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- 1. 1 Abstract— Linear induction motor has found many applications, from slow moving sliding doors to high-speed ground transportation around the world. Although various methods have been developed for the analysis of linear induction motor, only few verifiable comparisons between these methods have been done in literature. These methods solve the Maxwell’s equations in and around the geometry of the machine and take care of the physical dimensions of the machine. These methods which are proved accurate are based on Fourier transform, Fourier series and Space harmonic techniques. Out of different methods of analysis, only few methods have been reported into the literatures which are capable of taking care of topological changes in Linear induction motor. Here in this work different computer programs have been developed which are based on the above techniques. These computer programs have been validated using the published results. These have been predicting the single sided and double sided Linear induction motors. A comparison of these methods has been reported for finding their suitability for the use of these methods for predicting the characteristics of LIM under different operating conditions. Index Terms—Linear Induction Motor, Fourier Transform Analysis, Fourier series Analysis, Space Harmonic Method 1. INTRODUCTION Inear Induction Motor (LIM) have been used in many industrial applications including actuators, conveyer systems, sliding doors, material handling, pumping liquid metals, robot base movers, elevators etc. They have been implemented as propulsion systems for transit vehicles in a number of countries.LIM has many advantages, such as simple structures, easy mainainace,repair,replacement,ability to exerts thrust on secondary without mechanical contacts, convenient control of thrust and speed, less environmental pollution etc. Although conventional rotating machines have been largely used to drive underground transportation systems, Linear induction motors have also been considering for future applications owing to their certain advantages. With better appreciation of the capabilities of linear drives, new modifications for these machines have been developed. Consequently, there is a continuing need for effective CAD tools and accurate analysis methods which allow LIM to be designed to meet the specific requirements of transportation and other application. Many techniques have been developed To analyze LIM, but only few verifiable comparisons between these methods have been done in literature. Linear Induction motor (LIM) operates on the same principle as the conventional rotary induction motor. The Email:Chandan.kumar.eee07@itbhu.ac.in Mobile no:+91-9889918032 Rotary motor is cut out and laid flat to form the equivalent LIM.The reaction plate in LIM becomes the equivalent rotor as shown in Fig1.The induced field is maximized by backing up the reaction plate with the iron plate ,commonly known as ‘back iron’. This serves to amplify the magnetic field produced in the coil. Figure 1.Rotary and Linear Induction Motor The difference between the LIM and its rotary counterpart is due to the difference in air gap. The LIM has an open air gap with an entry-end and exit –end, while rotating induction motor has closed air gap. The openness of the air gap gives rise to the particular characteristics of LIM.In contrast with rotary machine; LIM has an open magnetic circuit. 2. ANALYSIS OF LIM LIM can be analyzed by different techniques like finite difference, finite element, Fourier series, equivalent circuit, quasi 1D method and many more. Fig.2 shows the summary of the different approaches for the analysis of LIM. Figure 2.Different methods for analysis of LIM The numero –analytical methods are based on Fourier methods in which longitudinal and transverse effects have been suitably considered. A large saving of computational efforts has been observed, because results of electromagnetic equations in closed form has been obtained and computers are only used for summation or integration. These methods can be further categorized as Fourier series methods [1], Fourier transform methods [2] and space harmonic methods [2, 3 and 4]. COMPARISON OF ANALYTICAL METHODS OF ANALYSING LINEAR INDUCTION MOTORS Chandan Kumar, R.K.Shrivastava, S.N.Mahendra, Subrahmanyam Maddula Department of Electrical Engineering, I.T.-B.H.U, Varanasi, India L
- 2. 2 In order to consider the longitudinal end effects and transverse edge effect, the Double Fourier series decomposition of primary m.m.f. Distribution has been used. It has been assumed that the primary members periodically separated from each other by a certain distance such that the end flux of one machine is not affecting the performance of the other machine. The optimum distance by which the primary members need to be separated increases for increasing speed. Also at higher operating speed, the computational time may be fairly due to slow convergence of Fourier series. The Fourier series methods of analyzing LIM have been developed by Obrettl [1]. Fourier transform method is normally suited for a non- periodic distribution. Due to this it does not require assumption mentioned earlier. Also the computation time has been found to be independent of the speed of operation. It has been developed by Yamamura [1].the method uses product solution technique for solving Maxwell’s equations and transforms the x-component of 3D/2D field solution in Fourier space. The analysis can be done either in real space by inverse transformation using Residue’s theorem or in Fourier space using Perseval’s identity. In the space harmonic analysis method, not only the end effects but also the primary design parameters can be taken into account. It has been assumed in this method an infinite array of adjacent primary winding at the surface of the primary core extended to infinity in both the longitudinal direction. The distance between two adjacent windings has been chosen to be long that the magnetic interaction between them can be neglected. Fourier analysis of the m.m.f distribution has been carried out. The m.m.f wave produced by the primary currents consists of the fundamental pace component of equivalent pole pitch and an infinite number of space harmonic components. It has been applied to short primary LIM [2].A study has been made on the design of the SLIM for the urban transit with maximum speed of 70kmph [5] using space harmonic analysis method. A method for computing the performance of linear induction motor has been presented in [4].This analysis has been derived from 2D electromagnetic analysis using space harmonic technique based on the Fourier series expansion. The resultant method would be an interactive force between the Fourier series and Fourier transform, and can make a substantial abbreviation in the calculation without spoiling the accuracy, taking directly into the account of end effect, skin effect and the primary winding arrangement. 3. PROBLEM IN ANALYSIS OF LIM There are certain effects in LIM, which are normally not present in rotary induction motors. Due to this, the analysis of LIM has to be done using different techniques. Various researchers in past have analyzed these effects using different techniques. These have been briefly discussed here: (a).Longitudinal end effects: One obvious difference between LIM and conventional rotary machines is that the fact LIM has ends. This means that travelling magnetic field cannot join up on itself, and introduces end effects. The end effect is clearly exhibited in the form of non-uniform flux density distribution along the length of motor. These effects call for analyzing LIM taking into account of the full machine dimension in the longitudinal direction. The longitudinal end effects have been known to be pronounced at high speed and result in the reduction in thrust, efficiency and power factor, when compared to the corresponding rotating machine. These phenomena have been interpreted physically to be due to airgap flux density waves propagating from the entry and exit ends of the stator winding with the entry wave being responsible for the major effects. (b) Transverse edge effect: This edge effect is generally described as the effect of having finite width for a linear motor. This effect is more evident with lower values of width to air gap ratio. It results from the forward component of the eddy current density in the reaction plate causes the equivalent impedance to be increased and exerts harmful influences on the various characteristics of LIM such as thrust,efficiency,power fator,lateral instability due to uneven secondary overhangs etc. (c)Saturation and skin effect: The saturation and skin effect the back iron of LIM have been considered by Boldea, Nasar [6].In these cases, a composite reaction rail has been used, in which the back iron is screened by a thin reaction plate of conducting material like aluminium or copper. Further, due to larger airgap in LIM, these are lesser chances of primary getting saturated. Therefore, in most of these analyses, the saturation in the primary has been neglected. As such, the choice of infinite permeability of primary member of LIM leads to acceptable results. However in comparison to primary core the back iron easily gets saturated. The usual approach is to numerically compute flux dependent permeability of back iron and later utilize it in analytical methods [6].In this way LIM can be accurately analyzed by analytical methods even in the presence of heavily saturated thick back iron. These effects must be minimized as much as possible when designing LIM, so that they do not drastically affect the performance. In this paper different analytical methods has been used for SLIM and comparison is done. 4. THEORETICAL AND ANALYTICAL RESULT VERIFICATION In this section the results from different methods which are Fourier series, Fourier transform and space harmonic are discussed. A. Comparison of thrust obtained by different methods from SLIM with back iron The longitudinal flux in SLIM and DSLIM is responsible for traction. This force, unlike the mechanical devices, is produced without any contact between primary and secondary. This characteristic of linear machine is responsible for its application in levitated high speed ground transportation. The specification of SLIM with back iron i.e. SLIM-A [3] and SLIM-B [7] have been shown in appendix. The thrust slip characteristic from all the three methods and the published data [3, 7] is shown in Fig.3 and Fig.4 respectively.
- 3. 3 Figure 3. Thrust vs. Slip characteristic of SLIM-A with back iron by numero- analytical methods Figure 4. Thrust vs. Slip characteristic of SLIM –B with back iron by numero- analytical methods From these two curves following conclusion can be drawn: 1. For SLIM-A (Vs=134.16m/s), the method of Fourier series slightly overestimates the breakdown thrust as that of published results. However for SLIM-B (Vs=12.1 m/s) it slightly underestimate breakdown thrust. 2.For SLIM-A, The method of Fourier transform slightly overestimate the breakdown thrust as that of published results but the it is less than that of Fourier series method. However in SLIM-B Fourier transform overestimate the breakdown thrust, but the breakdown thrust calculated using Fourier series is lowest among all. 3. For SLIM-B, the space harmonic technique slightly underestimate the breakdown torque but overestimate in SLIM-A. 4. For SLIM-A (f=220Hz), the critical linear slip as obtained by the method of Fourier series is almost same as that of published result but in SLIM-B (f=50Hz) is overestimate it. 5. For SLIM-A, the method of Fourier transform underestimate the critical linear slip but it is almost same for SLIM-B. 6.For SLIM-A, the critical linear slip obtained from method of space harmonic is almost same as that of published results but it underestimate it in SLIM-B. 7. In all the three methods the thrust obtained in stable portion of the characteristics shows comparable results of that to the published results. 8. For SLIM-A, the starting thrust as obtained from all methods is comparable to that of published results. However for SLIM-B the method of Fourier series and space harmonic slightly overestimate and Fourier transform underestimate it with respect to published data. B. Comparison of Normal Force obtained by different methods from SLIM with back iron The normal force calculation is important because though in rotary machine it gets cancelled, in linear machine it is present. The normal force in SLIM may be used for levitating the vehicle above rail, thereby reducing the friction between wheel and rail the comparison of SLIM-A and SLIM-B are shown in the Fig.5 and Fig.6 respectively. Figure 5. Normal force vs. Slip characteristic of SLIM-A with back iron by numero-analytical methods Figure 6. Normal force vs. Slip characteristic of SLIM-B with back iron by numero-analytical methods From these two curves following conclusion can be drawn: 1. In both cases, the method of Fourier series overestimates the normal force at synchronous speed as compared to other methods. 2. In both cases, the starting normal force as obtained from the method of Fourier series is comparable to that of published. 3. For repulsion case space harmonic gives result near to published one. While in attracting case it overestimates the published result.
- 4. 4 4. The space harmonic and Fourier series methods show slight waviness as compared to that of Fourier transform technique. 5. RESULTS A. Fourier Transform Method: It overestimate the breakdown thrust for high speed operation compared to lower speed .While it underestimates the critical linear slip for the higher frequency operation as compared to lower frequency operation. There is slight overestimation of the starting thrust for lower speed of operation but comparable at higher speed. It gives comparable result for both attracting as well as repulsive type for normal force calculation. B.Fourier Series Method: It slightly overestimates the breakdown thrust for high speed of operation as compared to low speed operation. It also overestimates the critical linear slip for lower frequency of operation while comparable in case of higher frequency operation. It gives comparable starting torque for high speed operation while overestimate in low speed working region. It overestimates the normal force at synchronous speed but the starting normal force as obtained from the method of Fourier series is comparable to that of published results. C.Space Harmonic Method: This method overestimate the breakdown thrust for high speed. The critical linear slip for lower frequency of operation is more but comparable in region of low speed operation. It underestimates the starting thrust in lower speed but gives comparable thrust in higher speed of operation. For repulsion case space harmonic gives normal force comparable to published one. While in attracting case it overestimates the published result. 6. CONCLUSION The analysis of two different SLIM using numero-analytical methods has been done and compared to the published results. All the three methods viz, Fourier transform, Fourier series and Space harmonic are capable of predicting the performance of SLIM within reasonable accuracy. The model developed here are particularly effective in predicting the constant thrust and normal forces with respect to slip. Among them Fourier series and space harmonic show good agreement for the higher speed of operation. While at lower speed the Fourier transform is good in predicting the forces in SLIM. REFERENCES [1].S.Yamamura, Theory of Linear Induction Motor, Halstead Press, 1972 [2]E.R.LAITHWAITE, Transport without wheel, Elec Science, London, 1977 [3]Kinjiro Yoshida,Sakutaro Nonaka, evitation forces in ingle- ided inear nduction otors Transactions on Magnetics,Vol.Mag-11,No.6,pp-1717-1719,1975 [4] Sakutaro Nonaka,N.Fujji Simplified two-dimensional analysis of Linear Induction Motors ,IEEE Transactions on Magnetics,Vol.Mag-23,No.5,pp-2832-2834,1987 [5] Sakutaro Nonaka Tsuyoshi iguchi lements of inear Induction Motor design for urban transit transactions on Magnetics, Vol.Mag-23, No.5, pp.3002-3004, 1987 [6]I.Boldea, S.A.Nasar, Linear motion electromagnetic systems, John Wiley & Sons . . ahendra R. . hrivastava rag plate-single sided linear induction motor International conference on Industrial Technology, 2000 APPENDIX Parameters SLIM A SLIM B Number of poles 12 8 Pole Pitch(mm) 304.8 121 Frequency(Hz) 220 50 Air Gap(mm) 50 10 Synchronous Speed(m/sec) 134.16 12.1 Number of Phases 3 3 Number of slots per phase per pole 4 5/6 Primary Length(mm) 2057 968 Primary width(mm) 32.8 128 Width of secondary sheet(mm) 545 200 Thickness of secondary sheet(mm) 15.9 3 Thickness of Back iron(mm) 2 6 Current(A) 1760 10 Slot pitch 25.4 1.34 Permeability of Back iron 500 *µo 350*µo

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