422 BELL SYSTEM TECHNICAL JOURNAL 



although relations of this form do not hold for Z and for W. Each 

 of the relations (8) and (D) can be inferred also from simple physical 

 considerations. 



Equations (4) and (o) show that IV and Z' are alike in form, and 

 also W and Z, when a and a' are regarded as corresponding to each 

 other; in fact, when a = a' , 



zz' = WW = wz' = w';z=KK'jk'=iiir, h- = \. (lo) 



Besides, there is the set of perfectly general relations (2.1), which, 

 of course, continue to hold when a — a'. 



Equations (4) and (5) show also the existence of the following more 

 special relations, holding when the relative terminations {a and a') 

 have the values and 1, as indicated by the subscripts: 



ZoZ, = Zo'Zi' = IFoH'i = Wo'W^ = 1 , (11) 



|ZoI=|Zi!=|Zo'l = lZ,'| = |H'o|=|WM = |W| = ll^i'! = l. (12) 



Graphical Representations 



Graphical representations of the relative impedances Z = X-\-iY 

 and Z' = X'+iY', based on equations (4) and (5), will be taken up 

 in the following paragraphs. Evidently it will not be necessary to 

 consider also the relative admittances W = U-\-iV a.nd W' = U' + iV 

 explicitly, since these are of the same functional forms as Z' and Z 

 respectively — as noted in connection with equation (10). 



One graphical method of representing the dependence of Z on r and 

 a is by means of a network of equi-r and equi-a curves of Z in the 

 Z-plane; likewise the dependence of Z' on r and a', by means of the 

 equi-r and equi-a' curves of Z'. The analytic-geometric properties 

 of these curves, as deduced from equations (4) and (5), may be formu- 

 lated as follows, for any (real) values of a and ct' but for r restricted 

 to the range to 1 : 



(a) r fixed, a \aried: Z moves on the circle 



(A' - l/2\/r^')=+ P = 1/4(1 -r"), 



of radius l/2\/l-r- with center at Z = l/2\/l — it"- 



(b) (T fixed, r varied: Z moves on the curve 



{X^+ Y^-X^- P/(2(7- l)-=0. 



(c) r fixed, a' varied: Z' moves on the straight line 



X'=Vl-r\ 



