3 Properties of small antennas

3.1Performance of small antennas

Small antennas exhibit some specific characteristics that differ from antennas of a size comparable to the wavelength. The input impedance of a small electric (magnetic) dipole antenna is highly capacitive (inductive), while its resistive component is very small. Difficulty in the perfect matching to the load of 50 ohms is mainly due to this impedance characteristic.

The radiation pattern of a small antenna, as its size becomes smaller, tends to approach that of the classical elemental vertical dipole which is omnidirectional in the horizontal plane, with a figure-of-eight pattern in the vertical plane, and with a directivity approaching 1.5.

Small antennas, particularly ESAs, need specific design techniques.

3.1.1Input impedance

A small dipole antenna of radius d and the half length a (Figure 3.1), when ka is less than 0.5 (k: 2π /λ), exhibits the input impedance Za = Ra + j Xa as follows [1];

where = ln (2a/d).

Numerical values are illustrated in Figure 3.2. This is an excellent approximation of the correct value for βa ≤ 0.2 and quite a good approximation for βa ≤ 0.5. The figure shows loss-free cases, where k = β with α = 0 in k = α + jβ.

Meanwhile, the radiation resistance Ra of a small circular loop antenna of radius a and wire radius d (loop area: A = a2π ) for βa ≤ 1 (Figure 3.3) [2][3] is

The reactance X can be found by calculating ωL, where L, the inductance of a small circular loop, is given by [3]

Figure 3.4(a) gives Ra with respect to βa, and Figure 3.4(b) depicts L against a/d.

2d

2a

βa

Figure 3.2 Impedance characteristics of a dipole [1].

2d

a

Figure 3.3 A small circular loop antenna [1]–[3].

Figure 3.4 Impedance characteristics of a loop antenna; (a) resistance component and

(b) inductance.

I(ω)

Ir

Ir

√2

Figure 3.7 Frequency characteristics of a LCR circuit near resonance.

given set of specifications, typically based on the impedance characteristics. In small antennas, either the antenna input impedance or the power spectra can be taken as the parameter to specify the bandwidth. The bandwidth so defined is the frequency difference f between the highest frequency fh and lowest frequency fl, both specified at a certain value of parameters such as VSWR S or the return loss RL, based on the input impedance Za, and by some level of power reduced from the maximum. These parameters, S and

RL, are given by

where (the reflection coefficient) = (Za − ZL )/(Za + ZL ).

Here ZL is the load impedance.

The load impedance ZL is usually taken to be a pure resistance of 50 ohms. Usually S = 2 is taken, which corresponds to RL ≈ –10 dB. for the impedance at the half-power point is 0.707 and S for this is 5.828. The half-power point BW1/2 is usually used in circuitry and not generally used in the antenna case.

For convenience, the relative bandwidth RBW, which is a ratio of BW to the center frequency f0 = (fh – fl)/2 or the resonance frequency fr, is generally used, instead of BW. Sometimes, a ratio of fh/fl is used as a measure of wideband capability.

Antenna Q is defined from the physical implication as

where Preact denotes the reactive power stored in the antenna structure and Prad is the radiated power from the antenna.

Originally, Q was used as a circuit parameter showing a measure of selectivity of a tuning circuit, generally when the BW is relatively narrow. Taking an LCR series circuit as an example, the impedance Z = R + j(ωL – 1/ωC). When a voltage V is applied to the circuit,

current I flows in the circuit, that is, I = V/|Z| = V/ R2 + (ωL − 1/ωC)2. The current I(ω) with respect to frequency ω is depicted in Figure 3.7. At resonance ω = ωr, the current Ir = Vr/R, because ωL = 1/ωC. Taking a ratio of I/Ir = G in dB, G is given by

16 Properties of small antennas

√

At the half-power point, I /Ir = 1/ 2, and G = –3 dB. The frequency bandwidth ω for the half-power point is (ωh – ωl), where ωh and ωl, respectively, are the higher and lower half-power point frequencies (Figure 3.7), and given by

This f is used when fh and fl respectively, are not far removed from the resonance frequency fr.

At the resonance, voltage Vc of the capacitance and VL of the inductance, respectively, are

Since at the resonance ωr L = 1/ωr C, dividing this by R will give Q by the definition.

This provides the relation of a RBW to an inverse of Q, when the BW is not very wide. In the antenna case, this is applicable only when Q is much greater than unity.

The term Q was originated to mean “quality” of the circuit component or the selectivity of the circuit and is abbreviated as “Q.” In order to attain a highly selective circuit, a coil of low loss R, thus high Q is desired. From this, Q is used to represent a measure of high quality of the coil that also stands for high selectivity of the circuit. However, the Q is generally understood from the physical implication rather than the quality of the circuit as described by (3.9).

The bandwidth of a small antenna can be improved by inserting a network to the matching circuit, whereas the size of antenna limits the bandwidth. Fano demonstrated bandwidth increase by means of matching with a network [5][6]. The Fano concept can be applied to matching of an antenna so as to achieve wider bandwidth. Matthaei et al. treated this problem and gave design parameters along with the improvement factors as follows [7]:

tanh a/cosh a = tanh (nb)/cosh b

and sinh b = sinh a − 2 (κ) sin (π /2n).

Here a and b are parameters to be determined, is the reflection coefficient, κ is 1/Q of the load at the band edge, and n is the number of additional matching circuits. Matthaei solved equations (3.15) and obtained numerical values of these parameters as shown in Table 3.1, where values of a, b, and κ are shown for VSWR ≤ 2. Hansen gave the

bandwidth improvement factor, which is defined as a ratio (κ for n = 1)/(κ for n > 1), as Table 3.2 shows. In the table, values are given for VSWR ≤ 2 [8].

A novel matching method to increase bandwidth is application of Non-Foster circuits. The Foster Reactance Theorem says that the frequency derivative of the reactance dX/dω > 0 for the circuit which is composed with only lossless reactance X. The NonFoster circuit violates this theorem and may include not only passive components, but also active components in the network. A typical Non-Foster circuit is an NIC (Negative Impedance Converter), which transforms positive impedance to negative impedance [9]. Sussman-Fort demonstrated bandwidth improvement by applying a negative capacitance circuit to matching for a short dipole and obtained a 10-dB improvement over more than an octave of frequency [10].

In small antennas, Q or bandwidth that can be attained with a given size is limited. The fundamental limitation of small antennas has so far been discussed by many researchers, first by Wheeler and then Chu and others. These will be described in the next chapter.

3.1.3Radiation efficiency

The radiation efficiency η is given by

where Rloss denotes the loss resistance, which includes RLa in the antenna structure and RLc in the matching circuit. Input impedance of a small electric dipole antenna has a large capacitive component, so for matching, a large conjugate impedance – i.e. an inductance – is required. When a small dipole, for example, is a short stub of 0.16 λ with radius r of 3.3 × 10–4 λ, Rrad is 0.1 and Xrad (input reactance component) is –2071. The matching condition requires an inductive component of +2071 . When QL of this inductance is 100, for example, RLc of the inductive component alone would be 20.7 . The radiation efficiency η in this case would be about 0.005 – that is 0.5%. With QL of 400, η increases to 0.019, but that’s still only 1.9%. This is because of the much greater loss resistance RLc in the inductive component over the radiation resistance Rrad. Practically, a transformer between the low radiation resistance and the load resistance 50 is necessary and the loss at that transformer should be taken into consideration.