TRAVELING-WAVE TUBES 29 



streams of diflferent currents, /i and lo , were coupled to the circuit through 

 transformers, so as to be acted on by fields Ei and Eo, but that the streams 

 did not interact directly with one another, we would find the effective value 

 of C^ to be given by 



C' = (£l//3-P)(/i/8Fo) + (El/ 13' P) (1 2/SV0) 



Thus, if we neglect the direct interaction of electron streams through fields 

 due to local space charge, we can obtain an effective value of C^ by integrat- 

 ing £Wo over the beam. If we assume a constant current density, we can 

 merely use the mean square value of E over the area occupied by electron 

 flow. 



The axial component of electric field at a distance r from the axis is Io{yr) 

 times the field on the axis. Hence, if we used a tubular beam of radius r, we 

 should multiply {E^/^'^PY^^ as obtained from Fig. 3.4 by[ I^iyr)]^'^. The quan- 

 tity [/o(7'')]^'* is plotted vs. 7a for several values of r/a as the dashed lines 

 in Fig. 3.5. 



Suppose the current density is uniform out to a radius r and zero beyond 

 this radius. The average value of £- is greater than the value on the axis by 

 a factor \Il{yr) — l]{'Yr)\ and {E-/fi'Py^ from Fig. 3.4 should in this case 

 be multiplied by this factor to the \ power. The appropriate factor is plotted 

 vs. ya as the solid lines of Fig. 3.5. 



We note from (2.39) that the gain contains a term proportional to CN , 

 where N is the number of wavelengths. For slow waves and usual values of 

 ya, very nearly, N will be proportional to the frequency and hence to 7, 

 while C is proportional to (E-/l3~Py'^. We can obtain {E~/^"Py'^ from Figs. 

 3.4 and 3.5. The gain of the increasing wave as a function of frequency will 

 I thus be very nearly proportional to this value of {E^/^'^Py^ times 7, or, 

 ; times ya if we prefer. 



In Fig. 3.6, yaF{ya) is plotted vs. ya for hollow beams of radius r for 



various values of r/a (dashed lines) and for uniform density beams of 



radius r for various values of r/a (solid lines). If we assume that the electron 



speed is adjusted to equal the phase velocity of the wave, we can take the 



[ordinate as proportional to gain and the abscissa as proportional to 



'frequency. 



We see that the larger is r/a, the larger is the value of ya for maximum 

 jgain. For one typical 7.5 cm wavelength traveling-wave tube, ya was about 

 '2.8. For this tube, the ratio of the inside radius of the helix to the mean radius 

 of the helix was 0.87. We see from Fig. 3.6 that, if a solid beam just filled 

 I this helix, the maximum gain should occur at about the operating wave- 

 jlength. As a matter of fact, the beam was somewhat smaller than the inside 

 diameter of the helix, and there was an observed increase of gain with an 

 increase in wavelength (a higher gain at a lower frequency). In a particular 



