DESIGN FEATURES AND GOVERNING PARAMETERS OF LINEAR INDUCTION MOTOR

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. 164

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) 165

(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) 166

(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 J 0 is the exciting current density. In the case of quasi-static approximation of reduced Maxwell equations is as follows (5.28) 167

(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. 168

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 R 1 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 l R1 (5.30b) w w Aw 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 lw N l w (5.31) where lw is the mean length of one turn of the stator winding per phase l w s ce 2 W l (5.32) where lce is the length of end connection given by 169

p lce (5.33) 180 is pole pitch and is phase angle (b) Leakage Reactance of Stator-slot per-phase X 1 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 statorslot leakage reactance X 1 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 X 1 20f s 1 3 Ws d p q1 p l e ce N 2 1 (5.34) where, is the permeance of slot and given as in Eq. 5.35 h 1 3k s p s (5.35) 12ws kp is the pitch factor which has relation with as expressed in Eq. 5.36,which is known as permeance of end connection 0.3 3k 1 and (5.36) e p g e 5 ws d (5.37) g0 5 4 w s 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]. 170

(c) Magnetizing Reactance per-phase X m The per-phase magnetizing reactance, X m, is given by Eq. 5.38. 2 240fWsekwN1 X m 2 (5.38) pge where k w is the winding factor, g e is the effective airgap and W se is the equivalent stator width which is given as in Eq. 5.39 W se W g (5.39) s m (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 R 2 The per-phase reaction plate resistance R 2 is a function of slip, as shown in Figure 5.1. The R 2 can be calculated from the goodness factor G and the per-phase magnetizing reactance X m as X m R2 (5.42) G 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 X m of the motor by increasing or by 171

the secondary circuit resistance R 2 decreasing. The goodness factor is G Xm for a R 2 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. iii. The secondary extends to infinity in the y direction. 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. vii. viii. ix. The ferromagnetic material does not saturate. Variations in the z direction are ignored. All the currents flow in z direction only The primary is constructed with such material, to ensure that conductivity in the z direction is negligible. 172

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 173

(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 174

(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 175

(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) 176

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) Manipulation of these four equations gives (5.76) (5.77) (5.78) (5.79) (5.80) with these values, expressions for and can be written (5.81) 177

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) 178

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]: h y = Φ p / 2B y max W s (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. 179

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) We get Figure 5.4 An idealized model of LIM 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 180

(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 g e is g e = k c g 0 (5.97) where g 0 is the magnetic airgap, which further be given as g e = g + d (5.98) where d is the thickness of the conducting layer on the reaction plate, as represented in Figure 5.6 and k c being the Carter s coefficient, given by 181

k c = λ / λ- γ g 0 (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 / mq 1 (5.100) Figure 5.6 Geometrical view of LIM model The quantity γ in Eq. 5.99 can be expressed as γ = 4/π [w s /2g 0 arctan(w s /2g 0 )- ln 1+ (w s /2g 0 ) 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 w s = λ w t (5.102) where, w t is the tooth width. To avoid magnetic saturation in the stator teeth, there is a minimum value of tooth width w t min, which depends on the maximum allowable tooth flux density, B t max. The quantity w t min can be determined from [242] Eq. 5.103 w t min = π/2 B g avg λ / B t max (5.103) The stator slot depth h s shown in Figure 5.6, can be calculated with help of Eq. 5.104 h s = A s / w s (5.104) 182

where, A s is the cross-sectional area of a slot. Generally, 30% of the area of the slot is filled with insulation material. Therefore, A s can be calculated from Eq. 5.105 A s = 10/7 (N c.a w ) (5.105) where N C is the number of turns per slot, determined from Eq. 5.106 N c = N 1 / pq 1 (5.106) The variable A w 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 A w = I 1 / J 1 (5.107) where, I 1 = rated input phase current ;J 1 = stator current density. The value of J 1, 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/m 2. The yoke height of the stator core h y 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. h y = Φ p / 2B y max W s (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 crosssectional 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 183

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 exitpart is proposed in the present work. The Mosebach model [174] brought the concept of chamfering of the core by an angle (chamfering angle) 4 o -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 30 0-50 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. 184

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, from above equation and the values of = effective thickness can be obtained 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 nonferrous 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 endeffects, which further reduces the output. Therefore, for better results it is required to maintain the balancing between R 2 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 185

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 endeffect, 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. k s = (5.110) where = Effective depth of penetration = Secondary back iron permeability, (5.111) (5.112) where, k s = 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 186

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) 187

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; I 1 = rms value of input current; L s = length of one section and L s = 2 R ; R = stator radius; k w = winding factor; k p = pitch factor; k d = distribution factor; = magnetising reactance where k p = sin ; k w = k p x k d (5.114) k d = (5.115) α = (5.116) = coil span in electrical degree; q 1 = 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 k d and k p 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. 188

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 secondary leakage Larger starting current reactance Secondary resistivity (ρ ) Smaller end effects Larger goodness factor Less secondary loss Primary core materials Increases efficiency Reduces thrust magnetic flux density ( ) Increases power factor Number of poles (P) Smaller end effects Larger secondary leakage Chamfering of primary core ) Reduces end-effect Reduces Dolphin effect Thrust improves reactance 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 189

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] The end effect factor is (1 f (Q)) Figure 5.8 Airgap flux density for an ideal LIM [214] where, and hence, f(q) = (5.119) where f(q)=end effect factor It is worth mentioning here that the primary length L s is inversely proportional to mover velocity V r. L s 190

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] L m = L m (1 f(q)) (5.120) where L m = magnetizing inductance at RP; L m = 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) 191

P in P ag P m P out P cup P Ale P Als P f&w In Eq. 5.122, Figure 5.9 Power flow in linear induction motor is the magnetizing branch resistance which represents the power loss due to end effect and is given by (5.125) In Eq. 5.119, Q which is known as the normalized motor length and its value is (5.126) where L s 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: 192

(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]: Also, the tooth flux density is obtained as: (5.134) (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) 193

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]. 194

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) 195

further, the λ, used in Eq. 5.148 can be find as λ tanh tanh π τ (5.148) π τ (5.149) sinh cosh sin 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) 196

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) λ s 1 p λ s s q s λ (5.160) The goodness factor which is given in the Eqs. 1.2 and 5.87 can also be given as g e (5.161) All specific phenomena are incorporated in g ei and ei, which are functions of primary current I 1 and slip frequency. Further, for low speed LIMs, the expression of thrust and normal force becomes simplified. Thus, the total thrust F s may be written as F s 2 I 2 2 = 2 I 2 2 2 S2τ 1 S2τ SG ei 1 (5.162) neglecting the iron losses, the efficiency and power factor as Eqs. 5.129 and 5.130 further given as F s 2τ 1 S F s 2τ 1I 1 2 (5.163) The Normal Force F n is composed of an attraction component and a repulsion component. The final expression is [22] F n se pτ π 2 2 m g2 ei 1 S 2 G ei 2 1 π τ g e SG ei (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 197

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 k s 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. k s 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. 198

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 199

(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. 200

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 201

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) 202

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 nonferromagnetic 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 ( 2 2 mi2 R2 S ) minus the rotor Copper loss ( mi R 2 2 ), 203

2 R2 2 2 1 S P mi mi R mi R2 (5.187) 0 2 2 2 2 S S 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 F 2 mi2 R2 s or (5.188) VsS The LIM input active power is the summation of the output power and the copper losses from the stator and rotor, P i 2 2 0 mi1 R1 mi1 R2 P (5.189) where, is the stator Copper loss. Substituting P 0 and F s in 2 1 (5.190) V mi R 1 (5.191) F s s The efficiency of the LIM is found by calculating the ratio of P 0 and P i 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. 204