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BELL SYSTEM TECHNICAL JOURNAL 



a cascade transit-time-bunching amplifier in which a series of three gaps is 

 used together with two drift spaces. The first gap velocity modulates the 

 beam; this modulation is converted into a current modulation in the first 

 drift space. The beam then excites the second cavity, which again velocity 

 modulates the beam in quadrature with the original modulation. This 

 action of course occurs in the output gap of a two-gap tube but it is not 

 there used. Here this second and larger velocity modulation is converted 

 to current modulation in the second drift space. The output is finally 

 taken ofif the beam by the third gap. A phase-focusing diagram of this 

 sort (again taken from Harrison's paper) is shown in Fig. 12. 



Space-Charge-Wave Analysis 



This phase-focusing approach is rather intriguing as one feels that one 

 has a physical picture of what is going on. The picture is, however, very 

 inexact except under certain highly specialized cases, as it completely ignores 



Fig. 11- 



-Diagram showing reflex bunching combined with field-free transit-time bunching 

 (Harrison) . 



space-charge effects. These space-charge effects are of two sorts: a d-c 

 effect, if you will, and an r-f effect; that is, the presence of the electrons of 

 the beam will alter the average velocity of the electrons at different parts of 

 the beam, and will tend to undo the bunching action. Because of this 

 second effect, the electrons are effectively prevented from passing each other 

 as the graphical solution suggests. Instead, as the density of the electrons 

 in the bunch becomes greater, the mutual repulsion forces tend to prevent a 

 further concentration of charge. The electron bunch then tends to disperse. 

 The action could be likened to the propagation of a sound wave in a moving 

 column of air. While there are several approximate ways to handle this 

 problem, Hahn was the first to propose a really satisfactor>^ theory. Inci- 

 dentally it should be noted that the Benham, MuUer, Llewellyn and Peter- 

 son type of theory is capable of treating this aspect of the problem in a 

 rigorous way and including all space-charge effects, but unfortunately these 

 theories are limited in that they have been applied only to the parallel- 

 plane case, and of course they are only small-signal theories. 



