Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1955 The effect of non-sinusoidal impressed voltages on A.C. servo motors Busey, Charles W. Monterey, California: U.S. Naval Postgraduate School ..hwh:''..' THE EFFECT OF NON-SINX JSOIDAL IMPRESSED VOLTAGES ON A. C SERVO MOTORS vCHARLES W, SUSEY :^il 4.,, •K''*'wi*i' Library n. S. NdVdl Posfgririuate School Monterey, Calif(.ruin 1 1-aG V0\-' V(9\-TA6-E. 'Penntf (^. ^€iaU bookbinding 2216 CLEMENT STREET San Francisco 21, Calif. BA. 1-6352 DIRECTIONS FOR B\H\>\HQ THE EPFE BUCKRAM LETTERING ON BACK 8854 COLOR NO TO BE EXACTLY AS PRINTED HERE. FABRIKOID COLOR BUSEY 1955 LEATHER COLOR THESIS B92 l»nw in Gold OTHER INSTRUCTIONS Letter on front cove|r IMPRESSED THE EFFECT OF NON SINUSOIDAL VOLTAGES ON A. C. SERVO MOTORS CHARLES W. BUSEY \ THE EFFECT OP NON-SINUSOIDAL IMPRESSED VOLTAGES ON A. C. SERVO MOTORS by Charles Webb Buaey Lieutenant, United States Navy and William Russell Bartow Lieutenant, United States Navy Submitted in partial fulfillment of the requirements for the degree of MASTER OP SCIENCE IN ELECTRICAL ENGINEERING United States Naval Postgraduate School Monterey, California 19 5 5 .„,..• Tlicsis J^n Ub ^' 'V fv* '''"'-''ZT.y"'''s. J9 This work is accepted as fulfilling the thesis requirements for the degree of MASTER OP SCIENCE IN ELECTRICAL ENGINEERING from the United States Naval Postgraduate School PREFACE In response to a suggestion by the Bureau of Ships, Navy Department, Washington, D. C, investigation as to the effects of non-sinusoidal impressed voltages on the opera- tion of alternating current servo motors was undertaken in October 1954 at the Electrical Engineering Laboratory, U.S. Naval Postgraduate School, Monterey, California. Inter-related with the effects of non-sinusoidal im- pressed voltages are the effects of space harmonics of mag- neto-motive force, the latter producing such phenomena as asynchronous crawling and sub-synchronous locking. Bearing this in mind, an effort was made to detect adverse effects upon the operation of several two-phase induction servo motors which might be attributable to either time or space harmonics, or possibly a combination of the two. The motors tested were standard production models except where one modification was made in an endeavor to specifically create an unfavorable design situation. The results of this investigation indicated that wave form, balanced or unbalanced, had no seriously adverse ef- fect upon the torque-speed characteristics of the motors tested. This might have been anticipated of production models where design features would certainly tend to elimi- nate or minimize such effects. The writers acknowledge and deeply appreciate the' ii assistance and encouragement given by Professor W. A. Stein, U. S. Naval Postgraduate School, throughout the entire period of this investigation. ^il TABLE OP CONTENTS I Item Page Certificate of Approval i Preface ii List of Illustrations V Table of Symbols and Abbreviations vli Chapter I Introduction 1 Chapter II Theoretical Analysis 1. Non-sinusoidal waves 4 2. Time harmonics 4 3. Space harmonics 5 4. HaiTOonics in practical systems 7 5. Development of torque 7 6. Time-harmonic torques 8 7. Heating effects of non-sinusoidal waves 13 8. Space-harmonic torques 14 9. Unbalanced complex waves 14 10. Summary of theory 17 Chapter III Investigative Proceedure and Results 1. Testing equipment 19 2. Circuitry and instrumentation 19 3. Selection of motors to be tested 22 4. Proposals 23 5. General precautions 23 6. Sinusoidal time-harmonic test data 24 7. Space-harmonic test data 25 8. Unbalanced time-harmonic test data 26 9. Appraisal of test data 28 Chapter IV Conclusions 29 Bibliography 31 Iv LIST OF ILLUSTRATIONS Illustration • Page !• Schematic diagram of circuitry for 32 early phases of investigation 2. Schematic diagram of frequency con- 33 trol and oscilloscope triggering circuits 3. Photograph of harmonic generator 34 4. Frequency tripler circuit 35 5. Photograph of dynamometer and tern- 36 perature control 6. Temperature characteristics of 37 Ketay motor 7. Temperature characteristics of 38 Kearfoot motor 8. Current oscillograms showing effects 39 of saturation for 40o/l200 cps in- vestigation 9. Current oscillograms of rated and 40 sub-rated currents for 400/l200 cps investigation 10. Family of torque-speed curves for 41 Ketay motor 11. Stall torque versus control voltage 42 for Ketay motor 12. Ketay torque-speed curves for complex 43 impressed voltages 13. Cutaway photograph of Kearfoot motor 44 14. Rated torque-speed curve of Kearfoot 45 motor Illustration Page 15, Kearfoot torque-speed curve for 46 400/1200 cpa investigation 16, Schematic diagram of wattmeter 47 transfer switch 17, Photograph of frequency tripler 48 3et-up VI I TABLE OP SYMBOLS AND ABBREVIATIONS (Listed Alphabetically) (X - Space angle a.c. - Alternating-Current d.c. - Direct-Current Cp3 - Cycles per second fn - Frequency of Nth harmonic H - Magnetic Intensity «Cn - Magnetic intensity of Nth harmonic of control phase H/5^ - Magnetic intensity of Nth harmonic of reference phase K - Any constant multiplier P - Number of poles Rl - Primary ohmic resistance R2 - Secondary ohmic resistance referred to primary terms rras - Root-mean-square rpra - Revolutions per minute a - Slip Sn - Slip in terms of Nth harmonic synchronous speed 'o^Ax - Maximum torque per phase To "* - Torque of Nth harmonic per phase 1st - stall torque per phase Vr? - Effective ( rms ) value of Nth harmonic voltage per phase ^ , - Angular velocity ..1 ^ ^t - Time angle XI - Primary leakage reactance per phase X2 - Secondary leakage reactance per phase referred to primary terms 4^P - Instantaneous height- of rectangular wave rU^ - Instantaneous height of triangular wave = - Equal by definition vill CHAPTER I INTRODUCTION Automation is a newly coined word tending to describe the modern trend toward greater and greater application of automatic control. A system by which automatic control is obtained is generally referred to as a servomechanism, and it may be one or a combination of several mechanical, hy- draulic, pneumatic, or electrical components. The tWo-phase induction motor is an electrical compo- nent widely used in control systems. For such application, characteristics such as bi-directional control, linear variation of torque with speed, linear variation of torque with amplitude of signal, and high starting torque are par- ticularly desirable, and much study has been devoted to the development of two-phase induction motors that possess these characteristics. A motor so designed is commonly re- ferred to as a servo motor. The classical analysis of the operation and character- istics of the induction motor Is based upon the assumption that the impressed voltages, the air gap and rotor fluxes, and the primary and secondary currents are all sinusoidal. Using this assumption. Professor Koopman /7/"- showed that the torque versus speed curve is essentially linear, and -"• Numerals in brackets refer to bibliography. See page 28« . that the torque versus control voltage Is precisely linear at stall. In practice, however, these assumptions are rarely valid, and the effect this deviation may have on the theoretical operation of the servo motor therefore becomes of vital interest to those charged with the specification, design, and development of servomechanisras The purpose of this investigation was to determine the effect of non-sinusoidal volta,^es upon the theoretical torque-speed characteristics of the servo motor. A general investigation would have been desirable, but such investiga- tion would have required the design features of the motors under test to be variable in order to show the effects of coil pitch, distribution of winding, skew of rotor slots, and rotor resistance. Unfortunately, access to such motors could not be had, and consequently the investigation was limited to production models in which the designers had pre- sumably utilized all means at their disposal to adjust the motor parameters so as to provide optimum characteristics. Thus the investigation became one to determine effects upon a few particular, but representative, models rather than an investigation of comprehensive scope. The plan of procedure was roughly divided into two parts In the first part, families of torque versus speed curves were obtained for balanced sinusoidal impressed voltages of various magnitudes and frequencies, and for balanced non- sinusoidal voltages of known harmonic content. The torque • versus speed characteristics of the non-sinusoidal impressed voltages should theoretically be the same as the resultant torque obtained by superposition of the individual harmonic torques The second part of the plan was to force sinusoidal currents of harmonic frequencies through one phase winding irtiile maintaining current in the other phase winding sinu- soidal and of fundamental frequency. It was recof:;nized that no torque would be produced by time harmonics in this instance, but the effect due to a possible combination of time and space harmonics was not obvious; indeed, it was intuitively felt that a torque would exist irrespective of the negative Indication given by the theoretical analysis of Chapter II, Ihe test results of both phases of the investigation tended to verify in a most convincing manner, the theoreti- cal predictions pertaining to the effects upon the operating characteristics of the motors tested. CHAPTER II THEORETICAL ANALYSIS 1. Non-slnu3oldal waves. Fourier demonstrated how certain periodic functions could be analyzed into an infinite series of sinusoidal functions of increasing frequency. The lowest frequency present in such an anal^/sis is generally taken as the refer- ence, or fundamental, while the higher order frequencies are termed harmonics, -«• As exaraplies , the Fourier analysis of the rectangular wave is: UJkaAH /l^~^ tiyrULa hjU^h4 o/ /iMCL^Oyy^C-^ and the analysis of the triangular wave is cJkiAji. Af = kjUoU-f <^ t^u ci^^Jcjb^ T[hus these non-sinusoidal waves may be considered to be the sum of their individual sinusoidal components. 2. Time harmonics.