WAVES IN ELECTRON STRE.\MS AND CIRCUITS 



647 



the r-f field. Energy is extracted from the stream only by a bunching process 

 in which in the emerging beam the charge density is higher when the veloc- 

 ity is below average than it is when the velocity is above average. In other 

 words, the kinetic energy averaged over electrons is reduced, even though 

 the time average of u^ is not changed. This means that the emerging beam 

 must be strongly bunched if much power is to be abstracted. 



In the conventional traveling-wave tube all is well. At the input the 

 r-f field is small and the beam is imbunched. At the output the r-f field is 

 high, and the beam is strongly bunched, having lost energy to the circuit. 



Imagine a tube using a backward wave, however. The electrons are in- 

 jected unbunched at the output, where the signal level is high. They emerge 

 at the input where the signal level is low. If the tube is to give high power, 

 the stream must emerge strongly bunched. The disturbance in the electron 

 stream cannot gradually increase as the field ampHtude increases. 



jx, 



JXt 



JX; 



jx, 



JX2 



jx, 



JXj 



Fig. 4.2 — A ladder network. 



We have seen that one cannot draw conclusions about gain just by look- 

 ing at the propagation constants of the waves. Waves are merely solutions 

 of a differential equation connected with a physical system. To find the 

 properties of the system one must examine, not various solutions of the dif- 

 ferential equation, but the particular solution (which may be a combina- 

 tion of simple solutions) which applies to the system in question. 



As a further example, we will examine another system whose differential 

 equations yield "growing" solutions which turn out to be backward waves. 

 Consider the ladder network of Fig. 4.2. This propagates an unattenuated 

 wave if Xi and Xz have opposite signs, {Xi inductive and Xz capacitive, 

 for instance). If, however, Xi and Xz are both capacitive or both inductive, 

 then a wave excited in the circuit decays exponentially with distance. If 

 we speak in terms of /3i , then 



ft = —jai 



where ai is a real number. 



