VACUUM TUBE ELECTRONICS 69 



We have 



Vi = /(?) + constant, 



Vi = M) + constant, 

 so that 



Vx- F/ ^M) -fin. (24-a) 



Since the potential difference is always required rather than the 

 absolute potential, (24-a) gives the means for applying (24) to actual 

 problems. 



III. Application to Diodes 



In the application of the fundamental relations to diodes where the 

 thermionic emitter forms the plane located at the origin and the anode 

 coincides with the other plane, the boundary condition is that Ui shall 

 be zero at the cathode. This means that both M and N are zero and 

 that ^1 is also zero. The resulting forms taken by (18) and (24-a), 

 respectively, are as follows: 



^1= S2 



P 



V, - F/ 

 ~ e9p* 



2 . \ . / 2 . 2 



1 + cos ^ — --sin ^ j -\- i i -r — sin ^ — 7 cos ^ 



(25) 



C(2cos^ + ^sin^-2)+^(?+|?^-2sin?-|-^cos?)]. (26) 



These two equations are identical with those obtained by Benham,^ 

 and graphs are given in Figs. 1 and 2 showing their variation as a 

 function of the transit angle ^. In particular, the equivalent impe- 

 dance between unit areas of the two parallel planes may be found from 

 (26). It must be remembered that the current. A, was assumed 

 positive when directed away from the origin. Hence, we may write 



Z=-Zl^. (27) 



Moreover, the coefficient outside the square brackets in the equation 

 may be expressed more simply when it is realized that the low-fre- 

 quency internal resistance of a diode is given by the expression 



^0 = "~ ^T" ' (28) 



the minus sign again appearing because of the assumed current direc- 

 tion. Consequently, under the condition of complete space charge, 



