470 BELL SYSTEM TECHNICAL JOURNAL 



From the discussion of the d-c. space charge it follows that the 

 d-c. current has an upper limit; i.e., the Kipp current with the value 



_ 4 \2e (VFoi + VFoa)^ 



To determine the relation between the Kipp current (4) and the 

 current (3) required for infinite capacitance, consider the d-c. equations: 



Uh = Ua -f- aai + T TT 



kme 2 . . 



2 kme 6 



where Ub and Ua are electron speeds in cm./sec. at planes h and a re- 

 spectively and aa is the acceleration in cm./sec.^ at plane a. Elimi- 

 nating aa and introducing the values of iih and Ua in terms of Vqi and 

 Foi we find 



\ ek 1q la 



(6) 



When the transit time T is eliminated between (3) and (6) the result is 



But this is precisely the Kipp current as given by (4). Equation (3), 

 therefore, expresses the Kipp relation between current and transit 

 time. It has thus been found that the series capacitance at the Kipp 

 point becomes infinite, w^hereas the impedance between the two planes 

 is a pure resistance. 



Consider next the possibility of Jq ^ attaining values larger than 



unity when the d-c. current /o is limited to values smaller than the 

 Kipp value (4). The only manner in which this could happen would 

 be for (6) to have one root T^ such that T^ > Tk where Tk is the 

 transit time at Kipp. To determine this (6) is transformed by intro- 

 ducing /ok and Tk as parameters. Thus, 



^)'-/^-3^ + 2 = 0. (8) 



1 K / J-QK J- K 



The discriminant D of (8) is 



Ikq \ ^ / 1 -^A'O 



^ = 'x ^-17' (') 



