APPENDIX VI 449 



and 



7o and yp can be obtained from eqs. (3) and (4) through the implicit equations 



(/3flC0t^) - (Toa) r— — — (11) 



Ii{yoa)Ki{yoa) 



Koiypb) Kliy^a) 



ypa 

 and l/K is found to be 



/i(7pa)A'i(7pa.) - h{ypa)Ko{ypa) , 



(12) 



_1^ ^ i/i /^l + ^Y^^ ^° hiyoa) hiyoa) _ /o(7oa) 

 A' ^ y fx\ yl) /o(Toft) Ao(7oa) L-^o(7ofl) /i(7off) 



iro(7og) _ A:i(7oa) 4 1 



ifi(7oa) Ao(7oa) To a J 



(13) 



The equations for 70 and A' are the same as those given by Appendix II, 

 evaluated by solving the iield equations for the helix without electrons pres- 

 ent. The evaluation of yp , and thus Q, represents a new contribution. Values 



(o2\ -1/2 

 1 + -li I are plotted in Fig. A6.1 as a function of 70a for various 

 75/ 

 ratios of h/a. (It should be noted that for most practical applications the 



(^2\ -1/2 

 1 + -;; ) is very close to unity, so that the ordinate is prac- 

 7o/ 

 tically the value of Q itself.) 



Appendix IV gives a method for estimating Q based on the solution of 

 the field equations for a conductor replacing the helix and considering the 



liKOV^ ... 

 resultant field to be ^-— ^ — i. This estimate of Q is plotted as the dashed 



Pe 



lines of Fig. A6.1. 



A6.2 Thick Beam Case 



For an electron beam which entirely fills the space out to the radius b, 

 the electronic equations of both the normal mode method and the field 

 method are altered in such a way as to considerably complicate the solution. 

 In order to find a solution for this case some simplifying assumptions must 

 be made. A convenient type of assumption is to replace the thick beam by 

 an "equivalent" thin beam, for which the solutions have already been 

 worked out. 



