330 BELL SYSTEM TECHNICAL JOURNAL 



tube. This tube, the velocity-modulation tubes of Hahn and Metcalf, and 

 the klystron tubes of the Varian Brothers, are alike in their use of transit- 

 time bunching in a relatively-field-free drift tube. Since this separation of 

 functions renders these devices much easier to analyze and since the struc- 

 tures are quite interesting in any case we will spend most of our time con- 

 sidering them and will, I fear, rather neglect some of the other types of tubes. 

 \\ e will, of course, keep our analysis as general as possible so that the 

 results may be applied to a variety of different devices. 



Input Gap Analysis 



Let us begin by a small-signal consideration of a uniform electron stream 

 entering a region in which there is a longitudinal field defined as some func- 

 tion of the distance and of time. This can be the entire Llewellyn diode 

 or it can be the input region of a klystron. We ask ourselves with what 

 velocity will the electrons leave this region and what will be the net exchange 

 of energy between the electrons and the field. At any point within the field 

 a typical electron will experience an acceleration given by 



y =lE-^r,f{y)m (1) 



where r) is proportional to the maximum amplitude of the h.f. field, but con- 

 tains a numerical constant so that y is expressed in centimeters per second 

 per second. Now in the usual case f(t) will be a simple sine function but 

 f(y) may assume a variety of forms. Again, by way of simplifying our 

 work we will assume that it is also a sine function. Let us consider how 

 we can go about solving this apparently simple equation. Unfortunately 

 this expression can not be solved directly because the value of / at any plane 

 (that is, the time of arrival of an electron at this plane) depends upon the 

 interchange of energy between the electron and the field. Here we are 

 forced back to the time-honored mathematical device of assuming a solu- 

 tion in the form of a series and then evaluating these coefficients. There is a 

 large number of ways in which this can be done, and consequently a large 

 number of different solutions which look very different but which all give 

 comparable answers. Usually when such solutions are published, the arith- 

 metical work is omitted leaving one with the feeling that there is something 

 involved that is not within the ken of ordinary mortals. The fact is that 

 the work is usually extremely tedious but actually very simple. It will be 

 instructive to follow through one form of such an analysis in just enough 

 detail to see the amount of work involved. 



Since we are interested in the energy which is proportional to y- we will 

 write at once 



{y%^a = K = Ko-^vK,-{- v'K2 + rj^K^ + ... 



