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An investigation of high speed, thin steel rotor, annular, double sided, linear induction motors Peabody, Frank Gerald 1988

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AN INVESTIGATION OF HIGH SPEED, THIN STEEL ROTOR, ANNULAR, DOUBLE SIDED, LINEAR INDUCTION MOTORS By FRANK GERALD PEABODY B.A-Sc., The University of British Columbia, 1978 M.A.Sc, The University of British Columbia, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Electrical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1988 ©Frank Gerald Peabody, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of £/f^Af</ i ^ t e ^ ^ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Itjy. }£,  D E - 6 G / 8 1 ) ii Abstract The objective of this dissertation is to analyse the performance of a linear induction motor suitable to drive a circular saw blade. A selection of analytical methods available from the field of electrical machine theory was used to investigate the particular type of motor. The theoretical analysis is supported by an extensive experimental investigation. Although LIMs have been designed, analyzed and applied in other applications, significant differences exist between those LIMs and the one used for the new application. These include: the annular shaped motor, the smaller air gap, and the rotor which is thin and made of steel. Because of these differences, the methods used by previous investigators were not sufficient to design the LIM required. The theoretical analysis used a selection of methods described in the literature to quantify the effect of the rotor material, the end effect and the edge effect. New methods are described to analyse the effect of the annular shape, the normal forces on the rotor and the coil connection. In addition, a new consideration in the optimisation of these type of motors is described. iii An extensive experimental program was undertaken. Six different linear motors were constructed with output powers ranging from one to fifty kWatts. In addition, inverters, dynamometers, flux measurement apparatus, speed measurement, thrust measurement and friction measurement apparatus were designed and constructed. The effects on performance of slot harmonics, winding connections, the end effect and the edge effect were measured. Several contributions to the field of electrical machine theory are presented. The first is a new annular disc motor resistivity correction factor. Second, is the analysis of the effects of poles in parallel versus in series in linear induction motors. Third, is the experimental comparison between odd and even pole designs. The fourth is a second optimum goodness consideration for LIMs, which had not previously been considered. The fifth is the analysis of the rotor/stator attractive force for magnetic rotor double sided motors and a description of the flux (crenelated flux) which causes the force. Finally, a criterion for when the re-entry effect may occur is presented. iv Table of Contents Page ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES viii NOMENCLATURE xi ACKNOWLEDGEMENT xiv 1. INTRODUCTION 1 1.1 Objectives of Thesis 1 1.2 Application of Thesis Results 2 1.3 Historical Background 4 1.4 Summary 8 2. GENERAL FORMULATION 10 2.1 Description of Linear Induction Motors 10 2.2 End Effects 11 2.3 Review of Analysis Techniques 11 2.4 One-Dimensional Analysis 16 2.5 Effect of Annular Stator 20 2.6 The Re-Entry Effect 31 2.7 Conclusions 34 3. NORMAL FORCES 35 3.1 Introduction to Normal Forces 35 3.2 Saturation Conditions 37 3.3 Normal Force Equations 41 3.4 Experimental Results 44 3.5 Conclusions 49 4. EXPERIMENTAL APPARATUS 51 4.1 Introduction 51 4.2 Motor Descriptions 51 4.3 Description of Experimental Apparatus 61 4.3.1 Power Supplies 61 V 4.3.2 Speed Measurement 61 4.3.3 Thrust Measurement 62 4.3.4 Current Measurement 62 4.3.5 Power Measurement 62 4.3.6 Flux Measurement 64 4.4 Experimental Accuracy 66 4.5 Conclusion 67 5. EXPERIMENTAL RESULTS AND DISCUSSION 69 5.1 Introduction 69 5.2 Verification of the Computer Model 72 5.3 Effect of Segmented Rotor 78 5.4 Effect of Rotor Variations 86 5.5 Effect of Odd or Even Number of Poles 93 5.6 Effect of Series and Parallel Connection 98 5.7 Edge Effect 103 5.8 Effect of Harmonics 112 5.8.1 Slot Harmonics 112 5.8.2 Time Harmonics 120 5.9 The Effect of the Annular Motor 120 5.10 Discussion 123 6. FACTORS WHICH AFFECT OPTIMIZATION 127 6.1 Introduction to Optimization 127 6.2 Goodness Factor 127 6.3 Optimum Goodness Factor 128 6.4 Analytical Results 130 6.5 Optimization By Scaling Up 132 6.6 Series and Parallel Connection Optimization 134 6.7 General Comments On Optimization 136 7. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 138 7.1 Summary 138 7.1 Conclusions 139 7.2 Recommendations For Further Research 145 REFERENCES 146 APPENDIX 1 - DERIVATION OF AIR GAP EQUATIONS 151 APPENDIX 2 - FORTRAN SIMULATION PROGRAM 157 APPENDIX 3 - SPREADSHEET SIMULATION PROGRAM 165 vi APPENDIX 4 - COMPUTER MODELLING 167 APPENDIX 5 - STATOR STEEL SPECIFICATIONS 184 APPEND LX 6 - FINITE ELEMENT ANALYSIS 185 APPEND LX 7 - RELATIVE END EFFECT SIMULATION PROGRAM 187 vii LIST OF TABLES Page Table 2.1 - Hypothetical Motor Parameters 29 Table 4.1 - Parameters of Experimental Motor #1 55 Table 4.2 - Parameters of Experimental Motor #2 56 Table 4.3 - Parameters of Experimental Motor #3 57 Table 4.4 - Parameters of Experimental Motor #4 58 Table 4.5 - Parameters of Experimental Motor #5 59 Table 4.6 - Parameters of Experimental Motor #6 60 Table 5.1 - Entry Coil Current 99 Table 5.2 - Measurements For Experimental Motor #4 With and Without Slots at 380 Hz Supply 112 Table 5.3 - The effect On the Performance Of Experimental Motor #2 With and Without Misaligned Teeth 118 Table 5.4 - The effect On the Performance Of Experimental Motor #4 With and Without Slotted Rotor 119 Table 5.5 - Harmonic Components of Current Wave Form to LIM 120 Table 6.1 - Comparison of Motor Parameters 133 viii LIST OF FIGURES Fig. 1.1 - Prototype LIM driven Saw 3 Fig. 1.2 - Entry and Exit Effect 7 Fig. 2.1 - Motor Rolled Out To Form A LIM 10 Fig. 2.2 - Pattern of currents in rotor to show entry effect 12 Fig. 2.3 - Equivalent Circuit Model 14 Fig. 2.4 - Mesh Matrix Derivation 14 Fig. 2.5 - Mesh Matrix Equations 15 Fig. 2.6 - Motor representation 16 Fig. 2.7 - Curvature of the annular LIM stator. 20 Fig. 2.8 - Simulation of the power and thrust produced at the inner, middle and outer radii of the stator. 27 Fig. 2.9 - Effect of curvature on rotor resistivity. 28 Fig. 2.10 - Annular Resistivity Correction Factor Applied To Hypothetical Motor #1. 30 Fig. 2.11 - Annular Resistivity Correction Factor 31 Applied To Experimental Motor #1. Fig. 2.12 - Annular Resistivity Correction Factor Applied To Experimental Motor #2. 32 Fig. 3.1 - Magnetic flux paths 36 Fig. 3.2 - Saturated Rotor Flux Paths 38 Fig. 3.3 - Instantaneous stator-rotor attractive force along the stator. 40 Fig. 3.4 - Attractive force vs rotor position in the air-gap. 43 Fig. 3.5 - Attractive Force Experimental Apparatus 45 Fig. 3.6 - Sample of Saw Steel Being Tested 46 Fig. 3.7 - Test Set Up To Measure Attractive Force 47 ix Fig. 3.8 - Coefficient of Friction vs Speed Test Set Up 48 Fig. 3.9 - Coefficient of Friction vs Speed For Delrin 48 Fig. 4.1 - Stator Core A 53 Fig. 4.2 - Stator Core B 54 Fig. 4.3 - Stator Core C 54 Fig. 5.1 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 3 mm Copper Rotor 73 Fig. 5.2 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 3 mm Stainless Steel Rotor 74 Fig. 5.3 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 2 mm Stainless Steel/Copper Rotor 75 Fig. 5.4 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 2 mm Copper Rotor 76 Fig. 5.5 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #3 With 3mm Steel Rotor 77 Fig. 5.6 - Experimental Motor #6 Derived By Disconnecting Two Fifths of Experimental Motor #5 73 Fig. 5.7 - Experimental Results of Steel Rotor Motor #6 With and Without End-Effects, Steel Rotor 74 Fig. 5.8 - Simulation Results For Experimental Tests of Experimental Motors #5 and #6 75 Fig. 5.9 - Experimental Results of Aluminum Rotor in Experimental Motors #5 and #6 Shown Diagrammatically 76 Fig. 5.10 - Simulation Results For Experimental Motor #6 With and Without End Effects, Aluminum Rotor 77 Fig. 5.11 - Flux Sensors Shown Wound On Experimental Motor #3 78 Fig. 5.12 - Magnetic Flux Density Measured Along Experimental Motor #3 79 Fig. 5.13 - Experimental Points and Approximating Curves Showing Efficiency 89 Fig. 5.14 - Experimental Points and Approximating Curves Showing Power-Factor 90 Fig. 5.15 - Experimental Points and Approximating Curves Showing Power-Factor Efficiency Product (Zeta) 91 Fig. 5.16 - Experimental Points and Approximating Curves Showing Power 92 Fig. 5.17 - Conceptualized LIM 93 Fig. 5.18 - Analysis of the Flux For a Three Pole Stator 94 Fig. 5.19 - Experimental Results For A Comparison Between Even And Odd Number of Poles 97 Fig. 5.20 - Power Produced By the Entry Pole of a Series and Parallel Connected Stator 100 Fig. 5.21 - Measured Flux at the Entry End of a Parallel Connected (E.M. #2) and a Series Connected Motor (E.M. #3) 101 Fig. 5.22 - Experimental Results for a Comparison of the Performance for a Parallel Connected and a Series Connected Motor 102 Fig. 5.23 -Pattern of Rotor Currents 103 Fig. 5.24 - Diagram of Flux Density 104 Fig. 5.25 - Reference Frame For the Edge-Effect Analysis 105 Fig. 5.26 - Flux Density Variation Factor 107 Fig. 5.27 - Flux Sensing Coils On Experimental Motor #1 109 Fig. 5.28 - Measured and Expected Value of Flux Across Stator Face For experimental Motor #1 110 Fig. 5.29 - Slots Cut Into the Rotor of Experimental Motor #4 111 Fig. 5.30 -Air Gap Flux Of Experimental Motor #4 114 Fig. 5.31 - Force Due to the Slot Harmonic and Effective Rotor Thickness 115 Fig. 5.32 -The Effect of Misaligning the Stator Teeth 117 xi Fig. 5.33 - Slots Cut In Rotor To Eliminate Rotor Slot Harmonics 119 Fig. 5.34 - The Stator Tooth And Slot Profile For the Annular Stator 121 Fig. 5.35 - Lateral Variation in Flux For Experimental Motor #1 122 Fig. 5.36 - Variation in Power and Thrust With Radius Including the Effect of The Stator Slots 123 Fig. 6.1 - Relative end effect force. 129 Fig. 6.2 - Power for various rotor resistances. 132 Fig. 6.3 - Possible Parallel Connected Stator Designs Compared To a Series Connected Stator 136 NOMENCLATURE A = stator surface area b = air gap flux b n = -iBk2/(k2+iswa) bV = [Va+k tantfl-iv/kyON -v-Va)]b_ F Q = thrust produce by a motor without end effects F ^  = thrust due to the end effect F r g n = stator/rotor net normal force F * = normal force on rotor due to stator 1 F r g = normal force on rotor due to stator 2 g = air gap between the stator and rotor i = square root of-1 I = current in the overhang regions of the rotor j = rotor current current (A/m ) J = stator surface current density (A/m) k = 2TZ1T\ K = [l-exp{(r-L+ik)Pn}]/[r1+ik] 1 = stator width ly = width of rotor L = stator width E = stator length P = number of stator pole pairs r = radial distance in cylindrical co-ordinates r-, = Va /2{l-[l+4(iwa+v2y(Var]} Rl = inner radius of the stator R2 = outer radius of the stator s = slip S ^ = distance on stator of one tooth pitch S„o = distance on stator to next adjacent stator tooth S r =c*dr-D/2 t = stator tooth width v = 4/stator width/total rotor overhang V = velocity w = stator frequency x = distance from entry end of stator y = distance from center line of stator a = p.0*c/(p*g) 2 i s w u B =k + ^— " g > 5 = Rg-R-j^  = c(lr-l)/2 xiii 1 + • V l + i s G tanh pa tanh k(c - a) 9 = angular location on motor in cylindrical co-ordinates 0 = total angular section of motor in cylindrical co-ordinates r\ = two pole pitches (one wavelength) u = permeability of free space K =3.1415 p = rotor resistivity a = magnetic reluctance 1 = saturation flux density of rotor/peak flux in the stator <|> = tan"1(swc</(k2+v2) O = magnetic flux ® c = crenelated flux (tooth-rotor-tooth leakage flux) w = frequency in radians per second Q. = volume resistivity or ohm Mathematical Operators + = addition = subtraction * = multiplication (used if equation would be ambiguous) / = division exp = the natural exponential ln = the natural logarithm db/dx = differential operator dVd'x = partial differential operator xiv Acknowledgement The author wishes to thank the Science Council of B.C., the Natural Sciences and Engineering Research Council and Cetec Engineering Ltd. which have all supported this work. In addition I would like to thank Jan Bridcko who conceived of the idea of the application for the LIM and to thank Don Nyberg who programed the spreadsheet. I have also had the opportunity to discuss this project with most of the experts in the field of linear induction motor analysis. In particular I would like to thank Professors Hugh Bolton, Graham Dawson, Tony Eastham, Eric Laithwaite, S. Nonaka, J. Pascal, Michel Poloujadoff and Sakae Yamamura. Finally I would like to express my thanks to my thesis super-visor, Dr. Dunford, for the confidence he showed in my ability. 1. INTRODUCTION 1 1.1 Objective of Thesis A new type of circular saw is being developed which uses a linear induction motor (LIM) to drive the blade. Although LIMs have been designed, analyzed and applied in other applications, significant differences exist between those LIMs and the one used for the new application. These include: the annular shaped motor, the smaller air gap, and the rotor which is thin and made of steel. Because of these differences, the methods used by previous investigators were not sufficient to design the LIM required. An experimental investigation of this unusual type of LIM was undertaken aided by the use of various simulation and analysis methods. Specifically, in order to meet this objective, the following were undertaken: 1. Theoretical analysis was developed, one aspect of which involved applying conventional theories to a steady state electromagnetic model. The resulting equations were then set up for annular motors by using cylindrical coordinates and a resistivity correction factor. As a tool in developing this theoretical analysis, a spreadsheet program for simulating the performance of LIMs was developed. This spreadsheet method proved to have significant advantages over previous design methods, such as conventional Fortran simulation methods. 2. The normal forces between the rotor and the stator in this 2 type of machine were modelled and an experimental investi-gation undertaken in order to determine the factors affecting their magnitude. 3. Simulations and experiments were undertaken in order to analyze the motor parameters which affect efficiency, power-factor and power density since the maximizing of these factors is an important objective in the final design. The simulations and experiments consisted of using various rotor resistances, air gaps, frequencies and flux densities in machines with six different stators (These differed in terms of stator size, number of poles, type of connection and windings). These experiments required the construction of a variable frequency three-phase inverter, the various experimental machines and the measurement apparatus. 1.2 Application of Thesis Results The Linear Induction Motor (LIM) has found applications in tran-sportation and material handling. New applications now exist in sawmill and mining equipment for double sided annular induction motors with thin steel rotors. One new application for LIMs which is being considered involves using a double-sided LIM to drive circular saw blades in wood cutting mills. The advantage of such a machine, the prototype of which is shown in Fig. 1.1, is that a thinner, straighter cut can 3 be made which in turn reduces wastage. Conventional large circular saws require a thick plate of 4-5 mm in order to support Fig. 1.1 - Prototype LIM driven Saw the thermal and mechanical stresses without the blade distorting. The proposed LIM driven saw has a design goal of a 2 mm blade thickness. Band saws, which use a comparable thickness of blade, are presently used in sawmills but they are very large and costly machines and expensive to maintain. An additional advantage of 4 the proposed LIM driven saw is that it has a higher blade speed than present day band saws. The objectives for the LIM are: 1) that it produce 50 kW at 140 m/s and an overload power of 70 kW at 110 m/s. 2) the power-factor/efficiency product should be above .25. 3) the motor should be no longer than 1 m and no wider than .3 m. The primary application for the type of motor being investigated in this thesis is to drive circular saw blades in wood cutting mills. Other applications may exist in mining equipment to drive rock crushers or to drive sonic vibrators which enhance metal recovery in chemical reaction leaching tanks. 1.3 Historical Background One of the first patents for a linear motor was filed by Page [1] in 1854 to use magnets to produce linear motion similar to that produced by a linear synchronous motor. The Electric Shuttle Company applied for a patent [2] in 1859 for a linear motor supplied by a mechanical inverter which energized sequential magnets. The first linear induction motor was most likely conceived during the time when Galilao Ferraris and Nicola Tesla first demonstrated working models of rotary induction motors 5 during 1885. The first patent which closely describes the linear induction motor of today was given to Zehden [3] in 1902 for an electric traction system using a short primary and a long secondary, which is the configuration now used in most commercial applications. Although patents were filed, no significant applications of LIMs were attempted until 1945 when Westinghouse Electric developed a catapult launcher for use on aircraft carriers [4]. The LIM developed a peak power of 7,000 kW and was successful at launch-ing planes. However, the cost of the system made it impractical. Linear motors have also found applications in liquid metal pumps for atomic reactors [5] and some attempts have been made to use them in MHD generators [6]. The most active area for recent applications is in ground trans-portation (the automated light rapid transit system such as that used in Vancouver, Canada). When using a LIM the adhesion between the wheel and rail is not of concern at lower speeds and at higher speeds, above 300 km/h, where wheels cannot be used, then a LIM is the only electrical method of producing thrust (the Japanese HSST). Of all the possible LIM applications the Vancouver system is probably the greatest commercial success. The idea of using a LIM in certain applications occurred almost simultaneously with the use of the rotary induction motor. An 6 obvious disadvantage of the LIM, when compared to the rotary induction motor, is its open configuration which is the opposite of what is required for a good electric motor. It is important that an electric motor has a "tight" structure so that the electric and magnetic circuits are tightly coupled and thus the magnetic flux does not leak. Flux leakage results in motors with poorer efficiency, power-factor and power density. An apparent advantage of the Linear Induction Motor is that it does not require any gears, belts, pulleys or wheels to produce linear motion. However, these components are very durable and usually of negligible cost, so that their elimination alone does not justify the application of a Linear Induction Motor. This is the reason why so many attempts at using LIMs have failed to be a commercial success. The real advantage of a LIM is in applica-tions which require high speed or where no mechanical slip in the drive system can be tolerated. Although linear induction motors have been available since before 1905, they have not been extensively investigated until the last 40 years. It was only recently discovered [7] that LIMs behave differently from rotary motors at high speeds and require more complex analysis than conventional rotary motors. It was originally assumed that a LIM could be analyzed in the same way as a rotary motor. However, when high speed experiments were conducted, it was found that there was an 7 important difference - the LIM was less efficient and produced less power than -Entry Effect Current •Standard Currents X Fig. 1.2 - Entry and Exit Effect expected. A LIM behaves differently from a rotary motor at high speed due to the entry and exit effects. The entry effect is caused by new rotor material entering the stator magnetic field 8 and disturbing it (see Fig. 1.2). As the magnetic field in the rotor cannot change instantaneously (because energy must be transferred) the rotor in the entry area does not produce significant thrust. At the exit area a disturbing field is created by the exiting rotor material which also decreases the thrust but this effect is generally insignificant. The performance of the LIM was found to be strongly dependent on the resistance of the rotor. The optimum resistance for the rotor of a high speed LIM had not been investigated until recent work by Poloujadoff [8]. His results show that a higher resistance rotor may actually increase the efficiency of a high speed LIM. This is essentially the same conclusion previously reached by Nasar and Boldea [9]. They used the "goodness factor" as defined by Laithwaite [10] and determined an "optimum goodness factor" by finding an optimum resistance. 1.4 Summary Various areas must be investigated during the design of such an unusual type of double sided LIM. The high blade speed (140 m/s) makes end effects important. Also, because the blade is made of steel, the attractive forces may be large and, since the rotor is only 2 or 3 mm thick, the blade must be well supported to withstand these forces. The possibility of a re-entry effect, which occurs when currents are still flowing in the rotor when it enters the stator, thereby affecting the performance, must also be 9 considered. In addition, the LIM must have good power density in order to keep the saw small. Moreover, a high power-factor efficiency product is essential since the motor is powered by an expensive inverter. Finally, the annular stator will not produce exactly the same performance as a conventional LIM of the same mean length. The amount of variation which will occur between the straight and annular LIM should be calculated if accurate motors designs are to be undertaken. The above mentioned areas of analysis and design are set forth in the following chapters of the Thesis. Chapter 2 gives a descrip-tion of LIMs, the basic theory, and a new analysis technique for annular LIMs. The analysis of the normal forces, which either attract or repel the rotor to the stator, is detailed in Chapter 3. The experimental apparatus is described in Chapter 4. The experimental results, which are a major component of this thesis, are presented in Chapter 5. Based on the experimental results from Chapter 5 and the analysis given in Chapter 2, optimization criteria and considerations are derived in Chapter 6. Chapter 7 reviews the results of the research and highlights the original work in the field of motor design presented in this thesis. 2. GENERAL FORMULATION 10 2.1 Description of LIMs The linear induction motor can be considered as a rotary induction motor which has been cut and rolled out as shown in Fig. 2.1 [11,12]. The possible configurations are short stator or short rotor and single-sided or double-sided machines (The annular LIM to be analyzed is double-sided, short stator.). The stator is wound with sets of three-phase coils arranged so that a series of north and south magnetic poles sweep the stator and induce a current in the rotor. The thrust is produced by the interaction of the stator magnetic field and the rotor induced currents. Cage Rotor • Sheet Rotor Single Sided-»-Double Sided Fig. 2.1 - Motor rolled out to form a LIM 11 In addition to an electromagnetic thrust in the x-direction (see Fig. 1.2) there is an electromagnetic force in the y-direction, and if the rotor is made of magnetic material, a magnetic attractive force in the z-direction. If the LIM is double-sided and the rotor is centered in the air-gap the normal forces (z-direction) will be zero. 2.2 The Entry Effect The performance of LIMs is influenced by the entry effect, which causes a parasitic drag on the rotor. This effect occurs due to new rotor material continuously entering the stator area. The new rotor material has no current flowing in it and as a result of the finite time required to establish the flux linkages of the rotor, the current will not instantaneously begin to flow as the rotor material enters the stator. This results in a reduction of the motor thrust at the entry area because the rotor current that begins to flow at the stator entry area is in neither the right phase, amplitude nor shape to produce large amounts of thrust. In fact the rotor currents which do initially form can produce negative rather than positive thrust [7]. These entry effect currents are shown in Fig. 2.2. 2.3 Review of Analysis Techniques The development of a computer model for the annular LIM may be approached in a number of ways. Two methods are the simple 12 equivalent circuit model [13,14] and the mesh matrix equations model [15,16]. The most common method is the use of the field theory model [17,18,19,20]. Each of these is discussed below. Fig. 2.2 - Pattern of currents in rotor to show entry effect A simple equivalent circuit can be developed using the motor test parameters. Recently Duncan [14] obtained a reasonably accurate equivalent circuit model for performance modelling. The advant-ages of the equivalent circuit model are that it is mathematically 13 simple, requires no knowledge of the internal design of the LIM, and provides the motor terminal conditions (see Fig. 2.3). The disadvantage of this model is that it does not predict the internal conditions, such as the stator core flux density, of the motor and is thus less useful as a design tool. The mesh matrix equations model the motor as sets of coils on the stator and on the rotor. The relationships between the coils are obtained by analyzing the structure of the machine, and the coil parameters are obtained by motor tests. This form of analysis is called "mesh matrix" as the mesh equations are manipulated in matrix form (see Fig. 2.4 and 2.5). This type of model is useful for transient studies as the input variable is either the current or voltage at each time step in the simulation, and so transient events are handled in the same manner as steady-state operation. The disadvantage of this model is that the entry effect found in LIMs is difficult to simulate. The advantage of the field theory model is that it provides a good picture of the magnetic fields and current densities within the motor and thus provides the best model to use when designing new motors. The field theory model is also the only model which can be adapted to take into account the variation in current density and pole pitch which will occur over the face of the annular motor. There have been a number of theses written on field analysis [21-25], but to date none have modeled the annular LIM. The model proposed in this thesis will do this. Fig. 2.3 - Equivalent Circuit Model Fig. 2.4 - Mesh Matrix Derivation RS+pLSS pLSM pLSM pMSRcos8 pMSRcos(e+~) 2TS pMSRcos(0 ^ ) pLSM RS+pLSS pLSM pMSRcos ( 0-~) pMSRcosG pMSRcos (e*y^) pLSM pLSM RS+pLSS pMSRcos(e+|^) pMSRcos(0~) pMSRcos0 pMSRcosO pMSRcos ( O—11) pMSRcos( €Hj2) RR+pLRR pLRM pLRM pMSRcos( 0+y )^ pMSRcos0 pMSRcos ( ©-^) pLRM RR+pLRR pLRM 2 pMSRcos ( 0 ^ ) pMSRcos(9+|^) pMSRcos 0 pLRM pLRM RR+pLRR LSS- - per phase stator self inductance LSM - per phase mutual Inductance between stator windings LRR - equivalent per phase rotor self inductance LRM - equivalent per phase rotor mutual inductance MSR - maximum value of inductance between rotor and stator winding RS - per phase stator resistance RR - equivalent per phase rotor resistance 16 2.4 Qnfi-DiT^ensional Field Theory Equations Field theory models simulate the induction motor by modelling the stator as a block of material of negligible magnetic reluctance with an infinitely thin surface current, and the rotor as a thin plate with a specified volume resistivity. Other assumptions made about the motor in most field theory models are: the slotted air gap is replaced by an unslotted one of greater length which is determined - either by the Carter coefficient or by finite element analysis and does not include the thickness of the rotor if it is magnetic the stator windings are represented as surface currents the stator iron has negligible conductivity y i -+ x 7 stator rotor stator Fig. 2.6 - Motor representation 17 The equations which describe the electromagnetic conditions of the LIM will now be solved to obtain the expression for the thrust of the motor. The LIM is divided into areas as shown in Fig. 2.6. In one-di-mensional analysis it is assumed that the EMF induced in the secondary by the primary exists only directly under the stator (y 61 L/2) and is normal to the x-axis. Also, the currents directly under the stator flow only in the y-direction. In this model the current and flux densities are functions of the x component only. The following three equations describe these conditions, (all variables are steady-state values) - the change in flux along the motor is equal to the current density flowing in the stator and rotor i - C * J ( S - the change in current flowing along the return paths of the rotor is equal to the current density flowing in the rotor (2.1) dj 2*1 i*w*b + V*-.db ~5k (2.3) P 18 - the change in current density along the rotor minus the effect of the resistance is equal to the transformer and the speed induced voltage divided by the rotor resistivity When equations (2.1), (2.2) and (2.3) are combined, a third order differential equation (2.14) is obtained which can be solved from the boundary conditions and the given surface current density, (see Appendix 1) -db u*c*j*J dx g (2.1) di *• Hx = C J (2.2) dj Hx" 2*1 i*w*b + V*- db dx (2.3) A dx3 V n 0 c A P g dx2 1 w o^ c + P g 2 c T 2 1 db "dx s [d 2 J 2 c j 2 dx - 1 S r (2.14) Equation (2.14) is then solved to obtain the magnetic field value [29]. The thrust can then be calculated from the stator surface current 1 9 and the rotor magnetic field. The thrust developed by the motor is separated into two components. F Q is the thrust described by standard induction motor theory and F -^ is the thrust produced by the parasitic end effects found in linear induction motors as given by Poloujadoff [ 2 9 ] . F Q =^ *Jm*B0*l*p*L*cos<!)*sin<)) ( 2 . 1 5 ) F l = -^Jm*B0*l*Real[(b^p)*(bp/B0)*K] ( 2 . 1 6 ) b x = [V*a+k*tan(|)(l-i*v/k)/(r1-v-V*a)]*bp B Q = u.Q*J/(k*g) b p = -i*BQ*k2/(k2+i*s*w*a) i = sqrt ( - 1 ) r x = V*oc/2*{l-[l+4*(i*w*a+v2/(V*a)*2]} V = velocity v = 4/stator width/total rotor overhang a = \iQ*c/p/g p = rotor resistivity K = {l-exp[(r1+i*k)P*Ti]}/[r1+i*k] = tan"1(s*w*c</k2+v2) L = length of stator J m = maximum primary current density 1 = width of stator T| = wavelength k = 2*7i/ri 20 The total thrust is F Q plus F-^ . For rotary induction motors the value of F^ is zero. 2.5 Effect of Annular Stator Annular stators have been used to drive disc rotors for applica-tions in linear induction motor test apparatus [26,27] and commer-cial machines [11]. These machines have curvature effects which have in the past been assumed to be negligible in simulation models. However, no analysis has yet been done to show when this is true or how the effects of curvature should be accounted for. One author [28] does analyze the effect of driving a disc with a straight LIM but he does not consider the curved LIM situation. Figure 2.7 - Curvature of the annular LIM stator. 21 Curvature of an LIM (see Fig. 2.7) creates three factors which affect the performance. The first is the varying effective surface current density and the second is the varying pole pitch. The third is the increase in the effective resistance of the stator due to the wedge shaped current path. The first two of these three factors will now be analyzed. To analyze the annular LIM, the equations described in section 2.4 were set up using cylindrical co-ordinates and an annular resistivity correction factor was developed. The notation for these equations is shown in Fig. 2.7. First write out equations (2.1), (2.2) and (2.3) in cylindrical co-ordinates. -db/rdG = n0/g*(c*j+J) (2.17) dl/rdG = c*j (2.18) dj7rd9-(2/(8*Sr))*I = -{(i*w*b)+((r*d8/dt)*db/rd9)}/p (2.19) 22 Then write out the terms of equation (2.19) dj/rde-2/(5*Sr)*I = -i*w*b/p-(r*d9/dt)/p*db/rd9 (2.20) Taking the first derivative of (2.20) with respect to 9 d2j/rd92 - 2/(5*SJ*dI/rd9 = 2 2 ( 2 - 2 1 ) -i/p*w*db/rd9 - (r*d9/dt)/p*d^b/rd9/! Now taking the first derivative of equation (2.17) -d2b/rd82 = ]iQ*c/g * dj/rd9 + \iQ/g*dJ/rdQ (2.22) Taking the first derivative with respect to 9 -d3b/rd93 = uQ*c/g*d2j/rd92 + uQ/g*d2J/rd92 (2.23) 2 2 From (2.23) obtain an expression for d j/rd9 d2j/rd92 = -g/M.G/c*d3b/rd93 - l/c*d2J/rd92 (2.24) 23 and obtain an expression for j by rearranging (2.17) j = -g/u0/c*db/rd9 - J/c (2.25) Substitute the expressions (2.18) and (2.24) into equation (2.21) d2j/rd92 - 2/(8*Sr)*dI/rd9 = -l/p*i*w*db/rd9 - (r*d8/dt)/p*d2b/rd82 -g/u.0/c*d3b/rdG3 - l/c*d2J/rd92 - 2/(5*Sr)*c*j = -l/p*i*w*db/rd9 - (r*de/dt)/p*d2b/rd82 (2.26) (2.27) Substitute (2.25) into (2.27) -g/(u.o*c)*d3b/rd03 - l/c*d2J/rd92 - 2*c/(5*Sr)* {-g/(|X0*c)*db/rd9-l/c*J} =-l/p*i*w*db/rd9 -(r*d9/dt)/p*d2b/rd92 (2.28) Gathering together terms 24 -g/u0/c*d3b/rd93 - l/c*d2J/rd92 + 2*g/(5*Sr*u0)*db/rd9 + 2*J/5/S„ =-l/p*i*w*db/rd6 - (r*d9/dt)/p*d2b/rd92 Rearrange to get: -g/(M.Q*c)*d3b/rd93 + (r*d9/dt)/p d2b/rd92 + {2*g/(5*Sr*M-0) + i*w/p] =l/c*d2J/rd92 - 2*J/(8*Sr) Multiply through by (-u0*c/g) to get: d3b/rd93 - (r*d9/dt)*u.0*c/(g*p)*d2b/rd92 - {2*c/(8*Sr) + uG*c*i*w/(g*p)}db/rd9 = -u.Q/g*d2J/rd92 + 2u.0*c*J/87SI/g d3b/rd93-(r*d9/dt)*u.0*c/(p*g)d2b/rd92-[i*w*uo*c/(p*g)H (2*c/(5*Sr))]db/rd9=-u.0/g(d2J/rd92-(2*c/8*Sr)*J) (2.30) Equation (2.30) is then solved (using the same general solution 25 method found in [29] but for cylindrical coordinates) to obtain the magnetic field value. The thrust can then be calculated from the stator surface current and the rotor magnetic field. The torque developed by the motor is separated into two components. T Q is the torque described by standard rotary induction motor theory and Tj is the torque produced by the parasitic end effects found in annular linear induction motors. T Q = *r2*0*Jm*B0*(R2-R1)*cos(|)*sin(!) (2.31) T x = - *r*Jm*B0*(R2-R1)*Real[(b1/bpXbp/B0)*K] (2.32) B Q = >i0*J/(k*g) b p = -i*B0*k2/(k2+i*s*w*cc) bj = [(rWdt)*cc+k*tan<j)*(l-i*v/k)/ (r1-v-(r*de/dt)*a)]*bp K = [l-exp{(r1+i*k)P*Ti}]/[r1+i*k] k = 2*7i/ri r x = (r*d6/dt)*0(/2{l-[l+4*(i*w*a+v2)/(a*r*de/dt)2]} v = 4/stator width/total rotor overhang a = |j.0*c/(p*g) p = rotor resistivity P = number of pole pairs <|> = tan"1(s*w*ot/(k2+v2) T] = wavelength R^ = stator inner radius R2 = stator outer radius 8 = J^2"^l 26 The total torque is T Q plus T-^ . For continuous annular induction motors the value of T-^  is zero. The solution of the above equations will now enable the analysis of the first and second factors which affect the performance of the annular LIM. Over the surface of an annular LIM the current density varies inversely with radius, and as the surface current density increases so does the thrust. The pole pitch varies in proportion to the radius, and similarly the synchronous speed of the machine will also increase. These two effects were analyzed using the above derived analysis for annular LIMs and applied to Experimental Motor #1 (acting about a .25 m radius). The net effect (at 50% slip) was found to be a 25% increase in power and a 12% increase in thrust at the outside of the stator (due to the greater stator length at the outside) compared to the center of the stator. This is shown in Figure 2.8. The third factor, the increase in rotor resistivity, will now be analyzed. The effective increase in resistance can be calculated by integrating the resistance over the wedge shaped path of the rotor current (see Fig. 2.9). The correction factor to the rotor resistivity obtained is: k =[ln(R2/Rl)*(R2+Rl)]/[2*(R2-Rl)] a. (2.33) Q. O JZ 27 O cantor slip + inner o outer Figure 2.8 - Simulation of the power and thrust produced at the inner, middle and outer radii of the stator. In order to illustrate the effect on performance due to the rotor resistivity correction factor for annular motors, three cases will be considered. The first case will be that of a hypothetical disc positioning motor, the second case will be for the Experimental Motor #1 and the third case will be for Experimental Motor 28 #2. (Both the Experimental Motors, which have significantly larger mean radii, are described in Chapter 5.) j'R2 • j B. dr J Rl R e s i s t a n c e of Wedge = r i J o p l n (R2/R1) t e r de R e s i s t a n c e of Block = D <R2 -Rl) t 6 (R2 + RD/2 Ka = Re s i s t a n c e of Wedge Res i s t a n c e of Block ln<R2/Rl)«<R2+Rl) (R2-R1) «2 Figure 2.9 - Effect of curvature on rotor resistivity The hypothetical motor (see Table 2.1) has an outside radius of the stator (R2) which is four times greater than the inside radius (Rj). The calculation for the annular resistivity correction factor is shown below. k =[m(R2/Rl)*(R2+Rl)]/[2*(R2-Rl)] a. kQ=[ln(.0762/.0190)*(.0762+.0190)]/[2*(.0762-.0190)] Si k =1.15 MOTOR PARAMETERS Hypothetical Motor Number: 1 Type: Segmental Winding Parameters Number of Poles Number of Phases Pole Pitch Pitch Factor Coil Pitch Coil Span Distribution Factor Number of Coils Turns per Coil Turns in Series/Phase Mean Length of Turn Equivalent Wire Gauge Winding Connection Connection of Stators Together 4 3 6.19 cm .966 5.16 cm 1:6 .966 24 30 60 (4 poles in parallel) .254 m 18 Wye parallel Mechanical Parameters Primary Width 5.175 cm Primary Thickness 3.81 cm Tooth Width 5.16 mm Slot Width 5.16 mm Slot Depth 2.86 cm Total Primary Length .314 m Active Primary Length .298 m Number of Stator Slots (single layer) 24 Stator Slots (total) 29 Stator Slots (active) 29 Stator Slots (half-filled) 19 Secondary Thickness 2.5 mm Primary-Secondary Gap per side 1.4 mm Secondary Overhang per Side 2.54 cm Magnetic Gap (total) 2.8 mm Mean Radius 4.76 cm Electrical Parameters Primary Resistance per Phase .0569 Q Secondary Resistivity 26 uQ'cm Frequency 400 Hz Voltage 440 V Current Rating (continuous) 40 A Table 2-1 30 The annular rotor resistivity correction factors for Experimental Motor #1 and Experimental Motor #2 were calculated in the 0 . 1 0 . 3 0 . 5 0 . 7 0 . 9 1 . 1 1 . 3 - 1 . 5 1 . 7 1 . ; } 2 . 1 (Thousonds) RPM O Amps. + Torque ( N - m ) O Hp Fig. 2.10 - Annular Resistivity Correction Factor Applied ( ) and Not Applied ( ) To Hypothetical Motor #1. same manner and were found to be 1.0045 and 1.0003 respectively. These corrected values of rotor resistivity were used in the simulations shown in Figs. 2.10 to Fig. 2.12 for the three cases. The simulations show that only the Hypothetical Motor #1 shows a significant difference in performance when the annular resistivity correction factor is taken into account. It can be concluded that 31 the annular resistivity correction factor does not have to be used in order to obtain accurate simulations of the experimental motors that were constructed. RPM O Amps. + Torque (N-rn) o Hp Fig. 2.11 - Annular Resistivity Correction Factor Applied ( ) and Not Applied ( ) To Experimental Motor #1. 2.6 Re-Entrv Effect The re-entry effect could occur if the steel rotor still had currents flowing as it re-entered the stator area. In this case interference between the entry and exit effects «»could affect the performance of the motor. 32 • Am pr.. (Thousonds) R P M + Torque (N—m) Hp Fig. 2.12 - Annular Resistivity Correction Factor Applied ( ) and Not Applied (—-) To Experimental Motor #2. One way to determine whether re-entry will occur is to compare the length of the longitudinal entry wave to the distance that the currents would have to continue flowing before entering the stator again. If the entry wave is significantly shorter, then no interference will occur. The length of the longitudinal entry wave 33 is known and its value will have been calculated during the design and analysis of the LIM. For the Experimental Motor #2 these values were calculated using the analysis of Poloujadoff [29]. The value given for the entrance wave is: entrance wave = b^ *exp(r2*x) (2.34) where: b^ = [V*a+k*tan(|)(l-i*v/k)/(rj-v-V*a)]*bp B0= u0*J/(k*g) bp= -i*B0*k2/(k2+i*s*w*a) r 1 = V*0(/2*{l-[l+4*(i*w*a+v2/(V*a)*2]} V = velocity v = 4/stator width/total rotor overhang a = M-Q*c/p/g p = rotor resistivity w = stator electrical frequency Using the above formula the distance before the end effect wave attenuates to .368 (1/e) of its peak value was calculated to be .63 m for the Experimental Motor #2 (see Table 5.2 for motor par-ameters). This is the worst case condition because the entry wave is supported by the steel of the stator which increases the time constant of a perturbation far longer on the rotor while under the stator than outside of the stator. For the Experimental Motor #2, re-entry is clearly not a problem as the distance before the blade will re-enter the stator is approximately 2 meters. If the above 34 analysis does indicate that the exit wave may enter the stator then a more exact calculation can be performed using the boundary conditions for the rotor outside of the stator. 2.7 Conclusions The annular motor has been analyzed in this chapter using the electromagnetic field analysis equations in cylindrical co-ordin-ates. In addition an annular resistivity correction factor was developed and applied to one hypothetical and two experimental motors. It can be concluded that the annular stator will have a measurable effect on performance if the stator width is greater than half its mean radius. For the experimental motors construc-ted, the annular stator will have an almost negligible effect on performance. The possibility of a re-entry effect was analyzed and found not to occur for the experimental machines. When one reviews the analysis, it is clear that the only time that a re-entry effect will ever occur (for physically realizable machines) is when two sets of annular stators are used to drive a very high speed rotor (greater than 140 m/s) and the exit point of one stator is very close (within a few centimeters) to the entrance of the other. 3. NORMAL FORCES 35 3.1 Introduction to Normal Forces The analysis of the normal forces is important in the design of the steel rotor LIM because a large unbalanced normal force will create large friction losses which result in a low efficiency design. The large friction forces occur between the rotor and those pads which keep the rotor from touching the stator core. There are two normal forces acting on the rotor. These are: the magnetic attractive forces due to areas in the machine where the reluctance of flux paths can be reduced by movement of the blade ("reluctance normal forces"), and the normal force due to interac-tion of the magnetic field of the stator and currents in the rotor ("electromagnetic normal forces"). These flux paths are shown in Figure 3.1. There are also others (see Alger p.200 [30]). In addition there is a large attractive force between the two stators which should not be confused with the forces on the rotor. It is interesting first to compare the magnitude of the forces acting on the stators and rotor. The attractive force acting on the two stators due to the main flux is given by [31,32]: F g = B2A/(2*u0) (3.1) 36 where A is the stator surface area and B is the stator magnetic field. For the Experimental Machine #2 (parameters for this rf z 1' 1 i 1 i 1 n n Main Flux Path -Crenelated Flux Path "i_ i 1 Fig. 3.1 - Magnetic flux paths motor are found in Table 5.2), this was calculated to be 44.5 kN. The forces acting on the rotor will be shown to be approximately one to two orders of magnitude less. The reluctance normal forces will be analyzed by first looking at the flux paths and saturation conditions. This analysis gives the peak attractive force (for the worst case of the rotor positioned against one stator), calculated from equations (3.2) and (3.3). Then the modulation of the peak attractive force, in space, along the 37 stator is analyzed and the result is given in equation (3.4). Once these values are determined, the relationship between the maximum value of attraction and the rotor position is analyzed and the resulting relationship for the experimental motor is given in equation (3.5). Finally the maximum normal force due to the rotor currents (electromagnetic normal force) is given in equation (3.6) and shown to be small in relation to the attractive force. Following this, the analysis is first confirmed in the experimental results section, and then applied to obtain an expected value for drag during the full load operation of the motor. 3.2 Saturation Conditions The force of attraction occurs when the rotor moves from the center of the air-gap, resulting in greater attractive force being exerted by one stator than the other. Once the rotor is far from the center position in the air-gap the attractive force can be determined by looking at the flux path through the rotor and determining the total maximum flux which can take this path before the rotor saturates. If the rotor did not saturate (at approximately 2.0 T [33]) then the attractive force due to the pole-to-pole flux on the rotor would be as great as for the stator-to-stator force. However, the rotor does saturate, and is also relatively thin, so that the maximum flux that can pass through the rotor longitudinally is 38 only a small percentage of the main flux. The percentage of main flux carried by the blade is given by: % main flux= 4*;c*c*x/r|*100 (3.2) where c is the rotor thickness, T| is the length of two poles and i is ratio of saturation flux density of the rotor divided by the peak flux in the stator (see Fig. 3.2). For the Experimental r Tooth -Tooth Flux Limited By Rotor Thinness /Pole-Pole Flux Limited By Rotor Thinness Pn*1 n Fig. 3.2 - Saturated Rotor Flux Paths Machine #2 this value is 1.8 percent. When this value of flux is used to determine the rotor-to-stator attractive forces, a value of only 12 N is calculated. Experimental results, however, showed a much greater value. This greater attractive force can occur due to slot leakage flux which travels from tooth to tooth through the rotor blade. In a sheet rotor motor this flux will hereafter be 39 called "crenelated flux" as compared to zigzag flux as described by Alger [30] which changes pattern depending on the position of the rotor bars in the squirrel cage motor. The attractive force can be determined, once again by looking at the flux path through the rotor and determining the total maximum flux which can take this path before the rotor saturates. This is given by: where t is the width of one stator tooth. For the experimental machine (EM #2) this gives a maximum force of 2350 N for a 3 mm thick blade. This force is modulated along the length of the stator due to the sinusoidal current distribution (which is producing the mmf) along the stator. The mmf which drives the crenelated flux along this path is the difference of mmf between adjacent stator teeth. By using magnetic circuit analysis and finding the reluctance for the crenelated flux path an equation can be obtained for the value of this flux. For a stator with nine teeth per pole (20 degree spacing) the maximum crenelated flux is given by: %main flux= 2*c*t/t*100 (3.3) <J>C =[sin(w*t)-sin(w*t+20 degrees)]*mmf/(2*a) (3.4) and the force: F=(G> ,2 * i o 7 ) / ^ * 8n) neutons (3.5) 40 where o is the reluctance of the magnetic path from the stator tooth to the rotor and is the tooth area. The calculated stator-rotor attractive force along the stator length is shown in Figure 3.3. The per unit base is the maximum attractive force due to the pole to pole flux running through an infinitely thick rotor. 0 10 2 0 3 0 40 5 0 60 70 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 e l e c M c a l d e g r e e s Figure 3.3 - Instantaneous stator-rotor attractive force along the stator. 3.3 Normal Force Equations The previous analysis determined the force exerted on the blade in the worst case position, i.e. when it is against one stator. Now the relationship between the force and position from the center 41 will be analyzed. To determine the force on the blade dependent on the position in the gap, it should be remembered that, although the motor is supplied from a voltage source, the stator flux and the rotor currents set the stator current. The stator current is not affected by the position of the blade, so in effect the stator is supplied by a current source for this analysis. In other words, the current to the stator may increase due to greater load but the current will not increase due to the position (z - direction) of the rotor. This postulate is confirmed by the fact that the rotor is attracted to one stator or the other in the experimental motors. The slot-to-rotor leakage flux of opposing stator teeth is of opposite polarity in the rotor and cancels out when the blade is in the center of the air-gap. However, when the blade moves from the center position, the flux increases linearly (from magnetic circuit theory) since the driving force is a constant current source, and so the force on the rotor varies with the square of distance from the center position up to the saturation point. The reluctance normal force of one stator on the rotor is given by (see Fig. 3.1 for notation): (3.6) 42 where Brp = The net rotor to stator force for both stators is: rsn ~~ rsl" rs2 (3.7) The force on the blade, for Experimental Motor #2, is then given by the equation: F = F 1 - F o rsn rsl rs2 Frsn = (.37/(a+g))2-(.37/(g-a))2*4450 (3.7) (3.8) This is plotted for various air gaps and rotor positions as shown in Figure 3.4. These "V" shaped curves are an important result of this analysis because they show that if low normal forces are to be obtained, either the rotor must be accurately maintained in the center of the air-gap or a large air-gap must be used. The analysis for force on the rotor has so far ignored the effect of rotor current (electromagnetic normal force). This can cause a repulsive force on the rotor which will force it to the center of the air-gap. This effect and the complete analysis are 43 VARIOUS S T A T O R / R O T O R C L E A R A N C E S - 0 . 2 4 - 0 . 1 9 - 0 . 1 4 - 0 . 0 9 - 0 . 0 4 0 . 0 1 0 . 0 6 0 . 1 1 0 . 1 6 0 . 2 1 d i s t a n c e o f b l a d e o f f c e n t e r • . 2 5 m m c l e a r a n c e + . 5 m m o . 7 5 m m A 1 m m Figure 3.4 - Attractive force vs rotor position in the air-gap. given by Poloujadoff [8]. The maximum value of this force when it is repulsive is given by: F = u0*Jm2*A/(g*4) (3.9) Where J is the maximum stator surface current density. For the 44 experimental motor the formula predicts a peak centering force of 110 N, which is 5% of the previously analyzed reluctance normal force. This result is important because it shows that during normal operating conditions the total normal forces will always pull the rotor to one stator face or the other. 3.4 Experimental Results In the previous analysis it was assumed that all the flux which causes the reluctance magnetic attraction must pass longitudinally through the blade. In order to verify this assumption and to estimate the normal and drag forces in Experimental Motor #2, three experiments were conducted. First, a simple measurement device was constructed to test the electromagnetic force that could be exerted on a sample of saw blade material. Second, the coefficient of drag for the normal force pads in the machine was measured for different speeds. Third, the force required to move the rotor backwards, and the forward thrust, while energized, were measured in order to calculate the friction and the electro-magnetic thrust. (The term electromagnetic thrust refers to the thrust that would be measured if there was no friction in the machine). The experimental test apparatus, shown in Fig. 3.5, was con-structed and the specimen of saw steel is shown mounted in the jig in Fig. 3.6. The flux density in the saw steel was driven to saturation and the attractive force measured. The flux density in experimental conditions. A diagram of the experimental set-up is 45 Fig. 3.5 - Attractive Force Experimental Apparatus 4 6 Fig. 3 . 6 - Sample of Saw Steel Being Tested shown in Fig. 3 .7 . The measured force (356 N) under the test conditions compares favorably with the calculated value ( 3 6 0 N), given that fringing fields are not taken into account in the calculation method. The last experiment was designed to find the thrust and friction while at 1 0 0 Hz energization ( 0 . 2 7 % slip). These values can be 47 Fig. 3.7 - Test Set Up To Measure Attractive Force Shown Diagrammatically I Fig. 3.8 - Coefficient of Friction vs Speed Test Set Up 48 c c fr o o Coefficient of Friction for Delrin ( V a r i a t i o n With S p e e d . ) ( T h o u s a n d s ' ) S p o u d i n R P M Fig. 3.9 - Coefficient of Friction vs Speed For Delrin found by measuring the forward thrust and the force required to turn the rotor backward against both the friction force and the electromagnetic thrust. The friction occurs between the rotor and the low friction plastic (trade name Delrin) pads which are mounted in the stator slots in the space above the windings. The force required to turn the rotor backwards was 869 N and the forward thrust was 162 N. The calculation then gives an electromagnetic thrust of 507 N and a friction of 347 N. 49 If the 347 N of friction is accurate then a normal force of 2180 N (347/0.16 = 2180 N) is acting on the rotor of the four pole machine. The coefficient of friction for the stator pads was obtained with the disc brake apparatus shown in Fig. 3.8 to obtain the values shown in Fig. 3.9. The results of these three experiments can now be used to calculate the expected drag in the machine during operation. This drag on the rotor, at full speed, would then be 190 N (2180*0.07). This value would be an extra 21 kW in losses. This calculation does not include the electromagnetic repulsive force that would be acting on the blade under these conditions. 3.5 Conclusions The reluctance normal force has been analyzed by looking at the flux paths and saturation conditions. The analysis first identified the source of the large attractive force (the crenelated flux), then gave the peak attractive force and the modulation of the peak attractive force, in space, along the stator. Once these values were determined, the relationship between the maximum value of attraction and the rotor position was analyzed and the resulting relationship for the experimental motor was found. Finally the maximum normal force due to the rotor currents (electromagnetic normal force) was calculated and shown to be small in relation to the attractive force. 50 The analysis was then confirmed in the experimental results section and subsequently applied to obtain the expected value for drag during the full load operation of the motor. The normal force, which will always occur in the double sided steel rotor LIM, has the potential to create large losses even with low friction guiding surfaces. With the proper design of the LIM, as shown in the analysis which describes the "V" shaped curves, it should be possible (by increasing the air-gap) to reduce the drag to a more acceptable value than that measured in Experimental Motor #2. 4. EXPERIMENTAL APPARATUS 51 4.1 Introduction This chapter describes the experimental motors which were designed and constructed, the measurement apparatus and the possible error which may occur in the measurements. In Section 4.2 the parameters of the five motors are presented and a brief description of the construction techniques and materials is given. The equipment which was designed or obtained for use in the experiments is described in Section 4.3. The potential areas for errors in the experiments and the amount of expected error is discussed in Section 4.4. Section 4.5 concludes the chapter with a summary of the motors and the experimental apparatus. 4.2 Motor Descriptions Six experimental motors were designed and constructed during the course of the investigation. The parameters for these machines are presented in Tables 4.1 - 4.6. The experimental motors were wound on three different stator cores called Stator Cores A,B and C. Experimental Motor #1 was wound on Core A (one-third scale model of the Rimsaw motor). Experimental Motors #2, #3 and #4 were wound on Stator Core B (full scale Rimsaw motor) and Experimental Motors #5 and #6 were wound on Stator Core C (small scale completely circular motor). The stator cores are shown in Fig. 4.1-4.3. The details of each stator core can be found in the mechanical parameter section of the motor parameter tables. 52 Experimental Motor #1 was used to test the effect of different rotor materials and to measure the flux variation in the radial direction. Experimental Motors #2-#4 were used to determine the effect of odd and even number of poles and the effect of series and parallel connected poles and to measure full size motor performance. Experimental Motors #5 and #6 were used to deter-mine the magnitude of the end effect. All the motors were constructed of M-19, 29 Gauge, non-oriented electrical steel with a standard lamination finish. Further information on the specifications for the steel can be found in Appendix 5. The laminations were inserted into grooves machined into solid blocks of aluminum. The slots for Experimental Motors #1, #5 and #6 were machined after the laminations were placed in the grooves. The slots for Experimental Motors #2-#4 (stator core B, the full sized machine) were punched into the laminations before insertion into the aluminum blocks. For the case of motors designed for experimental purposes it was found that machined slots were less expensive to construct. Surprisingly, the cores with the machined slots did not have significantly higher losses then the punched slotted cores. There are two reasons for this. First, the laminations have very low pressure forcing them together since they are slid into the machined grooves of the aluminum blocks and secondly, very little flux leaves the machined surfaces which would cause circulating currents to flow (see finite element analysis in Appendix 6). If the tops of the teeth are machined, however, the 5 3 steel laminations will heat prohibitively (this was found experimentally) even with the very low lamination pressure. The only disadvantage of the machined slots for an experimental machine is that there are no notches at the top of the slot so that spacers usually used to hold down the copper windings can not be used. The active length of the machine is defined as that length of the stator steel encircled by active stator conductors. \ Fig. 4.1 - Stator Core A Fig. 4.3 - Stator Core C MOTOR PARAMETERS Experimental Motor Number: 1 Type: Segmental Stator Core: A Winding Parameters Number of Poles Number of Phases Pole Pitch Pitch Factor Coil Pitch Coil Span Distribution Factor Number of Coils Turns per Coil Turns in Series/Phase Mean Length of Turn Equivalent Wire Gauge Winding Connection Connection of Stators Together 4 3 6.19 cm .966 5.16 cm 1:6 .966 24 30 60 (4 poles in parallel) .254 m 18 Wye parallel Mechanical Parameters Primary Width 5.175 cm Primary Thickness 3.81 cm Tooth Width 5.16 mm Slot Width 5.16 mm Slot Depth 2.86 cm Total Primary Length .314 m Active Primary Length .299 m Number of Stator Slots (single layer) 24 Stator Slots (total) 29 Stator Slots (active) 29 Stator Slots (half-filled) 10 Secondary Thickness 2.5 mm Primary-Secondary Gap per side 1.4 mm Secondary Overhang per Side 2.54 cm Magnetic Gap (total) 2.8 mm Electrical Parameters Secondary Resistivity 26 (iflcm Frequency 400 Hz Voltage 440 V Current Rating (continuous) 40 A Table 4-1 MOTOR PARAMETERS Experimental Motor Number: 2 Type: Segmental Stator Core: B Winding Parameters Number of Poles 5 Number of Phases 3 Pole Pitch 17.1 cm Pitch Factor .966 Coil Pitch 15.2 cm Coil Span 1:9 Distribution Factor .9598 Number of Coils 45 Turns per Coil 5 Turns in Series/Phase 30 (5 poles in parallel) Mean Length of Turn .670 m Equivalent Wire Gauge #4 Winding Connection Wye Connection of Stators Together series Mechanical Parameters Primary Width 6.19 cm Primary Thickness 11.4 cm Tooth Width 9.52 mm Slot Width 9.52 mm Slot Depth 6.35 cm Total Primary Length 1.03 m Active Primary Length 1.01 m Number of Stator Slots (single layer) 45 Stator Slots (total) 53 Stator Slots (active) 53 Stator Slots (half-filled) 16 Secondary Thickness 3.0 mm Primary-Secondary Gap per side 1.4 mm Secondary Overhang per Side 2.54 cm Air-Gap (total) 2.8 mm Electrical Parameters Secondary Resistivity 26 |xQ"cm Frequency 360 Hz Voltage 440 V Current Rating (continuous) 600 A Table 4-2 MOTOR PARAMETERS Experimental Motor Number: 3 Type: Segmental Stator Core: B Winding Parameters Number of Poles Number of Phases Pole Pitch Pitch Factor Coil Pitch Coil Span Distribution Factor Number of Coils Turns per Coil Turns in Series/Phase Mean Length of Turn Equivalent Wire Gauge Winding Connection Connection of Stators Together 5 3 17.1 cm .966 15.2 cm 1:9 .9598 45 2 30 (5 poles in series) .670 m #1 Wye series Mechanical Parameters Primary Width 6.19 cm Primary Thickness 11.4 cm Tooth Width 9.52 mm Slot Width 9.52 mm Slot Depth 6.35 cm Total Primary Length 1.03 m Active Primary Length 1.01 m Number of Stator Slots (single layer) 45 Stator Slots (total) 53 Stator Slots (active) 53 Stator Slots (half-filled) 16 Secondary Thickness 3.0 mm Primary-Secondary Gap per side 1.4 mm Secondary Overhang per Side 2.54 cm Air-Gap (total) 2.8 mm Electrical Parameters Secondary Resistivity 26 ull'cm Frequency 360 Hz Voltage 440 V Current Rating (continuous) 600 A Table 4-3 58 MOTOR PARAMETERS Experimental Motor Number: 4 Type: Segmental Stator Core: B Winding Parameters Number of Poles Number of Phases Pole Pitch Pitch Factor Coil Pitch Coil Span Distribution Factor Number of Coils Turns per Coil Turns in Series/Phase Mean Length of Turn Equivalent Wire Gauge Wmding Connection Connection of Stators Together 4 3 17.1 cm .966 15.2 cm 1:9 .9598 45 2 24 (4 poles in series) .670 m #4 Wye parallel Mechanical Parameters Primary Width 6.19 cm Primary Thickness 11.4 cm Tooth Width 9.52 mm Slot Width 9.52 mm Slot Depth 6.35 cm Total Primary Length 1.03m Active Primary Length 0.836 m Number of Stator Slots (single layer) 36 Stator Slots (total) 53 Stator Slots (active) 44 Stator Slots (half-filled) 16 Secondary Thickness 3.0 mm Primary-Secondary Gap per side 1.4 mm Secondary Overhang per Side 2.54 cm Air-Gap (total) 2.8 mm Electrical Parameters Secondary Resistivity 26 ufl'cm Frequency 360 Hz Voltage 440 V Current Rating (continuous) 600 A Table 4-4 MOTOR PARAMETERS Experimental Motor Number: 5 Type: Annular Stator Core: C Winding Parameters Number of Poles Number of Phases Pole Pitch Pitch Factor Coil Pitch Coil Span Distribution Factor Number of Coils Turns per Coil Turns in Series/Phase Mean Length of Turn Equivalent Wire Gauge Winding Connection Connection of Stators Together Mechanical Parameters Primary Width 3.40 cm Primary Thickness 3.81 cm Tooth Width 7.3 mm Slot Width 7.3 mm Slot Depth 2.22 cm Total Primary Length .878 m Active Primary Length .878 m Number of Stator Slots (single layer) 60 Stator Slots (total) 60 Stator Slots (active) 60 Stator Slots (half-filled) 0 Secondary Thickness 3.8 mm Primary-Secondary Gap per side 1.4 mm Secondary Overhang per Side 2.54 cm Magnetic Gap (total) 2.8 mm Electrical Parameters Secondary Resistivity 26 |xQ'cm Frequency 400 Hz Voltage 440 V Current Rating (continuous) 40 A 10 3 8.76 cm .966 7.30 cm 1:6 .966 60 35 140 (5 pairs in parallel) .254 m 18 Wye parallel Table 4-5 MOTOR PARAMETERS Experimental Motor Number: 6 Type: Segmental Stator Core: C Winding Parameters Number of Poles Number of Phases Pole Pitch Pitch Factor Coil Pitch Coil Span Distribution Factor Number of Coils Turns per Coil Turns in Series/Phase Mean Length of Turn Equivalent Wire Gauge Winding Connection Connection of Stators Together 6 3 8.76 cm .966 7.30 cm 1:6 .966 60 35 140 (3 pairs in parallel) .254 m 18 Wye parallel Mechanical Parameters Primary Width 3.40 cm Primary Thickness 3.81 cm Tooth Width 7.3 mm Slot Width 7.3 mm Slot Depth 2.22 cm Total Primary Length .878 m Active Primary Length .527 m Number of Stator Slots (single layer) 36 Stator Slots (total) 60 Stator Slots (active) 46 Stator Slots (half-filled) 10 Secondary Thickness 3.8 mm Primary-Secondary Gap per side 1.4 mm Secondary Overhang per Side 2.54 cm Magnetic Gap (total) 2.8 mm Electrical Parameters Secondary Resistivity 26 nfl'cm Frequency 400 Hz Voltage 440 V Current Rating (continuous) 40 A Table 4-6 61 4.3 Description of Experimental Apparatus In addition to the linear induction motors various other equipment was constructed or obtained for use in the experiments. This equipment is described below. 4.3.1 Power Supplies Three different power supplies were used in the course of the experimental work. The 25 kVA supply was designed specifically for the experiments with Experimental Motors #1,#5 and #6. This power supply had programmable Volts per Hertz and digital readout of frequency to within one Hertz. Two larger power commercial units were used to run the larger motors #2, #3 and #5. The two commercial units did not have as advanced control circuitry however they did perform adequately. 1. Custom Made 8085 based, transistorized, 25 kVA 2. Yaskawa, Model # VS - 616 H 45B, 45 kVA 3. Yaskawa, Model # VS - 616 H 160B, 160 kVA 4.3.2 Speed Measurement The speed measurement system was based on a Commodore Computer 64. The speed of the disc was measured by counting the number of revolutions of the disc over an interval of time. This number of was found by putting a hole in the rotating disc and directing the output from an infrared LED so that it could be measured whenever the hole passed by a receiving unit. The time 62 was measured and averaged over a number of revolutions so that a highly accurate and stable reading of the disc RPM could be made. The general set up of this equipment is shown in Fig. 4.4. 4.3.3 Thrust Measurement The thrust was measured with an Omega DP-240 force transducer. The braking action was produced by an air activated friction disc brake unit. The disc pads were water cooled. This system proved to be very successful and allowed for very smooth operation and stable readings. The thrust measurement system is shown in Fig. 4.5. 4.3.4 Current Measurement The instantaneous current was measured using a Tektronix Current Probe Model A6302 (20 Amp.) or A6303 (100 Amp.) connected to an AM 503 current probe amplifier. The average current was measured using a conventional current transformer placed around the motor leads and the reading was made from a panel meter. 4.3.5 Power Measurement The power to the inverter was measured with a Paladin #256 -TWMU three-phase 60 Hz power measurement system. The power to the motor was measured with a Load Controls Inc. model PH-3A three-phase variable frequency power measurement system (response time 15 milliseconds, frequency to 1000 Hz). The two power measurement systems were used so that the efficiency of the Fig. 4.5 - Friction Brake Load Set Up 64 inverter could be measured and so that accurate power input measurements could be made when the motor was running at low power-factor. The accuracy of the motor power measurement is poor during low power-factor operation because the power is low and is very sensitive to the phase angle between the voltage and current. This is not a problem at higher power-factor operation where the accuracy will be within 5%. 4.3.6 Flux Measurement The flux was measured by flux sensing coils mounted on the stators. The voltage from the sensors was integrated to produce the actual flux value using a simply designed integrator and phase measurement system. The flux sensing coils specifications are: Experimental Motor #1: 100 turns of #38 AWG (see Fig. 4.6) Experimental Motor #2-#4: 20 turns of #28 AWG (see Fig. 4.7) Experimental Motor #5-#6: no sensors Fig. 4.7 - Flux Sensors Mounted on Experimental Motor #2-#4 66 4.4 Experimental Accuracy The measurement accuracy of the experiments is dependent on the accuracy of the instruments and the ability of the observer to read the values correctly. For the case of the frequency measurement of the inverters and the measurement of speed of the disc the accuracy will be greater than one percent as these values are measured referenced to crystal oscillators. The voltage and the current are measured with an accuracy of five percent from the output of the inverter (rms value measured). The motor input power is measured to an accuracy of five percent. The friction brake assembly, when properly calibrated and steady values are ob-served, will read to within five percent. All of the above are typical value for electrical machine experimental measurements. The difficulty in obtaining an accurate value of motor output power cannot be overstated. In order to obtain an accurate value for output power all friction and windage losses must be added to the measured value of output power. In normal rotary induction motors friction and windage losses are a very small fraction of the total output power, but as was shown in Chapter 3, the power lost due to friction in the induction motors under investigation may be very high. The difficulty in' obtaining an accurate measurement for the friction can only be appreciated when one considers the con-straints on the measuring device. The friction measuring 67 mechanism must fit between the face of the stator and the rotor which is less than 1 mm in height. The normal force that must be supported is approximately 2000 N and the mechanism cannot be made of magnetic or electrically conductive material. During the course of the investigation no such mechanism could be produced which worked properly. The most elaborate mechanism consisted of a ladder network which supported two rails alongside each side of the stator on which the rotor would slide. The steps of the ladder were placed in the slots of the stator and were designed to flex in the longitudinal direc-tion of the stator but not in the perpendicular direction. The amount of deflection for the whole ladder was to be measured with a position transducer. Unfortunately the amount of vibration in the motor during operation was far greater than the amount of deflection to be measured and no accurate results were obtained. 4.5 Conclusions Various experimental motors and apparatus were constructed in order to investigate the annular disc LIM motors operating characteristics. Experimental Motor #1 was used to test the effect of different rotor materials and to measure the flux variation in the radial direction. Experimental Motors #2-#4 were used to determine the effect of odd and even number of poles and the effect of series 68 and parallel connected poles and to measure full size motor performance. Experimental Motors #5 and #6 were used to deter-mine the magnitude of the end effect. The experimental apparatus which was used to load the machines and the instruments used to make the measurements was described in Section 4.3. The experimental accuracy, discussed in Section 4.4, of all measurements except electromagnetic thrust were accurately obtained. The apparatus described in this chapter was used in the experi-ments described in the following chapter. 5. RESULTS AND DISCUSSION 69 5.1 Introduction Experiments were conducted in which the motor parameters were varied in order to measure the effect on motor performance. In addition, those factors which reduce performance and may have been exacerbated by the unusual design of the thin steel rotor LIM were measured and analyzed. In order to clearly show the effect of different parameter variations, simulation results are also presented. In some cases the simulation results are used to highlight the effect of the parameter changes because in a simulation it is possible to make comparisons which cannot be made in the physical world. The results of these experiments and simulations, which were conducted on six experimental motors, are presented in this chapter. In Section 5.2 various experimental results are compared to the one dimensional model to verify that the model will accurately predict the power produced by the experimental motors. The end effect and the effect it has on the actual overall performance of a steel rotor machine is described in the experi-ments of Section 5.3. 70 Various rotor materials were considered and tested. The results of these experiments and simulations are presented in Section 5.4. Although none of the other rotor materials tested would produce a practical alternative to steel at the present time, the results of the experiments are very useful in that they show potential areas of improvements in the design of this type of machine. The choice of whether to use an odd or even number of poles does not occur in standard rotary induction motors which must all have an even number of poles. However, in the LIM there is no such requirement since the entry pole does not have to be of the opposite polarity of the exit pole. It is thus possible, and sometimes done, to have an odd number of poles. The result of an experiment to compare the effect of odd or even number of poles is presented in Section 5.5. An option in the design of any LIM is whether to connect the poles of the stator in series or parallel. The advantage of connecting the winding in series is that the current which flows is the same in all the coils whereas in parallel connected windings some of the coils carry more current than others. This is important if the current capability of the wire is near the limit that it can carry without overheating. However, more power may be produced if the windings are connected in parallel. An experiment was conducted to determine the actual effect of the 71 two alternative connection methods and these results are pre-sented in Section 5.6. The edge effect will cause the square pattern of rotor current shown in Fig. 1.1 to close in to form ovals. The result of the oval pattern (as compared to the more optimum square) is higher losses and lower output power. Three different experiments were conducted to determine the extent of this problem for a steel rotor LIM. These experimental results are described in Section 5.7. Space and time harmonics cause higher losses and reduced output. The effect of slot harmonics (the space harmonics) is expected to be more pronounced in the type of motor under investigation than large air gap single sided LIMs due to the small effective air gap and the non-laminated rotor. Space harmonics also exist due to the winding distribution but these harmonics will not be any greater for this type of machine than for other LIMs. Slot harmonic data was obtained from air gap flux measurements. An experiment was devised to compare the performance of a solid steel blade with and without the effect of space harmonics, in order to determine the magnitude of their effect. The time harmonics were also measured and calculations undertaken to determine the magnitude of their effect. This work is presented in Section 5.8. 72 It was shown analytically in Chapter 2 that the effect of the annular stator would be very small on the performance of the annular LIMs under investigation (i.e. the increased rotor resistivity and the variation in power produced over the face of the motor). To obtain further confirmation of this conclusion, flux plotting was conducted and these results are presented in Section 5.9. Finally, in Section 5.10 a summary of the experiments is present-ed and the important results are highlighted. 5.2 Verification of the Computer Model The model used to simulate the performance of the experimental machines is a one-dimensional current source model. This model has been used by previous investigators [29]. The advantages of using the one-dimensional model are: 1) intermediate calculated values are more meaningful, 2) the equations can be solved on personal computers and 3) the one-dimensional model can be more accurate than very complex models when used with correction factors obtained from previous experimental results [29]. As there are numerous experimental results from which to obtain the correction factors the one-dimensional model is the best model for use in this investigation. The five simulations (Figs. 5.1 - 5.5) presented in this section contain the correction factors for the air-gap due to the stator 73 teeth (the Carter coefficient [48]) and the effect on the rotor resistivity due to the finite width of the rotor [50,51]. The value for friction, which is included in these simulations, is obtained from coast down tests and friction measurements for the actual machines. 4 — 3.5 -Fig. 5.1 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 3 mm Copper Rotor 74 All of the simulation curves show reasonable accuracy in predicting the performance of the motors. Some discrepencies do occur and these are caused by changing friction in the machine. This will occur if the rotor goes through a mechanical resonance, the cooling water flow rate changes, or mechanical clearances change during the experiment due to thermal effects. 2 — 1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -0 0.2 0.4 0.6 0.8 1 synchronous speed • Power Fig. 5.2 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 3 mm Stainless Steel Rotor 75 In Fig. 5.3 the simulation and the experimental result show the greatest discrepancy. The rotor of this motor was poorly balanced which caused severe vibration during the test. This would result in the friction factor used in the simulation not being as accurate as for the other simulations. The large amount of vibration did not permit high speed operation of the motor and this is why the experimental points do not include the no load operating point. 0.5 0.7 0.9 synchronous speed • Power Fig. 5.3 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 2 mm Stainless Steel/Copper Rotor These results presented in this section will be referred to again in Section 5.4 which contains comparisons on the performance of different rotor materials. Fig. 5.4 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 2 mm Steel Rotor Fig. 5.5 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #3 With 3mm Steel Rotor 78 5.3 Effect of Segmented Rotor In this section various measurements and the results of experi-ments will be presented to show experimentally the effect of the end-effect on the performance of the thin steel rotor annular LIM. The analysis for the end-effect was presented in Chapter 2. The first experiment consisted of measuring the performance of an experimental motor (see Fig. 5.6) designed so that it could be simply converted from a continuous annular motor (thereby having no end-effects, Experimental Motor #5) to a segmental annular motor by disconnecting poles of the annular stator core (Experi-mental Motor #6). The two rotor materials chosen for this experiment were steel and aluminum. The steel rotor is expected to exhibit very little degradation in performance due to the end effect in this small prototype motor. The speed of the motor is slow so the rotor currents will have time to build up to the correct level. The aluminum rotor is five times more conductive than the steel rotor and should show the degradation in performance due to the end effect. Fig. 5.7 shows the experimental results for Experimental Motor #5 and Experimental Motor #6 with the steel rotor. In this case neither the efficiency or the power factor of the motor are degraded by the end effect. Fig. 5.6 - Experimental Motor #6 Derived By Disconnecting Two Fifths of Exrjerimental Motor #5 0.5 - i 0.5 0.7 0.9 X synchronous speed Fig. 5.7 - Experimental Results of Motor #5 ( and Motor #6 ( ), Steel Rotor ) 80 Fig. 5.8 shows the simulated power curves for Experimental Motor #5 (three-fifths of the actual power to make the result comparable to Experimental Motor #6) and for Experimental Motor #6. Plotted on the • figure are the actual experimental points (three-fifths of the actual power for Experimental Motor #5 to make the result comparable to Experimental Motor #6). The simulation curves for the two motors are so similar that it appears as one line. These simulations include the effect of friction. 0.5 0.7 0.9 % synchronous speed • Power Fig. 5.8 - Simulation Results and Experimental Points For Experimental Motor #5 With ( ) and Without ( ) End-Effects, Steel Rotor 81 Fig. 5.9 shows the experimental results for the aluminum rotor. It can be observed that the end effect is causing a significant reduction in efficiency for this motor. The peak efficiency drops from 0.48 to 0.27. The power factor, however, is almost identical for both motors at 0.31. This results in zeta (power factor times efficiency) being significantly worse for the aluminum rotor with end effects than without. Fig. 5.10 shows the simulation curves and the experimental points for the power produced by these two rotors. The aluminum rotor with end effects shows the significant loss of power that the simulation predicts. 82 0.5 0.4 -P o w e r F a c t o r 2eta efficiency 0.3 -0.2 -0.1 -% s y n c h r o n o u s speed Fig. 5.9 - Experimental Results of Motor #5 ( ) and Motor #6 ( ), Aluminum Rotor From these results it is clear that for the smaller annular motors (Experimental Motor #5) the end effect does not reduce the power or efficiency for the thin steel rotor machine signifi-cantly. However, the more conductive aluminum rotor does have a significant loss in efficiency due to the end effect and would not be a good rotor for a motor with the parameters of Experimental Motor #5. per unit of synch ronous speed • Power Fig. 5.10 - Simulation Results For Experimental Motor #5 With (- ) and Without (—-) End Effects, Aluminum Rotor The second experiment consisted of measuring the flux and phase of the steel rotor Experimental Motor #4 in order to determine the magnitude of the end-effect in the full size steel rotor ma-chine. This motor is the full size prototype for the sawing application. The flux was measured with flux sensing coils wrapped around each tooth of the stator as shown in Fig. 5.11. The measured values for flux for three different conditions (no rotor, 15% slip and 20% slip) are shown in Fig. 5.12. It can be Fig. 5.11 - Flux Sensors Shown Wound On Experimental Motor #3 seen that the flux near the entry area is reduced for the higher speed rotor but as a greater load is applied, and the rotor slows down, the flux moves more towards the no rotor shape. From the theory presented in Chapter 2 and from measuring the amount of reduction in flux at the entry of the motor, the ratio b-j/bp can be estimated (bp is the no load flux and b^  is the maximum value of the end-effect flux which reduces the air gap flux during operation of the motor). From Fig. 5.12 it can be estimated that bj is approximately (measured at the 12th stator tooth) .35 times bp at 1956 rpm and approximately .32 bp at 1850 rpm when the machine is under greater load. This significant reduction shows the effect of the end effect and why the performance of the LIM is reduced at high speed. Fig. 5.12 - Magnetic Flux Density Measured Along Experimental Motor #3 86 5.4 Effect of Rotor Variations Various rotor materials were tested on Experimental Motor #1 to find the optimum rotor material in terms of maximum efficiency and power-factor and also to produce the maximum power for the size of stator (power density). It is especially important that the power-factor efficiency product is maximized because the motor must run from an inverter, which is the most costly single component of the sawing system. These experiments could not be conducted on the full size machine (Experimental Motors #2-#5) because only steel was rigid enough to run smoothly through the stators at high speed. The experimental equipment consisted of a custom made 25 kVA, three-phase, voltage source inverter which could be accurately set to produce frequencies from 25 to 600 Hz. All of the experiments were conducted with Experimental Motor #1. The speed of the rotor was measured with a hand held digital tachometer and the torque was measured with a commercial strain gauge force transducer. A friction brake dynamometer was used as the load which means that only those operating points where the torque decreases with increasing speed could be measured. This is not an important disadvantage of the friction brake system since this the only operating region for most motors and is the operating region for the sawing application. The test results for four different rotor materials are presented in this section - they 87 are: copper, stainless steel, a stainless steel/copper layered blade, and steel. The experimental results for the efficiency, power-factor, power-factor*efficiency product (zeta) and power output are shown in Figures 5.13 to 5.16. The curves drawn in on these figures are to show the trend of the data points and are not simulated values. The Figure 5.16 is Figures 5.1 to 5.4 combined onto one graph but without the simulation results. The first rotor test result is for a 3 mm thick copper blade. This is a highly conductive rotor which results in higher losses due to the end-effect. In a normal rotary machine such a highly conductive rotor would reach maximum power very close to synchronous speed. In this case, however, maximum power is reached at 20% slip. Both the efficiency and power-factor are 50% at the maximum power point, resulting in a very high pow-er-factor efficiency product (zeta) of 0.25. The maximum power measured was the second best produced of the four blades tested. The next rotor material is a non-magnetic stainless steel blade (SS Type 3041). The stainless steel is a very resistive material (p= 68 uQ-cm) so it would be expected to have very little end-effect but would also require a very high slip frequency before much power is produced. This can be seen in the graph. The peak power was reached at 40% slip and this was also at a 88 high stator frequency of 600 Hz (to obtain good performance in the desired operating speed). The power-factor is not too unreasonable at 44% but the efficiency is very low, 24%, due to the high stator core losses which results from the high stator frequency. The output power was the lowest of the tested rotors at 1.07 kW. The third rotor material is a stainless steel/copper sandwich rotor consisted of two 0.5 mm sheets of stainless steel covering each side of a 1 mm thick copper sheet. The stainless steel sections of the rotor would have very little effect on the performance because the stainless steel is both non-magnetic and approximately fifty times more resistive than the copper. Due to a mechanical imbalance in the rotor it was not possible to run the blade faster than 0.8 of synchronous speed so the no-load speed was not measured. The sandwich blade proved to be the optimum of the four tested. This rotor produced the maximum power (3.1 kW) and also had the highest power-factor. The efficiency of 48%, although not the highest compared to the 3mm copper blade, when combined with the high power-factor resulted in the highest power-factor efficiency product of all the rotors tested. The last material tested was 2 mm, hardened steel with a resis-tivity of 26 ul2-cm. Since the rotor is made of a magnetic material the effective air-gap is approximately half of what it 89 would be with the other rotors. This results in half the magnetizing current and is expected to result in a higher power-factor which is what can be seen in the results. The high resistivity of the steel, approximately fifteen times that of copper, results in a blade that does not reach maximum power until 60% slip and thus has a low efficiency of 28%. The maximum power-factor efficiency product is only 0.16 for this rotor material. Efficiency Fig. 5.13 - Experimental Points and Approximating Curves Showing Efficiency 90 Fig. 5.14 - Experimental Points and Approximating Curves Showing Power-Factor Fig. 5.15 - Experimental Points and Approximating Curves Showing Power-Factor Efficiency Product (Zeta) 92 Power 4.00 - i : : — — — 3.50 -j 0.2 0.4 0.6 0.8 1 per unit of synchronous speed cu/stoinless steel + 3 mm cu o - 2 mm steel A 3 mm. stoin. steel Fig. 5.16 - Experimental Points and Approximating Curves Showing Power 93 From the test results and simulations presented in this section it can be concluded that a rotor material with the resistance of a 1 mm copper sheet would produce close to the optimum motor for a stator with the parameters of Experimental Motor #1. These experimental results may be extended to Experimental Motors #2-#4 if the proper scaling factors are used [49] to determine the optimum rotor material. This scaling to a larger size motor is presented in Chapter 6. 5.5 Effect of Odd or Even Number of Poles Unlike its rotating induction motor counterpart, the linear induction motor can have either an odd or even number of poles. This is due to the way in which the LIM is constructed. Fig. 5.17 shows how a LIM can be imagined as a rotary induction ; N S N S N Fig. 5.17 - Conceptualized LIM motor which has been split and unrolled. Once the LIM has been formed it is possible to add on an extra pole if it is desirable to increase the length of the motor. This would not be possible in 94 the rotary induction motor because there is no place to add in a pole which would not be adjacent to a pole of the same polar-ity. It would have also been possible to increase the length of the LIM by increasing the pole pitch. In theory, the odd number of poles may result in lower power-factor since when the magnetic circuit is analyzed, it is found that there is no path for the extra flux to return along, except around the sides and edge of the stator, as show in Fig. 5.18. In a real LIM, however, there are other paths available as described by Laithwaite [7] and as will be shown in the results of this section. Fig. 5.18 - Analysis of the Flux For a Three Pole Stator 95 When the flux is integrated across the face of the stator in equation 5.1 there is a net flow which must return through air and this is what should result in a lower power-factor. In the analysis presented by Yoshido et al. [17] it is stated that this extending flux will result in lower efficiency, although quant-itative comparisons are not made. In this section theoretical values for the expected increase in power and experimental values for the increase in power and the effect on efficiency and pow-er-factor are obtained. In addition, an experiment was conducted to observe the effect of an odd number of poles on the pow-er-factor and efficiency of the thin steel rotor, annular LIM. Designing an experiment which will accurately compare the effect of odd or even number of poles will always result in compromise. If the pole-pitch is increased to eliminate a pole, then the magnetizing current will decrease and the Goodness will increase (which will increase the power factor). On the other hand, the relative end effect force will increase which reduces the efficiency. If the other alternative - disconnecting a pole - is chosen, then the parasitic thrust of the end-effect will remain the same but the motor thrust is reduced because of the fewer poles x (5.1) 96 In this experiment, the second option was chosen. One pole was disconnected from Experimental Motor #3 (5 poles to four poles), which then became Experimental Motor #4 (this was the simplest method of obtaining the even number of poles), and then the ex-pected changes in efficiency and power-factor were taken into account. Another disadvantage of this method of obtaining the even number of poles is that since all the poles are connected in series, the voltage applied per pole of the four pole motor is greater than for the five pole motor. To compensate for this factor the voltage gain of the inverter supply was turned up by 25% for the Experimental Motor #3 tests. The experimental results for the odd and even comparison are shown in Fig. 5.19. From the results it can be seen that prior to increasing the voltage the four pole motor has a slightly greater power-factor but a lower efficiency. The resulting power-factor efficiency product is almost identical for the two motors over much of the operating range. When the output voltage of the inverter is now increased for the five pole motor (to produce the same voltage per pole as the four pole motor), then the efficiency and power-factor of both motors is about the same over the normal operating range. From these experimental results it can be concluded that whether a thin steel rotor, high speed LIM has an odd or even number of 97 0.7 -0.6 -0 .5 -0.4 -0.3 -0.2 -0.1 -O O A -0.7 -0.6 -0.5 0.4 -0.3 -0.2 -0.1 -0 .5 Voltage Increased By 1.25 For Odd — i — 0.7 0.9 Veloci ty Iper unit of s y n c h r o n o u s ] O Power Factor Pover + Efficiency o  oote  A Zeto Fig. 5.19-Experimental Results For A Comparison Between Four (——) And Five (— ) Number of Poles poles does not effect the overall performance of this type of motor operating under the conditions described in the table of Motor Parameters. 98 5.6 Effect of Series and Parallel Connection Very little has been written on the subject of connecting poles of a LIM in series or parallel. Laithwaite [11] discusses the flux pattern of parallel connected LIMs but does not discuss the effect of the parallel connection on the performance of the LIM. Yamamura [56] describes parallel connected compensation windings in his book but Dukowicz [57] shows in his paper that these type of windings do not improve the overall performance of a LIM. The reason that the series or parallel connection question is seldom considered is that most LIM stators have windings which are current limited and the entry coil would not be able to with-stand the greater current resulting from a parallel connection. The entry coil carries more current because in the parallel con-nection the full line voltage is applied across the coil and so the flux will be constant. At the same time the rotor material entering the stator has no flux associated with it and the stea-dy-state result is higher current in the entry coils and higher flux density at the entry end of the stator than for the series connection. For double sided LIMs with small air gaps, the sta-tor windings can be designed to carry the extra current. If power density of the motor is important then the parallel con-nection may be advantageous. In order to investigate the effect of the parallel connection on 99 the performance of the LIM, the foUowing tests were carried on a parallel machine: a) simulations of the power produced by the entry pole for both the parallel and series connected motor were made, b) the flux at the entry area of the motor was measured and c) an experiment was conducted to compare the power, power-factor and efficiency for the two different connections. First an experiment was conducted to measure the current of the entry coils of a parallel connected motor (Experimental Motor #2) to determine the increase in current density during different operating conditions. These values are given in Table 5.1. % of synchronous speed Motor Current (A) Coil Current (A) 90 285 120 84 300 124 . 80 320 106 75 345 106 Table 5.1 - Entry Coil Current The above values can then be used to simulate the power produced by the entry pole of the stator for the parallel connected ma-chine. The simulation for the entry pole of the series connected machine is obtained from the standard series connected simulation technique described in Chapter 2. The results of these simula-tions (Fig. 5.20) show that the power produced by the first pole of the stator is greater for the parallel connected stator at all 100 operating points and is 15 kW (150 %) greater at 75 % of synchronous speed. The next step of the experiment was to measure the flux at the entry end of both the parallel and the series connected stator. The results of these measurements are shown in Fig. 5.21. One can observe in the figure that the flux of the parallel connected SO - . •—; • 0.5 0.7 0.9 Veloci ty Iper unif of s y n c h r o n o u s ] • Force (M) + Power (kW) Fig. 5.20 - Power Produced By the Entry Pole of a Series ( ) and Parallel ( ) Connected Stator machine is not reduced as much as for the series connected 101 machine. This confirms that the parallel connected machine should produce more power for the same length of machine. Finally, direct measurements were made for the power produced, the efficiency and the power-factor for two motors which were wound on the same stator core, but differed in that one was FLUX DENSITY AT ENTRY OF THE MOTOR rollel 0.8 0.7 O.G -0.4 -0.3 0.2 0.) Cornportson Between Parallel and !>erles Tooth Number + EM « 3 , Sertes o EM « 3 , No Rotor Fig. 5.21 - Measured Flux at the Entry End of a Parallel Connected (E.M. #2) and a Series Connected Motor (E.M. #3) parallel connected (Experimental Motor #2) and one was series 102 connected (Experimental Motor #3). These experimental results, in Fig. 5.22, amfirm the simulation results shown in Fig. 5.20. The measured increase in power was 17.5 kW ( 70 % increase). The power-factor remains the same for both but the parallel connected machine has greater efficiency and thus has a greater efficiency power-factor product. Experimental Results of Test - . 1231 1 -I " 0.9 -0.5 0.7 0.9 Velocity Iper unit of synchronous] F'ower + Efficiency •> Power Factor Fig. 5.22 - Experimental Results for a Comparison of the Performance for a Parallel Connected ( ) and a Series Connected ( ) Motor These results are of importance for the design of the double-sid-103 ed, thin steel rotor LIM because one of the greatest difficulties of using a thin steel rotor is obtaining good power densities. The parallel connected LIM has been shown to have a significantly greater power density with no decrease in the power-factor effic-iency product. This increase in power for the parallel connected LIM also demonstrates the deterioration due to the end effect. 5.7 Edge Effect The current which flows in the rotor of a sheet rotor motor does not flow in a perfect rectangular pattern as in the rotor bars of a squirrel cage induction motor. (The sheet rotor motor in Fig. 1.1 shows a rectangular pattern only because it is the idealized case.) Rather, the pattern is more like that shown in Fig. 5.23. The curved current paths are caused by the edges of the rotor not being of zero resistance. These curved paths result in poorer than predicted performance of sheet rotor motors and this effect is called the edge effect. Fig. 5.23 - Pattern of Rotor Currents 104 The curved lines of current cause a disruption of the flux pat-tern, with higher flux density at the edge of the stator and lower flux density near the center as shown in Fig. 5.24 [50]. Fig. 5.24 - Diagram of Flux Density This effect was studied by Hugh Bolton [50] among others [51]. Although Bolton does not analyze the specific case of the solid iron rotor, he states at the conclusion of his paper: "Finally, pronounced transverse flux redistribution is known to take place in rotary induction motors with solid-iron rotors". As the motor types under investigation use solid-iron rotors this effect was 105 analyzed and experiments conducted in order to determine its magnitude. Bolton's paper describes how the various motor parameters can be used to decide whether or not severe edge effects can be expect-ed. The analysis of the edge effect and how it is affected by the motor parameters will now be presented. The analysis of the edge^ effect requires the two-dimensional analysis of the stator and rotor in the x-y plane. The definit-ions of the variables can be seen in Fig. 5.25 [50]. Fig. 5.25 - Reference Frame For the Edge-Effect Analysis 106 The air gap equation for the rotor and the stator is given by equation 5.2. (d'b/d'x refers to partial derivatives) d-b2 +d'_b_2 _ M ^ b _ = - M ^ _ ( 5 2 ) d'x2 d'y 2 fig d't g d'x Assuming sinusoidal functions, equation (5.2) can be written as: d B , 2 j s w Lt T J j a k J /c o\ - k + lo B = J o^ (5.3) dy2 fig g The solution of this equation is given by: B = 'to J Z 2 1 + ( 1 - Z } F C o s h ^ (5.4) g k Z 2 Cosh (3a where: 1 + Tanh pa Tanh k(c - a) V 1 + i s G p = k2 + i b w uQ Q g = k2 ( 1 + i s G) 7 - 1 z - 1 + i s G To find the ratio, U, of the flux density at the edge to the mean flux density, first calculate the mean flux density: B = 1 m e a n Ta J-a f a B dy / - a 107 n Side. 1 + i s G e (5-5) B ~. l + iaia_£Tanhpa mean a " From the above equations Bolton was able to develop a family of curves based on the stator width, rotor width, the pole pitch and the Goodness Factor for a sheet rotor induction motor. These curves are shown in Fig. 5.26 and give a good insight into the parameters of the motor which can be varied to reduce the edge-effect. The parameters for Experimental Motor #2 are plot-ted on the curves and it can be seen that the edge effect is expected to have minimal effect on the performance of the motor. 3.--Fig. 5.26 - Flux Density Variation Factor [50] 108 For the test motors the parameters were checked against these curves, which predicted that severe edge effects were not expect-ed. However, because the analysis is based on a non-magnetic rotor the effect of the steel rotor on the edge effect is included in the above analysis only partially (by a reduction in the effective air gap), so that the actual effect of the steel rotor is not certain. For this reason experiments were conducted to determine the magnitude of the edge effect, first by measuring the variation in flux density, and second, by decreasing the effect of the edge effect on the performance of the thin steel rotor annular LIM. In the first experiment, flux search coils were laid across the stator and the flux measured for Experimen-tal Motor #1. In the second, slots were cut into the rotor of Experimental Motor #4 to eliminate the possibility of the longi-tudinal currents flowing over the stator area and causing the edge effect to reduce the performance of sheet rotor motors. In the first experiment, flux sensing coils were laid across the face of Experimental Motor #1 to measure the variation in flux density due to the edge effect. The positioning of these sensors is shown in Fig. 5.27. It was not possible to measure the lateral variation in the flux of the full size prototype motor because the flux sensors are approximately 1.5 mm thick, which could not be made to fit in the air gap of Experimental Motor #4, so Experi-mental Motor #1 was used with a larger air gap (to accommodate 109 the flux sensors) but the rotor was coated with a copper layer so that the Goodness of the LIM would be the same as that for Experimental Motor #3. The Goodness partially determines the extent of the edge effect. Fig. 5.27 - Flux Sensing Coils On Experimental Motor #1 Figure 5.23 shows that the flux profile does not reduce signifi-cantly in the center (less than five to one difference; see Bolton [50]), indicating that the edge effect is not significant. The reason for the sloping lateral flux density curve is due to the 110 sloping flux density curve is discussed in Section 5.9) The second experiment consisted of cutting slots into the steel rotor of Experimental Motor #4. This was done to eliminate the possibility of longitudinally flowing currents over the stator area reducing the performance of the LIM (which is the edge effect). Approximately three slots per pole (3.14 slots per pole length) were cut with a laser through the steel plate rotor. A laser was used so that the slots could be cut thin enough that Stator ' ' Fig. 5.28 - Measured and Expected Value of Flux Across Stator Face For experimental Motor #1 the amount of rotor material removed would have a negligible b Y I l l effect on rotor resistivity. These slots are shown in Fig. 5.29. Experimental Motor #4 was tested with the rotor described above and the results of this experiment are given in Table 5.2. It can be seen from the table that the performance of the machine improved slightly after the slots were cut into the rotor. If the edge effect had been significant than the power input would have dropped significantly for the case where the rotor had slots cut into it. Fig. 5.29 - Slots Cut Into the Rotor of Experimental Motor #4 112 Power Input Speed Without Slots 82 kW 2024 With Slots 80 kW 2001 Table 5.2 - Measurements For Experimental Motor #4 With and Without Slots at 380 Hz Supply The results of these two experiments confirmed the analysis that the edge effect did not have a significant effect on the perfor-mance of the type of motors being analyzed. 5.8 Effect of Harmonics Space and time harmonics in the air gap flux can have detrimental effects on the performance of induction motors. Space harmonics occur in motors due to the slotted nature of the stators. Time harmonics are produced by the switching of the transistors which supply the variable frequency voltage source. These harmonic effects can be expected to be greater than for a convention in-duction motor due to the unlaminated rotor of the LIM. 5.8.1 Space Harmonics If the magnitudes of the space harmonics is large in relation to the fundamental, then a significant decrease in motor efficiency and power may be expected [53]. The slot harmonics will appear to the rotor at a frequency of at 113 least 36 times that of the fundamental flux (see Fig. 5.29). This is because the slot harmonic is due to the stator teeth and the number of stator teeth is usually at least twelve per fundamental and most motors run at no less than 75% of synchronous speed (.75*12/[l-.75] = 36). Due to skin effect this high frequency will usually be attenuated and the drag due to the slot harmonic will be quite low. Three experiments were conducted on Experimental Motor #4 to determine whether slot harmonics would decrease the efficiency. In order to determine the magnitude of the slot harmonic, the air gap flux was measured (along the center line of the stator) with the use of a Hall Effect flux probe. The measurement was done at low flux density so that the abnormal saturation effects of using DC would not occur. Saturation should not occur during normal operation because as is shown in Fig. 5.11 (the flux profile plot) the flux density is not into the saturation region for the steel (see Appendix 4 for the steel specifications). The air gap flux measurement for two poles of the stator is shown in Fig. 5.30. 114 Fig. 5.30 - Air Gap Flux Of Experimental Motor #4 From the figure it can be estimated that the first slot harmonic is one-fifth the amplitude of the fundamental flux. For a rotor running at 75% of synchronous speed fed with a 360 Hz supply the effective frequency that the rotor would see is given by: Freq. = .75 * 360 Hz * 18 slots/ two pole pitches (5.6) = 4860 Hz Using the value for the effective stator to rotor harmonic fre-quency, the magnitude of the flux slot harmonic, and the simula-tion technique described in Chapter 2, it is possible to obtain a relation between torque and rotor resistivity as shown in Fig. 5.31. 115 The rotor resistivity will be affected by skin depth [52] which is difficult to determine due to the non-linear effect of saturation in the steel rotor [27]. For this reason a possible range of force is shown in the graph. The figure shows that if the effective rotor thickness becomes 50 um then the slot harmonics 90 AO 70 -SO 50 -AO -50 20 -lO -SLOT HARMONIC TORQUE Operation at 360 H i . .25 Slip (4860 Hi ) O JfcZ ! 7 T T , (  1.00E-O7 1.00E--06 1 .OOE-05 1.00E-O4 Rotor Fffeetlvs Thickness (m) Fig. 5.31 - Force Due to the Slot Harmonic and Effective Rotor Thickness would produce the maximum drag of 90 N which is 10 % of the 116 full load power of the motor. From the above analysis it was shown that the slot rotor harmonics could contribute to significant losses in the LIM depending on the skin depth of the induced slot harmonic currents. In order to obtain an exact value for the slot harmonic loss the experiments described in the following were undertaken. The first experiment consisted of misaligning the teeth of the stator as shown in Fig. 5.32, in order to reduce the air gap flux harmonic. Two parameters will actually change in this experiment. The effective air gap will actually increase in addition to the re-duction in space harmonics. Discussions regarding the effect of misaligning the teeth have been made with some experimenters claiming improved performance [29, p. 220]. However, in the experiment undertaken no significant improvement was measured as is shown in Table 5.3. From the geometry of the motor shown in the above diagram one can conclude that very few motors would expect to have greater space harmonic to fundamental ratio than is found in Experimental Motor #2. This is because Experimental Motor #2 has a very small air gap and is a double sided motor. 117 Fig. 5.32 - The Effect of Misaligning the Stator Teeth The one problem with this experiment and the ones conducted by other researchers is that the slot harmonics may not be reduced significantly right at the surface of the rotor which is were the induced rotor slot harmonic currents will flow due to the skin effect. To eliminate the possibility that skin effect was causing the misaligned teeth experiment to produce incorrect conclusions a second experiment was conducted. 118 Power Speed Aligned Teeth 93 kW 1850 RPM Misaligned Teeth 93 kW 1850 RPM Table 5.3 - The effect On the Performance Of Experimental Motor #2 With and Without Misaligned Teeth A second experiment was conducted which involved cutting slots into the blade so that no harmonic currents could flow (see Fig. 5.33). By skewing the slots and having their pitch the same as the tooth pitch, no current path exists directly in the shape of / the slot harmonics. The resistance of the rotor current path is changed only slightly; by less than 1% when the end paths are included. This means that the slot harmonics can be totally eliminated with almost no effect on the fundamental. When this experiment was conducted a small decrease in power draw was observed (Table 5.4) which did indicate that the slot harmonics were decreasing the efficiency by approximately 5 %. 119 Fig. 5.33 - Slots Cut In Rotor To Eliminate Rotor Slot Harmonics Power Speed Unslotted Rotor 82 2024 Slotted Rotor 78 2001 Table 5.4 - The effect On the Performance Of Experimental Motor #4 With and Without Slotted Rotor 120 5.8.2 Time Harmonics The time harmonics of the current were measured with a spectrum analyzer to determine their magnitude and these are presented in table 5.5. The third order harmonics, which cancel out in a balanced supply, are present in the inverter supply due to timing inaccuracies in the control circuits. The fifth and seventh harmonics, which are the strongest ones in this particular case, will produce a pulsating force. These amplitudes of harmonics have been shown by George John [55] not to decrease the efficiency or thrust of the LIM. Frequency Order Amplitude 350 1 1 1050 3 0.044 1750 5 0.11 2100 6 0.068 2450 7 0.177 Table 5.5 - Harmonic Components of Current Wave Form to LIM 5.9 The Effect of the Annular Motor In the analysis presented in Chapter 2 the effect of the annular stator was simulated with a constant current source of varying 121 surface current density. The flux that was generated by the surface current sheet was assumed not to travel in a radial di-rection. This resulted in a higher flux density at the inner radius of the stator. However, due to the physical construction of the stator there is actually less stator area to carry the flux than was assumed in Chapter 2. This is because the conductors which carry the stator current are of constant diameter so that the sides of the stator slots must be parallel and the result is that the stator teeth are a wedged shape as shown in Fig. 5.34. Fig. 5.34 - The Stator Tooth And Slot Profile For the Annular Stator The analysis presented in Chapter 2 predicted that more torque and power would be generated at the outer edge of the stator. 122 The measured flux plot of the stator in Fig. 5.35 however, shows a greater air gap flux density at the outer edge of the stator so in actual fact the variation in power shown would actually be greater. Expected — i — 1 1 — i — — i 1 — — i — i — J — —• i —I Stator Fig. 5.35 - Lateral Variation in Flux For Experimental Motor #1 A simulation of the variation in flux and the power produced by the motor is shown in Fig. 5.36. The figure shows the extra increase power and torque due to the increased flux which is now accounted for. 123 variation of horsepower with radius 0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05 slip + Inner o outer A average X In, Teeth V 0 « V T K « I y Fig. 5.36 - Variation in Power and Thrust With Radius Including the Effect of The Stator Slots 5.10 Discussion Various experiments were conducted to determine the effect of different parameter variations and to experimentally determine the effect of the steel rotor on the performance of a high speed annular LIM. The results of these experiments, which were cond-ucted on six experimental motors, were presented in this chapter. 124 The end effect and the effect it has on the actual overall per-formance of a steel rotor machine is described in the experiments of Section 5.3. Various rotor materials were considered and tested. The results of these experiments and simulations are presented in Section 5.4. Although none of the other rotor materials tested would produce a practical alternative to steel at the present time, the results of the experiments are very useful in that they show potential areas of improvements in the design of this type of machine and the results of these experiments can used with scal-ing factors to find a more optimum design. This is presented in Chapter 6. This section also allowed for the direct comparison between simulated and actual results. The simulation when cor-rected for friction and rotor resistivity agreed well with the actual experimental results. The choice of whether to use an odd or even number of poles does not occur in standard rotary induction motors which must all have an even number of poles. However, in the LIM there is no such requirement since the entry pole does not have to be of the opposite polarity of the exit pole. It is thus possible, and sometimes done, to have an odd number of poles. The result of an experiment to compare the effect of odd or even number of poles was presented in Section 5.5 which showed that the an odd number of poles did not reduce the performance of the LIM. 125 An option in the design of any LIM is whether to connect the poles of the stator in series or parallel. The advantage of connecting the winding in series is that the current which flows is the same in all the coils whereas in parallel connected wind-ings some of the coils carry more current than others. This is important if the current capability of the wire is near the limit that it can carry without overheating. However, more power may be produced if the windings are connected in parallel. An experiment was conducted to determine the actual effect of the two alternative connection methods and these results showed that a parallel connected LIM has greater power density with no de-crease in the power-factor/efficiency product. The edge effect will cause the square pattern of rotor current shown in Fig. 1.1 to close in to form ovals. The result of the oval pattern (as compared to the more optimum square) is higher losses and lower output power. Three different experiments were conducted to determine the extent of this problem for a steel rotor LIM. These experimental results showed that the edge ef-fect did not decrease the efficiency or power output of the ex-perimental motors. Space and time harmonics cause higher losses and reduced output. The effect of slot harmonics (the space harmonics) was expected to be more pronounced in the type of motor under investigation 126 due to the small effective air gap and the non-laminated rotor. Slot harmonic data was obtained from flux measurements. An ex-periment was devised to compare the performance of a solid steel blade with and without the effect of space harmonics, in order to determine the magnitude of their effect. The space harmonics were found to have some effect on the efficiency of the double sided steel rotor LIM but not a great amount. The time harmonics were also measured and calculations undertaken to determine the magnitude of their effect. The time harmonics were found to have insufficient amplitude to have an effect on the performance of the LIM. It was shown analytically in Chapter 2 that the effect of the annular stator would be very small on the performance of the annular LIMs under investigation (i.e. the increased rotor resis-tivity and the variation in power produced over the face of the motor). To obtain further confirmation of this conclusion, flux plotting was conducted and these values were then used to conduct a simulation based on the measured value of flux. The result of this simulation showed that the effect of the annular motor was more pronounced than the initial analysis indicated but it still did not significantly affect the performance of the LIMs under investigation. 6. FACTORS AFFECTING OPTIMIZATION 127 6.1 Introduction to Optimization Optimization of electric motors generally means the maximizing of efficiency. However, in this application the size of the inverter supply (the most expensive component of the machine) is determined by the power-factor/efficiency product. If this value can be maximized while a high power density is obtained then the motor can be said to be optimized for this application. The following sections discuss various factors which can be modified to improve power density or the power-factor/efficiency product. 6.2 Goodness Factor One criterion which is often used in the design of induction motors is that of goodness factor. The goodness factor is a measure of how well the magnetic and electrical circuits are utilized. The goodness factor is defined as: where w is the stator frequency. The goodness factor increases with the velocity squared and the thickness of the rotor. It will decrease for greater rotor resistivity, larger air gap and higher frequency. The calculated value of goodness for a machine must also include correction terms to take into account the actual paths of the current and flux [42,51]. Generally speaking the higher the •2 (6.1) 128 goodness factor the better the power density and the power-factor/efficiency product. However, due to the end effects in LIMs, the highest goodness factor achievable is often not optimal. This fact led to the "optimum goodness factor" as defined by Nasar and Boldea [9]. 6.3 Optimum Goodness Factor Nasar and Boldea defined the optimum goodness factor as that value of goodness for a given LIM such that it will have zero thrust at zero slip. In this way the performance of a LIM and of a conventional rotary motor are the same. However, in order to make this one characteristic the same it is usually required that the rotor resistance must be increased, the number of poles increased or the air gap increased; in other words to do those things which are known to make for an inferior rotary motor. The reason why an inferior motor should be produced is to reduce the negative thrust of the longitudinal end effect. It has been shown by Nasar and Boldea and others, that if a LIM is produced with very good performance in the conventional sense then the low slip performance will be quite poor. For this reason a poorer machine will be more efficient and have better power-factor. However, it will also produce less power than it could. The optimum goodness factor for an eight pole motor, about the greatest number of poles usually found in a LIM, is twenty. This 129 is much lower than for conventional motors which have goodness factors greater than 100. A LIM using the optimum goodness design rule (zero thrust at zero slip) will have some improvement in efficiency but certainly will not require less material and the power density will be less. R e l a t i v e E n d E f f e c t Velocity Iper unit of synchronous] Fig. 6.1 - Relative end effect force. 130 6.4 Analytical Results It is important to observe the thrust due to the longitudinal end effect at higher slips. In Fig. 6.1 [9] it can be observed that, if the motor operates at higher than typical slips, say ten percent, then the negative thrust due to the longitudinal end-effect is actually less for a machine with a goodness factor of forty-two than for one with the optimal factor of twenty. If the analysis included a motor with goodness factor of an even higher value, then the end effect parasitic thrust will go to zero. From the above example it is clear that the "optimum goodness-factor criterion" is not optimum for all LIMs as implied in the analysis of Nasar and Boldea. The original hypothesis, however, is still true that if the motor is to run very close to synchronous speed then the motor should be designed according to the optimum-factor goodness criterion. From a design point of view, the double sided LIM will be flux limited; the stator steel will be driven to its magnetic saturation point. The other case is the single sided LIM which will be current limited due to the smaller area to place the copper windings on the stator (only one stator). If a constant current design is used then a higher resistivity rotor, for example, will produce greater power at lower slip [8]. The constant current limitation may be important from the design point of view for the large air gap single sided machine, which is admittedly a common and important machine. For that type of machine the stator 131 resistive losses and cooling problems can be significant. However, for the constant voltage double sided machine the saturation of the stator steel, and not the stator resistive losses, is the limitation on the power density. The double sided machine with small air gap can have very good power to active surface area ratio and has low leakage inductance and stator winding losses. For the thin steel rotor machine a double sided LIM is the only practical design due to the attractive forces and the requirement of a return path for the magnetic flux. Figure 6.2 shows the speed vs power curves for an optimally designed EM#1 (by goodness factor) and for the same LIM with a lower value of rotor resistivity. The optimized machine has such a high rotor resistivity that it produces very little power at low slip (although the end effect thrust is also low) and produces a peak power of only 50 kw at 50% slip and 20 kw at 20% slip. The goodness factor is at the optimum value of ten. However when a far more conductive rotor material is used (resistivity= 27 uohm-cm; alloyed steel) then the power peaks at 100 kw and is 40 kw at 20% slip. For an actual machine the core loss is 4 kW so that the optimized machine is actually less efficient than the motor with the high goodness factor. 132 E x p e r i m e n t a l M o t o r #1 Exp. Motor #1 100 - i — 0 0.2 0.4 0.6 0.8 1 % of synchronous speed + Power (kW) Fig. 6.2 - Power for various rotor resistances. 6.5 Optimization Bv Scaling Up In Section 5.3 various rotor materials were tested to find the optimum rotor material with respect to maximum power-factor/ef-ficiency product and maximum power density. Using scaling factors [54] it is possible to scale up the results of the smaller 133 motor experiments (Experimental Motor #1) and use them to predict some of the performance characteristics of the full size machines (Experimental Motors #2, #3 and #4). The most fundamental of these factors is the Goodness Factor described in Section 6.2. The equation for the Goodness Factor is shown again in (6.2). v 2 G = V-o0 v (6.2) pg w An example for a scaling up between the optimum conditions found for Experimental Motor #1 to the full size machine, Experimental Motor #4, will now be described. The Experimental Motors' parameters can be found in Tables 4.1 and Table 4.4. The parame-ters found in (6.2) are shown in Table 6.1. The objective EM#1 E M #4 c 1 mm 3 mm V 50 m/s 133 m/s P 1.7 uQ cm 26 uQ cm g 4 mm 2.5 mm freq 400 Hz 360 Hz G 18 44 Table 6.1 - Comparison of Motor Parameters 134 will be to obtain the same Goodness factor for the full size machine as for the smaller prototype. Using the above parameters the Goodness Factor was found to be approximately 2.4 times greater for the full size machine. To reduce the Goodness Factor, the most reasonable parameter to adjust is the rotor thickness since the final objective is to produce the thinnest rotor LIM. Experimental Motor #4 would produce more power and have a higher power-factor/efficiency product if it had a thinner rotor (of approximately 1.22 mm) in order to reduce the Goodness Factor to the more optimum value found in the scale model experiments. (Unfortunately the mechanical design of the full size machine did not allow for such a thin rotor to be tested.) The actual value of the efficiency and the power-factor for the Optimum Goodness LIM can be estimated from the experimental results conducted on the smaller scale model motor. Both of these values will increase slightly for the larger machine (as is the case for all larger electrical machines) but the exact value cannot be determined using the techniques presented in this thesis. 6.6 Series and Parallel Connection Optimization As was mentioned in the introduction of this chapter, optimiza-tion for the sawing application means obtaining a high power-fac-tor/efficiency product while also maintaining a high power density. The power density is of critical importance in the 135 sawing application because a longer motor means a smaller depth of cut and a less powerful motor means that the saw would have to either cut slower or again decrease the depth of wood which it cuts. The connection of the stator poles in parallel has been shown in Chapter 5 to produce more power and thus increase the power density at the same or greater power-factor/efficiency product as the series connection. Two approaches are possible for the design of the parallel connected stator. The first uses heavier copper windings and wider slots for the entry pole of the LIM and the second uses twice as heavy wire (and twice as deep slots) as would be required for the series connected machine. The first alternative would require the use of three different slot punches and two different coil wire diameters. The second alternative is to use an extra deep core and punch in deeper slots which would than be wound with a gauge of wire throughout the core which was heavy enough to carry the current which occurs in the entry coils. These are shown in Fig. 6.3 along with the equivalent series connected stator. Of the possible designs the extra deep slots would be the less costly to manufacture. 136 Series Connected ( Parallel Connected \ Graduated Slots Parallel Connected Deep Slots Fig. 6.3 - Possible Parallel Connected Stator Designs Compared To a Series Connected Stator 6.7 General Comments On Optimization The effects of different parameter variations have been analyzed and experimental results have been obtained, some of which have been shown to be important in the design of the annular, high speed, thin steel rotor, double sided LIM. 137 The most important of the factors are the magnetic attractive forces analyzed in Chapter 3, the parallel/series connection of the stators and the resistivity of the rotor. A less important factor is the air gap flux harmonic caused by the slots. On the other hand, some factors which were originally thought to affect the performance and therefore the design of this type of machine but did not in fact have a significant effect. These are: the edge effect, supply harmonics, re-entry and the annular stator. Scale model testing of the annular LIM was shown to be a valuable tool in determining the expected values for efficiency, power-factor and output power including frictional losses. The scale model LIM also allowed for testing of some rotor materials which was not possible on the full size machine. In addition, the optimum value of goodness found for the small scale machine can be applied to the design of the full size machine. The answer to the question as to what constitutes the optimum design of a LIM is still very difficult and that answer must be decided by looking at all the parameter restrictions which are being placed on the machine by the particular application before general optimization criteria are applied to the design. 138 7. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH 7.1 Summary A type of induction motor with unusual parameters has been analyzed and the results presented in this thesis. The induction motor has a thin steel rotor, a double sided stator, an annular shape and is operated at high speed. This type of motor has been analyzed for the purposes of developing new types of machines for use in sawmill and mining applications. There may also be other applications for this type of motor in material handling or metal processing industries. Previous theories and formulas developed by other authors are outlined in chapter two and have been used in the analysis of these machines. A one-dimensional linear induction motor model was developed which was implemented in both a Fortran program and a spreadsheet version. This analysis was supplemented with extensive experimental results including flux plotting both longitudinally and transversely for the stators. In addition, new and original work has been presented; a new annular disc motor resistivity correction factor (p. 24) analysis of the effects of poles in parallel or series in linear induction motors. (Chapt. 5) experimental comparisons between odd and even pole designs a second optimum goodness consideration for LIMs which has 139 not previously been considered (p.123) rotor/stator attractive force analysis for magnetic rotor double sided motors and description of the flux (crenelated flux) (p. 34) a criterion for when the re-entry effect may occur (p. 29) the introduction of the use of spreadsheets for machine design and analysis (Appendix 4) 7.2 Conclusions The annular motor was analyzed using the electromagnetic field analysis equations in cylindrical co-ordinates. In addition an annular resistivity correction factor was developed and applied to one hypothetical and two experimental motors. It can be concluded that the annular stator has a measurable effect on performance if the stator width is greater than half its mean radius. For the experimental motors constructed, the annular stator will have an almost negligible effect on performance. The possibility of a re-entry effect was analyzed and found not to occur for the experimental machines. When one reviews the analysis, it is clear that the only time that a re-entry effect will ever occur (for physically realizable machines) is when two sets of annular stators are used to drive a very high speed rotor (greater than 140 m/s) and the exit point of one stator is very close (within a few centimeters) to the entrance of the other. 140 The reluctance normal force was analyzed by looking at the flux paths and saturation conditions. The analysis first identified the source of the large attractive force (the crenelated flux), then gave the peak attractive force and the modulation of the peak attractive force, in space, along the stator. Once these values were determined, the relationship between the maximum value of attraction and the rotor position was analyzed and the resulting relationship for the experimental motor was found. Finally the maximum normal force due to the rotor currents (electromagnetic normal force) was calculated and shown to be small in relation to the attractive force. The analysis was then confirmed experimentally and applied to obtain the expected value for drag during the full load operation of the motor. The normal force, which will always occur in the double sided steel rotor LIM, has the potential to create large losses even with low friction guiding surfaces. With the proper design of the LIM , as shown in the analysis which describes the "V" shaped curves, it should be possible (by increasing the air-gap) to reduce the drag to a more acceptable value than that measured in the experiments. Various experimental motors and apparatus were constructed in 141 order to investigate the annular disc LIM motors operating characteristics. One motor was used to test the effect of different rotor materials and to measure the flux variation in the radial direction. Three other motors, based on the same stator core, were used to determine the effect of odd and even number of poles and the effect of series and parallel connected poles and to measure full size motor performance. A completely annular type motor was used to determine the magnitude of the end effect. The experimental apparatus which was used to load the machines and the instruments used to make the measurements was described. The experimental accuracy of all measurements except electro-magnetic thrust were accurately obtained. Various experiments were conducted to determine the effect of different parameter variations and to experimentally determine the effect of the steel rotor on the performance of a high speed annular LIM. One parameter that was investigated was the rotor material. Although none of the other rotor materials tested would produce a practical alternative to steel at the present time, the results of the experiments are very useful in that they show potential areas of improvements in the design of this type of machine and the re-sults of these experiments can used with scaling factors to find a 142 more optimum design. This section also allowed for the direct comparison between simulated and actual results. The simulation when corrected for friction and rotor resistivity agreed well with the actual experimental results. The choice of whether to use an odd or even number of poles does not occur in standard rotary induction motors which must all have an even number of poles. However, in the LIM there is no such requirement since the entry pole does not have to be of the opposite polarity of the exit pole. It is thus possible, and sometimes done, to have an odd number of poles. The result of an experiment to compare the effect of odd or even number of poles was presented which showed that the an odd number of poles did not reduce the performance of the LIM. An option in the design of any LIM is whether to connect the poles of the stator in series or parallel. The advantage of connecting the winding in series is that the current which flows is the same in all the coils whereas in parallel connected wind-ings some of the coils carry more current than others. This is important if the current capability of the wire is near the limit that it can carry without overheating. However, more power may be produced if the windings are connected in parallel. An experiment was conducted to determine the actual effect of the two alternative connection methods and these results showed that a parallel connected LIM has greater power density with no de-143 crease in the power-factor/efficiency product. The edge effect will cause the square pattern of rotor current to close in to form ovals. The result of the oval pattern (as compared to the more optimum square) is higher losses and lower output power. Three different experiments were conducted to determine the extent of this problem for a steel rotor LIM. These experimental results showed that the edge effect did not decrease the efficiency or power output of the experimental motors. Space and time harmonics cause higher losses and reduced output. The effect of slot harmonics (the space harmonics) was expected to be more pronounced in the type of motor under investigation due to the small effective air gap and the non-laminated rotor. Slot harmonic data was obtained from flux measurements. An ex-periment was devised to compare the performance of a solid steel blade with and without the effect of space harmonics, in order to determine the magnitude of their effect. The space harmonics were found to have some effect on the efficiency of the double sided steel rotor LIM but not a great amount. The time harmonics were also measured and calculations undertaken to determine the magnitude of their effect. The time harmonics were found to have insufficient amplitude to have an effect on the performance of the LIM. 144 It was shown analytically that the effect of the annular stator would be very small on the performance of the annular LIMs under investigation (i.e. the increased rotor resistivity and the variation in power produced over the face of the motor). To obtain further confirmation of this conclusion, flux plotting was conducted and these values were then used to conduct a simulation based on the measured value of flux. The result of this simulation showed that the effect of the annular motor was more pronounced than the initial analysis indicated but it still did not significantly affect the performance of the LIMs under investigation. The effects of different parameter variations was analyzed and experimental results obtained, some of which were shown to be important in the design of the annular, high speed, thin steel rotor, double sided LIM. The most important of the factors are the magnetic attractive forces, the parallel/series connection of the stators and the resistivity of the rotor. A less important factor is the air gap flux harmonic caused by the slots. On the other hand, some factors which were originally thought to affect the performance and therefore the design of this type of machine but did not in fact have a significant effect. These are: the edge effect, supply harmonics, re-entry and the annular stator. 145 Scale model testing of the annular LIM was shown to be a valuable tool in determining the expected values for efficiency, power-factor and output power including frictional losses. The scale model LIM also allowed for testing of some rotor materials which was not possible on the full size machine. In addition, the optimum value of goodness found for the small scale machine can be applied to the design of the full size machine. 7.3 Recommendations For Further Research The machines which were analyzed in this thesis may have many commercial applications. The analysis presented is the first to deal with these particular machines with these unusual parameters and so not all areas could be covered in sufficient depth. In particular the possibility that a second goodness value may exist would in itself be a complete PhD thesis topic and one which would be very critical to the design of future machines of the type described. In addition, the effect of slot harmonics on the performance of this type of machine and the analysis of tooth shape and stator cross gap alignment would be another suitable topic. This last topic has been debated by other authors but has never been analyzed in detail. A further topic area is in the metallurgical field were the properties of suitable high conduct-ivity and high strength materials for use in LIM driven saws should be studied. 146 REFERENCES [I] Page, C. G., U.S. Patent 10480, granted Jan. 31,1854. [2] Weaver Jacquard and Electric Shuttle Co., "Improvements in the Shuttle Mechanisms and Reeds of Looms", British Patent 12364, June 26, 1901. [3] A. Zehden, "Travelling Wave Electric Traction Equipment", U.S. Patent 732312,1905. [4] "A Wound Rotor, 1400 Feet Long", Westinghouse Engineer, pp. 160-161, Sept. 1946. [5] L.R. Blake, "Conduction and Induction Pumps for Liquid Metals", Proc. IEE, Pt. A, pp. 49-67, 1957. [6] Elliot and Alt, "Performance Capabilities of Liquid Metal MHD Induction Generators", Energy From MHD, Vol. 3, International Atomic Energy Agency, Vienna, Austria, pp. 1859-1877, 1968. [7] E. R. Laithwaite, "Linear Induction Motors", Proc. IEE, Pt. A, pp. 461-470, 1957. [8] A. Bolopion and M. 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B, No. 1, January, 1983. 147 [15] Boon-Teck Ooi, "A Generalized Machine Theory of the Linear Induction Motor", IEEE Trans. On PAS, Vol. PAS-92, 1973, pp. 1252-1259. [16] T.A. Nondahl and T.A. Lipo, "Transient Analysis of A Linear Induction Machine Using the D,Q Pole-By-Pole Model", IEEE, PAS-98, No. 4, July/Aug., 1979. [17] Kinkiro Yoshido, Koichi Harada and S. Nonaka, "Analysis of Short-Primary LIMs With Odd Poles Taking Into Account Ferromagnetic End Effects", Electrical Engineering In Japon, Vol. 99, No. 1,1979, pp. 43-50. [18] Sakae Yamamura, N. Ito and Hiroyuki Masuda, "Three Dimensional Analysis of Double-Sided Linear Induction Motor With Composite Secondary", Electrical Engineering In Japon, Vol. 99, No. 2,1979, pp. 100-104. [19] Sakutaro Nonaka and Morio Matsuzaki, "Analysis of Per-formances of Various Primary Windings of High-Speed Linear Induction Motors",Electrical Engineering in Japon, Vol. 99, No. 3, 1979, pp. 103-109. 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Arifur, "Analysis of Solid-Rotor Induction Motors", PH.D. Thesis, University of Kentucky, 1974. [28] A. Mor and M.S. Erlicki, "On the Performance of the Linear Induction Motor With Disk-Shaped Rotor", Proceedings of Melcon'81, The First Mediterranean Electrotechnical Conference, Tel-Aviv, Isreal, May, 1981, pp. 3.4.4, 1-4. [29] M. Poloujadoff, The Theory of Linear Induction  Machinery. Oxford University Press, London, England, 1980. [30] P.L. Alger, Induction Machines. Gordon and Breach Science Publishers, New York, 1970. [31] Carl A. Johnk, Engineering Electromagnetic Waves. John Wiley and Sons, New York, 1975. [32] Nunzio Tralli, Classical Electromagnetic Theory. Mc-Graw-Hill Book Company, Inc., 1963. [33] Armco Electrical Steels, Armco Nonoriented Electrical  Steels. Armco Inc., Middletown, Ohio, 1984. [34] H. Bolton, "An Electromagnetic Bearing", IEE, Proceedings of the Conference On Linear Electric Machines, London, England, 1974, pp. 45-50. [35] Roger M. Katz and Tony R. Eastham, "Single-Sided Linear Induction Motors With Cage and Solid-Steel Reaction Rails For Integrated Mgnetic Suspension and Propulsion of Guided Ground Transport", Proceedings of the IEEE IAS Annual Meeting, Cincinnati, IEEE Publ. No. 80CH1575-0, 1980, pp. 262-269. [36] E.M. Freeman and D.A. Lowther, "Normal Forces in Sin-gle-Sided Linear Induction Motors", IEE, Proceedings of the Conference On Linear Electric Machines, London, England, 1974, pp. 210-215. [37] L.M. Sweet, R.J. Caudill, P.G. Phelan and K. Oda, "Im-provement In Vehicle Performance Through Control of Linear Motor Lateral and Normal Forces", Conference Record of the IAS Annual Meeting, 1981, pp. 314-322. [38] T. Oktsuka and Y. Kyotani, "Recent Progress in Supercon-ducting Magnetic Levitation Tests in Japan", Conference Record of the IAS Annual Meeting, 1980, pp. 238-243. [39] F. Peabody, D. Nyberg and W.G. Dunford, "The Use of A Spreadsheet Program To Design Motors On A Personal 149 Computer", in Proceedings of the IAS Conference, 1985, pp. 840-845. [40] F. Peabody, D. Nyberg and W.G. Dunford, "The Use of A Spreadsheet Program To Design Motors On A Personal Computer", IEEE Trans. Ind. Appl., Vol. IA-23, no. 3, May/June, 1987. [41] C.G. Veinott, Computer Aided Design of Electric  Machinery. MIT Press, 1972. [42] P.L. Alger, Induction Machines. Gordon and Breach Science Publishers, 1970. [43] F.G. Peabody, 'Induction Motor Simulation For the Computer Aided Design of Induction Motor Drives', MASc. Thesis, The University of British Columbia, 1981. [44] K. Mauch and F. Peabody, 'A Compact Thruster Drive For Industrial Submarines', in Proceedings of the IAS Confer-ence, 1985, pp. 710-715. [45] B. Jeyasurya, T.A. Little and M.A. Rahman, 'Application of Personal Computers in Power Engineering Design and Analysis', in Proceedings of the IAS Conference, 1985, pp. 759-764. [46] F. Peabody, W.G. Dunford and J.Brdicko, 'An Analysis of a Thin Steel Rotor, Double Sided, Annular, Linear Induction Motor', in Proceedings of the International Conference on Maglev and Linear Drives, IEEE, 1986, pp. 193-198. [47] B. Nelson, Timing Studies of the RNA Series 32 Computer With the -Linpack- System of Subroutines', Technical Announcement From RNA Inc., Livermore, Ca., 1984. [48] F.W. Carter, 'The Magnetic Field of the Dynamo Electric Machine', The Journal of the IEE, Vol. 64, 1926, pp. 1115-1138. [49] G.R. Slemon, 'Scale Factors For Physical Modelling of Magnetic Devices', Electrical Machines and Electromag-netics: An International Quarterly, 1:1-9, 1976, pp. 1-9. [50] H. Bolton, "Transverse Edge Effect In Sheet-Rotor Induc-tion Motors", Proc. IEE, 116 (5), 1969, pp. 725-731. [51] R.L. Russel and K.H. Norsworthy, "Eddy Currents and Wall Losses In Screened-Rotor Induction Motors", The Institute of Electrical Engineers, Paper No. 2525U, April, 1958, pp. 163-175. 150 [52] John Davies and Peter Simpson, Induction Heating  Handbook. McGraw-Hill Book Company Ltd., London, 1979. [53] B. Heller and V. Hamata, Harmonic Field Effects In  Induction Machines. Elsvier Scientific Publishing Company, Oxford, 1977. [54] D.A. Lowther and E.M. Freeman, "Electromagnetic Scale Models of Linear Induction Motors", IEE, Proceedings of the Conference On Linear Electric Machines, London, England, 1974, pp. 167-172. [55] George John, "The Effect of Harmonic Voltage and Currents On LIMs", M.Sc. Thesis, Queens University, 1986. [56] S. Yamamura, Theory of Linear Induction Motors. Halsted Press, New York, 1978. [57] John K. Dukowicz, "Theory of Optimum Linear Induction Motors", Journal of Applied Physics, Vol. 47, No. 8, August 1976, pp. 3690-3696. 151 Appendix 1 Derivation of Field Theory Equations [29] The LIM is divided into areas as shown in Fig. 2.3. In one-di-mensional analysis it is assumed that the EMF induced in the secondary by the primary exists only directly under the stator (y*L/2) and is normal to the x-axis. Also, the currents directly under the stator flow only in the y-direction. In this model the current and flux densities are functions of the x component only. The following three equations describe these conditions. the change in flux along the motor is equal to the current density flowing in the stator and rotor the change in current flowing along the return paths of the rotor is equal to the current density flowing in the rotor -db/dx = n0/g*(c*j+J) (2.1) dl/dx =c*j (2.2) dj/dx - 21/(1 *SJ = -(i*w*b+Vdb/dx)/p (2.3) the change in current density along the rotor minus the effect of the resistance is equal to the transformer and the speed 152 induced voltage divided by the secondary resistivity When equations (2.1),(2.2) and (2.3) are combined, a third order differential equation (2.4) is obtained which can be solved from the boundary conditions and the given surface current density. First write out equations (2.1),(2.2) and (2.3) -db/dx = ii0/g*{(c*j)+J) (2.1) dl/dx = c*j (2.2) dj/dx-(2/(lg*Sr))*I = -{(i*w*b)+(V*db/dx)}/p (2.3) Then write out the terms of equation (2.3) dj/dx-2/(l *S_)*I = -i*w*b/p-V/p*db/dx (2.4) Taking the first derivative of (2.4) d2j/dx2 - 2/(1 *SJ*dI/dx = -i/p*w*db/dx - V/p*d2b/dx2 (2.5) 153 Taking the second derivative of equation (2.1) - d 2 b / d x 2 = n 0*c/g * dj/dx + |X 0/g*dJ/dx (2.6) Now taking the second derivative -d3b/dx3 = u*c/g*d2j/dx2 + un/g*d2J/dx2 (2.7) Obtain an expression from (2.7) for d j/dx d 2 j / d x 2 = -g /u 0 / c *d 3 b /dx 3 - l / c * d 2 J / d x 2 (2.8) Obtain an expression for j by rearranging (2.1) j = -g/u Q/c*db/dx - J/c (2.9) Substitute expressions (2.2) and (2.8) into equation (2.5) d2j/dx2 - 2/(ls*Sr)*dI/dx = -l/p*i*w*db/dx - V/p*d2b/dx2 (2.10) -g/(^ i0*c)*d3b/dx3 - l/c*d2J/dx2 - 2/(lg*Sr)*c*j = -l/p*i*w*db/dx - V/p*d2b/dx2 Substitute (2.9) into (2.11) -g/( 0^*c)*d3b/dx3 " l/c*d2J/dx2 - 2*c/(ls*Sr) *{-g/^ i0/c*db/dx -l/c*J) =-l/p*i*w*db/dx - V/p*d2b/dx2 Expand to get: -g4i0/c*d3b/dx3 - l/c*d2J/dx2 + 2*g/(lg*Sr* 0^)*db/dx + 2*J/1/Sr =l/p*i*w*db/dx - V/p*d2b/dx2 Gathering together terms: -g/^i0/c*d3b/dx3 + V/p d2b/dx2 + {2*g/l/SI/^ i() + i*w/p}*db/dx= l/c*d2J/dx2 - 2*J/1/Sr Multiply through by (-\iQ*c/g) to get: d3b/dx3 - V*^i0*c/g/p*d2b/dx2 - {2*c/(lg*Sr) + 0^*c*i*w/g/p}db/dx -^i0/g*d2J/dx2 + 2 0^*c*J/(ls*Sr)/g 155 d3b/obc3-V*u0*c/(p*g)d2b/a^2-[i*w* 0^*c/(p*g)+ (2*c/(l *SJ)]db/dx=-uo/g(d2J/dx2-(2c/l * )*J) (2.14) Equation (2.14) is then solved to obtain the magnetic field value. The general solution of this differential equation is of the form: bQ +bx exp(rj) and the particular form is: bp = -i|iGJ/(kg) cos<j) exp(i<t>) = - i B Q cos<j) exp(i(|)) The thrust can then be calculated from the stator surface current and the rotor magnetic field as given by the equation: P T | F = -£l s I Re (J*b) dx ' 0 The thrust developed by the motor is separated into two compon-ents. F Q is the thrust described by standard rotary induction motor theory and F^ is the thrust produced by the parasitic end effects found in linear induction motors. 156 F Q =^ *Jm*B0*ls*p*L*cos<i)*sin(!) (2.15) F l = -i*Jm*B0*ls*Real[(b1/bp)*(bp/B0)*k1] (2.16) b x = [V*a+k*tan<|)(l-i*v/k)/(r1-v-V*a)]*b B Q = u0*J/(k*g) b = -i*B *k2/(k2+i*s*w*a) r x = V*a/2*{l-[l+4*(i*w*a+v2)/(V*a)2]} V = velocity v = 4/stator width/total rotor overhang a =\i.0*c/p/g p = rotor resistivity K = [l-exp{(r1+ik)PT|}]/[r1+ik] (j) = tan"1(s*w*a/k2+v2) L = length of stator J m = maximum primary current density 1 = width of stator D Line# 1 7 Microsoft F0RTRAN7 V3.£0 1 *DEBUG cl c This is a linear induction motor (LIM) simulation program based 3 c the steady-state analyis which is sumarised in Dr. Poloutjadof f' s 4 c book. 5 c For this simulation the magnetic flux begins at the entry of the £ c motor and trails from the exit end. 7 c vers i on of: Aug. £1,1965. 6 c This section wil define all the variables used in the program. 9 c vnot •.. 10 c vnot sq 11 c ve 1 velocity i£ c velsyn synchronous velocity 13 c f req frequency of stator currents 14 c frad frequency of stator currents in radians 15 c bpr i rne 16 c bone 17 c btwo 18 c alpha 13 c rone r. £0 c r t wo r. £1 c J m stator surface current density in arnDS Der meter c' c' c kay -c c mewnot permitivity of the airgap £4 c cee roter thickness in meters £5 c row Cooper- 1.7£4d-8 Aluminum- £.9d-8 Steel 17.d-B £6 c • ao stator to stator effective airgap in meters £7 c e i stator width £6 c e 1 Dr rotor width £9 c bnot Page 09-09 £1 :4S 0  1 -65 : 16 /•8 4 o 3 to -a 30 c negeye - ' 31 c Define the types of variables for the program. 33 c Rea 1 *8 vnot, vnotsq, vel, ve 1 syn, f req, f rad, a 1 Dha, jrn, kay, mewnot, 34 +eee, row, gap, el, elpr, slip, 1 arnbda, psi, length, x, force. 35 +forcet,teste,dreal, forceO, force1, force£, forces,bnot, 36 + rcalcl, rcslc£, rcalcS, rcalc4, rcalcS, rcalc6, rcalc7,fibs, 37 +polen,forctO,forctl,forct£ 36 complex*16 bprime, bone, btwo, rone, rtwo, btotal,cdsqrt, 39 +cdexp,negeye,poseye, calcl, calc£, calc3, calc4,calcS,test 1,test3 40 character*64 result,greslt 41 c *****0pen a file for results***** 4£ open(10,file=' result' ) 43 open(11,file='gresIt' ) 44 c *****Set parameter values for temporary testing. * * * * * * 45 c These must all be specified. 46 freq=396. 47 jrn=8. 10£d4 48 cee=.00£00 49 row=35.Od-8 50 gap=.00300 51 el=.0570 5£ elpr=0.108 53 c lambda is the wavelngth 54 larnbda=. 1460 55 c polen is the number of poles 56 polen=4.0 57 c Set physical constants 58 mewnot=l£.566£8d-7 59 c Calculate other motor parameters not specified 00 Page £ 03-03-85 £1:46:18 D Line* 1 7 Microsoft F0RTRAN7  V3. £0 Q£/S4 60 al pha=mewnot*ce/row/gap 61 l e n g t h = D o l e r i / £ . 0*lambda 6£ ve1syn=frea*1ambda 63 frad=6.£8318*freq 64 c Calculate all other variables. 65 vriotsq = 4. 0/el* (elDr-el > 66 vnot =dsqrt < vriot SQ > 67 kay=6.£8314/1ambda 68 c Define math constants. 63 nepeye=(0. 0,-1.0) 70 poseye=(0.0,1.0) 71 c Set initial values to 0.0 7£ force=0.0 73 forcet=0.0 74 c Set u'j output table 75 write(10, 004) 7t, 004 format(lx.'THESE ARE THE RESULTS OF A 1—DIMENSIONAL. LINEAR', 77 + ' INDUCTION MOTOR SIMULATION') 78 write(10,005) 79 005 format (lx, ' Frea' , ' Arnps/M' , ' Rotor Thk' , ' Resist' , ' Gap' , 80 +' S.Width', R.Width', W.Len.', #poles') 6 1 write < 10, 006) f req, jrn, cee, row, gap, el, el pr, 1 ambda, polen 8£ 006 format (1 x, f5. 0, f7. 0, lx,f7.4, lx,e9.£, lx,f6.3, lx,f6.3, lx, 63 +f 6. 3, 3x, f6. 3, 4x, f4. 1 ) 8'+ write (*, 050) 65 write(10,050) 86 050 format (1x,'SIip' , V Horsepower' , ' F0 ' , ' Fl ' , ' F£ ' , M 87 +' Thrust(N) ' , ' Thrust(1bs)') g 88 c * * * * * * * start main program routine * * * * * * * * * 83 c 90 do 700 j=10,100,10 1 91 forcet=0.0 1 9£ forct0=0.0 1 93 forct1=0.0 1 94 forct£-0.0 i 95 vel=real(j)/100.*velsyn i 96 sip=l.0-kay*vel/frad 1 37 psi=dat an (si i p*f rad*al pha/ (kay**£+vriot sa ) ) 1 93 bnot =mewnot*jrn/kay / gap 1 99 bprime= (negeye*rnewnot*jm/gap/kay) * ( (kay**£+vnotsq) / 1 100 + (k.ay**£+vnotsq + (poseye*sl iD*frad*a1pha) ) > 1 101 rone=vel*alpha/£. 0* < 1. 0-cdsqrt(1. 0+4. 0*(poseye*frad*alcha 1 10£ ++vnotsq)/(<vel*alpha)**£)))) 1 103 rtwo=vel*alpha/£. 0*(1. 0+cdsqrt(1. 0+4. 0*(poseye*frad*alDha 1 104 ++vnotsq)/((vel*alpha)**£)))) 1 105 bone=bpr irne*(<(vel*alpha)+<kay*dtan(psi)*(1.+negeye*vnot 1 106 +/kay)/(rone-vnot-vel*alpha) 1 107 calcl=negeye*psi 1 108 calc£=poseye*psi 1 109 btwo=bprine* (rone+negeye*dsin (DSI ) * (rone-negeye*CEXP 1 110 +(calcl)))/(rtwo-rone)/dcos(psi)*cexp(calc£) 1 111 c WRITE(*,£05)BPRIME,BONE,BTWO 1 11£ c £05 FORMAT (IX, ' bprirne, bone, btwo' , 6E14. 7) 1 113 c write(*,£06)rone,rtwo 1 14 c £06 format(1x,'rone=',£el4.7, ' rtwo=', £el4.7) 1 115 do 500 i=l,39 £ 116 c write(*,£01)i £ 117 c £01 format<lx,i3) £ 118 x=real(i)/100.*length 03 o 161 m in co -r CO ^ CO I " \ cn KD ca OJ o 4- o at i •• •3* • X 7-1 * > 01 OJ ns „ a. h- IT. * X Z II 01 Oi d n >> fj ir u 01 * j — r—1 JZ •» or ns -p Ol a a c * u. C 01 Q +> * r—4 X r 4- • ,-, * * 01 + J M * i—i u Cli * in w • * * C * * 0 01 * in * r-; 01 * * r *- OJ * -p If) r-i * * U w Ol * u * 1 * * •r-4 r- •4" JZ * Oi o r^ X * £ II -p * • X * at OJ a; * * * * Li a f l c * OJ CU fl X-i—i -P ai C 2 UJ 0 as Ul i—t n TJ in o -p T-i 4- X o ai 01 c Q 4->- S- * ^ -p r-H 0 01 r: JZ •P n ns c D Q -p u U ns u X •a -P "O 0 •rH X X a i t J- Ol r—i 1-i c o 0 Ul 01 Oi C ns 4-r->. * ns m . as +> / *~ 0 U Ol 4- X Ol T^  0 J3 a r * * * r-4 01 * >, X • + 01 „. -p .—. Ol fl * >, 01 QJ * 01 OJ 1-1 -p >. OJ r-t ai •H r-4 c 2 r—4 ns 01 c •1 >. c u OJ -p Ul Ol -p CJ u 2 Ul l"l -p 01 2 U 0 ai ns fl QJ 0 .—. Ul Ql l/l 1/1 i . Q. XI A * i- t >- c r ns OJ r-l ns Ql -p CU (0 0 01 * * in •P <n * * * * +» u as -P -P • •p U »*- 0 in 01 01 o C r A- Ol X Ol 01 01 X •i »> 0 X 0 1/1 * !• *->> >. o 01 -P >. * >, >. >, * c *-« 11 •— * - -P •f* 1-1 T—i 3 r^ 01 0 u 01 Ol • S- i- CU Ol Ol 01 Ol Ol a *H Q J3 p -p •p >»- UJ Ul * 01 01 Ul c in cn Ul as i-H r-l (J X ns o * 1/1 Ul Ul Ul • -p * M x: fl r-* M- fl u fi QJ fl 1 Oi a -X r-1 Ol r-i 01 Ol <f 01 3 a -P a .-*** 4- QJ a a C Q. * X u as fj r—* * ns •P -p r p -p r-i CU 01 -p — Oi X •r-l -p ^ ~- %-X a * as 01 u U1 r-T. I - — 01 01 -p ns • i c ai * X) as Qi • i Ol • l >. * * If) «- 01 u 01 • I OJ M •3' r- ns r—4 •P ns 01 ns >, r—4 m ns ns ns as 01 01 0 OJ c Ol Q -p f"\ as ••- r-i "i 0 r-4 c 1— ni ns —i d c 3 3 •1 c 2 OJ X fl a • l X * CU X CU X OJ X fj !>. * •ri _* a Ql 0 -p r-i X oi ai 3 r-i -r-t y~i a r-4 J3 T) r-i X) * > --I X) X) X) X) X! C r A- i. * II Ql c u "~> T3 * * •-' ns ns II II Ol II 01 t as n n II II II II II II -P r-l CJ ii II II 4-> II -P +> u u 1-4 * OJ u TH OJ a) i> f'l rt OJ Oi as as * ro Ol ns OJ OJ 'IS Ol ns 01 Ol If) Ol Ql u u u u 0 u CJ U -p 4-> fi (J "P p -p •P •p 3 4-> * * U 0 u • l * * r-4 ,—i r-4 r—4 r-« r-i r-4 t 0 2 .-» 1/1 'Jl •ri A- Ul •ri < * * -w r >- r j— 01 * ns ns as as as ni OS t A- 0 -P -p m cu 01 >— 1*1 Ol f 0 S- 0 * * t'l fl • fl c * * 0 a a U u »> u 0 u 2 4-JD u *> -P 2 •p 2 2 * * 4- * 4-— * * r i* f f i* + * * + + * * If) ro OJ * * * * OJ T-4 * * * * OJ •XI CO OJ * * T-< 0 U U u u 0 U CJ a 0 u u u 0 0 u u fn ,-, T-4 OJ fO <J- in U) N CO cn .-, OJ -3* U'J o cn Q T-* OJ fO <i- u) r- CD Ql OJ OJ OJ co CO OJ CO CO OJ CU I'") 10 r-o (0 10 <f <f _J Q OJ OJ OJ OJ OJ CJ OJ OJ OJ CJ OJ CJ OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ Oj OJ Co OJ OJ OJ OJ OJ OJ 150 c rca1c4=d i mag(bone) c: 151 c rcalc5=datari£ (rcalc3, rcalc4) c 15£ c rcalc3=datan£ (rcalcl, r c a l c l ) -datan£ (rcalcG, rcalc7) +rcalc5 c 153 c force1 = . 005*el*length*cdabs(bone)*dexp(rcalcl) c' 154 c +*jm*dcos(rcalc3) c * * * * * * * * calculate exit efect force * * * * * * * * * c 156 c calcl=rtwo*(x-length) 157 c calc£=negeye*kay*x £ 158 c rcalcl=d irnag (poseye*calcl) 159 c realc£=d irnag ( c a l c l ) 160 c rca1c6=d i mag(DOseye*calc£) 161 c rcalc7=dirnag (calc£) £• 16£ c rcalc3=d irnag ( poseye*bt wo) u. 163 c rca 1 c4=d i rnag ( bt wo) C 164 c rcalc5=datan£(rcalc3, rcalc4) C 165 c rcalc3=datan£(rcalcl,rcalc£)-datan£ <rcalc6, rcalc7)+rcalc5 c! 166 c force£=.005*el*length*cdabs(btwo)*dexp(rcalcl) £' 167 c +*jm*dcos(rcalc3) 168 forces=forceO-force1-force£ £ 169 forcet=forcet+forces 170 forct 0= forct 0+forceO 171 forctl=forct1+foreel 172 forct £=forct£+force£ 173 c force=.005*el*test3*length 174 c forcet—forcet+force £ 175 c write (*,600)i,forces, forceO, force1, foree£ £ 176 c 600 format<lx,'1=',i3.' Force=', el3. 6. ' F0=',el3. 6, ' Fl = ',el3. £ 177 c + ' F£=',el3. 6) Pane 4 09-09-65 £1:46:18 Linet 1 7 Microsoft F0RTRAN7 V3. £0 0£/64 178 500 continue 179 f1bs=forcet/4.448 180 c f orct0=0. 5*el*length*bnot*dcos ( DS i ) *dsi n ( osi ) * jrn 181 c write(*,610)forcet, fibs 18£ c 610 format (1x,'Forcet = ',el3.6, ' Neut. ',el3.6,' 1bs.' ) 183 power=vel*3.£81*flbs/550. 0 184 wr ite(*,615)slip, power, f orct 0, f orct 1, f orct £, f orcet, f 1 bs 185 wri t e(10,615)si p, power,forctO, forct1, forct£,forcet,f1bs 186 615 format (lx,f4.£, lx, f10. 3,5x,f6. 1, lx, f6. 1, lx,f6. 1, f11.£, f1£.£) 187 write(11,650)si p,vel,power,f1bs, forcet 188 £50 format(1x,5f10.£) 189 c write(*,617)el,elpr,freq, lambda,alpha,vel 190 c 617 format(lx,6f10.3) 191 c write(*,6£0)forctO,forct1, forct£ 19£ c 6£0 format (lx,' Thrust0=',f7.3,' Entrance efect = ',f7.3. 193 c +' Exit effect=',f7.3) 194 700 centinue 195 close(10) 196 close(11) 197 stop 196 end CO Name Type Ofset P Class ALPHA R£AL*3 56 BN07 REAL*6 6&£ BONE COMPLEX*16 . 1086 BPRIME C0MPLEX*16 670 BTDT'AL C0MPLEX*16 1436 BTWO C0MPLEX*16 1246 CALC1 C0MPLEX*16 118£ CALC£ C0MPLEX*16 1£14 CALC3 C0MPLEX*16 1466 CALC4 C0MPLEX*16 * * * * * CftLCS C0MPLEX*16 * * * * * CDEXP C0MPLEX*16 * * * * * CDSQRT INTRINSIC CEE REAL*B 3£ CEXP INTRINSIC DATAN INTRINSIC DCOS INTRINSIC DIMAG INTRINSIC DREftL REP iL * 8 * * * * * DSIN INTRINSIC DSQRT INTRINSIC DTAN INTRINSIC EL R E A L * 8 56 ELPR R E A L * 8 64 FLBS REAL*8 £098 FORCE REAL*8 184 FORCEO REAL*S 1714 F0RCE1 REPiL* B 17££ F0RCE2 REAL*6 1906 FORCES REftL*8 £030 FORCET REAL*8 19£ FORCTO REAL*8 614 FORCT1 REAL*8 £££ ^ FORCT£ REAL*8 630 CR FRAD REAL*8 1£0 165 Appendix 3 Spreadsheet Simulation Program ANALYSIS OF LIN PERFORMANCE - MOTOR TYPE t MODEL NO. Exoerimental Motor t3 File: V0LTA6E 03-Har-87 Note: To obtain a printout of this f i l e enter "F7* then "Print". SPECIFIED PARAMETERS Ref: Effect of Annular Stator 1.0045 o Potter Supply o Motor 6eoaetry Rated RMS SUDDIV Current: Io (A) = 60.0 Rotor Thickness: c (•) = 3rO0E"O3 RMS Voltage per Phase E (V) = 254.0 Average Air-Gap on Each Side of Rotor (•) = 1.00E-03 Supply Frequency: fO (hz) = 360 Stator Lamination Width: 1 (•) = 3.40E-02 No. of Phases: a 3 Stator Back Iron: dl (•) = 1.60E-02 Type of Connection: 0 for Delta, 1 for V = 1 Rotor Width: 11 (•) = 8.4BE-02 Mean Diameter: D (•) = 0.279 o Electrical and Maonetic Specifications of Motor Angular Sector Occuoied by Motor: SEC (deg) 360.0 Rotor: 1 for nonmagnetic, 0 for Mimetic No. of Parallel Coil 6rouos/Stator/Phase: Np = 0 Winding Pitch (Coil Soan/Pole Pitch): Wo = 0.833 = 5 Mean Tooth Width: t (•) = 7.30E-O3 No. of Turns/Slot: N 70 Mean Slot Width: SM (*) = 7.30E-03 No. of Turns/Coil: Nc = 35 Tooth Deoth (•  2.20E-02 No. of Turns in Series: Ns = 140 No. of Stators: Sn = 2 Nuiber of Poles: P = 10 Rotor Electrical Resistivity: rho (ohr-a) = 3.50E-07 Diameter of Coooer Hire: dw (•) 1.00E-03 Penittivity of Free Space: muO (H/m> = 1.26E-06 CALCULATED PARAMETERS o Electrical Parameters RMS Voltage/Phase: E (V) Peak Stator Surface Current Density: Jmo (A/m) RMS Magnetizing Current/Phase: Iso Inductance/Phase: Li (H) Fringing Factor: fr Effective Stator-Stator Airgap: Ge (•) 6oodness Factor: 6 254 = 4.44tfOttte*ohiiK* 5.23E+04 = Hsqrt(2)/Np*KN*H/(t+sw) (2 stators) 41.3 = 4.42£5*PA2*6e*phi/(q*K»»Dtl#Ns)t(360/SEC) 2.93 ( 0.184 flloer) 2.85E-03 = gt(t+sw!7<l+fr*g) 7 = mu0*ctvs«2/(rho«6e*omega) (p.94 Alger) Geometrical Parameters Stator Length: Ds (•) Wavelength:"lambda (•) Pole Pitch: p (•) = 8.76E-02 No. of Slots/Phase Belt: n Synchronous Velocity: vs (m/s) Synchronous Frequency: fs Synchronous RPM: SRPM = 4320 = fs*60 0.876 = D*«*SEC/360 1.75E-01 = 2#Ds/P Ds/P 2.0 = o/(t+s«)/q 63 = laabdatfO 72.00 = vs/«/D Peak Flux and Flux Densities Peak Air Gap Maonetic Field: B0 (Tesla) Total Flux/Pole: phi (Wb) Average Flux Density: Bav (Wb/m*2) •• Average Tooth Flux Density: Bta (Wb/m*2) • Peak Tooth Flux Density: Btp (Wb/m*2) Peak Core Flux Density: Be (Wb/m*2) Coil Parameters Pitch Factor: Kp Distribution Factor: Kd Winding Factor: KM F i l l Factor: Ff Copper Losses in Windings Length of Coil/Turn: lc (•) Resistance/Winding of Ns Turns: RM (ohm) Current/Winding: IM (A) Total Copper Losses: Wcu (KW) 0.642 = muO#Jm/k/6e 1.22E-03 = (2/«)tB0*l*D 0.409 = (2/i)«B0 0.817 = phi/(n«l*o«t) 1.283 = i/2*Bta 1.118 = phi/l/dl (Divided by 2 for a 360 deg stator - Alner pl&8) 0.9659 = sin(Wpti/2) 0.9659 = sin<x/(2*q))/<n*sin(«/2/n/a>) 0.9330 = Kp*Kd 0.3423 = diT2/4*pi/N/SM/Tooth deoth o Other Constants alpha v02 k 2.87E-01 = 2tl+3*Wp*o 0.8819 = Ns*lc»rho«4/«/dMA2 4.1 = Ii/(Sn««o> 0.452 = IiT2*RM*Np*Sn#q/1000 3.8 = mu0«c/rho/6e 2315.9 = 4/1/(11-1) 35.9 = i/o (3 ohases) 166 THRUST AND POWER f^ULflTIONS o Main Thrust = FO FO = l«J.fr«0tD/2«cos(psi)*sin(psi) =kl»cos(osi)Hin(psi) kl = 499.4814 tan(psi) = 2«i»s«fo*alpha/(kAa+vo2) o Entrance Thrust = Fl Fl = J«»l/2*Re{blt(l-«p[,rl+i«k)#0])/(rl+i«k)} o Exit Thrust is not calculated, but is usually saall. o Optima Rotor Resistivity for Maxima Thrust as a Function of Slip s it) = 5 10 15 20 25 30 35 40 45 rho (ohM) = 4.15E-08 B.29E-08 1.24E-07 1.66E-07 2.07E-07 2.49E-07 2.90E-O7 3.32E-07 3.73E-07 OJNSTflNT FLUX MODEL Specific Slip v Thrust Outout Output Bo-=el+itfl (a/s) psi in (fl/i) I i (fl) F0 (NI Fl (N) Ft (N) Ft (lb) (lb/in*2) HP KU el f l OK 63.1 0.000 52275 41 0 0 0 0 0.00 0.00 0.00 0.000 -0.642 2* 61. B 0.047 52334 41 24 0 24 5 0.12 1.96 1.46 -O.030 -0.641 4* 60.5 0.095 52509 42 47 0 47 ' 11 0.23 3.84 2.87 -0.061 -0.639 6* 59.3 0.141 52801 42 71 0 71 16 0.35 5.65 4.21 -0.090 -0.635 8* 58.0 0.187 53206 42 95 0 95 21 0.46 7.37 5.50 -0.120 -0.630 10* 56.8 0.233 53723 42 118 0 118 27 0.58 9.01 6.72 -0.148 -0.624 12* 55.5 0.277 54348 43 142 0 142 32 0.69 10.57 7.B9 -0.176 -0.617 14* 54.2 0.320 55077 44 166 0 166 37 0.81 12.05 8.99 -0.202 -0.609 16* 53.0 0.362 55906 44 189 0 189 43 0.92 13.45 10.04 -0.227 -0.600 IB* 51.7 0.403 56832 45 213 0 213 48 1.04 14.78 11.02 -0.252 -0.590 20* 50.5 0.443 57849 46 237 0 237 53 1.15 16.02 11.95 -0.275 -0.560 22* 49.2 0.481 58952 47 260 0 260 59 1.27 17.18 12.81 -0.297 -0.569 24* 47.9 0.517 60138 48 284 0 284 64 1.3B 18.26 1162 -0.317 -0.558 26* 46.7 0.552 61400 49 308 0 308 69 1.50 19.26 14.37 -0.337 -0.546 28* 45.4 0.586 62735 50 331 0 331 75 1.61 20.18 15.06 -0.355 -0.535 30* 44.1 0.618 64138 51 355 0 355 80 1.73 21.02 15.68 -0.372 -0.523 35* 41.0 0.692 67915 54 414 0 414 93 2.01 22.77 16.99 -0.410 -0.494 40* 37.8 0.759 72027 57 473 0 473 106 2.30 24.03 17.92 -0.441 -0.466 45* 34.7 0.817 76420 60 533 0 533 120 2.59 24.78 18.48 -0.468 -0.439 50* 31.5 0.870 81049 64 592 0 592 133 2.88 25.03 18.67 -O.490 -0.414 55* 28.4 0.916 85875 68 651 0 651 146 3.17 24.78 18.48 -0.509 -0.391 60* 25.2 0.958 90867 72 710 0 710 160 3.45 24.03 17.92 -0.525 -0.369 65* 22.1 0.995 96000 76 769 0 769 173 3.74 22.77 16.99 -0.538 -0.349 70* 18.9 1.028 101251 80 829 0 829 186 4.03 21.02 15.68 -0.550 -0.331 75* 15.8 1.058 106603 84 888 0 888 200 4.32 18.77 14.00 -0.559 -0.315 80* 12.6 1.085 112042 89 947 0 947 213 4.60 16.02 11.95 -0.568 -0.299 85* 9.5 1.110 117556 93 1006 0 1006 226 4.89 12.76 9.52 -0.575 -0.285 90* 6.3 1.132 123135 97 1065 0 1065 239 5.18 9.01 6.72 -0.581 -0.272 95* 3.2 1.153 128769 102 1124 0 1124 253 5.47 4.76 3.55 -0.586 -0.261 CONSTANT CURRENT MODEL Specific Slip v Thrust Out out Out out Bo=el+i»fl (•/s) psi F0 (N) Fl (N) Ft (N) Ft (lb) (lb/in*2) HP KU el f l 0* 63.1 0.000 0 0 0 0 0.00 0.0 0.0 0.000 -0.642 Appendix 4 COMPUTER MODELLING 167 A4.1 Introduction to Computer Modelling This Chapter describes two methods of implementing the equations required to simulate the performance of the linear induction motor. A traditional method, using a Fortran program, is described in Section A4.2. The new method, using a spreadsheet program, is described in Section A4.3. This second method is also described in [39] and [40]. Computers are an important tool in the complex task of designing and analyzing the behavior of motors. Major manufacturers of motors have computer programs to aid in the design of motors and books have been written specifically on the topic [41] or have sections that discuss areas in which computers can be used [42]. Computers are also used to calculate the steady-state and transient performance of motors under unusual conditions [43,44]. These applications of computers for the design and analysis of motors have traditionally relied on the large mainframe computer and usually the Fortran programming lang-uage. Recently, with the introduction of more powerful personal computers, it has been possible to do some of the complex tasks of motor design on an inexpensive machine, especially since the Fortran programming language is now available on personal computers [45,46]. 168 Another new tool to emerge for the motor designer is the spread-sheet program. The advantages of using a spreadsheet program include the clear and logical organization of the input data, the ease with which the spreadsheet can be built up to include greater detail of analysis as required, the built in graphics capability, the low cost of the program and the high degree of portability since the program will run on any IBM-PC compatible computer. In addition, many design engineers will already be familiar with the use of a spreadsheet program and, if not, the programs are designed to be easy to learn, with many good manuals also available on the use of particular programs. For the analysis of induction motors, two versions of a worksheet have been developed; one does a design based on minimum input data, and the second does a simulation based on a complete set of motor parameters. The normal design procedure is to first use the design worksheet to obtain a starting point for choosing the motor parameters and to then use the analysis worksheet to determine if the motor performance can be improved. A4.2 Fortran Analysis Method A program to simulate the performance of Linear Induction Motors was written in the Fortran programming language. This program is found in Appendix 2 of the thesis. 169 The program uses the one-dimensional field theory equations presented in Chapt. 2. There are two versions of the program; one for constant current conditions and one for constant flux. The required input variables are the pole pitch, stator width, number of poles, surface current density, rotor width, rotor thickness, rotor resistivity frequency of the supply and the effective air-gap. The effective air-gap must be precalculated and may be found either by the Carter co-efficient [48] or finite element analysis. The output data is produced in two forms: tabular form to be read out, and in another form suited for the graphics routines used in the Lotus spreadsheet program. These graphics routines provide the basic X-Y plots very quickly and with a minimum amount of learning time required. The advantage of programming in Fortran is that it is the most widely used simulation program and has many powerful subroutines available. The disadvantages are that it is very difficult to read data and to output data, to write correct code and to debug. Some of these disadvantages were not found when a spreadsheet version of the simulation program was written as described below. A4.3 Spreadsheet Analysis Method The first spreadsheet program was introduced in 1978 under the 170 name of VisiCalc (the name is derived from "visible calculator"). The original application for the program was to do financial analysis. In 1983, the Lotus Development Corporation introduced Lotus 1-2-3 which had many more features than previous spreadsheets. The most important of these features, for the motor designer, are the graphics and trigonometric functions. The Symphony program is basically the 1-2-3 program with word processing, data base management and data communications. A spreadsheet program lays out the computer screen as an array of cells which are labeled by rows (1,2,3..) and columns (A,B,C..)-so that a particular cell will have an address, for example, of Al. Each cell can have a number, a statement or a formula entered into it. In this way information can be entered into some cells, documentation describing what different cells represent can be written into others, and the results of computa-tions performed by the formulas appear in others. The spread-sheets can become very large. The Symphony spreadsheet can accommodate up to two million cells but in actual fact the compu-ter's memory would limit the available number of cells to a much smaller number. A4.3.1 Problem Formulation As stated earlier, the advantage of the field theory model is that it provides a good picture of the magnetic fields and current 171 densities, and is the simplest method of representing the longitudinal end effect which occurs in high speed linear induction motors. The induction motor is approximated as a stator of negligible magnetic reluctance with an infinitely thin surface current and the rotor as a thin plate with a specified volume resistivity. The field theory equations described in Chapter 2 were implemented in the spreadsheet program as described in Section A4.3.2. A4.3.2 Organization and Documentation of the Calculations The design and analysis of electrical motors was systemized and documented in detail using the Symphony program. Figure A4.1 shows the design spreadsheet and Figure A4.2 shows the analysis spreadsheet. The section which calculates the parameters of the motor is common to both the analysis and the design spreadsheet and is shown in Figures A4.3 and A4.4. Figure A4.5 shows the table of output values, which each program generates. Both of the spreadsheets are organized into three sections as described below (see Appendix 3 to see the spreadsheet as it normally appears). ANALYSIS OF LlM PERFORMANCE - MOTOR TYPE 6. MODEL NO.: 3SSSZSSZSZB = SS = SSSS3=S = CXZZ = S = = Z = SSX = = = SSX3 = = S = S = = = = =.SS: F i l e : LIM_DES 21-Jun-86 SSSSSSSS = = = = 3 3SSZSS = SSS3SSS=S = S = SSS = SS3SSZSSeSS = 3 = SS = = SS = S = = = SSSX3 SPECIFIED PARAMETERS Ref. DESIGN A MOTOR #621 Note: To o b t a i n a p r i n t o u t of t h i s f: o Power Supply Rated RMS L i n e Voltage: E l <V> = 220 Supply Frequency: fO <hz) = 60 Type of Connection: 0 f o r D e l t a , 1 f o r Y = 1 No. of Phases: q = 3 o E l e c t r i c a l and Magnetic S p e c i f i c a t i o n s of Motor Rotor: 1 f o r non-magnetic, 0 f o r magnetic = 1 Peak A i r Gap Magnetic F i e l d <s=0>: BO <Wb/m~2> = 0.800 P e r m i t t i v i t y o f Free Space: muO <H/m) = 1.26E-06 Rotor E l e c t r i c a l R e s i s t i v i t y : rho (ohm-m) = 1.72E-08 Number of P o l e s : P = 6 o Motor Geometry Rotor T h i c k n e s s : c <m>~= 4.76E-03 Lamination Width: 1 (m) * 7.62E-02 Rotor Width: 11 (m> = 1.27E-01 No. o f S t a t o r s : Sn = 2 Tooth Depth: t d <nO « 5.56E-02 Each S i d e o f Rotor <m> = 1.81E-03 S t a t o r Back I r o n : d l <m> • 5.00E-02 Mean Diameter: D <m> = 0.470 : SEC (deg) = 360.0 : s s s = = = = s = i : SE s s s s : Fig. A4.1 - Design Spreadsheet *° ANALYSIS OF LIM PERTORMANCE - MOTOR TYPE 6. MODEL N O . : MOTOR #621 F i l e : CUR_ANAL 21-Jun-86> Note: To o b t a i n a p r i n t o u t o f t h i s f i = 3SZ3SX33S333SS** = 3BSC3S&3333£CS83333S3S3333C3S3 = 3Z333 = 3 = 333S3Z3 = S333S3S3 333333X333 3=33333= = = = S SPECIFIED PARAMETERS R e f : DESIGN B o Power Supply Rated RMS Supply C u r r e n t : Io <A> = RMS M a g n e t i z i n g C u r r e n t : I <A> = Supply Frequency: fO <hz> * No. o f Phaaes: q 8 Type o f C o n n e c t i o n : 0 f o r D e l t a , 1 f o r Y » o E l e c t r i c a l and Magnet ic S p e c i f i c a t i o n s o f Motor R o t o r : 1 f o r non-magne t i c , 0 f o r magnetic = No. o f P a r a l l e l C o i l G r o u p s / S t a t o r / P h a s e : Np = No. o f T u r n s / S l o t : N = No. o f T u r n s / C o i l : Nc = No. o f Turns i n S e r i e s : Ns = Number o f P o l e s : P = Rotor E l e c t r i c a l R e s i s t i v i t y : rho (ohm-m) = 1 Diameter o f Copper W i r e : dw <m) = 4 P e r m i t t i v i t y o f F r e e Space: muO <H/m) = 1 300.0 250.0 60 3 1 1 6 36 IS 54 6 .72E-08 . 12E-03 .26E-06 o Motor Geometry R o t o r T h i c k n e s s : c <m> = 4 .76E-03 Each S i d e o f R o t o r <m> = 1.81E-03 L a m i n a t i o n W i d t h : 1 <m> » 7 .62E-02 S t a t o r Back I r o n : d l <m> » 5 .08E-02 R o t o r W i d t h : 11 <m> = 1.27E-01 Mean D i a m e t e r : D <m> = 0.470 by M o t o r : SEC (deg) = 360.0 S p a n / P o l e P i t c h ) : Wp = 0.889 Mean Tooth W i d t h : t Cm) = 1.37E-02 Mean S l o t W i d t h : aw (m) = 1.37E-02 Tooth Depth <m> = 2 .54E-02 No. o f S t a t o r s : Sn 3 2 : = 3 s s s : t S S 3 3 3 : [ S 3 S 3 3 : : 3 3 3 S 3 3 : ! 3 3 = S 3 ! Fig. A4.2 - Analysis Spreadsheet CO CALCULATED PARAMETERS o E l e c t r i c a l Parameters Rated RMS Voltage/Phase: E <V) = 127 Peak S t a t o r S u r f a c e Current D e n s i t y : Jmo <A/m) * 7.84E+04 RMS Magnetizing Current/Phase: Imo = 267.3 Inductance/Phase: Lm (H) = 1.26E-03 F r i n g i n g F a c t o r : f r - 1.20 E f f e c t i v e S t a t o r - S t a t o r Airgap: Ge (m) =» 9.64E-03 Goodness F a c t o r : G = 83 = E l f o r D e l t a 6. El/sqrt<3) f o r V = k-»Ge»B0/mu0 (At s = 0> = 4.42E5«P' s2«Ge»phi/ (q«Kw«D«l»Ns) = E/(2«ir«f 0»lm) (p.184 A l g e r ) = g« ( t + s'w) / <l + f r*g) -= muO»c»va»«2/(rho»Ge«o«ega) Geometrical Parameters S t a t o r Length: Ds (m) = 1.476 Wavelength: lambda <m> = 4.92E-01 Pole P i t c h : p <m> = 2.46E-01 No. of Slots/Phase B e l t : n - 3 Mean Tooth Width: t <m) = 1.37E-02 Mean S l o t Width: sw (m) = 1.37E-02 Synchronous V e l o c i t y : vs (m/s) = 30 Synchronous Frequency: f s 3 20.0 Synchronous RPM: SRPM = 1200 a D « T T » S E C / 3 6 0 = 2«Ds/P = Ds/P = p/(t*sw)/q = p/(2»n«q> (sw = t ) = lambda»f0 = va/ic/D = fa«60 Peak Flux and Flux D e n s i t i e s T o t a l F l u x / P o l e : phi (Wb) * 9.55E-03 Average Flux D e n s i t y : Bav (Wb/m^2) = 0.509 Average Tooth Flux D e n s i t y : Bta (Wb/m's2) = 1.019 Peak Tooth Flux D e n s i t y : Btp (Wb/m/s2) = 1.600 Peak Core Flux D e n s i t y : Be (Wb/m^2) = 1.253 (2/T T) •B0«l«p (2/T T) «B0 p h i / (n«l«qi»t) Tr/2»Bta p h i / l / d l ( D i v i d e d by 2 f o r a 360 (Maximum a c c e p t a b l e =1.5 Wb/m~2) Fig. A4.3 - Calculated Parameters (First Half) ^ o C o i l Parameters Winding P i t c h : Wp * P i t c h F a c t o r : Kp * D i s t r i b u t i o n F a c t o r : Kd * Winding F a c t o r : Kw * No. of Turns i n S e r i e s : Ns * No. of T u r n s / C o i l : Nc = No. of T u r n s / S l o t : N * No. of P a r a l l e l C o i l Groupa/Stator/Phase: Np * o Wire Gauge & Copper Losses i n Windings S l o t C r o s s - S e c t i o n a l Area: Sa (n A2) * Useable Area/Conductor: Aw (m"2) 3 Diameter of Copper Wire: dw <m) = Length of C o i l / T u r n : l c (m> * Resistance/Winding of Ns Turns: Rw (ohm) = Current/Winding: Iw (A) » T o t a l Copper Losses due to Im: Wcu (KW) = Current Density i n Wire: Jw (A/cm"2) = 0.8889 = C o i l Span/Pole P i t c h 0.9848 = sin(Wp«*/2> 0.9598 « sindr/(2»q> > / (n»sin (ir/2/n/q) ) 0.9452 = Kp»Kd 54 * E/(4.44«Kw«f0«phi> 18 » Ns/n 36 * 2«Nc 6 7.60E-04 » td»sw 1.06E-05 = 0.5»Sa/N (50* Packing F a c t o r ) 4.12E-03 (from Look-Up Table) 8.09E-01 = 2»l*3»Wp«p 5.66E-02 » Ns»lc»rho»4/Tr/dw"2 22.27 » Im/(Sn»Np) 1.01E+00 = IW*2»Rw«Np«Sn«q/1000 (3 phases) 167 = 4«Iw/<ir»dw*2>. (Maximum Acceptable Fig. A4.A4 - Calculated Parameters (Second Half) -a cn THRUST AND POWER CALCULATIONS o Main T h r u s t * FO FO * l«Jm«B0»D/2«cos(psi)«ain(psi) *kl«coa(psi)*sin<psi) k l * 3075.007 t a n ( p s i ) 35 2« T T»s«f o»alpha/(k~2+vo2> o E n t r a n c e T h r u s t • F l F l = Jm«l/2«Re(bl«(1-expC(rl*i«k)«D1)/<rl+i«k)J o E x i t T h r u s t i s not c a l c u l a t e d , but i s u s u a l l y s m a l l , o Optimum Ro t o r R e s i s t i v i t y f o r Maximum T h r u s t as a F u n c t i o n o f S l i p a (X) = 5 10 15 20 25 30 35 40 rho (ohm-iO = 9.77E-09 1.95E-08 2.93E-08 3.91E-08 4.89E-08 5.86E-08 6.84E-08 7.82E-08 8.80E = . . 3 = = = * = =i = = = : J = = x = n = = = : J = = , : = ^ CONSTANT FLUX MODEL S p e c i f i c T h r u s t S l i p v (m/s) p s i Jm (A/m) Im (A) FO (N) F l (N) F t (N) F t ( l b ) ( l b / i n ~ 2 ) OX 29.5 0.000 73179 250 0 0 0 0 0.00 2X 28.9 0.223 75045 256 699 0 699 157 0.90 4X 28.3 0.427 60384 274 1398 0 1398 314 1.80 6X 27.8 0.598 88570 302 2097 0 2097 471 2.70 8X 27.2 0.738 98899 337 2795 0 2795 628 3.60 F i g . 3 - A n a l y s i s Worksheet Fig. A4.5 - Table of Output Values ^ 177 A4.3.2.1 Specified Parameters In the first section, there are three sub-sections for entering the parameters on which the analysis is based. Supply Power: The rated supply current and/or voltage, the frequency, the number of phases, and the type of connection (Y or Delta) are entered in this section. Electrical and Magnetic Specifications: The parameters entered in this section depend on whether the analysis is of an existing motor or a new motor. In the latter case, only the rotor material, the peak air gap magnetic field, the rotor resistivity and the number of poles are specified, while in the first case the various coil parameters are also entered, as illustrated in Figures A4.1 and A4.2, which are printouts of parts of the Design Worksheet and the Analysis Worksheet. Motor Geometry: The rotor and stator dimensions and geometry are specified in this area. Again, fewer parameters are needed if a new motor is being designed, as illustrated in Figures A4.1 and A4.2. Immediately to the left of the cell in which the data is entered, a description of the parameter is entered, along with its abbre-viation and the units used. A global cell protection feature can be used to restrict users to entering data only in the desired 178 cells, thus minimizing the chance of an inexperienced operator accidentally modifying the formulae or the documentation. A4.3.2.2 Calculated Parameters In this section of the spreadsheet, the remaining electrical and geometrical parameters and the coil parameters are calculated, along with magnetic fluxes and copper losses in the windings. In the case of a motor design, a wire gauge for the coils is recommended. In the cell immediately to the left of the parameter value, a description of the parameter, along with its abbreviation and the units used, is placed. Immediately to the right of the parameter value, the actual formula used to calculate the value is recorded, thus providing complete documentation in the spreadsheet and on the printout. A4.3.2.3 Thrust and Power Calculations In the final section of the spreadsheet, a complete table of thrust and power, under varying operating conditions, is generat-ed. The calculations required are involved since they contain complex numbers. Symphony does not have the capability of handling complex numbers directly; however, they can be handled as normal calculations by keeping track of the real and imaginary parts of the calculation in separate columns. 179 Once one set of calculations is formulated correctly, using relative and absolute cell addresses as required, the powerful copy features of Symphony are used to generate a complete table of parameters for printing or for generating graphs. (The terms "relative" and "absolute" refer to the manner in which cell addresses are treated in formulae. For example, if a set of formulae on one line is copied to the line below, all relative cell addresses remain unchanged.) An example of how the table generation is handled is shown in Table A4.1, where the formulae for two lines of the Thrust and Power Table are printed. Once the first line of the table has been formulated, the remaining lines are immediately generated by the Copy command. Note that when line 88 was copied to line 89, the cells modified with the $ signs are absolute references and are not incremented. On the other hand, cells without the $ are relative addresses and are incremented by one row. This is seen by referring to the formulae in cells F94 and F95. Note that more columns than expected are used to perform the calculations, because of the complex numbers. The intermediate calculations are used to keep track of the real and imaginary parts of the calculations. The intermediate calculation columns are kept to the right side of the worksheet where they do 180 ASS: (tt) 0 B881 (Fl) •»*42»(l-fl88) CSS: (F3) MTftN(«I«31tABB) DS8: (FO) •$M30/«C0S(C88) ESS: (FO) •«$31/K0S(C88) FBS: (FO) +*C*75««IN(C8B)/K0S(C8B) 688: (FO) *ftD88*«F«15 H6S: (FO) +F88+B88 IBS: (FO) 4H88M.U8 J8S: (F2> •I8B/t«ll/«B*38/39.*A2 K8S: (Fl) U-fl88)«3.281«I88/550*iB$42 LBS: (Fl) 0.746«KB8 M88: <F3) -*Bi47t«IN(C88) AB9: (tt) 6.02 BS9: (Fl) +«*42»(1-A89) C89: (F3) lflTfiN(«I«3Hflfl9) D89s (FO) HBt30/iCGS(CS9) .E89: (FO) «*W31/K0S(C89) F89: (FO) HC$75fKIN(C89)/eC0S(C89) S89i (FO) +flD89**F«15 H89: (FO) +F89+B89 189: (FO) +H89/4.448 J89: (F2) •I89/««11/$B<38/39.4A2 KB9: (Fl) (l-fl89)«3.281»I89/550«iB*42 L89: (Fl) 0.7*6»K89 W9: (F3) -$BiA7i*SIN(C89) Table A4.1 - Thrust and Power Formulae not appear as part of the normal viewing area or printout. The results are available in tabular or graphical form. One form of the graphical results is shown in Fig. A4.6. The graphical results can also be displayed simultaneously with the tabular results which is very helpful to students learning about the effect of motor parameter variation on motor performance. A4.3.3 Input/Output The spreadsheet programs have good graphics capabilities which can be used to obtain graphs from data produced by other computer programs, for example Fortran programs. The procedure is to write the output results to a separate file using the F10 format code. The resulting file is then imported into the spreadsheet program where the data can be used to produce graphs or be integrated into word processing documents. MOTOR #621 181 DESIGN B Fig. A4.6 - Graphical output of spreadsheet motor simulation. A4.4 Comparison With Experimental Result To verify that the computer model would simulate the performance of LIMs a comparison was made between the experimental results of motor tests and the simulation. One of these comparisons is shown in Fig. A4.7 where the test and simulation results for Experimental motor #3 (this motor is completely described in Chapter 5) are presented. Although the model does not appear to be as accurate as one would expect, the windage loss due to large amounts of water applied to the rotor results in a large unknown drag on the rotor. The model can be refined to obtain increased accuracy, if required, but this was not considered to be important as the model is used as a tool to analyse the effect of paramater variations in this thesis. 182 Experimental Results of Test U9S % synchronous speed • Power Fig. A4.7 - Comparison Between Simulation ( ) and Experimental Result ( ) For Experimental Motor #3 A4.5 Conclusion The spreadsheet analysis method has been shown to be a valuable new tool in the analysis of induction machines. The original 183 simulation program was developed using the Fortran programming language and took approximately two man months to write, debug, obtain graphical output and document. The spreadsheet analysis method required only five man days to obtain the same results. In addition, the spreadsheet program was better documented and could be used by other engineers with very little instruction. The computational time of both programs was the same at approxi-mately 23 seconds. The results from these two simulations are discussed in the next Chapter. The use of spreadsheet programs and personal computers has resulted in an analysis tool with many new applications. These may include the use of spreadsheets for finite element analysis and transient analysis of electrical machines. When newer versions of the spreadsheet programs with more mathematical functions become available, than even greater numbers of applica-tions will be possible. 1 3 I i i i I I I i i . iiiiiiii.iiiuiiiuiiiiiinhiiii»Miniiii.IIIR,II t" ARMCO D I - M A X M - 1 9 C R F P 2 9 GAGE ( . 3 5 MM) T H I C K T Y P I C A L CORE LOSS & E X C I T I N G POWER AT 6 0 AND '100 HERTZ 0 1.0 ^ .4 .7 CORE LOSS OR E X C I T I N G POWER - W/LB OR V A / L B ( X 2 . 2 0 5 = W/KG OR V A / K G ) CO TF-V 185 Appendix 6 Finite Element Analysis of Experimental Motor #2 (Courtesy of G.E. Dawson, Queens University) 186 Paqe 1 10-28-87 21:25:42 7 Microsoft F0RTRAN7 V3.20 02/84 «* This Program Does The Analysis of Relative End Efect * • * is is relendos. Real»8 RELEND,SLIP,GOOD.PAIR,PI,BONE, KN,KP,TEST2 COMPLEX*16 ALPHAOB,ALPHATB.NEGEYE.CALFAOB,CALFATB.TEST1.TEST3, *TEST4,TEST5,TEST6,CRELEND.TEST7.TEST8 NEGEYE=(0..-1.0) PI=3.1415 G0D=42.0 PAIR=4.0 DO 100 J=20.42.22 G0D=J WRITE < »,40)GOOD,PAIR 40 FORMAT(IX,'GOODNESS=',F7.1,' NUMBER OF POLE PAIRS='.F7.1) DO 100 1=0,99,1 SLIP=I/100.0 SLIPSQ=(1.O-SLIP)**2 BONE = SQRT(1.0+(4.0/(GOOD-SLIPSQ) *»2) KP=SQRT(0.5«(BONE+1.0) KN=SQRT(0.5«(BONE-1.0) TEST2=0.5*G00D«(1.O-SLIP) TEST3=CMPLX(KP,KN) TEST4=TEST3-1.0 ALPHAOB=0.5»G00D*<1.O-SLIP)*(TEST3+1.O) ALPHATB=-0.5«G00D»(1.0-SLIP)*(TEST4) CALFA0B=DC0NJG(ALPHAOB) CALFATB=DCONJG(ALPHATB) TEST5=2.0*PI*PAIR*(CALFATB+NEGEYE) 2 29 TEST6=CDEXP(TEST5) 2 30 TEST1=2.0»PI«PAIR*(CALFATB+NEGEYE) 2 31 C WRITE< *.50)PI,PAIR,BONE,KP,KN,TEST2,ALPHATB.CALFATB,NEGEYE.TEST1. 2 32 C -TEST3,TEST4.TEST5.TEST6 2 33 C 50 FORMAT(lx. 'REAL',6F10.3, COMPLEX'.8F14.7. .8F14.7) 2 34 CRELEND=(NEGEYE* <CALFAOB + SLIP*GOOD)*(TEST6-1.0) 2 35 •/(1.0+NEGEYE«SLIP*GOOD)*(CALFATB-CALFAOB)*(CALFATB*NEGEYE) 2 36 TEST7=(NEGEYE*(CALFAOB+SLIP*GOOD)*(TEST6-1.0) 2 37 TEST8=(1.0+NEGEYE*SLIP"GOOD)*(CALFATB-CALFAOB)*(CALFATB+NEGEYE) 2 38 RELEND=REAL(CRELEND) 2 39 WRITE(«,110)slip.relend 2 40 110 FORMAT(IX, 'SLIP=',F10.3,' RELATIVE END EFFECT='.F10.3) 2 41 C WRITE(*,15)TEST7.TEST8 2 42 C 115 FORMAT(IX,'TEST7=',2F10.3,' TEST8=',2F10.3) 2 43 100 CONTINUE 44 STOP 45 END Name Type Ofset P Class ALPHAO ALPHAT BONE CALFAO CALFAT CDEXP CMPLX CRELEN DCONJG GOD I C0MPLEX*16 C0MPLEX*16 REAL*8 C0MPLEX*16 C0MPLEX»16 C0MPLEX*16 REAL-8 INTEGER»4 178 210 114 242 258 370 26 98 INTRINSIC INTRINSIC INTRINSIC D Li n e J KN KP NEGEYE PAIR PI REAL RELEND SLIP SLIPSQ SORT TEST1 TEST2 TEST3 T E S T 4 TEST5 TEST6 TEST7 TEST8 # 1 7 INTEGER»4 REAL'S REAL»S C0MPLEX«16 REAL«8 REAL*S REAL'S REAL»8 REAL C0MPLEX«16 REAL»8 C0MPLEX«16 COMPLEX-16 C0MPLEX»16 C0MPL£X«16 C0MPLEX«16 C0MPLEX»16 42 130 122 2 34 18 626 102 110 338 138 146 162 274 306 498 546 Page 2 10-28-87 21:25:42 M i c r o s o f t F0RTRAN77 V3.20 02/84 INTRINSIC INTRINSIC Name MAIN Type S i z e C l a s s PROGRAM Pass One No E r r o r s Detected 45 Source L i n e s 

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