# 9.5 Effective Area of a Receiving Antenna

So far, antennas have been treated as directional radiators of energy. Now, we examine the reverse process, namely how a receiving antenna extracts energy from an incident wave and delivers it to a load. The ability of an antenna to capture energy from an incident wave of power density S i (W/m 2) and convert it into an intercepted power P int (W) for delivery to a matched load is characterized by the effective area A e:

Other commonly used names for A e include effective aperture and receiving cross section. The antenna receiving process may be modeled in terms of a Thévenin equivalent circuit (Fig. 9-18) consisting of a voltage V˜oc in series with the antenna input impedance Z in. Here, V˜oc is the open-circuit voltage induced by the incident wave at the antenna terminals, and Z L is the impedance of the load connected to the antenna (representing a receiver or some other circuit). In general, both Z in and Z L are complex:

Z in=R rad+j X in,(9.58a)

# Figure 9-18 Receiving antenna represented by an equivalent circuit.

(a) Receiving antenna

(b) Equivalent circuit

Figure 9 18 Full Alternative Text

Z L=R L+j X L, (9.58b)

where R rad denotes the radiation resistance of the antenna (assuming R loss≪R rad). To maximize power transfer to the load, the load impedance must be chosen so that either Z L=Z in∗, or equivalently, R L=R rad and X L=−X in. In that case, the circuit reduces to a source V˜oc that is connected across a resistance equal to 2 R rad. Since V˜oc is a sinusoidal voltage phasor, the time-average power delivered to the load is

P L=1 2|I˜L|2 R rad=1 2|V˜oc|2 R rad 2 R rad=|V˜oc|2 8 R rad,(9.59)

where I˜L=V˜oc/(2 R rad) is the phasor current flowing through the circuit. Since the antenna is lossless, all of the intercepted power P int ends up in the load resistance R L. Hence,

P int=P L=|V˜oc|2 8 R rad.(9.60)

For an incident wave with electric field E˜i parallel to the antenna polarization direction, the power density carried by the wave is

S i=|E˜i|2 2 η 0=|E˜i|2 240 π.(9.61)

The ratio of the results provided by Eqs. (9.60) and (9.61) gives

A e=P int S i=30 π|V˜oc|2 R rad|E˜i|2.(9.62)

The open-circuit voltage V˜oc induced in the receiving antenna is due to the incident field E˜i, but the relation between them depends on the specific antenna under consideration. By way of illustration, let us consider the case of the short-dipole antenna of Section 9-1. Because the length l of the short dipole is small compared with λ, the current induced by the incident field is uniform across its length, and the open-circuit voltage is simply V˜oc=E˜i l. Noting that R rad=80 π 2(l/λ)2 for the short dipole (see Eq. (9.38)) and using V˜oc=E˜i l, Eq. (9.62) simplifies to

In Example 9-2 it was shown that for the Hertzian dipole the directivity D=1.5. In terms of D, Eq. (9.63) can be rewritten in the form