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http://farside.ph.utexas.edu/teaching/em/lectures/node94.html states:

So, in the optimum case half of the power absorbed by the antenna is immediately re-radiated. Clearly, an antenna which is receiving electromagnetic radiation is also emitting it.

Suppose I have an ideal isotropic radiator radiating $P_x$ into empty space. Now every closed surface integral over the Poynting flux enclosing the radiator should give me the $P_x$. Take radii $r_1 < r_2 < r_3$, all much larger than the wavelength.

Now I place an ideal absorber (a material with wave impedance of $Z_0=377\Omega$ converting the incoming radiation into heat??) at $r_2$. The closed surface integral at $r_3$ should now give less than $P_x$ (because a fraction of the energy was absorbed inside). But to my understanding, the closed surface integral at $r_1$ should still be $P_x$.

Neglecting heat radiation, can anything at $r_1$ detect that radiation was absorbed at $r_2$?

Now I replace this ideal absorber with an ideal antenna. According to the text above, this antenna would re-radiate half of this power in most ideal conditions.

If I place another antenna at $r_1$, this antenna should be able to receive $\frac{P_x}{L_1} + \frac{P_x}{2 L_2}\frac{1}{L_2-L_1}$, where $L_n$ denotes the ideal free space path loss at radius $r_n$. The first term is from the ideal radiator and the second one the re-radiated power from the antenna at $r_2$. This implies that the mere presence of a receiving antenna with a perfect match can indeed be detected.

What is the difference between an ideal antenna and an ideal absorber? Why does an antenna re-radiate half of its power? Why can there be an ideal absorber but we cannot use it to detect signals? Why are we instead relying on something that throws away half of the power?

My vague explanation is that the electric field in the wave excites electrons in the conductor which accelerate. This acceleration not only generates the current/voltage (which is detected by the circuit) but also generates electromagnetic radiation. However, with a black-body, it is certainly possible to have something that absorbes 100% of radiation. (I have to note that the most efficient solar cell 47.1%, i.e. nearly half. Is this the same fundamental limit?)

From a wave propagation perspective, an electromagnetic wave can only be reflected if there is a change of impedance. If we assume the load perfectly matched with the antenna, where is this impedance change that causes half of the wave to be reflected? Is this somehow related with a change from $Z_0=377\Omega$ to the radiation resistance of the antenna? (mathematically this could make sense for a Hertz dipole where the radiation resistance represents a short as compared to $Z_0$ but not so much for a walf-wavelength dipole).

Similarly, assume the antenna is not terminated (open circuit). In this case, only 50% of the incoming power would be reflected by the open circuit?

PS: This question is inspired by https://electronics.stackexchange.com/questions/187681/can-a-radio-transmitter-somehow-detect-the-number-of-receivers-in-its-area, which leaves the actual answer controversial/open.

Any elaborate explanation is much appreciated.

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  • $\begingroup$ I don't know much about this, but I note that special cases have been worked out in atomic physics whereby an atom in a cavity manages exactly to absorb a photon that has been prepared in a special state. I wonder if some similar trick would apply to classical e.m. It would involve impedance matching among other things. $\endgroup$ – Andrew Steane Aug 21 at 17:26
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I will try to summarize the classic results from 1, 2, 3. In short, a receiving antenna does not scatter as much power as it absorbs except in a few very special cases such as a very short dipole or a small loop. For these small dipoles, when the incident wave induces the same current distribution as would be seen when the antenna is used and driven as a transmitting one, then the scattered field energy is the same as the absorbed energy when used as a receiver.

In general though the scattered and absorbed fields have no such relationship. To see this note that a receiver antenna cannot just be analyzed as an antenna by itself. In fact, its behavior depends both on its orientation and distance to the transmitter antenna. These can be described by a reciprocal 2-pole network and its corresponding $2\times2$ impedance (admittance) matrix

$$\mathbf Z =\begin{bmatrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{bmatrix}$$

that can be represented as a "T" (or a "$\Pi$") network. Assume that the transmitting generator has open-circuit $emf$ $V_g$ with internal impedance $Z_g$ and is connected via a transmission line to the antenna ports $2$, while the load impedance is $Z_L$ and it is attached to port $1$.

The $Z_{12}=Z_{21}$ transfer impedance represents the coupling between the antennas through air and its magnitude is proportional to the squared reciprocal distance between the transmitter and receiver antennas and, in practice, it is many orders smaller than either $|Z_{11}|$ or $|Z_{22}|$. The figures below taken from [3] shows the impedance matrix representation and, hence, the Thevenin equivalent. (While both can be used if instead a $T$ we would use a $\Pi$ network then the admittance matrix representation would be more convenient to get the Norton equivalent.)

In the Thevenin equivalent we have $$Z_T = Z_{11}-\frac{Z_{12}^2}{Z_{22}+Z_g}\\ V_{oc}=\frac{Z_{12}}{Z_{22}+Z_g} V_g $$

The Norton equivalent short circuit current is $I_{sc}=\frac {V_{oc}}{Z_g}$

The received absorbed and network dissipated powers are $$P_L = \frac{1}{2} \left | \frac{V_{oc}}{Z_L+Z_T} \right |^2 \Re \{Z_L\} \\ P_D ^t = \frac{1}{2} \left \lvert\frac{V_{oc}}{Z_L+Z_T}\right\rvert ^2 \Re \{Z_T\} $$

Interestingly, if we use the Norton equivalent we get the dissipated network power as $$P_D^n = P_D^t \left\lvert \frac{Z_L}{Z_T}\right \rvert^2= \frac{1}{2} \left\lvert\frac{V_{oc}}{Z_L+Z_T}\right\rvert ^2 \Re \{Z_T\} \left \lvert \frac{Z_L}{Z_T} \right\rvert^2$$

This result immediately shows that one cannot and should not interpret $P_D$ as scattered power when we use the Thevenin or Norton equivalents to calculate the dissipated power inside the network because they can be *different*. Instead, one may use these equivalent circuits to calculate the power dissipated in the load *only*.

For example, if the receive antenna is open circuited then there is no absorbed power nor internal dissipation, $P_D=0$, and get the completely nonsense result that the antenna does not even scatter. On the other hand, in the open circuit case the Norton equivalent shows a complete reflection from the antenna. When the load of the receive antenna is matched to the driving point impedance then $Z_L = Z_T^{*}$ and we do have $P_D^n=P_D^t$ and it could interpreted for a small dipole as the scattered power, see Collin for details.

Transmit-receive antenna pair represented as a 2-pole network

Thevenin equivalent

Another approach is that of Love who instead of using a Thevenin (or Norton) equivalent circuit that uses a constant voltage (or current) source synthesizes a constant power source representing the incident power wave directly. This is shown below

constant power source - the "Love" equivalent

For aperture antennas Love shows that the voltage and current sources of his equivalent circuit depend as in below where $\eta$ is the aperture efficiency. His analysis concentrating on aperture antennas, while not generally valid for arbitrary antennas, removes the apparent contradictions and is in agreement with normal physical (engineering) intuition.

Love equivalent for aperture antennas

1 Silver: Microwave Antenna Theory and Design, chapter 2, sec 10,11 and 12

2 Love, “Equivalent Circuit for Aperture Antennas,” Electronics Letters, 23, 13, June 1987, pp. 708-710.

3: Collin: Limitations of the Thevenin and Norton Equivalent Circuits for a Receiving Antenna, IEEE Antennas and Propagation Magazine, Vol. 45, No. 2, April 2003

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