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THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



for a flat helix. This is due to the fact that the large dielectric constant 

 reduces the circuit wavelength so much that the radius appears infinite 

 by comparison once it exceeds 1.5. In traveling-wave tube practice, 

 howe\'er, fo is generally below 1.5. 



The behavior of the ^ — fo curves can be understood from the be- 

 havior of the ratio 



and of the coefficient l/[)u//io (2x —I)]- Suppose first that x is positive. 

 When X exceeds zero only slightly, Zy^('u) behaves essentially like 

 Ko(u/2)/Ki(u/2). This function is always positive; it begins at when 

 u = vnih a vertical tangent and steadily increases to unity as u -^ x . 

 Z^(u) varies in the same way, see Figs. 14 and 15, in the range < x < 

 3-^. For ^ > X ^ 3^, ^x.o , and therefore Z^ , has a zero which increases 

 from w = 0atx = j^tow= latx =^'^- Accordingly Z^iu) in ^ > 



Fig. 14 — Schematic behavior of the function Zy 

 various ranges of x- 



W) 



Il'x.o(^0/W'x.i(w) in 



