EQUATIONS FOR TRAVELING-WAVE TUBE 



391 



ballistical equation (2.22), which gives the charge density in terms of the 

 voltage, to give (7.9), which is an equation for the propagation constant. 

 The attenuation, the difference between the electron velocity and the phase 

 velocity of the wave on the circuit in the absence of electrons and the dif- 

 ference between the propagation constant and that for a wave traveling 

 with the electron speed are specified by means of the gain parameter C 

 and the parameters d, b and b. It is then assumed that J, b and b are around 

 unity or smaller and that C is much smaller than unity. This makes it pos- 

 sible to neglect certain terms without serious error, and one obtains an 

 equation (7.13) for b. 



In connection with (7.7) and Fig. 7.1, it is important to distinguish be- 

 tween the circuit voltage Vc , corresponding to the first term of (7.7), and 

 the total voltage V acting on the electrons. These quantities are related 

 by (7.14). The a-c velocity v and the convection current i are given within 

 the approximation made (C « 1) by (7.15) and (7.16). 





C, PER 

 METER 



Fig. 7.1 



7.1 Approxim.^te Circuit Equation' 



From (6.47) we can write for a current / = / and a summation over n 

 modes 



£. = (l/2)(r -f I5l)i E 



(£V/3'i')„rl 



" (r; + ^oKK - n 



(6.47a) 



This has a number of poles at F = F,, . We shall be interested in cases 

 in which F is very near to a particular one of these, which we shall call 

 Fi . Thus the term in the expansion involving Fi will change rapidly with 

 small variations in F. Moreover, even if {Er/^-P)i and Fi have very small 

 real components, FI — F- can be almost or completely real for values of F 

 which have only small real components. Thus, one term of the expansion, 

 that involving Fi , can go through a wide range of phase angles and magni- 

 tudes for very small fractional variations in F, fractional variations, as it 

 turns out, which are of the order of C over the range of interest. 



The other modes are either passive modes, for which even in a lossy 

 circuit {E}/ff^P)n is almost purely imaginary, and F„ almost purely real, 



