VACUUM TUBE ELECTRONICS 



97 



transit angles on the two sides of the grid in the same proportion. A 

 modification of this generaHty occurs because the value of x in (65) 

 will shift as the effective position of the virtual cathode moves about. 

 Also, the complete space-charge condition near the plate becomes 

 modified, and so the general relations become extremely variable. The 

 partial space charge that exists with very negative values of plate 

 potential, or with very high values of grid potential does not lend itself 

 readily to mathematical treatment, so that intermediate conditions 

 between complete and negligible space charge can be treated only by 

 inference as to what happens between the two limiting conditions. 



With inappreciable space charge on both the plate and the cathode 

 sides of the grid, there can be no oscillations at all, since all impedances 

 then approach pure capacities, with no negative resistance components. 



A word concerning the so-called "dwarf" waves is in order before 

 this general theoretical discussion is completed. In the curves. Fig. 14 

 distinctly shows this possibility in curve A , since the resistance reaches 

 a large negative value at 2ir and again at 47r. Likewise Fig. 1 1 shows 

 the same possibility. On account of the resulting confusion in the 

 figures, the higher frequency portions have not been drawn in the 

 figures, but from (57), (59), (61), and (63) we can see what happens. 

 Thus, for very high frequencies, t] is large compared with unity, so 

 that the formulas may be written. 



Ro= - 



— -(t/ + smr], 



(57-a) 



Rq = ^ (1 + cos 7/ + sin 7}), 



R,= - 



i?n - 



12ro 

 12ro 

 12ro 



1 + xl sin ( r; + ^ j 



1 — - cos ^ 

 V 



-cos 17 



V 



(59-a) 



(61-a) 



(63-a) 



It is noteworthy that all of these exhibit the possibility of "dwarf" 

 waves separated by discrete frequency intervals except (63-a). On the 

 other hand, (63-a) gives possible conditions for operation at all high 

 frequencies provided that the proper external circuit may be secured. 



VIII. Postscript 



The extension of the electronics of vacuum tubes which was de- 

 scribed in the preceding pages must be regarded in the light of a tenta- 



