1284 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



(14) 



For a line of iterative serpentine bends we apply the rules of matrix 

 calculus. If £"10 = 1 and £"20 = at the input of the first section, then 

 the output amplitudes of the n section are : 



Eln = - 



1 + 



7^11 - T 



22 



V{Tn - T22y + 4ri2T2i_ 



Tn - T 



i J 11 ~ -t 22 



"^ 2 L ~ VCTii - 7^22)^ + 4^127^21. 



-"02 



, (15) 



Tn 



V(7^n - T,,y + 47^12^21 



[e-"«i _ e-""'], 



Eon = 



where 



e~''- = hiTn + 7^22 ± V(rn - 7^22)^ + 4^12^21]. (16) 



Two limiting cases for the expressions (15) and (16) are of special interest: 

 1. I 7^11 - 7^22 r» 4 I 7^12^21! 



7^12 7^21 



-01 



= 7^11 + 



-(72 _ 



i 11 — J 22 



TriTii 



Tr^T 



Tn ~ T22 

 2. 4 I r 12 T21 I » I Tn - 7^22 r 



E^n = 



12^21 ^^-„«i _ ^-na,j ^7) 



{Tn - T,,y 



Tn 



7 n — 7 22 



22 



-0,., ^ Tn + 7^22 ^ ^jT^ 



21 



£i„ = hie-"'' + e-"^^) 



(18) 



