WAVES IN ELECTRON STREAMS AND CIRCUITS 649 



The phase velocity may be either positive or negative, depending on the 

 sign of Xo . This added feature gives the solution a more "wavelike" quality, 

 but physically we have merely a slight perturbation of the disturbance 

 natural to the non-propagating ladder network. 



In the absence of electrons, there is no real power flow in the modes of 

 propagation of a purely reactive ladder network in which the shunt and 

 series reactances have the same sign. Such a network can of course transmit 

 power to a resistive load, but it transmits no power when terminated in its 

 (reactive) characteristic impedance. 



In the presence of electrons, there is a small power flow in the circuit. 

 We can easily evaluate this. If, in (1.11), we assume a variation of the quan- 

 tities with time and distance as 



we obtain 



a 



JU3L 



Here coL stands for the series reactance, which we may call Xi 



wL = Xi 



A positive value of Xi means series inductance. For non-propagating lad- 

 ders, Xi and the characteristic reactance Xq have the same sign. 

 We then have 



^ Xi 



The quantity —ja/Xi as evaluated in the presence of electrons will be 

 the "hot" characteristic admittance. 

 The complex power flow P is 



P= VI* 



So, in this case 



* ^ 



p = ^-L vv* 

 Xi 



Now, the "backward" wave, for which a is given by (4.12), "increases" 

 in the direction of electron flow. For it, the real part of the power Re F is 

 given by 



coeo-jSpOiiXo 



Note that Xo and Xi must have the same sign. Thus, the power flow for 

 the wave which "increases" in the direction of electron flow is always in 

 the direction opposite to the electron flow. The circuit power does not flow 

 in the direction of increasing amplitude for the wave which "grows" in the 



KeP/VV*= - ,T^;"\C^, (4.12) 



