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MODELING and SIMULATING of SINGLE SIDE SHORT STATOR LINEAR INDUCTION MOTOR with the END EFFECT ∗ Hamed HAMZEHBAHMANI

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MODELING and SIMULATING of SINGLE SIDE SHORT STATOR LINEAR INDUCTION MOTOR with the END EFFECT ∗ Hamed HAMZEHBAHMANI Journal of ELECTRICAL ENGINEERING, VOL. 62, NO. 5, 2011, 302{308 MODELING AND SIMULATING OF SINGLE SIDE SHORT STATOR LINEAR INDUCTION MOTOR WITH THE END EFFECT ∗ Hamed HAMZEHBAHMANI Linear induction motors are under development for a variety of demanding applications including high speed ground transportation and specific industrial applications. These applications require machines that can produce large forces, operate at high speeds, and can be controlled precisely to meet performance requirements. The design and implementation of these systems require fast and accurate techniques for performing system simulation and control system design. In this paper, a mathematical model for a single side short stator linear induction motor with a consideration of the end effects is presented; and to study the dynamic performance of this linear motor, MATLAB/SIMULINK based simulations are carried out, and finally, the experimental results are compared to simulation results. K e y w o r d s: linear induction motor (LIM), magnetic flux distribution, dynamic model, end effect, propulsion force 1 INTRODUCTION al [5] developed an equivalent circuit by superposing the synchronous wave and the pulsating wave caused by the The linear induction motor (LIM) is very useful at end effect. Nondahl et al [6] derived an equivalent cir- places requiring linear motion since it produces thrust cuit model from the pole-by- pole method based on the directly and has a simple structure, easy maintenance, winding functions of the primary windings. high acceleration/deceleration and low cost. Therefore nowadays, the linear induction motor is widely used in Table 1. Experimental model specifications a variety of applications such as transportation, conveyor systems, actuators, material handling, pumping of liquid Quantity Unit Value metal, sliding door closers, curtain pullers, robot base Primary length Cm 27 movers, office automation, drop towers, elevators, etc. In a Primary width Cm 9.5 linear induction motor, the primary winding corresponds Length of secondary sheet m 6 to the stator winding of a rotary induction motor (RIM), Thickness of Secondary Sheet mm 5 while the secondary corresponds to the rotor. Normally, Number of Slots - 7 the secondary, similar to the RIM rotor, often consists of Width of Slots mm 13 a sheet conductor, such as copper or aluminum, with a High of Slots mm 37 solid back iron acting as the return path for the magnetic Air gap length mm 15 flux. Based on the relative primary and secondary length, Number of poles { 2 linear induction motors are categorized into two types: Number of coils per phase { 2 long rotor and short rotor. The structures of these LIMs Number of turns per coil { 300 − − are shown in Figs. 1a and 1b, respectively [1{3]. The Conductivity of the rotor material Ω 1m 1 2:29 × 107 photograph of the experimental model of short stator LIM, constructed for this research is shown in Fig. 2. Duncan et al [7] developed the `per-phase' equivalent Table 1 shows its specifications. circuit by modifying the magnetizing branch of the equiv- The LIM special structure means that its performance alent circuit of the RIM. Noting that the air gap flux is a little different from that of a RIM. As we know, in the grows and decays gradually from the entry and the exit, RIM, an accurate equivalent circuit model can be derived respectively, he introduced a weighting function with the easily by simplifying the geometry per pole. The main use of a dimensionless factor Q. difference between LIM and RIM is related to the \End Then, the average values were taken to describe the Effects” which is explained in part 3. These effects should magnetizing inductance and the resistance of the eddy be considered in both the modelling and controlling of loss. Kang et al [8], according to the result in [7], deduced LIMs. the two-axis models (dq), which can be applied in vector In recent decades, a number of papers on the LIM per- control or direct force control to predict the LIM dynamic formance analysis, involving steady and dynamic states, performance. However, the derivation process of the re- have been available [1{15]. Gieras et al [4] and Faiz et vised function f(Q) is very coarse on the assumption that ∗ Islamic Azad University Sanandaj Branch Sanandaj, [email protected] DOI: 10.2478/v10187-011-0048-5, ISSN 1335-3632 ⃝c 2011 FEI STU Journal of ELECTRICAL ENGINEERING 62, NO. 5, 2011 303 I0(A) sults. A comparison of the experiments results with the v y calculation and simulation results show that the calcu- Air gap Primary core lation and simulation procedure and the obtained model for this type of linear motor have sufficient accuracy. Three phase primary winding Secondary sheet Back iron (a) 2 MAGNETIC FLUX DISTRIBUTION FOR STEADY CASES Back iron v y Secondary sheet Air gap Distribution of the magnetic flux density in the core and surroundings of the experimental model is investi- gated in this section. Two fundamental postulates of the magneto static that specify the divergence and the curl Three phase primary Primary core winding of B in free space are [9] r~ × H~ = J;~ (1) (b) r~ · B~ = 0 : (2) Fig. 1. Linear Induction Motor with (a) Short Stator (b) Short Rotor By using the equation B~ = µ0H~ , we obtain from (2): r~ × B~ = µ0J:~ (3) Where B is the magnetic flux density, H is the magnetic field intensity, J is the current density, and µ is the magnetic permeability of material. It follows from (1) that there exists a magnetic vector potential A~ such that B~ = r~ × A;~ (4) ( ) 1 r~ × r~ × A~ = J:~ (5) µ For 2-D case, the magnetic flux density B is calculated as ( ) @A @A B = ; ; 0 : (6) @x @y On the other hand Z Fig. 2. Experimental model of short stator linear induction motor Φ = B~ · d~s: (7) S From (4) we have: Z I the eddy current in the secondary sheet decreases from Φ = (r~ × A~) · d~s = A~ · d~` : (8) maximum to zero by exponential attenuation only in the S C primary length range. It only considers the mutual in- ductance influenced by the eddy current, disregarding its effect on the secondary resistance. In this paper, in order to investigate the magnetic flux distribution, first a 2-D model of LIM is simulated and the circumstance of the magnetic flux density and the equipotential lines of the magneto static potential in the core and its surrounding for steady cases are presented. Then, based on the per-phase equivalent circuit in [8], a mathematical model for the single side short stator linear induction motor with a consideration of the end effects is presented. In order to verify the mathematical model, a laboratory prototype of short stator LIM is developed and the results of experiments and simulations are presented in Section 7. In this section, first, the equivalent circuit parameters are obtained and compared with calculation results. Then, two experiments, a free acceleration and locked primary experiment (S = 1) have been done and Fig. 3. (a) Mesh shape (b) magnetic flux density (c) equipotential the results have been compared with the simulation re- lines of the magneto static potential 304 H. Hamzehbahmani: MODELING AND SIMULATING OF SINGLE SIDE SHORT STATOR LINEAR INDUCTION MOTOR WITH Hence, the resulting magneto motive force (MMF), and thereby air gap flux, looks like Fig. 4c [8]. The Q factor is associated with the length of the primary, and to a certain degree, quantifies the end effects as a function of the velocity v as described by equation (9) [11]. DR Q = y : (9) (Lm + Lly)v Where D; Ry;Lm;Lly and v are the primary length, secondary resistance, magnetizing inductance, secondary leakage inductance and velocity. Note that the primary's length is inversely dependent on the velocity, ie, for a zero velocity the primary's length may be considered in- finite, and the end effects may be ignored. As the velocity increases, the primary's length decreases, increasing the end effects, which causes a reduction of the LIM's mag- netization current. This effect may be quantified in terms of the magne- tization inductance with the equation [4] and [11] ( ) Fig. 4. Linear induction motor (a) Eddy-current generation at the L0 = L 1 − f(Q) (10) entry and exit of the air gap (b) Eddy-current density profile along m m the length of LIM (c) Air gap flux profile [8] where Using (1) to (8), the magnetic flux density and equipo- tential lines of the magneto static potential in the core 1 − e−Q f(Q) = : (11) and its surroundings of the experimental model is simu- Q lated and the results are shown in Fig. 3. Figure 3a shows the mesh shape of the experimental model and Figs. 3b According to (9) to (11), the spatial distribution of the and 3c show the magnetic flux density and equipotential magnetic flux density along the width of the primary de- lines of the magneto static potential, respectively. pends on the relative speed between primary and sec- ondary. For null speed of the primary, the LIM can be 3 END EFFECT OF LINEAR considered as having an infinite primary and in this case INDUCTION MOTOR the end effects can be neglected. In LIMs, as the primary moves, a new flux is contin- uously developed at the entry of the primary yoke, while 4 MATHEMATICAL MODEL existing flux disappears at the exit side. Sudden gener- OF SHORT STATOR LIM ation and disappearance of the penetrating field causes eddy currents in the secondary plate. This eddy current In the dynamic analysis of the electrical machines, a affects the air gap flux profile in the longitudinal direc- mathematical model should be extracted from their equa- tion.
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