SOME THOUGHTS ON
LOG-PERIODIC ANTENNAE



Perhaps a more suitable name (or description) would be a selective multiple dipole array; or an "Isbell" antenna, in honour of one of its main conceptors; much in accordance with the well known "Yagi-Uda" or "Yagi" antenna i.e. "Isbell" antenna (conceived at the University of Illinois in 1958)
The advantage of the "Isbell" is that it can operate over a range of frequencies where only one dipole is active at the operating frequency, the other elements being essentially dormant or; perhaps operating in a parasitic fashion much like that of the Yagi antenna?
The radiation pattern is somewhat broad, and the power gain somewhat modest, for the size of the structure.
This is because only a limited portion of the array is active at a given frequency.
"Isbell" (L-P) antenna's are nevertheless useful for applications where it is necessary to cover a wide frequency band without resorting to an antenna switching system. e.g. Television reception.


Adjacent half wave dipoles are traspose mounted on the feeder-boom.
Feedpoint impedance is reckoned to be approx' 100 ohm.
According to Ref 2 :- A 1/8 λ short-circuited matching stub; cut for the lowest operating frequency will provide a match to 300 ohm ribbon at the vertex/feedpoint of the array.
The performance is supposedly PERIODIC with the LOGarithm of the frequency of the lowest "cell"

ANTENNAE DESIGN


Despite all the nomograms and formulae usually associated with this antenna the physical design reduces to simple "plane" trigonometry!
Essentially the criterium for design of an isbell (L-P) is :-

1) Upper and lower operating frequencies
2) Number of elements
3) Apex angle of antennae
4) The requirement that each sucessive element is a scaled-down length of its immediate predecessor, and so on, down through the array.
This scaling factor is called τ (tau)
Derivation of this factor is shown later.
Scaling factor τ is also used to calculate the sucessive inter-element spacings.
5) Boom length is usually (but not mandatory) equal to one wavelength (1λ ) at the lowest design frequency.
This gives a reasonable looking structure
with an apex angle of approx' 15 to 30

Example:-

To design an "Isbell" (L-P) antennae to the following specs:-
Freq (low) 150 MHz
Freq (high) 300 MHz
Elements 10


where L L is a length = to λ of lower design freq'
...and L H is a length = to λ of higher design freq'
and of course frequency and wavelength are related as C = λ x F
where C = velocity of electromagnetic radiation reckoned at 3 x 10 8 m/s

LET THE FUN BEGIN ! . . . . .
By definition L 2 = L L x τ
AND L 3 = L 2 x τ
Therefore L 3 = L L x τ 2
So logically we can see that for an array of N elements LN = L L x τ (n-1)
LH = L L x τ (n-1) ....... (transposing)


as element length is inversley proportional to frequency we can say :-


In our example : -

τ 9 = 0.5 .........(take log of both sides)
9 log τ = log 0.5
(perhaps this is where the LOGarithm comes from?)
..........(antilog of both sides)
therefore τ = 0.92587
from this calculate the lengths of each successive element.
The relationship of the antenna's elements lengths from longest to smallest is in fact a "Geometric progression".
Perhaps this is where the PERIODICity comes from ?
We can also use the same factor to calculate the element spacing upon the boom i.e.
To calculate inter-element spacing: is shown below.
(more trigonometry and algebra I'm afraid !)


By similar triangles we can say ;
the ratio
....(transposing)
......therefore

........but L 2 = τ L L

........taking out the common factor of L L
therefore

where LL = λ of lower design freq'
and L H = λ of higher design freq'
Boom length usually 1 λ at lowest design frequency
X 1 = first inter-element spacing
The next inter-element spacing
(X 2 ) is then = X 1 x τ
and so on following the design requirement that each successive spacing is a scaled down dimension of its immediate predecessor.
Now consider this :
If we were to design an L-P antenna using a boom of 1 λ at the lowest operating frequency and we wished to cover a frequency ratio of 2:1 (as in our example of 150 MHz to 300 MHz) we can substitute values of wavelength in the above formula and simplify and reduce the equation further viz :
therefore Boom would be equal to λ
L L would be equal to 1/4 λ
L H would be equal to 1/8 λ (of 150 MHz wave)
substituting in the above formula we get:

therefore X 1 = 2 λ (1- τ ) ....(2:1 freq ratio).....EQUATION 3

and by similar reasoning we obtain:
X 1 = 1.5 λ (1- τ ) ....(3:1 freq ratio)
X 1 = 1.3 λ (1- τ ) ....(4:1 freq ratio)
One would wonder if the antenna would function the same should the spacing and element length not be logarithmically aperiodic, and dipoles be equally spaced?
Compare the operation of a multiple dipole for 20/15/10 metre ham bands all terminated at the feed point where the only active dipole is the one resonant at the feed frequency, the others remaining dormant.
Some published designs show the Log Periodic antenna with either boom/feeder half seperated at an angle; so called Pyramidal or trapezoid style.
This is supposed to produce a slight increase in gain over the conventional "flat" design ( all other parameters remaining equal).
Figs 1 & 2 show the authors construction in both styles:
Antenna design: 16 elements, frequency coverage 50 - 590 MHz
I could not discern any difference when used as a receiving antennae, except to say the "pyramidal" form is visually more striking
Fig 3 (photo-scan) shows this style in use by German broadcaster Deutsche Welle. (5.85 - 26.1 MHz 400 Kw power handling capability)


In summary:
Provided you know the upper and lower limit frequencies for your antenna and number of elements required, using formula 1 and formula 2 and perhaps formula 3 you can calculate all the various dimensions needed to construct an "LPA" (Log periodic array)

References:

Ref 1 C.T. Milner/W1FVY Log periodic antennas. QST magazine Nov 59 pg 11
Ref 2 K Woodward Log periodic antenna's Electronics Australia Dec65/Jan66
Ref 3 Geo Monser "Pyramidal " TV/FM antenna Popular Electronics July 69 pg 27

Addendum: Response from noted Antenna academic; L B Cebik (W4RNL) SK


Frank Hughes VK6FH Mar 2001, (update Oct 2009)
E-Mail: vk6fh@westnet.com.au