414 BELL SYSTEM TECHNICAL JOURNAL 



serious. For CN large (the break far from the input) the loss approaches 

 3.52 db. 



Beyond such a break, the total voltage increases with CN as shown in 

 Fig. 9.7, and from CN = 0.2 the circuit voltage is very nearly equal to the 

 voltage of the increasing wave. 



Often, for practical reasons loss is introduced over a considerable distance, 

 sometimes by j)utting lossy material near to a helix. Suppose we use CN 

 computed as if for a lossless section of circuit as a measure of length of 

 the lossy section, and assume that the loss is great enough so that the circuit 

 voltage (as opposed to that produced by space charge) can be taken as zero. 

 Such a lossy section acts as a drift space. Suppose that an increasing wave 

 only reaches this lossy section. The amplitude of the increasing wave ex- 

 cited beyond the lossy section in db with respect to the amplitude of the in- 

 creasing wave reaching the lossy section is shown vs. CN, which measures 

 the length of the lossy section, in Fig. 9.8. 



9.1 General Boundary Conditions 



We have already assumed that C is small, and when this is so the charac- 

 teristic impedance of the various waves is near to the circuit characteristic 

 impedance A'. We will neglect any reflections caused by differences among 

 the characteristic impedances of the various waves. 



We will consider cases in which the circuit is terminated in the -{-z direc- 

 tion, so as to give no backward wave. We will then be concerned with the 

 3 forward waves, for which 8 has the values 8i , 80 , Sj and the waves repre- 

 sented by these values of 8 have voltages Vi , V2 , V^ , electron velocities 

 Vi , '02 , V3 and convection currents ii , i-i , i-i . 



Let V, V, i be the total voltage, velocity and convection current at 2 = 0. 

 Then we have 



V, + V,+ V^= V (9.1) 



and from (7.15) and (7.16), 



Oi 62 03 



Fi V2 Vs 



T^ + J + tI= (-2FoCV/o)e (9.3) 



oi do 03 



These equations yield, when solved, 



Vi = \V - (60 -f 8,)(juuC v)v + 8M-2V0C- fo)i] 



[(1 - 62/5,)(l - 8,8,)r' 

 We can ol)tain the corresponding expressions for V^ and K^ sim[)ly by inter- 



