A rigorous derivation of the relation between the effective area and the directive gain of 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 effective area-directive gain relation is derived by constructing an equivalent two-port network involving the transfer impedance. Next, by a use of the Thevenin theorem, the received power is related with the open-circuit voltage, which in turn is proportional to the transfer impedance. From field analysis, the retrieved power is also related with 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.