Effect of boundaries on the space charge potential in coupled cavity travelling wave tubes

In the theory of microwave linear beam tubes, it has hitherto been assumed that the space charge potential for a cylindrical electron beam passing through a concentric, earthed, conducting tunnel broken by interaction gaps can be calculated by supposing that the tunnel is continuous. In this paper, the effects of the gaps are computed for both an electron beam and a disc of charge. It is shown that for a uniform electron beam these effects can be represented by replacing the actual boundary with an equivalent continuous tunnel, the radius of which is termed the effective tunnel radius. An empirical equation has been derived from which the effective tunnel radius can be calculated to an accuracy of better than ±3%. The effective tunnel radius is a function of the tunnel radius, the gap length and the gap separation but is independent of the beam radius. For a disc of charge, it is found that the effective tunnel radius depends on the position of the disc relative to the gaps but that the average value of the effective tunnel radius as the disc moves through the gapped tunnel is equal to the value computed for a uniform electron beam. This average value of the effective tunnel radius is independent of the disc thickness.