Traveling-Wave Tubes (Third Installment)

N CHAPTER VI we have expressed the properties of a circuit in terms of its normal modes of propagation rather than its physical dimensions. In this chapter we shall use this representation in justifying the circuit equation of Chapter II and in adding to it a term to take into account the local fields produced by a-c space charge. Then, a combined circuit and ballistical equation will be obtained, which will be used in the following chapters in deducing various properties of traveling-wave tubes. In doing this, the first thing to observe is that when the propagation constant T of the impressed current is near the propagation constant Ti of a particular active mode, the excitation of that mode is great and the excitation varies rapidly as T is changed, while, for passive modes or for active modes for which T is not near to the propagation constant T r i , the excitation varies more slowly as T is changed. It will be assumed that T is nearly equal to the propagation constant r x of one active mode, is not near to the propagation constant of any other mode and varies over a small fractional range only. Then the sum of terms due to all other modes will be regarded as a constant over the range of T considered. It will also be assumed that the phase velocities corresponding to T and Ti are small compared with the speed of light. Thus, (6.47) and (6.47a) are replaced by (7.1), where the first term represents the excitation of the Ti mode and the second term represents the excitation of passive and "non-synchronous" modes.