TRAVELING-WAVE TUBES 



27 



This is essentially the same as (3.5) for small pitch angles 4^. Thus, for large 

 values of the abscissa in Fig. 3.2, the phase velocity is just about that corre- 

 sponding to propagation along the sheet in the direction of conduction with 

 the speed of light and hence in the axial direction at a much reduced speed. 

 For helices of smaller radius compared with the wavelength, the speed is 

 greater. 



The bandwidth of a traveling-wave tube is in part determined by the 

 range over which the electrons keep in step with the wave. The abscissa of 

 Fig. 3.2 is proportional to frequency, but the ordinate is not strictly propor- 

 tional to phase velocity. Hence, it seems desirable to have a plot which does 

 show velocity directly. To obtain this we can assign various values to cot rp. 



1.0 

 0.8 



~b 0.4 



U- 0.3 



0.1 

 0.08 

 0.06 



0.04 

 0.03 



01 23456789 



7a 



Fig. 3.4 — A curve giving the impedance function F{ya) vs. ya. On the axis, {E^/0rPy^ = 

 (/3/^u)"KT//3)^"f(Ta). 



The ordinate (1S0/7) cot \p then gives us y/^Q and from (3.2) we see that 



v/c = /3o/^ = (1 + (y/^oY-)-'" (3.6) 



We have seen that, for large values of /3oo cot \}/, (/Sq/t) cot ^ approaches 

 unity, and v/c approaches a value 



v/c = (1 + cot2 ^py'" = sin ^ 



(3.7) 



To emphasize the change in velocity with frequency it seems best to plot the 

 difference between the actual velocity ratio and this asymptotic velocity 

 ratio on a semi-log scale. Accordingly, Fig. 3.3 shows (v/c) — sin ip vs. /3oO 

 cot 7 for tan \p = .05, .075, .1, .15, .2. 

 For large values of the abscissa the velocities are those corresponding to 



