MAGNETIC-AMPLIFIER ANALYSIS USING A GENERALIZED MODEL FOR THE SATURABLE REACTOR CORE by HERBERT HORACE WOODSON S.B., Massachusetts Institute of Technology (1952) S.M., Massachusetts Institute of Technology (1952) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY (1956) Signature of Author ............. ......... - - - - ---- - Department of Electrical Engineering, May 14, 1956 Certified by ........................... .... Accepted by .. Chairman, De mittee on Graduate Students CoI vs oWi OCT 22" '196 LIBe R AW DBPARTMENT OF KLBCTRICAL BNGINiRRING MASSACHUSETTS INSTITUTE OF TECHNOLOGY CA M 8 R I DGB 3 , MAS AC H U BTTS May 8, 1956 Mr. R. H. Woodson Room 10-098 M.I.T. Dear Mr. Woodson: This letter is to give you permission to print additional copies of your thesis by the multilith process, and to submit to the Department of Electrical Engineering copies thus produced, in lieu of the typed copies normally required. A copy of this letter is to be reproduced by the same process and is to be placed in each copy of the thesis immediately following its title page. Sincerely yours, SS. H. Caldwell for the Department Graduate Committee SHC:mm iii MAGNETIC-AMPLIFIER ANALYSIS USING A GENERALIZED MODEL FOR THE SATURABLE REACTOR CORE by HERBERT HORACE WOODSON Submitted to the Department of Electrical Engineering on May 14, 1956, in partial fulfillment of the requirements for the degree of Doctor of Science. ABSTRACT A mathematical representation for a polycrystalline, thin-tape, ferromagnetic metal is derived for the operating conditions found during flux resetting in a magnetic ampli- fier. The starting point for the derivation is the dynamic behavior of the magnetization process in ferromagnetic single crystals reported in the literature. The general reactor representation is simplified to a form which is suitable for magnetic-amplifier analysis, and the simplified representa- tion is verified experimentally. The constants which describe the reactor characteristics are obtained from a constant cur- rent switching characteristic of the type used for describing reactors in the digital-computer field. The simplified reactor representation is applied to the analysis of a single-core, self-saturating magnetic-amplifier circuit with a direct resetting voltage and arbitrary reset circuit resistance. The results of the analysis yield reason- ably accurate predictions of the amplifier input-output char- acteristic over wide ranges of supply frequency and reset cir- cuit resistance. In addition, the reset circuit parameters necessary for maximum power gain are obtained in terms of the reactor constants and supply frequency. At the same time, the analysis yields a measure of the relative reset circuit resistance necessary for operation with so-called current control and voltage control. This provides an answer to the problem of whether reset aircuit resistance is high or low. Thesis Supervisor: David C. White Title: Associate Professor of Electrical Engineering 0i ACKNOWLEDGMENTS A research of this type would be impossible without the encouragement and assistance of many people. The author would like to thank Professor David C. White for his support, encouragement, and patience in the supervision of this thesis. Thanks' are due to Professors Kusko and Epstein for their helpful comments on the work, and especially to Professor Epstein and his colleagues in the Lab- oratory for Insulation Research who helped the author over many rough spots in the theory, of ferromagnetism. Miss Evelyn Fraccastoro was very helpful with the typing of the first draft in record time. The superb skill of Mrs. Bertha Hornby made the task of typing the final draft an. easy one. Mr. Lund and Mr. Kelley, of the M.I.T. Illustration Service, have been most helpful with the figures. The author appreciates the support ex- tended by the Energy Conversion Group of the Servomechanisms Laboratory under Air Force Contract No. AF33(616)3242, without which this research would not have been possible. vi TABLE OF CONTENTS Page ABSTRACT iii .... 0000aaa000000†.0..0 006000 .. .*.. * .. . CHAPTER I INTRODUCTION 1.0.0 Objectives ................................... 1 1.1.0 Present Status of Magnetic-Amplifier Theory 2 1.1.1 Brief History of Magnetic Amplifiers . 2 1.1.2 Operation of the Single-Core Circuit 7 1.1.3 Previous Analyses of the Single- Core Circuit ......0..0..0................ 8 1.2.0 Theory of Ferromagnetism .................... 13 15 1.2.1 Processes of Nucleation ..,......... 18 1.2.2 Domain Wall Motion .................... CHAPTER II THE MATHEMATICAL REPRESENTATION FOR A REACTOR 2.0.0 Introduction ................................. 23 2.1.0 Single-Domain Behavior ....................... 24 2.1.1 Plane Wall in Rectangular Specimen .... 24 2.1.2 Cylindrical Wall in Cylindrical Specimen 2.1.3 General Form of Equations Describing Single-Domain Behavior ..... 31 2.2.0 Mathematical Representation for a Polycrystalline Material .................... 35 2.2.1 Distribution of Starting Fields 36 2.2.2 Definition of an Average Normalized Domain Dimension .......... 38 2.2.3 Rate of Change of Flux ............... . 40 2.3.0 Experimental Verification of the Representation ....................... ..... 43 2.3.1 Switching with Large Constant Field ... 43 2.3.2 Switching with Small Constant Field ... 45 2.3.3 Switching with Slowly Varying Field ... 51 2.4.0 Variation of ID/OD Ratio .................... 55 Svi vii TABLE OF CONTENTS (continued) Page CHAPTER III MAGNETIC-AMPLIFIER ANALYSIS USING THE REACTOR REPRESENTATION 3.0.0 Introduction ................................ 58 * 3.1.0 "Exact" Analysis, Constant-Current Reset . 63 * 3.1.1 Calculation of Reset Flux ............ 65 3.1.2 Calculation of Normalized Input-Output Characteristic .......... 66 3.1.3 Experimental Verification of "Exact" Analysis .................... . 66 3.2.0 Approximate Analysis .. 67 3.2.1 Constant-Current Reset ............... 70 3.2.2 Constant Voltage Reset with Finite Resistance 72 3.2.3 Maximization of the Power Gain ...... 74 3.2.4 Normalized Input-Output Characteristics, Finite Resistance 76 .. e CHAPTER IV RESULTS AND CONCLUSIONS 4.0.0 Introduction ............................ ... . 80 4.1.0 The Mathematical Representation for a Reactor ........ 0.. 000......... ....... 83 4.1.1 Switching with Large Constant Field .. 87 4.1.2 Switching with Small Fields .......... 88 4.2.0 Magnetic-Amplifier Analysis ................. 91 4.2.1 "Exact" Magnetic-Amplifier Analysis .. 92 4.2.2 Approximate Magnetic-Amplifier Analysis 94 4.3.0 Conclusions ................0. . 0..... 100 4.3.1 Material Quality ..................... 100 4.3.2 Application of Existing Reactors ..... 102 4.4.0 Areas of Future Work ........................ 104 APPENDIX DERIVATION OF SINGLE-DOMAIN BEHAVIOR ..... 106 APPENDIX II INPUT-OUTPUT CHARACTERISTIC, "EXACT" ANALYSIS, CONSTANT-CURRENT RESET ......... 112 viii TABLE OF CONTENTS (continued) Page APPENDIX III MAXIMIZATION OF THE POWER GAIN 116 · · · · · · · · · · · APPENDIX IV NORMALIZATION OF INPUT-OUTPUT CHARACTERISTICS .............. 119 · · · · · · · · · · · BIOGRAPHICAL NOTE ................. •. ....... 122 · · · · · · · ,··· BIBLIOGRAPHY 123 ............ ***********•.• . • · · · ~·,·~·~· ix ILLUSTRATIONS Fi No. Title Page .g. 1-1 Saturable reactor circuits ................... 4A 1-2 (?-F characteristic of ideal saturable reactor 4A 1-3 Series-connected saturable reactor with feedback 6A 1-4 Examples of self-saturating circuits 6A 1-5 Doubler circuit using two cores ................ 6B 1-6 Single-core self-saturating circuit ............ 6B 1-7 Single-core circuit .0000000.......................... 6B 1-8 Dynamic B-H loop for 2-mil Orthonol 60 cps sinusoidal flux ......... .............. 6C 1-9 Voltage-current characteristic of silicon junction rectifier ..................... 8A 1-10 qP-F loop of saturable reactor .................. 8A 1-11 Domain configurations .......................... 12A 1-12 Specimen shape for mathematical representation 12A 1-13 Specimen with surface imperfection ............. 16A 1-14 Processes of flux change o...................... 16A 1-15 Energy as a function of wall position .......... 16A 2-1 Structure for plane-wall movement .............. 24A 2-2 Definition of co-ordinates for plane- wall motion .............. ...................... 24A 2-3 Normalized rate of change of flux vs wall position for a plane wall in a rectangular specimen with eddy-current damping only ........ 26A 2-4 Structure for cylindrical domain dynamics ...... 26A 2-5 Definition of variables for cylindrical wall ... 28A 2-6 Voltage waveform for expanding cylindrical domain in cylindrical specimen with constant applied field and eddy-current damping ......... 28A