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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.