402 BELL SYSTEM TECHNICAL JOURNAL 



boundary conditions; for this we need to know the three x's and the three 



In a tube in which the total gain is large, a change in 6 of ± 1 about b = 

 can make a change of several db in gain. Such a change means a difference 

 between phase velocity of the undisturbed wave, i\ , and electron velocity 

 Uo by a fraction approximately ±C. Hence, the allowable difference between 

 phase velocity i\ of the undisturbed wave, which is a function of frequency, 

 and electron velocity, which is not, is of the order of C. 



8.2 Effect of Attenuation 



If we say that J ?^ but has some small positive value, we mean that the 

 circuit is lossy, and in the absence of electrons the voltage decays with 

 distance as 



Hence, the loss L in db/wavelength is 



L = 20(27r)(logioe)Cr/ 



(8.15) 

 L = 54.5C(/ db/wavelength 



or 



d = .01836 {L/C) (8.16) 



For instance, for C = .025, d — \ means a loss of \.^6 db wavelength. 

 If we assume d 9^ we obtain the equations 



(^2 _ y)(^ + 6) + 2.rv(.v + J) + 1 = (8.17) 



(x2 - /)(.v -{- d) - 2xy{y + b) = (8.18) 



The equations have been solved numerically for d = .5 and </ = 1, and the 

 curves which were obtained are shown in Figs. 8.2 and 8.3. We see that for 

 a circuit with attenuation there is an increasing wave for all values of b 

 (electron velocity). The velocity parameters yi and y-y are now distinct for 

 all values of b. 



We see that the ma.ximum value of Xi decreases as loss is increased. This 

 can be brought out more clearly by showing .Vi vs b on an expanded scale. 

 It is perhaps more convenient to plot B, the db gain per wavelength per 

 unit C, vs 6, and this has been done for various values of d in Fig. 8.4. 



We see that for small values of d the maximum value of .Vi occurs very 

 near to b = 0. If we let b = in (8.17) and (8.18) we obtain 



y{x^ - /) + 2xy{x + (/) -h 1 = (8.19) 



