626 BELL SYSTEM TECHNICAL JOURNAL 



Or the distribution curve of the return losses from the repeater section 

 with equipment is 



. S, + Sf> = (5i + Sy) X (5i + Se - 5i) 



= (5i + S,) X (Se) 

 p 



S,^> = (Sj.) X (Se - 5i). 

 p 



Curves of Sf' could be drawn for different values of Se — Si, since 

 Sf is known for a given value of F. These would apply either for 

 passive return losses of a single repeater section or for active return 

 losses of several repeater sections in tandem. 



For example, suppose Se = Si. For Sf = 0, the value of F is about 



0.368 and the value oi Sf' = 0X0 = - 3.01 db for F' = 0.368. Or 



p 



the same point could be derived from the earlier formulas as follows: 

 When Sf = - 3.01 db, F = 0.1353. The factor antilog ( 0.434 



antilog ( tt; ) ) equals 2.718 and the value of F' for Sf' = — 3.01 db 

 is F' = (0.1353)(2.718) = 0.368. 



APPENDIX V 



Distribution of Active Return Loss Made up of Intermediate 

 Paths and a Fixed End Path 



In a practical case, when an active singing point toward a circuit 

 terminal is measured for maintenance purposes, the termination at the 

 circuit drop will be a fixed known value of return loss rather than one 

 selected at random from a distribution curve. The distribution curve 

 of active singing points toward the drop will then be a curve obtained 

 at each percentage, by the current sum addition of the active singing 

 point with no current returned from beyond the terminal repeater and 

 the end path singing point. If the active intermediate path singing 

 point has a distribution curve Sn — 2.5 + Sq{2) = ^4 + Sq(2) and 

 the end path has a fixed loss S^, the following table may be used to find 

 the distribution curve of the combination ^4 + Si,{d), for different 

 values oi d = S;, — S4. The table was derived from the equation 



(54 + 5o(2)) X 55 = 54 + S,{d) 

 p 



