FILTER-TYPE CIRCUITS 213 



We will evaluate the coefficients by the usual means of Fourier analysis. 

 Suppose we let z = at the center of one of the gaps. We see that 



EE* dz = Z / A„,Alll{y„,r) dz 



til m=-oo J—Lhl 



(4.64) 

 = XI AmAtjlhmr)!^ 



All of the terms of the form E,„Ep , p ^ m integrate to zero because the 

 integral contains a term exp(-j2Tr{p — m)/L)z. 



Let us consider the field at the radius r. This is zero along the surface of 

 the tube. We will assume with fair accuracy that it is constant and has a 

 value —V/i' across the gap. Thus we have also at r = a, 



f EE*dz= - {V/O E f Ale-'^-' h{y,„a) dz 

 J— lIi »«=— 00 J— (hi 



{v/o z cf:)/o(7.a) ( 



m=— « \ 



g i?.^t|■2 _ ^i^,n(l'^ 



(4.65) 



I 



We can rewrite this 



"' EE- dz = - (V/() ± A:h{y„.a) "^^^^^ (4.66) 



L/2 m=-QO \Pmtl I 



By comparison with (4.64) we see that 



^,„ = - (F/L)( sin (/^„//2)/CS„//2))(l//o(7a)) (4.67) 



This is the magnitude of the wth field component on the axis. The magnitude 

 of the field at a radius r would be loic^r) times this. 



The quantity ^,J- is an angle which we will call dg , the gap angle. Usually 

 we are concerned with only a single field component, and hence can merely 

 write 7 instead of jm . Thus, we say that the magnitude E of the travelling 

 field produced by a voltage V acting at intervals L is 



E = -M{V/L) (4.68) 



^^sin(^/oW 

 {dg/2) lo(ya) 



dg = I3(. (4.70) 



The factor M is called the gap factor or the modulation coefficient*. 

 For slow waves, 7 is very nearly equal to ^, and we can replace yr and ya 

 by /3r and jSa. For unattenuated waves, ikf is a real positive number; and, 



* This factor is often designated by /3, but we have used /3 otherwise. 



