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
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(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.
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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
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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.