646 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



A sure way of distinguishing conditions which will allow amplification 

 from conditions which will not is through a solution of the differential 

 equations together with a fitting of the boundary conditions. In the case of 

 backward waves there are, however, considerations concerning the source 

 of energy and its transfer to the circuit (or field) which are useful. 



Suppose that an unmodulated electron stream enters a microwave ampU- 

 fier, travels for some distance through it, and emerges. If the electromag- 

 netic output power of the amplifier is greater than the input power, the 

 additional power must have come from the kinetic energy of the electron 

 stream. The average electron must leave the amplifier with less energy of 

 motion than it had on entering it. 



We can say a Uttle more. Let us call the total velocity of electrons, a-c. 

 and d-c, u. Then we have, corresponding to (1.1) 



du , du e ^ 



— + u — = — - E 

 ot dz m 



(4.5) 



ot dz m 



We will consider an amplifier in which the u and E at any z-position 

 are truly periodic. Let us integrate over the period of a cycle, r, and divide 

 by r 



1 



u+-~ u'dt= - E (4.6) 



t r oz J t T Jt 



As u will be the same at t and / + r, the first term on the left is zero, 

 and we have 



~u' = E (4.7) 



dz 



Here u^ and E are time averages. 



The field E is produced in a linear circuit by (1) the application of an 

 a-c. signal, (2) by the presence of the electron stream. Certainly, the applied 

 signal can produce no average field in a hnear circuit. Further, unless elec- 

 trons are turned back, the average electron convection current is inde- 

 pendent of r-f level. In a linear circuit the average field must be propor- 

 tional to the average impressed current, so the average field E must be 

 zero or independent of r-f level. Thus, the time average of u^ at a given 

 point must be independent of r-f level.* 



This means that the electron stream cannot be slowed down bodily by 



* L. A. MacColl pointed this out to the writer. 



