The Reciprocal Energy Theorem 



By JOHN R. CARSON 



This paper gives a simple theorem determining relative transmission 

 efficiencies in a two-way transducer, and showing that the conditions for 

 equal efficiencies of transmission in the two directions are simply those for 

 maximum output and maximum reception of energy. The theorem is then 

 applied to radio communication and a second theorem stated and proved by 

 which the ratio of the transmitting efficiences of any two antenna systems is 

 expressed in terms of their receiving efficiences. The paper closes with a 

 mathematical note on a generalization of Rayleigh's Reciprocal Theorem. 



THE Reciprocal Theorem, originally enunciated by Rayleigh, which 

 has proved so useful to communication engineers, may be stated, 

 with sufficient generality for engineering purposes, as follows: 



Let an e.m.f. E] , inserted in any branch, designated as No. 1, of a 

 transducer,^ produce a current I2 in any other branch No. 2; correspond- 

 ingly let an e.m.f. Ei" inserted in branch No. 2 produce a current I\' 

 in branch No. 1; then 



I\ E\ = 1-2 El 



■'1 



and when £/ = Eo" the currents in the two branches are equal. 



The engineer, however, is primarily interested in energy rather than 

 current relations, whereas the theorem says nothing explicitly regarding 

 energy relations and relative efficiencies in two-way transmission. 

 It is, however, a simple matter to deduce from it quite general and 

 useful formulas relating to relative transmission efficiencies. In the 

 present paper there will be formulated and proved a reciprocal energy 

 theorem for the general transducer, after which it will be applied to the 

 question of antenna transmission efficiency in radio communication. 



Consider a transducer having two sets of accessible terminals 1,1 



and 2,2. With terminals 2,2 closed by an impedance S2 = ^2 + ixt, 



let the driving point impedance, as measured from terminals 1,1 be 



denoted by Zn = Rn + iXn; similarly with terminals 1,1 closed by 



an impedance Zi = r^ -\- ixi, let the driving point impedance, as 



measured from terminals 2,2 be denoted by Z22 = -R22 + iX22. Now 



with the terminals closed by the impedances z-i and 22, let an e.m.f. 



£1 be inserted in series with the terminal impedance Zi; then the 



current In, delivered to the transducer at the sending terminals 1,1 is 



* A transducer is defined as a complete transmission system which may or may 

 not include a radio link, which has two accessible branches, either of which may act 

 as the transmitting branch while the other acts as the receiving branch. These 

 branches may be designated as operating branches. 



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