44 



BELL SYSTEM TECHNICAL JOURNAL 



We then let ^i = and find the value of Xi for which dV/dzi = 0. When Zi 

 0, (3.58) becomes 



A = s!nh 2x1 tanh 2.vi 



(1 — tanh" .Vi) 

 (1 + tanh2 xi) 



(3.59) 



As A is given by (3.53), we can obtain x, from (3.57), and the ratio of the 

 x-diameter d^ to the pitch is 



d-Jp = :ti/(7r/8) 

 Figure 3.16 shows di/d and d^/d vs. d/p. 



(3.60) 



^bi 



o 





0.1 0.2 0.3 0.4 0.5 0.6 7 0.8 0.9 



d/p 



Fig. 3.16 — Ratios of the wire diameters for the four turns per wavelength analysis. 



The ratios R and the impedance are obtained merely by comparing the 

 power flow for the developed sheet with a single sinusoidally distributed 

 component with the power flow for case II for the same distant field. In a 

 comparison with the helically conducting sheet, n = 2 is used in (3.50). The 

 results are shown in Figs. 3.13, 3.14, 3.15. We see that on the basis of the 

 largest available field, the best wire size is d/p = .19. 



3.4 Transmission Line Equations and Helices 



It is of course possible at any frequency to construct a transmission line 

 with a distributed shunt susceptance B per unit length and a distributed 

 shunt reactance X per unit length and, by adjusting B and X to make the 

 phase velocity and E-f^-P the same for the artificial line as for the heli.x. 

 In simulating the helix with the line, B and X must be changed as frequency 

 is changed. Indeed, it may be necessary to change B and .Y somewhat in 

 simulating a lieli.v with a forced wave on it, as, the wave forced by an elec- 

 tron stream. Nevertheless, a qualitative insight into some problems can bo 

 obtained by use of this type of circuit analogue. 



