APPENDIX VI 



EVALUATION OF IMPEDANCE AND Q FOR 

 THIN AND SOLID BEAMS' 



Let us first consider a thin beam whose breadth is small enough so that 

 the field acting on the electrons is essentially constant. The normal mode 

 solutions obtained in Chapters VI and VII apply only to this case. The more 

 practical situation of a thick beam will be considered later. The normal mode 

 method consists of simultaneously solving two equations, one relating the 

 r-f field produced on the circuit by an impressed r-f current from the electron 

 stream and the other relating r-f current produced in the electron stream by 

 an impressed r-f field from the circuit. 



We have the circuit equation 



and the electronic equation 



oK IjQKVn . 



i/3e h £ (2) 



The solution of these two equations gives T in terms of To , A', and Q, which 

 must be evaluated separately for the particular circuit being considered. 



The field solution is obtained by solving the field equations in various 

 regions and appropriately matching at the boundaries. For a hollow beam of 

 electrons of radius b traveling in the z direction inside a helix of radius a and 



pitch angle xp, the matching consists of finding the admittances ( W^ I inside 



and outside the beam and setting the difference equal to the admittance of 

 the beam. Thus the admittance just outside the beam for an idealized helix 

 will be- 



V =^ =. -"^ /i(7^>) - SKijyb) . . 



' E,o ^ y h{yb) + SKoiyb) ' 



' This appendix is taken from R. C. Fletcher, "Helix Parameters in Traveling- Wave 

 Tube Theory," Proc. I.R.E., Vol. 38, pp. 4l3-il7 (1950). 



^ L. J. Chu and J. D. Jackson, "Field Theory of TraveUng-Wave Tubes," I.R.E., 

 Proc, Vol. 36, pp. 853-863, July, 1948. 



O. E. H. Rydbeck, "Theory of the Traveling-Wave Tube," Ericsson Technics, No. 46 

 pp. 3-18, 1948. 



447 



