ELECTRON STREAMS IN A DIODE 841 



second order terms in the oscillating components. Each part of the general 

 solution then separates into two equations, one for the d-c. quantities, and 

 another for the a-c. quantities. The resulting equations for the d-c. com- 

 ponents are 



f-j+Sf (27) 



and the equations for the a-c. components are 



ue'"' = ^(5) - ^ [^ ? + -i] e"" (29) 



= B{S) - '-^ A{S) + i[-J q+ £-3 »] e'"' (30) 



where in the last equation g has been replaced by its value from (27). 



3.1 The D-c. Components of the Electron Stream 



We first consider the d-c. components in (27) and (28). It is easily shown 

 that they obey the primitive differential equations 



dx 



(31) 



f/ 1^ = -/A 



dx 



which are the static equations for a diode, when it is idling in the absence 

 of an a-c. signal. Their solutions are given in various pubHshed articles, and 

 they are available without further calculations.'^- ^ These d-c. components 

 are involved in the subsequent development of the a-c. theory, and the 

 latter requires certain d-c. relations. These relations are briefly presented 

 without derivations as follows: 



The current density / and the d-c. voltages at the two diode planes are 

 assumed to be known quantities. Then the d-c. velocities at those planes 

 are also known quantities, because their values are given by the simple 

 relations 



Ua = VWa , Ub = V2v^b (32) 



where it is assumed that the original source of electrons is at zero voltage. 

 The d-c. transit time plays an important role in the small signal theory. 



