Microwave Tubes

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  • Topic: Hyperbolic function, Radius
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  • Published : February 13, 2013
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A Model for the, Klystron .Cavity Gap

radius of curvature r,,. This was recognized by the workers during World War I1 [l], [2] ,who observed that the “perfectly sharp” gap could be solved by the method of conformal transformation, and yielded a sin-’ variation of the potentia2 across the gap when g/a was either very small or very large. The assumption that this solution was also reasonably good in the middle range of g/a led to the proposal [2] to use J0(S/2) instead of sin (8/2)/(8/2) in (1). This was equivalent to saying that the sharp gap case corresponded to a blunt gap about 25 percent longer, since Jo(x) is a good approximation to sin 1.25d1.25~ over the -1. INTRODUCTION range of values that occur (up to about 4 radians transit HE GAP of a klystron cavity is shownin Fig. 1. Fie.ds angle). developedby an RF voltage VRF across the gap of In reality, the gap is neither perfectly sharp nor perlength g will interact with an electron beam of radiuc. b fectly blunt. Kosmahl and Branch [3] adopted the cosh passing at velocity uo through thetunnel of radius a. Under function to represent the field across the gap in the form large signal conditions, an integral of the form 1E.J dv is E&, a) = Eo cash (mz), (-g/2 < z < g/2) required to compute the interaction accurately. But, uric-er small-signal conditions, which apply for all but the Past =0 ( 2 > g/2). 11 (2) few cavities of a multicavity tube, the interaction can be represented by a coupling factor M: this is the factor 3y The parameter m, which has the dimension of an inverse which the energy exchange is reduced from that due te a length, describes the sharpness of the gap noses, but asdc voltage change equal to the peak RF voltage. It is ba.- signing a value to it is a matter of some difficulty for the sically a transit-time factor, but the fringing of the fiel’js design engineer. Kosmahl and Branch obtained a value by in an ungridded gap causes the transit time to vary as a using a relaxation program to get a detailed map of the function of radius. This variation, in turn, depends on the cavity fields; the program LALA by Rich and McRoberts detailed distribution of field across the gap, since this is [5] is capable of carrying out this calculation, but costs part of the boundary condition from which the fields in the about $300 for a single case even at “overnight” rates, as interior must be derived. well as being fairly difficult to use. Kosmahl and Branch An accurate vdlue for M is of increasing importance be- also showed that the value obtained was consistent with cause klystrons with larger numbers of cavities are now classical pekurbation measurements made by drawing a being seriously considered; in the expression for the gain bead along the axis; but this does not provide a method of an n-cavity klystron, M appears to the 2nth power. for measuring the value of m, because the Gaussian hump The simplest assumption is the uniform field of strengllb obtained from measurements along the axis is consistent V&/g, the “blunt gap. ” This assumption leads to a thew with any reasonable distribution at the gapedge. Kosmahl retical coupling factor Mth,averaged overthe beam radius, and Branch also suggested that the value of cosh (mg/2) would normally lie in the range 1.25 to 3. The cosh function is in fact closely related to the earlier sin-’ solution for the potential: differentiating the latter to find the field, we have l / m , and expanding this by where 8 is the transit angle u g h o and the binomial theorem we find the first two terms to be y = J(o/uo)2 - (w/# identical with the series expansion of the cosh x ; the solutions differ only close to the gap edge(x -+ l ) , where l/ which has been the standard expression used formany l/1-;cz has a singularity appropriate to the sharp corner years. while cosh x has no singularity. This difference in the...
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