164

CHAPTER 5

DESIGN FEATURES AND GOVERNING PARAMETERS

OF LINEAR INDUCTION MOTOR

5.1 Introduction

To evaluate the performance of electrical machines, it is essential to study their

electromagnetic characteristics. For the optimal design of electric machine thorough the

knowledge of the internal distribution of following fields is required:

a) Electric field

b) Magnetic field

c) Thermal field

d) Geometrical field

The knowledge of above field distribution provides the efficient and economical

design of electrical machines. The derivation of the governing equations for such field

problems is not only difficult, but their solution by appropriate methods of analysis is

another challenge. The pre-hand knowledge of these fields and coupled fields may be

quite helpful for the design and analysis of linear induction motor. To design induction

machines requires accurate prediction of the machine behavior, e.g. magnetic flux

density, electromagnetic force, etc. These are based on magnetic flux distribution

passing the motor cross-sectional area. Before discussing the design aspects of linear

induction motor, the idea about stationary and quasi stationary form of Maxwell

equations is of great significance.

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5.2 Stationary Form of Maxwell Equations

Distribution of the magnetic flux density in the core and surroundings of the

experimental model is investigated in this section. Two fundamental postulates of the

magnetostatic that specify the divergence and the curl of B in free space are [9]

(5.1)

(5.2)

By using the Eq. , Eq. 5.2 becomes

(5.3)

where B is the magnetic flux density, H is the magnetic field intensity, J is the current

density, and is the magnetic permeability of material. It follows from Eq. 5.1 that

there exists a magnetic vector potential such that

(5.4)

(5.5)

For 2-D case, the magnetic flux density B is calculated as

(5.6)

On the other hand

(5.7)

From Eq. 5.4 we have:

(5.8)

5.3 Quasi-Stationary Form of Maxwell Equations

Maxwell’s four equations and the constitutive relations are as follows:

(5.9)

(5.10)

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(5.11)

(5.12)

(5.13)

(5.14)

(5.15)

From the Eqs. 5.9 and 5.12, the electrical continuity equation is obtained as

(5.16)

The quasi-static form of Maxwell equation is obtained by removing or neglecting

the term in the first term on the right hand side of Eq. 5.9. This term, changing in

an electric field, is called displacement current, since it affects the magnetic field in

exactly the same way as the conducting current J. But in the low frequency domain, the

displacement current can be neglected. To prove this, the intensities of the conduction

current and the displacement current are compared under the low frequency domain.

(5.17)

From the Eq. 5.17 and the approximation of electric conductivity as

and that of electric permittivity as if the

angular frequency satisfies, the effect of displacement current with the magnetic field is

sufficiently negligible compared to those of exciting current or eddy current. The

Eq. 5.18 is almost satisfied for the electric machines. Consequently, the eddy current

field can be treated as a quasi-static magnetic field [239].

(5.18)

The quasi-static form of the Maxwell equations are obtained with the help of basic

Eqs. 5.9 - 5.12

(5.19)

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(5.20)

(5.21)

(5.22)

From the Eq. 5.19, the electrical continuity equation is modified as

(5.23)

Note that the neglecting of the term in Eq. 5.19 allows decoupling of the

equations into two parts, i.e., the magneto dynamic Eqs. 5.19 - 5.21 and the electrostatic

in Eq. 5.22. Here, note that the electric field E in Eq. 5.21 and the electric flux density D

are induced in different nature. The former is generated by a time varying of magnetic

field, and the latter is a result of the presence of electric charges. Furthermore, we can

reduce the Maxwell’s equations can be reduced by introducing the concepts of a vector

potential A and a scalar potential . From Eq. 5.20, the field B can be expressed in

terms of a vector potential A as

(5.24)

substituting Eq. 5.24 into Faraday’s law as given in Eq. 5.21 further gives

(5.25)

substituting Eqs. 5.24-5.25 into Ampere’s law Eq. 5.9 and Gauss’s law Eq. 5.12, then

we obtain the following reduced equation of full Maxwell’s equations

(5.26)

And,

(5.27)

where J0 is the exciting current density. In the case of quasi-static approximation of

reduced Maxwell equations is as follows

(5.28)

168

(5.29)

As stated above, the quasi-static Maxwell equations can be decoupled, then

Eqs. 5.28 - 5.29 can be solved independently as magneto-static and electrostatic

respectively. As from Eq. 5.28, the scalar potentials are defined only in the conductive

materials while the vector potentials are defined in the whole analysis domain. It means

that the electric field only appears in the conductive regions. Therefore, it is not

required to solve the electric field in the non-conductive region simultaneously. After

solving Eq. 5.28, further calculation can be done for the electric field outside the

conductive regions by solving Eq. 5.29. While solving Eq. 5.29, the boundary

conditions of A and can be obtained from Eq. 5.28 and these must be applied at the

interface between conductive and non-conductive regions.

In most of the cases, there is need to solve Eq. 5.28 only for the analysis of eddy current

problems. In fact, almost all the commercial softwares in this area is based on Eq. 5.28

only. The electromagnetic quantities involved in Maxwell's equations are: electric field

intensity, electric flux density or electric induction, magnetic field intensity, magnetic

flux density, surface current density, volume charge density and these can be

computed with help of COMSOL Multiphysics simulation of model and in addition to

these the magnetic permeability, electric permittivity, electric conductivity, inductance,

resistivity etc. can be found with material properties and equivalent circuit of the LIM.

5.4 Equivalent Circuit

For the design of linear induction motor, the equivalent circuit model proposed by

Duncan Model [240] has been employed. The per phase equivalent model of linear

induction model is shown in Figure 5.1.

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Figure 5.1 Equivalent circuit of LIM

The core losses are neglected because a realistic airgap flux density which leads to

moderate flux densities in the core and hence, rather low core losses. The skin effect is

small at rated frequency for a flat linear induction motor with a thin reaction plate (RP).

Therefore, equivalent RP inductance is negligible [153, 263].

(a) Stator Resistance Per-phase R1

It is the resistance of each phase of the LIM stator windings as expressed in Eqs.

5.30 (a) – 5.30 (b).

(5.30a)

or w

ww

A

lR 1 (5.30b)

In Eq. 5.30(a), is the conductivity of the conductor used in the primary

winding, length of the Copper wire, primary slot width, N is the number of turns

per phase, Aw is the cross-sectional area of the wire. And in Eq. 5.30(b), ρw is the volume

resistivity of the Copper wire used in the stator winding. The length of the Copper wires

lw, may be calculated from

ww lNl (5.31)

where lw is the mean length of one turn of the stator winding per phase

cesw lWl 2 (5.32)

where lce is the length of end connection given by

170

180

p

cel (5.33)

is pole pitch and is phase angle

(b) Leakage Reactance of Stator-slot per-phase X1

The flux which is produced in the stator windings is not completely linked with

the reaction plate. There may be some leakage flux in the stator slots and hence stator-

slot leakage reactance X1 has to be taken into account. This leakage flux is generated

from an individual coil inside a stator slot and caused by the slot openings of the stator

iron core. In a LIM stator having open rectangular slots with a double-layer winding, X1

can be determined with the help of Eq. 5.34

p

Nlq

W

pf

X

cees

ds

2

1

1

0

1

312

(5.34)

where, is the permeance of slot and given as in Eq. 5.35

s

ps

sw

kh

12

31 (5.35)

kp is the pitch factor which has relation with as expressed in Eq. 5.36 ,which is

known as permeance of end connection

133.0 pe k and (5.36)

sw

g

w

g

s

e

d

045

5

(5.37)

Where, is the differential permeance. It has been noticed here, that the stator

winding is either single-layer windings or double layer windings. In the former case one

side of coil is known as a coil side which occupies the whole slot, whereas in later case

there are two different coil sides of different phases in any one slot [243].

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(c) Magnetizing Reactance per-phase Xm

The per-phase magnetizing reactance, Xm, is given by Eq. 5.38.

e

wsem

pg

NkfWX

2

2

1024

(5.38)

where kw is the winding factor, ge is the effective airgap and Wse is the equivalent stator

width which is given as in Eq. 5.39

msse gWW (5.39)

(5.40)

In Eq. 5.40 m is magnetic airgap, can be given as

m = + d (5.41)

where, is airgap length, d is reaction plate (secondary sheet) thickness.

(d) Secondary Resistance per-phase R2

The per-phase reaction plate resistance R2 is a function of slip, as shown in

Figure 5.1. The R2 can be calculated from the goodness factor G and the per-phase

magnetizing reactance Xm as

G

XR m2 (5.42)

In Eq. 5.42, goodness factor G can be substituted from Eq. 1.2. Induction motors

draw current from its primary source and then transfer it to the secondary circuit

crossing the airgap by induction. The difference between the power transferred across

the airgap and secondary losses is available as the mechanical energy to drive the load.

In perspective of energy conversion, the primary resistance and the leakage reactance of

the primary and the secondary circuit are not essential. The energy conversion

efficiency can be improved as the mutual reactance Xm of the motor by increasing or by

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the secondary circuit resistance R2 decreasing. The goodness factor is 2R

XmG for a

basic motor. As the value of G increases the performance of motor gets better [241].

5.5 Governing Equations for Machine Design

Traditionally, there are two approaches for the analysis of electrical machines,

namely

Lumped parameter circuit theory method

Distributed parameter field theory method

The second method is more convenient to be used in LIMs and has been

employed in the present analysis [22, 194]. The following geometrical configurations by

varying the length of reaction plate have been considered as follows:

5.5.1 Infinite Thick Reaction Plate

Consider an idealized LIM with an infinitely thick reaction plate (RP) as shown in

Figure 5.2. [22] The following assumptions are made to simplify the analysis:

i. All layers extend to infinity in the + x- direction

ii. The secondary extends to infinity in the y direction.

iii. The excitation windings are located in the slotted primary structure. For

convenience, the structure is smoothed to permit representation of the motor

excitation as a current sheet of negligible thickness and finite width.

iv. The motion of the secondary is in the x- direction

v. The physical constants of the layer are homogenous, isotropic and linear.

vi. The ferromagnetic material does not saturate.

vii. Variations in the z direction are ignored.

viii. All the currents flow in z direction only

ix. The primary is constructed with such material, to ensure that conductivity in the

z direction is negligible.

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x. Time and space variations are sinusoidal.

The Table 5.1 provides the justification for all above assumptions used here for deriving

the governing equations of LIM.

Figure 5.2 Two dimensional model of LIM

Table 5.1 Justification of assumptions made

Assumption Justification Assumption 1 & 2 Forms the starting point of analysis Assumption 3 Makes the model amenable to mathematical analysis Assumption 4 This is an obvious one, since the secondary consists of a

solid conductor moving in one direction only. Assumption 5 & 6 Valid in the light of the linearity assumption stated

earlier Assumption 7 & 8 To reduce the problem to two dimensional field problem Assumption 9 The laminated primary core justifies it Assumption 10 As source voltage, varies sinusoidally with time

Ohm’s Law for a moving medium is given by

(5.43)

Considering the Maxwell’s equations from Eqs. 5.9-5.12, are the basic governing

equation of the electromagnetic phenomenon for LIM. Since the displacement current

density

is negligible (at power frequencies) so, from Eq. 5.9, it comes to

(5.44)

Substituting the value of from Eq. 5.43, it appears to

(5.45)

The magnetic vector potential A is defined by

(5.46)

Substituting the value of B from Eq. 5.45 in Eq. 5.46 it becomes

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(5.47)

The expansion of Eq.5.47 yields to

(5.48)

But since there being no free charges and can also be assumed here.

Therefore,

(5.49)

When suitably excited the primary creates y directed traveling field in the airgap given

by:

(5.50)

which implies to

(5.51)

Since A is assumed to be Z directed, where, = Chording factor

(5.52)

Now, Eq. 5.49 can be rewritten as

(5.53)

where and U =

The Eq. 5.53 is the basic governing equation. The solution to this equation,

subject to the given boundary conditions, yields the quantitative information regarding

the electromagnetic phenomena in the machine. For the model under consideration, it

can be recalled that the airgap field, produced by the primary, travels at a synchronous

speed , which is related to the slip ‘S’ and the speed of the secondary by

(5.54)

Because

, becomes

175

(5.55)

If we put

(5.56)

For region 2, the airgap where =0, then Eq. 5.55 reduces to

(5.57)

For region 3, the secondary, Eq. 5.56 becomes

(5.58)

The solution of Eqs. 5.57 - 5.58 can be written as

(5.59)

And

(5.60)

where the subscript number identifies the region under consideration. To evaluate

the constants, the following boundary conditions can be employed:

i. y=0,

ii. y=g, =

and

iii. , =0

Resulting in the following equations

(5.61)

(5.62)

and (5.63)

(5.64)

From Eqs. 5.61 –5.64, the coefficients and can be obtained as

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(5.65)

(5.66)

(5.67)

(5.68)

5.5.2 Finite Thick Reaction Plate

Figure 5.3 illustrates the arrangement of a model having a reaction plate of finite

thickness d. The assumptions listed in Section 5.5.1 are applicable here also. In

addition, the airgap is assumed to be very small with no fringing or fester of the

magnetic field in the airgap [22].

The following layers are shown in Figure 5.3.

Layer 1: Primary; Layer 2: Airgap; Layer 3: RP; and Layer 4: Air below RP.

As in the previous Section, for region 3 Eq. 5.58 applies

And for region 4

(5.69)

So for region 3 the solution

(5.70)

may be assumed. For region 4

(5.71)

using the defining equation for vector potential A

(5.72)

gives for region 3

(5.73)

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and for region 4

(5.74)

Figure 5.3 Model of an idealized LIM with a RP of finite thickness

The following boundary conditions are employed here as

i. y=0,

ii. y=d,

and =

iii. , =0

The following results are obtained:

(5.75)

(5.76)

(5.77)

Manipulation of these four equations gives

(5.78)

(5.79)

(5.80)

with these values, expressions for and can be written

(5.81)

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and

(5.82)

where,

5.6 Significant Governing Parameters

For the analysis and optimum design of linear induction motor the following

parameters are essential to be known for reducing significant effects or losses like end

effects, longitudinal transverse edge effects, skin effects etc.

The most of performance parameters are influenced with magnetic and electric

characteristics of motor during static and dynamic states. The improved thrust can be

achieved with the help of equivalent circuit and simulated results of magnetic flux

density, magnetic potential, surface current density, energy and Lorentz force etc. These

computed values further determine the thrust, efficiency, power factor, etc. The design

of linear induction motor involves many parameters that can be varied to affect the

performance of the LIM.

5.6.1 The Goodness Factor

The overall quality of the linear induction motor can be accessed by the Goodness

factor G, introduced by Laithwaite, E. R., [1]. The goodness factor can be derived

with the help of governing equations of motor. Let the surface current density due to

primary current be given by

(5.83)

This current sheet may be transformed to the secondary coordinates by substituting

so that the at the secondary surface is

(5.84)

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from which it can be obtained as

(5.85)

where the value of G can be substituted from Eq. 1.2. It is assumed that the flux in the

yoke is one-half of the flux in the airgap, then it can be expressed as [242]:

hy = Φp / 2By max Ws (5.86)

The ratio given by Eq. 5.86 can be defined as the goodness factor because it is the

real part of the field and denotes the active component of the force-producing

component, in contrast to the reactive component of the field. The goodness factor may

also be given as

or

(5.87)

Finally, the fundamental definition of the goodness factor for the secondary, in terms of

an equivalent circuit, is given as Eq. 5.42 may be written as

(5.88)

where magnetizing reactance, has been given in Eq. 5.179 and

= secondary resistance, ν = frequency.

5.6.2 Mechanical Airgap

The length of the mechanical airgap is the very important parameter in the

machine design. A larger airgap needs large magnetizing current and gives the smaller

power factor. With larger airgap, exit-end area losses shoots up and due to this thrust

and efficiency of machine decreases as from Eq. 1.2 indicates the goodness factor

inversely proportional to the airgap. Therefore, for the low speed motors it is desired to

keep the minimum airgap as possible to obtain the larger goodness factor. The effective

airgap equation derived here for further use in performance evaluation of LIM.

180

Let the primary and secondary currents, respectively, be replaced by their current

sheets and having linear current densities. We assume the currents to flow in the

z-direction only and the permeability of the core material to be infinity. It’s assumed

here that there is no relative motion between the primary and the RP [22]. The idealized

model of LIM with their paths of integration is shown in Figure 5.4.

(5.89)

And

(5.90)

From Ampere’s law

(5.91)

Figure 5.4 An idealized model of LIM

We get

and (5.92)

(5.93)

But, from ohm’s law

(5.94)

where is the surface conductivity. Thus Eqs. 5.89 – 5.94, finally yield

(5.95)

or in terms of we have

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(5.96)

Eq. 5.96 is the effective airgap field resultant equation.

where, = magnetic potential at primary, y component; = magnetic potential of

secondary, y component; = surface energy as primary; = surface energy at

secondary relative to primary; H = magnetic potential; = surface current density of

primary core; = surface current density of conducting layer of reaction plate; =

absolute permeability; = magnetic flux density at y component; = Carter’s

coefficient; = mechanical airgap

The actual airgap of the machine is replaced by an effective airgap, which is

around 1.02 to 1.2 times larger than the original airgap [243]. The effective airgap

variation for a large airgap machine as drawn in Figure 5.5

Figure 5.5 Effective airgap for LIM [243]

Further, according to Gieras [34], the effective airgap ge is

ge = kc g0 (5.97)

where g0 is the magnetic airgap, which further be given as

ge = g + d (5.98)

where d is the thickness of the conducting layer on the reaction plate, as

represented in Figure 5.6 and kc being the Carter’s coefficient, given by

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kc= λ / λ- γ g0 (5.99)

The parameter λ used in Eq. 5.99 is the slot pitch, which is the distance between the

centers of two consecutive teeth, can be derived from Eq. 5.100

λ = τ / mq1 (5.100)

Figure 5.6 Geometrical view of LIM model

The quantity γ in Eq. 5.99 can be expressed as

γ = 4/π [ws /2g0 arctan(ws/2g0 )- ln√ 1+ (ws/2g0)2] (5.101)

Slot pitch is the sum of slot width and tooth width and hence the slot width can

be calculated with the help of Eq. 5.102

ws = λ − wt (5.102)

where, wt is the tooth width. To avoid magnetic saturation in the stator teeth, there

is a minimum value of tooth width wt min, which depends on the maximum allowable

tooth flux density, Bt max. The quantity wt min can be determined from [242] Eq. 5.103

wt min= π/2 Bg avg λ / Bt max (5.103)

The stator slot depth hs shown in Figure 5.6, can be calculated with help of Eq. 5.104

hs= As / ws (5.104)

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where, As is the cross-sectional area of a slot. Generally, 30% of the area of the slot is

filled with insulation material. Therefore, As can be calculated from Eq. 5.105

As = 10/7 (Nc.Aw) (5.105)

where NC is the number of turns per slot, determined from Eq. 5.106

Nc= N1 / pq1 (5.106)

The variable Aw in Eq. 5.97 is the area of a cross section of a conductor winding without

insulation, which can be obtained with the help of Eq. 5.107

Aw= I1 / J1 (5.107)

where, I1 = rated input phase current ;J1 = stator current density.

The value of J1, which depends on the machine output power and the type of cooling

system. In most of the cases, it has been assumed to be 6 A/m2.

The yoke height of the stator core hy is the portion of the core below the teeth, as

shown in Figure 5.5. If it is assumed that the flux in the yoke is one-half of the flux in

the airgap, then it can be expressed as in Eq. 5.108.

hy= Φp / 2By max Ws (5.108)

In the present work, the airgap of the model has been varied from 0.5mm to

8mm. The simulation result obtained in the form of magnetic flux density, magnetic

potential, surface current density of reaction plate is computed and further substituted in

the governing equations of LIM to find an effective airgap.

5.6.3 Primary Core

The core material also affects the performance of a linear induction motor. Even

the design features of the core also affect the motor’s performance. With constant cross-

sectional area of a slot with narrower teeth produces more force and has better

efficiency and a better power factor, than a motor with wider teeth. This is due to the

leakage reactance in stator and mover in smaller secondary time constant. It leads to

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produce an end effect travelling wave of less magnitude and this leads to larger machine

output. In case, where it is not feasible to vary the tooth width, the flux density of the

tooth can be changed with change of core material for limiting the tooth saturation. The

effective pole pitch can be decided by using pole pitch governing equation as given in

Eq. 5.97 [34].

In the present work various materials are assigned to the core to find the

optimized value of tooth’s magnetic flux density, which further may help to reduce the

end effects of the motor. Thus, the simulation results of prominent materials have been

discussed with their comparative analysis in next Chapter.

In the further analysis, it also has been observed that the end effect on the exit-part

is less as compared with the entry-part due to “Dolphin Effect” which cannot be

ignored. It is not feasible to put any additional hardware at the entry-part. The end effect

can be reduced to a certain level by modifying the tooth shape. The concept of virtual

primary core has also been included in the present work, in which, primary core

generates drag force and uneven normal force at the exit zone. Hence, Dolphin effect

reduces rapidly. The chamfering of the primary outlet teeth, at the entry and at the exit-

part is proposed in the present work. The Mosebach model [174] brought the concept

of chamfering of the core by an angle (chamfering angle) 4o-51

o. The value of the

angle may vary with respect to the airgap in the tooth length of the model. The number

of iterations made during simulation process by varying the angle of chamfering ranges

between 300-50

0. Although it was not an easy task for bringing a new geometry through

AutoCAD to COMSOL for further analysis, but the consistent efforts made here to find

the suitable choices of angles for core chamfering to make an optimal design. The

complete analysis and their results have been discussed in the next Chapter.

185

5.6.4 Thickness of Reaction Plate

The reaction plate thickness plays a vital role in the performance. The thicker the

reaction plate, goodness factor increases. Out of Aluminium or Copper, any one of the

material can constitute the conducting part of the secondary. It can be useful to calculate

resistance modeling of the eddy currents in the RP. The thickness can be decided with

the help of equivalent impedance of reaction plate as given

(5.109)

where Z = impedance of reaction plate, = effective thickness can be obtained

from above equation and the values of for a non- magnetic material. In the

present work thickness of RP ranging from 1mm to 8mm have been simulated for

finding the optimum value of thrust of LIM.

5.6.5 Reaction Plate Conducting Material

In case of a non ferrous secondary, a thicker material results in a larger airgap. It

may not be recommended for good performance of the motor. Therefore, for non-

ferrous sheets, by keeping small thickness, with strength material to withstand the

magnetic-forces present between the same time. The other benefit of selection of

reaction plate material is that, with less resistivity, goodness factor becomes higher as

evident from Eq. 5.88. In the converse to this low resistivity helps in falling of end-

effects, which further reduces the output. Therefore, for better results it is required to

maintain the balancing between R2 and G values. The ferromagnetic material when has

an advantage of high permeability, means less magnetising current, then other side the

disadvantage is the strong pull between mover and stator. Whereas, the non ferrous, and

electrically conducting material, reduces this large magnetic pull, but due to less

permeability of airgap, magnetizing current increases. From the exhaustive literature

survey [163, 175, 184, 244, 259, 262-263], it has been discovered that Copper and

186

Aluminium used as conducting material for the reaction plate back with an Iron is the

best suitable combination. The secondary back iron has two main advantages.

i. It is useful as magnetic flux pass produced by primary.

ii. It is a mechanical support for the secondary.

Since it is required that the magnetic field produced by the primary should

penetrate in the Copper/Aluminium (conducting layer of RP) as well as the secondary

back iron. This is due to the low value of the low permeability of the reaction plate.

However, the depth of the penetration is limited .

All these phenomena involved in the present case influence the longitudinal end-

effect, iron saturation, transverse edge effect and skin effect. The governing equations

described here to bring 1< s secondary iron saturation and other important coefficients

affect the thrust and efficiency of the motor. The following governing equations are

used for calculating the above factors.

ks = (5.110)

where = Effective depth of penetration = Secondary back iron permeability,

(5.111)

(5.112)

where, ks = secondary iron saturation factor; S = slip of the motor; = pole pitch

= Effective conductivity of RP; = Effective permeability of secondary back iron;

= Effective equivalent primary width; = Transversal edge effect factor;

ν = supply frequency.

The observation made with the help of simulation values of model is that with

conducting material of RP as Copper or Aluminium back iron provides better results as

187

compared to use of other materials and hence the present work is extended by selecting

this combination of materials.

5.6.6 Slip

As it is already discovered that the airgap field, produced by the primary, travels

at a synchronous speed , which is related to the slip ‘s’ and the speed of the secondary

as explained in Eq. 5.54. How the slip plays an important role in the performance of

the LIM can easily be understood with help of Eqs. 5.161-5.163. All the prominent

performance parameters are directly governed with this value, hence the present work

carries the evaluation of effect of thrust by varying the slip from 3% to 10% as

discussed in the next Chapter.

5.6.7 The Poles

The end-effects are reduced to increase in number of poles in linear induction

motor. This is because of the sharing of constant end-effect loss between them. In the

present work number of simulation iterations are done for pole configurations, but, the

reduction in the loss was not as per desired, hence this parameter was not included for

final result evaluation.

5.6.8 Pole Pitch

One of the other parameters influencing the performances of motor is pole pitch.

The amplitude of the current sheet is determine from the relationship

In this expression, the total winding factor takes into account the deviations from

a sinusoidal distribution because of chording slots, and so on. The equation can be

written as Eq. 5.112 as

(5.113)

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where = current density of the reaction plate; m = number of phases; Length

of the machine; p = number of pole pairs; = pole pitch; W = number of turns per

phase. = maximum phase current; I1 = rms value of input current; Ls = length of one

section and Ls = 2 R ; R = stator radius; kw = winding factor; kp = pitch factor;

kd = distribution factor; = magnetising reactance

kw = kp x kd (5.114)

where kp = sin

;

kd =

(5.115)

α =

(5.116)

= coil span in electrical degree; q1 = number of slots/pole/phase in stator iron core.

As it is known that one pole pitch is equal to 180 electrical degrees, therefore in a

full pitch coil where the coil span is equal to one pole pitch, the pitch factor becomes

unity. After substituting the values of kd and kp in Eq. 5.114

(5.117)

From the Eq. 1.2, it is clearly observed that for high goodness factor the pole pitch

should be as large as possible. But on the other side, to increase the pole pitch, back iron

thickness has to be increased. This further leads to many ill effects on LIM as:

The weight of the motor will increase

With increase in , efficiency decreases as per equation

(5.118)

It results in less active length of conductor (in the slot) to total length of

conductor (slot + end connections). Since, end connections have no useful

purpose and will produce very high leakages and losses.

189

Due to all above mentioned reasons, the idea of pole pitch variation has been

dropped for the present analysis of LIM. Since, by changing geometrical parameters

(pole pitch), other effects may increase, which will not allow the efficiency to improve

for better value. Some of the performance parameters and their effects are summarized

in Table 5.2 [9].

Table 5.2 Effects of parameters variations on LIM performance

Parameter In case of increasing In case of decreasing

Airgap (g) Larger magnetizing current

Larger exit-end losses

Larger goodness factor

Larger output force

Larger efficiency

Secondary thickness (d) Larger goodness factor

Larger starting current

Larger secondary leakage

reactance

Secondary resistivity (ρ ) Smaller end effects Larger goodness factor

Less secondary loss

Primary core materials

magnetic flux density ( )

Increases efficiency

Increases power factor

Reduces thrust

Number of poles (P) Smaller end effects Larger secondary leakage

reactance

Chamfering of primary core

) Reduces end-effect

Reduces Dolphin effect

Thrust improves

Increase in transversal edge

effect

Increase in end-effect

Tooth width (w) Larger leakage reactance Larger force

Larger efficiency

5.7 Important Effects and their Analysis

There are certain phenomena which account for major differences between

conventional rotary induction motors (RIM) and linear induction motors (LIM). Due to

change of its constructional features the different effects and losses are introduced to

evaluate for its performance. The analysis of the significant effects with the help of

equivalent circuit and governing equations are discussed.

5.7.1 End Effects Analysis

To evaluate the end effects of LIM, it is essential to understand eddy current

present in the reaction plate. In LIMs, as the primary moves, a new flux is continuously

developed at the entry of the primary yoke, while existing flux disappears at the exit

side. Sudden generation and disappearance of the penetrating field causes eddy currents

190

in the secondary plate. The end effects are not very noticeable in conventional

induction motors. On the other hand, in LIMs, these effects become increasingly

relevant with the increase in the relative velocity between the primary and the

secondary. Thus, the end effects will be analyzed as a function of the velocity. Both

generation and decay of the fields cause the eddy current in the reaction plate. The eddy

current in the entry grows very rapidly to mirror the magnetizing current, nullifying the

airgap flux at the entry. On the other hand, the eddy current at the exit generates a kind

of weak field, dragging the moving motion of the primary core. Eddy current density of

the LIM along the length illustrated in Figure 5.7 and the resulting airgap flux is

represented in Figure 5.8.

Figure 5.7 Eddy current for an ideal LIM [214]

Figure 5.8 Airgap flux density for an ideal LIM [214]

The end effect factor is (1 – f (Q))

where,

and hence,

f(Q) =

(5.119)

where f(Q)=end effect factor

It is worth mentioning here that the primary length Ls is inversely proportional to

mover velocity Vr.

Ls

191

As the velocity increases, the primary’s length decreases, increasing the end

effects, which causes a reduction of the LIM’s magnetization current. This change can

be computed with the help of magnetization inductance [214]

Lm’ = Lm (1 – f(Q)) (5.120)

where Lm’ = magnetizing inductance at RP; Lm = magnetizing inductance at primary

5.7.1.1 Power Loss Due to End Effects

To discuss the power loss due to end effect, consider the Duncan [240] equivalent

circuit model of linear induction motor. From the Bazghaleh [244], analytical equations

have been derived from efficiency and power factor; however, in calculations, the

power loss due to the end effect is supposed to occur prior to airgap. It is obvious that

the power loss due to end effect occurs in the secondary due to eddy currents produced

by the end effect. So, the developed airgap power is defined in such a way that

considers this phenomenon as shown in Figure 5.9 by keeping Aluminium as reaction

plate conducting layer material [172].

Thus, the following equation holds

(5.121)

In the Eq. 5.121, is the power loss due to the end effect, is the secondary

ohmic loss, and is the converted mechanical power. So, considering the Figure 5.9

can be written:

(5.122)

(5.123)

(5.124)

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Figure 5.9 Power flow in linear induction motor

In Eq. 5.122, is the magnetizing branch resistance which represents the power

loss due to end effect and is

(5.125)

In Eq. 5.119, Q which is known as the normalized motor length and its value is

given by

(5.126)

where Ls is the primary length, the magnetizing inductance, the secondary

leakage inductance which is equal to zero for sheet secondary, and is the motor

speed. It is seen that the value of Q inversely proportional to the motor speed, so, in

high speeds it becomes smaller. In addition to Eq. 5.121, the airgap power can be

written in terms of the developed thrust, :

τ (5.127)

where is the synchronous speed; is the primary supply frequency; and τ is the pole

pitch of the motor. The efficiency of the motor is defined as:

(5.128)

Where and are output and input power of the motor, respectively.

Referring to Figure 5.9 and replacing proper terms for input and output power, the

following equations for efficiency, power factor and developed thrust are derived:

Pin Pag Pm Pout

Pcup PAle PAls Pf&w

193

(5.129)

cos

(5.130)

(5.131)

From the Eqs 5.129 – 5.131, S is the motor slip and is the modified magnetizing

reactance considering the end effect gives in Eq. 5.132

(5.132)

It should be mentioned that in deriving the above equations, the mechanical

friction and windage loss of the motor are neglected. Airgap flux density is [245]

(5.133)

where, is the amplitude of the equivalent current sheet is calculated as follows [34]:

(5.134)

Also, the tooth flux density is obtained as:

(5.135)

5.7.1.2 End Effect Braking Force

As it is known that, the longitudinal end effect decreases the airgap flux density of

the LIM. The final effect of this phenomenon is producing a braking force which is

opposite to developed thrust in the airgap. This braking force can be considered as an

external mechanical load. The end effect braking force (EEBF) has not been dealt with

by researchers in designing the linear induction motors. Here the attempt has been made

to drive a new equation for the EEBF.

The net output force can be written as:

(5.136)

194

Where is the developed force of the motor in the airgap and is the end

effect braking force. The developed airgap power is obtained by Eq. 5.123 and the

converted mechanical power can be calculated by the following equation:

(5.137)

In the above equation, is the motor speed. Using Eqs. 5.122-5.124 the converted

mechanical power can be written as:

(5.138)

dividing Eq. 5.134 by (1-s) and using Eqs. 5.123 and 5.133, following relation is

derived:

(5.139)

replacing from Eq. 5.132 in the above equation and using Eq. 5.123 for , the

braking force produced by the end effect may be derived as:

(5.140)

using Eqs. 5.122 and 5.125, the final form of the end effect braking force may be

derived as:

(5.141)

During the stand still operation of LIM, the value of Q tends to infinity as

Eq. 5.126 and consequently, the end effect force becomes zero. Whereas, when speed

increases, the value of Q starts decreasing which gives rise to EEBF. The reason for the

latter is that when Q decreases, the modified magnetizing reactance of the motor as per

Eq. 5.132 also decreases and causes the magnetizing current to increase. Figure 5.10

represented the EEBF versus motor speed, from where it can be seen that increasing the

motor speed causes the EEBF to increase [172, 247].

195

Figure 5.10 EEBF versus speed [172]

5.7.2 Transversal Edge Effect and Dolphin Effect Analysis

The previous Section reveals that airgap field contains a forward component as

well as a backward component apart from the one un-attenuated wave, and these waves

are called as end effect waves.

(5.142)

(5.143)

In a LIM, the width of the primary stack is usually less than the width of the

secondary plate resulting in a physical feature called transverse edge effects [22]. Due

to this, transverse and longitudinal components of current densities exist, consequently

increasing the secondary resistance by a multiplicative factor , and reducing the

magnetizing reactance by a multiplicative factor

where

and (5.144)

(5.145)

the value of , in Eq. 5.145 is given

λ

tanh

(5.146)

λ

tanh

(5.147)

196

further, the λ , used in Eq. 5.148 can be find as

λ

tanh

tanh

π

τ

(5.148)

π

τ (5.149)

sinh sin

cosh sin (5.150)

(5.151)

The value of and can be calculated by using Eqs. 5.110 - 5.111.

(5.152)

(5.153)

(5.154)

(5.155)

(5.156)

In summary, the main consequences of transverse edge effects appear as:

An increase in secondary resistivity

A tendency toward lateral instability

A distortion of airgap fields, and

A deterioration of LIM performance, due to the first three factors.

Considering the edge effects, the equivalent circuit parameters of a LIM can be

written as follows [243].

The factor in the magnetizing reactance is replaced by and the

goodness factor G in the secondary resistance is replaced by so that

(5.157)

197

And the basic secondary resistance from Eq. 5.88 can be further be derived as

(5.158)

The primary phase resistance and leakage reactance can also be given by the

following expressions in addition to the Eqs. 5.30 (a) - 5.31.

(5.159)

λs 1

p λs

s

qs λ

(5.160)

The goodness factor which is given in the Eqs. 1.2 and 5.87 can also be given as

ge (5.161)

All specific phenomena are incorporated in gei and ei, which are functions of

primary current I1 and slip frequency. Further, for low speed LIMs, the expression of

thrust and normal force becomes simplified. Thus, the total thrust Fs may be written as

Fs I22 2

S2τ =

I22 2

S2τ 1

SGei 2

1

(5.162)

neglecting the iron losses, the efficiency and power factor as Eqs. 5.129 and 5.130

further given as

Fs2τ 1 S

Fs2τ 1I12 (5.163)

The Normal Force Fn is composed of an attraction component and a repulsion

component. The final expression is [22]

Fn sepτ

π2

m2

gei2 1 S2Gei

2 1

π

τgeSGei (5.164)

In the low speed region, the normal force is attractive (positive) but for high

speeds it may become repulsive (negative). The above equations also helps to determine

the Dolphin effect present in the LIM operations, especially low speed motors. It can be

198

reduced with control of magnetic fringes at the entry-exit point of the mover. The

effective airgap selection plays a vital role in the control of this effect.

5.7.3 Skin Effect and Saturation Effect Analysis

In very fast-changing fields, the magnetic field does not penetrate completely into

the interior of the material of RP. However, in any case increased frequency of the same

value of field will always increase eddy currents, with non-uniform field penetration.

The penetration depth δ in (m) for a conductor can be calculated as:

(5.165)

The skin effect of the linear induction motor can be analyzed with help of

secondary iron saturation factor ks as given in Eq. 5.109. The field penetration in the

secondary back iron as given in Eq. 5.110, which is reduced by the factor because

of edge effect. To control this effect, the conductivity of the reaction plate should be

modified as

(5.166)

Then, can be obtained with the help of Eq. 5.150. Where, = depth of

penetration in the reaction plate and it can be calculated as in Eq. 5.156. By knowing the

values of ν, , the value of can be computed. The model can be simulated by

changing airgap at different values to obtain the minimum skin effect of LIM [153].

Saturation Effect Analysis

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. This effect can be analyzed with the help of

i. ks saturation coefficient for the secondary back-iron which is nothing but the

ratio of back-iron reluctance to the sum of conductor and airgap reluctances as

given in Eq. 5.167.

199

ii. The depth of penetration in Iron, as given in Eq. 5.111 and

iii. The average length of flux path as .

where is chording(coil span factor)

.

(5.167)

In order to obtain the permeability of RP back iron (level of saturation), the

following iterative algorithm is used.

I. First, a logical value of is estimated.

II. Then and are used from Eqs. 5.110 – 5.11 and 5.167.

III. In the step, the edge effect factors, may be evaluated using Eq. 5.168.

(5.168)

The values of , , and λ can be obtained from Eqs. 5.146 – 5.149 and

finally the realistic goodness factor can be given as

(5.169)

IV. Then , , are calculated by using the following expressions:

(5.170)

Eq. 5.170 includes the saturation factor unlikely in Eq. 5.97

(5.171)

(5.172)

V. Next approximate value of the airgap flux density can be determined as in the

following [246]:

(5.173)

The effective goodness factor and conductivity can be written as

200

(5.174)

(5.175)

where is the amplitude of an equivalent current stator sheet which is also given in

Eq. 5.134 may also be given as

(5.176)

VI. Assuming an exponential form for the field distribution in back iron, the flux

density at the surface of Iron is given by

(5.177)

VII. With this flux density and using back iron saturation curve, a new value for the

back iron permeability, is calculated [153].

VIII. Using the following expression, a new iteration is commenced, and the

computation is carried out until sufficient convergence is attained.

(5.178)

5.8 Lorentz Force

The Lorentz force is one of the most significant performance parameter of LIM

which links with thrust, efficiency, power factor. There are four methods to numerically

compute as Lorentz force [210, 249]:

5.8.1 Lorentz Force Method

In this method, the total force on body is obtained by integrating the forces due to

magnetic field acting on each differential current carrying element,

(5.179)

where is the force density in conductor in Eq. 5.179.

201

5.8.2 Maxwell Stress Tensor

Maxwell stress tensor is widely used for the electromagnetic force computation. A

quantity called stress tensor is defined in this method whose divergence is actually the

force density throughout the volume of the body on which the force is to be determined.

Applying divergence theorem to the stress tensor can be considered Maxwell stress as

surface force density which when integrated over surface enclosing the body gives total

force acting on it. The choice of surface can be chosen so as to satisfy certain

performance criterion and to improve accuracy of results. The expression of stress

tensor is,

(5.180)

where (i; j) can take values (x; y; z). ij is 1 if i = j and zero otherwise. The Eq. 5.180

can be written in terms of force density vector as in Eq. 5.181,

(5.181)

where is the normal unit vector to the surface under consideration.

5.8.3 Virtual Work Method

The virtual work method for electromagnetic thrust calculation is based on the

generalized principle of virtual displacement. The mover of the LIM is assumed to be

displaced and change in stored magnetic energy divided by displacement gives the force

acting on the body as the displacement tends to be infinitesimal. The displacement is not

actual physical displacement of the mover; hence it is called as virtual displacement.

The point should be kept in mind while virtually displacing the body is that the flux

linkage has to be kept constant throughout the operation. The implementation can be

both at the level of displacement of the whole mover or at the level of displacement of

elements or nodes. If the nodes are displaced then the method is called local virtual

202

work method [249]. The expression for the magnetic energy stored in the field is given

in Eq. 5.182

(5.182)

where V is the volume of the field region, B is the flux density, and H is the magnetic

field intensity. The force acting on a node which is virtually displaced is given by,

(5.183)

where z is the amount of virtual displacement.

5.8.4 Equivalent Sources Method

The equivalent magnetizing currents are used in this method. The theory and

implementation of the method has been discussed in literature [250-251]. It uses the fact

that there is physical existence of microscopic atomic current loops in any material,

particularly ferromagnetic material, which experience the force in presence of magnetic

field, which eventually gets transferred to the machine. Conventionally, the field

intensity produced by these atomic current loops is taken care of by introducing concept

of relative permeability for isotropic material without hysteresis. The relative

permeability value is taken for calculating flux inside the ferromagnetic material and

hence the presence of atomic current loops are not required to be considered separately

for calculating saliency force. Otherwise by keeping permeability inside the material

same as that of air we can take into account the presence of atomic current loops

separately in the equivalent sources method. Thus, instead of considering the presence

of actual atomic current loops, we can find the total force acting on the body by

calculating the forces acting on these fictitious sources and they turn out to be the same

as the actual forces. The magnetic behaviour of a ferromagnetic material can generally

be described as in Eq. 5.184

(5.184)

203

where is the flux density, the field strength, the magnetic permeability of

vacuum and the magnetization. For soft magnetic materials, is induced due to

external field and is a function of .

(5.185)

where is the relative permeability which may be constant or a function of . In non-

ferromagnetic materials vanishes. The governing equation is:

(5.186)

where is the conduction current density.

The second term on the right side has the same effect as the conduction current, hence it

is called equivalent magnetizing current . The force can be calculated by formula

similar to Lorentz force formula. It should be noted that exists only on the

boundaries.

In another approach, the magnetic material with permeability is replaced by a

non-magnetic material having a superficial distribution of magnetic charges [252] and

the force density is calculated as the product of the superficial surface charge density

and calculated surface magnetic field intensity. In the present work the most popular and

feasible method i.e, Maxwell stress tensor has been used. Although in the initial stage of

the computation the Lorentz force method also been used and the results of the method

are compared for the selection of best method.

5.9 Thrust and Efficiency

As explained earlier, the input power to the stator windings is utilized in

producing useful mechanical power which is exerted on the mover and to account for

the rotor (mover) Copper losses. As the mechanical power transferred across the airgap

from the stator to the mover ( SRmI 2

2

2 ) minus the rotor Copper loss ( 2

2

2 RmI ),

204

S

SRmIRmI

S

RmIP

12

2

22

2

222

20 (5.187)

Using the Eqs. 5.187 and 5.54 the electromagnetic thrust generated by the LIM stator is

given as alternate to the Eq. 5.162

SV

RmIF

s

s2

2

2 or

(5.188)

The LIM input active power is the summation of the output power and the copper losses

from the stator and rotor,

2

2

11

2

10 RmIRmIPPi (5.189)

where, is the stator Copper loss. Substituting P0 and Fs in

(5.190)

1

2

1 RmIVF ss (5.191)

The efficiency of the LIM is found by calculating the ratio of P0 and Pi as given in

Eq. 5.128. The designed linear induction motor is simulated using finite element method

in the next Chapter to validate the analytical analysis. The significant governing

parameters have been taken into consideration for the optimal design of the motor. The

magnetic field analysis has been done and the post-processing results have been

analyzed using h-type refinement for the design optimization of LIM with the help of

COMSOL Multiphysics and MATLAB environment. The following algorithm depicts

the step by step procedure to achieve the proposed objectives:

5.10 Design Algorithm

I. Set the required or given specifications for the desired performance with the

boundary and environment conditions for selecting LIM model.

II. Identify parameters to be varied, the extent to which each should be allowed to

vary and the feasible increments of variations.

205

III. Assume an initial design from the available core and reaction plate variables.

IV. Execute the design analysis by

(a) Calculation of parameters and currents in the equivalent circuit.

(b) Complete the field variables with COMSOL Multiphysics.

V. Compare the computed performance with desired or reference performance with

help of MATLAB.

VI. (a) If design does not meet the goal, change the respective factor, within

permissible limit and continue iteration until the desired performance is met or

until it is confirmed that the desired performance cannot be met under given

refinements and tolerances.

(b) If a design does meet desired performance, combine the simulation and

iteration process to find whether any other variations also meet the desired

performance.

VII. If the desired result does not come, it can be concluded that the LIM design

cannot be further optimize with given specifications and conditions.