12 BELL SYSTEM TECHNICAL JOURNAL 



Now, if there were no impressed current, the righthand side of (2,6) would 

 be zero and (2.6) would be the usual transmission-line equation. In this case, 

 r assumes a value Fi , the natural propagation constant of the line, which 

 is given by 



Ti = jVBX (2.7) 



The forward wave on the line varies with distance as exp(— Fiz) and the 

 backward wave as exp(4-riz). 



Another important property of the line itself is the characteristic im- 

 pedance A', which is given by 



K = \^XjB (2.8) 



We can express the series reactance X in terms of Fi and K 



X = -jKT, (2.9) 



Here the sign has been chosen to assure that X is positive with the sign 

 given in (2.7). In terms of Fi and K, (2.6) may be written 



-VTiKi 

 V = (f.-ff) (2.10) 



In (2.10), the convection current i is assumed to vary sinusoidally with 

 time and as exp( — Fs) with distance. This current will produce the voltage 

 V in the line. The voltage of the line given by (2.10) also varies sinusoidally 

 with time and as exp(— Fs) with distance. 



2.4 Convection Current Produced by the Field 



The other part of the problem is to find the disturbance produced on the 

 electron stream by the fields of the line. In this analysis we will use the 

 quantities listed below, all expressed in M.K.S. units." 



■q — charge-to-mass ratio of electrons 



77 = 1.759 X 10" coulomb/kg 

 Wo — average velocity of electrons 

 Vq — voltage by which electrons are accelerated to give them the velocity 



«o. Mo = s/lriVQ 

 /o — average electron convection current 

 Po — average charge per unit length 

 po = —h/uo 

 V — a-c component of velocity 

 p — a-c component of linear charge density 

 i — a-c component of electron convection current 



* Various physical constants are listed in Appendix I. 



