By reference to a suitable book on electromagnetics or antennas we can discover that a wire of length L carrying a uniform current oscillation of amplitude I0 will radiate an oscillating electric field
where; r is the radial distance from the centre of the antenna to the position where we wish to determine the electric field; theta is the angle between the dipole wire and the line connecting the point and the centre of the dipole; and lambda is the free space wavelength of the radiation. (This result is only true for the ‘far field’ where
The behaviour of the field pattern nearer to the dipole is more complex.) The rms field at this position will therefore be
If we imagine the dipole being located at the centre of a sphere of radius r then the power density (power per unit area, or ‘flux’) flowing through a unit area on the sphere's surface normal to the radial direction will be
where Z is the impedance of free space (E/H ratio). For a Hertzian dipole can can therefore say that
The way in which the radiated power density varies with direction is called the Antenna Power Pattern. Some books also call this the Antenna Pattern, however this term is also used to indicate how E varies with direction, so we'll stick to the term power pattern to make things clearer.
Usually, an antenna will produce a direction dependent power pattern which can be specified in a spherical co-ordinate system as in terms of two direction angles. However, a simple dipole has rotational symmetry about its dipole line, hence the power it radiates doesn't depend on the “round the dipole” angle.
6.2 Gain and directionality.
A Hertzian dipole produces a directional power pattern — i.e. it radiates more power in some directions than others. Consider a pair of antennas, each radiating the same total power from a transmitter or source, “TX”. Figure 6.3 compares the Hertzian dipole's power pattern with that produced by an isotropic radiator (sometimes also called an Omnidirectional antenna) which radiates the same power in all directions. The dipole's power pattern means that it tends to radiate less than the isotropic radiator in directions along the line of the dipole ‘wire’. Since it radiates the same power overall, this means that it radiates more than the isotropic radiator in some other directions (those approximately perpendicular to the dipole wire).
A distant receiver or observer, “RX”, in the direction will therefore pick up more radiated power from a transmitter dipole than an isotropic radiator if both radiate the same total power. So far as the observer is concerned, using a dipole has the same effect as if the transmitter power had been boosted using an amplifier whose power gain was around ×1·5. Antenna designers call this effect Antenna Gain and it is usually specified in decibels. The Hertzian dipole is said to have a Gain of around 1·75 dBi where the subscript “i” means “compared with an isotropic radiator”. Note that, as with a transformer, this isn't a ‘real’ gain, we aren't getting more power out of the antenna that is being put into it. The dipole is simply redirecting the power it radiates, sending more in some directions at the expense of others. However, the observer “RX” doesn't know that.
In practice the gain over an omnidirectional antenna or isotropic radiator can't be measured since it is impossible to make a coherent isotropic radiator. (This is based on the topological proof that, “You can't comb a hairy billiard ball smooth!”) Antenna gains are therefore measured by comparing the antenna's power pattern with that of a dipole a half-wavelength long — a halfwave dipole. This kind of antenna is quite easy to make, and it works quite well. Theoretical calculations show that the halfwave dipole has a shape similar to the Hertzian dipole, but with a gain over an isotropic radiator in the direction of 2·15 dBi. The measured gains of other antennas are therefore usually specified in dBd where the subscript “d” means “compared with a halfwave dipole”. This means that the figures quoted in these two ways can be related using the formula, dBi = dBd + 2·15.
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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.