392 BELL SYSTEM TECHNICAL JOURNAL 



or they afe active modes which are considerably out of synchronism with 

 the electron velocity. Unless one of these other active modes has a propaga- 

 tion constant V,, such that ] {Vi — r2)/ri | is so small as to be of the order 

 of C, the terms forming the summation will not vary very rapidly over the 

 range of variation of T which is of interest. 



We will thus write the circuit equation in the approximate form 



E 



2(ri - r') ~ coC] 



(7.1) 



Here there has been a simplification of notation. E is the z component 

 of electric field, as in Chapter II, and is assumed to vary as exp(—Tz). 

 {E?/^^P) is taken to mean the value for the Fi mode. It has been assumed 

 that jSo is small compared with | Ti | and | F- |, and /So has been neglected 

 in comparison with these quantities. 



Further, it has been pointed out that for slightly lossy circuits, {E?/^"^?) 

 will have only a small imaginary component, and we will assume as a valid 

 approximation that (E^/^^P) is purely real. We cannot, however, safely 

 assume that Fi is purely imaginary, for a small real component of Fi can 

 aflfect the value of Fi — F- greatly when F is nearly equal to Fi . 



The first term on the right of (7.1) represents fields associated with the 

 active mode of the circuit, which is nearly in synchronism with the elec- 

 trons. We can think of these fields as summing up the effect of the elec- 

 trons on the circuit over a long distance, propagated to the point under 

 consideration. 



The term (— jTVcoCi) in (7,1) sums up the effect of all passive modes 

 and of any active modes which are far out of synchronism with the elec- 

 trons. It has been written in this form for a special purpose; the term will 

 be regarded as constant over the range of F considered, and Ci will be given 

 a simple physical meaning. 



This second term represents the field resulting from the local charge den- 

 sity, as opposed to that of the circuit wave which travels to the region 

 from remote points. Let us rewrite (7.1) in terms of voltage and charge 

 density 



dV 



E= -^ = TV (7.2) 



dz 



From the continuity equation 



i = (jWr)p (2.18) 



-M\{E?/^^'P) 



V = 



_ 2(F1 - r) ^ c 



'] 



+ n\p (7.3) 



