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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  degree at the  thesis in  partial  University of  fulfilment  of  the  requirements  for  an advanced  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  department  this or  thesis for by  his  scholarly purposes may be granted by the  or  her  representatives.  It  is  understood  that  head of copying  my 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  DE-6G/81)  Itjy. }£,  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  of  analytical  methods  electrical machine theory was type  of  motor.  available  from  the  field  used to investigate the particular  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. previous  Because of these differences, the methods used by  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  end effect and the edge effect.  of the  rotor material, the  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. measurement  In addition, inverters, dynamometers, flux  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. analysis  of the  rotor/stator attractive  The fifth is the  force for magnetic rotor  double sided motors and a description of the flux (crenelated flux) which  causes the  force.  Finally,  re-entry effect may occur is presented.  a criterion for when  the  iv Table of Contents Page ABSTRACT  ii  TABLE OF CONTENTS  iv  LIST OF TABLES  vii  LIST OF FIGURES  viii  NOMENCLATURE  xi  ACKNOWLEDGEMENT 1.  2.  3.  4.  xiv  INTRODUCTION  1  1.1 1.2 1.3 1.4  1 2 4 8  Objectives of Thesis Application of Thesis Results Historical Background Summary  GENERAL FORMULATION  10  2.1 2.2 2.3 2.4 2.5 2.6 2.7  10 11 11 16 20 31 34  Description of Linear Induction Motors End Effects Review of Analysis Techniques One-Dimensional Analysis Effect of Annular Stator The Re-Entry Effect Conclusions  NORMAL FORCES  35  3.1 3.2 3.3 3.4 3.5  35 37 41 44 49  Introduction to Normal Forces Saturation Conditions Normal Force Equations Experimental Results Conclusions  EXPERIMENTAL APPARATUS  51  4.1 4.2 4.3  51 51 61 61  Introduction Motor Descriptions Description of Experimental Apparatus 4.3.1 Power Supplies  V  4.4 4.5 5.  4.3.2 Speed Measurement 4.3.3 Thrust Measurement 4.3.4 Current Measurement 4.3.5 Power Measurement 4.3.6 Flux Measurement Experimental Accuracy Conclusion  EXPERIMENTAL RESULTS AND DISCUSSION 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8  Introduction Verification of the Computer Model Effect of Segmented Rotor Effect of Rotor Variations Effect of Odd or Even Number of Poles Effect of Series and Parallel Connection Edge Effect Effect of Harmonics 5.8.1 Slot Harmonics 5.8.2 Time Harmonics 5.9 The Effect of the Annular Motor 5.10 Discussion  6.  FACTORS WHICH AFFECT OPTIMIZATION 6.1 6.2 6.3 6.4 6.5 6.6 6.7  7.  Introduction to Optimization Goodness Factor Optimum Goodness Factor Analytical Results Optimization By Scaling Up Series and Parallel Connection Optimization General Comments On Optimization  61 62 62 62 64 66 67 69 69 72 78 86 93 98 103 112 112 120 120 123 127 127 127 128 130 132 134 136  CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER RESEARCH  138  7.1 7.1 7.2  138 139 145  Summary Conclusions Recommendations For Further Research  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  - The effect On the Performance Of Experimental Motor #2 With and Without Misaligned Teeth  118  Table 5.3 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  Fig. 2.2  - Pattern of currents in rotor to show entry effect  10  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 Fig. 2.8  - Curvature of the annular LIM stator. - Simulation of the power and thrust produced at the inner, middle and outer radii of the stator.  20 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 Applied To Experimental Motor #1.  Fig. 2.12  - Annular Resistivity Correction Factor Applied To Experimental Motor #2.  31  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. - Attractive force vs rotor position  40  Fig. 3.4  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  - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 3 mm Stainless Steel Rotor  74  - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 2 mm Stainless Steel/Copper Rotor  75  - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 2 mm Copper Rotor  76  - 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. 5.2  Fig. 5.3  Fig. 5.4  Fig. 5.5  Fig. 5.13 Fig. 5.14 Fig. 5.15 Fig. 5.16  - Experimental Points and Approximating Curves Showing Efficiency  89  - Experimental Points and Approximating Curves Showing Power-Factor  90  - Experimental Points and Approximating Curves Showing Power-Factor Efficiency Product (Zeta)  91  - 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 - Power Produced By the Entry Pole of a Series and Parallel Connected Stator 100  Fig. 5.20 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  - 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  - 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  -The Effect of Misaligning the Stator Teeth  117  Fig. 5.22  Fig. 5.29  Fig. 5.32  xi Fig. 5.33 Fig. 5.34 Fig. 5.35 Fig. 5.36  - Slots Cut In Rotor To Eliminate Rotor Slot Harmonics  119  - The Stator Tooth And Slot Profile For the Annular Stator  121  - Lateral Variation in Flux For Experimental Motor #1  122  - 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 Fig. 6.3  - Power for various rotor resistances. - Possible Parallel Connected Stator Designs Compared To a Series Connected Stator  132 136  NOMENCLATURE A = stator surface area b = air gap flux b = -iBk /(k +iswa) bV = [Va+k tantfl-iv/kyON -v-Va)]b_ 2  2  n  F = thrust produce by a motor without end effects F ^ = thrust due to the end effect F = stator/rotor net normal force F * = normal force on rotor due to stator 1 F = 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) Q  n  r g  r g  k  K 1 ly L E P r  r-,  = 2TZ1T\  = [l-exp{(r- +ik)Pn}]/[r+ik] = stator width = width of rotor = stator width = stator length = number of stator pole pairs = radial distance in cylindrical co-ordinates L  1  = Va /2{l-[l+4(iwa+v y(Var]} 2  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 r =c*d -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.*c/(p*g) S  r  0  2  isw u ^— " g  B  =k +  5  = c(l -l)/2 = Rg-R-j^  >  r  xiii  tanh pa tanh k(c - a)  1+ • V l + isG  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" (swc</(k +v ) O = magnetic flux ® = crenelated flux (tooth-rotor-tooth leakage flux) w = frequency in radians per second Q. = volume resistivity or ohm 1  2  2  c  Mathematical Operators + * / exp ln db/dx dVd'x  = addition = subtraction = multiplication (used if equation would be ambiguous) = division = the natural exponential = the natural logarithm = differential operator = partial differential operator  xiv  Acknowledgement The author wishes to thank the Natural  Science Council of B.C., the  Sciences and Engineering Research  Council  Engineering Ltd. which have all supported this work.  and Cetec 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 supervisor, Dr. Dunford, for the confidence he showed in my ability.  1 1. INTRODUCTION  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. been  designed,  analyzed  and  applied  Although LIMs have 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 involved  analysis  applying  was  developed,  conventional  electromagnetic model.  one  theories  aspect  to  a  of  steady  which state  The resulting equations were then set  up for annular motors by using cylindrical coordinates and a resistivity  correction  factor.  As  theoretical analysis, a spreadsheet  a  to  have  significant  in  developing  this  program for simulating the  performance of LIMs was developed. proved  tool  This spreadsheet method  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 investigation undertaken in order to determine the factors affecting their magnitude.  3. Simulations analyze  and  the  experiments  motor  were  parameters  undertaken which  in  affect  order to efficiency,  power-factor and power density since the maximizing of these factors is simulations  an important objective and experiments  in the  consisted  final  design.  The  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, experiments  type  of connection  and windings).  These  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 transportation 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.  circular saws require a thick plate of 4-5 mm  Conventional  large  i n order to support  Fig. 1.1 - Prototype L I M driven Saw the thermal The  and  proposed L I M  thickness.  mechanical  stresses without the blade  driven saw  distorting.  has a design goal of a 2 mm  blade  Band saws, which use a comparable thickness of blade,  are presently used i n sawmills machines and  expensive  but they are very large and costly  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 Company applied for a patent supplied magnets.  by  a  The  mechanical first  linear  motor.  The Electric Shuttle  [2] in 1859 for a linear motor  inverter  which  induction  energized  motor  was  sequential 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 launching 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 transportation (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  experiments  as  a  rotary  motor.  were conducted, it  was  However,  when  found that  high  there  speed  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 rotor  cannot  transferred)  Fig. 1.2).  change the  significant thrust.  As the magnetic field in the  instantaneously  rotor  in  the  (because  entry  area  energy does  must  not  be  produce  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. essentially Boldea  This is  the same conclusion previously reached by Nasar and  [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. makes end effects important.  The high blade speed (140 m/s)  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. order to  In addition, the LIM must have good power density in 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.  10 2.  GENERAL FORMULATION  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  magneticfieldand 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. new  This effect occurs due to  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 material enters the stator.  begin to flow as the rotor  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. fact  the  negative  rotor rather  currents than  which  positive  do initially thrust  [7].  form These  In  can produce 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  pLSM  RS+pLSS  pLSM  pLSM  pLSM  RS+pLSS  pMSRcos ( 0-~) pMSRcos(e+|^)  pMSRcos8  pMSRcos(e+~)  2TS  pMSRcos(0 ^ )  pMSRcosG  pMSRcos ( O— ) 11  pMSRcos0  2  pMSRcos ( 0 ^ ) pMSRcos(9+|^)  pMSRcos( €Hj2) pMSRcos ( © - ^ ) pMSRcos 0  RR+pLRR  pMSRcos(0~)  pMSRcos (e*y^)  pMSRcos( 0+y^)  pMSRcosO  pMSRcos0  pLRM  pLRM  pLRM  RR+pLRR  pLRM  pLRM  pLRM  RR+pLRR  LSS- - per phase stator s e l f inductance LSM - per phase mutual Inductance between stator windings LRR - equivalent  per phase rotor s e l f 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. model the component  current and flux only.  densities are functions  The following  three  equations  In this of the x  describe these  conditions, (all variables are steady-state values)  (2.1)  - the change in flux along the motor is equal to the current density flowing in the stator and rotor  i  - *J C  ( S  - the change in current flowing along  the return paths of the  rotor is equal to the current density flowing in the rotor  dj  2*1  .db i*w*b + V*~5k P  (2.3)  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 dx  u*c*j*J g  (2.1)  di *• Hx = J  (2.2)  C  A dx  db i*w*b + V*- dx  2*1  dj Hx"  Vn cA  1  0  3  P g  w  dx  2  ^o P g  c +  (2.3)  2 c T 2 db "dx 1 (2.14)  [d J j2 dx s  2c  2  -  1 S  r  Equation (2.14) is then solved to obtain the magnetic field value [29].  The thrust can then be calculated from the stator surface current  19  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 ] .  =^*J *B*l*p*L*cos<!)*sin<))  (2.15)  -^J *B *l*Real[(b^ )*(b /B )*K]  (2.16)  FQ  F  b B  =  m  0  0  p  p  0  = [V*a+k*tan(|)(l-i*v/k)/(r -v-V*a)]*b 1  p  = u. *J/(k*g)  Q  b  Q  = -i*B *k /(k +i*s*w*a) 2  p  2  Q  = sqrt ( - 1 )  i r  x  l  m  = V*oc/2*{l-[l+4*(i*w*a+v /(V*a)* ]} 2  x  2  V  = velocity  v  = 4/stator width/total rotor overhang  a  = \i *c/p/g  p  = rotor resistivity  K  = {l-exp[(r +i*k)P*Ti]}/[r +i*k]  Q  1  1  = tan" (s*w*c</k +v ) 1  L J  2  2  = length of stator 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 applications in linear induction motor test apparatus [26,27] and commercial 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  resistivity  up  using  correction  cylindrical factor  was  co-ordinates  and  an annular  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 = n/g*(c*j+J)  (2.17)  dl/rdG = c*j  (2.18)  0  dj7rd9-(2/(8*S))*I r  (2.19) = -{(i*w*b)+((r*d8/dt)*db/rd9)}/p  22 Then write out the terms of equation (2.19)  dj/rde-2/(5*S)*I = -i*w*b/p-(r*d9/dt)/p*db/rd9 r  (2.20)  Taking the first derivative of (2.20) with respect to 9  d j/rd9 - 2/(5*SJ*dI/rd9 = 2  2  2 2 -i/p*w*db/rd9 - (r*d9/dt)/p*d^b/rd9  ( 2  /!  -  2 1 )  Now taking the first derivative of equation (2.17)  -d b/rd8 = ]i *c/g * dj/rd9 + \i /g*dJ/rdQ 2  (2.22)  2  Q  Q  Taking the first derivative with respect to 9  -d b/rd9 = u *c/g*d j/rd9 + u /g*d J/rd9 3  3  2  2  2  Q  2  Q  2  (2.23)  2  From (2.23) obtain an expression for d j/rd9 d j/rd9 = -g/M. /c*d b/rd9 - l/c*d J/rd9 2  2  3  G  3  2  2  (2.24)  23 and obtain an expression for j by rearranging (2.17)  j = -g/u /c*db/rd9 - J/c  (2.25)  0  Substitute the expressions (2.18) and (2.24) into equation (2.21)  d j/rd9 - 2/(8*S)*dI/rd9 2  2  r  =  (2.26)  -l/p*i*w*db/rd9 - (r*d8/dt)/p*d b/rd8 2  2  -g/u. /c*d b/rdG - l/c*d J/rd9 - 2/(5*S)*c*j = 3  3  2  2  0  r  -l/p*i*w*db/rd9 - (r*de/dt)/p*d b/rd8 2  2  (2.27)  Substitute (2.25) into (2.27)  -g/(u. *c)*d b/rd0 - l/c*d J/rd9 - 2*c/(5*S)* 3  3  2  2  o  r  {-g/(|X *c)*db/rd9-l/c*J} =-l/p*i*w*db/rd9 0  (r*d9/dt)/p*d b/rd9 2  2  (2.28)  24 Gathering together terms  -g/u /c*d b/rd9 - l/c*d J/rd9 + 2*g/(5*S*u )*db/rd9 3  3  2  2  0  r  0  + 2*J/5/S„ =-l/p*i*w*db/rd6 - (r*d9/dt)/p*d b/rd9 2  2  Rearrange to get:  -g/(M. *c)*d b/rd9 + (r*d9/dt)/p d b/rd9 + {2*g/(5*S*M-) + 3  3  2  2  Q  r  0  i*w/p] =l/c*d J/rd9 - 2*J/(8*S ) 2  2  r  Multiply through by (-u *c/g) to get: 0  d b/rd9 - (r*d9/dt)*u. *c/(g*p)*d b/rd9 - {2*c/(8*S) 3  3  2  2  0  r  + u *c*i*w/(g*p)}db/rd9 = -u. /g*d J/rd9 + 2u. *c*J/87S/g 2  G  2  Q  0  I  d b/rd9 -(r*d9/dt)*u. *c/(p*g)d b/rd9 -[i*w*uo*c/(p*g)H 3  3  2  2  0  (2*c/(5*S ))db/rd9=-u. /g(d J/rd9 -(2*c/8*S )*J) ]  r  2  0  (2.30)  2  r  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 = *r *0*J *B *(R -R )*cos(|)*sin(!)  (2.31)  T = - *r*J *B *(R -R )*Real[(b /b Xb /B )*K]  (2.32)  2  Q  x  m  m  0  0  2  2  1  1  B  Q  = >i*J/(k*g)  b  p  = -i*B *k /(k +i*s*w*cc)  p  p  0  0  2  bj  1  2  0  = [(rWdt)*cc+k*tan<j)*(l-i*v/k)/ (r -v-(r*de/dt)*a)]*b 1  p  K  = [l-exp{(r +i*k)P*Ti}]/[r +i*k]  k  = 2*7i/ri  r  1  1  = (r*d6/dt)*0(/2{l-[l+4*(i*w*a+v )/(a*r*de/dt) ]} 2  x  v  = 4/stator width/total rotor overhang  a  = |j.*c/(p*g)  p  = rotor resistivity  P  = number of pole pairs  < > |  = tan" (s*w*ot/(k +v )  T]  = wavelength  R^  = stator inner radius  R2  = stator outer radius  8  = J^2"^l  0  1  2  2  2  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)  27  Q. O JZ  O cantor  slip + inn er  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 be considered.  for annular motors, three cases will  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. d r  J Rl Resistance  o f Wedge =  r i  p l n (R2/R1) r de  t e  Jo  Resistance  Ka  o f Block  = Resistance Resistance  =  D <R2 - R l ) t 6 (R2 + R D / 2  o f Wedge o f 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. k =[ln(.0762/.0190)*(.0762+.0190)]/[2*(.0762-.0190)] Q  Si  k =1.15  MOTOR PARAMETERS Hypothetical Motor Number: 1  Type: Segmental  Winding Parameters 4 3 6.19 cm .966 5.16 cm 1:6 .966 24 30 60 (4 poles in parallel) .254 m 18 Wye parallel  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 Primary Thickness Tooth Width Slot Width Slot Depth Total Primary Length Active Primary Length Number of Stator Slots (single layer) Stator Slots (total) Stator Slots (active) Stator Slots (half-filled) Secondary Thickness Primary-Secondary Gap per side Secondary Overhang per Side Magnetic Gap (total) Mean Radius  5.175 cm 3.81 cm 5.16 mm 5.16 mm 2.86 cm .314 m .298 m 24 29 29 19 2.5 mm 1.4 mm 2.54 cm 2.8 mm 4.76 cm  Electrical Parameters Primary Resistance per Phase Secondary Resistivity Frequency Voltage Current Rating (continuous)  .0569 Q 26 uQ'cm 400 Hz 440 V 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  O  0 . 3  0 . 5  Amps.  0 . 7  +  0 . 9  1  .  1  1.3-  T (h o u s o n d s ) R P M Torque ( N - m )  1  .  5 1  O  .  7 1 . ; } 2.1  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 The  Re-Entrv Effect 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  performance of the motor.  the  32  (Thousonds) •  RPM  Am pr..  +  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  interference will occur.  wave  is  significantly  shorter,  then  no  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 B = u *J/(k*g) 0  0  b = -i*B *k /(k +i*s*w*a) 2  p  r  2  0  V*0(/2*{l-[l+4*(i*w*a+v /(V*a)* ]} 2  1 =  2  V = velocity v = 4/stator width/total rotor overhang a = M- *c/p/g Q  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 parameters).  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 ates.  In  field  analysis  equations  in  addition an annular resistivity  cylindrical co-ordincorrection 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.  35 3. NORMAL FORCES  3.1 The  Introduction to Normal Forces analysis of the normal forces is important in the design of  the steel rotor LIM because a large unbalanced normal force will create design.  large  friction  losses  which  The large friction forces  result  in  a  low  efficiency  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 interaction of the magnetic field of the stator and currents in the rotor ("electromagnetic normal forces"). Figure 3.1.  These flux paths are shown in  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 = B A/(2*u ) 2  g  0  (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  Main Flux Path  nn  z  -Crenelated Flux Path "i_  1 ' 1  i 1 i  i  1  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  equation (3.5).  for  the  experimental  motor  is  given in  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. the  center position in the  Once the rotor is far from  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  flux  on the rotor  would be as great as for the stator-to-stator force.  However, the  rotor does saturate,  and is  pole-to-pole  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:  (3.2)  % mainflux=4*;c*c*x/r|*100  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. much greater value.  Experimental results, however, showed a  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:  %main flux= 2*c*t/t*100  where t is the width of one stator tooth.  (3.3)  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: <J> =[sin(w*t)-sin(w*t+20 degrees)]*mmf/(2*a) C  (3.4)  and the force: F=(G>,2 * i o ) / ^ * 8n) neutons 7  (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  20  30  40  50  60  70  80  90100110120130140150160170180  elecMcal  degrees  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 currents  set  the  stator  source, the current.  stator flux and the rotor 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. the  center  circuit  position,  theory)  since  the the  However, when the blade moves from flux  increases  driving  force  linearly (from magnetic 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:  (3.7)  rsn ~~ rsl" rs2  The force on the blade, for Experimental Motor #2, is then given by the equation:  F rsn = Frsl 1 - F o rs2 F  (3.7)  rsn = (.37/(a+g))-(.37/(g-a))*4450 2  2  (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  -0.24 •  .25mm  - 0 . 1 9 - 0 . 1 4 - 0 . 0 9 - 0 . 0 4 0.01 distance  clearance  of blade  +  CLEARANCES  0.06  off  . 5 m m  0.11  center  o  0 . 1 6 0.21  .75mm  A  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 = u *J *A/(g*4) 2  0  Where J  m  is the maximum stator surface current density.  (3.9)  For the  1 m m  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 device  was  were conducted.  constructed  to  test  First, a simple measurement the  electromagnetic  could be exerted on a sample of saw blade material.  force  that  Second, the  coefficient of drag for the normal force pads in the machine was measured for different speeds. the  rotor backwards, and the  Third, the force required to move forward thrust,  while  energized,  were measured in order to calculate the friction and the electromagnetic 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  45  experimental conditions. A diagram of the experimental set-up is  Fig. 3.5 - Attractive Force Experimental Apparatus  46  Fig. 3 . 6 - Sample of Saw Steel Being Tested  shown in Fig. 3 . 7 . The measured force ( 3 5 6 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  Coefficient of Friction for Delrin ( V a r i a t i o n With  Speed.)  c  c  fr  o o  (Thousands') Spoud in RPM  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  low  The friction occurs between the rotor and  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  electromagnetic thrust of 507 N and a friction of 347 N.  then  gives an  49 If the 347 N of friction is accurate then a normal force of 2180 N (347/0.16 = 2180 machine.  N) is acting on the rotor of the four pole  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  Motor #2.  acceptable  value  than that  measured in Experimental  51 4. EXPERIMENTAL APPARATUS  4.1 Introduction This  chapter  designed  describes  the  and constructed,  experimental  the  measurement  motors  which  apparatus  possible error which may occur in the measurements.  were  and the 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). in Fig. 4.1-4.3.  The stator cores are shown  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  performance.  connected poles  and to measure  full  size motor  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,  sized machine) were punched into the laminations  the full  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  53  steel  laminations  will  heat  prohibitively  (this  was  experimentally) even with the very low lamination pressure. only  disadvantage  of  the  machined  slots  for  found The  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 Type: Segmental Stator Core: A  Experimental Motor Number: 1 Winding Parameters  4 3 6.19 cm .966 5.16 cm 1:6 .966 24 30 60 (4 poles in parallel) .254 m 18 Wye parallel  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 Primary Thickness Tooth Width Slot Width Slot Depth Total Primary Length Active Primary Length Number of Stator Slots (single layer) Stator Slots (total) Stator Slots (active) Stator Slots (half-filled) Secondary Thickness Primary-Secondary Gap per side Secondary Overhang per Side Magnetic Gap (total)  5.175 cm 3.81 cm 5.16 mm 5.16 mm 2.86 cm .314 m .299 m 24 29 29 10 2.5 mm 1.4 mm 2.54 cm 2.8 mm  Electrical Parameters Secondary Resistivity Frequency Voltage Current Rating (continuous)  26 (iflcm 400 Hz 440 V 40 A Table 4-1  MOTOR PARAMETERS Experimental Motor Number: 2  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 5 30 (5 poles in parallel) .670 m #4 Wye series  Mechanical Parameters Primary Width Primary Thickness Tooth Width Slot Width Slot Depth Total Primary Length Active Primary Length Number of Stator Slots (single layer) Stator Slots (total) Stator Slots (active) Stator Slots (half-filled) Secondary Thickness Primary-Secondary Gap per side Secondary Overhang per Side Air-Gap (total)  6.19 cm 11.4 cm 9.52 mm 9.52 mm 6.35 cm 1.03 m 1.01 m 45 53 53 16 3.0 mm 1.4 mm 2.54 cm 2.8 mm  Electrical Parameters  Secondary Resistivity Frequency Voltage Current Rating (continuous)  26 |xQ"cm 360 Hz 440 V 600 A Table 4-2  MOTOR PARAMETERS Type: Segmental Stator Core: B  Experimental Motor Number: 3 Winding Parameters  5 3 17.1 cm .966 15.2 cm 1:9 .9598 45 2 30 (5 poles in series) .670 m #1 Wye series  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 Primary Thickness Tooth Width Slot Width Slot Depth Total Primary Length Active Primary Length Number of Stator Slots (single layer) Stator Slots (total) Stator Slots (active) Stator Slots (half-filled) Secondary Thickness Primary-Secondary Gap per side Secondary Overhang per Side Air-Gap (total)  6.19 cm 11.4 cm 9.52 mm 9.52 mm 6.35 cm 1.03 m 1.01 m 45 53 53 16 3.0 mm 1.4 mm 2.54 cm 2.8 mm  Electrical Parameters  Secondary Resistivity Frequency Voltage Current Rating (continuous)  26 ull'cm 360 Hz 440 V 600 A Table 4-3  58 MOTOR PARAMETERS Experimental Motor Number: 4  Type: Segmental Stator Core: B  Winding Parameters 4 3 17.1 cm .966 15.2 cm 1:9 .9598 45 2 24 (4 poles in series) .670 m #4 Wye parallel  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 Mechanical Parameters Primary Width Primary Thickness Tooth Width Slot Width Slot Depth Total Primary Length Active Primary Length Number of Stator Slots (single layer) Stator Slots (total) Stator Slots (active) Stator Slots (half-filled) Secondary Thickness Primary-Secondary Gap per side Secondary Overhang per Side Air-Gap (total)  6.19 cm 11.4 cm 9.52 mm 9.52 mm 6.35 cm 1.03m 0.836 m 36 53 44 16 3.0 mm 1.4 mm 2.54 cm 2.8 mm  Electrical Parameters  Secondary Resistivity Frequency Voltage Current Rating (continuous)  26 ufl'cm 360 Hz 440 V 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  10 3 8.76 cm .966 7.30 cm 1:6 .966 60 35 140 (5 pairs in parallel) .254 m 18 Wye parallel  Mechanical Parameters Primary Width Primary Thickness Tooth Width Slot Width Slot Depth Total Primary Length Active Primary Length Number of Stator Slots (single layer) Stator Slots (total) Stator Slots (active) Stator Slots (half-filled) Secondary Thickness Primary-Secondary Gap per side Secondary Overhang per Side Magnetic Gap (total)  3.40 cm 3.81 cm 7.3 mm 7.3 mm 2.22 cm .878 m .878 m 60 60 60 0 3.8 mm 1.4 mm 2.54 cm 2.8 mm  Electrical Parameters  Secondary Resistivity Frequency Voltage Current Rating (continuous)  26 |xQ'cm 400 Hz 440 V 40 A Table 4-5  MOTOR PARAMETERS Type: Segmental Stator Core: C  Experimental Motor Number: 6 Winding Parameters  6 3 8.76 cm .966 7.30 cm 1:6 .966 60 35 140 (3 pairs in parallel) .254 m 18 Wye parallel  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 Primary Thickness Tooth Width Slot Width Slot Depth Total Primary Length Active Primary Length Number of Stator Slots (single layer) Stator Slots (total) Stator Slots (active) Stator Slots (half-filled) Secondary Thickness Primary-Secondary Gap per side Secondary Overhang per Side Magnetic Gap (total)  3.40 cm 3.81 cm 7.3 mm 7.3 mm 2.22 cm .878 m .527 m 36 60 46 10 3.8 mm 1.4 mm 2.54 cm 2.8 mm  Electrical Parameters  26 nfl'cm 400 Hz 440 V 40 A  Secondary Resistivity Frequency Voltage Current Rating (continuous) 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 power  supply  had programmable Volts  readout of frequency  per  to within one Hertz.  and #6.  Hertz  This  and digital  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 The  Speed Measurement speed  Computer 64.  measurement  system  was  based  on  a Commodore  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 stators.  coils mounted on the  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 observed, 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  friction straints  in' obtaining an  can only be on  the  accurate  measurement  appreciated when one  measuring  device.  The  for the  considers the confriction  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 direction 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  performance.  connected  poles  and to  measure  full  size motor  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. 4.4,  of  all  The experimental accuracy, discussed in Section  measurements  except  electromagnetic  thrust  were  accurately obtained.  The apparatus described in this chapter was used in the experiments described in the following chapter.  69  5. RESULTS AND DISCUSSION  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. highlight  In some cases the the  effect  of the  simulation results  parameter  are used to  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 experiments 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.  harmonics were also measured and calculations determine the magnitude of their effect. in Section 5.8.  The time  undertaken to  This work is presented  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 presented 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. has been used by previous investigators [29].  This model  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  predicting the performance of the motors.  accuracy  in  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  synchronous speed Power  Fig. 5.2 - Comparison of Experimental Points and Simulation Curve For Experimental Motor #1 With 3 mm Stainless Steel Rotor  1  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 (Experimental Motor #6).  The two rotor materials chosen  experiment were steel and aluminum.  for this  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. than  the  The aluminum rotor is five times more conductive  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. neither  the  efficiency  degraded by the end effect.  or the  In this case  power factor of the  motor are  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 (threefifths 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. can be observed that  the end effect is causing a significant  reduction in efficiency for this motor. from 0.48 to 0.27.  It  The peak efficiency drops  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  Power Factor  2eta  0.4  -  0.3  -  0.2  -  0.1  -  efficiency  % synchronous 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) power cantly.  or efficiency  the  for the  end effect  does not  reduce the  thin steel rotor machine  signifi-  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.  p e r unit of s y n c h r o n o u s s p e e d • 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 machine.  This motor is  application.  The flux  the was  full  size prototype for the sawing  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 operation of the motor).  air gap flux during  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). power-factor efficiency  It is especially important that the  product is maximized because the motor  must run from an inverter, which is component of the sawing system.  the  most costly  single  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, set  voltage  source  to produce frequencies  experiments speed  inverter which could be accurately from 25 to 600  Hz.  All of the  were conducted with Experimental Motor #1.  of the  rotor was  measured with  The  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 factor*efficiency  for the efficiency,  power-factor, power-  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. conductive  In a normal rotary machine such a highly  rotor would reach  synchronous speed.  maximum power  very  close  to  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 power-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  end-effect  uQ-cm) so it would be expected  to have very little  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. sections  of  the  rotor would have  very  The stainless steel 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. power (3.1  This rotor produced the maximum  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 resistivity 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  results  in half the  result  in  a higher  power-factor which is what can be seen in the results.  The high  magnetizing  resistivity  the  current  of  the  other  rotors.  and is  steel,  This  expected  to  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  material. Efficiency  Fig. 5.13 - Experimental Points and Approximating Curves Showing Efficiency  rotor  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  c u / s t o i n l e s s steel  0.4  +  — — —  0.6  0.8  per unit of synchronous speed 3 m m cu o - 2 mm steel  1  A  3 m m . 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. experimental results  may be  extended  These  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 polarity.  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 x (5.1)  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 power-factor are obtained. In addition, an experiment was conducted to observe the effect of an odd number of poles on the power-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). relative efficiency.  end  effect  force  will  On the other hand, the  increase  which  reduces  If the other alternative - disconnecting a pole  the - 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  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 expected changes account.  in efficiency  and power-factor were taken into  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. efficiency  The resulting power-factor  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 OA  -  0.7  -  0.6  -  Voltage Increased By 1.25 For Odd  0.5 0.4  -  0.3  -  0.2 0.1  -  —i—  0.5 Pover  +  0.9  0.7 Efficiency  V e l o c i t y Iper unit o f s y n c h r o n o u s ] O o Power Factor Footer  A  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.  Zeto  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. pattern of parallel connected  Laithwaite [11] discusses the flux 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 withstand the greater current resulting from a parallel connection. The entry coil carries more current because in the parallel connection 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 steady-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 connection 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  .  90 84 80 75  Motor Current (A) 285 300 320 345  Coil Current (A) 120 124 106 106  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 machine.  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  V e l o c i t y Iper unif o f 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 Cornportson Between Parallel a n d !>erles 0.8  0.7  O.G  -  0.4  -  0.3  0.2  0.)  Tooth N u m b e r rollel  +  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). in  These experimental results,  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 - . 1 -I 0.9  1231  "  -  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 ( Series Connected (  ) and a  ) 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 efficiency 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  Fig. 1.1  motor. (The  sheet  rotor  motor in  shows a rectangular pattern only because it is  idealized case.) Fig. 5.23.  induction  the  Rather, the pattern is more like that shown in  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 pattern, 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". types under investigation  use  As the motor  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 expected.  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-b  d'_b_ _ M ^ b _  2  d'x  -M^_  2  +  d'y  2  fig  2  =  d't  (  5  2  )  g d'x  Assuming sinusoidal functions, equation (5.2) can be written as: d B- k , 2 + j s w lo Lt B TJ = j ^o a k J J  dy  fig  2  /c o\ (5.3)  g  The solution of this equation is given by: B = 'to J Z g k  1+  2  ( 1  Z Z  }  ^ Cosh (3a  C o s h F  2  where: 1 +  Tanh pa Tanh k(c - a) V 1 + i sG  p  =  k  2  i  +  b  wu Q g  Q  = k ( 1 + i s G) 2  7 z  -  1  1 + i sG  To find the ratio, U, of the flux density at the edge to the mean flux density, first calculate the mean flux density: f  B m  e  a  n  a  =1 B dy T a /J-a -a  (5.4)  107  n  Side.  1 + i  s  G  (5-5)  e  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  motor.  3.--  Fig. 5.26 - Flux Density Variation Factor [50]  of the  108  For the test motors the parameters were checked against these curves, which predicted that severe edge effects were not expected.  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 Experimental 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. is shown in Fig. 5.27.  The positioning of these sensors  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 Experimental 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 L I M 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 i n the center (less than five to one difference; see Bolton [50]), indicating that  the edge  effect is not significant.  reason for the sloping lateral flux density  The  curve is due to the  110 slopingfluxdensity 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  b  Y 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  Ill  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 performance 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 induction 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 frequency, the magnitude of the flux slot harmonic, and the simulation technique described in Chapter 2, it is possible to obtain a relation Fig. 5.31.  between  torque  and  rotor  resistivity  as  shown  in  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  SLOT HARMONIC TORQUE Operation at 3 6 0 H i . .25 Slip (4860 H i ) 90  AO  70  -  SO  50  -  AO  -  50  20  -  lO  -  O  JfcZ  !  1.00E-O7  7  T  1.00E--06  T  ,  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 reduction 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 Fig. 5.33).  (see  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. have  been  These amplitudes of harmonics  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 direction.  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  —  —  1 —  i  i  — i — i  —•  — J — —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 + Inner  0.85 o  0.75  outer  0.65 A  0.55  0.45  slip average  0.35 X  0.25  0.15 0.05  In, Teeth  V 0«VTK«I y  Fig. 5.36 - Variation in Power and Thrust With Radius Including the Effect of The Stator Slots  5.10  Discussion  Various experiments different  were conducted to determine the effect of  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 conducted on six experimental motors, were presented in this chapter.  124 The end effect and the effect it has on the actual overall performance 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 scaling factors to find a more optimum design. Chapter 6.  This is presented in  This section also allowed for the direct comparison  between simulated and actual results. rected for friction  The simulation when cor-  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 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. be  produced if the  windings  However, more power may  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 decrease 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 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  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 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.  127 6. FACTORS AFFECTING OPTIMIZATION  Introduction to Optimization  6.1  Optimization of electric motors generally means the maximizing of efficiency. supply  However, in this application the size of the inverter (the  most  expensive  component  determined by the power-factor/efficiency  of  the  product.  machine)  is  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 One  Goodness Factor criterion which is often  used in the  design of induction  motors is that of goodness factor.  The goodness factor is a  measure  and electrical circuits are  of how well the  magnetic  utilized. The goodness factor is defined as:  •2  where w is the stator frequency.  (6.1)  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  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.  Relative E n d Effect  Velocity Iper unit o f s y n c h r o n o u s ]  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. above  example  it  is  clear that  the  "optimum  From the  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  However,  for the  and  cooling  constant  problems  voltage  can  double  be  significant.  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. far  more  conductive  rotor  material  is  used  However when a (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 motor with the high goodness factor.  than the  132  Experimental Exp. 100  -  i  M o t o r #1  Motor #1  —  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  Using  product  and  maximum  power  density.  scaling  factors [54] it is possible to scale up the results of the smaller  133 motor  experiments (Experimental  Motor #1)  and use  predict some of the performance characteristics  them  to  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).  2 G = V-o pg w v  (6.2)  0 v  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. 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  44  18  Table 6.1 - Comparison of Motor Parameters  The parame-  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  Factor, the thickness  size machine.  most reasonable  since  rotor LIM.  full  the  To reduce  the  Goodness  parameter to adjust is the rotor  final objective  is to produce the  thinnest  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, optimization for the sawing application means obtaining a high power-factor/efficiency density.  product  while  The power density  also is  maintaining  a  high  power  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  connected stator.  are  possible  for the  design  of the  parallel  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. extensive  experimental  This analysis was supplemented with  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  characteristics.  One  the  annular disc  motor  was  LIM motors  used  to  test  the  operating 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 different  were conducted to determine the effect of  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 results 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 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. be  produced if the  However, more power may  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.  determine  the  Three different  experiments  were conducted to  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  undertaken to determine the magnitude of their effect.  calculations 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  resistivity of the rotor.  connection  of  the  stators  and  the  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 conductivity 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. 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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 Electromagnetics: An International Quarterly, 1:1-9, 1976, pp. 1-9.  [50]  H. Bolton, "Transverse Edge Effect In Sheet-Rotor Induction 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.  Design  of Electric  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  component  only.  The following three  equations  of the x  describe these  conditions.  -db/dx = n/g*(c*j+J) 0  (2.1)  the change in flux along the motor is equal to the current density flowing in the stator and rotor  dl/dx =c*j  the change in current flowing along  (2.2)  the return paths of the  rotor is equal to the current density flowing in the rotor  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 = ii/g*{(c*j)+J)  (2.1)  dl/dx = c*j  (2.2)  dj/dx-(2/(l *S ))*I = -{(i*w*b)+(V*db/dx)}/p  (2.3)  0  g  r  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)  d j/dx - 2/(1 *SJ*dI/dx = -i/p*w*db/dx - V/p*d b/dx 2  2  2  2  (2.5)  153  Taking the second derivative of equation (2.1)  -d b/dx 2  = n *c/g * dj/dx + |X /g*dJ/dx  2  0  (2.6)  0  Now taking the second derivative  -d b/dx = u*c/g*d j/dx + u /g*d J/dx 3  3  2  2  2  n  2  (2.7)  Obtain an expression from (2.7) for d j/dx  d j/dx = -g/u /c*d b/dx - l/c*d J/dx 2  2  3  3  2  2  0  (2.8)  Obtain an expression for j by rearranging (2.1)  j = -g/u /c*db/dx - J / c  (2.9)  Q  Substitute expressions (2.2) and (2.8) into equation (2.5)  d j/dx - 2/(l *S )*dI/dx 2  2  s  r  = -l/p*i*w*db/dx - V/p*d b/dx 2  2  (2.10)  -g/(^i *c)*d b/dx - l/c*d J/dx - 2/(l *S )*c*j = 3  3  2  2  0  g  r  -l/p*i*w*db/dx - V/p*d b/dx 2  2  Substitute (2.9) into (2.11)  -g/(^ *c)*d b/dx " l/c*d J/dx - 2*c/(l *S ) 3  3  2  2  0  s  r  *{-g/^i/c*db/dx -l/c*J) =-l/p*i*w*db/dx - V/p*d b/dx 2  2  0  Expand to get:  -g4i /c*d b/dx - l/c*d J/dx + 2*g/(l *S *^ )*db/dx 3  3  2  2  0  g  r  0  + 2*J/1/S =l/p*i*w*db/dx - V/p*d b/dx 2  2  r  Gathering together terms:  -g/^i /c*d b/dx + V/p d b/dx + {2*g/l/S/^i + i*w/p}*db/dx= 3  3  2  2  0  I  ()  l/c*d J/dx - 2*J/1/S 2  2  r  Multiply through by (-\i *c/g) to get: Q  d b/dx - V*^i *c/g/p*d b/dx - {2*c/(l *S) + ^*c*i*w/g/p}db/dx 3  3  2  2  0  g  r  0  -^i /g*d J/dx + 2^ *c*J/(l *S )/g 2  0  2  0  s  r  155  d b/obc -V*u *c/(p*g)d b/a^ -[i*w*^ *c/(p*g)+ 3  3  2  2  0  0  (2*c/(l *SJ)]db/dx=-uo/g(d J/dx -(2c/l * )*J) 2  (2.14)  2  Equation (2.14) is then solved to obtain the magnetic field value. The general solution of this differential equation is of the form:  b +b exp(rj) Q  x  and the particular form is:  bp = -i|i J/(kg) cos<j) exp(i<t>) G  = - i B cos<j) exp(i(|)) Q  The thrust can then be calculated from the stator surface current and the rotor magnetic field as given by the equation: PT|  F = -£l I Re (J*b) dx s  ' 0 The thrust developed by the motor is separated into two components.  F  Q  is the thrust described by standard rotary induction  motor theory and F^ is the thrust produced by the parasitic end  156 effects found in linear induction motors.  F =^*J *B*l*p*L*cos<i)*sin(!) Q  F  m  0  (2.15)  s  l = -i*J *B *l *Real[(b /b )*(b /B )*k ] m  0  s  1  p  p  0  b = [V*a+k*tan<|)(l-i*v/k)/(r-v-V*a)]*b x  1  B = u *J/(k*g) Q  0  b = -i*B *k /(k +i*s*w*a) 2  r  2  = V*a/2*{l-[l+4*(i*w*a+v )/(V*a) ]} 2  x  2  V = velocity v = 4/stator width/total rotor overhang a =\i. *c/p/g 0  p = rotor resistivity K = [l-exp{(r +ik)PT|}]/[r +ik] 1  1  (j) = tan" (s*w*a/k +v ) 1  L J  2  2  = length of stator m  = maximum primary current density  1 = width of stator  1  (2.16)  P a g e 1 0 9 0 9 -65 £1 :4S : 16 0 £0 F 0 R T R A N 7 7 V 3 £ . /•8 4  7 D L n ie #1 M i c r o s o f t 1 * D E B U G h i s is a l i n e a r i n d u c t i o n m o t o r L (IM) s i m u l a t i o n p r o g r a m b a s e d cl c T 3 c the s t e a d y s t a t e a n a l y i s w h c i h is s u m a r s i e d in Dr. P o o lu t j a d o f f' s 4 c b o o k . 5 c For t h i s s i m u l a t i o n the m a g n e t c i f l u xb e g i n s at the e n t r y of the o t o r and t r a i l s r f o m the e x i t end. £ c m 7 c v e r s i onof: Aug. £ 1 , 1 9 6 5 . o . 6 c T h i s s e c t i o n w i l ld e f i n e all the v a r i a b l e s u s e d in the p r o g r a m 9 c v n o t •.. 10 c v n o t sq 11 c ve 1 v e l o c i t y e l s y n s i£ c v y n c h r o n o u s v e l o c i t y 13 c f f r e q u e n c y of s t a t o r c u r r e n t s req 14 c f r a d f r e q u e n c y of s t a t o r c u r r e n t s in r a d i a n s 15 c bpr i rne to 16 c bone 3 17 c btwo 18 c a p lh a 13 c r o n e r. r. £0 c r t wo s t a t o r s u r f a c e c u r r e n t d e n s i t y in arnDS Der m e t e r £1 c Jm c' c' c kay c p e r m i t i v i t y of the a i r g a p c mewnot £4 r o t e r t h i c k n e s s in m e t e r s c cee £5 C o o p e r - 1 . 7 £ 4 d 8 A u l m n i u m - £ . 9 d 8 S t e e l 1 7 d .B c row c • ao s t a t o r to s t a t o r e f f e c t i v ea i r g a p in m e t e r s £6 s t a t o r w d it h £7 c ei £6 r o t o r w d it h c e 1 Dr £9 c bnot  -a  3 0 cn e g e y e -' cD e f n ie t h e t y p e s o f v a r i a b l e s 3 1  f o r t h e  p r o g r a m .  c R e a1* 8 v n o t , v n o t s q , v e l , v e1s y n , fr e q , fr a d , a 1 Dha, j r n , k a y 3 3 + e e e , r o w , g a p , e l , e l p r , s l i p , 1a r n b d a , p s i , l e n g t h , x , f o r c e . 3 4 3 5 + f o r c e t , t e s t e , d r e a l , f o r c e O , f o r c e 1 , f o r c e £ , f o r c e s , b n o t , +r c a l c l , r c s l c £ , r c a l c S , r c a l c 4 , r c a l c S , r c a l c 6 , r c a l c 7 , f i b s , 3 6 3 7 + p o l e n , f o r c t O , f o r c t l , f o r c t £ c o m p e l x * 1 6 b p r m i e , b o n e , b t w o , r o n e , r t w o , b t o t a l , c d s q r t , 3 6 + c d e x p n ,e g e y e p ,o s e y e , c a l c l , c a l c £ , c a l c 3 , c a l c 4 , c a l c S , t e s t 1 , t e s t 3 3 9 c h a r a c t e r * 6 4 r e s u l t , g r e s l t 4 0 4 1 * * * * 0 p e n a f i l e f o r r e s u l t s * * * * * c* o p e n ( 1 0 , f i l e = ' r e s u l t ' ) 4 £ o p e n ( 1 1 , f i l e = ' g r e s I t ' ) 4 3 * * * * S e t p a r a m e t e r v a l u e s f o r t e m p o r a r y t e s t i n g . ****** 4 4 c* c T h e s e m u s t a l l b e s p e c i f i e d . 4 5 4 6 f r e q = 3 9 6 . 4 7 j r n = 8 . 1 0 £ d 4 4 8 c e e = . 0 0 £ 0 0 4 9 r o w = 3 5 O . d 8 5 0 g a p = 0 . 0 3 0 0 5 1 e l = 0 .5 7 0 e l p r = 0 1 .0 8 5 £ 5 3 a l m b d a i s t h e w a v e e l n g t h c 5 4 a lr n b d a = . 1 4 6 0 5 5 cp o l e n i s t h e n u m b e r o f p o l e s 5 6 p o e l n = 4 0 . 5 7 cS e t p h y s i c a l c o n s t a n t s m e w n o t = l £ 5 .6 6 £ 8 d 7 5 8 a l c u l a t e o t h e r m o t o r p a r a m e t e r s n o t s p e c i f i e d 5 9 cC  00  D L i n e * 60 61 6£ 63 64 c 65 66 67 68 c 63 70 71 c 7£ 73 74 c 75 7t, 77 78 79 80 6 1 8£ 63 8+ ' 65 86 87 88 c 83 c  1  P a g e £ 0 3 0 3 8 5 £ 1 : 4 6 : 1 8 M i c r o s o f tF 0 R T R A N 7 7 V3. £0 Q £ S /4  7 al p h a = m e w n o t * c e e r / o w g / a p length=Doleri/£. 0 * a l m b d a v e 1 s y n = f r e a * 1 a m b d a f r a d = 6 . £ 8 3 1 8 * f r e q C a l c u l a t e all o t h e r v a r i a b l e s . v r i o t s q = 4. 0 / e l * (elDr-el > v n o t= d s q r t <v r i o t SQ > k a y = 6 £ .8 3 1 4 1 /a m b d a D e f n ie m a h t c o n s t a n t s . n e p e y e = ( 0 . 0 , 1 . 0 ) p o s e y e = ( 0 0 .1 ,0 .) Set i n i t i a lv a l u e s to 0.0 f o r c e = 0 0 . f o r c e t = 0 0 . Set u'j o u t p u t t a b l e w r i t e ( 1 0 , 004) 004 f o r m a t ( x lT ' .H E S E A R E T H E R E S U L T S O F A 1 — D M IE N S O IN A L . L I N E A R , ' +' N I D U C T O IN M O T O R S I M U L A T I O N ) ' w r i t e ( 1 0 0 ,0 5 ) 005 f o r m a t (lx, ' F r e a ' , ' A r n p M s / ' , ' R o t o r Thk' , ' R e s i s t ' , ' Gap' , +' S . W i d t h ' , ' R . W i d t h ' , ' W . L e n . ' , ' # p o l e s ' ) w r i t e < 10, 006) f req, jrn, cee, row, gap, el, el pr, 1a m b d a , p o l e n 006 f o r m a t (1 x, f5. 0, f7. 0, l x , f 7 . 4 , l x , e 9 . £ , l x , f 6 . 3 , l x , f 6 . 3 , lx, +f 6. 3, 3x, f6. 3, 4x, f4. 1 ) w r i t e (*, 050) w r i t e ( 1 0 0 ,5 0 ) 050 f o r m a t ( 1 x , ' S I i p ' , V H o r s e p o w e r ' , ' F0 ' , ' Fl ' , ' F£ ', +' T h r u s t ( N ) ' , ' T h r u s t ( 1 b s ) ' ) ******* s t a r t m a n i p r o g r a m r o u t i n e *********  M  g  1 1 1 1 i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  9 0 d o 7 0 0 j = 1 0 1 ,0 0 1 ,0 9 1 f o r c e t = 0 0 . 9 £ f o r c t 0 = 0 0 . 9 3 f o r c t 1 = 0 0 . 9 4 f o r c t £ 0 . 0 9 5 v e l = r e a l ( j ) / 1 0 0 . * v e l s y n 9 6 s i i p = l . 0 k a y * v e l / f r a d 3 7 p s = id a ta n ( s iip * fr a d * a lp h a / ( k a y * * £ + v r i o ts a) ) 9 3 b n o t= m e w n o t * j r n k /a y/g a p 9 9 b p r m i e = ( n e g e y e * r n e w n o t * j m g /a p k /a y ) * ( ( k a y * * £ + v n o t s q ) / 1 0 0 + ( k a .y * * £ + v n o t s q+ ( p o s e y e * s l i D * f r a d * a 1 p h a ) ) > 1 0 1 r o n e = v e l * a l p h a / £ . 0 * < 1 . 0 c d s q r t ( 1 . 0 + 4 . 0 * ( ( p o s e y e * f r a d * a l c h a 1 0 £ + + v n o t s q ) / ( < v e l * a l p h a ) * * £ ) ) ) ) 1 0 3 r t w o = v e l * a l p h a / £ . 0 * ( 1 .0 + c d s q r t ( 1 . 0 + 4 . 0 * ( ( p o s e y e * f r a d * a l D h a 1 0 4 + + v n o t s q ) / ( ( v e l * a l p h a ) * * £ ) ) ) ) 1 0 5 b o n e = b p ri r n e * ( < ( v e l * a l p h a ) + < k a y * d t a n ( p s i ) ) * ( 1 + .n e g e y e * v n o t 1 0 6 + / k a y ) ) / ( r o n e v n o t v e l * a l p h a ) ) 1 0 7 c a l c l = n e g e y e * p s i 1 0 8 c a l c £ = p o s e y e * p s i 1 0 9 b t w o = b p r r i n e * ( r o n e + n e g e y e * d s n i (DSI ) * ( r o n e n e g e y e * C E X P 1 1 0 + ( c a l c l ) ) ) / ( r t w o r o n e ) / d c o s ( p s i ) * c e x p ( c a l c £ ) 1 1 1 c W R I T E * £ ( , 0 5 B )P R I M E B ,O N E B ,T W O 1 1 £ c £ 0 5 F O R M A T ( I X , ' b p r i r n e , b o n e , b t w o ' , 6 E 1 4 . 7 ) 1 1 3 c w r i t e ( * , £ 0 6 ) r o n e , r t w o 1 1 4c £ 0 6 f o r m a t ( 1 x , ' r o n e = ' , £ e l 4 . 7 , ' r t w o = ' , £ e l 4 . 7 )  1  1 1 5  £ £ £  1 1 6 c w r i t e ( * , £ 0 1 ) i 1 1 7c £ 0 1 f o r m a t < l x i , 3 ) 1 1 8 x = r e a l ( i ) / 1 0 0 . * l e n g t h  d o 5 0 0 i = l , 3 9  03  o  161 m in co -r CO ^ CO I " \ cn KD ca OJ  o  4-  o  at i •••3* •  X  * ns *01  7-1  >  01 OJ  h-  IT.  „  Z d jir — or a u.  II u1 r— ns n  >>  01  JZ •»  -p Ol  a  +>  4 M-  C  ,-,  •  in  w  0  01 OJ  U  w r-  £  II OJ  r *-  •r-4  c  01 Q  * * * *  •  Ol  •4"  * *  JZ -p  * * n01  * r  +J  Cli  C  * * * * * * *  01 r-i  1  X  * * * * * at  Li * Xa fl i—i C -P UJ 2 0 i—t TJ X Ul as in o -p T-i 44o ai c Q S * r-H 0 -p 01 >JZ ^ r: -p u •P n X ns "O 0 •rH Q ns c D X X a i tJ- U -P •a Ol r->. u 1-i Ul 01 0 r—i . c o> *~ Oi C ns 4* + ns as O 0 *U r-4Ol 401- *X J3 X^ 0 >,l T a r * * .—. „. • + 01 01 01 >, -p O l fl * Q J * r-t •1 >. OJ r—4 OJ ai •H r-4 01 1-1 -p > . c n s c c 2 Ul 0 -p Ul u QJ OJ0 Ol -p1/1 CJ ui . 2 Q. XIl"l A-p 01 2 U ai .—. Ul Ql l/l i- t c ns fl >* r s OJ r-l ns -PQl -p • •pCU (0 »*-0 01 *01 *01 in •P <n * X * *01 *01 r +» n 0 -P A- O u a s U in o l Ol >, X X >> 0 0 o C 01 1/1 >. * T—i 3 r^ 01 0*- u 01 >. * >, >. •i *-« 11 •— •f* 1-1 S- -P i- CU -P * * c *H Ol 01 01 Ul Ol Ol 01 UlOl J3 >»UJ •p Ul O l a Q p -p * Mc in cn M r-* as i-H1 r-l (J X nsr-1o * 1/1 Ulr-i Ul01 Ul • -p 01 *3 x: fl .-*** fl u fi QJ %Q. fl -X Oi r— a* Ol r-i CU <f 01 QJ a ^ a ~-C — a -P X Oi a X 4 •P Ol -p * X u as fj * ns U1 -p Ir- — p -p •r-l -p Qi 01 01 Ol s 01 u 01 r-T. -p r— ns4 •P •i 01 c ai *>, X) r— as4 •i •l •3' «-a * 01 a > . * * If) • I OJ M n s n s n s m n s n s ns a u r-••- r-i r-4 c c 3 3s 01 01 0 OJ -p 0 O l Q f " \ a s "i ni n s d c —i •ri •1 OJ X X X OJ X !>. 1 — _* * X) c 2 r-i fl a •l X * CU CU fj J3 T) r-i a X) X) X) X) X! y~i a r-4 > --I Ql 0r -p * II X oi ai 3"~> r-i -r-t T3 01 t •-' ns ns II II Ol II * C IIA- i. -P Q l c u * * as THn OJn II II II -P +> 1-4 OJ II OJ II r-l CJ ii II II I I * a) i> f'l0 rt 4-> OJ'IS u u 01 u Oi as as * (J "P ro Ol ns OJ O u u ,—iu r-4u r—4 r-4 u CJ U -p u l Ql •P •p 3 Ol ns 0 l If) O -p U •l * * 4-> * r-« r-i r-4 t 4 0-> fi .-» 1/1 p 'Jl •ri A- Ul •ri * * * -w r>- • rj— 01 * * 0ns ns as as as f 2 — 1*1 0 OS t 0 -P -p 01 > 0 ni m c u O l S * * t' l fl fl c ** r ai* a U ui* f A»> u 0 u 2 4- JD + u *> -P 2 •p 2 2 ** * * 4- *+ 4- — f + * ** * OJ * * If) ro * OJ T-4 * * * * OJ * * •XI CO OJ 0 U U u u 0 U CJ a 0 u u u 0 0 u u T< T-4 OJ -3* U) N U'J CO cn fn ,-, fO <J- in o cn Q T-*OJ fO <i- u) r- CD .-, OJ Ql OJ OJ OJ <f 10 10 I'") co COOJ CO COOJ CU r-o ( 0 <f  fl  a; c ai  X * r—4 01 * i—i * u * * in * r-; -p u If) * Oi o • r^X * CU OJ  a. X Oi fj *  /  m  »>  -  !•  •  <  _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  c: c c c ' c  £  £•  u . C C c ! £'  £  £ £ £  1 5 0 1 5 1 1 5 £ 1 5 3 1 5 4 1 5 6 1 5 7 1 5 8 1 5 9 1 6 0 1 6 1 1 6 £ 1 6 3 1 6 4 1 6 5 1 6 6 1 6 7 1 6 8 1 6 9 1 7 0 1 7 1 1 7 2 1 7 3 1 7 4 1 7 5 1 7 6 1 7 7  r c a 1 c 4 = dim a g ( b o n e ) c c r c a l c 5 = d a t a r i £( r c a l c 3 , r c a l c 4 ) r c a l c 3 = d a t a n £( r c a l c l , rcalcl) d a t a n £( r c a l c G , r c a l c 7 ) + r c a l c 5 c f o r c e 1=. 0 0 5 * e l * l e n g t h * c d a b s ( b o n e ) * d e x p ( r c a l c l ) c +*jm*dcos(rcalc3) c a l c u l a t e e x i t e f f e c t f o r c e********* c ******** c c c a l c l = r t w o * ( x l e n g t h ) c c a l c £ = n e g e y e * k a y * x r c a l c l = dr i n a g ( p o s e y e * c a l c l ) c r e a l c £ = dr i n a g (calcl) c r c a 1 c 6 = d i mag(DOseye*calc£) c r c a c l 7 = d r i n a g ( c a l c £ ) c c r c a c l3 = dr i n a g (p o s e y e * b tw o ) r c a1c 4 = d ir n a g ( b t wo) c r c a l c 5 = d a t a n £ ( r c a l c 3 , r c a l c 4 ) c r c a l c 3 = d a t a n £ ( r c a l c l , r c a l c £ ) d a t a n £ < r c a l c 6 , r c a l c 7 ) + r c a l c 5 c f o r c e £ = . 0 0 5 * e l * l e n g t h * c d a b s ( b t w o ) * d e x p ( r c a l c l ) c + * j m * d c o s ( r c a l c 3 ) c f o r c e s = f o r c e O f o r c e 1 f o r c e £ f o r c e t = f o r c e t + f o r c e s f o r c t0 = f o r c t0 + f o r c e O f o r c t l = f o r c t 1 + f o r e e l f o r c t£ = f o r c t £ + f o r c e £ f o r c e = 0 .0 5 * e l * t e s t 3 * l e n g t h c c f o r c e t — f o r c e t + f o r c e w r i t e ( * , 6 0 0 ) i , f o r c e s , f o r c e O , f o r c e 1 , foree£ c 0 0 f o r m a t < l x , ' 1 = ' , i 3 . ' F o r c e = ' , e l 3 . 6 .' F 0 = ' , e l 3 . 6 , ' Fl = ' , e l 3 . c 6 +' F £ = ' , e l 3 . 6 ) c  P a n e 4 0 9 0 9 6 5 £ 1 : 4 6 : 1 8 M i c r o s o f tF 0 R T R A N 7 7 V 3 . £ 0 0 £ / 6 4  L i n e t t 1 7 1 7 8 5 0 0 c o n t i n u e 1 7 9 f 1 b s = f o r c e t / 4 4 .4 8 1 8 0 c fo r c t 0 = 0 . 5 * e l * l e n g t h * b n o t * d c o s ( DS i ) * d s in(o s i ) *j r n 1 8 1 c w r i t e ( * , 6 1 0 ) f o r c e t , f i b s 1 8 £ c 6 1 0 f o r m a t ( 1 x , ' F o r c e t=' , e l 3 . 6 , ' N e u t . ' , e l 3 . 6 , ' 1 b s . ' ) 1 8 3 p o w e r = v e l * 3 . £ 8 1 * f l b s / 5 5 0 . 0 1 8 4 w ri t e ( * , 6 1 5 ) s l i p , p o w e r , fo r c t0 , fo r c t 1 , fo r c t£ , fo r c e t , f1b s 1 8 5 w r ite ( 1 0 , 6 1 5 ) s i ip , p o w e r f , o r c t O , f o r c t 1 , f o r c t £ , f o r c e t , f 1 b s 1 8 6 6 1 5 f o r m a t ( l x , f 4 . £ , l x , f 1 0 . 3 , 5 x , f 6 . 1 , l x , f 6 . 1 , l x , f 6 . 1 , f 1 1 . £ , f 1 £ . £ ) 1 8 7 w r i t e ( 1 1 , 6 5 0 ) s i ip , v e l , p o w e r , f 1 b s , f o r c e t 1 8 8 £ 5 0 f o r m a t ( 1 x , 5 f 1 0 . £ ) 1 8 9 c w r i t e ( * , 6 1 7 ) e l , e l p r , f r e q , l a m b d a a ,l p h a v ,e l 1 9 0 c 6 1 7 f o r m a t ( l x , 6 f 1 0 . 3 ) 1 9 1 c w r i t e ( * , 6 £ 0 ) f o r c t O , f o r c t 1 , f o r c t £ 1 9 £ c 6 £ 0 f o r m a t ( l x , ' T h r u s t 0 = ' , f 7 . 3 , ' E n t r a n c e e f f e c t=' , f 7 . 3 . 1 9 3 c + ' E x i t e f f e c t = ' , f 7 . 3 ) 1 9 4 7 0 0 c e n t i n u e 1 9 5 c l o s e ( 1 0 ) 1 9 6 c l o s e ( 1 1 ) 1 9 7 s t o p 1 9 6 e n d  CO  N a m e  T y p e  O f f s e tP C l a s s  A L P H A R£AL*3 56 B N 0 7 R E A L * 6 6 & £ B O N E C O M P L E X * 1 6 . 1086 B P R M IE C 0 M P L E X * 1 6 670 BTDT'AL C 0 M P L E X * 1 6 1436 B T W O C 0 M P L E X * 1 6 1246 C A L C 1 C 0 M P L E X * 1 6 118£ C A L C £ C 0 M P L E X * 1 6 1£14 C A L C 3 C 0 M P L E X * 1 6 1466 C A L C 4 C 0 M P L E X * 1 6 ***** C L t fC S C 0 M P L E X * 1 6 ***** C D E X P C 0 M P L E X * 1 6 ***** C D S Q R T I N T R I N S I C C E E R E A L * B 3£ C E X P I N T R I N S I C D A T A N I N T R I N S I C D C O S I N T R I N S I C D M IA G I N T R I N S I C D R E L t f REPiL*8 ***** D S I N I N T R I N S I C D S Q R T I N T R I N S I C D T A N I N T R I N S I C E L  REAL*8  E L P R REAL*8 F L B S R E A L * 8 F O R C E R E A L * 8 F O R C E O R E A L * S F 0 R C E 1 REPiL*B F 0 R C E 2 R E A L * 6 F O R C E S R E L t f* 8 F O R C E T R E A L * 8 F O R C T O R E A L * 8 F O R C T 1 R E A L * 8 F O R C T £ R E A L * 8 F R A D R E A L * 8  56  64 £ 0 9 8 184 1714 17££ 1906 £ 0 3 0 19£ 614 £££ 630 1£0  ^ C R  165 Appendix 3 Spreadsheet Simulation Program ANALYSIS OF LIN PERFORMANCE - MOTOR TYPE t MODEL NO. File: V0LTA6E  03-Har-87  SPECIFIED PARAMETERS  Exoerimental Motor t3 Note: To obtain a printout of this f i l e enter "F7* then "Print".  Ref: Effect of Annular Stator 1.0045  o Potter Supply  = = Voltage perCurrent: Phase: E RatedRMS RMS Io(V) (A) Supply Frequency: fO (hz)=  SUDDV I  No. of Phases: a Type of Connection: 0 for Delta, 1 for V=  60.0 254.0 360 3 1  o Motor 6eoaetry  Rotor Thickness: c (•)= Average Air-Gap on Each Side of Rotor (•) = Stator Lamination Width: 1 (•)= Stator Back Iron: dl (•) = Rotor Width: 11 (•)= Mean Diameter: D (•)= Angular Sector Occuoied by Motor: SEC (deg) Winding Pitch (Coil Soan/Pole Pitch): Wo = Mean Tooth Width: t (•)=  o Electrical and Maonetic Specifications of Motor Rotor: 1 for nonmagnetic, 0 for Mimetic = 0 No. of Parallel Coil 6rouos/Stator/Phase: Np= 5 No. of Turns/Slot: N 70 No. of Turns/Coil: Nc= 35 No. of Turns in Series: Ns= 140 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  3rO0E"O3 1.00E-03 3.40E-02 1.60E-02 8.4BE-02 0.279 360.0 0.833 7.30E-O3 = 7.30E-03 2.20E-02 Deoth Mean SlotTooth Width: S M(• (*) ) 2 No. of Stators: Sn=  CALCULATED PARAMETERS o Electrical Parameters RMS Voltage/Phase: E (V) 254 = 4.44tfOttte*ohiiK* Peak Stator Surface Current Density: Jmo (A/m) 5.23E+04 = Hsqrt(2)/Np*KN*H/(t+sw) (2 stators) RMS Magnetizing Current/Phase: Iso 41.3 = 4.42£5*P2*6e*phi/(q*K»»Dtl#Ns)t(360/SEC) (p.94 Alger) Inductance/Phase: L i (H) Fringing Factor: f r 2.93 ( 0.184 flloer) Effective Stator-Stator Airgap: Ge (•) 2.85E-03 = gt(t+sw!7<l+fr*g) 6oodness Factor: 6 7 = mu0*ctvs«2/(rho«6e*omega) A  Geometrical Parameters Stator Length: Ds (•) 0.876 = D*«*SEC/360 Wavelength:"lambda (•) 1.75E-01 = 2#Ds/P Pole Pitch: p (•) = 8.76E-02 Ds/P 2.0 = o/(t+s«)/q No. of Slots/Phase Belt: n 63 = laabdatfO Synchronous Velocity: vs (m/s) 72.00 = vs/«/D Synchronous Frequency: fs Synchronous RPM: SRPM = 4320 = fs*60 Peak Flux and Flux Densities Peak Air Gap Maonetic Field: B0 (Tesla) 0.642 = muO#Jm/k/6e Total Flux/Pole: phi (Wb) 1.22E-03 = (2/«)tB0*l*D Average Flux Density: Bav (Wb/m*2) •• 0.409 = (2/i)«B0 Average Tooth Flux Density: Bta (Wb/m*2) •• 0.817 = phi/(n«l*o«t) Peak Tooth Flux Density: Btp (Wb/m*2) 1.283 = i/2*Bta Peak Core Flux Density: Be (Wb/m*2) 1.118 = phi/l/dl (Divided by 2 for a 360 deg stator - Alner pl&8) Coil Parameters  Pitch Factor: Kp 0.9659 = sin(Wpti/2) Distribution Factor: Kd 0.9659 = sin<x/(2*q))/<n*sin(«/2/n/a>) Winding Factor: KM 0.9330 = Kp*Kd F i l l Factor: Ff 0.3423 = diT2/4*pi/N/SM/Tooth deoth Copper Losses in Windings Length of Coil/Turn: lc (•) 2.87E-01 = 2tl+3*Wp*o Resistance/Winding of Ns Turns: R M (ohm) 0.8819 = Ns*lc»rho«4/«/dM2 Current/Winding: M I (A) 4.1 = Ii/(Sn««o> Total Copper Losses: Wcu (KW) 0.452 = IiT2*RM*Np*Sn#q/1000 (3 ohases) A  o Other Constants  alpha v02 k  3.8 = mu0«c/rho/6e 2315.9 = 4/1/(11-1) 35.9 = i/o  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/(ka+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 i s 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 A  OJNSTflNT FLUX MODEL Slip v (a/s) OK 2* 4* 6* 8* 10* 12* 14* 16* IB* 20* 22* 24* 26* 28* 30* 35* 40* 45* 50* 55* 60* 65* 70* 75* 80* 85* 90* 95*  63.1 61. B 60.5 59.3 58.0 56.8 55.5 54.2 53.0 51.7 50.5 49.2 47.9 46.7 45.4 44.1 41.0 37.8 34.7 31.5 28.4 25.2 22.1 18.9 15.8 12.6 9.5 6.3 3.2  psi in (fl/i) 0.000 0.047 0.095 0.141 0.187 0.233 0.277 0.320 0.362 0.403 0.443 0.481 0.517 0.552 0.586 0.618 0.692 0.759 0.817 0.870 0.916 0.958 0.995 1.028 1.058 1.085 1.110 1.132 1.153  52275 52334 52509 52801 53206 53723 54348 55077 55906 56832 57849 58952 60138 61400 62735 64138 67915 72027 76420 81049 85875 90867 96000 101251 106603 112042 117556 123135 128769  I i (fl)  F0 (NI  Fl (N)  41 41 42 42 42 42 43 44 44 45 46 47 48 49 50 51 54 57 60 64 68 72 76 80 84 89 93 97 102  0 24 47 71 95 118 142 166 189 213 237 260 284 308 331 355 414 473 533 592 651 710 769 829 888 947 1006 1065 1124  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  CONSTANT CURRENT MODEL Slip v (•/s) 0*  63.1  psi 0.000  F0 (N) Fl (N) 0  0  Specific Thrust Outout Ft (N) Ft (lb) (lb/in*2) HP 0 24 47 71 95 118 142 166 189 213 237 260 284 308 331 355 414 473 533 592 651 710 769 829 888 947 1006 1065 1124  0 5 ' 11 16 21 27 32 37 43 48 53 59 64 69 75 80 93 106 120 133 146 160 173 186 200 213 226 239 253  0.00 0.12 0.23 0.35 0.46 0.58 0.69 0.81 0.92 1.04 1.15 1.27 1.3B 1.50 1.61 1.73 2.01 2.30 2.59 2.88 3.17 3.45 3.74 4.03 4.32 4.60 4.89 5.18 5.47  Specific Thrust Out out Ft (N) Ft (lb) (lb/in*2) HP  Out out KU  0  0  0.00  0.0  0.0  0.00 1.96 3.84 5.65 7.37 9.01 10.57 12.05 13.45 14.78 16.02 17.18 18.26 19.26 20.18 21.02 22.77 24.03 24.78 25.03 24.78 24.03 22.77 21.02 18.77 16.02 12.76 9.01 4.76  Output KU 0.00 1.46 2.87 4.21 5.50 6.72 7.B9 8.99 10.04 11.02 11.95 12.81 1162 14.37 15.06 15.68 16.99 17.92 18.48 18.67 18.48 17.92 16.99 15.68 14.00 11.95 9.52 6.72 3.55  Bo=el+i»fl el fl 0.000  -0.642  Bo-=el+itfl el fl 0.000 -O.030 -0.061 -0.090 -0.120 -0.148 -0.176 -0.202 -0.227 -0.252 -0.275 -0.297 -0.317 -0.337 -0.355 -0.372 -0.410 -0.441 -0.468 -O.490 -0.509 -0.525 -0.538 -0.550 -0.559 -0.568 -0.575 -0.581 -0.586  -0.642 -0.641 -0.639 -0.635 -0.630 -0.624 -0.617 -0.609 -0.600 -0.590 -0.560 -0.569 -0.558 -0.546 -0.535 -0.523 -0.494 -0.466 -0.439 -0.414 -0.391 -0.369 -0.349 -0.331 -0.315 -0.299 -0.285 -0.272 -0.261  167 Appendix 4 COMPUTER MODELLING  A4.1  Introduction to Computer Modelling  This Chapter describes two methods of implementing the equations required  to  motor.  A  simulate  the  performance of the  traditional method,  described in Section A4.2.  using  a  linear induction  Fortran  program,  is  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 transient  are  also  performance  [43,44].  These  analysis  of  used of  applications  motors  have  to  calculate  motors  the  under  of computers traditionally  steady-state and  unusual for the  relied  on  conditions design and the  large  mainframe computer and usually the Fortran programming language.  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  computers [45,46].  is  now  available  on personal  168 Another new tool to emerge for the motor designer is the spreadsheet program.  The advantages of using a spreadsheet program  include the clear and logical organization of the input data, the ease with which the greater  detail  of  spreadsheet can be built up to include  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. is found in Appendix 2 of the thesis.  This program  169 The  program uses  the  presented in Chapt. 2.  one-dimensional  field  theory  equations  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  thickness, effective  poles, rotor  air-gap.  surface  resistivity  current density, frequency  The effective  of  rotor width, rotor  the  air-gap must  supply  and the  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 analysis. Lotus  program was  to do financial  In 1983, the Lotus Development Corporation introduced  1-2-3  which  spreadsheets.  had  many  more  features  than  previous  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. some  cells,  In this way information can be entered into  documentation  describing  what  different  cells  represent can be written into others, and the results of computations performed by the formulas appear in others. sheets can become very large.  The spread-  The Symphony spreadsheet can  accommodate up to two million cells but in actual fact the computer'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  longitudinal  is  end  induction motors.  the effect  simplest which  method occurs  of  in  The induction motor is  representing  high  speed  the  linear  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.  Chapter 2  were  The  field  theory  implemented in the  equations  described in  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. the spreadsheets below (see appears).  Both of  are organized into three sections as described  Appendix 3 to see the spreadsheet as it normally  ANALYSIS OF L l M PERFORMANCE - MOTOR TYPE 6. MODEL  NO.:  MOTOR #621  3 S S S Z S S Z S Z B = SS = SSSS3=S = CXZZ = S = = Z = SSX = = = SSX3 = = S = S = = = = =.SS:  File:  LIM_DES 21-Jun-86  Note:  To o b t a i n  aprintout  o f t h i s f:  S S S S S S S S = = = = 3 3 S S Z S S = SSS3SSS=S = S = SSS = S S 3 S S Z S S e S S = 3 = SS = = SS = S = = = SSSX3  S P E C I F I E D PARAMETERS o  Power  R e f . DESIGN A  Supply Rated  RMS L i n e V o l t a g e : E l <V> = S u p p l y F r e q u e n c y : f O <hz) = 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 = No. o f P h a s e s : q = o  220 60 1 3  E l e c t r i c a l and M a g n e t i c S p e c i f i c a t i o n s o f M o t o r R o t o r : 1 f o r non-magnetic, 0 f o r magnetic = 1 Peak A i r Gap M a g n e t i c 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 F r e e S p a c e : muO <H/m) = 1.26E-06 R o t o r E l e c t r i c a l R e s i s t i v i t y : r h o (ohm-m) = 1.72E-08 Number o f P o l e s : P = 6 :  s  s  s  =  = =  =  o Motor Geometry R o t o r T h i c k n e s s : c <m>~= 4.76E-03 L a m i n a t i o n W i d t h : 1 (m) * 7.62E-02 R o t o r W i d t h : 11 (m> = 1.27E-01 No. o f S t a t o r s : Sn = 2 T o o t h D e p t h : t d <nO « 5.56E-02 E a c h S i d e o f R o t o r <m> = 1.81E-03 S t a t o r Back I r o n : d l <m> • 5.00E-02 Mean D i a m e t e r : D <m> = 0.470 : SEC ( d e g ) = 360.0  s = i  Fig. A4.1 - Design Spreadsheet  : SE  s s s s :  *°  ANALYSIS OF LIM PERTORMANCE - MOTOR TYPE 6. MODEL N O . : File:  21-Jun-86>  Note: Ref:  Power  Supply R a t e d RMS S u p p l y C u r r e n t : I o <A> = RMS M a g n e t i z i n g C u r r e n t : I <A> = S u p p l y F r e q u e n c y : f O <hz> * No. o f P h a a e s : q 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 » 8  300.0 250.0 60 3 1  E l e c t r i c a l and M a g n e t i c S p e c i f i c a t i o n s o f M o t o r R o t o r : 1 f o r non-magnetic, 0 f o r magnetic = 1 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 = 6 No. o f T u r n s / S l o t : N = 36 No. o f T u r n s / C o i l : Nc = IS No. o f T u r n s i n S e r i e s : Ns = 54 Number o f P o l e s : P = 6 R o t o r E l e c t r i c a l R e s i s t i v i t y : r h o (ohm-m) = 1 .72E-08 D i a m e t e r o f C o p p e r W i r e : dw <m) = 4 . 12E-03 P e r m i t t i v i t y o f F r e e S p a c e : muO <H/m) = 1 .26E-06 : = 3 s s s :  t S S 3 3 3 :  To o b t a i n  a printout  of this f i  = 3Z333 = 3 = 3 3 3 S 3 Z 3 = S 3 3 3 S 3 S 3 3 3 3 3 3 3 X 3 3 3 3=33333= = = = S  S P E C I F I E D PARAMETERS  o  #621  CUR_ANAL  =3SZ3SX33S333SS** = 3 B S C 3 S & 3 3 3 3 £ C S 8 3 3 3 3 S 3 S 3 3 3 3 C 3 S 3  o  MOTOR  [ S 3 S 3 3 :  DESIGN B o M o t o r Geometry R o t o r T h i c k n e s s : c <m> = 4 . 7 6 E - 0 3 E a c h S i d e o f R o t o r <m> = 1 . 8 1 E - 0 3 L a m i n a t i o n W i d t h : 1 <m> » 7 . 6 2 E - 0 2 S t a t o r Back I r o n : d l <m> » 5 . 0 8 E - 0 2 R o t o r W i d t h : 11 <m> = 1 . 2 7 E - 0 1 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 T o o t h W i d t h : t Cm) = 1 . 3 7 E - 0 2 Mean S l o t W i d t h : aw (m) = 1 . 3 7 E - 0 2 T o o t h D e p t h <m> = 2 . 5 4 E - 0 2 No. o f S t a t o r s : Sn 2 3  : 3 3 3 S 3 3 :  Fig. A4.2 - Analysis Spreadsheet  ! 3 3 = S 3 !  CO  CALCULATED o  PARAMETERS  Electrical  Parameters R a t e d RMS V o l t a g e / P h a s e : E <V) Peak S t a t o r S u r f a c e C u r r e n t D e n s i t y : Jmo <A/m) RMS M a g n e t i z i n g C u r r e n t / P h a s e : Imo I n d u c t a n c e / P h a s e : Lm (H) Fringing Factor: f r E f f e c t i v e S t a t o r - S t a t o r A i r g a p : Ge (m) Goodness F a c t o r : G Geometrical  = 127 = E l f o r D e l t a 6. E l / s q r t < 3 ) f o r V * 7.84E+04 = k-»Ge»B0/mu0 ( A t s = 0> = 267.3 = 4 . 4 2 E 5 « P ' 2 « G e » p h i / ( q « K w « D « l » N s ) = 1.26E-03 = E/(2«ir«f 0 » l m ) 1.20 (p.184 A l g e r ) =» 9.64E-03 = g« ( t + s'w) / <l + f r * g ) = 83 -= m u O » c » v a » « 2 / ( r h o » G e « o « e g a ) s  Parameters S t a t o r L e n g t h : Ds (m) W a v e l e n g t h : lambda <m> P o l e P i t c h : p <m> No. o f S l o t s / P h a s e B e l t : n Mean T o o t h W i d t h : t <m) Mean S l o t W i d t h : sw (m) S y n c h r o n o u s V e l o c i t y : v s (m/s) Synchronous Frequency: f s S y n c h r o n o u s RPM: SRPM  3  and F l u x D e n s i t i e s Total Flux/Pole: A v e r a g e F l u x D e n s i t y : Bav Average Tooth F l u x D e n s i t y : B t a Peak T o o t h F l u x D e n s i t y : B t p Peak C o r e F l u x D e n s i t y : Be  = = = = = =  1.476 4.92E-01 2.46E-01 3 1.37E-02 1.37E-02 30 20.0 = 1200  a  D«TT»SEC/360  = 2«Ds/P = Ds/P = p/(t*sw)/q = p/(2»n«q> (sw = t ) = lambda»f0 = va/ic/D = fa«60  Peak F l u x  p h i (Wb) (Wb/m^2) (Wb/m' 2) (Wb/m 2) (Wb/m^2) s  /s  * 9.55E-03 = 0.509 = 1.019 = 1.600 = 1.253  (2/TT) • B 0 « l « p (2/TT) «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  Coil  Parameters  No. o f P a r a l l e l o  W i n d i n g 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 W i n d i n g F a c t o r : Kw No. o f T u r n s i n S e r i e s : Ns No. o f T u r n s / C o i l : Nc No. o f T u r n s / S l o t : N C o i l G r o u p a / S t a t o r / P h a s e : Np  W i r e Gauge & C o p p e r L o s s e s i n W i n d i n g s S l o t C r o s s - S e c t i o n a l A r e a : Sa ( n 2 ) U s e a b l e A r e a / C o n d u c t o r : Aw (m"2) D i a m e t e r o f Copper W i r e : dw <m) L e n g t h o f C o i l / T u r n : l c (m> R e s i s t a n c e / W i n d i n g o f Ns T u r n s : Rw (ohm) C u r r e n t / W i n d i n g : Iw (A) T o t a l C o p p e r L o s s e s due t o Im: Wcu (KW) C u r r e n t D e n s i t y i n W i r e : Jw (A/cm"2) A  * * * * * = * *  0.8889 0.9848 0.9598 0.9452 54 18 36 6  * 7.60E-04 1.06E-05 = 4.12E-03 * 8.09E-01 = 5.66E-02 22.27 » = 1.01E+00 = 167 3  = = « = * » *  C o i l Span/Pole P i t c h sin(Wp«*/2> sindr/(2»q> > / (n»sin ( i r / 2 / n / q ) ) Kp»Kd E/(4.44«Kw«f0«phi> Ns/n 2«Nc  » td»sw = 0.5»Sa/N ( 5 0 * P a c k i n g F a c t o r ) ( f r o m Look-Up T a b l e ) = 2»l*3»Wp«p » Ns»lc»rho»4/Tr/dw"2 » Im/(Sn»Np) = IW*2»Rw«Np«Sn«q/1000 (3 p h a s e s ) = 4«Iw/<ir»dw*2>. (Maximum A c c e p t a b l e  Fig. A4.A4 - Calculated Parameters (Second Half)  -a cn  THRUST AND POWER CALCULATIONS o  M a i n T h r u s t * FO FO * l«Jm«B0»D/2«cos(psi)«ain(psi) *kl«coa(psi)*sin<psi) kl * 3075.007 tan(psi) 2 « T T » s « f o»alpha/(k~2+vo2> Entrance Thrust • F l Fl = Jm«l/2«Re(bl«(1-expC(rl*i«k)«D1)/<rl+i«k)J E x i t Thrust i s not c a l c u l a t e d , but i s usually small, Optimum R o t o r R e s i s t i v i t y f o r Maximum T h r u s t a s 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 r h o (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 35  o o o =..  3  =  =  =  *  =  =i =  =  =  :  J  =  =  x  =  n  =  =  =  :  J  =  =  ,: =  ^  CONSTANT FLUX MODEL Slip OX 2X 4X 6X 8X  v (m/s) 29.5 28.9 28.3 27.8 27.2  p s i Jm (A/m) 0.000 0.223 0.427 0.598 0.738  73179 75045 60384 88570 98899  Im (A) 250 256 274 302 337  Fig.  FO (N)  F l (N)  0 699 1398 2097 2795  3 - Analysis  0 0 0 0 0  Specific Thrust F t (N) F t ( l b ) ( l b / i n ~ 2 ) 0 699 1398 2097 2795  0 157 314 471 628  0.00 0.90 1.80 2.70 3.60  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 abbreviation  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. the  case of a motor design,  a wire gauge for the  In  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 generated.  The calculations  complex numbers.  required are involved since  they contain  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  relative  set  of  and absolute  calculations  is  cell addresses  formulated  correctly, using  as required, the powerful  copy features of Symphony are used to generate a complete table of parameters for printing or for generating graphs. "relative" and "absolute" addresses  refer  to  are treated in formulae.  the  (The terms  manner in which cell  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.*2 K8S: (Fl) U-fl88)«3.281«I88/550*iB$42 LBS: (Fl) 0.746«KB8 M88: <F3) -*Bi47t«IN(C88)  AB9: BS9: C89: D89s .E89: F89: S89i H89: 189: J89: KB9: L89: W9:  A  (tt) 6.02 (Fl) +«*42»(1-A89) (F3) lflTfiN(«I«3Hflfl9) (FO) HBt30/iCGS(CS9) (FO) «*W31/K0S(C89) (FO) HC$75fKIN(C89)/eC0S(C89) (FO) +flD89**F«15 (FO) +F89+B89 (FO) +H89/4.448 (F2) •I89/««11/$B<38/39.42 (Fl) (l-fl89)«3.281»I89/550«iB*42 (Fl) 0.7*6»K89 (F3) -$BiA7i*SIN(C89) A  Table A4.1 - Thrust and Power Formulae  not appear as part of the normal viewing area or printout. results are available in tabular or graphical form. the graphical results is shown in Fig. A4.6.  The  One form of  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  spreadsheet program where the  is  then  imported  data can be used  graphs or be integrated into word processing documents.  into  the  to produce  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 Chapter 5) are presented.  (this motor is  completely  described in  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  182 variations in this thesis.  Experimental Results of Test U9S  •  % synchronous speed Power  Fig. A4.7 - Comparison Between Simulation ( Experimental Result ( A4.5  ) and  ) For Experimental Motor #3  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 approximately 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  versions of the spreadsheet programs  machines.  When  newer  with more mathematical  functions become available, than even greater numbers of applications will be possible.  13  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 29  GAGE  TYPICAL  ( . 3 5 MM) T H I C K CORE  & EXCITING  LOSS  POWER  AT 6 0 AND '100 HERTZ  0 1.0 ^  .4  C O TF V -  .7  CORE L O S S 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 )  185 Appendix 6 Finite Element Analysis of Experimental Motor #2 (Courtesy of G.E. Dawson, Queens University)  186  P a q e1 1 0 2 8 8 7 2 1 : 2 5 : 4 2 7 M i c r o s o f t F 0 R T R A N 7 7 V 3 2 .0 0 2 / 8 4 « * T h i s P r o g r a m D o e s T h e A n a l y s i s o f R e l a t i v e E n d E f f e c t *•* i s i s r e l e n d o s . R e a l » 8 R E L E N D S , L I P G ,O O D P .A I R P , B I ,O N E ,K N K ,P T ,E S T 2 C O M P L E X * 1 6 A L P H A O B A ,L P H A T B N .E G E Y E C .A L F A O B C ,A L F A T B T .E S T T 1 .E S T 3 , * T E S T 4 T ,E S T 5 T ,E S T 6 C ,R E L E N D T .E S T 7 T .E S T 8 N E G E Y E = ( 0 0 . 1 .0 .) P I = 3 1 .4 1 5 G 0 0 D = 4 2 0 . P A I R = 4 0 . D O 1 0 0 J = 2 0 4 .2 2 .2 G 0 0 D = J W R T I E<» 4 , 0 G )O O D P ,A I R 4 0 F O R M A T I ( X G ' ,O O D N E S S = F , ' 7 1 ' . ,N U M B E R O F P O L E P A I R S = F . '7 1 .) D O 1 0 0 1 = 0 9 ,9 1 , S L I P = I 1 /0 0 0 . S L I P S Q = ( 1 O . S L I P ) * * 2 B O N E=S Q R T ( 1 0 . + ( 4 0 . ( G / O O D S L I P S Q ) * » 2 ) K P = S Q R T ( 0 5 .« ( B O N E + 1 0 .) K N = S Q R T ( 0 5 .« ( B O N E 1 0 .) T E S T 2 = 0 5 .* G 0 0 D « ( 1 O . S L I P ) T E S T 3 = C M P L X K (P K ,N ) T E S T 4 = T E S T 3 1 0 . A L P H A O B = 0 5 . » G 0 0 D * < 1 O . S L I P ) * ( T E S T 3 + 1 O . ) A L P H A T B = 0 5 .« G 0 0 D » ( 1 0 .S L I P ) * ( T E S T 4 ) C A L F A 0 B = D C 0 N J G A (L P H A O B ) C A L F A T B = D C O N J G A (L P H A T B ) T E S T 5 = 2 0 . * P I * P A I R * C (A L F A T B + N E G E Y E )  2 2 2 2 2 2 2 2 2 2 2 2 2 2 2  T E S T 6 = C D E X P T (E S T 5 ) 2 9 T E S T 1 = 2 0 . » P I « P A I R * C (A L F A T B + N E G E Y E ) 3 0 W R T I E <* 5 . 0 P )P I ,A I R B ,O N E K ,P K ,N T ,E S T 2 A ,L P H A T B C .A L F A T B N ,E G E Y E T .E S T 1 . 3 1C T E S T 3 T ,E S T 4 T .E S T 5 T .E S T 6 3 2C 3 3C 5 0 F O R M A T x l ( . ' R E A L 6 ' ,F 1 0 3 ., C O M P L E X 8 . ' F 1 4 7 . . . 8 F 1 4 7 . ) 3 4 C R E L E N D = N (E G E Y E * < C A L F A O B+S L I P * G O O D ) * ( T E S T 6 1 0 . ) 1 0 ( ( . / + • N E G E Y E « S L I P * G O O D * ) C (A L F A T B C -A L F A O B * ) C (A L F A T B * N E G E Y E ) ) 3 5 3 6 T E S T 7 = N (E G E Y E * C (A L F A O B + S L I P * G O O D * ) T (E S T 6 1 0 ) . ) 3 7 T E S T 8 = 0 1 ( + .N E G E Y E * S L P I " G O O D * C ) (A L F A T B C -A L F A O B * C ) (A L F A T B + N E G E Y E ) 3 8 R E L E N D = R E A L C (R E L E N D ) W R I T E ( « , 1 1 0 ) s l i p . r e l e n d 3 9 4 0 1 1 0F O R M A T X I (, ' S L I P = ' , F 1 0 . 3 , ' R E L A T V IE E N D E F F E C T = F . '1 0 3 .) 4 1C W R I T E * ( 1 , 1 5 T )E S T 7 T .E S T 8 1 1 5F O R M A T ( I X T ' ,E S T 7 = 2 , ' F 1 0 3 . ' , T E S T 8 = 2 , ' F 1 0 3 .) 4 2C 4 3 1 0 0C O N T N I U E 4 4 S T O P 4 5 E N D  N a m e  T y p e  A L P H A O C 0 M P L E X * 1 6 A L P H A T C 0 M P L E X * 1 6 B O N E R E A L * 8 C A L F A O C 0 M P L E X * 1 6 C A L F A T C 0 M P L E X » 1 6 C D E X P C M P L X C 0 M P L E X * 1 6 C R E L E N D C O N J G E A L 8 G O O D R I N IT E G E R » 4  O f f s e t P C l a s s 1 7 8 2 1 0 1 1 4 2 4 2 2 5 8 I N T R I N S I C I N T R I N S I C 3 7 0 I N T R I N S I C 2 6 9 8  D L i n e# 1 7 J INTEGER»4 REAL'S KN REAL»S KP NEGEYE C 0 M P L E X « 1 6 REAL«8 PAIR REAL*S PI REAL RELEND REAL'S SLIP REAL»8 SLIPSQ REAL SORT C0MPLEX«16 TEST1 REAL»8 TEST2 C0MPLEX«16 TEST3 COMPLEX-16 TEST4 C0MPLEX»16 TEST5 C0MPL£X«16 TEST6 C0MPLEX«16 TEST7 C0MPLEX»16 TEST8  Name  Type  INTRINSIC 626 102 110 INTRINSIC 338 138 146 162 274 306 498 546  Size  Class PROGRAM  MAIN P a s s One  Microsoft 42 130 122 2 34 18  No E r r o r s 45 S o u r c e  Detected Lines  F0RTRAN77  Page 2 10-28-87 21:25:42 V3.20 02/84  

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