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Analysis of Induction Motor Torque

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Analysis of Induction Motor Torque ANALYSIS OF INDUCTION MOTOR TORQUE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Sakae Yamamura, B. E., M. S. The Ohio State University 1953 Approved by; Adviserv^ TABLE OF CONTENTS Page Chapter I. Introduction. 1 Chapter II. The Principle of the Analyzer. 6 Chapter III. The Component Parts of the Analyzer. 12 1. Homopolar generator. 12 2. Amplifiers. 15 3* Power supplies. 20 4. Differentiating circuit. 23 5. Frequency response of the analyzer. 27 6. Cathod-ray scope. 31 7. Adjustments of the complete set. 33 Chapter IV. Results Obtained. 38 1. Three-phase induction motor; 220 V, 1/4 HP, 4 poles, 60-cycle. 38 2. Single-phase operation of the three-phase Induction motor; 220 V, l/A HP, 4 poles, 60-cycle. 45 3. Three-phase induction motor; 220 V, 1/8 HP, 4 poles, 60—cycle. 48 4- Single-phase induction motor; 115 V, 1/6 HP, 4 poles, 60-cycle. 53 5. Shaded-pole motor; 115 V, 1/12 HP, 6 poles, 60-cycle. 58 Chapter V. Consideration of Steady States. 63 1. Three-phase induction motor. 63 2. Single-phase induction motor; 115 V, 1/6 HP, 4 poles, 60—cycle. 66 -i- A 0 U 8 6 1 Page 3. Consideration of braking torque of the three-phase induction motor; 220 V, 1/4 HP, 4 pOles, 60-cycle. 75 (a). Friction and windage. 79 (b). Iron loss try fundamental magnetic flux. 80 (c). Saturation of iron. 83 (d). Skin effect. 88 (e). Space harmonics of air gap magnetic flux. 92 (f). Iron loss due to zigzag leakage flux. 95 (g). Some additional considerations of the extraordinary braking torque. 99 4- Single-phase operation of the three-phase induction motor; 220 V, 1/4 HP, 4 poles, 60-cycle. 103 Chapter VI. Tr*isient Torque of Three-phase Induction Motors* 108 1. Differential equations of unbalanced two-phase induction motors. 109 2. Transient phenomena of a balanced two-phase induction motor. 112 3. Solution of the transient phenomena of the three-phase induction motor (220 V, 1/4 HP, 4 poles, 60-cycle) by the electronic analog computer. 126 Chapter VII. Conclusion and Acknowledgments. 133 Appendix I. Some experiments about the tooth-pulsation loss due to zigzag leakage flux. 135 -11- Page Appendix II. Derivation of the equivalent two-phase induction motor for a three-phase induction motor. 140 Appendix III. Solution of transient phenomena of a balanced three-phase induction motor by the ReeveB Analog Computer. 14-9 Bibliography. 159 Autobiography. 160 -iii- CHAPTER I IKTRODUCTIOli Speed-torque characteristics sre among the most important characteristics of induction motors. They enable us to judge, when the ■peed-torque curve of the load is also known, whether a motor under consideration can start and carry expeoted loads successfully or not. Here, we take the speed-torque characteristics to mean generally any relation between motor speed and motor torque. Usually they mean the relation between motor speed and time-average torque at steady state. But there are many aspects of speed-torque characteristics, as we shall see in this dissertation; and not all of these have been thoroughly investigated, mainly because they are difficult to determine experi­ mentally or to compute. In order to measure torque at any given speed with a dynamometer, the induction motor under test must be kept at constant speed under a certain load. This is difficult for certain speed ranges, although not impossible. Labor and cost involved in such measurements are prohibitively high in some oases, When attempts are made to oaloulate the torque, computations do not give accurate results for wide range of speed. This is probably due to variation of machine constants used in these computations. The machine constants are usually obtained from no load test and locked rotor test, or by calculations from formulas and design data. Because of magnetic saturation of the iron, these values are usually satisfactory for current up to or a little higher than rated current. Hence, for lower 2 speed, torque computed from these machine constants is different from the measured torque due to magnetic saturation by larger current. Several attemptshave been made to correct the machine con­ stants for magnetic saturation. But they are not accurate and are often awkward to use for practical purposes. Deep slot-oage motors or double-cate motors are of great practical importance, but the speed-torque curves involving deep slot or double cage effect are very difficult to oompute. It will be shown in Article 3 of Chapter V that stray load loss has a great influence on torque, sometimes doubling or tripling the torque that would be developed without it. But so far practioally no attempt has been made to com­ pute stray load loss. From the above considerations we can see the need for devices which will enable us to obtain the speed-torque characteristics of in­ duction motors with more accuracy and ease. In order to meet the need, many attempts have been made.^®^^^®^ The principles underly­ ing these attempts are about the same, and the devices developed by these investigations may be called speed-torque analyzers. These analyzers trace out speed-torque curves on an oscilloscope while the motors under test are accelerated from standstill to no load speed. Hence, speed-torque curves are determined with much less labor than by the other methods, such as the dynamometer method or computation. But in spite of their adequacy for certain purposes, these analyzers are not accurate enough and cannot indioate some of the fine details of the speed-torque curve. In the following section we shall briefly explain the main reasons why they are not satisfactory, if more details 5 of the speed-torque curve are desired. f 3 } Z. Set o'- J uses an ordinary d-c generator as a tachometer. In order to obtain instantaneous torque# the output voltage from the d-c generator is differentiated by a C-R series oirouit. But an ordinary d-c generator has commutation ripple or noise in its output voltage. Generally, when any voltage is differentiated# high frequency componente of the voltage become more pronounced, ^ence# in the torque signal the commutation noise becomes more predominant due to its high frequencies qnd masks the details of the speed-torque curves. YJhen the commutation noise is fil­ tered out# the higher harmonic components of the frequency spectrum of the speed-torque curves are also filtered out. Details of the curves are thus lost and the curves obtained do not give a complete or true picture. H. Kondo^) uses a vacuum tube osoillator to measure instantaneous torque. The frequency of the oscillator is controlled by slight change of capacitance resulting from the twist of the motor shaft by the torque transmitted. Due to meohanic&l transient phenomena of the motor shaft# frequency change of the oscillator does not always represent the true instantaneous torque# and error is introduced thereby# especially for rapidly changing portions of the curve. S. Chang^5^ uses a square wave generator consisting of a circular disc with evenly distributed slots on its periphery. Square wave voltage is generated photo-electrically and its frequency is proportional to instantaneous motor speed. By taking average value of the sequence of square waves by a filter# d-c voltage# which is proportional to motor speed# is obtained. But the filter# which takes the average value of the sequence of square wave voltages# suppresses the higher harmonics of the signal as well. Hence, curves obtained by Chang's analyzer are distorted and not accurate. In the present research a homopolar generator is used as a tachometer instead of an ordinary d-c generator. In a homopolar generator the electromotive force induced in a single oonductor is d-c while it is a-o in an ordinary d-c generator. So the commutator is not required for a homopolar generator. Thus, its output voltage does not contain any com­ mutation ripple or noise and is pure d-c voltage at any constant speed. It mig^it seem strange that a homopolar generator has not been used as a tachometer in previous analysers, but this can be explained by the fact that it introduces a new difficulty. A homopolar generator is inherently a lew voltage device. Its output voltage is therefore too low for indicators of the type such as an electromagnetic oscillograph or a oathode-ray oscillograph, as long as the generator size remains within practical dimensions. Hence, a high-gain d-c amplifier is needed to produce enough deflection on an indicator. As is well known, a stable high-gain d-c amplifier is difficult to build. This is the reason afcy a homopolar generator has not been used successfully thus far. But proper design of the homopolar generator and the d-c ampli­ fiers, which will be explained in Chapter III, has resulted in the construction of a satisfactory speed-torque analyzer. This analyzer reveals on its cathode-ray scope such details of speed-torque curves as switching transient torque, pulsating torque of single-phase induction motors, eto. In Chapters II and III of this dissertation the principle and the structure of the analyzer will be explained. The various interesting 5 results obtained "with it will be given in Chapter IV. This analyzer revealed many features of speed-torque curves which had not been obtained by the dynamometer method or by the analyzer developed heretofore. Some of the curvos are quite different from the familiar one. In order to show that these new curves give a satis­ factory representation of the actual speed-torque curves, they will be compared with measured or calculated values in Chapters V and VI.
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