FILTER-TYPE CIRCUITS 



217 



We see that, for this component, v is positive. 



The interaction of electrons with backward-travehng field components 

 will be considered later. Here it will merely be said that, in order to avoid 

 interaction with waves traveling in both directions, one must avoid having 

 the electron speed lie near both the speed of a forward component and the 

 speed of a backward component. 



In order that the fundamental component be slow, 6 must be large or L 

 must be small. The largest value of d is that near one edge of the band, where 

 d approaches tt. Thus, the largest fundamental value of /3 is tt/L, and to make 



377r 77 



FILTER 

 CHARACTERISTIC 



CONST X<*^- 



— '^'"" 



/'^^ CONST Xo; 



CO\ <^2 



Fig. 4.33 — The variation of phase with frequency for the fundamental (0 to ir over the 

 band) and a spatial harmonic {Itt to 37r over the band). The dotted lines show co divided 

 by the electron velocity for the two cases. For amplification over a broad band the dotted 

 curve should not depart much from the filter characteristic. 



j8 large with w = we must make L small and put the resonators very close 

 together. This may be physically difiicult or even impossible in tubes for 

 very high frequencies. The alternative is to use a harmonic component, 

 for which /3 = (2w7r + 0) L. 



Another reason for using harmonic operation is to achieve broad-band 

 operation. The phase of a filter-type circuit changes by tt radians between 

 the lower cutoff frequency coi and the upper cutoff frequency a;2t. Now, 

 for the wave velocity to be near to the electron velocity over a good part 

 of the band, /3 must be nearly a constant times w. Figure 4.33 shows how 

 this can be approximately true for the m = 1 component even when it ob- 

 viously won't be for the m = or fundamental component. Similarly, for 

 a filter with a narrower fractional bandwidth and hence a steeper curve of 

 6 vs CO, a. larger value of m might give a nearly constant value of v. 



t The phase of some filters changes more than this, but they don't seem good candidates 

 for traveling-wave tube circuits. 



