838 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



moreover contains two arbitrary functions, and it is therefore the general 

 and complete solution for an idealized stream flowing in a diode. 



As the solution stands, / is an arbitrary function of time, and Fi{S) and 

 F2{S) are perfectly arbitrary. They correspond to all possible determining 

 conditions: to all the d-c, a-c. and transient conditions that are possible 

 in the idealized diode, and to all the purely mathematical conditions that 

 cannot be realized in any physical sense. 



With this complete solution available, the situation is analogous in many 

 respects to that encountered in the solution of potential problems in two- 

 dimensional space. We can find a particular solution by merely assigning 

 definite functions to the three arbitrary functions /(/), Fi{S) and ^2(6*); 

 but we then encounter the difficult task of finding out just what problem 

 has been solved. 



As a simple example of the general method, the reader may be interested 

 in arbitrarily setting 7, Fi{S) and F2{S) equal to zero. He will then find that 

 the resulting expressions, (15) and (18), are actual solutions of the partial 

 differential equations, and that they represent a transient electron stream 

 that can flow for a short period of time in a diode space. 



2.1 The General Solution in the Presence of a Direct Current 



In the majority of circuits that are of practical interest, there is a continu" 

 ous direct current flowing in a diode, and the arbitrary functions then as" 

 sume a more restricted form. In such cases the total current density / may 

 be considered as the sum of a d-c. component, which for the time being is 

 indicated as Id, and a transient or alternating component Ia- 



Then we have the condition that 



Id>0 (19) 



and also the condition that U and x must be finite in any physical tube' 

 Now consider (15) for U and note that 



f I dt = lDt+ f Ia di 



(20) 

 jjlH^I^ + jjl^i, 



The bracketed factor in (15) thus contains power terms in /, which becomes 

 infinite as / approaches infinity. The function Fi{S) must therefore be of 

 such form that it cancels these terms and causes U to remain finite. Inspec- 

 tion shows that Fi(5) must consequently be of the form 



F,{S) ^ A(S) + g + g,S-\- g^ (21) 



