Talk:Log-periodic antenna

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Was " log-periodic antenna " in the first section a mistype? Is there anything generically called a periodic antenna? {Note that a the time of this edit, periodic antenna does not exist.}

According to the NTIA (http://www.its.bldrdoc.gov/fs-1037/dir-027/_3924.htm) it's "An antenna that has an approximately constant input impedance over a narrow range of frequencies. Note: An example of a periodic antenna is a dipole array antenna. Synonym resonant antenna." Colonies Chris 12:20, 9 June 2006 (UTC)

I´ll explain here WHY it is called a LOGARITHMIC PERIODIC ANTENNA

This antenna has a remarkable WIDE range of frequencies whith approximately constant gain and impedance. The relation between upper and lower frequency can be 10:1 and even more. The antenna has a vertex, which is an imaginary point where converge the imaginary straight lines that join the extremes of all the irradiating elements or dipoles. Those converging lines form an angle, wich is also a choice of design, called ALFA.

(I tried to add a drawing made with text characters here but the results were less than acceptable)

Every element is longer than its precedent by a constant factor known by greek letter TAU. So they DO NOT increase logarithmically but as a geometric progression of reason TAU. This factor TAU is a design choice, usually around 1.1 to 1.2.

The distance of every element to the vertex is also TAU times the distance of its precedent to the vertex. With smaller values of TAU, the distance between elements becomes smaller, and the impedance is lower, and more constant with frequency. With larger values of the angle ALFA, the elements become closer one to another, yielding a lower impedance.

Let be L(i) the length of the i-th element, and L(i-1) the length of its precedent,

and let be D(i), D(i-1) their respective distances to the vertex. Then

(1) L(i) / L(i − 1) = TAU

(2) D(i) / D(i − 1) = TAU

If we take the logarithm of this expressions, we have

(3) log[L(i)] − log[L(i − 1)] = log(TAU)

(4) log[D(i)] − log[D(i − 1)] = log(TAU)

it follows that

(5) log[L(i)] = log[L(i − 1)] + log[Tau]

and

(6) log[D(i)] = log[D(i − 1)] + log[Tau]

If we regard L(i) as the length of both halves of every dipole, and consider it as a halfe wave resonator, the resonant frequency of every element is aproximately

(7) f(i) = c / 2L(i)

where c is the speed of light. L(1)is a half wavelength for the highest frequency of operation,

and the last length, L(n), is a half wavelength for the lowest frequency of operation. The center of L(1) is the feeding point,

and all elements are linked to a line of a pair of conductors,

with alternately reversing polarity,

i.e., one line conductor is connected to the left half of the elements of odd order, and to the right half of the elements of even order.

Conversely, the other line conductor is connected to the right half of the element of odd order and to the left half of the elements of even order.

From (5) and (7) it follows that

(8) log[f(n)] = log[f(n − 1)] − log(TAU)

From (6) and (8) follows that if we plot D(1), D(2), etc. versus the logarithm of frequency, we have values of D(i) distributed with a period equal to log(TAU). Thus it is called a logarithmic periodic antenna. If the antenna has a reasonably large number of elements, namely 5 or more, the plot of [impedance] versus [logarithm of frequency] gives an almost periodic function, with its period being log(TAU), because within every period we have one dipole. This periodical variation is only notticeable for large values of TAU, say 2 or more, which is not practical. For the usual values of TAU there are rather small variations of impedance over the entire range of frequency, including the resonant frequencies of the first and last element. The gain is usually also constant within +/- 2db.