General Expressions for the Magnetic Flux Density Produced by Axially Magnetized Toroidal Permanent Magnets

This paper presents analytical-integral expressions which evaluate the magnetic flux density radial and axial components' on- and off-axis of axially magnetized toroidal permanent magnets. The pieces, when used in magnetic electron beam focusing structures called periodic-permanent magnets (PPMs), can be employed in microwave vacuum electronics devices. These expressions can be adapted to computational algorithms, especially those developed based on mathematical commercial codes, aiding the investigation of the effect of magnetic fields on electron beam dynamics. Comparisons between analytical-integral expressions and theoretical and experimental data are also presented and discussed.

I. Introduction

A magnetic electron beam focusing system is an essential constituent of power microwave devices, such as klystron amplifiers and traveling-wave tubes (TWTs), which avoids the electron stream spreading due to space-charge forces ensuring suitable electron beam propagation, with diameter approximately constant, through the microwave device drift tube. Thus, the beam constitutes an adequate material medium to be able to convert its own kinetic energy into electromagnetic radio-frequency (RF) power. Magnetic focusing systems can be composed of solenoids, permanent magnets, or both. Details of possible systems and design examples can be found in [1] and [2].