ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR A LARGE-SIGNAL ANALYSIS OF THE TRAVELING-WAVE AMPLIFIER Technical Report No. 19 Electron Tube Laboratory Department of Electrical Engineering By Joseph E. Rowe Approved by: W. G. Dow G. Hok Project 2112 NAVY DEPARTMENT BUREAU OF SHIPS ELECTRONICS DIVISION CONTRACT No. NObsr-63114 INDEX No. NE-071417 Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the University of Michigan April, 1955

ABSTRACT It is the purpose of this dissertation to derive the large-signal traveling-wave amplifier equations, to outline a method by which these equations may be solved on a high-speed digital computer, and to present the resulting equations in such form that the information will lead to a fuller understanding of the physical phenomena involved in the high-level operation of the travelingwave amplifier. Equations are derived describing the large-signal operation of the traveling-wave amplifier, including the effects of a-c space charge and attenuation along the helical slow-wave structure. The equations constitute a system of nonlinear partial-differential-integral equations and are valid for all values of the parameters which are encountered in typical high-power traveling-wave amplifiers. The parameters which appear in the equations are the relative injection velocity b, the gain parameter C, the large-signal space-charge parameters K and B, the loss parameter d, and the input-signal level A0. The working equations were programmed for and solved on the Michigan Digital Automatic Computer, MIDAC, located at the University of Michigan's Willow Run Research Center. The r-f voltage amplitude A(y), the phase lag of the r-f wave relative to the electron stream @(y), the electron phase g(y,0O), and the velocity deviation 2Cu(y,~o) were computed for several values of C, K, and b at B = 1, Ao = 0.0225, and d = 0. Zero-space-charge solutions are presented for C = 0.05, 0.1, and 0.2 with b as the parameter in order to determine the value of b which gives the maximum saturation gain and the optimum tube length. For C = 0.1 similar solutions are obtained for two values of the space-charge parameter K, 1.61 and 3.42. Then the input-signal level Ao is varied for a fixed C, K, and b. In addition, the effect of attenuation along the helix is determined by obtaining solutions for several values of the loss parameter d, namely 0.1, 0.25, 1.0, and 2.0, with fixed injection velocity and then the effect of attenuation in lowering the optimum injection velocity is shown through solutions for several values of b at d = 2.0. Finally, the effect on the saturation gain of the placement of the loss is studied: solutions are given for loss beginning at approximately 2, 3, and 4 wavelengths respectively from the input. The results of these various solutions are presented in graphical form and are very useful for designing and predicting the performance of largesignal traveling-wave amplifiers. iii

PREFACE The first chapter of this dissertation contains general material on the traveling-wave amplifier, background information on the small-signal analysis, and a discussion of why this approach is not suitable to be extended to describe the large-signal operation of the traveling-wave amplifier. In Chapter II and Appendices A and B the working equations describing the operation of the large-signal traveling-wave amplifier are derived. The boundary conditions are also discussed in Chapter II, and Appendix C presents an outline of the method used to obtain the roots of the small-signal equations needed to evaluate these boundary conditions. In Chapter III the working equations are programmed for solution on MIDAC and a discussion of the characteristics of MIDAC is presented. An experimental analysis of the presence of errors in the solutions and of the stability of the working equations relative to the propagation of errors is also discussed in detail, and a flow diagram for the solution of these equaions on MIDAC is presented. Appendix D contains the programs and operating instructions for the solutions. In Chapter IV the results are presented in graphical form and discussed. In Chapters V and VI these results are analyzed further and interpretive curves are presented for use in the design of large-signal traveling-wave amplifiers and also in predicting the performance of these tubes. A suggested program for future work is also outlined in Chapter VI. The author wishes to express his appreciation to the members of his committee for their assistance, and especially to Professors W. G. Dow and Gunnar Hok, who devoted considerable time and offered valuable suggestions during the course of the work. The Mathematics Group at the Willow Run Research Center, in particular Messrs. R. T. Dames and L. Razgunas, rendered invaluable assistance in the programming and running of the problem on MIDAC. Appreciation is also due to Miss Priscilla Woodhead for her excellent work in preparing the many figures. The research on which this dissertation is based was supported by funds from the U.S. Navy Bureau of Ships under Contract NObsr 63114 and the U.S. Signal Corps under Contract DA 36-039 sc-52654. iv

TABLE OF CONTENTS Page ABSTRACT iii PREFACE iv LIST OF FIGURES vii CHAPTER I. INTRODUCTION 1 1.1 Statement of the Problem 1 1.2 History of the Problem 2 1.3 Extension of the Eulerian Approach 11 CHAPTER II. THEORETICAL EQUATIONS FOR THE LARGE-SIGNAL ANALYSIS 16 2.1 Derivation of the Theoretical Equations 16 2.2 Relation to Nordsieck's Equations 30 2.3 Derivation of Alternate Theoretical Equations 31 2.4 Boundary Conditions 37 CHAPTER III. SOLUTION OF THE THEORETICAL EQUATIONS ON MIDAC 43 5.1 Introduction 43 3.2 Brief Description of MIDAC 43 3.3 Transformation of the Differential Equations into Difference Form 51 5.4 Details of the Computing Process 57 3.5 Experimental Study of Computation Errors 65 CHAPTER IV. GRAPHICAL PRESENTATION OF RESULTS 82 4.1 R-F Voltage and Power along the Helix 84.4.2 Phase Angle of the Wave Relative to the Electron Stream 92 4.3 Electron Phase vs. Initial Phase 103 4.4 Normalized Electron Velocity Deviation 113 4.5 Electron Flight-Line Diagrams 124 4.6 Normalized Linear Current Density vs. Electron Phase 146 4.7 Variable Input-Signal Level 167 4.8 Effect of Series Loss along the Helix 171 CHAPTER V. ANALYSIS OF THE RESULTS 185 5.1 Voltage Gain and Tube Length at Saturation 185 5.2 Efficiency and Tube Length at Saturation 188 5.3 Variable Input-Signal Level 196 5.4 Effect of Loss on Saturation Gain and Phase Shift 202 5.5 Comparison with Experimental Results 206 v

TABLE OF CONTENTS (cont.) Page CHAPTER VI. CONCLUSIONS 209 6.1 Introduction 209 6.2 Accuracy of the Results 209 6.3 Summary of the Results 210 6.4 Suggestions for Further Research 212 APPENDIX A. DERIVATION OF THE GENERAL TRANSMISSION-LINE EQUATION 214 A.1 Equivalent Circuit 214 A.2 Definition of Variables 214 A-3 Derivation of the Equation 216 APPENDIX B. DERIVATION OF THE SPACE-CHARGE FIELD EXPRESSION 220 B.1 Underlying Assumptions 220 B.2 Derivation of the Equation 221 APPENDIX C. CALCULATION OF THE SMALL-SIGNAL PROPAGATION CONSTANTS 235 APPENDIX D. MIDAC PROGRAM AND OPERATING INSTRUCTIONS 248 TRAVELING-WAVE TUBE: PROGRAM NO. 13MOm23,24 249 A. General Information 249 B. Initial Computations 249 C. Resuming Computations 252 TRAVELING-WAVE TUBE: PROGRAM NO. 13M13m7 295 A. General Information 295 B. Initial Computations 295 C. Resuming Computations 297 TRAVELING-WAVE TUBE: PROGRAM NO. 13M12m2 326 BIBLIOGRAPHY 330 Small-Signal Theory 330 Large-Signal Theory 330 Space-Charge Wave Analysis and Electron Interaction 331 Numerical Analysis 332 General Reference Books 333 LIST OF SYMBOLS 334 vi

LIST OF FIGURES Page Fig. 1-1. Schematic Diagram of a Typical Traveling-Wave Amplifier and a Sketch of the Electric and Magnetic Field Distribution about the Helix. 3 Fig. 1-2. A Helix-Type Traveling Wave Amplifier. 4 Fig. 1-3. Typical Phase Velocity Characteristic for a Helix. 5 Fig. 1-4. Equivalent Circuit of a Traveling-Wave Tube Assuming a Lossless Helix. 12 Fig. 2-1. Distance-Time Diagram for Electrons Entering the Helix Region, Indicating Starting Time and Entrance Time for Each Electron. Also the Multivaluedness of zo or yo as a Function of z or y Is Shown When Overtaking Occurs. Distance May be Measured in Units of z or y. 20 Fig. 3-1. Michigan Digital Automatic Computer Installation. 45 Fig. 3-2. Input Photoelectric Tape Reader. 46 Fig. 3-3. Functional Components of MIDAC. 48 Fig. 3-4. MIDAC High-Speed Acoustic Memory. 49 Fig. 3-5. MIDAC Low-Speed Magnetic Drum Memory. 50 Fig. 3-6. Integration Diagram Used in the Numerical Solution of the Working Equations. 52 Fig. 3-7. Flow Diagram for the Numerical Solution of the LargeSignal Traveling-Wave Amplifier Equations on MIDAC. 58 Fig. 3-8. Geometrical Interpretation of the Newton-Raphson Process for the Solution of f(e) = 0. 61 Fig. 3-9. Effect of the Number of Electrons Considered: Amplitude of the R-F Voltage along the Helix with Space Charge QC as the Parameter. C = 0.1, d = 0, b = 1.5, Ao = 0.0225. 69 Fig. 3-10. Effect of the Number of Electrons Considered: R-F Phase Lag of the Wave Relative to the Electron Stream with Space Charge QC as the Parameter. C = 0.1, d = 0, b = 1.5, Ao = 0.0225. 70 vii

LIST OF FIGURES (cont.) Page Fig. 3-11. Comparison of Solutions Using 32 and 64 Electrons: Percent Difference Values of the R-F Voltage Amplitude, A(y), vs. Distance. Solution Studied: C = 0.1, d = 0, b = 1.5, Ao = 0.0225. 71 Fig. 3-12. Comparison of Solutions Using 32 and 64 Electrons: Percent Difference in Values of the R-F Phase Lag, 9(y), vs. Distance. Solution Studied: C = 0.1, d = O, b = 1.5, Ao = 0.0225. 72 Fig. 3-13. Truncation Error Study: Amplitude of the R-F Voltage along the Helix vs. Distance. Solution Studied: C = 0.1, QC = 0, d = 0, b = 0, Ao = 0.01, 32 Electrons. 75 Fig. 3-14. Truncation Error Study: R-F Phase Lag of the Wave Relative to the Electron Stream vs. Distance. Solution Studied: C = 0.1, QC = 0, d = 0, b = 0, Ao = 0.01, 32 Electrons. 74 Fig. 3-15. Round-Off Error Study: First Significant Figure Affected in 9(y) When Rounded Multiplications Are Replaced by Unrounded Multiplications vs. Integration Increment. Solution Studied: C = 0.1, QC = 0, d = 0, b = O, Ao = 0.01. 79 Fig. 3-16. Propagation of Errors: Stable and Unstable Solutions of a Differential Equation. 81 Fig. 4-1. Amplitude of the R-F Voltage along the Helix with Relative Injection Velocity as the Parameter. C = 0.05, QC = 0, d = O, Ao = 0.0225. 85 Fig. 4-2. Amplitude of the R-F Voltage along the Helix with Relative Injection Velocity as the Parameter. C = 0.1, QC = O, d = O, Ao = 0.0225. 86 Fig. 4-3. Amplitude of the R-F Voltage along the Helix with Relative Injection Velocity as the Parameter. C = 0.2, QC = 0, d = 0, Ao = 0.0225. 87 Fig. 4-4. Amplitude of the R-F Voltage along the Helix with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0.125, d = 0, Ao = 0.0225. 88 Fig. 4-5. Amplitude of the R-F Voltage along the Helix with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0.25, d = 0, Ao = 0.0225. 89 viii

LIST OF FIGURES (cont.) Page Fig. 4-6. R-F Power along the Helix with Relative Injection Velocity as the Parameter. C = 0.05, QC = 0, d = O, Ao = 0.0225. 93 Fig. 4-7. R-F Power along the Helix with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0, d = 0, Ao = 0.0225. 94 Fig. 4-8. R-F Power along the Helix with Relative Injection Velocity as the Parameter. C = 0.2, QC = 0, d = 0, Ao = 0.0225. 95 Fig. 4-9. R-F Power along the Helix with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0.125, d = O, Ao = 0.0225. 96 Fig. 4-10. R-F Power along the Helix with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0.25, d = O0 Ao = 0.0225. 97 Fig. 4-11. R-F Phase Lag of the Wave Relative to the Electron Stream with Relative Injection Velocity as the Parameter. C = 0.05, QC = O, d = O, Ao = 0.0225. 98 Fig. 4-12. R-F Phase Lag of the Wave Relative to the Electron Stream with Relative Injection Velocity as the Parameter. C = 0.1. QC = O, d = O, Ao = 0.0225. 99 Fig. 4-13. R-F Phase Lag of the Wave Relative to the Electron Stream with Relative Injection Velocity as the Parameter. C = 0.2, QC = O, d = 0, Ao = 0.0225. 100 Fig. 4-14. R-F Phase Lag of the Wave Relative to the Electron Stream with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0.125, d = O, Ao = 0.0225. 101 Fig. 4-15. R-F Phase Lag of the Wave Relative to the Electron Stream with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0.25, d = 0, Ao = 0.0225. 102 Fig. 4-16. Electron Phase Relative to the R-F Wave vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.05, QC = O, d = 0, Ao = 0.0225, b = 0.076. 105 Fig. 4-17. Electron Phase Relative to the R-F Wave vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.05, C = O, d = O, Ao = 0.0225, b = 1.75. 106 ix

LIST OF FIGURES (cont.) Page Fig. 4-18. Electron Phase Relative to the R-F Wave vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0, Ao = 0.0225, b = 0.1525. 107 Fig. 4-19. Electron Phase Relative to the R-F Wave vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0, d = 0, Ao = 0.0225, b = 2.0. 108 Fig. 4-20. Electron Phase Relative to the R-F Wave vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.2, QC = O, d = O, Ao = 0.0225, b = 0.31. 109 Fig. 4-21. Electron Phase Relative to the R-F Wave vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.2, QC = 0, d = 0, Ao = 0.0225, b = 2.0. 110 Fig. 4-22. Electron Phase Relative to the R-F Wave vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.125, d = O, Ao = 0.0225, b = 0.65. 111 Fig. 4-23. Electron Phase Relative to the R-F Wave vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = 0, Ao = 0.0225, b = 1.0. 112 Fig. 4-24. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.05, QC = 0, d = 0, Ao = 0.0225, b = 0.076. 114 Fig. 4-25. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.05, QC = 0, d = 0, Ao = 0.0225, b = 1.75. 115 Fig. 4-26. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0, d = O, Ao = 0.0225, b = 0.1525. 116 Fig. 4-27. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = O, d = 0, Ao = 0.0225, b = 2.0. 117 Fig. 4-28. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.2, QC = O, d = 0, Ao = 0.0225, b = 0.51. 118 Fig. 4-29. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.2, QC = 0, d = 0, Ao = 0.0225, b = 2.0. 119 x

LIST OF FIGURES (cont.) Page Fig. 4-30. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.125, d = 0, Ao = 0.0225, b = 0.65. 120 Fig. 4-31a. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = 0, Ao = 0.0225, b = 1.0. 121 Fig. 4-31b. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = 0, Ao = 0.0225, b = 1.0. 122 Fig. 4-31c. Normalized Electron Velocity Deviation vs. Initial Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = 0, Ao = 0.0225, b = 1.0. 123 Fig. 4-32. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.05, QC = 0, d = O, Ao = 0.0225, b = 0.076. 125 Fig. 4-33. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.05, QC = 0, d = 0, Ao = 0.0225, b = 1.75. 126 Fig. 4-34. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = O, d ='0, Ao = 0.0225, b = 0.1525. 127 Fig. 4-35. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0, d = 0, Ao = 0.0225, b = 2.0. 128 Fig. 4-36. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.2, QC = 0, d = O, Ao = 0.0225, b = 0.31. 129 Fig. 4-37. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.2, QC = O, d = O, Ao = 0.0225, b = 2.0. 130 Fig. 4-38. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.125, d = O, Ao = 0.0225, b = 0.65. 131 Fig. 4-39a. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = O, Ao = 0.0225, b = 1.0. 132 xi

LIST OF FIGURES (cont.) Page Fig. 4-39b. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = O, Ao = 0.0225, b = 1.0. 133 Fig. 4-39c. Normalized Electron Velocity Deviation vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = O, Ao = 0.0225, b = 1.0. 134 Fig. 4-40. Electron Flight-Line Diagram: Distance along the Helix vs. Electron Phase with Initial Phase as the Parameter. C = 0.05, QC = O, d = O, Ao = 0.0225, b = 0.076. 136 Fig. 4-41. Electron Flight-Line Diagram: Distance along the Helix vs. Electron Phase with Initial Phase as the Parameter. C = 0.05, QC = O0 d = O, Ao = 0.0225, b = 1.75. 137 Fig. 4-42. Electron Flight-Line Diagram: Distance along the Helix vs. Electron Phase with Initial Phase as the Parameter. C = 0.1, QC = O, d = O, Ao = 0.0225, b = 0.1525. 138 Fig. 4-43. Electron Flight-Line Diagram: Distance along the Helix vs. Electron Phase with Initial Phase as the Parameter. C = 0.1, QC = 0, d = 0, Ao = 0.0225, b = 2.0. 139 Fig. 4-44. Electron Flight-Line Diagram: Distance along the Helix vs. Electron Phase with Initial Phase as the Parameter. C = 0.2, QC = O, d = O, Ao = 0.0225, b = 0.31. 140 Fig. 4-45. Electron Flight-Line Diagram: Distance along the Helix vs.Electron Phase with Initial Phase as the Parameter. C = 0.2, QC = O, d = 0, Ao = 0.0225, b = 2.0. 141 Fig. 4-46. Electron Flight-Line Diagram: Distance along the Helix vs. Electron Phase with Initial Phase as the Parameter. C = 0.1, QC = 0.125, d = O, Ao = 0.0225, b = 0.65. 142 Fig. 4-47. Electron Flight-Line Diagram: Distance along the Helix vs. Electron Phase with Initial Phase as the Parameter. C = 0.1, QC = 0.125, d = 0, Ao = 0.0225 b = 1.5. 145 xii

LIST OF FIGURES (cont.) Page Fig. 4-48. Electron Flight-Line Diagram: Distance along the Helix vs. Electron Phase with Initial Phase as the Parameter. C = 0.1, QC = 0.25, d = 0 Ao = 0.0225, b = 1.0. 144 Fig. 4-49. Electron Flight-Line Diagram: Distance along the Helix vs. Electron Phase with Initial Phase as the Parameter. C = 0.1, QC = 0.25, d = 0, Ao = 0.0225, b = 1.25. 145 Fig. 4-50a. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.05, QC = 0, d =, Ao = 0.0225, b = 0.076. 148 Fig. 4-50b. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.05, QC = 0, d = 0, Ao =,0.0225, b = 0.076. 149 Fig. 4-51a. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.05, QC = 0, d = 0, Ao = 0.0225, b = 1.75. 150 Fig. 4-51b. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.05, QC = O, d = O, Ao = 0.0225, b = 1.75. 151 Fig. 4-52a. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0, d = 0, Ao = 0.0225, b = 0.1525. 152 Fig. 4-52b. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = O, d = 0, Ao = 0.0225, b = 0.1525. 1553 Fig. 4-53a. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = O, d = O, Ao = 0.0225, b = 2.0. 154 Fig. 4-53b. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0, d = 0, Ao = 0.0225, b = 2.0. 155 xiii

LIST OF FIGURES (cont.) Page Fig. 4-54a. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.2, QC = 0, d = O, Ao = 0.0225, b = 0.31. 156 Fig. 4-54b. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.2, QC = 0, d = 0, Ao = 0.0225, b = 0.31. 157 Fig. 4-54c. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.2, QC = O, d = O, Ao = 0.0225, b = 0.31. 158 Fig. 4-55a. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.2, QC = 0, d = 0, Ao = 0.0225, b = 2.0. 159 Fig. 4-55b. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.2, QC = 0, d = O, Ao = 0.0225, b = 2.0. 160 Fig. 4-56a. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.125, d = O0 Ao = 0.0225, b = 0.65. 161 Fig. 4-56b. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.125, d = O, Ao = 0.0225, b = 0.65. 162 Fig. 4-56c. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.125, d = 0, Ao = 0.0225, b = o.65. 163 Fig. 4-57a. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = O, Ao = 0.0225, b = 1.0. 164 Fig. 4-57b. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = O, Ao = 0.0225, b = 1.0. 165 xiv

LIST OF FIGURES (cont.) Page Fig. 4-57c. Normalized Linear Current Density in the Stream vs. Electron Phase with Distance along the Helix as the Parameter. C = 0.1, QC = 0.25, d = 0, Ao = 0.0225, b = 1.0. 166 Fig. 4-58. Amplitude of the R-F Voltage along the Helix with *, the Input Power Level below CIoVo, as the Parameter. C = 0.1, QC = 0.125, d = 0, b = 1.5. 168 Fig. 4-59. R-F Power Relative to CIoVo vs. Distance with |, the Input Signal Level in db below CIoVo, as the Parameter. C = 0.1, QC = 0.125, d = 0, b = 1.5. 169 Fig. 4-60. R-F Phase Lag of the Wave Relative to the Electron Stream with 4, the Input Power Level below CIOVo, as the Parameter. C = 0.1, QC = 0.125, d = 0, b = 1.5. 170 Fig. 4-61. Amplitude of the R-F Voltage along the Helix with Loss as the Parameter. C = 0.1, QC = 0.125, Ao = 0.0225, b = 1.5, d = 0 for 0 4 y c 1.6 for all curves. 173 Fig. 4-62. R-F Power along the Helix with Loss as the Parameter. C = 0.1, QC = 0.125, Ao = 0.0225, b = 1.5, d = 0 for 0 y z 1.6 for all curves. 174 Fig. 4-63. R-F Phase Lag of the Wave Relative to the Electron Stream vs. Distance along the Helix with Loss as the Parameter. C = 0.1, QC = 0.125, Ao = 0.0225, b = 1.5, d = 0 for 0 4 y 1.6 for all curves. 175 Fig. 4-64. Expanded Portion of Fig. 4-61. 176 Fig. 4-65. Expanded Portion of Fig. 4-63. 177 Fig. 4-66. Amplitude of the R-F Voltage along the Helix for Fixed Loss with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0.125, Ao= 0.0225, d = 0 for 0 4 y 4 1.6; d = 2.0 for y > 1.6. 179 Fig. 4-67. R-F Power along the Helix for Fixed Loss with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0.125, Ao = 0.0225, d = 0 for 0 L y 4 1.6; d = 2.0 for y > 1.6. 180 Fig. 4-68. R-F Phase Lag of the Wave Relative to the Electron Stream for Fixed Loss with Relative Injection Velocity as the Parameter. C = 0.1, QC = 0.125, Ao = 0.0225, d = 0 for 0 ~- y i 1.6; d = 2.0 for y > 1.6. 181 xv

LIST OF FIGURES (cont.) Page Fig. 4-69. Amplitude of the R-F Voltage along the Helix for Fixed Loss with Point of Loss Initiation as the Parameter. C = 0.1, QC = 0.125, Ao = 0.0225, b = 0.65. a. d = 0 for 0' y' 1.2; d = 2.0 for y > 1.2. b. d = 0 for 0' y x 1.6; d = 2.0for. c. dfor y > 1.6. c. d = 0 for 0 z y _ 2.4; d = 2.0 for y > 2.4. 182 Fig. 4-70. R-F Power along the Helix for Fixed Loss with Point of Loss Initiation as the Parameter. C = 0.1, QC = 0.125, Ao = 0.0225, b = 0.65. a. d = 0 for 0 4 y' 1.2; d = 2.0 for y > 1.2. b. d = 0 for 0' y - 1.6; d = 2.0 for y > 1.6. c. d = 0 for 0 L y' 2.4; d = 2.0 for y > 2.4. 183 Fig. 4-71. R-F Phase Lag of the Wave Relative to the Electron Stream for Fixed Loss with Point of Loss Initiation as the Parameter. C = 0.1, QC = 0.125, Ao = 0.0225, b = 0.65. a. d = 0 for 0 K y - 1.2; d = 2.0 for y > 1.2. b. d = 0 for 0 y y L 1.6; d = 2.0 for y > 1.6. c. d = 0 for 0 L y 4 2.4; d = 2.0 for y > 2.4. 184 Fig. 5-1. Voltage Gain and Tube Length at Saturation vs. Relative Injection Velocity. QC = 0, d = 0, Ao = 0.0225. 186 Fig. 5-2. Voltage Gain and Tube Length at Saturation vs. Relative Injection Velocity. C = 0.1, d = 0, Ao = 0.0225. 187 Fig. 5-3a. Efficiency and Tube Length at Saturation vs. Relative Injection Velocity. QC = 0, d = 0, Ao = 0.0225. 190 Fig. 5-3b. Efficiency/C and Tube Length at Saturation vs. Relative Injection Velocity. QC = 0, d = 0, Ao = 0.0225. 191 Fig. 5-4. Efficiency/C and Tube Length at Saturation vs. Relative Injection Velocity. C = 0.1, d = 0, Ao = 0.0225. 192 Fig. 5-5. Maximum Saturation Efficiency vs. Small-Signal Gain Parameter. QC = 0, d = 0. 193 Fig. 5-6. Maximum Saturation Efficiency/C vs. Small-Signal Space-Charge Parameter. C = 0.1, d = 0. 194 Fig. 5-7. Maximum Saturation Efficiency/C vs. Large-Signal Space-Charge Amplitude Parameter, K. C = 0.1, d = 0. 195 Fig. 5-8. |, Input Signal Level in db below CIoVo, vs. Tube Length at Saturation in Undisturbed Wavelengths. Saturation Level is Approximately 7 db above CIoVo. C = 0.1, QC = 0.125, d = 0, b = 1.5. 197 xvi

LIST OF FIGURES (cont.) Page Fig. 5-9. Phase Shift vs. j for Fixed Tube Length with Variable Input-Signal Level. C = 0.1, QC = 0.125, d = 0, b = 1.5, Ng = 5.5. 198 Fig. 5-10a. Output Power Level Relative to CIoVo vs. Input Power Level Relative to CIoVo for Fixed Tube Length. C = 0.1, QC = 0.125, d = O, b = 1.5, Ng = 5.5. 199 Fig. 5-lOb. Output Power Level Relative to CIoVo vs. Input Power Level Relative to CIoVo for Fixed Tube Length. C = 0.1, QC = 0.125, d = O, b = 1.5, Ng = 5.5. (Plotted to Show Ratio of Power Levels at Saturation.) 200 Fig. 5-lOc. Output Power Level Relative to CIOVo, PO vs. Input Power Level below CIoVo, for Fixed Tube Length. C = 0.1, QC = 0.125, d = 0, b = 1.5, Ng = 5.5. 201 Fig. 5-11. Percent Reduction in Saturation Gain vs. Loss Factor for Fixed Injection Velocity. C = 0.1, QC = 0.125, Ao = 0.0225, b = 1.5, d = 0 for 0 i y z 1.6. 203 Fig. 5-12. Percent Reduction in Phase Shift at Saturation vs. Loss Factor for Fixed Injection Velocity. C = 0.1, QC = 0.125, Ao = 0.0225, b = 1.5, d = 0 for 0 c y _ 1.6. 204 Fig. 5-13. Voltage Gain at Saturation vs. Relative Injection Velocity for Fixed Loss Factor. C =.0.1, QC = 0.125, AO = 0.0225, d = 0 for 0 z y L 1.6, d = 2 for y > 1.6. 205 Fig. 5-14. Velocity and Current in the Stream at Saturation vs. Relative Electron Phase with Relative Injection Velocity as the Parameter. C = 0.075, QC = 0.22, Pb' = 0.388. 207 Fig. 5-15. Velocity and Current in the Stream vs. Relative Electron Phase with Input-Signal Level as the Parameter. C = 0.1, QC = 0.064, Pb' = 0.412, b = 0.26. 208 Fig. A-1. General Equivalent Circuit of a Traveling-Wave Tube. 215 Fig. B-l. Sketch of the Electron Stream and Drift Tube Used to Derive the Space-Charge Field Expression. 222 Fig. B-2. Nomograph for Obtaining Radian Electron-Plasma Frequency from the Voltage, Current, and Diameter of an Electron Stream. 226 Fig. B-3. Electron Plasma Frequency Reduction Factor for Axial Symmetry. 228 xvii

LIST OF FIGURES (concl.) Page Fig. B-4. Space-Charge Parameter vs. Normalized Effective Electron Plasma Frequency and Gain Parameter. 229 Fig. B-5. Normalized Velocity of the Slow-and-Growing Wave vs. Gain Parameter. Parameter: Space Charge. 231 Fig. B-6. Space-Charge Field Weighting Function. a'/b' = 2. 233 Fig. C-l. QC = O, d = 0, C = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5. 240 Fig. C-2. QC = 0, d = 0.025, C = 0.1, 0.2, 0.3. 241 Fig. C-3. QC = 0.5, d = 0.025, C = 0.1, 0.2, 0.3. 242 Fig. C-4. QC = 1.0, d = 0.025, C = 0.1, 0.2, 0.3. 243 Fig. C-5. QC = 2.0, d = 0.025, C = 0.1, 0.2. 244 Fig. C-6. QC = 0, d = 0.025, C = 0.20. 245 Fig. C-7. QC = 0.5, d = 0.025, C = 0.2. 246 Fig. C-8. QC = 1.0, d = 0.025, C = 0.2. 247 xviii

CHAPTER I. INTRODUCTION 1.1 Statement of the Problem The traveling-wave amplifier is an important and useful tube; its high gain and inherently large bandwidth compare favorably with the characteristics of other microwave devices. A detailed understanding of its behavior under various conditions is needed, however, if the tube is to be used with maximum effectiveness. Equations describing the behavior of an idealized traveling-wave amplifier transmitting small signals have been derived and studied extensively by Pierce8, Kompfner5'6'7, and others2. These equations for the small-signal case can be linearized by application of a few reasonable assumptions and hence are amenable to solution by straightforward methods. The large-signal case, however, has not been studied in any great detail up to the present time, largely due to the complexity of the equations involved and to the lack of adequate mathematical techniques for solving them. Since high-speed digital computers now afford a means by which complex systems of mathematical equations of the type encountered in the large-signal case may reasonably be handled, it has seemed worthwhile, in terms of both time and expense, to undertake an analysis of the operation of the traveling-wave amplifier at high output levels. It is therefore the purpose of this dissertation to derive the equations describing the large-signal operation of a traveling-wave amplifier, to outline a method by which these equations may be solved on a high-speed digital computer, and to present the resulting calculations in such form 1

2 that the information will lead to a fuller understanding of the physical phenomena involved in the high-level operation of the traveling-wave amplifier and assist in the construction of tubes that more closely realize a given set of specifications. 1.2 History of the Problem 1.2.1 General Description of the Traveling-Wave Amplifier. The essential parts of a traveling-wave amplifier, as shown in Fig. 1-1, are (1) the electron gun, which produces an electron stream*; (2) the slowwave structure, in this case a helix, which supports the propagation of a slow electromagnetic wave with a longitudinal electric field; and (3) an electron collector. Figure 1-2 is a picture of a helix-type travelingwave amplifier showing the helix and electron gun structure. The phase velocity of the radio-frequency wave as it travels longitudinally along the helix is of the order of one-tenth the velocity of light; the phase velocity variation with frequency for a typical helix is shown in Fig. 1-3. An r-f signal is applied to the helix, at the same point as the electron stream enters the helix, by means of a waveguide and an antenna or by a coaxial line. This signal appears, amplified, at the output end of the helix and is removed from the helix by similar means. The bandwidth of the transducers used to couple the signal into and out of the helix limits the overall bandwidth of the device, since unfortunately transducers have not yet been built which can utilize the full electronic bandwidth of *In this dissertation a differentiation is made between the words "beam" and "stream". "Beam" is used to apply to the gross movement of electrons; i.e., when Eulerian mechanics are applicable, hence only in the smallsignal theory. The term "stream", on the other hand, is used where the electrons must be considered as individual particles (Lagrangian mechanics) as in the large-signal theory.

5 HEATER ELECTRON GUN "INPUT WAVEGUIDE OUTPUT WAVEGUIDE - _ 1 I iE ENVELOPE HELIX COLLECTOR Vh l —-r -— I V0 CATHODE ANODE o MAGNETIC FIELD ELECTRIC FIELD /? / \ / /' 1 t I / *^/ ^ F \/, { a HELIX WIRE'~ w~~~~~~~~~~~AXIS OF HELIX DIRECTION OF PROPAGATION _ FIG. 1-I SCHEMATIC DIAGRAM OF A TYPICAL TRAVELING-WAVE AMPLIFIER AND A SKETCH OF THE ELECTRIC AND MAGNETIC FIELD DISTRIBUTION ABOUT THE HELIX

w w -J 0W 3 Iw )I 4

5 X )- IJ W: -J w n LU) 2 1 I i X \ X It I e0 > o oe w L-.: = < c u O C ) - -- ) - ----------— ) oo c o \ S \) ~ o.aD ~ ~ ~ ~ ~ ~ ~ ~ L. I1.cc:M: cc ~~0 ~ ~ ~ ~ ~ IL - - - 0 -_ I - - - — J I 0r C4 0r <D (O ^ CI Q ~ < 0- 0 6 I.H911 JO A11B003A: A1100"13A 3SVHd "A

6 the traveling-wave amplifier. Typically, the overall bandwidth of a traveling-wave amplifier operating at 5000 mcs, with coaxial-line-to-waveguide transducers, for example, may be as high as 30 percent. In addition, a longitudinal magnetic focusing field of several hundred gauss must be provided to confine the electron stream so that it will pass completely through the helix. Either electromagnets or permanent magnets may be used for this purpose. Recently the technique of periodic focusing has been applied to traveling-wave amplifiers, reducing the weight of the focusing magnets by a factor of approximately 10. For the analysis of the operation of the traveling-wave amplifier it is necessary to consider only the slow-wave structure and the electron stream. 1.2.2 Brief Description of Amplifier Operation. The operation of the traveling-wave amplifier described in the previous section depends on the interaction of the r-f wave propagating along the helix with the electron stream traveling along the axis of the helix at a velocity approximately, but not exactly, equal to the axial phase velocity of the r-f wave. The longitudinal electric field in the stream due to the r-f wave on the helix exerts forces which velocity-modulate the stream, resulting in current-amplitude modulation. Since as many electrons are accelerated as are retarded, no energy is transferred from the wave to the stream in this process. This current modulation induces a longitudinal r-f current in the helix, which increases the amplitude of the r-f wave on the helix. The phase of the induced r-f wave is such that more electrons are retarded than are accelerated and hence the electron stream on the average loses energy, which accounts for the increase in the amplitude of the r-f wave.

7 This cycle of operation is repeated many times along the length of the helix. The phase velocity of the complete wave resulting from this repeated interaction between the initial wave and the electron stream will be lower than that of the initial wave because each wave which the bunched electron stream induces onto the helix will lag its predecessor. Hence it is easily seen that interaction can occur even when the phase velocity of the wave is equal to the velocity of the electron stream, although this is not in general the condition for optimum gain. Thus, electrons injected at approximately the same velocity as the wave will produce a net increase in the amplitude of the wave. However, the amplitude of the r-f wave will not increase indefinitely, since the nonlinearities of an actual tube set an upper limit to the final amplitude. In short, there is a saturation power output for a particular device, regardless of further increase in power input. 1.2.3 Small-Signal Analyses. The small-signal theory as outlined by Pierce8 leads to a simple analysis of the operation of the travelingwave amplifier. In this analysis properties of various types of slow-wave structures are first discussed in detail. Then simplified equations describing the overall behavior of the traveling-wave amplifier are introduced and solved. Rather thorough discussions of overall gain, insertion loss of the helix, a-c space-charge effects, noise figure, and power output are included. The small-signal analysis is limited chiefly by the assumption that the a-c velocity of the electrons is small compared to their average velocity, so that the nonlinear terms in the equations may be neglected and a wave-type solution obtained. It is also assumed that all the electrons in the same cross section of the stream are acted on by the same electric

8 field, which is a reasonable approximation if the stream is small compared to the size of the helix or if an annular electron stream is used. Further, it is assumed that a strong magnetic focussing field is used and transverse motion may therefore be neglected. The small-signal analysis of Kompfner516?7, although obtained by a considerably more complicated procedure, is essentially the same as Pierce's and arrives at the same conclusions but employs the same assumptions and hence suffers from the same limitations. The field theory of traveling-wave tubes derived by Chu and Jackson2 is a more elegant and rigorous treatment of the problem than Pierce's network-and-beam theory, but it leads to more complicated expressions which are not so useful to the engineer designing a traveling-wave amplifier. It is interesting to note, however, that the two analyses are in quite good agreement. 1.2.4 Need for Large-Signal Analysis. In view of the many applications for a traveling-wave amplifier operating at high output levels, it seems desirable to study the large-signal equations governing its behavior in detail. Of particular interest are the variations of gain and phase shift at high output levels for different values of such parameters as space charge, electron injection velocity, loss along the helical transmission line, and the gain parameter. Also of interest are the effects of space charge and loss along the helical transmission line on the saturation gain and power output. The overall objective, of course, is to learn how to design a traveling-wave amplifier to operate at high power levels with maximum efficiency. 1.2.5 Previous Large-Signal Analyses. Partial large-signal theories of the traveling-wave amplifier have been presented by Brillouin11 and by

9 Nordsieck12 and a more complete theory has recently been presented by Poulter.13 Brillouin's analysis treats the large-signal case as an extension of the small-signal theory, using the hydrodynamical equation of continuity (Eulerian mechanics) as opposed to a particle-dynamics treatment (Lagrangian mechanics) where individual electron trajectories are followed. The fluiddynamics treatment of the electron stream is clearly inadequate when overtaking of electrons, i.e., electrons with different starting times appearing at a point together, causes crossing of electron flight lines. This problem is discussed more fully in Section 1.3. In addition, Brillouin's analysis neglects the effect of loss along the helical transmission line and the effects of the space-charge-induced r-f fields. The large-signal analysis of Nordsieck also neglects the effects of space-charge r-f fields and circuit loss. Furthermore, the equations are valid only for small (i.e., small compared to 0.1) values of the gain parameter C. But in traveling-wave amplifiers operating at high output levels, where the d-c stream currents encountered may be of the order of hundreds of milliamperes or even amperes, the effects of the space-charge-induced r-f fields may not with impunity be neglected. Furthermore the gain parameter in these large-signal devices may be of the order of 0.3 or 0.4 and hence ought not to be neglected. The work reported in this dissertation follows similar lines to those of Nordsieck's analysis, but it takes into consideration the effects of the space-charge-induced r-f fields, loss along the helical transmission line, and large values of the gain parameter. Another approach to the large-signal theory has been presented by Poulterl3 of Stanford University. His method is quite similar to Kompfner's

10 treatment of the small-signal problem in that he considers a unit of charge in the stream which excites both forward-traveling and backward-traveling energy on the helix; then the solution for the field on the helix is written as a convolution integral. The resulting system of equations has been solved for two non-space-charge cases, b = 1.5 and b = 2.2 at C = 0.2 and QC = 0, giving answers very close to those obtained for these cases in this dissertation. One disadvantage of his method is that it is difficult to interpret physically some of the intermediate steps and also the final working equations. In addition, the equations do not lend themselves to solution by numerical methods as readily as those derived here. Very recently additional large-signal calculations for the travelingwave amplifier have been presented by Tien, Walker, and Wolontis14. Nordsieck's equations were used for this work with a space-charge term added. The space-charge field was evaluated by considering an infinite array of unit-space-charge discs separated by a distance z and replacing the helix by a conducting cylinder of the same radius. This method of evaluating the space-charge field gives essentially the same results as those obtained for the space charge in this dissertation. However, the assumptions made in the Tien analysis are the same as those made by Nordsieck. The maximum value of the gain parameter C for which these results apply is approximately 0.02 and hence this theory, like Nordsieck's, does not include high-efficiency conditions. Also, the large values of the space-charge parameter QC investigated by Tien are not consistent with practical tubes, since the maximum applicable C is approximately 0.02. No solutions are included in the Tien work for the relative injection velocity b that gives maximum saturation gain and efficiency as a function of QC.

11 1.3 Extension of the Eulerian Approach The objections to extending the Eulerian approach to the large-signal case should perhaps be examined in more detail. Using the basic assumptions of Pierce's small-signal analysis, the continuity equation, Newton's force law, and the circuit equations are written in differential form for a helical transmission line composed of series inductance and shunt capacitance. For the sake of simplicity, the series loss along the line will be neglected and the equivalent circuit shown in Fig. 1-4, in which the distributed inductance and capacitance are chosen to match the phase velocity and field strength of the field acting on the electrons, will be used in the following development. From this simplified model*, t + p aZ +'v aO ~ (1-1) a + - Vz a - 0, (1-2);Vh Zo XaI + o, (1-3) az Vo az and 6Ih 1 6Vh ep + ZoVo 6t - = 0, (1-4) where = q/m, the ratio of charge to mass for the electron, coulombs/kg; Zo = qLh/Ch, the characteristic impedance of a lossless helix, ohms; vo = 1/VLhCh, the undisturbed axial phase velocity of the r-f wave along the helix, m/sec; *Symbols are defined where they are first introduced in the report. In addition, a list of symbols is given at the end of the text.

12 LINE CHARGEOFp I FI I I IF' I I I ELECTRON [,~ ||~ ~~II~ |~,i|| I STREAM Lh Lh Lh Lh =C =Ch Ch Ch Ch Ch FIG. 1 -4 EQUIVALENT CIRCUIT OFA TRAVELING-WAVE TUBE ASSUMING A LOSSLESS HELIX Lh AND Ch ARE THE EQUIVALENT SERIES INDUCTANCE AND SHUNT CAPACITANCE PER UNIT LENGTH.

13 v(z,t) = the axial velocity of the r-f wave along the helix, m/sec; Ch = the distributed shunt capacitance per unit length, farads/m; Lh = the distributed series inductance per unit length, henries/m; = the beam-to-circuit coupling coefficient; p(z,t) = the a-c space-charge density per unit length, coulombs/m; Vh(zt) = the r-f potential along the helix, volts; Ih(z,t) = the longitudinal r-f helix current, amp; and Ih, Vh, v, and p are dependent variables, functions of both the independent variables z and t. In order to put the above equations in dimensionless form, all the variables are normalized with respect to vo, the undisturbed phase velocity of the helix. Writing the equations in terms of these new normalized variables yields p + + - = o, (1-5) + v - 0, (1-6) a a =z 0, az at and 6ih 6Vh 6p. + -- - - = 0, (1-8) 6 T 6t t where p = p/Po = e Zo/vo, v = V/Vo

14 Vh = Vh/eo = tVh/vo2, and Ih = Ih/io = BTZoIh/Vo2 It will be noticed that Eqs. 1-5 and 1-6 are nonlinear, whereas Eqs. 1-7 and 1-8 (the circuit equations) are linear. To obtain the small-signal equations, Eqs. 1-5 and 1-6 are linearized by writing the velocity v and space-charge density p as the sum of their d-c and a-c components and then neglecting the squares and products of the a-c quantities in combining Eqs. 1-5 and 1-6. The resultant expression is then combined with Eqs. 1-7 and 1-8 to give a single fourth-degree differential equation in the r-f helix potential Vh. The solution of this equation is of the familiar exponential type. The next step would of course be application of the boundary conditions at the input and output ends of the tube. However, since the equations presented in this section are not those ultimately to be solved, discussion of the boundary conditions will be omitted. This approach yields valid results for the small-signal case, but difficulties arise when large signals are considered. Since the equations are based on the hydrodynamical equation of continuity, Eqs. 1-5, 1-6, 1-7, and 1-8 are valid only until the electron flight lines cross. This crossing of electron flight lines is due to overtaking of electrons, i.e., electrons that entered the tube at different times appearing at a point simultaneously. Overtaking of electrons by other electrons results in the velocity v, appearing in the above equations, becoming a multi-valued function of the independent variable z. This overtaking of electrons does unfortunately occur in a large-signal traveling-wave amplifier considerably before the point of saturation

15 is reached. The exact point in any particular tube at which overtaking begins to occur depends on such factors as the gain parameter, the space-charge density, the amplitude of the r-f wave initially impressed on the helix, and the relative injection velocity. Therefore these equations cannot be used for the large-signal case. Additional equations may be written to take this difficulty into account, but the number of equations to be solved soon becomes enormous as the number of electron-flight-line crossings increases. The number of these crossings will be seen to approach infinity as saturation is neared and the electrons, becoming trapped, oscillate back and forth in the regions of minimum potential. Therefore it seems desirable to reformulate the problem in terms of new variables so as to eliminate the difficulty introduced by the crossing of electron flight lines. The new independent variables to be used are entrance phase, i.e., the phase at which the electron enters the helix relative to the r-f wave at the input, and position along the tube. Thus the Eulerian approach is replaced by a Lagrangian treatment, which greatly simplifies the large-signal problem. The resulting equations are a system of second-order nonlinear partial-differential-integral equations. A detailed discussion of these new independent variables is presented in Chapter II.

CHAPTER II. THEORETICAL EQUATIONS FOR THE LARGE-SIGNAL ANALYSIS 2.1 Derivation of the Theoretical Equations It is anticipated that a large-signal study of the traveling-wave amplifier will yield information relating the various design parameters of any tube to the saturation power output and maximum obtainable efficiency. For this investigation the large-signal traveling-wave amplifier equations have therefore been derived, on the basis of Nordsieck's12 work but taking into account the influence of the a-c space-charge fields and circuit loss in the form of series loss along the helix. Also, the equations are valid for large values of the gain parameter C. The MKS rationalized system of units is used throughout the derivation. The general transmission-line equation, including the series loss, is derived in Appendix A as* *V(azt) - v02 V(zt) + 2c a d t) 6t2 o z2 6t = Z 2,t) + 2cJdZovo 6p(z,t) (2-1) where - V(z't) = the longitudinal radio-frequency electricaz field intensity, due to the circuit, at the electron stream, volts/m; z = distance measured along the tube, m; T(z,t) = the linear space-charge density of the electron stream, coulombs/m; *The bars over the dependent variables do not represent normalization as in Chapter I but serve only to differentiate the voltage and space-charge density as functions of z and t from the same quantities as necessarily different functions of y and 0. 16

17 v = = the undisturbed phase velocity of the line, JLC- m/sec; Z0 = L/C = the characteristic impedance of the lossless line, ohms; C = the gain parameter defined by C = |lZIo/2u * Io = the d-c stream current, amp; Uo = 2Vo the d-c stream velocity, m/sec; Vo = the d-c stream voltage, volts; n = q/m, the charge-to-mass ratio for the electron, coulombs/kg; c = angular frequency of the wave impressed on the helix, radians/sec; d = 1/20(2i)(log e)(C) = 0.01836 l/C, the loss factor;8 and i = the series loss expressed in db per undisturbed wavelength along the helix. The actual field along a helix is composed of an infinite number of components orthogonal to one another. The phase velocities of the spaceharmonic fields, however, are lower than that of the fundamental field, and she electron stream will interact appreciably only with the fundamental field, since in a typical traveling-wave amplifier the electron stream is approximately in synchronism with the phase velocity of the fundamental field. Consequently, the space-harmonic fields on a helix serve to carry power along the helix but do not contribute to the gain of the amplifier. Tien1o has calculated the impedance parameter for a tape helix model surrounded by *The impedance used in evaluating the gain parameter in the large-signal theory is that which the electron stream sees, not necessarily Z0. In the linear theory C is developed on the basis that the beam grazes the helix; however, in the large-signal theory when the ratio of helix diameter to stream diameter is not 1 the impedance Zo must be replaced by Z = 2Zo where t is the stream-to-circuit coupling coefficient defined on page 15.

18 a dielectric medium, and found that the results predicted by the theory agree quite well with measurements on several tubes. He reports that the presence of the space-harmonic fields lowers the actual helix impedance. Hence, even though the space-charge density T(z,t) in Eq. 2-1 will be.exceedingly rich in harmonics of the impressed frequency of the r-f wave, the longitudinal r-f voltage along the helix will be approximately all at the fundamental frequency since the helix impedance is very small for harmonics of the fundamental frequency. Thus the assumption that all but the fundamental frequency component of space-charge density may be neglected in determining the helix voltage V(z,t) in the transmission-line equation is not an appreciable limitation to the theory. On the basis of this assumption we may represent the r-f voltage along the helix V(z,t) in terms of two slowly varying functions of z, an amplitude and a phase, as shown in Eq. 2-12. The fields due to the longitudinal flow of energy on the helix and the field due to space charge are linearly superimposed to write the Newton equation of motion for an electron in a traveling-wave amplifier as dt2 a _s(z,t), (2-2) dt2 -z 6z where WVs(zt)/6z = -Es(z,t) = the space-charge field intensity, volts/m. The electron trajectories which are solutions of Eq. 2-2 will have the form z = F(zo,t) (2-3) where zo = the position of an electron at time t = 0. Equation 2-3 may of course be inverted and solved for z0 in terms of z and t as follows*: *It should be noted that the function G(z,t) is not unique since Zo is a multi-valued function of z.

19 Zo = G(z,t). (2-4) The electrons enter the helix at z = 0 but may be considered to originate at z = zo at t = 0, and each particular electron n has its own entrance position Zon as shown in Fig. 2-1. The solutions of Eq. 2-2 represented by Eq. 2-3 may be related to the space-charge density p(z,t) by remembering that conservation of charge requires the charge Podzo entering the tube unmodulated at the input, due to all electrons whose initial positions lie between zo and zo + dzo, to equal the charge pdz at some later position z. This requirement is expressed mathematically as T(z,t)dz = p(z,t) -- dzo = p(zo,0)dzo ~ (2-5) 6azo The space-charge density in the stream entering the tube may be related tothe stream current and voltage by Io'(zo,0) = - (2-6) Hence the desired relationship may be obtained using Eqs. 2-4, 2-5, and 2-6: P(z,t) = 1 |zo. (2-7) Considerably before saturation is reached, the electrons begin to overtake one another; i.e., electrons starting at different initial times appear simultaneously at the same point. When this occurs, zo is no longer a single-valued function of z and Eq. 2-7 must be modified by replacing the quantity (8zo/8z)tt by the sum of its values for all branches of the multi-valued

20 -9~ z or (Y 0ol// 2TT 00 FIG. 2-I' DISTANCE-TIME DIAGRAM FOR ELECTRONS ENTERING THE HELIX REGION,INDICATING STARTING TIME AND ENTRANCE TIME FOR EACH ELECTRON. ALSO THE MULTIVALUEDNESS OF Zo OR yo AS A FUNCTION OF z OR y ISWW N EANOUz DTE/ 03 IS SHOWN WEN VERTAKING OCCURS.2 O DISTANCE y MAz BE MEASURED IN UNITS OF z OR y.

21 function zo. The calculations are therefore made in terms of zo as the independent variable, since z is a single-valued function of zo. In order to simplify the calculations and put the equations in a form applicable to all traveling-wave amplifiers, whatever the frequency range, new variables are introduced. The new variables defined are usually normalized with respect to some constant velocity occurring in the problem; in the case of the traveling-wave amplifier either the stream velocity uo or the undisturbed phase velocity of the helix vo would be suitable. The definition of new variables amounts to changing to a coordinate system riding with the undisturbed phase velocity vo or with the stream velocity uo, the advantage being that the dependent variables vary more slowly with time and distance in these coordinate systems. The choice of which velocity to use is somewhat arbitrary, but it is believed that the numerical calculations will be simpler if the variables are normalized with respect to the initial stream velocity. The alternate set of equations is presented in Section 2.2. The normalized independent variables are therefore defined as follows:* A Cz 2iC y =- u z (2-8) Uo Xs and ~ 4 9 = cto, (2-9) ~o Uo where Xs = uo/f, the stream wavelength in meters. Physically, y is proportional to the position along the tube in stream wavelengths. 0o, the entrance phase, relative to the r-f wave at the input, of an electron in *The symbol A means that the left-hand side of the equation is by definition equal to the right-hand side.

22 radians, may be thought of as a "tag" for a particular electron entering the helix region*. In fact, o0 is a time standard in radians of the frequency of the input voltage. Thus in this Lagrangian formulation of the problem, particular electrons are followed through the interaction region rather than considering the electron stream as a "fluid". Introducing the new variable y into Eqs. 2-1 and 2-2 gives a2V(yt) - C2(vo2 2V(y,t) + 2dC d aV(y,t) at2 u0o/ ay2 at = oZ 2p(,t) + 2CdvoZo (yt (2-10) 0 0 6t2 6t 2 -(9 _ CD Es(yt)] (2-11) dt- - uo In the large-signal equations the space-charge density p(y,t),, which is a fluid constant, is interpreted as the electron charge qe times the number density of electrons Ne. In accordance with the discussion following Eq. 2-1 the following dependent variables are defined: V(z,t) = V(y,0) - Re 0o A(y)e-j. (2-12) Equation 2-12 serves to define A(y), the normalized voltage amplitude of the r-f wave along the helix, and 0(y,0o), the phase of the fundamentalfrequency r-f wave in the circuit relative to the phase of the wave at the input to the helix. A particular value of y not only defines a phase plane *Also, as may be seen in Fig. 2-1, 0o is in a sense a transit angle for a particular electron traveling from the point Zon to the helix boundary at z = 0.

23 of the r-f wave but also serves as the axial coordinate of one or more electrons; similarly M(y,0) can be used in a second sense as the phase of the displacement component of an electron at y that entered the helix at the moment 00 relative to the r-f wave. It is easily seen that the new time variable M(y,)o) in Eq. 2-13 is based on a coordinate system which moves with the initial stream velocity uo. The following additional reduced dependent variables are defined: 9(z) _ fu - t) - (zt) (2-13) and ut(Yo0) = d - uo [1 + 2Cu(y,0)]. (2-14) dt Equation 2-13 defines 9(z), the phase lag across the amplifier, at any point y, in radians. In particular, @(z) is the r-f phase angle (or lag) in radians of the wave on the helix relative to a hypothetical traveling wave whose phase velocity is the initial stream velocity uo. Equation 2-14, an expression for the total velocity of an electron, defines 2Cuou(y,o0), the velocity deviation of an electron relative to the initial d-c stream velocity. Since u is a function of o, the entrance phase of an electron, the velocity of an electron at any point y can be calculated. u(y,0o) is a dimensionless large-signal velocity parameter. Solving Eq. 2-13 for z and converting to the new independent variable y gives y = C [0(y,lo) + 9(y) + wt] (2-15) Equation 2-14 written in terms of y is -+ Co) [1 + 2Cu(y,0o)] * (2-16)

24 Taking the time derivative of Eq.?-15 and equating the result to Eq. 2-16 gives y dt d(y) dt + = [ + 2coCu(yo)] (2-17) Partial differentiation with respect to y and t is represented by and Partial differentiation with respect to y and t is represented by 6/6y and /6t, and partial differentiation with respect to 0o and y is represented by 5/50o and b/5y in the equations of this section. Obviously, 6/6y = 5/by and 6/!t = (l/w)(5/65o). Equation 2-17 may be simplified and written as 50(y,o) + d (y) 2u-(y o) 5y + dy 1 + 2Cuy,o) (2 ) Equation 2-18 is the first of the working equations, relating the dependent variables 0(y,0o), e(y), and u(y,0o). In terms of the new dependent variables, Eq. 2-10 may be rewritten with the space-charge density p(y,t) replaced by its fundamental-frequency component pi, in keeping with the previous discussion: (DV0 d2A(y) dE) ( u 2 \Uo / dy2 dy v0C) -(y)) ^ - ^(y - Uoo d 2( dA(y) d (y) 2 sin (y, + (dy dy () 2A(y sin(y.,~o v= oZ 2 + 2aCd voZo t. (2-19) Upon substitution from Eq. -12 for V/y the force equation 2-2 becomes Upon substitution from Eq. 2-12 for ~V/Sy, the force equation, 2-2, becomes

25 dv d^ 2 ZOIOW dA(y) dt:- = IT1 ud-o dy cos 0(y,0o) dt 0 A(y) sin 0(y,0o) (1 - dGy) - Es(Y0) (2-20) From Eq. 2-16, dv d~ 5u dy 2 b yO o) dt y dt =2Co by t 2C u(y,Co)] u(y. (2-21) The next step is to equate Eqs. 2-20 and 2-21 and simplify: [1 2Cu(y,0o)] = A(y) 1- C dyy] sin O(y,0o) byI1 dy dA(y_) u._o)uoC -, dy cos (Yo0) + Es(y, ). (2-22) In terms of the newly defined variables Eq. 2-7, expressing the conservation of charge, may be written as p(y,~) = 10 o 1 2 uO 0| 1 + 2Cu(y,po)- (2-2) Next, p(y,,) is expanded in a Fourier series in the time variable 0(y,0o). This operation introduces no additional assumptions in the analysis. Co 2it 2 P(y,0) = sin n /p sin n0 d0o + cos n PP cos no d0o. (2-24) n=l O n=l 0 Substituting Eq. 2-24 into Eq. 2-23 gives

26 00 2it 2i ~ p(y~,Zi) = Re |- Io Xfjn~ cos ns(y,~o)d j sin no(yj)d (2-25) = ( Re e- ~+ j 0 (2-25) l uO + tO 1 + 2Cu(y,0) O] 0 "!1 + 2Cu(y,0) 0 (In Eq. 2-25 the prime denotes the variable of integration. ) In terms of the new variables the first and second time derivatives of the fundamental component of space-charge density, pi of Eq. 2-19, may be expressed as follows: tP1 = - t, (2-26 2t2 62 Pi (6 ) 6 / 2 a P2 6t2 6- 6t2 6 t 2' (2-27 and P, = Plc cos + p1s sin (2-28 If Eqs. 2-26, 2-27, and 2-2& are substituted into the right-hand side of Eq. 2-19 with the aid of Eq. 2-25, the resulting expression may be written as two equations simply by equating the coefficients of sin 0 and of cos 0 on each side of the equation. This is a valid procedure because the coefficients are independent of 0 and the sine and cosine functions are orthogonal. In the two equations thus obtained, the constants are written in terms of the parameters previously defined and the additional parameter b u0- v, (2-29) where b = a relative injection velocity parameter. Thus where b = a relative injection velocity parameter. Thus

27 d2A(y) A() (y)2 (l+Cb)2 A(y)L(y -/(Cb) dy2 dy 2 2i 2: - l+Cb j cos, + 2Cd sin (y ) d (2-0) iC 1 + 2Cu(y,) 1+ c2Cuy,)2-30) and Ldy -+_ 2ir 21 1+Cb sin 0(y,) (y ) d C J 1 1 + 2Cu(y,0) J1 + 2Cu(y,o) 2 LP o y Finally, the expression for the space-charge field Es (derived in Appendix B), 2 2 2a F(O-') do' / OF(-~'9) d-2 0 (2-32) 71 7(i.Cb) \1. 1 + 2Cu(y,~'), (2 ) is substituted into the force equation, Eq. 2-22, to obtain the last of the working equations: u(>y~o0) [1 + 2Cu(yo)] = A(y) 1- C d sin 0(yo) by 2it 2n - C dA(y) coS o(Y, o) - l+Cb ( /() - ). (2-3) dy Cos O(Y,00) ]+cWC f + 2Cu(y, 23) 0 To summarize, the four equations 2-18, 2-30, 2-31, and 2-33 are the final working equations which will be solved for the dependent variables A(y), 9(y), u(y,0o), and 0(y,0o) subject to the boundary conditions discussed

28 in Section 2.4. This system of second-order nonlinear partial-differential-integral equations is valid for a wide range of the parameters C, d, b, CLq/M, and AO (the normalized amplitude of the r-f signal impressed on the helix at the input) and is limited only by the assumptions mentioned in their derivation: 1. The electrons will interact appreciably only with the fundamental field and hence all but the fundamental component of space-charge density may be neglected in determining the helix voltage. 2. There is no initial thermal velocity distribution in the electron stream. 3. The electric field is constant across the electron stream. 4. The radio-frequency wave impressed on the helix bunches the stream in such a manner that the space-charge density is constant in amplitude (i.e., has no radial variation) and varies sinusoidally with axial distance. (In calculating the space-charge forces it is assumed that the growth constant of the growing space-charge wave is small compared to unity.) 5. The relationships may be described using nonrelativistic mechanics; i.e., the squares of the ratios of the stream velocity uo and the wave velocity v to the velocity of light are small compared to unity, and hence the motion of the electrons is sufficiently slow that the formulas of electrostatics are valid. 6. The electron stream is in a strong axial d-c magnetic field (rectilinear flow), so that the electrons are constrained to follow linear paths. Consequently there is considered to be no transverse motion and the stream boundary is smooth. 7. A sufficient quantity of positive ions is present to neutralize the average space charge. Assumptions 3 and 4 are considered to be the chief limitations to the theory as presented here. For the small-signal theory one parameter, namely QC, is sufficient to specify the space charge; but in the large-signal theory it is necessary, for any given ratio of helix diameter to stream diameter, to specify two

29 parameters, a space range of effectiveness and an amplitude. The range parameter used is Pb', shown in Fig. B-6. The space-charge weighting function F(0-0') drops off more rapidly for smaller values of Pb' than for the larger values, and thus fb' determines the range over which the space-charge forces are effective. The coefficient (Wp/WC)2 of the space-charge field term in the force equation, Eq. 2-33, would seem to be a better choice for the amplitude parameter of the large-signal case than the QC appropriate to the linear theory. The relationship between (%p/aC)2 and QC is discussed in Appendix B. In the small-signal theory the current is essentially all at the fundamental frequency; hence QC need be evaluated only for the fundamental component of current. Correspondingly in the large-signal theory the reduction factor Rn of Fig. B-3 must necessarily, then, be evaluated only for the fundamental component of space charge (i.e., n = 1), even though the harmonic content of the stream current is high. The recommended space-charge parameters for the large-signal analysis are therefore Bb' and (>p/0C)2, which might be more briefly represented as K = (%)2 and B = Pb' For the purpose of calculation a mathematical model will be used in which the helix-stream interaction region of the traveling-wave amplifier is considered to be infinite in length and the helix is thus terminated in its characteristic impedance Zo. Hence the effect of reflections due to a mismatch at the output end is not considered, and the collector and helix d-c potentials are taken as equal. The small-signal conditions, corresponding

30 to the three waves, are applied at the input: a voltage is applied to the helix at y = 0, and the equations are integrated stepwise until a maximum occurs in the amplitude of the normalized r-f voltage A(y). The amplitude of the r-f signal initially impressed on the helix, Ao, is usually taken as 0.0225, which is approximately 30 db below CIoVo. However, solutions are also obtained for other values of Ao to provide data for a plot of power output vs. power input. In applying the results of these computations, therefore, to any existing finite-length traveling-wave amplifier, it must be assumed that there are no reflections at the input or output (which may be due to a mismatch in the coupling networks or to a discontinuity in the acceleration if the helix and the collector are operated at different d-c potentials), and that the interaction region is sufficiently long for small-signal conditions to prevail at the input. 2.2 Relation to Nordsieck's Equations The general working equations derived in the preceding section may be reduced to Nordsieck's12 "large-signal" equations, (yo~) + dG(y) - 2u(y, 6 )y y = 2u(y,00), (2-34) by dy u(y,) = A(y) sin 0(y,0), (2-35) 2ir dy 2y = - 2 f sin (y,o) d o, (2-36) 0 and 2t dO(y) 1 dy + b = 2A(y) cos 0(yo) d0 (2-57) 0

31 by the application of the appropriate assumptions. The necessary assumptions are (1) that the gain parameter C, a measure of stream-to-circuit coupling, is small compared to 1, i.e., C< 1; (2) that the a-c velocity term 2Cuou(y,0o) is a small fraction of the average velocity Uo; and (3) that terms involving second space derivatives of A(y) and 9(y) and products of the first space derivatives may be neglected. Nordsieck's equations are not valid for a C greater than about 0.02 or 0.035, which is much smaller than the C values of typical large-signal traveling-wave amplifiers. In addition, the assumptions listed above are valid only when the a-c quantities in the stream are small. Since efficiency is proportional to C, Nordsieck's theory does not cover high-efficiency operation and hence describes a transition region between that to which Pierce's8 small-signal equations apply and that to which the large-signal equations derived in Section 2.1 may be applied. 2.3 Derivation of Alternate Theoretical Equations As pointed out on page 21, there is an alternate possibility of defining new dependent variables relative to the undisturbed velocity of the wave, vo, so that the coordinate system moves with the velocity vo rather than traveling with the initial stream velocity uo. It is felt that the equations which result are sufficiently different and interesting to warrant presenting them here, although the mathematical manipulations involved are quite similar to those in Section 2.1. The general transmission-line equation, the force equation, and the conservation of charge relationship derived in Section 2.1 are repeated here for ease of reference:

32 62V(z t) _2V(z,t) 2 (z, t) - Vo2 + 2aC d t2 Vo- z2 Cd t = avZ (zt) 2C dvZ (zt) (2-1) p( ozt) = 6 5^ (2-7) dt2 6z )t Io 6zo The new normalized independent variables to be used are defined as y t = z (2-38) uo zg and (o Zo0 = oto (2-39) where kg = vo/f, the undisturbed helix wavelength in meters. Thus y is now proportional to the position along the tube in undisturbed helix wavelengths. Introducing these new variables into Eqs. 2-1 and 2-2 yields 2V(yt - C2c2 a V(yt) + 2uCd V(yt) at2 ay2 at = VoZO t2 + 2oC dvOZ, t (2-40) at' + 0 at2dv~Z~. and d- L( ) (y,t) (2-41) dt / y vo

33 Next the dependent variables O(z,t) A ZD (vZ - t) (2-42) VO and vt(y,'o) = dz a vo [1 + C v(Y,)] (2-4 dt are defined. Equation 2-42 transfers the problem to a coordinate system moving with the undisturbed velocity of the wave vo, while Eq. 2-45 expresses the velocity deviation of an electron Cv(y,0o) relative to the undisturbed phase velocity of the wave. In addition, a normalized r-f voltage is defined in terms of its vector components al(y) and a2(y) as follows: (zvt) = V(y,0o) -- [al(y) cos O(y,0o) - a2(y) sin O(yo). (2-44) Solving Eq. 2-42 for z and converting to the new independent variable y gives y = c [O(yo) + t] * (2-45) Equation 2-43, written in terms of y, is dy CW [1 + Cv(y,00)] (2-46) dt Taking the time derivative of Eq. 2-45, equating the result to Eq. 2-46, and simplifying yields __(y_,o) v(y, o) 6y 1 + Cv(y,o) (2-47) Equation 2-47, relating the dependent variables ~(y,0o) and v(y,0o), is the first of the alternate working equations.

34 In terms of the new dependent variables, Eq. 2-44 may be written with the space-charge density p(y,t) replaced by its fundamental-frequency component pl, as discussed in Section 2.1: d2aZ(y) da2(y) ZoIo32 0-C dy + 2 dy + 2da2(y) cos O(y,0) d2a2(y) dal(y) + C dy2 + 2 + 2 dal(y) sin o(Y,, dy2 d_ - 0oZ~0 -i + 2CdvoZo t ~ (2-48) 0 t2 t When aV/8y is written in terms of the new variables, the force equation, 2-2, becomes dv d2z - { daZ(y) _io a2(y) cos -(Ya2O) dt dt 4v L dy C / da((y)+ aly) sin d(y,0o) - Es(Y,0).(2-49) dy C / Also, from Eq. 2-46, dv ov dy v (y, o) dy C v[l (y ( dt 6y dt CVo0 y dt C Vo[l + Cv(Y0)] v(y (2-50) Equating Eqs. 2-49 and 2-50 and simplifying yields the second working equation: 8v(y glo) [1 + Cv~y,~o)~2 ~i-~,'ic~~~~,,Cdai(y,) cos'"(,",) "y [1 + Cv(y,) = C(+Cb)2(a2(y) - C ) os + (a(y) + Cda2(y) sin (Y,0) i4oCEs(Y, (2-51) dy / 03o o j

35 In terms of the newly defined variables Eq. 2-7, expressing the conservation of charge, may be written as p(Y,7) = 1 _ __ v a 1 + Cv (y, o) (2-52) Next p(y,O) is expanded in a Fourier series in the time variable 0: co 2it oo 2it p(y,~) = s p s in no d0o + cos nO n p cos nO d. (2-53) n=l 0 n=l 0 Substituting Eq. 2-53 into 2-52 gives ao 2it 2it 10 -~n~ cosn~(y,~)d~A P sin nO(y j')d) P(y, ) Re C e-j + oJ o (2-54) Vo 1 + cv(yo 1+ CV()' n=l 0 The next step is to substitute Eqs. 2-26, 2-27, 2-28, and 2-54 into Eq. 2-48. The resulting expression may be written as two equations simply by equating the coefficients of sin 0 and of cos ( on each side of the equation, since the coefficients are independent of < and the sine and cosine functions are orthogonal: 2it C d2al(y) da2(y), 2 cos nO(y 06) d6 2 dy dy da2(y) -= l+Cviy, ) 2 dy2 dy 2W/ ^ J 1 + Cv(yA) 2n7 + 2Cd/ sin n (y,$0)) d (2-55) 1 + Cv(y,o,

56 C da2 (y) dal (y) 2 - in n(y 2 y2 dy +dal(y) - inn (y,)d KI 1 + cCvy,0 Lo 2in - 2Cd cos nO(y, ) do' -22Cd 1 0 (2-56) Finally, the expression for the space-charge field Es, derived in Appendix B, ES(YO) = -Wp F(($-0,) d06 (2-31 os' ~ (I + Cb) Y1 J + Cv(y,g') 0 is substituted into the force equation, Eq. 2-51, to obtain the last working equation: bv (y,~o0v) [1 + Cv o)2 C( + b) a2( - C dal( cos (D(y,o) "y dy + &1() + C da(Y) sin O(Yo) 2 2it lCb)1 + Cv(y, 57 0 Equations 2-47, 2-55, 2-56, and 2-57 are the working equations in terms of the alternate coordinate system. The range of validity of these equations and the underlying assumptions are the same as those for the equations presented in Section 2.1.

37 The dependent variables al(y) and a2(y) of Section 2.2 are related to the dependent variables A(y) and 9(y) by the equations A(y) = a(y) + a(y) (2-58) and tan [-@(y) - by] =. (2-59) a1(y) Equations 2-47, 2-55, 2-56, and 2-57 reduce to Nordsieck's2 so-called "large C" equations if a-c space charge and series loss are neglected except for terms involving the second space derivative of the variables al(y) and a2(y), which clearly should not be neglected especially in the regions of rapidly changing slope, e.g., near the maximum value of A(y). The large-signal equations of Section 2.1 have been selected for numerical analysis because it is believed that the equations of Section 2.2 would require more time to evaluate, since in the latter case a square root and an arctangent would have to be evaluated to find the r-f voltage amplitude and the phase lag across the tube at each step in the integration procedure. This is a consequence of the manner in which A(y) was defined (i.e., in terms of its vector components) and not the fact that a coordinate system moving with the velocity vo was selected. 2.4 Boundary Conditions In Section 2.1 it was assumed that the r-f wave on the helix is of a single frequency. Hence the r-f voltage on the helix was defined by two slowly varying functions of distance, an amplitude and a phase, which mathematically assures the termination of the helix in its characteristic impedance. It was also assumed that the collector and helix are at the same d-c potential (i.e., there is no discontinuity in acceleration), and

38 in the Fourier analysis of the space-charge density only forward-traveling waves, not reflections, are considered. The three boundary conditions at the input, the amplitude and phase of the impressed voltage and the continuity of the electron velocity, are satisfied by linear superposition of the three waves of the small-signal solution. The initial values of the dependent variables A(y), @(y), 0(y,0o), and u(y,0o) can therefore be obtained from the small-signal conditions. The normalized amplitude of the r-f signal A(y) is by definition Ao, which is usually assigned a value approximately 30 db below CIoVo, or 0.0225. The phase lag 9(y) is of course initially zero. Since the electron stream enters the helix unmodulated, the velocity deviation u(y,0o) is initially zero for all electrons. The instantaneous phase 0(yyo) at y = 0 is equal to the initial phase 00, which is chosen to be 2ij/m where j = 0,1,...,m and m is the number of equally spaced electrons followed through the integration procedure. Since the storage space required in the, computer and the computation time increase several fold with each twofold increase in the number of electrons, it is desirable to use as few electrons as possible. The word electron is used here to mean not an individual particle but rather a finite amount of charge distributed over a finite cross-sectional area that enters the helix region at a phase 00 relative to the r-f wave at y = O. This concept is discussed further in interpreting the infinite peaks in the current-density curves of Section 4.6. It was found that 16 electrons are too few for this work, but 32 electrons are sufficient to insure the smoothness of the functions and this value has therefore been used in the computations reported in Chapter III. A more detailed discussion of the influence of the number of electrons on the solutions is presented in Section 3.5.1.

39 Mathematically the initial values of the dependent variables may be stated as follows: A(0) = Ao = 0.0225 (for most runs) e(o) = 0 Oj(000o) = 0oj =, where j = 0,1,2,...,32 and uj(O,o) = 0 for all electrons. At the output of the helix the boundary conditions are satisfied implicitly if the ratio of voltage to current on the helix is equal to the characteristic impedance Zo and the velocity is continuous. The derivatives of the functions A(y) and @(y) with respect to y at y = 0 are determined by the small-signal conditions at the input. The desired derivatives are obtained from the small-signal solutions in the following manner. Since the small-signal solutions are written as summations of three linearly independent, exponentially varying waves, the voltage on the helix near the input may be written as the sum of three components, for the three waves, each varying with distance and time in a manner described by the expression ej0t-rz = -ejt e-je(l - PiC)z e.e~iCz (2-60) where cQi and Pi are the growth and phase constants respectively of the individual wave. In terms of the new distance variable y, the right-hand side of Eq. 2-60 is written as e-jy/C + j (hi +-:i) Y + jot. (2-61)

4o If jG(y) is added and subtracted from the exponent, this expression may be written in terms of the new phase variable 0(y,00) as e(ai +~'i) y - j - j 9(y). (2-62) Hence, the voltage on the helix may be written as 3 Re A(y) eJ = AieBiY - [+(Y)] (2-63) i=l Cancelling e-ji and multiplying through by eje(y) in Eq. 2-63 yields 3 A(y) e (y) = Ai ei, (2-64) i=l where 5i = ai + jji Taking the derivative with respect to y in Eq. 2-64 and evaluating at y = 0 gives A(y) yo + j Ao 9() y=0 = Ai6i * (2-65) i=l Equation 2-65 may be separated into its real and imaginary parts to obtain explicit expressions for the derivatives of A(y) and 9(y) at the input: ~~eand(y e'(y) uyo = Im (2-67) i=l

41 Pierce's8 small-signal analysis may be used to write the total voltage (including the space-charge component) in terms of both the a-c velocity and the convection current as given by the equations A = -uo (J r) (2-68) and 2 A = 2Vo (Je - i ) (2-69) JIo~e-i i The quantity - F may be defined as - F = -je + PeC (2-70) which when substituted into Eqs. 2-68 and 2-69 gives the following expressions for the voltage on the line in terms of the a-c velocity, the convection current, and the propagation constant 5 for each wave: -2Vo C 282 A = i (2-71) Io and Uo C202 + jCF A = (- 1 + C282) v * (2-72) Since b is usually of the order of unity and C is small compared to 1, the terms involving C282 may be neglected, even for C as large as 0.1, and Eq. 2-72 may be simplified to juoC8 A = - v.(2-73) Hence at z = 0 the following relations apply:

42 AO = )Ai (2-74) i=l Ai_ -2VoC2 X f = IC i = O (2-75) Io i=l and 3 Ei~- = i~ v = O. (2-76) i=l Since the electron stream enters the interaction region unmodulated, the a-c velocity and convection current may be taken as zero at the input; i.e., i = v = 0 at z = O. Hence the normalized voltage amplitudes for each wave needed to evaluate A'(y) y=0 and i'(y) y in Eqs. 2-66 and 2-67 may be calculated from the various 8i's for the particular solution to be studied. The ci's and Pi's used here as growth and phase constants correspond to the x's and y's of Pierce's8 small-signal analysis. The values of the ai's and Pi's for various values of QC, C, and b when the loss factor d is- zero were obtained as indicated in Appendix C. For values of d other than zero, an extensive tabulation of the x's and y's for a wide range of QC, C, and b has been given by Brewer and Birdsall. Since, as noted previously, the stream is unmodulated when it enters the interaction region, the stream initially (at y = 0) has no convection current and therefore cannot act on the circuit. Hence for zero loss on the helix the voltage initially does not change in amplitude. If the loss parameter d is not zero, the initial change in voltage amplitude is determined by d. The value of i'(y) at y = 0 constitutes an apparent phase constant.a of the composite wave at the input.

CHAPTER III. SOLUTION OF THE THEORETICAL EQUATIONS ON MIDAC 3.1 Introduction Since the working equations derived in Section 2.1 form a nonlinear system of partial-differential-integral equations in two independent variables, they are not readily amenable to an analytical solution. The alternative, a numerical solution, is practical only with the aid of a highspeed electronic computer. The equations could be solved on an analogue computer, but a large amount of complex equipment would be needed. Since an analogue computer has only one independent variable (time), it would be necessary to scan each of the two independent variables in a specific manner on the time scale, which would require considerable storage space and many complex switching circuits.15 Since a large-scale high-speed digital computer was considered preferable and the Michigan Digital Automatic Computer (MIDAC), located at the University of Michigan's Willow Run Research Center, was available, the equations were solved on this machine. 3.2 Brief Description of MIDAC 3.2.1 General Characteristics. The Michigan Digital Automatic Computer is a high-speed electronic computer of considerable versatility. It uses the binary number system, in which the two digits "0" and "1" correspond to the absence and presence, respectively, of an electronic pulse in the machine. An input translation program can be used to convert decimal numbers automatically to the binary number system. 43

44 MIDAC operates sequentially on the digits of a "word", which consists of a sign and 44 binary digits. Such a word may convey operational instructions or it may represent a number of 44 binary digits, equivalent to approximately 13 decimal digits. Storage elements provide space for 512 words of both types in a high-speed acoustic memory and 6144 additional words in a slower-speed magnetic-drum memory. Instructions may be stored in" any desired portion of the memory; they are of the "three-address" type, using two operand locations and a result location. Since numbers are stored in the same type of memory elements and are handled in the same way as the instruction words, instructions may be modified on the basis of results as computation progresses. This ability to make "decisions" in the course of a program is an important property of MIDAC. Since the computer can operate only on numbers with an absolute value less than 1, programs must be designed either with a fixed binary point and a scaling factor selected to keep the results below 1, or with an automatic scaling factor or "floating" binary point. Figure 3-1 is an overall view of the MIDAC components, which are discussed in detail in the following paragraphs. 3.2.2 Input Photoelectric Tape Reader. The problem to be solved is first programmed and the resulting instructions and numbers, in the hexadecimal system, are transferred to a paper tape with six levels of punched holes. The information is presented to the machine by a high-speed Photoelectric Paper Tape Reader, shown in Fig. 3-2, at a speed of 200 characters/second. Light shining through the holes strikes photoelectric cells which translate it into electronic pulses. These pulses are stored in the memory in accordance with the prepared program.

45 _J IP 0I I IJ: o ~8~8~ al w

46 w 0 w::: i::: I::1:: ~ ~~~~~~:::::: i - <~a: i1::::::: - 0 w -J C S w 0 0 a=_ I o X:a:~~i:Qf As C, _ ~~~N -:': 0: X 0 D;::r -::: ~ ~ ~ ~ ~ ~

47 3.2.3 Control and Arithmetic Units. The control unit of Fig. 3-3 superintends the entire operation, calling in instructions from the memory, analyzing them, and sending each individual portion to the arithmetic unit, also shown in Fig. 3-3, in proper sequence. The arithmetic unit performs the actual addition, subtraction, multiplication, and division and sends the results back to storage. Computation times, including access to the acoustic memory, are listed below for the various operations. Operation Time, microseconds Minimum Maximum Addition 192 1536 Subtraction 192 1536 Multiplication 2304 3168 Division 2304 3168 Comparison 192 1200 Number conversion 768 1776 Larger operations, such as making "decisions" on the basis of the results, are handled by the control unit. 3.2.4 Storage Units. The primary storage unit is the high-speed acoustic memory, shown in Fig. 3-4, which has a capacity of 512 words stored in the form of sound pulses in a column of mercury. These pulses must be recirculated constantly if the memory is to retain its information; hence, the acoustic memory is not a permanent memory unit. Its access time averages 192 microseconds, with a maximum of 384 microseconds. Additional storage for some 6144 words is available on a magnetic-drum memory, shown in Fig. 3-5, which holds the information as magnetization of one polarity or another on the surface of a cylindrical drum coated with iron oxide. The magnetized "spots" on the drum retain their information

48..........~~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~...:.: R I~as','f 0~ a~~~~~~~~~~~~~~~~~~~~ ~:'.,: ~:.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....... ~~~~~~~~E 0 iLL 00~~~~~~ I — z 0 0 CD _~~~~~ Z _ Q~~~~~~~: Q I r O~~~~b - - A_ C)~~~~~~~~~~~~~~~

49,:,;, ~;:-" - ti 4;CI -"'i26 s a'I~qJ; ai " 1 r I r Psaassa8aisaarpa3iurar.il- aegjBplBkl ......................_.._........ __........... -i - r a *-;:c: ?IL ZI C::: i ~::w a ii:aai~:~-i;i:3: p:::::i;::;;--:i i:::: iB'j i" 1 i-li P:21:a x:::: 1 -::: :::::::(- a3i:~I)T";ilA ;_::1::::: ,:: i: " T i:8:-:::::ili K: B~ , iX:j$~*i ;::::: n:::::'':: i ~i i:i: ~ x c il' i ~" a~ ~-i _:1:::: w I ~: i " i;P: I::::: L $ :d "; - iiil: i: d: r:-::i: i a~:: -i.e:1:::*~ 1 airu:i '::::~ ' i :. ii:~;; *~ si -:* CI i1 i ~;~ iR~ "' -~ t~;i ::::i 7 91 1 FIG. 3-4 MIDAC HIGH-SPEED ACOUSTIC MEMORY

50 FIG. 35-5 MIDAC LOW-SPEED MAGNETIC DRUM MEMORY...... l......................................................... - - l l -! - - w~~~~~~~~~~~~~~~~~~~~~~o

51 indefinitely, even when the power is turned off, unless disturbed by new magnetization; hence the drum is a permanent memory unit. This memory has an average access time of 9 milliseconds and a maximum of 17 milliseconds. Information can be shifted between the two memories by the control unit. 3.2.5 Output Flexowriter. The computational results are printed on paper at a rate of 10 characters/second by an automatic typewriter actuated by sequences of electronic pulses from the acoustic memory. 3.3 Transformation of the Differential Equations into Difference Form To solve the working equations for a given set of parameters and initial conditions, electron trajectories are followed from the input boundary, and Integration is carried out along the tube in y, using 32 points in 0o (corresponding to 32 representative electrons) as indicated in Fig. 3-6. The effect of the number of electrons is discussed in detail in Section 3.5.1. To facilitate this process the working equations are transformed into difference equations. For this purpose first-order difference formulas are obtained using the first two terms of a Taylor series expansion of a function about a point. The general Taylor series expansion of the function f(x) about the point a may be written as f(x) = f(a) + f'(a) (x-a) + f (a) (x-a) +. 2.' f(n-1) a) n-l + (-l (x-a) +. (3-1) Using the first two terms of this series expansion, expressions for the dependent variables A(y), @(y), 0(y0o), and u(yo) at the (i+l)st row may

52 w W I 5 0 Z oLJ 2 cn, PAil Bi + j U+,j _, +i.,..........'>J..........__ ___ _,, _,_., _. ___ __*_ __L _. 0 32 +i~~~. ~2TT oj IDENT I FICATION COORDINATE OF AN ELECTRON FIG. 3-6 INTEGRATION DIAGRAM USED IN THE NUMERICAL SOLUTION OF THE WORKING EQUATIONS

53 be written in terms of their values at the ith row and their first derivatives with respect to y at the ith row, as follows: Ai+l = Ai + y A, (3-2) Gi+j = i + Ay, (3-3) Oi+l,j = Xij + AYy 0j (3-4) and U-i+,j = ij + Ay j (5-5) where Ay = the increment of integration in y. In terms of the dependent variables' second derivatives with respect to y at the ith row, the equations for the (i+l)st row are 2 t Ai+1 2 Ai - Ai + (Ay) Ai (3-6) and i+i = 2 Gi - i-1 + (Ay)2 i. (3-7) The corresponding expressions for 0(y,00) and u(yio) are not written because their second derivatives do not occur in the working equations. Higher-order difference formulas such as the classic Runge-Kutta fourthorder process, which uses more terms of the Taylor series, would give more accurate results but would also increase the storage space and computation time required. It was found that considerable accuracy can be obtained using first-order difference equations if the integration increment Ay is selected carefully. A detailed discussion of this point is included in Section 3.5 on sources of error in the numerical integration.

54 Using Eqs. 3-2, 3-3, 3-4, and 3-5, the working equations, Eqs. 2-18, 2-30, 2-31, and 2-33, are written as d(Yi+loj) - O(Yi,oj) + G(Yi+l- (Yi) - 2 (Yi yoj) (3-8) 1 + 2Cu(yj,1 0 A(Yil) + A(Yi+l) - 2A(yi) - A(Yi) - (yi+l)-e(yi)) 2 (l+Cb) 2it 2 c (Ay)1+Cb) 1cos +i, )d 2Cd sin 0(Yi.i)d0do C tC 1 + 2Cu(yi,,3 0 1 + 2Cu(yioj) - 0 0 0 A-1 C Ayi) 9 (yi.i) + ~(yi^) - 2~(yi) 2d(Ay) (1+Cb)2 + 2[A(y+1) - A(i)] [G(yi+l) - (yi) - A] - 2< 2:a (Ay) (l+Cb) sin F(yilj)d0ij 2Cd cos 0(yi,2.)do, (3-10 C sJ 1 + 2Cu(yiXoj) 1 + 2Cu(yi, - oj l 10 o 0 o and [u(Yi+ilooj) - u(Yi,0oj)] [1 + 2Cu(yi,0oj)] A(yi) (Ay - C[9(y+l) - e(yi)] sin 0(Yi,oj) 2 2it - C[A(yi+)-A(yi)] cos 0(Yioj) - b ( ) p FC* (-)o l1+Cb n\Cx 1 + 2Cu(yi,oj (3-11) The next step is to solve Eq. 5-9 for A(Yi+l). Let

55 t2it~~~ 2Tc (Ay)2(+Cb) +Ccos d(Yi,si)d oj 3 12 ( iC ) 1 +2Cu(yiC, ) + 2Cd 1 + 2Cu(yi,0) J (3-12) Solving Eq. 3-12 for A(yi+l) and simplifying gives A(yi+1) = (A Y - A(yi.) + A(Yi) (i+l) - 2 {(yi) + y) 9(Yi+) 92(yi) + y (Yi) - (Ay)2b(2+Cb) 1 (-1) C + The quantity f1i = - A(yi_) + A(Yi) 2(yi) + 2 (i) - (Y)(2+Cb) +2 (3-14) is then defined and substituted into Eq. 3-13 to give A(yi+) = A(yi)Q2(yi+l) - 2A(Yi) e(Yi) + C] (Yi+l) + (3-15) Next the quantities i~ 2it (Ay)2(l+Cb) sin 0(yij 6)d j f cos 0(yi'jj)dOj sin (Yi,0o - 2Cd - O 1 Jtc 1 + 2Cu(yi.Oj) JO- 1 +- 2Cu(y j ) (3-l6) ircC 1 iecu~yi+2Cu(yi.,~ and ~ = A(Yi) 9(yil) + 2A- 2d(A) (l+Cb)2 (3-17) are defined, whereupon Eq. 3-10 can be written as A(~yi)(yi+l) - 2A(yi+l) o(Yi+l) - (Yi) - - = T * (3-18) L~~~~~~

56 Finally, the expression for A(yi+l) in Eq. 3-15 is substituted into Eq. 3-18 to give, after simplification, 2A(yi) 3(i+l) - 6A(yi) () (Yi + A 2(Yi+l) + 211 + A(yi) [4[9(yi) + 2 l (yi+) +4 - 2l(yi) + Y = (3-19 The three equations 3-8, 3-11, and 3-19 represent the difference form of the working equations derived in Section 2.1. These three equations can of course be derived symbolically from the original working equations. At the (i+l)st level the working equations are written in terms of their values at the ith level symbolically as follows:, + 9 = C1, (3-20, A - Q2 + = C2, (-21; 9 + 2A = 3, (3-22: and u + = C4, (3-235 where C1, C2, C3, and C4 represent the right-hand sides of the equations, which have constant values for any particular row in the integration procedure. Solving Eq. 3-21 for A and substituting into Eq. 3-22 gives 293 - 292 + (1 + 2C2)G = C3. (3-24 Equations 3-20, 3-23, and 3-24 are equivalent symbolically to Eqs. 3-8, 3-11, and 3-19.

57 3.4 Details of the Computing Process Figure 3-7 is a flow diagram for the numerical solution of Eqs. 3-8, 3-11, and 3-19. Most of the computation involves straightforward addition, subtraction, multiplication, and division. However, some of the steps, such as the evaluation of the integrals, the solution of a cubic equation to find the new value of G at the (i+l)st row, and the numerical evaluation of the space-charge-field weighting function, warrant a somewhat detailed discussion. 3.4.1 Numerical Evaluation of Integrals. For the evaluation of the integrals in Eqs. 3-12 and 3-16, 32 electrons are carried through the integration procedure. As mentioned in Section 2.3, it is believed that consideration of 32 electrons will give results which are not markedly different from those which would be obtained if a larger number of electrons were considered. This point is taken up in more detail in Section 3.5.1. Therefore the functions under the integral signs are evaluated over the interval from 0 to 2t for 52 equally spaced values of oj However, the equations employ integral signs rather than summation signs; the answers are desired not to the 32-point problem but to the continuous problem. The integrals are evaluated from the 32 equally spaced values of each function between 0 and 2Ai using the familiar Simpson Rule37, which may be stated as follows: If Yo, Yl, Y2,... are the values of y = f(x) at equally spaced points xo, x1, X2,... with interval h, then X2m ydx = (yo + 4yl + 2 y + 4y3 + 2y4 + 4y5 +... 4y2m-. x ~3 xo + Y - m(4) h (3-25) -211 90

58 x r x m N r nIN 1+ I _ W - m 0om:4 - -1:*~~~~ z - ~~~~~~~~~ m 0o I x m -41+ n + h ~ ~~~~~~~ 0 v, a O mo~~~~~~~~~~~~~~~~~e, mo~~~ -U) I+ ~~~~~~~~~~~~~~~~~ m~~~~~a + O + z 0 x -. N G)N m C + x 4 x En 0 K u zU N PO 0 o~~~~~~~~~z~~~~~~~~~ I~~~~~~~~ o~~i I h mCo > I v, Y1E 0 to m P1PI I I' + 10 0k I + 4 + X -n I It o 0 P G o C "o m 0 D + IrO I) V, g~~~~~~~~~~~ P) B.Z L ~~~~~~~~~~~~~~ r 9 m s 8~~~~~~~~~~~~~~~~> r rr + _ b + - o W ri 0 +; mrr m 0 m m c m1 ~ 9D m cc Crr + 0 +;)a o + m rum~~~~~~~m m I L-j I I " u-I u, -? u, ru h) h) O m v, u, O r~~~~~~~~~~~~~~~~~~~~~~~~~) G: m m 2 a +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~D;o, < C o M 0 ~ rn x o 1 - m,~ a~~~~~~~~~~~~~~~~zC m- -nm >4b+! ro O cZ~~~~~~~~~~~~~~~~~~~~r ~ ~ ~ >+ e Zx +;, r4 z P 0 CL~~0 Yh O O " a V~~~ ~ ~~~~~~~~~~~~~ ~ ~~~~~~~~) w N +'I(f'II IIh 3~~~~~~~~~~( + 0 N G) MN + 0 + OO O -r h) Q) El 9~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+14 -E m 0 N-G- rje -0I m + V *- Ce-ej l I z N 0 0 -t. ~ ~ ~ t~OY

59 where h = the interval in x and y(4) = the value of the fourth derivative at some point between xo and xnm. Simpson's Rule is simple in form and quite accurate. It does have the disadvantage that it can be applied only to an even number of intervals, but that requirement causes no difficulty in this problem. The accuracy of Simpson's Rule for the evaluation of these integrals may be estimated by evaluating the remainder term, my(4)h5/90, of Eq. 3-25. The increment h is t/16 here, and the value of the fourth derivative is taken as unity. Thus the error in applying Simpson's Rule to this problem will amount to less than 10. The evaluation of the integral of the space-charge weighting function F(0-0') is discussed in Section 3.4.3. 3.4.2 Newton-Raphson Iterative Process. As indicated in the flow diagram, Fig. 3-7, it is necessary to solve for the real root of a cubic at each level of the integration procedure in order to find the phase lag 9(y). A general second-order iterative process called the Newton-Raphson Process is used for this purpose. The solution of f(x) = 0 is to be found at a point not in the neighborhood of a maximum or minimum of f(x): f(xK) XK+1 = XK - rK) (3-26) In terms of 9(y), Eq. 3-26 becomes (K) (K) O (3-27 =+l ) - - f(Gi f'i' J)

6o where f(gi) = X~i3 - YGi2 + ZEi + U. As shown in Fig. 3-8, this process may be interpreted geometrically as (K) linear interpolation along the tangent to the curve at (K) In order to find the solution by this procedure, it is necessary to guess at the root for the first iteration. The value of 9(y) at the preceding row is used for this first guess, and the process is carried on until the second term of Eq. 3-27 is less than 2-26. 3.4.3 Numerical Evaluation of the Space-Charge Weighting Function. The space-charge-field weighting function F(-01'), derived in Appendix B, is 00 F(0-t) = sin n(-') R2 (B-2) L 27cn n (B-22) n=l Since (l-Rn) varies exponentially with nPb', ln(l-Rn) vs. nPb' will be a straight line. Thus Eq. B-22 may be written as F(0) 0-') i[1 n nnbt') a ) -e (3-28) n=l where f(a'/b') = the slope of ln(l-Rn) vs. nPb'. For a'/b' = 2, this slope is approximately unity. Since the series represented by the first term of Eq. 3-28 is not uniformly convergent as (0-0') approaches zero, the point of discontinuity, it would be necessary to sum an infinite number of terms in order to avoid the familiar Gibbs phenomenon encountered in Fourier series: at (X-0') = 0, the summation required is of an infinite number of zeros, which is undefined. This last difficulty is cleared up if the first term is written as

61 (0) Ij () / /' 8(2) FIG. 3-8 GEOMETRICAL INTERPRETATION OF THE NEWTON- RAPHSON PROCESS FOR THE SOLUTION OF f(8) =

62 N F(-) = rlim sin n(-') (-29) N+oo 2itn n=l which clearly has a finite limit as (0-0') approaches zero. Thus it is desirable to write Eq. 3-28 in closed form. Consider the following general term of Eq. 3-28: 00 7 sin n(-') - L sin n(n e for a > 0 (3-30) n=l Replacing sin n(0-0') by its exponential form, Eq. 3-30 may be written as 00 00 _ esin n(_-0') 1-nC! 1__ Z 1 i-n[a-i(-_')]_ - en[Ca+i(_-0)]j. (3-31) i n 2i ne (331) n=l n=l The relation 00 n = Cln iz for Z < 1 (3-32) n=l where Z may be complex and the locus of I Z = 1 is called the circle of convergence, may be used to rewrite Eq. 3-31 as sin n(0-') e -ncx 1 n 1. (e n ~ (3-33) = 2i n=l - e Writing the exponentials of Eq. 3-33 in terms of their real and imaginary parts gives 00 \ sin n(0-0') -nca 1 [e~ - cos (0-0')] + i sin (*-(') n= nl e = 2i [e - cos (0-0')] - i sin (0-0')

63 Since the logarithm of a complex number is given by in Z = ln(x + iy) = in Jx2 + i tan-1, (3-35) x Eq. 3-34 can be simplified to 00 sin n(- ) e-na tan - cosin (0-0) 3-36) n l ece - cos (~-~') n=l It is easily seen that this expression is also valid when a = 0, i.e., on the circle of convergence. Applying the form of Eq. 3-36 to Eq. 3-28 gives;1 tan-' 1 sin (00 )- -_ 2 tan1 btf(ab'sin') F(-0') = - tan-1 - cos /b') - 2 tan C os (-0) sin (0-C,). + tan e'') cosi (0-') ] (3-37) Since, if 0 < (<-') < 2A, tan-1 sin (- ) (= -' (3-38) c - os (-~') 2 an alternate form of Eq. 3-37 is F(-0 = 1 - 0 ) - 2 tan-1 sin (-$ (0 t) + tane 2b'f(a/b') - cos (s * j') Equation 3-39 is presented in graphical form in Fig. B-6.

64 In order to shorten the computation time, the function F(0-0') is approximated by a polynomial in 0-0' for each of five values of the parameter Pb' where f(a'/b') = 1. The following expression is seen to satisfy the end conditions: F(x) = [1 + f(x) x] (16 - x) (4) ~(0)64 where f(x) = a polynomial in x and x - 0-'. By the method of least squares39 it is found that for each value of the parameter Pb' a cubic, f(x) = A + Bx + Cx2 + Dx3, (3-41) can be found which will satisfy the accuracy requirements of the problem. Hence Eq. 3-40 may be written as F(x) = - (16 - x)(l + Ax+Bx + Cx + Dx4). (-42) 64 Evaluating the coefficients in Eq. 3-42 gives the following polynomials for Pb' = 0.5, 0.75, 1.0, 1.5, and 2.0 respectively: F(x) = 0.25 - 0.54288x + 0.48836x2 - 0.21903x3 + O.o48302x4 - 0.0041769x5 (3-43) F(x) = 0.25 - 0.34891x + 0.17590x2 - 0.026466x3 - 0.0044749x4 + 0.0019788x5 (3-44) F(x) = 0.25 - 0.25227x + 0.063644x2 + 0.023688x3 - 0.014493x4 + 0.0019333x5 (3-45)

65 F(x) = 0.25 - 0.16432x + 0.0064676x2 + 0.027203x3 - 0.010045x4 + 0.0011026x5 (3-46) F(x) = 0.25 - 0.12640x - 0.00091601x2 + 0.015709x3 - 0.0048276x4 + 0.00045523x5. (3-47) The integral of the space-charge weighting function is evaluated using the Trapezoidal Rule, which can be stated as: If yO, yl, Y2, *.. are the values of y = f(x) at equally spaced points xo, xl, x2,... with interval h, then Xm y dx = h( y + yl + Y2 +... + Ymi- + y - myh (3-48) xO The Trapezoidal Rule is not as accurate as Simpson's Rule, as may be seen by comparing the remainder terms of Eqs. 3-25 and 5-48, but the difference was considered to be more than offset in this case by the lessening of storage capacity and computation time required. 3.5 Experimental Study of Computation Errors In order for the numerical results of this work to be applied intelligently to the analysis and design of traveling-wave amplifiers, it is necessary to consider the accuracy of the solutions. This section will discuss only the accuracy of the solutions to the differential equations; the exactness with which the equations describe the physical phenomena occurring in the tube and the physical limitations of their application are considered in Chapter II as the working equations are derived.

66 As yet no theory has been advanced for error analysis on solutions of nonlinear partial-differential-integral equations. The possibility of making an analytical study of error in a nonlinear system for application to this problem was explored but proved to be so complex that an experimental approach was chosen instead. An experimental investigation was therefore made into the presence of errors in the dependent variables A(y) and Q(y) and the question of stability of the solution. It should be noted here that the experimental error analysis described below has been carried out for the special case where the space-charge field Es is zero in order to shorten the computation time required. Although the space-charge force is of course not zero in the general problem, it is believed that the errors involved are of the same type and order of magnitude as those encountered in the special case considered. 3.5.1 Effect of the Number of Electrons Considered. The number of electrons which must be followed through the computations was determined by running the same solution for different numbers of electrons and examining the magnitude of the error resulting from use of the smaller number. It was established in this way that for the solutions where the space-charge parameter QC is taken as zero, consideration of 32 electrons is sufficient for the accuracy desired since non-space-charge solutions are insensitive to the number of electrons followed above that number. However, when the space-charge term is not zero the accuracy of the solutions is highly dependent on the number of electrons followed. The difficulty arises in the evaluation of the space-charge weighting function F(~-~'). (The function F(0-.') vs. 0-0' measures the space-charge force between each electron and its neighbors.) This function, as seen in Fig. B-6, is an odd periodic function and consequently has a discontinuity

67 at 2tn for n = 0,1,2,...; therefore the accuracy of the integration of F(0-0')/[l + 2Cu(y,0o)] over the period 2t is determined by the number of values of 0-0' between 0 and 2i considered. This number of points is in turn determined by the number of electrons considered. Inaccuracies in evaluating the space-charge weighting function are reflected immediately in determination of the a-c velocity term at the (i+l)st row, as shown in the following equation taken from the flow diagram of Fig. 3-7: -KI + K2 + Ks - -K+ +, (p-49) 1 + 2Cuij where Ay 2 F(0) do K1 1 + Cb =C) z + 2Cu(y,o) 0 The terms K2 and K3 are related to the force on the electron due to the field on the helix, while K1 is due to the space-charge force. Clearly these two forces act in opposite directions. If the space-charge force is small compared to the other force on the electron, i.e., if K1 is small compared to K2 + K3, then an error in evaluating K1 will have only a small effect on the new a-c electron velocities. However, if the space-charge force is of the same order of magnitude as the helix-field force, then an error in K1 will have a pronounced effect on the new a-c velocities. A study of this effect has shown that for QC d 0.25 the difference in the solutions when 32 electrons and a larger number are used is not appreciable. For values of QC around 0.5, however, it is necessary to use at least 64 electrons should be followed. The difference between solutions for 32 and 64 electrons

68 for a particular case is shown in Figs. 3-9 and 3-10. The two solutions are compared on a percentage basis in Figs. 3-11 and 5-12. The dependent variables 0(y,lo) and u(y,o0) are more sensitive to the number of electrons considered than the variables A(y) and 9(y), which is fortunate since only qualitative information is needed on O(y,0o) and u(y,0o) while more exact, quantitative information is needed about A(y) and 9(y). 3.5.2 Truncation Error. The differential equations were solved using a difference-equation approximation of the first order. That is, only the first two terms of the Taylor series expansion were used, which means that the first term neglected was of relative order (Ay)2. This difference-equation approximation amounts to replacing the function by a straight line drawn tangent to the function over the interval Ay. Hence, if the interval taken is too large, the first-order difference equation is a relatively poor approximation to the function over that interval. The resulting error is known as the truncation error. Truncation error can be lessened by either of two means. First, higherorder differences may be taken into account; i.e., more terms in the Taylor series expansion may be included. Considerably more machine storage space is required, however, when higher-order difference equations are used, and the computation time is usually extended also. The alternative is to decrease the interval Ay to the point where the straight-line approximation to the function is quite accurate. Unfortunately, this procedure leads, in the limit, to round-off error, which is discussed in the following section. A compromise value of Ay must therefore be selected to yield a minimum in the combined effect of the two types of error. As can be seen in Figs. 3-13 and 3-14, solutions for intervals greater than 0.025 differed markedly from those for Ay = 0.025 in any region where

69 -o I - 0 cl In 00 ww U J X X rOW o~ I ww r I H Nnz z - z CO L a: o z 0 a- Na _ _: 0 W 0 — W CD o o, w Z _ _ _ _ _ _ _ _ H - I - -. I w H — CY Ln UK LI -W " -o ( I 3vo 0 >, W 0 -- i "~,, o c~ ao, (r(~'Onl~d tl 9~10A -\

70 u~u zz 00 u (..)- - - - -' _1 -_ wwr w ~ ~ ~ ~ ~ ~ w CMD l i Ii J i I i IlI l ii T I I I I I l I I I i I I Ik Cf) z!,~~~~~~~~~~~~~~~~~~~~~~~~~~ w ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~, z W 4 1:: (.~ 0:. C — ) 113 0 -- <: o Wew o ro U > 8- 5 - -- - -- - ---- (/ h L Q: S - - _ _ _ _ _ _ _ _ _ _ _ - _ w o ~-: 03 |O fe? n: i m m mvm iz z - z <o w - / - io w w F. - ~ - w) F _ < w LL Il " 0 a. — w':~ -r W F- u=w LL. LL W n-' i C 0.c -- SNVIOVW'(X) e'9v 3SVHd j-1

71 FIG. 3-11 COMPARISON OF SOLUTIONS USING 32 AND 64 ELECTRONS: PERCENT DIFFERENCE VALUES OF THE R-F VOLTAGE AMPLITUDE, A(y),VS DISTANCE 4 | l l [SOLUTION STUDIED: C O.I, d 0, b 1.5, Ao 0.0225 (0 c~ _ +10.0' +8.0 - + 8.0 z + 6.0 li __ d +4.0 +C wQC =0.257 5 +2.0 _ POINT OF FIRST ui ELECTRON OVERTAKING CL I -2C0 MAX. -8.0 -10.0 0 2 3 4 5 6 Y -'

72 FIG. 3-12 COMPARISON OF SOLUTIONS USING 32 AND 64 ELECTRONS: PERCENT DIFFERENCE IN VALUES OF THE R-F PHASE LAG, 0(y),VS DISTANCE 5 J> l SOLUTION STUDIED: C O.I,d=O, b=1.5, AO 0.0225 + 10.0 8 + 8.0 z +6.0 w IL: w L + 4.0 D + 2.0 POINT OF FIRST ____ ELECTRON OVERTAKING MAX, z.I i ni- 0 w a-4.0 -6.0 -8.0 -- -10.0 0 I 2 3 4 5 6 y —

753 - - - - ----- -ZS5^\ - -" "'x'9 — -- - _ - -" o. w 0, w c TI 11I —-~z o.J _ o i0 _ I-LL 0 ^ m=c 0 0 C) - o nj, _< o o-r \\ < z o o w — [ F-, do n-~~~~~~~~t'3nldt'l111\Jt

74 — _ —Sv, \\ \\. > I, \ \^ - 0 - __- _ -- -- r- W LL W, P1 u, z o Z~,, \ Cle 00 N -0 0 0 0 > ow w wI - - z > z. —- ---- - -^Z * ~r-U. 0, -O..

75 the second derivatives of the function were of comparable magnitude to the function and its first derivative; whereas with intervals smaller than 0.025, down to Ay = 0.00625, there is little difference from the solutions for Ay = 0.025. Thus, Ay = 0.025 seems to be the upper limit of the region in which truncation error is insignificant in this problem. 3.5.3 Round-off Error. Round-off error occurs when the machine register is not large enough to store a number to the required number of significant figures, or when the shifting necessary to combine numbers in an iteration process results in moving a round-off originally beyond the point of interest into a significant place. As numbers are shifted to the left, the numbers used to start the next iteration are accurate to fewer significant figures, and thus the error increases in proportion to the number of iterations carried out. Round-off error could be minimized by using double-precision arithmetic. However, this has the disadvantage of requiring twice as much machine storage space, so that it is more practical to control the round-off error by choosing Ay sufficiently large (thus requiring sufficiently few iterations) to obtain the accuracy required. The limitation on Ay mentioned in Section 3.5.2 must of course be kept in mind. Solutions to the difference equations were therefore obtained using successively smaller increments in y, down to Ay = 0.00625. No appreciable difference between the solutions for Ay = 0.00625 and those for Ay = 0.025 was found out to five or six decimal places, from which it was concluded that round-off error is not present in the solutions to the accuracy desired. It was felt that continuing to decrease the integration interval in order to find the value of Ay at which the round-off error is so great that it completely dominates the solution (i.e., affects the first significant figure)

76 would require so much computation time on the machine as to make this approach highly impractical. A different type of test was used to find out something about the amount of round-off error present in the solutions for Ay = 0.025. The method was to determine the sensitivity of the solutions to an error introduced in the 44th bit of the machine, which corresponds approximately to the 13th decimal place in a number. The machine normally runs, and this problem was run, using rounded multiplication; i.e., the last digit for which there is room in the register is increased by 1 if the following decimal digit is equal to or greater than 5. Rounded multiplication in general produces some products which are greater and some which are smaller in absolute magnitude than the true values It is for this reason that the rounded multiplication order is used since it is assumed that working with both too large and too small numbers will tend to reduce the overall round-off error. When the unrounded multiplication order is used, the products formed must be either less than or equal to the true product in absolute magnitude. It should be noted that division and shifting have the same round-off effect as unrounded multiplication, while addition and subtraction are theoretically exact if the operands are exact. The propagation of round-off error was therefore examined by running a solution using the rounded multiplication order and then running the same solution with the same Ay using the unrounded multiplication order. Since multiplication of two 44-bit numbers by these two different methods will result in differing 44th binary digits an average of half the time and since a deviation in the 44th bit of an intermediate number is necessarily shifted to a more significant place as iterations are carried out, an error may be expected to propagate and show up in the solutions if enough iterations occur after the difference appears.

77 When intermediate numbers in a program are small or are scaled so that they are represented by only a few significant binary digits, it is quite obvious that round-off is of greater importance; i.e., a change in the last binary bit of a small number produces a larger relative round-off error than in a large number. Even in problems where the magnitudes of intermediate numbers are unknown initially it is often possible to rescale these numbers on the basis of trial solutions in order to retain more significance. However, it is not always possible to avoid small numbers by rescaling, as in the case of a number obtained by addition of small numbers or subtraction of nearly equal numbers. For example, in the problem which is the subject of this thesis the first derivatives are replaced by the approximation dx _ xi+ - Xi dy Ay in which it is clear that if Ay is small xi+1 - xi will be a small number. Hence when this number is used in forming a product the round-off error will become increasingly significant with decreasing Ay. In addition, the smaller Ay is the greater will be the number of iterations required to go a unit distance along the y axis, and this increased number of arithmetic operations will increase the frequency of round-off error. Thus it would be expected that some lower bound B on Ay exists such that for Ay < B roundoff error will completely dominate the problem and give diverging results. Some intuitive insight into the phenomenon of round-off error may be gained by comparing the solutions obtained by using rounded and unrounded multiplications. When the differences in corresponding solutions are slight it may be concluded that the problem is not sensitive to errors in

78 the 44th bit, if the program involves sufficient multiplication to insure reasonably frequent introduction of errors. When, on the other hand, corresponding ordinates of two solutions using rounded and unrounded multiplications differ in the jth significant figure, the accuracy of the results to j significant figures is certainly questionable since if changes in the rounding procedure can produce diverging results it is reasonable to assume that the true solution differs from the exact solution in at least the jth significant figure. Thus we have bounded the round-off error in a greater-than-or-equal-to sense. Solutions for A(y) and 9(y) using both rounded and unrounded multiplication were obtained for Ay = 0.025, 0.00625, and 0.003125. The solutions differed by approximately one unit in the seventh decimal place for Ay = 0.02i while the difference showed up in the fourth and fifth decimal places for Ay = 0.00625 and in the second decimal place for Ay = 0.003125. The results of this investigation are presented graphically in Fig. 3-15. 3.5.4 Selection of the Integration Interval. Two factors were considered in the selection of the integration increment Ay to be used for the solutions: the degree of accuracy desired and the computation time required. Obviously the larger the value of Ay the better from the standpoint of minimizing the computation time. As pointed out in Sections 3.5.2 and 35.53, accuracy considerations suggest a small value to minimize truncation error and a large value to minimize round-off error. The two types of error are independent of one another; what is desired is the smallest rms error. The value selected for Ay was 0.025. Computation can be carried out in a reasonable time with this interval, truncation error is insignificant at this level, and round-off error appears in approximately the seventh

79 FIG. 3-15 ROUND-OFF ERROR STUDY: FIRST SIGNIFICANT FIGURE AFFECTED IN 6(y) WHEN ROUNDED MULTIPLICATIONS ARE REPLACED BY UNROUNDED MULTIPLICATIONS VS. INTEGRATION INCREMENT Il SOLUTION STUDIED: C-O., QC -, d O, b= O, AO-~.OI I0 LU r_. CD 13 o.005.010.015.020.025.030 INTEGRATION INCREMENT, Ay

80 decimal place for the required iterations, so that the results are believed to be accurate to approximately six decimal places. 3.5.5 Stability of the Solution. If the solution of a differential equation is stable with respect to the propagation of errors, then an error introduced into one of the dependent variables at some point in the iteration procedure will result in a solution which either is parallel to the exact solution or converges to the exact solution. If the solution is unstable with respect to error propagation, then a solution containing an error will tend to diverge from the exact solution. In Fig. 3-16, line AB represents the exact solution; solutions 1, 2, and 6 are seen to be unstable; and solutions 3, 4, and 5 are considered stable. In order to test the stability of the solutions of this work relative to errors in the dependent variables, errors of 5, 10, and 15 percent were introduced in A(y) at several values of y (for example, at y = 3, 4, and 6) and the solution carried forward. A similar test was conducted on e(y). In all cases the resulting solutions proved be parallel to the sore olued ttion into which no error had been introduced. It was therefore concluded that this type of solution is stable with respect to the propagation of errors.

81 / / / / /.,1 / _ - 6 FIG. 3-16 PROPAGATION OF ERRORS: STABLE AND UNSTABLE SOLUTIONS OF A DIFFERENTIAL EQUATI ON.

CHAPTER IV. GRAPHICAL PRESENTATION OF RESULTS Solutions of the general large-signal equations derived in Chapter II may be obtained for a wide range of the parameters introduced there. For convenience the normalized variables, in addition to the parameters necessary for a particular solution, are listed below. The normalized variables are: Cm 2iC y = uT Z = -- z, distance measured from the input to the helix; ozo.~ = -- = to, entrance phase of an electron relative to Uo the r-f wave at the input, radians; jA(y)| = C — |V(y,0O)l, normalized amplitude of the r-f 0 ~ wave on the helix; Q(y) = y/C - owt - 0(y,0o), the r-f phase angle (or lag) of the wave on the helix relative to a hypothetical traveling wave whose phase velocity is the initial stream velocity uo, radians; 0(y,0o) = the phase of an electron relative to the r-f wave at a particular y plane, radians; 2Cu(y,0) = the normalized velocity deviation of an electron at a particular y plane, defined by ut(y,0o) = uo[l + 2Cu(y,0o)]; and ut(y,0o) = Uo[l + 2Cu(y,0o)], the total velocity of an electron at a particular y plane, m/sec. The parameters which must be prescribed for a particular solution are: C = the gain parameter defined by C3 = |IIZIo/2uo2; QC = 1,/ Cq/CD )2, small-signal space-charge 4C2 [1 + Wq/W0 parameter*; *The solutions were actually carried out for values of the large-signal spacecharge parameters K and B discussed in Section 2.4. However, since QC is more familiar to workers in the field the curves were plotted in terms of QC. Equivalent values of K and QC may be computed from Fig. B-4. 82

83 B = Pb' = the large-signal space-charge range parameter, radians; UO-vO b = C, the relative injection velocity; d = 2/20(2i)(log e)(C) = 0.01836 2/C, the loss factor; 2 = the series loss expressed in db per undisturbed wavelength along the helix; Ao = A(0) = the input-signal level, usually taken to be 30 db below CIoVo; and a'/b' = the ratio of mean helix diameter to stream diameter. The range of parameters to be investigated was selected so that the results would apply to high-power traveling-wave amplifiers in general, whatever their frequency range. As stated earlier, the object of this investigation was to study the high-level operation of the traveling-wave amplifier, and in particular how the various parameters listed above affect the saturation power output and efficiency of the amplifier. Clearly this objective requires determination of the optimum injection velocity (i.e., that value of b which gives maximum saturation gain) for each value of C and QC. In general, the b for maximum saturation gain will be greater than that for maximum small-signal gain. In order to determine the injection velocity for maximum saturation gain at various values of C and QC, the input-signal level AO was fixed for these calculations. The value selected was approximately 50 db below C times the stream power IoVo, or precisely Ao = 0.0225, which insures.that the operation is initially linear and the boundary conditions may/be calculated as indicated in Section 2.4. In addition, for all the calculations the ratio of helix to stream diameter a'/b' is taken as 2 and the space-charge range parameter fb' has the

84 typical value of 1. Except in the loss studies presented in Section 4.8, the loss factor d was considered to be zero. Since the effect of space charge is of particular interest, solutions were obtained both with the space-charge parameter QC taken as 0 and for two other values of QC, namely 0.125 and 0.25. (For the values of a'/b' and Pb' used, the corresponding values of ( p/wC)2 are 1.61 and 3.42 respectively.) The non-space-charge solutions were carried out for three values of the gain parameter C, 0.05, 0.1, and 0.2, which are considered to be representative of large-signal tubes in general. The value C = 0.1 was selected for the space-charge calculations as being nearest the C values most frequently encountered. The results of the computations carried out on the Michigan Digital Automatic Computer are presented and discussed in the following sections of this chapter. The curves are presented in historical order; i.e., the curves plotted directly from the data taken from the computer are presented first and are followed by cross plots of the data. The flight-line diagrams, which are perhaps the most useful curves from the standpoint of understanding the physical phenomena involved in a large-signal traveling-wave amplifier, are presented in Section 4.5. 4.1 R-F Voltage and Power along the Helix The r-f voltage amplitude A(y) is plotted vs. the tube length both in units of y and in units of the stream wavelength Xs (the latter is denoted by Ns) for a number of values of b in Figs. 4-1 through 4-5. One of the values of b selected for each set of calculations was that value which gives maximum small-signal gain, i.e., that b for which the growth constant xl of the slow-and-growing wave is a maximum. As was expected, in each case the

85': - F- IC)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,. - / f - - -"_ _ _ _ / __ _,, _ _ __ _ _ _ _ _ _ __ _I__ _ ff ^^^^^^"ZIZZZ IZZ^ZZZZZZ <~~~~~~~~~~~~ >, I~~~~~~~~ Ir IA N I I l1 I~i~ii Ii I 0 —....~ _ o ______/ _^ _____^^^~~~ ~~~'~__ ___ _" _ \ ~,.J y ^o - 7~~~~~~~~~~~~~'> Z0 > 0 L <~ > ~- U".I ~~~~~~~ —~ — - - ^ ^- -^ — f I Ic I.^- ^ z z z z z z z ^ - -S z 2 n-',,\ 3 u _ _.go__.,-_ _ _ _ _ _ _ _ _ _ _ _ L 1 _ w: g I'0 0. _ ~~~~~~~~~~~~~~~~~~~~~~~~~~~0 -~ -,,o >, I:|=-=^ ^^z:===zzz-z- =-|_. I"; > " "~EEEEi^Er^^E w I < W -- ~-~-d ~ ~ ~ ~ ~ () 3niiwv ~on~

86'* —-<, _ _, = = t(D' I 3c ~- a O t z W- - x la - -V E3 -,-:___ - - __',, \ __ _I 1- - 1 I T I -II I I T I T I I T 1Wo, 0 W ---— __^ - - - L - o d o (n. 1d: o -I-m

87 >5 o z - -- - -- - -- - - -- -- -- - -- -- -- -- -- -- -- -- -- -- -- -- -- i -- --- -- --- -- ( -^ -..x -- -- - ^ ^ -- -- -- - - - -- - - I I — I I~ ~ ~ ~ ~~~d I 1 Ir: rI r\I I I II r - II r 1 ( I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~IT ^ >. _ _ - _ _ _ _ _ _ _ _ _-__ _ _ _ _^ ^ ^N Y V _ o u u ^^ \ I W C m\ > W a 4 D m1 LU >r ~ - o _ _ _ _ _ _ _ _ _ _ _ _ _ ^ _ 0 c cl W L 1-;o o - ^- -- - -- > > LL~~~~~~~~~~~~~~~~L z a- s - --- - - I LL LL W 0 LL w LL It W w cr > c N w E, a --- -d- "- -- - ^ - Z LL 0 0 a. 2 --— __-o —7;J I I 1 I I I i i 1~~~~~~01 _* - - - 6 d d (^)v'3anin~drV~ 39vnoIId-d

88 "IN X X X XWA~f f f1X x' ~W _ 1 _ ___ 1 _ __ _ 1 1\aI 1 1\ \ 1 1 1 1 z J [_ I C 0 ZD w ( O. 0D i )t' 3tl49-0 -,> z __ gn I u <~ _ -- - -- | - - -- ^ -- -- -- -- S N~: -,.I o w 0 x ~ _ _ _ _ _ _ _ _ _ _ _ _ L0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 W I- I 0. - - - -- - - - - - -- - -- - - - -- - - - - -- - - - - - - -- - - -- - -- — r > (AV'3anindlv 39vi-iOA J-8 J~~~~~~~~~~~r)l'0l~~l 3t10\Jt

89 - (2, I -- - - ~J~~i W I z r I",,\U I- —. w U'f'' I 0_ -— > -_ E - I- w J 0 2 oL1 - 1 1 ". -- -- LL ^._ _____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _s. _ _ _ _ _ _ _ _ _ a, I;.s = I zzz^^zIzzz~zzzzzzE'x\ (A) v'aanindwv aovno~~~~~~~~~~~~~~d-y~

90 value of b for maximum saturation gain turns out to be greater than that for maximum small-signal gain. On each of the curves it will be noticed that the saturation tube length is largest for that b which gives maximum saturation gain, which is also to be expected since the low-level gain is in general less as b is increased. The point of first electron overtaking is noted on each of the graphs. From the flight-line diagrams presented in Section 4.5 it can be seen that this point along the tube at which electrons first overtake one another is in general dependent on all the parameters and occurs well into the nonlinear region. The first maximum of the r-f voltage amplitude is also marked on the curves when it cannot be located quickly by inspection. It can be seen from the working equations that if the solutions are carried beyond this first maximum the amplitude will drop to a minimum and then rise to a peak again. Since the later maxima are in general lower than the first maximum, the solutions were not computed beyond that point. In Fig. 4-1, for C = 0.05 and QC = 0, the b = 2.0 curve is plotted only for the region near saturation because the tube is quite long, some 65 stream wavelengths, for this case. It should be noted that the value of b for this curve is greater than that for which maximum saturation gain is obtained. Hence the interaction is less and the signal level on the helix builds up slowly with distance. Figure 4-2, for C = 0.1 and QC = 0, includes a solution for b = 0 (i.e., u = v) in order to show that gain may also be obtained when the wave and the electron stream are initially in synchronism. Clearly this is not an optimum value of b.

91 In Fig. 4-3, for C = 0.2 and QC = 0, the b = 2.0 curve has not been carried beyond saturation. Computation was suspended at this point because at saturation one electron has very nearly stopped; i.e., its total velocity as given by Ut(yo) = Uo[l + 2Cu(y,o)l (2-14) is approaching zero. One electron has in fact dropped back one period in time, as may be seen in the velocity-deviation curve of Fig. 4-37. The electron-velocity curves of Sections 4.4.1 and 4.4.2 show the approach of 2Cu(y,0o) to -1 for the case where C = 0.2; i.e., u(y,0o) approaches -2.5. Integration must be stopped when this condition is approached, since several terms in the working equations are divided by the quantity [1 + 2Cu(y,0o)]. The resultant large a-c velocities are a result of the "strongness of coupling" between the stream and the circuit as C becomes significant compared to 1. This situation illustrates the great inaccuracy introduced by neglecting the space-charge forces then C is large. Consideration of the space charge prevents this condition from arising and in general reduces the velocity deviation. The r-f power level along the helix is of considerable interest also because on a plot of power level vs. distance along the tube the point of departure of the signal from the small-signal value is clearly recognizable. The small-signal power output is a straight-line function of the input power; hence the deviation from a straight line drawn tangent to the power-level curve near the input is a measure of the nonlinearities present. The power along the lossless helix is given by Lv(z,t p = =l g, 2CIoVoA2(y). (4-1) avg.

92 The power-level curves for the solutions of Figs. 4-1 through 4-5 are presented in Figs. 4-6 through 4-10. The power level is plotted in db relative to the input signal, which is taken to be 30 db below CIoVo. The CIoVo level is marked for reference on each figure. 4.2 Phase Angle of the Wave Relative to the Electron Stream As pointed out in Section 2.1, @(y) is the r-f phase angle (or lag) between the actual wave on the helix and a hypothetical traveling wave whose phase velocity is the initial electron velocity uo. In Figs. 4-11 through 4-15 the r-f phase lag is plotted vs. distance along the helix, for the same parameter values as the Section 4.1 solutions. In all cases @(y) increases negatively with distance and also with the relative injection velocit In the small-signal region sufficiently far removed from the input so that only the slow-and-growing wave is present, 4(y) vs. y is a straight line with slope equal to P1, the phase constant of the slow-and-growing wave. In this region where the solutions are of exponential form and the slow-and-growing wave of the small-signal theory predominates, the phase velocity of the wave is given by v = Uo(l + CP1) (4-2) However, in the large-signal region where the variation of the dependent variables with time and distance can no longer be expressed in the familiar exponential form, the wave picture developed with the linear theory loses much of its significance and consequently its usefulness. Nevertheless, some information on the wave velocity may be obtained from these phase curves. At any point y along the tube the phase velocity of the helix wave is given by

93?i —--- -- - - - 2:> c N w!w 0'~ I "' r~zr rr- IJ U.'.'UJ JW NN O~ O ==^zzrr=^=^====^=^^^rz~j:=l I | | o~~J~~ ^ \; H- < LUJ, ---- ~" I -- - - -~ jIi~~ _\ l _ I 1 _ _ _ \ _ _ _ _ _ _ _ _ _-1J j < o -: ID o.CQI oo ZI - - - -i - I 10 N N - _ _ _ i_ ^ o_ -- - __ __ - _ ~_ _ — q_ _d -o qP'_3_Od..- _

94 -- -. —-f~ — - — r - 0 x o I I N - - - - - ^ - \ - - -- -- - --- --- ^ - ^ $ 0 w 0 -- - -- -4 ---- --— ~ — ---- j --- u- <t ^ ^ 0 ->CD 0 I < 0 OL. W 0 cr > -Cc~~~i n CD - a. - LL W cr F a ^ ~f *" ~ ^ ~ \, I I I I i I I I i I I I I 4 - _ __ _ _7 \ >-/ _ _ _ _ _ _ _ _ _ -- _s, _ _ _ _ _~ ~ ~~~~__ __ _ 0 ___ ____:LS^ - - - - - -S -- - - - -~c~~i - - ^ ^ — ^ —-— 0C \.\' T _ \ 1 1 1 I I I Y T I I I I I I I I I I I LC - - - - - _^ _^ i _ - _I > - - - - - - - - - - -I I - - - - - - - - - - - - - - - - - - _______^>\s,___^.-___________V —-- — ** —----- -~* __ _ _ ^_ _ _ _ _ _ _ _ _ L _ _ _ _ I n izzzzzz -rs-m~zrz^^:^^^^^^._ _ _ _ _ _ < ^.\ i _ _ _ _ _ _ _ _ - _ _ _^ - ^ _ _ _ - _ _v - _un _ - - -~~~~~~~~ _ _ _ _ _ _ _ _ _ _ _ _ _? ^ _ N^ _ _ _ _ _ _V - -'F - - - -'~~~~~~~~~~~~~~~~~~~~~~~~~~( ^s. x, v I~~~~~~~~~~~c mzz^=zi_______^^ _^ __^ _c_ —-~~~~~~~~~~~~~~~~~~~~~~~~~~~~, _ _ _ _ _ _ _ _ _ _ _- _ _ _ _ _ _ _^ ^ - _^ - c _ _ _ _ _ _ _ _ _,~~~~~~~~~~~~~~~~~~~~~~~~a'^s._\,',__ J_~ ~ ~ ~ i~~~izz~~~zzzrzzizzzzizz'~^$_:m~~~~~~~~~~~~~~rrLZ~~~~zzizzz~~r _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ^ ~ S _ _ \ _! _ _ _ _ _ _ _ _~~~~~~~~~~~~~~~~~~~~~c ^S'^s;^.' V" I" ~~~I —1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ^ ^;^ - _.V _ -L - - -".- - - - - ~~~~~~~~~~~~~~~~< n~~~~~~~^^N -- - - - - - - - - - - - - - - ^ ^ - \ - - - x -- - _ _ _ _ _ _ _ _ _ _ _ _, 4 i__ _ s O Y) 4L- - __ __ _ ^5^^ cT ~ ^^ ~~~~~~~ ~ ~ ~~~~P'13~ ~ - ^ ^^

95 20% z~ - - - -- 4 - -- -- - - - -- - ~ I —^ x 0 m w 0u - Z o 0 W OD z w cr la --- -- --- -- -- ---- ---- -- -- --- --- -- ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~ ) <r C -t Z a-~ LLJ w L 2 I w Fcf) /^ I ^ I -- a0 w < LL — f 4 1 —-- I I —- " ~ V~~~~~~~~~~~~~~~~~~Q, M I r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I =m ^%K~^ z===z====z I Y iIIIIIIIIirI( - - ^ — ^- - - - - - -o- ^ _ _ _ _ _ _ _ _ ------- I p --- ^ - --- __ _ _ _ _ _ _ _ -~ \^ "^ \u ^^^^SS^-S^-^lll^^^ZllZ^Z^ZZZZZ3MZd'-t

96,,.,- -J _ _ _ ---- _! _ I _ _ _ _ _ _ _ _ _ _ _ _ / x o X 0 -- -- l -- -- --- ^ - -- --- -- -- ---- - - _.J- c uN _ _ _ \ _ _ - _ S — _ _ _ _ _W- o o.. —I NI, I., \' Z W I:1::::' 6 o o _ -- /,~~~~~~~~~ I W O 7, ____^^ s^ -'^ - ^ - - -t - - - - $~~~~~~~~~~~t 0._1 <1 d!~ ~ ~ ~ ~~o n I,~ d~~~~~~~~~~~~~~~~~~~~~~~'__-d:E'~ I:Z:: —---— ^^ —— N^____C___ -- Io! \'. I I 122i S-] - ~.'..'^. \.-.,______________ — -'-<. —..~~~^ ".'\ \?' I~~~~~q'3M -t 121 I - I -e. \' Y -',, i "zzizzmzzzzizzi~~~~~s^-s^Yiz^-zzzizz^^^~~~~ 1_ 1_ _ _ _ _ - _ /^_ S. _L - -- ______________ —-^^^S5__\ —— ____.-"~ _ _ _ _ _ _ -- -- ------ - ^~~"',."x x,' - I _ _ _ _ _ _ - \ ^ _ _ - ----- ^^\\ f -~~~~~~~~~~~~~~~~~~~~~~ _ _ -- - _ _ _~',~.J s~~~~~~~~P' _______d.- I-I

97 I, __ __ __o_ _o: r-t Act~ — X n- - 2Lz in i X0w,, _ - o ~ 5 ~ i -~ -- ~ -''3 o (" I > ( c " o_ CJ I --- ^ -- ^ -- ^: -- -- --- -- --—, S I —' C. _ _ _ ___ _ ___ ______~ ^ ^-^^ -- ^^ -^^ — -- - - I^^__^ Y ^ I —-^^^^-^-'^-^-^^^^^^^^^^^^^^^z" _ _ _ _ _ _ _ _ _ _s ^ ^ ^ _ _ _^ _ _ _^ _ _ _ _ _ _ _ _ _ _ _ _ _~~~~~~ ___________^s ^ ^-_^ ^__s ^~~~~\\ __ _ _ _ _ _ " ^ ^ _S _ _ _ ^ ^ Z ^ ^ < _ _ _ _ _ _ _ _ _ - _ _ _^ ^;^ _\ _ _s _ _ _ _^ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _^ ^ ^ ^ ^ _s _^ _ _ _ _ _ _ _ _ _ _ _ _^~~~~~~ o_ __ _ _ _ _ _ _ _ o _ __ o_ _ __ o_ _o o_ _ _ _ ss ^ ^ - -_ -^ _ _ _ L_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ S ^~ P'3~Od.-I-_ _ _ _ _ _

98 ~ 0 0 0 Z ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ to CD 8 z ~ ~ ~ ~ o I | | | |\ | | | | 1 1 1 1 1 1 1 1 1 1 1 T TTI I I I I I I I I co a cr,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l MiC 0-_____-______ --- C~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C 0^\ d cV~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l M~~~~ a~~~~~~~~~~~a f ~0 - - - - - -<-' ^ - - - - - - - - - - -' - - - - -.\ a~~~~~ 0 m CD - -^ —-- -0 W~~ cli uc r CM..__ __ _ __ _ _,, \ 0 Ti w( ( IIU) ~ ~ ~ ~ ~ ~~~)l > IW ~i;i —$-C —C —J- ---- s -- - - -- - - -- - -- -- I \ -- - O~~~~~ 0 W I I I O > O~~~~~ w <j >_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ a I~, 1I II II\ w I N c ^ ^ ZI ZI W' "^"" " " ^ "" - W~~~~~~~d, ~ ~ ~ ~, 0 LL ~li -,o il I I - - " a.~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. oI- o-%-\ 0~~~~~~~ _- w-, _W, %, \ ^n z S - - --— ^ ^ - -- - -,,_ -,-' --,.,< -', \ \ ct ~~~~~~~~, LL. ). 0 " --- _ ~ o ~ xx__ _ _ _ _ _ _ _ _ ^ ^, ^ _ - o — d -- _ _ - _ _ ^ V o_,, =' >' —-- " —, -5 "X _ ^_ _ - -— = o'~^i (0 <> <> 0Q. I- — ~ SN~Iavu~(.4'r e'9'" 3SalHd:l-u

99 IO N'> L \ tZ 0Y F.\ \> V<>_IoQ u0 \w _ __ O ___ w cf\ ___ _ _ _ ~- w' - \ ---- -- - -- - - A-. -- - g - - - - z -^ ^ = = z Tnzz z~ LL ^^ ^^ ^ *-<uZ\ V <T <3 ~i o _ - - - ^ - A - -- -^ - - -1 _J < ^ N _ _ U: 0 0 -- - - --- - -^ - - - -- ^- - V4 --- O dr < - - A - - L _ _L L -- W. f (\j - - - - - - - -- - - - LJ I- 0 \ _ Y,! 11)-^^^?^1 Q, i J < o_ _ _ _ _ _ _ _ _ \ _ _ _ _ _. _ _ d ^ > -__ __ _. ^ V - - -? - "

100 t=-t.m N ~ ~ ~ ~ ~ ~ ~ > z - 0 iJ d I I I I I X 0 _- > SNVIaVs p()e'gV-:SVHd _ — _._ _ _ _ _ _ _ _ _ _ _ _ _ _ _\ _ V ^ _ _ A _ _ _ _ -' _ _ _ _ _ _ _ _ _ J-t

101 - - - - --- - - - -- ---- - I I I I I I I1 \-' I- -- _SS W "__-_ __ W < > > - I I I 1\ k _ _ _-_ mO~sta\ \ \ VN \ 1 I Z o) W 4 S 25 ^; """""""""""""" ""S'^S^^^"" ""

102 >T 0 I i W~~~~~~~~~ > _d W ~ ~ W - - - - - - - - - -o - - - - - - - - - - — d > __ __ __ __ _ JQ __ __ __ M - w i u_ 0 > ~ < 3:^ w u LL o z w z I _ w~ ~~~~~~SVO~ L() e g-I3Vl -Ii w LL. r 0 __ _ _ _ _ _ _ _ _ _ __ __ __ __ __ __ __ _ _ __ __ __ __ __ __ __ __ __ __ _ _ _ _ _ _ _ _ _ _ u, ~r o ao (D o__ __ __ __ __ __ __ __ __ __ __ __ __ _ __ --- _ __ \ __ __ __ _ __ __ __ __ __ __ __ __ __ __ __ -- _ ___ ___ ___ _ _ __ _ ~ J- t

103 v = Uo + Vpr = e +'(y) (4 Hence the phase velocity vpr relative to the stream velocity uo is - uoe (y) ( pr = e + 4'(y) (4-4) Hence when @'(y) is zero as at y = 3.5 and 4.4 on the b = 2.0 curve of Fig. 4-15, the phase velocity of the wave is equal to uO. This is the only solution obtained in which this condition occurred, and it should be noted that the b for this case is greater than that for maximum saturation gain. In these regions where the slope of the 9(y) vs. y curve is zero, the phase velocity of the wave has increased to uO, since vpr is zero, and as might be expected the interaction between wave and stream accordingly decreases and a low-level saturation is nearly reached, as indicated in the plot of power level vs. distance for this case in Fig. 4-10. 4.3 Electron Phase vs. Initial Phase Considerable information on the bunching of the electrons in the stream may be obtained from a plot of the dependent variable O(y,0o) vs. COP The information contained in these curves is the same as that in the flight-line diagrams of Section 4-5 but is plotted in a different manner. O(y,0oj) is the phase of a particular electron relative to the r-f wave on the helix and hence is proportional to the time a particular electron (which entered at a time 0o relative to the r-f wave) arrives at some point y along the tube. Thus the ~ plots show a distribution of the electrons in time (in electrical radians) for a constant y, and the distribution in y for constant t is the same.

104 The V(y,0o) vs. 0o data could be plotted for all the cases presented in Figs. 4-1 through 4-5, but since the various curves are of a similar nature, a few will serve to indicate the pattern and trends. Graphs of electron phase vs. initial phase are therefore presented as Figs. 4-16 through 4-23 for (1) at QC = 0, the two values of b which are considered to be most significant, namely the b for which the small-signal gain is a maximum at the particular C and QC involved and that value of b for which the saturation gain is a maximum, and (2) at QC = 0.125 and 0.25, only for the b that gives maximum small-signal gain. Note that only electrons whose initial phases are within the interval 0 to 2t need be considered, since electrons of the same phase or phases which differ by an integral multiple of 2it are acted upon by the same fields and hence move in the same way. Instead of the curves beginning at the origin as is customary for periodic functions, the data have been plotted in the computed sequence. A feature of this system is that a particular electron may be followed through the tube simply by selecting a value of 0o and remaining on that vertical line. The parameter is taken as distance from the input, and for each value of y the r-f signal level on the helix is given in db referenced to the saturation level. Negative values of db indicate power levels below saturation, while positive values indicate that saturation has been passed and the power has decreased by the amount given. The phenomenon of electron overtaking is depicted in these 0 vs. 00 curves by regions of zero or negative slope. When this occurs it is evident that o0 is a multi-valued function of 0. In the space-charge cases, some curves had to be faired through the indicated points. The deviations of these points from the smooth curve are due to the inaccuracy involved in evaluating the space-charge weighting function F(I-0') as indicated in Section 3.5.1.

105 z (6o aU L) 0) P' - N 0) N2. o i 0 o N o i- - 0 1-Z — z WQ~~~~~~~~~~~~ 0UJ x x, z CM) z - < I > \ x o I A o do _z __ -, \ u

lo6 106 -- - -- -- -- -- -- - r g - I 1 - 1 r 1 1 1 1 1 1 1 —------- w > w > U J y I i r I I a: In z,. o w w I o —-, I -- 0 U. w ~ c z,, u3 tl- 3 0 oz --- W c i,7~~~~~~~~~~~~~~~~~~~~~C W~~~~ ~ ~~~~- )0 o Uw S I I - - - - - - - - - - - - - - a z > < ( Q I- c -2^ —s —---— ~ —-— ~~~~~~~~~~~i- - -I — - -— ^ - _ b <1[ ( Q)- - -.- -- -^- - -- -i- -- -. —~ — _ ~. ~'... IN,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t. __o___ _ o o o 2 A -^zi^ -^ ^^, ^ —- I ^II I SQ I c N I W o, —- \ -- -- -- --. -- --- -, — --- — v \ - \~ Z o <: o -w _ _^ -0^8- -O#-A^-_ 3SV-d - _O33-13- - x^^ ^ N _ _ -, _ --- -1 i' -- -._' - _ _ - \ "' _ s - _ = -— _ _ _ _I_ S__ 1 " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. _ _ _j \' _ _ _ _ _ \ _ _ _ _ _ _^ _ _ _, _ _ _ _ v _ _ _ olrj k t t w~t c ~lIT.,, N ^ A ZZZZZ^ZZZTZ'ZSZZISZZ:v~ZZZ_' —--- - - - -^ - ^ -- Y ~~~~~ —- \ — - ^ - - _^ __ I _ _ ^ - _ _S <D~ ^'^0~<0X \,, sNvlovu'(~,e(x),'SVHd N~aZO~'~

107 w 3: o3 I-. ~ n- t II I 0 X 2 IC d_ —-— _ W zo - 0 0 w w _ w i? < " ^ ~ /- 0 U. I -~,, z,,,- ----- -,oa —t —t- 0~~~~~~~~~ ~~~~~ U)CYo -- S i -s i -^ - -- -5S - - - - - I —:f > a.oI < I-~ ~~- I; ~ —-/-~~"~~~~~~ I ~ ~~a. w CY CD t6 I <V \_ -r^S % (.... I I ~,, d I z E L_ of-tI q " 1 o'. ^A ^ ^ - S- - O_____ - - - -iio O (u=_4 O -= cU O CU=5 0 el) ---— NIOVB,,'3SVt- - -- NOB z N,, - - - -I _L_ I-L I r r - tlzizzzr~zzzzz —-- -^^^^ — _ _ _ _ - _ K _ V4 - - W p~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ II ==m~~~m~z~~m^==-== —^^^J: —^~~~~ ---------------------------— s —---------— l-^ —/ —-~c \ Ku \-r --— ~ _________________ JL\LSJLVS —-33~

108 w - -- - - - - - - - - - - 0 W o W 0 4 No.0 - 0 ~ Il - 0 k- - - 2 a. —--: c ID 6 ) cr( ) Vz o o' -- ---- - o,,.; ~~-::i i: ^^ ~ = - _ d W >. 4 q N " o. o " 3:: w — l: ^. _ L _ _ _ - ^ - -- ^ _ -- _ N ^ - -- - ^ --. _ _ __OD- -- cn a >! - LI " _" " " " ". 3::.._J "r' ""a O z _\ > 0 SNvia V'(_ 0 )'3SVHd N_ O _33-13

log I 109 a o _ < =__ ___ ___ I-. I. u Q - o n, O O0 X o o z F U I I 1\1 1 \ 1\1 _ _ - I _ I I U.I I I I I _.. "o a:- w I I? C!- - - - - -- - ^ Q - | u I.-I —- -- -._:: X o - I. — - _" - LIJ I II D T- T \ l M. / cfj ad -'i OD _ 0 U) z tS-X 0 W I I- I I I _ I ___ — t N o --- / o F- X(~0: ) 3 N 3 \ x' L(U —N IOyAI.I I \ ~~~J cr~~~\ \ \\~~~s SN~lC)VU,(o,~,(),'3SVHd NOU.L03-13

110 ^~~~~~ " ~~~~~~~~~~~~~~~~> CD < z - ~ - - - - - - + - - j ~ - -- -; < i ^ 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~j Cf ~~~~~~~~~~~~~~~~~~~LL W i () W C ~~~~~~,I~~~~~~~~~r P-,, cr~ ~ ~ Er -- nn-" 0 C n o <::, u n - - " qa. o o ~ ~~~~~~~~~~~~~~~~~~W W 0 o __lsY _^^ssss —-S -^ ----- - ^ —S ---- -— S ---- -- 2: - 6 z |, - z ~~~~~~~L~~~~~u. O~~~~~C o,~ o n J ry S; N, \ < \ \ \ \ LJ > ~~~~~~~~~~~..5 ~~~~~~~~~~~~~~ ~,n~~,~ oL o n, CDcl ( a w c -i d ~ - n _ _ _ _ ^^ S ^ -^ - N N u I~~~~~~~~~~~~~~~~~~~~~~~~~~z F I~. ~ ~ ~ ~ ~ I~ Z C/i w >? W I N I__ AmIr - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~u,,,OD ODO D D ~ SNVI;,V8';.'A):-'3S"Hd N..133 3 =ZZ=_ ---- ---- __ ---- I-^-^ ^ ---- — ^ — S — ---- ^~~ ~ ~~~~ — -- ~ _ _ _ - _ J _ _ " _ - ^ — _ --- - VII Z II - -- t ^S -^ - - -— LN - ___ _ __ _ _ ___ \X v' - ^ - ^ /\ - - __ -- -- -- -- -L N^ N -- S -- \_____\ __^ --- _^ -_^~~~~~~~~~~~~~~~~~~~~~~~~~~~1, SNYlaVU,(oj~')^,~ 3SVHd NOyiJ.3139

111 W~~~~~~~~~~~~~~~~~~ (3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~' > oC <z n' W W c~O. 0 U Iu I --- " I cc 2 D 10 - - - - - - - - - - - - - - - - i- i - < <M 0az ow z w -- - - - - - - -- 0 - - - - W - v_, 3 Ir' U*o q ^5 ^- 0( n'. ~.5 T T ~ w _ _S __ _ _t^ _ _ _ _ _ _ _ _ _ __. < X o - - - - - ^ - () l i 2 o _ _ " _ _ " _ _ _ to* _ _ _ _ _ _ _ _ x 6 0 Z UW Z I, W O i "n "w C 1= 0=D' [L Z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Li O~ q' ~n,n ~, Oj ~ ~~,n,~ o~ dL =3=s==== ^=^^ =========== ___________ _- - __ In 0 I I I --- -^'-..S^. *> —— 0 —si0 Wn W, C~ ========_=== —-~ - — ^ ~-; o Zz,==zi===_ — _o-q t_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __i i i i i _S _ Vi i - - _ _cl I- _ _ \_ \ _________________ 7 7~~ ~~\, \! I l 1,~~~~~~~~~~~~u \~~~~~~ Uj~~~~~~~ )\ -~~~~~ ~F I \ ( I\ \ \ I I r r r I r I I I I I I r 1 r a (~ ^ CM 0 0 ~ ~ ~ ~ - 0 <0 q' o, 0 SNVIOYB,('to,0) fS'3SVHd NOl0.3393

112 -------------- _ --- _ _ _ --- _ _ _ _. - _: " = t z W Z - _ - -,a. _: " "! Ii w o I I 1)C w Xo SitldVS',.'*. - c < 0.I - I I ii i ^l T 7 I --- l - I I — X,, II I I - 5U -1- 1 1-1 1 -1 ---— 1-V-r-^i-^^^^=^^ ^ I~ll111 111 EEEEEE^EEEEEEEEEEEEEEE^SSEEEES~~~~~~~~~z (O^-NQO~~~~~~~~~~~~~tD^CM~~~~~~~ 7 SNVIOVU'^ A~~~~~~~~~~aSV ~ ~ ~cd N(1l3 1

113 4.4 Normalized Electron Velocity Deviation 4.4.1 Velocity Deviation vs. Initial Phase. The total velocity of an electron as a function of position is given by the expression Ut(yo0) = Uo[l + 2Cu(y,0)] (2-14) In this equation it can be seen that the quantity 2Cu(y,00) is the normalized fractional velocity deviation from uo for a particular electron. This velocity deviation term is plotted against the initial phase Po in Figs. 4-24 through 4-31 for the same values of b as those for which the electronphase curves are presented in Section 4.3. The parameter is again distance measured from the input, and the signal level for each y is given in db relative to the saturation level. In these curves the maximum percent velocity deviation increases as saturation is approached and has its largest value there. The maximum percent deviation at saturation increases with increasing C. With b adjusted for maximum saturation gain, C values of 0.05, 0.1, and 0.2 correspond to maximum saturation velocity deviations of 24, 50, and 86 percent respectively at QC = 0. At QC = 0.125 and 0.25 the maximum saturation velocity deviations are 55 and 40 percent respectively. For still larger QC values the maximum percent velocity deviation at saturation should continue to decrease. For the cases in which QC / 0 the results were somewhat inaccurate, especially in regions where overtaking occurs, due to errors in evaluating the space-charge forces as discussed in Section 3.5.1. When this occurs the computed points are indicated on the graphs and a curve faired through them. As pointed out earlier, the dependent variables A(y) and @(y) are relatively insensitive to errors in the variables 0(y,o0) and u(y,0o).

114 _ z 0 0 C - --- ------ ---— W I —,, I - > w 6 _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ - -- 0 W-. - -___ _8 oW _ _ < o - Q Qa ~ 0s. N > r L o - - -- -- --- — 0 -.I: - _ _ _ _ _ -. 0 40 L~- z - I 6 ( - ) a., H w -N ~ - I ofc c. 0 0- " I Iz: 6_ _ _ _ _ _ _ _ _ _I I (~~~ ~ W'" Z NoV~OA10"3 3IIMO

115 w too z - a W QwP~~~~~~~~~~ L W to ~ o,,, o _ _- _ N_w -Z,-, ~ w ----'-?'"-,- -- - -- - - -- - - O 0 0 N' <2 S - -- > w0 o ---- 0 > ir -i Ci OD~ - l~ fP -- - -^ -- - ^ -- -- w to O a ----- )r) 0 ~ 0 ) (L X U J to a ~ III Ia 0 0 — =Z==m^=IIyL=^^Z===ZZI==Z=Z==m m _ _ _ _ _ _ _ _ _ ^^ ".^ _ _ _ 0/~'_ X ) no'IAt IO a I = _ = I. -. -_z - -_ _ -_ - ^ ^^ ^^^ ^ === ____-______t/ ________q - - ___ - ^ - - a. _ _ _ _ _ _ _ _ _ _ _ _ _ - -^ - - - - -- - - - - -4 - - - - -- - - - — ^ - -- ---- - - ---— ^E —-Z —==zzz~==== arzz -z~ m z'zzzz ~ z" _ _ _ _ _ _ _ j i _ _ _ _ _ _ _ _ ^ _ S _ =z==^Z= S Z^===_==_= ~ ~ ^. IN / / 7 ~~~~~~~~~~~~~~~~~~~~~IN8 _ _ _ _ _ _ __ \ JCT: _Z _ _ _ _ _ _ _^ _ _ _ _ _ _ _Z _ _ _ IZ -- _- - -- AJLLZ - -- -- ^_ H D^^_^~lI~~~~~~~~~~~~~~~~~~~~~~~~~c ~ ~ 8 0 ~ I? I s~~~~~~~~~~~~ (o^<A~n^^N011VIA~aA1DO~13A 3ZI~lV~dO

116 cr.~~~~~~Q rc o Cc c Iml; a: w O ~~~~~~~~~~~~~~~~~~~~n o~~: 1c W. o..) 0. -, 00 J o ~~b I~W >qW)of _ _ _ _ _ _ _ _ _ C g_ _: 2 S 0 2~,'o rr 03.. t OD OU C 0 i, V~~'a'h- W: X I - - - -- - - - -- ~ ~~ ~ z- -...... - - o m- m, a ~ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ - q y _ _ - _ _ _ o co,,,q Cy \]SO>CO Q:. -o. ou _____ —____-I ~~~ ~ ~~16 ^ ^ _J ~_ _~ - -U. ~~~~~~~~~~~~~~~~~~~~~~............. J I~~~~~~~~~~~~~~~~~t - -- - - - - - - - - <t >,,^ o -=.,- - - - - —. -. a, C.. cu~~ ~ u N _,, ==zmzz=^^:t==,^-"===^^==~//, / ^^ ^ m ^ ^. I — o (~~~~? > I./' ~ ~ I ^^^-^"^1/ / / [,4// _ _ _ _ _ _, L ( 11, /'N V A a3ZIoV ON ci o; s o ~~III,/', O d d d d/ /mix ~ /~1'~llh0~103 3ltWO

117 0 n)- W zj wzE Z 0 U W o OD 0Z I Q < <o t - o I-' -:,__ _ _ _ _ > z'l- M /) 0 - w I~ _ _ -- --- --- -- -- "o, o; --— __ --- w <~ z, -r X " ~,. o' <f g I > 0 E 1 ~ Ui d I. _8 0 I? | s I 8 i~ l o 0~'' o o z W~~~~~~~~~~~ - N (3 o I —^ a - - - - T - -'.o? -- - 0 - - -m- - 0 - - ^ - - - - - - - - - - ^ - - - - - D - - - - - - u.tid 0~ C oi o ^ ~~,,,1 -,. - -g 1 - N- -- -- - -- -- - - - - -M —- - — M — - - - -- y ~ d o -Tro-~ — o. - - - -o - ^ - - -. I > 0 z __ \ / l _ _ - __ _ ^:_ _ _ _S — g~1'* (/) I L —^-I - _ _ _ - 1 _ _ _ _ _ _ _ I i i _ _ _ __ _ I I _ _II I Ii_ _ I i_ _l-Jzz _ _ _l_ l z_ _ _ _ _ _ _ _ _ _ I, / I, IZ7I_ _ _ Z _ _ z _ _ _ _ I_ _ _ " I ^ ~, /'I I I I~ l l I, /'d, // / 7 cy -.~ 0, "; cy *1 ^ ~> ~ ci0~~~~- 0 - odo q (~/')noZ N I I A (O,(3"X) no 3'NOII~~~lA3Q XllD~q3A Q3Z~~q~lNBO N

118 z W z w - jz:~ ___ I < z _ CDo,T () W d ~- "j ~U L ool >u > I Z ""0 1 — -' — " ~z - >' z I- -- -- -- -- -- -- -- -- — 0. 0 0 6 o ( o-L- — (_ q: - o- u - _.'1 --:- - 0 - 0 U, ~ZZZZZ'^IZZZ JZZZ'S^^ZZIZZI^/ ~/~~ \* __^^===|~p=~^=====/z =====^=^ —^^^=/F===/== /

119 z _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ___ _5 _ --- -- -- _ _I > o -- -- -- - -- -- -- - -- -- -- - -- -- -- - -- -- -- - _ __ — _ t I- llJ ~ ~ C 0 oQ T " oz 3 W, D l1 > U (- ~ — 0'__-_ -- -- - - M n z' 0 0 < X lO - - - ^ li-jslj L -.~; - - - = ~ ~!?-:~ n -o a. o,0 -- _J -r o *" ^ *" I i^ T I < 0 -l, l:- C U) z.^. ^^J -^ -- -- -- — 0 -ji < Z-!_[-!! —!! t t! /- A l Izzz — I

120 I r c5 I 0 W Q if)~~~ 0: w I z < 0 U- - M < ~- Q c' 0 - - -- -- -- - -- -S - I 1 -- ~ - Ia I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ --- -- -- -- -—. <U:. — -' W z~~~~~~~~~ I -.. - r " I - ~~~o o LUL F o U~~~~~~~~~~~~~~~~~~~~~~~.;. zz - i^tP C:!!!:CD < 0 - (O n t c c U) 7 ) m3 c ~ U)N M OO ) I ci to U) W ~~~~~~~~~~~~~~~~U~~~1. o^ ^ io i * < o <~~~~(1 cnC/ ~N - ----- ----- ^ ~~-I — -I --- - - 0 ^ - --- - --- Ld < cli~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i. OCM O O (D < I - ~~~~~ I a~~~~~~~~U)wa w~~ CfI > o - - - - > >- T - -- - - - - -Q i i --- i ~~~~~~~ n ^ _ _ __') 11' N O I _ _ _ _ _ _ _.11 01^ Q'!II~^O -- - - - - - --- =^ w ^=============^ z 4 ^ ^zz^^.zzmzzrrmirzzzzrzzzzz5" _=======^._^==^ ================~~~~~ ^zzzz^-^.^z^^ ^^zzzzzzzzzzzz-~~~~~~ z~~~zzzzzzife~~~~~~~~~~zzzz~~~~~~zz^-rm~~~~~~~z^^z~~c __ __ __ __ __ __ __ _ __ - If - - / - -- -- - -- -- -- -- -- -- ^ __ __ __ __ - ^ ^ - _ __ __ __ __ __ __ __ _ __ __ __ fc = | c v m zz~~~~~iz~~~~lizzzzm zzzm -^^^zzzzm zzr~~~~~~~~~~~~~~~~~~0 (Old'TIQ Z'N011VIA3a A1130-13A a3ZI~IVWdON

121 - - - - - - - - - - - - - - - - - - _ _W W - _ _ _ _ _< _ z > < ( o Z ~~~W 0 L I) CJ )I~~~~ (. n,- 2 I - o -----— t-!-^ —— s —--— = _ $x o aw In~~~ W cu z O~~~~~ c~~~~~~~~~~~~~~~~~, I W N F — W d --- ----- ----- -----? -, -S o ---- ---- ---- --- --- -— ~~ — ~ -'. I I~~~~I 0 - - ^^^^^^^^^^t'*''" 0 o -- - _^ ^ z ^I j^- - - -- - - ^ E-~ lu 6 _ _ _ _ _ _ _ _ _ _ cr _ _ _ _ _ _ _ in o' <.I Od U, LL~~~~~~~~~~~~~~~~~~~~~~~~L 0~~~ d d I I c 6 - -- - - - - - - - - U) II0 C w X X O -- - - - -- - - N 0- I W 6 r X Di z > z Wai z Mil (~^') "32'N011VIA~a A113003A 03ZI~~IV~dO

122 z _____ o O> o -L > -< I' — <I: I I Sn P e rcj "o _ ____ ___o, __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ jt ) Z' _ z -r o', LjJ p L JIJc or o f o - - _ _ _ _ - _ —_d~~~~~ -___ -_ ^^ - - - - - - - - - - - - - - - - ^d o~ __ _ __ _ __ __ _ __ _ __ __ __ __ _ __ __ __ _ __ __ __ __ _ __ __ __ _ __ _ __ _ __ _ __ - ^ - _ __ _ __ _ __ _ __ _ __ _ __ _ __ _ __ _ __ _ ____._______________,~.^ —^ ^ <c - ---- ^ - ^ - - - --- --- --- --- --- --- --- -- - - -- ^ $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,' Q-~~~

123 z > Z 2 Z _Z__ _ I I I I I L 0 I- 0 --- -- i I — T ~~z>- 2z 2 w ] j __a_._._ __ __ ______________ zz W d o dn o0 ^ LJ ( n. c - 0 -- O 0 ri 0.. -—: ILd <I: - " N z W d,

124 4.4.2 Velocity Deviation vs. Electron Phase. The normalized electron velocity deviation may also be plotted against 0(y,0o), the electron phase, to give the electron velocity distribution vs. arrival phase at a particular position y along the tube. These curves are presented as Figs. 4-32 through 4-59. Regions of electron overtaking are characterized by vertical tangents to the velocity curves, since a vertical tangent can first be drawn just as overtaking occurs, and beyond that point the velocity is a multi-valued function of the electron phase 0(y,0o). In all cases, the electron velocity distribution is essentially sinusoidal with 0 near the input, but as y increases more and more electrons are decelerated and the velocity curve twists around the point 0 = AE further and further as saturation is neared. The electrons followed through the interaction region are indicated by circles on the curves so as to give an idea of the percentage of the total number of electrons that are decelerated or accelerated at any point along the tube. The corresponding drop in the average velocity of the stream may be obtained by performing a Fourier analysis on the velocity-distribution curve to determine its average value and subtracting this quantity from uo to get the final d-c velocity value. Clear uo decreases with distance, since when the electrons are in a retarding field they give up energy to the r-f wave on the helix. As with the space-charge solutions of Sections 4.3 and 4.4.1, the spacecharge curves of this group do not vary as smoothly as those for QC = 0 due to the inaccuracies introduced in evaluating the space-charge weighting function F(0-0') as pointed out in Section 3.5.1. 4.5 Electron Flight-Line Diagrams One of the most interesting and useful graphs that can be plotted from the data obtained for a particular set of conditions is an electron

125 z o 0 z (D ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~( o I __ _ _ ___ _ _ _ _ I __________ III rI II >__ dz W U Y-.0 z 1E 0W Q - III >s 0 U Q - - >- Z 2d i = I - < < 1 CM Ff5 - U - -- --- --- - -- - -- ^ -- --- --- -;t z L i- i _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I 8 _ _ _ _ _ _ _ _ _ _' - - o 1. w W z LL C) E n," c - - -- - ^ - -- - ___ __ __ L CLL_ 0 ^ ^ ^ _ ~~~~~~~~~~~~~~~0 03,, ~n,,. a O ~~~~~~~~o b.I o I UO "o -~ ^ ( > - <r 0 IN- a -- Z o - f) 0 ~ ~~~~~~ c F- t~~~~~~~~~o ~. W I n 0 o'~_X _ _ _ _ _ _ _- - Er i _ _ s ^ _ _ \ _ __ _ -, ^ r h a~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a W ~~~~~~~~~~~~~~~~~_.-T____-o-F-^ j-^ ^ —- - - - - - - - - -Z.^ - - -K i~~~~~~~~~~~~~~ -rr- ^ ^ ^"^ ^ s~cr ~,,_ f__ ~_~ ^ _ s^ V w00o d~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~ -o:~ ________ o __ — -- o 0 o I_ - - - - - - — ^. —- - - - B --- - -- ~ - -~ - -\ - ^- - - - - ^ -- - z' ^^^ ^^ ^ -, -^ ^I - - Li / ^ v[{~~,B N~'~/i ~/f~ —\ - -,, - J zo o dcu' / x A \ t _' _ _ _ _ _ _ 1 ~~0 0o 0 0to 0 to~ 0 ~3 (~|Z'X)n3 ~'NOI1VlA30 AX1IOO3A a3ZI9)VI~aON

126 z cD O j z===== 1 = = == = ==== ==r^ ==== 1== 0 z 0 OD_~~~~~~~~~~ 0 a 0- 0 — I W - - z > | I S - -- -- --- -- --- - - -- --. s" Z 0 0 C.Dd,. u- SE -- ~ OD < 0 w Io w O OP o ao. o rr>~~~~~~~~~~~~~~~~~~~~~~( I-(y)a ~ I- - 0 0 X I ram 2 I I l- _ Z > o me K'~ o ~n ~ Qd o b CW o - ~.. N I~~~~~~~ --'ri,, & "^ \ \ ~":"':"' ___ -? ^^- ~ -^^^ —-- _ _ _ __ _ _ _ _ _ _ _ _ __ ~-...- ~- I\ " _ _ _ _=~ ~_ ^ - - _ _ _ _ _ I~ ~ ~.m.Io> ~ ~ 0 -^-^^ ~s^' <~ ~' d-'"' S,, i U) ~ ~ ~ ~ ~ _ >o 7n n' ^^." o o Z > X -~.. —,-,.,,,. ) - -- A ^ -] - J - - o -. - " —L', r~ a b\ N u,~~~~~~~~~~~~~b N~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I _I ~~ I/' o ~ ~ ~ ~~ ~ 0 il 0 (~~(*~)n3~'NOIll.IA3a A.110"13A G3zr'1~UON

127 0 Z( 00 O - o 0-! N. W _ _ 0 -w -- * 0 <^ (K ~ --------------------— ^ —------,I z 0o s 8U o.V -- - l —hJ- ui -- w ~ I I o- z W N I CY + - U- 0 —----- 1 -- 0 0 1N I <-l O - " ~ a lli z d "cN ci CM W I w I O_ _ _,. - -| — I I_- o _o _ C WI- 0z, w o_. o - - - r- - - - - - --- 0 ____ A.w N W~ ~ ~ ~ ~~~~ 0S 01 z > I1\ ___ —1 __ --- s;(P aO (D g>O (~~X) no )'N01OVIA30 A.110013A 03ZI'IVION

128 oo;i IX cr I W F T L _F - _- _ - _ _ - - _ _ - <-_) -_ - H z ~< 0 w I Z _ o. Z. I - 0 7 A I.cre. > _ - _ - _ - - - - _ _ _ _ _ _ _- - - _ - - - _ IH- -n,_ _ oN j xCM T-i I — I 1 I 1 I'I 1- H- Q: -. O - - - - -^ sa.W Z 0 Ii N 0 09~~~ 0 - 0 - I' I - I-.'NOI. I to 0U O - ~ — o CO, I I I 7? —? —' —-W-Z —--. o.^h Idoo'k "'zzizzm

129 Z (.0 F- o 0 - D Uu Io >? ^ "_ ---- ---- <[a0.0 0 _ ~ o- ~________ Zw w n H ) cr _ 4" 0 -- - 2 0 ~ o o N oN < ia (",Z'A ) fl3'NOLLVIAQG AJOO13A G3ZI1Vk z w ^sl -- o o o o o o'" j "7 7~ "^ ol (0___n-OI.1A.I: 2 1'iON _ _ _ _ _'S S - 3 _ - _ ^- - _ _ _' _ _ _ _ _________^ ^ -B —-- S^ _____ ^ ___ -- __^~~~~~~~' _ _ _ _ _ _ _ _ _ _..^.^.l-^ _ _ _ _ _ _ _ _ ^ _ _ _ ^;" _ _ _ _ _ ___ _ _ _ _ _ _~~~~~~~~~~~CI ~ - - - - ^iS ====== - ^E - - = - -~~~~~~~~ z z - - -z - - - - ^ @ - -^ ^ - - -_ -^- _ _ _ _ _ _ _ _ _ _- _ _ - _ _ _ SSS'^Sit~~~~~~~~~~~~go0~~~ 6 6 d d d d d i~~~~~~~~~~~~ (^'^ "O Z'N011VIA30 A110013A G3ZI"1V~~~~~~~~~dON

150 z (o 0 z ~~~~~~~~~~0 n.' o ~ _j3 - - -- -- -— d —.9> w (K P W W CD - Z d 0 a. -n _J O r z F - U. z _ 0. U. -- -- -I i I --- - - - c- o a o I z ~~~~~~~~~~~~~~~~~~~~'I U _O I U < ~ _ _ - _- -- — \ —- — gj — - --- -- C w O OX.. ~_~ ~ ~_~_~_~ I. Q N1 Z S v! <~, " 0 ^ y ^:| ^_:_^ _./i', —., I,! \1^ —N I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I Z~~ ~I O i~i c " I~IilllQ i 1I / / ^!^ -^^R-f-^X^ —----- x, O^ i l II b lu Plii\_ V _ _s^*^ _ _ _ _ _ _ _ _ I g ~N cr_, ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~~~I W / // / J t \. _ l.-* \/ / - -_- V\\ \'~==y,^7B==========-==-===Aa===a* ===== _gg::m===========^/ to ~ ~ ~ ~t o ~u t. (R. a, q <0 0 0 M 0 0 M t 0 00 T o o d o o d d (~,') no 2'N011VIA3a A11.00O3A 03ZI9~VNBON

131 z D 0 Z J: U. -r-_-,, w W o --- /'-' z: _ t U~ <D 3 /0 Q < z CO.LJ "I, w z ii r) > W U ^- Z ^ H-0 — X -! --.0 T LI. H () - --- -- -— o- - -- a- 0 to:~~~~ 0 Q O 1 0 4~~~~~- ~c" O.c> N X z 6 CMN N |IT I............. rs A f Q ZZZZZ^^J ^Z^I sZZ.,^1-' - - - ^iw a__ _ -- _ v - _ _ _ zzzzz~z ^z-zzm ~ z~z^, ~^~-^zH^:====z~~~~~~~~~~~~~~~~~rr^_^r=$4^^^~~~~~~~~~~a / / i I,1^s > _ - -z- - -E -- - - - /^ - ^ - - L "4 - " - 7_^ 7 X4 4- s —^ AT_~~~~~~~ -- -- - -- \. - L -\ - -- - -- - - - - - - - - - - - - I - _-O- - - - - ^ - - _ - - - -; o _ - - _^-X ^ ~~ - __ _ _V _ _3 L __ \ _ _ ^ ^ = r ^ ^ _ _ --- _45^\~ ~~~~cu -^3.- -- _ ^ _^ -- _ - ^'- - - _ z~zzz^^S ziz^ ^^zrz z _ - _ _ - - _^ S ^ S y - - - - - - - — I I I I I \ I I I _ _ _ _ _ _ _ _ _ _ _ _ ^ ^I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I I - - - - - - - - - - - ~ - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I I I ""Ngo.Nio~~~~~~~~~~~~~~~~r~1 r I I I R In 00~~~,, 0u o 0 0 I 1 I I 1 r (~^'<)"3 ~ ~ ~ ~ ~ ~ n 1'01I3 A13"3A0Z"1HO

132 z 0 -I- 4 r o z W 1) cr N 0 0 W 6 >.. o d zF-~~~~~~~~~z O*~ > t:I - _ _ _ - -- - -- -- ";- - -- u i xli i 8 tto r~~~~~~~o4 W I ^-: 0 C (o0,x)n3'NOIlVlIA~ A3.13013A G3ZIgVWaON

133 z O 0 z _ o W W > Q o w 0 w _ ZW o 3 o~ ~ ~o- - - - - - - - - -j a 6 a) LJ -.(7 W LJ "o ) > W o 0O < w 0 W:X w -J Q -,, d N ~ W O ~ 0 Z < W I - --- __ —-- _ < 0 W <IZ:: Z > - - -o — - - ^ ^ ^ -; _ _ _ _ -.~ ~ C o J - - - - - ~, —(N 2^~s - +~-* —? -,- -^t, -.< - - - - - - - - - -- _3Z _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - \ \ o zCzz^^ir -zzzzzmmizzzzm^.-i~zj I/l I I I [ I'~ -~~~~~~~~~ a. w % s %~~~~~~~~~~~~~~~~~~~~~~~~~~~% I| \'~. 0'~' O Jd^ -c.0 _ _ _ _ _ _~ ~ ~~~ ~ ~ ~ \ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _' ~ o o ~ u ~ ~ I I I Ic (O~'x)n0 E'NOIIVIA3a AIIOO93A a3ZI9V~BON

134 Z ( 0 Z UJ W LU Z W u O CI d > W -- Om0:~ l n <i L I < CM - 7- - - - - - - - - - - - - - U_ c o~ zoo W o < O~a 0 W 7 ( 1o > L i - ~ - - _ _- - - - ~_ - __Z_ _ _ _ ^~~ ~ ~~~~~~~~ 0 ii o X -' rf d r0- o _1 O. J. d __ 0- o + z cm~ ~ ~ ~~ ~ ~~~~~~~~~~~~~~~~~~~~~~c N y I- - d _ _ _ _' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~~~~~~~~~~~~~~~~~ ---- n ^ i i^._Z^m ~ rj -'ZZ^ZZII~a ~E~~~~~~~~~~~ II. z o0 ~~~~~~~~~~~~~~~~~~~~~~z~ X~~~~~~~~~~~~~~ _ -- -- -- / -- _ _ —--— S\ —--—:-...s - - / o ^ rr _ -- - ^ -- -- ------- ^ -- _ ----- ^ --- a^~~~~~~~~~~~~~~~nCC —I ao~ I I I \ _ -- ^ - -- - -3 Op L- 7s, 7 _ _ _ -- _ - _ -- — \ o j _ (~~'X~n3 E'N0ll1lA30 AII3093A O3Zn~gV~ON _ _ _ _ _ _\ -- - _ _ ^ s. _ _^ - - ^ -^ _ - -v _ - _z --- ^~~~~~~~~~~~~~~~~tt - - N ss ss^ ^ _ _ _ _~~~~~~~~~~~~~~ 00~ ~ 0 0 0 o~~~~~~~~~~~~~~~~~~~01 (^ cl NjlIl A 103AaZ~IlO

155 flight-line diagram, which is similar to the well-known "Applegate diagram" associated with klystrons. The corresponding flight-line diagrams for the traveling-wave amplifier are plots of distance along the tube vs. electron phase (i.e., arrival phase) with entrance phase as the parameter. This "space-phase" diagram affords an opportunity to follow the electron trajectories through the interaction region and see just how their motions are affected by conditions in the stream. Figures 4-40 through 4-49 are the electron flight-line diagrams for the cases studied. Each flight line represents 1 electron; 32 electrons were followed through the integrations, but only those which are particularly interesting have been plotted. On each figure the saturation level and point of first electron overtaking are noted. The point of first overtaking is clearly the point y at which the flight lines begin to cross. Hence the closeness of the curves indicates the degree of bunching in the stream. The entrance phase of an electron is given by oj = 32i J = 0,1,2... 52 (4-5) and hence the numbers identifying individual curves represent the initial phases in units of t/16 radians. Over the interval in which 0(y,0o) varies from 0 to A, the circuit electric potential gradient is positive; hence the electrons are accelerated in the direction of increasing y and they accept energy from the circuit. On the other hand, between t and 2i the circuit electric potential gradient is negative; accordingly the electrons are decelerated in the direction of increasing y and they are delivering energy to the circuit. Initially and for some distance the average velocity of the electrons is equal to or greater than the phase velocity of the wave and hence the electrons are seen

136 CM - - - - 5 J w Z _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ 2 ~~~~~~~~~~~~~~~~~~~~~~~~~~- OD* W W cn cn 0- 03 Z 03 __________ ~~,, z 13::o - - - - - - - - - - 0 I D n,' o N 0 I 0 (~. 5 C w W. oz - _cli LL > L x 5: t Z - -6-' ~~~~~~~~~~~~~~~~~~~~~~~~- ze zII I I ~ ~ ~ ~ LI WL. _"j. — 4 N L CJ N ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C I~~~~~~~~~~~~~~~~~~~ or - O w w~~~ O -:. \, W^ - 7_ _ _ _ _ _ _ _ _ _ _ _______________ Cm: IJ I I s, > w IN OJ~~~~~~~~~~~ Z W cr ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ a _;- -I' ~'1 -~ n r~'~L-L_ i I I I 1I ~ I~ ~ l 1~~ ZZIZ^ ^ -^ -^ -^^ - - ^ -^ -- - _ - -- — ~~~~~n'n. - > Z zw U-r....? I - I _____^^.-___-^ —-— ^^,-^^___ o O ~ t ~ ~ tO"O^"

137 ( I z W W iI jIll Ill a. - - o?, -,., -. z - 0 0: o a, - - - - - 3: C - w 0 _j — - - - -to _ N= == _ _ I z ===_^5 = =^ =^ = =^ -= =^ = ==== = = = = = -^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ - - X - o - - ^-I - -- - - - - - - - - - - - - - - - - - - - - - - - - - - h - ^ ^ 0 ^^^-tiisz^-s'iiij i^^'miiziziizmzz. Wii, s, \ ~. %.^^ ^^ L- I W1 U)IN,~~ -- U --- — I:'' -IN - - — <- j _ - _ - _ 1 _ _ _ _ __~ 1 ^ _ ^. ^ _N\,% - - - - - - - _ - - _ - _ - - _ - _ - - -.. - ^ _ - I I 1 I I I I I 1 I I I 1 I I 1 \ ^1 ~ _ _ _ _ _ i _ _ _ _ _ _ _ _ _ _ _ _ _ ^ _ s I - _ S 5 ^

138 ~ 0 a.~~ I tt 0 w w w i ------------------— ~ ~ ~ ~ ~ ~~~~~~~~~~~~-; i4 > W,, - -J w or IC) b_ -1- -J zI, - 0 a: 0 O- 6 W Z,~' ^z~~~~~zzzzzzzzzzzzzzzzzzzz m izziz~~~~~~~~~~~~~~~~~~~~~~~~~LL C1 t ^ < -- - - -- - - -- - Z I'^ -J 0~a I'. ---------------- -- ------------- I r, I i = ^ E = = E = == == == = = == = = = —-— ~~~~~~~~~~~~~~~~~~~~~~C I ri I - LL >d I 5 I x _ -- _ _ _ a^ 4, —--------------- < ^ > CD - Z z Zwt 0 C OC\,W ili zzzzzziS^'^^Ssazzzzzozzzzzzz^ o -a _ _ _ _ _ _ _ _ _ _ _I \ \ ^ _ _ _ _ _ _ I I I I I I< a -- -- ~ — "Y' - ^ ^ ^ ---- _ _ _ -- -- _ I I I I I I I I L _ —^ —^v-^-^-^^^^- --- --- i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o/- _ _ _ _ _. /..-_.-....._ _-_ _ -- _ - - - - _i- _ --- - _'_- - - _ -- -' — - - - ^ r oo h- <o 10 <* M) CM.-.0 ~ >s~~~~~~~~~~~~~~~~~~~~~~~~~~~~O N~~~~~~~~~~~~~~~~~~~~~~~~~ o o Q', ~ ~ ~'- — I%)~~~a, (D ~ ~ ~ ~ 0 I'- Z)i 13O w ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~( I I I' ~ ~t~~lI I Ir~-~TinItcr`t' IDz~ 0 i~ ~ 1 0:>wrrl I I I I I I- I I T~T~-~T-~

139 0 No _____ "71 \ ~VW N V I\ N Y R; I II I jlZ I \I I I I I I I~~~~~~ ~~ I N I z:_t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Q w w \\N IN N N\ CD~~~o W Cf) w~~~~~~~~~~~~~~~~~~~~~~~~~~~~~nC z3 w z w __L > < __^^-_ \~^ X ^^ N^ - - - &Ni w Z LLJ -_ _ __ w_ _ _ _ | S ^ ~ ^ ^_ _ _ ^ cr uoQ- - -S$ ^^~ S-^ - st (3 [3 ^l^^ ^l^l ^^' C-) Lr) W j I o (E (3a^: -: -- - -- - - - ~ S ^ - ^ - I; I!3 LL ~ - - - - - - -- - - ^iS w TL I ^" > z ar^^^^^^^^^^^^^^'ls^^e^ <t ~ ~ z u $ o _ _. _ _ _ _ _ __ - t S S A z w w _ L _ - - _ _ _ _ _ - S ^ V \ O x ^ _ _ - _ _ _ _ - _ ^ S 5 s o Z I L ^^ _ _ _ - - - _ _ _ _t 7 ^J u~- ~- _ _c —-- - -- IJ Q o z- o- --- - - ^ "jl a- <o c o o <o ^- cJ o ~ >v

l4o W -- C4 < 0 0 z 2 - W W <[ w a'r d cn ~~~~~n Ir, ~ ~ c o~wo d -E =============== ======= ======= ======= O Q 0 ~ -- a. ~~~~~ua sf " LJ I- - o _ _ _ _ _ ^ I u S3 o wO W o3 ~. o W~~~~~ - ---- - - - - - ---- - - -- - -- - - -- - - - - - -: x < ~ Wz ~ ~ ~ Z~ ID —---- -- _ -- -- _ _ I -' n T W ~ ~ ~ ~ ~ ~ ~ - Z> 2== = === = =:W o > 0 " oll 1 4 - ---- Z Iv -J 0 W r ~o ==^^==^c^^m =^ u - ~.~~~~~~~~~~~~~~~~~~~~~~~, ___ I \ _\\ *-.-. ~~ \I'" (\)Y S. < -- ~^ ~- -- -- ^^ ^ ""^^ -- -- -- -- -- -- -- -- -- -- -- -- -- --, ^~..... s~ ^, ^z *~^, ^'-.. -; —. /L//i o ^^ ^^ ^-_ _ _ _ _ _~~~ ~ ~ ~~~ ~~~~~~~~_ --- - ^ ^ ^ _ l{~ _ _'-.. *^ _ _ _ _ ^ _ _- ^ ^ ~~~~~I-,/,-~ _o,. ia > ~,~~~~~~~~ 00 ^ <0 10^ 10 CM ~ ~ ~~~~~~~~~~~~~~~~~~- 0 3~

141 0 W r 0 Hz - cr~~~~r W - a-. 0 ~~~~ ~~~~~~~ cr a IC)~~~~~~C 0 z o wL1 a. N o Q L~~~~~~~~~~~~~~~~~~~~< F W 0 U.) E C): ~ ~ ~ - - - ( - 4 - - - - - _ _ _ _ _ __ _ I l l I " " ~ Q < i ~ W H- Is ~~j. - - 1 -^ W-= = = = = =: - - - - y Q UJ?! 4^^~~~~~~~~~~~~~~~~~~~~~~~~~ 4 j 0 ~_j I ~ cr LL I> w X~q - - - - ^J5^,^.,,],^'^ ^ ^;;^ - - ~ ~ - - - - - - -- - -- 3 uJ~ ~~ a- cy 2 N -1 I 2 >LL m - -- Z W < 0 0 X H z w z 0Q 0~~~~~~ N~~ 0 w~~~~~ z 0 aD~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i ID~~~~~ cr. w N Ci, I(: z cr0~~~~~~~~~~~~~~~~~~~~~~~~~~~~c w ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -J ( NZ- 0. LL I 1 i I I I I I I \ I I I L I r 0 X I I\ \ I I I I' zt

142 0 I I — Z _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _W ^ l Q 7 tSX ii _t L_ _- < = X u - o i I _ I l ~ I I Iz 1 1 ---- --- -- " —' 0 - - (D e * e N^[ - O CjI o z -,o < U, <-/i N..\ -... I. z. - -~u - - - - - - - - - - _ _ _ _ _ _ i _ _ _ _ _ _ _ _ _, ^ ^ ^ ^ ^ ~ _ 00 ^e Do It ( - 0

143 0 w - -— w w cn cn CD w 0 — 0 -I — - 2 W I - - - - - - - - - - - 0 CD - (/ c) N'9,, en o > (U - - - - - - - - - - - - - - - - --- - - - - - - - - I ~ ~~ ~ ~~~~x cmo _1 o z w _ - - --- - - - - - - - - --- - _ _ _ _ _ _ _ _ _ _ - _ _ -0 - j I -,, ^ —. n _ o < W in~~~~~~~~c (j Q~~~~CjE ~~~~~~~~l T u I x <r d CD <oA ^ ——,,,.,0 -. ^^%i? ^s^::^-^::::^^:^____ ^''" ~ ~ ~ ~ ~ ~ O C3 I I' ~ -~ ~-I -' I --' " >~~~~~~~~ S) I/[ L I I \`O o~~ ~~ cu~~~~~ N'~~~~~~~~~~~~~~~~~~~~~~~ e z ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~ ~~~/~-,-_7 — — J —. — ^,-^S^S~ S _ _ _ _ -. a z _ _ _ - s _ _ _ _ _ _ _ _ _ _ _ _ _ \. _.-..:.-^, "' - - - W~~~~~~~~~~~~~ /II' I /~'''.. hl I.\ I~,',,,I\'C 2< i \ ^^< ~ 7 ~'Y//' ~ ~ u I,.. r1 rIl \rl[.f~' - I I'"" I o^ w i^^ ^ r cs. - --- 4|~I,. ~~~'I d — 0 ^ - - -- - - ---- - - ^ |g o ^ N _ _ - ^ -- - _ ^ _ _ _ _ - -- _ _ _ _ _ _ _ _ _ ^ ~ -, g^ — - ---— _ 00 ^C <0t)^ ( S

144 o m C z W W ~n cn O I!Z.. z n,'.. n,'0 a N 2 a- W CD 0 3: CD cr o o y - - - W- ~ L I- -^. w LL I > w -ti - o 0 - - o =========^^~^-^ =====::n.'=I — — " " <I _ _ _ _ _ _ _ _ _ _. 4 S ^ _^ _ _ _ z _ _ o ~~~~~~~~~~~~-...... (..) (.~ -- - - < ~~~~~~~~~~~~~~~~~~~C x/_ \ x, S........ IZZZIZZZZZZZ II^^^^^^^-^^^^II In / /!s v ^ s,,,,"^ Jl, ^ s's s s ~~~~~~3 I k~~~~~~~~ o cr.r Z u k il 1'" - - -^^- - - - ^. —I ^^" cu ~ ~ ~ ~ / TV oui's~~~~~~~~~. -. - - - - - - - -,/ -j l'- -' ~ - I I ~-1 ^~ OD I'- qD~0t m~~p)O- 0

145 - <;ii t. = iii i - X __- __a.______ E.0 Cm o - -- _ _ _ _ - _ _ _ - ----- _ _ _ ------ _ _ _ _ _b. oi W F b -_1 R'~ C - ~ ~ ~~ _ _ ] ~~ 0 _ ~~U -, r _ C __ ___ ____s V _ - _____ ________ Ei *- z I - W,, z....'........ - o. 41, I -- -- -- ^ - s.^ ^ I - ^ ^ -- -- -- -- ": ~ I=, (n 0 0.-, __ ___ _ _ _ _ ____ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ __ _ _ _ _ - _ _ _ -_ _ _ _ _ _ ^ <> ^ <o m ^- ~> M - o ~~~~~~~~~~ >t~ ~ ~ ~ ~ ~~~~~~

146 to advance continually in phase until some of them are slowed down and become trapped in the negative-potential wells of the field. At saturation some of the electrons have not yet been trapped, but if the solution were carried beyond the first saturation point all the electrons would eventually become trapped in the negative-potential wells. As would be expected, the larger the value of b the faster the electrons advance in phase with respect to the circuit wave; this effect is indicated by the decreased initial slope of the curves, i.e., dy/dV is less. Flight-line diagrams are presented for various values of C and QC so that the electron motions may be compared for the different cases. As a typical set of curves consider Fig. 4-42, where C = 0.1, QC = 0, and b = 0.1525. At y = 4 electrons 4 through 22 are in a retarding-field region and hence are decelerated and must have velocities less than the wave velocity. These electrons are giving up energy to the circuit wave. Electron 25, on the other hand, has sufficient velocity with respect to the wave not to be trapped, so it continues to advance in phase. Electrons of this latter class become trapped after they cross the V = 53 boundary. Electrons in the category of No. 22 have velocities comparable to the wave velocity and hence continue to advance with the wave, forming the core of the bunch contributing energy to the circuit. Electrons 4 through 21 become trapped and near y = 5. they cross back over the A = t boundary from a decelerating to an accelerating region. Between 0 = 0 and V = A the electrons are accelerated, gaining in kinetic energy due to transfer of energy from the circuit to the stream. 4.6 Normalized Linear Current Density vs. Electron Phase Information on the current distribution in the eleurctrent distribution stream at any position y may be obtained by evaluating the expression

147 III - p(y o)uo - 46 1 I Io 1 + 2Cu(y,0o) (4-6) for a particular value of y. The summation sign in Eq. 4-6 is necessary because when overtaking occurs then there are in general three terms contributing to the total current density, as can be seen clearly from the electronphase curves of Section 4.4. The current-density curves for the same parameters and the same values of b as were used previously are presented in Figs. 4-50 through 4-57. The data for these curves are easily obtained from the curves of electron phase and velocity deviation of Sections 4.3 and 4.4 respectively. In this group the current density is plotted against one period of the electron phase. The signal level on the helix is given in db referenced to the saturation level for convenience. The infinite peaks in the current-density curves arise from the regions of zero slope in the 0 vs. 0o curves of Section 4.3. These infinite peaks do not indicate infinite current density, since the charge is conserved. An estimate of the actual current density at these points may be obtained from the flight-line diagrams of Section 4.5. In each case it is noticed that the initial electron bunch forms between 0 = 5i/4 and 0 = 35/2 and remains invariant with distance. As saturation is approached a second bunch is formed which at saturation is displaced approximately ic radians from the first bunch. At saturation one bunch is in a decelerating region and the other is in an accelerating region of the field, so that one bunch contributes energy to the helix and the other receives energy from the circuit wave. Clearly at saturation these energy transfers balance one another, since by definition there is no net transfer of energy between the wave and the stream.

148 FIG. 4-50o NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE 6ZZ ZZZZ_ _ _ _ _ _ 0 WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C =0.05, QC = 0, d O, A,= 0.0225, b = 0.076 ^ __ __ __ __ __ __ __ __ __ __ __ __ __ _ _ -- - - -- -- - POINT OF FIRST ELECTRON OVERTAKING y 5.2 - - -- 3 z y =1.2 (-33.57 db) ------------------------- -y = -.2 (-19.0 db) y z 4.0 (- 12.76 db) 3 —--------- ------- ----------- ---—.y 5.2 (-4.41 db) o E_ w w s 3~T? 2'

149 FIG. 4-50 b NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C = 0.05, QC =0, d =0, Ao= 0.0225, b =0.076 ____^______ I________,I. -__ __ ___ _6. 0 (. db) o! n 3' 5 I 3 7 f 2 4ELECTRON - -- --- - -- ---- -- — PH A - (SATURATION) EL~ - - - - - - - - - - -RON6 -- - (-y0.60 db- ) RA IN UJ 5 ~ 3 ~ 7n 2~ Q ~ ~ ~ ~ ~ ~ ~ ~~EE T O __ __ __ _ __ __ _ __ __ __ — __ _ _ _ __ _ __ _._ _ _ __ _ _ _ _ _ _ __ _ _ _ _ _ _

150 FIG. 4-51a I I / I/ I Z Z Z ZZ ZZZNORMALIZED LINEAR CURRENT DENSITY J| ~ ~ ( |I [ 1 _ _IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER!=115 QCS =0, dz0, A= 0.0225 b= 1.75 ELECTRON O T AK, =OG, RADIA 13: U --- ---- - -- - - - - - - - - — i l POIN T OFFIRST EALC T RON OVERTAKING y P A8. _J ELJ - - - - - - - - - - - - - - - - - - - - - - - C = 0. 0 5, Q C = 0, d = A = 0.0 2 4 b 1. y — 6.0 (-14.63 db) ELECTRON PHEASE,I( y=8,/' ) RA DIANS

151 FIG. 4-51b NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C=0.05, QC= 0, d =0, Ao = 0.0225, b=1.75 0\ Oj SATURATION OCCURS AT y = 9.4 o _ = _ __ _ _ __ I___ _ I_ _ I_ I^ I_ I_ _ I_ I_ I_ I__ I_ 1_ 1. I_ I_ I_ __ I I 1 1 >E ELECTRON _ __ __ __ __ __ _ _ ___y 9.2 ( 0.2 3 d ob)R I 0~ -~ -~ - L III Q ~ ~ ~ ~ ~ ~ ~ ~ ~~EE O -- -- -- -- -- - -- ------ -- - -- -- -- -- -- - -- -- -- --- -- -- -- -- -- - -- -- -- -- -- - -- -- -- -- -

152 FIG. 4-52a NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS.ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C = 0.I, OC O, d O, Ao= 0.0225, b 0.1525 IRI /i POINT OF FIRST ELECTRON OVERTAKING y = 5.0 hi Z 0 z i L 0 (- -3.78 db) y x $. 2(-17.92 db) y = 4.0 (-11.57. db),_ _ _ _ _.^ _ _ _ _, J - -... - ~ ~~~~~~~~y _ 5.2 (- 3.01 dIb) y 6.0 (-0.05 db) ^o —-— n — — $'it 5r 31T?....-. - 2 1T ELECTRON PHASEAI y (Y'.0 -— IANS

153 FIG. 4-52 b NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C -.1, QC - O, d = 0, AO= 0.0225, b = 0.1525 SATURATION OCCURS AT y X 6.1 4 6.4 4+ 26 db) _ _ _ _ _ _ _ _ _ _ _ _ _ - - z -- -- - ^rr^r _r=%=_zr===xr==

154 FIG. 4-53a NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C=0.1, QC= 0, d=O, Ao = 0.0225, b = 2.0 i 7 ITI 1 | S L f f D t 36 6 d b) crO. r = 6n (-18.0 db) -/y = 8.0 (-12.5 db) y = 9.2 (-8.0 db) y L P0.0S(- 4.0 db) 3If ~ 5__ 31 _ __ 2__

155 FIG. 4-53b NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C =0.1, QC 0 0, d=0, AO= 0.0225, b = 2.0 0 zzz^ zzzz zzzz~zzzzzz z SATURATION OCCURS AT y ~10. 85 z z w cr C r- I^ z -z zmr:^ 0 z y i 1.4 (-1.6 db) y 10.8 (- SATURATION) —y~ ~ n503 ~2 LJ~ ~~~~ EETO HSE;~~:IN

156 FIG. 4-54a NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C = 0.2, QC =0, d =0, Ao =0.0225, b=0.31 0 \ Ii DT / ELECTR — - - - -- O- POINT OF FIRST ELECTRON OVERTAKING y -- 4.6R 0c 2 n,, - - - - - - - - -^ ^ ^ ^ ^ ^ ^ - - - - - y O(-32.80 db) y._ 4. 8 "- 2.19 db) 0 w_ _ _ 3S_ 5T 3T 7T 2n ELECTRON PHASE,; (y,), RADIANS

157 FIG. 4-54b _____= __ ____ X 3 NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C= 0.2, QC =0, d = 0, AO= 0.0225, b = 0.31 ____2_____\.~~~~~~~~~~~~~~~~~~~~~~~ -__ -5.2 -0. 26 db) o rr! — - r —- -- -r 5- 3 rr- - 7.. 4]2 l4 4 24

158 o — o FIG. 4-54 c NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX 5 I 1 1 I- IJI__ _ _ _ _ _ _ ___-1._ _1AS THE PARAMETER C = 0.2, QC = 0, d = ~t Ao= 0.0225, b d 0.31::3 Z \ O - f -- 1 1 1 1 111 T T-2- - - -ELECTRON — - - — SATURATION OCCURS AT y 5.4INS _ _ _ ~ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __- y 5.6 (+ 0.16 db)

159 FIG. 4-550 6 NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C=0.2, QC =0, d= 0, Ao =0.0225, b= 2.0 _. ---------- --------- y - 0(-34.0 db) ~'""'"~ ~'"" — F~~~~~~~~~~~~y 5.2 (-10.3 db) - - - - - - - - - - — 4..y z 6.0 (-5.4 db) 30 TI TT 3T E TR 5ON 3'A 7TI 21) ELECTRON PHASE, / (y,Jo ), RADIANS

16o FIG. 4-55b NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C 0.2, QC - 0, d= 0, Ao= 0.0225, b=2.0 -U)I I O Z 4 2 4 4 2 4b) ELECTRON PHASE,,0 (Ty,Io), RADIANS

161 FIG. 4-56a NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C - 0.1, QC' 0.125, d - O, AO 0.0225, b 0.65 I luFl i 1 I / I I I I I i I I I I I I I I iPOINT OF FIRST E LECTRON OVERTAKING y 4.1 W 2 - - __....__ _ ___ _ cr 2 ——'""~ ~ - ~'""~-~ —- ~4- y = 4.0 (9.15 db nI ll' 131 1T 5n1 31 71 2.1 ELECTRON PHASE,;( y, ~o) RAD^IANS

162 FIG. 4-56 b NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C = 0.1, QC = 0.125, d=0, Ao= 0.0225, b=0.65 — _ q. z /_y = 5.2(- 0.z74 db) T[' 5'f 3' 7T 2'E ELECTRON PHASE,,'( y, ~o ), RADIANS

163 FIG. 4-56c NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C = 0.1, QC =0.125, d = 0, Ao=0.0225, b = 0.65 Il l 11[111:1 11 or I: 11; 11!111 f: 111 1 f I f I I I I I I I I I o --- — I -^ -- --- -- -- -- -— t - - -- --- -— y 56(SATURATION) — 0 ff l 3~ W 5~ 3- 71T 2ELECTRON PHASE, ( y,,o), RADIANS

164 FIG. 4-57a NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C = 0.1, QC 0.25, d =0 AO= 0.0225, b= 1.0 I -7.~ S Z w 3 cr. cI z Q::,,- i i t t | S e g S~~~~~g X X f 0 g 2 t ~~~~ y 2.0(- 26 70 db)\ 8 - -. ___ _ _ _ I/ —- Z_ -4 I I b) — - - - - - --- NELECTRON EPHASEAI(G y, ) RADI)NS =3 0 __ __ __ __ __ __ __ __ __ __ __ __ __ _ __ _ __ __ __ _. _ _ __ __ __ _ __ __ __ __ /y_~_4. 0 (- 7.61 db =1^ --- ~ ~ ~ j-*~~~~~~~~~~~~ ~~~~~~ - -- - -- -- -I- - - - - - - - - - - - - - -' - - - - - - - - - - - - - - - ^ - - - -- - - -- - - -- - - -- - - --. -- - - - - - - -- - - -- - - -- - - -- - - -- *y~~~~~~~~''f 3 T f' 3 f f2i iii~~~~~~~~ ~EETO PHASE, - - - - -- - - - - - -- - - * — - -- - y -- - -- - -- - - -- - -- - -- - -- -- - -

165 FIG. 4-57b NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER -C =0.1, QC C 0.25, d =, Ao= 0.0225, b = 1.0 5 2 0.27 d b ELECTRON PHA*SE, t __ _ _ __ _ _ _ _ _ _ s^ _ __ _ _ r _ _ _ _ __ _ _ _ _ _ _ __ _ _ _ / _ _ _ -f -- -- - --- - - -- - --- - - - -- - - -- - - --- - -/ - - - I - - - - — l lr - - - IT - - - — l- - - - - - =z _- __:^ -: -= =^:: ^^:\= i Tlll,,l^: -- — ~^'~ / y ~ 5 2 ( 0. 2 d b _ __ _ _ _ _ _ __ _ _ _, _ _ _ _ _ __ _ \ _ _ _ _ _ _ I_ _ _ _ _

166 FIG. 4-57c NORMALIZED LINEAR CURRENT DENSITY IN THE STREAM VS. ELECTRON PHASE WITH DISTANCE ALONG THE HELIX AS THE PARAMETER C =0.1, QC = 0.25, d = 0, Ao 0.0225, b =1.0 s W 0 IT 02 —-s-__- -- -- t_. — -- _- -.o 1- L_ L_ _L________ o3 0 SATURATION OCCURS AT y = 5.5 I — z W 2 N 0

167 4.7 Variable Input-Signal Level The input-signal level for all calculations heretofore presented has been the same, i.e., 30 db below CIoVo. This value of Ao was selected so that the input boundary conditions could be calculated from the linear theory. The purpose of these calculations, however, is to analyze the operation of high-power amplifiers. For the value of Ao used here the saturation gain is between 30 and 40 db, but it has been found experimentally that the maximum gain in a loss-free region must be limited to approximately 20 db if the standing-wave ratio at the output transducer is of the order of 1.5:1. Hence where higher gains are necessary added external attenuation must be placed on the helix and consequently the saturation gain and power output are reduced. Thus it is not practical to build high gain and high power output in the same tube; instead, a short low-gain loss-free tube should be used when high power output is desired. To shed some light on the tube characteristics under such conditions, solutions are presented for larger values of the input-signal level Ao. These solutions are shown in Figs. 4-58, 4-59, and 4-60 with j, the inputpower level below CIoVo, having values of 5, 7, 10, 15, 20, 25, 30, and 40 db. The case selected for this investigation is C = 0.1, QC = 0.125, and b = 1.5. The input boundary conditions on the initial phase 0(0,0)o and the a-c velocity parameter u(O,0o) are taken to be the same as in the previous calculations. The value of A'(y) at the input of a lossless helix is still zero since, as was pointed out in Section 2.4, the stream cannot influence the wave amplitude until it has become bunched. However, the value of i'(y) at the input for large input-signal levels is open to question. The value of' (0) gives an apparent phase velocity of the total wave at the

168 Z'0 "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It ( 0 I L X r X~~~~~~~~~~~~~~- 14:.F — _ —_4_ —I — __ _________I_ -- - t- __ ___ — l --- ^ll S~~~~~~~~~t~~~~~~~~sf f it~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~tfi$Et00XX~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t FXC WX XX Xn X 0 t __. a... < ( I ~ =t~f ~ z i OD <__ _____ - _ I i__o I _ I U. O ~X. _1FI _I;1 I~ ~ I~L J X X C F ^^^^^-^^ m ^ ^ —^zm m::^^^ _:I^^ZI"' —. -- _ _ _ _ _ _ _ _ _f — ( ) -_ "_ _ _ _ _ I —- =^i U a I_-.L_...i w ",- -. —- -.. s t -- ~ t:E (~. _= /:5:,.;^::::^ =^5 ='=s -f U. b is' —. x \ \ > 3r. IU.>^ I ----— lI —---- - ----- — I — 0' 0 _ \ \ * 0e U. - 0^ ^ ^ s X a ~ ~ - w- La- - -- - - IC — Y-^ - - - _ - - _ -i w ~~0 ~5g"~. ~ ~ x-~~ - - ^\ ^ \:^1- >' o -i - - I " I (^)v'aonindwv X5 x ~ ~ ~ ~ ~ ~ ~~~~ o3nrdY..170 I1

169 H Z ---? —- - — ~ — - - - - -- - - - ---- ----- - cn -- v- - - - - - - - - - - - - - - -------- --- -n o - 04 0-0 I,,,: " - 4 ~ H ~ z -- --- - - - - -- - - - - --— I- - - --- Li.2 5 0- ~ - W -,,, *I -— ^ ~ >, w o I ~'0) ~~~-^~ ~~~~ 4 — — ^- -^^- o - ~~~~~~~~~~~~~~1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ L' 0 -_ _: _ _ _ _ _ - ] - -- - - - - - - - - - — *- - - - - — I - - - - \ - - - - ----- --- --- -J'^ _ 1-,, —-I -~~~~~jlj~~~~I -_1 IL IIS -;" N\_ _ _ _ _ I -~ "I I'N,,~~~q "'A I&0.~~V~~~ ~-J -1 < s^ V \ w -, I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~U i "0, N N~~,;I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I ]I~~~~~~~~~~~~~~~~~~~~~~~~~~._, T T N cm,, qP'~^~ID 0.L 3^I.LV'13EI 83M~d.-i-E

170 Z, - fo to 0J~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 ~~~~~~~~o w 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,-t = a. ==_ I-J 0:: W Ii 0>. -co r~ 0,, \l - - -—, — - -- — _ LL i ----- --- _ - _ _ -- ~_ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - - -- — ~~~~~~N~(~ (, -- -' - - — I H.: — - 1

171 input which results when the three small-signal-theory waves are combined. Since no theory is available to calculate a more accurate value of i'(0), the value given by the linear theory is used. This is tantamount to assuming that G'(0) is independent of signal level, which is not considered to be a serious deviation from the truth. The insensitivity of the solutions to small errors in @(y) and A(y) further justifies this assumption. Additional analysis of these solutions is presented in Chapter V. 4.8 Effect of Series Loss along the Helix As was pointed out in Section 4.7, it is necessary to add series loss along the helix in high-gain amplifiers in order to insure stable operation. To assist in determining the effect of the amount and placement of this loss on the saturation gain and efficiency, solutions are presented for several values of the loss factor d at C = 0.1 and QC = 0.125. The values of d selected are 0.1, 0.25, 1.0, and 2.0, which, for C = 0.1, correspond to losses of approximately 0.56, 1.36, 5.45, and 10.9 db per undisturbed wavelength Xg respectively. In addition, the effect of the placement of loss and the reduction in the optimum value of b in the presence of loss are discussed for d = 2.0. The power flow along a lossy helix is given by PoL(Y) = Re VI* 2 W* Re, (4-7) 2ZoL where ZoL = Zo(l - jCd), the characteristic impedance of the lossy helix. Substitution of the defining expression for ZoL and Eq. 2-12 for the voltage give, for the power flow along the lossy helix,

172 PoL(y) = 2CIoVo A2(y).(4-8) 1 + jCd 4.8.1 Variable Loss Factor d for Fixed Injection Velocity b. The solution selected for investigation is C = 0.1, AC = 0.125, and that value of b which gives maximum saturation gain when d = 0, i.e., b = 1.5. The method is to integrate for a distance of approximately three undisturbed wavelengths from the input with d = 0 and then to carry out solutions from this point to saturation for the values of d listed above. The results are presented as Figs. 4-61, 4-62, and 4-63. Since the region in the vicinity of the point where the loss is first applied is of particular interest, this region has been expanded in Figs. 4-64 and 4-65 for the variables A(y) and @(y). It can be seen that in the cases where d = 1.0 and 2.0 the signal level on the helix diminishes in amplitude for approximately one undisturbed wavelength after the loss is applied and then as the stream is continually being bunched resumes increasing in amplitude. This temporary decrease in r-f signal amplitude indicates that energy is being put back into the stream and appears as an increase in the kinetic energy of the stream. Hence the electrons are accelerated, which lessens the interaction between the wave and the stream so that the signal level on the helix drops until the stream becomes rebunched sufficiently to give energy back to the helix wave. The slowing of the r-f wave due to the presence of series loss also contributes to a decreased interaction between the wave and the stream. The axial phase velocity of the wave in the presence of small amounts of loss is VoL = = r ). (4-9) voL = (1 +~~~~~~~~~-~Cdf

173 o > rZ 00 < -t- 7 1 -O Z o o o LO 0 _^/_D U._ _ _ CS_ c__o O v! - w 0s \ — 0 (.) ~~'33N.Li'i —di 3m~l'10.-I- -

174 o z ~~0~~~ ~ 0 N XJ W w O1~ _ \ __ W U. I I0 L' N _ ----- - I,, I, a % *X % X x ( ) i ci w,',,' " <N% N ~ 0._ ~ V'r... x qp'lj3M~~~~~d "%t

175 cm 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Z rO ~~~~~~~~0w do w~~~~~~ k o w~~~~~~~~~~~~~~~~~~~~~~~~~~~~> 2 cr ~ ~ ~ k k \ k \I \I~ t~ -j z o< wO w~ =E_= I w > 0 r wQ a, % "o I - v W o~ LL 0 0: 0 VI ~ h-'~x ~ ~ ~ n Jr —-- - r0\ -_ n- _j o MW o (D LLLL WI I J 0: \ r <~ o \\ -- - - O..'r _~ ~ _ g> <~ <0, —0- - - - - - - -- -L -_ _ _ ~~~~~~~~~~SlO l ()e'9vI3Sd:11 < S ~ ~ ~ —- ------ i-~ —------ --- -- -- -- | ~ d~ ~~~~~St/at "()'91 3Sl- - --- - -- - --

176 0 - i) 0 o a \I'L La L. CD 0: 0 a _ Q_ 0 _ 0 _ 0 C_ 0 0 I) 0\ 0 0 <' io >i = 0 0~ ~ i o o o o o o o o o d d c 0 c0 d d c c d (O)v'30nildWV 39V90OA j-a

177:me I I I I I 1S0' Ir~~~~~~~~~~~~~~~~~~~~~~~~~i. ~ ~ ~ ~ ~ ~vi 0.t__ 0I.L SL \

178 4.8.2 Variable Injection Velocity b for Fixed Loss Factor d. As discussed in Section 4.8.1, the effect of series loss along the helix is to decrease the wave velocity and thus reduce the interaction between the wave and the stream. The larger the value of d, the greater of course the decrease in wave velocity. In order to determine the amount by which the velocity is decreased and also the b associated with maximum saturation gain in the presence of loss, solutions have been carried out for several values of b with fixed loss. The value of d selected is 2.0 (10.9 db/Xg). The results of these computations are presented in Figs. 4-66, 4-67, and 4-68. These curves are discussed in more detail in Chapter V. 4.8.3 Effect of Position of Loss for Fixed Loss Factor d and Fixed Injection Velocity b. In his development of the linear theory Pierce* found that the loss should be applied at least a distance corresponding to CNg = 0.2 from the input, since at that value of CNg the signal on the helix is just beginning to increase. It should be noted that Pierce's curve is plotted for the special case QC = b = 0. Tn order to determine the effect of the placement of loss on a largesignal amplifier, the loss was introduced at CN "0.2 and CN 2 0.4 for the case C = 0.1, QC = 0.125, b = 1.5, and d = 2.0 (10.9 db/kg). A solution has already been obtained for the curves of Section 4.8.2 using these same conditions with the loss applied at CN - 0.3. The three solutions are presented for comparison in Figs. 4-69, 4-70, and 4-71. *Reference 8, p. 135, Fig. 9.1.

179 >Z I,- _. - - - - - - - - - - - - - - - 1 1 X i F 3W, W i M I0 0 -l cu I-: I > - c. i i _ i /r 1/~~~-.-o -' o, r^ > i^ If- 0S >-t 2!t -t t —NtU _ _ _ _ _ _ _ ^. L ^ _ ^ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I X 3 _l~~~~~~~z: _ j. o... ___ ___:L^ __^ _ I I I- - I r — LL 11 o 1 (' l d l. __ _: > o 2 07 j _ - _ L _Z_^ _ — _ _ o

180 m )rZ cr~~~~~~~~~~~~~~~~~c -- x~~~~~~~~~~ 0 o - -- - - - - - - - ~ -__ o c LL W 0 OI w n w W \X \", w --'x a o. -w o.,, ~,~' z V -. i — o o cj..- W < <, o v I I __ cr- c,..J bJ ) d o <I: 13: <IU,. \\\ \ \ \ - - - - - - - I -- o w 0. a N - - - -ss s- -- - I 13. I 0 ~o 0 mn 0 S 0 mo 0!< 1*3 ~~ ~. -- --- -^? qP'B3MOd J-I

181 0~~~~~~~~~~~~~~~~~~~ I 0 I )- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - W 0 L cr, a: Q. W X i QO a W W L > LLW 0 cr UJI on LL Q IjwLL W I W= =: ~ mzm^zmr^ ^mma:7 LL~~~~ WO' -- -- ---- -- - - - - -d 0 - - - - -- - - - -- - -- - - - - - -- - - - - - - - _ __ _ _ _ _ _ _ _ _ __ ^. _. - _ __ _ __ _ _ __ __ _ __ _ _ _ __ __ _ _ __ __ _ __ _ _ _ _ _ _ _ _ _ - L - - -- -- - -- mZ ~=^=:========== _ _ _ _ _ _ _ _ \ _, _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ^ \ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ __ _ S ^ ^ — --- aocr m 11 ===== ^ =x ======= s s =========^^^^=15=1========== 3S~H J-l

182 0> Z o~~~~~~~~~~~~c >~2 w - 0 ~. 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~U C -0AI 0 0: U,) O r O w d c 30 0 Z; 2" (^ - < o- o U. U --- --- --- --- -- --- --- --- -- - ~. a - ^ -;., _ Q10~~~-Lrl dlL0 U <o > >, zzm zzi^'z^ ^ zzr~~~~~~~~~~~~~~~~~~~~~~~~~~~zziim:~~~~~~~~~U 0 cm'0 o I o o< _j CY c; /.^~~~~~~~ cn S ^~ l v!v __~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ — 1._____1____. i i2 i.g U=>=: _., o o a LL <U (O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ W I Z U ~cu IoO ~~~vl v - - ----- - — J —-I- - W - - v l w vI f/",.. X < " S^ L.L VI VI vl / 0 - 6 o o o rr~z z~rrzzzzzzzmmzzs crm __ -- -- -- -- -- ________~~~~~~~~~~~~~~~~~~~~~n' —,, a:: n- _ - a.~~~~~~~~~~~~~~~: oX~L W 10 Z o o o o -- - w Ad 6 L "Iz IN " C o to o ~~~~~~Co 6 o doI ~ ( a w l d- u (r(tl30lldlV ~ 39lnI\" -

183 C)~~~~~~~~~~~~~C X Z~~~~~~ O~~~~~~ LL>,2 - - - - - - - - - - -- -_______ X W --- - - - - - - LL - - - - - - - - - - - - - - - - - - - - - - - - - - - x 0 0>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_j0 0 0 OM <ni ^" <L II 11 Xj z- 5e'- A A - - - - - - - - - - - - - - - - - - - - - - - - - - - - x r_ 0 0' W v 0 cliZ I r c ~~~~~~o~~~s VI VI VI ol I I I I I I I I I I I I II Q a. U ~~~~~~~~~~~~~~~~~~~~~~~~ )r I V V _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -J W G S! g ~: a: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~LJ c c c _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0 1- O Q - ^.,,,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 3: U U. U.L LL \f Q; I -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 0 0 _ n ~ ~ ~ ~ ~ c O O " o o - - - -- - ^ - -- -- -- - - 2 in ~I ~ ^ _ _ _ _ _ _ _ _ _ _ _I5 __, In __ _ __ __ __ __ __ __ __ __ __ __ __ ^ ^ __ __ __ __ __ __ __ __ __ __ __ _ __c == = = = = = ^= = = =, 1 11 11 11 11 _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ \ s ^~~9't _ _ _ _ __M_ _ _ _ _ _ -a

184 o~ 20 -I I t I II I I il l I I I I I 1 1 \ l\kl 1 1 I I 1 - I l I Fi181 1 —\i l I I I l\-\ _ Z w o -J cnS A) 3 _ _ __ _ __- o, -- VI —, vl vlI._ n/) ci o o o o oo _ _ -- _ —-_ _ _ _ _\ _^ s. U. I l__ ~) I t~ w - -- o $NFlb~o'(op)8'gV' 3$-Hd -I —1

CHAPTER V. ANALYSIS OF THE RESULTS In addition to the basic curves drawn from the computed data on r-f voltage and phase lag and presented in Chapter IV, a number of curves may be drawn from the same data to emphasize certain especially interesting relationships and examine phenomena of particular importance in design applications. 5.1 Voltage Gain and Tube Length at Saturation In order to determine the optimum injection velocity for particular C and QC values, the voltage gain and tube length in undisturbed wavelengths at saturation are plotted vs. b. The resulting curves are shown in Figs. 5-1 and 5-2. The drop-out point of the voltage-gain curve is quite sharp, as it is in the linear case, and the value of b at which this drop of the gain to a very small value occurs is approximately that value of b for which the small-signal gain goes to zero. As can be seen from the figures of Section 4.1, for these large values of b the saturation tube length is very short, which is to be expected since the electron velocity is much greater than the wave speed and hence the interaction between the wave and the stream is negligible. In comparing the curves of this section with those of Section 4.1 it should be remembered that Xs/kg = 1 + Cb. In Fig. 5-1 the gain curve for C = 0.2 is shown dotted in the vicinity of b = 2.0 because the total velocity ut(y,00) approaches zero near saturation for this case. As pointed out in Chapter II, for values of C as large as 0.2 the neglect of space charge leads to appreciable errors and hence these particular results are questionable. 185

186 FIG. 5-1 VOLTAGE GAIN AND TUBE LENGTH AT SATURATION VS. RELATIVE INJECTION VELOCITY QC=, d =0, A0o 0.0225 80 1 I I 3 r I,,,, 80 VOLTAGE GAIN -- TUBE LENGTH 70 0. —70 C 0.05 2. a l eo AT —- -- - I —-h — --- --- --- --- I/ C=0.\0^^ I I UI ii I i I Ii 00060( 1 I I L~ I I 1 ~11 1 1 I 1 0 < <c~~~~ <\-w.t.. C"-T^rC = 0.20 t —1 040 Z. / z= ~~~~< ^~~~0 i 0> s40r --- - - -- -I 01 0 I I-'' w CsO.05 0.10 0 Cm 0.20 RELATIVE INJECTION VELOCITY, b= u~-~ Cvo

187 FIG. 5-2 VOLTAGE GAIN AND TUBE LENGTH AT SATURATION VS. RELATIVE INJECTION VELOCITY C 0.1, d=O, Ao= 0.0225 80 VOLTAGE GAIN. —--- TUBE LENGTH 70 120 0QC 0 \ ~ z 1 =0 ---- -- 02QC - 0.12 5 w L I I I\II I I rQC=-0.25 X (0:: 2 Ii I IN I \ITb- ~ ~ 40 -- -- -- -- - )z 0 z z 0 a: QC=O 20 --- -- -- -- -- -- -- /\ -- 20 ( QC=0.125 Ii. 0 H RELATIVE INJEC10TI —- 0.5 - -- --- -- -— O- Cv00 RELATIVE INJECTION VELOCITY, b =. ~ Cvo

t88 The curves of Fig. 5-1 indicate that saturation gain decreases with increasing C. The nonlinearities which largely determine the gain in the large-signal region and which increase with increased coupling (i.e., larger C), are responsible for this decreased gain. Maximum saturation gain for C = 0.1 and QC = 0 occurs at b = 2.0, which amounts to an 18.2-percent increase in velocity or 40-percent increase in the stream voltage Vo over that for maximum small-signal gain. As b is increased the tube length at saturation would be expected to be greater, since the electrons advance in phase with respect to the wave faster and hence it takes a greater distance for them to build up comparable a-c velocities. Also, the expected saturation gain should be greater for larger b, since the kinetic energy in the stream is greater by an amount proportional to b2 In Fig. 5-2 gain curves are shown for C = 0.1 and three values of the space-charge parameter QC. For QC = 0.125 and 0.25 it is noticed that the b for maximum saturation gain is less than that for QC = 0. However, for larger values of QC it is expected that the b for maximum saturation gain will increase as in the linear theory. Also, the saturation gain with small values of QC has increased over that for QC = 0, but as QC is increased further the gain decreases as is expected for large values of space charge. A qualitative explanation of this phenomenon is given in the following section. 5.2 Efficiency and Tube Length at Saturation For the same data as discussed in Section 5.1, saturation efficiency and tube length may be plotted vs. the velocity parameter b. The saturation efficiency is is, from Eq. 4-1,

189.s = 2C a (5-1) These curves are plotted with C and QC as the respective parameters in Figs. 5-3 and 5-4. For the QC = 0 case the efficiency is plotted both in percent and as %s/C. The efficiency curves are seen to be quite similar to the voltage-gain curves, except that for increasing C the maximum efficiency increases whereas the saturation gain decreased. Obviously this increase is to be expected, since larger C values correspond to greater coupling between the circuit and the stream. In Fig. 5-5 the maximum saturation efficiency is plotted vs. C for QC = O. For each value of C the efficiency is maximized by optimizing the velocity parameter b. Figure 5-5 indicates that not much is gained in efficiency by increasing the gain parameter C beyond 0.10 or 0.12 for small values of space charge. The maximum saturation efficiency divided by C is plotted against the small-signal space-charge parameter QC in Fig. 5-6 and against the largesignal space-charge amplitude parameter K in Fig. 5-7. Figures 5-6 and 5-7 might be interpreted qualitatively as follows. For small values of QC, up to approximately 0.125, the maximum saturation efficiency increases with QC, as might be expected since the charge density and the space-charge forces in the stream are very small and the increase in charge density associated with increasing QC appears as increased power output. However, for larger values of QC it is probable that the strong short-range spacecharge forces inhibit the bunching severely enough to overcome the effect of the increased charge density.

190 FIG. 5-3a EFFICIENCY AND TUBE LENGTH AT SATURATION VS. RELATIVE INJECTION VELOCITY QC= 0, d = O, A =0.0225 80 ---—..... —- -- 80 EFFICIENCY ---— TUBE LENGTH U, 2 Ior:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~" 60 0 - z < IW -J 0: z - 50 - -50 n z 0 I I I I I I I II \ I I I I I I cw/)C=020 Z Lcr u- D0 50 v1 -100W~~~0 40003 I0 So 1 Cf 0= 20 ~ ~ Cv3 1. o 30:30 Cv.

191 FIG. 5-3b EFFICIENCY/C AND TUBE LENGTH AT SATURATION VS. RELATIVE INJECTION VELOCITY QC 0, d 0, A0 0.0225 80 EFFICIENCY -TUBE LENGTH 70 -< u) Z 6 5.0 -50 I0 RELATIVE INJECTION VELOCITYJb - ~ 0 0 I- - CE C = 0,05 nn Z 4.0 -40 C3 z 3.0 0__ 2.01 1 -- 7 1 I.I 1 / \ 0..1 O - V0. CT.O.IO L b = Cvo

192 FIG. 5-4 EFFICIENCY/C AND TUBE LENGTH AT SATURATION VS. RELATIVE INJECTION VELOCITY C =0.1, d= O, Ao =0.0225 -— _ — EFFICIENCY ----- TUBE LENGTH ol 0 6.0 C bJ LU Z Q/OC= 0.125 0H~~~~~ x C5.0 m < Y I I I —W 03 4.0 -— 030' ~~~__o_ — -- -- - _ ___z U-) — - -- - 03 LIi H QC = 0.25" 0 1.0 2.0 3.0 RELATIVE INJECTION VELOCITY, b = 0 Cvo

193 FIG.5-5 MAXIMUM SATURATION EFFICIENCY VS. SMALL-SIGNAL GAIN PARAMETER QC =O,d=O z w 0 rn w 60 Q-? 0 z ~-' U O. — 0.2 0.3 LL.SMALL-SIGNAL GA z 40 30 20 0 O.I 0.2 0.3 SMALL-SIGNAL GAIN PARAMETERC

194 FIG.5-6 MAXIMUM SATURATION EFFICIENCY/C VS SMALL-SIGNAL SPACE-CHARGE PARAMETER C =.I, d1 = x 6.0 5.0 0 I^ 4.0 3.0 2.0 1.0 0 0.1 0.2 0.3 SPACE-CHARGE PARAMETER, QC

195 FIG. 5-7 MAXIMUM SATURATION EFFICIENCY/C VS. LARGE-SIGNAL SPACE-CHARGE AMPLITUDE PARAMETER,K C = 0.1, d=O 0~_ il <0 I2 I 0 I 2 3 4 5 6 SPACE - CHARGE PARAMETER, K

196 5.3 Variable Input-Signal Level Some additional information on the operation of large-signal amplifiers when the input-signal level is varied may be obtained from the figures of Section 4.6. Since the saturation level is relatively independent of 4, one such curve is a plot of the input-signal level vs. the saturation length, shown in Fig. 5-8. It can be seen that the optimum tube length is very nearly a linear function of the input-signal level for a wide range of 4. Figure 5-9 is a plot of the phase shift at saturation, relative to that for a very small input-signal level, vs. the input-signal level for a fixed tube length. The tube length chosen is Ng = 5.5 (i.e., y = 3); from the figure it can be seen that the tube would saturate at approximately y = 3 for 4 = 7 db. This curve indicates the amount of incidental phase modulation that would accompany a change in the input-signal level or amplitude modulation of the input signal. Clearly if the tube is to be amplitude-modulated it must be operated well below saturation and for small amplitudes of modulating voltage if linearity is desired. For the same tube length, i.e., Ng = 5.5, a power-output vs. powerinput curve prepared from the data of Fig. 4-66 is shown as Figs. 5-10a, 5-lOb, and 5-lOc. The power levels are plotted on a linear scale in Figs. 5-lOa and 5-lOb and on a logarithmic scale in Fig. 5-lOc. In the smallsignal region the power output is seen to be a linear function of the input power as expected; however, as the input power is increased still further the output power reaches a maximum and then decreases. Figure 5-lOc indicates that for this case the saturation power output is approximately 5.75 db below that which would be obtained if the output were still linearly related to the input and it is seen from Fig. 5-lOa that the power

197 FIG. 5-8 4, INPUT SIGNAL LEVEL IN db BELOW C IoVo, VS.TUBE LENGTH AT SATURATION IN UNDISTURBED WAVELENGTHS. SATURATION LEVEL IS APPROXIMATELY 7 db ABOVE CIoVo C =0.1, QC 0.125, d= 0, b =1.5 0 0 0 0 co.0 20 60 _____X < P z - 30 40 50 0 2 4 6 8 10 12 14 16 18 20 22 24 26 TUBE LENGTH, Ng,IN UNDISTURBED WAVELENGTHS

198 FIG. 5-9 PHASE SHIFT VS. FOR FIXED TUBE LENGTH WITH VARIABLE INPUT-SIGNAL LEVEL C=0.1, QC=0.125, dO=, b =1.5, Ng 5.5 1.0 0.9.L 0 0.8 ( = 0.6 / 0.5 0.4 0.3 0.2 0.1 50 40 30 20 10 0 INPUT POWER LEVEL BELOW CIoV0,, db

199 ol I / /V I FIG. 5-IOa OUTPUT POWER LEVEL RELATIVE TO CI V 2___ ___// ___ VS. INPUT POWER LEVEL RELATIVE TO ClI^Qo FOR FIXED TUBE LENGTH >-0.1, QC= 0.125, d=-0 bz1.5l Ng 5.5 C IoVI, 00

200.~| _ __ _ | |_|| I _ 111 RATIO OF POWER LEVELS IS 2.4:1.^i7 00 3 - -A __ _____ —FIG. 5-lIOb 2 - OUTPUT POWER LEVEL RELATIVE TO CIoVo VS. INPUT POWER LEVEL RELATIVE TO C IVo FOR FIXED TUBE LENGTH C= 0.1, QC 0.125, d =, b = 1.5, Ng 5.5 CI IV CIV.l2 I 04 6 8 1 ~ 2 I0 154.8 02 22 5. 0.260,3 C loV

201 20 FIG. 5-IOc OUTPUT POWER LEVEL RELATIVE TO C IoVo, Po VS. INPUT POWER LEVEL BELOW CIoVo,,FOR FIXED TUBE LENGTH C O.1, QC =0.125, d O, b l.5, Ng 5.5 10 40 // ___- P____ L __/ L 3.75 db -)?~~ I so I 0 aI-J 10: ____ -- -'C 0 -20 50 40 30 20 10 0 INPUT POWER LEVEL BELOW CIoV,4, db

202 output departs slowly from the straight-line value. When QC = 0.125 it was found that there is an increase in the output due to the presence of space charge, but for larger QC the output decreases. Hence for larger values of QC it is expected that the saturation power output should be more than 4 db below the extrapolated linear output value. Experimentally it has been found that when C = 0.1 and QC = 0.25 the best that can be achieved for saturation power is 5 to 6 db below the value predicted by the linear relationship. 5.4 Effect of Loss on Saturation Gain and Phase Shift From the loss curves for constant injection velocity shown in Fig. 4-61, the percent reduction in saturation gain and phase shift are calculated and plotted in Figs. 5-11 and 5-12. For low values of loss the percent reduction in saturation gain increases rapidly, but for values of d greater than 0.2 (1.09 db/Xg) the percent reduction increases more slowly. The optimum injection velocity is reduced by the presence of loss as pointed out in Section 4.8.2. The saturation voltage gain is plotted vs. the velocity parameter for the case C = 0.1 and QC = 0.125 in Fig. 5-13, from which it is seen that the optimum b at d = 2 is 0.65 as compared to 1.5 for d = 0, a reduction of 8.5 percent in velocity above synchronism. The saturation gain is seen to vary only slightly for b between 0 and 0.65. Hence it would seem desirable to change the helix pitch under the attenuator of a traveling-wave amplifier in order to realize an increased gain. This is especially true if the loss per wavelength is large. In the above-mentioned case an improvement in the saturation gain in the lossy region of 1 db would be realized. The reduction in optimum b would obviously be less for smaller values of loss.

203 FIG. 5-11 PERCENT REDUCTION IN SATURATION GAIN VS. LOSS FACTOR FOR FIXED INJECTION VELOCITY |_____ C 0.1, QC =0.125, A = 0.0225, b 1.5 d 0 FOR O< y<1.6 Z 100 z 0 - 90 a 0 IW 0 70 cr W 60 50 40 30 20 10 0 0.2 0.4 0.6 0.8 1.0 1.2 1 4 1.6 1.8 2.0 LOSS FACTOR, d

204 FIG. 5-12 PERCENT REDUCTION IN PHASE SHIFT AT SATURATION VS. LOSS FACTOR FOR FIXED INJECTION VELOCITY C- 0.1, QC: 0.125, Ao - 0.0225, b - 1.5 d = FOR 0 < y < 1.6 z 0 IIE w 50 5O s0 I. 0 0 w U 30 0 wO 20 i0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 18 2.0 LOSS FACTOR,d

205 FIG. 5-13 VOLTAGE GAIN AT SATURATION VS. RELATIVE INJECTION VELOCITY FOR FIXED LOSS FACTOR C 0.1, QC -0.125, Ao 0.0225 d O FOR 0- y < 1.6 d - 2 FOR y >1.6 z 0 15 c1 t,) z -J i —0 1.0 2.0 3.0 RELATIVE INJECTION VELOCITY,b = oCvo

206 5.5 Comparison with Experimental Results C. C. Cutler of the Bell Telephone Laboratories has made some rather interesting measurements on the velocity distribution in the electron stream of a traveling-wave amplifier as a function of the signal level at the output end of the tube. Cutler's tube is a scale-model traveling-wave amplifier operating at 100 me. Figures 5-14 and 5-15 are some typical results obtained by Cutler. These curves are similar to the plots of electron velocity deviation vs. electron phase of Section 4.4.2. Although the parameters (i.e., C, QC, Pb' and b) are not exactly those of any curves presented in Section 4.4.2 the similarity between Cutler's experimental results and the theoretical curves is striking.

207 2 (a) b = o.5 1 - _ -vI Vw <u 3 JRELATIVE PHASE IN DEGREES z 2 <[: (b) b o.9 0-FIG. 5 C - 0.075, OC =0.22, fb = 0.388 b. b = 0.9 L)I GAIN) 3U -4 _J, RELATIE PE IN D S (c) b1.35 0 I VI, VW 240 180 120 60 0 60 120 180 240 FIG. 5-14 VELOCITY AND CURRENT IN THE STREAM AT SATURATION VS. RELATIVE ELECTRON PHASE WITH RELATIVE INJECTION VELOCITY AS THE PARAMETER C =0.075, QC = 0.22, pb' 0.388 a. b = 0.5 (ADJUSTED FOR X.max)

208 2 2 -22 DB (a) ----. - -2PDB|12 -2 -"' 2 -3 21^ I I I /' 1_-,.,,:,-17 DB (b) e~ 2j iJ L -1 DB>|()-4, C -J T J| |. - LEVEL FOR MAX POWERj R — - I I _________ -2 "~ — l 2 5 -- /~ -?u~RELATIVE PHASE IN DEGREES 1'IFIG. 5-15DB -3 INPUT-SIGNAL LEVEL AS THE PARAMETER C, 0.064, fb 0.412, b- 0 HEAVY LINE -ELECTRON VELO _____ - V::t-71Li -~Jd -'""' -2 -12 DB I J\~yrM)\~ _ /2 - - - 3,LIGHT DASHED LINEVECIRCUIT VOLTAGEES (COURTESY OF CC. - VELOCITY AND CURRENT IN THE STREAM INPUT-SIGNAL LEVEL AS THE PARAMETER HEAVY LINE- ELECTRON VELOCITY LIGHT DASHED LINE - CIRCUIT VOLTAGE

CHAPTER VI. CONCLUSIONS 6.1 Introduction In the preceding chapters the large-signal traveling-wave amplifier equations have been derived and solved for several values of the gain parameter C, the space-charge parameter QC, the loss factor d, and the velocity parameter b. Solutions were also obtained for variable input-signal level. The assumptions made in deriving the large-signal equations are listed in Section 2.1. It is believed the chief limiting assumptions are that the el-ectric field is constant across the electron stream and that the spacecharge density has a sinusoidal variation with z but does not vary with radius. 6.2 Accuracy of the Results As discussed in Chapter III, the various errors which arise in the course of computation have been analyzed and found to affect the results as follows: 1. Truncation error was found to be insignificant for a Ay of 0.025 or less. 2. Round-off error, on the other hand, is insignificant for a Ay of 0.025 or more; 3. The value Ay = 0.025 was therefore selected as the optimum value to give a minimum rms error and require the least computation time for minimum error. 4. The analysis of truncation and round-off error showed the results to be accurate solutions of the differential equations to approximately 5 decimal places. 209

210 5. The solutions were found to be stable with respect to the propagation of error. 6. The number of representative electrons that must be followed through the interaction region depends on the magnitude of the space-charge parameter QC. For QC less than 0.5, 32 electrons are sufficient; but for 0.5 < QC < 1.0, 64 electrons are required. For QC > 1.0 it is necessary to follow an even larger number of electrons to obtain reasonably accurate results. 6.3 Sumary of the Results From the graphical analysis of the resulting data the following characteristics are notable: 1. The optimum relative injection velocity b increases with C for zero space charge (QC = 0). Typically, for C = 0.1 and QC = 0 the optimum b is 2.0. 2. When QC f 0 and C = 0.1 the optimum b decreases with increasing QC up to QC = 0.25, for which the optimum b is 1.25. However, it is expected that the optimum b will increase with QC beyond that point. If the experimentally determined optimum value of b is larger than that predicted by the curves of Chapter V, a possible explanation is that the potential depression in the electron stream due to the average space charge lowers the stream voltage significantly below the helix voltage. 3. In the absence of space-charge forces, the saturation gain was found to decrease with increasing C, but the saturation efficiency increases with C.

211 4. For zero space charge the maximum saturation efficiency increases rapidly with C up to approximately C = 0.1; beyond this value the increase in efficiency is slower. Hence there is a value of C around 0.1 to 0.12 beyond which the increase in maximum efficiency probably does not justify the corresponding loss in gain noted in (3) above. 5. The drop-out of each voltage gain vs. relative injection velocity curve is quite sharp, as in the small-signal case, and the value of b at which this drop of the gain to a very small value occurs is approximately that value of b for which the small-signal gain goes to zero. 6. At C = 0.1 the optimum tube length for QC between 0 and 0.25 is between 10 and 20 undisturbed wavelengths. 7. For small values of space charge and C = 0.1, the saturation gain and efficiency both increase with increasing QC up to about QC = 0.125 and then both decrease as QC increases further. 8. For a typical case, C = 0.1 and QC = 0.125, the saturation power output is approximately 3.75 db below the value predicted by the linear theory. The difference would be still greater for larger QC (see Figs. 5-lOa and 5-10b). 9. The presence of series loss along the helix has two important effects. First, the loss reduces the value of b for optimum gain, for example from 1.5 to 0.65 in the case of d = 2.0 (10.9 db/kg) for C = 0.1 and QC = 0.125.

212 Second, a series loss as high as d = 2.0 reduces the saturation gain to 18.3 percent of its value for d = 0 where C = 0.1, QC = 0.125, and b = 1.5. 10. The point at which loss is introduced at a distance of CN C 0.2, 0.3, or 0.4 from the input has little effect on the saturation gain for C = 0.1, QC = 0.125, and b = 0.65. 11. The curves of Chapters IV and V are very useful for designing and predicting the performance of largesignal traveling-wave amplifiers. 6.4 Suggestions for Further Research During the course of this investigation a number of points were touched upon which were aside from the main direction of this research but which seemed to hold promise for fruitful work. Several of the most interesting of these ideas are listed below. 1. In tubes with especially large stream diameters the effect on the saturation power output of variation of the field across the stream diameter is of particular interest. Equations can be derived to express this effect by dividing the electron stream up into several sections, but the resulting expressions are even more complex than the set solved for this dissertation and the computation time would be correspondingly great. Perhaps improved computer techniques will make this problem feasible in the near future, however. 2. Investigations of the effect on A(y) and @(y) of varying the stream diameter sb' and the ratio of the helix diameter to the stream diameter should produce worthwhile results.

213 3. Solutions for additional values of the gain parameter C and the space-charge parameter QC beyond the range investigated here would be of interest and usefulness as higher-power traveling-wave amplifiers are developed. 4. Loss calculations for many other values of the loss parameter d and various configurations of loss along the helix constitute an interesting area for further investigation. 5. Plots of power output Po vs. power input Pi for additional values of QC, C, and d would also provide useful information.

APPENDIX A. DERIVATION OF THE GENERAL TRANSMISSION-LINE EQUATION A. 1 Equivalent Circuit The interaction of an electron stream and a radio-frequency wave propagating along a helical slow-wave structure can be studied in terms of the simplified model shown in Fig. A-1. The electron velocity and the phase velocity of the radio-frequency wave are approximately equal and about onetenth the velocity of light; hence, relativistic effects may be neglected. The helical slow-wave structure is represented by a distributed-constant transmission line made up of series inductance and resistance and shunt capacitance. The inductance and capacitance of the equivalent line are chosen to match the phase velocity of the radio-frequency wave and the field strength of the longitudinal electric field acting on the electron stream. The transmission line is then divided into a number of lumpedconstant sections which are short compared to the helix wavelength Xg. Probably the best justification for using this simplified equivalent circuit for the helical slow-wave structure is that results obtained with such a circuit agree well with experimental evidence. A.2 Definition of Variables The following variables are used in the derivation of the general transmission-line equation: L = Ldo, the self induction per section Az = do, henries/m; Ro = Rdo, the resistance per section, ohms/m; Co = Cdo, the capacitance per section, farads/m; In = the current in Lo and Ro in section n, amp; qn = Pndo, the electric charge of the stream in section n, coulombs; 214

215 LINE CHARGE OF qn $n __I L. Lo I Ro o Ro Lo o o n-1 0 0 n 0 _ n + 0, Qn-2C C Qn Qn+ i V^ Azdg o o FIG. A-I GENERAL EQUIVALENT CIRCUIT OF A TRAVELING-WAVE TUBE

216 uo = the electron velocity in the stream, m/sec; n = the charge on the capacitance of section n, coulombs; Vn = the potential on the capacitance of section n, volts; qn = the charge brought on the capacitance of section n by the currents occurring in the line, coulombs; and Gn = a guiding section of the electron stream. A.3 Derivation of the Equation From the equivalent circuit described in section A.1, the following equations are obtained along the transmission line: n - Iqn (A-1) oIn Vni - Vn = Lo + RoIn (A-2) ot An -= n + qn = qn + Pndo (A-3) and n' V = CO (A-4) where RoI2 represents the dissipation in the transmission line. The electron stream has a charge of q' in the section Gn and thus induces a charge -qi on the helix wall of that section, which produces an additional charge qn on the capacitance of Gn. The time derivative of Eq. A-l is n aIn+l a an at at. t2 * (A-5)

217 Solving Eq. A-2 for aIn/6t gives aIn _ Vn-1 - Vn Ro at = L- L o In. n (A-6) By analogy, aIn+l Vn - Vn+1 Ro -5 t 1no -- T-; (A-7) Cdt L L, ~ LL In+1 Substituting Eqs. A-6 and A-7 into Eq. A-5 and collecting terms gives Vn-i + Vn+ - 2Vn Ro 2qn Lo + Lo (In+ - In) = t2 (A-8) Substituting Eq. A-4 into Eq. A-3 and taking the second time derivative yields 2 62 qn 62V pn ____ 9-n 2( dvn n = C- do (A-9) 6t2 6t2 6t2 Equating Eq. A-9 to Eq. A-8 and rearranging terms results in the following: V + Vn+ - 2Vn 2n Pn n-1 — - LC d -*2n I -Ln + R) ( (A-10) do2 at2 - ta do Equations A-l and A-5 may be utilized to rewrite the last term of Eq. A-10 as qn + + -2V In - In -L- - -. (A-11) dot a2 t at

218 The sections of the equivalent transmission line shown in Fig. A-1 are considered to be very short compared to the helix wavelength Xg. Thus in Eq. A-12 do is allowed to approach zero and the limit taken V(z,,t) LC (z,t) -L a2(zt) + RC (zt) T(z,t) 2 - LC 2 =-L at2 + RC R at (A-13) +RC't' at' (A-13) 6t 3t Substituting vo = = the undisturbed phase velocity of the line, m/sec, and Ez2(o) 7Z = L/C = the characteristic impedance of the 22 P lossless line, ohms, where Ez2(O) = the maximum value of the longitudinal electric field intensity on the axis of the helix carrying power P with a phase constant P, volts/m, reduces A-15 to 2 22 V(z,t) v 2 V(z,t) + R aV(z,t) at2 ~ az2 L at v Zo ( ) + Lp(zt. (A-14) 0 -0 t2 L OZ~ t In terms of the notation introduced by J. R. Pierce in Traveling-Wave Tubes8, the quantity R/L in the above equation may be replaced by 2a)Cd, where = angular frequency of the wave impressed on the helix, radians; C = the gain parameter defined by C3 = Irn ZoIo/2Uo2; Io = the d-c stream current, amp; r = q/m, the charge-to-mass ratio for the electron, coulombs/kg;

219 d = 0.01836 O/C, the loss factor; and I = the series loss expressed in db per undisturbed wavelength along the helix. Thus Eq. A-14 may be written V(z t) -v~ 2 V(zt) C V(zt) -..2 + 2oC d Ot2 6z2 6t = voZ 2(z + 2, dVZ P(z,t (A-15) =o v 6t2 + 2t For the case of a lossless line, Eq. A-15 reduces to v(zt) v2 V(zt) = Z o(zt) (A-16) tA2 0e2 t2 A similar expression for this special case has been derived by Brillouin.

APPENDIX B. DERIVATION OF THE SPACE-CHARGE FIELD EXPRESSION B.1 Underlying Assumptions A number of basic assumptions are essential to the derivation of a usable expression for the space-charge field in a traveling-wave amplifier. The assumptions underlying the derivation in section B.2 are as follows: 1. The radio-frequency wave impressed on the helix bunches the stream in such a manner that the space-charge density is constant in amplitude (i.e., has no radial variation) and varies sinusoidally with axial distance. (In calculating the space-charge forces it is assumed that the growth constant of the growing space-charge wave is small compared to unity.) 2. The relationships may be described using nonrelativistic mechanics; i.e., the squares of the ratios of the stream velocity uo and the wave velocity v to the velocity of light are small compared to unity, and hence the motion of the electrons is sufficiently slow that the formulas of electrostatics are valid. Thus, the solution will be found for the static stream of electrons with the same a-c charge distribution as the moving stream of electrons. 3. The electron stream is in a strong axial d-c magnetic field (rectilinear flow), so that the electrons are constrained to follow linear paths. Consequently there is considered to be no transverse motion and the stream boundary is smooth. 220

221 4. A sufficient quantity of positive ions is present to neutralize the average space charge. Assumption 1 is considered to be a reasonable supposition, the sine variation having been selected for its mathematical convenience. Assumption 2 is valid even for accelerating voltages of the order of 20 kv, while Assumption 4 does not appreciably limit the generality of the derivation. Assumption 3 is exact only for an infinite focusing field. However, this is one of the usual assumptions made in dealing with space-charge waves in electron streams and is probably valid in practice where the magnetic field is such that the cyclotron frequency is several times the electron plasma frequency. The effect of considering transverse motion of the electrons in wave propagation along an electron stream has been investigated by Rigrod and Lewis29 who found that the growth constant is somewhat larger for Brillouin flow (finite magnetic field) than for rectilinear flow (infinite magnetic field). B.2 Derivation of the Equation For purposes of calculating the space-charge field, the helical slowwave structure is replaced by a drift tube related to the helix as shown in Fig. B-l. The equations satisfied by a stationary charge distribution may be obtained directly from Maxwell's equations by setting all the derivatives with respect to time equal to zero. Hence, we may write at all regular points of the electrostatic field: V x E = (B-l) and v D = p. (B-2)

222 w w 0 w CU. QZ 0 z 0 0( u. Z e U,) r. U IC,) ~$ a ~ _c C- Cj______ vCD ~^\ ~ E

225 The conservative nature of the electrostatic field as indicated in Eq. B-1 is a necessary and sufficient condition for the existence of a scalar electric potential whose gradient is E. Hence, E = -V (B-3) Since the electron stream is homogeneous, V must be a solution of Poisson's equation within the stream; and between the stream boundary and the drift tube Laplace's equation must be satisfied. Within the electron stream Poisson's equation may be written as follows for the case of cylindrical geometry: 2 2V = 1 (lol V\ 1 6aV 62V -p(z) 72V r r \ r /+r Eb'2 r (B-4) where V(r,z) = scalar electric potential within the electron stream, volts; o(z) = instantaneous linear space-charge density, coulombs/m; and b' = radius of the electron stream, m. If it is assumed that there is no variation with 0, the polar angle about the axis, and the first term is expanded, Eq. B-4 becomes V72V(rz) 2V(rz) 1 6V( r,z) 1 ) V(r,z) p(z) V= r2 +r r z2 = - eb'2 = Ap(z)e-jpz (B-5) As discussed above, in the region between the electron stream and the drift tube Eq. B-5 reduces to Laplace's equation, 2V' (r,z) = 0. (B-6)

224 The solution of Poisson's equation (Eq. B-5) may be written directly as follows: V(r) ejZ = ap( + BIo(Pr e-jz (B-7) where Io(Pr) = the zero-order modified Bessel function, first kind, of argument fr, and = = phase constant determined by the impressed r-f wave on the helix, radians/m. The electric field intensity within the electron stream is given by Ez-(rz) V = - [V(r) e ] = Jo V(r) e-jPZ (B-8) Substituting for V(r) from Eq. B-7 gives Ez(rz) = j A (z) + BIo(Pr) e-z. (B-9) Equation B-9 represents the electric field intensity within the stream in terms of the space-charge density and the constants A and B, which are to be determined from the boundary conditions. The boundary conditions to be satisfied are that the radial and longitudinal components of the electric field intensity are continuous across the stream boundary and that the longitudinal component of electric field intensity is zero at the drift tube. B in Eq. B-9 can be evaluated from these boundary conditions, and Eq. B-9 written as

225 Jp(z)A____ I0(~r) Ez(r,z) = jp(z)A 1 -b' (I1(Pb') Ko(pa') + Io(Pa') Kl(pb')) e jPz (B-10) The space-charge field is obtained from Eq. B-10 by evaluating the field on the axis as follows: Ez(r=O) jp(z)A R2 e-jpz (B-ll) where R2 A 1 - jb' (IlblKoa' + IoaKlb') (B-12) Ioa Iub' - li(b') and Koa, - KO(Pa') For convenience in computation, the radian electron-plasma frequency ap is introduced. Physically cp is the natural oscillation frequency of an infinite plasma of uniform electron distribution30. The plasma radian frequency for an electron is defined as Mpa2 A I oli I (B-13) iECb'2u For typical electron streams in traveling-wave amplifiers, the plasma frequency is of the order of several hundred megacycles per second. Figure B-2 is a nomograph for computing ~cp from the stream parameters.

226 FIG. B- 2 NOMOGRAPH FOR OBTAINING RADIAN ELECTRONPLASMA FREQUENCY FROM THE VOLTAGE, CURRENT, AND DIAMETER OF AN ELECTRON STREAM. 300 290 22801 -25,000 270 260 250 240 -20,000 230 220 00 210 — 00 =10 (1~- X -19ot200 15,000 - --— 190 -50 180 -40 -30 10,000 ~~170 — 20 ~ 160 - 20 -150 - 10,000 10. 140 -9000 130 8000 5.0 1000- - -4.0 _ 120 -7000 - 3.0 2.-30_ — I 10 -- 6000 2.0 -_100 _5500 100 91. Q 100-^ -95 ^ -5000 4._i \ 0 t80 1 0- -- 90 -4500.50 1- -85.30 - - 75 - 3500 7.00 -70 3000.10 765 -6 0 ^^^ ^ ~ — 2500.05 -.04 -.03 1.0 I —.'03 1-0/ -2000.02 / -50.01 45 1500 -.005.10- -.004 -.003 -35.002 - 1000.001.01 _ 30 V0(kv) l (ma) 2b' (o.oo0 inch) T p (106 rad.)

227 Also defined is an effective plasma frequency, 2 A 2 n2 qn A (p2 Rn (B-14) where Rn 1 - (Il nb1,Kona + naKnb ) (B-) ona The quantity Rn is defined as the electron-plasma-frequency reduction factor for axial symmetry, and a plot of Eq. B-15 is shown in Fig. B-3. The reduction factor as calculated here for a stream within a helix is in good agreement with the calculations of Fletcher7 With the value of c q/) thus calculated, the small-signal space-charge parameter QC may be found for a particular value of C from Fig. B-4 or it may be calculated from the following relationship::=4QC3,3 (B-16) 1 + aq/( which when solved for QC gives 1 / 1 ~q/ co QC = 4 ____) * (B-17) Suggested new space-charge parameters are discussed in Section 2.1. Substituting Eqs. B-13 and B-14 into Eq. B-ll gives the space-charge field as o0 2 Es E en ( D + <t/2) |r'0 (B-18)

228 F 0 o Icr: - F Q r cr r) _ — --- --- --- --- _ w w I 0 H M- - J 0 I H W H' tL ^~0 a: l\rr~~, (Kz \ \ ITc 0 L) 0 a. C) w. I ~cx~~~c 0: n-o II C

229 1.0 2.0 3.0 2.0 1.5 L/ C= 1 C 0. I /C 0.2 C _ _0.3 0.5 _T_ -- --- FIG. B-4 l~o/!~~ /SPACE CHARGE PARAMETER VS. NORMALIZED EFFECTIVE ELECTRON PLASMA FREQUENCY \1'- --- --- ~ AND GAIN PARAMETER - 1.0 q 2.0 3.0 */w

230 In Eq. B-18, P, the phase constant of the space-charge waves in the stream, is written as v \Uo ( v(B-l9) From the small-signal calculations shown in Appendix C and Fig. B-5 it is seen that v = vo, particularly for large values of the space-charge parameter QC; therefore an approximate expression for P is () ( uo) - -u (1 + Cb). (B-20) The Fourier expansion of space-charge density given in Eq. 2-24 may be used to rewrite one factor of Eq. B-18 as j p e-j(no + t/2) ( /_n n/ n n=l.= I (/) Rn 1 ej(n2 + t/2) eJn(Y,,,)d~6 (B-21) UO ^ Cn 23 J 1 + 2Cu(y,' (B-21) n=l 0 Thus the space-charge field has been expanded into a Fourier series in the time variable ^(y,0o) at a particular point in space and will be evaluated for electrons within one cycle of 0. Actually the space-charge field pattern for the nearest neighboring cycle will be very like its own, but the ones further away may be very different. However it is believed that this assumption is sufficiently accurate for practical purposes, as the influence of space charge does not extend further than two or three cycles in either direction.

231 FIG. B-5 NORMALIZED VELOCITY OF THE SLOW-AND-GROWING WAVE VS. GAIN PARAMETER PARAMETER: SPACE CHARGE I.I 3.0 > \ 2.0 0.9 1.0 0.8 0.7 o /cuG 0.6 0 0.1 0.2 0.3 0.4 0.5

232 The distribution of electrons in space for constant time is very nearly the same as their distribution in time for a small interval of y even for relatively large a-c velocities, since it is the closely spaced electrons about 0 which are important in evaluating the space-charge force at 0 as indicated in Fig. B-6. In Eq. B-21, since the function under the integral sign converges uniformly to the sum, the order of summation and integration may be interchanged, as follows: oo 2' Pne -j(no + /2) ) n 1 n=l 2 2t~ oo uI C ljn(O-0' - j/2 2 F 1 do' P) - /2 Rn2 1 +d. (B-22) 0 0 n 2mcn 1+ 2Cu(y,0) 0 n=l Since a sinusoidal variation of space-charge density was postulated, this may be obtained by taking the real part of Eq. B-22, which gives Re pe-j(no + /2) (in)2 () Re y pne [I~ ~ ~~~nl OU P 27~n j1 + 2Cu(ya() (B-23) U(Po sin n('-n) Rn | 21 The space-charge-field weighting function F(0-0'), which weights the influence of an electron at 0' on the electron at 0 in determining the space-charge-field force, is shown in Fig. B-6 and defined as

253 z Z 0 I WI (j j t h iJ <D _ _ / /1 / / a r~ I U) CJ 2/ // 0 0// ^ / / / 6 ---- u /// /~~ a - - - // / - - - -~~~~~~~ <, B// / o s> _ _j//__ _ _ -- ^1 —

234 00 2 F(- ) 7A sin n(0-0') Rn (B-24) 2rtn n=l The form of the space-charge weighting function indicates that for (0-0') very much different from 0 or 2At the influence of space charge decreases rapidly. The closed form of Eq. B-24 is derived in Section 5.4.3. Equation B-23 may then be rewritten as follows: Re pn e-j (n + t/2)(qn 1 = -2o f F( o (B-25) 1n-, " " ~,~ n = J 1 + 2Cu(yo)' jn=l ) 0 Substitution of the right-hand side of Eq. B-25 into Eq. B-18 yields the desired expression for the space-charge field within the electron stream: 2 2 j in (l+Cb)2 J 1 + 2Cu(y,d ) (B-26) 0 An expression for the space-charge field has been derived in a similar manner by Poulter3.

APPENDIX C. CALCULATION OF THE SMALL-SIGNAL PROPAGATION CONSTANTS The initial conditions for the large-signal traveling-wave-amplifier problem are given by the solutions to the small-signal equations. The small-signal equations are therefore presented here and the method of solution is indicated. The notation used is that of Pierces, and it will be assumed that the reader is sufficiently familiar with the notation and the basic assumptions underlying the fundamental equations that a detailed derivation is unnecessary. The method of solving the small-signal determinantal equation will be outlined; in addition solutions will be given in terms of some newly defined parameters in order to convert the small-signal solutions to a more convenient form. In the small-signal analysis the following determinantal equation in the propagation constant r is obtained when the circuit and stream equations, which describe the action of the circuit on the stream and in turn the stream on the circuit, are combined: JI'oe rr (E2/p2P) jr 1: 2 Vo(Je - r )2 2(r - r2) 1 (C-1) The propagation constants -r and -r describing propagation respectively in the presence and absence of the electron stream are -r = - Jpe( + jcs) (C-2) and -r. = - ije(1 + Cb - jCd) * (C-3) 255

236 Substitution of Eqs. C-2 and C-3 into Eq. C-1 results in the expression 62 [1 + C(2jb - C62)][1 + c(b - jd)] 4C[ + C(2j8-C2)] (C-4) -b + jd + j5 + C(jbd - 2 +2 2 ) where = x + jy; x = growth constant of the wave; y = propagation constant of the wave; C = gain parameter; b = ~ - v = relative injection velocity; Cvo Uo = d-c beam velocity, m/sec; vo = undisturbed phase velocity of the wave, m/sec; d = loss factor; and QC= space-charge parameter defined by Q = Pe/ WC (E2/i32P) The four roots of the quartic equation, C-4, are the propagation constants of the four waves which are identifiable in a traveling-wave tube where the interaction is negligible. In order to show how the four wave solutions arise, Eq. C-4 has been rewritten by Brewer and Birdsall as follows: ( + f (6 - 7s (6 + Jb + d)(S - Jb - d j) v- 1 - *AQC^ 1 + _QC3/ C (6 -j)2 2C(l+Cb-jCd) c' 1 - 41Q3 (c-5) Solutions of Eq. C-5 for the growing wave, the unchanging wave, and the fast wave are given by Brewer and Birdsalll for a wide range of the parameters QC, C, b, and d.

237 Sensiper9 has derived explicit expressions for the small-signal propagation constants by expanding about the electron-beam propagation constant. From these expressions, values of the propagation constants may be computed readily, but the expressions given are valid only over a somewhat limited range of the parameters QC, C, b, and d. The initial conditions for the large-signal traveling-wave-tube analysis of this investigation were obtained, for the most part, from the solutions presented in References 1 and 4, quoted previously. In all cases where data were not available the solutions were found directly from the quartic determinantal equation C-4. In order to present the data obtained from the small-signal solutions in a more convenient form, gain curves have been constructed for several values of the parameters QC, C, and d with newly defined variables as follows: Vd = o - vo (C-6) and vo = v - v s (c-7) where vsg = phase velocity of the slow-and-growing wave. In terms of Pierce's notation these variables are O/vsg = 1-Cy (-8) UO/Vo = l+Cb (C-9) - lCy-1 (c-10)

238 Vd/vo = Cb.(C-ll) The small-signal gain of the traveling-wave tube is given by Gdb = A + BCN, (C-12) where A = total transition loss suffered by the signal at the input and output, db; N = length of tube in electron wavelengths; and B = 54.6x.. Hence Eq. C-12 may be written as db = A + 54.6x CN. (C-13) Dividing both sides of Eq. C-13 by 54.6N(l+Cb) gives Gdb A Cx 54.6N(l+Cb) 54.6N(1l+Cb) 1+Cb but N = N(l+Cb) = o (C-15) g 2i( where N = number of circuit wavelengths. Hence Eq. C-14 may now be rewritten as Gdb = ( + (C-16) 8.68 o 8.6P8'0 where = Cx (xc-17) and gngewl+Cb and ac = O: = gain constant of the growing wave. When new variables

239 Gdb g = 8.G (c-18) and a = 68 (C-18) are defined and substituted in Eq. C-16, it becomes! a= r + a X (c-19) Po Po Po Figures C-1 through C-6 show the variation of Ca/Po with vd/vo, using QC and C as parameters. Figures C-7 and C-8 similarly indicate the variation of g/Po with vd/vo for three values of Ng, using QC and C as parameters.

240 d i 0 0 _ _ 0 " — IS~~~~~~II ~ ~ ~ ~++ 0 —-'e - O ~~~__ __ __ __ __ __ __ ~~__ __^ ^^ ~~ c_ i --— ^^^^^^-^-,l"/ I i - -' = o i --- -- -- - 7 --- A' I-f —— t —f-t-t-+ —f —\-t —-t —-~t-r^z~" ci s+ j

241 N r00 d o NL Ir <J 0 + ~~~~0~~~~~~~~~~~~ ~0 ^ + O'O -- _ -— N —--- o. 0 a) _ _ _ _ _ _ __~ _ _ _ __^:.~-: _ I'' — 0 0 __ _ 0_ - ___ ___ ___ N (6 0 -0 0 01

242 r cu O ~ ~ ~ ~ ~ ~ ~ ~ ~~~~>') O ii 0 L' o 0,, 0 II ou~~~~~~ \ / N /\\ O~ 0+'/" \ \ /__\.,/?\ \ <o o \qy7~ a^q'I -, -7' 2 0i 0 -0

243 /7 CJ rl) + 0 O _ + 1\ o -— A —T -4 o \\ - -.- -s]^^ -- - _ __ _\ +\ CD f o ~ o \ \ \ \ +,.0 \ - 0+^ a\ + c^\ \\ 2 6 o q

244 N N o4o C) G iin~~ f " < 5 _ _ _ _ _ _ _ _ ____ _ I 0 + \ -, \ II I' 0 o o ~ i \* + __ _ _ _ __ _ _ _ _ __ _ _ _ _ _ O- \

245 0 (b 0 (r, 0 o (0 T >I 0 II II 00 0 N 0O ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ O 0P od o> o> I __ __ _ __ _ __ _ __ __ _ __ _ __zzzzzzzzziz * _ _ _ __ _ __ _ __ 1 7 _ _ _ _ _ _ _ _ _ _ _ _ _ - ^-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r ^zz z mII l z Iz 1r I? "zzzzz ^^zz"^. ~ _ __ _ _ _ _ ^_ _ _ _ _ _ _ _,~~~~~~~~~~~~~~~~~~~~~c _ _ _ _ _ _ _^_ _ _ _ _ _ _ _ _ -~~~~~~~ ~ -- -- - - - - -- - -- -- - -- - -- -

246 Ic o -- O C)~ —-------— _ - II __U 0_ ~o o' 0 z Z — Zo.,, 0 d 0- oo d o> o -I od 0 ======zz~~~~~f==z====.~~~~ --- -- -- 0 -- -- -- --- -- --- -- --- -- --- -- \ -- - -- *

247 C LO tI N ) 0 LLI 00 S ~~0 0X [XU~~~~t1 1 1 1 1 1 1 1 7~~~~~~~1 1 I l -T~ 0 NC6~ ~ ~ ~ ~~~~o [ —- -^^^ ^----- J —- ~ I c _ ____ \ \__ _ __ __ __<_ ao^ -- -- - -- - -- - -- -- - -- - - A -' - - L L n \~~~~~~~~~~~~ CM J

APPENDIX D. MIDAC PROGRAM AND OPERATING INSTRUCTIONS For the sake of completeness the various programs that were used in obtaining the solution to the problem on MIDAC are presented in this appendix. The following programs are included: 1. For 52 electrons (3l5MQm25,24): a) The operating instructions. b) The program used in the beginning of the problem with the following scaling: A.2-1.2"2, I2-4, and u-23 c) The program used from the point where G reached an absolute value greater than 5.75. This program has the following scaling: A.24, Q22-6, Z.2-6, and u.2-3 d) The program for the parameter tape. 2. For 64 electrons (3l5M15m7): a) The operating instructions. b) The program used, with the following scaling: A.2-4, Q*26, 0.2-6, and u-2-3 c) The program for the parameter tape. 5. The program and operating instructions for converting voltage gain into power gain in db (l5M12m2). 248

249 TRAVELING-WAVE TUBE: PROGRAM NO. 13MOm23,24 Operating Instructions for Run No. A. General Information The output of this program is a sequence of 4 groups of 3 words and 2 groups of 33 words. It can be determined if the program is proceeding correctly by examining the last 2 words in the groups of 3: the second word in each group of 3 should be a positive number and these numbers should increase continuously until a maximum is reached and the problem is terminated (see below), while the third word in each group of 5 should be a negative number and these numbers should increase negatively throughout the entire problem. If at any stage of the problem these numbers do not follow the right pattern, then the computer should be halted and computation resumed at a point where the numbers are of the correct form. B. Initial Computations 1. Insert 13MOml4. The memory sums should agree. The entire tape does not read in. Read in the last word on run. Halt the computer while the last words on this tape read in. 2. Place 13MOm25 on the Ferranti and press the start. The tape reads in and the memory sums should agree. Do not read in the last word. 3. Place the parameter tape 13MOp_ on the Ferranti and press the start button. Read in the last word on halt. 4. Patch Panel Settings: Decimal point at 1 Tab at 7 Carriage return at 11 5. Put the output switch on patch.

250 6. Put overflow on automatic. Put on run and press start. If everything works all right, run this problem until the second word in a group of 3 in the print-out reaches a maximum and starts to decrease. Let a final set of 2 groups of 33 words print out. 7. Place tape 13MOmlO (in the form of a loop) on the Ferranti and, starting after the first 2 groups of 33 in the print-out, do the following: While the last word in the second group of 33 words in the printout is being typed out, halt the computer. Clear the counters and insert one word from 001. Then put the computer on run and press the start button. This reads in tape no. 13MOmlO. This tape dumps the internal memory on the drum and then automatically resumes computation. Repeat this process every 10 or 15 minutes throughout the program. 8. When the last word in a group of 3 gets close to 5.75, the computer will halt after printing out the word f.fffff. When this happens, take the output off of patch and put on 1 column. Then clear the counters and place 13MOm24 on the Ferranti. Take overflow off automatic. Insert this tape in the standard manner. The memory sums should agree. Put overflow back on automatic and the output to patch. Read in the last word on run. This resumes computation again. After the next 2 groups of 33 print out, repeat 7. 9. Print-out should occur at least every 5 minutes. If the computer gets stuck in a loop (i.e., print-out does not occur as often as it should), halt the computer and read out the counter. If the counter is in the range of 08d to Oa5 then step through until

251 the counter reads Oa2 or Oa3. Then read out one word from 009. Insert into Of3 a positive word which is just larger (in the first significant hex digit) than the word just read out of 009. Put on run and press the start button. If the counter is not in this range, then run a changed-word post mortem on 13MOm20 or 13MOm24, depending on which program is currently in the memory. 10. If overflow occurs the computer will halt. Take the output off patch and read out the counter. Put back on patch and place 13MOmll on the Ferranti. Clear the counters and insert one word from 001. Put the computer on run and press the start button. Computation should resume. However, if the computer immediately prints out O.ObadO and halts, then reset the carriage, press the format reset button and press the start button again. If the same thing happens, try once more. If the same thing happens again, put the punch on and tape feed, insert, and process with 1-column output + lfl OOe 000 Of, which dumps the memory on tape. Then use a changed-word post mortem on either 13MOm23 or 13MOm24, depending on which is in the memory. IF OVERFLOW OCCURS MORE THAN 3 TIMES AT ANY ONE POINT IN THE PROGRAM, THEN DO STEP 11 BELOW AND QUIT. 11. If at any time during the program a short word prints out (except the f.fffff word mentioned in step 8) and the computer halts, then clear the counters, put 13MOmll on the Ferranti and insert the first word. Leave the computer on halt and step through until the counter reads 004. Then put on 1-column-output format, put the punch on and some tape feed, and insert and process the

252 word + lfl OOe 000 Of which punches out the mercury memory. Then take the punch off and use a changed-word post mortem on either 13MOm23 or 13MOm24, depending on which is the last program in the memory. 12. Halting the Computer for Purposes of Time: To store the program on the drum overnight, insert and process Ifl OOe 6c0 Of C. Resuming Computations 1. Read in tape 13MOm14 with the last word on run. 2. Insert and process lfl 6c0 OOe 00. 3. Put the overflow switch on automatic and set the patch panel as in section B4. 4. Jump to 068 and begin computing.

253 CONTROL CONTROL TRAVELING-WAVE AMPLIFIER PROGRAM NO. 1 EXPLANATION COMBINATION 32 ELECTRONS Scaling: A2 1, @2 2, -24and u-2'3 Program Tape No. 13MOnm23 AC 006 cd 238 _____T FA Bll 00 000, 000 su set up 000 ________ -6c01 -6c 01 -su clear tallys -001 003 B11 ba " B10 001 B09 fi Find constants FA B12 le -OeOS 2c05 -mr 2c05= 2Cu 2-4 _13d, 2c05 lc05 ad lc05=(1+2Cu)2'4 lc05, 27d -0e07 -sn Oe07=(1+2Cu)2-2 -001 033 B12 ba FA B17 I7e 8c01 8c01 ad |y + y _- y le06 3eO6 5c su 5c(Q1i+l-i) 2-2 5c______ 25d 8c sn8 2-1 I T le 8c 4c mr 4c=C( ) 2-1 _Oe06 2e066c s 6c (Ai+ Ai)2'1 6c le 6c mr 6c= C(" )2'1 10e 4c 7c su 7c-3 [y-C(Qji+1-i) j 2-1 FA AOO 5e | -OeO5 2c -mr 2c- 24yu2-4 -Oe07 2c cl -dv lc 1+2Cu 2 lc 5c lc SU lc= (ii+++2Cu 2). -- ----— l 1.. _ --- -- -,..... | lc 5l d cl sn lc- 24 lc -Oe02 -OeO1 -ad Store new 2 2-4| I 2-4 -001 033 AOO ba " FA B19 -0e02 4d le -ad lca 01 a T 2-4 B22 001 B25 - Fle base counter B22 -001 B25 -fi File base counter.________ i ________ ____________

2254 CONTROL COMBITEXPLANATION FA B25 llc lc llc su clear lic FA B20 Ic -Oe02 2c -su -2c=x=T-0,o 2 A37 001 A30 fi 2c=F (x) 2-6 -Oe07 2c 2c -dv - 2c= F() = Io 2-4 ___, i1+2Cuf 2c llc lc ad llc= h 2-4 -052 064 B20 ba llc= f 2-4 _ _ I _ _ _ _ _I -4 FA B21 lc -le02 2c -su 2c= x 2-4 A37 001 A30 fi 2c= F(x) 2-6 F(x) 2 -leO7' 2c 2c -dv- 2c= l2 2c 1lic l ad Ilc= / 2-4 -001 031 B21 ba lc= " 1lc 50d lc mr- llc= 1 2-4 FA B22 [o000 511 B23 ba eset C FA B23 9cl lc mr - lc= K.2-8 6c -Oe05 Ic -mr lc=c(Ai+ - Ai)cos 2-2 2e06 7c 2c mr 2c=Ai [4 —C(i-@) 22 i 2c -0e04 2c -mr 2c=A I t ] a in25-3 lc lid Ic sn lc= C(AI+l- Ai)cos0 2-8 2c 1 c c su- c= K2. 2-8 -Oe7 I -Oe03 2c -mr c= u(l+2Cu) 2-5 2 K -8 ___ 2c 21d 2c sn -c,,,, 2c c ad i _+ - Ai )o 2 -_____________I_........._________________________________

_~__________________ 255 CONTROL ~CO~MBINATIONTROL 255| EXPLANATION COMBINA TION lc L lc lc su lc= ( -K1 + K2 + K5) 2-8 lc 26d lc sn lc= " 2-5 -Oe07 lc -Oe03 -dv store u 2'3 -001 033 B19 ba FA B214 -OeOl 4d -Oe02 -ad tore new. "s le -OeO3 2c -mr 2c 2Cu 2-4 15d 2c Ic ad lc= (1+2Cu) 2-4 lc 27d -Oe07 -sn e07=( l+2Cu) 2-2 -001 033 B24 ba tore new (1+2Cu)2-2 6c01 25d 6c01 ad TA+ 1 -. T |7cOl! 25d 7cOl ad T+1 Tg 6c01 9e A04 cn jump if no print out 6c01, 6c01 6c01 su Clear tally 4d, 30d 000 ad set up 000 FA B27 064 1024m A05 ri all in P. 0. routine -001 064 B26 ba 002 35d 002 su 02 = A(m.s.) 4d 002 B28 cm s 0 < IA 0020 000 u reset 000 _ 000 000!003_ 003 su Clear 003 _______ Od 4 511 ad 511= 4 2 8c01 I4d 510 ad 10 y 2I A05 001 1A05 fia 002 510 000 ro out y _0020_ _ 0__ 051-0 -000 su reset 000 ______________________________ ___________________ _________________ _____________ ______________________________ ___________________I ___________

256 CONTROL COMBINATION___ | | ________EXPLANATION 0E06 4d 510 ad A - 510, i -i -..,.... 25d' 4d 511 ad 1 - 511 A05 001 1A05 fi _ 002, 510 000 ro read out A le06, 4d 510 ad 510 = Q 4d, 5d 511 su 511 = 2 A05 001 1A05 fi 002 I 510 000 ro read out @ -001 f 22d 000 ro 1 carriage return 7cO_______ 8e A04 cn Is TO < C 7c01 1 7c01 7c01 su clear tally FA B31 -0e02 1 4d 510 -ad 510 = 2 24 Od 1 4d 511 ad 511 = 4 A05 001 1A05 1 l fi 002 510 000 ro print out.,,, i.,, i. -001 055 B51 ba -001 f 22d 000 ro 1 carriage return FA B32 -Oe03 510 -ad 510 1 u _26d 4d 511 ad 511 = 3 A05 I 001 lA05 fi --- - — i i. i 002: 510 000 ro print out u -001' 033 B32 ba _ -0035 22d 000 ro 3 carriage returns FA A04 le06 i 6e51 2A04 cm Is I 01 < 3.75 001 28d 000 ro- Halt _i II _____I__III________I___________ t9____________ _________ _________ ____________________________ I9

257 CONTROL COMBINATION |. EXPLANATION FA A04 d1 4d 000ad set up 000 061 102m A05 ri- Call in sin-cos routine FA B39 -003 -003 -003 -su Clear 003.....010 -001 008 B39 ba FA B33 -A05 003 003 -ad Form memory sum _-_001, 061 B33 ba in 003 003 34d 003 su 003 = A(ms) 4d 1 003 B34 cm Is 0 < I I 000 000 000 su reset 000 FA A21 -OE02 4d lc -ad lc = 24 ____ I Ilc, 4d 2c ex 2c = sign of Od 4d lc ex lc = 10| 2-4 FAA2 I A03 cm Is I < 2 Ic t 19d lc su lc ( - 2 24 _______ 4d, Od A02 cn Jump FA A03 2c 4d lc ex lc - + 2-4 lc 25d 511 sn 511 23 _______ A05 001 1A05 fi 510usin021 511coso2 __________ 510 4d -0e04 -ad Store sino 2-1 511 1 4d -OE05 -ad Store coso 2'1 i -001 033 A21 ba d32 4d 000 ad setup 000 FA B35 064 1054m A05 ri- read in 64 inst. from drum.!* I_-. __ _ _ __ _ _ _ __ _ _ _ __ _ _ _ __ _ _ _ __ _ _ _ 1... _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _r_ _ _ _ _ _ _ u | ~~~~~~~~~~~~~~

258 CONTROL COMBINATION |_____________ EXPLANATION 003 003 003 su clear 003 __004 004 004 su clear 004 FA B36 -A05 003 003 -ad form memory sum -001, 064 B36 ba in 002, 003 _____ 003 35d 003 su 003 = A(m.s.) _4d, 003 B37 cm Is 0 < IAL 00000 000 000 su reset 000 FA A05 -leO4 lid 2c -sn 2c=sing 2-7 -leO5 I lid 3c -sn 3c=coso 2-7 i___n_____________ ______^______v S'in 2i5 -le07 3d 2c 2c -dv FI A 2A60c 5d | 2c sn |2c 27 2c 21d~~~~~~Cos 22n -e0 5d 3c -sn 3c O u 2-5 _i __ __ 2c 5c 5c ad 5c= 4( ~ yisin) 2-9 3c ___ 6 6cc _ ad 6c= 4( yjcos) 2-9 _____-002 032 A05 ba FA Ao6_| -2eO4 lid 2c -sn 2csin 2-7 _______ -2e0O5 l ld c -sn 3c-co80 2-7 ____-2eO7__-5___________2e0 2 2__c _c__ _ _________ ~~~~~i | 2c'__21d_ _|__2c_ _|__sn_ _|__2c= sn 2_ _5 -2eO7 3c 2 3c -dv | 3c + 2-5 ~2c ~ 21d 2c an 2c 28 -2e07 5c 3c -dv 5c" ^ __ 2-5 | 3c, 21d 3c sn 3c- n 2-8+ 2c 7c 7c ad 7c= 2( yisin) 2-9 3c 8c 8c ad 8c6 2(. yicos) 2-9 -002, 050 A06 ba ______________-2e05__ 5c_____ -----------------._____________ i __________________________~~~

~__________________ 259 CONTROL ~~~~~COMBINATION______EXPLANATION C OM BINA TI ON FA A07 -0e04 lid 2c -an 2ce sin$ 2-7 ~__________ -Oe05 lid 3c -sn 3cm coso 2"7 r ~~~~~~~~sin' ~__________ -Oe07 2c 2c -dv 2c 25 2c 3d 2c sn 2cw 2" 9 -0e07 3c 5c -dv 3c C ____________ __3 _______ lcan +20u "____ _c 3d 3c sn 3cm 2" 2c 9c 9c ad 9c= X yiin 2-9 ~___________ 3c _I e Ic ad 10c= yicos 2"9 -032 0614 A07 ba 1 ~~~~~~~~~h >C 2'^7 FA A08 -5c 12d -5c -mr 5cm3 2' -6c 12d -6c -mr 6cm ~________ ~-002 006 A08 ba 5c 7c 5c ad c (4L Xsin +2 X ) 2" ___________ 5c 9c 9c ad 9c (4Z +2 + X ) 2-7 6c 8c 6c ad 6cm (4 cos +2 ) 2-7 6c OIc i c ad 1Ocm(4Z+2X +Z ) 2-7 FA A09 IcOl -9c -Ic -mr c= 2Cd / sin 2-9 -9c 5d -9c -sn- 9/sin 2-9 -001 002 A09 ba cm2Cddfcos2"9,0cmf/cos2S lc l Oc 3c ad c=(/cos+2Cdfsin) 2-9 9c 2c 2c su C=( /sin-2Cdf/cos)2"9 2c01 3c ic mr- cL A 2-5 ~_________ 2c01 2c 2c mr- c=T 2-5 3e06~~~~~~~~~~ -6 ____ 53e06 3d 3c sn c.2 3e06 3c01 5c ad c(mil + ])2-6 Oe06 5c 5c mr- 5 3c 3 2-7-.___________________I_____________________________________

260 ____________ CONTROL COMBINATION EXPLANATION le06 le06 4c mr c 2 5c0ol le06 5c mr 5c~ ^ Gi 2-2 5c 8d 5c sn 5cm 2 24 410o 17d l 9c sn 9c~ (4c01) 2-4 2 2 ~~~ -4 4c 9c 4c Su 4c_ [ -_Y2 2 c ____4c 1 5c 4c ad 4c[" [N @j 24 Oe06 4c 4c mr 4c A1[ J25 2e06 3d 6c sn 6cm A1_, 2-5 Ic c __ Ic su lc_ (-A - ) 2-5 Ic 4c Ic ad lc-fl 2-5 5c01 5d 4c sn 4c~ 2-2 le06 4c 4c ad 4c~ [ i + I2-2 _Ic 4c 5c mr 5c" n [ A+ -J 2-7 Ic r~~~~5 c I Q 5c 8d 5c sn 5c 2 2-9 5c 5d 5e sn 3cm T.2-9 2c 3d 2c sn 2c= T.2-9 3c 2c 2c su 2c= ( - T) 29 FA A10 2c 5c 2c su- 2cm U 29 _____________~~~~~~~~_________________________ ____I_____________________ —-----------------.________~~~~~~~~~~~~~_________________________ i__________ —-----

261 CONTROL COMBINATION EXPLANATION D_3 3 4d 000 ad set up 000 FA E35 042 1062m A05 ri call in instructions lOc, lOc lOc su clear lOc llc lie llc su clear lle FA E31 -A05 1 lOc lOc -ad form memory sum -001, 042 E31 ba in lOc 10c 1 58d lOc su 10c= m.s. _____ 4d lOc E30 cm Is 0 < | A 000 g 000 000 su reset 000 4d' Od A05 cm jump to A05 3840m, 000 000 ri Dummy inst. FA D30 4d Od lb26 cn constant instr. FA D51 4d, Od lb33 cn const. instr. FA D32 4d Od lb36 cn const. instr. FA D33 4d Od le51 cn const. instr., _______ ^^__ _ __ ~~ ~._ _. i _ _ _ ii_ t_... I - I 11I.... 1i....... i..................... —-----------

262 CONTROL COMBINATION BEGINNING OF INSTRUCTIONS FOR DRUM EXPLANATION FA A05 4c 4c 5c mr 5c=. 4 + )2 26 _5_c l6a 3c su 5c= [ " _ -1] 2-6 Oe06 5c 5c mr 5c7= Ai [27 lc _ 8d 3c sn 3c= 211 27 3c 5c 5c ad- 3c= Z 2'7 __________ Oe06 99d 6c mr 6c 6Ai 2_ 6c 4c 6c mr- 6c= Y 26 T I ______________________________ ____________ ~ ~~~~~~ ________,_ _ _^ _ _.____________________________________ __________________________________ ________________________________ --------- I

265 CONTROL COMBINATION NEWTON RAPHSON METHOD EXPLANATION leO6 4d 5c ad 5c. G 2'2 5c OeO6 7c mr 7c= XG 2-4 5, 5c 8c mr 8ce G2 2'4 7c 8c 7c mr 7c= x3 2-8 -6c 8c 99c l T 9cm Y2 2-10I 7c 9c c 7c su 7c= ( xa3 - Y@2) 2-10 __7c 25d 7c sn 7c=.2-9 ___c, c 9c mr 10c= Z 2-9 ___| 7c I 1c 7c ad |7c= (X3 - yo2 + zO) 2-9 7c 2c 7c ad 7c= f(Q) 2-9 5c 6c lOc mr l0c 2YQ 2-9 OeO6 9d 9c mr 9cm 3X 2-4 9c 8c 9c mr 9c= 3XG2 228 9c 8d 9c sn 9c= " 2-9 _ 9 _c ClOc 9c su 9c= (5XQ2 -2YQ) 2-9 |_____ 9 ___ 9 927d 9c n9c n 2-72 9c c 9c ad 9cm f (Q) 2-7 9c 7c 7c dv 7cm f/f' 2-2 I i+l 5c 7c 7c | u 7c- new @ (i+l) 2-2 7c @ 5c e D i_,~........ _7c 1 4d 5c ad 5c= fQu 29._____, _____c'. 1AO ___ UMP'~~~~~~~~~ Oe O 6 ____ t 9d 9 c r 9= 32i~ ~ ~ ~ ~ ~~C.... cn,jump,,......_________________________.......__________________' ______...._________ ______________............__________________________________ ___________________________ ~ ~ ~ ~ ~ i _____________________.. _____ ______________________ I ___________ __________ 9c___________. e9c _____________ 9c=_______ __X______ 2'8_

264 CONTROL ~~~~~~~COMBINATION___ lEXPLANATION COMBINATION Oe06 7c 9c mr 9c=- AiQi+1 2-3 |_____ 9c c 7 ClOc mr lOc= Ai2 2-5 2 4c__________C 9c 3-c Siu 30cl [A^Q 2] 2-5 | 4c, 9c 9c mr |9c Ai(Gi + c)i+ ) 2 ___ lOc 9c 10c su 10c= [ AjQii+ -2"] 2-5 10c I c 10c ad 10c= Ai+ 2'5 lOc Od 6c sn 6c= Ai+l 231 -Oe06 4d -2e06 -ad Ai+l Ai _____ _. 1, i =~I |-6c 4d -Oe06 -ad A+2 — A -001 3 002 38AO5 ba 9 i+ - Qi i+2- -9i 1 4d, Od B17 cn Jump I I______1__1__1__1____ I T __________ i __________________I_____I_ c X r. X ] X l E~~~~~~~~~~~~~~~~~~~~~~~~~~~~ [ T 1 L I 1 L~~~~~~~~~~~~~~~~~~~ [ I, I I I L~~~~~~~~~~~~~~~~~~~ I 1 1 4 r x L~~~~~~~~~~, I F T L~~~~~~~~~~~~~~~~~ 4 I! I l I I~~~~~~~~~~~~~~~~~ I I T I I j~~~~~~~~~~~~~~~~~~~~ I I, -T 1 1 j~~~~~~~~~~~~~~~ F - 1 —- 1 1 1 X~~~~~~~~~~~~~~~~~,,...~~~~~~~~~~~~~ [ [ 2 T T l I~~~~~~~~~~~~~~~~~~~ L H, l l l,~~~~~~~~~~~~~~~~~ L-_ [ ~~~~~~~~ ~ ~~~~~~~~~i 1 1 Z r T [ r j~~~~~~~~~~~~~~~~

265 CONTROL COMBINATION |_____________ EXPLANATION FA A30 3c 353c su- clear 3c 2c 4d 28d ex sign of x onto fffff FA A31 ld 4d 2c ex- I x 2-4 FA A52 2c 19d A33 cn Is x < 2:r 2c 19d 2c su x - 2A - x FA A33 2c 530d A54 cn is x < t 24 19d 2c 2c su- 2t - x -- x 28d 29d 28d mr- 28d= + 1 FA A54 2c 37d A37 cm Is xI 2-4 < 2c 27d 2c sn 2c X 22 _______ A36 -001 A35 -fi File CB FA A35 -lle 5c 5c -ad- c 26 _35c 2c 5c mr- 5c= Xa=24 -001 | _~5 005 A55 ba 5c=(a5x5+a4X4+.alX) 2-4 __r___ _ 3_5c 16e 3c ad- 3c= F(X) 2-4 5c 28c& 3c mr 3c= + F(X) ] T 5 ~3c 5d 2c sn 2c= F(X) 2-6 4d4' 29d 28d su Reset fff FA A56 [-000] | 511 A57 ba Reset CB FA A37 4 04d Od [000] cn jump I I ________________________________T__I____________I_______________I_______________________ _______________________________________ I__________________ __________________ __________________________________________________________

~__________________ 266_________ CONTROL COMBINATION EXPLANATION FA B28 25d 003 003 ad 1 - Tpo ~__________005 26d B27 cn IsT0 < 5 FA B29 001 B27 000 ro- error halt FA B54 25d 004 004 ad 1- Tg 004 26d A04 cn Is Tsc < 5 001 A04 000 ro- error halt FA Bj7 25d 1 04 004 ad 1-Tj 004 26d B35 cn Is T < 5 001 B55 000 ro- error halt FA E30 25d lic lic ad 1- T lie 26d E35 cn Is Tj < 5 001 E30 000 ro- Error halt FA A19 000 1 000 100 ri N.R. comp. const. T I _________________________ _____________________________________ ____________________________ B___ ___ _________.9

267 CONTROL CONTROL EXPLANATION COMBINATION CONSTANTS EXPLANATION FA DOO DA 036 000 000 000 04 Od 02 ld -000 000 oob 08 2d -000 000 Dc 04 3d -o 4d -ooo ooo ooo 05 |d -000 000 000 02 5d 000 000 0ooo[6 ____ ______ 6d__________________ 000 000 000105 7d -000 ooo oo o01 8d cO g 9d= 3.2-2 -000 000 oo~~ 05 10d -000 000 00oo 06 lld. _450 547 9e9i58 12d nt/48 22 10 1 ||13d _ _ | c-90 fda a22' l6|| 14d- i 2-2 -20! I |15d I ________04 ______________ 16d _______1 000 000 00~c0 8 ||_________ 17d_ 000 000 000Oc 18d [___-_____ OO O6487ed511ob ___ 19d = 2n 2 0004 0 20d ___-000_____ I _....0004.o......____________2_________________ 00000 oo0 o0 25d | _~~~-000 ~000 0o 06 26d i -- ___________OP OP OO; 01......................_______ i ~ ~ ~ ~ 00000 0052d_______

268 _________________ CONTROL COMBINATION EXPLANATION 000 000 000 02 27d ____________ fff fff fff: ff____ 28d( ~~___~___ _-Ifff fff Ifff ff ____ 29d ~_________3_24 5f6 a88 86 ____ 50d3 ^ 2 4 h -000 000 00) 09 31d ~___~~__OC9 OFd aa _2212d _________________ 2'6 _____-3ad 330619_62 3_____ 53d = M.S. for P.O. 644ed9_abfa9 54d(1= M.S. for s-c ~~d_______do060b2449ed____ 5d= M.S. for 64 instr. 00 000 000Oa __________ 56d 00 000 00010 c ___ 57d e092246adfc ___________ ______ 8 = M.S. for 42 instr. ~~~~~~~~~~~ _____________ ____________________ ________ ______ ___________________ __________________________________ ______ i _____________________________

269 CONTROL COMBINATION EXPLANATION FA B09 (E07) le 3e lc mr lc= Cd 2-2 lc lcOl ad lcOl = 2Cd 2-2 _6e I 6e lc mr lc= (Ay)228 l4d le c mr c= C 2-2 2c 5d 4c sn 4c= (1+Cb) 2'6 3c 4c 3c dv 3c= 2 4 Ic 3c 2c01 mr- 2c01= (Ay)2" 24 2e 3d 35c sn c b 28 le 5c 5c dv 3c= [b/C] 2-8 _|13d 2c 5c ad 5c= (2+Cb) 2'4 3c I 3c 3 r 3c= (4)2 b/C 2~ 3c 5c 3c mr 35c n (2+Cb) 24 3c 15d 4c01 su 4c0l1 [ = -22-4 4c01 2d 4cO1 sn 4c01 = " 212 Xr 4 2c 5e 7c dv 7c= 2,3d' 2c 5c ad 5c=.(.2I+Cb2 __ __ _ __ ____ _ __________ _________ ____ I___________________ I ______) I __0 - i~~ ~ ~~~ l~ 4cr su4ol -2] ~~........................

270 __________ CONTROL TI COMBINATION EXPLNATION le I le 89 mr 8c= c 2~ 4e e 9c mr 9c [2p/o ] 2 8c 9c 9c dv 9cV [ P/CIo j 2 2-8 ~_________ _7c 9c 9cOl mr 9cOl= - [wp/co] 2 2-4 le 5e 5c01 dv 5c01= Ay/C 20 2c 2c 2c mr 2c~ (1+Cb)2 28 5c 6e 5c mr 5cP dAy22 2c 3c 2c mr 2c= dAy(l+Cb) 226 16d 2c 2c su 2cm [ 1 ]2__________5c01 2c 2c mr 2cm Ay/C [ 2-6 2c 2c 3c01 ad 2c01 a6y/C[ ]26 FA B10 4d Od [0001] cn Jump 000 000 000 ri Dummy order 000 000 000 ri 000 000 000 ri ~__________ 000 i 000 000 ri _____________________________~~I__________________________________.____________~~~~~~~~~~~__________________________ ____I _________________________________________I

271 CONTROL COMBINATION___ __________EXPLANATION FA e ol +1 DA 033 DA 0535 _________________ ________________________________ FA eO2 | 2-4 DA 033______________________ ____ FA eOT sino 2-1 DA 033 FA eO5 _cos08 2-1 DA 033 AC B09 _ FA e07 (l+2cu) 22 AC 33eO3 FA eO6 1 Ai2 122-_______________ 1O____A FA 5e061 _1 _o _ _o AC 3eO6, FA eOO I Parameters AC 16e00 _ _ FA cOl | __Tr_ temporaries AC 9c.01 __l__,___ FA c05 temporaries AC 000 FA cOO, BC B ll____ i T'i ______' 2-1__ _ _2_ _ - ______________________I_____________ ___________________________________________

272 CONTROL COMBINATION PARAMETERS EXPLANATION. I le00 C 2 2e00 b 2 3e00 d 2-2'I __ 04eoo |'::p/c) 2 -4___ I 5e00 Ay 20 6e00 Ay 24 7e00 4y 2-4 8eOO', u comparison const 2-44 9e000 A, u comparison constat 2 lOeOO Ay 2-1 lleOO a 26 12e00 84 24 2 13eOO a 22 a2 2 l4eoo., 15e00 a 2 2 ______ _ _____ _ 1______________ ___________ I0.2-2; |l__ Ao.2 *2-4 -----------------— ________ ___________________ I.i'' I I

___________________ 273____________ CONTROL COMBINATION TEMPORARIESEXPLANATION IcOl 2Cd 2'2 I (2cC] ()2 4)1 2c~~~~~l ~~2 5c01 24y/C [i - dAy(l+Cb) 2) 2"6 4c01l (_)2(b/C) (2+Cb) -2 22_______ 5c0l 4Y/C 2 I_________ 6c01 Tally for A, Q print ot________ 7c01 Tally for ll, u print o t 8c01]. _y 2-4____________________ _________________________ 8c01 7 p O 224 lco5 ____ ______________________________ _____________ gc~l T~AV /iC 2 2-4 gcol w 2co5 ____________ lcOO _______________________ _________________________________ 1coo 3cOO___________ ___________________________ 4+c00_____________ ______________________________________ 5c00 6c00 ________ ___ _________________________ 8c00 _ _ __ _ _ _ 6coo j 9c00 9c00 _____________________________________________________ llcCOO __________________________________________________ -----------— ____________________ I _______________ —--------— ________________________________________ 11c1________________________________ _ ______________________________________________________________.________________________ _________________________________________ ________

274 CONTROL TRAVELING-WAVE AMPLIFIER PROGRAM NO. 2 EL COMBINATION 32 ELECTRONS EXPLANATION Scaling: A.2k, 0.2-6, 2"6 and u.2-' Program Tape No. 15MOm24 AC 006 CD 238 _ _ _ _ _ _ _ FA BU 000 000 000 su set p 000 ~_________ _ —6col -6col -6col -BU clear tallys ~___________ -001 003 Bl ba n _____________ B10 0031 B09 fi Find constants FA B12 le* -OeO 2c05 -mr 2c0 Cu 24 _ ~___________ ___15_ __2co5 lc05 ad lc5O(1+2Cu)2'4 ___________lc01c05 27d O-e07 -n eO7(1+2Cu)2-2 ~___________-001 033 B12 ba_________ FA B17 7e 8col01 8col01 ad y + YY _le06 r3e06 5c su 5=(Qi+l -i) 2'6 S_____________ 5c 17d 8c sn 8c " 2-1 lIe 8c 4e mr 4cC( )2-1 e06 2e06 6c su 6cm (Ai+l-Ai)2-4 6c I 26d 6c sn 6c=( )21 __~_________6c le 6c mr 6c= C ( )2-1 10Oe 4c 7c su 7c= [-C(Qi+-Qi)] 2-1 FA AOO 5e -e05 2c -mr 2cm 24y12-4 -0e07 2c le -dv lc. 2-2 _______- ~-~_^~ ___ _ ^1+2Cu ~___________ iC Ic sn -cI 2-6 le~ ~ ~ ~ ~ ~~~~~/ Q,i2Ayu \ 6 Ic _ fc_____ Ic Su _C___at_ _ 2_ ____________ 000 000 000 s Dummy order _Ic _ -0e02 -0e01 -ad tore new 0 2-6 -00). 033 AOO ba ________________0____1_____________ _______ AO a " _______

275 CONTROL COMBINATION EXPLANATION FA B19 -0e02 4d lc -ad lc= iJ j T B22 -001 B25 -fi File base counter FA B25_ lHlce lec l c su clear llc FA B20 lc -Oe02 2c -su- 2ci x- T-ij,0 2 A37' 001 A30 fi 2cm F(x) 2-6 I' F(x) -4 ______ -Oe07 i 2c 2c -dv - 2c= mj = Io 2 ____I cl la2c 1 1c l ad | 2-4 -032 064 B20 ba llcC 24 FA B21 1c \ -le02 2c -su 2cw x 2'6 A37 i 001 A30 fi 2c- F (x) 2-6 |-_le07! 2c 2c -dv | 2c- F(x) 2_-4 I' 1 r 2-4 ______________ | -001! 031 B21 ba lc-m n __________ lic 30d 11c mr lc 2-4 FA B22 |[oo00] I 511 B2 ba Reset CB FA B23 9c01 c l lle mr - llce K1 2 6c I -e0O5 Ic -mr Ic0 (Aji1- Aj) cosi 2-2 __________ t2e06 i 7c 2c mr 2cm Ai [~y-C(~i+-i ) ]25 __________ 2c -0e04 2c -mr 2cm Aj [ n ] sin2 6 2c j d 2c sn 2cm 2-8..____..I lc l ld I. c sn lcm C(A+ Aj)cosX 2-8 ____ _ 2c Ic Ic su lc K2 2-8 ___._.,_-Oe07 -OeO3 2c -mr 2c_ u(1+2Cu) 2 _2c 21d 2 c 2 2c n 2c= K3 2-8 le 2c lc ad Le. R + K3) 2-8 __________. lIc, 2c Ic ad lc- (K2 K) - I A B 3 9 0 1 1 r —--— 11c=_____________K1_________2 _

276 CONTROL COMBINATION EXPLANATION lc' 1 1 lc.. l. lic su lc= ( -K1+K2+K5) 2-8.lc, 26d lc sn lc 2 I.., I.. -0e07 I lc -Oe03 -dv store u 2-3; {___ ____-001 033 B19 ba FA B24 -OE01 4d -Oe02 -ad Store new i's le i -OeO3 2c -mr 2c= 2Cu 2-4 13d 2c lc ad lc= (l+2Cu) 2-4 ______ _ I Ic I 27d -Oe07 -sn Oe07=( 1+2Cu) 2-2 -001 055 B24 ba Store new (1+2Cu) 2-2 6cO01 25d 6c01 ad TA+1 TA:i I T ____ _ __ 7c01 25d 7c01 ad! — T 6c01 I 9e A04 cn jump if no print out ___________ 6c01 6c01 6cl0 su Clear tally 4d 50d 000 ad Set up 000 FA B27 064 1024m A05 ri Call in P. 0. routine 002 002 002 su Clear 002 003 003 003 su Clear 003 FA B26 -A05 002 002 -ad Form memory sum in -001 064 B26 ba 002 002 5 3d 002 su 002 = A(m.s.) 4d 002 B28 cm Is 0 < IA1 ______ 000 000 000 su reset 000 r___..... OId 4d 511 ad 511. 4 2-44 8c01 4d 510 ad 10 = y 2-4 AO05 001 1A05 fi ______... _______________.... ------—.. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __._ _ _ _ _ _ _ _ _ _ __I_ _ _ I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __I_ _ _ _ _ _ _ _ _ _ _ __I_ _ _ _ _ _ _ _ _ _ _ __I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

277 CONTROL COMBINATION___ | | ________EXPLANATION 002 510 000 ro read out y O________ E06 4d 510 ad A -IL 510 Od, 4d 511 ad 4 - 511 A05, 001 1A05 fi [_____ _O002 5 10 000 ro read out A leO6 1 4d 510 ad 510 = O __ A05 1 001 1A05 fi___ _ 002 1 510 000 ro read out 9 -001 T 22d 000 ro 1 carriage return 7c01' 8e A04 cn Is To < C 7c01 7cO1 7c01 su clear tally FA B351 -Oe02 I 4d 510 -ad 510 2-6 6d 4d 511 ad 511 = 6 A05 001 1A05 fi,I' 1 - _-I ____ _-_-__ 002, 510 000 ro print out u -001 033 B31 ba -001 22d 000 ro 1 carriage return FA B32 -OeO3 4d 510 ad 510 = u | 26d _ 4d 511 ad 511 = 3__ A05 I 001 1A05 fi _____ 002 510 000 ro print out u -001 033 B32 ba -003 22d 000 ro 3 carriage returns ____________ r_______ L _______ ______ ______ __________________:_____ ____r___r__

278 CONTROL COMBINATION _|_EXPLANATION FA A04 31d 4d 000 ad set up 000 061 1032m A05 ri- Call in sin-cos routine FA B59 -003 -003 -003 -su clear 003..... 010 -001, 008 B39 ba FA B33 -A05 0 0033 -ad Form memory sum _-001 061 B33 ba in 003 003 54d 005 su 003 = A(ms) 4d 003 B34 cm Is O < |A 000 000 000 su reset 000 FA A21 -OE02 4d lc -ad lc = 0 2'6 lc 4d 2c ex 2c = sign of -Od 4d lc ex lc 10 | 2-6 FA A02 c 19d A05 cm Is I S I < 2x __{__ _lc 19 1 lc su lc = ( 0 - 2A ) 2-6 4d Od A02 cn jump FA A053 2c 4d lc ex lc = + 0 2'6 _-i. _Ilc I 26d 511 sn 11= 2-3 A05 001 1A05 fi 510 sin02-l 511=cos02-1 __________ 10 I 4d -0e04 -ad Store sin0 2-1 5113 4d -OE05 -ad Store cos0 2-1 -001 033 A21 ba d32 4d 000 ad setup 000 FA B35 064 1040m A05 ri- read in 64 inst. from dru

1 __________________279 ___ CONTROL EXP LANA TION COMBINATION EXPLANATION 003 0 003 005 su clear 003 004 004 004 su clear 004 FA B56 -A05 0 0000 -ad form memory sum -001 o64 B36 ba in 002, 003 003 5 35d 003 su 003 - A(m.s.) 4d 005 B37 cm Is 0 < |IA 000 000 000 su reset 000 FA A05 -leO4 31d 2c -sn 2c= sino 2-10 -leO5 31d 3c -sn 5c= cos2 2-10 T_-le07_7' 2c 2c -dv 2c Si 2-8 -leO7' c 3c -dv c 1+2Cu 2c 5d 2c sn 2c = 2-10 cos 2-10 3c, 5d 3c sn 3c =n 2-10u 2c 5c 5c ad 5c = 4( yisin) 2-12._.3c 66c 6c ad 6c = 4( yjcos) 2-12 ____ -002 032 A05 ba FA A06 -2e04 31d 2c -sn 2csin( 2-10 -2e05 31d 3c -sn c= coso 2-10 ____ -2e07' 2 c 2c -dv 2c 2in 2-8 2c 21d 2c sn 2p 2-11 _-2eI 2c 3c -dv c= 2c 21d 2c sn 2c " 2-11 __ 53c 8c 8c ad = 2( C y cos) 2-12 3c 2 1d 3c s c 2-11 -002 8 030 A06 ba.: io h - _ — - -_, ______________________________________I __________________ __________________ ________________________________________________________

280 CONTROL COMBINATION EXPLANATION FA A07 -Oe04 31d 2c -sn 2c= sine 210 |___ ___-Oe05' 31d 3c -sn 3c= cos_ 2-10, ______________ 2c 3d 2c an 2c= 212 -Oe07 3c 5c -dv 3c l+S 2-8 c' 3d 3c sn 3c= " 212 _2c 9c 9c ad 9c ysin 212 i-i1 3c 1lc I c ad 10c= yicos 212 ____i_______ _ 4 -032 o 064 A07 ba ffI~ | | l ~ ~ h 4 2-10 FA A08 -5c 12d -5c -mr 5c 2-10s J__^___ _ 9-6c 12d -6c -mr 6c i________|__ -002 006 A08 ba 5c 7c 5c ad 5cZ (4 Z sin +2 X ) 2___ _c_ _9c __9c ad 9c=(4 E +2 + ) 2-1i _6c I 8c 6c ad 6c= (4X cos +2 ) 2-1 ___6c CIc I| c | ad 10c= (4 Z +2 X + Z ) 2 10 FA A09 lcOl -9c -Ic -mr Ic a 2cd sin 2-12 _-9c 5d -9c -sn- 9C= I sin 2-12 ______ -001 002 A09 ba 2ca2cd Isin2-12,10c-. __________________, ____f I.../cos2-12,j.5__|_lc I lOc 3c ad 3c= ( cos+2cd/sin) 2-c L____.... 9c 2c T 2c su 2c= ( sin-2c4 /cos) 2" 2c01 3c Ic mr- lc= A 28 2c01Ol 2c 2c mr- 2c- T 2-8 |___" __|_e06 _c1 _|_c |ead 3c= ( Z-l +2T,, i 1)2-6 LI___ ___Oe06' 3c 3c mr- 3c= 2-10o. - - -__ -_ i__________________________

281 CONTROL COMBINATION EXPLANATION le06 le06 4c mr 4cm Q 212,__,___ | _5cO1 1le06 5c mr 5c= 2-6 5fc _I 10dc 5c sn 5cm 2 " 2-12 4c, 4co, 4c su 4c [ @2 ().. ] 212 ____4c 5| 4cc ad 4cm [ n"+ i ] 2-12 Oe06, 4c 4c mr 4cm A [ ] 2-16 ___Ic 3d Ilc sn lc= A 2-12 2e06 2d 6c sn 6c= Ai-1 2'12 Ic, 6c Ic su lc- (A Ai-1) 2-12 Ic 3d Ic sn lc= (A -Ai.1) 2'16 _lc, 4c lc ad lc= I 2-16,._______ lc 1 27d lc05 sn lc05 II 2'14,.._, L 2-6 5c01 lid 4c sn f 4c= C |__ _ |,le06 4c 4c ad 4cm [ + ] 26.__.__!clc05, 4c 5c mr 5c n [ + ] 2-20..c. 5c 5c ad 5c= 2 " 2-20 3c 36d 36 c sn 3c/ ___2-20 ___1____ 2c, 37d 2c sn 2c= T 2'20 3___c 2c 2c cu 2c ( - T ) 2-20 FA A10 2c 5c 2c su - 2c U 220 _______ D33 4d 000 ad set up 000 FA E35 043 1048m A05 ri call in instructions lOc I lc Ilc su clear lOc llc llc llc su clear lic _.. 1_____________________________________ I, I_, I.II__ t _____________ ___________ ___________

282 CONTROL COMBINATION EXPLANATION FA E31 -A05 10 c 1c -ad form memory sum -001 043 E31 ba in 10c 10c, 38d 10c su 10c= m.s. 14d I L c E30 cm 0 < | A 000 00 000 0 su reset 000 _4d, Od A05 cm Jump to A05 FA D30 4d I d lb26 cn constant instr. FA D31 4d OOd lb33 cn const. instr. FA D32 4d Od 1B56 cn const. instr. I i [ T_! I____ [I F A D,) II1' i_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ BI._________________ t ________________________________I

283 CONTROL COMBINATION BEGINNING OF INSTRUCTIONS FOR DRUM EXPLANATION FA AO5 4c 4c 5c mr 5c4( 2-1 | 5c 20d 5c su 5 [ -1] 2-14 OeO6 I 5c 5c mr 5c= 4[ ]2-" 5c Od 5c an 5c n 2'14 _lc0' lc5 3 c ad 21 2-14 3_c 5c 5c ad - 3c_ Z 2-14 _ _ Oe06 9d 6c mr 6c- 6Aj 2-7 6c 4_4c 6c mr- 6c= Y 2-13 i T I i'~ ~~~ I 1 1 ~I I i —--- I I - I I — [ I I [_____________________________ T_____1_____1____T_______.______T______ i ________________T_________ t T I I I 0X~~~~~~~~~~~~~~~~~~~ [ I l I I 1 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.. L l l l l ~~~~~~~~~~~~~~~~~~~~I t I | T I I j~~~~~~~~~~~~

284 CONTROL CONTROL l N~wl~l: APESON EIEOD | EXPLANATION COMBINATION NEWTON RAPHSON METHOD EXPLANATION le06 4d 5c ad 5c= @ 2-6 5c oe06 7c mr 7c" XG 2-11 5c 5c 8c mr 8c. 92 2-12 7c t8c 7c mr 7cXQ3 23 7c, 26d 7c sn 7cm n 2-20 6c I 8c 9c mr,9c yG2 2-25 9c - 1 7d 9c sn 9c n 2-20 7c_ 9c 7c su 7cm (XQ3 - Y2) 2-20 3 ________ 3_ 5 1c 5c 10c mr 10c= ZQ 2-20 _7c' 10c 7c ad 7c= (XQ3 - ye2 + Z@ ) 2-20 7c 2c 7c ad 7c= f(Q) 220;7 — -....... _ ___ _ _____5c 1 6c lOc mr 10c2 2YQ 2-20 _________ Oe06 9d 9c mr 9cm 3X 2-7 L9c I 8d 9c sn 9cm "X 220 9c I 8d 9c sn - 2,. i 9.9c I lOc 9c su 9c= (3XQ2 - 2YQ) 2-20 9c, 6d 9c sn 9c 4 9c 3c 9c ad 9c= f' () 2-14 9c 7c 7c dv 7c f/f' 2-6 5c' 7c 7c su 7cm new (i+l) 26 _ 9c A19 32A05 cm a |kn| < 2-17 __4d 1 Od 9A05 -cn jump ), ______7c ___I.._____________ — ___ —

__85 CONTROL CONTROL ~~~~~~~EXPLANATION COMBINATION EXPLANATION Oe06 " 9c mr 9c3 Ai. 2-10 2. -16 9c 7C lOc mr 10c= Ai2 2 _______________________ 9____; 7 cmr1cAQJ 26_______Ajj126 4c c 9c c m 9c A"i(i+ )Qig+2'16 10c 9c c10 su 10cc AiQ + 2"16 1~~~~~~~~~~~ -16 1c; 49c 10c su 10c- AiQ+1 -2 " 2-16 10c Ic 10c ad 10cm Ai+, 2-16 10c 1 8d 6c sn 6c- Ai+ 2 -OeC06 4d -2e06 -ad Ai+l Aj -6c I 4d -0e06 -ad Ai+ -001 002 39A05 ba Qil Qj, 9 i+2' gi+1 4d I Od B17 cn Jump FA A30 3c I 3c 5c su clear 3c 2c? 4d 28d ex sign of x onto fffff FA A)1 ld 4d 2c ex- lx 26 FA A52 2c 19d A3 en Is x < 2 2c 19d 2c su x - 27- x FA A3'A2c 5 2d A54 cn is x<ng 2-6 1_____ 2c 2c su - 2 -x --- x 28d 29d 28d mr - 28d +1 FA A54 __ 2c ____ A37 cm Is |x| 2' < c 2c' Od 2c sn 2cm X 2'2 ~__________ A36' -001 A35 -fi File Cg 6 FA A3. -lie 3c c -ad - c= a5 2 3c 2c 5c mr - 5cmXa5 2 _____________~~~~~~~~~~~ _____________________ _____ ____________________ _________________________________ 1________ _____i_________ —----------— _________________________________

^__________________ 286________ CONTROL COMBINATION ______________________EXPLANATION -001 005 A55 bae 5c(a5X5+aX4 +...a1X)24 _________ cl 6e 3c ad- 5c= F(X) 2-4 3___c _ 28d _c mr 3c=+ F(X) 3c 54d 2c sn 2c= F(X) 2'6 4d 29d 284 su Reset ffff FA A36 [-ooo0001] 511 A37 ba Reset CB FA A57 4d Od [000] cn juimp FA B28 254 003 003 ad 1 Tpo 005 26d B27 cn Is Tpo<5 ____________ 00;_____26 B2T ___ enI ^<3 _____ FA B29 001 B27 000 ro- error halt FA B34 2542 004 004 ad 1 - T ~_____ _oo_ r004 264 A04 cn Is T5<<3 001 A04 000 ro- error halt FA B37 25d 004 004 ad 1-Ti _004+ 264 355 cn Is Tj<3 oo4 26(i`35 001 B55 000 ro- error halt FA E50 __ 25d lle lie ad 1 - T llec 26d E35 cn Is Tj < 3 001 E50 000 ro- Error halt FA A19 000 001 000 ri N. R. comp. const. _________________ _____J.___ _______ —-------------.____________________ ~~~~~~~I _______________ _ ____________________________________ —

287 ____________ CONTROL COMBINATION COiSTA-TS EXPLANATION FA DOO DA 036 ooo000 000 000 O4 Od 02 Id -000 00 oo 08 24 -000 000 o00oo 04 3d -0 4d -000 000;000 02 5d 000 000 1000 06 ________ 000 000 ooo00 05 Ad -000 000 DOOO 01 8d eQj __ __ 94 -000 000 P00 05 10d -000 OOQ OOO 06 114d 430 54- e9 58 ___________12d /48 22 10 ____15d c90 fda la22 16 14d4 n 2-2 20 154 04 i6d 000 000 poo 8 17d 000 000 000 Oc 18d 192 lfb h44 44 194 = 2 2-6 20d. --------- -H —------- --- -------------—. 0004 _______204 -000 000 000 05 21d 44 224 44 254 44 244 000 000'000 01 265 _________________________________ ________________________________ _________________________________________ _________1____________________ —---------

288 CONTROL COMBINATION |_________ EXPLANATION COMBINATION 000 000 000 02 27d fff fff:fff ff 28d,fff fffifff ff ___ _ 29d 524 3f6:a88 86 30d= i 2-4 = h -000 000 000 09 ___ 531d Oc9 Ofd aa2 21 52d =- 2-6 -3ad 35501962 5543= M. S. for P. 0. ________ 64+4 ed iabI a,9_ _33_4d= M. S. for s-c ceof4Q557f2 3_ _ 5 M. S. for 64 instr. -000 000o 000 Oa 36d -000 000'000 Oc 37d. 5. OOd 41e i6ba 01 _8d =. S. for 43 instr. ri-~ - T I IlI i II 1 i i I I I * I j IL _i~~~~~~~~~~~~~~~~~~ I. I____________________ I......I — _ [ = 1 ~~~~~~~~ _I I I

289 CONTROL COMBINATION EXPLANATION FA B09 le 3e Ic mr lc= Cd 2'2 Ic lc lcOl ad lc01 2Cd 2 2,6e 6e Ic mr lc= (y)228 le' 2e 2c mr 2c= Cb 2'4 13d, 2c 2c ad 2c ( l+Cb) 2-4 1,14d e 3c mr 5c= C 2'2 2c d 4c sn 4c ( l+Cb) 2-6 [3c c 14c 3 dv c +Cb 2Ic 3 2c01 mra- 2c01 = (y) 2 24 2e 3d 5c sn 5c_ b 2-8 le 53c 3c dv 3c= [b/C] 2-8 13d 2c 5c ad 5c ( 2Cb) 2-4!__ ____ _ 3c Ic 53c |mr 5cu (MAy)2 b/C ] 20 I.... - 3c 15d 4cOl Su 4cO [ " -2 2-4 4cOl 2d 4c01 s n 4colr ]~ 212 ___c! 5e 7c dv 7c: i 24 __ _|__ le _le l 8c | mr 8cC C2 2~ 4e 4e 9c mr 9c- [,p/ a>32 2_8 8c 9c |9c dv 9c- [(p/CC] 22-8 _____7c: 9c 9col mr 9cOlm- /ccoC 2-4 le 5e 5c01 dv 5cOl Z4r/ C 2~ 2c I 2c 2c mr 2c ( +Cb) 2-8 3e 6e 3c mr 3cm dAy 2 2c 3c 2c mr 2c- d4y(1+Cb)2 226 16d I 2c 2c u 2c [1 - "] 2-6 5_cl e 2 e2c m_ mr2cu',/C. "J26 l~ ~~~~~~~ 8c C2 20l l

290 CONTROL COMBINATION EXPLANATION 2c 2c 3col ad 5c01= 2Y/4C " 2-6 FA B10 4d Od [000] cn jump 000 000 000 ri Dummy order 000 000 000 ri 000 000 000 ri 000 000 000 ri I _____________ ~~~~~~~~~~~~~_______________________________ ---- --- ---------- __________________________________ ______________________I ____________________________ —-------.___________________________________i_____________ —----- - _____________________ _____________ ____________ __________ _________ _____________________________

~~___~_____________291___________ CONTROL COMBINATION EXPLANATION FA eol________ ______ i+l 2-6 DA 033 FA e02 2DA 033 _______ ________ FA eO4 ____ sino 2'1 DA 033___ ______ FA e05 cosB 2-1 DA 033 _______ FA e03 ____ u 2'3 AC B09 __ ___________ FA e07 __ ___ (1+2Cu) 2-2 AC 33e03 ___________________ FA e06___ 1 ____ _2,_ A,2.2_^ 26 4^ AC 3e06 __________ ____________ FA eOO Parameters AC 16e00 FA cOl temporaries AC 9c01 ______________ ____________ __________________ FA c05 temporaries AC 000 FA cOO F AC O ___ ________ FA______________________________ I________________ __________________ ___________________ __________________ ~ ___ ____________________ ----------------------------------------------------------— _ ___ ______ -----------— ~ -------------- --— _ --— _ —---------

292_ _ _ _ _ _ _ CONTROL COMBINATION PARAMETERS EXPLANATION leOO C 20 -4 4e00___^ t2"'2_____________ _____ __________________ 6e00 y 2.2 Ano d 2' 4eoo P/w 2-4 " 0 5e00 Ay 2 -4 6eOO 24~ 7e00 _ Ly2" 4_ _8.00 _ u comarison cons t2 9eoo000 A,Q ccup4rison cons t 2" lOeOO _ LAY 2___ 1__________ __________I__ _________ ____________________ 6 ll4e00 2 _____________________________ 12e00 2 _______ ____________ ____________________ ______ ____________________ _______l26__________________________A12 Q2 1 ___________ 26 ______________ _______ _____________________ ___________Qo.2"6~~~~~~~~~~~~~~~~~~~I_________________________ ___________________________________ 1~3eOO a3 15eoo a 2 -4 i6eoo 0o 2 n.-4.61,.2" 1 ______________2__________________________________________ 90.2 -6 gl~2-6

293 CONTROL C OMBINATION TEPORARIES EXPLANATION lc01 2Cd 2-2' 2col,, ~c,-........., (~)2 (+ 24 2c01 __ _ _ _ 3cOl 2y-/C [1 - dAy(l1+Cb)2] 2-6 4c01 [ (y )2(b /c) (2+Cb) -> ] 2-12 5c01 Ay/C 20 _ 6c01 Tally for,A, @ print out 7c01 Tally for 1, u print out 8c01 y 2-4 j 9cOl (+Cb) I lco5 _ 2co5, lcOO a __ ___ ___ 2c00 3cOOI 4coo I 5c00o__________ 6c00 7c00 B - cOO 1,cOO I. 8c00 9c00. __,_ _.____ _,.____ _ _______ _____.-____i_ i. lo 6 c0 0 I 1 I I I 0

294 CONTROL PARAMETER TAPE FOR TRAVELING-WAVE TUBE COMBINATION PROBLEM EXPLANATION TRANSLATE by INTERIM HEX ITP AC 481 Program Tape No. 13 MO de 020 df _ ___________ -lb A 2-1 _________________ 2b Ql 2-2 I_________ Ob C 2 __ _____-4b b 2-4 -2b d 2-2 __ ___*___ -4b ('P/ C) 2-4.25 -Id Ob Ly 20.25 i -Id 4b ny 24.25 T -1i -4b ny 24.16 _ 2d -44b $, u comp. const. 2-44.4 I id -44b A, @ comp. const. 2-44 __25 I -Id -lb ny 2-1! | _ | - - 7 " 6b a5 26 t ____________ ______ _____r 4b a4 24~ _ i ~I | ~O-2b a 22 — i —--- -2b a1 2-2 ~___~_________ _1_______________- ______ ---- -a 2~ [ Bc 00 _______ L ______ _ _, I-l4b Iy Bc~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ OM

295 TRAVELING-WAVE TUBE: PROGRAM NO. 13M13m7 Operating Instructions for Run No. A. General Information The output of this program is a sequence of 4 groups of 3 words and 2 groups of 65 words. It can be determined if the program is proceeding correctly by examining the last 2 words in the groups of 3: the second word in each group of 3 should be a positive number and these numbers should increase continuously until a maximum is reached and the problem is terminated (see below), while the third word in each group of 3 should be a negative number and these numbers should increase negatively throughout the entire problem. If at any stage of the problem these numbers do not follow the right pattern, then the computer should be halted and computation resumed at a point where the numbers are of the correct form (see B.7). B. Initial Computations 1. Tape 13M13m7: Insert and process the first word on tape no. 13M13m7. With all the switches set on normal, begin computing. As the tape is read in, 4 groups of 5 identical memory sums are read out for drum transfers and 1 group of 2 for transfers to the acoustic memory only. 2. After 13M13m7 is in the machine, place 13m13p in the Ferranti and begin computing. This will read in the tape and read out 2 memory sums, halting the process. Read in the last word on halt. 35. Patch Panel Settings: Decimal point at 2 Carriage return at 8 4. With overflow switch on halt and format on patch, begin computing.

296 5. Intermittent Read-out without Machine Halting: Occasionally several intermittent read-outs will occur during computations. Usually this read-out is less than 5 words; however, if it should be much greater halt the machine and proceed to the section on program malfunction (B.7). 6. If everything is working correctly, run the program until the second number in the groups of 3 begins to decrease. After this point let the computations continue until a print-out of 2 sets of 65 words occurs. After each set of 3 words the computer halts to permit the operator to check the output with previous outputs. Computations are resumed by pushing the start button. 7. Program Malfunction: A program malfunction is here understood to be any difficulty arising from rapid, unexpected changes in the second and third words in the groups of 3 words, overflow, or cessation of computation other than those described earlier. In the event of a program malfunction: a. Read out the counter, gamma, and instruction. Also read out register OOc. b. Clear the acoustic memory. c. Insert and process the first word of 13M13plO. d. With the overflow switch on halt, begin computing. e. Repeat a-d several times if necessary. If the malfunction repeats successively at the same point, read out 088 (hex) words from OOc. That is all. 8. Halting the Computer for Purposes of Time: The operator should halt the computer approximately 15 seconds after the computer halts on a 3-word output and the start button is pushed.

297!C. Resuming Computations 1. Tape 13M13m7: Insert and process the first word of 13M13m7. With all the switches on normal, begin computing. As the tape is read in, 4 groups of 3 identical memory sums are read out for drum transfers and 1 group of 2 for transfers to the acoustic memory only. 2. After 13M13m7 is in the machine, clear the acoustic memory. 3. Insert and process the first word of 13M13plO. 4. Patch Panel Settings: Decimal point 2 Carriage return at 8 5. With overflow switch on halt and format on patch, begin computing. Proceed with instructions B.6-B.8.

.___.__.____.___298____ CONTROL TRAVELING-WAVE TUBE PROBLEM COMBINATION 64 EECTS EXPLANATION RmAD IN INSTR TIONS (READS IN SIN-COS BLOCK) AC 001 _PROGRAM TAPE NO. 13113 CD 014 _ 014 1 001 001 ri Read in 14 words [000] O 001 015 ri Read in X words to 015 000 i 000 000 su Clear 000 -016 000 000 -ad Obtain Memory _-001 [000 003 ba Sum 001 000 001 ro Read out computed M. S. ______001 I 015 001 ro Read out correct M. S. [0ooo] [016] [ooo] ro Transfer to Drum i____ [___[000], [000] 160 ri Read in from Drum 000 00 000 000 su Clear 000 000, -160 000 -ad Obtain drum __ _-001 [000o 010 ba Memory sum 001 1 000 001 ro Read out M. S. 001 001 001 ri Read in 1-word to 001 -....._... _-,..... —-- 1024m 001 001 fi Transfer Control to 001 Sin-Cos Block 013 001 002 ri Read in 13 words to 002. __! __________________., ___________________________________ j _______________ ____ ____ ___ ____ ____ ___ ____ ____ _ j ____ ____ ___ ____ ___

299 _ CONTROL Cont. -- EXPLANATION COMBINATION (READS IN BLOCKS 2, 3, and 4) EXPLANATION AC 002 CD 013 [000] 001 015 ri Read in X words to 015 000 000 000 su Clear 000 000 -016 000 -ad Obtain -00 [001 [004 ba lemory sum 001 00 0 01 ro Read out computed M. S. 001 01 001 ro Read out correct M. S. [000ooo] 016 [000] ro Transfer to Drum [000] [000] 160 ri Read in from drum 000 000 000 su Clear 000 000 160 000 -ad Obtain _______ __-001 [ 000 ] 011 ba $emory sum 001 000 000 ro Read out M. S. 1024m 001 001 fi Transfer control to 001 i.______ I _ -ad- 1b1in

300 CONTROL Cont. -- COMBINATION (Reads in Block 1) EXPLANATION AC 002 CD 013 [o000] 001 011 ri Read in X words 000 000 000 su Clear 000 000 -012 000 -ad Obtain -001 [000] 004 ba Memory sum 001 1000 001 ro Read out computed M. S. 001 011 001 ro- Read out correct M. S. 001 001 000 ri Read in 1 word to 000 1024m 001 000 fi Transfer control to 000 000 000 000 ri 0 000 000 000 ri 0 000 000 000 ri 0 000 000 000 ri 0 000 000 000 ri 0 i _____________________________________ __ i _= S g __ _.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 00 0000r I T - I T X -X~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

301 CONTROL COMBINATION DIRECTORY EXPLANATION Register' Hex. No. Hex Location of Hex Location b erLocation Number' Re t on Drum $i+l.. 65 41 1024m 1052m 400 Sin -.2-1 65 41 1042m 1050m 412 Sin-cos sub- 61 3 D 1051m 1058m 4lB routine Block No. 1 58 3A 12 70 Not stored on drum Block No. 2 84' 53 1059m 1069m 423 Block No. 3 98 62 1070m 1082m 42E - ____._ __ -- Block No. 4 136 88 1083m 1099m 43B Dead Storage ______ 65: 41 1728m 1736m 6C0 ui 65, 41 1737m 1745m 6C9 Sections A, B, 31 IF 1746m 1749m 6D2 and C.______.______ ___ _________._______ —-----------

302 CONTROL SCALING: A2-4 2-6, 2-6 u EXPLANATION COMBINATION AC 012 Block No. 1 CD 048 FA BOO 083, 1059m BOO ri Read in next block 061, 1051m E07 ri Read in Sin-cos Routine 512, D17 000 su o.2-8 - A FA B01 D17 I 000 000 ad n.2-8 + A- 512 1 000 511 ad A- 511.... 001 l O f ____E07 ___001 1E07 fi Sin (.21, Qos -.21 000 D18 -E02 -sn A i26 512 3 510 -E04 -ad Store Sin 0.2'1 512 3 511 -E05 -ad Store Cos 0'2-1 -001 065 B01 ba _____________ 065 E04 1042m ro Store Sin ~ on drum ____________ 065 E02 1728m ro Store 0.2-6 on drum FA B12 512 D19 -EO7 -ad (1+2Cu).2-2 -Ei -E0, -E-03 -u Clear ui -001 065 B12 ba 065 [ E05 1737m ro Store ui.2' 3 FA B09 -6C01 -6CO1 -6C01 -su Clear tallys -001 000 B09 ba le 5 3e lc mr lc= 2Cda2-2 lc i c l cO1 ad cOl- 2Cd-2"2 __ 6e 6e c mr lc () )2.28 le f 2e 2c mr c= Cb 2-4 D13 2c 2c ad 2c= (+Cb) 2'4 D14 le 3c mr c ()2 2c, D05 4c sn c= (l+Cb) 2-6 ^______c____ _ _ _ _ 4c 3c fdv c= L ^^J *2-4,,,

505 CONTROL COMBI OEXPLANATION C OMEBINATION ic' 3c 2c01 mr- 2c01= (y)2 [ C 24 2e D05 3c sn 3c= b-28 le 1 3c 3c dv 3c (b/C)2-8 D13 2c 5c ad 5c= (2+Cb) 2-4 5c Ic 53c mr 53c= (qy)2 (b/c)'2~ 3c 5 mr 3c= (Ay)2 (2+Cb) 2-4 3c I D15 4c01 su 4c01= [" -2] *2-4 4c01 D02 4C01 sn 4c01= [ " 2-12 _ 2c 5e 7c dv 7c= bh' 24 j ~~le le 8c mr 8c= C2.2~ __4e _ 4e 9c mr 9-c [=p/a] 2'2_-8 8c 9c 9c dv 9c= [p/CwO] 2 2.-8 ______________ T7c [9c 9c01 mr -9cOl= [2p/C]2-4 | le 5e 5c01 dv 5c01= [, 2/C] 2~ 2c: 2c 2c mr 2c= (1+Cb)2.2-8 r _3e _, 6e 5c mr 3c== (dLy) 22 2 2c 3c 5 2c mr 2c= day (1+Cb)2 2-6 D16 2c 2c su 2c= [ 1 2-6 5c01 2c 2c mr 2c y/C [1 2-6._______ 2 2c 3c01 ad 3c01= 24y/C [" 2-6 031 E06 1746m ro Read A, B, and C onto drum FA B10 512 001 BOO fi Jump Constants FA D13 DA 001 10 D1 FA D14 I DA 001 | C90 FDA'A22 16 14 = I t.2-2 I I T l I __I_________ ______________________________

3o4 CONTROL COMBINATION ________ EXPLANATION FA D05 DA 001 -000 000 000 02 D05= -2-2-44 FA D03 r FA D15 I DA 001 -000 000 000 08 DO2= -842-44 FA D16 DA 001 204 1 D16 2 FA D17 ________________ _______ DA 001 032 43F 6A8 88 D17- n.2-8 FA D18 DA 001 -000 000 boo 03 D18- 3 24I FA D19 DA 001 1 4000 000 0000 ________ _ D19- 2-2 -..! i —------------------- j -------- - - -i ----------- _ I DA O01 0I I D16 -----------—. —----- j -------------------- -------------------!. FA D17~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ FA D18 ________ I ____________ —----------

305._ CONTROL COMBINATION EXPLANATION AC 136 BOO FA E02.2-6 AC 65 E02 FA E04 Sin'.2'1 AC 65 E04 _ FA E05, Cos _. 2-1 AC 65E05 3 FA E03 u 2-3 AC 65 E03 FA E07 (1+2Cu) 2'2 AC 65 E07 I FA E06 SECTION A Ai12 A2, 22Ao24 AC 3E06 FA EOO SECTION B Parameters AC 16EOO;FA C01 SECTION C Temporaries i _ c _ AC 9C01o _ FA C05 Temporaries AC 000 FA COO _______ _________ BC 000,

________________________ 506_________.___ _ 306 CONTROL COMBINATION EXPLANATION AC 012 Block No. 2 CD 068 FA BOO 098 1070m BOO ri Read in next block FA B17 7'8c FA BIT _____ _7e 8cOl 8c01 ad 4 + y -- y leO6' e06 5c su 5c= (Oi+l-Qi) 2-6 5c, DO7 8c sn 8c= " 21 le 8c 4c mr 4cC ( ") 2-1 e06 2e06 6c su 6c= (Ai+l- A) 2-4 6c D26 6c sn 6c= ( ) 2-1 6c le 6c mr 6c= C (") 2'1 1Oe i 4c 7c su 7c=[Ay-C(@Q+l-Qi) 2-1 FA AOO 5e -e03 2c -mr 2c= 24y u-24 -e07 2c ic -dv lc= [L u] 2-2 065 e04 024m ro Read out Ii+l' to Drum I 065 1042m eo4 ri Read in Sin I's FA B19 -e02 D04 lc -ad lc= i " j T B22 -001 B25 -fi File Base Counter -FA B25 lc llc llc su lear liu FA B20 lc -e02 2c - ad Store new 2-6 06 e042c 2c -dv- ead out + to Dru __065 __ 104 e04 ri |1R0ead in Sin x+'s2C FA B -e02 Dll4 llc -ad lc= 2/hij T B22 -001 B20 -fi le Base / C FA B25'1 llc llc su Clear llc FA B20 | lc, -e02 2c -su- 2c=x=T-~i o2-6 A37 001 A30 fi 2c= F(x)'2-6 -e07' 2c 2c -dv- c= F (x) = io.2-4 2c 1 nlo nlo ad nCo 2/h |'2-4 -o06' 128 B20 ba llc= 2/h /.2-4 I llc' DO8 Ill |n

307___ CONTROL COMBINATIONEXPLANATION FA B21 lc, -le02 2c -u 2c= x-26 A37' 001 A30 fi 2c= F(x)'2'6 F(x) -4 -le07 2c 2c -dv- 2c= +u'-24 __________2c____ l _l___lc 1*2Cu 2e lc 1c ad lc= 1/h f'2-4 -001 063 B21 ba 1lc= " le 1 D19 lie mr — 11c /'2-4 FA B22 -[000] 511 B23 ba Reset Cb FA B25 9c01 11c llc mr- llc= K126c -e05 c -mr lc=C (Ai+i-AI) Cos *2'-2 2e06 7c 2c mr 2c= Ai [y-C(Qi+l-Qi)] 2' 2c -e04 2c -mr 2c= Ai [" ] Sin $-26 2c D05 2c sn 2c= " 2-8 c, Dll lc sn lc=C (Ai+i-Ai) Cos 0'2-8 | 2e 1 c 2lc su- lc= K2 2'8 -e07 -e03 2c -mr 2c= u(1+2Cu) 2-5 -8 2c D21 2c sn- 2c= K3.2 lc, 2c Ic ad l (K+K( ) 2 -8 Ic lie Ic su 1|= (-K1+K+K3+) 2'8 1 C D26 1C sn lc=' 2'5 -e07 c -e03 -dv ur 2-3 -001 065 B19 ba 512 001 BOO fi Jump FA A30 35 3c 3c su- Clear 3c 2c DOi D28 ex Extract sign of x onto fff FA A31 DO 1 B D0o 2c ex- IxI ~2-6 FA A32 2c I D19 A33 cn x < 2n 2c D19 2c su - 2 - x _A _ 7 ___2e _ D_ 2 A.4 en _.

308 CONTROL COMBINATION | EXPLANATION D19, 2c 2c su- 2n -x -- x D28 D29 D28 mr- D28 = + 1 FA A34 2c D57 A37 cm Is |x| *2-6 < < 2c, DOO 2c sn 2c= x'2-2 A36' -001 A35 -fi File Cb FA A35 -lie 5 3c 3c -ad- 3c= a'2 3c 2c 3c mr- 3c= x a-24 -001 1 005 A35 ba 3c aixi 2-5 3___________ c 16e 3c ad- 3c= F(x)2'-4 |__ 5__3c' D28 3c mr- 3c= + F(x) ________ 3c D05 2c sn 2c= F(x)'2-6 D04 D29 D28 su Reset fff FA A36 [-000] 511 A37 ba Reset Cb FAA37 FA A37 D04 DOO [0ooo cn Jump Constants FA D07_ DA 001 000 000,000 05 D07 FA D26 DA 001 000 000 000 03 D26 FA D03 I DA 001 -000 000 000 oo4 _________ D0 FA D04____ ___ ___ ___ ____ DA 001 -0 I D04 FA Do8 DA 001 -000 000 000 01 D08 FA D05 DA 001 -000 000 000 02 0 DA 001 -000 000 1000 02 D0 F A DO5 ________ I ____________________

____________________ 09 CONTROL CONRLMBINATION _________EXPLANATION COMBINATION FA Dl1 DA 001 -00 000 oOd 06 Dl1 FA D21 DA 001 -000 000 00o 03 D21 FA D28 ___________ DA 001 fff fff fff ffL D28 FA D29 DA 001 ff fff fffL ff_____ D29 FA D01 ____________________________________ DA 001 02 1 D01 FA D19_____________ ____ ________ DA 001 192 1FB 5144144 D19=h=(2n)26 FA D32 _______ DA 001 OC9 OFD AA2j21 D32 = ()2-6 FAD37 ____________________ DA 001 000 000 000 OC D37 FA DOO DA 001 000 ooo000 000 ooo4 ______ OO _____________________ ________. ____________________________ _________________________________________ ___ ____________I ________________________ -------------- ____________________ ____________I____________ ________________________________ _____i_____________ —-----

310 CONTROL COMBINATION EXPLANATION AC 16 BOO I FA E02 |1.2-6 AC 65 E02 FA E04 Sin 21 AC 65 E04 _ AC 65 E05 FA E07 (1+2Cu) 2AC 65 E07 I FA E06 4 u'2'-6-2 P FA E067, (_+Cu ____2SECTIO A A1-21 4.2-6,Ao'2 4,Q2 AC 3Eo6 _ _ __ _ AC 15E00 I FA EO, SECTIO1 B Parameters FA C01 SECTIO C Temporaries AC9C01 91 FA C05 _ Temporaries AC 000 FA COO r BC______________ _000 BC 000 _,____________ ________________ ______ _______ -------------------,i. FA C O SETO C Tepoare

311 CONTROL COMBINATIONEXPLANATION' COMBINATION AC 012 Block No. 3 CD 060 FA BOO 136 1083m BOO ri Read in next block 065 1024m e02 ri Read in i+l 6c01l D25 6c01 ad TA+1 TA 7cOl, D25 7cOl ad T0,1- T 6c01 9e A04 cn 6c01 6c01 6c01 su Clear Tally -44 DOO DO4 510 ad 510= 4- 2 8col Do4 511 ad 511= y2-44 B09 001 1B09 fi Read out y e06 Do4 | 511 ad A - 511 B09 001 1B09 fi Read out A D04 Dll 510 su 510=6.001 D22 001 ro- Read out 1-. R. 7c01 8e Ao4 cn is T < Co 7c01 7cO1 7cO1 su Clear Tally D06 D04 510 ad 510 = 6 -001 065 B51 ba I -I001 D22 000 ro Read out 1 C. R. D26 D04 510 ad 510 = 3 FA B32 -eO3 5 D04 511 -ad 511 = u._____B09' 001 1B09 f i Read out u. -001 065 B52 ba

312 CONTROL COMBINATION EXPLANATION FA A04 065' e02 1728m ro ead out I 1 to drum o_______65 eO3 1737m ro Read out ui+.J to drum 031 e06 1746m ro ead out Sec.A,B andC to Drum 061 1051m e07 ri aall in Sin-cos routine FA B39 -003, -003 -003 su lear 003 --- 010 -001 008 B39 ba FA A21 -e02 D04 lc -ad lc=.-2-6 Ic D04 2c ex c= Sign of D OO D04 lc ex lc= I *2-6 FA A02 lc' D19 A03 cm s I1 < 2n Ic D D19 1C su Ic= (0-2g) 2-6 D04 _ DOO A02 cn ump FA A03 2c, D04 lc ex Lc= + 02-6 lc D26 511 sn 511= 2'2-3 e07 001 le07 fi 10=Sin(.2'1;511=Cos' 2'1 510 D04 -e04 -ad toreSin (*21 511 D4 -e05 -ad 3tore Cos 2'2-1 Ji -001 o 065 A21 ba FA B24 le -e03 2c -mr.c= 2Cu.2-4 D13' 2c lc ad (1+2Cu) -2'4 lc D27 -e07 -sn.07= (1+2Cu) 2-2 -001' 065 B24 ba 065 -e04 1042m ro 3tore Sin 0'2-1 512 001 BOO fi ump Printout Subrc utine FA B09 512 _ 001 000 fi _xit 510' DO1 18D06 su K- 44 - KI - ___ _____511' 511 D05 n_

313__ CONTROL COMBINATION EXPLANATION 511 18D06 511 sn Ni.2'44 D29 001 1D29 fi 511 D02 511 sn Ni'2-12 D05 001 D05 bd D05 D03 D05 sn Nf-212 _D05 511 511 ad (Ni+Nf) 2-12 001 511 001 ro 512 001 B09 fi Jump Constants DA 001 000 000 000 2C D01= 442-44 FA D02 DA 001 000 000 O00 20 D02= 3242-44 DA 001_ D___________ ____________05 N2_______ FA D25 _ DA 001 000 000 ) ______ ______ D25 FA DOO_ DA 001 000 000 oo00 01 D02= OO FA D00' FA D15 DA 001 -000 000 oo 06 D11 FA D22 I DA 0010 000 Poo I ID2 DA 001 d40 _____ D22 DA 0_d4 __________ ____ D22A01D u~- d4...........

314__ CONTROL COMBINATION EXPLANATION d4, D22 FA D06 DA 001 000 000 000 06 D06 FA D26 I___ DA 001 000 000 000 03 ______D26 FA D19 | I_ DA 001 192 FB,544 44 D19 = 2n2-8 FA D13 ____ DA 001 10 D13 FA D27 __ DA 001 000 000 000 02 D27 FA D29 DA 022 Tape 14,D2m4 Integer Subroutine t ~ II

315 CONTROL COMBINATION | EXPLANATION AC 136 BOO_ ____ FA E02' ___.2'6 AC 65 E02 I FA E04, Sin.2-1 AC 65 Eo4_ FA E05 _ os _' 2'1 AC 65 E05 FA A03 __2-3 AC 65 E03 ______ FA E07 _(1+2Cu) 2'2 AC 65 E07 ___ FA E06' SECTIO A h2-4 l262-24 -4. 2 AC 3E06 FA EOO, SECTIO B emporaries AC 16 EOO FA COO FA CO1 SECTIO~ C Cemporaries AC 9CO1' I FA C05 _emporaries BC 000 BCO 000 IIlII L_____________ -r___! _________________________________I

316 CONTROL COMBINATION EXPLANATION C OM BINAT I ON AC 012 Block No. 4 CD 113 FA BOO 083 1059m BOO ri Read in next block FA A05 -leO4 D31 2c -sn 2c=Sin'2-10 FA A05 -le4 D31 5c -sn 3c=Cos 02-10 1.Sin -leO7, 2c 2c -dv 2c=. u'2-2 I,3c' D05 3c sn | +u 210 _________________ n1 2 CU! 2c 5c 5c ad 5c= 4 ( yiSin ) 2-12 3c 6c 6c ad 6c= 4 ( y Cos ) 2-12 -002 064 A05 ba FA A06 -2e04 I D31 2c -sn 2c= Sin'2-10 -2e05 D31 3c -sn 3c= Cos 0 2'10 _ -Sin 0. 28 -2e07 2c 2c -dv 2c= +2Cu 2c D21 2c sn 2c= n 2-11 -2e07 3c 3c -dv 3c= +c 2-8 1+ 2Cu 35c' D21 3c sn 3c= "'211 2c 7c 7c ad 7c= 2(2 yiSin ) 2-12 35 8C 8c 8 ad 8c= 2(Y yiCos ) 2-12 -002 062 AO6 ba FA A07 -e04, D31 2c -sn 2c= Sin'2-10 _-e05 D31 3c -sn 3c= Cos 2-10 2c D03 2c sn 2c='212, -8 -e07 3c 3 -dv 3c= ^'2 ___c' D03 3c sn 3c= "'2-12 2c D9c 9c ad 9c=iSin2'122

317____ CONTROL COMBINATION EXPLANATION 3c lOc lOc ad 10c= YiCos-2-12 -064 128 A07 ba__ FA A08 -5c D12 -5c -mr 5c= 21 -6c D12 -6c -mr 6c= " -002 006 A08 ba 5c 7c 5c ad 5c= (4Z Sin + 2.) 2'10 5c 9c 9c ad 9c= (42+ 2 + ) 2-10 6c 8c 6c ad 6c= ( 4 2 Cos+2 X )2-10 6c lOc lOc ad loc= (4 +2 Z + 2 ) 2-10 FA A09 lcOl -9c -lc -mr lc= 2CdfSin-2 12 -_____9c, D05 -9c -sn- 9c= /Sin2-12 -001 002 A09 ba _-20diCos_2-12;10c=fCos_2'-1 |___l__ _ I_ c l ClOc _3c ad |3c= ( Cos+2Cdf Sin )'2 12 [T___ ________ 9c 2c 2c su 2c= (/Sin-2Cd /Cos )2-12 2c01 5c Ic mr- lc= A'2'8 2c01' 2C 2c mr- 2c = T'2-8 e0o6 3c01 )c ad 3c=Qi-+ [ ] 2-6 Oe06 3c c mr- 3c= a.2-10 le06 lOe06 4c mr 4c= 2-2 5c01 le06 5c mr 5c= Ay/Cei-2- 6 5c D10 5c sn 5c= 2 " 2-12 2 2 -12 _4c 414c01 4 su c= [Q_-(4y2b/C.]'212 _4c_ 5c 4c ad 4C= [ C+2 Qi ]32-12 e06 4c 4c mr 4c= Ai[ 1*216 _Ic 1, D5O Ic srn c= A. 2-12___ 2e06 D02 6c sn s c= Ai2'212 Ic 6c Ic su (= (A -Aij. ) 2-12 eI0c' D03 Ic r sn c=i['2-1 6c ( _____________________________ ____________________________ _____________________

318 CONTROL COMBINATION EXPLANATION lc 4c lc ad lc= 1 -216 Ic D27 lc05 sn lc05= I 2-14 5c01 Dll 4c sn 4c= Ay/C'2-6 le06. 4c 4c ad [ [i+ y/C 1'2-6 lc05 4c 5c mr 5c= n [@i+&y/C]'2-6 5c, 5c 5c ad 5c= 2 " *2-20 3c D36 3c sn 3c= * 220 2c D37 2c sn 2c= T'2-20 3,5c 2c 2c su 2c= (T' - T) *2-2 FA A10.2c 5c 2c su- 2c= U'2-20.....i.........,,2.... FA E35 4c I 4 5c mr 5c= 4 (@i+C )'214 5c D20 5c su 5c= [ -1] 214 e06 I 5c mr 5c= Ai [ ] 2-18 |5c DOO 5c sn 5c= " 2-12 lc05 lc05 3ad 35c 2 II *214 5c 5c 5c ad- 3c Z Z2-14 eO6 D09 6c mr 6c= 6Ai"2"7 FA C20 D01 D01 D01 su Clear tally 5c e06 mr 7c= X9'24 5c 5c 8c mr 8c= e2 2-12 __ _ _ 7c 8c 7c mr 7c= X52-23 n' -2-20 7c D26 7c sn 7c= " 2 6c 8c 9c mr c= Y2.2-25 ________ __9c__ ID079c_____. _2___________. 9c I D07 9c sn 9c= 2 -20 |_____ _ 37c 9c 7c su 7c= (X6i-Y@2) 2-20 L 6 13c, 5c 6 lOc mr 1Oc=Z''2120 ______________________________________ i

319____ CONTROL ~~COMBINATION~~ | | EXPLANATION C OMBINA TION 7c 1c 7c ad 7c= (Xg3-Y@2+Ze).2-20 7c 2c 7c ad 7c= f()' 2-20 5c 6c lc mr 10c= 2YGe2'20 |__ _____ _ e0O6 _ D09 c r 9c= X2 9c i 8c 9c Mir 9 Xr 2 219c __ 9c, D08 9c sn 9c= n2-20 _9c' 10c 9c su 9c= (3xQ2-2Ye)'220 _9c DO6 9c sn 9= "'2-14 ____9c 3c 9c ad 9c= fO)214 9c__ c 7c dv 7c= f ()/fr26 ___5____c u 77c c su 7c New Q26 7c 3c 9c su 9c=.2-6! | 7c 9 5c L c9 | su 9c [Q(i+l)-Q(i)]'2-6 i _______9c 9 D19 C21 Cm Is I| A<2-17 __1___?'7c D04 5c adc Q (_____il) D08, D01 DO1 ad Cycle Tally DO1,1DO 1C20 cm Is tally - 20 |___ ___D19 D19 D19 ad D19= 2'216 F ___001 _ _ r __ 9c | 001 ro Read out (AQ)'2-6 C000 1 000 000 ex 512 001 C20 fi Jump FA C21 e06 7c 9c mr 9c= Aii+1'20 ---------------------- --- 9c2 -16 -T___-__ _ r9c_ 7c 10c mr 10c= Ai+l2 162 _4c ___9c 9c mr 9c=Ai(Qi+/y/C)@il'2-16 1c I9C 10c su 1 SU Cc= [Aiei+1i - n 2 16 1Occ 9c_ lOc su 0c= [A @O - 2"] 2-16 _______________lc______, 9c loc su oc= Amji+2"16-_"]'2lOc I lc lc ad Oc= Ai+1*2 16 lCOc lI D18 6c sn 6 Ai 4 FA C22 -eO6 I D04 -2e06 -ad i+l Ai

520 CONTROL |COMBINATION |_____________ | EXPLANATION -6c D04 -e06 -ad Ai+2 - Ai+l -001' 002 C22 ba Qi+ -- Gi,-Gi+2 -- i+l D04, DOO BOO cn Jump Constants FA D51 _ DA 001 -000 000 1000 09 D31 FA D05 DA 001 -000 000 1000 02 D05 FA D21 DA 001 -000 000!000 03 ___ _ D21 FA D03 I_ ____III DA 001 -ooo ooo',ooo 04 D03 --.- -. —DA 001 -000 000:000 05 D10 FA D02 ________ ______ ______ ____________________ FA D27___ _____rrI DA 001 000 000 2 000 02 __4 AC D127 2 FA D19 DA 001 -000 000 0 0 D1= 22 1 DA 001o -000 000 I00 06____________ D62 FA D710 DA 001 -000 000 P000 05 D37 FA D02 DA 001 -000 000bO 08 D12 DA 001 -o000 000 1)00 A02. D27 FA D11 I _ _ _ _ DA 001 -000 000 p0O oC D117 FA D35 I DA 001 -000 O00 1000 08 D36 FA D27 I DA 001 000 000 PCO 0C D 27

321 CONTROL ~~~~~~~COMBINATION |__ EXPLANATION C OMBINATI ON FA D20 DA 001 000 4 D20 FA DOO DA 001 000 000 000 04 DOO FA D09 DA 001 C, D09 FA D04 _ DA 001 -0 D04 FA D26____________ DA 001 000 000 000 03 D26 FA D07 DA 001 000 000 000 05 _ _D07 FA Do8 __ DA 001 -000 000 o000 01 _____D08 * A Do6 = DA 001 000 000'000 06 _ _D06 I FA D18 DA 001 000 000 1000 OC D18 FA D01 DA 002 0 0__ _ _ DOl= Cycle tally 000 000 000 14 1D0= Upper count....... j................

..~_ ~_ __. 5~322 CONTROL COMBINATION EXPLANATION AC 156 BOO FA E02 _ __2-6 AC 65E02 _ AC 65Eo04 FA E05 Cos-2'-1 AC 65E05 FA E035 u C 21 AC 65E03 _____ FA E07' (1+2Cu) 2-2 AC 65E07 ______ FA E06 I2, SECTI A 124, 2-6,AO-2-4JOO2AC 3EO6 I FA EOO SECTIO B Parameters AC 16E00_ FA CO1 I SECTIO C Te oraries AC 9cOl ____ FA CO5 I Temporaries AC 000__ FA COO _______ _ BC 000 _ - I ll I I I X~~~~~~~~~~~~

323 CONTROL COMBINATION TEMPORARIES EXPLANATION lcOl 2Cd-2-2 2c01 ()2 [l+ b]2 2 3c01 24y/c [l -d 4y (1+Cb)2 ] 2-6 4c01o [(: )2 (b/C) (2+Cb - 2 122 5col Ay/c'2~ 5c01 y/C 2~__ _ _ _ 6c01 Tally for A,g Print out. 7cOl Tally for iu Print o ut. ___ 8c01 y'2-4 _ 9c01 o (+Cb) [ p/C]22 - _____ lc05 I I I I 2c05 2c00 5c00 l 5coo I 6c00 6c00 7cOO 8c00 lOcOO I____________________ llcOO _______ _ _ 1_______ ________________________________ lc00 I 2c I! I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _...... _ _ _ _ _ _ _ _ _..... _ _ _ _ _ ___ __ __ __ __ __ __ __ __ __ ___ __ __ __ __ __ __ __ __ __ _,............... _ _ _ _ _ _ _ _ __ _ _ _ _ __ _ _ _ _ _ _ _ _ _ __ _ _ _ _ __ _ _ _ _ __ _ _ _ _ __ _ _ _ _ __ _ _ _ _ __ _ _ _ _ __ _ _ _ _ __ _ _ _ _ _

324 CONTROL COMBINATION PARMETERS EXPLANATION leOO C-2~ -4 2e00 b-2 5eO0 d*' 2-2 I_ 4e00 Dp/C 24 _________ 4eoO | / 2-4' 5e00 Ay 2~ 6e00 AY 24 7eOO 0 y 24 8e00 |, u comnarison cons ant 2-44 9e00 A, 9 comparison " 2 r....- i-,..... —--- lOeOO j _Ay 2_____ i ile00'a5* lleOO'26_____________________________ 4 12e00 a4 2 13eOO a, 22 _____ l4e00_________a_ 23 | 14eOO r a2 2~ 02 2 16eoo a'2 1 _ __ I ___________________________ ______I_____ --— I —-I —-I —— _ ___________...... _____________ _______ L- ______ ____________ ------------------

CONTROL PARAMETER TAPE FOR TRAVELING-IWAVE COMBINATION TUBE PROBLEM EXPLANATION TRANSLATE BY H E.NI_ HEX _ TP_ _______ __ _ __ _ _ __ __, __Program Tape No. 13113 AC 475 de 020 df...'....A 2-4 ____________________b A1 2-4b Ao 2,I | | l | -6b Q 2-6 Ob C 20 -4b b 2-4'||-2b " d 2-24|.25.2 -Id Ob 4y 20.25 -Id 4b 4y 24 _.25 -Id -4b BAy 2-4.16' 2d -44b |, u comp. const. 2-44.4 Id -44b A, @ comp. const. 2-44 _____l ___ _.25 -Id | -lb |a 2_1' _ t - 6b a5 2 [ ______ _______________ ____4b a424 i 1-70 - --— ===-t=- ffi = I 1 4b a4 2 == | | __|__ 2b a 22 J Ob a2 2~ __________________ ___]__ -2b |a 2-2, | -4b a 2-4 BC 013_______ ___ -___......i.

326 TRAVELING-WAVE TUBE: PROGRAM NO. 13M12m2 Operating Instructions for Run No. 1. Place tape 13M12m2 in the reader and insert and process the first word on halt. 2. With all switches on normal, push the start button; this will read in the remainder of the tape. 3. Read in the last word on halt. 4. Place the parameter tape in the reader and with the overflow switch on halt begin computing. 5. End of computation is signified by a print-out of the word fff.

. _______.__ ____._527 CONTROL COMBINATION CALCULATION OF POWER GAIN EXPLANATION A(Y)i | db='20 log10 A Program Tape No. 13 M12m2 AC 006 CD 022!I I ~~~~~~~~~~~~Read in two words from FA A02 002 i 001 DOO ris from FA A03 DOO 1COO A04 ex. _____________DOO 5 3COO DOO sn DOO 2C00 A08 ex FA A04 000 001 D01 ri Read in Ais.2-8 000 1 000 000 ba Clear Cb FA AO1 EOO 4COO EOO ad AY + Y Y EOO G COO 511 ad 511 = y2 _10COO COO 51 ad 510 = 4.2e A06 001 1Ao6 fi Print out y value.2-11 _______ IOO I -DOl 511 -dv Ai/Ao 2 A05 001 lA05 fi Transfer control ____.00 5COO 6coo lEOO mr [11 In 2J 2-5 i____ |__ _ 5*111 1 lEOO lEO ead t Lln Ai/AoJ.2-5 _7C00 8COO 2EOO mr [20 loglo0e 1____________lEOOE _ 2EOO 511 mr (db).21G __ _9_____QCOOa COO 510 ad 510 = 10 2-44 AQ6 Q01 1A06 fi Print out (db) -002 llC 00 000 ro Read out 2 C. R.'s FAAQ8 -001 000 A01 ba ___001,1C00 000 ro- _ _____________________________________________11 = y'2-4____________________________________________

328 CONTROL EXPLANATION COMBINATION EXPLANATION FA A05 dA 059 Subroutine P4d 9m3 for in x __ FA A06 dA 96_ Subroutine 14d 18 m5 ____ FA COO _____ 1___IT dA 013 0r _T____I' =n = L= FFFI_ II_;_____ 000 F__ FFF ___________ -000 000 000 0 C 12.2-44 j__ T01Q 999 999 199 1____ _______I__ B17217F7 DId ln2 = 0.693. 11.2-5 l____________6F2DEC59B9 __ loglae 0.4)4 + l ~ooo ooo ooo j~o A l 2-^44 000 000 000'0 A - -44 d4_ _____ C. R. _d_4 __ _ _ _ C. R. i I ________________________ _______________ r ______________ ___________ ___________ _______________________ ______________ I___________ _................___________________________ I ____________ __________ __________

_________________________ 5^9___________29 CONTROL (PARAMETER TAPE) EXPLANATION COM BINATI ON CMBINATION Translate Using Subroutine ITP de OOOdf _________ ________ (Number of Ijnits to be Converted) ____ -12b _____________.225 I -Id 3b (List Number's Beginning Here Witl -8b Proper Scale Facto s.) bc be~~ ~ ~ ~ ~ ~~~~~~~~ ________________________________ ---------— ___ —----------------— _____ —-— __ —-------------— _ ______________________________________ ________________________ I___________________ ~______________________ —---— ____

BIBLIOGRAPHY Small-Signal Theory 1. Brewer, G. R., and Birdsall, C. K., "Normalized Propagation Constants for a Traveling-Wave Tube for Finite Values of C", Technical Memorandum No. 33551, Hughes Research and Development Laboratories, October, 1953 2. Chu, L. J., and Jackson, J. D., "Field Theory of Traveling Wave Tubes", Proc. IRE, 56, No. 7, 855-865 (July, 1948). 5. Fletcher, R. C., "Helix Parameters Used in Traveling-Wave-Tube Theory", Proc. IRE, 58, No. 4, 413-417 (April, 1950). 4. Hines, M. E., Unpublished Bell Telephone Laboratory curves of the small-i signal propagation constants vs. b for several values of QC and C. 5. Kompfner, R., "The Traveling-Wave Tube as an Amplifier at Microwaves", Proc. IRE, 35, No. 2, 124-127 (February, 1947). 6. Kompfner, R., "Traveling-Wave Tube; Centimeter-Wave Amplifier", Wireless Engineering, 24, 255-266 (September, 1947). 7. Kompfner, R., "Traveling-Wave Tubes", Reports on Progress in Physics, 15, 278-528 (1952). 8. Pierce, J. R., Traveling-Wave Tubes, D. Van Nostrand, New York, 1950. 9. Sensiper, S., "Explicit Expressions for the Traveling-Wave-Tube Propagation Constants", Memorandum No. 55-16, Hughes Research and Development Laboratories, December, 1955. 10. Tien, Ping King, "Traveling-Wave Tube Helix Impedance", Proc. IRE, 41, 1617-1624 (November, 1955). Large-Signal Theory 11. Brillouin, L., "The Traveling-Wave Tube (discussion of waves for large amplitudes)", Jour. App. Phys., 20, No. 12, 1196-1206 (December, 1949). 12. Nordsieck, A. T., "Theory of the Large-Signal Behavior of TravelingWave Amplifiers", Proc. IRE, 41, No. 5, 650-657 (May, 1955). 550

331 BIBLIOGRAPHY (cont.) 13. Poulter, H. C., "Large Signal Theory of the Traveling-Wave Tube", Technical Report No. 73, ONR Contract No. N6onr 251(07), Electronics Research Laboratory, Stanford University, January 28, 1954. 14. Tien, P. K., Walker, L. R., and Wolontis, V. M., "A Large Signal Theory of Traveling-Wave Amplifiers", Proc. IRE, 43, No. 3, 260-277 (March, 1955 15. Wang, C. C., "Solution of Partial Integral-Differential Equations of Electron Dynamics Using Analogue Computers with Storage Devices", Project Cyclone, Symposium II on Simulation and Computing Techniques, Part 2, Reeves Instrument Corporation, April 28 - May 2, 1952. Space-Charge Wave Analysis and Electron Interaction 16. Bohm, D., and Pines, D., "A Collective Description of Electron Interactions: II. Collective vs. Individual Particle Aspects of the Interactions", Phys. Rev., 85, 338-353 (January 15, 1952). 17. Bohm, D., and Gross, E. P., "Theory of Plasma Oscillation: A. Origin of Medium-Like Behavior", Phys. Rev., 75, 1851-1864 (June 15, 1949). 18. Bohm, D., and Gross, E. P., "Theory of Plasma Oscillations: B. Excitation and Damping of Oscillations", Phys. Rev., 75, 1864-1876 (June 15, 1949). 19. Gabor, D., "Dynamics of Electron Beams", Proc. IRE, 33, No. 11, 792805 (November, 1945). 20. Haeff, A. V., "Space-Charge Effects in Electron Beams", Proc. IRE, 27, No. 9, 586-602 (September, 1952). 21. Hahn, W. C., "Small Signal Theory of Velocity-Modulated Electron Beams", General Electric Review, 42, 258-270 (June, 1939). 22. Hahn, W. C., "Wave Energy and Trans-Conductance of Velocity-Modulated Electron Beams", General Electric Review, 42, 497-502 (November; 1939). 23. Kent, Gordon, "Space-Charge Waves in Inhomogeneous Electron Beams", Jour. App. Phys., 25, 32-41 (January, 1954). 24. Pierce, J. R., "Increasing Space-Charge Waves", Jour. App. Phys., 20, No. 11, 1060-1066 (November, 1949).

332 BIBLIOGRAPHY (cont.) 25. Pierce, J. R., "Note on Stability of Electron Flow in the Presence of Positive Ions", Jour. App. Phys., 21, No. 10, 1063 (October, 1950). 26. Pierce, J. R., "Possible Fluctuations in Electron Streams Due to Ions", Jour. App. Phys., 19, No. 3, 231-236 (March, 1948). 27. Ramo, Simon, "Currents Induced by Electron Motion", Proc. IRE, 27, No. 9, 584-586 (September, 1939). 28. Ramo, Simon, "Space Charge and Field Waves in an Electron Beam", Phys. Rev., 56, 276-283 (August, 1939). 29. Rigrod, W. W., and Lewis, J. A., "Wave Propagation along a Magnetically6 Focused Cylindrical Electron Beam", Bell System Technical Journal, 33, No. 2, 399-416 (March, 1954). 50. Tonks, L., and Langmuir, I., "Oscillations in Ionized Gases", Phys. Rev., 33, 195 (February, 1929). 31. Wang, Chao-Chen, "Large-Signal High-Frequency Electronics of Thermionic Vacuum Tubes", Proc. IRE, 29, 200-214 (April, 1941). 32. Watkins, D. A., "The Effect of Velocity Distribution in a Modulated Electron Stream", Jour. App. Phys., 23, No. 5, 568-573 (May, 1952). Numerical Analysis 35. Bellman, R., Stability Theory of Differential Equations, McGraw-Hill Book Co., Inc., New York, 1953. 34. Fort, Tomilson, Finite Differences and Difference Equations in the Real Domain, Oxford University Press, London, 1948. 355 Hartree, D. R., Numerical Analysis, Oxford University Press, London, 1952. 36. Householder, A. S., Principles of Numerical Analysis, McGraw-Hill Book Co., Inc., New York, 1953. 37. Milne, W. E., Numerical Calculus, Princeton University Press, Princeton, New Jersey, 1949.

333 BIBLIOGRAPHY (concl.) 38. Milne, W. E., Numerical Solution of Differential Equations, John Wiley and Sons, New York, 1953. 39. Wilks, Elementary Statistical Analysis, Princeton University Press, Princeton, New Jersey, 1949. General Reference Books 40. Churchill, R. V., Fourier Series and Boundary Value Problems, McGrawHill, New York, 1941. 41. Churchill, R. V., Introduction to Complex Variables and Applications, McGraw-Hill, New York, 1948. 42. Dow, W. G., Fundamentals of Engineering Electronics, John Wiley and Sons, New York, 1952, Chapter 11. 45. Gray, A., Mathews, G. B., and MacRoberts, T. M., Bessel Functions and Their Applications to Physics, Macmillan and Company, London, 1931. 44. Jeans, J. H., The Mathematical Theory of Electricity and Magnetism, Cambridge University Press, London, 1951. 45. Pierce, J. R., Theory and Design of Electron Beams, D. Van Nostrand Inc., New York, 1949. 46. Slater, J. C., Microwave Electronics, D. Van Nostrand Inc., New York, 1950. 47. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941. 48. Watson, G. N., A Treatise on the Theory of Bessel Functions, Macmillan and Company, New York, 1944. 49. Whittaker, E. T., and Watson, G. N., A Course of Modern Analysis, Cambridge University Press, London, 1952.

LIST OF SYMBOLS A Total transition loss suffered by the signal at the input and output ends, db A(y) Normalized amplitude of the r-f wave along the helix Ai Amplitude of the r-f wave at the ith row of the integration procedure Ao Normalized amplitude of the r-f wave impressed on the helix a A/8.68 a' Mean radius of the helix, m ai(y), Vector components of the r-f voltage amplitude A(y), defined a2(y) by A2(y) = al2(y) + a22(y) B 54.6 x B Large-signal space-charge range parameter, Pb' b Relative injection velocity of the electron, (uo-Vo)/Cvo b' Radius of the electron stream, m C Gain parameter, defined by C3 = rZIo/2u02 C0 Capacitance per section of the helical transmission line, farads/m d Loss factor, 0.01836 i/C E(z,t) Longitudinal r-f electric field intensity at the stream, -aV(z,t)/az, volts/m Es(z,t) Space-charge field intensity in the stream, volts/m F(0-0') Space-charge weighting function Gdb Small-signal gain, A + BCN, db Gn A guiding section of the electron stream I Instantaneous linear current density in the stream, amp/m I(z,t) Longitudinal r-f helix current, amp In Current in section n of the helical transmission line, amp 354

355 LIST OF SYMBOLS (cont.) Io D-c stream current, amp Io(pr) Zero-order modified Bessel function, first kind, of argument Pr Il(Pr) First-order modified Bessel function, first kind, of argument Pr i A-c convection current, amp K Large-signal space-charge amplitude parameter, (op/oC)2 K,(Pr) Zero-order modified Bessel function, second kind, of argument Pr Kl(Pr) First-order modified Bessel function, second kind, of argument Pr Lo Self induction per section of the helical transmission line, henries/m ~ Series loss per undisturbed wavelength along the helix, db/m N Length of the tube in electron wavelengths P Poynting vector, Wvg, watts/m2 P Power output along the lossless helix, watts Pi Power input to the helix, watts P (y) Power output at any point y along the lossless helix, watts?oL(Y) Power output at any point y along the lossy helix, watts QC Small-signal space-charge parameter, E2C/p2P Qn Charge on the capacitance of section n, coulombs qn Charge brought on the capacitance of section n by the currents occurring in the line, coulombs tq Electric charge in the stream, in section n of the line, coulombs R Resistance per section of the helical transmission line, ohms/m Rn Electron-plasma-frequency reduction factor for axial symmetry t Clock time, sec to Electron entrance time, sec

336 LIST OF SYMBOLS (cont.) u(y,0o) A-c velocity parameter Ujj Total a-c velocity of the jth electron in the ith row, m/sec uo Average velocity of the electrons in the stream, m/sec ut(Y,00) Total a-c velocity of an electron relative to the electron stream, m/sec V(z,t) R-f voltage along the line, volts Vn Potential on the capacitance of section n, volts V0 D-c stream voltage, volts V5(z,t) Space-charge voltage, volts v Velocity of propagation of the r-f wave on} the helix, m/sec v(y,0o) A-c velocity parameter vd Difference between the stream velocity and the undisturbed phase velocity of the wave, m/sec Vg Group velocity, m/sec vo Undisturbed axial phase velocity of the wave, l/4LC, m/sec voL Axial phase velocity for the lossy helix, m/sec vpr Phase velocity of the wave relative to the stream velocity U 0, m/sec v Propagation velocity of the slow-and-growing wave in the smallsignal analysis, m/sec vt-(y,0o) Total a-c velocity of an electron relative to the undisturbed velocity, m/sec v,, Difference between the slow-and-growing wave velocity and the undisturbed velocity of the wave, m/sec W Stored energy per unit length, watts/m Xn Growth constant of the wave n in the small-signal solutions, nepers/m y Normalized distance variable, m IY Propagation constant of the wave n in the small-signal solutions, radians/m

337 LIST OF SYMBOLS (cont.) ZO Helix impedance, 7L/C or E2/2P2P, ohms ZoL Characteristic impedance of the lossy helix, ohms z Distance along the tube, m Zo Entrance position of an electron, m a, Growth constant of the slow-and-growing wave in the small-signal a0 analysis, nepers/m Growth constant of any of the three waves in the small-signal analysis, nepers/m Bf3 Phase constant of the wave on the helix, co/v, radians/m Pe Phase constant of the electron stream, (/uo, radians/m Pji Phase constant of any of the three waves in the small-signal analysis, radians/m PO Free-space phase constant, c)/c, radians/m p Propagation constant describing propagation in the presence of an electron stream Propagation constant describing propagation in the absence of r1 an electron stream r n Xn + jYn' where n = 1,2,3,4 6i Propagation constant, ai + JPi ~e Dielectric constant of vacuum, 8.854x10 12 farads/m Ratio of charge to mass for an electron, q/m, coulombs/kg ns. Saturation efficiency, percent G(y) Phase lag across the tube relative to the electron stream, radians.i Phase lag of the wave at the ith row of the integration procedure, radians g Wavelength along the helix axis, vo/f, m Xg Wavelength along the electron stream, uo/f, m Beam-to-circuit coupling coefficient

338 LIST OF SYMBOLS (concl.) p(z,t) Instantaneous linear space-charge density of the stream, coulombs/m Plc(Y0lo) Fundamental cosine component of space-charge density, coulombs/n P1s(Y,00) Fundamental sine component of space-charge density, coulombs/m Pn(Y,0o) nth component of space-charge density, coulombs/m O(ylo) Instantaneous phase of the fundamental r-f wave relative to vo, radians Polar angle in cylindrical coordinates (used only in Appendix B) A(yoo) Instantaneous phase of the fundamental r-f wave relative to the stream, radians Oij Instantaneous r-f phase of the jth electron at the ith row, radians 00 Entrance phase of an electron, radians 4t ~ Input-signal level in db below CIoVo w Angular frequency of the longitudinal r-f wave, radians/sec U) Electron-plasma frequency, defined by p = Io0/tebt2uo, radians/sec WOqn Effective electron-plasma frequency, EpRn, radians/sec

DISTRIBUTION LIST No. of No. of Copies ency Copies nc 35 Chief, Bureau of Ships 1 Air Force Cambridge Research Laboratories Navy Department Library of Radiophysics Directorate Washington 25, D. C. 230 Albany Street ATTENTION: Code 527 Cambridge, Massachusetts 20 Director, Evans Signal Laboratory 1 Bell Telephone Laboratories Belmar, New Jersey Murray Hill Laboratory FOR: Chief, Thermionics Branch Murray Hill, New Jersey ATTENTION: S. Millman 10 Chief, Engineering and Technical Service Office of the Chief Signal Officer 1 Bell Telephone Laboratories Washington 25, D. C. Murray Hill Laboratory Murray Hill, New Jersey 10 Director, Air Materiel Command ATTENTION: Dr. J. P. Molnar Wright-Patterson Air Force Base Ohio 1 Bell Telephone Laboratories ATTENTION: Electron Tube Section Murray Hill Laboratory Murray Hill, New Jersey 2 Mr. John Keto ATTENTION: Dr. J. R. Pierce Director, Aircraft Radiation Laboratory Air Materiel Command 1 Bell Telephone Laboratories Wright-Patterson Air Force Base Murray Hill Laboratory Ohio Murray Hill, New Jersey ATTENTION: Dr. P. K. Tien 2 Document File Electronic Defense Group 1 Department of Electrical Engineering Engineering Research Institute California Institute of Technology University of Michigan Pasadena, California Ann Arbor, Michigan ATTENTION: Professor L. M. Field 1 J. R. Black, Research Engineer 1 Collins Radio Company Engineering Research Institute Cedar Rapids, Iowa University of Michigan ATTENTION: Robert M. Mitchell Ann Arbor, Michigan 1 Columbia Radiation Laboratory 1 W. G. Dow, Professor Columbia University Department of Electrical Engineering Department of Physics University of Michigan New York 27, New York Ann Arbor, Michigan 1 Department of Physics Engineering Research Institute File Cornell University University of Michigan Ithaca, New York Ann Arbor, Michigan ATTENTION: Dr. L. P. Smith 1 Gunnar Hok, Professor 1 Electronic Products Company Department of Electrical Engineering 111 E. 3rd Street University of Michigan Mount Vernon, New York Ann Arbor, Michigan ATTENTION: Dr. J. H. Findlay 1 J. S. Needle, Assistant Professor 1 Vacuum Tube Department Department of Electrical Engineering Federal Telecommunication Laboratories, Inc. University of Michigan 500 Washington Avenue Ann Arbor, Michigan Nutley 10, New Jersey ATTENTION: A. K. Wing, Jr. 1 H. W. Welch, Jr., Project Supervisor Electronic Defense Group 1 General Electric Company Engineering Research Institute General Engineering Laboratory Library University of Michigan Building 5, Room 130 Ann Arbor, Michigan 1 River Road Schenectady 5, New York 1 Air Force Cambridge Research Laboratories Library of Geophysics Directorate 1 Mr. A. C. Gable 230 Albany Street Ind. and Trans. Tube Dept. Cambridge, Massachusetts General Electric Co. (Bldg. 269) ATTENTION: Dr. E. W. Beth Schenectady, New York

DISTRIBUTION LIST (cont.) No. of No. of Copies gencopies Agency 1 General Electric Research Laboratory 1 Department of Electrical Engineering Schenectady, New York University of Kentucky ATTENTION: P. H. Peters Lexington, Kentucky ATTENTION: Professor H. Alexander Romanowit 1 General Electric Research Laboratory Schenectady, New York 1 Gift and Exchange Division ATTENTION: S. E. Webber University of Kentucky Libraries University of Kentucky 1 General Electric Research Laboratory Lexington, Kentucky Schenectady, New York ATTENTION: D. A. Wilbur 1 Document Office - Room 20B-221 Research Laboratory of Electronics 1 Mrs. Marjorie L. Cox, Librarian Massachusetts Institute of Technology G-16, Littauer Center Cambridge 39, Massachusetts Harvard University ATTENTION: John H. Hewitt Cambridge 38, Massachusetts 1 Electronic Defense Group 1 Cruft Laboratory Engineering Research Institute Harvard University University of Michigan Cambridge 38, Massachusetts Ann Arbor, Michigan ATTENTION: Professor E. L. Chaffee ATTENTION: Dr. J. A. Boyd 1 Electron Tube Laboratory 1 Mr. R. E. Harrell, Librarian Research and Development Laboratory West Engineering Library Hughes Aircraft Company University of Michigan Culver City, California Ann Arbor, Michigan ATTENTION: Dr. C. K. Birdsall 1 Microwave Research Laboratory Electron Tube Laboratory University of California Research and Development Laboratory Berkeley, California Hughes Aircraft Company ATTENTION: Professor D. Sloan Culver City, California ATTENTION: Dr. G. R. Brewer 1 Microwave Research Laboratory University of California 1 Electron Tube Laboratory Berkeley, California Research and Development Laboratory ATTENTION: Dr. J. Whinnery Hughes Aircraft Company Culver City, California 1. Department of Electrical Engineering ATTENTION: Dr. A. V. Haeff University of Minnesota Minneapolis, Minnesota 1 Electron Tube Laboratory ATTENTION: Professor W. G. Shepherd Research and Development Laboratory Hughes Aircraft Company 1 National Bureau of Standards Library Culver City, California Room 263, Northwest Building ATTENTION: Dr. H. R. Johnson Washington 25, D. C. 1 Electronics Research Laboratory 1 Dr. D. L. Marton Electrical Engineering Department Chief, Electron Physics Section Illinois Institute of Technology National Bureau of Standards Chicago 16, Illinois Washington 25, D. C. ATTENTION: Dr. George I. Cohn 1 Mr. Stanley Ruthberg 1 Electron Tube Laboratory Electron Tube Laboratory Department of Electrical Engineering Bldg. 83 University of Illinois National Bureau of Standards Urbana, Illinois Washington 25, D. C. 1 Industry and Science Department 1 National Research Council of Canada Enoch Pratt Free Library Radio and Electrical Engineering Division Baltimore 1, Maryland Ottawa, Ontario Canada 1 Mr. R. Konigsberg Radiation Laboratory 1 Department of Electrical Engineering Johns Hopkins University Ohio State University 1315 St. Paul's Street Columbus, Ohio Baltimore, Maryland ATTENTION: Professor George E. Mueller

DISTRIBUTION LIST (concl.) No. of Agency No. of Copies Agency Copies 1 Department of Electrical Engineering 1 Department of Electrical Engineering Pennsylvania State University Stanford University State College, Pennsylvania Stanford, California ATTENTION: Professor A. H. Waynick ATTENTION: Dr. D. A. Watkins 1 Dr. O. S. Duffendack, Director 1 Sylvania Electric Products, Inc. Phillips Laboratories, Inc. 70 Forsyth Street Irvington-on-Hudson, New York Boston 15, Massachusetts ATTENTION: Mrs. Mary Timmins, Librarian 1 Polytechnic Institute of Brooklyn 55 Johnson Street 1 Sylvania Electric Products, Inc. Brooklyn 1, New York Woburn, Massachusetts ATTENTION: Dr. E. Webber ATTENTION: Mr. Marshall C. Pease 1 Radio Corporation of America 1 Department of Physics RCA Laboratories Division Syracuse University Princeton, New Jersey 102 Steele Hall ATTENTION: Fern Cloak, Librarian Syracuse 10, New York ATTENTION: Dr. E. P. Gross 1 Radio Corporation of America RCA Laboratories Division 1 Department of Electrical Engineering Princeton, New Jersey Yale University ATTENTION: Mr. J. S. Donal, Jr. New Haven, Connecticut ATTENTION: Dr. L. P. Smith 1 Radio Corporation of America RCA Victor Division 415 South 5th Street Harrison, New Jersey ATTENTION: W. J. Dodds 1 Radio Corporation of America RCA Victor Division 415 South 5th Street Harrison, New Jersey ATTENTION: Hans K. Jenny 1 Magnetron Development Laboratory Power Tube Division Raytheon Manufacturing Company Waltham 54, Massachusetts ATTENTION: W. C. Brown 1 Magnetron Development Laboratory Power Tube Division Raytheon Manufacturing Company Waltham 54, Massachusetts ATTENTION: Edward C. Dench 1 Raytheon Manufacturing Company Research Division Waltham 54, Massachusetts ATTENTION: W. M. Gottschalk 1 Sanders Associates, Inc. 155 Bacon Street Waltham 54, Massachusetts ATTENTION: Mr. James D. LeVan 1 Sperry Gyroscope Company Library Division Great Neck, Long Island, New York 1 Department of Electrical Engineering Stanford University Stanford, California ATTENTION: Dr. S. Kaisel

39016 36 6624