An Analytic Theory of Multi-stream Electron Beams in Traveling Wave Tubes (498 Pages)

"The Traveling Wave Tubes (TWT) is a powerful vacuum electronic device used to amplify radio-frequency (RF) signals as well as numerous applications such as radar, television and telephone satellite communications. This monograph is devoted to the author's original theoretical developments in the theory of a traveling wave tube (TWT). Most of the monograph is the author's original work on an analytical theory of TWTs. It is a constructive Lagrangian field theory of TWT in which the electron beam (e-beam) is represented by one-dimensional multi-stream electron flow and the guiding slow-wave structure is represented by possibly non-uniform multi-transmission line (MTL). The proposed analytic theory accounts for a number of electron plasma phenomena including space-charge effects such as electron-to-electron repulsion (debunching), convective instabilities, wave-particle interaction, amplifying waves and more. It allows, in particular, to (i) identify origins of the wave-particle interaction and the system convective instability (exponential growth); (ii) evaluate the energy transfer rate from the e-beam to the electromagnetic radiation; (iii) identify instability modal branches which under condition of sufficiently strong coupling between the e-beam and the MTL can cover ideally all frequencies."-- ProQuest Ebook Central resource page, viewed March 4, 2021 Contents Dedication Preface List of Symbols and Acronyms I Review of the Theory and Its Key Elements 1 Introduction 2 Summary of the TWT-system Features and Effects 3 e-Beam and Multi-transmission Line Parameters 3.1 e-Beam Parameters 3.2 The MTL Parameters 3.3 Recovering the MTL Matrices from Its Significant Parameters 4 System Lagrangian, Field equations, Characteristic Equation and Eigenmodes 4.1 Matrix Polynomial Eigenvalue Problem 4.2 Characteristic Functions and Equations 4.3 Characteristic Equations for Uncoupled e-Beam 4.4 Eigenmodes 4.5 Characteristic and Admissible Velocities 4.6 Values of the TWT Parameters Used for Plots 4.7 Infinite-Frequency Limit Approximation 5 Wave-particle Interactions and Origins of Instability 5.1 Origins of Instability and Amplification 5.1.1 Slow Wave and Its Negative Energy 5.2 Our Theory on Wave-Particle Interactions and Origins of Instability 5.3 Our Views on Non-Fluid Aspects of Plasma 6 Energy Transfer and Stream Velocities 6.1 Energy Transfer from the e-Beam to the MTL and Power Gain Factor 6.2 Energy Balance and Stream Velocities 7 Instability Concepts and Their Graphical Representation 7.1 Dispersion-Instability Graph 7.2 Instability Concepts and Quantities 7.3 Space Charge, Debunching Effects 8 Instability Branches of the Characteristic Function 9 Circular Approximations to Sets of Admissible Phase Velocities 9.1 Type I Circular Approximations 9.2 Type II, III and IV circular approximations 10 All-frequency Modal Branches 10.1 afm-Branches Involving Instability 10.1.1 afm-branches with negative wavenumbers 10.1.2 afm-branches with positive wavenumbers 11 Instability Phases via the Dispersion-Instability Graph 11.1 Instability Phases 11.2 Instability Phase Transitions 12 Instability Structure of All-Frequency Modal Branches 12.1 Instability Structure for Negative Wavenumbers 12.2 Instability Structure for Positive Wavenumbers 12.3 Instability for Large Values of TWT-System Parameter 12.3.1 Negative wavenumbers 12.3.2 Positive wavenumbers 13 Instability Nodes and the Degeneracy of the Characteristic Function 13.1 Onset of and Transition to Instability at its Nodes 13.2 Instability for Small Values of TWT System Parameter 13.2.1 Positive phase velocities 13.2.2 Negative phase velocities 14 Instability at Critical States - The Third-Order Degeneracy 14.1 Third-Order Degeneracy of the Dispersion Relations at Critical States 14.2 Dispersion-Instability Graphs at CriticalStates 15 Almost-Linear Unstable Modal Branches 16 Wave-packet Propagation and Amplification 16.1 Wave-Packet Representation in Dimensionless Variables 16.2 Wave-Packet Representation in Frequency Domain 16.3 Features Special to an Exponentially Growing Wave-Packet 17 Single-Stream e-Beam and MTL 18 Multi-Stream e-Beam Coupled to a Single TL 19 MTL as an Approximation and TWT Observables 19.1 Recovering the MTL Coefficients through Nodal Velocities 19.2 Recovering the MTL Coefficients through Modal Frequencies and Velocities 20 Instability - Possibilities and Limitations 20.1 Effect of TWT Features on the Instability 20.2 Multi-Stream e-Beam TWT-System for small and large values of its parameter II Traveling Wave Tube Components 21 Multi-transmission Line 22 Multi-Stream e-Beam 22.1 Justification of the One-Dimensional Model 22.2 Potential Form of the One-Dimensional Linear Model 22.3 Multi-stream e-Beam as a Dielectric Medium 22.4 Electron Beam Lagrangian 22.5 e-Beam Eigenmodes, Dispersion Relations and Instability 22.6 Bounds on e-Beam Characteristic Velocities 22.7 e-Beam Energy Balance 22.8 Energy Balance for Eigenmodes 22.9 e-Beam Energy Balance when the Electric Potential is an Independent Variable 23 Single-Stream Uncoupled e-Beam III TWT Composed of a Single-Stream e-Beam and a Single Transmission Line 24 Lagrangian, Field Equations and Power Transfer 25 Characteristic Function 26 Characteristic Velocities and Wavenumbers 27 Dispersion-Instability Graphs 28 Asymptotic Expansions for Infinite-Frequency Limit 28.1 Expansions for Small Values of TWT Principal Parameter 28.2 Expansions for Large Values of TWT Principal Parameter 29 Pierce Model as the Infinite-Frequency Limit Approximation 30 Nodal Phase Velocities 30.1 Nodal Positive Phase Velocity 30.2 Nodal Negative Phase Velocities IV TWT with two-stream e-beam and a single TL 31 TWT Lagrangian and the Characteristic Equation 31.1 Characteristic Functions and Equations 31.1.1 Characteristic equations for dimensionless velocities 31.2 Eigenmodes and Power Transfer 31.3 Characteristic Function with Instability Branches Plots 31.4 Dispersion-Instability Graphs 32 Uncoupled Two-Stream e-Beam 32.1 Complex-valued Characteristic Velocities 32.2 Nodal Velocities for an Uncoupled e-Beam 32.3 Zero-Frequency Velocities 33 Infinite-Frequency Limit Approximation 33.1 Expansions for Small Values of TWT Principal Parameter 33.2 Expansions for Large Values of TWT Principal Parameter 34 Instability Nodes and Nodal Velocities 34.1 Nodal Velocities Asymptotics 35 Complex-Valued Characteristic Velocities 36 Complex-Valued characteristic Wavenumbers 37 Critical Value of the TWT Principal Parameter 38 All-frequency Instability and Transition to it 38.1 Characteristic Function in Dimensionless Variables 38.2 Dispersion Relations in Dimensionless Variables 39 All-frequency Almost-Linear Unstable Branch and Its Gain 40 Single-Stream Approximation for Negative Phase Velocities V Lagrangian Field Theory of TWTs 41 The e-Beam Interacting with Multi-Transmission Line 41.1 Matrix Form Representation of the System 41.2 Voltages and Currents as Canonical variables 41.3 Circuit and Transmission Line Theories Point of View on the e-Beam and the Charge Wave 41.4 Matrix Hamiltonian Form of Equations 42 Homogeneous Multi-Transmission Line Interacting with Multi-Stream e-Beam 42.1 The Euler-Lagrange Equations and the Dispersion Relations 42.2 Admissible Phase Velocities 43 Detailed Energy Balance 43.1 Energy Balance for Homogeneous System Eigenmodes 43.2 Summary of Important Properties 43.3 Verification of the Power Flow and Energy Flux Formulas 44 Dielectric Medium Point of View VI Mathematical Subjects 45 Characteristic functions and equations 45.1 Derivatives of the Characteristic Functions 45.2 Imaginary and Real parts of the Characteristic Functions 45.3 Basic Properties of the Characteristic Velocities 46 MTL Characteristic Function Properties 47 Uncoupled e-Beam Related Quantities 47.1 e-beam Related Functions for Real Values of Their Variables 47.1.1 Characteristic function and its derivative 47.2 Bounds on the Characteristic Velocities 47.3 Zeros of the e-Beam Function 47.4 Solutions to the Level Equation for the e-Beam Function 47.4.1 One dominant stream 47.4.2 One pole (stream) case 47.4.3 Two poles (streams) case 47.4.4 Many poles (streams): small c case 47.4.5 Many poles (streams): large c case 48 Uncoupled Two-Stream e-Beam 48.1 Complex-Valued Characteristic Velocities 48.2 Zero-Frequency Complex-valued Velocities 48.3 Instability Node and Complex Velocities 48.4 Quasi-Concavity and Related Subjects 49 Nodal Function and Its Critical Points 50 Nodal and Critical Velocities 50.1 Single-Stream e-Beam and a Single TL Nodal Function 50.2 Two-Stream e-Beam and a Single TL Nodal Function 51 Bounds on the Characteristic Velocities for TWT 52 Series Expansions for the MTL and the e-Beam Functions 52.1 Series Expansions for the MTL Function 52.2 Series Expansions for the e-Beam Function 53 Circular Approximations for Admissible Non-Real Phase Velocities 53.1 Circular Type I Approximation 53.2 Accuracy of Circular type I Approximation and Higher-Order Approximations 53.3 Circular Type II Approximation 53.4 Circular Type III Approximation 53.5 Circular Type IV Approximation 54 Dispersion Relation at Instability Nodes - the Second-Order Degeneracy 55 Dispersion Relations at Critical States - Third-order Degeneracy 56 Real-Valued Functions of Real Variable and Their Zeros 57 Pseudo-Parts of a Complex Number VII Some Plasma Physics Subjects: Concise Review 58 Kinetic Theory Basics for Cold Collisionless Plasma 59 Macroscopic Fluid Model of Plasma 60 Electron-Beam Steady States for Macroscopic Fluid Model 61 Linearized Equations for the Cold Collisionless Plasma 62 The Pierce Model 63 Plasma Frequency Reduction Factor for Confined e-Beam 64 Velocity as a Function of the Gap Potential 65 Child’s Law and Fedosov’s Solution VIII Appendices A Fourier Transforms B Wave-Packet C Nevanlinna Functions D The Rouche Theorem E Reversion, Inversion and Other Operations on Power Series F Fixed Point Theorem and the Contraction Principle G Inverse of an Analytic Function at Critical Points H Moebius (Linear-Fractional) Transformations I Dielectric Properties and Material Relations J Lagrangian Field Theory, the Euler-Lagrange Equations and Energy Conservation Laws K Block-Matrices - their Inverse and Determinants L Matrix Polynomials M Time Averaging and Energy Conservation N Convexity, Quasi-Convexity and Log-Convexity O Polar Representation of a Curve in a Plane P Important Expressions and Equations P.1 MTL P.2 e-Beam P.3 TWT References Index Dedication -- Preface -- List of symbols and acronyms -- Review of the theory and its key elements. Introduction. Summary of the TWT-system features and effects. E-beam and multi-transmission line parameters. System lagrangian, field equations, characteristic equation and eigenmodes. Wave-particle interactions and origins of instability. Energy transfer and stream velocities. Instability concepts and their graphical representation. Instability branches of the characteristic function. Circular approximations to sets of admissible phase velocities. All-frequency modal branches. Instability phases via the dispersion-instability graph. Instability structure of all-frequency modal branches. Instability nodes and the degeneracy of the characteristic function. Instability at critical states : the third-order degeneracy. Almost-linear unstable modal branches. Wave-packet propagation and amplification. Single stream e-beam and MTL. Multi-stream e-beam coupled to a single TL. MTL as an approximation and TWT observables. Instability : possibilities and limitations -- Traveling wave tube components. Multi-transmission line. Multi-stream e-beam. Single-stream uncoupled e-beam -- TWT composed of a single-stream e-beam and a single transmission line. Lagrangian, field equations and power transfer. Characteristic function. Characteristic velocities and wavenumbers. Dispersion-instability graphs. Asymptotic expansions for infinite-frequency limit. Pierce model as the infinite-frequency limit approximation. Nodal phase velocities -- TWT with two-stream e-beam and a single TL. TWT lagrangian and the characteristic equation. Uncoupled two-stream e-beam. Infinite-frequency limit approximation. Instability nodes and nodal velocities. Complex-valued characteristic velocities. Complex-valued characteristic wavenumbers. Critical value of the TWT principal parameter. All-frequency instability and transition to it. All-frequency almost-linear unstable branch and its gain. Single-stream approximation for negative phase velocities -- Lagrangian field theory of TWTs. The e-beam interacting with multi-transmission line. Homogeneous multi-transmission line interacting with multi-stream e-beam. Detailed energy balance. Dielectric medium point of view -- Mathematical subjects. Characteristic functions and equations. MTL characteristic function properties. Uncoupled e-beam related quantities. Uncoupled two-stream e-beam. Nodal function and its critical points. Nodal and critical velocities. Bounds on the characteristic velocities for TWT. Series expansions for the mtl and the e-beam functions. Circular approximations for admissible non-real phase velocities. Dispersion relation at instability nodes : the second-order degeneracy. Dispersion relations at critical states : third-order degenerac. Real-valued functions of real variable and their zeros. Pseudo-parts of a complex number -- Some plasma physics subjects : concise review. Kinetic theory basics for cold collisionless plasma. Macroscopic fluid model of plasma. Electron-beam steady states for macroscopic fluid model. Linearized equations for the cold collisionless plasma. The pierce model. Plasma frequency reduction factor for confined e-beam. Velocity as a function of the gap potential. Child's law and fedosov's solution -- Appendices. Fourier transforms. Wave-packet. Nevanlinna functions. The rouche theorem. Reversion, inversion and other operations on power series. Fixed point theorem and the contraction principle. Inverse of an analytic function at critical points. Moebius (linear-fractional) transformations. Dielectric properties and material relations. Lagrangian field theory, the Euler-Lagrange equations and energy conservation laws. Block-matrices: their inverse and determinants. Matrix polynomials. Time averaging and energy conservation. Convexity, quasi-convexity and log-convexity. Polar representation of a curve in a plane. Important expressions and equations -- References -- Index "The Traveling Wave Tubes (TWT) is a powerful vacuum electronic device used to amplify radio-frequency (RF) signals as well as numerous applications such as radar, television and telephone satellite communications. This monograph is devoted to the author's original theoretical developments in the theory of a traveling wave tube (TWT). Most of the monograph is the author's original work on an analytical theory of TWTs. It is a constructive Lagrangian field theory of TWT in which the electron beam (e-beam) is represented by one-dimensional multi-stream electron flow and the guiding slow-wave structure is represented by possibly non-uniform multi-transmission line (MTL). The proposed analytic theory accounts for a number of electron plasma phenomena including space-charge effects such as electron-to-electron repulsion (debunching), convective instabilities, wave-particle interaction, amplifying waves and more. It allows, in particular, to (i) identify origins of the wave-particle interaction and the system convective instability (exponential growth); (ii) evaluate the energy transfer rate from the e-beam to the electromagnetic radiation; (iii) identify instability modal branches which under condition of sufficiently strong coupling between the e-beam and the MTL can cover ideally all frequencies"--Publisher's website