DISCONTINUITIES 415 



changing subscripts; to obtain Vn , for instance, we substitute subscript 

 2 for 1 and subscript 1 for 2 in (9.4). 



9.2 Lossless Helix, Synchronous Velocity, No Space Change 



Suppose we consider the case in which b = d = Q = 0, so that we have 

 the values of 5 obtained in Chapter II 



6, = e-'""' = V3/2 - il/2 



5, = e~''''" - - -v/3/2 - ./1/2 (9.5) 



§3 = e/T/2 ^ j 



Suppose we inject an unmodulated electron stream into the helix and 

 apply a voltage V. The obvious thing is to say that, at ^ = 0, r = / = 0. 

 It is not quite clear, however, that v = Sit z — (the beginning of the 

 circuit). Whether or not there is a stray field, which will give an initial 

 velocity modulation, depends on the type of circuit. Two things are true, 

 however. For the small values of C usually encountered such a velocity 

 modulation constitutes a small efTect. Also, the fields of the first part of 

 the helix act essentially to velocity modulate the electron stream, and hence 

 a neglect of any small initial velocity modulation will be about equivalent 

 to a small displacement of the origin. 



If, then, we let v = i = Q and use (9.4) we obtain 



V, = V[{1 - V5i)(l - 5,,/5:)]-' (9.6) 



Fi = V/3 (9.7) 



Similarly, we tind that 



F2 = Vs= V/3 (9.8) 



We have used T' to denote the voltage at 2 — 0. Let V^ be the voltage at z. 

 We have 



V.= {V/3)e'-'^'''-''' (1 + 2 cosh ({V3/2)0eCz)e-''"''^''') ^"^""^^ 



From this we obtain 



I V,/V [' = (1/9)[1 + 4 cosh2(V3/2)j8,Cz 



+ 4 cos {3/2)l3eCz cosh {\/3/2)l3eCz] 



(9.10) 



We can express gain in db as 10 logio | V^/V |-, and, in Fig. 9.1, gain in db 

 is plotted vs CN, where N is the number of cycles. 



We see that initially the voltage does not change with distance. This is 

 natural, because the electron stream initially has no convection current, 



