Modification of klystron beam loading by initial velocity modulation of the beam

The authors discuss the correction of klystron cavity beam-loading formulas for the effect of velocity modulation on the beam. These effects were small for older klystrons, in which the quarter-plasma-wavelength spacing minimized the velocity modulation at each gap. It is noted that they appear to be significant for recent designs using larger numbers of cavities with irregular spacings. Accurate formulas are given for the transit time and exit velocity of individual particles, and analytic solutions are presented which appear to have some merit but are incomplete.<>