SINGING ON TWO-WIRE CABLE CIRCUITS 611 



approximately 



(Sn - 2.5) X (521 - 1) + Sq{2.0), (9) 



i 



where X means that quantities so connected are combined as if their 

 currents added directly, e.g., 5ii X 5'2i == ^si means that 



IQ-Sul^o ^ IQ-S21I20 _ lO""^" 



/20 



When two end paths are connected in series, say around a four-wire 

 cable circuit, it is believed that singing points in the 2000-3000-cycle 

 range will add directly, because (1) due to the large amount of phase 

 shift around the loop, there will be a large number of frequencies which 

 will have the proper phase shift to permit singing and, (2) the 

 return loss in the frequency range of interest on a given line will 

 change slowly with frequency, which will cause the gain required to 

 produce singing to change only slowly with frequency. On this 

 assumption and assuming the normal law distribution given above, it 

 follows directly from the mathematics of such functions that the dis- 

 tribution of the singing points of two such end paths in series is 



12 + £i + £2 + 5q(2.8) = 521 - 1 + ^22 - 1 + 5q(2.8). (10) 



On the other hand, if only intermediate paths were present on the 

 two sides of a two-wire repeater, the singing points in the two direc- 

 tions would not, in general, add directly, because they would generally 

 be at different frequencies and when singing occurs at one of these 

 frequencies (or at some new frequency) the sum of the return losses 

 will generally be greater than would be indicated by the sum of the 

 singing points. Tests show that the internal singing margin (the 

 22-test value corrected for frequency characteristic of the repeater, 

 with only intermediate paths) is about 2.5 db higher on the average 

 than would be indicated by the sum of singing points in the two direc- 

 tions. Similar tests show that when the end path is added so that it 

 is very important compared with the intermediate paths, this average 

 difference becomes about zero. 



Also, the average one-way singing point on intermediate paths is 

 about B = 2.5 db below the reference single-frequency return loss, 

 as outlined above. Considering the internal singing margin as 

 effectively a 21-test ^ measurement through two return losses in series, 



« A 21-test is a singing test made by increasing the gain of a repeater until singing 

 occurs, with a line and network connected to the line and network terminals, re- 

 spectively, of one hybrid coil and a fixed known return loss connected to the other 

 hybrid coil. 



