TRAVELING-WAVE TUBES 21 



00 cot ^ and /3 thus approaches /3o/sin ^; this means that the wave travels 

 with the velocity of Hght around the sheet in the direction of conduction. 

 In the case of an actual helix, the wave travels along the wire with the 

 velocity of light. 



The gain parameter C is given by 



C = (ro/8V,y'\E''/l3'Py" 



Values of (E^/0'^PY'^ on the axis may be obtained through the use of Fig. 3.4, 

 where an impedance parameter F(ya) is plotted vs. ya, and by use of (3.9). 

 For a given helix, {E?/0^PY ^ is approximately proportional to F(ya). F{ya) 

 falls as frequency increases. This is partly because at high frequencies and 

 short wavelengths, for which the sign of the field alternates rapidly with 

 distance, the field is strong near the helix but falls ofif rapidly away from the 

 helix and so the field is weak near the axis. At very high frequencies the field 

 falls off away from the helix approximately as exp(— 7Af), where Ar is dis- 

 tance from the helix, and we remember that y is very nearly proportional to 

 frequency. (E^/0^P) measured at the helix also falls with increasing 

 frequency. 



In many cases, a hollow beam of radius r (the dashed lines of Fig. 3.5 

 refer to such a beam) or a solid beam of radius r (the solid lines of Fig. 3.5 

 refer to such a beam) is used. For a hollow beam we should evaluate £- in 

 {E-/0"Py ^ at the beam radius, and for a solid beam we should use the mean 

 square value of E averaged over the beam. 



The ordinate in Fig. 3.5 is a factor by which (E^/ff^Py^ as obtained from 

 Fig. 3.4 and (3.9) should be multiplied to give (E-/0^Py'^ for a hollow or 

 solid beam. 



The gain of the increasing wave is proportional to F{ya) times a factor 

 from Fig. 3.5, and times the length of the tube in wavelengths, N. N is very 

 nearly proportional to frequency. Also y, and hence ya, are nearly propor- 

 tional to frequency. Thus, F(ya) from Fig. 3.4 times the appropriate factor 

 from Fig. 3.5 times ya gives approximately the gain vs. frequency, (if we 

 assume that the electron speed matches the phase velocity over the fre- 

 quency range). This product is plotted in Fig. 3.6. We see that for a given 

 helix size the maximum gain occurs at a higher frequency and the band- 

 width is broader as r/a, the ratio of the beam radius to the helix radius, 

 is made larger. 



It is usually desirable, especially at very short wavelengths, to make the 

 helix as large as possible. If we wish to design the tube so that gain is a maxi- 

 mum at the operating frequency, we will choose a so that the appropriate 

 curve of Fig. 3.6 has its maximum at the value of ya corresponding to the 

 operating frequency. We see that this value of a will be larger the larger is 

 r/a. In an actual helix, the maximum possible value of r/a is less than unity, 



