86 BELL SYSTEM TECHNICAL JOURNAL 



and inserting the boundary conditions, (14) and (15), we have, in com- 

 plex form, 



£/, = (.« + iN) -I a^ - 1-, ( 1 - f; aJi|). (51) 



^ smh ^1 ^- \ ^ smh |i / 



which is a simpler equation than its analogue (18). The potential is 

 obtained as in (24) giving. 



]', = - 



- (i1/ + iN) -——r [^ sinh t + 7(^ cosh ^ - sinh ^)] 

 -e I smh ^1 -" 



9p 



+ 



9p'e 



__ ^1^ sinh ^ 

 sinh ^1 



+ M ^-J - • u . (^ cosh ^ - sinh i) 

 ' ^ smh ^1 



+ constant. (52) 



The alternating-current potential difference between the grid and 

 the virtual cathode where all of the electrons are turned back may be 

 obtained immediately from (52). As before, the variable f will be sub- 

 stituted for ^ to show that the grid-plate region is considered, and 

 currents and velocities will be considered positive when directed 

 towards the origin at the virtual cathode. Thus, from (52) 



yp~e 





~j - r coth i' + i' 



(53) 



The velocity, (AI + iN) may be expressed in terms of the alternating- 

 current grid potential, Vg, so that the path between grid plane and 

 virtual cathode may be represented by an effective generator in series 

 with an impedance, as was done in (34), (35), and (36). 



VII. Oscillation Properties of Positive Grid Triodes ^ 



The oscillation properties of the positive grid triode are next to 

 be investigated. In the usual experimental procedure, an external 

 high-frequency circuit is connected between the grid and the plate of 

 the tube. It is unfortunate that this particular arrangement greatly 

 complicates the theoretical relations. Accordingly, a slightly modified 

 experimental set-up will be considered. This modification consists in 

 connecting the external circuit between the cathode and plate of the 

 tube, rather than between grid and plate. Experimental tests have 

 shown that the modified circuit exhibits the same general phenomena 



'" Loc. cit. 



