66 BELL SYSTEM TECHNICAL JOURNAL 



cathodes, but also with virtual - cathodes where the assumption of zero 

 alternating-current velocity and acceleration is unwarranted. Such a 

 virtual cathode might occur, for instance, between a grid operated at 

 a positive direct-current potential and a plate nearly at cathode 

 potential. If enough electrons came through the mesh of the grid 

 to depress the potential until it became practically zero at some point 

 in the space between grid and plate, the direct-current boundary con- 

 ditions of zero velocity and acceleration of electrons would be fulfilled 

 at that point. The general equations for the alternating current will 

 therefore apply when the origin is taken at the point of direct-current 

 potential minimum which forms the virtual cathode, and when all of 

 the electrons which are emitted by the actual cathode pass by the 

 virtual cathode and reach the plate. In the event that some of the 

 electrons are turned back at the virtual cathode and move again 

 toward the grid, as indeed they all do when the plate is at a negative 

 potential, a change in the form of the general equation is necessary, 

 and will be described in the sections dealing particularly with positive 

 grid triodes. This change, however, affects merely the form of the 

 equations and not the physical arguments underlying the selection of 

 boundary conditions, which are the same whether all the electrons 

 reach the plate or whether some or all of them turn back toward the 

 grid. 



If the alternating-current velocity is determined by small varia- 

 tions in grid potential, let us say, it is evident that no additional 

 assumptions save the requirement that the velocity must not become 

 infinite may be made concerning its value at the virtual cathode. 

 Consequently, a quite general set of boundary conditions will suffice 

 to determine the quantities, a, b, c, d, which appear in (13) and thus 

 completely determine Ui. 



Since there are two arbitrary functions in (12), two boundary con- 

 ditions will be needed. Further inspection shows that the stipulation 

 that the alternating-current velocity be finite at the origin is sufficient 

 to furnish one of these boundary conditions. For the other, a knowl- 

 edge of the value of the alternating-current velocity at any point 

 between the two reference planes is sufficient. Thus, if at a particular 

 value of ^, say ^i, we know that Ui is equal to M sin pt -\- N cos pt, 

 we have enough information to calculate its value at all other points 

 between the two planes. For example, the two reference planes might 

 be the grid and plate of a positive grid triode. In this event, the 

 alternating-current velocity at the grid could be calculated at the 

 grid plane by means of conditions between there and the cathode. 



= E. W. B. Gill, "A Space-Charge Effect," Phil. Mag., Vol. 49, p. 993 (1925). 



I 



