722 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1953 



vector from Ye* to Ye according to the Bessel function plot of Fig. 2. 

 For 6 = 2t{N + ^i) radians where iV = 0, 1, 2, 3 . . . the electronic 

 admittance becomes a pure negative conductance as may be seen by 

 putting this relation into equations (2.1) or (2.2). 



Summarizing, we have seen that the presence of an electron stream 

 bunched in accordance with the arrangement of Fig. 1 gives rise to an 

 admittance appearing across the grids bounding the interaction space. 

 The phase angle of this admittance is a function only of the repeller drift 

 angle, d, while its magnitude depends on both 6 and the RF gap voltage, 

 F. For values of 6 in the vicinity of 2x(iV + ^) radians the conductance 

 component is negative, a necessary condition for the production of sus- 

 tained oscillations. 



2.2 Passive Circuit Admittance of Single Resonator 



Any resonant cavity may be represented by a simple parallel G-C-L- 

 combination provided the desired resonance is sufficiently far removed 

 from adjacent ones. In some cases such as in cylindrical or waveguide 

 cavities, to name two, it is difficult to ascribe physical significance to 

 the lumped elements appearing in the equivalent circuit representation. 

 This is not so in the case of a conventional reflex klystron cavity. Since 

 the latter always consists of a re-entrant type resonator, most of the 

 electric field is concentrated in the interaction gap, i.e., the narrow region 

 traversed by the outgoing and returning electrons and bounded by two 

 parallel grids, while the major portion of the magnetic flux resides in 

 the outer cylindrical section. Thus, the effective shunt capacitance ap- 

 pearing in the equivalent circuit is associated primarily with the above 

 grid planes, a minor contribution originating in the fringing field close 

 to the re-entrant post and the residual electric field in the outer cylindri- 

 cal part of the cavity. 



The input admittance, F, of a high Q resonator when represented by 

 C, L, and G connected in parallel is given by, 



Y = G{\ -\-j2Qb) (2.3) 



= G + j2CAcu, (2.4) 



where Q = 03qC/G and 5 = Aw/coq = A///o = (/ - /o)//o , and G represents 

 all internal resonator losses plus the external load referred to the gap. 

 Plotted in the complex admittance plane, the locus of the admittance 

 vector with varying frequency is a straight line parallel to the imaginary 

 axis and spaced a distance corresponding to G to its right. The var- 

 iation in susceptance is directly proportional to the frequency deviation 



