Q factor and efficiency of the loop antenna. Encyclopedia of radio electronics and electrical engineering

Encyclopedia of radio electronics and electrical engineering / VHF antennas

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Small-sized loop antennas have been of interest to radio amateurs for many years. Recently, commercially available antennas have appeared that are remotely tuned using motor-driven KPIs.

It is well known that the quality factor Q of a small (compared to the wavelength X) frame is high, and therefore the bandwidth 2Δf \u0d f 0 / Q is no more than a few percent of its resonant frequency f XNUMX. There are either legends about the efficiency (efficiency) of the frames, or completely inaccurate data.

In amateur radio conditions, it is easy to make a loop antenna, tune it to the desired frequency f 0 and match it with the feeder. It is also easy to determine the bandwidth 2Δf, at least by increasing the modulus of the input resistance by 1,4 times with a detuning by Δf. The SWR in the feeder then changes from unity at frequency f 0 to about 2,6 at frequencies f 0 ± Δf. It turns out that these data are quite enough to evaluate the efficiency of the constructed loop antenna, the dimensions of which, of course, are also known. Let's derive some simple formulas together with the author that will allow us to evaluate the efficiency. If someone does not like mathematics, he can simply look at the end of the article with the conclusions and results. I hope they don't seem complicated to him (hi).

So, we will talk about a "round" frame with a perimeter p = πD, noticeably smaller than λ/2 (Fig. 1). The inductance of the loop is compensated by the capacitance of the tuning capacitor C, thus, the antenna is tuned into resonance, and its resistance becomes purely active and equal to R∑ + Rn, where R∑ is the radiation resistance; Rn is the loss resistance. In this case, a ring current of maximum amplitude is established in the frame with an almost uniform distribution along the perimeter. Coordination with the cable is done in different ways: a communication loop, a ferrite transformer or a gamma loop, as in fig. 1.

Let us first give the formulas known from the theory of antennas that are useful in calculating the loop antenna. Its radiation resistance R∑ = 20π2р4/λ4, as we can see, is quite small and rapidly decreases with decreasing perimeter. They also try to make the loss resistance small, since the efficiency = R∑/(R∑ + Rn).

The quality factor of the frame, as well as a conventional oscillatory circuit, is equal to the ratio of the inductive resistance at the resonant frequency X = 2πf 0L to the active one: Q = X/2(R∑ + Rn). The two in the denominator is introduced to take into account the output impedance of the transmitter or the input impedance of the receiver transformed to the frame, equal (by matching conditions) to the active impedance of the antenna. For a lossless antenna (Rn = O, efficiency = 100%), the quality factor is a finite value, since useful radiation losses remain: Q0 = X / 2R∑. It is also easy to show that efficiency = Q/Q0.

To calculate the loop inductance, quite a lot of formulas have been proposed in the literature that differ slightly in numerical coefficients (there is no absolutely exact formula, since it is difficult to take into account small effects: the difference in shape from a circle, the final diameter of the wire, the current distribution over its surface, the skin effect, etc.). d.). The author prefers to use the simplest and most accurate formula - L = μ0R ln(R/r), where μ0=4π10-7 H/m is the magnetic constant; R = D/2 and r = d/2 are the radii of the frame and wire, respectively. All dimensions here are in the SI system of units. We see that the inductance is directly proportional to the loop diameter D multiplied by the shape factor β = ln(D/d). Its values ​​are shown on the graph (Fig. 2)

We calculate the inductive reactance X = 2πf0L = πf0Lμ0Dβ and move from frequency to wavelength, taking into account that f0 = c/λ. c = 1√μ0/e0(speed of light) and √μ0/e0 = 120π (wave impedance of free space):

It remains to find the quality factor:

As for other small antennas (see the author's previous articles on this topic), the quality factor turned out to be inversely proportional to the cube of the linear dimensions or the volume of the near field of the antenna. Let's simplify the formula: since π ≈ 3 with an error of no more than 5%, we can finally write down:

Q0 = β(λ/p)3.

This should be the quality factor of the loop antenna with efficiency = 100%. If the measured value of Q is less (and it theoretically cannot be more), then efficiency = Q / Q0. Now, radio amateurs will be able, by calculating the required quality factor Q0 from the known perimeter of the frame and measuring the real quality factor Q, to determine the efficiency of their antenna.

Author: Vladimir Polyakov (RA3AAE), Moscow

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