To: Paul Goldsmith

From: Jon Hagen

Date: 7-13-01

Subject: More on the antenna theorem

Paul - It still seems to me you can get the antenna theorem without invoking the reciprocity theorem twice. And your first invocation seems somewhat indirect (that the normalized transmitting pattern is the same as the normalized receiving pattern). Let's see if I can defend this claim, but first let me write down what I think are the definitions and the theorem.

From a receiving standpoint, the concept of effective area is straightforward - the equivalent projected 'rain gauge" area such that area times an incident plane wave flux equals the power available at the antenna terminals. An antenna used for receiving has its greatest effective area in the boresight direction and smaller effective areas in other directions. A_eff is a function of theta and phi, and the normalized effective area, A_eff(theta,phi)/A_eff(0,0) is, by definition, the receiving pattern. Any losses in the antenna will reduce A_eff.

Since the effective area function totally describes an antenna, the antenna theorem must be expressable in terms of just A_eff. That expression is

<A_eff > LT or EQ lamba^2/4pi

where < > denotes averaging over 4 pi steradians and the EQ holds for a lossless antenna. The theorem shows that if A_eff is huge in any direction, it will have to be almost blind in other directions.

A more arbitrary characterization of a receiving antenna is its "antenna solid angle", Omega_a , which describes its beam size. Beam size is interest even if we are receiving a signal from a distant point source: we want to know how critical the pointing is and also how well the antenna suppresses signals from directions other than the pointing direction (the antenna's performance as a spatial filter). Since the antenna's directional characteristics are entirely specified by A_eff(theta,phi), beam solid angle must be a function of A_eff(theta,phi). The definition chosen for beam solid angle is

Omega_a = Integral [ receiving pattern] dOmega

where the receiving pattern is as defined above, A_eff(theta,phi)/A_eff(0,0). We see that an antenna with a sharp pattern will have a small Omega_a. We also note that this definition of

antenna solid angle is equivalent to

Omega_a = 4 pi <A_eff> / A_eff(0,0),

which allows us to express the antenna theorem in the mixed form:

A_eff Omega_a LT or EQ lamba^2.

Proof of the antenna theorem:

Let U(theta, phi) be the power radiated per unit solid angle when a power of 1 Watt enters the antenna terminals, i.e. U accounts for losses. (U is called usually called "gain"). We invoke the reciprocity theorem argument (transmitter/receiver swap) and conclude that

U(theta,phi)/A_eff(theta,phi) = universal constant

Analysis of any particular antenna, e.g. square aperture antenna, shows that the constant is

4pi/lambda^2.

Thus

U(theta,phi) = 4pi A_eff(theta,phi) / lambda^2.

for any theta and phi. (This seems to me the strongest version of the theorem).

Conservation of transmitter energy requires that <U> LT or EQ 1. It follows immediately that

<A_eff> LT EQ lambda^2/4pi. (antenna theorem)

Substituting the definition of Omega_a, we have

Omega_a A(0,0) LT EQ lambda^2 (theorem in alternate 'mixed' form).

QED, I think.

.........................

What about the LT EQ vs. simplyEQ? I'm looking at the text by Balanis. On page 63 he boxes the expression

A_em = lambda^2/4pi D_0

where A_em is the maximum effective aperature and D_0 is the directivity. Checking his definitions of these, A_e is power (W) delivered to the load divided by power density of the incident wave (W/m^2), so his A_e does allow a lossy antenna. But D_0 , the directivity, is calculated from just the radiated power. It seems to me he has been sloppy and that he should have written LT or EQ rather than "=".

But Slater, on page 264, writes <A_e> = lambda^2/4pi. Again, no LT, only EQ!?

Silver writes the same equation, quoting Slater. I'm perplexed to think that such illustrious authors would be sloppy. As an antenna gets lossier and lossier, the power at the load has to vanish. Yet, if this loss is contained in the transmission line, the directivity doesn't change.

What is the correct statement of the theorem?

(These authors seem to agree that U(theta,phi) = 4pi A_eff(theta,phi) / lambda^2 , where U and A allow for loss.)