NORMAL MODE BENDS FOR CIRCULAR ELECTRIC WAVES 1303 



in which 



- " o 

 6 1 — vqi 



(28) 



u = 



I 



Here again the summation signs indicate that all coupled modes have 

 to be taken into account. The factors S, B, and C do not depend on the 

 bend geometry, but only on the total bending angle, the waveguide prop- 

 erties, and the frequency. Necessary conditions for A(u, I) to be a mini- 

 mum are : 



dA _2(l - 2m) 

 du 



B 



C 



with the two roots 



Z(l - u') L3 /V_ 

 1 



= 0, 



u = 



2' 



and 





'-i"i 



4C 



— = S - 73 



dl (1 - uY r- (u - u')- I' 



If u = h, the solutions of (32) are the roots of 



S f - iBf - 64C = 0. 

 If {III) = ?){C/B), the solutions of (32) are the roots of 



Sufficient conditions for A{u, I) to be a minimum are: 



d'A ^ ^ d'A 

 T^ > 0- ^ > 0> 



= 0. 



6w2 



dl' 



(29) 



(30) 

 (31) 



(32) 



(33) 

 (34) 



(35) 

 (3()) 



h 



