Small antennas: physical limitations. Encyclopedia of radio electronics and electrical engineering

Encyclopedia of radio electronics and electrical engineering / Antennas. Theory

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Antennas are considered electrically small if their dimensions do not exceed 10 ... 20% of the wavelength λ. These include a dipole shortened by capacitive loads at the ends and inductors located near the capacitive "hats" (Fig. 1), and an annular frame (Fig. 2). It is advisable to turn on the coils in the dipole exactly as shown in the figure, since the current in the vertical part is maximum and more evenly distributed, which ensures the maximum effective height of the dipole, which is practically equal to its geometric height hd = h (Hertzian dipole). The inclusion of one coil in the center is worse - the current to the ends of the dipole drops, and the effective height decreases. The effective height of the frame is hd = 2πSр/λ, where S is the area of ​​the frame.

Both the dipole and the frame are tuned to the operating frequency in resonance: the first - with coils, the second - with a capacitor included in the wire break. This provides compensation for their reactances, which is necessary according to the conditions of matching with the load (during reception) or with the generator (during transmission). Recall that, according to the reciprocity theorem, the properties of antennas are the same when transmitting and receiving.

An important parameter of antennas is the radiation resistance, for small antennas it is equal to RΣ = 80π2 (hd / λ) 2- It is on this resistance R = RΣ that the receiving antenna must be loaded so that it gives maximum power, and it is this resistance that the generator will "see" if it connect instead of R (see pictures). We see that the radiation resistance decreases sharply with a decrease in size, and, consequently, the effective height - in proportion to the square of h for the dipole and S for the frame. Difficulties arise in agreement. If we now take into account that the antenna efficiency η = RΣ/(RΣ + Rn), where Rn is the loss resistance, we can draw the following conclusion.

Conclusion 1. The smaller the antenna, the less ohmic losses should be in it. The resistance of the antenna conductors Rn must be reduced in proportion to the square of the length for the dipole and the square of the area for the loop. Small antennas made of thin wires cannot work effectively - "thick" conductors are needed, or better - volumetric bodies with a developed surface (skin effect!) And low surface resistance.

Let us assume that we have constructed such a "bulk" antenna conditionally in the form of a cylinder with radius r and height h, radiating through the side surface (Fig. 3). Even without considering what is inside this cylinder, that is, what is the design of the antenna, it is possible to draw the following important conclusion. All radiated power P is equal to the integral of its flux density (Poynting vector) P over any closed surface surrounding the antenna.

For simplicity, we replace the integration by multiplying П by the area of ​​the side surface Sside = 2πrh: P=П·Sside = EH·2Kπrh. Hence we get EH = P/2πrh. Assuming the radiated power to be constant, we see that a decrease in the size of the antenna (product rh) leads to an increase in the strength of both the electric E and magnetic H fields of the antenna. Which of them increases more strongly depends on the specific design of the antenna. In addition, taking into account the near field (quasi-static) can give even higher field strengths.

Conclusion 2. Reducing the size of the antenna leads to an increase in the field strength near it, according to the minimum estimate, the field strength is inversely proportional to the size of the antenna. Since the fields are generated by voltages and currents, overvoltages and overcurrents are inevitable in small antennas.

The above conclusions explain why, for example, a short dipole in the form of a volumetric bicone and a frame made of a wide copper tape are effective, but the same antennas made of thin wire are not. Elma already with an input power of 136 W, and the same electrically small antenna of the detector receiver develops (without load) a voltage of tens of volts.

Let us now consider the issue of the quality factor of the antenna Q, which determines its broadband 2Δf = f0/Q using the antenna shown in Fig. 1 as an example. 2. Since the dimensions of the antenna are small compared to the wavelength, almost all the inductance L is concentrated in the "extending" coils, and the capacitance C is between the "shortening" end disks. Just like with an oscillatory circuit, the quality factor of the antenna is equal to the ratio of the reactive capacitive or inductive resistance (they are equal at the resonant frequency) to the active one. The latter, in the absence of losses, is made up of the radiation resistance RΣ and equal to it, according to the matching condition, the output impedance of the transmitter or the input impedance of the receiver R. Thus, Q = Xc/XNUMXRΣ.

We find the capacitance using the formula for the capacitance of a flat capacitor: С = ε0S/h, Хс = 1/ωС = h/ωε0S. Expressing the angular frequency in terms of the wavelength ω = 2πс/λ and using the relations known from the Maxwell equations for the wave propagation velocity (the speed of light) c = 1/(μ0ε0)1/2 and the wave resistance of free space W = 1/(μ0ε0)1/ 2 = 120π, we get Хс = 60λh/S. Substituting this formula and the expression for the radiation resistance into the formula for the quality factor, we finally obtain Q = 3λ3/8π2Sh = λ3/26V. Here V = Sh is the volume occupied by the antenna. Thus, the quality factor of the antenna turned out to be inversely proportional to its volume.

But what about the case of a short linear vibrator, in which the capacitive "hats" at the ends (see Fig. 1) are replaced by vertical wire segments (Fig. 4)? After all, the volume of such a dipole is practically zero. However, there is a capacitance between the end segments, which tunes the antenna, together with the inductance L, into resonance.

The lines of force of the electric field associated with this "capacitor" are shown as dashed lines. It decreases very quickly with distance from the dipole, so we can talk about some effective volume in which this field is concentrated. It has a shape close to an ellipsoid of revolution (Fig. 4, thin solid lines). In fact, this is the volume of the near quasi-static field of the antenna. For a dipole, it is predominantly electrical, which is why it is called an electrical antenna. It is also possible to estimate the volume of the field of the wire frame. It is predominantly magnetic. For a frame, the inductive reactance is proportional to the first power of the diameter, and the radiation resistance is the fourth, as a result, the quality factor turns out to be proportional to the cube of the diameter. Now it is possible to formulate one more conclusion.

Conclusion 3. The quality factor of a small antenna is inversely proportional to the volume occupied by its near, quasi-static field. The quality factor cannot be reduced by varying the design of the antenna, since in any case, with a decrease in size, the active radiation resistance decreases very quickly with respect to the reactive one.

Let us make approximate estimates, considering the volume of the antenna equal to the cube of its linear dimensions. With antenna dimensions of the order of λ/3, the formula we derived gives Q = 1, i.e., such a (large) antenna can be broadband. But reducing the dimensions to λ/10, we get a quality factor of about 40 and a relative bandwidth of no more than 2,5%, and reducing the dimensions to λ/20 gives a quality factor of more than 300 and narrows the band to 0,3%.

If a small antenna has a wide bandwidth and a low quality factor, then this can only mean the following: either the antenna is not small and some of its parts that are clearly not included in the design (cable braid, support elements, etc.) radiate, or the antenna has a high loss resistance and its efficiency is low.

Low efficiency is not such a big obstacle for amateur radio communications. Assume that we have expanded the bandwidth of an antenna with dimensions λ/20 up to 10% (by a factor of 30), introducing losses and reducing the efficiency also by a factor of 30, i.e., to 3%. By connecting a hundred-watt transmitter and emitting a power of 3 W, it is quite possible to carry out even long-distance radio communications, which, perhaps, explains the rave reviews about the operation of small-sized antennas.

Author: V.Polyakov (RA3AAE)

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