IK.IXSMIS.'^IOX CH.lR.tCTl-RlSTlCS Of WAVE-riLTRRS 613 



Although the transducer has four indepeiideiU parameters, it will 

 be seen that the sending end current involves but three effective trans- 

 ducer parameters, the sum (7,,6+7"i„,), U'^, and UV. As a resuh, 

 the four one-point impedance measurements which can be made upon 

 the transtlucer itself, the open-circuit and short-circuit driving-point 

 im^KHlances at both ends, must have a relation between them. Let 

 Xa and Ya denote the dri\inK-point impedances across terminals a 

 when terminals b are open-circuited and short-circuited, respectively. 

 Then if in (.">!) Za = and terminals b are ojx^n-circuited by putting 

 Zi, = x. . the impedance at terminals a, the open-circuit impedance, is 



A'a = ^" = H-« COth .1 ( Ta, + na) ■ (52) 



Similarly for the short-circuit impedance, when Zu = U and Zt,=0, 



y. = ir, tanh.\(7"a^+n„). (.53) 



For the other end we gel 1)\" interchanging subscripts 



.V^= n't coih \{'I'ab+Tba), (54) 



and 



y,= \\\ tanh M7'«6+no). (55) 



These give the necessar\' relation as 



Xa_X, 



Y, Y,- 



(56) 



Hence, in the most general linear transducer the ratio of the open-circuit 

 to short-circuit impedances at one end is equal to the corresponding ratio 

 at the other end. 



Other important derived formulae are 



7a»+7-M = 2 tanh' J 1^ = 2 tanh"' |2jl, (.57) 



\Va = VXTVI. (58) 



n\ = v/xa^. (59) 



H ',. - W, = V{X,-X,){Ya-Y,), (60) 



\VaWi = XaYi = X^V^. (61) 



and 



Thus the open-circuit and short-circuit impedance measurements 

 determine the sum of the directional transfer constants and both of 

 the image impedances. 



