An analysis of the halfwave dipole shows that the field from a single dipole radiated at an angle, 
, to the plane normal to the dipole axies will have a pattern of the form
Using exactly the same arguments as before we can therefore show that, in the 
plane, the antenna field pattern will be
The 
and 
planes are said to be the principle planes of the antenna.
Figure 7.3 illustrates the power pattern in the principle planes of an array of five driven elements of the kind considered above.
Most dipole arrays use just one driven element and all the others are passive or parasitic elements. (Note that the term, ‘driven’ is often still used when considering receiving antennas to mean the one(s) connected to the wires or waveguide.) The passive elements are dipoles which are simple strips of metal with no central break. Despite this, they behave as if they were linked to the wires/guide connected to the driven element. In effect, they are linked together by the fields radiated backwards and forwards between elements. 7.4 shows two examples from the wide range of possible designs.
The Yagi-Uder array is named after its inventors (in practice it's usually just called a “Yagi”). It consists of one driven element plus one or more Directors on one side and a Reflector on the other. 7.4a shows a 5-element Yagi. 
, 
, and 
and the directors, R is the reflector. An incoming field sets up resonant currents on all the dipole elements. This causes the passive elements to re-radiate signals. These re-radiated fields are then picked up by the driven element. Hence the total current induced in the driven dipole is a combination of the direct field striking it and the re-radiated contributions from the directors and reflector.
The relative phases of the contributions reaching the driven element depend upon three factors:—
- The direction the incident signal is coming from.
- The relative positions of the passive elements.
- The lengths of the passive elements.
This last effect is because the dipoles are resonators. The phase delay between an incident and a reflected wave depends on how long the dipole is in wavelengths. A shorter dipole tends to reflect ‘more quickly’.
The resonances tend to ‘tune’ the antenna to a specific frequency band. The more elements there are of the same length, the narrower the band of frequencies over which the antenna will work and the higher the peak gain. The elements of a Yagi all have different lengths. The directors are all slightly shorter that the driven elements and the reflector is longer. This spread in sizes widens the antenna's working bandwidth at the expense of reducing the peak gain slightly. The spacing between elements can also be altered a little to improve the performance of a practical antenna. For the 5-element example the optimum gain-bandwidth product is when R = 0·525l, 
, 
, and 
. The director spacing is around 
and the reflector is about 
from the driven element. This produces an antenna with a peak ‘front’ gain value of around 15dBi and a bandwidth of ±10%.
In principle, we can make Yagi antennas with as many elements as we like, but in practice Log Period antennas tend to be better when using more than about 8 elements. In these, the element spacing and size varies logarithmically along the antenna. Most VHF (100MHz) antennas are Yagi arrays. UHF (600MHz - 1 GHz) TV antennas are normally log periods with 12-24 elements. They give peak gains of 20 - 25dBi but have wider bandwidths than a Yagi, typically around ±20%.
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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.