678 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



axis. The space variation of the wave ampHtude may be written 



'^-^ = -(ri + kn)E, + k-nE, , (17) 



and 



'^ = k,,E, - (r, + /:22)^2 , (18) 



ax 



in which kn , A-22 represent the reaction of the coupUng mechanism on 

 lines 1 and 2 respectively 

 ^21 , ki2 represent the transfer effects of the coupHng mechanism 

 ri,2 are the uncoupled propagation constants of line 1 and 2 



respectively ; 

 -E'1,2 are the complex wave amplitudes on lines 1 and 2, 

 and are so chosen that \ Eif and | E2 \ represent the powder carried by 

 lines 1 and 2 respectively at the input or output of the coupling region. 

 The usual transmission-line equations are of this general form, except 

 for second derivatives iii place of the first. The first derivatives appear 

 here because we deal only with the forward travelling waves, which the 

 preceding section has shown are the only significant waves when small 

 coupling per wave length is employed. Limiting our interest to the cases 

 for Avhich reciprocity holds and noting that there is alwaj^s a transverse 

 plane of symmetry midway between the ends of any pair of uniforml}^ 

 coupled lines, we ma}^ transform the wave amplitudes to make ki2 = 

 kii = k. We may further simplify the equations without loss of essential 

 generality by submerging the differences (kn — k) and (k22 — k) into a 

 modified propagation constant for lines 1 and 2 respectively, yielding 



and 



in which 



dEi 

 dx 



dE2 

 dx 



= -(71 + k)E, + kE2, (19) 



= kE, - (y, + k)E2 , (20) 



71 = Ti + An - k, and . 



72 - To + A-22 - k. 



For some cases kn = kn = k and for all cases of interest here 7„ differs 

 very little from r„ since we are concerned only with loose coupling per 

 wavelength. 



