450 



BELL SYSTEM TECHXICAL JOURNAL 



TJ 



Two beams will be equivalent if the value of -=^ is the same outside the 

 beams, since the matching to the circuit depends only on this admittance. 



1000 

 800 

 600 



400 



100 

 80 

 60 



+ 10 



fN|<Q^ 6 



o 



1.0 

 0.8 

 0.6 



7o^ 



Fig. A6.1 — Passive mode parameter Q for a hollow beam of electrons of radius h inside 

 a helix of radius a and natural propagation constant yo . The solid line was obtained by 

 equating the circuit efjuation of the normal mode method, which defines Q, with a cor- 

 responding circuit equation found from the field theory method. The dashed line was 

 obtained in Appendix IV from a solution of the field equations for a conductor replacing 

 the helix. 



The problem, then, of making a thin beam the equivalent of a thick beam 

 is the problem of arranging the position and current of a thin beam to give 

 the same admittance at the radius h of the thick beam. This is of course 

 impossil)lc for all values of 7. Tt is desirable therefore that the admittances 



