648 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



The characteristic impedance K is reactive, inductive if both impedances 

 are inductive and capacitive if both impedances are capacitive. Let 



K = jX, 



where Xq is a real number. Then a positive value of Xq means inductive 

 elements, and a negative value capacitive elements. 

 The space-charge waves are given by (2.17) 



^ = ^o±^p i/ -^^^^o«i go (4.8) 



y al + ^l 



We see that ,the waves are unattenuated for negative, capacitive values 

 of Xq , and are increasing and decreasing for positive, inductive values 

 of Xq . It can be shown that the increasing space-charge waves can be used 

 to obtain gain. 



The forward circuit waves are given by using K = jXq , /5i = —jai in 

 (2.16), /3 = —jai on the left and in the numerator on the right and j8 = —ja 

 in the denominator on the right. 



" = •■(■ + £?#■)■" 



As a = y/3, the variation with distance is as 



The backward wave is given by using /3 = -\-Joli on the left of (2.16) and in 

 the numerator on the right 



If a differs little from ±ai , we can expand the square root in (4.6) and 

 (4.7), separate real and imaginary parts, and write: 

 Forward wave: 



" "'V^ 2(Bl + a\) (3l + air ) 



(4.10) 



Backward wave: 



"A'"^ 2(^S + a?) ^ (^S + a?)7 ^ ^ 



The circuit "waves" which were rapidly attenuated in the absence of 

 electrons (/3p = 0) are a little more or less rapidly attenuated in the presence 

 of electrons (more or less depending on whether Xq is positive or negative, 

 and on the relative magnitudes of /3o and ai), and they now have a phase 

 constant, that is, an imaginary component of the propagation constant. 



