Lines in Telephonic Transmission. 317 



If terminal transformers are added and the transformer 

 impedances are J\ for the set side winding, J 2 for the line 

 side winding, and J 12 between the two windings, then (12) 

 becomes 



e^'y ikj. y j„(*+j.) \« , 

 '"- 2jA(^+j s ) 2 A(^+j 2 )(Ji+j 8 )-Ji 3 v 



(13) 



where the four factors are, respectively, the value of the 

 current for a circuit consisting of the two sets alone, the 

 effect of terminal reflexion, the effect of transformers, and 

 the effect of transmission over the line. For transformers of 

 high inductance, negligible resistance, and negligible magnetic 

 leakage (13) becomes 



Ee l P f J 9 s 



2J S '/ 7 Ji 



C'W 



e-y\ . . . (13a) 



and the transformers are equivalent to a change in the line 

 impedance from k to k Ji/J 2 . 



The impedance of a circuit consisting of length I of uniform 

 line (ky) closed through an impedance J at the further end 

 is by formula (11) 



where x=(k— J )/(& + Jo) is the reflexion coefficient from 

 line to terminal impedance. 



Diagram I. (PL V.) shows the value of the reflexion factor 

 in equation (12). The factor involves the two impedances 

 symmetrically, is a function of their ratio only, and becomes 

 unity if they are equal. Let the absolute value and angle of 

 the impedance ratio be r, 6, and of the reflexion factor be 

 e~ b , (f>, then 



— = v cis 0, 



t>=.log{l(r+l)+lcose}, . . (15) 



<b=t:xn-> —1L (16) 



2 + (r+±jcos0 



