Why exactly is a Hertzian dipole inefficient?

Question

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

Note that the formula (1100) is only valid for \$l\ll \lambda\$. This suggests that \$R_{\rm rad} \ll R\$ for most Hertzian dipole antennas: i.e., the radiated power is swamped by the ohmic losses. Thus, antennas whose lengths are much less than that of the emitted radiation tend to be extremely inefficient.

Of course, this is a well-known fact. But I don't see the reasoning for that.

Let's plug in some numbers (10 kHz, wavelength 300km; length of dipole \$l=300\mathrm{m}\$ (=100x less than wavelength).

\$ R_{\rm rad} = 789 \left(\frac{l}{\lambda}\right)^2 = 78.9m\Omega . \$

The text above does not clarify what exactly is meant by the swamp of ohmic losses. But on the back of some envelope, let's assume that the wire can't be longer than \$l\$ (by definition) but that also implies that it can't wider (otherwise it would get longer). So as an upper bound we have a metallic cube of length \$l\$. Resistivity of silver is 1e-8:

\$ R = 1\cdot 10^{-8} \frac{l}{l^2} = 33p\Omega . \$

Orders of magnitude smaller than the radiation resistance!

Even if I make the cross section 1000x smaller than the length, the ohmic losses are still just \$33\mu\Omega\$ ... orders of magnitude smaller than the radiation resistance.

I also take the Skin effect into account but it does not change the result significantly:

\$ R_{\rm skin} = \frac{\rho}{2\pi r \delta} = \frac{\rho}{2\pi r \sqrt{\frac{2\rho}{\mu 2\pi f}}} \approx 27\mu\Omega . \$

Answer 1

1. You haven't considered the loss in the matching network - practical inductors have a Q of only a few thousand, these would add significantly to the loss.

2. You haven't considered the skin effect - at 900 MHz only the outside few microns of the metal are effectively carrying any current.

The most efficient-for-their-size antennas are a bit like two hemispheres fed in the middle, so your idea is as good as it gets.

Edit to add: Since matching networks have limited Q, the unmatched Q of the antenna is of little practical interest.
The main problem with high Q antennas and matching networks is that the available bandwidth is reduced, until they become useless. This is where there has been some interesting research.

Look up the Chu-Harrington Limit - relating the maximum efficiency, bandwidth and dimensions of electrically small antennas. This paper has a graph.

I have in my head a more recent graph with the same limit line, but various newer antennas like the two-hemisphere and others, but I can't find it.