For an antenna when used for transmittiing, what is the optimum radiation resistance?

Question

For a 100 % efficient half wave dipole antenna, the real part of the impedance is equal to the radiation resistance.

If the antenna were a dead short with zero impedance then the radiation resistance would be zero.

And at the other end of the scale an antenna with an infinite impedance (ie: no antenna) and so infinite radiation resistance also probably won't work that well.

So somewhere in between those two extremes there must be an optimal radiation resistance.

Assuming a matched transmission line and transmitter, and ignoring the transmission line impedance values available (eg: 50, 75, 300 ohms), and all other things equal, what is that optimum radiation resistance for an antenna ?

Answer 1

Any non-zero radiation resistance will do. What matters is not the radiation resistance, but the ratio of energy radiated to energy lost to other means, such as ohmic losses in the feedline, antenna, or soil.

If the antenna system is made of ideal, lossless components, then any nonzero radiation resistance is 100% efficient. While somewhat counterintuitive it's easy to prove: the antenna must be 100% efficient because by definition there is no loss, so by the law of conservation of energy 100% of the energy must go to radiation.

There is this equation:

$$ \text{efficiency} = {R_\text{radiation} \over R_\text{radiation} + R_\text{loss}} $$

which would suggest that higher radiation resistance means higher efficiency. But it is important when using this equation to pick a single point to measure radiation resistance and loss resistance, and normalize all values to that point.

Consider that a typical antenna system consists of numerous devices which transform impedances. They may be simple transformers, but also transmission lines and even antennas themselves (which are effectively transformers between the feedpoint impedance and the impedance of free space) do transformer impedances. So a resistance of a particular value may appear to have a different value when viewed from another perspective.

For example, a folded dipole has 4x the impedance of a similar not-folded dipole, and thus 4x the radiation efficiency if your definition for radiation resistance is to measure it at the feedpoint. (There are other definitions!) But that does not mean a folded dipole is 4x as efficient, because the ohmic losses in the antenna conductors are also 4x bigger. And if this folded dipole is fed with coax it will be a poor match, and the antenna system is subject to additional SWR loss as well.

Answer 2

No. What matters is if the radiation resistance (or do you mean feedpoint impedance?) of the antenna is matched to the feedline.

Answer 3

The antenna's radiation resistance (in a modal view, which is feedpoint-agnostic) is generally unrelated to the feedpoint impedance. They may agree in certain limited but common cases, but not equal in general situations.

That is because the radiation resistance is a relationship between the total power radiated in the form of a propagating wave in the far field and the average antenna current on the radiating element ($R = P / \bf{I}^{2}$ where $\bf{I}^2$ is a phasor current magnitude squared).

As such, even if the antenna element is made longer than 1/4WL monopole or 1/2WL dipole, the radiation resistance does not go up as rapidly as the real part of the feedpoint impedance (it's close to flat).

However, when the antenna element is shortened, the average antenna current increases and the radiation resistance drops rather rapidly (it drops like the real part of the feedpoint impedance). This leads to the antenna's internal loss as well as ground loss in the case of monopoles. If you care about antenna efficiency or performance, you would avoid this situation.

In case of simple monopole or dipole antennas, the antenna's efficiency or performance is more strongly influenced by the directivity (especially the takeoff angle in the vertical plane), effective length/aperture, the near field reaction with the ground, support structure, and the surrounding environment than the exact element lengths as long as they are not shorter than 1/4WL (monopole) or 1/2WL (dipole) as long as the conductor and ground losses are well controlled. It shouldn't be attributed to the radiation resistance.

"Radiation" resistance

In analyzing and optimizing the radiator performance, what matters is the overall current distribution, not the feedpoint impedance. So, the most useful way to look at the radiation resistance is the "modal view," which factors in the current distribution, rather than the "port view" that treats the antenna as a one-port network at the feedpoint (where the relation to the "radiation" is abstracted to invisible level), as has been done in this answer.

For example, when analyzing how capacity hats improve the antenna efficiency through enhanced current distribution (and extending the antenna's effective length), it is only appreciated in the modal view of the radiation resistance. When analyzing folded dipoles, again, the modal view would factor in the overall antenna operation, while the port view would abbreviate the matter to a level where you don't appreciate the true operation of the overall antenna.

That is my summary of the comment exchanges with tomnexus.

Incidentally, one big advantage of folded dipole configuration over plain dipole with fat radiators (to match the bandwidths) is that folded version concentrates the most reactive part of the near field between and near the radiators thereby making the antenna less susceptible to the environmental interactions, i.e. more predictable performance. Things like this are not fully appreciated by any version of radiation resistance, despite a significant practical advantage... another reason why radiation resistance should not be an optimization parameter.

Addendum: response to Tomnexus

You can read the rest in your copy of Kraus... this is from 2nd edition.