4 Antenna Fundamentals
4.4 PRACTICAL DIPOLES .1 Dipole Structure
2 ð4:25Þ
where. denotes the vector dot product. The Friis formula (4.22) must be multiplied by this formula whenever pa6¼ pw.
In free space, the polarisation state of the received wave is the same as that of the transmitter antenna. In more complicated media, which may involve polarisation-sensitive phenomena such as reflection, refraction and diffraction, the wave polarisation is modified during propagation in accordance with the principles in Chapter 3.
4.4 PRACTICAL DIPOLES 4.4.1 Dipole Structure
In Section 4.2.4, the Hertzian dipole was assumed to have a uniform current distribution. In this section the current distribution of dipoles of varying lengths is examined and shown to modify the radiation pattern.
If a length of two-wire transmission line is fed from a source at one end and left open-circuit at the other, then a wave is reflected from the far end of the line. This returns along the line, interfering with the forward wave. The resulting interference produces a standing wave pattern on the line, with peaks and troughs at fixed points on the line (upper half of Figure 4.10). The current is zero at the open-circuit end and varies sinusoidally, with zeros of current spaced half a wavelength apart. The current flows in opposite directions in the two wires, so the radiation from the two elements is almost exactly cancelled, yielding no far-field radiation.
Lobe Null
I(0) 2 L λ/2
Figure 4.10: From transmission line to dipole
Antenna Fundamentals 73
If a short section of length L/2 at the end of the transmission line is bent outwards, it forms a dipole perpendicular to the original line and of length L (lower half of Figure 4.10). The currents on the bent section are now in the same direction, and radiation occurs. Although this radiation does change the current distribution slightly, the general shape of the current distribution remains the same and a sinusoidal approximation may be used to analyse the resulting radiation pattern.
Some qualitative results may be deduced before the full analysis:
As the dipole is rotationally symmetric around its axis, it must be omnidirectional, whatever the current distribution.
In a plane through the transmission line and perpendicular to the plane of the page in Figure 4.10, the distance from the arms of the dipole to all points is equal. Hence the radiation contributions from all parts of the dipole will add in phase and a lobe will always be produced.
The current always points directly towards or away from all points on the axis of the dipole, so no radiation is produced and a null appears at all such points.
For the Hertzian dipole, the radiated field was proportional to the dipole length in Eq. (4.8). It is desirable to increase the overall field of the antenna by increasing its directivity. Two steps are required in order to calculate the radiated field of a longer dipole; first an expression for the current distribution is determined (Section 4.4.2), then the effects of short sections ( l) of the dipole are summed in the far field as if they were individual Hertzian dipoles to determine the total field (Section 4.4.3).
4.4.2 Current Distribution
The uniform transmission line illustrated in Figure 4.10 has a sinusoidal current distribution before it is bent. An exact calculation of the current distribution on a dipole must account for the variation in capacitance and inductance along the line as well as the effect of the radiation away from the dipole. However, the current distribution on the dipole may initially be assumed unchanged from the transmission line case. Standard transmission line theory then gives
IðzÞ ¼ Ið0Þ sin k L 2 jzj
ð4:26Þ
where I(0) is the current at the feed point, assuming that the dipole is aligned with the z-axis and centred on the origin. This distribution is shown in Figure 4.11 for various dipole lengths.
It turns out that these results are exact if the wire forming the dipole is infinitesimally thin, and they are good approximations if the wire thickness is small compared with its length.
4.4.3 Radiation Pattern
The current distributions of Figure 4.11 can be used to calculate the corresponding dipole radiation patterns by summing the small field contributions dE, dErand dHfrom a series of
74 Antennas and Propagation for Wireless Communication Systems
Hertzian dipoles of length dL, with current contributions set by their position on the dipole in accordance with Eq. (4.26) and their radiation by Eq. (4.3). Thus the far-field contribution from a Hertzian dipole located on the z-axis at position z is
dE¼ jZ0
where the ejkz cosphase term accounts for the extra path length associated with an element at z compared with one at the origin.
The total Efield is therefore
E¼
The result of these calculations is
E¼jZ0Ið0Þejkr
Figure 4.11: Dipole current distribution for various lengths; y-axis is the current normalised to the feed-point current, I(0)
Antenna Fundamentals 75
This pattern is omnidirectional with nulls along the z-axis, just as for the Hertzian dipole. A similar calculation can be performed for H, but it is simpler to use the relation established in Section 4.2.4.
E
H¼ Z0 ð4:30Þ
Equation (4.29) is plotted in Figure 4.12 for the same dipole lengths as Figure 4.11. In the case of L¼ 0:01l the radiation pattern is essentially identical to the Hertzian dipole:
despite the current distribution being non-uniform, the radiation contributions are so closely located that they sum as if the current was indeed constant along the dipole length.
As the length increases through 0:5l to l, the HPBW increases through 78 to 47 . Above around L¼ 1:2l, however, the pattern becomes multilobed, which is rarely useful in practice. Note also that the magnitude of the field is itself strongly dependent upon the
0.0001
Figure 4.12: Pattern factors for dipoles of various lengths
76 Antennas and Propagation for Wireless Communication Systems
antenna length. The most common length is the half-wave dipole (L¼ 0:5l), which has significant directivity, high efficiency and relatively compact size. The directivity of the half-wave dipole is 1.64 or 2.15 dBi. As the half-wave dipole is an easily controlled, realisable antenna, whereas the isotropic antenna is not, it is common for antenna manufacturers to measure and quote antenna gains and directivities referenced to the half-wave dipole, in which case a suffix ‘d’ is used. Thus
0 dBd¼ 2:15 dBi ð4:31Þ
The above analysis of the dipole illustrates one general means of analysing antennas: first the current distribution is determined, and then contributions from infinitesimal elements are summed to find the radiation pattern. The first step is usually the most complicated, as simple approximations like the transmission line analogy above cannot always be used. Specific methods are beyond the scope of this book; see [Balanis, 97] for more details.
4.4.4 Input Impedance
A thin, lossless dipole, exactly half a wavelength long, has an input impedance Za¼ 73 þ j42:5. It is desirable to make it exactly resonant, which is usually achieved in practice by reducing its length to around 0:48l, depending slightly on the exact conductor radius and on the size of the feed gap. This also reduces the radiation resistance.