38

CHAPTER 2

LITERATURE REVIEW

2.1 History of Linear Motors

Patent literature first mentioned linear induction machines in 1890, only two

years after the discovery of the rotating induction principle. Around 1891, the

“photoelectric system”, intended for luggage transportation, was tested in

Dorchester, Massachusetts [19]. The electric shuttle used in a weaving loom, and

which employed a LIM, was developed by the Waver Jacquard and Electric Shuttle

Company in England and was patented in 1895.

The idea of a train system originated with Korda, but was reinvented by

Rosenfeld, Zelenay and Dulait, who actually experimented with it. The test track was

electrified over a length of 400 meters and had 20 primaries, which were 2.8 meters

long and were gramme-wound. In 1902, Zehden applied for a French patent on an

“electric traction system” and for a similar patent in the U.S. in 1907, in which he

suggested dragging a train with a short primary mounted under the cars, and using a

long secondary with a configuration quite similar to those that are now tested for high-

speed ground transportation [20] and for the mechatronic systems [255].

During the late 1940’s, the Westinghouse electric corporation carried out two

large-scale experiments for the U.S. Navy. The apparatus known as ‘Electropult’ was

built for the U.S. Navy to launch aircraft. The primary winding was mounted to a

wheel-supported shuttle while the slotted iron secondary served as a track on the ship

deck. The secondary consisted of a squirrel cage, set flush with the ship deck. Its length

was 425 meters, the useful core was 30 cm wide, and the required AC power amounted

to 12000 kW. The launching run was 300 meters long, and it lasted 4 to 15 minutes,

bringing the plane from rest to a take-off speed that could be as high as 360 km/h. The

39

remaining 125 meters were used for stopping the shuttle with a reversal of the magnetic

field, after the aircraft took off. Although its originators claimed that the launcher did

not have the limitations in speed or capacity of the mechanical types of launching

devices, the prohibitive expense prevented further developments of the Electropult [21].

E. R. Laithwaite et al. (1966) became quite enthusiastic about the linear induction

motor. He directed many theses on the subject in his laboratory at Manchester

University, and the first results of his work were published in the proceedings of the

IEE London. Since then, he has published related papers in many professional journals,

attracting the attention of many engineers to the subject. He analyzed linear induction

motors according to their configurations and associated applications and published them

in his book [1].

Nasar & Bodea (1976) published a book on linear induction motor covering all the

design features, effects, applications etc. Most of the researchers have cited this book in

their research publication as benchmark for the governing equations of linear motion

electric motors [22].

2.1.1 Linear Synchronous Machines

Nobuo Fujii et al. (1987) attempted to realize the theory of compensation, the magnet

rotator with permanent magnets as a mechanical rotating type and AC coil type with

concentrated winding as a static compensator. The proposed design with 10 m long

linear induction motor, recorded efficiencies of about 85 % at 40 Km/h and about 90 %

at 360 km/h respectively as compared to the traditional linear induction motor [23].

R. De Weerdt et al. (1995) described the calculation of the end effect parameters of

squirrel cage induction motors using the finite element method. A two dimensional

axisymetric problem analyzed for the end ring parameters (ring resistance and

inductance) using finite element method [24].

40

R.J. Cruise, et al. (1999) discussed the cogging forces in linear synchronous motors,

which are due to the interaction between the edges of the permanent magnet and the

teeth of the primary core. Various techniques have been employed to reduce the

cogging force, namely, the optimization of the magnet length, the use of semi-closed

slots, the variation of the airgap length and skewing. With the help of finite element

methods (FEM) assessing the various schemes for reduction of cogging force in linear

synchronous motor has been discussed in this paper [6].

A. Tenconi et al. (2000) described that the flat type linear motors that were permanent

magnet, can also be considered for standstill applications. These machines have been

developed from a cylindrical brushless D.C. machine by cutting along a rolling plane.

These machines employed skewed permanent magnet poles to eliminate the magnetic

cogging effects [25].

K. Shima (2003) made an understanding to the relationship between Iron core

configurations and leakage flux distributions. Paper presented a method for calculating

the steady and transient-state leakage flux distributions in salient-pole synchronous

machines. This method provided hands-on information which cannot be obtained by the

use of terminal quantities. This method made an analysis of the leakage inductances that

properly represented the corresponding leakage fluxes. These are calculated using the

gap flux distributions by finite elements with magnetic saturation. These obtained

inductances of the machine were then calculated for various loads [26].

Zhu Yu-wu et al. (2008) presented a uniform analytical model by energy method and

Fourier series expansion to analyze detent force in uneven magnetic field for permanent

magnet linear synchronous motor (PMLSM). The model revealed the detent force in

long-primary type was mainly influenced by non-ideal distribution of permanent

magnet magnetic motive force, while no unified airgap presence makes any great

41

impact on detent force of short-primary type. Hence, magnetic field similarity of motor

design techniques referring rotary counterpart were adopted. For long-primary type

novel method of splitting edge magnets is proposed to reduce end effects force, and

optimal widths of edge tooth in short-primary type verified the effectiveness of

magnetic field similarity. The experimental results were validated with finite element

analysis results [7].

Gilardi G et. al (2011) presented a paper in which, a finite element analysis of the

cogging force generated by an array of linear synchronous motors (LSM) moving on a

curvilinear track was presented. In this the driving system for the two main axes, thirty

meter telescope that was built in Mauna Kea, Hawaii was proposed. The finite element

analysis was carried out using planar cross sections, and the results found were used to

find an approximate solution for the model. The results of the work showed that the

presence of the curvilinear–configured track increases the cogging force of a single

LSM significantly, while the presence of the array of LSMs interacts in limiting the

increase in the cogging force. A geometric optimization in regards to the relative

positions of the LSMs along the curvilinear tracks was subsequently carried out in order

to reduce the total cogging force [3].

Fan Zhi1 (2013) made an analysis on ironless permanent magnet synchronous linear

motor (ILPMLSM) of a Halbach magnet array. An analytical model presented by using

the magnetic scalar potential. The effect of magnetizing patterns of the permanent

magnets was investigated. The effect of other design parameters, such as thickness of

windings and magnet array, thickness of airgap, pole pitch, was investigated in details.

The results were validated by comparison of the ILPMLSM with finite element analysis

and experiments [27].

42

2.1.2 Linear Induction Motor

The LIM has a beginning and an end in the direction of travel. This feature

produces an end effect that adversely influences the performance of the linear motor.

This is one of the most important differences between the linear induction motor and the

rotary induction motor. Hence, the longitudinal end effects that are of concern in a LIM

are not a consideration for rotary motors.

Hermant, C (1960) given a history in LIM that this type of motor came up in 1890, just

two years after the development of rotary induction principle. The annular linear

induction pump and the flat linear induction pump are similar to linear induction

motors. The search for new ways of generating electricity led to the

investigation of liquid-phase magneto hydro dynamic (MHD) generators. These include

the liquid induction pumps, operating above synchronous speed [28].

Yamamura and Poloujadoff (1980) studied the theoretical aspects of LIM in detail,

especially the influence of longitudinal end effects. The work was primarily concerned

with theoretical analysis of single and double-sided linear induction motors [29-30].

Nasar and Boldea (1987) did an extensive research on linear electric motors. The

cocept of single-sided linear induction motor was discussed in their book [9].

R. M. Pai and I. Boldea (1988) developed a complete equivalent circuit of a linear

induction motor with secondary sheet. They derived the steady state performance

characteristics of linear induction motors using 2D/3D dimensional analyses, including

longitudinal end effects, transverse edge effects and skin effects in the secondary.

Furthermore, they obtained an equivalent circuit of a LIM from the field analysis, where

the various effects of LIM were taken into account [31].

Anthony R. Eastham and J. F. Gieras (1988) studied causes and consequences of

phase imbalance in single-sided linear induction motor were demonstrated by them. The

two methods of evaluating phase imbalance were discussed. The first was analytical,

43

based on an equivalent circuit model. The second was numerical using the finite-

element method. The computational results were validated with test results. It was also

shown that phase imbalance produces a reduction in both thrust and normal forces, but

this effect is likely to be significant only for high-speed LIM’s [32].

Zhang et al. (1993) investigated LIMs dynamic performance assessment based on

parameter identification. They presented a reliable method of obtaining nominal

parameters of an induction motor based on conventional tests, and then proposed a

dynamic performance assessment scheme for linear induction motors, based on an

on-line parameter updating algorithm. The experiments with a static LIM test device

demonstrated its effectiveness. Verified scheme was then applied to an operational

transportation LIM and this computed real-time information for control purposes [33].

J. F. Gieras et al. (1994) covered all the aspects, including constructional features,

applications, electromagnetic effects, and design of linear induction motors of single-

sided, double- sided, and tubular. The configurations were analyzed in terms of

equivalent circuits and their components [34].

E. R. Laithwaite (1995) made an efforts for applying linear induction motors to the

acceleration for large masses to high velocities. The paper described the features needed

in the design of a linear induction motor that may accelerate a mass of 200 kg to 1200

m/s at a distance of 1500 m. Attention was confined to the accelerated part of the

motion. Deceleration to rest in the second 1500 m of track could be approached with the

same techniques. Paper mentioned that many novel features needed to be embodied in

the design and that these features extended in the practice of making LIMs [15].

Gieras (2002) in his book on LIM covered all aspects including constructional

features, applications, electromagnetic effects, and design. The different types of linear

44

motion motors and their configurations were analyzed in terms of equivalent circuits

and their components in his text [16].

2.2 Prominent Numerical Methods for Solution of Low Frequency Field Problems

The numerical methods for the solution of strongly coupled problems with finite

elements are studied extensively in the literature. In the time-stepping analysis of FEM

based nonlinear differential equations; the solution process requires methods for

modeling the time-dependence, handling the nonlinearity and solving the resulting

system of equations. The simple difference methods, like backward Euler, Galerkin or

Crank-Nicholson, were the most commonly used methods for the time-stepping

simulation. While these utilize results from two adjacent time steps, there were

numerous multi-step methods performing numerical integration over several time steps

and providing higher accuracy.

Kawakami, T (1966) discussed in his publication the numerical methods have long

been recognized as accurate and practical methods of field computation to aid in

electrical design [35].

When the phenomena of substantially different time scales are coupled together,

the problem is mathematically considered as stiff. Most of the multi-step methods

usually fail for such problems, but the implicit difference methods often converge.

Further discussions on stiff problems have been presented by Gear [36] who given a

backward-difference formulae (BDF) method, which was known as Gear algorithms

implemented for highly oscillatory problems which have eigenvalue close to the

imaginary axis. It was the middle sixties when events necessitated the development of

advanced solution methods to enable designers to build with confidence. Such an event

was the increasing demand on electrical energy at that time. To meet the demand the

power per unit volume of large turbine generators had to be increased. Up to this time

45

analytic design methods prevailed, but were limited due to the need to model the non-

linearity of magnetic materials, irregular geometries and many other features.

Haase G. et al. (1997) described the use of the FEM and BEM in different sub domains

of non-overlapping domain decomposition (DD) and their coupling over the coupling

boundaries (interfaces) brings about several advantages in many practical applications.

The algorithm applied to the direct current motor (DCM) with permanent excitation.

The BEM used here for the air sub domains including the exterior of the motor and

FEM which was preferred was ferromagnetic materials where non-linearity can occur in

partial differential equation. The main aim of the work was to design and analysis of the

well adapted parallel solvers for the large scale coupled FE/BE equations approximating

plane linear and non-linear magnetic problems [37].

Bruce Klimpke (2003) described a method to solve the magnetic field problems with

finite element method. The less common method of solving magnetic field problems

was using the boundary element method. The purpose of this paper was to illustrate the

best numerical method to be use depends on the type of problem to be solved. In general

the best strategy discussed here to use a hybrid solver in which the advantages of each

method can be applied to each region for the entire problem [17].

2.2.1 Boundary Element Method

Y. Bulent Yildir (1988) presented the use of BEM in computer-aided magnetic field

analysis. The method has been shown to be an efficient technique also for the solution

of Poisson’s equation for non-linear media. This is mainly due to the reduction of one in

dimensionality as all the unknowns are located only on the boundaries and interfaces.

This differs from the FDM in which the whole domain must be discretized. The

unknown, computed using the boundary element method, is the equivalent source that

sustained the field. Once the equivalent source is known any parameter can be derived.

46

The BEM, combined with a highly interactive user interface, automates the computation

and analysis of magnetic field distributions for motor, generator, solenoid, speaker,

actuator and other magnetic device design [38].

Nakata, T et al. (1988) examined the periodic boundary condition in analysis and

presented a new method for analyzing the magnetic fields and currents in electrical

machines excited from voltage sources using the A-∅ method. The usefulness of the

periodic boundary conditions is shown by analyzing some models and observing that

the CPU time is considerably decreased [39].

Brebbia, C.A. et al. (1994) discussed the applications of the boundary element method,

which has been successfully applied to a variety of problems in engineering, particularly

to analyse electrostatics problems. Work presented applications of the BEM to study the

electromagnetic problems present in synchronous motors. In many electromagnetic

applications its required to consider the domain extending to infinity [40].

Y. Bulent Yildir et al. (1995) described a frequency domain boundary element method

(BEM) for simulating and analyzing polyphase induction machines with steady-state

sinusoidal time-varying current source excitation. Advantages of the BEM technique,

such as ease of use, accurate solutions, and significant reduction in computation time,

were addressed [41].

Poljak D. & Brebbia C.A. (2005) discussed many ceases of electromagnetic fields at

low frequencies including eddy currents analysis based on the boundary element

method in their book published by WIT press, UK [42].

Johannes J. H. Paulides et al. (2011) presented a general mesh-free description of the

magnetic field distribution in multiple airgap electromagnetic machines based on

magnetic field solution for a wide class of 2-D boundary element method. The

technique discussed in this paper applied to the rotary multiple airgap machine with slot

47

less (without slots but with and without rotor back-iron) armature. The presented

analysis further compared with finite element analysis for the multiple-layer winding,

which showed the applicability of this method for future optimization [43].

2.2.2 Finite Difference Method

K. S. Kunz et al. (1993) discussed the applications of finite difference time domain

method in the electromagnetic field problems in their book. They emphasized on the

low frequency domain problems in their text [44].

Moerloose Jan De (1997) presented the use of the standard finite difference time

domain (FDTD) algorithm extended to quasi-static electromagnetic field problems.

Although, the straightforward application of the standard FDTD algorithm at very low

frequencies leads to excessively long simulation times, discussed here for linear

structures which can be circumvented by using a ramp excitation function. The method

found well suited for the modeling of fields in and around complex-shaped,

heterogeneous, conductive bodies [45].

D. M. Sullivan (2000) discussed various kinds of simulation techniques based on

frequency difference time domain method for the electromagnetic problems in his book.

He also discussed some examples related to electrical machines [46].

Daniel Nahum Zmood (2001) presented a method of analyzing VSI-based current

regulators in the frequency domain, so that these can be transformed between the

stationary and the rotating frame of reference for comparison and performance

assessment. The approach has been used to identify the primary reason why a stationary

frame regulator cannot eliminate steady-state ac error, and was then used to develop a

new resonant stationary frame current regulator, which overcomes this limitation. The

new regulator offered significant advantages over synchronous frame regulators in

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terms of complexity and ease of implementation, while achieving equivalent transient

and steady-state performance [47].

Katsumi Yamazaki (2002) presented a procedure to obtain equivalent circuit

parameters of induction motors considering skin effect, harmonics and magnetic

saturation, movement of the rotor and the end effects of the motor using the finite-

element method. The frequency domain analysis applied in advance to obtain the

primary value of the magnetic vector potential. The nonlinear time-stepping finite-

element analysis was carried out with this primary value. A method, which reduces the

calculation term by utilizing the symmetry shape of the rotor, was also applied. It

obtained the whole time-variation of not only the stator, but also the rotor

electromagnetic fields at steady rotation with small calculation time [48].

Johan Gyselinck et al. (2003) have presented a general method to include time

periodic movement in the frequency domain finite- element simulation of rotating

electromagnetic devices. In the particular case of a two-dimensional finite element

model of a rotating machine, it only requires elementary manipulations of the static,

moving band stiffness matrix, for a number of discrete rotor positions in a fundamental

period. The no-load operation of an induction motor was simulated [49].

R. Horvath et al. (2005) showed that stable time stepping schemes for the Maxwell

equations are important in connection with finite difference spatial discretization [50].

Yuichiro Nozaki et al.(2005) presented an analysis of linear induction motor for HSST

and linear metro using two-dimensional finite difference method was presented. The

biggest problem of a linear induction motor was the end-effect which appeared in high-

speed operation and deteriorated performance. The influence of end effect was analyzed

in their work [51].

49

Mayank Gupta (2013) elaborated implementation of the FDTD on a GPU and

demonstrated the ability of low cost GPU’s to accelerate real life problems of interest in

the electromagnetic field simulation domain [52].

2.2.3 Finite Element Method

The finite-element method originated from the need for solving complex elasticity

and structural analysis problems in civil and aeronautical engineering. As per

Wikipedia, Its development can be traced back to the work by Alexander Hrennikoff

(1941) and Richard Courant (1942). While the approaches used by these pioneers are

dramatically different, they share one essential characteristic: mesh discretization of a

continuous domain into a set of discrete sub-domains, usually called elements.

Hrennikoff's work discretizes the domain by using a lattice analogy while Courant's

approach divides the domain into finite triangular sub regions for solution of second

order elliptic partial differential equations (PDEs) that arise from the problem of torsion

of a cylinder. Courant's contribution was evolutionary, drawing on a large body of

earlier results for PDEs developed by Rayleigh, Ritz, and Galerkin.

Development of the finite element method began in earnest in the middle to late

1950s for airframe and structural analysis and gathered momentum at the University of

Stuttgart through the work of John Argyris and at Berkeley through the work of Ray

W. Clough in the 1960s for use in civil engineering. By late 1950s, the key concepts of

stiffness matrix and element assembly existed essentially in the form used today. The

first book on FEM by Zienkiewics and Chung was published in 1967. Late 1960’s and

early 1970’s, FEM was applied to a wide variety of engineering problems. NASA

issued a request for proposals for the development of the finite element software

NASTRAN in 1965. The method was provided with a rigorous mathematical

foundation in 1973 with the publication of Strang and Fix's. An analysis of the finite

50

element method has since been generalized into a branch of applied mathematics for

numerical modeling of physical systems in a wide variety of engineering disciplines,

e.g., electromagnetism and fluid dynamics [53].

Chari, M.V.K (1977) analyzed the skin effect phenomena using the triangular finite

element method. The current density distribution in a conductor of finite size is affected

by the presence of eddy currents in the conductor was discovered. This phenomenon,

generally known as skin effect, causes ohmic losses in the conductors and alters their

magnetic induction. The results obtained by this method were compared with those of

classical and dimensional analysis for [53].

Sadowski N et al. (1995) presented the methodology for solving simultaneously the

equations of magnetic fields and electric circuits of electrical machines. To evaluate the

magnetic fields the FEM was used. The permanent magnet and induction motors were

fed by different electric circuits. The time stepping technique was employed to simulate

the steady and transient states. The results were obtained in magnetic vector potential

form for describing the magnetic behaviour of the machine [54].

D. Rodger et al. (1997) presented some of the numerical techniques, which can help in

modelling electrical machines using finite elements. The sliding interface technique

allows a machine to rotate in a realistic manner, while connected to an external circuit

were explained here. The use of periodic boundary conditions can sometimes yield

more economic solutions. Here some features of finite element method package

MEGA’s were described [55].

2.3 Self Adaptive Finite Element Method

As the power of computers increased the finite element method came more into its

own and was in the early seventies the preferred choice. Industries used it as a research

tool. The boundary element method still had its place in design, especially in

51

electrostatics, but the need to solve a large unsymmetrical and full matrix was still its

major drawback. Before the finite element method, finite differences and integral

equation techniques were being used.

With the advent of computers, programming was attaining heights in the

design of electromagnetic devices. Hence flexible computer programming was used by

Silvister P [56]. Computer techniques have revolutionized the way in which

electromagnetic problems are analyzed.

Lynch D. R [57] solved the vector parasite problem as an increasing availability of

computer resources coupled with a desire to model more complex electromagnetic

problems has resulted in a wave of renewed interest in finite element methods for

solving EM field problems. Specific implementations of three-dimensional

electromagnetic finite element codes were described in his Ph.D. dissertations by Maile

[59] and Webb [60].

Chellamuthu K.C et al. (1997) discussed the adaptive finite element mesh refinement

combined with a robust and functionally reliable error estimation to get optimal solution

accuracy. The performance of the error estimates applied for the adaptive modelling of

the non-linear problem involved in the permanent magnet synchronous machine [61].

F. Bellina et al. (2002) developed a method, based on the finite formulation approach

for the solution of a quasi-static eddy-current problem (ECP). The method was tested

against a two-dimensional axis symmetric ECP and the obtained results have been

compared with those from well-established methods. The finite formulation presented a

new formulation, complementary to the traditional differential formulation [62].

Awadhya et al. (2003) published the finite element method, which was explored by

them in high frequency engineering design. A survey focused on microwave

applications of FEM was explored [14].

52

Marcel Janda et al. (2007) described possibilities of interoperability between modern

computer aided design (CAD) systems and ANSYS workbench. In their work modern

CAD systems represented by Autodesk Inventor 10 Professional, was presented which

were used for creating models of electric machines and devices. The program facilitated

the creation of professional technical documentation, which further can be used in the

production process in particular. Another benefit of its functions was the creation of

animated presentations. Proposed program was used for the creation of models of

physical fields with the use of finite element method. Interconnecting these programs

was relatively simple in a way to create realist model electric machine [63].

Jaun J. Rodenas et al. (2008) attempted to compare 2D problems between the XMLS

(extended moving least squares) technique and an enhanced version of the SPRXFEM

(super convergent patch recovery extended finite element method) technique.

Sequences of uniformly refined meshes for problems with known exact solution have

been analyzed in order to evaluate the accuracy of the error estimators. The advantages

and disadvantages of the each method were also discussed in their paper [64].

A. Tessarolo et al. (2008) had proposed an approach for an automated finite element

analyses to define a lumped-parameter machine model, which was analytically solved in

all the motor operating points of interest. At each optimization step, the machine

efficiency was computed. The proposed program was assessed as a design platform for

customized induction motor optimal dimensioning. As the need to enhance the

efficiency of induction motors is an important task due to the large diffusion of these

machines in variable-frequency drives and to their massive contribution to the global

energy consumption [65].

53

2.3.1 Mesh Generation Trends

Shephard M. S. (1988) discussed in his review paper the algorithmic approaches being

used in the development of finite element mesh generators capable of automatically

discretizing general domains without the need for user intervention. Consideration was

also given to both a priori and a posteriori mesh control device for automatic mesh

generators as well as their integration with geometric modeling and adaptive analysis

procedures [66].

Lo S. H. et al. (1992) proposed an algorithm for the generation of ‘hybrid’ finite

element meshes composed of quadrilateral and triangular elements over any arbitrary

planar domain. The new meshing strategy did not require more input data than a correct

representation of the domain boundary. Mesh generation started by constructing as

many square elements as possible in the interior part of the domain, leaving a relatively

small region around the boundary which is to be discretized into triangular elements by

standard triangulation procedures. As a result, only the quality of the triangular

elements near the boundary got affected by the irregularity of the domain. The

extraction of the interior part and forming of square elements of an irregular domain

was discussed in this paper [67].

F. Sheperd Jason et al. (2010) presented the mesh adapting methods to improve the

efficiency and accuracy of solutions to computational modeling problems. The methods

presented provided coarsening, while maintaining confirming all-quadrilateral and

hexahedral meshes [68].

Weizhang Huang et al. (2010) demonstrated the anisotropic mesh adaption, which

significantly improved computational efficiency over isotropic mesh adaption,

especially for problems with strong anisotropic features. The M-uniform mesh approach

of mesh adaption was considered to define a proper metric tensor mesh adaption bound

54

for the first variation of general functional derived, which is semi- a posteriori in the

sense that it involves the residual and an edge [69].

Houman Borouchaki et al. (2010) discussed the concept of ‘unit meshes’ (whose

elements are of size unity) in an appropriate non-euclidean metric. They proposed a

remeshing method that makes this mesh construction possible with the approach of

mesh refinement and mesh coarsening using the particular Delaunay kernel. The

improved adaptive remeshing strategy was based on the a posteriori error indicator

estimation and it accelerated drastically the convergence of computing time and

minimized the memory requirement. It was also applicable to the most complex shapes,

involving FEM resolution of Multiphysics problems [70].

Persson, Per-olof, et al. (2010) proposed a method for generating well-shaped curved

unstructured meshes using a nonlinear elasticity analogy. The un-deformed geometry is

the initial mesh of linear triangular or tetrahedral elements. The external loading result

of prescribing a boundary displacement to be that of the curved geometry, and the final

configuration was determined by solving for the equilibrium configuration. The

deformations were represented using piecewise polynomials within each element of the

original mesh. When the mesh is sufficiently fine to resolve the solid deformation, this

method guarantees non-intersecting elements, even for highly distorted or anisotropic

initial meshes. This method is highly resistant to inverted elements and typically

produces very well-shaped elements. Another benefit is that it curves the original mesh,

without introducing new nodes or modifying the mesh topology [71].

2.3.2 Error Analysis

I. Babuska et al. (1978) introduced error estimators and adaptive refinement

procedures. The estimation was based on evaluating the residuals of the approximate

solution and using them to obtain a more accurate solution. Both the error estimation

55

and the refinement process proved to be costly for practical use, since the final accuracy

is reached only after many resolution steps [72].

Song-yop Hahn (1988) presented a method to derive error estimates for adaptive finite

element mesh generation. In this, the elements which violate many of the conditions as

local errors were refined. Procedures to derive error estimations and numerical

examples to verify the validity of the proposed method were presented [73].

Darcherif et al. (1990) worked in the direction to improve the trial function by

discretizing the mesh into smaller elements and by increasing order of trial functions

while using the same mesh. The advantage of accuracy associated with higher orders is

partly retained with this method used [74].

Ackermann, J. (1994) proposed an error controlled finite element method (FEM) for

solving stationary Schrödinger equations in three space dimensions. The method is

based on an adaptive space discretization. The adaptive FEM turns out to be an efficient

and rather universal tool for an accurate and error controlled treatment of the stationary

Schrödinger equation in three space dimensions. The final accuracy is limited only by

the requirements on the CPU-time and the core of the computer being used [75].

L. Lebensztajn et al. (2003) presented different error estimator to improve accuracy in

linear and nonlinear self- adaptive finite element field calculations. The first estimator

was based on the polynomial theory, the second one made an estimation of the flux

density divergence, the third one was linked to a magneto motive force associated with

elements sides, and the fourth one was based on the use of the bilinear element [76].

Alberto Bertoldo (2007) obtained high-level adaptivity on complex scientific

applications such as finite element (FE) simulators by building an adaptive version of

their computational kernel, which consisted of a sparse linear system solver. FEMS

relied on a simple model-driven mesh partitioning strategy, which makes it possible to

56

perform efficient static load balancing on both homogeneous and heterogeneous

machines. Model was used at run time in a preprocessing phase to achieve good static

load balancing that reduces synchronization and communication volume involved in the

numerical solution process [77].

2.3.3 Mesh Refinement for Electromagnetic Field Problems

Williamson et al. (1982 and 1983) connected to conventional calculation methods and

numerical field analysis in their work. The current densities in the conductors were

assumed to be constants so that skin effect was not taken into account while solving the

field equations in their work. The rotor currents were solved from the airgap flux

(vector potential) induced by the stator currents with an aid of rotor bar impedances.

The method was found to be quite versatile, in which, the effects of harmonic

components and skewed rotor slots were considered [78].

Luomi, J et al. (1988) presented adaptive mesh refinement for the finite element

solution of two-dimensional magnetic fields. Attention was given to problems with

saturable ferromagnetic regions, high and abrupt variations of the magnetic permeability

and current density distributions [79].

Todd H. Hubing (1991) demonstrated comparative study of different computational

techniques employed for the solution of electromagnetic field problems. [80].

Lombard, P. et al. (1993) presented an approach for automatic adaptive mesh

refinement in the FE computation of magnetic fields and estimated the errors [81].

Wang, J. S. et al. (1993) explained that edge type finite elements are very useful in

computation of electromagnetic fields. They provided continuity of triangle components

of the field variables across the element interfaces while allowing discontinuity in

normal components. Paper described systematic construction of first and second order

‘edge’ and ‘facet’ finite element based on the nodal-based conventional elements and

57

higher order elements allowing an easy implementation in existed nodal-based finite

element computer programs [82].

Francis Piriou et al. (1993) presented two approaches for the numerical simulation of

electromagnetic systems accounting for electric circuit equations. Firstly, they have

considered the indirect coupled model which permits the simulation of moderate

calculation time. Secondly, they have developed the direct coupled model where the

magnetic and electric circuit equations are solved simultaneously [58].

Taylor et al. (1999) presented a novel technique for adaption where the field

distribution is iteratatively used to determine the element size, orientation and shape in

order to create an optimal mesh. At each step the area of the model is packed with

specified number of ellipses whose shape varies according to local field level and

gradient obtained from the previous solution and an isotropic mesh was created. This

adaption process is particularly efficient when rapid field variations occur in one

direction. The choice of ellipse sizing algorithm was however much harder to determine

the errors within the solution arising due to poor modelling of field and of induced

currents [83].

K. Hameyer et al. (2002) have revealed the modeling and optimization of different

electromagnetic field problems, including a magnetic circuit of penny motor [84].

K. Yamazaki (2003) proposed a novel algorithm of adaptive mesh generation for the

nonlinear finite-element analysis of electric machines. An auto-mesh generation

technique was applied at each nonlinear iterative step with the error estimation of the

field distribution obtained by each nonlinear iterative calculation. The flux density and

the permeability of the new elements were defined by interpolating calculations from

the previous mesh in order to continue the nonlinear iteration [85]. They also presented

a novel algorithm of adaptive mesh generation for the non-linear finite element analysis

58

of electric machines. It was applied at each non-linear iterative step with the error

estimation of the field distribution obtained by each non-linear iterative calculation. The

calculation time was reduced significantly in several analyses [86].

2.3.3.1 Various Types of Refinement Techniques

Meyer F.J.C. et al. (1996) have discussed the use of adaptive methods to refine the

mesh as needed, for a more efficient use of the computational resources. Both

h-adaption (smaller) and p-adaption (higher-order elements) have been described. Good

results were obtained for energy-related errors within the FEM mesh [87].

Marcel G. Vanti et al. (1997) implemented p and h-p adaptive versions of finite

element method. The different procedures of adaptivity and optimal mesh concept were

discussed and a methodology to obtain optimal or quasi optimal meshes with

hierarchical elements was presented [88].

Mark Dorica et al. (2004) described new mesh smoothing techniques for adaptive

finite element analysis which was significantly improved the mesh quality. The

proposed method fails to establish the relation between field quality and solution

efficiency [89].

S. H. Lo et al. (2010) described that h-type refinement can be achieved in two ways.

Firstly, smaller elements are connected directly to large elements with full compatibility

at element interfaces. Secondly, the transition elements are employed to link up

elements of different sizes. They developed a series of versatile transition elements

based on the hybrid stress approach [90].

2.3.3.2 Delauney Triangulation Method

Boris Nikolaevich Delaunay (1934) introduced the Delaunay triangulation grid

concept for the two-dimensional grids for the analysis of symmetrical regions [91].

59

Peraire et al. (1987) described that the initial uses of iterative Delaunay refinement

have been motivated by Frey [92], who used the circumcentre of triangle for selecting

new vertex on the basis that circumcentre, a new vertex that was equidistant from the

vertices of a triangle, and a longer distance from any other nodes of mesh covered in his

contributions [93].

Mavriplis D.J. (1990) extended the ideas of past literature to anisotropic errors. These

errors were minimized by meshes in appropriate stretched coordinate systems [94].

P. Olszewske et al. (1993) presented a simple algorithm yielding unique face

subdivision patterns of locally refined tetrahedral elements by the Delaunay tessellation

process. The performance of the presented algorithm and the effectiveness of the error

estimator were demonstrated by means of problems [95].

Baker T. J. (1994) noted that a characteristic of Delaunay meshes was that those

contained the most equiangular. This is due to the benefit of a shape as a triangle which

is not bound to any circumstances for the choice of circumcentre for insertion [96].

B. Edwin (1994) developed an algorithm based on the Delaunay technique grid

generation. Delaunay triangularlisation used a particularly simple criterion for

connecting points to form conforming, non-intersecting elements [97].

Rivara MC et al. (1997) explored a new mathematical tool for dealing with the

refinement and/or the improvement of unstructured triangulations, named as the

‘Longest-Edge Propagation Path’ was associated with each triangle, which was to be

either refined and/or improved in the mesh. It was defined as the (finite) ordered list of

successive neighbor triangles having longest-edge greater than the longest edge of the

preceding triangle in the path. This idea was used to introduce two kinds of algorithms.

These algorithms and practical issues related to their implementation were discussed in

this paper [98].

60

Chernikov N. Andrey et al. (2010) described the complete solution for both sequential

and parallel construction of guaranteed quality Delaunay meshes for simple geometries.

They presented a parallel algorithm which can be used in conjunction with any

sequential point placement strategy that chooses points within the selected disks. The

two-dimensional regions are derived and known to be here as selection disks [99].

2.4 Finite Element in Electrical Machines

Ito, M et al. (1981) computed the nonlinear field of an induction motor using an eddy-

current formulation and assuming sinusoidal time variation. The end rings of the rotor

cage were assumed to form ideal short circuits so that the ends of rotor bars were all in

the same potential. The rotor was assumed to be pseudo stationary. The method was

used to study various properties of induction motors and some suggestions were made

to improve the design of the motors [100].

Chari and Konrad (1982) performed finite element computations for three

dimensional electrostatic and magneto static field problems. Paper included new

computational technique employing mixed scalar and vector formulations was

introduced [101].

Bouillault et al. (1983) used time-stepping methods to calculate the time variation of

magnetic fields in induction motors. The paper dealt with solid rotor induction motors.

It is assumed that the ends of the rotor are equi-potential surfaces of the scalar potential

so that the rotor field can be solved with the potential differences equal to zero. The

voltage equations of the windings of an induction motor were coupled to the field

equation [102].

Ashtiani C.N. (1983) presented a direct FEM for the simulation of the steady state load

operating point of a synchronous generator. The terminal constraints of the generator

under load were conveniently expressed in terms of the system state vector. This made

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it possible to solve the FE equations along with the terminal constraints in a single run,

hence avoiding unnecessary iterations. The method presented may be applied to both

turbo and waterwheel generators so for as saliency is concerned [103].

S. Mukherjee et al. (1989) discussed a direct method to simulate the steady-state stator

and rotor currents in an inverter-induction machine of known geometry and winding

design. The electromagnetic analysis of the skin effect problem was used to determine

the variation of rotor resistance with rotor current frequency. The rotor conduction loss

calculated from these two sets of information [104].

J. Salon Sheppard (1993) presented two-dimensional analysis, emphasizing the use of

finite elements to perform the most common calculations required for machine

designers and analysts. He explained a finite element program to determine the

behavior of electric machines [105].

Ho and Fu (1995) described a method of evaluating harmonic stray losses in non-

skewed induction motors using adaptive time–stepping finite element method together

with externally coupled circuits [106].

Since the FEM has advantages over integral equation methods for certain

electromagnetic problems such as those involving material inhomogeneity and

nonlinearity, it is preferable to use the FEM in conjunction with a special

technique to solve these electromagnetic problems as presented by Chen Q [107].

D. Dolinr et al. (1997) proposed a finite element method for determining the

parameters of a two axis model of a three phase cage induction motor taking into

account the saturation of the motor while calculating the flux distribution by using a

static nonlinear vector potential solution [108].

Y. K. Wong et al. (1997) proposed an algorithm to study the steady state performance

of induction motors using time stepping finite element model. The paper described that

62

only incremental values instead of original unknown are required to be solved in this

method [18].

Zhao Zhengming et al. (1997) presented a novel adaptive algorithm for on-line

estimation of variable parameters of a synchronous machine (SM) as a function of the

operating conditions. The computer simulation results were presented to highlight the

advantages of the novel adaptive algorithm over conventional least mean square and

recurrence least square algorithm [109].

Zhou Ping et al. (1998) discussed a method for accurately predicting the steady-state

performance of induction motors. The approach was based on the use of complex two-

dimensional finite element solutions to deduce per-phase equivalent circuit parameters

for any operating condition. Core saturation and skin effect were directly considered in

the field calculation. An application example demonstrated the effectiveness of the

proposed approach [110].

Zhou Ping. et al. (1999) presented a method for predicting the steady state performance

of induction motor based on complex two-dimensional finite element solutions for

deducing per phase equivalent parameters considering core saturation [111].

Winslow Ray (1999) made the first application of FEM in electrical engineering in the

analysis of saturation effects in accelerator magnets for the nonlinear variational

formulation of electromagnetic field problems [112].

Do-Wan Kim et al. (1999) made an attempt to analyze the traction motors for

transient analysis. Equivalent circuit modeling was performed with the parameters

calculated from FEM. In the transient state, the equivalent parameters such as mutual

and leakage inductance were changed, because of the change of magnetic permeability

and current density. The motor model was analyzed and inductances were calculated

from the resultant flux linkage. The d-q transform analysis was accomplished where

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the parameters are dependent on permeability of Iron core and slots [113].

Chang-Hoo et al. (1999) analyzed the mechanical deformations and stresses by finite

element method (FEM) on the rotor bars of a small squirrel cage induction motor. This

considered the magnetic forces and the centrifugal forces which provoked the

deformations and stresses on the rotor bars [114].

Aldo Canova et al. (2001) presented the use of a three dimensional eddy current FE

procedure for the analysis and design of two different induction machine structures: a

radial and an axial machine. In the first case, attention was devoted to the simulation of

locked rotor conditions. The simulation results have been compared with 2D analysis

and experimental data. The axial flux machine was analyzed under fixed speed

conditions [115].

Busch T. et al. (2002) presented four different examples of the way finite-element

models, being used at the Institute for electrical machines at Aachen institute of

technology. The bandwidth of the applications covered the static torque and force

calculations for electrical machines, transient calculation of eddy current losses, and a

procedure to determine the mechanical and acoustic behavior of electrical machines and

coupled simulations to calculate the dynamic behavior of electrical machines [117].

Gersem Herbert De et al. (2002) presented a time-harmonic finite-element

formulation, which was strongly coupled to fast Fourier transforms discretion, a

particular kind of interface conditions. The approach enabled the splitting of the

elliptical airgap field, typical for single-phase induction machines, into two contra-

rotating parts that are applied to two distinct rotor models. Motional eddy-current

affected by slip transformation. The simulation of a capacitor-run motor demonstrated

the coupled discretization scheme. The coupled time harmonic finite element

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formulation with airgap flux splitting resulted in a fast simulation tool for steady-state

operations of rotating single-phase induction motors [118].

Koen Delaere et al. (2003) described the stator currents and the stator vibration

spectrum of induction machine, which was computed using a transient 2D finite element

analysis. The analysis was performed for two different rotor geometries: with open and

with closed rotor slots. Open rotor slots increased the harmonic content of the airgap

field, and also the harmonic content of the stator currents. Open rotor slots, thus

increased the stator vibration level for frequencies higher than 1kHz [119].

Bugeza Miroslav et al. (2003) presented the calculation of linear induction motor and

linear synchronous motor steady- state characteristics using finite element method.

Two-dimensional non-linear magneto static and eddy-current solver were used for

magnetic field and force calculation. Calculations were performed for different

excitation currents and different airgap lengths for both types of linear motor. The

results were obtained from the magnetic field calculations used for determination of

motor parameters (e.g. Inductance) and also were useful for mathematical modeling and

control system simulations [120].

Theodoropoulos Ioannis (2003) studied optimization of solid rotor induction machine

by using analytical and finite element technique. Design optimization was performed

based on the finite element method and analytical solutions in case of regular

geometries combined with sensitivity analysis techniques. The model precision was

checked by comparing their results for the same machine configuration. This method

was applied to optimize the rotor geometry of a solid rotor induction motor, presenting

important advantages for small scale applications due to its structure simplicity [121].

Mezani S. et al. (2003) described computation method for steady state analysis of

induction motors accounting for space harmonics including space harmonic fields using

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complex finite elements. The principle of the method included a spectral decomposition

of the source field in the airgap, to compute each elementary problem

corresponding to a harmonic order and then to use superposition to determine the

total field. The presented results permitted to highlight the effect of the space harmonics

on the rotor Joule losses which increased in normal operation of the machine. The

computations also illustrated the harmonic torques that may cause asynchronous

crawling of induction machines, revealing the effectiveness of the method when

tracking steady state operation [122].

Gersem H. De et al. (2003) presented the motional eddy current effects in a shaded-

pole motor due to higher harmonic airgap fields. The method incorporated time-

harmonic finite element machine models by a spectral decomposition of the airgap field

and the distribution of higher harmonic components to additional rotor models [84].

Park J. T. et al. (2003) presented design of bracing to support the windings. The force

of the end winding should be calculated more correctly. A force calculation program

based on Biot-Savart's law was developed because of its high rapidity, and finite

element method was implemented for analysis [123].

Katsumi Yamazaki et al. (2003) specified the main component of the stray load loss of

induction motor from both results of measurement and analysis. As such it provided

loss segregation and other performance values including operating temperature, power

factor, speed and line currents as a function of load. The losses generated at the stator

core, the rotor core and the rotor cage were calculated directly by the finite element

method considering the magnetic saturation and the harmonic fields, which vary due to

the load condition [86].

Oriano Bottauscio et al. (2004) attempted for analysis of electric and magnetic

quantities of an induction motor. The measured results compared with the by an

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advanced FEM code, achieved satisfactory agreement. The presentation was

emphasized on accurate numerical models for the analysis of the electromagnetic

phenomena occurring inside electrical induction motors [124].

D. Spalek (2004) dealt the problem of force and torque calculation for linear,

cylindrical and spherical electromechanical converter. The electromagnetic field was

determined analytically with the help of separation method for each problem. The

proposed method for electromagnetic torque/force calculation described in an analytical

way enabled to calculate the component values [125].

Dobai Jeno Barna et al. (2004) presented FE model of three phase squirrel-cage

induction motor to analyse the behavior of the machine having broken rotor bars [126].

Nicola Bianchi (2005) published a book on electrical machine analysis using finite

element method covering all the electromagnetic problems with proper examples and

equations [127].

Baltazar Parreira et al. (2005) described the step by step procedure for modeling an

8/6 switched reluctance machine (SRM). Series of experimental tests were performed to

obtained the magnetic characteristics of the machine, and results were computed by

finite element method [128].

Lee Hyung-Woo et al. (2005) presented the electromagnetic field distribution of the

end winding part of induction motor using 3-D FEM to analyze end winding forces

more precisely. Stress analysis of the end windings was performed by inputting the

force distribution calculated by FEM for the electromagnetic field [129].

Y. Ouazir et al. (2006) discussed a method for coupling the stator and rotor fields of

induction motors in time harmonic finite element analysis. The coupling scheme was

derived from Fourier series of the magnetic vector potential at airgap interface

conditions. The obtained results were compared with experimental results in critical

67

situations. The effects of inter-bar currents, which caused important stray torques in the

back rotating operation, were discussed [130].

Bhoj Raj Singla et al. (2006) described a new method for the analysis of transient

phenomena in cage induction machines. The procedure described was implemented

directly into design routines by automating the complete finite element calculation. The

extensive use of circuit techniques were allowed the effects of axial skew and spatially

distributed airgap flux harmonics incorporated [131].

Mehmet Çunkaş et al. (2006) compared three different optimally designed three-

phase squirrel-cage induction motors with an existing motor of the same rating. An

optimization technique based on genetic algorithms was applied to the design of 30 HP

three-phase induction motor. A package program that analyzed and optimized

induction motors and evaluated the cost and performance of the design. Comparison of

the final optimum designs to the existing design indicated that the gain of the proposed

performance is even better than expected [132].

Darabi Ahmad et al. (2007) presented a finite element based simulation coupled with

a well known hysteresis model called ‘Preisach model’ that was used to calculate local

Iron loss of a machine. Calculation of Iron loss, particularly local loss with some

accuracy, offered an opportunity to the designer for reducing losses where loss

densities were high. This lead to an optimum design of cooling system [133].

Chin-I Huang et al. (2007) proposed a non-linear adaptive controller and an adaptive

back stepping controller for linear induction motor to achieve position tracking. A non-

linear transformation was proposed to facilitate controller design. A stability analysis

based on Lyapunov theory was performed to guarantee that the controller have been

designed to stabilize the system. The computer simulations and experiments were

conducted to demonstrate the performance of controller designs [134].

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Lorand Szabo et al. (2008) discussed the structural analysis or electromagnetics. The

MegNet program was used to analyze the magnetic field of three phase modular double

salient linear motors. This modularly built up linear motor eliminated several

disadvantages of the ‘classical’ hybrid linear stepper motor. Which replaced the ball

screws, gear trains, belts and pulleys etc. For improvement in the linear drive system’s

performance [135].

B. Vaseghi, et al. (2009) performed a time stepping two-dimensional FEM for

modeling and analysis of a induction machines with insulation failure inter-turn fault.

Study helped the machine designers to improve the fault tolerance aspect as well the

overall design of the machine drive system [136] .

Sadegh Vaez - Zadeh et al. (2010) presented design optimizations, were performed on

a wound rotor linear synchronous motor (WRLSM) to achieve high thrust, reduced

motor weight, and a high product of efficiency and power factor. A layer model and a

d-q model of the machine were used in defining the optimization problem in each case.

The genetic algorithm was employed here to search the optimal design variables. Due to

lower bound constraints the motor developed thrust, the thrust to weight ratio and

efficiency were optimized independently [137].

Skalka A, M. et al. (2011) published paper on contained torque components

identification of 3-phase induction machine. Electromagnetic torque and pulsating

torque was calculated via a circular path integral of the Maxwell stress tensor which

provided a convenient way of computing forces acting on bodies by evaluating a surface

integral. Torque calculations were done by finite element method in ANSYS and

MATLAB. The torque identification verified experimentally on a laboratory machine

using a special torque measurement system in the dynamic state [138].

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2.5 Past Trends of Finite Element Implementation in Linear Induction Motor

Now days, the Finite element method has become a popular tool for analyzing

various kinds of electrical machines which came in existence in early sixties. The first

application of FEM in electrical engineering was made by Winslow in the analysis of

saturation effects in accelerator magnets. In case of nonlinear variational formulation of

electromagnetic field problems, the main research was carried by Silvester and Chari

[139, 140]. Since then the problems have been solved by many other researchers. In

1973 Silvister, Browne and Cabayan presented the highly efficient methods for

implementing the finite element method in the analysis of electrical machines; including

improved techniques for generating Jacobian matrices, implicit handling of boundary

conditions, in addition they carried out a study for enlarging the scope of FEM [56].

Poloujadoff, M (1971) introduced the concept of turbine-propelled locomotives as

perfect examples of systems that use LIMs. The greatest effort involving linear motors

to attain ultrahigh speeds for mass transportation was studied at Rensselaer Polytechnic

Institute, New York State. Finally, Freight Pipeline Company (FPC) located in

Columbia, Missouri, USA, was awarded a project sponsored by the New York State

Energy Research and Development Authority (NYSERDA) to study the feasibility

of using the vast underground network of tunnels in New York City for freight

transport by pneumatic capsule pipelines (PCP) propelled by LIMs [141].

E. A. Mendrela et al. (1982) presented the Fourier’s series method for evaluating the

magnetic flux density, secondary current and power losses in single sided linear

induction motor with finite primary length and two layers secondary. The proposed

technique was used to describe electromagnetic phenomena in the LIM. However, the

considered computational model of the LIM takes only into account finite length of the

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excitation (neglecting finite core length) but it allows considering currents in the back

Iron of the two layer secondary [142].

Mizuno, T (1992) discussed the linear induction motors, which are generally used in

the open-loop control mode that generates problems relating to efficiency and

controllability. This led to a limited application in the field of factory automation and

conveyance. Hence, for the purpose of increasing the accuracy of such motors, the

magnetic circuit analysis of permanent magnet linear synchronous motors was done

using FEM. Moreover, the thrust and normal force characteristics were investigated

from the analysis and experimental aspects. The effect of the size of the permanent

magnets on the static thrust was analyzed [143].

J. F. Eastham (1992) emphasized that the linear motors may provide a major

breakthrough in applications of linear positioning that demand high quality in the

performance parameters of accuracy, thrust and acceleration. The motor was

designed to work with complete contactless transportation systems, and consisted

two stators and an ironless cursor. A test bench of the prototype model was constructed

and experimentally validated. The results were performed using 2D and 3D

formulations of FEM [144]. In addition to this, finite element modeling and analysis

schemes for electrical machines was also presented by D. Rodger et al. (1997). They

described the FE formulation; that yielded more economic solution [55].

J. Faiz et al. (1999) presented the 2D field analysis to predict the performance

characteristics for the single sided linear induction motor (SLIM). In this technique a

new idea introduced to account for the longitudinal end effect [147].

Byung-2 Kwon et al. (2000) described a control algorithm to compensate the joint

effect (junction between Aluminium sheets) creates discontinuity in the secondary eddy

currents non-uniform and cause fluctuations of thrust and normal force of LIM [148].

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Sung Chan Ahn et al. (2000) analyzed the phenomenon of static end effects due to the

motor structure while analyzing the dynamic characteristics of linear induction motor

using coupled FEM [149].

Jawad Faiz (2000) described the accurate modeling of a single-sided linear induction

motor considering end effect and equivalent thickness, which is the thickness of

Aluminium sheet with a solid back Iron plate. An analytical technique based on the

equivalent circuit model, one and two dimensional field analysis is used to predict

performance characteristics for the single-sided linear induction motor. In this technique

a new idea was introduced to account for the longitudinal end effect. Simulation results

produced by this analytical technique have a better agreement with the experimental

results than those reported in the literature. This analytical model may be used to predict

the sensitivity of the performance characteristics to various parameters [150].

Chung, M. J. (2001) presented the FEM analysis which may be also useful for

minimizing the force ripple in a linear brushless PM motor. In such a motor,

there is a force ripple that was detrimental for the smooth positioning of the motor.

An analytic model, represented the force ripple in terms of geometric parameters of the

motor . Subsequently, an optimal design of such a model was developed [151].

E.B. Dos Santos et al. (2002) aimed to present a mathematical model for the linear

induction motor, which allowed estimating its efficiency, considering the winding space

harmonics. Experimental procedures with a LIM prototype made viable the theoretical

against experimental tests to make the validation of the proposed model. The static and

dynamic end effects were modeled by the characteristic and unbalanced factors making

clear the dependence of the speed for these factors. The inclusion of winding space

harmonics produced good results [152].

72

Mirsalim, M. (2002), suggested the model here simulation of linear induction motors

by means of the finite-element method (FEM). The modified model covers special

phenomena in linear motors such as transverse edge effect, longitudinal end effect, and

saturation of back Iron. The modified model first computes the level of saturation by

both simple calculations and an iterative method. Then, using the FEM, it computed the

equivalent circuit parameters. Finally, it used the Duncan model to account for the end

effect [153].

A comparison of results were based on the proposed model with

experimental measurements proved the accuracy of the model. The efforts made to

maximize improvements on conventional trains using the refinements of modern

technology at the Bay area Rapid Transit System of San Francisco [154] and the New

Tokaido Line [35].

R. K. Srivastava et al. (2003) described the five-layer Fourier transform model of

SLIM to evaluate the effect of double airgaps. The performance of the model predicted

in Fourier-space using Parseval’s theorem. Further the effect of the additional gap on

thrust and normal force have been investigated. The study was to gauge the effect of

double gap such that its effect on the constant current performance of SLIM initially.

which suggested that preparation of reaction rail can be done by anyone of the methods

cited earlier. The performance of the machine improved with larger additional gap (with

constant magnetic gap) which is possible in case of the transport system [155].

Cui Jiefan et al. (2004) presented an investigation of direct thrust force control

(DTFC) for permanent magnet linear synchronous motor. Because end-open structure

of linear motor causes the special end-effect, which leads new problems when the direct

torque control (DTFC) was applied to linear motor. It was theoretically illustrated that

principle of the DTC can be applied to linear motor control, and clear concepts

73

about different reference frames, physical variables and action of voltage vectors

were explained. The special end-effect was taken so as to get thrust force response

performance as well as possible. Simulation results were shown that the DTFC was

effective in a linear motor control system [156].

Tae-Won Lee et al. (2005) presented the finite element analysis to calculate thrust of

linear motor. The temperature of the conductor was calculated from the experimentally

determined thermal resistance. The ANSYS for the optimum design process. Design

variables and constraints were chosen based on manufacturing feasibility and existing

products. As a result, it was showed that temperature of linear motor plays an important

role in determining the optimum design. The 2D electromagnetic analysis was used here

to evaluate thrust of the linear motor [157].

I. K. Bousserhan et al. (2006) proposed the position control of a linear induction motor

using adaptive fuzzy back stepping design with integral action. The applications of a

hybrid control system to the periodic motion control of a LIM were demonstrated. The

control dynamics of the proposed hierarchical structure was investigated by numerical

simulation. The results concluded that fuzzy-integral back stepping controller was

robust with regard to parameter variations and external load disturbance [158].

Griffiths et al. (2006) presented the characteristics of a two-phase levitated linear

induction motor. The system is AC induction type, and its secondary conductor

completely levitates and moves above a stator by non- contact. The secondary

conductor using the self-shading effect is also proposed for stable levitation. The

validity of the proposed method was demonstrated by comparing the calculated results

from the finite element method with the results of experiments [116].

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T. Sadauskas et al. (2007) investigated the distribution of the magnetic field of LIM

within the airgap and outside of the inductor’s limits in cases of three-phase

asynchronous linear induction motor and single-phase linear capacitance motor [159].

Qinfen Lu et al. (2008) presented the large airgap linear induction motor, with

introduction of new application explored. The construction and traditional design

method were described. Based on FEM model, the performance of LIM's normal force

and thrust were investigated in detail. Results showed that the ratio of normal force to

thrust is smaller than normal motor so that LIM can directly substitute for old rotate

motor without much changing system construction. This application was successfully

verified by experimental results [160].

D. Roy et al. (2008) reported the eddy current and secondary core flux density

distributions of a flux-concentration type linear induction motor (FCLIM) based on the

effect of flux concentration by eddy currents. In the proposed FCLIM, the slot leakage

flux can be reduced and as a result more flux will concentrate into the airgap. Thrust

characteristics can also be determined with the help of the equivalent circuit. By

comparison made between computed and experimental results it was found that the

FCLIM has better performance than the normal type of TLIM [161].

Haroutuon A. Hairik et al. (2009) discussed the dynamic behavior of linear induction

motor by a mathematical model taking into account the end effects and the core losses.

The end effects by introducing speed dependent scale factor to the magnetizing

inductance and series resistance in the d-axis equivalent circuit were incorporated.

Simulation results were presented to show the validity of the model during both no-load

and sudden load change intervals. The thrust force oscillations were presented in the

LIM caused by end effects [162].

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J. H. Alwash et al. (2010) presented finite element analysis applicable to all forms of

sheet rotor, cylindrical, linear induction motor as well as, helical motion induction

motor. The analysis accounted for longitudinal and transverse end effects, skin depth

and finite sheet thickness; furthermore, there was practically no restriction on the shape

of magnetic circuit that can be considered. It yields detailed space profile of the state

variables. The formulation results in a set of linear equations which were solved by

point relaxation method. The solution algorithm employed power mismatch in the

machine to indicate accuracy [164].

2.5.1 Magnetic Lavitation (MAGLEV)

Boldea, I. (1978) described the complete equivalent circuit of the linear induction

motor. They developed the steady-state characteristics of the LIM by employing

1/2/3-D analysis. This included the longitudinal and transverse-end effects and the

skin effects in the secondary winding of the LIM [165].

Tevan, Gy (1979) described is an extension of his previous work "Computation of a

double-sided linear induction motor by undamped travelling waves". The computation

was by neglecting skin effect as an approximation. The edge-effect and the end-effect

are taken into consideration one-by-one for investigating the optimum performance of

linear motors [166].

Eastham and Gieras (1988) investigated the causes and effects of phase

imbalance of the in single-sided LIM. They presented two methods for evaluation

of phase imbalance: the analytical one based on the equivalent circuit model and

the numerical one employed the finite element method. The validation of the

computational results showed that phase imbalance produces a reduction in both

thrust and normal forces, and these are restricted to only for a high-speed LIM [167].

Tachino, K. Amei (1999) mentioned that in recent times, the LIM has been employed

76

for industrial purposes in the form of transportation systems. The paper threw light on

the performance characteristics of two-phase levitated LIM. This movement is non-

contact based and the experimental results obtained are validated using FEM

calculations [146].

2.5.2 Governing Parameters for Efficiency of LIM

Sandeep Bala (2002) mentioned that the LIMs are preferred choice for high speed

propulsion, in comparison to other conventional motors, as the rotary motor depends

on friction and is limited by the maximum value of friction achieved. In case of

LIMs, there is an open airgap due to the finite length and width of its elements. Hence,

it was greater than the airgap length of conventional motors. The end effect wave travels

at the same speed as the LIM. The end effect casts a shadow on the effect of large

airgap that was earlier considered as the major reason for the poor performance of

LIMs. Another non-ideal effect, in the LIM, is the transverse edge effect. This stems

from the finite width of the primary and the secondary of the LIM. This effect be

detrimental for the power factor and efficiency, but does affect the thrust-speed

characteristics of the LIM [168].

Lee, Hyung-Woo (2008) presented a paper on performance characteristic, analysis

methodology for a linear induction motor used for a lightweight train. They suggested

methodology which is a kind of hybrid solution with an analytical method. FEM

analyses for various design schemes of the secondary reaction plate has been carried out

and from the analysis results, the best configuration for an urban railway transit is

selected [169].

Rolf Hellinger (2009) explained the operation of various types of linear motors used in

Maglev systems. He discussed and compared their suitability, and described the scope

of worldwide Maglev developments. The current status of the different linear motors

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used in the transportation sector was analyzed. Finally, a look at worldwide activities

and future prospects were presented [170].

Ahmad, H. A. (2010) presented an analysis and operation of linear induction motor

using finite element method. The solution of the magnetic field problem was performed

for both two and three dimensional approach. Magnetic vector potential, flux density,

field intensity, induced rotor current, and propulsion force were computed for LIM

model. The effect of velocity is taken into consideration. The primary winding self and

mutual inductances were computed for three dimensional analysis [163].

Lingamurty, K. S. (2011) suggested and described the methodology for the design of a

linear induction motor which may accelerate the rotor (Aluminium sheet) with a

specified mass with the required acceleration to the target distance. A single sided linear

induction motor (SLIM) of specified parameters was designed using a user-interactive

MATLAB program. The SLIM design and performance equations and design

procedures were developed and its performance was predicted using equivalent circuit

models. Optimum design parameters were obtained by the iterative procedure of the

design algorithm by choosing various design parameters. The performance of the SLIM

for values of thrust, rotor acceleration, rotor thickness and slip was analyzed [171].

Abbas Shiri (2012) described how the end effect influences performance of motor,

especially at high speeds also deteriorates the performance of the motor by producing

braking force. In this paper for Duncan equivalent circuit model, a new analytical

equation was proposed to model end effect braking force. Employing the proposed

equation and considering all phenomena involved in the single-sided linear induction

motor, a simple design procedure is presented and the effect of different design

variables on the performance of the motor was analyzed. A multi-objective optimization

method based on genetic algorithm is introduced to maximize efficiency and power

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factor, as well as to minimize the end effect braking force. Finally, to validate the

optimal results, 2D finite element method was employed [172].

G. A. Simone et al. (1997) showed a new way to establish the thrust of a linear

induction machine. A new factor named the relation factor KR was established, which

provided the conditions necessary to establish the thrust and other important variables

of the linear induction machines [145].

E.B. Dos Santos (2000) worked on a mathematical model for the linear induction motor

to estimate its efficiency, considering the winding space harmonics. This work followed

the strategy of a research among rotary machines to find the one which could be the

base of all the studies. The experimental tests were also validated [173].

Lee, Hyung-Woo (2011) presented the end effect of a linear induction motor,

especially in high speed operation. The exit part of the primary is not dealt as

extensively as the entry part because of its minor effect. However, the exit part is one of

the keys to weaken the dolphin effect, which occurs in high speed operation. In this

paper, the concept of the virtual primary core is introduced, and chamfering of the

primary outlet teeth is proposed to minimize the longitudinal end effect at the exit zone.

For this, LIM for the high-speed train designed and analyzed by using finite element

method. The results confirmed the chamfering for improvement in thrust [174].

Zai Qiang Jiang (2011) presented the thrust characteristics of a linear induction motor.

A 2D finite element model was built by using the Ansoft’s Maxwell. Based on the

developed model, simulations were made by considering different materials and

excitations. As the result, the optimal design has been identified, for higher thrust [175].

F. Korkmaz (2014) described a simulink model of direct thrust controlled linear

induction motor with end effect for electromagnetic launcher system and presented a

simple and effective scheme for direct thrust control of a linear induction motor. The

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simulink model of single sided linear induction motor with end effect was developed to

realize the direct thrust control. The simulink model of single sided LIM was also

developed to realize direct thrust control (DTC) scheme on electromagnetic launcher

system (ELS). Thrust and speed response were realized by numerical simulations by

using the simulink interface of the MATLAB environment. It has been observed that the

dynamic response of ELS faster and torque ripples were reduced [176].

Huilai Li (2014) discussed the two types of linear motors, including parallel magnetic

circuit linear permanent magnet brushless DC motor (PMC-LPMBDCM) and serial

magnetic circuit linear permanent magnet brushless DC motor (SMC-LPMBDCM),

presented for electromagnetic catapult. Performance of PMC-LPMBDCM and SMC-

LPMBDCM was investigated. The electromagnetic properties and thrust characteristics

were analyzed by finite element method [177].

Kumin Ludvik et al. (1999) presented the dynamic model of LIM using Lagrange

method. The input-output linearizing control was designed in order to achieve the

highest possible tracking performance and high disturbance rejection which was based

on the derived model. Performances of the drive were investigated by simulations [178].

Gary L. Skibinski (2003) explained the finite elements, used to predict the losses and

temperature of two different inductors used in AC drives. One of the inductors core

was made of thin steel laminations while the second one was made of new soft magnetic

composite core material. In this paper, the electromagnetic finite element software is

used for computing different parameters that include inductance, core loss, Copper

loss and power loss for various amplitudes of current and frequencies that range

from 1 Hz to 200 kHz. The Iron losses were computed using Steimetz formulas. In the

end, the thermal FEM software was used for computing typical temperature

distributions. The readings of these were found to be in reasonable agreement with the

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measurements made on both the inductors [179].

Kuang-Yow Lian et al. (2008) proposed a novel robust adaptive speed/ position

tracking control for a linear induction motor with both end-effect and secondary

resistance. The practical, current-fed control LIM, with residual current error, was

considered, i.e., the traditional ideal current-loop assumption was relaxed. More

practical conditions, such as bounded primary voltage and a finite absolute-integral of

current tracking error was considered. To overcome the high nonlinearity and nonzero

current error, a back stepping method, combining virtual desired variable synthesis was

developed for the speed and position tracking. To achieve asymptotic speed and

position tracking with unknown parameters and immeasurable secondary flux [180].

I. K. Bousserhane et al. (2009) presented an adaptive back stepping control system

with a fuzzy integral action proposed to control the mover position of a linear induction

motor drive. First, the indirect field oriented control LIM was derived. Then, an integral

back stepping design for indirect field oriented control of LIM was proposed to

compensate the uncertainties which occurred in the control. Finally, the adaptive back

stepping controller with fuzzy integral actions was investigated where a simple fuzzy

inference mechanism was used to achieve the mover position tracking objective under

the mechanical parameter uncertainties. The performance of the proposed control

scheme was demonstrated through simulations. The numerical validation results of

the proposed scheme have presented good performances compared to the adaptive

back stepping control with integral action [181].