TRAVELING-WAVE TUBES 



39 



In Fig. 3.12, di/d is plotted vs. d/p by means of (3.35) and (3.36). This 

 shows that for wire diameters up to d/p — .5 (open space equal to wire diam- 

 eter) the equipotentials representing the wire are very nearly round. 



The total electric flux from each wire is lire and the potential of a wire of 

 2-diameter 2zisV= — In tan z. Hence, the stored energy Wi per unit length 

 per wire, half the product of the charge and the voltage, is 



Wi = -7re In tan ^ (3.37) 



0.1 02 0.3 0.4 0.5 06 0.7 0.8 0.9 



d/p 



Fig. 3.12 — Ratio of the two diameters of the wire of a hehx for two turns per wave- 

 length (see Fig. 3.9) vs. the ratio of one of the diameters to the pitch. 



The total distant field and the useful field component are given by ex- 

 panding (3.31) in Fourier series and taking the fundamental component, 

 giving 



V - -2cos22g^'^ (3.38) 



The — sign applies for x > and the -f sign for x < 0. Half of this can be 

 regarded as belonging to a field moving to the right and half to a field moving 

 to the left. 



For a field equal to half that specified by (3.38), which might be part of 

 the field of a developed helically conducting sheet, the stored energy Wo 

 per unit depth can be obtained by integrating {El + Ex) e/2 from .v = 

 — 00 to .T =: -f ^ and from z = — 7r/4 to +7r/4, and it turns out to be 



W2= hire (3.39) 



If we add another field component similar to half of (3.38), but in quadra- 

 ture with respect to z and /, we will have the traveling wave of a helically 

 conducting sheet with the same distant traveling field component as given 

 by (3.31). Hence, the ratio R of the stored energy for the developed sheet 

 to the stored energy for the developed helix is 



1 



R = 2W2/W1 = - 



In tan z 



(3.40) 



