Certain Mean Values in the Theory of the Traveling-Wave Amplifier

Some aimplc results relating to certain mean values are given. It is not assumed that the signals are necessarily small; hence nonlinear effects arc taken into account. The purpose of this note is to give a few simple results relating to certain mean values that occur in the theory of the traveling-wave amplifier. Whereas all of the previous theory of the amplifier has been based on the assumption that the signals are small, so that the system behaves effectively as a linear system, no such assumption is involved in the results given here. After some idealization of the physical system, the fundamental equations of the traveling-wave amplifer can be written as follows: ' =- V (1) dt dx ' , clV d/ dp L d + R I d r ( ( 2 ) df dt dM dx = 0 ( 3 ) dt 2 dx dx The independent variables t and x represent, respectively, time and distance measured in the axial direction from the driving point; I, V, p and v denote the instantaneous local values of the current in the conductor, the potential of the conductor, the linear charge density of the p 1 * This material was prepared as a memorandum or report in 1946, but was never published. It has, however, been known to people working in the field, and it has been mentioned in published work. It seems desirable that it be made generally available in its original form. 365 3GG THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1960 electron stream, and the velocity of the electrons, respectively; L, /?, C, a and (3 are constants. Now suppose that, we have a state of the system in which, for each value of x, the variables I, V, p and v are all periodic functions of time, with the period T.