A Large Signal Theory of Traveling 

 Wave Amplifiers 



Including the Effects of Space Charge and Finite 

 Coupling Between the Beam and the Circuit 



By PING KING TIEN 



Manuscript received October 11, 1955) 



The non-linear behavior of the traveling-wave amplifier is calculated in 

 this paper by numericalhj integrating the motion of the electrons in the 

 presence of the circuit and the space charge fields. The calculation extends 

 the earlier work by Nordsieck and the srnall-C theory by Tien, Walker and 

 Wolontis, to include the space charge repulsion between the electrons and 

 the effect of a finite coupling between the circuit and the electron beam. It 

 however differs from Poulter's and Rowers works in the methods of calcu- 

 lating the space charge and the effect of the backward wave. 



The numerical work was done using 701 -type I.B.M. equipment. Re- 

 sults of calcidation covering QC from 0.1 to 0.4, b from 0.46 to 2.56 and k 

 from 1.25 to 2.50, indicate that the saturation efficiency varies between 

 23 per cent and 37 per cent for C equal to 0.1 and between 33 per cent and 

 Jf.0 per cent for C equal to 0.15. The voltage and the phase of the circuit wave, 

 the velocity spread of the electrons and the fundamental component of the 

 charge-density modidation are either tabulated or presented in curves. A 

 method of calculating the backward wave is provided and its effect fully 

 discussed. 



1. INTRODUCTION 



Theoretical evaluation of the maximum efficiency attainable in a 

 traveling-wave amplifier requires an understanding of the non-linear 

 behavior of the device at various working conditions. The problem has 

 been approached in many ways. Pierce/ and later Hess,^ and Birdsalf 

 and Caldwell investigated the efficiency or the output power, using cer- 

 tain specific assumptions about the highly bunched electron beam. They 

 either assume a beam in the form of short pulses of electrons, or, specify 



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