DISCONTINUITIES 419 



and 8.3 we see that the initial loss A becomes higher as d is increased. This 

 is natural, because the electron stream can act to cause gain only after it is 

 bunched, and if the initial section of the circuit is lossy, the signal decays 

 before the stream becomes strongly bunched. 



Let us consider a section of a lossless helix which is far enough from the 

 input so that the increasing wave predominates and the total voltage V can 

 be taken as that corresponding to the increasing wave 



F = Fi (9.16) 



Then, at this point 



{ju,CH)v = Fi/Si (9.17) 



(-2FoCV/o)i- V,/b\. (9.18) 



Here 5i is the value for J = (and, we assume, 6 = 0). If we substitute the 

 values from (9.16) in (9.4), and use in (9.4) the values of 5 corresponding to 

 b = Q = 0, d 9^ 0, and call the value of Vi we obtain Fi , we obtain the 

 ratio of the initial amplitude of the increasing wave in the lossy section to 

 the value of the increasing wave just to the left of the lossy section. Thus, 

 the loss in the amplitude of the increasing wave in going from a lossless to a 

 lossy section is 20 logio | V\/Vi | . This loss is plotted vs d in Fig. 8.5. 



This loss is accounted for by the fact that | ii/Vi \ becomes larger as the 

 loss parameter d is increased. Thus, the convection current injected into 

 the lossy section is insufficient to go with the voltage, and the volt- 

 age must fall. 



If we go from a lossy section {d ^ 0, b — 0) to a lossless section 

 {d = 0, b = 0) we start with an excess of convection current and | Fi | , 

 the initial amplitude of the increasing wave to the right of the discontinuity 

 is greater than the amplitude | Fi | of the increasing wave to the left. In 

 Fig. 9.5, 20 logio I Fi/Fi | is plotted vs d for this case also. 



We see that if we go from a lossless section to a lossy section, and if the 

 lossy section is long enough so that the increasing wave predominates at 

 the end of it, and if we go back to a lossless section at the end of it, the net 

 loss and gain at the discontinuities almost compensate, and even for d — i 

 the net discontinuity loss is less than 1 db. This does not consider the re- 

 duction of gain of the increasing wave in the lossy section. 



9.6 Severed Helix 



If the loss introduced is distributed over the length of the helix, the gain 

 will decrease as the loss is increased (Fig. 8.5). If, however, the loss is dis- 

 tributed over a very short section, we easily see that as the loss is increased 

 more and more, the gain must approach a constant value. The circuit will 



