THE RECIPROCAL ENERGY THEOREM 



327 



rj 

 r 



2 \ / JV22 



i/Un 



Ri2 + r^X Zu + zi 



+ n 



^22 I 22 



Pu" 



22 



i? 



22 



Ru 



Zn + 2i 

 Z22 + Z2 



P22 



(10) 

 (11) 



From (10) it follows that for equal total generated powers, the relative 

 transmission efficiency in the two directions is given by 



12 



P21 



R22 + ^2 

 Ru + r, 



Zn + 



Z22 + S2 



(12) 



while on the basis of equal powers delivered to the transducer, the 

 relative transmission efficiency is, by (11) 



V = 



12 



P21 





R22 

 Ru 



Zu + Zi 

 Z22 + Z2 



(13) 



Now in correctly designed communication transmission systems, 

 the terminal impedances are so proportioned with reference to the 

 characteristics of the transducer itself as to secure maximum output 

 and maximum transfer of power from generator to load; the required 

 condition is that the terminal impedances 2) and 22 be the 'conjugate 

 image impedances' of the transducer; analytically stated 



Zi = Ru — iX 



u 



and 



Z2 = R22 — iX 



22- 



Introducing these relations into (12) and (13), we have 



^^ = r/ = 1 (14) 



and the relative transmission efficiencies are the same in the two 

 directions. We thus have the following propositions: — 



If a transducer is terminated in its conjugate image impedances — the 

 condition for maximum output and maximum transfer of power — the 

 efficiency of transmission is the same in the two directions. 



We shall now apply the preceding to the derivation of a simple 

 formula which enables us to determine the relative transmission 

 efficiencies of any two long wave radio antennas.- 



Consider any antenna, designated as No. 1, and let it be acting as 



* As pointed out in the paper on "Reciprocal Tlieorems in Radio Transmission " 

 Proc. I. R. E., the Reciprocal Theorem does not hold rigorously in radio transmission 

 if the earth's magnetic field plays an appreciable part in the transmission phenomena. 

 Consequently the formula and proposition which follow apply rigorously only to' 

 long wave transmission; they are probably, however, approximately correct for short 

 wave transmission except in the neighborhood of the critical wave-length 214 meters. 

 See a paper by Nichols and Shelling, "Propagation of Electric Waves over the Earth '' 

 B. S. T. J., April 1925. 



