Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
41,
No.
1,
January
2026,
pp.
3
∼
17
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v41.i1.pp3-17
❒
3
Dynamic
beha
vior
of
induction
machines
in
A
TP-EMTP
with
space
harmonics
J
ose
Manuel
Aller
,
Ruben
Nicolas
Gue
v
ara,
Bryam
Ste
v
en
Pulla
Department
of
Electrical
Ener
gy
,
Uni
v
ersidad
Politecnica
Salesiana
(UPS),
Cuenca,
Ecuador
Article
Inf
o
Article
history:
Recei
v
ed
Apr
4,
2025
Re
vised
Dec
5,
2025
Accepted
Dec
14,
2025
K
eyw
ords:
A
TP-EMTP
Induction
machine
Space
harmonics
T
ransient
analysis
VBR
model
ABSTRA
CT
This
w
ork
de
v
elops
a
space-v
ector
model
of
a
squirrel-cage
induction
machine
that
incorporates
the
ef
fects
of
spatial
harmonics
arising
from
the
windi
ng
distri-
b
ution.
The
modeli
ng
approach
includes
the
rst,
fth,
and
se
v
enth
spatial
har
-
monics,
which
are
the
components
with
the
greatest
inuence
on
the
machine’
s
magnetic
eld.
Simulation
results
highlight
the
impact
of
these
harmonics
on
the
stator
and
rotor
currents,
the
electromagnetic
torque,
and
the
machine’
s
speed.
T
o
b
uild
the
model,
the
v
oltage
behind
reactances
(VBR)
technique
is
emplo
yed,
enabling
a
h
ybrid
strate
gy
that
combines
circuit-based
modeling
tools—such
as
A
TP-EMTP—with
computational
programming
in
models
t
o
complement
the
solution
of
the
dif
ferential
equations
go
v
erning
the
beha
vior
of
the
electrome-
chanical
system.
This
methodology
ef
fecti
v
ely
transforms
the
induction
ma-
chine
into
a
dynamic
Th
`
ev
enin-equi
v
alent
circuit
for
each
phas
e
of
the
con
v
erter
.
This
study
pro
vides
a
useful
frame
w
ork
for
e
v
aluating
ho
w
space
harmonics
af-
fect
t
he
performance
and
operating
characteristics
of
induction
machines.
The
models
were
implemented
using
the
A
TP-EMTP
softw
a
re
and
its
graphical
in-
terf
ace,
A
TPDra
w
.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Jose
Manuel
Aller
Department
of
Electrical
Ener
gy
,
Uni
v
ersidad
Politecnica
Salesiana
(UPS)
Sede
Cuenca,
Calle
V
ieja
y
Elia
Liut,
010105
Cuenca,
Ecuador
Email:
jaller@ups.edu.ec
1.
INTR
ODUCTION
The
induction
machine
is
currently
the
most
widely
used
electromechanical
con
v
erter
.
Its
rob
ustnes
s,
reliability
,
and
lo
w
maintenance
requirements
enable
it
to
operate
e
v
en
under
v
ery
harsh
conditions
[1],
[2].
Its
in
v
ention
is
attrib
uted
to
Nicola
T
esla,
who
de
v
eloped
the
concept
of
the
rotating
magnetic
eld
in
1888
[3].
Ho
we
v
er
,
during
the
same
period,
the
Italian
Galilera
Ferraris
presented
similar
w
ork
between
1885
and
1888
[4],
[5].
Later
,
in
Doli
v
o-Dobro
v
olsk
y
[6]
and
Ferraris
and
Arno
[7]
introduced
the
use
of
three-phase
coils
in
the
induction
machine,
which
is
no
w
standard
in
electrical
distrib
ution
systems.
This
de
v
elopment
laid
the
foundation
for
modern
electrical
industry
,
pro
viding
signicant
performance,
v
ersatility
,
and
reliability
across
v
arious
sectors
[8].
The
space
harmonics
generated
by
the
space
distrib
ution
of
the
stator
and
rotor
windings
af
fect
the
machine’
s
performance,
which
has
been
e
xtensi
v
ely
studied
[9],
[10].
These
harmonics
alter
the
distrib
ution
of
the
magnetic
eld
and
can
generate
uctuations
in
electrical
torque,
distorted
v
oltage
w
a
v
eforms,
losses
and
noise
that
af
fect
the
ef
cienc
y
and
operational
characteristics
of
the
machine
[11]–[13].
This
article
models
the
induction
machine,
considering
the
rst,
fth,
and
se
v
enth
space
harm
o
ni
cs,
which
are
the
most
important
components
of
the
eld
in
the
air
g
ap.
This
type
of
model
allo
ws
for
the
de
v
el-
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
4
❒
ISSN:
2502-4752
opment
of
speed
measurement
strate
gies
u
s
ing
the
harmonic
spectrum
of
the
currents
[14],
[15].
The
approach
used
consists
of
representing
the
machine
through
space
v
ectors
where
both
the
stator
and
rotor
refer
to
their
respecti
v
e
reference
systems
[16],
[17],
due
to
the
presence
of
space
harmonics
that
pre
v
ent
simplifying
the
dependence
on
the
angular
position
θ
of
the
rotor
when
transforming
the
equations
to
the
stator
reference.
The
v
oltage
behind
reactance
(VBR)
model
is
a
widely
used
method
for
representing
induction
ma-
chines
due
to
its
impro
v
ed
numerical
ef
cienc
y
and
e
xibility
compared
to
traditional
models
[18],
[19].
In
the
modeling,
the
VBR
approach
is
applied,
which
f
acilitates
analysis
by
representing
the
internal
v
oltage
of
the
machine
as
a
combination
of
electromoti
v
e
force,
r
esistance,
and
inductance
whose
parameters
v
ary
o
v
er
time
with
the
rotor’
s
position.
This
approach
has
been
utilized
by
se
v
eral
authors
for
the
inte
gration
of
the
machine
with
its
electronic
dri
v
es,
b
ut
the
inclusion
of
space
harmonics
in
this
type
of
modeling
is
unprecedented,
e
v
en
though
[20]
has
proposed
an
approach
based
on
the
VBR
technique
to
include
the
ef
fects
of
saturation
in
the
induction
machine.
Furthermore,
this
model
is
implemented
using
the
A
TP-EMTP
tool
and
its
graphical
en
vironment
A
TPDra
w
[21],
[22]
to
study
the
impact
of
space
harmonics
on
k
e
y
v
ariables
such
as
angular
speed,
electrical
torque,
and
currents.
This
approach
allo
ws
for
obtaining
a
more
accurate
model
of
the
induction
machine,
which
f
acilitates
the
e
v
aluation
of
losses,
ef
cienc
y
,
and
its
operational
characteristics
[23]–[25].
2.
METHOD
2.1.
Statement
of
the
theor
etical
pr
oblem
The
analysis
of
the
dynamic
beha
vior
of
induction
machines
is
fundamental
to
ensuring
the
st
ability
and
ef
cienc
y
of
modern
electrical
systems
[26],
[27].
These
machines
represent
an
essential
component
in
industrial,
commercial,
and
po
wer
generation
applications,
where
their
proper
modeling
allo
ws
for
predicting
their
response
under
dif
ferent
operating
conditions.
Ho
we
v
er
,
the
presence
of
space
harmonics
in
their
opera-
tion
introduces
comple
x
phenomena
that
af
fect
their
performance,
generating
current
distortions,
v
ariat
ions
in
electromagnetic
torque,
and
additional
losses
in
the
system
[11]-[13].
Space
har
monics
are
produced
by
the
geometry
of
the
air
g
ap,
the
winding
distrib
ution,
and
magnetic
saturation,
generating
adv
erse
ef
fects
on
the
operation
of
induction
machines.
Consequently
,
the
precise
sim-
ulation
of
these
ef
fects
is
crucial
for
e
v
aluating
their
impact
on
the
stability
of
the
electrical
system
and
the
ef
cienc
y
of
de
vices
connected
to
the
grid
[20].
In
this
conte
xt,
there
is
a
need
to
de
v
elop
a
model
that
realistically
represents
the
dynamic
beha
vior
of
induction
machines,
considering
the
ef
fects
of
space
harmonics
and
v
alidating
it
through
simulations
in
A
TP-
EMTP
.
This
study
aims
to
bridge
the
e
xisting
g
ap
between
theory
and
practice
by
implementing
more
accurate
models.
The
v
alidation
of
these
models
will
impro
v
e
the
design
of
induction
machines
by
implem
enting
ne
w
technologies
to
lter
the
harmonics
produced
by
the
induction
machine.
Furthermore,
this
model
will
allo
w
for
observing
the
impact
of
these
harmonics
on
electrical
parameters
suc
h
as
torque,
angular
speed,
currents
in
the
stator
,
among
others.
In
this
w
ay
,
it
will
contrib
ute
to
mitig
ating
these
ef
fects
and
aid
in
the
constructability
of
the
machine.
2.2.
VBR
model
of
the
IM
considering
rst
harmonic
space
distrib
ution
The
v
oltage
equations
of
the
IM
in
space
v
ectors
can
be
represented
as
[28]:
⃗
v
s
0
=
R
s
0
0
R
r
⃗
i
s
⃗
i
r
+
p
"
⃗
λ
s
⃗
λ
r
#
(1)
where,
"
⃗
λ
s
⃗
λ
r
#
=
L
s
M
1
e
j
θ
M
1
e
−
j
θ
L
r
⃗
i
s
⃗
i
r
(2)
From
(1)
and
(2),
we
obtain,
p
⃗
λ
r
=
−
R
r
L
r
⃗
λ
r
−
M
1
e
−
j
θ
⃗
i
s
(3)
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
41,
No.
1,
January
2026:
3–17
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
5
and,
⃗
ir
=
1
Lr
⃗
λ
r
−
M
1
e
−
j
θ
⃗
i
s
(4)
Introducing
(3)
and
(4)
into
the
stator
v
oltage
equation
⃗
v
s
(1),
we
obtain
the
VBR
model,
which
is
a
dynamic
Th
`
ev
enin
equi
v
alent:
⃗
v
s
=
R
eq
,
⃗
i
s
+
L
eq
,
p
⃗
i
eq
+
⃗
e
eq
,
(5)
where:
R
eq
=
R
s
+
R
r
M
2
1
L
2
r
(6)
L
eq
=
L
s
−
M
2
1
L
r
(7)
⃗
e
eq
=
M
1
L
r
j
˙
θ
−
R
r
L
r
e
j
θ
⃗
λ
r
(8)
The
electrical
torque
in
space
v
ectors
is
obtained
from,
T
e
1
=
M
1
ℑ
m
n
⃗
i
s
⃗
i
r
e
j
θ
∗
o
=
M
1
L
r
ℑ
m
n
⃗
i
s
⃗
λ
r
e
j
θ
∗
o
(9)
In
Figure
1,
the
VBR
model
of
the
IM
in
space
v
ectors
is
presented,
referenced
to
the
stator
and
rotor
ax
es,
considering
that
the
space
distrib
ution
is
of
the
rst
harmonic.
⃗
i
s
R
eq
=
R
s
+
R
r
M
2
1
L
2
r
L
eq
=
L
s
−
L
r
M
2
1
L
2
r
−
+
⃗
e
eq
⃗
v
s
p
⃗
λ
r
=
−
R
r
L
r
⃗
λ
r
−
M
1
⃗
i
s
⃗
e
eq
=
M
1
L
r
j
˙
θ
−
R
r
L
r
e
j
θ
⃗
λ
r
T
e
=
M
1
L
r
Im
n
⃗
i
s
⃗
λ
r
e
j
θ
∗
o
Figure
1.
VBR
model
of
the
IM
corresponding
to
the
rst
space
harmonic
in
stator
-rotor
coordinates
2.3.
VBR
model
of
the
IM
considering
higher
space
harmonics
T
o
assess
the
ef
fect
of
space
harmonics
in
the
IM
model,
the
contents
of
the
rst,
fth,
and
se
v
enth
space
harmonics
are
included,
as
the
y
ha
v
e
the
most
signicant
magnitudes.
Ho
we
v
er
,
the
proposed
method
can
be
generalized
to
an
y
number
of
harmonics
in
space.
The
third
harmonic
in
balanced
three-phase
machines
is
generally
ne
gligible
and,
for
this
reason,
it
has
not
been
considered
in
this
model.
The
(1)
is
v
alid
for
an
y
number
of
harmonics.
The
fundamental
dif
ference
lies
in
the
determination
of
the
inductance
matrix
and
its
relationship
with
the
ux
links:
"
⃗
λ
s
⃗
λ
r
#
=
L
s
P
M
sr
i
(
θ
)
P
M
∗
r
s
i
(
θ
)
L
r
⃗
i
s
⃗
i
r
(10)
Dynamic
behavior
of
induction
mac
hines
in
A
TP-EMTP
with
space
harmonics
(J
ose
Manuel
Aller)
Evaluation Warning : The document was created with Spire.PDF for Python.
6
❒
ISSN:
2502-4752
where:
X
M
sr
i
(
θ
)
=
M
1
e
j
θ
+
M
5
e
−
j
5
θ
+
M
7
e
j
7
θ
+
M
11
e
−
j
11
θ
+
M
13
e
j
13
θ
+
·
·
·
From
e
xpression
(10),
we
obtain:
⃗
ir
=
1
Lr
⃗
λr
−
X
M
∗
r
s
i
(
θ
)
⃗
i
s
(11)
and
replacing
(11)
in
(1),
we
ha
v
e:
p
⃗
λ
r
=
−
R
r
L
r
⃗
λ
r
−
X
M
∗
r
s
i
(
θ
)
⃗
i
s
(12)
The
stator
v
oltage
equation,
obtained
from
(1),
including
the
ef
fect
of
the
space
harmonics,
is:
⃗
v
s
=
R
s
⃗
i
s
+
p
L
s
⃗
i
s
+
X
M
sr
i
(
θ
)
⃗
i
r
(13)
By
substituting
(11)
and
(12)
into
(13),
we
obtain:
R
eq
=
R
s
+
R
r
L
2
r
M
2
1
+
M
2
5
+
M
2
7
+
.
·
·
·
+
2
(
M
1
M
5
+
M
1
M
7
+
M
5
M
7
)
cos
6
θ
+
·
·
·
(14)
The
equi
v
alent
inductance
L
eq
is:
L
eq
=
L
s
−
1
L
r
M
2
1
+
M
2
5
+
M
2
7
+
.
·
·
·
+
2
(
M
1
M
5
+
M
1
M
7
+
M
5
M
7
)
cos
6
θ
+
.
·
·
·
(15)
The
electromoti
v
e
force
⃗
e
e
is:
⃗
e
eq
=
−
R
r
L
r
A
+
j
˙
θ
∂
A
∂
θ
⃗
λ
r
L
r
+
12
˙
θ
B
L
r
sin
6
θ
⃗
i
s
(16)
where:
A
=
M
1
e
j
θ
+
M
5
e
−
j
5
θ
+
M
7
e
j
7
θ
B
=
M
1
M
5
+
M
1
M
7
+
4
M
5
M
7
cos
6
θ
Finally
,
the
electrical
torque
can
be
determined
as:
T
e
=
T
e
1
+
T
e
5
+
T
e
7
(17)
where,
T
e
1
=
M
1
L
r
ℑ
m
n
⃗
i
s
⃗
λ
r
e
j
θ
∗
o
(18)
T
e
5
=
−
5
M
5
L
r
ℑ
m
n
⃗
i
s
⃗
λ
r
e
j
5
θ
∗
o
(19)
T
e
7
=
7
M
7
L
r
ℑ
m
n
⃗
i
s
⃗
λ
r
e
j
7
θ
∗
o
(20)
The
VBR
model
is
a
dynamic
equi
v
alent
of
the
dif
ferential
equations
of
the
machine
that
allo
ws
it
to
be
represented
by
an
electrical
circuit
with
R
eq
,
L
eq
,
and
an
electromoti
v
e
force
⃗
e
eq
.
The
v
ariables
used
are
space
v
ectors
obtained
from
the
general
e
xpression:
⃗
x
s
=
r
2
3
n
x
a
(
t
)
+
e
j
2
π
3
x
b
(
t
)
+
e
j
4
π
3
x
c
(
t
)
o
(21)
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
41,
No.
1,
January
2026:
3–17
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&
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Sci
ISSN:
2502-4752
❒
7
The
equi
v
alent
resistance
R
eq
and
inductance
L
eq
can
be
directly
used
in
each
of
the
phases
of
an
electrical
circuit.
The
rotor’
s
ux
link
can
be
obtained
by
numerically
inte
grating
the
dif
ferential
(3).
W
ith
this
information,
it
is
possible
to
determine
the
space
v
ector
of
the
electromoti
v
e
force
⃗
e
eq
.
T
o
determine
the
electromoti
v
e
forces
of
each
phase
of
the
IM,
the
in
v
erse
transformations
are
used:
x
a
(
t
)
=
r
2
3
ℜ
e
{
⃗
x
s
}
(22)
x
b
(
t
)
=
r
2
3
ℜ
e
n
⃗
x
s
e
j
4
π
3
o
(23)
x
a
(
t
)
=
r
2
3
ℜ
e
n
⃗
x
s
e
j
2
π
3
o
(24)
2.4.
VBR
model
of
the
IM
in
A
TPDraw
Figure
2
represents
the
dynamic
model
of
an
induction
machine
de
v
eloped
in
the
graphical
en
viron-
ment
of
A
TPDra
w
from
the
A
TP-EMTP
program,
including
the
inuence
of
space
harmonics.
The
model
includes
the
three
phases
of
the
IM,
calculating
within
the
simulation
models
block
the
rotor’
s
ux
link,
which
allo
ws
for
the
determination
of
instantaneous
v
alues
of
R
eq
,
L
eq
,
e
a
,
e
b
y
e
c
.
The
dependent
electromoti
v
e
forces
must
be
simulated
using
Norton
equi
v
alents
because
the
tool
does
not
ha
v
e
isolated
dependent
v
oltage
sources
from
ground.
One
of
the
main
dif
culties
with
programming
in
models
is
that
it
requires
separating
the
v
ariable
s
into
real
and
imaginary
parts,
as
this
tool
does
not
handle
comple
x
numbers.
The
programming
used
to
create
the
model
can
be
found
in
section
4.2.1,
where
the
v
ariables
denoted
with
x
represent
the
real
part
and
those
with
y
represent
the
imaginary
part
of
the
space
v
ector
.
Figure
2.
Model
in
A
TPDra
w
of
a
VBR
induction
machine
for
space
harmonic
analysis
Dynamic
behavior
of
induction
mac
hines
in
A
TP-EMTP
with
space
harmonics
(J
ose
Manuel
Aller)
Evaluation Warning : The document was created with Spire.PDF for Python.
8
❒
ISSN:
2502-4752
3.
RESUL
TS
3.1.
Results
of
the
VBR
model
of
the
IM
In
T
able
1,
the
data
used
to
analyze
the
beha
vior
of
the
IM
is
presented.
T
able
1.
Data
of
the
IM
used
in
the
VBR
modeling
P
arameter
V
alue
R
s
0
.
353
Ω
R
r
0
.
424
Ω
L
ls
2
.
59
mH
L
lr
3
.
88
mH
p
2
pair
M
sr1
67
.
47
mH
M
sr5
0
.
60
mH
M
sr7
0
.
60
mH
J
0
.
163
k
g
m
2
k
r
0
.
002
W
/
(
r
a
d/s
)
2
3.1.1.
Results
corr
esponding
to
the
simulation
with
fundamental
harmonic
in
space
In
Figure
3,
the
results
obtained
with
the
VBR
model
of
the
IM
are
sho
wn,
considering
the
beha
vior
of
the
rst,
fth,
and
se
v
enth
space
harmonics.
In
Figure
4,
the
beha
vior
of
the
currents
obtained
is
represented
for
the
same
case,
considering
in
the
simulation
only
the
rst
space
harmonic.
Figure
5
presents
a
detail
of
the
machine’
s
currents
in
steady
state
when
only
the
rst
harmonic
is
considered
in
the
spac
e
distrib
ution.
In
this
case,
the
stator
currents
also
e
xhibit
a
rst
temporal
harmonic.
Figure
3.
Electrical
torque
and
angular
speed
of
the
IM
considering
the
fundamental
harmonic
Figure
4.
Currents
in
phases
a,
b,
and
c
of
the
IM
stator
,
considering
the
fundamental
harmonic
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
41,
No.
1,
January
2026:
3–17
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ISSN:
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❒
9
Figure
6
sho
ws
the
F
ouri
er
spectrum
of
the
currents
obtained
in
t
he
simulation,
where
it
is
observ
ed
that
only
the
presence
of
the
rst
temporal
harmonic
e
xists
due
to
the
space
distrib
ution,
which
includes
only
the
fundamental
component.
Finally
,
Figure
7
presents
the
de
v
elopment
of
the
space
v
ectors
of
the
rotor
ux
of
the
IM
during
startup.
Figure
5.
Currents
in
phases
a,
b,
and
c
of
the
IM
stator
,
considering
the
fundamental
harmonic
Figure
6.
F
ourier
spectrum
of
the
currents
obtained
in
the
VBR
simulation
of
the
IM
with
rst
harmonic
space
distrib
ution
Dynamic
behavior
of
induction
mac
hines
in
A
TP-EMTP
with
space
harmonics
(J
ose
Manuel
Aller)
Evaluation Warning : The document was created with Spire.PDF for Python.
10
❒
ISSN:
2502-4752
Figure
7.
Ev
olution
of
the
space
v
ector
of
the
rotor
ux
linkages
⃗
λ
r
during
the
startup
of
the
IM
3.1.2.
Results
corr
esponding
to
the
simulation
considering
rst,
fth,
and
se
v
enth
space
harmonics
In
Figure
8,
the
results
obtained
with
the
VBR
model
of
the
IM
are
sho
wn,
considering
the
beha
vior
of
the
rst
three
space
harmonics.
In
Figure
9,
the
beha
vior
of
the
currents
obtained
is
represented
for
the
same
case,
considering
the
simulation
of
the
rst
three
space
harmonics.
Figure
10
sho
ws
a
detail
of
the
steady-state
currents
where
the
presence
of
temporal
harmonics
caused
by
the
space
harmonic
distrib
ution
in
the
air
g
ap
is
observ
ed.
Figure
8.
Electrical
torque
and
angular
speed
of
the
IM
considering
the
rst,
fth,
and
se
v
enth
space
harmonics
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
41,
No.
1,
January
2026:
3–17
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Indonesian
J
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Eng
&
Comp
Sci
ISSN:
2502-4752
❒
11
Figure
9.
Currents
in
phases
a,
b,
and
c
of
the
IM
stator
,
considering
the
rst,
fth,
and
se
v
enth
harmonics
Figure
11
presents
the
F
ourier
spectrum
of
t
h
e
currents
obtained
in
the
simulati
on
,
where
the
pres
ence
of
the
rst,
fth,
and
se
v
enth
temporal
harmonics
in
the
currents
can
be
observ
ed,
along
with
their
multiples,
due
to
the
space
distrib
ution
that
includes
the
rst,
fth,
and
se
v
enth
space
harmonics.
Finally
,
Figure
12
presents
the
de
v
elopment
of
the
s
pace
v
ectors
of
the
rotor
ux
of
the
IM
during
startup,
considering
the
rst
three
space
harmonics.
Figure
10.
Detail
of
the
steady-state
currents
in
phases
a,
b,
and
c
of
the
IM
stator
,
considering
the
rst,
fth,
and
se
v
enth
harmonics
Dynamic
behavior
of
induction
mac
hines
in
A
TP-EMTP
with
space
harmonics
(J
ose
Manuel
Aller)
Evaluation Warning : The document was created with Spire.PDF for Python.
12
❒
ISSN:
2502-4752
Figure
11.
F
ourier
spectrum
of
the
currents
obtained
in
the
VBR
simulation
of
the
IM
with
space
distrib
ution
of
the
rst,
fth,
and
se
v
enth
space
harmonics
Figure
12.
Ev
olution
of
the
space
v
ector
of
the
rotor
ux
linkages
⃗
λ
r
during
the
startup
of
the
IM
considering
the
rst
three
space
harmonics
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
41,
No.
1,
January
2026:
3–17
Evaluation Warning : The document was created with Spire.PDF for Python.