WAVE PROPAGATION ALONG AN ELECTION BEAM 



413 



From (68), (72), and (74), we find: 



(Ys\ ^ ( d}\/dy \ 



yb hiyh) 

 LT/i(76)J,=,„ 



(79) 



Kb = Kh 



(80) 



The impedance i)arameters for the two beams are therefore related to 

 each other by: 



"2^ hiyhy 



_76 /o(7&). 



Both Pierce^*^ and Fletcher ha\e found the impedance parameter of 

 the hollow beam to be related to that of a thin beam along the axis of a 

 helix, Kr , as follows: 



Kh = /Vr[/o'(7&)]7=7o 



The gain parameter C is defined by: C^ = {2K){Io/SVo) 



(81) 

 (82) 



Thus, for gi^'en In and Vo , the factor by which the gain parameter of a 

 thin beam should be multiplied to give that of a hollow beam, is: 



(K^/KrY" = [h"\yb)],^,, (83) 



Tliis "impedance reduction factor" can similarly be evaluated for the 

 finite cylindrical beam with Brillouin flow: 



1/3 



{Ks/KrY" = 



2 



_76 



h{yh)-U{yh) 



(84) 



Cutler, who calls this quantity F^ , has described how it and the 

 parameter Q can be used to compute the gain of traveling-wave tubes. 

 The procedure depends upon the evaluation of C and QC. The expres- 

 sion for Cb , in Cutler's notation, is: 



(lu/Kf FiF^ih/SVof" 



Cb 



(85) 



Here K2/K is a factor, of the order of 0.5, which corrects the impedance 

 of the ideal sheath helix for the physical dimensions and support ele- 

 ments of the actual hehx. It is best found by measurement. The factor 

 Fi is plotted in Fig. 3.4 of Reference 1, and obeys the empirical relation: 



F,{ya) = 7.154 exp (-0.66G4 ya) 



(86) 



Finalh% the factor F2 is the impedance reduction factor (84), which is 

 plotted in Fig. 3 of this paper for various ratios of the radii, b/a. 



It is of interest to compare the relative gain of beams with rectilinear 

 and with Brillouin flow, respectively. Pierce " has computed a first 

 approximation to the impedance reduction factor for the sohd-cyhndrical 



