400 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



normal modes of propagation. For any particular tijpe of beam and cir- 

 cuit, three circuit parameters must be evaluated from the field solution. 

 The performance of the traveling-wave tube is then described quite ac- 

 curately by a cubic equation containing these parameters, over a wide 

 range of dimensions and operating conditions. The usefulness of this 

 normal-mode method has been further enhanced by publication of a 

 nomograph for the calculation of the gain constant. 



In its initial form, the normal-modes solution for a helix travehng- 

 wave tube was greatly simplified by the assumption that the electron 

 beam is so thin that the electric field acting on it is constant. Employing 

 the field solution for a beam of finite thickness in a helix, Fletcher'' was 

 able to compute the circuit parameters for the solid and hollow cylindri- 

 cal electron beams, respectively, confined to rectifinear flow. 



This procedure will now be extended to cylindrical beams in Brill ouin 

 flow, in which transverse electron motion occurs. First, it will be neces- 

 sary to solve the field equations for this type of beam in a helix. As a 

 by-product of this computation, the solution of the field equations for 

 the beam in a concentric drift tube will briefly be given. Finally, with 

 some restrictions, the helix parameters will be evaluated, and the gain of 

 helix amplifiers with such beams compared with that obtained with 

 otherwise identical rectilinear beams. 



FIELD EQUATIONS IN THE ELECTRON BEAM 



When a small ac field is impressed upon a short length of electron beam, 

 the electrons respond by executing small ac excursions about their steady- 

 state trajectories. These ac motions of charged particles constitute a 

 transverse distribution of ac currents, which in turn excites an ac field 

 distribution. The propagation of an ac signal along a beam depends upon 

 the reciprocal action of these currents and fields. 



To find the propagation constants for a particular configuration of 

 electron stream and enclosure, we must therefore solve Maxwell's equa- 

 tions in the presence of the ac driving currents in the beam, subject to 

 the external boundary conditions. When the fields and currents possess 

 circular symmetry, these equations may be formally separated into TE 

 and TM groups." In addition, as we are concerned only with "slow" 

 waves, the equations may be simplified by neglecting all terms of rela- 

 tive magnitude k /y , where k is the wave number in free space, and 7 

 the propagation wave number. 



TM WAVE 



Id/ dE\ 2-n T't-iTI^/t-n f^\ 



-^V-^)-'^^^^ ^'^-+--T- ^rJr) (1) 



r or \ or I jcoe coe r or 



