Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
System
(IJPEDS)
V
ol.
11,
No.
4,
December
2020,
pp.
1737
1749
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v11.i4.pp1737-1749
r
1737
Discr
ete
and
continuous
model
of
thr
ee-phase
linear
induction
motors
considering
attraction
f
or
ce
and
end-effects
Nicol
´
as
T
or
o-Gar
c
´
ıa
1
,
Y
eison
Alberto
Gar
c
´
es-G
´
omez
2
,
Fr
edy
E.
Hoy
os
3
1
Department
of
Electrical
and
Electronics
Engineering
and
Computer
Sciences,
Uni
v
ersidad
Nacional
de
Colombia
-
Sede
Manizales,
Colombia
2
Unidad
Acad
´
emica
de
F
ormaci
´
on
en
Ciencias
Naturales
y
Matem
´
aticas,
Uni
v
ersidad
Cat
´
olica
de
Manizales,
Colombia
3
F
acultad
de
Ciencias,
Escuela
de
F
´
ısica,
Uni
v
ersidad
Nacional
de
Colombia
-
Sede
Medell
´
ın,
Colombia
Article
Inf
o
Article
history:
Recei
v
ed
Mar
12,
2020
Re
vised
Jun
10,
2020
Accepted
Jun
26,
2020
K
eyw
ords:
Continuous
time
model
Discrete
control
systems
Discrete
time
model
Linear
induction
motors
Non-linear
beha
viors
ABSTRA
CT
The
continuous
model
of
the
linear
induction
motor
(LIM)
has
been
made
considering
the
edge
ef
fects
and
the
attraction
force.
T
aking
the
attraction
force
into
account
is
im-
portant
when
considering
dynamic
analysis
when
the
motor
operates
under
mechanical
load.
A
laboratory
prototype
has
been
implemented
from
which
the
parameters
of
the
equi
v
alent
LIM
circuit
ha
v
e
been
obtained.
The
discrete
model
has
been
de
v
eloped
to
quickly
obtain
computational
solutions
and
to
analyze
non-linear
beha
viors
through
the
application
of
discrete
control
systems.
In
order
to
obtain
the
discrete
model
of
the
LIM
we
ha
v
e
started
from
the
solution
of
the
continuous
model.
T
o
de
v
elop
the
model,
the
magnetizing
inductance
has
been
considered,
which
reflects
the
edge
ef
fects.
In
the
results,
the
model
is
compared
without
considering
the
edge
ef
fects
or
the
attraction
force
with
the
proposed
model.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Y
.
A.
Garc
´
es-G
´
omez,
Unidad
Acad
´
emica
de
F
ormaci
´
on
en
Ciencias
Naturales
y
Matem
´
aticas,
Uni
v
ersidad
Cat
´
olica
de
Manizales,
Cra
23
No
60-63,
Manizales,
Colombia.
Email:
yg
arces@ucm.edu.co
1.
INTR
ODUCTION
The
linear
induction
motor
(LIM)
w
as
in
v
ented
and
patented
more
than
a
hundred
years
ago
being
impractical
due
to
the
dif
ficulties
in
its
construction
by
not
being
able
to
ha
v
e
small
air
space
without
roughness
in
addition
to
not
being
able
to
achie
v
e
good
ef
ficienc
y
f
actors.
No
w
adays,
tec
h
nol
ogical
adv
ances
ha
v
e
allo
wed
the
LIM
to
ha
v
e
greater
importance,
e
xtending
i
ts
use
to
important
industrial
and
research
applications
[1-7].
Linear
induction
motors
are
three-phase
A
C
de
vices
that
w
ork
by
the
general
principles
of
electromechanical
ener
gy
transformation
lik
e
other
induction
motors
and
are
constructed
for
to
produce
mo
v
ement
on
a
straight
line.
Although
are
named
“Linear”
the
mathematical
models
are
nonlinear
and
due
to
symmetrical
missing
in
their
construction
is
necessary
to
consider
ef
fects
that
not
are
present
in
the
rotary
electric
machines.
When
the
topology
of
a
machine
is
modified,
which
is
the
case
of
the
LIM
with
the
RIM
(ro-
tary
induction
machine),
the
design
and
operating
conditions
are
also
modified.
Specifically
,
dif
ferent
phe-
nomena
appear
in
the
magnetic
circuit
that
must
be
re-modelled.
This
leads
to
the
de
v
elopment
of
ne
w
theories
[1,
2,
4].
J
ournal
homepage:
http://ijpeds.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
1738
r
ISSN:
2088-8694
When
it
is
required
to
generate
a
linear
mo
v
ement
from
RIM,
the
use
of
mechanical
elements
is
necessary
,
this
can
be
a
v
oided
with
the
use
of
LIM.
In
addition
to
eliminating
the
use
of
mechanical
elements,
the
latter
ha
v
e
the
adv
antages
of
high
acceleration
and
deceleration
capacity
,
use
in
le
vit
ation
systems
by
normal
magnetic
forces,
lo
wer
maintenance
costs,
lo
w
noise,
possibility
of
use
in
systems
with
curv
es
and
slopes,
braking
that
does
not
depend
on
the
system
conditions,
among
others
[8-12].
There
is
little
w
ork
on
sampling
LIM
dynamics;
therefore,
it
is
of
great
importance
to
in
v
estig
ate
an
accurate
representation
of
the
sampled
data
of
the
complete
dynamics
of
linear
induction
motors,
and
to
design
slide
controll
ers
at
discrete
time
[7,
13].
W
ith
respect
to
the
non-linear
models
of
control
strate
gies
applied
to
LIMs,
an
in-depth
re
vie
w
is
made
of
in
[7,
14,
15],
also
in
the
terms
of
the
mathematical
model.
The
ph
ysical
model
of
the
LIM
has
been
de
v
eloped
to
model
the
figure
system
as
sho
wn
in
Figure
1.
The
construction
aspects
of
the
LIM
ha
v
e
been
fully
de
v
eloped
in
[7,
16].
The
or
g
anization
of
the
document
is
as
follo
ws.
Section
2
de
v
elops
the
modelling
of
the
linear
induction
motor
taking
into
account
the
ef
fects
of
edges
and
forces
in
the
equi
v
alent
circuit
which
is
then
discretized
for
comparison
with
the
continuous
model.
Section
three
implements
the
whole
system
and
compares
the
results
to
conclude
with
the
conclusions
of
the
w
ork.
Figure
1.
Ph
ysical
system
implemented
to
obtain
the
linear
induction
motor
(LIM)
model
parameters
2.
LIM
MODEL
CONSIDERING
A
TTRA
CTION
FORCE
AND
END-EFFECTS
Based
on
the
d
q
theory
,
the
LIM
model
has
been
made
with
its
equi
v
alent
circuit
starting
from
[date13].
It
is
tak
en
into
account
that
the
q
axis
of
the
linear
induction
motor
is
equi
v
alent
to
the
rotary
motor
so
the
parameters
are
in
v
ariable.
Ho
we
v
er
,
if
the
currents
of
the
d
axis
are
analyzed,
the
y
af
fect
the
flo
w
of
the
air
g
ap
causing
a
decrease
in
dr
.
Thus
the
equi
v
alent
circuit
of
the
rotary
motor
in
the
d
axis
is
not
applicable
to
the
linear
motor
if
the
edge
ef
fects
are
tak
en
into
account.
In
rotary
motors,
the
edge
ef
fects
ar
e
not
appreciable,
which
is
the
case
with
linear
motors.
Furthermore,
these
ef
fects
increase
as
the
motor
speed
increases,
which
leads
to
an
analysis
of
these
ef
fects
as
a
function
of
speed,
taking
into
account
that
the
y
also
ha
v
e
dif
ferent
beha
viour
at
the
output
and
arri
v
al
ends
of
the
linear
motor
,
since
the
y
decrease
more
sl
o
wly
at
the
input
than
at
the
output
due
to
the
increase
in
the
time
constant
that
modifies
the
deri
v
ati
v
e
of
the
function.
2.1.
Equi
v
alent
cir
cuit
f
or
LIM
The
construction
model
of
the
linear
motor
is
illustrated
in
Figure
2(a).
As
it
can
be
seen,
as
the
primary
mo
v
es,
it
interacts
with
another
re
gion
of
the
liner
dif
ferent
from
the
pre
vious
one
and
that
also
op-
poses
the
increase
of
penetrating
magnetic
flux
and
accumulating
more
flux
in
the
air
g
ap
which
af
fects
the
performance
of
the
linear
motor
as
reported
[17–19].
This
ef
fect
can
be
analyzed
in
Figure
2(c).
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
4,
December
2020
:
1737
–
1749
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
1739
Primary
Entry rail
eddy current
Secondary sheet
Secondary back iron
Exit rail
eddy current
Eddy current
by the end effect
Motor lenght
Airgap average Flux
D
Motor lenght
(a)
(b)
(c)
V
Figure
2.
(a)
Motion
ef
fect
of
the
primary
coil
generating
eddy
currents,
(b)
Input
and
output
current
w
a
v
eforms,
(c)
Flo
w
w
a
v
eform
in
the
air
g
ap
As
the
coils
of
the
primary
mo
v
e,
the
ne
wly
generated
field
enters
the
secondary
as
the
pre
vious
field
disappears
at
the
output
of
t
he
primary
creating
eddy
currents
in
the
primary
[20]
(see
Figure
3(b)).
Aligning
the
reference
frame
with
the
reaction
linor
flux
and
call
it
d
axis
,
it
results
in
q
r
=
0
.
Noting
that
as
f
ar
as
q
r
=
0
and
dr
does
not
change,
the
end
ef
fect
does
not
play
an
y
role
in
equi
v
alent
circuit.
Since
i
q
"
=
i
q
s
the
entry
q
axis
eddy
current
k
eeps
q
r
=
0
.
Hence,
the
q
axis
equi
v
alent
circuit
is
identical
to
the
case
of
the
rotary
induction
motor
.
Ho
we
v
er
,
the
d
axis
air
g
ap
flux
is
af
fected
much
by
the
eddy
current
since
d-axis
entry
eddy
current
in
linear
induction
motor
,
i
d"
,
reduces
dr
.
Normalized motor length Q
d-axis Airgap MMF
Q
(a)
(b)
0
i
e
ds
i [1-exp(-x)]
e
ds
1
2
3
4
Secondary eddy current
due to end effect
0
Q
1
2
3
4
-i
e
ds
-i exp(-x)]
e
ds
entry rail current
exit rail current
Normalized time x
Figure
3.
(a)
Ef
fecti
v
e
air
g
ap
MMF
and
(b)
eddy
current
profile
in
normalized
time
scale.
Discr
ete
and
continuous
model
of
thr
ee-phase
linear
...
(N.
T
or
o-Gar
c
´
ıa)
Evaluation Warning : The document was created with Spire.PDF for Python.
1740
r
ISSN:
2088-8694
2.2.
Magnetizing
inductance
r
eflecting
the
end
effects
When
the
primary
is
mo
ving,
the
primary
MMF
observ
ed
by
the
rail
will
be
decreased
at
the
entry
and
be
reflected
in
the
output
rail
to
k
eep
the
air
g
ap
in
flo
w
(
continuous).
In
particular
,
the
polarity
of
the
input
eddy
current
is
contrary
to
that
of
the
output
eddy
current,
as
the
y
are
naturally
opposed
to
the
generation
and
e
xtinction
of
the
fields,
specifically
.
Note
that
the
input
eddy
current
has
a
higher
decay
period
relati
v
e
to
the
output
eddy
current,
since
the
inductance
is
greater
in
the
air
g
ap
than
in
the
free
air
.
The
pattern
of
the
eddy
currents
is
dra
wn
in
Figure
4
which
is
based
on
the
standard
time
scale.
[20].
Figure
4.
The
equi
v
alent
linear
motor
circuits
taking
into
account
the
end-ef
fects,
(a)
The
equi
v
alent
d-axis
circuit,
(b)
The
equi
v
alent
q-axis
circuit
Observing
that
the
input
of
the
d-axis
of
the
eddy
currents
decreases
with
the
time
dif
ferential
T
r
,
the
mean
v
alue
of
the
eddy
current
input
from
the
d-axis
i
d"
is
gi
v
en
by
1
i
"
=
i
ds
T
v
Z
T
v
0
e
t=T
r
dt
(1)
where
T
v
=
D
=v
,
and
D
,
v
are
the
motor
e
xtension
and
v
elocity
.
Noting
that
T
v
=
D
=v
is
the
time
for
the
motor
to
tra
v
els
a
point.
Because
the
tra
v
el
length
for
the
period
T
r
is
eqi
v
alent
to
v
T
r
and
normalizing
the
motor
size
with
v
T
r
as
2
[21].
Q
=
v
T
v
v
T
r
=
D
R
r
(
L
m
+
L
l
r
)
v
(2)
Notice
that
Q
is
non-dimensional
yet
it
represents
the
length
of
the
motor
on
the
standardized
time
scale.Based
on
this,
the
length
of
the
motor
is
strongly
influenced
by
the
speed
of
the
motor
,
s
o
that
at
zero
speed,
the
length
of
the
motor
is
infinitely
long.
As
the
speed
increases,
the
length
of
the
motor
will
ef
fecti
v
ely
decrease.
Using
2,
(1)
can
be
re
written
as
follo
ws:
i
"
=
i
ds
Q
Z
Q
0
e
x
dx
=
i
ds
1
e
Q
Q
(3)
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
4,
December
2020
:
1737
–
1749
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
1741
The
ef
fecti
v
e
magnetizing
current
is
thus
decreased
in
such
a
manner
that:
i
ds
i
"
=
i
ds
1
1
e
Q
Q
(4)
The
reduction
of
the
magnetizing
current
caused
by
the
eddy
current,
can,
ho
we
v
er
,
be
justified
by
changing
the
magnetizing
inductance
in
a
w
ay
that:
L
0
m
=
L
m
(1
f
(
Q
))
(5)
where
f
(
Q
)
=
(1
e
Q
)
=Q
[20].
As
v
elocity
tends
to
zero,
L
0
m
con
v
er
ges
to
L
m
i.e.,
the
LIM
dynamics
becomes
equi
v
alent
to
the
RIM
dynamics
as
the
end
ef
fect
disappears.
Figure
4
sho
ws
the
ef
fecti
v
e
air
g
ap
MMF
and
the
eddy
current
profile
in
normalized
time
scale.
2.3.
Equi
v
alent
series
r
esistor
r
eflecting
rail
eddy
curr
ent
losses
When
inflo
w
and
outflo
w
eddy
currents
flo
w
along
the
rail
,
an
ohmic
loss
of
R
r
will
occur
.
Note
that
the
a
v
erage
square
v
alue
of
the
input
eddy
current
o
v
er
the
length
of
the
motor
is
gi
v
en
by:
i
"R
M
S
=
"
i
2
ds
Q
Z
Q
0
e
2
x
dx
#
1
2
=
i
ds
1
e
2
Q
2
Q
1
2
(6)
Hence,
the
loss
caused
by
the
entry
eddy
current
is
e
v
aluated
as
[22]
in
7:
P
entr
y
=
i
2
"R
M
S
R
r
=
i
2
ds
R
r
1
e
2
Q
2
Q
(7)
Using
the
methodology
of
[22],
we
can
assess
the
losses
due
to
the
eddy
current
by
the
temporal
rate
of
the
magnetic
ener
gy
change
when
e
xiting
the
air
space
of
the
motor
.
Note
from
3
that
the
total
eddy
current
in
the
air
g
ap
is
equal
to
i
d
s
(1
e
Q
)
.
This
flo
w
must
be
eliminated
in
the
e
xit
rail
for
T
v
to
satisfy
the
steady
flo
w
condition
of
the
air
g
ap.
Thus,
the
loss
due
to
the
output
eddy
current
is
pro
vided
by
8:
P
exit
=
L
r
i
2
ds
(1
e
Q
)
2
2
T
v
=
i
2
ds
R
r
(1
e
Q
)
2
2
Q
(8)
Adding
(7)
and
(8),
the
total
ohmic
losses
due
to
eddy
currents
in
the
rail
are
gi
v
en
by
this
loss
of
po
wer
can
be
sho
wn
as
a
resistance
wired
in
a
series
R
r
f
(
Q
)
in
the
magnetizing
current
branch.
4
the
total
ohmic
losses
due
to
eddy
currents
in
the
rail
are
gi
v
en
by
this
loss
of
po
wer
can
be
sho
wn
as
a
resistance
wired
in
a
series.
Duncan’
s
circuit
has
been
de
v
eloped
consideri
ng
v
elocity
and
po
wer
loss.
It
supposes
uniform
wind-
ing
and
materials,
symmetric
impedances
per
phase
and
equal
mutual
inductances.
It’
s
based
on
traditional
model
of
three-phase,
Y
-connected
rotatory
induction
motor
whit
linear
magnetic
circuit
in
a
synchronous
reference
system
(superscript
“e”)
aligned
with
the
linor
flux.
Also
only
longitudinal
end
ef
fects
ha
v
e
been
considered.
Duncan’
s
model
has
been
adopted
in
order
to
obtain
a
space
state
representation
both
continuous-t
ime
and
discrete-time.
Se
v
eral
techniques
ha
v
e
been
de
v
eloped
for
non
linear
dynamics
analysis
in
the
state
space.
P
arameter
Q
,
function
f
(
Q
)
,
Magnetizing
Inductance
Reflecting
the
End
Ef
fects
and
Equi
v
alent
Se-
ries
Resistor
Reflecting
Rail
Eddy
Current
Losses
ha
v
e
been
deri
v
ed
from
circuit
theory
.
The
Q
f
actor
is
associated
with
the
length
of
the
primary
,
and
to
a
certain
de
gree,
quantifies
the
end
ef
fects
as
a
function
of
the
v
elocity
v
as
described
by
(9).
Q
=
D
R
r
L
r
v
(9)
Note
that
the
Q
f
actor
is
in
v
ersely
dependent
on
the
v
elocity
,
i.e.,
for
a
zero
v
elocity
the
Q
f
actor
may
be
considered
infinite,
and
the
end
ef
fects
may
be
ignored.
As
the
v
elocity
increases
the
end
ef
fects
increases,
which
causes
a
reduction
of
the
LIM’
s
magnetization
current.
This
ef
fect
may
be
quantified
in
terms
of
the
magnetization
inductance
with
the
equation:
Discr
ete
and
continuous
model
of
thr
ee-phase
linear
...
(N.
T
or
o-Gar
c
´
ıa)
Evaluation Warning : The document was created with Spire.PDF for Python.
1742
r
ISSN:
2088-8694
L
0
m
=
L
m
(1
f
(
Q
))
where
f
(
Q
)
=
1
e
Q
Q
.
The
resistance
in
series
with
the
inductance
L
0
m
in
the
magnetization
branch
of
the
equi
v
alent
electri-
calcircuitofthe
d
axis
,
is
determined
in
relation
to
the
increase
in
losses
occurring
with
the
increase
of
the
currents
induced
at
the
entry
and
e
xit
ends
of
the
linor
.
These
losses
may
be
represented
as
the
product
of
the
linor
resistance
R
r
by
the
f
actor
f
(
Q
)
,
ie,
R
r
f
(
Q
)
[23,
24].
From
the
d
q
equi
v
alent
circuit
of
the
LIM,
the
primary
and
linor
v
oltage
equations
in
a
stationary
reference
system
aligned
with
the
linor
flux
are
gi
v
en
by:
u
ds
=
R
s
i
ds
+
R
r
f
(
Q
)(
i
ds
+
i
dr
)
+
d
ds
dt
u
q
s
=
R
s
i
q
s
+
d
q
s
dt
u
dr
=
R
r
i
dr
+
R
r
f
(
Q
)(
i
ds
+
i
dr
)
+
d
dr
dt
+
v
q
r
u
q
r
=
R
r
i
dr
+
d
q
r
dt
v
dr
(10)
Due
to
the
short-circuited
secondary
their
v
oltages
are
zero,
that
is,
u
dr
=
u
q
r
=
0
.
The
linkage
flux
es
are
gi
v
en
by
the
follo
wing
equations:
ds
=
L
s
i
ds
+
L
m
i
dr
L
m
f
(
Q
)(
i
ds
+
idr
)
q
s
=
L
s
i
q
s
+
L
m
i
q
r
dr
=
L
r
i
dr
+
L
m
i
ds
L
m
f
(
Q
)(
i
ds
+
idr
)
q
r
=
L
r
i
q
r
+
L
m
i
q
s
(11)
T
o
de
v
elop
a
state
space
LIM
model
from
10
and
11
is
necessary
to
combine
both
equations.
Be
cause
q
axis
equi
v
alent
circuit
of
the
LIM
is
identical
to
the
q
axis
equi
v
alent
circuit
of
the
induction
motor
(RIM),
the
parameters
do
not
v
ary
with
the
end
ef
fects
and
so
d
q
r
dt
and
di
q
s
dt
in
12
remaind
it
equals
to
13
[6].
di
q
s
dt
=
R
s
L
s
+
1
T
r
i
q
s
L
m
L
s
L
r
v
dr
+
L
m
L
s
L
r
T
r
q
r
+
1
L
s
u
q
s
di
ds
dt
=
R
s
L
s
+
1
T
r
i
ds
+
L
m
L
s
L
r
T
r
dr
+
L
m
L
s
L
r
v
q
r
+
1
L
s
u
ds
d
q
r
dt
=
L
m
T
r
i
q
s
+
v
dr
1
T
r
q
r
d
dr
dt
=
L
m
T
r
i
ds
1
T
r
dr
v
q
r
dv
dt
=
K
f
M
(
dr
i
q
s
q
r
i
ds
)
B
M
v
F
L
M
(12)
di
q
s
dt
=
h
R
s
L
s
+
1
T
r
i
i
q
s
L
m
L
s
L
r
v
dr
+
L
m
L
s
L
r
T
r
q
r
+
1
L
s
u
q
s
d
q
r
dt
=
L
m
T
r
i
q
s
+
v
dr
1
T
r
q
r
(13)
The
RIM
electrical
torque
in
an
arbitrary
reference
frame
is
gi
ving
by
[19],
and
modifying
it
with
relation
v
=
!
1
=
2
f
1
we
obtain
follo
wing
LIM
thrust
force.
F
e
=
3
2
!
r
[
!
(
ds
i
q
s
q
s
i
ds
)
+
(
!
!
r
)
(
dr
i
q
r
q
r
i
dr
)]
in
a
stationary
reference
frame
(
!
=
0)
the
thrust
force
becomes:
F
e
=
3
2
[
q
r
i
dr
dr
i
q
r
]
(14)
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
4,
December
2020
:
1737
–
1749
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
1743
clearing
i
dr
from
d
r
in
11
i
dr
=
dr
L
m
(1
f
(
Q
))
i
ds
L
r
L
m
f
(
Q
)
(15)
clearing
i
q
r
from
q
r
in
11
i
q
r
=
q
r
L
m
i
q
s
Lr
(16)
Substituting
i
dr
and
i
q
r
into
14
results
in:
F
e
=
3
2
L
m
L
r
dr
i
q
s
+
f
(
Q
)
L
r
L
m
f
(
Q
)
q
r
dr
1
f
(
Q
)
L
r
q
r
i
ds
(17)
Then
space
state
mechanical
equation
is
gi
ving
by
18
dv
dt
=
K
f
M
"
dr
i
q
s
+
f
(
Q
)
L
r
L
m
f
(
Q
)
q
r
dr
1
f
(
Q
)
1
Lm
L
r
f
(
Q
)
q
r
i
ds
#
B
M
v
F
L
M
(18)
Considering
short-circuited
linor
circuit
(
u
dr
=
0)
and
solving
for
d
dr
dt
gets
d
dr
dt
=
R
r
(1
+
f
(
Q
))
L
r
L
m
f
(
Q
)
dr
v
q
r
+
R
r
(
L
m
L
r
f
(
Q
))
L
r
L
m
f
(
Q
)
i
ds
(19)
Substituting
the
first
equation
of
11
into
first
equation
of
10
results:
u
ds
=
R
s
+
R
r
f
(
Q
)
L
m
d
f
(
Q
)
dt
i
ds
+
[
L
s
L
m
f
(
Q
)]
di
ds
dt
+
L
m
[1
f
(
Q
)]
di
dr
dt
L
m
d
f
(
Q
)
dt
i
dr
(20)
Clearing
i
dr
from
dr
in
11
i
dr
=
1
L
r
L
m
f
(
Q
)
dr
L
m
(1
f
(
Q
))
L
r
L
m
f
(
Q
)
i
ds
and
substituting
into
20
results
u
ds
=
"
R
s
+
R
r
f
(
Q
)
(
L
r
L
m
)
2
(
L
r
L
m
f
(
Q
))
2
L
m
d
f
(
Q
)
dt
#
i
ds
+
L
s
L
m
f
(
Q
)
L
2
m
(1
f
(
Q
))
2
L
r
L
m
f
(
Q
)
di
ds
dt
+
L
m
(
L
m
L
r
)
(
L
r
L
m
f
(
Q
))
2
d
f
(
Q
)
dt
dr
+
L
m
(1
f
(
Q
))
L
r
L
m
f
(
Q
)
d
dr
dt
substituting
d
dr
dt
in
last
term
into
abo
v
e
equation
we
obtain:
u
ds
=
"
R
s
+
R
r
f
(
Q
)
(
L
r
L
m
)
2
(
L
r
L
m
f
(
Q
))
2
L
m
d
f
(
Q
)
dt
+
R
r
L
m
(1
f
(
Q
))
L
r
L
m
f
(
Q
)
(
L
m
L
r
f
(
Q
))
L
r
L
m
f
(
Q
)
#
i
ds
+
L
s
L
m
f
(
Q
)
L
2
m
(1
f
(
Q
))
2
L
r
L
m
f
(
Q
)
di
ds
dt
+
"
L
m
(
L
m
L
r
)
(
L
r
L
m
f
(
Q
))
2
d
f
(
Q
)
dt
R
r
L
m
1
f
2
(
Q
)
(
L
r
L
m
f
(
Q
))
2
#
dr
L
m
(1
f
(
Q
))
L
r
L
m
f
(
Q
)
v
q
r
Discr
ete
and
continuous
model
of
thr
ee-phase
linear
...
(N.
T
or
o-Gar
c
´
ıa)
Evaluation Warning : The document was created with Spire.PDF for Python.
1744
r
ISSN:
2088-8694
Solving
for
di
ds
dt
di
ds
dt
=
[
R
s
+
R
r
f
(
Q
)]
[
L
r
L
m
f
(
Q
)]
2
L
m
(
L
r
L
m
)
2
d
f
(
Q
)
dt
+
R
r
L
m
[1
f
(
Q
)]
[
L
m
L
r
f
(
Q
)]
[
L
S
L
r
L
s
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)]
[
L
m
f
(
Q
)
L
r
]
l
ds
+
L
m
(
L
m
L
r
)
d
f
(
Q
)
dt
R
r
L
m
1
f
2
(
Q
)
[
L
s
L
r
L
s
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)]
[
L
m
f
(
Q
)
L
r
]
dr
+
L
m
[1
f
(
Q
)]
L
s
L
r
L
S
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)
v
q
r
+
L
r
L
m
f
(
Q
)
L
s
L
r
L
S
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)
u
d
s
(21)
Grouping
the
state
equations
and
changing
the
inde
x
d
and
q
by
and
respecti
v
ely
,
and
omitting
the
primary
and
secondary
(linor)
inde
x
es
because
the
v
oltages
and
currents
are
with
respect
to
primary
and
the
flux
es
are
with
respect
to
secondary
,
we
obtain
22:
di
dt
=
R
S
L
S
+
1
T
r
i
L
m
L
S
L
r
v
+
L
m
L
S
L
r
T
r
+
1
L
S
u
di
dt
=
[
R
S
+
R
r
f
(
Q
)]
[
L
r
L
m
f
(
Q
)]
2
L
m
(
L
r
L
m
)
2
d
f
(
Z
)
dt
+
R
r
L
m
[1
f
(
Q
)]
[
L
m
L
r
f
(
Q
)]
[
L
S
L
r
L
S
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)]
[
L
m
f
(
Q
)
L
r
]
i
+
L
m
(
L
m
L
r
)
d
f
(
Q
)
dt
R
r
L
m
1
f
2
(
Q
)
[
L
S
L
r
L
S
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)]
[
L
m
f
(
Q
)
L
r
]
+
L
m
[1
f
(
Q
)]
L
s
L
r
L
s
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)
v
+
L
r
L
m
f
(
Q
)
L
S
L
r
L
S
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)
u
d
dt
=
L
m
T
r
i
+
v
1
T
r
d
dt
=
R
r
(1
+
f
(
Q
))
L
r
L
m
f
(
Q
)
v
+
R
r
(
L
m
L
r
f
(
Q
))
L
r
L
m
f
(
Q
)
i
dv
dt
=
K
f
M
"
i
+
f
(
Q
)
L
r
L
m
f
(
Q
)
1
f
(
Q
)
1
L
m
L
r
f
(
Q
)
i
#
B
M
v
F
L
M
dx
dt
=
v
(22)
where
v
is
the
mo
v
er
linear
v
elocity;
and
are
the
d
axis
an
q
axis
secondary
flux;
i
and
i
are
the
d
axis
and
q
axis
primary
current;
u
and
u
are
the
d
axis
and
q
axis
primary
v
oltage;
T
r
=
L
r
R
r
is
the
secondary
time
constant;
=
1
L
2
m
L
s
L
r
is
the
leakage
coef
ficient;
K
f
=
3
2
L
m
L
r
is
the
force
constant;
R
s
is
the
winding
resistance
per
phase;
R
r
is
the
secondary
resistance
per
phase
referred
primary;
L
m
is
the
magnetizing
inductance
per
phase;
L
r
is
the
secondary
inductance
per
phase
referred
primary;
L
s
is
the
primary
inductance
per
phase;
F
L
is
the
e
xternal
force
dis
turbance;
M
is
the
total
mass
of
the
mo
v
er;
B
is
the
viscous
friction
and
iron-loss
coef
fici
ent;
is
the
pole
pitch;
D
is
the
primary
length
in
meters;
Q
=
D
R
r
L
r
v
is
a
f
actor
related
to
the
primary
length,
which
quatifies
t
he
end
ef
fects
as
a
function
of
the
speed
and
f
(
Q
)
=
1
e
Q
Q
is
the
f
actor
related
to
the
loss
es
in
the
magnetization
branch.
T
o
discretize
the
st
ate
LIM
model
with
end
ef
fects
we
use
the
backw
ard
dif
ference
method
[25]
and
finally
we
obt
ain
an
approximate
discrete
time
v
ersion
of
the
LIM
model
23
taking
into
account
end
ef
fects.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
4,
December
2020
:
1737
–
1749
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
1745
i
k
+1
=
i
k
R
S
L
S
+
1
T
r
T
i
k
L
m
L
S
L
r
T
v
k
k
+
L
m
L
S
L
r
T
r
T
k
+
1
L
S
T
u
k
i
k
+1
=
i
k
+
[
R
s
+
R
r
f
(
Q
)]
[
L
r
L
m
f
(
Q
)]
2
L
m
(
L
r
L
m
)
2
f
(
Q
)
T
+
R
r
L
m
[1
f
(
Q
)]
[
L
m
L
r
f
(
Q
)]
[
L
s
L
r
L
s
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)]
[
L
m
f
(
Q
)
L
r
]
T
i
k
+
L
m
(
L
m
L
r
)
f
(
Q
)
T
R
r
L
m
1
f
2
(
Q
)
[
L
S
L
r
L
S
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)]
[
L
m
f
(
Q
)
L
r
]
T
k
+
L
m
[1
f
(
Q
)]
L
s
L
r
L
s
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)
T
v
k
k
+
L
r
L
m
f
(
Q
)
L
s
L
r
L
s
L
m
f
(
Q
)
L
r
L
m
f
(
Q
)
L
2
m
+
2
L
2
m
f
(
Q
)
T
u
k
k
+1
=
k
+
L
m
T
r
T
i
k
+
T
v
k
k
1
T
r
T
k
k
+1
=
k
R
r
(1
+
f
(
Q
))
L
r
L
m
f
(
Q
)
T
k
T
v
k
k
+
R
r
(
L
m
L
r
f
(
Q
))
L
r
L
m
f
(
Q
)
T
i
k
v
k
+1
=
v
k
+
K
f
M
T
"
k
i
k
+
f
(
Q
)
L
r
L
m
f
(
Q
)
k
k
1
f
(
Q
)
1
Lm
L
r
f
(
Q
)
k
i
k
#
B
M
T
v
k
F
L
M
T
x
k
+1
=
x
k
+
v
k
T
(23)
where
v
k
=
v
(
k
T
)
is
the
mo
v
er
linear
v
elocity;
k
=
(
k
T
)
and
k
=
(
k
T
)
are
the
d
axis
an
q
axis
secondary
flux;
i
k
=
i
(
k
T
)
and
i
k
=
i
(
k
T
)
are
the
d
axis
and
q
axis
primary
current;
u
k
=
u
(
k
T
)
and
u
k
=
u
(
k
T
)
are
the
d
axis
and
q
axis
primary
v
oltage;
T
r
=
L
r
R
r
is
the
secondary
time
constant;
=
1
L
2
m
L
s
L
r
is
the
leakage
coef
ficient;
K
f
=
3
2
L
m
L
r
is
the
force
constant;
R
s
is
the
winding
resistance
per
phase;
R
r
is
the
secondary
resistance
per
phase
referred
primary;
L
m
is
the
magnetizing
inductance
per
phase;
L
r
is
the
secondary
inductance
per
phase
referred
primary;
L
s
is
the
primary
inductance
per
phase;
F
L
is
the
e
xternal
force
disturbance;
M
is
the
total
mass
of
the
mo
v
er;
B
is
the
viscous
friction
and
iron-loss
coef
ficient;
is
the
pole
pitch;
D
is
the
primary
length
in
meters;
Q
=
D
R
r
L
r
v
k
is
a
f
actor
related
to
the
primary
length,
which
quantifies
the
end
ef
fects
as
a
function
of
the
speed;
f
(
Q
)
=
1
e
Q
Q
is
the
f
actor
related
to
the
losses
in
the
magnetization
branch
and
f
(
Q
)
T
=
d
f
(
Q
)
dt
t
=
k
T
.
3.
RESUL
TS
Figure
5
sho
ws
the
end
ef
fects
on
mo
v
er
v
elocity
,
flux
es
and
currents.
Figures
6,
7
and
8
sho
w
the
system
22
beha
vior
when
the
frequenc
y
of
input
v
oltage
v
ary
.
The
steady
state
v
elocity
is
a
periodic
w
a
v
e
in
all
ca
ses,
b
ut
when
the
fed
frequenc
y
is
lo
wer
,
higher
output
frequenc
y
components
appear
.
Phase
portraits
in
subfigures
6b,
6c,
7b,
7c,
8b
and
8c
with
attracti
v
e
limit
c
ycles
are
sho
wn.
(a)
(b)
Discr
ete
and
continuous
model
of
thr
ee-phase
linear
...
(N.
T
or
o-Gar
c
´
ıa)
Evaluation Warning : The document was created with Spire.PDF for Python.
1746
r
ISSN:
2088-8694
(c)
(d)
Figure
5.
Mo
v
er
v
elocity
,
v
elocity
dif
ferences,
currents
and
flux
es
resulting
from
the
model
simulation
using
ODE45
function
of
Matlab,
taking
into
account
end
ef
fects
in
model
of
LIM
(22)
and
without
end-ef
fects
model
(3),
(a)
Mo
v
er
v
elocity
of
LIM
with
and
without
end-ef
fects,
(b)
V
elocity
dif
ference
vs
mo
v
er
v
elocity
without
end-ef
fects,
(c)
axis
W
ith
and
without
end-ef
fects
currents,
(d)
axis
W
ith
and
without
end-ef
fects
flux
es
(continue)
(a)
(b)
(c)
Figure
6.
LIM
beha
vior
with
30
H
z
input
frequenc
y
.
Mo
v
er
v
elocity
and
phase
portraits
of
some
state
v
ariables,
(a)
Mo
v
er
v
elocity
of
LIM
with
end-ef
fects,
(b)
Phase
portrait
of
mo
v
er
v
elocity
vs
i
,
(c)
Phase
portrait
of
vs
i
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
11,
No.
4,
December
2020
:
1737
–
1749
Evaluation Warning : The document was created with Spire.PDF for Python.