132 THE BELL SYSTEM TECHNICAI- JOURNAL, JANUARY 1956 



functions; this is made possible by the freedom with which couplers 

 and attenuators can be placed at any chosen point along the tube. 



Considerable work in this field has been done elsewhere. Reference 

 will be made to it wherever possible. However, only that work with 

 which the authors have been intimately connected will be fully reported 

 here. In particular, the effect of the electron beam on the wave propaga- 

 tion phenomena will not be considered. 



2. THEORY OF COUPLED HELICES 



2.1 Introduction 



In the past, considerable success has been attained in the under- 

 standing of traveling wave tube behavior by means of the so-called 

 "transmission-line" approach to the theory. In particular, J. R, Pierce 

 used it in his initial analysis and was thus able to present the solution 

 of the so-called traveling-wave tube equations in the form of 4 waves, 

 one of which is an exponentially growing forward traveling wave basic 

 to the operation of the tube as an amplifier. 



This transmission-line approach considers the helix — or any slow- 

 wave circuit for that matter — as a transmission line with distributed 

 capacitance and inductance with which an electron beam interacts. 

 As the first approximation, the beam is assumed to be moving in an RF 

 field of uniform intensity across the beam. 



In this way very simple expressions for the coupling parameter and 

 gain, etc., are obtained, which give one a good appreciation of the 

 physically relevant quantities. 



A number of factors, such as the effect of space charge, the non-uniform 

 distribution of the electric field, the variation of circuit impedance with 

 frequency, etc., can, in principle, be calculated and their effects can be 

 superimposed, so to speak, on the relatively simple expressions deriving 

 from the simple transmission line theory. This has, in fact, been done and 

 is, from the design engineer's point of view, quite satisfactory. 



However, phj^sicists are bound to be unhappy over this state of 

 affairs. In the beginning was Maxwell, and therefore the proper point to 

 start from is Maxwell. 



So-called "Field" theories of traveling-w^ave tubes, based on Maxwell's 

 equation, solved with the appropriate boundary conditions, have been 

 worked out and their main importance is that they largely confirm the 

 results obtained by the inexact transmission line theory. It is, however, 

 in the nature of things that field theories cannot give answers in terms of 



