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DESIGN AND OPTIMIZATION OF ELECTRICALLY SMALL ANTENNAS
FOR HIGH FREQUENCY (HF) APPLICATIONS
A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI‘I AT MĀNOA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN
ELECTRICAL ENGINEERING
DECEMBER 2014
By
James M. Baker
Dissertation Committee:
Magdy F. Iskander, Chairperson Zhengqing Yun David Garmire Victor Lubecke
John Madey
Keywords: Compact HF, Coastal Radar, Electrically Small Antennas
Copyright
By
James M. Baker
2014
ii
ABSTRACT
This dissertation presents new concepts and design approaches for the development and
optimization of electrically small antennas (ESA) suitable for high frequency (HF) radio
communications and coastal surface wave radar applications. For many ESA
applications, the primary characteristics of interest (and limiting factors) are lowest self-
resonant frequency achieved, input impedance, radiation resistance, and maximum
bandwidth achieved. The trade-offs between these characteristics must be balanced when
reducing antenna size in order to retain acceptable performance. The concept of “inner
toploading” is introduced and utilized in traditional and new designs to reduce antenna ka
and resonant frequencies without increasing physical size. Two different design
approaches for implementing the new concept were pursued and results presented. The
first design approach investigated toroidal and helical designs, including combinations of
toroidal helical antennas, helical meandering line antennas, and additional designs
incorporating toploading and folding to improve performance. The other approach
investigated fractal-based designs in two and three dimensions to improve performance,
reduce size, and lower resonant frequency. The performance characteristics of fractal
geometries were analyzed and compared with non-fractal designs of similar height, total
wire length, and ka. Inner toploading was also applied in the two design approaches and
shown to reduce antenna Q by up to a factor of 4 with a corresponding increase in input
resistance by up to a factor of 10, when properly applied. When folded arms were applied
to various designs, Q was further decreased by a factor of 2 with a corresponding increase
in input resistance proportional to the number of arms. Genetic algorithms were
developed for optimizing antenna designs and used in custom programs, including a new
iii
cost function for better comparison of ESA performance. Antenna performance was
modeled, analyzed, and optimized using set performance criteria. Several unique antenna
designs were simulated and experimentally tested in field measurements.
Experimentation was conducted using full-size prototypes with performance measured
using vector network analyzers and HF transceivers. Experimental performance
measurements were reproduced in simulation models with a high degree of correlation.
Successful two-way radio communications were established with amateur radio stations
around the world using prototype antennas.
iv
TABLE OF CONTENTS
ABSTRACT ................................................................................................................................................. iii LIST OF TABLES ......................................................................................................................................vii
LIST OF FIGURES .................................................................................................................................. viii
ACRONYMS ...............................................................................................................................................xii
CHAPTER 1 INTRODUCTION ................................................................................................................. 1
A. BACKGROUND ....................................................................................................................................... 1
B. OBJECTIVE ............................................................................................................................................ 3
C. ORGANIZATION...................................................................................................................................... 4
CHAPTER 2 ELECTRICALLY SMALL ANTENNAS............................................................................ 6
A. BACKGROUND ....................................................................................................................................... 6
B. PROPERTIES .......................................................................................................................................... 7
C. DESIGN PRINCIPLES .............................................................................................................................. 14
CHAPTER 3 METHODS OF SOLUTION .............................................................................................. 17
A. NUMERICAL ELECTROMAGNETICS CODE (NEC) .......................................................................................... 17
B. LABVIEW .......................................................................................................................................... 18
C. FEKO ................................................................................................................................................ 20
CHAPTER 4 EVALUATION OF ESTABLISHED DESIGNS AND METHODS ............................... 21
A. ESTABLISHED DESIGNS .......................................................................................................................... 21
B. TOPLOADING ....................................................................................................................................... 23
C. FOLDING ............................................................................................................................................ 28
D. SUMMARY .......................................................................................................................................... 31
CHAPTER 5 NEW CONCEPT AND DESIGN APPROACHES ........................................................... 32
A. BACKGROUND ..................................................................................................................................... 32
B. INNER TOPLOADING.............................................................................................................................. 33
C. NEW DESIGN METHODOLOGY ................................................................................................................ 38
D. NOVEL DESIGNS FOR ELECTRICALLY SMALL HF ANTENNAS ........................................................................... 39
v
E. INVESTIGATION OF FRACTAL GEOMETRIES ................................................................................................. 53
F. SUMMARY .......................................................................................................................................... 72
CHAPTER 6 ALGORITHMS FOR DESIGN OPTIMIZATION .......................................................... 74
A. RANDOM SEARCH ................................................................................................................................ 74
B. NELDER-MEAD DOWNHILL SIMPLEX ALGORITHM ....................................................................................... 74
C. SIMULATED ANNEALING (SA) ................................................................................................................. 75
D. GENETIC ALGORITHMS (GA) .................................................................................................................. 75
E. SUMMARY .......................................................................................................................................... 83
CHAPTER 7 EXPERIMENTAL VERIFICATION ................................................................................ 84
A. FIELD TEST CONFIGURATIONS ................................................................................................................. 84
B. FIELD MEASUREMENTS ......................................................................................................................... 85
CHAPTER 8 SUMMARY AND CONCLUSIONS .................................................................................. 98
CHAPTER 9 FUTURE WORK ............................................................................................................... 101 REFERENCES .......................................................................................................................................... 102
APPENDIX A – ENGLISH TRANSLATION OF HILBERT (1891) ...................................................A-1
APPENDIX B – FRACTAL GEOMETRY ............................................................................................. B-1
vi
List of Tables
Table 1. Shortened Monopole Performance ..................................................................... 14
Table 2. Toploaded λ/4 Monopole Performance .............................................................. 25
Table 3. MLA Performance .............................................................................................. 30
Table 4. Design Analysis for Inner Toploading................................................................ 34
Table 5. Helical MLA Performance.................................................................................. 41
Table 6. Performance for three-arm HMLA, direction of helical coils modified ............. 45
Table 7. Helical MLA and Toroidal Helical Performance ............................................... 52
Table 8. Fractal Tree Performance at 20 MHz, one meter height ..................................... 63
Table 9. Fractal Tree and Helical Fractal Tree Performance ............................................ 66
Table 10. Hilbert Curve Simulated Performance .............................................................. 70
Table 11. Baseline and GA Optimized Performance ........................................................ 80
vii
List of Figures
Figure 1: Landing Craft Air Cushion (LCAC) .................................................................. 2
Figure 2: Normalized wave resistance ............................................................................. 12
Figure 3: Normalized wave reactance .............................................................................. 12
Figure 4: Current distribution for λ/4, λ/8, and λ/20 monopole antennas ....................... 15
Figure 5: Chu and Hansen/Collin limits with shortened monopoles from Table 1 ......... 16
Figure 6: LabVIEW program for Genetic Algorithms .................................................... 19
Figure 7: LabVIEW program for controlling HP8753B Network Analyzer ................... 19
Figure 8: FEKO display for early ESA prototype ........................................................... 20
Figure 9: Performance of established designs, Q(ka) ...................................................... 22
Figure 10: NEC model of Marconi’s 1904 toploaded antenna ........................................ 23
Figure 11: Monopole antenna over PEC ground (left) and with mesh toploading (right)25
Figure 12: Current distribution for monopole with and without toploading ................... 26
Figure 13: Two-arm and three-arm folded monopole antennas ....................................... 29
Figure 14: Folded meandering line antenna with three arms ........................................... 30
Figure 15: The concept of inner toploading ..................................................................... 33
Figure 16: One-turn helical with inner toploading. ......................................................... 35
Figure 17: Current Magnitude, helical antenna with and without inner toploading. ....... 35
Figure 18: One-turn helical with inner toploading. ......................................................... 36
Figure 20: Simulated and measured S11with and without inner toploading ................... 37
Figure 19: Prototype helical antenna with inner toploading ............................................ 37
Figure 21: Helical meandering line antenna .................................................................... 39
Figure 22: Impedance, helical MLA, 3 – 30 MHz ........................................................... 40
viii
Figure 23: Impedance, helical MLA, 30 – 100 MHz ....................................................... 40
Figure 24: Far-field radiation pattern ............................................................................... 42
Figure 25: Current magnitude in three-arm helical MLA ................................................ 43
Figure 26: Current Magnitude in one arm ....................................................................... 44
Figure 27: Current Phase in one arm ............................................................................... 44
Figure 28: HMLA single arm, original (left), modified with alternating turns (right) .... 45
Figure 29: One-turn toroidal helical antenna ................................................................... 46
Figure 30: Impedance for one-turn toroidal helical antenna ............................................ 47
Figure 31: Gain for one-turn toroidal helical antenna ..................................................... 47
Figure 32: One-turn toroidal helical antenna with two-turn inner toploading ................. 48
Figure 33: Impedance for one-turn toroidal helical antenna with inner toploading ........ 49
Figure 34: Gain for one-turn toroidal helical antenna with inner toploading .................. 49
Figure 35: Toroidal helical antenna with four half-turn folded arms .............................. 50
Figure 36: Impedance for half-turn toroidal helical antenna, four folded arms............... 51
Figure 37: Gain for half-turn toroidal helical antenna, four folded arms ........................ 51
Figure 38: Generator for a Koch curve ............................................................................ 54
Figure 39: Koch antennas after one and two iterations .................................................... 54
Figure 40: Sierpinski Triangle from IFS .......................................................................... 55
Figure 41: Fractal tree from IFS ...................................................................................... 55
Figure 42: Fractal tree antennas, iteration #2 and #3 ....................................................... 56
Figure 43: Fractal tree antenna with two arms, iteration #4 ............................................ 57
Figure 44: Fractal tree antenna with three arms, iteration #4 .......................................... 58
Figure 45: Fractal tree antenna with four arms, iteration #4............................................ 59
ix
Figure 46: Helical fractal tree with two arms, iteration #1 .............................................. 60
Figure 47: Helical fractal tree with two arms, iteration #2 .............................................. 61
Figure 48: Comparison of fractal tree geometries ........................................................... 62
Figure 49: Impedance for four-arm fractal tree ............................................................... 64
Figure 50: Current Magnitude for four-arm fractal tree .................................................. 64
Figure 51: Gain pattern for four-arm fractal tree ............................................................. 65
Figure 52: Hilbert curves ................................................................................................. 67
Figure 53: Antenna, Hilbert curve, one iteration ............................................................. 68
Figure 54: Antenna, Hilbert curve, second iteration ........................................................ 69
Figure 55: Antenna Prototype, Hilbert curve, second iteration ....................................... 70
Figure 56: Simulated and measured S11 for Hilbert prototype ....................................... 71
Figure 57: Simulated and measured Impedance for Hilbert prototype ............................ 71
Figure 58: GA optimized model ...................................................................................... 79
Figure 59: Q and input resistance for four-arm GA ......................................................... 80
Figure 60: Impedance, baseline design ............................................................................ 81
Figure 61: Impedance, GA optimized design .................................................................. 81
Figure 62: Improvement in Q .......................................................................................... 82
Figure 63: Improvement in effective radius..................................................................... 82
Figure 64: Simulated and measured S11, open circuit mode over lossy ground ............. 87
Figure 65: Simulated and measured S11, short circuit mode over lossy ground ............. 87
Figure 66: Simulated and measured HPBW .................................................................... 88
Figure 67: Measuring antenna patterns near Hanauma Bay ............................................ 89
Figure 68: Received power measured over azimuth ........................................................ 90
x
Figure 69: Simulated and measured gain at 16 MHz ....................................................... 90
Figure 70: Smith chart of measured impedance, open circuit mode................................ 91
Figure 71: Smith chart of measured impedance, short circuit mode ............................... 92
Figure 72: Field testing on beach near the Makai Research Pier ..................................... 93
Figure 73: Amateur Radio Communications – Field Testing .......................................... 94
Figure 74: Measuring two-element array properties at Waimanalo Park ........................ 95
Figure 75: Power Spectral Density, with and without filtering ....................................... 96
Figure 76: Field testing at Sandy Beach .......................................................................... 97
Figure 77: Q(ka) in this dissertation .............................................................................. 100
Figure 78: Hilbert (1891) Figs. 1, 2, and 3 .................................................................... A-2
Figure 79: Riemann cosine function R(t) for n = 1 ........................................................ B-2
Figure 80: Riemann cosine function R(t) for n = 5 ........................................................ B-2
Figure 81: Riemann cosine function R(t) for n = 200 .................................................... B-2
Figure 82: Riemann R(x,y) cosine function ................................................................... B-2
Figure 83: Riemann R(x,y) sine function ...................................................................... B-2
Figure 84: Riemann function 0 to pi .............................................................................. B-2
Figure 85: Scaling the Riemann function ...................................................................... B-2
xi
Acronyms
CW Continuous Wave
ηr Radiation Efficiency
ESA Electrically Small Antenna
GA Genetic Algorithms
HCAC Hawaii Center for Advanced Communications
HF High Frequency (3 - 30 MHz)
HFSWR High Frequency Surface Wave Radar
HP Hewlett-Packard
hn Spherical Hankel function of the second kind for mode n
HPBW Half-power Bandwidth
IFS Iterated Function System (applies to fractal generation)
LCAC Landing Craft Air Cushion
MLA Meandering Line Antenna
MoM Method of Moments
NEC Numerical Electromagnetics Code
PEC Perfect Electric Conductor
Q Radiation Quality Factor
RF Radio Frequency
RLC Resistor, Inductor, and Capacitor
RPF Radiation Power Factor
TE Transverse Electric mode
TM Transverse Magnetic mode
UHF Ultra High Frequency (300 MHz - 3 GHz)
VHF Very High Frequency (30 - 300 MHz)
VNA Vector Network Analyzer
VSWR Voltage Standing Wave Ratio
xii
Chapter 1 Introduction
A. Background
The design of electrically small antennas (ESA) presents a wide variety of challenges,
primarily due to inherently low impedance and narrow bandwidths. Improving these
performance characteristics is especially challenging in the HF band (3 – 30 MHz) due to
the longer wavelengths (10 – 100 meters) and corresponding antenna physical
dimensions. These challenges are amplified for applications such as coastal HF surface
wave radar (HFSWR) systems which also require vertical polarization for long range
surface wave propagation over the ocean, and military applications that require mobile,
rapidly deployable, covert systems.
A typical coastal HFSWR antenna system involves arrays of quarter-wave monopole
structures with antenna heights of up to 25 meters and even larger ground radial
networks. As a result, current HFSWR and over the horizon radar (OTHR) antenna
systems tend to be located at fixed sites with extensive infrastructure and site preparation
requirements. These radar arrays can extend for several kilometers with semi-permanent
structures and significant environmental impact. Many current systems use quarter-wave
monopole antennas which are omnidirectional, requiring extensive arrays to accomplish
the beam-forming required to minimize clutter and back-scatter from the surrounding
terrain. Coastal HF radar system performance is also affected by ionospheric conditions
which are constantly changing and impact the useable frequencies available. These
systems may operate as intended in the geographic region in which they are constructed,
but are not suitable for mobile operations or rapid deployment to remote, desolate, or
1
otherwise unprepared locations. For these types of applications, the primary
characteristics of interest (and limiting factors) in antenna design are self-resonant
frequencies, impedance, gain, bandwidth, polarization, and phase stability. For military
and homeland security applications, the antenna may also be required to consist of a low
or otherwise compact physical profile to camouflage its purpose. However, disguising a
25 meter high antenna without limiting performance or mobility can be challenging.
The implementation of HF radios on military mobile platforms is also problematic with
the antenna size being restricted by operational factors such vehicle size, tactical profile
requirements, or logistical issues. In Naval applications, traditional HF antennas can also
be an issue due to size and space limitations on seaworthy platforms such as the Landing
Craft Air Cushion (LCAC). The current HF antenna installed on the LCAC is a
horizontal center-fed dipole, aligned longitudinally with the vessel, mounted just a few
feet above the metal structure. This antenna configuration is roughly depicted by the solid
black line in Figure 1.
Figure 1: Landing Craft Air Cushion (LCAC)
2
The reality remains though that many LCAC radio operators rely on VHF and UHF
radio systems because the HF communications are unreliable. This is one of many
examples of the need for a low-profile HF antenna system that retains acceptable radio
frequency (RF) performance while satisfying non-RF design requirements. An additional
consideration for practical applications is the requirement for low-profile antennas to
exhibit broadband or multi-resonance characteristics to support system operations at
different frequencies as atmospheric and other propagation conditions change. For these
and other HF systems (particularly those suitable for mobile surveillance,
communications, and homeland security applications) antenna size is considered a critical
factor. Clearly, there is still room for improvement in traditional HF antenna design and
the requirement remains for compact, low profile HF antenna designs that are suitable for
mobile, tactical, and other mission-specific applications.
B. Objective
The objective for this research was to find solutions to the current challenges and
limitations in designing electrically small antennas in the HF band for homeland security
and military applications through the investigation of new design methodologies and
approaches. This dissertation presents a new design methodology that represents a
paradigm shift from traditional and currently accepted practices. Two different innovative
antenna design approaches, developed using the new methodology, are presented.
3
The specific tasks identified for this effort were:
1) study established designs and review their characteristics and limitations
2) explore new avenues for more effectively utilizing an antenna’s enclosed volume
3) explore methods for design optimization
4) select promising designs and build prototypes for field experimentation
The desired outcomes include a study of the trade-offs involved in performance
optimization, producing new and functional designs for HF ESA, and validation of
predicted performance through field experimentation with full-size antenna prototypes.
C. Organization
Chapter 2 provides a review of the fundamental characteristics and limitations of
electrically small antennas. These fundamental characteristics are used to develop
consistent and practical measures and metrics for comparing antenna performance,
regardless of specific physical features. Chapter 3 presents an overview of the tools and
methods used for developing solutions and analyzing results. The primary tools used
were Numerical Electromagnetics Code (NEC) and National Instruments LabVIEW
application development environment. Several custom applications were specifically
developed which integrated NEC and LabVIEW into single applications for the
automation of design and analysis. LabVIEW was primarily used to develop user
interfaces for auto-generating antenna designs based on user input (e.g., designs
generated using genetic algorithms) and NEC was used to simulate antenna designs and
provide performance data for further analysis. FEKO was also used for simulating
various antenna designs during the early phases of research. Chapter 4 reviews traditional
4
ESA antenna designs along with methods for improving performance, specifically the
effects of toploading and folding when applied to canonical geometries. Benefits and
trade-offs for each design methodology are evaluated and compared. Chapters 5 presents
a new design methodology for optimizing antenna geometries to better utilize the inner
volume along with two design approaches for implementing the new concept. The first
design approach examines toroidal and helical designs which utilize their total inner
volume. Innovative designs and geometries were developed using this design approach.
Designs presented include helical meandering line antenna (MLA) geometries, designs
combining toroidal and helical elements, and fractal geometries. Fractal geometries are
reviewed with the focus on Hilbert curves and fractal tree and helical fractal tree designs.
Computer applications were developed to generate fractal images for use in antenna
design and performance analysis, and for evaluating the benefits of this approach. These
antennas were designed to explore different methods (e.g., inner toploading) for using
their inner space to reduce total antenna volume and achieve lower resonant frequencies.
Chapter 6 investigates various techniques for performance optimization using methods
such as random search, simulated annealing, and genetic algorithms. The chapter finishes
with a comprehensive investigation into the use of genetic algorithms for optimizing a
helical MLA design. Chapter 7 describes the extensive field testing conducted using full-
size antenna prototypes and provides detailed results and design comparisons. Chapter 8
provides a summary of observations and conclusions developed throughout this research
effort. Chapter 9 describes areas of opportunity for future work building on the designs
and concepts presented.
5
Chapter 2 Electrically Small Antennas
A. Background
Over the years there have been many contributors to the theory and design of antennas
that are considered electrically small. Wheeler [1] [2] and Chu [3] pioneered the field by
developing fundamental properties of electrically small antennas. The generally accepted
criteria for what makes an antenna “electrically small” is based on the spherical volume
occupied by the antenna. Using the free-space wave number k = 2π/λ and the radius a of
the sphere enclosing an antenna, it is considered to be electrically small when the value of
ka is less than or equal to 0.5 [4]. In the early days of radio communications all antennas
were electrically small. Hansen provides a chronology of electrically small antennas
beginning in 1889 with the Hertz loop antenna, followed by Marconi’s long wire
antennas [5].
It is important to identify the relevant metrics for meaningful comparison of various
dissimilar antenna geometries. The following sections review fundamental and practical
performance properties that have been well-established for antennas in general, the
fundamental limitations to these properties for electrically small antennas, and the
derivation of practical metrics for characterizing performance of antenna geometries.
This is necessary to provide for consistent and practical analysis and comparison for all
designs presented herein, as well as any other published designs, regardless of specific
geometry.
6
B. Properties
1) Wheeler’s Radiation Power Factor
Wheeler [1] developed radiation power factor (RPF) as a figure of merit for electrically
small antennas, provided here for electric (1) and magnetic (2) antennas.
Wheeler also developed the concept of a radian sphere: a sphere with radius equal to
one radian length (r = λ/2π). The radian sphere defines the transition between the near-
field and far-field regions of an electrically small antenna [2]. The radian sphere is used
to determine the effective antenna volume and its associated effective radius. The volume
of the radian sphere and the equivalent radian cube is stated in (3).
From this expression and the RPF, represented by p, the effective antenna volume can be
calculated (4).
3
23
34
λπ
ωba
CGp e
e ==
3
23
34
λπ
ωba
LRp m
m ==
(1)
cs VππλπV
34
234 3
=
=
sceff VpVpV296 == π
(3)
(4)
(2)
7
From (4), the effective radius can be stated in terms of RPF (5).
This provides a very useful metric for direct comparison of performance improvement
for different antenna structures [2]. If the antenna structures being compared are of the
same height, the effective radii can be compared directly. If the structures have different
heights, the ratio of their respective effective radius to height can be compared.
The constraints developed by Wheeler are considered to be practical rather than
fundamental in that they were developed for dipole (electric) and loop (magnetic)
antennas [6]. It is also worth noting Wheeler established that the RPF is a function of
antenna dimensions and resonant wavelength alone and is the same for electric and
magnetic antennas.
2) Chu’s limit on Q
Chu derived a radiation quality factor (Q) for a hypothetical antenna enclosed in a
sphere of radius a. The electromagnetic fields outside the sphere could be described in
terms of infinite series of transverse electric (TE) and transverse magnetic (TM) modes.
Chu defined Q at the input terminals of an antenna structure based on its equivalent RLC
circuit and expressed as a function of the energy stored beyond the input terminals and
the power dissipated in radiation [3]. Chu expanded the mode wave impedance into a
continued fraction that could be interpreted as a ladder network of capacitors and
inductors. This minimum Q is considered to be a fundamental limitation for electrically
small antennas because it applies to any antenna configuration that fits within a sphere of
31
29
2
= preff π
λ(5)
8
radius a and excites a single TM mode [3]. This fundamental limit was later re-examined
by McLean [7] who derived an expression for Q based on antenna ka (6). Chu’s Q did not
include stored energy inside the sphere and so is considered the lower bound for
lossless antennas. Many papers over the years have refined and modified this expression
to increase the value for predicted Q in search of better agreement with measured antenna
performance. One estimate for Q was developed by Hansen and Collin [6] using numeric
methods and curve fitting to better predict measured performance for the lowest order
TM mode as depicted in (7). This approximation was said to be accurate within 0.5%
for ka ranging from 0.1 to 0.5 and is useful when comparing simulation results for loss-
less antennas in free space or over PEC ground planes. More recent publications continue
to refer to (6) while adding an efficiency term as shown in (8), where ηr represents
radiation efficiency [8].
( )300 23
)(21
akakQapprox +=
( )
+=
kakaQ rlb
113η
( ) kakaQChu
113 +=
(7)
(6)
(8)
9
Wheeler’s RPF has been shown to be related to Chu’s Q as p = 1/Q and so the effective
radius (5) can be re-expressed in terms of Q (9).
Yaghjian and Best developed an expression (10) for Q derived directly from antenna
impedance and radian frequency [9] which provides an additional metric, independent of
ka, for comparing the quality factor of different antenna geometries.
This approximation of Qz(ω), with R'(ωo) and X'(ωo) representing the derivative of the
resistance and reactance with respect to the radian frequency (ω), is valid for antennas
with a single resonance within the resonant bandwidth. This expression can be calculated
and analyzed over ranges including anti-resonant as well as resonant frequencies [8]. It
also provides an additional parameter for comparing antenna performance between
designs with differing geometries and dimensions as well as for comparing performance
of a specific design to fundamental limitations.
( ) ( ) ( ) ( ) ( ) 2
0
00
20
0
00 2
+′+′=
ωω
ωωω
ωωX
XRR
QZ
31
29
2
=
Qreff π
λ (9)
(10)
10
3) Wave impedance
Chu defined voltage (11), current (12), and impedance (13) of an equivalent circuit for
the spherical waves outside the radian sphere for the TMn mode [3], where hn is the
spherical Hankel function of the second kind for mode n, permeability µο and
permittivity εο for free space, and ρ = ka.
Harrington further expanded these definitions for the TEn modes [10], showing that
impedance for the TEn mode is equal to the admittance of the TMn mode (14).
These equations for TE and TM mode impedances are significant as they are only
dependent on the ka of an antenna’s geometry and they represent the normalized
impedances of the spherical waves that will couple to free space for far-field radiation.
The resistive component of the TMn mode (normalized) is presented in Figure 2 where a
significant decrease in resistance is readily observed as the value of ka is decreased from
( )ρ
ρρπεµ
dhd
jnnn
kA
V nn
o
on
21
21
12)1(4
++
=
)(12
)1(4 21
21
ρρπεµ
nn
o
on h
nnn
kAI
++
=
( ) )(/ ρρρ
ρρn
nn h
dhdjZ =
( )ρ
ρρρρ
dhdjhZ n
nn /)(=
(11)
(12)
(13)
(14)
11
0.5 down towards 0.0. It is also observed that for ka < 0.5 only the first mode has any
useable resistance for supporting propagation. The normalized reactive component of
wave impedance is plotted in Figure 3 where the negative reactance increases (in
magnitude) significantly as ka is decreased below 0.5. For comparison purposes the ka of
a quarter-wave monopole, ka ≈ 1.5, is also depicted in the two figures.
Figure 2: Normalized wave resistance
Figure 3: Normalized wave reactance
12
4) Radiation Resistance
The term “radiation resistance” is defined in the IEEE Standard Definition of Terms for
Antennas (IEEE Std 145-1993) as “the ratio of the power radiated by an antenna to the
square of the RMS antenna current referred to a specified point”. In the general case for
arbitrary current distribution on a lossless dipole, radiated power [11] can be expressed in
terms of average current amplitude and antenna length, as shown in (15), where β = 2π/λ,
and L = antenna length [12].
Radiation resistance for an electrically small dipole antenna can then be expressed (16) in
terms of the ratio of average to peak current and the ratio of physical length to
wavelength (Lλ = L/λ) [12].
This approximation is valid for antennas with L << λ. Antennas with dimension ka <
0.5 (equivalently, Lλ < .08) easily satisfy this limitation.
5) Effects of Height, Volume, and Wire Length
The antenna radiation resistance is proportional to (L/λ)2, therefore wire antennas of the
same height and resonant frequency generally exhibit the same properties, independent of
their geometries. Lower resonant frequencies can be achieved when the geometry
effectively utilizes available height and overall volume to meet design goals and
(15)
(16)
πβ
εµ
12
222 LIP av
o
o=
22
790 λLIIR
o
avr
≅
13
requirements. The challenge is to minimize the impact on performance when reducing the
height or volume. The length of the wire inside the antenna volume is the dominant factor
in establishing resonant frequency. For folded arm geometries, the wire length of one arm
is used to determine the wavelength of the resonant frequency because the current
distribution is symmetrical in all the arms. Increasing wire length is one way to increase
antenna inductance, but may also decrease the resonant frequency if done
indiscriminately. Decreasing the wire diameter is one method for decreasing the resonant
frequency; this will reduce the antenna’s radiation efficiency, which may or may not be
an acceptable trade-off.
C. Design Principles
In order to capture the impact of the limitations on performance for electrically small
antennas, a simple quarter-wave (λ/4) monopole was designed for resonance at 6 MHz.
The wire length was progressively reduced from λ/4 down to λ/20 with the impacts on
performance analyzed, compared, and results reported in Table 1, where r’/h represents
the effective radius normalized by the physical height. This metric provides a useful
parameter for comparing performance changes between antennas independent of specific
geometries.
Table 1. Shortened Monopole Performance
Antenna ka Height (meters)
Resistance (Ω)
Reactance (Ω) Qz(ω) r’/h
λ/4 1.49 11.8 36.0 0 6.4 0.60 λ/8 0.75 5.9 6.5 -249 50.2 0.60 λ/10 0.60 4.7 3.9 -322 95.5 0.60 λ/12 0.496 4.0 2.7 -385 160.0 0.61 λ/20 0.298 2.4 0.9 -580 657.9 0.63
14
This preliminary analysis demonstrated the unavoidable impact on performance
characteristics as the length of a lossless monopole over PEC ground plane was reduced.
The geometries were modeled in NEC version 4.2 [13] using 15 segments to calculate the
distribution of electric current. The current distributions for λ/4, λ/8, and λ/20
configurations were calculated at the center of each segment and are plotted in Figure 4.
It can be seen from Table 1 that for a simple monopole antenna to be considered
“electrically small” its height must be ~λ/12 or less, the radiation resistance is then less
than 3Ω, and the Qz(ωο) over 300. The effect of size reduction on antenna current
distribution is apparent in Figure 4 where the sinusoidal distribution for the λ/4 (resonant)
antenna quickly converges to a triangular distribution as the height is reduced.
Figure 4: Current distribution for λ/4, λ/8, and λ/20 monopole antennas
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Curr
ent (
amps
)
Segment #
Current Distribution - Monopole Antennas
one-fourth lambda one-eighth lambda one-twentieth lambda
15
Specific performance characteristics for the shortened monopole antennas listed in
Table 1 are depicted in Figure 5 together with Chu’s limitation and the Hansen/Collin
revision to demonstrate the rapid increase in Q as size is reduced. The individual
monopole configurations may be identified in the figure by their ka values from Table 1.
Figure 5: Chu and Hansen/Collin limits with shortened monopoles from Table 1
16
Chapter 3 Methods of Solution
A. Numerical Electromagnetics Code (NEC)
The Numerical Electromagnetics Code [13] is a software application used for analyzing
the electromagnetic response of antennas and scatterers. It is based on the numerical
solution of integral equations using the Method of Moments (MoM), combining electric-
field and magnetic-field integral equations for modeling thin wires for closed surfaces.
NEC was selected as the engine for modeling and simulating antenna designs because it
is optimized for wire and electrically small antennas, and provides command line
executables for 32-bit and 64-bit Windows operating systems. This functionality was
used to automate many of the analyses performed as part of the research for this
dissertation, and to facilitate integration with other applications developed in LabVIEW.
NEC also supports modeling and simulation using lossy antenna materials and ground
planes for comparative analysis with field experimentation data.
The user interface application selected was 4NEC2 version 5.8.14 [14]. This
application is available online at no cost and provides an efficient interface to NEC
engines for generating properly formatted NEC input files and for graphic analysis of
NEC output files. 4NEC2 was used for rapid prototyping and analysis of three-
dimensional wire antenna designs and for analysis of impedance, gain, and electric field
components.
When modeling antennas in NEC, it is very important to follow the rules and guidelines
published in the NEC User’s Manual [15], especially those related to the relationship
between wire radius and segment length. For example, NEC version 4.2 uses a thin-wire
17
approximation which limits the minimum separation allowed between parallel wires. This
recommended minimum separation is two to three times the wire diameter. If parallel
wires are placed too closely together errors may be significant, as seen in some published
performance results that violate fundamental limits such as Chu’s limit on Q. Model
accuracy should be assessed by varying parameters and checking for convergence, and by
slightly varying the frequency and observing the sensitivity of the results.
B. LabVIEW
National Instruments LabVIEW [16] is an advanced graphical programming
environment for developing custom applications in science and engineering. LabVIEW
was selected based on its extensive function libraries, instrument driver libraries, and 3D
graphing tools which allowed for rapid software prototyping, automated instrument
control, and development of comprehensive user interfaces. A wide range of custom
software applications were developed in LabVIEW to support the research and analysis
efforts in completing this dissertation. These applications ranged from solving and
plotting solutions for advanced mathematical equations to controlling external
instrumentation and sensors for test automation, data recording, analysis, and reporting.
Specific examples include automated analysis of NEC output files, control and data
recording with Anritsu and HP network analyzers, and the implementation of genetic
algorithms for antenna optimization. LabVIEW programs were also developed to solve
spherical Hankel functions of the second kind for plotting TM and TE mode impedances.
18
User interfaces for two of the applications developed in LabVIEW are provided below.
Figure 6 displays the genetic algorithm application interface including parameters, design
goals, and chromosome binary patterns. Figure 7 displays the interface for an HP 8753B
network analyzer application, developed for recording and analyzing data during field
experimentation.
Figure 7: LabVIEW program for controlling HP8753B Network Analyzer
Figure 6: LabVIEW program for Genetic Algorithms
19
C. FEKO
FEKO [17] is a commercial application for modeling and simulating complex designs
for a variety of applications in electromagnetics. FEKO combines a selection of
numerical methods for analyzing complex structures including antennas, waveguides, and
other electromagnetic devices. Numerical methods include Method of Moments, Physical
Optics, Geometrical Optics, Uniform Theory of Diffraction, and Finite Element Method.
This application is also capable of combining various methods to develop hybrid
solutions.
FEKO was used during the early stages of research for simulation of electrically small
antennas, providing insight into design methodology, geometry development, and
predicted performance characteristics. The graphical analysis products were of very high
quality, especially three-dimensional plots of antenna structures combined with far-field
radiation patterns. A screenshot is provided in Figure 8 for an early ESA prototype.
Figure 8: FEKO display for early ESA prototype
20
Chapter 4 Evaluation of Established Designs and Methods
A. Established Designs
Numerous designs were evaluated and compared to determine their limitations and
examine potential areas for improving performance, reducing overall size, and achieving
multiple resonances and lower frequencies within the HF band. Also examined were two
common methods for improving performance: toploading and folding. Designs were
selected for inclusion in this review based on several factors:
1) the design was the result of an intentional effort to reduce antenna size or improve
performance
2) the design was at least close to meeting the criteria of ka ≤ 0.5
3) the published design performance characteristics had to be reproducible using
NEC
Designs which violated Chu’s fundamental limit or did not provide all the dimensions or
other information required for modeling were not included. The designs selected for
evaluation are well-documented in [4], [5], [8], [12], [18], [19], and [20], and range from
shortened monopoles to multi-arm spherical helical structures. Since most of the
traditional designs considered were developed for VHF and UHF frequencies, their
performance was compared by evaluating Q as a function of ka, as depicted in Figure 9.
21
Two common methods used for reducing antenna size or improving the performance
for antenna designs depicted in Figure 9 are toploading and folding. These techniques are
well-documented and are discussed here to provide a baseline for expected performance
improvements and also to demonstrate how current design methods and approaches
restrict the antenna elements to the surface of the enclosed volume or in some cases
increase the overall size of the antenna.
Figure 9: Performance of established designs, Q(ka)
22
B. Toploading
1) Designs
Toploading is one of the earliest mechanisms employed for reducing the physical size
of an HF antenna without giving up too much performance. Marconi described a radial
network for reducing the height of vertical HF antennas, documented in his 1904 patent
[21]. This antenna design consisted of a reduced height vertical element with a system of
8 radial arms extending outward from the top. Marconi’s design, as modeled in NEC, is
depicted in Figure 10.
The purpose of toploading is to modify the current distribution on the vertical
(radiating) element from being triangular (as previously depicted in Figure 4) to being
more uniform in magnitude along the entire length. As the current distribution is made
more uniform, the radiation resistance and corresponding radiated power are also
Figure 10: NEC model of Marconi’s 1904 toploaded antenna
23
increased. A sampling of toploaded designs described by Best and Hanna [8] were
modeled and simulated in NEC to validate the antenna metrics and algorithms
implemented herein and to provide a baseline for comparing antenna performance. A
comparative analysis of mesh versus radial toploading was conducted to determine the
impacts on performance for both structures. The term mesh is used here to describe a
web-like wire structure with radial and cross radial components providing multiple
symmetric paths for current flow.
The current distribution in a normal mode helical antenna at resonance is similar to
other wire antennas, with maximum current at the feed point, zero current at the opposite
(open) end of the wire, and continuous distribution along the length of the wire.
Traditional toploading techniques would place a disk or radial network at the top of the
helical antenna to provide for more uniform current distribution and for lowering the
resonant frequency [8]. The drawbacks of this method include the increase in volume and
ka of the antenna due to the additional external components.
2) Simulation Results
NEC models for monopole antennas, with and without mesh toploading, are depicted in
Figure 11. These antennas were simulated using a perfect electric conductor (PEC)
ground plane; current distribution is represented by color. The NEC simulation results for
these two models were comparable to the results in [8] and are listed in Table 2 along
with the results for designs scaled to 6 MHz.
24
Table 2. Toploaded λ/4 Monopole Performance
Antenna Frequency (MHz) ka Height
(meters) Resistance
(Ω) Reactance
(Ω) Qz(ω) r’/h
Monopole A 300 0.52 .0848 3.0 -461.4 173 0.56
- Mesh 300 0.59 .0848 10.7 0.8 16 1.2 Monopole
B 6 0.52 4.22 3.0 -463.7 176 0.56
- Mesh 6 0.59 4.22 10.6 -0.6 16 1.2
- Radial 6 0.63 4.22 10.6 -0.2 16 1.2
Figure 11: Monopole antenna over PEC ground (left) and with mesh toploading (right)
25
Figure 12 plots the current distribution for a monopole antenna with and without
toploading along with a comparison of two different toploading geometries, radial and
mesh. The effects of radial and mesh toploading appear to be virtually the same.
The improvement in monopole antenna performance through toploading has been
demonstrated. This technique improves performance by providing a more uniform
current distribution along the vertical (radiating) element of the antenna due to the
relationship between the radiation resistance and the ratio of average to maximum
current, as described earlier in (15) and (16). Given that the ratio of average to maximum
current (Iavg/Imax) for a triangular distribution is about 0.5 and the ratio for uniform
distribution is about 1.0, the resulting increase for uniform over triangular distribution is
2:1. The radiation resistance is proportional to the square of the current ratio (16) so the
Figure 12: Current distribution for monopole with and without toploading
00.10.20.30.40.50.60.70.80.9
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Curr
ent (
Amps
)
Segment #
Current Distribution - Monopole Antenna with and without toploading
Mesh toploading Radial toploading No toploading
26
maximum achievable improvement in radiation resistance is roughly a factor of four.
This property was verified by adding toploading to the design depicted in Figure 11,
resulting in an increase of input resistance by a factor of approximately 3.5:1 as observed
in Table 2 data. The trade-off however, is an increase in ka resulting from the additional
width added by the toploading structure. This may or may not be acceptable depending
on the specific design requirements for the antenna physical dimensions.
During the analysis for toploaded designs, two different structures for achieving
uniform current distribution were compared. The radial design depicted in Figure 10 was
scaled to the same height as the mesh structure shown in Figure 11 and the width adjusted
to achieve self-resonance at 6 MHz. The performance of the radial design was very
similar to the results of the mesh structure with the exception of ka which was slightly
larger due to the increased radius required to achieve resonance at the same frequency.
27
C. Folding
1) Designs
One popular method for increasing the radiation resistance is commonly referred to as
“folding” or “folded arms”. The fundamental properties of folding have been well-
documented in books and journals including [8], [19], [20], and [22]. The addition of
folded elements to a basic antenna is one method for improving the radiation resistance
without expanding the height or volume of the antenna. The geometry of a folded antenna
typically consists of a symmetrically repeating pattern based on the initial element (e.g., a
quarter-wavelength, straight-wire monopole). The additional components are positioned
at a specified distance and connected at the ends. The antenna feed point is connected at
the base of the initial element while the bases of additional elements are short-circuited to
ground. When properly connected, this structure forms a “folded arm” geometry.
Additional arms may be added by short-circuiting the base of the additional elements to
ground and connecting all elements to each other at the top. Balanis [19] provides a
derivation of the input impedance for folded antennas, which for a simple straight-wire
design is proportional to the square of the total number of arms. For example, the input
impedance of a quarter-wave monopole antenna (single arm) is about 36 Ω while the
input impedance with two folded arms is about 4 times that, or 144 Ω. This property can
be very useful when trying to increase input impedance of an electrically small antenna
for matching purposes and/or improving radiation performance.
28
Figure 13 depicts folded monopole antennas with two and three arms, as modeled in
NEC. The vertical elements are connected together at the top and connected to the PEC
ground at the bottom. The small red circle indicates the feed point.
2) Simulation Results
A variety of folded designs were simulated in NEC with results comparable to
previously published antenna performance. An analysis was conducted on the effects of
folding for a meandering line antenna (MLA) [20] designed for resonance at 6 MHz and
ka < 0.5. The three arm variant of a folded MLA is depicted in Figure 14 and analysis
results for two through six arms are provided in Table 3.
.
Figure 13: Two-arm and three-arm folded monopole antennas
29
Table 3. MLA Performance
# of arms ka Height
(meters) Resistance
(Ω) Reactance
(Ω) Qz(ω) r’/h
2 0.419 2.17 8.5 0.0 100.2 1.29
3 0.424 2.17 17.4 0.0 75.5 1.42
4 0.429 2.17 29.3 0.0 62.2 1.51
5 0.436 2.17 44.6 0.0 54.0 1.59
6 0.443 2.17 63.5 0.0 48.7 1.64
The results listed in Table 3 were obtained through modeling of the design in NEC with
#10 copper wire over PEC ground plane. The width of the meandering line component
was adjusted for each configuration to maintain resonance at 6 MHz, constant height, and
ka < 0.5. The small increases in the width of the meandering elements are reflected in a
slight increase in ka.
Figure 14: Folded meandering line antenna with three arms
30
D. Summary
The primary benefit of toploading is the uniform current distribution in the radiating
elements of the antenna, resulting in a lower resonant frequency. This improvement in the
current distribution is realized by providing additional paths for current flow at the end of
the monopole element. Different methods for implementing toploading include radial
arms, mesh networks, and solid disks, however an important trade-off is the increase in
physical size due to the addition of the toploading components.
The value of folded arms is realized in the corresponding increase in input resistance
due to the cumulative effects of current distribution in all the arms. The relationship
between the number of folded arms and the radiation resistance of an antenna provides a
method for optimization of antenna impedance for input matching and improvement of
radiation properties. This can be extremely useful for improving the performance of
electrically small antennas which typically exhibit input impedances well under 10 Ω.
During the evaluation of established designs, it was observed that many designs
published as ESA had ka > 0.5 when scaled for the HF frequencies, heights ranged from
two to six meters, and many were only resonant at a single frequency in the HF band. It
was also observed that a majority of the designs only placed wire on the outer surface of
the enclosing volume and many required additional tuning and matching networks. This
establishes the need for a new design methodology for electrically small antennas that
enables design approaches to effectively utilize the inner volume of an antenna’s
geometry.
31
Chapter 5 New Concept and Design Approaches
A. Background
A currently prevalent theme in ESA methodologies is that “optimization” involves
maximizing the placement of wire on the surface of the sphere enclosing the antenna
volume [19]. This section presents a new alternative design methodology utilizing the
inner space of the enclosed volume to achieve self-resonances at much lower frequencies.
This methodology offers designers an alternative when the design requirements and
restrictions on maximum height and volume would otherwise not support self-resonance
at a lower required frequency. Several alternative designs have been simulated and
analyzed, comparing performance parameters including radiation resistance, Q,
bandwidth, and the minimum operating frequency. Results from these simulations are
presented and trade-offs discussed. The design approaches presented herein provide
innovative methods to utilize the entire volume of the space enclosing the antenna [23], a
departure from methods typically described in current publications. The concept of “inner
toploading” is introduced here, followed by design approaches for implementing the new
concept. Notional requirements were established (for the purposes of this research) for an
antenna that is less than one meter high, one meter wide, and resonant within the HF
band. The concepts, methodologies, and designs presented herein are targeted for
producing an antenna that satisfies those requirements.
32
B. Inner Toploading
The concept of inner toploading is a modification of traditional methods by moving the
toploading elements from the exterior of the antenna geometry to the interior, as depicted
in Figure 15. This provides the toploading effects on current distribution without
increasing the size of the antenna. In this method the radius of the enclosing sphere “a” is
kept constant, yet the overall ka is reduced due to the longer wavelengths of the lower
resonant frequencies achieved.
The resonant frequency of this design may be tuned by changing the length of the
toploading radials inside the volume, similar to traditional toploading. Although helical in
nature, the far-field pattern is omnidirectional in azimuth and vertically polarized. A 25%
reduction in resonant frequency was achieved in designs using this concept with no
increase in antenna enclosed volume.
Figure 15: The concept of inner toploading
33
Table 4 provides an initial analysis of the effects of inner toploading on the self-
resonant frequency, resistance, Qz, and ka for helical antennas with various helical
configurations with and without inner toploading. The values provided in Table 4
represent the initial resonance for each design.
Table 4. Design Analysis for Inner Toploading
Helical Antenna
Parameters Frequency
(MHz) Resistance
(Ohms) ka Qz Wire Length
(meters) Turns: 0.75 26.1 6.917 0.62 44 3.09 Turns: 1.0 21.5 4.572 0.51 67 3.83 Turns: 1.5 15.8 2.674 0.38 123 5.35 Turns: 2.0 12.5 1.864 0.30 187 6.90
Turns: 0.75 with toploading 18.3 4.491 0.43 72 6.19
Turns: 1.0 with toploading 16.0 3.433 0.38 97 6.93
Turns: 1.5 with toploading 12.9 2.381 0.31 152 8.45
Turns: 2.0 with toploading 10.7 1.780 0.26 214 10.0
The reduction in self-resonant frequency and the matching reduction in ka were
achieved within the same occupied volume due to the improved current distribution
provided by the toploading elements. The current magnitude for the one-turn helical
antenna depicted in Figure 16 was analyzed with and without inner toploading; results are
plotted in Figure 17. This methodology offers designers an alternative when the system
design requirements and restrictions on maximum height and volume would otherwise
not support self-resonance at a lower (yet required) frequency.
34
Figure 17: Current Magnitude, helical antenna with and without inner toploading.
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60Curr
ent M
agni
tude
(nor
mal
ized)
Segment #
with toploading without toploading
Figure 16: One-turn helical with inner toploading.
35
The effects of inner toploading on Q as a function of ka for the antenna configurations
listed in Table 4 are plotted in Figure 18.
Figure 18: One-turn helical with inner toploading.
0
50
100
150
200
250
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Q
ka
without toploading with toploading
36
A prototype two-turn helical antenna was constructed as depicted in Figure 19. A
comparison of simulated and measured S11 for the antenna with and without toploading
is provided in Figure 20. A frequency reduction of 15% was achieved for this design
configuration with no increase in antenna size.
Figure 19: Simulated and measured S11with and without inner toploading
-10.0-9.0-8.0-7.0-6.0-5.0-4.0-3.0-2.0-1.00.0
140 160 180 200 220 240 260 280
S11
Log-
Mag
(dB)
Frequency (MHz) without toploading - simulated with toploading - simulatedwithout toploading - measured with toploading - measured
Figure 20: Prototype helical antenna with inner toploading
37
C. New Design Methodology
There are a variety of methods for adding elements to the surface of an antenna’s
enclosed volume to improve performance [8]. In one example, straight wire
components of antenna geometries are replaced with helical components to lower the
resonant frequency within a given volume [24]. The methodology and design
approaches presented herein build upon those initial concepts and provide a
framework for more efficient use of the entire volume enclosed by an antenna. This is
a departure from methods typically described in current publications which focus on
using the outer surface. This methodology is based on first determining any
limitations and requirements for height, volume, and frequency in order to establish
the maximum value for ka allowed by those requirements. The preliminary steps
required are:
1. Establish the maximum height limitation for the physical antenna
2. Establish the maximum volume limitation for the physical antenna
3. Establish the requirement for lowest operating frequency
4. Determine the ka for the above parameters
5. Determine whether the expected performance for this ka is acceptable (e.g., input
impedance and Q)
6. Effectively use the antenna’s inner volume for placing elements in order to
achieve required performance (e.g., inner toploading)
Several new design approaches developed using this methodology are presented,
including variations on toroidal and helical geometries, helical meandering line
geometries, and fractal geometries.
38
D. Novel Designs for Electrically Small HF Antennas
1) Helical Meandering Line Antenna (MLA)
A new design combining helical and meandering line elements for compact high
frequency (HF) antennas was presented in [25] and [26]. This antenna was designed to be
portable and rapidly deployable, while maintaining a comparatively small profile for
mobile radar applications. The Helical MLA design is low-profile (less than one meter
high) and provides for effective performance, improved antenna radiation resistance, and
strong vertical polarization while maintaining an omnidirectional radiation pattern with
low take-off angle. The Helical MLA, depicted in Figure 21, also provides for multiple
self-resonance frequencies allowing for selectable channels without requiring additional
matching networks.
Figure 21: Helical meandering line antenna
39
Figure 22 depicts the impedance of the helical MLA in the HF band. Figure 23 depicts
the impedance for the range 30 – 100 MHz, showing multiple self-resonances above the
HF band.
Figure 23: Impedance, helical MLA, 30 – 100 MHz
Figure 22: Impedance, helical MLA, 3 – 30 MHz
40
One of the arms of the helical MLA serves as the input or feed port. The other arms
may be either connected to the copper ground disk (folded arm configuration, referred to
as the “short” mode) or left open circuit (inner toploading configuration, referred to as the
“open” mode). Table 5 provides a comparison of the various effects of number of arms
on performance for the helical MLA. This analysis was performed using NEC 4.2 with an
infinite PEC ground plane and AWG #10 copper wire. The different effects of toploading
(open-circuit mode) versus folding (short-circuit mode) are observed in comparing the
performance for various configurations.
Table 5. Helical MLA Performance
# of arms
Frequency (MHz) Resistance (Ω) ka Q Efficiency
(%) Mode
1 12.20 2.57 0.28 254 63.6 NA
2 13.33 9.97 0.30 154 79.2 SHORT
3 14.47 24.9 0.33 109 86.2 SHORT
4 15.55 49.8 0.35 81 90.0 SHORT
5 16.55 59.6 0.39 52 91.4 SHORT
2 6.47 1.77 0.15 415 17.1 OPEN
3 5.57 1.55 0.13 400 13.4 OPEN
4 5.09 1.49 0.12 391 11.2 OPEN
5 4.80 1.47 0.11 385 9.3 OPEN
41
Figure 24 displays the far-field radiation pattern for the three-arm helical MLA as
modeled in FEKO. The pattern is vertically polarized and omnidirectional with low take-
off angle. Simulation depicted is over lossy ground plane with permittivity εr = 13 and
conductivity σ = 0.01 to emulate a coastal environment with saltwater on one side of the
antenna and regions of iron rich soil on the other.
Figure 24: Far-field radiation pattern
42
Figure 25 displays the current distribution for the first resonant frequency as modeled in
FEKO. At first resonance, the current density is maximum (red) at the base and minimum
(blue) at the intersection of the folded arms at the top of the antenna.
Figure 25: Current magnitude in three-arm helical MLA
43
Figure 26 plots the current magnitude through the segments of one arm from bottom to
top of the antenna. Figure 27 plots the corresponding current phase. The current
magnitude is symmetrical for the three arms at the first resonance as shown in Figure 25.
Figure 27: Current Phase in one arm
Figure 26: Current Magnitude in one arm
Segment
Am
ps
Segment
Deg
rees
44
Additional analysis was performed on the orientation of the horizontal helical elements
in each arm. The initial design depicted in Figure 21 was modified so that the direction of
the coil turns in each helical element were opposite of the previous element. A single arm
is depicted in Figure 28 for clarity; the analysis was performed with the three-arm
HMLA. Table 6 provides a comparison of the two configurations and shows the
similarity in performance.
Table 6. Performance for three-arm HMLA, direction of helical coils modified
HMLA Mode ka Qz(ω) Resistance (Ω)
Efficiency(%) r’/h
original SHORT 0.33 107 24.74 86.2 1.20
modified SHORT 0.32 112 23.68 85.8 1.22
original OPEN 0.19 402 1.55 13.4 2.02
modified OPEN 0.19 403 1.54 13.2 2.06
Figure 28: HMLA single arm, original (left), modified with alternating turns (right)
45
2) Toroidal Helical Antenna and Variations
The new method can also be applied to canonical antenna designs such as the helical
antenna. The helical geometry can be modified by replacing the straight wire sections
with helical elements creating a toroidal helical antenna. An example of this geometry is
presented in Figure 29. This design also provides additional options for inserting
toploading elements inside the volume for achieving lower self-resonant frequencies
without increasing the overall volume of the antenna, and provides additional options for
increasing wire length by modifying the radius and/or pitch of the helical coils. Figure 30
depicts the impedance and Figure 31 depicts the gain for a one-turn toroidal helical
antenna as simulated using NEC.
Figure 29: One-turn toroidal helical antenna
46
Figure 30: Impedance for one-turn toroidal helical antenna
Figure 31: Gain for one-turn toroidal helical antenna
47
The self-resonant frequency of this design was further reduced by adding toploading in
the form of a two-turn toroidal helical element on the interior of the antenna volume as
depicted in Figure 32. These designs also allow for the use of folded arms to increase the
antenna impedance in order to offset the reduction in resistance due to the increase in
resonant wavelength. Figure 33 depicts the impedance and Figure 34 depicts the gain
pattern for a one-turn toroidal helical antenna with inner toploading as simulated in NEC.
Figure 32: One-turn toroidal helical antenna with two-turn inner toploading
48
Figure 33: Impedance for one-turn toroidal helical antenna with inner toploading
Figure 34: Gain for one-turn toroidal helical antenna with inner toploading
49
A variation on the design theme is depicted in Figure 35 with four arms of one-half turn
each and toploaded with a single circular wire. The base of each arm is short-circuited to
the PEC ground plane to provide for symmetric current distribution. The circular wire
provides for toploading without increasing the enclosed volume of the antenna. The
combination of toploading for improving current distribution and folding for increasing
input impedance result in optimized Q for a given ka. Figure 36 depicts the impedance
and Figure 37 depicts the gain pattern for a four-arm toroidal helical antenna as simulated
in NEC.
Figure 35: Toroidal helical antenna with four half-turn folded arms
50
Figure 36: Impedance for half-turn toroidal helical antenna, four folded arms
Figure 37: Gain for half-turn toroidal helical antenna, four folded arms
51
A comparison of toroidal antenna designs is listed in Table 7 where it is observed that
folded arms (Figure 35) satisfied design requirements in terms of input resistance and Q.
Table 7. Helical MLA and Toroidal Helical Performance
Antenna Frequency (MHz)
Resistance (Ω) ka Q
Wire Length (meters)
Helical MLA (folded arm) Figure 21 15.6 20 0.36 111 34.1
Toroidal Helical Figure 29 15.0 2.5 0.32 160 8.5
Toroidal Helical (toploaded) Figure 32 8.5 2 0.19 355 48.0
Toroidal Helical (folded arm) Figure 35 14.4 24 0.32 82 32.9
3) Observations
After analyzing numerous helical meandering line and toroidal helical designs, it was
observed that both approaches provided for significant decreases in ka for HF band
antennas. The concept of inner toploading was also demonstrated and experimentally
verified using both design approaches. Resonant frequency, ka, and Q were all reduced
by as much as 50% depending on the design approach selected, providing options when
requirements or other size limitations restrict the allowable height or volume of the
antenna.
52
E. Investigation of Fractal Geometries
The term “fractal” has almost as many definitions as it does applications across various
fields of study. One popular definition, published by Benoit Mandelbrot, is "a rough or
fragmented geometric shape that can be split into parts, each of which is (at least
approximately) a reduced-size copy of the whole" [27]. In simpler terms, “fractal” can be
used to describe patterns with a self-similar or repeating nature. The concept of self-
similar nature is also applicable when using the terminology “fractal antenna” which is
defined in the IEEE standard as “A multiband antenna having a self-similar shape at
several different scales” [28]. A fractal antenna is usually designed through successive
iterations of applying a generator function to a basis (or “initiator”) shape.
One practical use for fractal geometries is they provide a method for generating
complex structures within a bounded region. This property, originally investigated by
mathematicians such as Hilbert [29], has been applied successfully to the design of many
types of antennas.
A particularly interesting parameter for analyzing fractals is the “fractal dimension”,
advocated by Mandelbrot [27] and defined as the dimension D = log N / log(1/r)
representing a bent line with N equal sides of length r. Calculation of the fractal
dimension for a Koch fractal structure is straightforward. The Koch geometry depicted in
Figure 38 is observed to have four equal sections (N = 4) with lengths equal to one-third
of the total length (r = 1/3). Using these values the fractal dimension is calculated to be D
= 1.26186, in contrast to the Euclidean spatial dimensions of lines (1 dimension), planes
(2 dimensions), and volumes (3 dimensions). This is relevant because one definition for
53
the term fractal requires that the fractal dimension “strictly exceed” the topological
dimension in Euclidean space [27].
The image shown in Figure 38 can be used to generate a Koch monopole antenna
design by replacing each unbroken line segment with the generator. The Koch fractal
pattern emerges as observed in the NEC models shown in Figure 39 for one and two
iterations.
Figure 38: Generator for a Koch curve
N = 4 r = 1/3
Figure 39: Koch antennas after one and two iterations
54
Other interesting examples of fractal designs, developed using the Iterated Function
System (IFS) [30], include the Sierpinski triangle depicted in Figure 40 and the fractal
tree presented in Figure 41. These images were generated using a custom LabVIEW
program developed for generating fractal images.
The Sierpinski triangle is interesting in that the transform function will always generate
this triangular form, regardless of the basis image used. The fractal tree, on the other
hand, is generated from a single straight line yet it evolves into an infinitely complex
geometry. It is also worth noting that both of these examples, as designed, have finite
boundaries as the number of iterations increases towards infinity. The Sierpinski triangle
is more suited for patch antennas so the fractal tree geometry was chosen for further
analysis.
Figure 41: Fractal tree from IFS
Figure 40: Sierpinski Triangle from IFS
55
1) Fractal Tree Geometries
The fractal tree seemed an appropriate choice for designing novel HF antennas and
warranted further investigation. This fractal pattern provides for longer unperturbed
segments while also offering improved current distribution over the Koch geometry. A
single straight wire was chosen for the fractal basis due to its simplicity and ease of
modeling in NEC. A transformation algorithm was developed to reproduce the basis
image using a scaling factor of 0.5 and a user-specified rotation relative to the z-axis.
Transformations were also developed to allow for multiple arms distributed
symmetrically around the z-axis in the fractal geometry. Self-similarity is maintained
throughout the process and the angular relationships are preserved as well. Fractal tree
antenna designs for one and two iterations, with two symmetric arms and 45 degree
rotation, are depicted in Figure 42.
Figure 42: Fractal tree antennas, iteration #2 and #3
56
These examples depict a two-arm configuration where the transformation involves two
new wires with lengths of one-half the previous length, rotated 45 degrees, and attached
at the midpoint of the generating wire. It is also noted that these parameters were chosen
so that the fractal geometry does not grow beyond the maximum height of the initial
vertical wire after successive fractal iterations and also does not expand the enclosed
spherical volume. The two-arm configuration is planar in nature, however three-arm and
higher configurations more effectively utilize the inner volume of the antenna. In the
fourth iteration the new segments are one-eighth of the original length, the self-similar
nature is clearly visible, and all angles are preserved. The fractal tree with 45 degree
angles and four iterations is depicted in Figure 43.
Figure 43: Fractal tree antenna with two arms, iteration #4
57
The three-arm configuration after four iterations is displayed in Figure 44. The three-
dimensional nature is clearly evolving and in the context of antenna design, each branch
provides additional current paths, thus providing a toploading effect.
Figure 44: Fractal tree antenna with three arms, iteration #4
58
The fractal tree depicted in Figure 45 demonstrates the 30 degree angle configuration
for four arms after four iterations. Issues such as intersecting wires start to become
prevalent with continued iterations if the number of arms and/or angles are chosen
indiscriminately.
Figure 45: Fractal tree antenna with four arms, iteration #4
59
It was observed that the fractal trees presented thus far, when constrained to a
maximum height of one meter, have ka values well above 0.5 and lowest resonant
frequencies above the HF band. These designs were then modified to include helical
elements in the fractal geometry. Figure 46 depicts the first iteration of a helical fractal
tree with two arms. The branch angle is 45 degrees and branch arms are scaled by 50%
and positioned at the mid-point of the first (vertical) helical element.
Figure 46: Helical fractal tree with two arms, iteration #1
60
Figure 47 depicts a helical fractal tree with two arms after two iterations. The scaling
factor of 50% is continuously applied at each iteration.
Figure 47: Helical fractal tree with two arms, iteration #2
61
2) Fractal Tree Antenna Performance
Fractal parameters were selected to maintain designs that converged to physical
dimensions within the specified maximum height (one meter) to allow for direct
comparison of antenna performance for each configuration. Performance was evaluated
at one constant frequency (20 MHz) for better comparison of different designs with
identical ka and heights. The ka for these geometries at 20 Mhz is 0.42, however when
constricted to a height of less than one meter, none of the designs were resonant within
the HF band. Performance characteristics for fractal tree antenna designs were evaluated
for multiple arm configurations over two, three, and four fractal iterations. Simulation
results are provided in the comparative plots depicted in Figure 48 and tabular data listed
in Table 8. Of the fractal tree geometries evaluated, the optimum configuration was
determined to be four iterations with four arms and a fractal angle of 45 degrees, however
within the given height constraints, the lowest achievable resonance was 32.8 MHz, just
above the HF band. Configurations with greater than four iterations became problematic
due to intersecting wire segments and decreasing separation between elements.
Figure 48: Comparison of fractal tree geometries
0
50
100
150
200
250
30 45 60 75
Q
Fractal Angle (degrees)
Q for Fractal Tree Geometries Iterations 4 Arms 4
Iterations 4 Arms 3
Iterations 4 Arms 2
Iterations 3 Arms 4
Iterations 3 Arms 3
Iterations 3 Arms 2
Iterations 2 Arms 4
Iterations 2 Arms 3
Iterations 2 Arms 2
62
Table 8. Fractal Tree Performance at 20 MHz, one meter height
Fractal Iterations
# of arms
Angle (degrees)
Resistance (Ω)
Reactance (Ω) Qz(ω) r’/h
2 2 30 2.86 -453.8 195 0.679 2 2 45 2.56 -436.8 198 0.676 2 2 60 2.35 -428.6 211 0.661 2 2 75 2.11 -425.4 233 0.639 2 3 30 2.89 -388.9 160 0.726 2 3 45 2.73 -369 161 0.724 2 3 60 2.47 -356 173 0.707 2 3 75 2.17 -352 194 0.68 2 4 30 3.04 -347 139 0.76 2 4 45 2.85 -321 139 0.76 2 4 60 2.56 -308 149 0.742 2 4 75 2.21 -303 169 0.712 3 2 30 3.15 -347 135 0.768 3 2 45 2.91 -322 138 0.763 3 2 60 2.54 -311 152 0.738 3 2 75 2.17 -301 176 0.703 3 3 30 3.53 -259 96.9 0.857 3 3 45 3.20 -228 96.9 0.857 3 3 60 2.73 -214 108 0.828 3 3 75 2.25 -210 128 0.782 3 4 30 3.74 -213 80.2 0.913 3 4 45 3.36 -178 78.6 0.920 3 4 60 2.81 -163 87.2 0.888 3 4 75 2.28 -159 105 0.836 4 2 30 3.40 -299 112 0.817 4 2 45 3.08 -270 115 0.811 4 2 60 2.63 -258 129 0.780 4 2 75 2.20 -257 152 0.738 4 3 30 3.89 -201 74.6 0.936 4 3 45 3.44 -166 74.0 0.938 4 3 60 2.83 -153 83.9 0.900 4 3 75 2.27 -151 102 0.842 4 4 30 4.07 -160 61.8 0.996 4 4 45 3.55 -123 60.0 1.006 4 4 60 2.87 -108 67.9 0.966 4 4 75 2.27 -113 85.7 0.893
63
Figure 49 plots the input impedance for the four-arm fractal tree depicted in Figure 45.
The plot shows that this design only has a single resonance at 34.7 MHz when the height
is constrained to one meter. Figure 50 depicts the current magnitude at 20 MHz.
Figure 49: Impedance for four-arm fractal tree
Figure 50: Current Magnitude for four-arm fractal tree
64
Figure 51 is the gain pattern for this design evaluated at 20 MHz, showing the strong
vertical polarization.
Figure 51: Gain pattern for four-arm fractal tree
65
Table 9 lists the performance comparison between straight wire and helical fractal
trees. All designs are constrained to a maximum height and width of one meter, with the
design goal of resonances in the HF band. Applying helical elements within the fractal
tree structures enabled antenna designs to achieve resonances in the HF band while
meeting the restrictions on height and width.
Table 9. Fractal Tree and Helical Fractal Tree Performance
Antenna arms Frequency (MHz) ka Qz(ω) Resistance
(Ω)
Straight Wire - Initial - 71.9 1.51 7.9 36.26
Straight wire fractal tree, iteration 1
2 51.9 1.09 10.3 21.48
iteration 1 3 47.6 1.00 11 18.59
iteration 1 4 44.6 0.94 11.6 16.72
iteration 2 2 44.0 0.92 12.1 16.51
iteration 2 3 38.0 0.80 13.6 12.92
iteration 2 4 34.7 0.73 14.6 11.01
Helical fractal tree - initial - 26.4 0.55 69.9 5.6
iteration 1 2 27.0 0.54 51.2 5.6
iteration 1 3 24.5 0.51 52.9 5.12
iteration 1 4 23.5 0.49 54.1 4.83
iteration 2 2 23.8 0.50 55.5 4.9
iteration 2 3 21.5 0.45 58.6 4.5
iteration 2 4 20.5 0.42 60.9 3.82
66
3) Hilbert Curve Geometries
Hilbert space-filling curves were also investigated as they are often included in
discussions on fractal designs. These curves were first described by Hilbert in a two-page
article published in 1891 [29], “Ueber die stetige Abbildung einer Linie auf ein
Flächenstück”. This article was written in German; further research did not locate a
suitable English translation so it was translated by the author (J.M. Baker) and provided
in Appendix A. The initiator for the Hilbert curve is a simple three-sided square.
Successive iterations involve scaling and rotating the initiator following Hilbert’s
methodology, producing a continuous space-filling curve that is bounded by a finite
square. The first three iterations are depicted in Figure 52.
Figure 52: Hilbert curves
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16iteration 1 iteration 2 iteration 3
67
A monopole antenna based on the first iteration of the Hilbert curve is depicted in
Figure 53. The antenna was rotated 45̊ to minimize coupling effects between the open
end of the wire and the ground plane. For this configuration the antenna height required
to achieve resonance at 6 MHz was over 6 meters.
Figure 53: Antenna, Hilbert curve, one iteration
68
This initial design was then modified to incorporate the second iteration of space-filling
curves as depicted in Figure 54. The overall height required for resonance at 6 MHz was
reduced by two meters.
Figure 54: Antenna, Hilbert curve, second iteration
69
4) Hilbert Curve Antenna Performance
The antenna models depicted in Figure 53 and Figure 54 were analyzed using NEC
with results presented in Table 10. These antennas were designed for resonance at 6 MHz
using AWG #10 copper wire over a PEC ground plane.
Table 10. Hilbert Curve Simulated Performance
Iteration ka Height (meters)
Wire Length (meters)
Resistance (Ω) Qz(ω) r_eff/h
First 0.79 6.29 13.25 12.1 30.5 0.67
Second 0.53 4.25 14.9 5.0 78.9 0.72
A scaled down prototype antenna for the second iteration Hilbert curve was constructed
and measured using a VNA. This prototype, depicted in Figure 55 was designed for
resonance above 300 MHz using AWG #18 copper wire over a copper plated ground
plane.
Figure 55: Antenna Prototype, Hilbert curve, second iteration
70
A comparison of simulated and measured S11 and input impedance parameters are
provided in Figure 56 and Figure 57 respectively.
Figure 56: Simulated and measured S11 for Hilbert prototype
-5
-4
-3
-2
-1
0
250 270 290 310 330 350
S11
logm
ag (d
B)
Frequency (MHz)
S11 (dB) - Simulated S11 (dB) - Measured
Figure 57: Simulated and measured Impedance for Hilbert prototype
-50
-30
-10
10
30
50
0
20
40
60
80
100
250 260 270 280 290 300 310 320 330 340 350
Reac
tanc
e (O
hms)
Resi
stan
ce (O
hms)
Frequency (MHz)
Resistance - Simulated Resistance - Measured
Reactance - Simulated Reactance - Measured
71
F. Summary
The debate continues as to whether antenna performance is improved due to the fractal
nature of the antenna or due to other inherent antenna properties [31] [32], however most
research to date has focused on planar designs such as Koch fractals [33]. The Koch
fractal monopole design was investigated and used to validate models and simulations
against current publications. A fractal antenna design based on Hilbert’s space-filling
curves was also investigated. Although initial iterations did achieve a significant
reduction in antenna size for a given frequency, this was offset by a corresponding
reduction in overall performance, as verified in experimental measurements.
New fractal designs that better utilize the inner volume occupied by antenna structure
were developed and simulated with detailed performance characteristics provided for
comparison. Performance was significantly improved when geometries made efficient
use of enclosed spherical volumes, as demonstrated in the helical fractal tree
configurations. It was also observed that some fractal geometries do support improved
current distributions for enhancing antenna performance.
It is interesting to contrast the fundamental properties of fractal geometries as defined
by Mandelbrot with the electromagnetic field properties established by Maxwell [34].
Fractals by definition are nondifferentiable, yet Maxwell established the fundamental
laws of electromagnetism in the form of differential equations. In the course of research
for this dissertation it was observed that the term “fractal” is often applied in a more
general sense and that regardless of the fractal methodology used, the physical
geometries of wire antennas are differentiable by the nature of their physical dimensions
such as wire thickness and the reality that wire angles are rounded to some degree. The
72
primary observed value of using fractal designs was in the mathematical nature of their
geometries. It was found to be much easier to design a fractal antenna with complex,
multidimensional geometries which satisfied simple rules (e.g., no intersections of wires)
than it was to design a multidimensional random wire antenna with the same size and
wire length. It was also observed that some fractal geometries that are too complex for
wire antennas may be suitable for 2-D and 3-D printing.
73
Chapter 6 Algorithms for Design Optimization
A. Random Search
Random search is a method that seeks to identify local minima and the global minimum
for a given function over the selected interval. This method may be employed when it is
not feasible to do an exhaustive search by checking all possible combinations of input
variables [35]. The random search method may be conducted by defining a function with
the parameters to be optimized as input variables. Initial values are randomly selected
from within the global range and then analyzed to identify parameter values that result in
a function minimum. This process may be repeated with random values selected from
narrower regions in the vicinity of identified local minima. A limitation of this method is
there are no guarantees of identifying all local minima or the global minimum.
B. Nelder-Mead Downhill Simplex Algorithm
The Nelder-Mead Downhill Simplex algorithm is a numerical search method for
locating the minima of a specified objective function. The “simplex” is defined as a
simple geometry with N+1 sides for N-dimensional space (e.g., the simplex for a two-
dimensional space is a triangle). For this example, the method begins by selecting a
starting point and then deriving the other two points of the triangle. The objective
function is analyzed and the point with the largest objective function value is replaced by
a new point with a lower value. This results in either an expansion or a reflection of the
triangle until a minimum is detected within the triangle which results in a contraction.
This process repeats until the triangle surrounds a minimum and is contracted until a
specified tolerance is achieved. This method has two limitations: it is sensitive to the
74
initial start location and it has difficulty locating a global minimum when the objective
function has more than one minimum [35].
C. Simulated Annealing (SA)
Simulated Annealing is a numeric and probabilistic method for finding the global
optimum of an objective function over a large region or solution space. The term
“annealing” refers to the metallurgical process of heating a material and then controlling
the cooling process in order to maximize the size of crystals and minimize their defects.
This method has been effectively applied in areas ranging from signal processing to
operational research. It provides an alternative method for nonlinear and multimodal
systems [36]. The advantage of SA is its probabilistic nature enabling it to break away
from solutions converging to local minima when searching for a global minimum [35].
D. Genetic Algorithms (GA)
A genetic algorithm is a numerical and probabilistic method used to search for the
minima (or maxima) of a specified objective function. GA methodology is based on
biological processes: pairs of parents produce children, those children possess
characteristics similar to their parents, children become parents and produce new children
with new characteristics, and the possibility exists for random mutation. GA offers
several advantages over traditional numerical optimization methods, including the ability
to analyze multiple points distributed over the entire objective function space. This
enables GA to adjust more quickly than serial optimization methods. Other advantages
include optimization with continuous and discrete input parameters of mixed data types
and the use of random mutations to enable persistent searching for better solutions in
75
other regions of the objective function space. The GA method is similar to the SA method
with one primary difference. The SA method is a serial process starting with a single
point and solving iteratively for an optimized solution, whereas GA begins with multiple
starting points and solves them in parallel to achieve optimization.
The optimization of an electrically small antenna presents a unique set of challenges
including the dependency of performance on effective height and volume [18]. The
complex nature of these dependencies and their effects on the development of a cost
function made GA the most suitable method for optimizing antenna design. The GA
process was implemented by first identifying those input parameters which could be
controlled: combining and converting input parameter values into a binary string to create
chromosomes, selecting measurable output parameters, and combining those output
parameters into a single cost function for analysis. This was accomplished by developing
a custom program in LabVIEW [16] following the methodology described by Haupt [35].
The selected design goals used to establish the cost function were: resonant frequency
near 16 MHz, quality factor Qz(ω) < 40, and input resistance R near 50 Ω at resonance.
The resonant frequency was determined from the reactance values and the Qz(ω) was
calculated from the impedance data using (10). The effective spherical radius of the
overall geometry was calculated and used for comparing the various antenna geometries
generated in the GA process.
76
The specific procedures used in implementing the GA process were:
1. Select antenna geometry parameters (inputs)
2. Encode parameters into binary genes
3. Generate initial population of eight random chromosomes
4. Generate antenna model using parameters contained in each chromosome
5. Analyze performance for each antenna model (input impedance)
6. Calculate Q and effective spherical radius from input impedance
7. Determine score (cost functions) for each configuration (chromosome)
8. Rank chromosomes and select two mother/father pairs, discard bottom pairs
9. Generate children chromosomes from mother/father pairs, replace discarded pairs
10. Repeat process with new set of chromosomes until exit criteria satisfied
The antenna geometry parameters selected for GA optimization included the number of
folded arms (3-bits), the helical radius (7-bits), wire radius (5-bits), and number of turns
(4-bits) of the helical elements. These parameters were chosen in order to constrain the
overall geometry to a specified height and enclosed spherical radius. Parameters were
encoded into binary genes with established upper and lower limits for each variable and
combined to form a chromosome with 19 bits. A population of eight chromosomes was
then randomly generated for the initial configuration. This population was used to
generate an NEC input file for each chromosome with the appropriate parameters
77
decoded and placed in the variable definition section of each file. The program then sent
each file in turn to NEC using a command line executable for analysis of each respective
model and generation of an output file containing geometry parameters and antenna
performance characteristics. The LabVIEW program then extracted input impedance data
from the NEC output file, calculated Qz(ω) and effective radius, and calculated a score
using the cost function.
The cost function was calculated for eight chromosomes which were then sorted by
score. Following the principle of natural selection, the bottom four members of the
population were discarded and the top four members were selected for mating. Each
chromosome in the top four was ranked for mating via the roulette wheel selection
process and paired to create two sets of parents. A binary mask was generated for each
set of parents and uniform crossover applied to generate two offspring for each set. The
four new offspring chromosomes were added to the population to replace those
previously discarded. Finally, mutation was induced in the population by randomly
selecting a specified number of bit locations within the population and inverting those
bits to produce random variations. The selected mutation rate for this GA program was
10%, with the best scoring chromosome exempt from mutation. This resulted in 13 of
133 bits being randomly changed throughout the population, which completed the
process in preparing for the next generation. The newly generated and mutated
population was then used to build an NEC input file for each chromosome and the overall
GA process was repeated.
78
After four generations, a noticeable degree of improvement was observed in reduced Q
and increased input resistance. Figure 58 depicts the NEC model for one of the GA
optimized designs, a configuration with four arms which provided discernable
improvement over the baseline three-arm design. Other changes in this geometry
included reducing the helical element radius from 5 cm to 3 cm while increasing the
number of turns from 9 to 11.
Table 11 provides a comparison of the original design and the GA optimized designs
for the selected parameters. The baseline in the table refers to a three-arm helical MLA;
the tuned baseline is the helical MLA resized to achieve resonance at 16 MHz. GA-1 and
GA-2 refer to the new configurations generated using genetic algorithms. The first GA
model came closest to the desired frequency of 16 MHz while reducing Q from 101 down
to 90 and raising the input resistance from ~21 Ω to ~40 Ω. Although the total wire
length also increased, the effective radius was similar. The second GA model
Figure 58: GA optimized model
79
Figure 59: Q and input resistance for four-arm GA
significantly reduced Q, however this was at the expense of increasing the resonant
frequency from 16 MHz to 19 MHz. Figure 59 depicts the Q and corresponding input
resistance for the four-arm designs generated using GA.
Table 11. Baseline and GA Optimized Performance
Antenna Frequency (MHz) Q(ωο)
Resistance (Ω)
Effective Radius
(meters)
Wire Length (meters)
Baseline 14.3 156 17.8 1.06 36.8
Baseline (tuned) 15.9 101 21.5 1.07 35.5
GA-1 16.1 90 39.6 1.09 44.7
GA-2 19.2 54 54.6 1.08 37.9
80
Figure 60 plots the resistance (Z_Real) and reactance (Z_Imag) for the baseline three-
arm configuration. Figure 61 plots the resistance and reactance for the GA optimized
four-arm configuration. Both models were simulated in NEC using a PEC ground plane.
Figure 60: Impedance, baseline design
Figure 61: Impedance, GA optimized design
81
The plots in Figure 62 and Figure 63 show the improvement achieved in Q and
effective radius obtained through use of genetic algorithms as compared to the baseline
geometry and to a normal mode helical antenna of the same physical height and width.
Figure 62: Improvement in Q
Figure 63: Improvement in effective radius
82
E. Summary
A LabVIEW program was developed to implement the genetic algorithm process with
NEC modeling software used to analyze design performance. The use of the GA process
for optimization provided insight into design changes for improving performance over
the original design and highlighted the necessity of modifying more than one design
parameter at a time. Significant optimization was achieved when all of the variables
(including the number of arms, the number of helical turns within each arm, and the
radius of helical components) were optimized together instead of separately. It is noted
that within the limits placed on the physical height and width of the antenna, the resonant
frequency was primarily determined by the total wire length enclosed within the volume.
The parallel nature of genetic algorithms make it ideally suited for solving this type of
complex optimization problem. The main issue observed in using GA for optimization,
due to the stochastic nature of the algorithms, is the lack of confidence that the global
optimum will be found in every case. It was also observed that for a given set of input
parameters, GA may produce completely different solutions after different runs. Overall
though, the GA process did produce design configurations that had not been previously
considered, and also provided a substantial amount of performance data for a broad range
of design configurations which helped assess and compare many different design options.
83
Chapter 7 Experimental Verification
A. Field Test Configurations
Two prototypes of the helical MLA design were constructed and experimentally
verified through field testing, demonstrating functional input impedance and vertical
polarization while maintaining an omnidirectional antenna pattern with low take-off
angle [26]. Prototype antennas were constructed using copper wire with cardboard tubes
and wood dowels to form the helical elements and to provide structural support and
spacing. The base was constructed using a 24 inch round of ¾ inch plywood covered with
aluminum sheeting. The antenna feed is an N-type connector mounted directly to the
aluminum ground plate. An inline 1:1 RF isolator was also installed underneath the
ground plate at the feed point to isolate the coax cable and prevent it from radiating RF
energy reflected from the antenna. RF isolation is very important, yet often overlooked,
when measuring performance characteristics of electrically small antennas. RF energy
radiated from the outside of the coax is not measured by a network analyzer, making the
antenna appear to have better performance. The input resistance, measured with a
properly calibrated network analyzer, can appear to be as much as 10 times higher than
the true value. There are numerous examples of published experimental results, which
upon closer examination, are only achieved if the feed line is also radiating. This was
observed in the early stages of prototype development and field testing where initial field
performance seemed too good to be true.
84
Open and short circuit mode selection was implemented using switches installed at the
interfaces between antenna wire and the ground plane. These switches provided for mode
selection between the available frequency channels by providing a mechanism for
switching the two non-input arms between open circuit and short-circuit to the ground
plate. Performance was primarily analyzed without the use of matching networks or other
tuning mechanisms in order to determine antenna self-resonant characteristics, however
during communications testing an external antenna tuner was used to ensure transmit
operations were only conducted at authorized frequencies. These operations were
conducted at frequencies between 7.1 MHz and 28 MHz as allocated for amateur radio
and all transmit operations were conducted by licensed operators: AH6SU, WH6R, and
AH6TW.
B. Field Measurements
Antenna performance was measured using vector network analyzers and RF power
meters. Basic antenna measurements recorded during field testing included S11, input
impedance, and voltage standing wave ratio (VSWR). Gain patterns were measured at 16
MHz in the open-circuit mode by transmitting a 1 mW continuous wave (CW) signal
from a quarter-wave vertical reference antenna and then measuring received power at the
test antenna as it was manually rotated in 15° increments through 360° of azimuth. Phase
linearity was also verified within each resonant bandwidth for open and closed
configurations.
85
Communications testing was conducted within authorized HF amateur radio bands for
voice and CW modes of operation. Test setup consisted of one helical MLA antenna
prototype connected to an amateur radio transceiver via 100’ of coax cable and RF
isolation at the antenna feed point. Transmit power was maintained between 50 to 75
watts due to limitations on the portable battery used for system operations on the beach.
86
Figure 64 shows a comparison of simulated and measured S11 (dB) data for the open
circuit configuration over lossy ground. Figure 65 depicts these measurements for the
short circuit configuration.
Figure 64: Simulated and measured S11, open circuit mode over lossy ground
Figure 65: Simulated and measured S11, short circuit mode over lossy ground
87
A comparison of simulated versus measured half-power bandwidth (HPBW) is depicted
in Figure 66. The simulations used for this comparison accounted for system losses
including a lossy ground plane (sandy beach and saltwater) and copper wire. The larger
HPBW at 5.7 MHz is primarily due to the losses of the small (30 cm diameter) aluminum
ground plane disk and the real earth environment.
Figure 66: Simulated and measured HPBW
0%
2%
4%
6%
8%
10%
12%
14%
5.7 15.1 16.1 18.5 20.6 26.7 28.1
HPBW
Frequency (MHz)
Simulated Measured
88
The photo in Figure 67 shows a prototype antenna on the ridgeline above Hanauma
Bay, one of the locations chosen for measuring received power. The measured power is
plotted in Figure 68. A comparison of simulated and measured gain (normalized) is
depicted in Figure 69. Measurements were made using a portable Anritsu MS2036B
vector network analyzer connected to an omnidirectional vertical quarter-wave monopole
antenna. The far-field gain pattern was observed to be omnidirectional with +/- 0.5 dBm
at all azimuth angles.
Figure 67: Measuring antenna patterns near Hanauma Bay
89
Figure 68: Received power measured over azimuth
-60-59.5
-59-58.5
-58-57.5
-57-56.5
-56-55.5
-55
0 30 60 90 120 150 180 210 240 270 300 330 360
Pow
er (d
Bm)
Azimuth of RCV antenna relative to TX antenna (degrees)
Compact HF @ 16 MHz Reference @ 16 MHz
Figure 69: Simulated and measured gain at 16 MHz
90
During field testing the impedance was measured using the portable vector network
analyzer. The measured impedance for the open circuit mode is displayed in the Smith
chart depicted in Figure 70. The resonant frequencies and impedances are listed in the
table at the bottom of the figure and indicated by the green dots in the Smith chart.
Figure 70: Smith chart of measured impedance, open circuit mode
91
The measured impedances and resonant frequencies for the short circuit mode are
displayed in Figure 71.
Figure 71: Smith chart of measured impedance, short circuit mode
92
The photo in Figure 72 shows the helical MLA antenna prototype on a beach near the
Makai Research Pier during field testing with a Kenwood TS-570D HF transceiver. It is
worth noting that this antenna did not require any site preparation and could be easily
moved as the tide rolled in.
Figure 72: Field testing on beach near the Makai Research Pier
93
Figure 73 shows the location of amateur radio stations around the world with positive
contact (or at least received) using a helical MLA prototype during one night of field
testing at Waimanalo beach on the windward side of Oahu. The red pins on the map
indicate stations where positive two-way communications were established and the blue
pins indicate the location of radio stations only received. The levels of gray shading on
the map indicate the regions of night time and the twilight zone (transition between day
and night), and no shading indicates day time regions.
Figure 73: Amateur Radio Communications – Field Testing
94
Figure 74 shows the setup used during field testing of two prototype antennas for
measuring the beam-forming and phase properties of a two-element array. A digital,
multi-channel receiver developed at HCAC was used to record and analyze signal
characteristics.
Figure 74: Measuring two-element array properties at Waimanalo Park
95
During this event, a remote high power transmitter was radiating at a frequency below
and near the test frequency range of 5 – 6 MHz. The HCAC digital receiver was able to
filter out the interfering signal as shown in the Power Spectral Density charts, Figure 75.
Figure 75: Power Spectral Density, with and without filtering
0 2 4 6 8 10 12-140
-130
-120
-110
-100
-90
Frequency (MHz)
Pow
er/fr
eque
ncy
(dB
/Hz)
Power Spectral Density Estimate w/o Filter
0 2 4 6 8 10 12-140
-130
-120
-110
-100
-90
-80
Frequency (MHz)
Pow
er/fr
eque
ncy
(dB
/Hz)
Power Spectral Density Estimate with Filter
Ant1Ant2
Ant1Ant2
96
The photo in Figure 76 depicts the author and a prototype antenna on Sandy Beach
while setting up for field testing. This photo provides a visual indication of the relatively
low height of the HF antenna.
Figure 76: Field testing at Sandy Beach
97
Chapter 8 Summary and Conclusions
A new design methodology for reducing the size of electrically small antennas has been
presented and demonstrated. Innovative designs developed using the new methodology
were also presented. These compact antennas present very low profiles compared to
traditional antenna designs in the HF band (3 – 30 MHz) with heights less than one
meter. Antennas which are self-resonant at multiple frequencies throughout the HF band
have been developed, simulated, and experimentally verified. These designs are low cost,
light-weight, with minimal to no environmental impact making them suitable for military
and homeland security applications.
This design methodology provides additional options when antenna design
requirements restrict the volume and height to minimal limits. Methods for using the
inner volume of the antenna geometry are described as an alternative to traditional
designs that only use the outer surface. The concept of inner toploading was introduced
along with several design approaches for implementation. One method of accomplishing
this was the introduction of helical elements into meandering line and toroidal geometries
to achieve lower resonant frequencies while maintaining a low profile. Another
innovation is related to the combination of open and closed circuit termination on
multiple arms to combine the benefits of toploading (open arms) and folding (shorted
arms) which achieved lower resonant frequencies for a given volume while maintaining
good impedance matching characteristics.
Design optimization was achieved through the implementation of genetic algorithms in
a program developed in LabVIEW for parallel optimization of multiple antenna
98
parameters. This parallel nature of GA made it ideally suited for solving complex
optimization problems; there is no guarantee that the global optimum will be found in
every case due to their stochastic nature. It was also observed that for a constant set of
input parameters, GA may produce completely different solutions after different runs.
Another area investigated during this research was the feasibility of using fractal
geometries for design optimization. Several fractal antennas were designed and analyzed,
with performance observed to be as good as non-fractal antennas of similar enclosed
spherical volumes. The primary value in using fractal designs was in the mathematical
expressions which clearly defined the geometries. It was much easier to design a fractal
antenna with complex geometry which satisfied simple rules (e.g., no intersections of
wires) than a random wire antenna with the same physical size and wire length. It was
also observed that although straight wire fractal trees investigated (restricted to one meter
in height) could not achieve resonance within the HF band, helical fractal trees were
resonant in the HF band.
The three-arm configuration of the helical MLA was selected for further analysis and
field experimentation. Prototypes of the three-arm helical MLA design were constructed
and tested on beaches, parks and mountain ridges around Oahu. Measured results were
used to validate software models and simulations with excellent correlation. Successful
long-range communications were demonstrated using amateur radio for distances ranging
from 1.5 miles to greater than 6,000 miles on less than 75 watts of transmit power. The
antennas exhibited vertical polarization, omnidirectional radiation patterns, and linear
phase shift in their resonant bandwidths. The three-arm helical MLA was also self-
resonant at seven different frequencies within the HF band. The antenna structures were
99
easily and rapidly transportable, required no site preparation, and left no environmental
footprint when removed. The measured performance indicates this design offers a much
more suitable HF antenna solution for the LCAC and other military platforms as well as
for homeland security systems requiring rapid response to remote or unprepared coastal
radar sites. For example, the compact nature of the helical MLA antenna and its reduced
volume make it ideally suited for deployment on small platforms such as buoys or barges,
without the need for external support structures. They could also be deployed to remote
locations such as arctic regions where shifting ice layers may not be stable enough to
support larger antenna towers. Figure 77 provides a summary of all the designs evaluated
or developed during this research and demonstrates the success of the new concepts and
design approaches in producing electrically small antennas with ka <= 0.5.
Figure 77: Q(ka) in this dissertation
0
50
100
150
200
250
300
350
400
450
500
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Q
ka
Traditional Designs
HMLA (prototypes)
HMLA (Genetic Algorithms)
Toroidal Helicals
Fractal (Trees)
Fractal (Helical Trees)
Fractal (Hilbert curves)
Chu Limit
ka = 0.5
100
Chapter 9 Future Work
Suggested areas of future work include the following:
• Further simplify presented designs
– Simplify construction for efficient manufacturing
– Minimize non-conductive structural components such as wood dowels and
cylindrical cardboard forms
• Examine implementation on printed circuit boards and using 3-D printing for 3-D
antennas
– Fractal tree design beyond three iterations becomes problematic for wire
antennas due to the irregular nature of fractal geometries yet may be
achievable with 2-D and 3-D printing
– The fractal leaf patterns (XY plots of Riemann’s sine and cosine functions)
provided in Appendix B are potential candidates for 2-D and 3-D printing
• Examine feasibility of using foldable structures for easy transport and deployment
– Explore the use of “antenna origami” for complex structures that fold into
minimal size/space for storage or transport, then expand to full size when
deployed
– Explore implementation of other paper folding mechanisms for generating
complex 3-D structures from 2-D printed designs
• Examine feasibility of using the Lindenmayer system (L-system) to design antennas
– Investigate whether the L-system can be used to generate fractal designs that
also satisfy Maxwell’s equations
101
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104
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105
Appendix A – English Translation of Hilbert (1891)
During the investigation into fractal geometries it was observed that numerous authors
refer to “Hilbert curves” yet many of the designs don’t appear to be space-filling at all.
David Hilbert’s article, “Ueber die stetige Abbildung einer Linie auf ein Flächenstück”
was published in the German periodical Mathematische Annalen in 1891. Hilbert’s paper
built upon the work of Giuseppe Peano published the year prior. Peano’s article was
based on mathematics whereas Hilbert used geometry to describe his variation on space-
filling curves. Hilbert’s article was written in German and further research did not locate
a suitable English translation so I translated it.
On the mapping of a continuous line onto a flat surface *)
David Hilbert, 1891 [Mathematische Annalen, Vol. 38, Issue 3, pages 459-460]
Peano has recently shown in Mathematische Annalen **) through mathematical
observations, how the points on a line can be mapped continuously onto a flat surface.
The required functions for such a mapping can be produced in a clear manner if one
makes use of the following geometric insight. The depicted line – a straight line of length
1 – is first divided into 4 equal parts 1, 2, 3, 4 and the surface is made in the form of a
square of side 1 and then divided by two mutually perpendicular lines into 4 equal
squares 1, 2, 3, 4 (Fig. 1).
*) See, Proceedings of the Society of German Natural Scientists and Physicians.
**) Vol. 36, page 157 [Mathematische Annalen, Vol.36, Issue 1, pp. 157-160]
A-1
Next, we divide each part of sections 1, 2, 3, 4 again in 4 equal parts, so that we have
16 sections on the straight line, sections 1, 2, 3, ..., 16; at the same time each of the 4
squares 1, 2, 3, 4 will be divided into 4 equal squares and thus form 16 squares, labeled as
1, 2 ... 16, but the order of the squares has to be chosen so that, following the line, each
[consecutively numbered] square has one side adjacent to the previous one (Fig. 2).
Let us imagine this process is continued – Fig. 3 illustrates the next step – it is obvious
how one can assign a particular point on each segment to the corresponding squares. It is
only necessary to determine the specific path of the line which falls on a given point. The
squares labeled with the same sequence of numbers are connected to each other and form
a closed boundary around the points in the line. This is valid for the mapped and
numbered points. The resulting image is unique and continuous, and reciprocal to each
point of the squares corresponding to one, two or four points of the line. It is noteworthy
that through suitable modification of the segments in the squares, it is possible to easily
find a clear and continuous mapping whose inverse is unambiguous.
Figure 78: Hilbert (1891) Figs. 1, 2, and 3
A-2
The previously discussed mapping functions are also simple examples for everywhere
continuous and nowhere differentiable functions.
The practical significance of the discussed mapping is as follows: a point may be
constantly moving so that during a finite time it crosses all the points on a surface. At the
same time, you can also (through suitable modification of the line segments in the
squares) cause an infinitely dense arrangement of the points within the square so that a
designated path exists to move forwards as well as backwards.
With respect to the analytical representation of the mapping functions, it immediately
follows from their continuity (a general theorem proven by K. Weierstrass *), that these
selected functions can be progressively generated to infinity, and the complete intervals
converge absolutely and uniformly for all successive iterations.
*) Refer to Proceedings of the Academy of Sciences in Berlin, July 9, 1885.
END OF TRANSLATION
A-3
Appendix B – Fractal Geometry
Mandelbrot describes the beginnings of “fractal geometry” as:
“…a new branch born belatedly of the crisis of mathematics that started when
duBois Reymond 1875 first reported on a continuous nondifferentiable function
constructed by Weierstrass…” [27].
Mandelbrot coined the term “fractal” from the Latin adjective “fractus”, to indicate the
“broken” and “irregular” nature of fractal geometries. He went on to provide historical
examples of continuous yet nondifferentiable functions, several of which were credited to
Riemann around 1861. Riemann’s functions, e.g., R(t) = Σ n-2 cos(n2t) , were found to be
very interesting due to the simplicity of their mathematical expressions as contrasted with
the complexity of the fractal lines generated. In the example R(t) just mentioned, the
cosine function can be replaced with a sine function to achieve very different results.
The following figures display the Riemann function R(t) plotted in LabVIEW for
selected values of n over the range of 0 ≤ t ≤ 2π to better examine the fractal properties.
The solution for n = 1 reduces to the simple expression R(t) = cos(t) as depicted in Figure
78. The fractal nature begins emerging around n = 5 as depicted in Figure 79. This
pattern continues to emerge as n is increased, as shown in Figure 80 for n = 200. This
discussion of the Riemann summation function is provided to offer an example of fractal
line geometry very different from the space-filling Hilbert lines discussed previously, and
to provide real examples for fractal geometries.
B-1
Figure 81: Riemann cosine function R(t) for n = 200
Figure 79: Riemann cosine function R(t) for n = 1
Figure 80: Riemann cosine function R(t) for n = 5
B-2
In Figure 81 and Figure 82 the Riemann function plots are converted to XY coordinates
for the cosine and sine functions respectively.
Figure 82: Riemann R(x,y) cosine function
Figure 83: Riemann R(x,y) sine function
B-3
Figure 83 and Figure 84 display the self-similar nature viewed at different scales.
Figure 84: Riemann function 0 to pi
Figure 85: Scaling the Riemann function
B-4
