Traveling-Wave Tubes 



By J. R. PIERCE 



Coinright, 1950, D. Van Nostrand Company, Inc. 



[THIRD INSTALLMENT] 



CHAPTER VII 

 EQUATIONS FOR TRAVELING- WAVE TUBE 



Synopsis of Chapter 



IN CHAPTER VI we have expressed the properties of a circuit in terms 

 of its normal modes of propagation rather than its physical dimensions. 

 In this chapter we shall use this representation in justifying the circuit 

 equation of Chapter II and in adding to it a term to take into account the 

 local fields produced by a-c space charge. Then, a combined circuit and 

 ballistical equation will be obtained, which will be used in the following 

 chapters in deducing various properties of traveling-wave tubes. 



In doing this, the lirst thing to observe is that when the propagation con- 

 stant r of the impressed current is near the propagation constant Pj of a 

 particular active mode, the excitation of that mode is great and the excita- 

 tion varies rapidly as P is changed, while, for passive modes or for active 

 modes for which P is not near to the propagation constant P„ , the excita- 

 tion varies more slowly as P is changed. It will be assumed that P is nearly 

 equal to the propagation constant Px of one active mode, is not near to the 

 propagation constant of any other mode and varies over a small fractional 

 range only. Then the sum of terms due to all other modes will be regarded 

 as a constant over the range of P considered. It will also be assumed that 

 the phase velocities corresponding to P and Pi are small compared with 

 the speed of light. Thus, (6.47) and (6.47a) are replaced by (7.1), where the 

 first term represents the excitation of the Pi mode and the second term repre- 

 sents the excitation of passive and "non-synchronous" modes. In another 

 sense, this second term gives the field produced by the electrons in the ab- 

 sence of a wave propagating on the circuit, or, the field due to the "space 

 charge" of the bunched electron stream. Equation (7.1) is the equation for 

 the distributed circuit of Fig. 7.1. This is like the circuit of Fig. 2.3 save for 

 the addition of the cajmcitances C'l between the transmission circuit and 

 the electron beam. We see that, because of the presence of these capaci- 

 tances, the charge of a bunched electron beam will produce a field in addi- 

 tion to the field of a wave traveling down the circuit. This circuit is intui- 

 tively so appealing that it was originally thought of by guess and justified 

 later. 



Equation (7.1), or rather its alternative form, (7.7), which gives the volt- 

 age in terms of the impressed charge density, can be combined with the 



390 



