798 BELL SYSTEM TECHNICAL JOURNAL 



impedance correction, since it normally means an improvement in the 

 match between generator and load impedances, should result in a 

 slight increase of transmission efficiency. 



Reciprocal Impedance Relatioyis at Terminals of a Reactive Network 



Our restriction to networks of pure reactances allows us to make use 

 of a principle by means of which the impedances measured at the two 

 ends of the network under certain terminal conditions can be recipro- 

 cally related to one another. The theorem will be given here since it is 

 of frequent application in further discussion. Referring to Fig. 1, let 

 us assurne that the impedance measured at terminals cd, with an im- 

 pedance Zi connected to terminals ab, is equal to Z2, as is shown in the 

 diagram. The theorem is concerned with the impedance Z looking into 

 terminals ab when Z2, the conjugate of Z2, is connected across cd. Let 

 us suppose that the generator e in Z2 produces a current i in Zi. Then, 

 by the usual principle of reciprocity, the generator e when inserted in 

 Zi will produce the current i in Z2. In the first case the power entering 



e^ - 



the network is obviously -r^ and the power flowing from it into Zi is 



4iV2 



e^R 

 i^Ri. In the second case these powers are 



(i?i + Ry + (Xi + xy 



and i^R2. Since the network is non-dissipative the power entering the 

 network equals the power leaving it in both cases. 



4i?2 



e^R 



= i^Ri 



= i'R, 



{R^ + Ry + (Xx + xy 



Upon dividing the two equations and simplifying we find : 



(R, - Ry + {X, + xy=o 



which can be true only if: 



R = RiandX = - Xi 



In other words, Z is the conjugate of Zi. We can state this result in 

 the following words : 



A network composed of pure reactatices will have a given impedance, 

 Z\ at one pair of terminals when an impedance Zi is connected to a second 

 set of terminals if, when the conjugate of Z\ is connected to the first pair of 

 terminals, the impedance measured at the second pair of terminals is the 

 conjugate of Zi. 



