BELL SYSTEM TECHNICAL JOURNAL 



CHAPTER II 



SIMPLE THEORY OF 

 TRAVELING-WAVE TUBE GAIN 



Synopsis of Chapter 



IT IS difficult to describe general circuit or electronic features of traveling- 

 wave tubes without some picture of a traveling-wave tube and traveling- 

 wave gain. In this chapter a typical tube is described, and a simple theoret- 

 ical treatment is carried far enough to describe traveling-wave gain in terms 

 of an increasing electromagnetic and space-charge wave and to express the 

 rate of increase in terms of electronic and circuit parameters. 



In particular, Fig. 2.1 shows a typical traveling- wave tube. The parts of 

 this (or of any other traveling-wave tube) which are discussed are the elec- 

 tron beam and the slow-wave circuit, represented in Fig. 2.2 by an electron 

 beam and a helix. 



In order to derive equations covering this portion of the tube, the proper- 

 ties of the helix are simulated by the simple delay line or network of Fig. 22, 

 and ordinary network equations are applied. The electrons are assumed to 

 flow very close to the line, so that all displacement current due to the pres- 

 ence of electrons flows directly into the line as an impressed current 



For small signals a wave- type solution of the equations is known to exist, 

 in which all a-c electronic and circuit quantities vary with time and dis- 

 tance as exp(_;co/ — Tz). Thus, it is possible to assume this from the start. 



On this basis the excitation of the circuit by a beam current of this form is 

 evaluated (equation (2.10)). Conversely, the beam current due to a circuit 

 voltage of this form is calculated (equation (2.22)). If these are to be con- 

 sistent, the propagation constant V must satisfy a combined equation (2.2vS). 



The equation for the propagation constant is of the fourth degree in F, 

 so that any disturbance of the circuit and electron stream may be expressed 

 as a sum of four waves. 



Because some quantities are in j)ractical cases small compared with others, 

 it is p()ssil)le to obtain good values of the roots by making an approximation. 

 This reduces the cciuation to the third degree. The solutions are expressed 

 in the form 



