440 BELL SYSTEM TECHNICAL JOURNAL 



is a solution of (11.3) for k = —b. We see that a solution of (11.3) is the con- 

 jugate of a solution of (8.1) if we put b in (8.1) equal to k in (11.3). Thus, a 

 solution of (11.2) is the conjugate of a solution of (8.1) in which b in (8.1) 

 is made the negative of the value of b for which it is desired to solve (11.2). 



We can use the solutions of Fig. 8.1 in connection with the circuit of 

 Fig. 11.1 in the following way: wherever in Fig. 8.1 we see b, we write in 

 instead —b, and wherever we see yi , y-i or y^ we write in instead —y\ , 

 —yi or —yz . 



Thus, for synchronous velocity, we have 



5i = V3/2 + jY^ 



h= -Vs/2-\-jy2 



^3 = -j 



We can determine what will happen in a physical case only by fitting 

 boundary conditions so that at z = the electron stream, as it must, enters 

 unmodulated. 



Let us, for convenience, write $ for the quantity (3Cz 



^Cz = $ (11.4) 



We will have for the total voltage Vz at z in terms of the voltage F at 2 = 



V, = Ve~'^'([{l - 52/50(1 - 53/50]~V 



+ [(1 - 53/50(1 - 5x/50r'^"'*''^*'' (11.5) 



+ [(1 - 5i/53)(l - 52/53)]-i^-^*^'/^') 



We must remember that in using values from an unaltered Fig. 8.1 we use 

 in the 5's and as the y's the negative of the y's shown in the figure (the sign 

 of the x's is unchanged), and for a given value of b we enter Fig. 8.1 at —b. 



In Fig. 11.3, I Vz/V I has been plotted vs 6 for $ = 2. We see that, for 

 several values of 6, | Fz | (the input voltage) is less than | V \ (the output 

 voltage) and hence there can be "backward" gain. 



We note that as $ is made very large, the wave which increases with 

 increasing $ will eventually predominate, and | Vz | will be greater than 

 I F |. "Backward gain" occurs not through a "growing wave" but rather 

 through a sort of interference between wave components, as exhibited in 

 Fig. 11.3. 



Fig. 11.3 is for a lossless circuit; the presence of circuit attenuation would 

 alter the situation somewhat. 



