612 BELL SYSTEM TECHXICAL JOURNAL 



It is desired to obtain, among others, expressions for the sending end 

 and receiving end currents h and h which contain the image para- 

 meters. 



Each terminal impedance will be considered as cqui\alent to the 

 image impedance at that end plus another impedance whose potential 

 drop is to be replaced in the usual manner by an equal opposing 

 electromotive force. In effect this equivalent electromotive force 

 substitution reduces the system to one in which the transducer is 

 terminated by its image impedances and in which determinate electro- 

 motive forces are acting at both ends. From this viewpoint, the total 

 effective electromotive forces acting at the ends a and b of the trans- 

 ducer terminated by its image impedances Wa and Wi are, respectively, 



Ea + (\Va-Za)I„, 



and (49) 



(\Vi-Zb)h. 



Superposing the currents due to these electromotive forces at both 

 ends we may write the current expressions immediately from the 

 definitions of the parameters invoked. 

 Thus 



_ Ea + iWa-Z,)Ia {Wj-Z,)!, _j., 

 2Wa "^ 2Wt ^ '' 



and (50) 



_ Ea + {\Va-Za)Ia ^^^ (\\\-Zi)h 



' 2Wa ^ '^ 2H'» • 



Their solution gi\'es the explicit formulae '" 



/. = ; 



Ea {l+ne-^T-'-+^''-^) 



and (51) 



Ea (l + ri,)e-^»» 



h = 



W„ + Za (\-r„ne-^T-"'+^i»^) 



where r„ and ii.. ilic current reflection coefficients at terniin.iisi; .uui b, 

 are 



^ Wa-Z„ 



''" Wa+Za 



and 



_ Wb-Zb 

 " W,+Zo 



'"These formiilii- may also l>e (Icrlvcd syntlu-lically l)y tlic iiinciil rcllcctioii 

 method. 



