TRAVELING-WAVE TUBES 45 



3.4a Effect of Dielectric on Helix Impedance Parameter 



One possible application of the transmission line equivalent is in estimating 

 the lowering of the helix impedance parameter {E?/ff^Py^^. 



In the case of a transmission line of susceptance B and reactance X per 

 unit length, we have for the phase constant /3 and the characteristic imped- 

 ance K 



~^ (3.61) 



(3.62) 



Now, suppose that B is increased by capacitive loading so that /3 has a 

 larger value /3d. Then we see that A' will have a value Ka 



Kd = (l3/l3d}K (3.63) 



Where should K be measured? It is reasonable to take the field at the 

 surface of the helix or the helically conducting sheet as the point at which 

 the field should be evaluated. The field at the axis will, then, be changed 

 by a different amount, for the field at the surface of the helix is h{ya) times 

 the field at the axis. 



Suppose, then, we design a helix to have a phase constant /3 (a phase 

 velocity co/)3) and, in building it, find that the dielectric supports increase 

 the phase constant to a value ^d giving a smaller phase velocity aj/(8d. Sup- 

 pose |S//So is large, so that 7 is nearly equal to /8. How will we estimate the 

 actual axial value of {E-f^-PY'^} We make the following estimate: 



(£'/.'P)r = (£)""(gg)^"(i^V.= «- (3.64) 



Here the factor (13/ ^dY'^ is concerned with the reduction of impedance 

 measured at the helix surface, and the other factor is concerned with the 

 greater falling-off of the field toward the center of the helix because of the 

 larger value of 7 (taken equal to 13 and /3d in the two cases). 

 The writer does not know how good this estimate may be. 



3.4b Coupled Helices 



Another case in which the equivalent transmission line approach is par- 

 ticularly useful is in considering the problem of concentric helices. Such 

 configurations have been particularly suggested for producing slow trans- 

 verse fields. They can be analyzed in terms of helically conducting cylinders 

 or in terms of developed cylinders. A certain insight can be gained very 

 quickly, however, by the approach indicated above. 



We will simulate the helices by two transmission lines of series impedances 

 jXi and JX2, of shunt admittances jB^ and JB2 coupled by series mutual 



