WAVES IN ELECTRON STREAMS AND CIRCUITS 635 



Suppose we consider a beam of area o-. We can write the total electron 

 convection current /« in the form 



le = (Ti = Y^ (2.6) 



We will call Yg the electronic admittance; it is measured in mho meters. 

 Later we will deal with waves in which the electron stream transfers 

 power to the circuit, and it is interesting to see under what conditions 

 this can take place. Let the amplitude of the wave under consideration vary 

 with distance as 



We may take the complex nature of the propagation constant into account 

 by substituting in (2.7) 



This leads to 



-j^ = Qfl - j^i 



Y = -j<^^^^l 



(2.8) 



Ye = 



coe(r/3^[2ai(i3o - /Si) - jQgo - ^if] 

 [(/3o - ^lY + al] 



The electron stream can transfer energy to the circuit only if the real 

 part of Ye is negative (a negative conductance). For a wave which in- 

 creases in the direction of electron flow (the +z direction), ai is positive 

 and the electronic conductance wiU be negative if jSi > j^o ; that is, if the 

 electron velocity is greater than the phase velocity of the wave.^ 



For a wave which decreases in the +z direction, the conductance will 

 be negative if the electron velocity is smaller than the phase velocity of 

 the wave. 



Let us now consider the interaction of our thin electron stream with the 

 circuit. Here there is some possibility of confusion. In (1.12) the field caused 

 by impressing a current on a circuit was calculated. This may be likened 

 to the voltage along an impedance Z caused by an impressed current 7. 



8 This is indicated by very elementary arguments (J. R. Pierce and L. M. 

 Field, "Traveling Wave Tubes," Proc. I.R.E., Vol. 35, pp. 108-111, Feb. 1947). It is easy 

 to forget, however, and was recently pointed out to me, to my consternation, by Dr. L. 

 J. Chu. 



