Log Periodic Antenna Theory, Operation, & Formula

- notes and overview about the theory of operation of a log periodic dipole array antenna..

As with any form of RF antenna design, the theory of operation of the log periodic dipole array can become complicated if all the mathematics is investigated deeply.

It is however possible to look at the basics of log periodic antenna theory without a complete mathematical proof.

It is useful to look at the basic mathematics and the log periodic theory to understand the basics of the design and operation.

Log periodic antenna operation basics

It is possible to explain the operation of a log periodic array in straightforward terms.

The feeder polarity is reversed between successive elements. Take the condition when this RF antenna is approximately in the middle of its operating range. When the signal meets the first few elements it will be found that they are spaced quite close together in terms of the operating wavelength. This means that the fields from these elements will cancel one another out as the feeder sense is reversed between the elements.

A log periodic dipole array showing the way in which the element lengths shorten in the direction of the main beam,
Basic log periodic dipole array

Then as the signal progresses down the antenna a point is reached where the feeder reversal and the distance between the elements gives a total phase shift of about 360 degrees. At this point the effect which is seen is that of two phased dipoles. The region in which this occurs is called the active region of the RF antenna. Although the example of only two dipoles is given, in reality the active region can consist of more elements. The actual number depends upon the angle α and a design constant.

The elements outside the active region receive little direct power. Despite this it is found that the larger elements are resonant below the operational frequency and appear inductive. Those in front resonate above the operational frequency and are capacitive. These are exactly the same criteria that are found in the Yagi. Accordingly the element immediately behind the active region acts as a reflector and those in front act as directors. This means that the direction of maximum radiation is towards the feed point.

Log periodic theory

There are several dimensions that determine the operational characteristics of the antenna, and there are some straightforward formulas that can be used to calculate the lengths.

A log periodic dipole array showing the way in which the element lengths shorten in the direction of the main beam and the key dimensions and angles.
Key dimensions of a log periodic dipole array

As can be seen from the diagram there are several dimensions that are marked:

    Lx = length of element x.
    dp,q = distance between elements p and q.
    τ = the design constant.
    α = the angle of the line of the elements to the line drawn through the centre of the elements (see diagram).
    σ = relative spacing constant - ratio of is the ratio of the length of one element to its next longest neighbour..

From the definition of the factor τ it is possible to see the relationship between the sizes and spacing of the different elements.

Log periodic dipole array formula for calculating element length and spacing: tau = Ln+1/Ln = Dn+1/Ln.

It is also possible to determine the reason for the name of the log periodic from the mathematics associated with the antenna.

The features of the antenna grow by a constant geometric multiple. As result of all the elements growing by a constant multiple then the ratios of the logarithm will be constant

Log periodic dipole array formula for showing the logarithmic periodicity of the antenna.

It is also possible to relate the three main figures together.

Log periodic dipole array formula cot α = 4 x σ / (1 - τ)

It is also possible to relate the distance between two elements and the length of each one using the angle that the element lengths form at the apex.

Log periodic dipole array formula relating the successive element lengths via the angle subtended by the element lengths at the apex  dx,y = 1/2  [ Lx - Ly ] cot α

These are some of the basic formulas and equations that relate the basic parameters for the log periodic antenna.

By Ian Poole

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