BACKWARD WAVES 



439 



(7.9) and (7.12), by changing the sign of the left-hand side. From Chapter 

 VIII, the equation for a lossless circuit with no space charge is 



Hd^jb) = -j (8.1) 



The corresponding moditication is to change the sign preceding 5'\ giving 



S--{S+jb) = +j (11.2) 



-H ( I l ( I l ( I l ( I [ (-- 





Fig. 11.1 Fig. 11.2 



Fig. 11.1 — A circuit with a negative phase velocity. The electrons can be in synchron- 

 ism with the field only if they travel in a direction opposite to that of electromagnetic 

 energy flow. 



Fig. 11.2 — A circuit with a positive phase velocity. 



Fig. 11.3 — Suppose we have a tube with a circuit such as that of Fig. 11.1, in which 

 the circuit energy is really flowing in the opposite direction from the electron motion. 

 Here, for QC = rf = 0, we have the ratio of the magnitude of the voltage Vz a distance 

 z from the point of injection of electrons to the magnitude of the voltage V at the point 

 of injection of electrons. V, is reallv the input voltage, and there will be gain at values of 

 6 for which | V^IV\ < 1. 



In (11.2), h and 5 have the usual meaning in terms of electron velocity and 

 propagation constant. 



Now consider the equation 



^\h-jk) =j (11.3) 



Equations (11.2) and (8.1) apply to different systems. We have solutions 

 of (8.1) and we want solutions of (11.2). We see that a solution of (11.2) 



