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CHAPTER 1
INTRODUCTION
1.1 Introduction to Linear Induction Motor
Michael Faraday, the Father of Electrical Engineering discovered in Aug. 29,
1831 that Copper disk spinning within a horseshoe magnet generates electricity, which
he termed his discovery as “Induction”. This Induction effect is the fundamental of
many electrical technologies, among them the recent is linear induction motor (LIM).
The linear induction motor, as the name reveals produces linear motion as
compared to traditional rotary motors. LIM was invented by Prof Eric Laithwaite [1],
the British electrical engineer and described his invention as “no more than an ordinary
electric motor, spread out”. Because of its applications and endless merits, linear
induction motor has gained popularity. LIM can be obtained from its rotary complement
incising by a radial plane and unrolled unlike as demonstrated in Figure 1.1.
Figure 1.1 Linear motor
Rectilinear motion is obtained in a linear induction motor, that “Whenever a
relative motion occurs between the field and short circuited conductors, currents are
induced in them, which results in electro-magnetic forces and under the influence of
these forces, according to Lenz’s law the conductors try to move in such a way so as to
eliminate the induced currents”. In this way the field movement is linear and so is the
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conductor movement. The force producing this, horizontal motion is called as thrust or
propulsion force or traction force, while a force perpendicular to the direction of thrust
is called as normal force. In rotary induction motors, the magnetic field is a travelling
magnetic field which consists of forward component, backward component and
pulsating component. Out of this, forward component is predominant and therefore
accounts for the useful force production by interacting with the currents induced in the
secondary. These induced currents can be maximized by backing up the reaction plate
(secondary) with an Iron plate. The geometrical approach from rotary to flat linear
induction motor is shown in Figure 1.2.
Figure 1.2 Rotary motor laid out flat as linear motor
1.1.1 Linear Induction Motor as against its Rotary Counterpart
As the magnetic circuit is open at the two longitudinal ends along the travelling
field direction, the flux law has to be observed. The airgap field will contain additional
waves whose negative influence on performance is called dynamic longitudinal end
effect as illustrated in Figure 1.3(a). Further due to current asymmetries between phases,
one phase has a position of the core longitudinal ends, which is different from those of
the other two. This is called static longitudinal effect as shown in Figure 1.3(b). Because
of the primary core length, the back iron flux density tends to include an additional non
travelling component which should be taken when sizing the back iron of LIMs as can
be seen in Figure 1.3(c). This normal force may be put to use to compensate for part of
the weight of the moving primary and thus reduce the wheel wearing and noise level as
shown in Figure 1.3(d) where Fn is difference of attraction normal force (Fna) to
repulsion normal force (Fnr).
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(a) Dynamic longitudinal effect (b) Static longitudinal effect
(c) Back core flux density distribution (d) Non-zero normal force Fn
(e) Transverse edge effect (f) Non-zero lateral force Fn
Figure 1.3 Paranomic views of LIM and RIM [1]
For secondary with Aluminium, Copper sheet with or without solid back iron, the
induced currents in general at a slip frequency Sf1 that have part of their closed path
contained in the active primary core shown in Figure 1.3(c). They have additional-
longitudinal components along x-axis, which produce additional losses in the secondary
and a distortion in the airgap flux density along the transverse direction along y-axis.
This is called the transverse edge effect as shown in Figure 1.3(e) [1].
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When the primary is placed off centre along y-axis, the longitudinal components
of the current density in the active zone produce an ejection type lateral force. At the
same time, the secondary back core ends to realign the primary along y-axis.
So the resultant lateral force may be either decentralizing or centralizing in character as
shown in Figure 1.3(f). On the basis of the above differences between the LIM and
rotary induction motor (RIM), following merits and demerits can be stated below:
Merits
Direct electromagnetic thrust propulsion without mechanical transmission or wheel
adhesion limitation for propulsion
Ruggedness; very low maintenance costs
Easy topological adaptation to direct linear motion applications
Precision linear positioning as no backlash with any mechanical transmission
Separate cooling of primary and secondary
All advanced drive technologies for RIMs may be applied without changes to LIMs
Demerits
Due to the large airgap to pole pitch ratios the power factor and efficiency tend to be
lower than with RIM. However, the efficiency is to be compared with the combined
efficiency of rotary motor and mechanical transmission counterpart. Larger
mechanical clearance is required for medium and high speeds. The Aluminium sheet
(if any) in the secondary contributes an additional magnetic airgap.
Efficiency and power factor are further reduced by longitudinal end effects.
Fortunately, these effects are notable only in high speed, low pole count LIMs and
they may be somewhat limited by pertinent design measures.
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Additional noise and vibrations due to uncompensated normal force continues,
unless the latter is put to use to suspend the mover either partially or totally by adequate
close loop control.
Figure 1.4 Application tree of linear induction motors
1.2 Applications of Linear Induction Motor
In engineering, especially in electrical and mechanical engineering, automated
structures such as industrial robots, fast manipulators and machine tools, linear
movements are very usual. In case where high speed and accuracy are required of these
mechanisms, serious problems may arise with stiffness, mass, friction and backlash.
Obviously a linear actuator will overcome many of these problems. Unfortunately, as
with direct drive rotating motors, mechanical fixation of the actuator load in a power off
Applications of linear
induction motor
Semiconductors
and electronics
industry
Explosion
localizing
systems
Industry
robots
Medical
instruments
Machine
tools
Protection
and
control
systems
of power
energetic
Semiconductor
plate
positioning,
screwing and
transporting
Breadstuffs
and mixture are
explosive,
fast speed
LIM used
Automatic
mounting
system
Plotter,
printer
Lens
production,
microscope
platform
positioning
Laser
cutting
machine,
diamond
polishing
machine
High
voltage
circuit
board
drives
High-speed
transport
Factory
transportation
systems
Batching
systems
Vertical
transport
systems
Fast train
drives, metro
train drives
Pneumatic
transport
system,
flexible
transport system
Package,
luggage
sorting
machine
Elevators,
buildings
construct
robots
Transport
systems
Computer
engineering
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situation, such as that obtained from self braking screw spindle mechanisms is not
available. This is a drawback that results from the above-mentioned advantages, thus
adding to many applications indeed [2]. The complete linear induction motor
application tree is illustrated in Figure1.4.
1.3 Classification of Linear Induction Motors
Linear induction motors can be classified according to their constructional
features and working principle as described in Figure 1.5.
Figure 1.5 Classification of linear induction motors
All the types mentioned above are possible for nearly all types of excitation
systems. Further, Linear induction motors can also be classified based on their working
principle of operation of their travelling magnetic field or excitation systems as:
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I. Permanent magnets in the reaction rail
II. Permanent magnets in the armature (passive reaction rail)
III. Electromagnetic excitation system (with winding)
1.3.1 Cylindrical Moving Magnet Linear Motor
In this motor, the forcer is cylindrical in construction and moves up and down in a
cylindrical bar which houses the magnets. This motor was among the first to find
commercial applications, but does not exploit all of the space saving characteristics of
its flat and U channel counterparts. The magnetic circuit of the cylindrical moving
magnet linear motor is similar to that of a moving magnet actuator. The difference is
that the coils are replicated to increase the stroke.
The coil winding typically consists of three phases, with brushless commutation
using Hall effect devices. The forcer is circular and moves up and down the magnetic
rod. This rod is not suitable for applications sensitive to magnetic flux leakage and care
must be taken to make sure that fingers do not get trapped between magnetic rod and an
attracted surface. A major problem with the design of tubular motor is shown up when
the length of travel increases. Due to the fact that the motor is completely circular and
travels up and down the rod, the only point of support for this design is at the ends. This
means that there will always be a limit to length before the deflection in the bar causes
the magnets to contact the forcer.
1.3.2 U Channel Linear Motor
This type of linear motor has two parallel magnet tracks facing each other with the
forcer between the plates. The forcer is supported in the magnet track by a bearing
system. The forcers are ironless, which means that there is no attractive force and no
disturbance forces generated between forcer and magnet track. The ironless coil
assembly as shown in Figure 1.6, has low mass, allowing for very high acceleration.
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Figure 1.6 Ironless coil assembly
Typically, the coil winding is three phase, with brushless commutation. Increased
performance can be achieved by adding air cooling to the motor. This design of the
linear motor is better suited to reduce magnetic flux leakage, due to the magnets facing
each other and being housed in a ‘U’ shaped channel. This also minimizes the risks of
being trapped by powerful magnets.
Due to the design of the magnet track, they can be added together to increase the
length of travel, with the only limit to operating length being the length of cable
management system, encoder length available and the ability to machine large flat
structures.
1.3.3 Flat Type Linear Motor
Flat type linear motors as shown in Figure 1.7 can be classified into three types
based on the design of these motors: (a) Slot-less ironless (b) Slot-less iron (c) Slotted
iron. Again, all types of constructions are brushless. To choose between these types of
motors requires an understanding of the application.
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Figure 1.7 Flat type linear motor
1.3.3.1 Slot-less Ironless Flat Motor
The slot-less ironless flat motor is a series of coils mounted to an Aluminium base
as shown in Figure 1.8. Due to the lack of iron in the forcer, the motor has no attractive
force or cogging as same with U channel motors. This will help with bearing life in
certain applications. Forcers can be mounted from the top or sides to suit most
applications. Ideal for smooth velocity control, such as scanning applications, this type
of design yields the lowest force output of flat track designs. Generally, flat magnet
tracks have high magnetic flux leakage, and as such, care should be taken while
handling these to prevent injury from magnets trapping between human being and other
attracted materials.
Figure 1.8 Slot-less ironless flat type linear motor
1.3.3.2 Slot-less Iron Flat Motor
The slot-less iron flat motor is similar in construction to the slot-less ironless
motor except the coils are mounted to Iron laminations and then to the Aluminium base.
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Iron laminations are used to direct the magnetic field and increase the force. Due to the
Iron laminations in the forcer, an attractive force is now present between the forcer and
the track and is proportional to force produced by the motor. As a result of the
laminations, a cogging force is now present on the motor. Care must also be taken when
presenting the forcer to the magnet track as they will attract each other and may cause
injury. This design of motor produces more force than the ironless designs.
1.3.3.3 Slotted Iron Flat Motor
In this type of linear motor, the coil windings are inserted into a steel structure to
create the coil assembly as described in Figure 1.9. The Iron core significantly increases
the force output of the motor due to focusing the magnetic field created by the winding.
There is a strong, attractive force between the Iron-core armature and the magnet track,
which can be used advantageously as a preload for an air bearing system. However,
these forces can cause increased bearing wear at the same time. There will also be
cogging forces, which can be reduced by skewing the magnets.
Before the advent of practical and affordable linear motors, all linear movement
had to be created from rotary machines by using a ball or roller screws or belts and
pulleys. For many applications, for instance, where high loads are encountered and
where the driven axis is in the vertical plane, these methods remain the best solution.
However, linear motors offer many distinct advantages over mechanical systems, such
as very high and very low speeds, high acceleration, almost zero maintenance because
there are no contacting parts and have high accuracy without backlash. Achieving linear
motion with a motor that needs no gears, couplings or pulleys makes sense for many
applications, where unnecessary components, that diminish performance and reduce the
life of a machine, can be removed.
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Figure 1.9 Slotted iron flat type motor
1.4 Forces in Linear Induction Motor
The three main forces that are involved within the LIM, are: Thrust, Normal force
and Lateral force. Although some other forces are also involved in the operation, with
less impact direct impact on the operational performance of the LIM. In this thesis,
attention is focussed on Thrust force. Lateral forces are undesirable one, that is
developed within the LIM due to the stator orientations. The normal force is
perpendicular to the stator as discussed.
1.4.1 Normal Forces
As stated earlier, the forces perpendicular to the thrust are normal forces.
Explicitly, LIM experiences two types of normal forces, an attractive and a repulsive
force. In the case of single sided linear induction motor (SLIM), the attraction force
appears between the primary core and the back iron of the secondary increases the
apparent weight of the “vehicle” eventually stressing the secondary mounting frame. In
double sided linear induction motor (DLIM), this force occurs between the two
primaries independent of the position of reaction plate, producing stresses in the frame
design. Two magnetized Iron surfaces experience this force. Repulsive force, in SLIM
appears between the currents in the primary and induced currents in the reaction plate. If
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the back iron of SLIM is not laminated then also repulsion force appears because of the
induced currents in the back iron plate.
In SLIM configuration, due to the asymmetrical topology, there is a large net
normal force between the primary and the secondary. During the synchronous speed of
the machine, this force is attractive and its magnitude is reduced as the speed reduced.
At certain other speeds, the force becomes repulsive, especially at high frequency
operations. In DLIM configuration, the resultant net force is zero at the reaction plate is
centrally located between the two primaries. Thus the net normal force will occur only
if the reaction plate (RP) is placed asymmetrically between the two primaries.
1.4.2 Thrust
The LIM develops the thrust force proportional to the square of the applied
voltage. This force is responsible for the linear motion of the LIM. This force reduces as
slip reduced in a LIM. The speed-thrust curve of a LIM is drawn in Figure 1.10
Figure 1.10 Speed - thrust curve of a LIM [9]
1.4.3 Lateral Forces
Lateral Forces act in the Y-direction, perpendicular to the movement of the stator.
Due to these forces, the system becomes unstable. Due to the asymmetric positioning of
the stator, these forces are developed within the LIM. Small displacement results in a
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very small lateral force, that can also be negligible. In high frequency operations which
is more than 50 Hz, these forces become the matter of concern.
1.4.4 Cogging Forces
During the no load condition, when currents are zero, the only force that exists is
the force of attraction between the Iron-core of primary and magnets of the mover. This
force is termed as ‘cogging force’ and occurs in both rotational and linear machines [3].
Cogging force and end effects can together influence the speed and position control,
producing oscillations along the way. Therefore, it is important to reduce such forces
which can be done by:
(a) Skewing of magnets or slots of the primary. But skewing is not recommended in
case of concentrated windings as the winding factor is already less than one.
Skewing further reduces the fundamental winding factor compromising the
maximum thrust and power density [4-5].
(b) Cogging force is produced by both leading and trailing edges of the magnet. It is
therefore possible to optimize the length of the magnet such that the two
waveforms of cogging force produced by both edges are cancelled out by each
other. However, the magnet length is limited by the length of slot pitch and
therefore there is little freedom to make changes [6].
(c) Semi-closed slots or use of magnetic wedges in slot openings has also been found
useful to suppress cogging force. As cogging force occurs due to the variation in
magnetic permeance of slot and iron-teeth, it aims to reduce the variation in the
reluctance of the magnetic path. However the wedges may get highly saturated
and yet the effect of inclusion of the wedge is still not large enough [7].
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1.4.5 Detent Forces
The one of the major problem which occurs in linear machines is detent force.
Which is a result of magnetic attraction between permanent magnets (PM) mounted on
the translator and the stator teeth. It is the attractive force component that attempts to
maintain the alignment between the stator teeth and the reaction plate with PMs [8]. The
ripples of the detent force produce both vibrations and noise, which are limiting factors
for any machine. The detent force is summation of cogging force and end effect force as
given:
F Detent = F Cogging + F End effect (1.1)
1.5 Effects of Linear Induction Motor
There are several effects involved in the operation of the linear induction motor,
which influence the performance of the motor. The significant effects are the end-
effects and longitudinal edge transversal effects. The brief details of all effects have
been discussed here.
1.5.1 End Effects
It is found that primary core and windings of linear induction motor has finite
length called as the active length of the motor and due to this, LIM has two ends, the
phenomenon so introduced is called as end effect. Owing to the finite length of the
stator or primary, the magnetic field density longitudinal distribution presents an entry
and exit perturbation as the secondary enters or leaves the airgap during the motor
operation as shown in Figure 1.11. Therefore, unlike the rotating induction machines,
entry and exit longitudinal end effects are specific characteristics of linear machines.
These effects are absent when the motor is at standstill. Static end effects are introduced
because of an inevitable unbalance in the phase winding impedances.
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Figure 1.11 End effect forces in linear induction motor
Due to this, unbalanced current flows in the various phases even when, the
balanced voltages are applied at the terminals. These effects lead to additional losses
causing reduction in the thrust of the LIM, which further deteriorates the performance
of motor.
1.5.2 Transverse Edge (Finite Width) Effects
The primary and secondary of a LIM have finite lengths. In general, the secondary
is wider than the primary. The consequences of this physical feature of a LIM are called
transverse edge effects. These currents have a longitudinal component jx and a
transverse component jz. The component jz is the source for these effects. The LIM
having equal secondary and primary widths will show more transverse edge effects than
having wide secondary.
1.5.3 Skin Effects
There exists an appreciable distance between the primary and secondary of the
single sided LIM or between two primaries of double sided LIM due to mechanical
constraints. Moreover, when the input frequency rating is greater than 100 Hz and the
slip is 10%, skin effect cannot be neglected. Thus an appropriate correction factor
should be used to consider this effect. The skin effect is more prominent in high speed
LIM’s than of low speed.
1.5.4 Winding Peculiarities
There are two possible arrangements for primary winding layout in linear
induction motor which exhibits good performance.
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Layout 1: This is a unique feature wherein number poles in a LIM can be added
(or may be fractional). This layout can be obtained by not changing the coils at the ends
in a double layer winding with full pitched coils provided the divergence equation is
satisfied.
Layout 2: This layout has an even number of poles and can be obtained from
single layer winding with concentrated coils as in the armature of a hydroelectric
generator.
Thus, it can be stated that there exists a number of possible primary winding
layouts for LIMs. For example, a double layer winding with half filled end slots often
results in good performance of LIM.
1.5.5 Effects of Unrolling
At the instant, when the rotary machine is split and unrolled into a linear
equivalent circuit, the magnetic patterns are disturbed. The situation changes with
passage of time. This effect is caused by the core of the machine having impenetrable
flux barriers at each end, from which travelling wave is reflected to create standing
wave patterns.
1.5.6 Airgap Effect
The conventional rotary machine has a very small airgap, which allows a high gap
flux density.
Figure 1.12 Effect of airgap on attraction force [9]
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The magnetic circuit reluctance is much higher for larger airgaps, in which
magnetizing current is also higher. There is a rather large leakage flux that further
reduces the operating power factor. The gap density is less than for a rotary counterpart,
and consequently Iron losses form a smaller part of the total loss. Figure 1.12 shows the
effect of airgap on attraction force [9].
1.5.7 Saturation Effect
In the linear induction motor, back iron material is made of Steel or Iron. So, at
certain transient conditions saturation appears which has to be pre-determined for
performance evaluation. The saturation level is governed by the depth of field
penetration in Iron. This can be determined and controlled with help of saturation
coefficient, which is the ratio of back iron reluctance to the sum of the conductor and
the airgap reluctances.
1.5.8 Dolphin Effect
An asymmetric airgap magnetic flux density at the end parts of a linear motor
produces an unbalanced normal force, due to which, the airgap tends to increase at the
entry and decrease at the exit part. This is called the “Dolphin Effect.”
Figure 1.13 Forces graph of LIM [9]
It is clear from the graph publicized in Figure 1.13 that the force of attraction and
the force of repulsion are not uniformly distributed, therefore primary experiences a drag
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in addition to normal forces called Dolphin Effect, due to which the entry end airgap
becomes non-uniform at the ends of LIM, which is found very prominent in high speed
double sided linear induction motor (DLIM) [1].
1.5.9 Goodness Factor
The concept of the realistic goodness factor is important for the study of LIM. It is
a convenient measure for assessing the quality of LIM. This factor takes into account
the airgap leakage, the reaction plate skin effect, and transverse edge effects. It is shown
that the end effect can be expressed as a function of this factor, the slip and the number
of poles. The Goodness factor is a useful index in the preliminary design of linear
induction motors [9].
Mathematically, it can be written as:
G = 2 (1.2)
Where, = Electrical angular velocity, σ = Electrical conductivity of mover
= Magnetic permeability, g = Length of airgap = Face constant rad. / mt.
1.6 Review of Solution Methods for Electromagnetic Field Problems
The theory of the electromagnetic field (EMF) actually took long time to be
established owing to the fact that these are intangible in nature, i.e. cannot be touched or
felt like other mechanical and thermal quantities. EM can be best described by the work
presented by Maxwell (1831-1879), Ampere (1175-1836), Coulomb, Gauss (1777-
1855), Lorentz, Faraday (1791-1857), and a few more [252]. There are different
approaches that have been applied in the recent past for formulating numerous
electromagnetic problems. As the number of these problems is varied, there can be
umpteen types of approaches that have been found to give reasonable results since their
time of occurrence.
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The analytic approaches suffer from constraints governed by the boundary and
interface conditions. In fact, it has been practically impossible to model in-
homogeneities, non-linearties and so on, by means of a media different from the ones
involved in the real problem. They are capable of solving only a limited class of
problems involving basic homogeneous and linear media with simple geometric
configurations. Whenever exact analytical solutions are not available, approximate
methods are sought. It should be emphasized that the "numerical approach" is not
automatically equivalent to the "approach with use of computer", although one usually
use a numerical approach to find the solution with use of computers. That is because of
the high computer performance incomparable to abilities of the human brain. Numerical
approach enables solution of a complex problem with a great number of very simple
operations. It is perfect for the computer which is basically a very fast moron. There are
several numerical techniques that can be used to overcome the drawbacks of the
analytical methods.
1.6.1 Numerical Methods
Electromagnetic field calculation has been revolutionized over the last three
decades by the rapid development of the digital computer with progressively greater
capacity and speed and a decreasing cost of arithmetic operations.
Any numerical field computation involves, among others, the following steps:-
1. Definition of the region over which the governing equation is to be solved. This is
the approximate problem domain (discretization of the field region).
2. Formulation of the discrete problem.
3. Derivation of a system of equations whose solution gives the numerically
computed approximate field [10].
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Various numerical methods that can be used for solution of electromagnetic field
problems are as follows:
1.6.1.1 Transmission Line Method
The transmission line method (TLM) performs analysis in the time domain and
the entire region of the analysis is gridded. Instead of interleaving E-field and H-field
grids, however, a single grid is established and the nodes of this grid are interconnected
by virtual transmission lines. Excitations at the source nodes propagate to adjacent
nodes through these transmission lines at each time step. The symmetrical condensed
node formulation introduced by Johns [12], has become the standard for three-
dimensional TLM analysis. The basic structure of the symmetrical condensed node is
illustrated in Figure 1.14. Each node is connected to its neighboring nodes by a pair of
orthogonally polarized transmission lines. Generally, dielectric loading is accomplished
by loading nodes with reactive stubs. These stubs are usually half the length of the mesh
spacing and have characteristic impedance appropriate for the amount of loading
desired. Lossy media can be modeled by introducing loss into the transmission line
equations or by loading the nodes with lossy stubs. Absorbing boundaries are easily
constructed in TLM meshes by terminating each boundary node transmission line with
its characteristic impedance.
Figure 1.14 Symmetrical condensed nodes
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1.6.1.2 Method of Moments
The method of moments or moment method is a technique for solving complex
integral equations by reducing them to a system of simple linear equations. In contrast
to the variational approach of the finite element method, however, moment method
employs a technique known as the method of weighted residuals. Actually, the terms
method of moments and method of weighted residuals are synonymous. Harrington
[11] was largely responsible for popularizing the term method of moments in the field
of electrical engineering.
All weighted residual techniques begin by establishing a set of trial solution
functions with one or more variable parameters. The residuals are a measure of the
difference between the trial solution and the true solution. The variable parameters are
determined in a manner that guarantees a best fit of the trial functions based on a
minimization of the residuals.
Equation solved by moment method techniques is generally a form of the electric
field integral equation (EFIE) or the magnetic field integral equation (MFIE). Both of
these Eqs. 1.3 and 1.4 can be derived from Maxwell’s equations by considering the
problem of a field scattered by a perfect conductor or a lossless dielectric. These
equations are,
EFIE: (1.3)
MFIE: (1.4)
Where the terms on the left-hand side of these equations are incident field
quantities and J is the induced current. The form of the integral equation used
determines which types of problems a moment-method technique is best suited to solve.
For example, one form of the EFIE may be particularly well suited for modeling thin-
wire structures, while another form is better suited for analyzing metal plates.
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1.6.1.3 Charge Simulation Method
The charge simulation method (CSM) is very useful to solve partial differential
equations in electrical engineering, and has been studied and developed by a lot of
researchers. The method is easy to understand, and can be applied only by solving a
system of simultaneous linear equations. Many examples show that the method makes
possible to get rather precise solutions or the boundary value problems with respect to
domains bounded by smooth curves.
Accurate computations of electric field intensities between non-uniform field
configuration electrodes are useful in the design of high voltage equipment insulation.
In view of innumerable possibilities of electrode geometry configurations in
equipments, and the electric fields being complex in these regions, analytical solutions
for electric field intensity are extremely difficult. For more accurate calculations of
electric field intensities, numerical methods are suitable. The fundamentals of CSM and
calculations of electric field intensities for models having rotational symmetry have
been presented by Singer, Steinbigler & Weiss [13].
The basic idea of charge simulation method is the introduction of fictitious
charges in the region other than where the field solutions are desired, so as to simulate
the field in the region of interest with the boundary conditions met as appropriate to the
problem. The position and magnitude of charges are selected to compute the potential
and field distribution in the region using classical theory of electric fields.
1.6.1.4 Finite Difference Time Domain Method
The finite difference time domain (FDTD) method is a direct solution of
Maxwell’s time dependent curl equations as given in Eqs. 1.5-1.6.
(1.5)
(1.6)
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It uses simple central-difference approximations to evaluate the space and time
derivatives. The FDTD method is a time stepping procedure. Inputs are time-sampled
analog signals. The region being modeled is represented by two interleaved grids of
discrete points, one for representing magnetic field and other for electric field
evaluation.
The main disadvantage of this technique is that the problem size can easily get out
of hand for some configurations. The fineness of the grid is generally determined by the
dimensions of the smallest features that need to be modeled. The volume of the grid
must be great enough to encompass the entire object and most of the near field. Objects
with large regions that contain small, complex geometries may require large, dense
grids. When this is the case, other numerical techniques may be much more efficient
than the FDTD method.
1.6.1.5 Finite Difference Frequency Domain Method
Although conceptually the finite difference frequency domain (FDFD) method is
similar to the finite difference time domain method, from a practical standpoint it is
more closely related to the finite element method. Like FDTD, this technique results
from a finite difference approximation of Maxwell’s curl equations. However, in this
case the time-harmonic versions of these equations are employed,
(1.7)
(1.8)
Since, there is no time stepping it is not necessary to keep the mesh spacing
uniform. Therefore, optimal FDFD meshes generally resemble optimal finite element
meshes. Like the moment-method and finite-element techniques, the FDFD technique
generates a system of linear equations and the corresponding matrix is sparse. Although
it is conceptually much simpler than the finite element method, very little attention has
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been devoted to this technique in the literature. Perhaps this is due to the head start that
finite element technique achieved in the field of structural mechanics.
1.6.1.6 Finite Difference Method
The finite difference method (FDM) is one of the oldest and yet decreasingly
used numerical methods. In essence, it consists in superimposing a grid on the space-
time domain of the problem and assigning discrete values of the unknown field
quantities at the nodes of the grid. Then, the governing equation of the system is
replaced by a set of finite difference equations relating the value of the field variable
at a node to the value of the neighboring nodes. It has certain limitations as: -
a) Lack of geometrical flexibility in fitting irregular boundary shapes.
b) Large points are needed in regions where the field quantities change very
rapidly.
c) The treatment of singular points and boundary interfaces that do not coincide
with constant coordinate surfaces [14].
1.6.1.7 Boundary Element Method
To formulate the eddy-current problem as a boundary element method (BEM),
an integral needs to be taken at the boundary points. To avoid the singularity which
occurs in the integrand when the field point corresponds to the source point, the
volume is enlarged by a very small hemisphere, whose radius tends to zero, with the
boundary point being the center of the sphere. The usage of the boundary element
method reduces the dimensionality of the problem from three to two or from two to
one. It is found to be useful in open boundary problems where it strongly challenges
the finite element method [14].
1.6.1.8 Finite Element Method
The most powerful numerical method appears to be the finite element method
(FEM), which from the mathematical point of view can be considered as an extension of
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the Rayleigh-Ritz / Galerkin technique of constructing coordinate functions whose
linear combination approximate the unknown solutions. In this method, the field region
is subdivided into elements i.e. into sub-regions where the unknown quantities , such as,
for instance, a scalar or vector potential, are represented by suitable interpolation
functions that contain, as unknowns, the values of the potential at the respective nodes
of each element. The potential values at the nodes can be determined by direct or
iterative methods. The normal procedure in a field computation by the FEM involves,
basically, the following steps:-
a. Discretization of the field region into a number of node points and finite
elements.
b. Derivation of the element equation: The unknown field quantity is represented
within each element as a linear combination of the shape functions of the
element and in the entire domain as a linear combination of the basic functions.
A relationship involving the unknown field quantity at the nodal points is then
obtained from the problem formulation for a typical element. The accuracy of
the approximation can be improved either by subdividing the region in a finer
way or by using higher order elements [15].
c. Assembly of element equations to obtain the equations of the overall system.
The imposition of the boundary conditions leads to the final system of equations,
which is then solved by iterative or elimination methods.
d. Post-processing of the results: To compute other desired quantities and to
represent the results in tabular form or graphical form, etc.
Limitations- (i) As the number of elements is increased, a colossal amount of time is
required for the execution of the program.
(ii) The time required to prepare the input data and to interpret the results.
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This numerical technique can be applied to a wide class of problems. In general,
by increasing the number of elements, improved results are obtained. In order to reduce
computational and manpower costs, self-adaptive finite element mesh generators have
been developed and are widely used today. These generators produce mesh structures
from the outline of the problem with the minimum of user-specified information and
lead to an acceptable accuracy of the resulting solution in the minimum time.
1.7 Adaptivity and Error Analysis
The real time evaluation is expensive, and when it can be avoided, it should be.
Some applications do not need exact results, but require the absolute error of a result to
bring under certain value. If this threshold is known before the computation is
performed, it is economical to employ adaptivity by prediction. The healthy literature
found in this context [72-76, 79], approximates the result with a different degree of
precision, and with a correspondingly different speed. Error bounds are derived for each
of these procedures mentioned in literature [87-90]. These bounds are typically much
cheaper to compute than the approximations themselves, except for the least precise
approximation.
The term adaptivity means a procedure in which finite element model is created
simultaneously with the solution. In the adaptive procedures, the FE model is generated
iteratively, beginning with a coarse approximation to the problem, refining the
approximation successively to minimize the error in the solution. It is observed that by
increasing the number of degree of freedom (DOF) in the vicinities of higher solution
error only, it is possible to make the most significant improvement in the global
accuracy of finite element solution, for the minimum additional computational cost. The
solution is to tailor the mesh to the particular problem, with many small elements in
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areas of high variations and few large elements in areas with little change. This is where
mesh adaptivity comes in picture.
To enhance the quality of the discretization, different strategies can be used to find
errors in the solution at different stages of the field analysis as:
a) A priori- before the field is computed
b) Posteriori- using the field solution to estimate the error
The error estimators indicate which element in the mesh is to be refined. Based on
these indicators, various steps like, node displacement, edge swapping etc. can be
performed to bring refinement in the mesh for more accuracy of results [90, 238].
1.8 Trends of Numerical Methods in Low Frequency Electromagnetic Field
Problems
For exposures to electromagnetic fields from low-frequency sources such as
power lines at 50/60 Hz, induction heaters and electrical machines etc., several
numerical techniques have been developed. Regarding the formulation employed, they
can be classified either as time- or frequency-domain methods. Alternatively, they can
be divided between differential and integral methods. These methods are the admittance
and impedance methods, the finite-element method, the scalar potential finite-difference
method, and the finite difference time-domain method with frequency scaling. In the
modern techniques is the class of moment of methods (MOM), which is typically
formulated in the frequency domain and is based on an integral equation approach. A
further class of methods uses an approach based on differentiation of Maxwell’s
equations leading to FDM. The FDTD and as a further type the TLM are typical
differential time-domain approaches. Each of these approaches has its particular
advantages for low frequency EMF problems and the proper choice for the actual
problem depends on the subjected conditions, such as the necessity for considering
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nonlinearity or magnetic analysis. Thus, the objective of the model and its
corresponding conditions are responsible for selection of appropriate numerical method.
Basically it can be stated that during the past years the power of numerical
methods has continuously improved and nearly every electromagnetic field problems
can be treated by means of such simulations. However, the chosen method might not
have the capability to investigate all aspects of an analysis area and several different
methods might have to be applied. The remaining challenge often involves the
application of the most appropriate and effective method to deal with an actual problem
and to find a user-friendly implementation through numerical simulation method.
When carrying out an EMF analysis of electric drive situations the following facts
have been kept in mind in order to gain confidence in achieving the results:
Analytical methods and the data used for the analysis have been observed from a
conservative viewpoint.
An iterative approach has been applied during the entire analysis by incorporating
error convergence.
1.9 Comparison of Finite Element Method with BEM and FDM
The comparison of FEM with BEM and FDM is a delicate matter. Indeed the
merit of any method depends essentially on its particular implementation. The basic
difference between BEM and FEM is that the BEM only needs to solve the unknowns
on the boundaries, whereas FEM solves for a chosen region of space and requires a
boundary condition bounding that region. The finite element method is more powerful
for large class of problems with complicated geometries, whereas for simpler
geometries such as domain made up of rectangles in a suitable coordinate system, the
two methods are similar and preference can be based on particular conditions
depending upon the type of boundary and interface conditions, materials and so on.
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Whereas, in case of FDM, the main disadvantage of is that the process of solution
must be carried out for each set of parameters which defines the problem. The boundary
conditions also create problems that should be considered for such calculations. The
solution obtained by finite difference method is approximate through any desired degree
of accuracy can be achieved if the computation is continued for a sufficient length of
time. In many problems it is also possible to achieve a given accuracy with a relatively
very small number of elements and consequently less computational time. It is generally
felt that for irregular domains, FEM is often easier to use and that for regular domains
FDM is more easily programmed. The choice between FDM and FEM in field
computation is finely balanced, depending on experience, skill in programming,
computer software availability and storage capacity, and of course the problem which is
being investigated [16].
FEM vs. BEM
In general both methods have some major strength and some major drawbacks.
These can be easily summarized [17].
For BEM
a) Extremely high accuracy is attainable for electromagnetic fields. This is due to the
field being calculated by integrating the solution.
b) The problem does not have to be artificially truncated and a boundary condition
applied to the artificial boundary.
c) For linear problems unknowns are only located on the boundaries of the problem.
This radically reduces mesh generation time and storage requirements.
d) The biggest disadvantage of boundary elements is the inability to handle nonlinear
problems efficiently. For weakly nonlinear problems the method is satisfactory but
for highly nonlinear problems the solution time is excessive.
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e) For some classes of problems, the solution time for post-processing calculations
can be longer than those for finite elements.
For FEM
i. The method is easily applied to all types of problems unlike the boundary element
method where a different kernel or Green's function may be required for each
domain of the problem.
ii. Nonlinear problems of all types are readily handled. The nonlinear regions have to
be discretized for both differential and integral formulations. Thus the advantage
of only discretizing the boundaries of the problem for boundary elements is lost.
Very efficient nonlinear methods such as Newton-Rhapson are easily employed
within the finite element system of equations.
iii. Provided a good mesh can be created the finite element implementation is quite
straightforward. Inherently integral equations are very difficult to implement for
general curved surfaces due to the singular kernel.
iv. The field is trivial to calculate once the problem has been solved. As the solution
is known at the nodes of the finite element mesh, it is easily interpolated to any
point within the finite element mesh.
v. For problems where the surface area is large compared to the volume it is
contained in the solution time can be very good compared to boundary element
formulations.
vi. For problems where a huge amount of volume must be discretized relative to the
total surface area the finite element method can be extremely inefficient if it can
be applied at all.
vii. Differentiating the solution to get the field is a numerically poor process. So, for
example, to calculate we have to take the curl. The plots of will appear smooth but
have discontinuities which are artefacts of the method due to differentiation.
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viii. The "action at a distance" concept is only applicable to equivalent source methods.
Thus, for example, forces and torques cannot be calculated by Ampere currents
using finite elements.
FEM vs. FDM
The FDM solution contains an immense amount of numerical information
from which it is possible to obtain some useful quantities of interest. A major
drawback is that it attempts too much by filling all space with its net. Whereas in the
finite element method, there is a possibility of a completely free topology for a mesh
and it is based on the energy distribution rather than on the differential equation
describing the equilibrium condition. This is characterized by distinctive features:-
a) The domain of the problem is viewed as a collection of non-intersecting
simple sub-domains called finite elements.
b) Over each finite element, the solution is approximated by a linear combination
of undetermined parameters and pre-selected algebraic polynomials.
Figure 1.15 Tree of applications of finite element method
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1.10 Applications of Finite Element Method
The variety of applications of FEM under the umbrella of the various engineering
fields such as, Aeronautical, Biomechanical, Automotive industries, Electrical,
Chemical, Mechanical and structural engineering etc are available as given in
Figure 1.15. Several modern FEM packages include specific components such as
thermal, electromagnetic, fluid, and structural working environments suitable for
coupled problems. In an electrical simulation, FEM helps tremendously in producing
electric and magnetic field visualizations and also assists in minimizing losses due to
flux leakage, thermal runaway, material compositions etc.
1.11 Motivation for Selecting Finite Element Method for Present Work
In the numerical solution of practical problems of engineering such as
computational fluid dynamics, elasticity, or semiconductor device simulation one often
encounters the difficulty that the overall accuracy of the numerical approximation is
deteriorated by local singularities such as, e.g., singularities arising from re-entrant
corners, interior or boundary layers, or sharp shock-like fronts. An obvious remedy is to
refine the discretization near the critical regions, i.e., to place more grid-points where
the solution is less regular. The question then is how to identify those regions and how
to obtain a good balance between the refined and un-refined regions such that the
overall accuracy is optimal. Now days, the fast and accurate analysis of electrical
machines becomes possible and, therefore, numerical simulation has become an
important tool for the analysis and the design of such devices in electrical engineering.
It is obvious that the computed solution is subjected to an error, which is as usual a sum
of several independent contributions due to the several parameters of model, phases of
modeling, discretization and computation. Regarding this, the challenge is to measure
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the error and to carry out the numerical simulation in a way that as a result of solution is
obtained more accurate with minimum computational cost.
The finite element method (FEM) currently represents the state-of-the-art in the
numerical magnetic field computation relating to electrical machines. A given domain
can be viewed as an assemblage of simple geometric shapes, called finite elements, for
which it is possible to systematically generate the approximation functions. The
collection of these finite elements is called as mesh. The usefulness of FEM method is
hampered by the need to generate a mesh which accurately defines the geometry of the
motor. Mesh generation is a critical step in computer simulation because the accuracy of
the analysis depends upon the size of mesh elements.
The adaptivity flexibility and versatility being the main advantage of finite
element method with respect to other numerical methods motivated the use in the
present work. In this method the subdivisions may consist of triangles of first/higher
orders or their combinations with or without curved sides. These can be fitted very
easily to the profile of any complex shaped domain. The grid can be made fine or coarse
in different regions of the solution domain in a very flexible way as and when required.
Another advantage lies in the form of algebraic system of equations obtained which
generally has a symmetric positive definite matrix of coefficients. Inclusion of complex
boundary conditions is the inherent capability of the method, which further simplifies
the problem. The solutions obtained by finite element method as with other numerical
methods, are approximate, through any degree of accuracy can be achieved provided
sufficient numbers of elements are used [18].
1.12 Thesis Objectives
The aim of present work is electromagnetic field analysis of linear induction motor
by self adaptive finite element method. The pre-hand information of the flux
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distribution in various parts of machine assists the designer to make an efficient and
compact design of machine. The electromagnetic field solution therefore plays a vital
role in computing the performance parameters of the motor. When the solution is
plagued by presence of domain singularities such as boundary layers, re-entrant corners,
sharp bands and multiple material discontinuities, it is necessary to selectively add more
degrees of freedom where the solution varies abruptly. Under such circumstances,
adaptivity helps to optimally improve the accuracy by selective spatial decomposition of
a problem domain. In adaptive finite element method the efficiency of adaptivity
depends upon the effectiveness of mesh refinement. Keeping in view the above
important aspects the objectives of the present work have been defined as follows:
a. To explore the Delaunay triangulation technique and mesh refinement strategies
for electromagnetic field problems.
b. To develop self adaptive computational tool for field analysis of linear induction
motor with the help of FEMLAB (COMSOL Multiphysics), MATLAB and
AutoCAD.
Thus, the main motivation of the work presented in the thesis is the optimal design of
linear induction motor by calculating partial differential equations (PDEs), also develop
self adaptive tool based on numerical techniques. The implementation of self adaptive
FE procedure has been done by interfacing the AutoCAD with FEM based COMSOL
Multiphysics software for computation of field parameters. The optimetric analysis is
further performed by exporting the basic field values on MATLAB environment for
optimization of LIM design. Figure 1.16 shows systematic block diagram of self
adaptive finite element modeling that operates as follows:
a. The geometry of the LIM has to be drawn in the AutoCAD with respect to its
geometrical dimensions.
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b. The model is imported on the COMSOL Multiphysics (FEMLAB) domain by from
AutoCAD platform, then attributes such as boundary conditions, material properties,
input supplies, coefficients and constants are assigned.
c. The mesh generator produces a discretized model by Delaunay triangular mesh.
Further adaptive refinement has to be done by Delaunay refinement algorithm to
reduce the convergence error of the model.
Figure 1.16 Self adaptive finite element systems for LIM analysis
d. The processing stage runs on COMSOL evaluating the prominent field variables at
primary core, airgap and secondary of the LIM.
e. The error evaluator compares the error estimates derived from the analysis output
with pre-specified tolerances, and either accepts the results or requests a new analysis
of refined mesh. In the later case, the error evaluator indicates the regions in the LIM
model that require further refinement by increasing the degree of freedom (DOF).
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f. The final performance parameters are evaluated in terms of thrust, efficiency and
power factor for the optimized design of the model on the MATLAB platform by
comparing and validating the results with existing model.
1.13 Thesis Organization
Chapter-1 introduces the classification and applications of linear induction motor, the
forces and effects which determine its performance, overview of various methods of
solution for electromagnetic field problems. The selection of appropriate numerical
technique which can be applied for the design and analysis of linear induction motor
has also been elaborated.
Chapter-2 surveys the work done in each of the components of the FEM analysis
process, namely- linear induction motor forces, effects and other related features,
numerical methods, mesh analysis, Delaunay mesh refinements, and error analysis.
FEM based analysis of induction machines, magnetic levitation and various
performance parameters of LIM have been explored.
Chapter-3 illustrates the elements of the electromagnetic field problems, framework of
the finite element method, integral formulations for numerical solutions with
variational methods, shape of elements, transformations, mesh generation concepts,
optimality of the Delaunay meshing algorithms and mesh quality measures.
Chapter-4 emphasizes the importance of mesh refinement procedures, classification of
element refinement, Delaunay refinement algorithms, mesh smoothening, adaptive
mesh refinement its implementation on various parts of LIM, elaborated self adaptive
mesh refinement with error indicators and analysis stages.
Chapter-5 describes the electromagnetic field solutions, equivalent circuit, significant
governing equations and parameters, analysis of the prominent effects in LIM like,
end effects and power loss due to it. Transversal edge effect, dolphin effect, skin
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effect with saturation level analysis for deriving the results in terms of thrust,
efficiency and power factor of motor.
Chapter-6 includes all the post processing results of modeling and simulation
incorporating adaptive mesh refinements, computation of field variables, and error
analysis through surface plots, contour plots and comparative line graph
representations. LIM model simulation re-excised to evaluate its performance by
changing parameters like, effect of geometrical modification of core, effect of
mechanical air gap variation, effect of material variation of core as well as reaction
plate’s conducting layer, effect with change in thickness of reaction plate, change in
number of poles etc. The COMSOL environment based computed values of field and
circuit variables further used for comparative analysis with reference and analytical
results in terms of efficiency, thrust and power factor with the help of MATLAB
programming for validation and conclusions.
Chapter-7 summarizes the contributions of the thesis and outlines further work
necessary in the same direction for the DLIM and TLIM in industry and rapid
transportation applications.