350 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



an optimum ratio of the fundamental component of convection current 

 to the average or d-c current. The method, although an abstract one, 

 generally gives the right order of the magnitude. When the usual wave 

 concept fails for a beam in which overtaking of the electrons arises, we 

 may either overlook effects from overtaking, or, using the Boltzman's 

 transport equation search for solutions in series form. This attack has 

 been pursued by Parzen and Kiel, although their work is far from com- 

 plete. The most satisfying approach to date is Nordsieck's analysis.' 

 Nordsieck followed a typical set of "electrons" and calculated their 

 velocities and positions by numerically integrating a set of equations of 

 motion. Poulter has extended Nordsieck equations to include space 

 charge, finite C and circuit loss, although he has not perfectly taken into 

 account the space charge and the backward wave. Recently Tien, 

 Walker, and Wolontis have published a small C theory in which "elec- 

 trons" are considered in the form of uniformly charged discs and the 

 space charge field is calculated by computing the force exerted on one 

 disc by the others. Results extended to finite C, have been reported by 

 Rowe,^*^ and also by Tien and Walker.^^ Rowe, using a space charge 

 expression similar to Poulter's, computed the space charge field based on 

 the electron distribution in time instead of the distribution in space. This 

 may lead to appreciable error in his space charge term, although its 

 influence on the final results cannot be easily evaluated. 



In the present analysis, we shall adopt the model described by Tien, 

 Walker and Wolontis, but wish to add to it the effect of a finite beam to 

 circuit coupling. A space charge expression is derived taking into account 

 the fact that the a-c velocities of the electrons are no longer small com- 

 pared with the average velocity. Equations are rewritten to retain terms 

 involving C. As the backward wave becomes appreciable when C in- 

 creases, a method of calculating the backward wave is provided and the 

 effect of the backward wave is studied. Finally, results of the calculation 

 covering useful ranges of design and operating parameters are presented 

 and analyzed. 



2. ASSUMPTIONS 



To recapitulate, the major assumptions which we have made are: 



1. The problem is considered to be one dimensional, in the sense that 

 the transverse motions of the electrons are prohibited, and the current, 

 velocity, and fields, are functions only of the distance along the tube and 

 of the time. 



2. Only the fundamental component of the current excites waves on 

 the circuit. 



