Behavior of traveling-wave tubes near circuit cutoff

A theory is developed which explains the operation of a traveling-wave tube when operated near the cutoff frequency of the slow-wave circuit, including the effect of two circuit waves instead of the usual one. The theory is normalized in a manner analogous to that used in more conventional analyses and a first-order expansion about the cutoff frequency is used, making a relatively small number of curves applicable to a large number of cases. The relationship between this theory and the three-wave theory usually used in traveling-wave-tube analysis is shown, and they are in agreement when the system is operated far from the cutoff frequency. Numerical results are given for a range of parameters which might be useful in traveling-wave-tube design, and an excellent agreement with published experimental results is shown.