A Large Signal Theory of Traveling Wave Applifiers: Including the Effects of Space Charge and Finite Coupling Between the Beam and the Circuit

Theoretical evaluation of the maximum efficiency attainable in a traveling-wave amplifier requires an understanding of the non-linear behavior of the device at various working conditions. T h e problem has been approached in many ways. Pierce, 1 and later Hess, 2 and Birdsall 3 and Caldwell 1 investigated the efficiency or the o u t p u t power, using certain specific assumptions about the highly bunched electron beam. T h e y either assume a beam in the form of short pulses of electrons, or, specify 349 350 TIIE B E L L SYSTEM TECHNICAL J O U R N A L , MARCH 1 9 5 6 an optimum ratio of the fundamental component of convection current to the average or d-c current. T h e 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 a t t a c k has been pursued by Parzen 5 and Kiel, 6 although their work is far from complete. T h e most satisfying approach to date is Nordsieck's analysis. 7 Nordsieck followed a typical set of "electrons" and calculated their velocities and positions by numerically integrating a set of equations of motion. Poulter 8 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 9 have published a small C theoiy in which "elect r o n s " 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.