This document summarizes the design process of a 10:1 frequency-independent logarithmically periodic dipole antenna operating from 500 MHz to 5 GHz. It describes the theoretical background of log-periodic antennas and outlines the design steps taken. These include calculating tooth radii and angles to achieve the desired performance. Simulations were conducted and a quarter-wave transformer was added to match the impedance. Both simulated and measured results are presented, although the bandwidth goals were not fully achieved. Potential sources of error are discussed.

1. UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE, ECE DEPARTMENT, ECGR 4121/5121 PROJECT II 1
Design of a Frequency-Independent Logarithmically
Periodic Antenna
Joshua S. LaPlant
Abstract—The intent of this paper is to cover the design
process of a 10:1 frequency independent logarithmically periodic
dipole (toothed) antenna that operates with the frequency range
of 500 MHz to 5 GHz. The Log-periodic toothed antenna is
not necessarily logarithmically periodic as it pertains to its
geometrical structure; rather, it is a function of frequency,
varying periodically according to the logarithm over frequency.
I. INTRODUCTION
Frequency independent antennas... As suggested by the
name of this class of antennas (Frequency Independent), its
performance is independent of frequency and wavelength. In
other words, a frequency independent antenna is specified
solely based on angles, resulting in a structure that extends
infinitely. However this is impractical, giving rise to the need
for at least one wavelength based parameter to constrain the
antenna to a finite geometry. In theory, it is desired that the
wavelength based parameter be selected so that the pattern
and impedance performance of the finite structure converges
approximately to that of the infinite structure across the
intended frequency band [2].
According to Duhamel and Ore [1], logarithmically periodic
antenna structures are defined such that their pattern and
impedance characteristics will periodically repeat with respect
to the logarithm of frequency. Geometrically speaking, the
log-periodic toothed antenna (LPTA) can be realized as a
combination of a Bow-tie antenna with an Equiangular Spiral
antenna. The reasoning behind combining a broadband dipole
(Bow-tie) and a frequency independent antenna (Equiangular
Spiral) is crucial to the make-up of the LPTA. Typically, a
bow-tie antenna will not suffice the performance requirements
for a 10:1 frequency independent antenna design. This is in
part due to the actual behavior of the current, which terminates
somewhere along the side of the bow-tie, causing the current
to resonate at an undesired length, thus inducing higher-
order modes. Since the far-field patterns are directly related
to the current, these higher order modes significantly limit
and narrow the bandwidth of the antenna. To overcome this,
an equiangular spiral...*Finish Explanation*. The longer the
currents can travel, the lower the frequency of operation
II. THEORY
Initially, it was decided that the size of the teeth and the
gaps would be the same. To accomplish this, the relationship
Joshua S. LaPlant is an undergraduate Electrical Engineer in the Depart-
ment of Electrical and Computer Engineering, University of North Carolina
Charlotte, Charlotte, North Carolina (e-mail: jlaplant@uncc.edu).
shown in Equation 1 was applied; τ is the scale factor and σ
is the slot width.
τ =
√
σ (1)
Fig. 1. Logarithmically Periodic Antenna Model with Design Parameters
In order to solve for these crucial variables, τ was initially
selected to be 0.5, which was then adjusted later by performing
an iterative process to achieve a 10:1 ratio between the longest
and shortest teeth (discussed later). The next order of business
was to choose how many teeth the antenna would have. For the
purposes of this antenna, five teeth were chosen. According to
Adams [3], the shortest tooth’s (r5) arc-length was designed
to be λ
8 at the highest frequency of interest (5 GHz). Also the
distance (r5) was designed to be the same as the arc-length,
resulting in a distance of approximately λ
4 at the highest
frequency. Similarly, the arc-length and distance to the largest
tooth (R1) was designed to be λ
8 at the lowest frequency of
interest. In reference to the model shown in 1,the angle δ
was calculated by applying trigonometric identities. Since the
arc-length and distance to the teeth were the same, δ was
equal to 1 radian (57.3◦
). For this particular LPTA design, a
self-complimentary structure was desired. To accomplish this,
Equations 2, 3, and 4 where applied simultaneously to solve
for the remaining angles.

2. UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE, ECE DEPARTMENT, ECGR 4121/5121 PROJECT II 2
α + γ = 180◦
(2)
β + 2δ = α (3)
β = γ (4)
These angles where found to be: β = γ = 32.7042◦
, α =
147.296◦
, and δ = 57.2958◦
. β fell within an acceptable range
(between 10◦
and 40◦
), noting that the impedance is smoother
as β approaches 40◦
[3]. Finally, the remaining teeth radii were
solved for by applying Equations 5 and 6 .
Rn+1 =
Rn
τ
(5)
rn = σRn (6)
As stated previously, τ was originally chosen to be 0.5, then
tweaked slightly to achieve an approximate 10:1 ratio between
teeth r1:R5 and R1:r4. This can be seen below in Table 1. τ
was adjusted to be 0.5218, resulting in a value of 0.7224 for σ.
Table 1. LPTA Tooth Radii
III. SIMULATION
Using HFSS, the LPTA was constructed based off of the
previous calculations and simulated. Upon properly simulating
the antenna, it was observed that the antenna was in need of a
quarter-wave transformer to match the impedance to a 50-ohm
line, in hopes of lowering the S11 below -10 dB for designed
frequency range. This can be observed in Figure 2 and 3.
Fig. 2. Input Impedance Before λ
4
-Transformer
A λ
4 -Transformer was then designed and added to the model
(Figure 4). It was designed to match an impedance of 100-
ohms for the frequency range of 500 MHz to 5 GHz.
Fig. 3. S11 before λ
4
-Transformer
Fig. 4. Logarithmically Periodic Antenna HFSS Model with λ
4
-Transformer
Fig. 5. S11 with λ
4
-Transformer
A. Reflection Coefficient (S11)
After the λ
4 -Transformer was added, the model was simu-
lated again, resulting in what is shown in Figure 5.
B. Input Impedance
The input impedance, post λ
4 -Transformer, can be seen
in Figure 6. Notice that the average input impedance is
approximately 50-ohms.
C. Gain E-Plane Field Patterns
Included in this section is the 3-D polar plots and E-Plane
Field Patters for the lower, center, and upper frequencies.

3. UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE, ECE DEPARTMENT, ECGR 4121/5121 PROJECT II 3
Fig. 6. Input Impedance with λ
4
-Transformer
Fig. 7. 3-D Polar Plot of Total Gain (dB) at fl (500 MHz)
Fig. 8. Simulated E-Plane Field Pattern at fl (500 MHz)
IV. MEASURED RESULTS
Included in this section is the fabricated LPTA. This was
fabricated by using double-clad Copper FR4-Epoxy board. The
antenna was measured using a network analyzer.
Fig. 9. 3-D Polar Plot of Total Gain (dB) at fo (2.75 GHz)
Fig. 10. Simulated E-Plane Field Pattern at fo (2.75 GHz)
Fig. 11. 3-D Polar Plot of Total Gain (dB) at fu (5 GHz)

4. UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE, ECE DEPARTMENT, ECGR 4121/5121 PROJECT II 4
Fig. 12. Simulated E-Plane Field Pattern at fu (5 GHz)
Fig. 13. Fabricated Logarithmically Periodic Antenna with λ
4
-Transformer
A. Fabrication
The fabricated LPTA has been included in Figure 13 for
viewing purposes. Making note that this antenna was double-
sided; connections from the top to bottom were made by indi-
vidually placing copper vias at strategically located positions.
The antenna was then fed by a quarter-wave transformer with
an extra trace added to match an impedance of 50-ohms.
B. Reflection Coefficient (S11)
Included below in Figure 14 is the measured results for
the fabricated LPTA. The blue curve represents the results
measured by the network analyzer, whereas the orange curve
is the simulated results for purpose of compares
Fig. 14. Measured S11 of Final LPTA Design (Blue)
V. CONCLUSION
In conclusion, the simulated and measured results of this
antenna do not meet specifications as expected. The measured
and simulated results go under -10 dB at approximately 750
MHz. The simulated results go above -10 dB around 3.7
GHz; the measured results go above -10 dB in a few places
before 3.8 GHz, but stay below -10dB for the majority of the
interval. A few sources of error can be attributed to incorrect
assumptions made when theoretically designing this antenna.
Instead of guessing and tweaking for τ, it would have been
better to solve for τ by dividing the center frequency by the
upper frequency. This would have resulted in a τ value of 0.55,
leading to a truly ”self-complimentary structure.” Another
source of error, for the measurements, could be attributed to
the fact that I was unable to use the quarter-wave transformer
that I designed for my specific impedance and frequency range.
This was because of the unavailability of the milling station
before the time of testing.
VI. REFERENCES
REFERENCES
[1] R.H. DuHamel and F.R. Ore, ”Logarithmically Periodic Antenna Design,”
Collins Radio Company, Cedar Rapids, I.A., USA, 1957
[2] V.H. Rumsey, ”Frequency Independent Antennas,” University of Illinois,
Urbana, I.L., USA, 1957
[3] R.S. Adams, ”ECGR 4121/5121- Lecture 13,” University of North Car-
olina at Charlotte, Charlotte, N.C., USA, Oct. 17, 2016