A rigorous derivation of the relation between the effective area and the directive gain and its extension to lossy antennas

It is well known that the effective area of a receiving antenna is linearly related to the directive gain of the same antenna in the transmitting mode. Conventionally, this relation is derived by constructing a two-port network involving the transfer impedance between a pair of antennas. Next, by use of the Thevenin theorem, the received power is related with the open-circuit voltage, which in turn is proportional to the transfer impedance. Alternatively, from field analysis, the received power is related to the effective area and the directive gain. Then, the reciprocity theorem is used to equate the transfer impedances between a pair of antennas. Based on these, the area-directivity relation is derived. However, it is indicated that the circuit model with the open-circuit voltage is not rigorous. In this investigation, the reciprocity theorem is used in a different way to directly derive the power absorbed at the load in terms of the radiated electric field in the transmitting mode. Thereby, the area-directivity relation is derived in a straightforward and rigorous manner. In the derivation of the relation, it is seen that the current in the terminal gap should be uniform. Thus, one limitation of this well-known relation is found. The effect of ohmic loss due to finite conductivity of the antenna material can be taken into consideration in the derivation. Thus, the area-directivity relation is extended such that it is applicable to lossy antennas.