Designing Log Periodic Antennas
Copyright 2000,
2005 Ampyx LLC
Lightweight and precise, the log periodic has become a
favorite among EMC engineers.
In 1957, R.H. DuHamel and
D.E. Isbell published the first work on what was to become known as the log
periodic array. These remarkable
antennas exhibit relatively uniform input impedances, VSWR, and radiation
characteristics over a wide range of frequencies. The design is so simple that
in retrospect it is remarkable that it was not invented earlier. In essence, log periodic arrays are a group
of dipole antennas of varying sizes strung together and fed alternately through
a common transmission line. Still,
despite its simplicity, the log periodic antenna remains a subject of
considerable study even today.
The log periodic antenna
works the way one intuitively would expect.
Its “active region,” -- that portion of the antenna which is actually
radiating or receiving radiation efficiently -- shifts with frequency. The longest element is active at the
antenna’s lowest usable frequency where it acts as a half wave dipole. As the frequency shifts upward, the active
region shifts forward. The upper
frequency limit of the antenna is a function of the shortest elements.
Figure
1. Basic arrangement of a Log Periodic Dipole
Array (LPDA). From Ref. 1.
Log periodic designs vary,
but the one most commonly used for EMC work is the Log Periodic Dipole Array
(LPDA) of Figure 1 invented by D.E. Isbell of the University of Illinois. The LPDA we will discuss in this article
covers a frequency range of 200 to 1000 MHz.
We did not actually build it, but we did simulate its operation on a
Method of Moments simulator.
Figure
2: A closer look at
the LPDA. Note that adjacent elements
are fed out of phase.
The basic geometry is that
shown in Figure 2.. Each element is
shorter than the element to its left.
Ratio of each element to each adjacent element is constant, and is
referred to as tau (t). The other critical dimension is the spacing between elements,
designated “d” in Figure 2. Distance d_{1,2}
for example, is the distance between the left most element and its nearest
neighbor. The distance between two adjacent elements is equal to:
_{}
Two factors, tau (t) and sigma (s), are for the most part the only factors we need to
consider. Tau, as mentioned, is the
ratio of the length of one element to its next longest neighbor. Sigma is known as the “relative spacing
constant” and along with t determines the angle of the antenna’s apex, a.
_{}
Tau and sigma can be selected
by using Figure 3 (Ref. 1, 3). For EMC
work, we would like to keep the antenna as compact as possible, and we can do
so by selecting a low tau. We also
would like to keep the gain fairly low so as to avoid too narrow a beam
width. We will choose a sigma of .12
and a tau of .8, which should produce an antenna with a gain of approximately
6.5 db over isotropic.
Figure 3: The parameters tau and
sigma can be chosen from this graph. We
chose a tau of .8 and a sigma of .12 for a predicted gain of 6.5 dBi. The line for optimum sigma is for those
designers who want maximum gain. [Ref. 1, 3].
In operation, the LPDA works
as follows. Referring to Figure 2,
assume that we are operating at a frequency in which the third (middle) element
is resonant. Elements 2 and 4 are
slightly longer and shorter, respectively, than element 3. Their spacing, combined with the fact that
the transmission line flips 180 degrees in phase between elements allows these
two elements to be in phase and nearly (but not quite) resonant with element
3. Element 4, being slightly shorter
that element 3 acts as a “director” shifting the radiation pattern slightly
forward. Element 2, being slightly
longer, acts as a “reflector” further shifting the pattern forward. The net result is an antenna with gain over
a simple dipole. As the frequency
shifts, the active region (those elements that are receiving or transmitting
most of the power) shifts along the array.
Having chosen tau and sigma,
we simply plug in the numbers. The result is Table 1.
Now comes the tricky
part. We have to select the
characteristic impedance of the transmission line that feeds the elements, a
transmission line that also acts as the boom of the antenna. We will call this transmission line
impedance Z_{b} (for boom) and Reference 1 tells us it should be:
_{}
This equation deserves some
explanation. Z_{d} is the
average characteristic impedance of a simple dipole antenna. As stated previously, Z_{b }is the
characteristic impedance of the boom, itself a transmission line (referred to
as the “antenna feeder” line in Figure 2). Z_{i } is the impedance of the antenna as seen from
its input terminals. Those terminals
are usually connected to some kind of balun which performs the balanced to
unbalanced transformation and steps down the impedance Z_{i} to match
the impedance of the signal source (when transmitting) or receiver input (when
receiving). The impedance is of this
line, referred to as the “coax feed” line in Figure 2, is usually 50 ohms and
we will refer to it as Z_{0}.
Figure 4: The antenna we have
chosen to model uses two booms acting as a transmission line.
We have chosen to make the
antenna elements from 1/4 inch rods.
That makes the impedance Z_{d }of the longest element:
_{}
Next we choose the spacing
between the two booms. The booms will
consist of one-quarter inch rods to which the left and right elements are
alternately attached (Figure 4). Note
that each element is fed 180 degrees out of phase with the elements adjacent to
it. For reasons that will become
apparent shortly, we will need to choose a relatively high impedance to feed
the booms (Z_{i} ). We will
choose 200 ohms. This impedance is four
times the characteristic impedance of our coaxial line (Z_{0} = 50
ohms) and is readily produced through the use of a balun with a 2:1 ratio of windings. The spacing between the two booms needs to
be (after Ref. 1):
_{}
Where:
diam
= diameter of each boom in inches.
S =
center-to-center spacing between the booms in inches.
We will not be able to hold
each element’s ratio of the length to diameter constant since metal stock is
not available in all sizes. Instead we
will choose diameters which attempt to preserve a length to diameter ratio of
approximately 120 (Table 3).
The last design element we
will have to design is the terminating stub shown in Figure 1. After Ref. 1, this should be l_{max} / 8 or
7.4 inches long.
Next we will plug in these
numbers into a Methods of Moments program.
We will choose EZNEC, being mindful of some its limitations (Table
2). Log periodic antennas can be
particularly challenging to model. Through the use of a “Guidelines Check”
built into the software, allegiance to the limitations in Table 2 is
automatically checked. After some
compromises, we settle on the values listed in Table 3 and then press enter to
start the simulation.
Figure 5(a) shows the predicted VSWR for our antenna over the frequency range of 200 to 1000 MHz. The VSWR is acceptable below 700 MHz but above that drifts upward. We can make a reasonable guess that the problem at the high end is due to the shortest elements being too long, and so we trim .5 inch off these and run the simulation again. Now the VSWR barely rises above the 1.5:1 range over the entire range of frequencies.
The radiation patterns and
gain at 200, 500, 700 and 1000 MHz are shown both as elevation and azimuth
patterns in Figure 6. The elevation
pattern is the field produced in a plane that slices through the boom of the
antenna and is perpendicular to the antenna elements. The azimuth pattern is the field strength in the plane that the
antenna elements lie in. Both predict a
forward gain that varies from 5 dBi to 7 dBi, reasonably close to our design
target of 6.5 dBi.
Our antenna can now be built
and should perform well. One drawback
to the design, though, is the 200:50 balun that has to be used. Such baluns can introduce losses and may
have difficulty in handling the high powers that are sometimes used for
susceptibility testing. For this
reason, some commercial designs aim at setting the characteristic impedance of
the boom at something close to 50 ohms, 75 ohms being a typical target. (Setting the impedance of the boom elements
at 50 ohms is not possible when round booms are in use.)
To get such a low impedance,
we will have to use thick booms and antenna elements. For the purposes of our example, we will choose .75 inch diameter
pipe. First we compute Z_{d}
for .75 inch elements:
_{}
Then we can calculate the
boom center-to-center spacing:
_{}
Unfortunately, this log
periodic antenna will have to be built to be tested. Its large elements and close spacings (.044 inches between booms)
will cause it to exceed the parameters of Table 2. Yes, even in the 21^{st} century there are some things
best left to craftspeople.
Those craftspeople also know
a few other tricks. One is to dispense
with the 50:75 ohm balun and just suffer with a slightly higher VSWR. In order to allow for a balanced to
unbalanced transformation, ferrite beads can be slipped over the 50 ohm coaxial
cable, producing what is known as a W2DU balun (Ref. 1).
Log periodic antennas are
rugged, simple and versatile. They will
remain one of the best tools available to the EMC engineer for years to come.
The antenna is fed through a 200:50 ohm balun.
Tau = .8
Sigma = .12. Gain = 6.5 dBi.
l_{1} = 492/(200 MHz) = 2.46 feet = 29.52 inches.
l_{min} = 492/(1000 MHz) = .492 feet = 5.9 inches.
Element |
Formula |
Length (inches) |
l_{1} |
(492/200) ft. |
29.52 |
l_{2} |
l_{1}t |
23.64 |
l_{3} |
l_{2}t |
18.84 |
l_{4} |
l_{3}t |
15.11 |
l_{5} |
l_{4}t |
12.09 |
l_{6} |
l_{5}t |
9.67 |
l_{7} |
l_{6}t |
7.74 |
l_{8} |
l_{7}t |
6.19 |
l_{9} |
l_{8}t |
4.95 |
Spacing Between Elements |
Formula |
Distance (inches) |
d_{1,2} |
.5(l_{1} - l_{2})
cot a |
7.08 |
d_{2,3} |
d_{1,2} t |
5.64 |
d_{3,4} |
d_{2,3} t |
4.56 |
d_{4,5} |
d_{3,4} t |
3.60 |
d_{5,6} |
d_{4,5} t |
2.88 |
d_{6,7} |
d_{5,6} t |
2.28 |
d_{7,8} |
d_{6,7} t |
1.86 |
d_{8,9} |
d_{7,8} t |
1.49 |
EZNEC
warns us to observe the following limitations, least it produce untrustworthy
answers:
Diameters of all wires should be < .02 l.
Use at least 20 segments per each half wavelength.
Wire spacing should be > .0015 l.
Transmission line spacings should be greater than
several wire diameters.
Voltage sources feeding transmission lines should be
on a wire 3 segments long and > .02 l.
At their connection point, the ratio of any two wire
diameters should be < 2:1.
Table 3: The parameters fed into EZNEC to simulate our antenna. The numbers are in inches.
Figure 5: Figure 5a shows the
predicted VSWR for the antenna described in Tables 1 and 3. Shortening the shortest elements of the
array by .5 inch each yields the VSWR plot of Figure 5b.
Figure 6: Azimuth and elevation
plots for our modified antenna.
References
1. The ARRL Antenna Book, The American Radio Relay League,
Newington, CT, 1991.
2. R. H. DuHamel and D. E. Isbell, “Broadband Logarithmically
Periodic Antenna Structures,” 1957 IRE National Convention Record, Part 1.
3. R. L. Carrell, “The Design of Log-Periodic Dipole Antennas,” 1961
IRE International Convention Record, Part 1.