398 BELL SYSTEM TECH MCA L JOLRSAL 



velocity of the third wave approaches the velocity of the circuit wave in the 

 absence of electrons (y approaches minus b). For large negative values of 

 b, Xi , Vi and .vo , y-i become the "electron" waves and Vs becomes the "cir- 

 cuit" wave. For large values of b, Vi and y^ become the "electron" waves and 

 yo becomes the "circuit" wave. The "circuit" wave is essentially the wave 

 in the absence of electrons, modified slightly by the presence of a non-syn- 

 chronous electron stream. The "electron waves" represent the motion of 

 "bunches" along the electron stream, slightly affected by the presence of 

 the circuit. 



Figures 8.2 and 8.3 indicate the effect of loss. Loss decreases the gain of 

 the increasing wave, adds to the attenuation of the decreasing wave and 

 adds attenuation to the wave which was unattenuated in the lossless case. 

 For large positive and negative values of b, the attenuation of the circuit 

 wave (given by .V3 for negative values of b and .V2 for positive values of b) 

 approaches the attenuation in the absence of electrons. 



Figure 8.4 shows B, the gain of the increasing wave in db per wavelength 

 per unit C. Figure 8.5 shows, for b = 0, how B varies with d. The dashed 

 line shows a common approximation: that the gain of the increasing wave 

 is reduced by ^ of the circuit loss. Figure 8.6 shows how, for b = 0, Xi , 

 X2 and .V3 vary with d. We see that, for large values of d, the wave described 

 by .V2 has almost the same attenuation as the wave on the circuit in the 

 absence of electrons. 



Figures 8.7-8.9 show .v, y for the three waves with no loss ((/ = 0) but 

 with a-c space charge taken into account {QC 7^ 0). The immediately 

 striking feature is that there is now a minimum value of b below which 

 there is no increasing wave. 



We further note that, for large negative and positive values of 6, y for 

 the electron waves approaches ±2 \/QC. In these ranges of b the electron 

 waves are dependent on the electron inertia and the field produced by a-c 

 space charge, and have nothing to do with the active mode of the circuit. 



As QC is made larger, the value of b for which the gain of the increasing 

 wave is a maximum increases. Now, C is proportional to the cube root of 

 current. Thus, as current is increased, the voltage for maximum gain of the 

 increasing wave increases. An increase in optimum operating voltage with 

 an increase in current is observed in some tubes, and this is at least i)artly 

 explained by these curves.* There is also some decrease in the maximum 

 value of X\ and hence of B as QC is increased. This is shown more clearly in 

 Fig. 8.10. 



If X and B remained constant when the current is varied, then tlie gain 

 per wavelength would rise as C, or, as the \ power of current. However, 



* Other factors include a possible lowering of electron speed because of d-c space 

 charge, and boundary condition eflects. 



