312 BELL SYSTEM TECHNICAL JOURNAL 



and at their I'mal terminals the image impedance relations are functions 

 of ;;/ and tu' , namely, 



^ Wn{m., m') ^ R 



^ ~ R IVuim, m') 



^ [1 + aiU, + iV,)']<\ + Uk + iV, .^j. 

 [l + a'(t/, + jFA.)] ' ^" ^ 



where a = 1 — nr, and a' = I — m-m'". Since m and m' lie between 

 zero and unity, it follows that ^ a ^ a' < 1. 



When there is no dissipation in the network elements, Vk = and 

 all these image impedances are pure resistances in all transmitting 

 hands. Then the image impedance ratio y is there real and it can be 

 given a variety of characteristics depending upon the choice of parame- 

 ters a and a'. For the range Uk- = to — 1, 7 as a function of Ui, 

 may have no maximum or minimum, one maximum, or one maximum 

 and one minimum; at Uh = 0, y = I and at U,c = — 1, y = 0. 



The parameters corresponding to any such physical characteristic 

 can be determined from the values of y at two non-zero values of Uic, 



where now 



_ [1 + a^,]Vl+ U^- 

 ^~ [1 + a't/,] 



This, when rewTitten, yields the general linear equation in a and a' 



— 11a + va' = %v, (28) 



where 



« = - t/,Vl + Uu, 



V = — ylJk, 

 and 



IV = y — Vl + Uk. 



For generality, let the data be 



J = ji at {Uk)u 

 and 



y = y-i at (Uk)-!. 



Substitution of these values in (28) gives two simultaneous linear 

 equations in a and a' whose solution is 



_ ViW-y — V2IV1 

 U1V2 - llfVl ' 



and ^^^^ 



, UiW-2 — n-fci\ 

 a = • 



iliV2. — ll-lVl 



