# Accession Number:

## AD0260403

# Title:

## PHASE RELATIONSHIPS IN A NONLINEAR VELOCITY-MODULATED ELECTRON BEAM

# Descriptive Note:

# Corporate Author:

## CORNELL UNIV ITHACA N Y

# Personal Author(s):

# Report Date:

## 1961-01-31

# Pagination or Media Count:

## 1.0

# Abstract:

The fundamental differential equation describing the polarization of an electron in an infinitely wide electron beam is derived. The modified fundamental differential equation is formulated into a fourth-degree non-linear partial differential equation. Terms involving the polarization raised to powers higher than the third are neglected and the remaining third-degree equation is separated into 3 linear differential equations by successive approximations. The solution to the differential equation for the first-degree polarization is substituted into the equation for the second-degree polarization to yield the driving terms of that equation. The second-degree polarization is composed of dc and first-harmonic frequency terms. The procedure is repeated and the fundamental-frequency terms of the third-degree polarization are computed. The component of the third-degree fundamental-frequency polarization in quadrature with the first-degree polarization describes the nonlinear phase shift behavior of the electron beam. Author

# Descriptors:

- *ELECTRON BEAMS
- *PHASE MEASUREMENT
- DIFFERENTIAL EQUATIONS
- ELECTRIC FIELDS
- ELECTRONS
- HARMONIC ANALYSIS
- KLYSTRONS
- MAGNETIC FIELDS
- MAGNETOHYDRODYNAMICS
- MICROWAVE AMPLIFIERS
- MICROWAVE FREQUENCY
- MODULATION
- NONLINEAR DIFFERENTIAL EQUATIONS
- NONLINEAR SYSTEMS
- NUMERICAL ANALYSIS
- PARTIAL DIFFERENTIAL EQUATIONS
- PHASE SHIFT CIRCUITS
- PLASMAS(PHYSICS)
- POLARIZATION
- SPACE CHARGE
- STABILITY
- TRAVELING WAVE TUBES
- VELOCITY