7.3 Antennas with dimensions << wavelength.
At low frequencies the free space wavelength becomes very large. For example, the BBC Radio 4 LW transmissions use a carrier frequency of 198 kHz. This corresponds to a free space wavelength of 1514 metres. Even lower frequencies are used for special purposes (e.g. submarine communications use frequencies below 10 kHz!). It isn't practical to make halfwave dipoles at these frequencies, but Hertzian dipoles are a possibility. Figure 7.6 shows some examples of the use of Hertzian dipoles at long wavelengths.
Most car radios use a ‘whip’ antenna. This visible rod of metal forms one half of the antenna dipole. The other half is the rod's image reflected in the metal body of the car. At medium wave (MW = 526 to 1606 kHz) and long wave (LW = 140 to 283 kHz) frequencies this acts as a Hertzian dipole.
MW & LW transmitters usually consist of a set of tall masts whose tops are linked with a network of metal cables. At first glance it may appear that it is the cables which are the antenna and that the masts are purely to hold the antenna well above the ground.
In fact, it is the masts which are the antenna! As with the car whip, the masts form one half of a Hertzian dipole. The other half being their reflection in the ground. The top cables (and their ground image) form an end capacitance which helps the dipole operate.
Going back to our explanation of a Hertzian dipole we see that it is assumed that we need somewhere for the moving charges to ‘go to & come from’ at the ends of the dipole. Initially we assumed that this was a convenient (invisible!) pair of spheres. More precisely, the charge is stored in the capacitance between the ends of the antenna. The cables at the top of a MW/LW transmitter make one ‘plate’ of a capacitor — the other being their ground plane image. The charge pumped up and down the antenna masts can now be moved from plate to plate of this very large capacitor. It is this need for an end capacitance which is one of the main problems of a Hertzian dipole. To see why, consider the example of the car whip antenna. We can model any Hertzian dipole as the the simple circuit shown in figure 7.6. This shows the antenna's real (dissipation) resistance, R, its radiation resistance, , and its end capacitance, C.
We know from our analysis of a Hertzian dipole that it radiates a power density of
The total radiated power will therefore be
Working out this integral leads to the result
where P is in Watts, is the peak current on the dipole in Amps, L is it's length in metres, and is the wavelength in metres. The definition of radiation resistance means we can also say that
Hence, for a Hertzian dipole,
A typical car dipole will be around 1 metre long, hence at metres it will have a radiation resistance of Ohms. Quite a small resistance value! This has two practical implications. Firstly, the dissipation resistance must be much lower than this if we want to avoid wasting most of the signal warming up the dipole.
Secondly, the antenna's driving impedance will be
The capacitance of a 1 metre long dipole will be quite small. For the sake of example we can assume that C = 1 pF. This means that
at 198 kHz. i.e. the antenna impedance is completely dominated by the end capacitance. This high terminal impedance makes the antenna difficult to use since it means that a very high voltage or electric field is required to be able to induce a significant amount of current along the antenna. The low value of the radiation resistance means we need quite a lot of current to transmit a significant amount of power. The real (dissipation) resistance is also likely to be a much larger value than . As a result we find it difficult to get a voltage large enough to set up the current required to radiate power... and when we do, we find that quite a lot of the power is wasted simply warming up the antenna!
Although high-power transmitter antennas are usually larger than car whip antennas, and hence will have a higher , they may also have a low end capacitance even when their top finishes in a great spray of wires. This may mean that very high voltages are required in order to transmit significant powers. We can therefore conclude that simple Hertzian dipoles are not usually a very efficient way to transmit & receive power.
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