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What is the best way to impedance-match a simple, resonant loop antenna (intended for transmission) to a 50 ohm amplifier?

Assume that the antenna is:

  • circular, with a circumference equal to the design frequency
  • resonant at 2 GHz
  • constructed from copper magnet wire, side-fed by semi-rigid coax

I have read that when the perimeter matches the signal wavelength, the antenna impedance is entirely real. But what is it? Simply the resistance of the antenna wire? If that's the case, should I measure the resistance of the antenna wire, and then insert a series resistor to bring the total real impedance to 50 ohms?

Or must I use a stub tuner and a signal analyzer? Or something else?

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  • $\begingroup$ Why a loop antenna at 2 GHz? That's rather uncommon, and relatively small and hard to manufacture accurately. At these frequencies, we typically go for completely different kinds of antennas. $\endgroup$ – Marcus Müller Aug 23 '18 at 19:40
  • $\begingroup$ (hint: what's the actual value of the circumference in your first bullet point? If that loop is a perfect circle, what's it's diameter?) $\endgroup$ – Marcus Müller Aug 23 '18 at 19:42
  • $\begingroup$ (the skin effect isn't really much of a friend of high-turn-count coil antenna at microwave frequencies.) $\endgroup$ – Marcus Müller Aug 23 '18 at 19:49
  • $\begingroup$ @MarcusMüller I rounded up to keep the question general, yet still useful to my application. I'm building a quick-and-dirty antenna to drive transitions between ground state sodium atoms at 1.772 GHz. (So circ = 17 cm, diam = 5.39 cm.) The atoms will be located 4 cm out-of-plane from the center of the loop. The signal is generated with an old HP8657B and amplified to 44 dBm. The loop geometry is to allow optical access for laser beams. $\endgroup$ – higgy Aug 23 '18 at 20:41
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    $\begingroup$ Ahh nice, yeah, OK, for solid state physics scales of things, this definitely is a humongous antenna :D. But hint, telling us you're a physicist would have made us answer differently :D $\endgroup$ – Marcus Müller Aug 23 '18 at 20:58
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I have read that when the perimeter matches the signal wavelength, the antenna impedance is entirely real.

Note that that is the same as saying it is resonant.

But what is it? Simply the resistance of the antenna wire?

No. The (real) impedance of a resonant antenna consists of two components: the loss resistance you have already thought of, and the radiation resistance. The loss resistance accounts for the signal being turned into heat in the wire; the radiation resistance accounts for the signal actually being radiated.

In order to find out the radiation resistance, you must calculate, simulate, or measure it. In general, the impedance of a full-wavelength loop antenna is in the vicinity of 100 ohms (citation: assorted web sites that all agreed).

The internet generally seems to recommend a quarter-wave transformer for matching a loop — this is likely because 75 ohm coax is readily available and a fairly good approximation to the needed intermediate impedance (the geometric mean of the two other impedances).

If you need a better match than those approximations, then you will want to measure the impedance of the loop, and possibly construct a matching circuit on a custom PCB (using discrete components or a PCB transmission line).

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The precise impedance depends on the particular geometry, but 100 ohms is a ballpark estimate for a full-wave loop. Antenna modelling can generate a more precise estimate. Or the design can be prototyped and the impedance determined empirically.

Since the impedance will almost certainly be greater than 50 ohms, no amount of series resistance will match the antenna to 50 ohms. There are few applications where you'd want to do that anyhow: such a matching technique would waste a significant fraction of the transmitter power in simply making the resistor warm.

Thus, most RF impedance matching is performed with reactive components: inductors, capacitors, and transmission lines. There are infinite ways to combine such components to achieve a match, the best solution really depends on the requirements, like if it needs to be cheap to mass-manufacture or easy for a hobbyist to assemble, if it needs to be small, or wideband, etc.

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