# Is an antenna always matched to free space impedance?

In this thesis the following model for an antenna is proposed:

It is seen as a 2 port network, in which only port 1 is accessible (port 2 is an imaginary connection between the antenna and the free space). Precisely:

• $$Z_0$$ is the feed line impedance.
• $$jX_a$$ is the antenna reactance.
• $$R_l$$ is the antenna parasitic resistance (which accounts for losses).
• $$Z_{fs}$$ is the free space impedance (377 Ohm)
• The transformer represents what an antenna really does: it transforms the free space impedance $$Z_{fs}$$ to a radiation resistance (which is the impedance seen at the primary of the transformer.

This description is quite clear. But then, the author says:

Since free space represents a matched termination, there would not be any reflection occurring ($$\Gamma_L = 0$$).

It does not seem too obvious for me.

As seen above, an antenna may be seen as a two - port network in which port 2 is closed on 377 Ohm. I'd say that usually there exists a mismatch on port 2, and so there is reflection at port 2. This doesn't mean there will be reflection at port 1 too, it depends if the antenna is able to match those 377 Ohm to the feed line impedance.

But I don't understand why there shouldn't be reflection at port 2. Maybe the author is considering a specific assumption he hasn't written, even because I've read somewhere that reflection along (not at the feeding port) some antennas exists and is a problem.

For instance, an infinite biconical antenna is perfectly matched to free space because it's like an infinite transmission line

A real biconical antenna is not matched to free space because it's truncated.

• Does it matter? Is it even possible with a passive 2 port device in general to know if the mismatch is at port 1 or port 2? Jan 14, 2021 at 18:37
• @ Phil Frost - W8II I'm not an expert but I think that it may be useful to design the geometry and the materials of the antenna. For instance, I've been told that a horn antenna, which as a gradually increasing transverse section, is better matched to free space than an abrupt truncation. So, I'm curious about how an antenna transforms the free space impedance and if there is or not reflection there. Jan 14, 2021 at 20:07

I don't think the "matching to free space" should be taken so literally. I've never seen an actual equivalent circuit in a textbook or used it to derive any property of an antenna.

Sure an antenna is the interface between the transmission line and free space. And they both have an impedance with units of ohms, but they're of a completely different nature.
On the transmission line, impedance Z=V/I, the ratio of the voltage and currents, while the impedance of a travelling wave, in free space or something else, is Z=E/B the ratio of the electric and magnetic field strengths of the TEM wave).

But the behaviour of an antenna is well described by solving the field equations near the antenna. Some antennas can be solved or approximated analytically, and then the solutions contain the permittivity and permeability of free space and the speed of light.

As for solving transmission line equations - you generally work from right to left, transforming the impedance, until the input impedance is found. So in this model you'd take 377 $$\Omega$$, then apply some transformer and some RLC network, and that would yield the input impedance of the antenna. There's no transmission line, so no need to talk about reflections, it's just impedances being transformed by circuit elements. (until you connect it to a piece of 50 $$\Omega$$ coax, then you can calculate the reflected power etc. if that's what you need).

Let's assume the answer to your question is yes: an antenna can be mismatched to free space. What would that look like?

Well, the feedline, let's say it's 50 ohms, is matched to the antenna's feedline port, so no reflection there. But at the mismatch to free space there'd be a mismatch, generating some reflection.

Is this any different than the antenna being mismatched to the feedline? I think it's not.

Consider the same thing in reverse: say you begin with an antenna that provides a good 50 ohm match to the feedline. Let's assume this antenna is matched to free space.

Now, say we fill the area around the antenna in PTFE foam or some other material with a dielectric constant significantly different from air out to several wavelengths away. Do you think this antenna will still present a good 50 ohm match to the feedline?

Probably not. So if one port is mismatched, then the antenna transforms that to a mismatch as seen from the other port. The mismatch isn't a property of one port or another, but rather is an error in the transformation function inside the antenna, which works in either direction.

I think what the author meant to convey is you might as well just assume the antenna is matched to free space, then calculate what the feedpoint impedance is. And of course that can be mismatched.

• If port 2 is mismatched, will port 1 be mismatched too? I've always thought it's not necessarily true. For instance, a lambda/4 transformer is mismatched with its load impedance, but will be matched at its input port. Jan 15, 2021 at 6:57
• @Kinka-Byo That's not true. The input port sees an impedance of $Z_0^2/Z_L$. So the input port is matched only for one $Z_L$. For any other load impedance, the input port is mismatched. Jan 15, 2021 at 15:10
• But isn't the quarter wave transformer mismatched at the position of ZL? Jan 15, 2021 at 16:56
• The quarter wave transformer has Z0, so it is mismatched to ZL. But the input port, won't be mismatched (if the quarter wave transformer is properly chosen). So, why cannot be the same for an antenna? Mismatch with free space, match at the input port Jan 15, 2021 at 22:07
• @Kinka-Byo Consider an example: say we have a 50 ohm source and a 200 ohm load. To match that, we use a quarter-wave length of 100 ohm transmission line between them. The source sees a matched load that looks like 50 ohms, and with an ideal transmission line S11=0, S21=1, S22=0, S12=1. This is "matched", but neither the source nor load impedances are equal to the Z0 of the quarter-wave transmission line. Jan 16, 2021 at 15:51

$$Z_{fs}$$ (space) is resonant at all frequencies due to $$\varepsilon_0$$ (farads/meter) and $$\mu_0$$ (henries/meter). Thus it will have no reflection. Free space is lossless and the signal only gets weaker because the energy spreads out over a larger and larger sphere. The /meter part causes space to develop a resonant velocity (speed of light). But all frequencies resonate in space with no reflection. The antenna matches space with the transmission line.

• Hello Lutz, and welcome to ham.stackexchange.com! Sep 27, 2021 at 16:25

Matching an antenna to free space can be read as "radiation efficiency".

Dipoles occur in nature; so a dipole becomes the most common element of most antennae, for that reason. A resonant dipole ... even better.

And half wavelength dipoles still better, because electrically they are well behaved, ... meaning their radiation efficiency is about as good as you can get. A dose of physics [that part about a current in a wire produces a magnetic field around that wire, applies.]