I
nte
rna
t
io
na
l
J
o
urna
l
of
Ro
bo
t
ics
a
nd
Aut
o
m
a
t
io
n
(
I
J
R
A)
Vo
l.
9
,
No
.
3
,
Sep
tem
b
er
2020,
pp.
1
71
~
1
7
7
I
SS
N:
2089
-
4856,
DOI
:
1
0
.
1
1
5
9
1
/
i
j
r
a
.
v9
i
3
.
pp
1
71
-
177
171
J
o
ur
na
l
ho
m
ep
a
g
e
:
h
ttp
:
//ij
r
a
.
ia
esco
r
e.
co
m
Derivi
ng
t
he
sy
ste
m
eq
u
a
tions
of
un
ba
la
nced
tw
o
-
pha
se
inducti
o
n
m
o
tor
H
a
ny
I
bra
hi
m
Sh
o
u
s
ha
,
Ab
dels
a
m
ie
B
.
K
o
t
b
De
p
a
rtme
n
t
o
f
El
e
c
tri
c
a
l
En
g
in
e
e
rin
g
,
A
l
-
Az
h
a
r
Un
iv
e
rsit
y
,
Eg
y
p
t
Art
icle
I
nfo
AB
ST
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Au
g
2
0
,
2
0
19
R
ev
i
s
ed
Oct
06
,
2
0
19
A
cc
ep
ted
Feb
1
8
,
2
0
20
A
s th
e
re
is
n
o
s
y
ste
m
d
riv
e
n
e
sp
e
c
iall
y
f
o
r
th
e
tw
o
-
p
h
a
se
in
d
u
c
ti
o
n
m
o
to
r
f
e
d
f
ro
m
u
n
b
a
lan
c
e
d
tw
o
-
p
h
a
se
su
p
p
ly
y
e
t,
so
w
e
sta
rt
f
o
r
d
e
riv
a
ti
o
n
th
e
sy
ste
m
e
q
u
a
ti
o
n
s
f
o
r
th
e
sa
id
m
o
to
r
to
b
e
g
e
n
e
ra
ll
y
u
se
d
e
v
e
n
f
o
r
th
e
b
a
lan
c
e
d
o
r
u
n
b
a
la
n
c
e
d
tw
o
-
p
h
a
se
su
p
p
ly
.
In
th
is
p
a
p
e
r,
w
e
w
il
l
d
e
riv
e
a
s
y
ste
m
e
q
u
a
ti
o
n
sta
rti
n
g
f
ro
m
th
e
se
q
u
e
n
c
e
e
q
u
iv
a
len
t
c
ircu
it
f
o
r
th
e
f
o
rw
a
rd
a
n
d
b
a
c
k
w
a
rd
s
e
q
u
iv
a
len
t
c
ircu
it
s,
t
h
e
n
w
e
w
il
l
re
-
a
rra
n
g
e
th
e
e
q
u
a
ti
o
n
s
w
it
h
so
m
e
m
a
th
e
m
a
ti
c
a
l
a
ss
u
m
p
ti
o
n
s
w
h
ich
w
il
l
lea
d
u
s
to
th
e
n
e
w
s
y
ste
m
e
q
u
a
ti
o
n
s.
f
irst
f
o
r
th
e
v
o
lt
a
g
e
e
q
u
a
ti
o
n
s
th
e
n
f
o
r
th
e
c
u
rre
n
t
e
q
u
a
ti
o
n
s
a
n
d
f
in
a
ll
y
f
o
r
b
o
t
h
p
o
w
e
r
a
n
d
t
o
rq
u
e
e
q
u
a
ti
o
n
s
.
M
o
re
o
v
e
r,
w
e
w
il
l
p
u
t
a
n
e
x
a
m
p
le
w
h
ich
w
il
l
c
o
v
e
r
a
ll
c
a
s
e
s
w
it
h
sp
e
c
if
ic
v
a
lu
e
s an
d
re
latio
n
s ch
a
rts.
K
ey
w
o
r
d
s
:
Fo
r
w
ar
d
a
n
d
b
ac
k
w
a
r
d
eq
u
iv
ale
n
t c
ir
c
u
its
P
er
f
o
r
m
a
n
ce
o
f
t
w
o
-
p
h
a
s
e
m
o
to
r
u
n
d
er
u
n
b
alan
ce
d
s
u
p
p
ly
Seq
u
en
ce
cir
c
u
its
T
h
is
is
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Han
y
I
b
r
ah
i
m
S
h
o
u
s
h
a
,
Dep
ar
t
m
en
t o
f
E
lectr
ical
E
n
g
i
n
ee
r
in
g
,
A
l
-
A
z
h
ar
Un
i
v
er
s
it
y
,
E
g
y
p
t
.
E
m
ail:
h
.
i.s
h
o
u
s
h
a
@
g
m
ai
l.c
o
m
1.
I
NT
RO
D
UCT
I
O
N
W
h
en
t
h
e
m
o
to
r
w
o
r
k
s
u
n
d
er
a
g
en
er
al
o
p
er
atin
g
co
n
d
i
tio
n
s
o
f
u
n
b
ala
n
ce
d
s
tato
r
v
o
ltag
es
o
r
w
i
n
d
i
n
g
s
,
an
d
i
f
w
e
s
tar
t
to
d
er
iv
e
th
e
s
y
s
te
m
eq
u
a
tio
n
s
,
it
is
n
ec
e
s
s
ar
y
to
s
tar
t
f
r
o
m
t
h
e
k
n
o
w
n
eq
u
i
v
ale
n
t
cir
cu
its
w
h
ich
ar
e
o
b
tain
ed
f
o
r
b
o
th
f
o
r
w
ar
d
an
d
b
ac
k
w
o
r
d
co
m
p
o
n
e
n
ts
[
1
]
.
Us
in
g
a
s
p
e
cial
an
al
y
s
is
w
h
ich
w
il
l
b
e
r
eq
u
ir
ed
in
o
r
d
er
to
co
m
p
letel
y
d
escr
ib
e
t
h
e
t
w
o
-
p
h
a
s
e
eq
u
i
v
ale
n
t
c
ir
cu
it
o
r
t
h
e
s
y
s
te
m
o
f
t
h
e
v
o
ltag
e
eq
u
atio
n
s
f
o
r
th
e
t
w
o
-
p
h
a
s
e
m
o
to
r
[
2
]
.
Usi
n
g
t
h
e
r
elatio
n
s
b
et
w
ee
n
t
h
e
s
eq
u
e
n
ce
an
d
p
h
ase
v
al
u
es,
to
o
b
tain
th
e
i
m
p
o
r
tan
t
s
y
s
te
m
o
f
v
o
lt
ag
e
eq
u
atio
n
s
f
o
r
th
e
ca
s
e
o
f
u
n
b
ala
n
ce
d
s
y
s
te
m
[
3
]
.
Af
te
r
th
e
d
er
iv
atio
n
o
f
th
e
s
y
s
te
m
eq
u
at
io
n
s
w
e
ca
n
lear
n
t
h
e
p
er
f
o
r
m
an
ce
o
f
t
h
e
u
n
b
alan
ce
d
t
w
o
-
p
h
a
s
e
elec
tr
ical
m
o
to
r
,
u
s
i
n
g
th
e
s
y
m
m
etr
ical
co
m
p
o
n
en
t
s
r
ep
r
esen
tatio
n
f
o
r
b
o
th
v
o
lt
ag
es
a
n
d
cu
r
r
e
n
ts
[
4
]
.
T
h
ese
s
y
s
te
m
eq
u
atio
n
s
w
il
l
b
e
u
s
ed
f
o
r
all
ca
s
es
f
o
r
th
e
t
w
o
-
p
h
ase
i
n
d
u
ctio
n
m
o
to
r
w
h
ic
h
f
ed
f
r
o
m
t
w
o
-
p
h
ase
p
o
w
er
s
u
p
p
l
y
u
n
d
er
all
co
n
d
itio
n
s
[
5
]
.
2.
RE
S
E
ARCH
M
E
T
H
O
D
2
.
1
.
E
qu
iv
a
lent
circ
uit
s
o
f
f
o
r
w
a
rd
a
nd
ba
c
k
w
a
rd
co
m
po
ne
n
t
s
Fro
m
t
h
e
u
s
u
al
eq
u
i
v
ale
n
t
cir
cu
its
o
f
f
o
r
w
ar
d
an
d
b
ac
k
w
ar
d
co
m
p
o
n
en
t
s
,
t
h
e
f
o
llo
w
i
n
g
eq
u
atio
n
s
f
o
r
b
o
th
co
m
p
o
n
e
n
ts
ar
e
s
h
o
w
n
i
n
Fi
g
u
r
e
1.
Fro
m
t
h
e
f
i
g
u
r
e,
w
e
ca
n
e
x
p
r
ess
t
h
e
v
o
lta
g
e
eq
u
a
tio
n
s
as
th
e
f
o
llo
w
i
n
g
:
V
sf
=
(
R
s
+
j
x
LS
+
j
x
m
)
I
sf
+
jx
m
I
rf
V
rf
=
Jx
m
I
sf
+
(
j
x
m
+
R
r
/
s
)
I
rf
V
sb
=
(
R
s
+
j
x
LS
+
j
x
m
)
I
sb
+
j
x
m
I
rb
V
rb
=
j
x
m
I
sb
+
(
j
x
m
+
R
r
/
(
2
–
s
))
I
rb
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
I
nt
J
R
ob
&
A
u
to
m
,
Vo
l.
9
,
No
.
3
,
Sep
tem
b
er
2020
:
1
71
–
177
172
(
a)
(
b
)
Fig
u
r
e
1
.
Sin
g
le
-
p
h
ase
eq
u
iv
al
en
t c
ir
cu
i
ts
f
o
r
a
t
w
o
-
p
h
ase
m
o
to
r
u
n
d
er
u
n
b
alan
ce
d
co
n
d
iti
o
n
(
a)
f
o
r
w
ar
d
f
ield
an
d
(
b
)
b
ac
k
w
ar
d
f
ield
Fo
r
th
e
ab
o
v
e
eq
u
iv
ale
n
t c
ir
c
u
its
th
er
e
ar
e
s
o
m
e
a
s
s
u
m
p
t
io
n
s
Z
m
=
jX
m
Z
s
=
R
s
+
j
Xs.
,
Z
sm
=
Z
m
+
Z
s
Z
mr
=
Z
m
+
jX
r
+
(
R
r
/
S
f
)
S
f
=
S
,
S
b
=
(
2
–
S
)
T
h
e
s
y
m
m
etr
ical
co
m
p
o
n
en
t
s
r
elatio
n
s
f
o
r
th
e
v
o
lta
g
es a
r
e
V
sf
=
(
V
s1
+
j
V
s2
)
/
2
,
V
sb
=
(
V
s1
–
j
V
s2
)
/
2
,
V
rf
=
(
V
r1
+
j
V
r2
)
/
2
,
V
rb
=
(
V
r1
–
j
V
r2
)
/
2
an
d
f
o
r
th
e
cu
r
r
e
n
ts
ar
e
I
sf
=
(
I
s1
+
j
I
s2
)
/
2,
I
sb
=
(
I
s1
–
j
I
s2
)
/
2
,
I
rf
=
(
I
r1
+
j
I
r2
)
/
2,
I
rb
=
(
I
r1
–
j
I
r2
)
/
2.
2
.
2
.
So
lutio
n o
f
t
h
e
f
o
r
w
a
rd
a
nd
ba
ck
w
a
rd
curr
ent
s
Ma
th
e
m
atica
l
an
al
y
s
is
f
o
r
d
er
iv
atio
n
o
f
t
h
e
s
y
s
te
m
eq
u
atio
n
s
is
t
o
o
b
tain
th
e
s
y
s
te
m
eq
u
atio
n
s
o
f
th
e
v
o
lta
g
es a
n
d
cu
r
r
en
t
s
.
2
.
2
.
1
.
F
o
r
v
o
lt
a
g
e
s
y
s
t
e
m
equa
t
io
n
s
us
i
ng
t
he
s
y
mm
et
rica
l c
o
mp
o
nents
Fo
r
v
o
ltag
e
s
y
s
te
m
eq
u
a
tio
n
s
u
s
i
n
g
th
e
s
y
m
m
etr
ical
co
m
p
o
n
en
t
s
,
w
e
u
s
e:
V
s1
=
(
V
sf
+
V
sb
)
,
V
s2
=
–
j
(
V
sf
–
V
sb
)
,
V
r1
=
(
V
rf
+
V
rb
)
,
V
r2
=
–
j
(
V
rf
–
V
rb
)
,
T
h
e
s
i
m
ilar
r
elatio
n
s
f
o
r
th
e
c
u
r
r
en
ts
ar
e:
I
s1
=
(
I
sf
+
I
sb
)
,
I
s2
=
–
j
(
I
sf
–
I
sb
)
,
I
r1
=
(
I
rf
+
I
rb
)
,
I
r2
=
–
j
(
I
rf
–
I
rb
)
,
Fro
m
th
e
p
r
ev
io
u
s
an
al
y
s
i
s
,
th
e
ab
o
v
e
s
y
s
te
m
o
f
eq
u
atio
n
s
,
ca
n
b
e
u
s
ed
to
o
b
tain
th
e
f
o
llo
w
i
n
g
g
e
n
er
al
s
y
s
te
m
o
f
eq
u
atio
n
s
f
o
r
th
e
p
h
ase
v
o
lta
g
es o
f
t
h
e
t
w
o
-
p
h
ase
in
d
u
ctio
n
m
ac
h
i
n
e
i
n
th
e
g
en
e
r
al
f
o
r
m
s
.
V
s1
=
(
V
sf
+
V
sb
)
=
(
R
s
+
j
x
LS
+
j
x
m
)
I
sf
+
jx
m
I
rf
+
(
R
s
+
j
x
LS
+
j
x
m
)
I
sb
+
j
x
m
I
rb
=
(
R
s
+
j
x
LS
+
j
x
m
)
.
(
I
sf
+
I
sb
)
+
jx
m
(
I
rf
+
I
rb
)
u
s
i
n
g
th
e
p
r
ev
io
u
s
c
u
r
r
en
t e
q
u
atio
n
s
w
e
g
et
V
s1
=
(
R
s
+
j
x
LS
+
j
x
m
)
. I
s1
+
jx
m
.
I
r1
B
y
t
h
e
s
a
m
e
m
an
n
er
,
V
s2
=
–
j
(
V
sf
–
V
sb
)
=
–
j
((
R
s
+
j
x
LS
+
j
x
m
)
I
sf
+
jx
m
I
rf
–
(
R
s
+
j
x
LS
+
j
x
m
)
I
sb
–
j
x
m
I
rb
=
(((
R
s
+
j x
LS
+
j
x
m
)
.
–
j
(
I
sf
–
I
sb
)
+
j
x
m
(
–
j
(
I
rf
–
I
rb
))
V
s2
=
(
R
s
+
j
x
LS
+
j
x
m
)
. I
s2
+
j
x
m
I
r2
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
R
ob
&
A
u
to
m
I
SS
N:
2089
-
4856
Derivin
g
th
e
s
ystem
eq
u
a
tio
n
s
of
u
n
b
a
la
n
ce
d
2
-
p
h
a
s
e
in
d
u
ct
io
n
mo
to
r
(
Ha
n
y
I
b
r
a
h
im
S
h
o
u
s
h
a
)
173
Fo
r
th
e
o
th
er
v
o
lta
g
e
eq
u
atio
n
s
V
r1
=
(
V
rf
+
V
rb
)
V
rf
=
Jx
m
I
sf
+
(
j
x
m
+
R
r
/
s
)
I
rf
V
rb
=
j
x
m
I
sb
+
(
j
x
m
+
(
R
r
/
(
2
–
s
)
)
)
I
rb
V
r1
=
jx
m
I
sf
+
(
j
x
m
+
R
r
/
s
)
I
rf
+
j
x
m
I
sb
+
(
j
x
m
+
(
R
r
/
(
2
–
s
)
)
)
I
rb
W
e
w
ill d
iv
id
e
th
e
p
r
ev
io
u
s
e
q
u
atio
n
i
n
to
t
w
o
p
ar
ts
to
s
o
l
v
e
th
e
m
s
ep
ar
atel
y
t
h
en
w
e
w
i
ll
co
llect
th
e
m
a
g
ai
n
:
V
r1
=
[
jx
m
I
sf
+
j
x
m
I
sb.
]
+
[(
j
x
m
+
R
r
/
s
)
I
rf
+
(
j x
m
+
(
R
r
/
(
2
–
s
)))
I
rb
]
Fo
r
th
e
f
ir
s
t p
ar
t
[
jx
m
I
sf
+
j
x
m
I
sb.
]
=
jx
m
(
I
sf
+
I
sb
)
=
jx
m
I
s1
Fo
r
th
e
s
ec
o
n
s
d
p
ar
t
[(
j
x
m
+
R
r
/
s
)
I
rf
+
(
j
x
m
+
(
R
r
/
(
2
–
s
)))
I
rb
]
=
=
j
x
m
(
I
rf
+
I
rb
)
+
(
R
r
/
s
)
I
rf
+
(
R
r
/
(
2
–
s
))
I
rb
=
j
x
m
I
r1
+
((
2
–
s
)
R
r
/
(
s
(
2
–
s
)))
I
rf
+
(
sR
r
/
(
s
(
2
–
s
)
)
)
I
rb
=
j
x
m
I
r1
+
[(
R
r
/
(
s
(
2
–
s
)))
.
((
2
–
s
)
I
rf
+
s
I
rb
)]
=
j
x
m
I
r1
+
(
R
r
/
(
s
(
2
–
s
)
)
)
.
[(
1
+
1
–
s
)
I
rf
–
(
–
1
+
1
–
s
)
I
rb
]
=
j
x
m
I
r1
+
(
R
r
/
(
s
(
2
–
s
)
)
)
.
[
I
rf
+
I
rb
+
(
1
–
s
)
.
(
I
rf
–
I
rb
)]
=
j
x
m
I
r1
+
(
R
r
/
(
s
(
2
–
s
)
)
)
.
[
I
r1
+
j
(
1
–
s
)
I
r2
]
=
[
j
x
m
I
r1
+
(
R
r
/
(
s
(
2
–
s
)
)
)
]
I
r1
+
j
(
1
–
s
)
[(
R
r
/
(
s
(
2
–
s
)))
.
I
r2
T
h
en
f
o
r
b
o
th
p
ar
ts
V
r1
=
jx
m
I
s1
+
[
j
x
m
+
(
R
r
/
(
s
(
2
–
s
)))]
I
r1
+
j
(
1
–
s
)
[(
R
r
/
(
s
(
2
–
s
)
)
)
. I
r2
f
o
r
th
e
f
o
llo
w
i
n
g
eq
u
atio
n
V
r2
=
–
j
(
V
rf
–
V
rb
)
,
V
rf
=
jx
m
I
sf
+
(
j
x
m
+
R
r
/
s
)
I
rf
,
V
rb
=
j
x
m
I
sb
+
(
j
x
m
+
(
R
r
/
(
2
–
s
)
)
)
I
rb
as th
e
s
a
m
e
f
o
r
t
h
e
p
r
ev
io
u
s
eq
u
atio
n
o
f
t
h
e
V
r1
V
r2
=
–
j
[
jx
m
I
sf
+
(
j
x
m
+
R
r
/
s
)
I
rf
–
j
x
m
I
sb
–
(
j
x
m
+
(
R
r
/
(
2
–
s
)))
I
rb
=
–
j
[
jx
m
.
(
I
sf
–
I
sb
)
+
j
x
m
.
(
I
rf
–
I
rb
)
+
R
r
/
s
.
I
rf
–
(
R
r
/
(
2
–
s
))
I
rb
=
–
j
[(
–
x
m
.
I
s2
)
–
(
x
m
.
I
r2
)
+
[
R
r
/
s
.
I
rf
–
(
R
r
/
(
2
–
s
))
I
rb
]
]
f
o
r
th
e
s
ec
o
n
d
p
ar
t o
f
th
e
eq
u
a
tio
n
R
r
/
s
.
I
rf
–
(
R
r
/
(
2
–
s
))
I
rb
]
=
((
2
–
s
)
R
r
/
(
s
(
2
–
s
)))
I
rf
–
(
sR
r
/
(
s
(
2
–
s
)
)
)
I
rb
=
(
R
r
/
(
s
(
2
–
s
)))
.
((
2
–
s
)
I
rf
–
s
I
rb
)
=
(
R
r
/
(
s
(
2
–
s
)))
.
[(
1
+
1
–
s
)
I
rf
+
(
–
1
+
1
–
s
)
I
rb
]
=
(
R
r
/
(
s
(
2
–
s
)))
.
[
j
(
–
j
)
(
I
rf
–
I
rb
)
+
(
1
–
s
)
.
(
I
rf
+
I
rb
)
]
=
(
R
r
/
(
s
(
2
–
s
)
)
)
[
j I
r2
+
j
(
1
–
s
)
. I
r1
]
T
h
en
f
o
r
b
o
th
p
ar
ts
j
V
r2
=
[(
1
–
s
)
. R
r
/
(
s
(
2
–
s
)
)
)
. I
r1
+
(
R
r
/
(
s
(
2
–
s
))
+
j
x
m
)
. I
r1
+
j
(
R
r
.
(
1
–
s
)
/
(
s
(
2
–
s
)))
. I
r2
T
h
en
w
e
d
r
iv
e
t
h
e
f
o
llo
w
in
g
s
y
s
te
m
eq
u
at
io
n
s
f
o
r
v
o
ltag
e
s
V
s1
=
(
R
s
+
j
x
LS
+
j
x
m
)
. I
s1
+
jx
m
.
I
r1
j
V
s2
=
(
R
s
+
j
x
LS
+
j
x
m
)
. j
I
s2
+
j
x
m
j
I
r2
V
r1
=
jx
m
I
s1
+
[
j
x
m
+
(
R
r
/
(
s
(
2
–
s
)))]
I
r1
+
j
(
1
–
s
)
[(
R
r
/
(
s
(
2
–
s
)
)
)
. I
r2
j
V
r2
=
[(
1
–
s
)
. R
r
/
(
s
(
2
–
s
)
)
)
. I
r1
+
(
R
r
/
(
s
(
2
–
s
))
+
j
x
m
)
. I
r1
+
j
(
R
r
.
(
1
–
s
)
/
(
s
(
2
–
s
)))
. I
r2
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
I
nt
J
R
ob
&
A
u
to
m
,
Vo
l.
9
,
No
.
3
,
Sep
tem
b
er
2020
:
1
71
–
177
174
2
.
2
.
2
.
F
o
r
curr
ent
s
y
s
t
e
m
e
qu
a
t
io
ns
Usi
n
g
t
h
e
p
r
ev
io
u
s
eq
u
atio
n
s
f
o
r
th
e
s
in
g
le
-
p
h
ase
eq
u
iv
a
len
t
cir
cu
its
w
i
ll g
i
v
e
as t
h
e
f
o
llo
w
in
g
:
V
sf
=
Z
sm
. I
sf
–
Z
m
.
I
rf
V
rf
=
Z
mr
.
I
rf
–
Z
m
. I
sf
I
rf
=
(
Z
sm
. I
sf
)
/
Z
m
–
V
sf
/
Z
m
A
l
s
o
,
ac
co
r
d
in
g
to
th
e
r
elatio
n
b
et
w
ee
n
c
u
r
r
en
t,
i
m
p
ed
an
ce
a
n
d
v
o
ltag
e
I
rf
=
(
Z
m
/
Z
mr
)
.
I
sf
T
h
en
,
I
rf
=
(
Z
m
/
Z
mr
)
. I
sf
=
(
Z
sm
. I
sf
)
/
Z
m
–
V
sf
/
Z
m
V
sf
/
Z
m
=
[(
Zsm
/
Z
m
)
–
(
Z
m
/
Z
mr
)]
. I
sf
I
sf
=
[
Z
mr
/
((
Z
sm
. Z
mr
)
–
Z
m
2
)]
.
V
sf
No
w
w
e
g
et
th
e
f
ir
s
t c
u
r
r
en
t e
q
u
atio
n
as
f
o
llo
w
in
g
I
sf
=
[
Z
mr
/
((
Z
sm
. Z
mr
)
–
Z
m
2
)]
.
V
sf
I
sf
=
[(
Z
m
+
jX
r
+
(
R
r
/
S
f
))
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
f
)))
–
Z
m
2
)]
. V
sf
Usi
n
g
t
h
e
f
o
llo
w
i
n
g
eq
u
atio
n
V
rf
=
Z
mr
.
I
rf
–
Z
m
. I
sf
=
0
T
h
en
w
e
g
et
I
rf
=
[
Z
m
/
((
Z
sm
. Z
mr
)
–
Z
m
2
)]
. V
sf
I
rf
=
[
Z
m
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
f
)))
–
Z
m
2
)]
. V
sf
T
h
en
w
e
ca
n
ex
p
r
es
s
f
o
r
th
e
f
o
r
w
ar
d
as th
e
f
o
llo
w
in
g
I
sf
=
[(
Z
m
+
jX
r
+
(
R
r
/
S
f
))
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
f
)))
–
Z
m
2
)]
. V
sf
I
rf
=
[
Z
m
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
f
)))
–
Z
m
2
)]
. V
sf
Fo
r
th
e
b
ac
k
w
ar
d
s
,
I
sb
=
[(
Z
m
+
jX
r
+
(
R
r
/
S
b
))
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
b
)))
–
Z
m
2
)]
. V
sb
I
rb
=
[
Z
m
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
b
)))
–
Z
m
2
)]
. V
sb
T
o
o
b
tain
th
e
air
g
ab
p
o
w
er
eq
u
atio
n
I
rf
=
(
I
r1
+
j
I
r2
)
/
2,
I
rb
=
(
I
r1
–
j
I
r2
)
/
2
Fo
r
th
e
P
gf
w
e
u
s
e
th
e
e
x
p
r
ess
i
o
n
P
gf
=
m
s
I
rf
I
rf
⃰
R
r
/
s
f
,
2
I
rf
=
I
r1r
+
j
I
r1i
+
j
I
r2r
–
I
r2i
=
(
I
r1r
–
I
r2i
)
+
j
(
I
r1i
+
I
r2r
)
T
h
en
2
I
rf
=
(
I
r1r
–
I
r2i
)
+
j
(
I
r1i
+
I
r2r
)
,
2
I
rf
*
=
(
I
r1r
–
I
r2i
)
–
j
(
I
r1i
+
I
r2r
)
P
gf
=
m
s
I
rf
I
rf
⃰
R
r
/
s
f
=
[((
m
s
.
R
r
)
/
4
s
)
.
(((
I
r1r
–
I
r2i
)
+
j
(
I
r1i
+
I
r2r
))
.
((
I
r1r
–
I
r2i
)
–
j
(
I
r1i
+
I
r2r
))
=
[((
m
s
.
R
r
)
/
4
s
)
.
((
I
r1r
–
I
r2i
)
2
+
(
I
r1i
+
I
r2r
)
2
)
=
[((
m
s
.
R
r
)
/
4
s
)
.
(
I
r1r
2
–
2
I
r1r
. I
r2i
+
I
r2i
2
+
I
r1i
2
+
2
I
r1i
.
I
r2r
+
I
r2r
2
))]
An
d
f
o
r
,
I
rb
P
gb
=
m
s
I
rb
I
rb
⃰
R
r
/
s
b
,
I
rb
=
(
I
r1
–
j
I
r2
)
/
2.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
R
ob
&
A
u
to
m
I
SS
N:
2089
-
4856
Derivin
g
th
e
s
ystem
eq
u
a
tio
n
s
of
u
n
b
a
la
n
ce
d
2
-
p
h
a
s
e
in
d
u
ct
io
n
mo
to
r
(
Ha
n
y
I
b
r
a
h
im
S
h
o
u
s
h
a
)
175
W
ith
th
e
s
a
m
e
m
a
n
n
er
2
I
rb
=
I
r1r
+
j
I
r1i
–
j
I
r2r
+
I
r2i
=
(
I
r1r
+
I
r2i
)
+
j
(
I
r1i
–
I
r2r
)
t
h
en
:
2
I
rb
=
(
I
r1r
+
I
r2i
)
+
j
(
I
r1i
–
I
r2r
)
,
2
I
rb
*
=
(
I
r1r
+
I
r2i
)
–
j
(
I
r1i
–
I
r2r
)
P
gb
=
m
s
I
rb
I
rb
⃰
R
r
/
s
b
,
=
[((
m
s
.
R
r
)
/
4
s
b
)
.
((
I
r1r
+
I
r2i
)
+
j
(
I
r1i
–
I
r2r
))
.
((
I
r1r
+
I
r2i
)
–
j
(
I
r1i
–
I
r2r
))
=
[((
m
s
.
R
r
)
/
4
s
b
)
.
((
I
r1r
+
I
r2i
)
2
+
((
I
r1i
–
I
r2r
)
2
)
=
[((
m
s
.
R
r
)
/
4
s
b
)
.
(
I
r1r
2
+
2
I
r1r
. I
r2i
+
I
r2i
2
+
I
r1i
2
–
2
I
r1i
.
I
r2r
+
I
r2r
2
))]
w
h
er
e
P
g
=
P
gf
–
P
gb
=
[((
m
s
.
R
r
)
/
4
s
)
.
(
I
r1r
2
–
2
I
r1r
. I
r2i
+
I
r2i
2
+
I
r1i
2
+
2
I
r1i
.
I
r2r
+
I
r2r
2
))]
–
[((
m
s
.
R
r
)
/
4
s
b
)
.
(
I
r1r
2
+
2
I
r1r
. I
r2i
+
I
r2i
2
+
I
r1i
2
–
2
I
r1i
.
I
r2r
+
I
r2r
2
))]
=
[((
m
s
.
R
r
)
/
(
2
s
.
(
2
–
s
)
)
)
.
(
I
r1r
2
+
I
r2i
2
+
I
r2r
2
+
(
1
–
s
)
.
(
2
I
r1r
. I
r2i
+
2
I
r1i
.
I
r2r
+
I
r1i
2
))
Fo
r
th
e
p
o
w
er
eq
u
atio
n
s
P
gf
=
m
s
I
rf
I
rf
⃰
R
r
/
s
f
,
P
gb
=
m
s
I
rb
I
rb
⃰
R
r
/
s
b
,
w
h
er
e
, s
f
=
s
an
d
s
b
=
2
-
s
P
g
=
P
gf
–
P
gb
Pm
=
Pg
.
(
1
–
s
)
T
=
Pg
/
ω
s
w
h
er
e
ω
s
=
ω
/ p
,
w
it
h
p
=
n
u
m
b
er
o
f
p
o
le
p
air
.
2
.
3
.
P
er
f
o
r
m
a
nce
o
f
t
wo
-
ph
a
s
e
m
o
t
o
r
un
der
un
ba
la
nced
s
u
pp
ly
C
ase
s
t
u
d
y
f
o
r
d
if
f
er
en
t
v
alu
e
s
o
f
th
e
p
h
a
s
e
an
g
le
o
f
t
h
e
s
ec
o
n
d
v
o
ltag
e
s
o
u
r
ce
u
s
e
s
Ma
tla
b
s
o
f
t
w
ar
e
f
o
r
s
i
m
u
la
tio
n
t
h
e
r
es
u
lts
ac
c
o
r
d
in
g
to
d
if
f
er
en
t
v
al
u
es
o
f
t
h
e
v
o
lta
g
e
a
n
g
le
v
al
u
es
(
9
0
o
60
o
30
o
15
o
0
o
)
an
d
o
th
e
r
ass
u
m
p
tio
n
s
as t
h
e
f
o
llo
w
i
n
g
V=220 v ac
,
f=50 Hz
, p=2, w=2.*
.*f,ws=w./p,ms=2,V
s1
=v,
x
m
=40 Ω
,
Z
m
=j.*x
m
Ω
,
r
s
=2
Ω
,
X
s
=2 Ω,X
r
=2 Ω,r
r
=2 Ω
,
no=1.0,Bv2=(pi./2.).*1./no,vs2=v.*((cos(Bv2))
-
(j.*sin(Bv2))),
s=0.00:0.001:1,
s
f
=s,s
b
=2.
-
s
F
r
o
m
th
e
p
r
ev
io
u
s
v
alu
e
s
an
d
u
s
i
n
g
Ma
tlab
w
e
ca
n
g
e
t th
e
r
e
s
u
lt a
s
in
Fig
u
r
es 2
–
5.
Fig
u
r
e
2
.
T
h
e
co
n
tr
o
l o
f
th
e
to
r
q
u
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4856
I
nt
J
R
ob
&
A
u
to
m
,
Vo
l.
9
,
No
.
3
,
Sep
tem
b
er
2020
:
1
71
–
177
176
Fig
u
r
e
3
.
T
h
e
co
n
tr
o
l o
f
Pm
Fig
u
r
e
4
.
T
h
e
co
n
tr
o
l o
f
ls
1
Fig
u
r
e
5
.
T
h
e
co
n
tr
o
l o
f
ls
2
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
e
tar
g
et
ac
h
ie
v
ed
,
as
w
e
d
er
iv
e
a
s
y
s
te
m
eq
u
atio
n
s
o
f
u
n
b
ala
n
ce
d
t
w
o
-
p
h
ase
in
d
u
ct
io
n
m
o
to
r
co
u
ld
b
e
ap
p
lied
f
o
r
an
y
r
u
n
n
i
n
g
co
n
d
itio
n
s
o
f
th
e
m
o
to
r
,
ev
en
it
r
u
n
s
i
n
b
alan
ce
d
o
r
u
n
-
b
alan
ce
d
co
n
d
itio
n
s
ac
co
r
d
i
n
g
to
th
e
f
o
llo
w
i
n
g
eq
u
atio
n
s
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
nt
J
R
ob
&
A
u
to
m
I
SS
N:
2089
-
4856
Derivin
g
th
e
s
ystem
eq
u
a
tio
n
s
of
u
n
b
a
la
n
ce
d
2
-
p
h
a
s
e
in
d
u
ct
io
n
mo
to
r
(
Ha
n
y
I
b
r
a
h
im
S
h
o
u
s
h
a
)
177
V
s1
=
(
R
s
+
j
x
LS
+
j
x
m
)
. I
s1
+
jx
m
.
I
r1
V
s2
=
(
R
s
+
j
x
LS
+
j
x
m
)
. I
s2
+
j
x
m
I
r2
V
r1
=
jx
m
I
s1
+
[
j
x
m
+
(
R
r
/
(
s
(
2
–
s
)))]
I
r1
+
j
(
1
–
s
)
[(
R
r
/
(
s
(
2
–
s
)
)
)
. I
r2
j
V
r2
=
[(
1
–
s
)
. R
r
/
(
s
(
2
–
s
)
)
)
. I
r1
+
(
R
r
/
(
s
(
2
–
s
))
+
j
x
m
)
. I
r1
+
j
(
R
r
.
(
1
–
s
)
/
(
s
(
2
–
s
)))
. I
r2
I
sf
=
[(
Z
m
+
jX
r
+
(
R
r
/
S
f
))
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
f
)))
–
Z
m
2
)]
. V
sf
I
rf
=
[
Z
m
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
f
)))
–
Z
m
2
)]
. V
sf
I
sb
=
[(
Z
m
+
jX
r
+
(
R
r
/
S
b
))
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
b
)))
–
Z
m
2
)]
. V
sb
I
rb
=
[
Z
m
/
(((
Z
m
+
Z
s
)
.
(
Z
m
+
jX
r
+
(
R
r
/
S
b
)))
–
Z
m
2
)]
. V
sb
I
rf
=
(
I
r1
+
j
I
r2
)
/
2,
I
rb
=
(
I
r1
–
j
I
r2
)
/
2.
P
gf
=
m
s
I
rf
I
rf
⃰
R
r
/
s
f
,
P
gb
=
m
s
I
rb
I
rb
⃰
R
r
/
s
b
,
P
g
=
[((
m
s
.
R
r
)
/
(
2
s
.
(
2
–
s
)
)
)
.
(
I
r1r
2
+
I
r2i
2
+
I
r2r
2
+
(
1
–
s
)
.
(
2
I
r1r
. I
r2i
+
2
I
r1i
.
I
r2r
+
I
r1i
2
))
Pm
=
Pg
.
(
1
–
s
)
T
=
Pg
/
ω
s
4.
CO
NCLU
SI
O
N
As
t
h
er
e
i
s
n
o
s
y
s
te
m
d
r
iv
e
n
esp
ec
iall
y
f
o
r
t
h
e
t
w
o
-
p
h
a
s
e
in
d
u
ctio
n
m
o
to
r
f
ed
f
r
o
m
u
n
b
alan
ce
d
t
w
o
-
p
h
a
s
e
s
u
p
p
l
y
,
s
o
w
e
s
tar
t
f
o
r
d
er
iv
atio
n
t
h
e
s
y
s
te
m
eq
u
a
tio
n
s
f
o
r
th
e
s
a
id
m
o
to
r
to
b
e
g
en
er
all
y
u
s
ed
ev
e
n
f
o
r
th
e
b
alan
ce
d
o
r
u
n
b
alan
ce
d
t
w
o
-
p
h
ase
s
u
p
p
l
y
.
I
n
th
is
p
ap
er
,
w
e
d
er
i
v
e
a
s
y
s
te
m
eq
u
atio
n
s
tar
ti
n
g
f
r
o
m
th
e
s
eq
u
en
ce
eq
u
i
v
alen
t
cir
c
u
it
f
o
r
th
e
f
o
r
w
ar
d
an
d
b
ac
k
w
ar
d
s
eq
u
i
v
ale
n
t
cir
c
u
its
,
t
h
en
w
e
re
ar
r
an
g
e
th
e
eq
u
atio
n
s
w
it
h
s
o
m
e
m
at
h
e
m
a
tical
a
s
s
u
m
p
t
io
n
s
w
h
ich
lead
u
s
to
t
h
e
n
e
w
s
y
s
te
m
eq
u
atio
n
s
.
f
ir
s
t
f
o
r
th
e
v
o
ltag
e
eq
u
atio
n
s
t
h
e
n
f
o
r
th
e
cu
r
r
en
t
eq
u
atio
n
s
a
n
d
f
i
n
all
y
f
o
r
b
o
th
p
o
w
er
a
n
d
to
r
q
u
e
eq
u
atio
n
s
.
Mo
r
eo
v
er
,
w
e
p
u
t a
n
e
x
a
m
p
le
w
h
ic
h
co
v
er
all
ca
s
es
w
i
th
s
p
ec
if
ic
v
alu
e
s
an
d
r
elatio
n
s
ch
ar
t
s
.
RE
F
E
R
E
NC
E
S
[1
]
H.
M
.
El
Zo
g
h
b
y
,
S
.
M
.
S
h
a
ra
f
,
a
n
d
M
.
A
.
G
h
a
z
y
,
“
Co
n
tro
ll
in
g
o
f
Tw
o
-
P
h
a
se
S
e
rv
o
m
o
to
r
b
y
Ch
a
n
g
in
g
T
h
e
P
h
a
se
Diff
e
r
e
n
c
e
A
n
g
le
Us
in
g
P
o
ly
n
o
m
ial
–
P
ro
p
o
rti
o
n
a
l
P
l
u
s
In
teg
ra
l
(P
P
I)
Co
n
tro
l
le
,
”
Pro
c
e
e
d
in
g
s
o
f
th
e
1
4
t
h
In
ter
n
a
t
io
n
a
l
M
i
d
d
le E
a
st P
o
we
r
S
y
ste
ms
Co
n
fe
re
n
c
e
(
M
EP
CON'
1
0
)
,
2
0
1
0
.
[2
]
M
.
P
o
p
e
sc
u
,
E
.
De
m
e
ter,
D.
M
ic
u
,
V
.
Na
v
ra
p
e
sc
u
a
n
d
T
.
Jo
k
in
e
n
,
“
A
n
a
l
y
sis
o
f
a
v
o
lt
a
g
e
re
g
u
lato
r
f
o
r
a
t
wo
-
p
h
a
se
in
d
u
c
ti
o
n
m
o
to
r
d
riv
e
,
”
IEE
E
I
n
ter
n
a
ti
o
n
a
l
El
e
c
tric
M
a
c
h
in
e
s
a
n
d
Dr
ive
s
Co
n
fer
e
n
c
e
.
IEM
DC'9
9
.
Pr
o
c
e
e
d
in
g
s
(
Ca
t.
No
.
9
9
EX
2
7
2
)
,
S
e
a
tt
le,
W
A
,
USA
,
1
9
9
9
,
p
p
.
6
5
8
-
6
6
0
.
[3
]
H.
S
.
P
a
tel
a
n
d
R.
G
.
Ho
f
t,
"
Ge
n
e
ra
li
z
e
d
T
e
c
h
n
iq
u
e
s
o
f
Ha
r
m
o
n
ic
El
im
in
a
ti
o
n
a
n
d
Vo
lt
a
g
e
Co
n
t
ro
l
i
n
T
h
y
risto
r
In
v
e
rters
:
P
a
rt
I
--
Ha
rm
o
n
ic
El
im
in
a
ti
o
n
,
"
in
IEE
E
T
ra
n
sa
c
ti
o
n
s
o
n
In
d
u
stry
A
p
p
li
c
a
ti
o
n
s
,
v
o
l.
IA
-
9
,
n
o
.
3
,
p
p
.
3
1
0
-
3
1
7
,
M
a
y
1
9
7
3
.
[4
]
V
.
D.
C
h
ik
a
te,
A
.
Y.
F
a
d
n
is,
G
.
C.
G
o
n
d
h
a
lek
a
r,
“
S
p
e
e
d
Co
n
tr
o
l
o
f
Tw
o
P
h
a
se
In
d
u
c
ti
o
n
M
o
t
o
r
,
”
In
ter
n
a
t
io
n
a
l
J
o
u
rn
a
l
o
f
E
n
g
i
n
e
e
rin
g
Res
e
a
rc
h
in
El
e
c
trica
l
a
n
d
E
lec
tro
n
ic E
n
g
i
n
e
e
rin
g
(
IJ
ER
EE
E)
, v
ol
.
4
,
n
o
.
3
,
2
0
1
8
.
[5
]
P.
Zás
k
a
li
c
k
ý
,
“
T
o
rq
u
e
rip
p
le
c
a
lcu
latio
n
o
f
tw
o
-
p
h
a
se
im
su
p
p
l
ied
b
y
th
re
e
-
leg
V
S
I
in
v
e
rter
,”
M
a
sz
y
n
y
El
e
k
try
c
zn
e
:
ze
sz
y
ty p
ro
b
lem
o
we
,
2
0
1
7
.
Evaluation Warning : The document was created with Spire.PDF for Python.