The Reciprocal Energy Theorem

HE 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.j. E , inserted in any branch, designated as No. 1, of a transducer,l produce a current /·/ in any other branch No. 2; correspondingly let an e.m.j. E>>" inserted in branch No. 2 produce a current I" in branch No. 1; then Ii'Ei = h'E2" and when = E->" 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 s 2 = r 2 + ix2, let the driving point impedance, as measured from terminals 1,1 be denoted by Zu = Rn + iXn; similarly with terminals 1,1 closed by an impedance Z = r + ixif let the driving point impedance, as measured from terminals 2,2 be denoted by Z 22 = R-22 + iXTM. Now with the terminals closed by the impedances and z», let an e.m.f.