APPENDIX VI 



451 



be the same close to the complex values of 7 which will eventually solve 

 the equations. 

 The solution of the field equations for the solid beam yields the value for 



7/(1) 



11 ^ 



where 



«' = 1 + 



1 /m ^eL 



i3o y e 2x62 



1 



(14) 



(15) 



Thus the electronic equation for the solid beam which must be solved simul- 

 taneously with the circuit equation (given above by either the normal mode 

 approximation or the field solution) must be 



Y = ^ - Y = i^ 



nliinyb) h{yb) 



h(yb)_ 



(16) 



Complex roots for 7 will be expected in the vicinity of real values of 7 



By plotting Ye and Yc vs. real values of 



dYe dY 



for which Ye ^^ Ye and -^ ^ ~ 

 dy 07 



7, it is found that the two curves become tangent close to the value of 7 for 

 which n = 0, using typical operating conditions (Fig. A6.2). Our procedure 

 for choosing a hollow beam equivalent of the solid beam, then, will be to 



equate the values of Ye and ~-^ at n — 0. This will give us two equations 



dy 



from which to solve for the electron beam diameter and d-c current for the 



equivalent hollow beam. 



If the hollow beam is placed at the radius sb with a current of th , the 



TT 



value of -^ at the radius b gives the value for Yen as 



F.ff = 



I', = 



jcoeb -^ (1 



y'b'll{syb)-i\ 



n) 



lljsyb) 



n{yb) 



Koisyb) 



K.iyb) 



llo(syb) Io{yb)_\ 

 Equating this with eq. (16) at n = yields the equation 



- = ie'n{sd) 



'Ko(se) KM 



(17) 



(18) 



