630 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



We now have three approximate equations, which are good approxima- 

 tions at small signal levels 



(1.2) 



(1.4) 



(1.7) 



We can eliminate p and v from these equations and obtain an equation re- 

 lating i and E. To do this we solve (1.7) for v 



differentiate (1.2) with respect to /, 



dt \dtj dz \di) ~ m~di 



and substitute for dv/dt , obtaining 



d'i , ^ dH , 2 dH e dE ,, „, 



;r7^ + 2Mo-— +Wo— = --Po— (1.8) 



dt^ dzdt dz^ m dt 



This is an equation relating i and its derivatives with E. It is a linear equa- 

 tion; that is, i and its derivatives, and E appear to the first power only. 

 This is because we have neglected non-linear terms, saying that at low levels 

 they are small compared with the linear terms. 



Now, the electron flow interacts with surroundings of some sort, or, we 

 shall say, with a circuit. Let us consider as an example of a circuit a trans- 

 mission line with a distributed capacitance C per unit length and a dis- 

 tributed inductance L per unit length, which will transmit a slow wave. 

 Suppose that the electron stream flows along very close to the line. Then 



