612 BELL SYSTEM TECHNICAL JOURNAL 



it seems reasonable to expect this singing point to be about 2.5 db on 

 the average below the reference single-frequency return loss of the 

 two in series. 



From either or both of the above two paragraphs, it may be assumed 

 that the internal singing margin is about 



M/ - 5i - F{N,, T) - Gc + Si 



- F{N2, T) - gc- 2.5 + 5o(2.8). (11) 



The distribution function Sq{2.^) is an assumption which seems 

 reasonable from general considerations. 



From the above discussion the singing margin of the entire circuit 

 will be a function of all the intermediate and end paths which meets 

 the following conditions: 



1. When the end paths are unimportant compared to the 

 intermediate paths, the singing margin is 



Mi = 5n + 5i2 - 2.5 + 5q(2.8). (12) 



2. When the intermediate paths are unimportant compared 

 to the end paths, the singing margin is 



ilf^ = 521 - 1 + 522 - 1 + 5q(2.8). (13) 



3. When both kinds of paths are fairly important, the singing 

 margin is some compromise between 



(5n) X (52i) + (5n) X (52i) + 5q(2.8), 

 p p 



which would have an average value equal to the reference 

 value of the circulating current margin and about 2.5 db 

 less than this. The singing margin is obviously greater than 



(5ii - 2.5) X (521 -1)4- (5i2 - 2.5) X (^22 - 1) + 5q(2.8) 



because the active singing point in one direction is almost cer- 

 tain to be at a different frequency from that of the active 

 singing point in the other direction. 



A reasonable empirical compromise which satisfies all of these con- 

 ditions and preserves the symmetry of the equation is to say that the 

 total singing margin of the circuit is 



M, = {Sn - 1.25) X (521 - 1) 



+ (5i2 - 1.25) X (52. - 1) + 5o(2.8). (14) 



