Log Periodic Antenna Formulas & Basic Theory

Some of the key details, formulas, equations and theory of operation of a log periodic dipole array antenna.

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Log periodic antenna includes:
Log periodic antenna basics Log periodic theory & equations

There are some relatively straightforward equations of formulas then enable some of the basic parameters for a log periodic antenna to be defined.

Also looking at these log periodic antenna equations or formulas it is possible to see why the antenna gain its name.These simple formulas give an insight into the log periodic antenna theory.

Log periodic antenna theory basics

It is possible to explain the theory behind the log periodic antenna in a qualitative basic form. This provides an understanding of how the log periodic antenna operates.

As an introduction it is necessary to understand that the antenna elements diminish in size as does the spacing between them from back to font. Additionally the feeder polarity is reversed between adjacent elements.

Looking at the operation of the log periodic antenna, take the condition when this antenna is approximately in the middle of its operating range.

When the signal meets the first elements on the antenna (i.e. those closest to the font that are the smallest) it will be found that they are spaced close together in terms of the operating wavelength. As the feeder sense is reversed between elements, the fields from these elements will tend to cancel out and no radiation will occur from these elements.

Log periodic dipole array concept — Log periodic dipole array, LPDA concept

As the RF signal travels along the feeder in the antenna it reaches a point where the feeder reversal and the distance between the elements gives a total phase shift of about 360°. At this point the effect which is seen is that of two phased dipoles. The signal from adjacent dipoles is in phase.

The region in which this occurs is called the active region of the log periodic antenna. Although the example of only two dipoles is given, in reality the active region can consist of more elements – it may be three or more - the actual number depends upon the angle α and a design constant.

Behind the active region, the signal again falls out of phase and no radiation occurs.

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 formulas

There are several relationships or formulas that describe the characteristics of the log periodic antenna. In particular these can be sued to calculate the lengths and spacings of the elements within the antenna.

Log periodic dipole array – key dimensions — Key dimensions for a log periodic dipole array

Several distances and angles are used within the diagram and also the various formulas used:

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 σ it is possible to see the relationship between the sizes and spacing of the different elements.

σ = \frac{L_{n + 1}}{L_{n}}

σ = \frac{d_{n + 1}}{d_{n}}

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. This is expressed in the formula below.

\log (σ) = \frac{\log (f_{n + 1})}{\log (f_{n})}

It is also possible to relate the three main figures together using the formula or equation given below.

\cot (α) = \frac{4 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 within the formula below.