COUPLKD WAVK THKOin AND W A \ i;( ; T 1 1)|; A I'l'I.K Al'K ).\S (170 



The solution, for Ei = ].0 and 7^2 = at x = 0, is 

 1 (Ti - T2) 



Ei = 



and 



2\/(7i -72)2 + 4fc2_ 

 + 



1 , (ji - 72) 



2 2v^(7i - 72)2 + 4/o2_ 



(21) 



k k 



."here 



n = -H(2A- + 71 + 72) + }W(7i - 72)^ + 4ib2, (23) 



r2 = -1/^(2/.- + 71 + 72) - i.i\/(7i - 72)- + 4A-2. (24) 



The nature of the couphng eoefhcient A- is the first thing to investigate. 

 Assume no dissipation in either the transmission Hne or in the coupling 

 mechanism. Then it follows that for any value of x, 



1 ^1 I" + I -^2 I" = constant (25) 



on the basis of energy conservation. It may be determined that (25) 

 leads to the requirement that the coupling constant k be purely imagin- 

 ary. This is a very important result. In all of the following discussion A- 

 is taken to be purely imaginary. Even where dissipation in the trans- 

 mission lines themselves is important, it is still assumed that the coupling 

 mechanism is non-dissipative. 



The simplest case is 71 = 72 = 7, coupling between identical trans- 

 mission lines. Then (21) and (22) reduce to 



El = cos ex e-^''^'^\ (26) 



and 



E. = i sin ex r^''+'^\ (27) 



where k = ic. The exponential of (26) and (27) shows that the coupling 

 modifies the average phase constant, and that the attenuation in the 

 driven line (Ei) is the same as in the uncoupled case for ex (coupling 

 length times coupling strength) eciual to nw radians. The amplitude and 

 phase variations due to the coupling are plotted in Fig. 16. Complete 

 power transfer between lines takes place cyclically, A\ith a jjeriod of 

 ex — IT, and with suitable choice of the product ex, an arljitrary di\'ision 

 of power between the lines may be selected. 



