SINGING ON TWO-WIRE CABLE CIRCUITS 625 



Columns (5) and (6) show computed values corresponding: to column 

 (4), on different assumptions as to the method of addition of the 

 intermediate and end paths. The values in column (5) are generally 

 much closer to the values in column (4) than are those in column (6). 



APPENDIX IV 



Distribution of Active Return Losses with Intermediate 

 Paths Including Equipment 



From a consideration of the paper, " Irregularities in Loaded 

 Telephone Circuits," by G. Crisson,^ it appears that it is reasonable to 

 say that the total current returned from a cable section with near-end 

 equipment will be 



it = V^lM^^, 



where ii is the current returned from the line without equipment and ie 

 is the current returned from the equipment. (It is assumed here that 

 it and ie are referred to the line side of the coil and not the drop side.) 

 Then the probability that the current returned from the cable alone 

 will exceed / is 



1 r°° ir P 



^(^) = Tl i^e - ^rT^ ^^"i = ^ - 973 

 1 J J 



2/1^""'"" 2/i2 



The probability that the total current will exceed I3 — V/" + ie" is 



(r.2 _ { 2) 1 2 



or 



P{it > h) = antilogio ( .434 antilogio ( ^^Jq^ )) ^ih) = F', 



where 



Si ^ — 10 logio 2/r and Se = — 20 logio ie. 



This equation says that the probability of getting more than a given 

 value of return current I3 from the combined cable section and near end 

 equipment is equal to the product of the probability of getting I3 

 from the cable section alone by the factor e{ie'^J2Ii^}. Or by analogy 

 with the method used in the paper referred to above, letting the 

 distribution curve of the combination return loss be Si + Sf, 



Sp, = - log 10 ( loge-^ + 2^^ 



= - logio ( loge -^ + antilogio ^ ^^ j 



Sf' ^^ Sf yQ {Se — Si), 

 p 



'' Loc, cit. 



