410 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



F = {kh cot ^) i"^) l'(yWr(ya) (gl) 



\ c / Koiya) 



In (01), c is the velocity of light. 



The right side of (58), which is the admittance Hg/E^ looking away 

 from the l)eam surface, contains a term 5 which depends on the hehx 

 geometry and the ampUtude of the TE fields excited by the surface 

 current Ge . Thus, although the TE fields do not affect the electron paths, 

 they are excited by the beam, and coupled to the TM fields at the helix. 

 If Ge were zero, 8 would reduce to 5o , and the circuit admittance in (58) 

 would then be the same as for a cylindrical beam with rectilinear flow. 

 In (59), 5 is expressed in terms of 5o and the product, RF, where R is the 

 beam propagation function, and F a factor dependent on the magnetic 

 field and the geometry. 



This is the complete field solution of the problem. Equation (58) has 

 four roots: two complex and two real propagation wave numbers, one of 

 the latter representing a backward wave. In addition, there are two un- 

 attenuated space-charge waves, given by (34); or a total of 6 waves in 

 all. 



EQUIVALENT THIN-BEAM SOLUTION 



Pierce^ has expressed the admittance equation for an ideally thin 

 beam, interacting with an arbitrary distributed circuit, as follows: 



9 ;/3. A^_ .^ 





E ij^. - r)^2n V^ ^^Lr' - Tl'^ ^e 



where q = total convection current 



E = longitudinal electric field 



r = propagation constant = V'— 7- — A;^ 



/o = dc beam ciUTent 



Vo = dc beam potential 



To , K, Q = normal-mode circuit parameters. 



For slow waves, F c^ jj. For moderate values of perveance, the ac- 

 celerating voltage may be replaced by the beam potential at the axis: 



■Mo ^^ \/2r]Vo 



