# Is a resonant antenna inherently more efficient because it couples to free space more efficiently?

I saw an interesting assertion recently, that resonant antennas are more efficient because they provide a better impedance match to free space at the resonant frequency. I believe that to be untrue, and that resonant antennas are just easier to match to the output impedance of a transmitter, which is almost always non-reactive. The consensus opinion of this group, as expressed by the votes on answers to two related questions, is clearly that generally resonant antennas are not more efficient.

What do I mean by "efficient", you ask? I suppose I mean efficient in terms of receiving or transmitting power. I'm talking about just the antenna, and not about any matching network or feed line that might be required to connect a real antenna to a radio.

But maybe I'm wrong, or maybe I'm oversimplifying. (There is a lot more that I don't know than I do know, and oversimplifying is how we humans make sense of the universe.) I investigated the Friis Transmission Formula, which I learned about thanks to this site, but it's no help. It introduces the idea that an antenna has an "effective aperture", which is unrelated to its physical size, but the effective aperture is a theoretical construct that is just a stand-in for the gain of the antenna. The formula isn't helpful here because the equation says nothing about gain, which is what this question is about.

So what about the assertion that the impedance of free space has something to do with the gain of an antenna? "Free space", i.e. vacuum, has an impedance, about 377 Ω, that relates the strength of the electric-field component to the strength of the magnetic-field component of an electromagnetic wave passing through the vacuum. When the electromagnetic wave encounters a conductive object then voltages and currents are induced in the object, which is how radio waves are received. I doubt that how close the impedance of the object is to (377 + 0j) Ω has anything to do with the efficiency of the reception, but I don't have enough Calculus or Physics to pick apart and understand everything about Maxwell's equations, so again I could be wrong.

So, is a resonant antenna inherently more efficient because it couples to free space more efficiently?

• Going out on a limb here... I think this is an identity question. If 1. efficiency here is defined as maximal power xfer, and 2. maximal power xfer from circuit to circuit only happens at matched impedances, and 3. resonant antenna is defined as one having no reactive component with respect to free space at a given frequency, then isn't a resonant antenna "most efficient" by definition? Apr 27 at 17:24
• Can you define "free space coupling efficiency"? Apr 27 at 19:13
• I think the idea of one antenna coupling to free space more efficiently than another is bunk, @PhilFrost-W8II, which is why I'm asking the question. I'm asking if a resonant antenna has an inherently higher gain, or effective aperture, than a non-resonant antenna, just because it's resonant. Apr 27 at 20:14
• Is that what you're asking? Because it's not what's in the title of the question. It seems to me the crux of your inquiry is defining what it means to "couple to free space more efficiently". And it could be that some people make that claim without ever defining what it means. Is that the answer you're looking for? Apr 27 at 21:30
• Thanks everyone for the answers and comments. I need some time to do more research. I'll come back to this question. Apr 28 at 19:25

You could view an antenna as a two-port device. One port is the feedpoint, and the other port is "free space". The antenna's job is to transform the free space impedance of 377 ohms to the specified feedpoint impedance such as 50 ohms.

Ideally we want an antenna with scattering parameters like so:

$$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$

In other words, all the power coming in from free space makes it to the feedpoint, and all the power coming from the feedpoint radiates to free space.

If port 1 is the feedpoint and port 2 is free space, then we would say the antenna is a "good match" when $$S_{11} = 0$$, meaning, no reflected power.

Likewise, we could say the antenna is a "good match to free space" when $$S_{22} = 0$$, since port 2 is free space.

So the question then becomes: does the antenna being non-resonant prevent $$S_{22}$$ from being zero?

And the answer is: no. If you can provide a matched termination at the feedpoint then the free space port will also be matched.

How do we determine if an antenna is resonant? We connect port 2 to free space (no choice), and then we measure the impedance looking into port 1. If that impedance has zero reactance, it's resonant. Effectively we are discovering the impedance-transforming properties of the antenna by connecting one port to a matched and known load (free space) and seeing what impedance appears at the other port (the feedpoint).

Assume for now we find that feedpoint impedance to be (50+0j) ohms. We know that if we want to capture the maximum power from free space we should put a 50 ohm resistor across the feedpoint. The antenna, being an impedance transforming network, makes this 50 ohm resistor look like 377 ohms from free space (looking in port 2).

If we instead put a 25 ohm resistor there, looking in port 2 we won't see 377 ohms. Since this doesn't match the impedance of free space it means the antenna will scatter some power rather than absorbing it.

If the antenna isn't resonant, this just means we need some reactive component at the feedpoint in addition to a resistor to achieve a complex conjugate match. So if we measure the feedpoint impedance to be (30+15j) ohms, this means simply we need to place a (30-15j) ohm impedance on the feedpoint if we want the other port to look like 377 ohms.

The problem in practice is real capacitors and inductors which we would use to come up with that reactance at the feedpoint have loss. That means some of the power will be lost in those components before it can make it to the receiver, reducing overall efficiency. Or we can choose to omit the reactive components, in which case some of the power from free space will be scattered by the antenna rather than transferred to the feedpoint.

• Hi Phil, isn't the function of a conjugate match to bring the antenna element to resonance? The analogy I have in my head is like adding weights to the prongs of a tuning fork to achieve desired resonant frequency. Apr 28 at 3:15
• @webmarc I suppose if you consider the antenna and the conjugate match together, you could consider them to be a resonant system. It's merely a question of where you consider "the antenna" to stop, and "everything else" to begin. In some cases it's pretty clear, in others, maybe not so much. Ultimately I'm not sure it matters much in the general case: the behavior is the same no matter how we name the things. Apr 28 at 4:03
• @hotpaw2 The phase of each element is important if you want an antenna with directivity. Apr 28 at 12:27
• @hotpaw2 You asked why the length matters in a Yagi. It matters because if you change the length, you change the phase of the reflection. I would certainly hope a Yagi would be designed to minimize losses, and this is why the reflectors and directors don't have little 72 ohm resistors in the middle. Apr 28 at 18:04
• @hotpaw2 Because not just any spacing will work equally well. This is why you wont find a 20 element 2 meter Yagi with a boom that's 10 mm or 1 km long. Apr 28 at 19:17

Not really answering the question but requesting more clarity in terms.

From the IEEE definition of standard terms for antennas: I redraw the diagram of power flow in an antenna, (ignoring the polarisation mismatch branch)

Where:

$$P_A$$ = power available from the generator
$$P_M$$ = power to matched transmission line
$$P_O$$ = power accepted by antenna
$$P_R$$ = power radiated by antenna
$$I$$ = radiation intensity

and

$$M_1$$ = mismatch between generator and line
$$M_2$$ = mismatch between line and antenna
$$G_R$$ = realised gain
$$G$$ = gain
$$D$$ = directivity
$$\eta$$ = radiation efficiency

The above "loss" terms on the straight line each have a physical explanation:
Directivity is only affected by the geometry of the antenna; how it concentrates power.
Radiation efficiency is only affected by losses in the metal and the dielectric.

The first part of the question is really about what type of antenna has the highest $$\eta$$ given real-world materials. I can think of a few but would have to simulate to be sure. A dipole is a good start. A travelling wave antenna like an (unterminated) rhombic or very long dipole might be better, they have smaller standing waves.

The second part of the question, "why", is harder to answer, because it might not be possible to reduce the physics of the antenna to a simple explanation. It will certainly involve $$Z_0$$ of free space but I don't think there will be a direct relationship between that and antenna impedance.

Another example of an antenna without a connected feed point is a broadcast AM radio enhancer (MF band). A multi-turn loop of wire with a capacitor to tune it can be placed around or near, but electrically disconnected from a small AM pocket radio. The received signal will become stronger when the disconnected loop antenna circuit is tuned to (or very near) the receiver frequency.

Here's an example of one, the Eton AN-200 Tunable Passive AM Radio Antenna Loop: https://www.hamradio.com/detail.cfm?pid=H0-015461 , but there are construction articles on how to build one DIY with just wire and a variable capacitor, similar to making a DIY crystal radio.

Effective coupling can result in either higher gain (into some feed point impedance) or to higher heating losses (due to the antenna's elements own real impedance == resistance). Either way, energy is being removed from the RF field (to maintain conservation of energy).

I hypothesize (needs an NEC model to confirm the current maxima?) that a resistor of some very low value in the middle of a tuned Yagi reflector or director element (close to the same length as the driven element) will get warmer than one in the middle of an antenna element much farther (especially much shorter) from any near resonant lengths. Thus that element must be more efficiently coupled to the RF field to extract that energy, and turn it into heating.

Whether any energy extracted from the RF field can be fed somewhere else is a separate issue.