Discrete and continuous model of three-phase linear induction motors considering attraction force and end-effects

International Journal of Power Electronics and Drive System (IJPEDS)
Vol. 11, No. 4, December 2020, pp. 1737∼1749
ISSN: 2088-8694, DOI: 10.11591/ijpeds.v11.i4.pp1737-1749 r 1737
Discrete and continuous model of three-phase linear
induction motors considering attraction force and
end-effects
Nicolás Toro-Garcı́a1
, Yeison Alberto Garcés-Gómez2
, Fredy E. Hoyos3
1
Department of Electrical and Electronics Engineering and Computer Sciences, Universidad Nacional de Colombia - Sede
Manizales, Colombia
2
Unidad Académica de Formación en Ciencias Naturales y Matemáticas, Universidad Católica de Manizales, Colombia
3
Facultad de Ciencias, Escuela de Fı́sica, Universidad Nacional de Colombia - Sede Medellı́n, Colombia
Article Info
Article history:
Received Mar 12, 2020
Revised Jun 10, 2020
Accepted Jun 26, 2020
Keywords:
Continuous time model
Discrete control systems
Discrete time model
Linear induction motors
Non-linear behaviors
ABSTRACT
The continuous model of the linear induction motor (LIM) has been made considering
the edge effects and the attraction force. Taking 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
equivalent LIM circuit have been obtained. The discrete model has been developed
to quickly obtain computational solutions and to analyze non-linear behaviors through
the application of discrete control systems. In order to obtain the discrete model of the
LIM we have started from the solution of the continuous model. To develop the model,
the magnetizing inductance has been considered, which reflects the edge effects. In the
results, the model is compared without considering the edge effects or the attraction
force with the proposed model.
This is an open access article under the CC BY-SA license.
Corresponding Author:
Y. A. Garcés-Gómez,
Unidad Académica de Formación en Ciencias Naturales y Matemáticas,
Universidad Católica de Manizales,
Cra 23 No 60-63, Manizales, Colombia.
Email: ygarces@ucm.edu.co
1. INTRODUCTION
The linear induction motor (LIM) was invented and patented more than a hundred years ago being
impractical due to the difficulties in its construction by not being able to have small air space without roughness
in addition to not being able to achieve good efficiency factors. Nowadays, technological advances have allowed
the LIM to have greater importance, extending its use to important industrial and research applications [1-7].
Linear induction motors are three-phase AC devices that work by the general principles of electromechanical
energy transformation like other induction motors and are constructed for to produce movement 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 effects 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, different phe-
nomena appear in the magnetic circuit that must be re-modelled. This leads to the development of new
theories [1, 2, 4].
Journal homepage: http://ijpeds.iaescore.com

1738 r ISSN: 2088-8694
When it is required to generate a linear movement from RIM, the use of mechanical elements is
necessary, this can be avoided with the use of LIM. In addition to eliminating the use of mechanical elements,
the latter have the advantages of high acceleration and deceleration capacity, use in levitation systems by
normal magnetic forces, lower maintenance costs, low noise, possibility of use in systems with curves and
slopes, braking that does not depend on the system conditions, among others [8-12].
There is little work on sampling LIM dynamics; therefore, it is of great importance to investigate an
accurate representation of the sampled data of the complete dynamics of linear induction motors, and to design
slide controllers at discrete time [7, 13]. With respect to the non-linear models of control strategies applied to
LIMs, an in-depth review is made of in [7, 14, 15], also in the terms of the mathematical model. The physical
model of the LIM has been developed to model the figure system as shown in Figure 1. The construction
aspects of the LIM have been fully developed in [7, 16]. The organization of the document is as follows.
Section 2 develops the modelling of the linear induction motor taking into account the effects of edges and
forces in the equivalent 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 work.
Figure 1. Physical system implemented to obtain the linear induction motor (LIM) model parameters
2. LIM MODEL CONSIDERING ATTRACTION FORCE AND END-EFFECTS
Based on the d − q theory, the LIM model has been made with its equivalent circuit starting from
[date13]. It is taken into account that the q axis of the linear induction motor is equivalent to the rotary motor
so the parameters are invariable. However, if the currents of the d axis are analyzed, they affect the flow of the
air gap causing a decrease in λdr. Thus the equivalent circuit of the rotary motor in the d axis is not applicable
to the linear motor if the edge effects are taken into account.
In rotary motors, the edge effects are not appreciable, which is the case with linear motors.
Furthermore, these effects increase as the motor speed increases, which leads to an analysis of these effects
as a function of speed, taking into account that they also have different behaviour at the output and arrival ends
of the linear motor, since they decrease more slowly at the input than at the output due to the increase in the
time constant that modifies the derivative of the function.
2.1. Equivalent circuit for LIM
The construction model of the linear motor is illustrated in Figure 2(a). As it can be seen, as the
primary moves, it interacts with another region of the liner different from the previous one and that also op-
poses the increase of penetrating magnetic flux and accumulating more flux in the air gap which affects the
performance of the linear motor as reported [17–19]. This effect can be analyzed in Figure 2(c).
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Int J Pow 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 effect of the primary coil generating eddy currents,
(b) Input and output current waveforms, (c) Flow waveform in the air gap
As the coils of the primary move, the newly generated field enters the secondary as the previous field
disappears at the output of the 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 λqr = 0. Noting that as far as
λqr = 0 and λdr does not change, the end effect does not play any role in equivalent circuit. Since iqε = −iqs
the entry q axis eddy current keeps λqr = 0. Hence, the q − axis equivalent circuit is identical to the case
of the rotary induction motor. However, the d − axis air gap flux is affected much by the eddy current since
d-axis entry eddy current in linear induction motor, idε, reduces λdr.
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) Effective air gap MMF and (b) eddy current profile in normalized time scale.
Discrete and continuous model of three-phase linear ... (N. Toro-Garcı́a)

1740 r ISSN: 2088-8694
2.2. Magnetizing inductance reflecting the end effects
When the primary is moving, the primary MMF observed by the rail will be decreased at the entry and
be reflected in the output rail to keep the air gap in flow ( continuous). In particular, the polarity of the input
eddy current is contrary to that of the output eddy current, as they are naturally opposed to the generation and
extinction of the fields, specifically. Note that the input eddy current has a higher decay period relative to the
output eddy current, since the inductance is greater in the air gap than in the free air. The pattern of the eddy
currents is drawn in Figure 4 which is based on the standard time scale. [20].
Figure 4. The equivalent linear motor circuits taking into account the end-effects, (a) The equivalent d-axis
circuit, (b) The equivalent q-axis circuit
Observing that the input of the d-axis of the eddy currents decreases with the time differential Tr, the
mean value of the eddy current input from the d-axis idε is given by 1
iε =
ids
Tv
Z Tv
0
e−t/Tr
dt (1)
where Tv = D/v, and D, v are the motor extension and velocity. Noting that Tv = D/v is the time
for the motor to travels a point. Because the travel length for the period Tr is eqivalent to vTr and normalizing
the motor size with vTr as 2 [21].
Q =
vTv
vTr
=
DRr
(Lm + Llr)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, so that at zero
speed, the length of the motor is infinitely long. As the speed increases, the length of the motor will effectively
decrease. Using 2, (1) can be rewritten as follows:
iε =
ids
Q
Z Q
0
e−x
dx = ids
1 − e−Q
Q
(3)
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Int J Pow Elec & Dri Syst ISSN: 2088-8694 r 1741
The effective magnetizing current is thus decreased in such a manner that:
ids − iε = ids

1 −
1 − e−Q
Q

(4)
The reduction of the magnetizing current caused by the eddy current, can, however, be justified by
changing the magnetizing inductance in a way that:
L0
m = Lm(1 − f(Q)) (5)
where f(Q) = (1 − e−Q
)/Q [20]. As velocity tends to zero, L0
m converges to Lm i.e., the LIM dynamics
becomes equivalent to the RIM dynamics as the end effect disappears. Figure 4 shows the effective air gap
MMF and the eddy current profile in normalized time scale.
2.3. Equivalent series resistor reflecting rail eddy current losses
When inflow and outflow eddy currents flow along the rail, an ohmic loss of Rr will occur. Note that
the average square value of the input eddy current over the length of the motor is given by:
iεRMS =

i2
ds
Q
Z Q
0
e−2x
dx
#1
2
= ids

1 − e−2Q
2Q
1
2
(6)
Hence, the loss caused by the entry eddy current is evaluated as [22] in 7:
Pentry = i2
εRMSRr = i2
dsRr
1 − e−2Q
2Q
(7)
Using the methodology of [22], we can assess the losses due to the eddy current by the temporal rate
of the magnetic energy change when exiting the air space of the motor. Note from 3 that the total eddy current
in the air gap is equal to ids(1 − e−Q
). This flow must be eliminated in the exit rail for Tv to satisfy the steady
flow condition of the air gap. Thus, the loss due to the output eddy current is provided by 8:
Pexit =
Lri2
ds(1 − e−Q
)2
2Tv
= i2
dsRr
(1 − e−Q
)2
2Q
(8)
Adding (7) and (8), the total ohmic losses due to eddy currents in the rail are given by this loss of
power can be shown as a resistance wired in a series Rrf(Q) in the magnetizing current branch. 4 the total
ohmic losses due to eddy currents in the rail are given by this loss of power can be shown as a resistance wired
in a series.
Duncan’s circuit has been developed considering velocity and power 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 effects have been
considered.
Duncan’s model has been adopted in order to obtain a space state representation both continuous-time
and discrete-time. Several techniques have been developed for non linear dynamics analysis in the state space.
Parameter Q, function f(Q), Magnetizing Inductance Reflecting the End Effects and Equivalent Se-
ries Resistor Reflecting Rail Eddy Current Losses have been derived from circuit theory.
The Q factor is associated with the length of the primary, and to a certain degree, quantifies the end
effects as a function of the velocity v as described by (9).
Q =
DRr
Lrv
(9)
Note that the Q factor is inversely dependent on the velocity, i.e., for a zero velocity the Q factor may
be considered infinite, and the end effects may be ignored. As the velocity increases the end effects increases,
which causes a reduction of the LIM’s magnetization current. This effect may be quantified in terms of the
magnetization inductance with the equation:

1742 r ISSN: 2088-8694
L0
m = Lm(1 − f(Q))
where f(Q) = 1−e−Q
Q .
The resistance in series with the inductance L0
m in the magnetization branch of the equivalent 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 exit ends of the linor. These losses may be represented as the product of the
linor resistance Rr by the factor f(Q), ie, Rrf(Q) [23, 24].
From the d − q equivalent circuit of the LIM, the primary and linor voltage equations in a stationary
reference system aligned with the linor flux are given by:
uds = Rsids + Rrf(Q)(ids + idr) +
dαds
dt
uqs = Rsiqs +
dαqs
dt
udr = Rridr + Rrf(Q)(ids + idr) +
dαdr
dt
+
π
τ
vλqr
uqr = Rridr +
dαqr
dt
−
π
τ
vλdr
(10)
Due to the short-circuited secondary their voltages are zero, that is, udr = uqr = 0.
The linkage fluxes are given by the following equations:
λds = Lsids + Lmidr − Lmf(Q)(ids + idr)
λqs = Lsiqs + Lmiqr
λdr = Lridr + Lmids − Lmf(Q)(ids + idr)
λqr = Lriqr + Lmiqs
(11)
To develop a state space LIM model from 10 and 11 is necessary to combine both equations. Because
q − axis equivalent circuit of the LIM is identical to the q − axis equivalent circuit of the induction motor
(RIM), the parameters do not vary with the end effects and so
dλqr
dt and
diqs
dt in 12 remaind it equals to 13 [6].
diqs
dt
= −

Rs
ρLs
+
1 − ρ
ρTr

iqs −
Lmπ
ρLsLrτ
vλdr +
Lm
ρLsLrTr
λqr +
1
ρLs
uqs
dids
dt
= −

Rs
ρLs
+
1 − ρ
ρTr

ids +
Lm
ρLsLrTr
λdr +
Lmπ
ρLsLrτ
vλqr +
1
ρLs
uds
dλqr
dt
=
Lm
Tr
iqs +
π
τ
vλdr −
1
Tr
λqr
dλdr
dt
=
Lm
Tr
ids −
1
Tr
λdr −
π
τ
vλqr
dv
dt
=
Kf
M
(λdriqs − λqrids) −
B
M
v −
FL
M
(12)
diqs
dt = −
h
Rs
ρLs
+ 1−ρ
ρTr
i
iqs − Lmπ
ρLsLrτ vλdr + Lm
ρLsLrTr
λqr + 1
ρLs
uqs
dλqr
dt = Lm
Tr
iqs + π
τ vλdr − 1
Tr
λqr
(13)
The RIM electrical torque in an arbitrary reference frame is giving by [19], and modifying it with
relation v = τ ω1
π = 2τf1 we obtain following LIM thrust force.
Fe =
3
2
π
τωr
[ω (λdsiqs − λqsids) + (ω − ωr) (λdriqr − λqridr)]
in a stationary reference frame (ω = 0) the thrust force becomes:
Fe =
3
2
π
τ
[λqridr − λdriqr] (14)
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Int J Pow Elec Dri Syst ISSN: 2088-8694 r 1743
clearing idr from λdr in 11
idr =
λdr − Lm(1 − f(Q))ids
Lr − Lmf(Q)
(15)
clearing iqr from λqr in 11
iqr =
λqr − Lmiqs
Lr
(16)
Substituting idr and iqr into 14 results in:
Fe =
3
2
π
τ
Lm
Lr

λdriqs +
f(Q)
Lr − Lmf(Q)
λqrλdr −
1 − f(Q)
Lr
λqrids

(17)
Then space state mechanical equation is giving by 18
dv
dt
=
Kf
M

λdriqs +
f(Q)
Lr − Lmf(Q)
λqrλdr −
1 − f(Q)
1 − Lm
Lr
f(Q)
λqrids
#
−
B
M
v −
FL
M
(18)
Considering short-circuited linor circuit (udr = 0) and solving for dλdr
dt gets
dλdr
dt
= −
Rr(1 + f(Q))
Lr − Lmf(Q)
λdr −
π
τ
vλqr +
Rr (Lm − Lrf(Q))
Lr − Lmf(Q)
ids (19)
Substituting the first equation of 11 into first equation of 10 results:
uds =

Rs + Rrf(Q) − Lm
df(Q)
dt

ids + [Ls − Lmf(Q)]
dids
dt
+ Lm[1 − f(Q)]
didr
dt
− Lm
df(Q)
dt
idr
(20)
Clearing idr from λdr in 11
idr =
1
Lr − Lmf(Q)
λdr −
Lm(1 − f(Q))
Lr − Lmf(Q)
ids
and substituting into 20 results
uds =

Rs + Rrf(Q) −
(Lr − Lm)
2
(Lr − Lmf(Q))
2 Lm
df(Q)
dt
#
ids
+

Ls − Lmf(Q) −
L2
m(1 − f(Q))2
Lr − Lmf(Q)

dids
dt
+
Lm (Lm − Lr)
(Lr − Lmf(Q))
2
df(Q)
dt
λdr +
Lm(1 − f(Q))
Lr − Lmf(Q)
dλdr
dt
substituting dλdr
dt in last term into above equation we obtain:
uds =

Rs + Rrf(Q) −
(Lr − Lm)
2
(Lr − Lmf(Q))
2 Lm
df(Q)
dt
+
RrLm(1 − f(Q))
Lr − Lmf(Q)
(Lm − Lrf(Q))
Lr − Lmf(Q)
#
ids
+

Ls − Lmf(Q) −
L2
m(1 − f(Q))2
Lr − Lmf(Q)

dids
dt
+

Lm (Lm − Lr)
(Lr − Lmf(Q))
2
df(Q)
dt
−
RrLm 1 − f2
(Q)

(Lr − Lmf(Q))
2
#
λdr −
Lm(1 − f(Q))
Lr − Lmf(Q)
π
τ
vλqr

1744 r ISSN: 2088-8694
Solving for dids
dt
dids
dt
=
[Rs + Rrf(Q)] [Lr − Lmf(Q)]
2
− Lm (Lr − Lm)
2 df(Q)
dt + RrLm[1 − f(Q)] [Lm − Lrf(Q)]
[LSLr − LsLmf(Q) − LrLmf(Q) − L2
m + 2L2
mf(Q)] [Lmf(Q) − Lr]
lds
+
Lm (Lm − Lr) df(Q)
dt − RrLm

1 − f2
(Q)

[LsLr − LsLmf(Q) − LrLmf(Q) − L2
m + 2L2
λdr
+
Lm[1 − f(Q)]
LsLr − LSLmf(Q) − LrLmf(Q) − L2
m + 2L2
mf(Q)
π
τ
vλqr
+
Lr − Lmf(Q)
LsLr − LSLmf(Q) − LrLmf(Q) − L2
m + 2L2
mf(Q)
uds
(21)
Grouping the state equations and changing the index d and q by α and β respectively, and omitting
the primary and secondary (linor) indexes because the voltages and currents are with respect to primary and the
fluxes are with respect to secondary, we obtain 22:
diβ
dt
= −

RS
ρLS
+
1 − ρ
ρTr

iβ −
Lmπ
ρLSLrτ
vλα +
Lm
ρLSLrTr
λβ +
1
ρLS
uβ
diα
dt
=
[RS + Rrf(Q)] [Lr − Lmf(Q)]
2
− Lm (Lr − Lm)
2 df(Z)
dt + RrLm[1 − f(Q)] [Lm − Lrf(Q)]
[LSLr − LSLmf(Q) − LrLmf(Q) − L2
m + 2L2
iα
+
Lm (Lm − Lr) df(Q)
dt − RrLm

1 − f2
(Q)

m + 2L2
λα
+
Lm[1 − f(Q)]
LsLr − LsLmf(Q) − LrLmf(Q) − L2
m + 2L2
mf(Q)
π
τ
vλβ
+
Lr − Lmf(Q)
LSLr − LSLmf(Q) − LrLmf(Q) − L2
m + 2L2
mf(Q)
uα
dλβ
dt
=
Lm
Tr
iβ +
π
τ
vλα −
1
Tr
λβ
dλα
dt
= −
Rr(1 + f(Q))
Lr − Lmf(Q)
λα −
π
τ
vλβ +
Rr (Lm − Lrf(Q))
Lr − Lmf(Q)
iα
dv
dt
=
Kf
M

λαiβ +
f(Q)
Lr − Lmf(Q)
λβλα −
1 − f(Q)
1 − Lm
Lr
f(Q)
λβiα
#
−
B
M
v −
FL
M
dx
dt
= v
(22)
where v is the mover linear velocity; λα and λβ are the d−axis an q −axis secondary flux; iα and iβ
are the d−axis and q−axis primary current; uα and uβ are the d−axis and q−axis primary voltage; Tr = Lr
Rr
is the secondary time constant; ρ = 1 −
L2
m
LsLr
is the leakage coefficient; Kf = 3
2
πLm
τLr
is the force constant;
Rs is the winding resistance per phase; Rr is the secondary resistance per phase referred primary; Lm is the
magnetizing inductance per phase; Lr is the secondary inductance per phase referred primary; Ls is the primary
inductance per phase; FL is the external force disturbance; M is the total mass of the mover; B is the viscous
friction and iron-loss coefficient; τ is the pole pitch; D is the primary length in meters; Q = DRr
Lrv is a factor
related to the primary length, which quatifies the end effects as a function of the speed and f(Q) = 1−e−Q
Q is
the factor related to the losses in the magnetization branch. To discretize the state LIM model with end effects
we use the backward difference method [25] and finally we obtain an approximate discrete time version of the
LIM model 23 taking into account end effects.

Int J Pow Elec Dri Syst ISSN: 2088-8694 r 1745
iβ
k+1 = iβ
k −

RS
ρLS
+
1 − ρ
ρTr

Tiβ
k −
Lmπ
ρLSLrτ
Tvkλα
k +
Lm
ρLSLrTr
Tλβ
k +
1
ρLS
Tuβ
k
iα
k+1 = iα
k +
[Rs + Rrf(Q)] [Lr − Lmf(Q)]
2
− Lm (Lr − Lm)
2 ∆f(Q)
T + RrLm[1 − f(Q)] [Lm − Lrf(Q)]
[LsLr − LsLmf(Q) − LrLmf(Q) − L2
m + 2L2
Tiα
k
+
Lm (Lm − Lr) ∆f(Q)
T − RrLm

1 − f2
(Q)

m + 2L2
Tλα
k
+
Lm[1 − f(Q)]
m + 2L2
mf(Q)
π
τ
Tvkλβ
k
+
Lr − Lmf(Q)
m + 2L2
mf(Q)
Tuα
k
λβ
k+1 = λβ
k +
Lm
Tr
Tiβ
k +
π
τ
Tvkλα
k −
1
Tr
Tλβ
k
λα
k+1 = λα
k −
Rr(1 + f(Q))
Lr − Lmf(Q)
Tλα
k −
π
τ
Tvkλβ
k +
Rr (Lm − Lrf(Q))
Lr − Lmf(Q)
Tiα
k
vk+1 = vk +
Kf
M
T

λα
k iβ
k +
f(Q)
Lr − Lmf(Q)
λβ
k λα
k −
1 − f(Q)
1 − Lm
Lr
f(Q)
λβ
k iα
k
#
−
B
M
Tvk −
FL
M
T
xk+1 = xk + vkT
(23)
where vk = v(kT) is the mover linear velocity; λα
k = λα(kT) and λβ
k = λβ(kT) are the d − axis
an q − axis secondary flux; iα
k = iα(kT) and iβ
k = iβ(kT) are the d − axis and q − axis primary current;
uα
k = uα(kT) and uβ
k = uβ(kT) are the d − axis and q − axis primary voltage; Tr = Lr
Rr
is the secondary
time constant; ρ = 1 −

L2
m
LsLr

is the leakage coefficient; Kf = 3
2
πLm
τLr
is the force constant; Rs is the
winding resistance per phase; Rr is the secondary resistance per phase referred primary; Lm is the magnetizing
inductance per phase; Lr is the secondary inductance per phase referred primary; Ls is the primary inductance
per phase; FL is the external force disturbance; M is the total mass of the mover; B is the viscous friction and
iron-loss coefficient; τ is the pole pitch; D is the primary length in meters; Q = DRr
Lrvk
is a factor related to the
primary length, which quantifies the end effects as a function of the speed; f(Q) = 1−e−Q
Q is the factor related
to the losses in the magnetization branch and ∆f(Q)
T = df(Q)
dt

Discrete and continuous model of three-phase linear induction motors considering attraction force and end-effects

t=kT
.
3. RESULTS
Figure 5 shows the end effects on mover velocity, fluxes and currents. Figures 6, 7 and 8 show the
system 22 behavior when the frequency of input voltage vary. The steady state velocity is a periodic wave in
all cases, but when the fed frequency is lower, higher output frequency components appear. Phase portraits in
subfigures 6b, 6c, 7b, 7c, 8b and 8c with attractive limit cycles are shown.
(a) (b)

1746 r ISSN: 2088-8694
(c) (d)
Figure 5. Mover velocity, velocity differences, currents and fluxes resulting from the model simulation using
ODE45 function of Matlab, taking into account end effects in model of LIM (22) and without end-effects
model (3), (a) Mover velocity of LIM with and without end-effects, (b) Velocity difference vs mover velocity
without end-effects, (c) β − axis With and without end-effects currents,
(d) β − axis With and without end-effects fluxes (continue)
(a) (b)
(c)
Figure 6. LIM behavior with 30Hz input frequency. Mover velocity and phase portraits of some state
variables, (a) Mover velocity of LIM with end-effects, (b) Phase portrait of mover velocity vs iα,
(c) Phase portrait of λα vs iα

Discrete and continuous model of three-phase linear induction motors considering attraction force and end-effects

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