FILTER-TYPE CIRCUITS 191 



if suitable lumped-admittance networks are used to represent the admit- 

 tances Bi and B2, the frequency-dependent behavior of the structures of 

 Figs. 4.10 and 4.11 can be approximated. 



It is, for instance, convenient to represent the shunt admittances B2 and 

 the series admittances Bi in terms of a "longitudinal" admittance Bl and 

 a "transverse" admittance Bt . Bl and Bt are admittances of shunt resonant 

 circuits, as shown in Fig. 4.15, where their relation to Bi and B2 and ap- 

 proximate expressions for their frequency dependence are given. The res- 

 onant frequencies of Bl and Br , that is, wl and cot , have simple physical 

 meanings. Thus, in Fig. 4.10, ojz, is the frequency corresponding to equal 

 and opposite voltages across successive slots, that is, the x mode frequency. 

 wr is the frequency corresponding to zero slot voltage and no phase change 

 along the filter, that is, the zero mode frequency. 



If w I, is greater than cot , the phase characteristic of this lumped-circuit 

 analogue is as shown in Fig. 4.17. The phase shift is zero at the lower cutoff 

 frequency cor and rises to t at the upper cutoff frequency col . If oot is greater 

 than col , the phase shift starts at — tt at the lower cutoff frequency wi, and 

 rises to zero at the upper cutoff frequency cor, as shown in Fig. 4.19. In this 

 case the phase velocity is negative. Figure 4.20 shows a measure of (Er/jS^P) 

 plotted vs. CO for col > ojt ■ This impedance parameter is zero at cor and rises 

 to infinity a,t wl . 



The structure of Fig. 4.11 can be given a lumped-circuit equivalent in a 

 similar manner. In this case the representation should be quite accurate. 

 We find that coz, is always greater than ojt- and that one universal phase curve, 

 shown in Fig. 4.27, applies. A curve giving a measure of (E^/^^P) vs. fre- 

 quency is shown in Fig. 4.28. In this case the impedance parameter goes to 

 infinity at both cutoff frequencies. 



The electric field associated with iterated structures does not vary sinus- 

 oidally with distance but it can be analyzed into sinusoidal components. 

 The electron stream will interact strongly with the circuit only if the elec- 

 tron velocity is nearly equal to the phase velocity of one of these field com- 

 ponents. If 6 is the phase shift per section and L is the section length, the 

 phase constant ^m of a typical component is 



/3„ = (^ -f- 2m7r)/i: 



where m is a positive or negative integer. The field component for which 

 m = is called the fundamental; for other values of m the components are 

 called spatial harmonics. Some of these components have negative phase 

 velocities and some have positive phase velocities. 

 The peak field strength of any field component may be expressed 



E = -M(V/L) 



■ Here V is the peak gap voltage, L is the section spacing and M is a function 

 ! of /8 (or /3m) and of various dimensions. For the electrode systems of Figs. 



