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… (more)

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.**Contents:**

- 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

**Readership:** Graduate students and researchers in plasma physics, electrical engineering and mathematical physicist interested in vacuum electronics. As well as those interested in scope of subjects related to physics of electron beams. Amplifier;TWT;Traveling Wave Tube;Electron Beam;Multi-Stream Electron Beam;Two-Stream Electron Beam;Multiple Beams;Double-Stream Amplifier;Instability;Two-Stream Instability;Vacuum Electronics;Wave-Particle Interaction0**Key Features:**

- Unique mathematical physics prospective on the theory of TWTs. All analytical developments are illustrated extensively by plots and graphs
- Deep analytical insight into the physics of electron beams including origins of plasma instabilities and the "wave-particle interaction"
- The theory provides tools for analytical assessment of regimes of TWT operation as well for conceiving new amplification regimes

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