GENERAL CIRCUIT CONSIDERATIONS 223 



\ velocity is of course the same for all components, as they are all aspects of 

 i one wave. Relation (4.61) is consistent with this: 



/3. = (^ + 2w7r)/L (4.61) 



1 1/t'ff = d^Jdo: = {dd/doi)/L (5.13) 



5.2 Gain and Bandwidth in a Traveling- W.ave Tube 



We can rewrite the impedance parameter E^/0^P in terms of stored 

 energy per unit length TT' for a field strength £, and a group velocity Vg . 

 If ir is the stored energy per unit length, the power flow P is 



I P = WVg (5.14) 



and, accordingly, we have 



j £2//32P = pr-/^Wvg (5.15) 



And, for the gain parameter, we will have 



C = (P?/l3'Wvg)'''ih/8Vo}"' (5.16) 



For example, we see from Fig. 4.20 that E-/^'~P for the circuit of Fig. 4.10 

 t^^oes to infinity at the upper cut-off. From Fig. 4.17 we see that dd/do}, 

 and hence 1 '^'g , go to infinity at the upper cutoff, accounting for the infinite 

 impedance. We see also that dd do: goes to infinity at the lower cutoff, but 

 there the slot voltage and hence the longitudinal field also go to zero and 

 hence E-fff-P does not go to infinity but to zero instead. 



In the case of the circuit of Fig. 4.11, the gap voltage and hence the longi- 

 tudinal field are finite for unit stored energy at both cutoffs. As dd do: is 

 infinite at both cutoffs, V- P and hence E~ff-P go to infinity at both cut- 

 ; offs, as shown in Fig. 4.28. 



i To get high gain in a traveling-wave tube at a given frequency and volt- 

 age (the phase velocity is specified by voltage) we see from (5.16) that we 

 must have either a small stored energy per unit length for unit longitudinal 

 field, or a small group velocity, v g . 



To have ampUfication over a broad band of frequencies we must have the 



phase velocity v substantially equal to the electron velocity over a broad 



band of frequencies. This means that for very broad-band operation, v 



j must be substantially constant and hence in a broad-band tube the group 



velocity will be substantially the same as the phase velocity. 



If the group velocity is made smaller, so that the gain is Increased, the 



I range of frequencies over which the phase velocity is near to the electron 



velocity is necessarily decreased. Thus, for a given phase velocity, as the 



group velocity is made less the gain increases but the bandwidth decreases. 



Particular circuits can be compared on the basis of (E-/l3''P) and band- 



