Comparison of analytical methods of analysing linear induction motors

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
Abstract— Linear induction motor has found many applications,
from slow moving sliding doors to high-speed ground
transportation around the world. Although various methods have
been developed for the analysis of linear induction motor, only
few verifiable comparisons between these methods have been
done in literature. These methods solve the Maxwell’s equations
in and around the geometry of the machine and take care of the
physical dimensions of the machine. These methods which are
proved accurate are based on Fourier transform, Fourier series
and Space harmonic techniques. Out of different methods of
analysis, only few methods have been reported into the literatures
which are capable of taking care of topological changes in Linear
induction motor. Here in this work different computer programs
have been developed which are based on the above techniques.
These computer programs have been validated using the
published results. These have been predicting the single sided and
double sided Linear induction motors. A comparison of these
methods has been reported for finding their suitability for the use
of these methods for predicting the characteristics of LIM under
different operating conditions.
Index Terms—Linear Induction Motor, Fourier Transform
Analysis, Fourier series Analysis, Space Harmonic Method
1. INTRODUCTION
Inear Induction Motor (LIM) have been used in many
industrial applications including actuators, conveyer
systems, sliding doors, material handling, pumping liquid
metals, robot base movers, elevators etc. They have been
implemented as propulsion systems for transit vehicles in a
number of countries.LIM has many advantages, such as simple
structures, easy mainainace,repair,replacement,ability to exerts
thrust on secondary without mechanical contacts, convenient
control of thrust and speed, less environmental pollution etc.
Although conventional rotating machines have been largely
used to drive underground transportation systems, Linear
induction motors have also been considering for future
applications owing to their certain advantages. With better
appreciation of the capabilities of linear drives, new
modifications for these machines have been developed.
Consequently, there is a continuing need for effective CAD
tools and accurate analysis methods which allow LIM to be
designed to meet the specific requirements of transportation
and other application. Many techniques have been developed
To analyze LIM, but only few verifiable comparisons between
these methods have been done in literature.
Linear Induction motor (LIM) operates on the
same principle as the conventional rotary induction motor. The
Email:Chandan.kumar.eee07@itbhu.ac.in
Mobile no:+91-9889918032
Rotary motor is cut out and laid flat to form the equivalent
LIM.The reaction plate in LIM becomes the equivalent rotor
as shown in Fig1.The induced field is maximized by backing
up the reaction plate with the iron plate ,commonly known as
‘back iron’. This serves to amplify the magnetic field produced
in the coil.
Figure 1.Rotary and Linear Induction Motor
The difference between the LIM and its rotary counterpart is
due to the difference in air gap. The LIM has an open air gap
with an entry-end and exit –end, while rotating induction
motor has closed air gap. The openness of the air gap gives
rise to the particular characteristics of LIM.In contrast with
rotary machine; LIM has an open magnetic circuit.
2. ANALYSIS OF LIM
LIM can be analyzed by different techniques like finite
difference, finite element, Fourier series, equivalent circuit,
quasi 1D method and many more. Fig.2 shows the summary of
the different approaches for the analysis of LIM.
Figure 2.Different methods for analysis of LIM
The numero –analytical methods are based on Fourier methods
in which longitudinal and transverse effects have been suitably
considered. A large saving of computational efforts has been
observed, because results of electromagnetic equations in
closed form has been obtained and computers are only used for
summation or integration. These methods can be further
categorized as Fourier series methods [1], Fourier transform
methods [2] and space harmonic methods [2, 3 and 4].
COMPARISON OF ANALYTICAL METHODS OF
ANALYSING LINEAR INDUCTION MOTORS
Chandan Kumar, R.K.Shrivastava, S.N.Mahendra, Subrahmanyam Maddula
Department of Electrical Engineering, I.T.-B.H.U, Varanasi, India
L

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In order to consider the longitudinal end effects and transverse
edge effect, the Double Fourier series decomposition of
primary m.m.f. Distribution has been used. It has been
assumed that the primary members periodically separated from
each other by a certain distance such that the end flux of one
machine is not affecting the performance of the other machine.
The optimum distance by which the primary members need to
be separated increases for increasing speed. Also at higher
operating speed, the computational time may be fairly due to
slow convergence of Fourier series. The Fourier series
methods of analyzing LIM have been developed by Obrettl
[1].
Fourier transform method is normally suited for a non-
periodic distribution. Due to this it does not require
assumption mentioned earlier. Also the computation time has
been found to be independent of the speed of operation. It has
been developed by Yamamura [1].the method uses product
solution technique for solving Maxwell’s equations and
transforms the x-component of 3D/2D field solution in Fourier
space. The analysis can be done either in real space by inverse
transformation using Residue’s theorem or in Fourier space
using Perseval’s identity.
In the space harmonic analysis method, not only the end
effects but also the primary design parameters can be taken
into account. It has been assumed in this method an infinite
array of adjacent primary winding at the surface of the primary
core extended to infinity in both the longitudinal direction. The
distance between two adjacent windings has been chosen to be
long that the magnetic interaction between them can be
neglected. Fourier analysis of the m.m.f distribution has been
carried out. The m.m.f wave produced by the primary currents
consists of the fundamental pace component of equivalent pole
pitch and an infinite number of space harmonic components. It
has been applied to short primary LIM [2].A study has been
made on the design of the SLIM for the urban transit with
maximum speed of 70kmph [5] using space harmonic analysis
method. A method for computing the performance of linear
induction motor has been presented in [4].This analysis has
been derived from 2D electromagnetic analysis using space
harmonic technique based on the Fourier series expansion. The
resultant method would be an interactive force between the
Fourier series and Fourier transform, and can make a
substantial abbreviation in the calculation without spoiling the
accuracy, taking directly into the account of end effect, skin
effect and the primary winding arrangement.
3. PROBLEM IN ANALYSIS OF LIM
There are certain effects in LIM, which are normally not
present in rotary induction motors. Due to this, the analysis of
LIM has to be done using different techniques. Various
researchers in past have analyzed these effects using different
techniques. These have been briefly discussed here:
(a).Longitudinal end effects: One obvious difference between
LIM and conventional rotary machines is that the fact LIM has
ends. This means that travelling magnetic field cannot join up
on itself, and introduces end effects. The end effect is clearly
exhibited in the form of non-uniform flux density distribution
along the length of motor. These effects call for analyzing LIM
taking into account of the full machine dimension in the
longitudinal direction. The longitudinal end effects have been
known to be pronounced at high speed and result in the
reduction in thrust, efficiency and power factor, when
compared to the corresponding rotating machine. These
phenomena have been interpreted physically to be due to
airgap flux density waves propagating from the entry and exit
ends of the stator winding with the entry wave being
responsible for the major effects.
(b) Transverse edge effect: This edge effect is generally
described as the effect of having finite width for a linear
motor. This effect is more evident with lower values of width
to air gap ratio. It results from the forward component of the
eddy current density in the reaction plate causes the equivalent
impedance to be increased and exerts harmful influences on
the various characteristics of LIM such as
thrust,efficiency,power fator,lateral instability due to uneven
secondary overhangs etc.
(c)Saturation and skin effect: The saturation and skin effect the
back iron of LIM have been considered by Boldea, Nasar
[6].In these cases, a composite reaction rail has been used, in
which the back iron is screened by a thin reaction plate of
conducting material like aluminium or copper. Further, due to
larger airgap in LIM, these are lesser chances of primary
getting saturated. Therefore, in most of these analyses, the
saturation in the primary has been neglected. As such, the
choice of infinite permeability of primary member of LIM
leads to acceptable results. However in comparison to primary
core the back iron easily gets saturated. The usual approach is
to numerically compute flux dependent permeability of back
iron and later utilize it in analytical methods [6].In this way
LIM can be accurately analyzed by analytical methods even in
the presence of heavily saturated thick back iron. These effects
must be minimized as much as possible when designing LIM,
so that they do not drastically affect the performance.
In this paper different analytical methods has been used
for SLIM and comparison is done.
4. THEORETICAL AND ANALYTICAL RESULT
VERIFICATION
In this section the results from different methods which are
Fourier series, Fourier transform and space harmonic are
discussed.
A. Comparison of thrust obtained by different methods from
SLIM with back iron
The longitudinal flux in SLIM and DSLIM is
responsible for traction. This force, unlike the mechanical
devices, is produced without any contact between primary and
secondary. This characteristic of linear machine is responsible
for its application in levitated high speed ground
transportation. The specification of SLIM with back iron i.e.
SLIM-A [3] and SLIM-B [7] have been shown in appendix.
The thrust slip characteristic from all the three methods and
the published data [3, 7] is shown in Fig.3 and Fig.4
respectively.

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Figure 3. Thrust vs. Slip characteristic of SLIM-A with back iron by numero-
analytical methods
Figure 4. Thrust vs. Slip characteristic of SLIM –B with back iron by numero-
analytical methods
From these two curves following conclusion can be drawn:
1. For SLIM-A (Vs=134.16m/s), the method of Fourier series
slightly overestimates the breakdown thrust as that of
published results. However for SLIM-B (Vs=12.1 m/s) it
slightly underestimate breakdown thrust.
2.For SLIM-A, The method of Fourier transform slightly
overestimate the breakdown thrust as that of published results
but the it is less than that of Fourier series method. However in
SLIM-B Fourier transform overestimate the breakdown thrust,
but the breakdown thrust calculated using Fourier series is
lowest among all.
3. For SLIM-B, the space harmonic technique slightly
underestimate the breakdown torque but overestimate in
SLIM-A.
4. For SLIM-A (f=220Hz), the critical linear slip as obtained
by the method of Fourier series is almost same as that of
published result but in SLIM-B (f=50Hz) is overestimate it.
5. For SLIM-A, the method of Fourier transform underestimate
the critical linear slip but it is almost same for SLIM-B.
6.For SLIM-A, the critical linear slip obtained from method of
space harmonic is almost same as that of published results but
it underestimate it in SLIM-B.
7. In all the three methods the thrust obtained in stable portion
of the characteristics shows comparable results of that to the
published results.
8. For SLIM-A, the starting thrust as obtained from all
methods is comparable to that of published results. However
for SLIM-B the method of Fourier series and space harmonic
slightly overestimate and Fourier transform underestimate it
with respect to published data.
B. Comparison of Normal Force obtained by different
methods from SLIM with back iron
The normal force calculation is important because though in
rotary machine it gets cancelled, in linear machine it is present.
The normal force in SLIM may be used for levitating the
vehicle above rail, thereby reducing the friction between wheel
and rail the comparison of SLIM-A and SLIM-B are shown in
the Fig.5 and Fig.6 respectively.
Figure 5. Normal force vs. Slip characteristic of SLIM-A with back iron by
numero-analytical methods
Figure 6. Normal force vs. Slip characteristic of SLIM-B with back iron by
numero-analytical methods
From these two curves following conclusion can be drawn:
1. In both cases, the method of Fourier series overestimates the
normal force at synchronous speed as compared to other
methods.
2. In both cases, the starting normal force as obtained from the
method of Fourier series is comparable to that of published.
3. For repulsion case space harmonic gives result near to
published one. While in attracting case it overestimates the
published result.

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4. The space harmonic and Fourier series methods show slight
waviness as compared to that of Fourier transform technique.
5. RESULTS
A. Fourier Transform Method: It overestimate the breakdown
thrust for high speed operation compared to lower speed
.While it underestimates the critical linear slip for the higher
frequency operation as compared to lower frequency
operation. There is slight overestimation of the starting thrust
for lower speed of operation but comparable at higher speed. It
gives comparable result for both attracting as well as repulsive
type for normal force calculation.
B.Fourier Series Method: It slightly overestimates the
breakdown thrust for high speed of operation as compared to
low speed operation. It also overestimates the critical linear
slip for lower frequency of operation while comparable in case
of higher frequency operation. It gives comparable starting
torque for high speed operation while overestimate in low
speed working region. It overestimates the normal force at
synchronous speed but the starting normal force as obtained
from the method of Fourier series is comparable to that of
published results.
C.Space Harmonic Method: This method overestimate the
breakdown thrust for high speed. The critical linear slip for
lower frequency of operation is more but comparable in region
of low speed operation. It underestimates the starting thrust in
lower speed but gives comparable thrust in higher speed of
operation. For repulsion case space harmonic gives normal
force comparable to published one. While in attracting case it
overestimates the published result.
6. CONCLUSION
The analysis of two different SLIM using numero-analytical
methods has been done and compared to the published results.
All the three methods viz, Fourier transform, Fourier series
and Space harmonic are capable of predicting the performance
of SLIM within reasonable accuracy. The model developed
here are particularly effective in predicting the constant thrust
and normal forces with respect to slip.
Among them Fourier series and space harmonic show good
agreement for the higher speed of operation. While at lower
speed the Fourier transform is good in predicting the forces in
SLIM.
REFERENCES
[1].S.Yamamura, Theory of Linear Induction Motor, Halstead
Press, 1972
[2]E.R.LAITHWAITE, Transport without wheel, Elec
Science, London, 1977
[3]Kinjiro Yoshida,Sakutaro Nonaka, evitation forces in
ingle- ided inear nduction otors Transactions on
Magnetics,Vol.Mag-11,No.6,pp-1717-1719,1975
[4] Sakutaro Nonaka,N.Fujji Simplified two-dimensional
analysis of Linear Induction Motors ,IEEE Transactions on
Magnetics,Vol.Mag-23,No.5,pp-2832-2834,1987
[5] Sakutaro Nonaka Tsuyoshi iguchi lements of inear
Induction Motor design for urban transit transactions
on Magnetics, Vol.Mag-23, No.5, pp.3002-3004, 1987
[6]I.Boldea, S.A.Nasar, Linear motion electromagnetic
systems, John Wiley & Sons
. . ahendra R. . hrivastava rag plate-single sided
linear induction motor International conference on
Industrial Technology, 2000
APPENDIX
Parameters SLIM A SLIM B
Number of poles 12 8
Pole Pitch(mm) 304.8 121
Frequency(Hz) 220 50
Air Gap(mm) 50 10
Synchronous Speed(m/sec) 134.16 12.1
Number of Phases 3 3
Number of slots per phase per pole 4 5/6
Primary Length(mm) 2057 968
Primary width(mm) 32.8 128
Width of secondary sheet(mm) 545 200
Thickness of secondary sheet(mm) 15.9 3
Thickness of Back iron(mm) 2 6
Current(A) 1760 10
Slot pitch 25.4 1.34
Permeability of Back iron 500 *µo 350*µo

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