
wi
ndi
ng
, v
o
l
t
a
ge
,
and c
u
rre
nt
can be
refl
ect
e
d
i
n
t
h
e
harm
oni
c s
p
ect
ra [
7
]
.
A
m
i
nim
u
m
ai
r gap
fl
ux l
i
nka
ge i
s
r
e
qui
red
f
o
r t
h
e
sel
f-e
xci
t
a
t
i
o
n
an
d st
abl
e
ope
rat
i
o
n
of
an
i
s
ol
at
ed
i
n
d
u
ct
i
o
n
ge
ner
a
t
o
r
feedi
n
g a
n
im
pedan
ce l
o
a
d
.
The
m
i
nim
u
m
ai
r ga
p
fl
u
x
l
i
nkage
re
q
u
i
r
e
m
ent
is th
e v
a
lu
e at wh
ich
th
e d
e
riv
a
tiv
e of th
e
mag
n
e
tizin
g
ind
u
c
tan
ce with
resp
ect to
th
e
air g
a
p
flux
lin
k
a
g
e
i
s
zero
.
T
h
i
s
m
i
ni
m
u
m
ai
r gap fl
u
x
l
i
n
kage
det
e
rm
i
n
es t
h
e m
i
nim
u
m
or m
a
xi
m
u
m
l
o
ad i
m
pedan
c
e an
d
minim
u
m excitation capacitance requir
em
ents. This res
u
l
t
is de
m
onstrated using single-phase and three
-
pha
se i
n
d
u
ct
i
o
n
gene
rat
o
rs
[1
]
.
C
o
n
n
ect
i
o
n
of i
n
duct
i
o
n
ge
nerat
o
rs
t
o
l
a
r
g
e p
o
we
r sy
st
e
m
s t
o
sup
p
l
y
el
ect
ri
c
powe
r ca
n als
o
be ac
hieve
d
whe
n
t
h
e
rot
o
r spee
d
of a
n
i
n
du
ction
g
e
n
e
rato
r
is gr
eater
th
an
t
h
e sy
n
c
hr
ono
us
sp
eed
of th
e air-g
a
p
revo
lv
i
n
g
field
[2
]. The
m
a
g
n
e
tizatio
n
curv
e
o
f
the in
du
ctio
n
m
o
to
r
was id
en
tified
and
com
p
ared wit
h
the one obta
ined by the
no-loa
d test. The m
e
thod se
nsitivity to the load torque
and the
transient i
n
duc
t
ance ha
s also
been consi
d
ere
d
.
A
very
good
acc
uracy of the
m
a
g
n
e
tizatio
n
curv
e estimatio
n
has bee
n
al
so
obt
ai
ne
d at
bi
g
g
er l
o
a
d
t
o
r
que
s [3]
.
T
w
o m
odes o
f
o
p
e
r
at
i
o
n can
be em
ploy
ed f
o
r an i
n
duct
i
o
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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:
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-86
94
I
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S
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l.
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,
No
.
3
,
Feb
r
uar
y
201
5 :
3
55 –
36
5
35
6
g
e
n
e
rator. On
e is th
ro
ugh
self-ex
c
itatio
n
an
d o
t
h
e
r is th
ro
ug
h
ex
tern
al-excitatio
n
.
In
fi
rst
m
o
d
e
, th
e in
du
ction
g
e
n
e
rator tak
e
s its ex
citatio
n fro
m
VAR
g
e
n
e
rating
un
its, g
e
n
e
rally real
ized
in th
e
fo
rm
o
f
cap
acitor b
a
n
k
s
[
4
],
[6
].
The asy
n
chronous
m
achine is m
achin
e, wh
at it h
a
s th
e ro
t
a
tin
g
ro
t
o
r and ro
tating
stator flux
’s are
diffe
re
nt.
The asynchronous machine
is kno
wn
also
as the in
du
ctio
n
m
achine,
what it doe
s as a ge
nerator
neede
d
.
One
of s
p
eci
al
t
y
the i
n
d
u
ct
i
o
n gene
rat
o
r fr
o
m
t
h
e sy
nchr
on
o
u
s ge
nerat
o
r ca
n ope
rat
e
d ab
ove
syn
c
hrono
us sp
eed, wh
ice kno
wn
as Sel
f-Excitatio
n
.
In
th
is co
nd
itio
n, th
e
g
e
n
e
rator will u
s
e th
e en
erg
y
, th
at
it is gene
rated
from
rotor rot
a
tion
fo
r to ge
nerate stator
flux and
rot
o
r
fl
ux usi
n
g reacti
v
e
power. T
h
e
reactive
po
we
r i
s
gi
ve
n
l
o
cal
ban
k
ca
paci
t
o
r
,
that
it conected to t
h
e stator.
W
i
t
h
suitable ca
paci
tors c
o
nne
cted
acros
s
th
e termin
als an
d
with
ro
tor d
r
iv
en
in
eith
er
d
i
r
ect
i
o
n by
a pri
m
e
move
r,
vol
t
a
ge
bui
l
d
s u
p
acr
o
ss t
h
e
t
e
rm
i
n
al
s of t
h
e gene
rat
o
r d
u
e
t
o
sel
f
exci
t
a
t
i
on p
h
e
nom
enon l
e
a
v
i
n
g t
h
e
gene
rat
o
r o
p
e
r
at
i
ng u
n
d
er m
a
gnet
i
c
sat
u
rat
i
o
n at
s
o
m
e
st
abl
e
p
o
i
nt
. S
u
c
h
gene
rat
o
r
i
s
k
n
o
w
n
as sel
f-e
xci
t
e
d i
n
d
u
ct
i
o
n
ge
nerat
o
r
(S
EI
G
)
[
4
]
.
Using
t
h
e sim
u
latio
n
will b
e
d
o
n
e
t
h
e m
a
th
ematical ap
p
r
oa
ch
for ho
p
e
to
ach
iev
e
a d
e
scrib
e
abou
t
all SIEG
responses, in
dq a
x
is.
2.
METH
ODOLOG
Y
The t
h
ree
p
h
as
e i
n
d
u
ct
i
o
n
ge
nerat
o
r
has
so
m
e
equat
i
o
n.
T
h
e e
quat
i
o
n
fl
u
x
a
v
era
g
e
is the flux
as
ti
m
e
fu
n
c
tion
[
1
1
]
.
The e
q
uat
i
o
n
s
st
at
or
vol
t
a
ge:
v
i
r
λ
(1
)
v
i
r
λ
(
2
)
v
i
r
λ
(
3
)
Th
e
equ
a
tio
ns
ro
t
o
r vo
ltag
e
:
v
i
r
λ
(
4
)
v
i
r
λ
(
5
)
v
i
r
λ
(
6
)
Th
e
stator
and
ro
t
o
r
t
u
rn
s flux
are written
as b
e
low:
λ
λ
L
L
L
L
i
i
(
7
)
λ
λ
,
λ
,
λ
(
8
)
λ
λ
,
λ
,
λ
(9)
Th
e stator and
ro
t
o
r curren
t
are written
as
b
e
l
o
w:
i
i
,i
,i
(
1
0
)
i
i
,i
,i
(11)
The
Inductanc
e
stator to
stator:
L
L
L
L
L
L
L
L
L
L
L
L
L
(
1
2
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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S
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8-8
6
9
4
Th
e
S
e
lf Excited
Indu
ctio
n Gen
e
ra
to
r with Ob
serva
tio
n Mag
n
e
tizing
C
h
ara
c
teristic in
…
(Rid
wa
n Gun
a
wa
n
)
35
7
The Inductanc
e
ro
tor to ro
tor:
L
L
L
L
L
L
L
L
L
L
L
L
L
(
1
3
)
The
Inductanc
e
stator to
ro
tor an
d ro
tor t
o
stato
r
:
L
L
(
1
4
)
L
cos
θ
cos
θ
π
cos
θ
π
cos
θ
π
cos
θ
cos
θ
π
cos
θ
π
cos
θ
π
c
o
s
θ
(
1
5
)
Whe
r
e:
stator self inductance
:
L
N
P
(1)
rot
o
r self
induc
t
ance
:
L
N
P
(2)
stator m
u
tual inductance
:
L
N
P
cos
2
π
3
(18)
ro
t
o
r m
u
tu
al ind
u
c
tan
c
e
:
L
N
P
cos
2
π
3
(19)
stato
r
to ro
tor
p
eak m
u
tu
al ind
u
c
tan
ce
L
N
N
P
(20)
N
: stato
r
t
o
tal turn
s lilitan
stato
r
N
: ro
t
o
r t
o
tal tu
rn
s
P
: air
g
a
p p
e
rmeab
ility
The e
q
uat
i
o
n t
r
ans
f
orm
a
t
i
on
fr
om
st
at
or a
n
d
rot
o
r
i
n
0
axis
is
obtaine
d from
the Clark a
n
d Pa
rk
t
r
ans
f
o
r
m
a
ti
on,
Figure
1. The
“
v
ector a-a
x
is”
at
stato
r
and ro
tor
an
d dq
ax
i
s
[1
0
]
f
d
f
q
f
o
T
θ
fa
b
fc
(21)
Th
e equ
a
tio
n of stator an
d ro
to
r
p
o
s
ition
:
θ
t
ω
t
dt
θ
0
(
2
2
)
θ
t
ω
t
dt
θ
0
(
2
3
)
q -
a
xi
s
d
-
axi
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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:
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I
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,
No
.
3
,
Feb
r
uar
y
201
5 :
3
55 –
36
5
35
8
Th
e m
a
tric tran
sfo
r
m
a
tio
n
i
n
dq
0 ax
is, is sho
w
n
as
b
e
low:
T
θ
2
3
cos
θ
cos
θ
π
cos
θ
π
sin
θ
sin
θ
π
s
i
n
θ
π
(
2
4
)
An
d,
T
θ
cos
θ
sin
θ
1
cos
θ
π
s
i
n
θ
π
1
cos
θ
π
s
i
n
θ
π
1
(
2
5
)
The st
at
o
r
a
n
d
rot
o
r
v
o
l
t
a
ge i
n
dq
0 a
x
i
s
i
s
s
h
o
w
n:
v
λ
ω
λ
r
i
(
2
6
)
v
λ
ω
λ
r
i
(
2
7
)
v
λ
r
i
(
2
8
)
v
λ
ω
ω
λ
r
i
(
2
9
)
v
λ
ω
ω
λ
r
i
(
3
0
)
v
λ
r
i
(
3
1
)
Th
e
f
l
ux
eq
u
a
ti
o
n
in
dq
0 ax
is:
λ
λ
λ
λ
λ
λ
L
L
00
L
00
0L
L
00
L
0
00
L
00
0
L
00
L
L
00
0L
00
L
L
0
00
0
0
0
L
i
i
i
i
i
i
(
3
2
)
Th
e stator
and
r
o
t
o
r
f
l
ux
equ
a
tio
n
s
in
dq
0 ax
is:
λ
L
i
L
i
(
3
3)
λ
L
i
L
i
(
3
4
)
λ
L
i
L
i
(
3
5
)
λ
L
i
L
i
(36)
Whe
r
e:
L
L
L
and
L
L
L
(
3
7
)
An
alysis
h
a
s
b
een ex
tend
ed to
iden
tify effectiv
enes
s
of the m
achine
param
e
ters to
im
prove the
ope
rat
i
n
g
per
f
o
rm
ance o
f
t
h
e ge
nerat
o
r
.
It
i
s
f
o
u
n
d
t
h
at operating
pe
rform
a
n
ce of the m
achine may be
im
pro
v
ed
by
p
r
o
p
er
desi
g
n
o
f
st
at
o
r
a
n
d ro
t
o
r pa
ram
e
t
e
rs
[
4
]
.
Whe
n
an
i
n
duct
i
o
n
m
achi
n
e
i
s
dri
v
e
n
by
a
p
r
im
e
m
o
v
e
r, th
e resi
d
u
a
l m
a
g
n
e
tism
in
th
e ro
t
o
r
p
r
o
d
u
ces
a s
m
all v
o
ltag
e
th
at cau
ses a cap
acitiv
e curren
t
to
fl
o
w
.
The
res
u
l
t
i
ng c
u
r
r
e
n
t
p
r
ovi
des
fee
dbac
k
a
n
d
fu
rt
he
r i
n
crease
s
t
h
e
vo
l
t
a
ge. It
i
s
e
v
e
n
t
u
al
l
y
l
i
m
i
t
e
d by
t
h
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Th
e
S
e
lf Excited
Indu
ctio
n Gen
e
ra
to
r with Ob
serva
tio
n Mag
n
e
tizing
C
h
ara
c
teristic in
…
(Rid
wa
n Gun
a
wa
n
)
35
9
mag
n
e
tic satu
ratio
n
in
t
h
e
ro
t
o
r. Variab
le cap
acitan
ce is
req
u
i
red
fo
r sel
f
-ex
c
ited
indu
ctio
n g
e
n
e
rat
o
r
[2]. Th
e
Self Excited
In
du
ction
Gen
e
rato
r
(SEIG)
usin
g
cap
acito
rs, is th
e indu
ctio
n
g
e
n
e
rat
o
r as no
lo
ad
op
eratio
n
.
Th
is syste
m
is
d
e
scrib
e
d
as a th
ree p
h
ase in
du
ctio
n
m
a
c
h
in
e sym
e
trica
lly an
d
co
n
ect
ed
to
id
en
tic b
a
nk
cap
acito
r. The u
s
ing
m
o
d
e
l in
du
ctio
n
m
ach
in
e station
e
ry, th
an
to
ob
tain
equ
i
v
a
len
t
circu
it o
f
t
h
e self
exci
t
a
t
e
d i
n
d
u
c
t
i
on
gene
rat
o
r
SEI
G
i
n
d-a
x
i
s
, as
Fi
g
u
re
2
a
s
bel
o
w
[
5
]
:
Fi
gu
re
2.
St
asi
one
ry
ci
rc
ui
t
at
d-ax
is with ex
cited
cap
acitor [5
]
Fro
m
th
e eq
u
i
valen
t
circu
it as
Fig
u
re
2
,
is ob
t
a
in
ed
vo
ltag
e
eq
u
a
tion
s
in
dq ax
is:
v
v
v
v
r
L
p
0L
p0
0r
L
p
0L
p
L
p
ω
L
r
L
p
ω
L
ω
L
L
p
ω
L
r
L
p
i
i
i
i
(
3
8
)
The three ext
e
rnal elem
ents
that
can
ch
an
g
e
the vo
ltag
e
profile
of
SEIG are s
p
e
e
d, term
inal
capaci
t
a
nce a
n
d t
h
e l
o
a
d
i
m
pedance
.
B
y
var
y
i
ng t
h
e el
em
e
n
ts, one at a time the
performance charact
eristics
of t
h
e s
qui
r
r
el
-
cage i
n
d
u
ct
i
o
n
gene
rat
o
r
obt
a
i
ned.
In m
o
st
of SEI
G
ap
pl
i
cat
i
ons, t
h
e r
o
t
a
t
i
onal
sp
eed i
s
r
a
rel
y
cont
rollable. T
h
ere
f
ore, t
h
e load see
n
by the gene
rato
r or
termin
al cap
acitan
ce h
a
s to
be co
n
t
ro
lled
[9]. Th
e
load RL
series
, is conected pa
rallel with
th
e
b
a
nk
cap
acitor.
Fi
gu
re
3.
The
SEI
G
R
L
C
l
o
a
d
[5]
and
(3
9)
Whe
r
e:
i
i
i
,
a
n
d
i
r
pC
L
p
C
i
i
V
r
L
pi
or
V
i
(40)
Usi
n
g t
h
e t
h
e
e
qui
val
e
nt
ci
rc
u
i
t
for
q
-
a
x
i
s
i
s
obt
ai
ne
d:
(4
1)
r
r
qr
i
dr
i
ds
v
ds
C
e
v
cd
V
dr
r
s
L
1
s
L
1
r
+
L
m
r
b
L
b
i
Ldq
i
dqs
i
Cd
q
C
e
v
Ld
q
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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94
I
J
PED
S
Vo
l.
5
,
No
.
3
,
Feb
r
uar
y
201
5 :
3
55 –
36
5
36
0
Th
e sub
s
titu
tion
v
o
ltag
e
v
,
v
and
V
,
V
t
o
E
quat
i
on
(
2
6
)
-
(
2
7
)
a
n
d
(
2
9)
-(
30
),
a
n
d
t
h
e
n
:
v
v
v
v
Z
i
i
i
i
(42)
r
p
L
0
0r
p
L
pL
0
0p
L
pL
ω
L
ω
L
pL
r
p
L
ω
L
ω
L
r
p
L
(43)
The Equation (33)
until (36) is written
in t
h
e
state space m
o
del, as bel
o
w:
(4
4)
Whe
r
e:
0
0
0
0
0
0
0
0
(
4
5
)
(4
6)
K1
L
L
.L
∶
⁄
(47)
A
K
A
A
A
A
(
4
8
)
A
r
L
ω
L
ω
L
r
L
r
L
ω
L
L
ω
L
L
r
L
r
L
ω
L
L
ω
L
L
r
L
r
L
ω
L
L
ω
L
L
r
L
A
L
0
0L
00
00
L
0
0
L
00
00
A
1
C
K
0
0
1
C
K
00
00
00
00
00
00
A
00
00
1
C
K
0
0
1
C
K
1
L
K
0
0
1
L
K
00
00
(49)
The
reactance
of inductance
m
a
gnetizing
Xm
is
determined usi
n
g tec
hnical a
p
proac
h
with t
h
e
ex
pon
en
tial equ
a
tio
n as
Equ
a
tio
n
(50
)
:
⁄
.
(50)
Using t
h
e equations of
reactance, is done an al
gorithm of sim
u
lation the
self excited induction
generat
o
r
usi
n
g R
u
n
g
e Kut
t
a
m
e
t
hod. Fi
rst
usi
ng th
e po
lyn
o
m
ial
eq
u
a
tio
n
and
s
e
co
nd u
s
ing
t
h
e e
x
p
one
nt
equat
i
o
n
.
Sim
u
l
a
t
i
on usi
ng
Li
nea
r
Ti
m
e
-Vari
y
i
n
g
S
t
at
e M
odel
i
s
use
d
di
sc
ret
e
com
put
at
i
on
ru
n
g
e k
u
t
t
a
m
e
t
hod
o
f
fo
u
r
t
h
or
de
r, i
n
t
h
e st
at
e space
e
quat
i
o
n a
s
bel
o
w [
8
]
:
x
t
f
x,
t
(51)
1
1
and
≅
(52)
Th
e sam
p
lin
g
ti
m
e
T is
ste
p
i
n
terval
. T
h
e state s
p
ace c
o
unting
progra
mme is using
the
functi
on
,
f
o
r
determ
ine
,
alon
g
x
an
d
t
.
And t
h
en is
determ
ine every s
t
ep as
below:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Th
e
S
e
lf Excited
Indu
ctio
n Gen
e
ra
to
r with Ob
serva
tio
n Mag
n
e
tizing
C
h
ara
c
teristic in
…
(Rid
wa
n Gun
a
wa
n
)
36
1
For
x
n
1
≅
x
n
1
T
(53)
f
x,
t
A
x
t
x
t
B
x
t
u
t
(54)
An
d,
AnT
A
n
A
x
n
BnT
B
n
B
x
n
(55)
A
n
1
T
A
n1
A
x
n1
B
n
1
T
B
n1
B
x
n1
(56)
g
≡f
x
n
g
≡f
x
n
g
g
≡f
x
n
g
g
≡f
x
n
g
T
g
≡
g
2
g
2
g
g
6
⁄
(57)
Renew the stat
e equation a
n
d
tim
e
:
x
n
1
x
n
g
T
(58)
A
n1
A
x
n1
B
n1
B
x
n
1
(59)
n
n1
a
n
d
t
n
T
(60)
x
n
1
T
f
x
n,
nT
(61)
f
x
n,
nT
A
x
n
x
n
B
x
n
u
n
(62)
3. RES
U
LTS AN
D A
NAL
Y
S
IS
The
dat
a
o
f
t
h
e sel
f
exci
t
e
d i
n
d
u
ct
i
o
n
gene
r
a
t
o
r S
E
I
G
, t
h
r
ee p
h
ase
38
0
vol
t
,
5
0
he
rt
z,
7.
5 k
W
,
a
n
d
4 pol
es
[
5
]
.
Tab
e
l 1
.
Data o
f
Self Ex
itated
Indu
ctio
n
Gen
e
rat
o
r [5
]
m
a
gnitude unit
m
a
gnitude
unit
r
s
1
Oh
m
Ce
180
μ
F
L
s
1
m
H
r
b
180
Oh
m
r
r
0.
77
Oh
m
L
b
20
m
H
L
r
1
m
H
J
0.
23
Kg
m
2
3.
1
Si
mul
a
ti
on usi
n
g the Pol
y
n
o
mi
al
E
q
u
a
ti
on
In th
is sim
u
lati
o
n
is
u
s
ed th
e
mag
n
e
tizin
g ind
u
c
tan
ce equ
a
tio
n
[5
]:
0
.1407
0
.0014
0
.0012
0
.00005
(6
3)
Usi
n
g t
h
e par
a
m
e
t
e
r i
n
Tabl
e 1, i
s
done s
o
m
e
of sim
u
l
a
t
i
ons usi
ng Eq
uat
i
on (
5
3) a
n
d sam
p
l
i
n
g
ti
m
e
10
4
second
.
Th
e l
o
ad vo
ltag
e
r
e
spon
se i
n
d
q
ax
is is sh
own
as Figu
r
e
4 as b
e
l
o
w
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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-86
94
I
J
PED
S
Vo
l.
5
,
No
.
3
,
Feb
r
uar
y
201
5 :
3
55 –
36
5
36
2
(a)
(b
)
Fi
gu
re
4.
Loa
d
v
o
l
t
a
ge
usi
n
g
(a)
sam
p
l
i
ng t
i
m
e
10
second
,
(b) sam
p
lin
g
time
10
second
In
Fi
g
u
re
4(a
)
t
h
e l
o
ad
v
o
l
t
a
ge, t
h
at
i
t
t
h
e
sim
u
l
a
t
i
on d
o
e
s
n
o
t
gi
ve
g
o
o
d
resp
o
n
se a
n
d
not
occ
u
re
o
f
t
h
e
exci
t
a
t
i
on beca
use t
h
e sam
p
l
i
ng t
i
m
e very
hi
gh. T
h
e sec
o
n
d
si
m
u
l
a
ti
on i
s
used t
h
e sam
p
l
i
ng t
i
m
e
redu
ce t
o
becom
e
10
sec
o
nd,
and the
re
s
u
lt is
shown in Figure
4(
b). T
h
e
accuracy c
h
oice
of sam
p
ling tim
e gives
a
best re
sponse.
After the e
x
citation s
u
ccee
d,
then
using sam
p
ling tim
e
10
seco
nd
, is ob
tained
the l
o
a
d
vol
t
age as
Fi
gu
re
5 as
bel
o
w:
(a)
(b
)
Fig
u
re
5
.
Lo
ad vo
ltag
e
at
d ax
is an
d
q
ax
is
(a)
un
til 7
secon
d
.
(b
)
ti
m
e
fro
m
6
.
50
u
n
til 6.55
secon
d
The load voltage rises begin a
t
tim
e
is 3 second until 5
second and after that its
is constant at voltag
e
24
0 v
o
l
t
,
and
t
h
e form
of w
a
ve i
s
pure t
h
e si
ne form
wi
t
h
th
e
f
r
e
qu
ency
is
5
0
cycles
p
e
r
seco
nd
,
as t
h
e
concl
u
si
on
t
h
i
s
m
e
t
hod
usi
ng
polynom
i
al equation gi
ve
s the accuracy response.
The magnetizing curve from
pol
y
n
o
m
i
al
equat
i
o
n
i
s
s
h
o
w
n i
n
Fi
g
u
r
e
6, a
s
bel
o
w:
q
a
x
is
lo
a
d
vo
lta
g
e
secon
d
secon
d
V
d
(volt
)
V
q
(volt
)
d
a
x
is
l
o
a
d
volt
a
g
e
d
axi
s
load
volt
age
V
q
(v
o
l
t
)
V
d
(v
o
l
t
)
q
axis
load
volt
age
seco
n
d
seco
n
d
s
eco
n
d
s
eco
n
d
V
d
(
volt
)
d
axis
loa
d
v
o
lt
ag
e
q
a
x
i
s
l
o
ad
vo
l
t
a
g
e
V
d
(
vol
t
)
s
eco
n
d
d
axi
s
lo
ad
vo
lt
a
g
e
q
axis
load vol
t
a
ge
V
d
(
vol
t)
V
q
(v
ol
t)
s
eco
n
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
Th
e
S
e
lf Excited
Indu
ctio
n Gen
e
ra
to
r with Ob
serva
tio
n Mag
n
e
tizing
C
h
ara
c
teristic in
…
(Rid
wa
n Gun
a
wa
n
)
36
3
Fi
gu
re
6.
The
m
a
gnet
i
z
i
n
g
cu
rve
f
r
om
pol
y
n
o
m
i
al
equat
i
o
n
3.
2.
Simulati
on usi
n
g the
E
x
p
o
ne
nt
Eq
uation
Base on
u
s
ing po
lino
m
ia
l equ
a
tio
n t
h
at it i
s
iterrated
b
y
mag
n
e
tizin
g cu
rren
t
in
i
n
te
r
v
a
l
0
.
0
1
am
pere a
n
d
t
h
en
i
s
det
e
rm
ined
t
h
e e
x
po
n
e
nt
eq
uat
i
o
n
u
s
i
ng t
h
e
pr
o
g
ra
m
m
e
“
co
n
s
tant Kij d
e
termine
”, so
t
h
at
i
s
obt
ai
ne
d t
h
e
cu
r
v
e as
Fi
gu
re
7,
as
be
l
o
w:
Fi
gu
re
7.
T
h
e
pol
y
n
o
m
i
al
equat
i
o
n
u
s
i
n
g t
h
e ex
po
ne
nt
eq
u
a
t
i
on a
p
p
r
oach
Using
th
e si
m
u
la
tio
n
and
m
a
tlab
p
r
og
ramm
e
is d
e
termin
ed
th
e co
n
s
tan
t
K
1
= 0.
10
2
7
Ohm
.
seco
nd/
ra
di
an , K
2
= -
0
.0
08
1
1/
am
pere
2
a
n
d
K
3
=
0.
0
3
9
5
O
h
m
.
seco
nd/
ra
di
an
.
T
h
e m
a
gne
t
i
z
i
n
g
inductance
curve
L
m
i
s
s
h
ow
n
i
n
Fi
g
u
re
7
has
ex
p
one
nt
equat
i
o
n a
s
E
q
uat
i
o
n
(
6
4):
0
.
1027
∗
.
∗
∗
0
.0395
(6
4)
B
a
se o
n
Fi
g
u
r
e
6,
t
h
e
m
a
gn
et
i
z
i
ng c
u
r
r
ent
st
art
s
fr
om
n
u
l
l
am
pere u
n
t
i
l
9 am
pere.
T
h
e c
o
nst
a
nt
val
u
e
,
and
is ch
osen, cause
h
a
s a m
o
st p
r
ecise v
a
lu
e, th
at it n
earst th
e
p
o
l
yno
m
i
al eq
u
a
tio
n un
til 9
am
pere i
s
s
h
o
w
n
as Fi
gu
re
7
.
Usi
n
g t
h
e
dat
a
pa
ram
e
t
e
r
m
o
t
o
r a
nd t
h
e e
x
po
ne
nt
eq
uat
i
o
n i
s
d
one
si
m
u
l
a
t
i
on ,
u
s
e t
h
e sam
p
lin
g tim
e
10
4
sec
o
n
d
a
n
d t
h
e
l
o
a
d
v
o
l
t
a
ge
cu
rve
i
s
sh
ow
n
as
Fi
gu
re
8(a
)
.
A
n
d t
h
en
u
s
i
n
g t
h
e
sam
p
lin
g
ti
m
e
10
5
secon
d
and
do
th
e sim
u
latio
n
,
is ob
tain
ed
t
h
e lo
ad
vo
ltage ach
iev
e
th
e
n
o
m
in
al v
o
ltage
2
4
0
vo
lt, is sh
ow
n as
Figu
r
e
8
(
b
)
:
The ex
p
one
nt
equat
i
o
n o
f
si
nus
oi
d l
o
ad
vol
t
a
ge at
o
n
e
peri
o
d
e i
s
2
0
m
i
li
second
, i
t
m
eans t
h
e
f
r
e
q
u
e
n
c
y o
f
sin
e
w
a
v
e
is 50
cycles p
e
r
secon
d
, and
th
e p
e
ak
lo
ad
v
o
ltag
e
ach
iev
e
s th
e no
m
i
n
a
l v
o
ltag
e
240
v
o
lt is shown in
Fi
g
u
re
9. The resu
lt of
simu
latio
n u
s
i
n
g t
h
e exp
o
n
e
n
t
equ
a
tio
n is d
e
termin
e th
e m
a
g
n
etizin
g
cur
v
e
i
s
s
h
o
w
n a
s
Fi
gu
re
10
.
M
agnet
iz
at
i
o
n
c
u
r
v
e
m
a
g
n
et
i
z
a
t
i
o
n
cu
rr
en
t
I
m
a
m
p
e
r
e
Xm
oh
m
M
a
gn
et
i
z
a
t
i
o
n
cu
rv
e
m
agne
t
i
z
at
io
n
c
u
rr
e
n
t I
m
am
pe
r
e
X
m
o
h
m
Bl
a
c
k
p
o
l
y
nom
i
a
l
Re
d
e
xpon
e
n
t
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
.
3
,
Feb
r
uar
y
201
5 :
3
55 –
36
5
36
4
(a)
(b
)
Fi
gu
re 8.
Loa
d
v
o
l
t
a
ge usi
n
g (a) sam
p
l
i
ng
t
i
m
e
10
second
,
(b
) sam
p
lin
g
time
10
second
Fig
u
re
9
.
Th
e
lo
ad
vo
ltag
e
at 6
.
50
u
n
til 6.55secon
d
Fig
u
re
10
. Th
e m
a
g
n
e
tizin
g
cu
rv
e
3.
3.
Co
mpa
r
iso
n
Results Between The Po
ly
nomial And The
Exponen
t
Eq
uati
ons
(a)
(b
)
Figure
11. Com
p
arison
resul
t
s using t
h
e
pol
ynom
ial
and the expone
nt equation
(a) t
h
e m
a
gnetism
reactance
Xm
vs t
h
e m
a
get
i
z
i
ng c
u
r
r
ent
Im
, (b)
The
l
o
a
d
s
vol
t
a
ge at
q
axi
s
usi
n
g t
h
e
pol
y
n
o
m
i
al
and t
h
e
ex
p
one
nt
equat
i
o
n
Fo
r to
o
b
serv
e th
e d
i
fferen
ce b
e
tween
th
ese resu
lts, is
done to com
p
are t
h
e data
of it, th
at it are th
e
magnetizing
reactance
Xm
as f
unct
i
o
n
of
an
d t
h
e m
a
gn
et
i
z
i
ng c
u
r
r
en
t
i
m
is sh
ow
n
as Figu
r
e
11
(
a
)
.
The
secon
d
, to
ob
serv
e th
e
d
i
fferen
ce resu
lts b
e
tween
th
e loads v
o
ltag
e
u
s
ing
of bo
th
th
e
eq
u
a
tion
s
. The lo
ad
vol
t
a
ge at
q
-
ax
i
s
usi
ng t
h
e
p
o
l
y
nom
i
a
l
l
a
ggs 64
0
μ
s t
o
t
h
e exp
one
nt
eq
uat
i
o
n
a
nd t
h
e pol
y
n
o
m
i
al
vol
t
a
g
e
mag
n
itu
d
e
is less th
an
0
.
60
68 vo
lt fro
m
th
e
ex
pon
en
t
vo
lt
ag
e m
a
g
n
itud
e
. is shown as Fi
g
u
re
11
(b
).
4.
CO
NCL
USI
O
N
The
res
u
l
t
s
ha
ve bee
n
det
e
r
m
i
n
ed fo
r
SE
I
G
w
ith using
t
h
e iterration
with
sam
p
lin
g
t
i
m
e
,
so m
u
ch
the sm
a
ller of sa
m
p
ling time, that the error value
bec
o
mes very s
m
all. The accurac
y
choice of sa
m
p
ling
s
econ
d
d
axis
load
volt
a
ge
q
axis
load volt
age
s
econ
d
V
d
(
volt
)
V
q
(
volt
)
seco
n
d
seco
n
d
d
axis
load volt
age
q
axis
load volt
age
V
d
(
volt
)
V
q
(
volt
)
d
ax
is
l
oad
v
o
lt
age
q
axis
load
volt
a
g
e
se
c
o
n
d
se
c
o
n
d
V
d
(vo
l
t)
V
q
(
vol
t
)
q
axis
load volt
a
g
e
sec
o
n
d
V
q
(
volt
)
Evaluation Warning : The document was created with Spire.PDF for Python.