TRAVELING-WAVE TUBES 



41 



The maximum available field for the developed helically conducting sheet 

 (equation (3.38)) is that for x = 0. The maximum available field for the 

 developed helix (equation (3.31)) is that for an electron grazing the helix 

 inner or outer diameter, that is, an electron at a value of x given by (3,35). 

 The fundamental sinusoidal component of the field varies as exp(— 2x) 

 for both the sheet and the helix, and hence there is a loss in £'- by a factor 

 e.xp(— 4.v) because of this. We wish to make a comparison on the basis of 

 E^ and power or energy. Hence, on basis of maximum available field squared 

 we would obtain from (3.40) 



^ -'^ (3.41) 



R = - 



In tan z 



where x is obtained from (3.35). Figure 3.14 was obtained from (3.32), 

 (5.35) and (3.41). 



3.0 



2.0 



0.8 



0.6 

 0.5 



0.4 

 0.3 



01 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9 



d/p 

 Fig. 3.15 — The transverse impedance of helices with two and four turns per wavelength 

 vs. the ratio of wire diameter to pitch. 



In a transmission line the characteristic impedance is given by 



K = y^l (3.42) 



Here L and C are the inductance and capacitance per unit length. This im- 

 pedance should be identified with the transverse impedance of the helix. 

 We also have for the velocity of propagation, which will be the velocity of 

 fight, c, 



c = —^^^ = —^ (3.43) 



Vlc V\ 



He 



