62 BELL SYSTEM TECHNICAL JOURNAL 



regions are of course the same in the two plots (i.e., Z = 1) while the 

 boundaries of the B region are changed, since only the portion Zy of 

 the injected current is transmitted. The A region, which corresponds 

 to negative values of (p for all values of 7 and gives Zy = 0, is not 

 indicated. It may be of interest to note that for any given value of (p 

 it is impossible to transmit more than a certain maximum current, 

 shown by curve a, and that this maximum occurs for a C type distri- 

 bution. For injected currents greater than this amount only type B 

 distributions are possible. 



An enlargment of the B region is shown in Fig. 9a and a portion of the 

 region to a still larger scale in Fig. 9b. Lines for constant injected 

 current 7 are plotted with the potential (p and the transmitted current 

 Zy as the coordinates. These plots correspond to plots frequently 

 used to describe vacuum tube characteristics and may be used ac- 

 cordingly. For values of 7 less than four and values of ip less than 

 unity, double values of Zy appear. In the region lying between the lines 

 b and c on Fig. 9, the slopes of the constant 7 lines are negative corre- 

 sponding to conditions which are unstable unless sufficient resistance is 

 included in the external circuit. Conditions outside of this region are 

 always stable and need no further comment. 



Solutions of the C type are shown in Fig. 10a. For these solutions 

 the injected and transmitted currents are equal. The depth of the 

 potential minimum between the two planes is, however, of interest 

 and is shown in the figure in terms of the lesser of the two boundary 

 potentials. For values of (p greater than unity the value of cp min. 

 (i.e., the minimum potential in units of Vi) is indicated while for 

 values of ^ less than unity the value of ipmin. (i.e., the minimum po- 

 tential in units of F2) is shown. In the enlargement. Fig. 10&, both 

 values are given. For values of (pmin. = 1, curve e, and <pmin. = 1, 

 curve d, the potential minimum is equal to the potential at one of the 

 planes and is located at this plane. For currents greater than the 

 value indicated by these limiting curves the potential minimum be- 

 comes deeper and moves away from the plane while for lesser currents 

 it disappears entirely. The minimum value to which the potential 

 may be forced by increasing the injected current before the distribu- 

 tion changes precipitously to one of the B type is indicated by the 

 values on curve a. 



Again to avoid confusion the overlap type C region is shown sepa- 

 rately in Fig. 11. Conditions are somewhat different for this region 

 in that with any assumed value oi <p a minimum potential of (but 

 with complete transmission of injected current) occurs for a certain 

 value of 7. With increasing injected current, the potential minimum 



