EQUATIONS FOR TRAVELING-WAVE TUBE 395 



The quantity wCj has the dimensions of admittance per unit length, 

 ^e has the dimensions of (length)"^ and (E-/'iS'P) has the dimensions of 

 impedance. Thus, () is a dimensionless parameter (the space-charge param- 

 eter) which may be thought of as relating to the impedance parameter 

 {E^/^-P) associated with the synchronous mode the impedance (/3,./coCi), 

 attributable to all modes but the synchronous mode. 



At this point it is important to remember that there are not only two im- 

 pedances, but two voltage components as well. Thus, in (7.8), the first 

 term in the brackets times the current represents the "circuit voltage", 

 which we may call \\ . The second term in the brackets represents the 

 voltage due to space charge, the voltage across the capacitances Ci . The 

 two terms in the brackets are in the same ratio as the two terms on the right 

 of (7.14), which came from them. Thus, we can express the circuit com- 

 ponent of voltage Vc in terms of the total voltage V acting on the beam either 

 from (7.8) as 



-['- 



MTl - r^) V y (7.16) 



a,Ciri(£2/^2p)J 



or, alternatively, from (7.14) as 



Fc = [1 - 4QCi-b + jd + j8)]-' V (7.17) 



From Chapter II we have relations for the electron velocity (2.15) and 

 electron convection current (2.22). If we make the same approximations 

 which were made in obtaining (7.14), we have 



{juoC/v)v = J (7.18) 







(-2VoC'/I)i = '- (7.19) 



We should remember also that the variation of all quantities with z 

 is as 



^-;^.y,c«. (720) 



The relations (7.18)-(7.19) together with (2.36), which tells us that the 

 characteristic impedance of the circuit changes little in the presence of 

 electrons if C is small, sum up in terms of the more important parameters 

 the linear operation of traveling-wave tubes in which C is small. The param- 

 eters are: the gain parameter C, relative electron velocity parameter b, 

 circuit attenuation parameter d and space-charge parameter Q. In follow- 



