ELECTRON STREAMS IN A DIODE 845 



The formal solution for small signals is really completed with the deriva- 

 tion of the preceding general equations for the a-c. amplitudes. But there 

 still remains the rather tedious process of deriving the relations for circuit 

 calculations as outlined in the following section, and they give a direct 

 comparison with previous theories of electron streams. 



3.3 Small Signal Equations for Circuit Calculations 



Llewellyn^ has shown that the treatment of a diode as a circuit element 

 requires certain variables at the second plane to be expressed in terms of their 

 values at the first plane ; that is, the circuit theory requires three equations 

 of the form 



Vb — Va = AH -\- B*qa + C*Ua 



qj, = DH + E*qa + F*Ua (53) 



Ub = G*i + H*qa + I*Ua 



where the starred coelB&cients are known functions of the d-c. components. 

 The derivation of these relations from the preceding general equations is 

 outUned as follows : the first step is the evaluation of the arbitrary constants 

 A and B. This is done by substituting the values at the first plane in (44) 

 and (45), and then solving for A and B, which gives 



(54) 



(55) 



These expressions, and the values at the second plane, are then substituted 

 in the equations for the a-c. amplitudes (45), (46) and (49); and they im- 

 mediately give the desired relations. They do, however, involve the incon- 

 venient electric intensities Ea and £&, and these quantities are replaced by 

 their values from (39). 



These simple but rather tedious substitutions are illustrated by the fol- 

 lowing derivation of qb—, which is brief enough to be included for that 

 purpose. The first step is the substitution of the values at the second plane 

 in (45); this gives 



where j8 isjcoT. It is now advantageous to replace B by its value from (55), 

 and 



,. = >e {^) Ae-^ + ^; e-V + J,^ (e^ - l)i (57) 



