458 BELL SYSTEM TECHNICAL JOURNAL 



the helix 



X. = X 2? (4) 



C 



N = i = il (5) 



K 7 «o 

 Also, if ^w is the total length of wire in the helix, approximately 



N = ^^ (6) 



A7.3 The Gain Parameter C 



The gain parameter can be expressed 



1/3 /^Xl/3 



-(f;s¥^^^"V 



Here A' is the helix impedance properly defined, h is the beam current in 

 amperes and Vq is the beam voltage. 



A 7. 4 Helix Impedance K 



In Fig. 5 of Appendix \T, A'(— j(l+f — jj is plotted vs. 70a for 



values of b/a. Kg is the effective value of K for a solid beam of radius b, and 

 a is the radius of the helix. 70 is to be identified with 7 for present purposes, 

 and is given by 



1/2 



27r 



1 - . 



X 



2n 

 7 



(8) 



where X^ is given in terms of X by (4). We see that in most cases (for voltages 

 up to several thousand) 



(X«/A)'- « 1 (9) 



and we may usually use as a valid approximation 



27r 



7o = 

 and 



(10) 



^0 

 As /So = 27r/X, this approximation gives 



7.a = ^-^ (11) 



(S)' = 



1 + - = 1 + 



