improving the performance of antennas with...
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IMPROVING THE PERFORMANCE OF ANTENNASWITH METAMATERIAL CONSTRUCTS
Item Type text; Electronic Dissertation
Authors Jin, Peng
Publisher The University of Arizona.
Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.
Download date 16/04/2018 17:46:09
Link to Item http://hdl.handle.net/10150/193567
IMPROVING THE PERFORMANCE OF ANTENNAS WITHMETAMATERIAL CONSTRUCTS
by
Peng Jin
Copyright c© Peng Jin 2010
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF ELECTRIC AND COMPUTER ENGINEERING
In Partial Fulfillment of the RequirementsFor the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2010
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THE UNIVERSITY OF ARIZONAGRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the dis-sertation prepared by Peng Jinentitled Improving the Performance of Antennas with Metamaterial Constructsand recommend that it be accepted as fulfilling the dissertation requirement for theDegree of Doctor of Philosophy.
Date: 18 March 2010Richard W. Ziolkowski
Date: 18 March 2010Hao Xin
Date: 18 March 2010Nathan Goodman
Date: 18 March 2010
Date: 18 March 2010
Final approval and acceptance of this dissertation is contingent upon the candidate’ssubmission of the final copies of the dissertation to the Graduate College.I hereby certify that I have read this dissertation prepared under my direction andrecommend that it be accepted as fulfilling the dissertation requirement.
Date: 18 March 2010Dissertation Director: Richard W. Ziolkowski
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STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for anadvanced degree at the University of Arizona and is deposited in the UniversityLibrary to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission,provided that accurate acknowledgment of source is made. Requests for permissionfor extended quotation from or reproduction of this manuscript in whole or in partmay be granted by the copyright holder.
SIGNED: Peng Jin
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ACKNOWLEDGEMENTS
I would like to express my sincere gratitude towards my dissertation adviser, Profes-sor Richard W. Ziolkowski for his continuous support, guidance and encouragement.Professor Ziolkowski is foremost, a teacher with great enthusiasm, patience, andkindness, not only in classroom, but also on personal level. He challenged me fromtime to time with interesting yet difficult topics and offered all the help I needed toget them done.
I would also like to express my appreciation to Professor Nathan Goodman forhis mentorship to me in my early graduate student period. It was him who gave methe interest in research and the necessary preparation. I would also want to thankProfessor Hao Xin. The problems raised by him and discussions with him greatlyenchanced my research. I would also like to thank both Professor Goodman andHao for being members on my dissertation committee.
Last but not the least, I would like to thank my parents, my wife Lingling, andmy son Ethan. It is their endless and unconditional love that encourages me allalong.
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DEDICATION
This dissertation is dedicated to my wife Lingling.
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TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 141.1 Metamaterial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2 Transmission Line Theory and Applications of MTMs . . . . . . . . . 161.3 MTM-based antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1 Antenna Metrics . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.2 MTM-based Electrically Small Antennas . . . . . . . . . . . . 191.3.3 MTM-inspired Electrically Small Antennas . . . . . . . . . . . 20
1.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
CHAPTER 2 METAMATERIAL-BASED DISPERSION ENGINEERINGTO ACHIEVE HIGH FIDELITY OUTPUT PULSES FROM A LOG-PERIODIC DIPOLE ARRAY . . . . . . . . . . . . . . . . . . . . . . . . . 242.1 LPDA Antenna Dispersion . . . . . . . . . . . . . . . . . . . . . . . . 242.2 MTM Phase Shifter Corrections . . . . . . . . . . . . . . . . . . . . 312.3 MATLAB Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4 Phase Shifters Included on the Radiating Elements . . . . . . . . . 482.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
CHAPTER 3 LOW-Q, ELECTRICALLY SMALL, EFFICIENT NEAR-FIELD RESONANT PARASITIC ANTENNAS . . . . . . . . . . . . . . . 533.1 Chu-limit and Electrically small antennas . . . . . . . . . . . . . . . . 533.2 Z Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.1 Fabrication and Measurement . . . . . . . . . . . . . . . . . . 613.2.2 Matching Methods Comparison . . . . . . . . . . . . . . . . . 67
3.3 Stub Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4 Canopy antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1 Parameter Studies . . . . . . . . . . . . . . . . . . . . . . . . 793.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5 Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.6 Metamaterials Within the Minimum Enclosing Hemisphere . . . . . . 963.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
TABLE OF CONTENTS – Continued
7
CHAPTER 4 BROADBAND, EFFICIENT, ELECTRICALLY SMALLMETAMATERIAL-INSPIRED ANTENNAS FACILITATED BY ACTIVENEAR-FIELD RESONANT PARASITIC ELEMENTS . . . . . . . . . . . 1014.1 Eletrically small antennas bandwidth limits . . . . . . . . . . . . . . 1014.2 ANSOFT HFSS and Designer Simulations of the Z Antenna . . . . . 1034.3 Inductor Versus Resonant Frequency . . . . . . . . . . . . . . . . . . 1084.4 Bandwidth Enhancement for Metamaterial-inspired ESAs . . . . . . . 109
4.4.1 Z antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.4.2 Stub Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.4.3 Canopy Antenna . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
CHAPTER 5 CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . 125
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8
LIST OF FIGURES
1.1 Material Classification . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2 Common structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3 Infinitesimal electric dipole of surrounded by DNG shell . . . . . . . . 19
2.1 Bipolar Pulse Time History . . . . . . . . . . . . . . . . . . . . . . . 252.2 Bipolar Pulse Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Bandlimited Far Field from Infinitesmal Dipole . . . . . . . . . . . . 272.4 Printed log-periodic dipole array antenna . . . . . . . . . . . . . . . . 272.5 S11 of Printed LPDA . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.6 Far-zone Electric Field of Printed LPDA . . . . . . . . . . . . . . . . 302.7 Phase Shifters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.8 Modified logperiodic dipole array antenna . . . . . . . . . . . . . . . 332.9 Dipole current phase w/o phase shifter . . . . . . . . . . . . . . . . . 332.10 Far-zone electric field with perfect phase compensation . . . . . . . . 342.11 Modified log-periodic eight element array output . . . . . . . . . . . . 372.12 Modified log-periodic 10 element array output . . . . . . . . . . . . . 382.13 Log-Periodic cylindrical antenna . . . . . . . . . . . . . . . . . . . . . 392.14 Phase-shifter in transmission line . . . . . . . . . . . . . . . . . . . . 392.15 S11 of Modified LPDA . . . . . . . . . . . . . . . . . . . . . . . . . . 442.16 Phase Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.17 Modified LPDA and Unmodified LPDA E-Plane Antenna Patterns . 512.18 Phase center locations . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1 Three dimensional view of the Z Antenna with Inductor=1000nH . . 603.2 Three dimensional view of the Z Antenna, Duroid design . . . . . . . 613.3 S11 values for the Duroid design Z antenna predicted by HFSS . . . . 623.4 The bottom half of a broken fabricated Z antenna . . . . . . . . . . . 633.5 Fabricated Duroid Z antenna with large ground plane . . . . . . . . . 643.6 Fabricated Duroid Z antenna with large ground plane . . . . . . . . . 643.7 Horn Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.8 Monpole Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.9 Measured S11 values for the Duroid Z antenna . . . . . . . . . . . . . 663.10 Measured radiated power for the Duroid Z antenna . . . . . . . . . . 663.11 Monopole and monopole with external MFJ tuner . . . . . . . . . . . 673.12 Measured S11 for monopole with MFJ tuner . . . . . . . . . . . . . . 68
LIST OF FIGURES – Continued
9
3.13 Radiated power by bare monopole and monopole with external MFJtuner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.14 Monopole antenna with the double stub tuner . . . . . . . . . . . . . 693.15 S11 predicted by HFSS for the monopole antenna with the double
stub tuner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.16 Three dimensional view of the stub antenna . . . . . . . . . . . . . . 713.17 Stub antenna with four curve stubs . . . . . . . . . . . . . . . . . . . 733.18 Canopy antenna configurations (a) one-leg version, (b) four-leg version 763.19 S11 values of the one-leg canopy antenna . . . . . . . . . . . . . . . . 773.20 Complex impedance of the one-leg canopy antenna . . . . . . . . . . 773.21 Radiation pattern of the one-leg canopy antenna . . . . . . . . . . . . 783.22 Electric field distribution of the one-leg canopy antenna . . . . . . . . 783.23 Qratio values versus Ratioarea values for the one-leg canopy antenna . 813.24 Current distribution on the shell of the four-leg canopy antenna . . . 833.25 E field for the one-leg canopy antenna with RInd = 0.1 mm . . . . . . 863.26 Current density on the ka = 0.46 spherical cap . . . . . . . . . . . . . 863.27 Lax and Meshed-shell Canopy antennas . . . . . . . . . . . . . . . . . 873.28 Canopy antenna two-port model . . . . . . . . . . . . . . . . . . . . . 903.29 Canopy antenna equivalent two-port T-circuit model . . . . . . . . . 903.30 Comparison of Zin from HFSS and T-circuit model . . . . . . . . . . 913.31 Circuit models for the one-leg canopy antenna. . . . . . . . . . . . . . 913.32 Input impedance, Zin, of the spherical shell antenna . . . . . . . . . . 943.33 Canopy antenna HFSS model with metamaterial interior sphere . . . 983.34 Canopy antenna with a metamaterial interior sphere results . . . . . 98
4.1 The Z antenna configuration . . . . . . . . . . . . . . . . . . . . . . . 1044.2 The HFSS-predicted S11 values of the Z antenna . . . . . . . . . . . . 1054.3 Z Antenna Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4 ANSOFT Designer Circuit Model for the Z Antenna . . . . . . . . . . 1064.5 ANSOFT Designer predicted S11 values for the Z antenna . . . . . . 1064.6 Antenna circuit model with NET load . . . . . . . . . . . . . . . . . 1074.7 ANSOFT Designer circuit representation of the IMN-based Z antenna 1074.8 ANSOFT Designer predicted S11 vales for the IMN-based Z antenna 1084.9 Antenna circuit model with equivalent load . . . . . . . . . . . . . . . 1084.10 Z antenna resonant frequency vs lumped element inductor value . . . 1104.11 Z Antenna with ka = 0.266 . . . . . . . . . . . . . . . . . . . . . . . 1114.12 Results obtained by curve fitting of the inductor values . . . . . . . . 1124.13 Negative lumped element circuit model . . . . . . . . . . . . . . . . . 1134.14 Equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
LIST OF FIGURES – Continued
10
4.15 Floating negative impedance converter circuit . . . . . . . . . . . . . 1144.16 One-leg stub antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.17 Inductor-frequency (L-F) sweep for the stub antenna cases . . . . . . 1184.18 FBW10dB vs curve fitting error for stub antenna . . . . . . . . . . . . 1194.19 Four-leg stub antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.20 One-leg canopy antenna . . . . . . . . . . . . . . . . . . . . . . . . . 1214.21 Four-leg canopy antenna . . . . . . . . . . . . . . . . . . . . . . . . . 1224.22 L-F sweep for the one- and four-leg canopy antennas . . . . . . . . . 1224.23 FBW10dB vs curve fitting error for Canopy antenna . . . . . . . . . . 123
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LIST OF TABLES
2.1 Phase-shifter Parameters Values for the Eight Element LPDA Antenna 422.2 Phase-shifter Parameters Values for the Ten Element LPDA Antenna 422.3 Eight Element Log-Periodic Antenna Dimensions . . . . . . . . . . . 422.4 Ten Element Log-Periodic Antenna Dimensions . . . . . . . . . . . . 422.5 Phase Shifts at the Dominant Radiation Frequencies . . . . . . . . . 43
3.1 ka = 0.0467 Stub Antenna Comparisons . . . . . . . . . . . . . . . . 733.2 QRatio versus Copper Shell Thickness . . . . . . . . . . . . . . . . . . 803.3 QRatio versus PEC Shell Thickness . . . . . . . . . . . . . . . . . . . . 803.4 Copper Canopy with Multiple Inductors . . . . . . . . . . . . . . . . 82
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ABSTRACT
Metamaterials (MTMs) are artificial materials that can be designed to have exotic
properties. Because their unit cells are much smaller than a wavelength, homog-
enization leads to effective, macroscopic permittivity ε and permeability µ values
that can be used to determine the MTM behavior for applications. There are four
possible combinations of the signs of ε and µ. The desired choice of sign depends on
the particular application. Inspired by these MTM concepts, several MTM-inspired
structures are adopted in this dissertation to improve various performance character-
istics of several different classes of antennas. Three different metamaterial-inspired
engineering approaches are introduced to achieve enhanced antenna designs. First,
the transmission-line (TL) type of MTM is used to modify the dispersion charac-
teristics of a log-periodic dipole array (LPDA) antenna. When LPDA antennas are
used for wideband pulse applications, they suffer from severe frequency dispersion
because the phase center location of each element is frequency dependent. By incor-
porating MTM-based phase shifters, the LPDA frequency dispersion properties are
improved significantly. Both eight and ten element MTM-modified LPDA antennas
are designed to enhance the fidelity of the resulting output pulses. Second, epsilon-
negative unit cells are used to design several types of electrically small, resonant
parasitic elements which, when placed in the very near field of a driven element, lead
to nearly complete matching (i.e., reactance and resistance) of the resulting electri-
cally small antenna system to the source and to an enhanced radiation efficiency.
However, despite these MTM-inspired electrically small antennas being very effi-
cient radiators, their bandwidth remains very narrow, being constrained by physical
limitations. Third, we introduce an active parasitic element to enhance the band-
width performance of the MTM-inspired antennas. The required active parasitic
element is derived and an implementation methodology is developed. Electrically
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small active Z, stub, and canopy antennas are designed. It is demonstrated that
an electrically small antenna with ka ∼ 0.046 and over a 10% bandwidth can be
realized, in principle.
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CHAPTER 1
INTRODUCTION
An antenna is an element that converts electric current to electromagnetic waves
and vice versa. For a long time, antennas have been widely adopted for signal
transmission and receiving in applications where electromagnetic (EM) waves are
involved. These include radar, TV and radio broadcasting, satellite communications,
point to point communications, and current wireless communications. Design tech-
niques have been developed to produce antennas with different properties that fulfill
the requirements in these different applications. For example, in current personal
wireless communication systems, there continues to be a need for smaller devices,
longer battery life, and higher data bit rates. Technically, this means that there is
a desire for smaller, more efficient, and of broader bandwidth antennas. In some
cases, current antenna designs can meet those requirements, while in others they
can not. Often the requirements are extremely difficult or physically impossible
to achieve. Nonetheless, antenna design technique keep evolving to meet the ap-
plication requirements. Recently, new types of fabricated structures or composite
materials that mimic media with non-natural EM properties were introduced in the
microwave and optics fields. These new types of materials are known as metamate-
rials. With the flexibility and new properties provided by metamaterials, new type
of antennas have been conceived, while making their designs more straightforward.
1.1 Metamaterial
Metamaterials (MTMs) are artificial EM materials that have unusual properties not
available in nature. When their EM structures are effectively homogeneous, these
MTMs behave like real materials and exhibit the effective, macroscopic constitutive
parameters: permittivity ε and permeability µ. There are four possible combina-
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Figure 1.1: Material Classification
tions of the signs of ε and µ as shown in Fig. 1.1, where the double positive (DPS)
metamaterials have both ε > 0 and µ > 0. The epsilon-negative (ENG) metamate-
rial have ε < 0 and µ > 0. The mu-negative (MNG) metamaterial have ε > 0 and
µ < 0. The double negative (DNG) metamaterial have both ε < 0 and µ < 0. As
will be shown below, depending on the designs, these various types of MTMs can
be utilized to improve an antenna’s performance characteristics.
In 1968, Veselago [1] theoretically predicted several fundamental phenomena
when EM waves propagate in MTMs. About 30 years later, a MTM was experi-
mentally demonstrated by Smith, Schultz, and their UCSD group [2]. Since then,
MTMs have gained a great deal of attention and significant progress has been made
in both their theories and applications. Examples include transmission line (TL)
MTMs and their applications, as well as MTM-based antennas.
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1.2 Transmission Line Theory and Applications of MTMs
The theory and applications of TL MTMs is an approach based on generalized TL
theory. Instead of a physics point of view of MTMs, the TL MTM approach is on
engineering path that focuses on guided-wave, radiated-wave, and refracted-wave
structures. Caloz and Itoh gave a thorough review in [3] of the research work on
MTM TL theory done by their group, where 1-D and 2-D TL models of MTMs
were discussed along with applications. In [3], the traditional TL is called a right-
handed (RH) material, whereas the DNG MTM in Figure 1.1 is referred to as a
left-handed MTM, and the combination of these two is called a composite right-
/left-handed (CRLH) MTM. Unlike a RH material, which is composed of periodic
cells of serial inductors and shunt capacitors, the LH MTM is composed of periodic
cells of shunt inductors and serial capacitors. As a result, the LH MTM can produce
a negative phase delay. The CRLH TL MTM is a realistic representation of a LH
MTM and leads to better modeling of them in real applications and to performance
improvements in their transmission bands. By adjusting its inductor and capacitor
values, a CRLH MTM can exhibit either RH or LH properties in a certain frequency
range. With the already well developed TL theory, EM waves traveling through the
TL MTMs are readily analyzed and modeled. Moreover, the modeling directly
results in a TL MTM implementation that uses traditional components such as
microstrip lines and interdigital capacitors. Based on the TL MTM theory, many
applications have been developed such as delay lines [4] and backfire-to-end fire leaky
wave antennas (LWAs) with CRLH phase shifters [5]. A LWA with an active CRLH
was introduced in [6]. Since this kind of MTM is constructed with serial/shunt
capacitors (C) and inductors (L), it is referred to as a C-L configuration in some
references.
When EM waves travel in the C-L configuration, the relation between the prop-
agation number β and angular frequency ω in a LH MTM is
β = − 1
ω√LC
. (1.1)
It should be noted that although β < 0 in (1.1), which is the same as a positive
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phase delay in a DNG MTM, they are not the same in terms of the group delay. In
DNG MTMs, the relation between the propagation number β and angular frequency
ω is
β = −ω√εµ. (1.2)
It is clear that the group delay in (1.2) is negative and the group delay in (1.1)
is positive. Consequently, DNG TL structures were proposed in [7] by using lossy
resonators and in [8] with amplifiers. These active TL MTMs compensate for their
various losses and, as a result, lead to lossless DNG MTMs.
1.3 MTM-based antennas
Figure 1.2: Common structure
Compared to the planar TL MTMs, general MTMs are volumetric and behave
like a medium when they are interacting with an EM wave. Two well-known ex-
amples are shown in Fig. 1.2. They are constructed from unit cells of conducting
cylinders (wires) and split resonant rings (SRRs) [2]. They have been shown experi-
mentally to exhibit a negative index of refraction and to produce negative refraction.
It is well known that antennas designed for operation in free space will act
differently when other media are present. Because of the exotic properties of various
types of MTMs, it was attractive to introduce them into the designs of antennas and
to determine whether or not they could lead to better performance characteristics.
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Both electric and magnetic types of antennas have been considered. Interesting
performance improvements have been realized with a wide variety of designs.
1.3.1 Antenna Metrics
Because of the nature of the antenna applications of metamaterials investigated
in this dissertation, it is useful to introduce some basic antenna metrics that are
appropriate for electrically small antennas. Consider an antenna operating at the
resonance frequency f0. The corresponding free space wavelength is λ0 = c/f0,
where c is the free space speed of light. The ka value of an antenna is then defined
by the free space wave number k = 2π/λ0 and the radius, a, of the smallest sphere
that would enclose that antenna. If its ka value is small, for example, less than 1
in free space or 0.5 in the presence of a ground plane, an antenna is said to be an
electrically small antenna. In this dissertation, ka < 0.5 is chosen for the electrically
small standard since those designs involve a ground plane. For any antenna, it has a
Q value that is defined as the ratio: ω = 2πf0 times the energy stored in the smallest
enclosing sphere divided by the average power loss. The radiation efficiency, RE,
is the ratio of the total power radiated by the antenna to the total power accepted
at its terminals. The overall efficiency (OE) is the ratio of the total power radiated
to the power delivered by the source. The Q value can be related to the lower
theoretical bound, which is defined by the product of the radiation efficiency RE
and the Chu limit [9], which will be denoted as Qchu and is given by the expression
QChu =1
ka+
1
ka3. (1.3)
The ratio between Q and the lower bound Qlb = RE ×QChu will be denoted Qratio.
The bandwidth of the antenna as discussed in this dissertation is the impedance
bandwidth, which is defined by the frequency range over which the reflection co-
efficient |S11| is lower than certain value. When it is defined by −10 dB values,
the bandwidth is labeled as BW10dB and is called the 10dB bandwidth. When it is
defined by the −3 dB values, the bandwidth is labeled as BW3dB and is called the
3dB or VSWR half-power bandwidth. Then, for example, the VSWR half-power
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fractional bandwidth is defined by FBWV SWR = BW3dB/f0. The Q-factor and the
fractional bandwidth are related as
Q =2
FBWV SWR
=2 f0
BW3dB
. (1.4)
1.3.2 MTM-based Electrically Small Antennas
Electrical small antennas (ESAs) have been studied extensively in the past and are
still of great research interest. Because of its compact dimension, an ESA has many
potential wireless applications, including unmanned devices, personal communica-
tions, and wireless sensor networks. An ESA with high overall efficiency and broad
bandwidth is always preferred in these types of applications. However, as pointed
out in many works, for example [9–11], there are constraints amongst the antenna’s
physical dimensions, its bandwidth, and its radiation efficiency.
Figure 1.3: Infinitesimal electric dipole of surrounded by DNG shell
Electrically small antennas radiating in the presence of MTMs were studied
20
analytically in [12–16]. It was shown that by introducing a metamaterial-based ho-
mogeneous, isotropic, dispersive, electrically-small shell around an electrically-small
radiator as shown in Figure 1.3, the overall efficiency, OE, of the composite antenna
system could be close to one (assuming low loss MTMs) with Q values near the lower
bound and much lower than it with the introduction of an active metamaterial. One
important consequence of this approach was that a properly designed MTM shell
eliminated the need for an external matching network between the ESA and the
source, i.e., despite of its electrically small size, the antenna systems were shown
to have an input impedance Zin ≈ 50Ω at the resonant frequency, when the source
resistance was assumed to be Zs = 50Ω. More specifically, for an electrically small
electric dipole, which by nature is highly capacitive, an electrically small epsilon-
negative (ENG) MTM shell, which is inductive in nature, can be applied to form
a composite resonant antenna system. Similarly, an electrically small mu-negative
(MNG) shell, which is capacitive in nature, is applied for an electrically small loop
antenna. Although the above analytical conclusions were based on idealized elec-
trically small MTM shells, which have not yet been realized physically, they shine
light on a potential design approach which may be easily implementable to achieve
ESAs with high OE’s.
1.3.3 MTM-inspired Electrically Small Antennas
Influenced by these theoretical results, metamaterial-inspired efficient ESAs were
introduced in [17–19]. In the three-dimensional (3D) magnetic-based EZ antenna,
a 3D extrusion of a planer capacitive load loop (CLL) was introduced to provide a
resonant match for the coax-fed, electrically small, inductive semi-loop antenna. In
the 3D electric-based EZ antenna, a helical inclusion was introduced to provide a
resonant match for a coax-fed electrically small monopole antenna.
The resonant parasitic CLL extrusion and helical elements were located in very
near field of their coax-fed radiators. These elements, being unit cells of the corre-
sponding MTMs, acted as simplified versions of the requisite electrically small MNG
shell and ENG shells, respectively. They led to the predicted nearly complete re-
21
actance and resistance matching. These 3D EZ antennas were further simplified to
2D planar versions. The 3D CLL extrusion was simplified to a printed planar CLL
element, and the helix was simplified to a printed planer meander-line element. It
was shown further that the EZ magnetic-based antenna can be simplified even fur-
ther by incorporating a lumped element capacitor to replace the printed interdigited
capacitor. Again, these resonant near-field parasitic metamaterial-inspired elements
led to EZ antenna designs with high overall efficiencies for which no external match-
ing network was needed. In particular, in [19], it was experimentally verified that a
2D electric-based EZ antenna at 1.37GHz with ka = 0.49 had an OE ∼ 94% with
a fractional bandwidth of 4.1%. In a related manner, it is pointed out in [20] that
the spherical coupled resonator antenna behaves like a negative permittivity sphere.
The electric Z antenna was introduced in [21]; it is yet a further simplification
of the 2D electric-based EZ antenna. The meander-line is replaced by a split Z-
shaped element; the two halves of this element being connected with a lumped
element inductor. The relationship between the resonant frequency and the lumped
inductor value was revealed and the radiation mechanism was better understood.
The resulting resonant near-field parasitic element interpretation was extended to
the realization of the stub antenna introduced in [22]. The electrically small stub
antenna is an attractively simple structure, consisting only of a parasitic located
in the very near-field of a coax-fed monopole and normal to the ground plane that
is constructed from a cylindrical lumped element inductor and a series connected
cylindrical piece of wire, both having the same radius.
1.4 Dissertation Outline
In this dissertation, several antennas with MTMs are investigated. For the TL
based MTMs, it is adopted by a log-periodic dipole antenna for broadband pulse
transmission. The modified LPDA is analyzed using triditional microwave network
theory and an circuit model is developed. The rest of the dissertation is related to the
MTM-inspired eletrically small antennas. Antennas with low Q ratios are designed
22
and circuit model is developed for better understanding. For the Z antenna. Active
elements are adopted to significantly expand the bandwidth of the MTM-inspired
electrically small antenna.
The dissertation is composed by five chapters. The first chapter briefly reviewed
MTM history and related research. TL based MTMs and MTM related electrically
small antennas were discussed.
In Chapter 2, A metamaterial-enabled approach is presented that allows one
to engineer the dispersion of a log-periodic dipole array antenna (LPDA) to make
it more suitable for wide bandwidth pulse transmission. By modifying the LPDA
with electrically small transmission line metamaterial-based negative and positive
phase shifters, the phase of each element of the LPDA are adjusted such that in
the main beam direction, the phase shifts between each element approximates a
linear phase variation. The performance characteristics of the resulting dispersion-
engineered LPDA are obtained numerically with HFSS and MATLAB simulations.
By measuring in the far field the fidelity between the actual transmitted pulse and
the idealized output waveform, the required component values of the phase shifters
are optimized. Significant improvements in the fidelity of the pulses transmitted are
demonstrated with eight and ten element LPDAs.
Chapter 3 reports Metamaterial-inspired electrically small Z, stub and canopy
antennas are reported. They are near-field, resonant parasitic designs. Different Z
and stub antenna configurations and the effect on their Q values are studied. Their
behavior led to the canopy antenna design. At the size of ka ∼ 0.046, the canopy
antenna is an electric-based antenna with high overall efficiency (over 90%) and low
Q-ratio value and whose input resistance is almost completely matched to a 50Ω
source. The resonant frequency, ∼ 300MHz, in the UHF band is selected for the
designs. The canopy antenna is studied extensively to explore the lowest achievable
Q values. Various coupling configurations, canopy shapes, and metal-air ratios are
investigated. Circuit models are also introduced to explain the radiation mechanism.
Numerical simulation results are analyzed and compared with previously derived Q
value limits for electrically small antennas that are based the standard circuit models
23
of spherical wave multipoles. The Q value of the canopy antenna for the lowest order,
single electric resonance is shown to reach a fundamental limit of approximately 1.75
times the Chu value.
Chapter 4 investgates the possibility of using an active internal matching ele-
ment in several types of metamaterial-inspired, electrically small antennas (ESAs)
to overcome their inherent narrow bandwidths is demonstrated. Beginning with the
Z antenna, which is frequency tunable through its internal lumped element induc-
tor, a circuit model is developed to determine an internal matching network, i.e., a
frequency dependent inductor, which leads to the desired enhanced bandwidth per-
formance. An analytical relation between the resonant frequency and the inductor
value is determined via curve fitting of the associated HFSS simulation results. With
this inductance-frequency relation defining the inductor values, a broad bandwidth,
electrically small Z antenna is established. This internal matching network paradigm
is then confirmed by applying it to the electrically small stub and canopy anten-
nas. An electrically small canopy antenna with ka = 0.0467 that has over a 10%
bandwidth is finally demonstrated. The potential implementation of the required
frequency dependent inductor is also explored with a well-defined active negative
impedance converter circuit that reproduces the requisite inductance-frequency re-
lations.
Chapter 5 concluds the dissertation, sumarizing the works and suggesting pos-
sible direction in the future.
24
CHAPTER 2
METAMATERIAL-BASED DISPERSION ENGINEERING TO ACHIEVE HIGH
FIDELITY OUTPUT PULSES FROM A LOG-PERIODIC DIPOLE ARRAY
2.1 LPDA Antenna Dispersion
With the recent interest in ultra-wide bandwidth (UWB) systems for communi-
cations applications, there has been a surge of interest in UWB antennas. These
systems use UWB pulses rather than narrow bandwidth signals to propagate the
information. Unfortunately, the log-periodic dipole array (LPDA) antenna, which
is well-known for wide bandwidth applications, is not suitable for these pulsed ap-
plications. Because of the frequency dependent phase shifts that exist between the
elements of this antenna, the log-periodic array is known to be a very dispersive en-
vironment for a pulsed excitation; and, consequently, its output signal is a severely
distorted version of the input pulse. While there have been many novel antenna
designs introduced to satisfy UWB application criteria, the prevalence, simplicity
and familiarity of the log-periodic array would make it an appealing choice if one
could suggest a means to overcome these phase shift issues with properly centered
phases.
In this chapter we consider transmission line-based metamaterials (MTMs) to
achieve the appropriate phase shift elements and their introduction into a log-
periodic array to correct for the detrimental phase shifts associated with it. This
idea was introduced in [23]. We report data to support the fact that this approach
leads to a modified log-periodic array that has a sufficiently flat spectral response
over a wide frequency band and that produces the requisite phase values at the
dominant frequencies to achieve time alignment of the output signals to achieve an
overall high fidelity output pulse. Moreover, the design is straightforward and sug-
gests the possibility of retrofitting existing log-periodic systems to take advantage of
25
this MTM technology. Consequently, this dispersion-engineered log-periodic array
may have an important impact on UWB system designs and applications.
0 0.5 1 1.5 2 2.5 3 3.5 4−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (ns)
Exc
itatio
n C
urre
nt
Figure 2.1: Current source excitation: Bipolar pulse time history
To demonstrate the frequency dispersion observed in the output pulses generated
by an LPDA antenna, we adopt the differentiated Gaussian pulse as the waveform
that is used to excite the current sources driving the LPDA elements. In time
domain, the differentiated Gaussian pulse is described as
x(t) =At√2πσ3
e−t2
2σ2 , (2.1)
and its frequency spectrum is
X(f) = −A(j2πf)e−(2πfσ)2
2 , (2.2)
where σ is known as a characteristic time of the pulse. It should be noted that such
pulse has infinite duration in time. Effective pulse duration T can be defined by the
interval containing 99.99% of the total pulse energy and it can be shown that
T ≈ 7σ (2.3)
26
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (GHz)
Mag
nitu
de
Figure 2.2: Current source excitation:Bipolar pulse spectrum
. The pulse center frequency fc which corresponds to the spectrum peak is
fc ≈0.17
σ, (2.4)
and the half power low frequency fL and high frequency fH are
fL =0.09
σ;
fL =0.29
σ. (2.5)
This bipolar pulse waveform is shown in Fig. 2.1; it removes the DC components
from the input spectrum as shown in in Fig. 2.2. Because its behavior can be
treated analytically, we also adopt an infinitesimal electric dipole as our basic time
domain reference antenna. It has a highly capacitive nature and, as a result, the far
zone electric field generated by this infinitesimal dipole antenna is proportional to
the time derivative of the current pulse that excites it. While an infinitesimal dipole
can thus radiate theoretically all of the frequencies in the bipolar pulse driving it,
it must be noted that, in practice, it is extremely electrically small and can not
27
be matched directly to a real source over a wide range of frequencies. Nonetheless,
because one can calculate its far field response with little difficulty, it provides an
efficient analytical means of representing the overall time domain response of an
LPDA, which can be matched to a realistic source over a wide range of frequencies.
0 1 2 3 4 5 6−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time(ns)
Ele
ctric
Fie
ld(V
/m)
Figure 2.3: Far zone electric field radiated by an infinitesimal dipole that is drivenwith a band-limited version of the source excitation pulse shown in Fig. 2.1
z
y
Figure 2.4: Printed log-periodic dipole array geometry
28
For demonstration purposes only, we assume the frequency range of interest
in this paper to be covered by the LPDA antenna is 1GHz ≤ f ≤ 4GHz. For
the purpose of performing frequency domain ANSOFT High Frequency Structure
Simulator (HFSS) and MATLAB simulations, we band-limit the actual excitation
pulse to that frequency range. To illustrate the form of the electric field radiated by
a broadband antenna driven with such a band limited bipolar pulse, the electric field
signal radiated into the far field by the infinitesimal dipole antenna was calculated.
This signal is shown in Fig. 2.3, normalized by its total output energy. A diagram of
the log-periodic printed-dipole array antenna [24] considered in this paper is shown
in Fig. 2.4. A standard crisscross connection was assumed and implemented with
a two parallel layer structure that is represented in Fig. 2.4 by the black and white
colors. This LPDA antenna is assumed to be located in the x = 0 plane; its dipole
elements are oriented parallel to z axis with the largest element being the furthest
away from the origin along the +y direction.
To begin, an eight-element log-periodic printed dipole antenna was analyzed
analytically with MATLAB using the circuit model given in [25] and numerically
with HFSS for the indicated 1GHz ≤ f ≤ 4GHz frequency range. It should be
pointed out that the HFSS simulator solves the curl-curl - k2 form of the electric
field equation obtained from Maxwell’s equations in the frequency domain with the
finite element method. The HFSS-predicted |S11| values are shown in Fig. 2.5.
These magnitude of the S11 values are well below −10dB throughout the frequency
band of interest. The feed point current If,n induced on each printed dipole in the
LPDA antenna was measured using the HFSS Fields Calculator, where n denotes
the n-th dipole and f denotes the frequency. For each feed-point current If,n, the
far zone electric fields can be calculated with the expression [26]:
Eθ ' jη If,ne−jkrn
4πrn
[
2cos(khn
2cos θ)− cos khn
2
sin θ
] [
1
sin(khn
2)
]
, (2.6)
where hn is the length of the n-th dipole and rn is the physical distance between
the n-th dipole and the observation point. With this expression the far zone electric
field resulting from each individual element in the array is obtained at a specified
29
1 1.5 2 2.5 3 3.5 4−40
−35
−30
−25
−20
−15
−10
Frequency (GHz)
S11
(dB
)
Figure 2.5: Log-periodic dipole array response: S11
observation point for a single frequency value. The total far zone electric field is
then obtained as the superposition of all of these individual far zone electric fields.
To obtain the time domain signal observed at that far field point, this frequency
domain calculation is repeated for enough frequency points to resolve the frequency
interval of interest and an inverse Fourier transform is applied to all of these results.
It should be noted that for the n-th radiating element, which has the length hn, the
electric field component |Eθ| at the frequency f is affected by |If,n| and f itself. We
choose the frequency fn at which |Eθ| is maximized as the dominant frequency of
the n-th element.
Based on the currents predicted by the HFSS simulations and the far zone
electric fields calculated with (2.6), the far zone electric field radiated by the
30
0 1 2 3 4 5 6−4
−3
−2
−1
0
1
2
3
4
5E
lect
ric F
ield
(V/m
)× 1
05
Time(ns)
Figure 2.6: Log-periodic dipole array response:Far-zone Electric Field
eight element log-periodic antenna illustrated in Fig. 2.4 was obtained with this
combined analytical-numerical approach. The results are shown in Fig. 2.6 for
θ = π/2, φ = 3π/2. Compared with the waveform in Fig. 2.3, the effects of fre-
quency dispersion introduced by the LPDA antenna are readily observed. We note
that it was found to be necessary to calculate the far field with the indicated com-
bined analytical-numerical approach because a calculation with HFSS alone was not
feasible computationally because of the extremely large memory and time require-
ments. A direct time domain numerical calculation for all of the cases considered
also proved to be computationally challenging with our computer resources. The
combined approach was determined to be computationally efficient, and it was val-
idated with a few direct time domain simulations using CST’s Microwave Studio.
31
We introduce the concept of the fidelity of a radiated output pulse to measure the
likelihood that this pulse agrees with some ideal output pulse. For our discussion, we
have selected the band-limited output pulse generated by the idealized infinitesimal
dipole antenna, which is given in Fig. 2.3, to be that ideal pulse. The fidelity of any
output pulse is calculated by the expression
FD(%) = 100×MAX
(
s(t)
‖s(t)‖ ⊗ r(t)
‖r(t)‖
)
, (2.7)
where s(t) is the idealized output signal, r(t) is the actual output signal, ⊗ is the
correlation operator, and ‖ · ‖ means to calculate the total signal power. For the
output signal produced by the dispersive eight-element LPDA, which is shown in
Fig. 2.6, the fidelity is 75.36%.
As pointed out in [27], when a broad bandwidth pulse is radiated by an LPDA
antenna, its low frequency components are mainly radiated by the longer dipoles,
which are located furthest from the feed point, while the higher frequencies are radi-
ated by the shorter dipoles, which are located nearest that feed point. In addition,
the longer dipoles are also further from the observation points in the main beam
direction than the shorter ones are. These two facts combine to tell us that the
time delay is larger in the low frequency regime and, consequently, there is more
frequency dispersion introduced into the output pulse from that range.
2.2 MTM Phase Shifter Corrections
CL
(a)
L
C
(b)
Figure 2.7: (a) Left-Hand Phase-Shifter, (b) Right-Hand Phase-Shifter
Based on these observations, we propose to use a set of electrically-small
metamaterial-based phase shifters to adjust the time delays associated with each
32
element, particularly at the low frequencies. As the currents propagate along a nor-
mal transmission line, they acquire a negative phase shift. The MTM phase shifter
is essentially a left-handed transmission line; one cell of this left-handed transmis-
sion line structure is composed of a series capacitor C and a shunt inductor L as
shown in Fig. 2.7a. When the angular frequency ω 1
2√
(LC), this MTM phase
shifter produces a positive phase shift given by the relation [3]
∆φ ≈ 1
ω√
(LC). (2.8)
The set of MTM phase shifters introduced into the LPDA antenna is designed to
produce the phase shifts, mod 2π, required to align the phases of all of the elements
appropriately to achieve the desired time alignments. Both positive and negative
phase shifters are actually necessary to achieve the desired phase compensation. A
negative phase shift is simply obtained with a length of normal transmission line
composed of shunt capacitor C and a series inductor L as shown in Fig. 2.7b.
It must be noted, of course, that the positive phase shift ∆φ in Eq. (2.8) can
also be obtained by the negative phase shift −(2π −∆φ). Nonetheless, this would
require introducing long segments of transmission line and, hence, would impact the
balanced distribution of currents driving the radiating elements. The compactness of
the MTM phase shifter is very attractive in this regards. Moreover, we have found
that effective dispersion engineering in this LPDA case requires one to minimize
the amount of phase shift associated with each radiating element to minimize the
corresponding change in the magnitude of its current distributions.
Conceptually, we allow for the application of the MTM phase shifters along
the transmission lines or at the feed points of the radiating elements. A diagram
of the proposed modified LPDA antenna, including the MTM phase shifters, is
shown in Fig. 2.8, where the blocks between the printed dipoles and the feed line
represent the MTM-phase shifters. From the HFSS simulations of the performance
of this MTM-phase shifter modified LPDA antenna, it was observed for a given
dipole, with or without the phase shifter, that the current magnitude distribution
33
z
y
Figure 2.8: Modified logperiodic dipole array antenna
2.1 2.2 2.3 2.4 2.5 2.6 2.7−4
−3
−2
−1
0
1
2
3
4
Frequency(GHz)
Pha
se(R
ad.)
Without Phase ShifterWith Phase Shifter
Figure 2.9: Dipole current w/o phase shifter
remains essentially the same while the current phase is modified. For example, as
shown in Fig 2.9, the positive phase shift produced by an MTM-phase shifter is
clearly shown by the blue dash and red solid curves. In this paper, the antenna
performance is calculated using a MATLAB simulation model of the MTM-phase
shifter modified LPDA antenna. For the designs considered below, we have restricted
their location, because of the ease of construction of the required phase shifters, to
the centers of the transmission line segments between the printed dipoles. In the
34
MATLAB simulations, an MTM phase shifter is applied to the transmission line
segment between every printed dipole; the current phase at the subsequent dipole is
thus modified according to Eq. (2.8). The far zone electric field is then calculated
according to Eq. (2.6).
0 1 2 3 4 5 6−4
−3
−2
−1
0
1
2
3
4
5
Ele
ctric
Fie
ld(V
/m)×
105
Time(ns)
Figure 2.10: Far-zone electric field with perfect phase compensation
For the purpose of comparison, we first give the ideal LPDA result in Fig. 2.10.
For this case the phases φf,n of each of the current elements If,n were artificially
linearized with respect to the reference phase point of the LPDA, which was selected
as explained below. In this manner, with respect to the far field phase in the endfire
direction, the LPDA looks like a single element located at the phase reference point
throughout the frequency range of interest. For this reason, this phase reference
point is also referred to as the equivalent radiation point in this paper. The fidelity
of this ideal LPDA output pulse was approximately 97%.
To determine the phase adjustment for each individual radiating element along
the entire feed line of the LPDA antenna, we first choose a point along the feed line as
the phase reference point and then find the phase shift for each element with respect
to that reference point. The current phases are then artificially linearized with
respect to it. In particular, for every antenna element, at its dominant frequency
35
fn, we find the phase shift for Ifn,n such that its phase is equal to the artificially
linearized phase value relative to the reference value. This phase shift becomes the
target phase shift for the n-th antenna element. Then, all of the phase shifters are
designed to achieve these target values at those frequencies. The phase shift values
at the other frequencies follow from these design specifications. It should be noted
that the current phase at other frequencies is usually not equal to the artificially
linearized phase. As a result, there will be differences between the actual dispersion
engineered LPDA antenna and the idealized linear phase shift version that generates
the output waveform shown in Fig. 2.10. The output waveform is finally calculated.
Further adjustments of the phase shifters are made to ensure both a good fidelity
value and target phase shift values that are not too large so they can, in fact, be
implemented physically.
For the simulation results presented here, we choose the reference phase point as
((y5 + λf5), 0, 0), where y5 is the location of the fifth shortest antenna element and
λf5 is the wavelength corresponding to the resonant frequency, f5, of that element.
Thus, for the LPDA arrays considered in this paper, we have elected to have the
equivalent radiation point located at one of the radiating elements nearest to the
center of the array. We note that with this choice, there is a closer agreement
between the slopes of the response of the actual phase shifter and the linearized
values not just at the dominant frequency, but also for neighboring frequencies as
well.
Along the endfire direction, (−y), the phases of all the elements in the far field
are adjusted to the same point, i.e., to that reference phase point. We note that
this phase adjustment is made only for this endfire (main beam) direction. If high
fidelity were desired away from the endfire direction, different phase adjustments
would be required. They would need to account for the different time delays that
occur from each of the radiating dipole elements to the observation point in the
desired direction. This would only be possible if the new observation point was in a
direction supported by the patterns of each radiating element. Nonetheless, because
the variances in the time delay differences with respect to a change in direction are
36
continuous, the modified LPDA reported here should perform reasonably well for
directions near to the mainbeam (endfire) direction. We also note that by electing
to have the phase reference point near the middle of the array, the necessary phase
shift magnitudes were minimized. This choice also further decreases the variances
and helps improve the fidelity of the output pulse for directions near to the endfire
direction.
In the dispersion engineering procedure, we have also assumed that the current
element magnitudes were fixed. However, the magnitude distributions of the cur-
rents along the array are, in fact, affected by the phase shifters because of the mutual
couplings between the elements in the LPDA. These changes in the magnitudes of
the currents are taken into account in our simulations. Phase shifters introducing
large phase shifts change the magnitude distribution dramatically. For this reason,
we basically could not always obtain the target phase adjustment without destroying
the desired magnitude distribution. This was particularly true for the last adjust-
ment made to the LPDA. After adjusting all of the previous elements, we found that
some flexibility was needed in the choice of the last phase shift value to maximize
the resulting fidelity. In addition to choosing the reference point to help minimize
the size of the phase shifts, we also made the compromise to emphasize modifying
the behavior of the longer elements; that is, the phase shifters were designed so that
the phase adjustments for the longer antennas were matched to the target phase
adjustment first rather than for the shorter antenna elements. Note that we also
tried the obvious variation where the shorter antennas were emphasized first. It was
discovered that because the phase variations are larger for the longer dipoles, the
desired outcome was the best when the compensations for the dispersion behaviors
of the longer dipole elements were achieved first. Moreover, we have found that even
though we emphasize the longer elements first, the current distributions along the
shorter elements nevertheless remain very close to their ideal counterparts.
The resulting far-zone electric fields for an eight element LPDA antenna and a
ten element LPDA antenna are shown, respectively, in Figs. 2.11 and 2.12. The
fidelity of the waveforms shown in Fig. 2.11b and Fig. 2.12b are 92.59% and 90.08%,
37
0 1 2 3 4 5 6−4
−3
−2
−1
0
1
2
3
4
5
Time(ns)
Ele
ctric
Fie
ld(V
/m)×
105
(a)
0 1 2 3 4 5 6−4
−3
−2
−1
0
1
2
3
4
5
Ele
ctric
Fie
ld(V
/m)×
105
Time(ns)
(b)
Figure 2.11: Modified log-periodic eight element array output: (a) Far-zone electricfield without phase compensation, (b) Far-zone electric field with designed phasecompensation
respectively. One can see that the modified LPDA response is approaching the ideal
result of 97% corresponding to Fig. 2.10. Details of the phase shifters we designed
to achieve these results are given below. We note that while the modified current
distributions do not fully recover the peak amplitude of the idealized output signal,
they do reproduce a majority of the signal modulations well enough to achieve a
high fidelity.
2.3 MATLAB Simulations
The MATLAB simulator was used to obtain the desired phase shifts. These MAT-
LAB simulations are based on the model Carrel introduced in [25] for the log-periodic
cylindrical antenna shown in Fig. 2.13. Left-hand and right-hand phase-shifters are
added to this model antenna as needed to obtain the desired phase compensated
LPDA antenna. For the left-hand phase-shifter shown in Fig. 2.7a, its ABCD
matrix [28] is
A B
C D
L
=
1 1jωC
0 1
1 0
1jωL
1
=
1− 1ω2 L C
1jωC
1jωL
1
. (2.9)
Similarly, for the right-hand phase-shifter shown in Fig. 2.7b, its ABCD matrix is
38
0 1 2 3 4 5 6−4
−3
−2
−1
0
1
2
3
4
5
Time(ns)
Ele
ctric
Fie
ld(V
/m)×
105
(a)
0 1 2 3 4 5 6−4
−3
−2
−1
0
1
2
3
4
5
Ele
ctric
Fie
ld(V
/m)×
105
Time(ns)
(b)
Figure 2.12: Modified log-periodic 10 element array output: (a) Far-zone electricfield without phase compensation, (b) Far-zone electric field with designed phasecompensation
A B
C D
R
=
1 jωL
0 1
1 0
jωC 1
=
1− ω2 L C jωL
jωC 1
. (2.10)
The phase-shifter between the (n−1)-th and n-th elements of the LPDA antenna
is added at the middle of the transmission line of length ln that connects those two
elements as shown in Fig. 2.14. The ABCD matrix representation of this phase-
shifter modified transmission line is
An Bn
Cn Dn
=
cos(β ln2) jZ0 sin(β
ln2)
jY0 sin(βln2) cos(β ln
2)
Ap Bp
Cp Dp
×
cos(β ln2) jZ0 sin(β
ln2)
jY0 sin(βln2) cos(β ln
2)
, (2.11)
where
Ap Bp
Cp Dp
is the matrix that represents the phase-shifter, and Z0 and Y0
are the characteristic impedance and admittance, respectively, of the transmission
line. Referring to Fig. 2.14, for the phase shifter between the (n − 1)-th and n-th
39
n
hn
s
Rn
Figure 2.13: Log-Periodic cylindrical antenna
I ′n−1 I ′′n−1 I ′n I ′′n
−Vn−1
+In−1 −
Vn
+In
Figure 2.14: Phase-shifter in transmission line
radiating element, the relation between the input voltage and current of the phase
shifter, respectively, Vn−1 and I ′′n−1, and its output voltage and current, respectively,
Vn and I ′n, are
Vn−1
I ′′n−1
=
An Bn
Cn Dn
Vn
I ′n
. (2.12)
Equation (2.12) can be rearranged in the form
I ′n =1
Bn
Vn−1 −An
Bn
Vn
I ′′n−1 =Dn
Bn
Vn−1 + (Cn −AnDn
Bn
)Vn. (2.13)
Then, according to Fig. 2.14, the current In applied to the n-th radiating element
is given by the expression
40
In = I ′′n − I ′n = − 1
Bn
Vn−1 +
(
An
Bn
+Dn+1
Bn+1
)
Vn +
(
Cn+1 −An+1Dn+1
Bn+1
)
Vn+1. (2.14)
Because of the crisscross connections of the LPDA feed line, when n is odd, V tn = Vn
and Itn = In, and when n is even, V tn = −Vn and Itn = −In, where V tn and Itn
are, respectively, the voltage and current at the terminals of the n-th antenna. Thus
the current at the terminals of the antenna, Eq. (2.14), can then be written in the
form
Itn =1
Bn
V tn−1 +
(
An
Bn
+Dn+1
Bn+1
)
V tn +
(
−Cn+1 +An+1Dn+1
Bn+1
)
V tn+1. (2.15)
For n = 1 and n = N , the terminal currents are explicitly
It1 = I ′′1 =D2
B2V t1 +
(
−C2 +A2 D2
B2
)
V t2,
ItN =1
BN
V tN−1 +(
1
ZL
+AN
BN
)
V tN , (2.16)
where ZL is the terminating impedance of the LPDA antenna. The admittance
matrix [YT ] of the LPDA antenna driven with the phase shifter modified transmission
lines can be derived immediately from Eqs. (2.15) and (2.16). The currents on the
radiating elements, needed to calculate the far field output pulse, are then obtained
from the relation
IA = [YA] ( [YT ] + [YA] )−1 I , (2.17)
where [YA] is the admittance matrix of the radiating elements and I is the current
source driving the LPDA antenna as defined in [25]. The MATLAB simulations of
the far-field output waveform generated by the dispersion engineered LPDA antenna
were then calculated with Eq. (2.6) using the feed point currents, If,n, i.e., the
elements of the array, IA.
As noted previously, for ease of fabrication of the dispersion engineered LPDA
antenna, only phase shifters located in the middle of transmission lines were used,
41
as described above. The formulation, which includes phase shifters applied directly
to the radiating elements, is summarized for completeness in the last section of this
chapter. The transmission-line-based phase shifters were designed to dispersion en-
gineer both the eight element and ten element LPDA antennas. The parameters
that define the phase-shifters introduced in each case to achieve the highest fidelity
values are listed, respectively, in Tables 2.1 and 2.2, where n means the n-th an-
tenna, the C column gives the capacitor values used in the phase-shifters, and the
Type&Number column gives the number of phase-shifters and whether they are
right-hand (R) or left-hand (L). The corresponding inductor values are determined
for each phase-shifter by the expression
L = Z20 C, (2.18)
where Z0 = 77.23Ω is the characteristic impedance of the feed line in a dipole-
based LPDA antenna. The dimensions of the dipole LPDA antenna shown in Fig.
2.13 are given explicitly in Table 2.3 for each of the cases considered here, where
τ = Rn
Rn+1= hn
hn+1and σ = Rn+1−Rn
2hn. In Table 2.3, hmax is the length of the longest
antenna, Rmax is the distance between the longest antenna and the terminating
apex. The gap, sn, between the feed lines is set to a constant value: sn = 2.159mm.
The diameter of each transmission line segment is set to 1.778mm. The diameter dn
of the n-th dipole antenna is determined by dn = hn/LD, where LD is the ratio of
the length of that dipole to its diameter. The LPDA antenna is assumed to be fed
at the shortest antenna; it is terminated with a matched resistor, i.e., ZL = 73Ω.
As noted above, the performance of the dispersion engineered LPDA antenna was
measured by the fidelity of the bandlimited actual output waveform with respect to
the bandlimited reference output waveform generated by the idealized infinitesimal
dipole antenna. Taking into account the total power normalizations of the actual
and reference output signals, the fidelity of the actual output waveform is a measure
of how well the bandlimited ideal derivative of the bipolar input pulse is recovered.
For the eight element antenna, the result was bandlimited to the interval 1.0GHz ≤f ≤ 4.0GHz. On the other hand, for the ten element antenna, because of its
42
Table 2.1: Phase-shifter Parameters Values for the Eight Element LPDA Antenna
n Type&Number C(pF )1 N N2 N N3 R,1 0.034 L,1 95 R,1 0.146 R,3 0.147 R,4 0.278 R,4 0.44
Table 2.2: Phase-shifter Parameters Values for the Ten Element LPDA Antenna
n Type&Number C(pF )1 N N2 L,1 503 R,1 0.0114 L,1 4.45 N N6 R,3 0.187 R,3 0.328 R,4 0.49 R,4 0.4710 R,4 0.65
Table 2.3: Eight Element Log-Periodic Antenna Dimensions
n τ σ Rmax(mm) hmax(mm) LD8 0.867 0.152 157.7 138 117
Table 2.4: Ten Element Log-Periodic Antenna Dimensions
n τ σ Rmax(mm) hmax(mm) LD10 0.867 0.152 198.9 174 117
43
Table 2.5: Phase Shifts at the Dominant Radiation Frequencies
n Eight Elemant Ten ElementFreq.(GHz) Phase-shift(Rad) Freq.(GHz) Phase-shift(Rad)
1 4.0 -0.1048 3.85 -0.19392 4.0 -0.0694 4.43 -0.02763 3.85 -0.1815 4.1 -0.08534 3.32 0.1244 3.59 0.25975 2.94 -0.0984 3.11 0.48966 2.62 -0.2273 2.77 0.01247 2.28 -1.6479 2.38 -1.24388 1.98 3.1633 2.04 4.06489 1.8 -4.137710 1.52 1.3999
broader bandwidth, the excitation pulse was band limited to the frequency interval:
0.5GHz ≤ f ≤ 4.5GHz. To ensure a large tolerance factor in the results for any
future fabrication and measurement efforts, this 4.0GHz frequency bandwidth was
larger than the desired operating interval of 1.723GHz ≤ f ≤ 4.06GHz. We note
that there is a significant increase in the phase offset as the number of elements in
an LPDA antenna is increased. This was the main reason that we studied both the
eight and ten element LPDA antennas in detail. Similar fidelity improvements were
realized with LPDA antennas with even more elements.
For the configurations specified by Tables 2.1 and 2.2, the required phase-shifts
are given in Table 2.5. The output waveforms with and without these designed phase
compensations are shown, respectively, in Figs. 2.11 and 2.12. For the eight element
antenna, the fidelity was 73.39% without phase compensation and 92.6% with phase
compensation. For the 10 element antenna, the fidelity was 65.73% without phase
compensation and 90.08% with phase compensation. These fidelity results hold for
all observation points in the endfire direction as long as those points are in the far
field of the entire LPDA. Note the decrease of the fidelity of the output waveform
between the uncompensated ten and eight element systems. If more bandwidth is
desired, more elements have to be added to an LPDA antenna. As more elements
44
are added, the dispersion effects become larger. While the introduction of the MTM
phase shifters produces a significant improvement in the fidelity of the output wave-
form even for a modest number of radiating elements, the improvement becomes
even more significant as the number of elements is increased.
0.5 1 1.5 2 2.5 3 3.5 4 4.5−40
−35
−30
−25
−20
−15
−10
−5
Frequency(GHz)
S11
(dB
)
Unmodified LPDAModified LPDA
Figure 2.15: The S11 values for the dispersion-engineered 10 element LPDA as afunction of the frequency.
For the dispersion compensated and uncompensated 10 element LPDAs we show,
respectively, in Figs. 2.15 and 2.16, the S11 values over the frequency band of
interest and their phase values in comparison to the corresponding values in the ideal
linearized case. According to the S11 values, the dispersion compensated 10 element
LPDA still has a wide bandwidth even though the impact of the introduction of
the phase shifters at the lower frequencies is noticeable, particularly at the resonant
frequencies of the radiating elements for which the phase shifters were designed.
This maintenance of the bandwidth is further confirmed by the overall fidelity of
the output signal. On the other hand, the small drop in the peak amplitude of the
modified LPDA’s output time signal in comparison to the ideal result is attributable
to this small increase in the insertion losses caused by the presence of the phase
shifters at the lower frequencies. The impact on the phase distribution achieved by
introducing the phase shifters along the LPDA is clearly seen in Fig. 2.16. The
phase distribution is matched to the target phases, particularly in the low frequency
45
1 2 3 4 5 6 7 8 9 10−4
−3
−2
−1
0
1
2
3
Element Index
Pha
se(R
ad)
Target PhaseWithout Phase ShifterWith Phase Shifter
Figure 2.16: The phase distribution along the dispersion-engineered 10 elementLPDA.
range, except for the last element, which, as noted above, requires some flexibility
in its phase value to optimize the overall fidelity.
It should be noted that the fidelity in Eq. (2.7) could also have been defined as
FD(%) = 100×MAX
( ∣
∣
∣
∣
∣
s(t)
‖s(t)‖ ⊗ r(t)
‖r(t)‖
∣
∣
∣
∣
∣
)
. (2.19)
By adding the magnitude operator (i.e., the |·|), this definition would allow a sign
difference between the signals s(t) and r(t), that is, it would allow for the introduc-
tion of an extra π phase shift. Such a π phase shift could occur, for instance, if one
elected to feed the LPDA antenna differently. Based on this magnitude definition,
the fidelities of the pulses without phase adjustment in Figs. 2.11a and 2.12a are
85.03% and 68.50%, respectively. There would be no changes in the fidelity values
associated with the dispersion compensated LPDA antenna results since they have
46
the correct signs already. However, because our dispersion engineering is focussed
on phase compensation, we elected to emphasize Eq. (2.7) in our results, i.e., Eq.
(2.7) more accurately accounts for all of the differences in the phases between the
actual and reference output pulses.
Representative E-plane antenna patterns produced by the modified LPDA and
by the unmodified LPDA at 2.04GHz, 3.59GHz, and 4.5GHz, are shown, respec-
tively, in Figs. 2.17(a)-2.17(c). As seen in Fig. 2.17(a), it was found that without
the additional tweaking to achieve a high fidelity, the modified LPDA does not main-
tain the requisite endfire antenna pattern originally obtained in the low frequency
range. On the other hand, the antenna patterns shown in Figs. 2.17(b) and 2.17(c)
show that it does in the mid and high frequency ranges. The optimum solution,
which was based on the best obtainable value of the time domain fidelity value,
does recover the desired endfire antenna patterns throughout the frequency range of
operation. The reason that the basic modified LPDA failed to maintain this highly
desirable frequency domain antenna pattern property in the low frequency range is
that although the current phase was adjusted to the optimum solution value at each
of the resonant frequencies, at other frequencies, the differences between the phases
of the modified LPDA and the optimum solution were significant, particularly in the
lower frequency range. By further adjusting the phase values to achieve a broader
matching of the responses, the endfire radiation pattern behavior was recovered over
the operational band.
The phase reference point or the equivalent radiation point is a simple approach
to understand the optimum solution. To find the equivalent radiation point in the
far field main beam direction, we define an equivalent current for the LPDA to be
Iequv =N∑
n=1
αnIne−jkdn = |Iequv|e−j(kd+φ0) = |Iequv|e−jφequv , (2.20)
where
αn =1
sin(kln)
cos(kln2cos(θ))− cos(kln
2)
sin(θ)(2.21)
is the current amplitude weighted dipole element pattern function, ln is the length
of the n-th element, and dn is the distance between the n-th antenna and the apex
47
point, the apex point being taken to be the coordinate system origin. This equivalent
current is to be understood as the single current source which generates the LPDA
far field at the observation point at a specific frequency.
This source is located at the “equivalent radiation point”, a distance d from
the apex, and has a relative initial phase φ0 with respect to it. It should be noted
that in general the equivalent radiation point is different from the usual phase center
concept. The phase center is an equivalent phase reference point at a given frequency
that is defined by the curved wavefront that passes through the far field observation
point. However, it can be proved that in the main beam direction, the radiation
point and the phase center coincide. Since the fidelity is obtained for the output
pulse in the main beam (endfire) direction relative to the idealized output pulse in
the same direction, the phase center and the equivalent radiation point coincide for
our dispersion engineering application.
The phase center calculation for the optimum solution shows that its phase center
does not change with respect to the frequency, i.e., a fixed equivalent radiation
point is obtained for the optimum solution. Thus, it can be used as a reference to
describe the phase evolution in the main beam direction. Consequently, according
to (2.20), the equivalent radiation point distance and, hence, the phase center at
each frequency of interest can be calculated as
d =dφequvdf /(2πc), (2.22)
where c is the light speed in vacuum. The calculated phase centers for the unmodified
and the modified LPDAs are shown in Fig. 2.18. They are compared there to
the fixed point value of the optimum solution. Figure 2.18 illustrates that as the
frequency changes, the phase centers of the modified LPDA vary less from the
optimum solution value than the original LPDA antenna phase centers do. Thus, the
dispersion engineering reduces the phase center variation from the optimum solution.
In particular, the variance from the optimum solution values of the phase centers
for the modified LPDA is 0.164, while it is 0.288 for the original unmodified LPDA.
This is yet another confirmation that the modified LPDA is indeed approaching the
48
optimum solution.
2.4 Phase Shifters Included on the Radiating Elements
In the above section, we calculated the admittance matrix [YT ] for the transmission-
line-based phase shifters. As noted, we can also add phase shifters at the connection
between the antenna elements and the transmission line. In this case, the admittance
matrix [YA] for the element must be calculated. For a phase shifter added to the
n-th antenna, the relation between its input voltage V ′
n and current I ′n and its output
voltage Vn and current In is
V ′
n
I ′n
=
An Bn
Cn Dn
Vn
In
. (2.23)
Then, one has immediately
Vn
In
=1
AnDn − BnCn
Dn −Bn
−Cn An
V ′
n
I ′n
=
A′
n B′
n
C ′
n D′
n
V ′
n
I ′n
. (2.24)
In [25], the antenna voltage vector, VA [V1, V2, · · · , Vn]T, and the antenna current
vector, IA = [I1, I2, · · · , In]T, are related by the expression
IA = [YA] VA, (2.25)
where [YA] is the admittance matrix of the radiating element. According to Eq.
(2.24),
VA = A′V ′ +B
′I ′
IA = C′V ′ +D
′I ′, (2.26)
where the matrices A′, B′, C′, D′ are diagonal, their diagonal elements being the
terms A′
n, B′
n, C′
n, and D′
n give in Eq. 2.24, respectively. The relation between V ′
and I ′ after the phase-shifter is added can then be calculated as
49
C′V ′ +D
′I ′ = [YA](A′V ′ +B
′I ′). (2.27)
Rearranging Eq. (2.27), one obtains
I ′ = (D′ − [YA]B′)−1([YA]A
′ −C′)V ′ = [Y ′
A]V′, (2.28)
where [Y ′
A] is the admittance matrix of the phase shifter-modified radiating elements.
The output waveform follows immediately with the MATLAB simulator.
2.5 Conclusions
In this chapter, we have demonstrated that the frequency dispersion associated with
an LPDA antenna can be improved by applying left-hand and right-hand phase
shifters to adjust the relative phases of the radiating elements to achieve a better
time alignment of the individual frequency components. The improvement was
measured by comparing the fidelity of the dispersion-engineered LPDA antenna’s far
field output pulse to an idealized output pulse generated by driving an infinitesimal
dipole with the input excitation current pulse. The performance characteristics of
the dispersion-engineered LPDA antenna were obtained with MATLAB and HFSS
simulations. In the MATLAB simulations, the LPDA antenna model reported in [25]
was extended in this work to accommodate the MTM-based phase shifters. The
procedures to find the suitable phase shifters, both in the transmission lines between
the radiating elements and at the terminals of the radiating elements, were provided.
Significant improvements in the output pulse fidelity were achieved for a dispersion-
engineered LPDA antenna with a large number of elements.
The HFSS simulations were performed for an eight element printed dipole LPDA
antenna. The frequency dispersion associated with an LPDA antenna is readily
observed in its output pulse. Because of the high complexity of these simulations,
the MATLAB simulations were performed rather using a known cylindrical dipole
LPDA antenna model. In particular, the MATLAB simulator was applied to the
phase-shifter modified eight element LPDA antenna and the ten element LPDA
50
antenna with and without phase shifters. We note that, as is pointed out in [26], the
printed dipole and cylindrical dipole can be made to be equivalent. In fact, in both
the HFSS and MATLAB simulations for the eight element LPDA antenna without
phase shifters, we obtain currents whose magnitudes and phases are very similar to
each other. Thus, we have found that the MATLAB simulation results using an
appropriately designed cylindrical dipole LPDA antenna are very consistent with
those generated by the equivalent printed dipole LPDA antenna HFSS simulations.
51
0.5
1
30
210
60
240
90
270
120
300
150
330
180 0
Modified LPDA
1
2
30
210
60
240
90
270
120
300
150
330
180 0
Unmodified LPDA
(a) 2.04GHz
1
2
30
210
60
240
90
270
120
300
150
330
180 0
Modified LPDA
1
2
30
210
60
240
90
270
120
300
150
330
180 0
Unmodified LPDA
(b) 3.59GHz
1
2
30
210
60
240
90
270
120
300
150
330
180 0
Modified LPDA
1
2
30
210
60
240
90
270
120
300
150
330
180 0
Unmodifled LPDA
(c) 4.5GHz
Figure 2.17: Modified LPDA and Unmodified LPDA E-Plane Antenna Patterns
52
1.5 2 2.5 3 3.5 4 4.50.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency(GHz)
d(m
)
Unmodified LPDAModified LPDAOptimum
Figure 2.18: Phase center locations
53
CHAPTER 3
LOW-Q, ELECTRICALLY SMALL, EFFICIENT NEAR-FIELD RESONANT
PARASITIC ANTENNAS
3.1 Chu-limit and Electrically small antennas
Electrically small antennas (ESAs) have been studied extensively in the past and
are still of great research interest. Because of its compact dimension, an ESA has
many potential wireless applications, including unmanned devices, personal com-
munications, and wireless sensor networks. An ESA with high overall efficiency and
broad bandwidth is always preferred in these types of applications. However, as
pointed out in many works, for example [9–11], there are constraints amongst the
antenna’s physical dimensions, its bandwidth, and its radiation efficiency. For an
ESA surrounded by the smallest enclosing sphere of radius a, its fractional band-
width at its resonance frequency f0 is approximately limited by 2(ka)3/RE, where
the free space wave number k = 2π/λ0, λ0 = c/f0 being the free space wavelength
at the resonance frequency with c being the free space speed of light, and RE is its
radiation efficiency, i.e., the ratio of the total power it radiates to the total power it
accepts at its terminals. We note that for an antenna in free space, it is said to be
electrically small if ka ≤ 1, which means, according to [10], that it is contained in
the Wheeler radiansphere. If there is an infinite perfect electric conductor (PEC)
plane present, this ESA criterion is reduced to ka ≤ 0.5 since only half of the radi-
ansphere is involved. However, in reality, if a PEC ground plane is involved with an
antenna, it will be finite. This will be the case in all of the designs presented below.
Nonetheless, even if the ground plane is relatively small, we will continue to invoke
the ka ≤ 0.5 ESA definition based on the smallest sphere enclosing the radiating
structure without the ground plane in our discussions.
Because of its compact size in terms of the wavelength, an ESA usually has a
54
highly reactive impedance, which requires specific attention in its design. Even if
this reactance is compensated properly, the corresponding radiation resistance of the
ESA is also usually very small. In real applications an ESA is connected to a source
with a certain source impedance, which is usually 50Ω. Although an ESA could be
made to be resonant, i.e., the total reactance equals zero, and could have a high
radiation efficiency, the overall efficiency will remain poor because of the huge mis-
match between the input resistance of the ESA and the source resistance. Matching
networks can be used to achieve the remaining resistive match to the source. For
instance, a quarter wavelength transformer is one of those matching networks. Ob-
viously, the extra quarter wavelength will break the ESA criterion if it is included in
the overall size of the antenna. Moreover, if an extreme impedance transformation
is required, the quarter wavelength transformer bandwidth itself will be extremely
narrow. Compact matching circuits constructed from L-sections of capacitors and
inductors are another kind of matching network. These matching circuits will in-
troduce additional losses into the antenna system and like the quarter wavelength
transformer, can even diminish the ESA bandwidth. On the other hand, losses
and incomplete matching may be an acceptable practical approach to increasing
the bandwidth. Despite of their disadvantages, these kind of small antennas with
external matching networks exist in many real applications because of their simple
design methodology.
Tremendous effort has been made to understand the limits of ESAs and to pro-
vide guidelines for the best possible designs. As an important metric for ESA per-
formance, the quality factor Q of a resonant antenna is defined as the ratio between
the bigger of the stored electric and magnetic energy and the radiated energy [9,11].
In [9], a minimum Q value was derived. A more exact value of this lower bound was
obtained in [29], labeled herein as QChu:
QChu =1
(ka)3+
1
ka. (3.1)
For a high Q value, which is now clearly the case for an ESA, the frequency band-
width of an antenna is approximately equal to two times the reciprocal of Q. Hence,
55
the lower bound on Q also defines the maximum possible achievable bandwidth for
an ESA.
Taking into account the radiation efficiency of the antenna RE = Prad/Pacc,
where Prad and Pacc are, respectively, the radiated and accepted powers, the lower
bound on the quality factor was given in [30] as
Qlb = RE ×QChu . (3.2)
In this chapter, we will focus on antennas that have both low Q and high overall
efficiencies. In particular, if Pinp is the total input power of the source, then the
overall efficiency of the antenna is OE = Prad/Pinp. For this reason, we obtain
nearly complete matching in all of the reported designs so that Pacc ≈ Pinp. More-
over, all designs will have the highest possible RE values, i.e., very close to unity.
Consequently, our designs will have OE ≈ RE. Note that because all the designs
are electrically small, their directivities are those of a small electric dipole. Because
the realized gain of an antenna is given by the expression G = OE × D, where D
is the directivity of the antenna, our designs also maximize the realized gain of an
ESA.
It should be noted that Qlb is not related to the source driving the antenna; it
simply reflects a property of the electrical size of the antenna and its radiation and
conductive losses. Antennas with low Q values can be achieved easily with low RE
values, i.e., since RE = Rrad/(Rrad +Rloss), where Rrad and Rloss are, respectively,
the radiation and conduction resistance of the antenna, low Q values can be obtained
with poor radiation performance in relation to high conductive losses. Given certain
practical bandwidth requirements, this has been an acceptable tradeoff for a variety
of real applications.
All of the Q values will be reported relative to the lower bound, i.e., the Q-ratio:
Qratio =Q
Qlb
. (3.3)
Moreover, the Q values will be obtained either from the half-power matched VSWR
56
bandwidth of the antenna
QV SWR = Q3dB =2
FBW3dB=
2f0f+ − f−
, (3.4)
where f+ and f− are, respectively, the frequencies for which the S11 values become
−3 dB above and below the resonant frequency of the antenna, f0, where in principle
X(f0) = 0; or from the corresponding −10 dB points as Q10dB = 23×FBW10dB
[11]; or
from the analytical result given in [11]:
QY B =f0
2 Rinp(f0)
√
√
√
√ [ (∂fRinp) (f0) ]2 +
[
(∂fXinp) (f0) +|Xinp(f0)|
f0
]2
, (3.5)
where Rin and Xin are, respectively, the resistance and reactance of the input
impedance of the antenna, i.e, Zin = Rin + j Xin.
Several values for the actual realizable lower bound limits on the Q value have
been reported [10, 31–34]. Moreover, several ESA designs approaching these limits
have been introduced. For instance, the spherical-cap dipole antenna considered
in [10] was re-considered in [31] and [35]. With the spherical cap diameter being
0.05λ, that is ka = 0.17, and assuming the overall efficiency to be one, the Q ratio
was calculated in [31] to be 1.75 and was claimed to be the lowest Q ratio for an
electric-based ESA. On the other hand, it was claimed in [34] and again in [35] that
the lower limit of electric-based ESAs should be 1.5. In contrast, the limit for a
magnetic-based ESA with free space in the interior of the smallest sphere enclosing
should be 3.0, but would reduce to 1.0 if this sphere were loaded with a magnetic
material having µ → ∞. In fact, an example of a magnetic-based ESA that could
achieve a Q ratio of one was originally given in [36]; this limit being re-affirmed
in [31, 34, 35]. Thus, one could reach lower Q ratios with magnetic-based antennas
than with their electric-based counter-parts. We note that since the Q ratios were
the primary results of interest in [31], it was assumed that the antenna was matched
externally in some manner to the source and, thus, that the input resistance at the
57
resonance frequency, which is in fact rather small, was not exposed. Similar external
matching conditions were also invoked in [34, 35].
Another Chu-limit ESA was introduced by Best in [37], i.e., the spherical folded
helix antenna. With ka ∼ 0.263, it has a measured Q ratio of 1.52. The spherical
helix is an electric-based antenna, i.e., it radiates the TM10 mode. Its first reso-
nance is an anti-resonance, i.e., at the resonance frequency where Xant(fres) = 0,
∂f Xant(fres) < 0. The noted results are measured at its second resonance fre-
quency, f0 = 299.99MHz, which is a resonance, i.e., ∂f Xant(f0) > 0. The measured
four-arm folded spherical helix antenna has an input resistance approximately equal
to 47.5Ω at that resonant frequency. The number of arms provided the ability to
achieve favorable impedance matching to the source. A small spherical coupling
resonator array antenna was introduced in [20,38], which for ka 0.54 and eight-ring
resonators, also approached a Q ratio of 1.5. In the same sense it too is an electric-
based antenna and the indicated Q ratio was also obtained at its second resonance
frequency where it has a resonance behavior. External matching networks to the
source were needed for this ESA (no ground plane) that depended on the number
of coupling resonator elements included in the antenna. Additional analysis [39,40]
into the multiple resonances associated with these antennas have been considered
in relation to Q values near to the lower bound.
The emergence of metamaterials (MTMs), i.e., artificial materials whose per-
mittivity and permeability values can be designed − for either positive or negative
values − and their use in the design of ESAs leads to an alternative point of view to
achieve matching, high overall efficiencies, and low Q ratio values. Electrically small
antennas radiating in the presence of MTMs were studied analytically in [12–16].
It was shown that by introducing a metamaterial-based homogeneous, isotropic,
dispersive, electrically-small shell around an electrically-small radiator, the overall
efficiency, OE, of the composite antenna system could be close to one (assuming low
loss MTMs) with Q values near the lower bound and much lower than it with the in-
troduction of an active metamaterial. One important consequence of this approach
was that a properly designed MTM shell eliminated the need for an external match-
58
ing network between the ESA and the source, i.e., despite of its electrically small
size, the antenna systems were shown to have an input impedance Zin ≈ 50Ω at the
resonant frequency, when the source resistance was assumed to be Zs = 50Ω. More
specifically, for an electrically small electric dipole, which by nature is highly capac-
itive, an electrically small epsilon-negative (ENG) MTM shell, which is inductive in
nature, can be applied to form a composite resonant antenna system. Similarly, an
electrically small mu-negative (MNG) shell, which capacitive in nature, is applied
for an electrically small loop antenna. Although the above analytical conclusions
were based on idealized electrically small MTM shells, which have not yet been real-
ized physically, they shine light on a potential design approach which may be easily
implementable to achieve ESAs with high OE’s.
Influenced by these theoretical results, metamaterial-inspired efficient ESAs were
introduced in [17–19]. In the three-dimensional (3D) magnetic-based EZ antenna,
a 3D extrusion of a planer capacitive load loop (CLL) was introduced to provide
a resonant match for the coax-fed, electrically small, inductive semi-loop antenna.
In the 3D electric-based EZ antenna, a helical inclusion was introduced to provide
a resonant match for a coax-fed electrically small monopole antenna. The reso-
nant parasitic CLL extrusion and helical elements were located in very near field
of their coax-fed radiators. These elements, being unit cells of the corresponding
MTMs, acted as simplified versions of the requisite electrically small MNG shell and
ENG shells, respectively. They led to the predicted nearly complete reactance and
resistance matching. These 3D EZ antennas were further simplified to 2D planar
versions. The 3D CLL extrusion was simplified to a printed planar CLL element,
and the helix was simplified to a printed planer meander-line element. It was shown
further that the EZ magnetic-based antenna can be simplified even further by incor-
porating a lumped element capacitor to replace the printed interdigited capacitor.
Again, these resonant near-field parasitic metamaterial-inspired elements led to EZ
antenna designs with high overall efficiencies for which no external matching net-
work was needed. In particular, in [19], it was experimentally verified that a 2D
electric-based EZ antenna at 1.37GHz with ka = 0.49 had an OE ∼ 94% with a
59
fractional bandwidth of 4.1%. In a related manner, it is pointed out in [20] that the
spherical coupled resonator antenna behaves like a negative permittivity sphere.
The electric Z antenna was introduced in [21]; it is yet a further simplification
of the 2D electric-based EZ antenna. The meander-line is replaced by a split Z-
shaped element; the two halves of this element being connected with a lumped
element inductor. The relationship between the resonant frequency and the lumped
inductor value was revealed and the radiation mechanism was better understood.
The resulting resonant near-field parasitic element interpretation was extended to
the realization of the stub antenna introduced in [22]. The electrically small stub
antenna is an attractively simple structure, consisting only of a parasitic located
in the very near-field of a coax-fed monopole and normal to the ground plane that
is constructed from a cylindrical lumped element inductor and a series connected
cylindrical piece of wire, both having the same radius.
For all of these metamaterial-inspired and near-field resonant parasitic antenna
designs, the Q values and Q ratios were given. For instance, the Z and stub antennas
had reported Q ratios approximately equal to 7 and 4, respectively. However, the
emphasis in this prior research has been on achieving high OE values rather than
pushing the associated Q values to the Chu-based limits. Based on the simple struc-
ture of the stub antennas and a further understanding of the radiation mechanism
of the various metamaterial-inspired antennas, an investigation of these resonant
near-field parasitic antennas was undertaken to see how close to the fundamental
limits their Q values could be pushed. Various configurations of the stub antenna
will be introduced in this chapter with different Q ratios and explanations of their
performance will be provided. This investigation leads to the canopy antenna de-
signs also introduced in this chapter, which are an electrically small electric-based
antennas having high OE and low Q ratio values.
The Z antenna design is re-evaluated with some very recent measurement results
and several stub antenna configurations, which incorporate multiple parasitic ele-
ments, are introduced in Section II. The proposed canopy antennas are discussed in
Section 3.3 and their equivalent circuit models are presented in Section 3.4. All of
60
the antenna designs were numerically simulated using ANSOFT’s High Frequency
Structure Simulator (HFSS) 11.1.2. Simulation results for the canopy antennas,
which include the presence of a metamaterial hemisphere within its interior whose
permittivities are less than one, are given in Section 3.5. In all cases, their Q ra-
tios, which are obtained for the first and single resonance state, are compared to
the reported fundamental lower bounds on the Q ratios for the electric-based ESAs,
as well as to those of other known ESA designs. Finally, in Section 3.6, it will be
demonstrated that by replacing the passive inductors with active equivalents, one
can achieve a very electrically small antenna with high OE values and frequency
bandwidths significantly larger than the fundamental upper bound. Section 3.7 will
summarize our conclusions.
3.2 Z Antennas
Figure 3.1: Three dimensional view of the Z Antenna with Inductor=1000nH
61
The Z antenna shown in 3.1 originally reported in [21] was designed with a copper
Z element in a 10mm×10mm×2mm volume, which has a minimum-enclosing sphere
with a = 11.25mm, and with a lumped element inductor of value L = 1000nH to
have a resonance at f0 = 193.921643MHz so that ka ∼ 0.046. The thickness
of the Z element was selected to achieve a large radiation efficiency at this VHF
frequency and below. A discrete sweep of designs from 67.399MHz to 1015.19MHz
was obtained and demonstrated that the resonance frequency is given by the relation
f0 =1
2π
√
1
LeffCeff
, (3.6)
where Leff and Ceff are, respectively, the effective inductance and capacitance of
the antenna system. The effective inductance is linearly proportional to the lumped
inductor value; the effective capacitance remains almost the same value in this
frequency sweep band, being determined primarily by the monopole.
3.2.1 Fabrication and Measurement
Figure 3.2: Three dimensional view of the Z Antenna, Duroid design
62
510 520 530 540 550 560 570 580 590 600−40
−35
−30
−25
−20
−15
−10
−5
0
Freq(MHz)
S11
(dB
)
Lossy(B)Lossless(A)
Figure 3.3: S11 values for the Duroid design Z antenna predicted by HFSS
Because of the weight of one half of the Z element, the first attempt at fabrication
of this design by Boeing Phantom Works, now called Boeing Research & Technology,
in Seattle, WA and its shipment to NIST-Boulder for measurement ended in a
catastrophic structural failure at the solder joint of the inductor as shown in Fig.
3.4.
As a result of this fabrication-shipping issue, it was decided that the Z antenna
needed to redesigned for fabrication with Rogers 5880 DuroidTM to provide struc-
tural integrity. This Duroid-based design is shown in Fig. 3.2. The inductor was
a 47nH element (COILCRAFT 1008HQ − 47NX LB) whose minimum circuit Q
was estimated to be over 100 at the resonant frequency. Based on this Q value, the
conductive loss of the inductor was estimated to be Rloss = 1.6242Ω. The minimum
enclosing sphere had a = 33.4840mm. It should be noted that Rloss is the estimated
maximum conductive loss and the the real loss should be smaller than this value.
For this reason, the lossless design was also provided for comparison purposes. For
the lossless design, which is refereed as design A, the antenna resonance frequency
was f0 = 580.32MHz and, consequently, ka ∼ 0.4077. For the lossy design, which
is refereed as design B, the antenna resonance frequency was f0 = 570.38MHz
63
Figure 3.4: The bottom half of a broken fabricated Z antenna
and, consequently, ka ∼ 0.4. The lossless and lossy designs have different monopole
lengths for good matching to the 50Ω source. The HFSS predicted S11 values for
the Z antenna with a lossy and lossless inductor are shown in Fig. 3.3. The lossy
inductor was modeled in HFSS using two serially connected lumped RLC boundary
elements, one to represent its L = 47nH inductance and the other to represent its
R = 1.6242Ω resistance.
This Z antenna was fabricated by Boeing and was shipped to NIST-Boulder for
total power measurements in their reverberation chamber. To explore the effect of
the ground plane size, designs with large ground plane and small ground plane were
fabricated and are shown in Fig. 3.5 and Fig. 3.6, respectively. The horn antenna
shown in Fig. 3.7 was used for the reference antenna in all of the radiated power
measurements and the bare monopole shown in Fig. 3.8 was also fabricated and
measured for comparison purposes. As shown in Fig. 3.9, the measured and HFSS
predicted S11 values agree very well. Moreover, they confirm that this Z antenna is
well matched to the Rs = 50Ω source. The measured overall efficiency is shown in
comparison with the reference horn antenna in Fig. 3.10, where in this figure, no
ground means small ground plane and ground plane means large ground plane. The
64
Figure 3.5: Fabricated Duroid Z antenna with large ground plane
Figure 3.6: Fabricated Duroid Z antenna with large ground plane
improvement of the Z antennas compared to the bare monopole is over 25dB at the
resonant frequency.
It is noted that because of its larger volume, 30mm × 30mm × 0.17mm, the
inductance of the copper portion of the Duroid-based Z antenna can not be ignored
65
Figure 3.7: Horn Antenna
Figure 3.8: Monpole Antenna
when compared to the lumped element inductor. Thus, while Leff is specifically the
summation of the split Z element inductance and the inductance of the connecting
lumped inductor, the parasitic element still determines Leff and, hence, the resonant
frequency of the Duroid-based Z antenna. Further HFSS simulations have shown
66
510 520 530 540 550 560 570 580 590 600−40
−35
−30
−25
−20
−15
−10
−5
0
Freq(MHz)
S11
(dB
)
Z−Antenna A:Small Ground PlaneZ−Antenna A:Large Ground PlaneZ−Antenna B:Small Ground PlaneZ−Antenna B:Large Ground PlaneHFSS Lossy(B)HFSS Lossless(A)
Figure 3.9: Measured S11 values for the Duroid Z antenna
Figure 3.10: Measured radiated power for the Duroid Z antenna
that the effective resistance of the inductor was closer to 0.7Ω.
67
Figure 3.11: Monopole and monopole with external MFJ tuner
3.2.2 Matching Methods Comparison
As was mentioned in the beginning of this chapter, ESAs usually need matching
networks for resistance and reactance adjustments. The matching networks usually
have side effects such as extra loss or extra bandwidth limitations. When the an-
tennas size is very small compared to its working wavelength, the adjustments are
significant and the side effects could dominate the results.
In Fig. 3.11, the bare monopole is tuned by a MFJ Travel Tuner and the mea-
sured S11 is shown in Fig. 3.12. It can be seen that at several frequencies, low S11
values are achieved with the MFJ tuner. From the radiated power shown in Fig.
3.13, the radiated powers at these frequencies are almost the same for the tuned
monopole and the bare monopole. It is well known that the radiated power Prad is
68
200 220 240 260 280 300 320 340 360 380 400−35
−30
−25
−20
−15
−10
−5
0
Frequency(Mhz)
S11
(dB
)
Figure 3.12: Measured S11 for monopole with MFJ tuner
Figure 3.13: Radiated power by bare monopole and monopole with external MFJtuner
determined from the input power Pin by
Prad = Pin(1− |S11|2)Rrad
Rrad +RL
= Pin(1− |S11|2)η, (3.7)
69
where Rrad and RL are the radiation and conductive loss resistance, respectively.
When S11 is small, the low radiated power is due to the dominant conductive loss
in the external matching network.
Figure 3.14: Monopole antenna with the double stub tuner
299.5 300 300.5 301−18
−16
−14
−12
−10
−8
−6
−4
−2
0
Frequency(MHz)
S11
(dB
)
Figure 3.15: S11 predicted by HFSS for the monopole antenna with the double stubtuner
To show the loss effect better, a lossless double stub tuner was simulated in HFSS
70
as shown in Fig. 3.14. The corresponding HFSS-predicted S11 values are shown in
Fig. 3.15. The HFSS-predicted radiation efficiency is over 98% which means that
with the double stub tuner, the monopole could radiate very well. However, the
fabrication and measurement of the tuner augmented monopole turned out to be
unsuccessful, i.e., a very low radiated power was measured in spite of the very good
HFSS prediction. It was discovered that the problem was that in the HFSS simula-
tions, in contrast to the real fabricated system, the double stub tuner was modeled
originally as a perfect conductor (PEC). Changing the double stub tuner from a
PEC to a copper element, the radiation efficiency predicted by the HFSS simula-
tions became 1.18%, which coincides very well with the actual measured results.
The above reported experiments with external matching networks illustrate some
of their limitations when they are used with the very electrically small antennas, par-
ticularly the resulting S11 values and the realized radiation efficiencies. In contrast
to the external matching networks, we refer to our parasitic structures as internal
matching networks. The internal matching network concept will be explained in
more detail later in this chapter and in the next one.
3.3 Stub Antennas
As noted above, another resonant near-field parasitic element-based antenna is the
stub antenna shown in Fig. 3.16. The driven element is a coaxial-fed monopole
assumed to be located here at the origin of the ground plane. The monopole has
radius equal to 0.5mm and a height equal to 0.78mm. The parasitic element is
oriented vertical to the ground plane and is composed of a cylindrical wire element
connected on top of and in series with a lumped element inductor of the same
radius which is connected to the ground plane at a point located here along the
x axis. The radius of the inductor and the wire equals 1.205mm. The length of
the inductor and the wire equal, respectively, 3.35mm and 2.5mm. The axes of
the inductor and the wire are both oriented along the z-axis and are centered at
the point (3.3555mm, 0, 0). The wire and monopole are treated as copper. The
71
Figure 3.16: Three dimensional view of the stub antenna
inductor is modeled as an epoxy cylinder, specified by the HFSS material, epoxy-
Kevlar-xy, which has the relative permittivity ε = 3.6 and relative permeability
µr = 1.0. This epoxy material encloses a planar lumped RLC boundary element
which defines the inductor. This RLC element lies in the zx plane; it has a height
that is the same as the inductor and a width equal to its diameter. The inductor
has the value L = 1196nH ; it is treated as a lossless element. This stub antenna
resonates at f0 = 299.6074MHz. The HFSS-predicted overall efficiency equals
OE = 97.207% with a fractional bandwidth equal to FBW = 2.2 × 10−3%. The
radius of the smallest enclosing sphere equals a = 5.9728mm giving ka = 0.0375
and, hence, Qratio = 4.94. However, in order to make comparisons with the two
and four element stub antennas, i.e., to compare the Qratio values for the same ka
value, we also take the minimum sphere relative to the center of the monopole,
which gives a = 7.4173mm, ka = 0.0466, and Qratio = 9.4287. On the other
hand, if the parasitic height is adjusted so that the enclosing sphere is centered on
the parasitic element, the ka value will also be matched to the multiple parasitic
element cases. This was achieved by making the following changes: the length of the
72
wire equals 3.969mm, the height of the monopole equals 1.02mm, and the inductor
has the value L = 1122nH . The HFSS-predicted resonance frequency becomes
f0 = 300.3501MHz with an overall efficiency equal to OE = 97.247% and with a
fractional bandwidth equal to FBW = 3.5 × 10−3%. Thus, ka = 0.0467 and the
Qratio = 5.91.
Because of the nature of the radiansphere description, many different antenna
configurations can be considered within a sphere of radius a that result in very
different behaviors in regards to their overall efficiency and quality factors, but
remain limited by the lower bound on Q defined by ka and RE. Nonetheless, it
is generally believed that if one designs an antenna that makes the most complete
use of the volume within the radian sphere, it will have a Q value that most closely
approaches that lower bound. It was clear from the single stub antenna case given
above that it does not optimally fill the minimum-enclosing sphere with respect to
the origin. To better fill this minimum-enclosing sphere, stub antennas with multiple
parasitic elements were studied. In the case of two parasitic elements, each has the
same configuration as the parasitic element shown in Fig. 3.16, but with the second
parasitic being symmetrically located along the −x direction. In the case of four
parasitics elements, the four are located symmetrically along the x, −x, y, and −y
directions. Because of these symmetric locations, the stub antennas with one, two,
and four parasitic elements can be enclosed by the same sphere of radius a. Then, by
adjusting their inductor values and monopole heights to achieve their fundamental
resonance and nearly complete matching to the 50Ω source in very close proximity
to 300MHz, their Qratio behaviors can be compared directly. We note that the
parasitics are in parallel and, consequently, the inductor values have to be increased
to maintain the resonance frequency at 300MHz (Note that this parallel effect is
even more apparent for the canopy antenna below.). Thus, compared to the one
parasitic element case, the inductor values have to be increased relatively larger for
the 2 and 4 parasitic element stub antennas. The results, including the varied design
parameters, for the one, two and four straight stub cases are summarized in Table
3.1.
73
Table 3.1: ka = 0.0467 Stub Antenna ComparisonsType Monopole Wire L(nH) f0(MHz) OE(%) Qratio
length(mm) length(mm)One stub 1.02 3.969 1122 300.3501 97.247 5.91Two stubs 0.76 2.500 1305 300.5898 98.609 5.45Four stubs 0.80 2.500 1664 300.3143 99.135 3.88
Figure 3.17: Stub antenna with four curve stubs
These results clearly demonstrate that by filling the minimum-enclosing sphere
more efficiently, the Qratio value is lowered. For instance, the Qratio value based on
the same ka value is nearly cut in half from the one to the two parasitic element
case. Introducing four parasitic elements further reduces the Qratio value. How-
ever, the improvement begins to saturate. For instance, introducing six parasitic
elements further reduces the Qratio value, but only slightly. Moreover, it is also clear
that straight parasitics are not the best elements for this purpose. Parasitics with
their tops being curved would more completely fill the radiansphere. To illustrate
this point theoretically, the metal cylinders of the parasitic elements of the four-
74
stub antenna shown in Fig. 3.17 were curved from the inductor top to touch the
a = 7.4173mm sphere and the monopole height was adjusted slightly. The HFSS
simulations for this ka = 0.0467 antenna system at 300MHz yielded Qratio = 2.689,
which brings the Q value yet closer to the lower bound.
In summary, from the HFSS simulation and measurement results for the Z
antennas and the simulation results for the stub antennas, the electrically small
metamaterial-inspired antenna systems, which are composed of a driven element
and one or more resonant parasitic elements, share the following properties:
• These near-field resonant parasitic element-based ESAs, for which ka 0.5,
can maintain good matching to a real source.
• Their resonant near-field parasitic element provides both the requisite reac-
tance and resistive compensation.
• If their parasitic element is designed with a lumped element reactance, their
effective reactances and, hence, the resonant frequency of this ESA can be
adjusted.
• When their reactances are varied, the height of their driven monopole can be
adjusted to re-establish good matching at the resonant frequency.
• Small changes in these reactance or height values lead to small changes in the
resonance frequency.
• By filling the enclosing sphere more efficiently, their Qratio can be significantly
improved;
3.4 Canopy antenna
The recognition that the Qratio value was lowered by introducing the curved (rather
than straight) parasitics into the stub antenna naturally led to the canopy antenna
designs shown in Fig. 3.18. Note that the parasitic “canopy” structure “shades”
the coax-fed monopole. Each leg of the canopy contains an inductor; the wires
75
of the parasitics of the corresponding stub antenna have been morphed into the
spherical shell of the canopy antenna. Thus the shell interconnects all of the induc-
tors while it helps the resulting near-field parasitic structure more efficiently fill the
minimum-enclosing sphere. The shell and the monopole are modeled using copper
with conductivity σ = 5.8× 107s/m. Note that because of this finite conductivity,
the radiation efficiency RE and, hence, the OE will be less than 100%. For some
cases below, unless specified otherwise, the monopole and the shell will be modeled
as perfect electric conductors (PECs) and the inductors will be treated as lossless to
rule out any conductive or dielectric loss effects on the predicted Qratio values, i.e.,
the radiation and overall efficiencies in all such cases will be OE ≈ RE ∼ 100%.
In Fig. 3.18a, the monopole’s radius is 0.5mm. The inductor’s radius is 1.205mm
and its height is 2.5mm. These three dimensions are the same as those in the stub
antennas reported above. The monopole height is now 1.57mm. The shell’s outer
radius is a = 7.417575345mm. Taking into account that the shell is cut by the plane
z = 2.5mm, the thickness of the shell is 2.2053mm so that the outer and inner edges
of the shell are flush with the outer and inner edges of the inductor. As shown in
Fig. 3.19, the HFSS-predicted S11 values for this canopy antenna indicate that it
is nearly completely matched to the coax feedline and the assumed 50Ω source at
the resonance frequency f0 = 298.3149MHz. Thus, ka = 0.0463. At this resonance
frequency the HFSS-predicted efficiencies are OE ≈ RE = 90.615% and the Q ratio
is Qratio = 2.013.
The complex input impedance, radiation patterns, and E field on the xz plane
are shown in Fig. 3.20, Fig. 3.21, and Fig. 3.22, respectively. The imaginary
part of the input impedance of the canopy antenna goes through zero at f0 as a
resonance mode. A broader frequency sweep (not shown here) confirms that it is
the fundamental mode. Note that without the canopy structure, the short monopole
exhibits a large capacitive reactance. The parasitic shell and the ground plane also
form a capacitive element. This negative reactance is compensated by the inductance
(positive reactance) of the lumped element inductor. The real part of the reactance
shows that the resistance is near 50Ω at f0. The canopy structure thus acts as
76
(a)
(b)
Figure 3.18: Canopy antenna configurations (a) one-leg version, (b) four-leg version
both a reactance and resistance matching element. We note that from 3.5 and from
Fig. 3.20 and the fact that all the canopy antennas will have Rin(f0) ≈ 50Ω and
Xin(f0) ≈ 0, theirQ values will be determined primarily by the slope of the reactance
at f0, i.e, ∂fRin(f0) ∂fXin(f0). We also note that since the radiation efficiency
77
is over 90%, the antenna input resistance is mainly determined by the radiation
resistance of the shell and its mutual coupling with the monopole. The radiation
patterns of the canopy antenna are just like those produced by the metamaterial-
inspired electric EZ, Z, and stub antennas and, hence, are comparable to those
produced by a finite-ground plane, electric monopole. The electric field distribution
is seen to be quite uniform within the confines of the canopy structure. As expected,
it has a high concentration near the edges of the monopole and the shell. Moreover,
it is normal to the smooth parts of the shell as it must be.
297.92 297.93 297.94 297.95 297.96 297.97 297.98−60
−50
−40
−30
−20
−10
0
Frequency(MHz)
S11
dB
Figure 3.19: S11 values of the one-leg canopy antenna
297.92 297.93 297.94 297.95 297.96 297.97 297.98−150
−100
−50
0
50
100
150
200
Frequency(MHz)
Zin
(Ω)
ReactanceResistance
Figure 3.20: Complex impedance of the one-leg canopy antenna
78
-40
-30
-20
-10
0
0
30
60
90
120
150
180
210
240
270
300
330
-40
-30
-20
-10
0
xy-plane xz-plane
Figure 3.21: Radiation pattern of the one-leg canopy antenna
Figure 3.22: The electric field distribution of the one-leg canopy antenna on the xzplane
79
3.4.1 Parameter Studies
The first investigation to lower the Q value for the canopy antenna closer to the
lower limits involved the determination of the effect of the shell thickness on it. The
simulation results for the shell thickness variations are given in Tables 3.2 and 3.3.
For the single inductor version with height 3.35mm, as shown in Fig. 3.18a, the
HFSS simulation results for different inductor radii are given in Table 3.2. The units
for the inductor radius, RInd, and the monopole height, HMono, are millimeters; they
are nH for the inductor value, L; and they are MHz for the resonant frequency f0.
We note from Fig. 3.18a that different inductor radii correspond to different shell
thicknesses. In these cases, the shell and the monopole were modeled using copper.
The same set of simulations were also performed for a PEC shell and monopole in
order to rule out any conductive loss effects. The results for the PEC shell and
monopole are shown in Table 3.3. We note that the RE values in that table are
nearly 100% because there are no conductive losses and, consequently, the OE values
are also almost 100% because the canopy antenna is well matched to the 50Ω source.
Since a smaller RInd causes the shell to be thinner, the effective capacitance, Ceff ,
becomes smaller. Consequently, a larger inductor value is needed to maintain the
same resonance frequency f0. Compared to a thicker shell, a thinner shell causes
a lower OE value and a broader bandwidth. Even though the Qratio values are
obtained with respect to the realized lower bound, they are decreasing as the shell
thickness decreases because the bandwidth is increasing faster than the OE value
is decreasing. Comparing the results between Tables 3.2 and 3.3, one finds that
changing from copper to PEC has almost no effect on the Qratio values. Although
the PEC shell canopy antennas have OE = 1, their bandwidths are narrower than
the copper cases. Therefore, one finds that the Qratio values, as expected from
introducing the realized lower bound values, show only minor differences between
the corresponding PEC and lossy cases.
The similarity between the canopy antenna and a coax-fed ground plane version
of the spherical-cap dipole antenna emphasized in [31]is easily recognized: there
80
Table 3.2: QRatio versus Copper Shell ThicknessRInd f0 BW3dB(%) HMono OE(%) L(nH) Qratio
1.205 298.3149 0.0105 1.57 90.615 281 2.0130.6 298.5612 0.0117 1.76 87.495 296 1.9120.3 298.7695 0.0130 1.94 82.957 304 1.8620.1 298.6910 0.0141 2.08 77.851 310 1.825
Table 3.3: QRatio versus PEC Shell ThicknessRInd f0 BW3dB(%) HMono L(nH) Qratio
1.205 298.6135 0.01 1.5 281 2.0200.6 298.4132 0.0105 1.7 296 1.9150.3 298.6216 0.0108 1.8 309 1.8620.1 298.5724 0.0109 1.85 317 1.822
are spherical cap structures in both antennas. The difference, however, is that the
spherical cap is a parasitic element in the former, while it is directly driven in the
latter. Moreover, the former is matched by design to the source while the latter
requires an external resistive matching network. Nonetheless, led by the discussion
in [31], we investigated if there is an optimum value of the ratio of the spherical
cap area to the area of the corresponding hemispherical region that leads to the
smallest Qratio value. A set of HFSS simulations were performed with the change
in the shell area for the case with RInd = 0.1mm. The height of the inductor, Hind,
was adjusted so that it was always connected to the bottom edge of the shell. The
value of the inductor and the height of the inductor were adjusted to ensure that
the resonance frequency remained at f0 = 300MHz and that the canopy antenna
was matched to the source at f0. The ratio of the area of the copper zone to the
area of the hemisphere is
Ratioarea =2πa(a−Hind)
2πa2=
a−Hind
a(3.8)
With a = 7.417575345mm, the predicted Qratio values versus this Ratioarea param-
eter are shown in Fig. 3.23. The optimal Qratio = 1.75 value given in [31] for the
spherical cap dipole was found to occur when Ratioarea,opt = 0.444.
Many other versions of the canopy antenna were explored to try to push the
81
0.2 0.25 0.3 0.35 0.4 0.45 0.51.7
1.75
1.8
1.85
1.9
1.95
Area ratio
Q r
atio
Figure 3.23: Qratio values versus Ratioarea values for the one-leg canopy antenna
Qratio value below 1.75. For instance, the effect of the number of inductors on the
canopy antenna and its operating characteristics was investigated. In particular,
HFSS simulations were performed for canopy antennas with two and four inductors
whose heights were set equal to Hind = 4.4mm, i.e, the optimum Arearatio value,
and whose radii were all set equal to Rind = 0.1mm. As shown in Fig. 3.18b, the
four inductors are located symmetrically on the x, y, −x, and −y axes in the four-
leg canopy antenna. In the two-leg canopy antenna, the two inductors are located
symmetrically, for instance, on the x and −x axes. In these HFSS simulations, the
inductor values and monopole heights were adjusted to ensure the same resonance
frequency f0 ∼ 300MHz and nearly complete matching to the source at f0. In all
of these multiple-leg canopy antennas, the HFSS simulation results show the same
input impedance behavior near the resonance frequency and the same finite ground
plane monopole antenna patterns. The other operating characteristics are given in
Table 3.4. Compared to the one-leg canopy antenna, the two-leg and four-leg canopy
antennas have increasing radiation and, hence, overall efficiencies but decreasing
bandwidths to give essentially identical Qratio = 1.75 values. It must be noted again
that because the inductors are in parallel in this configuration, their values must be
approximately doubled and quadrupled for two-leg and four-leg canopy antennas,
respectively, to maintain the same resonance frequency f0 ∼ 300MHz.
82
Table 3.4: Copper Canopy with Multiple InductorsInductorNumber f0 BW3dB(%) HMono OE(%) Inductor(nH) Q ratio
1 297.9400 0.0133 1.98 85.007 408 1.7502 298.4126 0.0122 1.90 93.140 816 1.7524 297.9679 0.0118 1.88 97.219 1632 1.749
To illustrate the current distributions on the spherical shell, a contour and vector
plot of the currents on the four-leg canopy antenna are given in Figs. 3.24a and 3.24b
respectively. One can clearly see that the largest currents are localized near the join
of the inductors to the shell. One can also see that the net current flow is θ-directed
with an effective null at the top of the shell.
3.4.2 Discussion
Wheeler in [41,42] argued that a small antenna that optimally occupies the radian-
sphere will have the optimum bandwidth. Wheeler claimed that the spherical cap
antenna represented an example of an electric-based antenna that could achieve the
lowest possible Q value. This was also supported by the discussions in [31] and [32].
The value Qratio = 1.75 was obtained numerically there as the lowest Qratio value
for the spherical cap antenna and, hence, it was argued that this was true for any
electric-based antenna. On the other hand, the spherical resonator antenna [20], the
folded spherical helix antenna [37], and the negative permittivity sphere antenna are
discussed in relation to the lower limits on Q in [35,38,39] and in [34] and [32]. All
of these antennas effectively achieve spherical configurations that could approach
Wheeler’s optimal one. They all have Qratio values approximately equal to 1.5.
Thal claims in [34], as does Stuart in [35, 38, 39], that this Qratio = 1.5 value is the
minimum quality factor that can be obtained from an electric based ESA. On the
other hand, Lopez claims in [31] that it is 1.75 and then makes a distinction between
lumped element electric designs with a limit of 1.75 and self-resonant designs with
a limit of 1.5 in [32]. These different claims were important to us because they sug-
gested that we might be able to drive the Qratio value of the canopy antenna to at
83
(a)
(b)
Figure 3.24: Current distribution on the shell of the four-leg canopy antenna. (a)Magnitude contour plot, (b) Vector plot
least 1.5. We also wanted to test whether one could go beyond that value and reach
the magnetic-based antenna limit which was shown in [41, 42] to be Qratio = 1.0
To test some of these issues, we first considered HFSS simulations of the spherical
cap dipole and the corresponding coax-fed spherical cap monopole. Because we
were using HFSS for all our calculations, this seemed to be the simplest test case to
84
examine the reported limits. Our HFSS simulations of the coax-fed PEC spherical
cap monopole with the Ratioarea = 0.44, the outer radius of the cap a = 73.7mm,
a shell thickness 3.65mm, and a 4mm radius of the center conductor of the coax
showed that it is resonant at f0 = 299.42MHz with an input resistance equal to
3.67Ω. Renormalizing the S11 values for this ka = 0.46 ESA to a 3.67Ω source, the
HFSS-predicted value was Qratio = 1.60, which is significantly below the 1.75 value.
With testing various sizes of the cap and, hence, different resonance frequencies, we
conclude that the Qratio = 1.50 limit is most likely the correct one for the spherical
cap monopole or dipole.
We note that there is a major difference between the folded spherical helix, the
spherical resonator, and the negative permittivity sphere (as discussed in [35]) an-
tennas, and the canopy and the spherical cap antennas. In particular, their first
resonance is an anti-resonance. The noted Qratio ∼ 1.5 values all occur at the sec-
ond resonance of the system. The canopy antenna, like the spherical cap monopole
(dipole) antenna, exhibits a resonance at its first resonance frequency. This is con-
firmed, for instance, by the input impedance behavior shown in Fig. 3.20 for the
one-leg canopy antenna, which is also representative of the behaviors for the two-leg
and four-leg versions. Nonetheless, all of these antennas are fundamentally electric
based, i.e., they radiate essentially the TM10 mode. Then according to [34], their
Qratio ∼ 1.5 values are indeed the best attainable. We note that, in contrast to
this simple description, the folded spherical helix antenna is actually described as
a combination of an electric and magnetic antenna in [34] and as a self-resonant
antenna in [32] to distinguish how it can reach the 1.5 Qratio limit.
As noted above, we found that the ka = 0.46 PEC spherical cap monopole
antenna has an input resistance of only Rin(f0) ∼ 3.67Ω at f0 = 299.42MHz,
which is substantially below the source resistance value. In reality, an external
matching network would have to be included to match it to the source resistance.
The ka = 0.046 version was also modeled to make further comparisons. It still had
the Ratioarea = 0.44, but the outer radius of the cap, the shell thickness and the
radius of the coax center conductor were scaled by ten, respectively, to a = 7.37mm,
85
0.365mm, and 0.4mm. As expected, the resonance was upshifted by the same
scaling value to f0 = 3.001GHz while the input resistance stayed approximately
the same and was equal to 3.90Ω. These HFSS simulations yielded Qratio = 1.59.
On the other hand, the copper one-leg canopy antenna with ka ∼ 0.046 produced
its minimum value Qratio = 1.75 with Rin(f0) ∼ 50Ω and OE ∼ 85% when the
Ratioarea,opt = 0.4068 (i.e., Hind = 4.4mm). As a consequence, it has the interesting
advantage over the spherical cap monopole antenna in that no external matching
network is needed to match it to the source whether it electrically small or very
electrically small. To emphasize this contrast, we say that the canopy antenna has
an internal matching network.
The question for the canopy antenna still remains: why can its Qratio value
not reach 1.5? For the spherical antennas with Qratio = 1.5 noted in [35], it
was shown that they generate essentially the same very uniform interior electric
fields and exterior electric fields which correspond to a dipole located at their cen-
ters. Thus, one might expect that this uniform interior field is characteristic of
this optimal Qratio,min = 1.5 configuration. For the one-leg canopy antenna with
RInd = 1.205mm, the field distribution shown in Fig. 3.22 is definitely not uni-
form in the minimum sphere. The corresponding electric field distribution for the
RInd = 0.1mm one-leg canopy antenna is shown in Fig. 3.25. While this interior
field is more uniform, it still does not approach the uniformity exhibited in the cases
discussed in [35]. Consequently, there must be an issue with the current distribution
on the canopy that causes this behavior.
From [34] one anticipates that θ-only directed current distributions on the shell
should lead to the minimum Qratio value for the fundamental resonance of the purely
electric-based spherical cap monopole antenna. The HFSS-predicted current density
on the spherical cap of the Qratio = 1.6 coax-fed spherical cap monopole described
above is shown in Fig. 3.26. One observes the θ-only behavior of this current
distribution. In particular, one sees that the current density near the edge of the
cap is directed orthogonal to it. On the other hand, one finds from Fig. 3.24 that
the presence of the inductors causes the current distribution to have a component
86
Figure 3.25: E field on the xz plane for the one-leg canopy antenna with RInd = 0.1mm
along the edge of the canopy. While the currents that flow along that edge cancel
out, its presence is distinctly different from the spherical cap monopole antenna
behavior. We believe that this current difference causes the canopy antenna to be
limited to the Qratio = 1.75.
Figure 3.26: Current density on the ka = 0.46 spherical cap monopole antenna at299.42MHz.
87
(a)
(b)
Figure 3.27: Alternate four-leg canopy antennas. (a) LAX version, (b) Meshed-shellversion
Given the reasoning that one would like the currents to be more purely θ-only
directed, we tried several other canopy configurations to significantly vary the cur-
rent distributions in an attempt to reach or even breach the Qratio = 1.5 limit. The
LAX version shown in Fig. 3.27a attempted to make a smoother transition of the
currents on the shell into the inductors. The inductors were set to L = 1601nH .
88
The HFSS simulations predict an OE ≈ RE = 94.186% at f0 = 300.0859MHz
with a FBW = 0.0117% to give Qratio = 1.82. In an attempt to make the currents
on the shell have φ-directed components as they do in a magnetic-based antenna,
the mesh shell version shown in Fig. 3.27b was considered. The inductors were
set to L = 1599nH . The HFSS simulations predict an OE ≈ RE = 96.398% at
f0 = 300.0420MHz with a FBW = 0.0124% to give Qratio = 1.755. Many other
such variations were considered. We tried varying the inductor values themselves
in an attempt to try to produce several resonances that overlapped. This proved
unsuccessful. Because the inductors are symmetrically located and are connected
through the shell, the net inductance defined a single resonance only. Because the
current flow on the shell is well localized near the inductors, we also tried split-
ting the shell to isolate each inductor and then varied the inductor values. If the
resonance frequency was kept at f0 ∼ 300MHz, no matter what we did to the con-
figurations, the Qratio = 1.75 value was determined to be the lowest limit. Thus, we
have found that there is a trade-off between almost reaching the fundamental Qratio
limit and having an internal matching network so no external one is needed.
3.5 Circuit Model
The canopy antenna, despite its electrically small size, e.g., ka ∼ 0.046 at f0 =
300MHz, is matched to the source, i.e., Rin(f0) ∼ 50Ω, and is found to have a
low Qratio value equal to 1.75. Because of the time costs involved with the HFSS
simulations and to better understand limits of its performance, we have developed
several passive circuit models for the canopy antenna. We have used it to try to
improve its design in order to achieve a larger operational frequency bandwidth. All
of the circuit models were obtained by analyzing the outcomes of numerous HFSS
simulations, which explored the variations of the parameters in the antenna’s design.
In addition to explaining the performance behaviors that have been observed in the
HFSS simulations, the model suggests a potential design to achieve the desired
increase in its bandwidth.
89
To check the mutual coupling between the directly driven monopole and the
parasitic shell in the presence of the lumped element inductor, a two-port model of
the one-leg canopy antenna was constructed. It is shown in Fig. 3.28. There are two
wave ports, which are both de-embedded to their respective feed point. One wave
port excites the monopole and the other excites the lumped element inductor and,
hence, the shell. The Z matrix for this two-port model was obtained with HFSS
simulations. Using this HFSS-predicted Z matrix, a T-circuit model [28] equivalent
of the two-port network, shown in Fig. 3.29, was generated. Wave port 1 is the one
driving the monopole and wave port 2 is one driving the lumped element inductor.
The canopy antenna itself can then be represented by the T-circuit with wave port
2 shortened. According to the physical configuration, the input impedance of the
electrically small monopole, Z11, is taken to be a capacitor C1. The impedance Z22
is modeled as a series RLC sub-circuit consisting of an inductor L, a capacitor C2,
and a resistor RShell = Rr +RL, where Rr is the radiation resistance and RL is the
conductive loss. Again, because of the electrically small sizes involved, the mutual
coupling impedance, Z12 = Z21, is modeled by a capacitor C12. All of the element
values, except L, which is defined by the lumped RLC boundary condition in HFSS,
were obtained by performing curve fitting at the resonant frequency. The obtained
values were: C1 = 1.0123×10−13 F , C12 = 3.029×10−12 F , C2 = 6.2313×10−13 F ,
RShell = 0.054952Ω, and L = 455nH . The input impedances obtained from the
HFSS model, Zin, and from the shorted T-circuit model, Zin,Circ, are shown in Fig.
3.30 for a range of frequencies surrounding f0. The agreement between the two
input impedances is excellent, both for the resistance and the reactance.
Although the T circuit model can predict the input impedance with high ac-
curacy, it is not capable of providing insightful information about the radiation
mechanism and guidelines to push the Qratio lower. For these two purposes, the
transformer circuit model shown in Fig. 3.31(a) was developed. The middle portion
of this transformer-based circuit model represents the shell and the lumped element
inductor. The value of the inductor, L, which is essentially the value of the lumped
element inductor, can be adjusted easily. The first transformer, which has a turn
90
Figure 3.28: Canopy antenna two-port model
Figure 3.29: Canopy antenna equivalent two-port T-circuit model
ratio N1, represents the coupling between the monopole and the shell. Because
the source is directly driving only the electrically small monopole via the coaxial
transmission line, the initial capacitive component, C0, is defined to represent the
capacitance of the electrically small monopole. It should be noted, however, that C0
is not simply the capacitance of the short monopole but is actually the impedance
of the monopole in the presence of the coax feed and the other structures. The
value of C0 is obtained from curve fitting the HFSS simulation results. The resis-
tance associated with the monopole is Rmono = Rrad,mono + RL,mono, a combination
of the radiation loss of the monopole and its conductive loss. Since the monopole
91
2.9782 2.9783 2.9784 2.9785 2.9786 2.9787 2.9788 2.9789 2.979 2.9791 2.9792
x 108
−250
−200
−150
−100
−50
0
50
100
150
200
250
Zin
Real
Zin,Circ
Real
Zin
Imag
Zin,Circ
Imag
Figure 3.30: Comparison of Zin obtained from the HFSS results and from the T-circuit model of the one-leg canopy antenna
C0L
R0
N1 N2
Zin
(a) Full version
C0L
R
N
Zin
(b) Simplified transformer version
Figure 3.31: Circuit models for the one-leg canopy antenna.
92
antenna considered here is electrically small, the radiation resistance, Rrad,mono,
would be very small, i.e., Rrad,mono 50Ω, if the inductor and the shell structure
were not present. To achieve an antenna system with a high overall efficiency, it
is obvious that N1 needs to be large enough to transform the small Rrad,mono to
the source impedance value, i.e., to 50Ω as we assumed it to be. Then, the con-
ductive loss of the monopole will be small in comparison. The transformer then
represents the resistive behavior of the monopole. The resistance associated with
the shell is labeled R0 = Rrad,shell + RL,shell, where Rrad,shell is the radiation resis-
tance of the shell and RL,shell is its conductive loss. Since the radiation efficiency
RE = Rrad,shell/(Rrad,shell+RL,shell, a high RE value means RL,shell Rrad,shell and
R0 ≈ Rrad,shell. The second transformer, which has a turn ratio N2, represents the
coupling between the antenna and free space. The radiation resistance is then calcu-
lated as Rrad,shell = N2R0, where the resistor R0 = 377Ω represents the impedance
of free space.
Because one can calculate the Q value from the S11 values using Eq. 3.5 and
because of the already noted properties of the input impedance of matched, reso-
nant antennas, our circuit model considerations have been focused on the frequency
derivative behavior of Xin. In particular, the circuit component values shown in
Fig. 3.31a can determined by curve fitting the input reactance, Xin(f), values in a
frequency band surrounding the resonant frequency. Note, however, that this cir-
cuit model can be readily reduced to the more common single transformer model
shown in Fig. 3.31b. The resistor value is now simply given as N × R = 50Ω, the
value of the source impedance to which the entire antenna system is designed to
be matched. In this reduced circuit model, the resistance R = Rrad + Rloss. The
coupling between the monopole and the shell is now represented by a transformer,
which has the turn-ratio N . From our simulations we have found that the capaci-
tance C0 is affected significantly by the monopole length. The longer the monopole,
the smaller the value of C0 (i.e., the reactance, being negative, acquires a smaller
negative value as the value of C0 becomes smaller). The inductor L is defined by the
lumped element component in the antenna and its value can be adjusted easily. We
93
also note that when various shell structures were tested, one constraint was keep-
ing the resonance frequency near the target value f0 = 300MHz. From our HFSS
simulations, it was then found that the transformer N is affected by the monopole
length. With the other dimensions being the same, a longer monopole corresponds
to a smaller N . We also found that C1 is very small and can be ignored basically in
the circuit model.
For a one-leg canopy antenna case with ka = 0.04618 and resonant frequency
fres = 297.2414MHz, the HFSS predicted input resistance and reactance are shown
as solid lines, respectfully, in Figs. 3.32(a) and 3.32(b). With C0 = 6.9934×10−4pF ,
L = 455 nH , and N =√900.9937 = 30.0165, the circuit model predicted input
resistance, Rin,circuit, and reactance, Xin,circuit, are also shown in those figures as
dashed lines. The agreement is very good. One finds that the input resistance of
the shell resistance is approximately constant over this narrow band of frequencies.
On the other hand, the input reactance varies significantly.
The circuit model input reactance is
Xin,circuit(ω) = ωNL− 1
ωC0
. (3.9)
Since the resonance frequency is defined by Xin,circuit(ω0) = 0, this gives
f0 =ω0
2π=
1
2π
1√NLC0
(3.10)
For the indicated circuit values corresponding to Fig. 3.32, one confirms that f0 =
297.2414MHz. Additionally, from (3.9) one finds
∂ωXin,circuit(ω) = NL+1
C0ω2. (3.11)
Because NL = ( C0 ω20)
−1at the resonance frequency, the derivative of the input
reactance (3.11) then becomes
∂ωXin,circuit(ω0) =2
C0ω20
= 2NL. (3.12)
94
2.9721 2.9722 2.9723 2.9724 2.9725 2.9726 2.9727
x 108
−200
−150
−100
−50
0
50
100
150
200
Freq.(Hz)
Xin
(Ω)
HFSSCircuit Model
(a)
2.9721 2.9722 2.9723 2.9724 2.9725 2.9726 2.9727
x 108
46
47
48
49
50
51
52
53
Freq.(Hz)
Rin
(Ω)
HFSSCircuit Model
(b)
Figure 3.32: Input impedance, Zin, of the spherical shell antenna. (a) Reactance,(b) Resistance
It is clear that ∂ωXin,circuit(ω0) is inversely related to C0 and directly proportional to
L. To achieve a smaller Q value, one would want a smaller value of ∂ωXin,circuit(ω0)
and thus a larger C0 value or smaller value of NL.
On the other hand, with the lower bound on the quality factor being given by
Eq. (3.2) and with ∂fRin(f0) ∂fXin(f0), one has
Q ≈ f02 Rin(f0)
|(∂fXin(f0)|. (3.13)
95
Thus, the value of the Qratio at the resonance frequency obtained from the circuit
model is
Qratio ≈ 2ω0
2 Rin(ω0)C0ω20
(ka)3
OE=
1
Rin(ω0) C0 ω0
(ka)3
OE
=1
Rin(ω0)
√
NL
C0
(ka)3
OE. (3.14)
Consequently, for a fixed ka value, one can decrease the Qratio by decreasing NL or
increasing C0. For the canopy antenna case with ka = 0.04618, the input resistance
at the resonance frequency Rin(ω0) = 49.6322Ω and the efficiencies OE ≈ RE =
0.8618. Thus (3.14) gives Qratio = 1.742, in very good agreement with the HFSS
simulation result value Qratio = 1.759.
To explore the possibility of pushing theQratio value closer to the limit and to test
the validity of this circuit model, HFSS simulations with an additional capacitor C1
introduced in parallel with the inductance L were performed. This was accomplished
simply by introducing a capacitance value, as well as the inductance value, into the
HFSS RLC element. With the capacitor C1 being present, one finds immediately
that the input reactance becomes
Xinp,circuit = − 1
ωC0+
ωNL
1− ω2C1L(3.15)
and, hence, its derivative becomes
∂ωXin,circuit(ω) =1
ω2C0
+NL
1− ω2LC1
− ωNL (−2ωLC1)
(1− ω2LC1)2
=1
C0ω2+NL
1 + ω2LC1
(1− ω2LC1)2 . (3.16)
Since the resonant frequency is now given by the expression
1
ω0C0+
ω0NL
1− ω20C1L
= 0, (3.17)
96
one has
ω20C1L+ ω2
0C0NL = 1 (3.18)
and, hence,
f0 =ω0
2π=
1√
NL(
C0 +C1
N
)
(3.19)
Therefore, at the resonance frequency
∂ωXin,circuit(ω0) =1
C0ω20
+NLω20NLC0 + 2ω2
0LC1
(ω20NLC0)
2
=1
C0ω20
+1
C0ω20
[
ω20C0 + 2ω2
0C1/N
ω20C0
]
=2
C0ω20
(
1 +C1/N
C0
)
. (3.20)
Consequently, the derivative is always slightly larger when the capacitance C1 is
present; and, hence, the Qratio value will always be a bit larger than when it is not
present. The HFSS simulations were completed with C1 = 0.1NC0, and L = 4551.1
nH .
According to (3.19), the resonant frequency f0 should then be the same as that
obtained without C1. Moreover, Rin should maintain the same value at the resonance
frequency. The HFSS simulation predicted the values f0 = 297.2508 MHz and
Rin(f0) = 49.6522 Ω with C1 being present in comparison to the values: f0 =
297.2414 MHz and Rin(f0) = 49.6322 Ω, without C1 being present. These results
confirm the expected outcomes from the circuit model.
These circuit models were explored thoroughly with several other variations and
with optimization routines. The best obtained Qratio value remained 1.75 (1.17
times the Thal-based limiting value).
3.6 Metamaterials Within the Minimum Enclosing Hemisphere
One interesting observation from [10] is how the limit Qratio = 1.0 is obtained in the
magnetic antenna case. By including a magnetic material with relative permeability
97
µr within the entire radiansphere and by letting µr → ∞, it was shown that the
Q ∼ (ka)−3, i.e., the lower bound was reached for this idealized case. The dual
result would be to fill the entire radiansphere with an electric material with relative
permittivity εr and let εr → 0. These conclusions were emphasized with Eqs. (8)
of [34], i.e., the Qratio values obtained when the radiansphere is filled with an electric
or magnetic medium were given, respectively, by the expressions: Qratio,electric ∼1+ εr/2 or Qratio,magnetic ∼ 1+2/µr. The electric-based effects were tested with the
canopy antenna model.
The canopy antenna HFSS model with a metamaterial hemisphere essentially
filling the minimum enclosing hemisphere is shown in Fig. 3.33. A hemisphere of
radius a−tshell, where tshell is the shell thickness, fills the interior of the canopy. Per-
mittivity values were considered in the interval 0 < εr < 1. In all cases, the overall
efficiency and resonance frequency were maintained as close as possible to 100% and
f0 = 300 MHz, respectively. First, as was the case in [34], the hemispherical filling
was treated as isotropic, homogeneous and dispersion free. Second, a Drude-model
dispersive hemispherical filling was considered, i.e., if the desired value of the relative
permittivity at the resonance frequency f0 was εres, then the relative permittivity
was given by εr(f) = 1− (1− εres) ∗ f 20 /f
2. Since the interval 0 < εr < 1 can only
be realized with dispersive materials and since dispersion is known to impact the
bandwidth performance of antennas integrated with metamaterials [15], the second
case gives more realistic outcomes while the first allows comparison to the idealized
theoretical predictions.
The HFSS-predicted Qratio values for these two configurations are given in Fig.
3.34 and are compared to their idealized limits. The HFSS results for the non-
dispersive fillings follow the predicted values rather closely. Because of the initial
offset value associated with the canopy antenna, i.e., Qratio = 1.75 for an interior
sphere with εr = 1.0, the idealized-filled canopy antenna results are consistently
0.25 greater than the predicted results given in [34]. Consequently, we obtained
Qratio → 1.25 as εr → 0. On the other hand, the results for the dispersive fillings
show a increase of the Qratio values as εr → 0. This is expected from [15] because
98
Figure 3.33: Canopy antenna HFSS model with metamaterial interior sphere
one finds that [∂f (fεr)](f0) = 2− εres, i.e., the derivative increases as εr → 0.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
εr ( f
res )
Q r
atio
No dispersionDrude dispersionThal equation
Figure 3.34: HFSS-predicted results for the canopy antenna with a metamaterialinterior sphere
99
3.7 Conclusions
In this chapter, the performances of several electrically small, efficient near field res-
onant parasitic antennas were studied and compared. These included the Z antenna,
the stub antenna, and the canopy antenna. In all of these antennas, a lumped ele-
ment inductor was integrated internally into the antenna structure and was used to
achieve reactance and some resistance compensation. Nearly complete impedance
matching was obtained by the coupling between the parasitics and the directly
driven, coax-fed monopole. At even a very electrically small size, e.g., ka ∼ 0.046,
the input resistance is nearly equal to 50Ω, which means that all of these antennas
can be used with a real source without any (usually lossy and bandwidth limiting)
external matching network.
The Q value and its variations were investigated for all of these antennas in
relation to the theoretical lower bounds for the fundamental mode of an electric-
based antenna. For the Z antenna, the Q ratio was on the level of 7+. For the stub
antenna, the Q ratio was on the level of 4+, but could be reduced by filling the
minimum enclosing hemisphere with more resonant parasitics to the 3+ level and
could be further lowered to 2+ level if those parasitics were curved to yet better
fill the enclosing hemisphere. The canopy antenna design was then proposed to
further improve the Q ratio. The effect of its shell thickness, inductor number,
and metal-air ratio on the Q ratio were presented. It was found that the lowest Q
ratio for any air-filled version of the canopy antenna was 1.75, which was obtained
for a thin shell (thickness less than 0.2mm) and an optimized metal-air ratio of
0.4068. This lowest Q ratio value was maintained with the one-, two- and four-leg
canopy antenna designs. With a single inductor, the RE and thus the OE values
of the canopy antenna were the lowest(∼ 85%), but its bandwidth was the largest.
With more inductors, the RE (OE) value was higher (over 97% for the four-leg
canopy antenna), but with a proportionately reduced bandwidth. The resulting
compromise between the RE and thus the OE values and the bandwidth produced
the same Q ratio for these three different designs. Moreover, extensive studies of
100
variations in the canopy design to change the current distributions responsible for
the radiation process while maintaining the same resonance frequency and high
RE value failed to lower the Q ratio below the 1.75 value. It was shown that the
current distribution associated with having the internal matching network led to this
fundamental limit of the canopy antenna. Several circuit model representations of
the canopy antenna were presented to explain its radiation mechanism and to provide
a guide to possible lower Q-ratio designs. Furthermore, by adding an idealized
homogeneous, isotropic, dispersionless metamaterial hemisphere into the minimum-
enclosing hemisphere, whose permittivity 0 < εr < 1, it was shown that the Q value
of the canopy antenna could be lowered further to 1.25 as εr → 0. However, if the
dispersion of these mematerials are taken into account, this lower Q ratio value was
shown to increase.
All of the reported Qratio results were shown to be in favorable agreement with
several predictions on the lower limits of the Q value for an electric-based ESA.
In particular, the Qratio values for the canopy antenna were 1.75 times the Chu
limit [9] and 1.17 times the Thal limit [34]. Finally, as discussed in the next chapter,
the bandwidths of the Z, stub and canopy antennas can be enlarged significantly,
exceeding even the Chu upper bound, with the introduction of an active internal
matching network, i.e., an active frequency dependent inductor, into these antennas.
101
CHAPTER 4
BROADBAND, EFFICIENT, ELECTRICALLY SMALL
METAMATERIAL-INSPIRED ANTENNAS FACILITATED BY ACTIVE
NEAR-FIELD RESONANT PARASITIC ELEMENTS
4.1 Eletrically small antennas bandwidth limits
Electrical small antennas (ESAs) have been studied extensively in the past and
have many potential applications in all wireless communication and sensor systems
because of their compact dimensions. It is well known that the performance charac-
teristics of an ESA are limited by its physical dimensions [9–11]. For instance, the
bandwidth performance of an ESA can be estimated by its Q value in relation to the
Chu-based lower bound. In particular, if FBW3dB is its half power VSWR fractional
bandwidth, its Q value is given by Q = 2/FBW3dB. If its radiation efficiency is ηrad,
then the Chu-based lower bound is Qchu = ηrad ( 1/ka3 + 1/ka ), where k is the free
space wavenumber and a is the minimum radius of a sphere that completely encloses
the antenna. Then the natural figure of merit associated with the bandwidth is the
Q ratio, i.e., QRatio = Q/QChu. An antenna is generally classified as an ESA if
ka ≤ 1. However, if ka 1, its compact electrical dimension comes at the cost of a
very narrow bandwidth, which is limited approximately by 2(ka)3/ηrad. For exam-
ple, when ka = 0.1, the bandwidth can at most be 0.2%/ηrad. Moreover, a resonant
ESA usually has an associated low radiation resistance and usually requires an ex-
ternal matching network to achieve a high accepted power level. Such a matching
network will add additional size to the ESA, and usually, it will further limit the
overall system bandwidth. To surpass the Chu limit, non-Foster (NF) matching net-
works have been proposed, e.g., see [43]. A NF matching network realizes negative
inductance and capacitance values with active elements; these values are designed
to bring the antenna into resonance (reactance matching) and to optimize the power
102
delivered to its terminals from the source (resistance matching). As depicted, for
instance, in [43], the NF matching network is implemented between the source and
the antenna. We will refer to it as an external matching network. The internal
matching network, which we introduce below, is internal to and part of the actual
radiating element.
Metamaterial-inspired, efficient ESAs have been introduced in [17–19]. These
ESAs are constructed as a driven element and a resonant parasitic element in the
very near field of the driven element. These ESAs are nearly completely matched to
a real source and have a very high overall efficiency. These properties are achieved
through the parasitic element, which replaces the need for an external matching
network and which works with the driven element to enhance the radiation process.
Based on these works, the Z antenna, which uses an internal lumped element, was
then introduced in [44] and [21]. In these works, the Z antenna was tuned to resonate
at different frequencies by changing the value of the lumped element, but without
changing the overall dimensions of the antenna system. From these results, we
realized that if one could develop self-tuned lumped elements fulfilling the resonance
requirements at all frequencies in a certain frequency band of interest, i.e., for f1 ≤f ≤ f2, the Z antenna would have an instantaneous bandwidth of f1 ≤ f ≤ f2.
In this chapter, we develop such a self-tuned lumped element, its frequency de-
pendent behavior, and ways to implement the resulting frequency dependent internal
matching element, to achieve an active, broad band ESA. Our work is assembled
as follows. In Section 4.2, the results for ANSOFT HFSS-Designer co-simulations
of the Z antenna are detailed and a circuit model equivalent is developed. The
impedance of the lumped element required to achieve a broad bandwidth is then
revealed numerically. The relation between the lumped element and the resonant
frequency of the antenna is obtained in Section 4.3. It is used to define a circuit
model that could be used to implement the desired self-tuning lumped element, i.e.,
the internal matching network. In Section 4.4, this self-tuning lumped element de-
sign is applied to several ESAs, including the stub and canopy antennas introduced,
respectively, in [22] and [45]. Approximately a 10% fractional bandwidth is achieved
103
for each case. Our conclusions are given in Section 4.5.
4.2 ANSOFT HFSS and Designer Simulations of the Z Antenna
Figure 4.1 shows the Z antenna loaded with a lumped element, 1000nH inductor. Its
HFSS-predicted (version 11.1.3) S11 values for a 50Ω source are shown in Fig. 4.2.
All of the materials are treated as lossless to simplify the bandwidth considerations.
The minimum enclosing sphere for this Z antenna has a radius a = 11.18mm so
that ka = 0.0461, where k = 2πc/fr, c being the speed of light in vacuum and
fr = 195.3292MHz being its resonant frequency. The overall efficiency, as expected,
was approximately ηrad = 100%. The 3dB fractional bandwidth was FBW3dB =
0.0027%, and the 10dB fractional bandwidth was FBW10dB = 8.86 × 10−4. Thus,
one finds QRatio ≈ 7.3. This value is rather far from the Chu-based lower bound
because the Z antenna physically occupies only a small portion of its minimum
enclosing sphere.
While these HFSS simulations show that this very electrically small Z antenna
is well matched to the 50Ω source and has a high overall efficiency, its potential for
applications is limited by its narrow fractional bandwidth. In fact, even if this very
small ka Z antenna could achieve the Chu limit with a similar overall efficiency,
its 3dB bandwidth would remain less than 0.02%. Thus, a means to increase its
bandwidth was sought.
Consistent results among HFSS simulations, HFSS-Designer co-simulations, and
measurements for the two-dimensional magnetic EZ antenna with a lumped capac-
itor were described in [46]. Consequently, we decided to employ the HFSS-Designer
co-simulation approach, which relies on a circuit model of the antenna system, to
study the bandwidth behavior of the Z antenna. The antenna block in Designer is
treated as an N-port sub-circuit which is imported as an S matrix from the HFSS
simulation in which the lumped LRC element is replaced with a lumped port. One
port is treated as the source wave port. The lumped RLC element is reintroduced
into the Designer model as a circuit element with the corresponding combination
104
Figure 4.1: The Z antenna configuration
of R, L, and C. The frequency can then be swept in the Designer simulation to
find the resonance frequency. Unfortunately, we have found that for our very narrow
bandwidth antenna systems, the resonant frequencies predicted by the co-simulation
approach and by HFSS for the same lumped RLC element values are consistently off-
set by 1% ∼ 2%. For instance, for the antenna shown in Fig. 4.1, the co-simulation
with ANSOFT Designer 3.5.2 was performed with the model shown in Fig. 4.4; and,
as shown in Fig. 4.5, predicted the resonance frequency to be 197.16. As noted, the
resonance frequency, predicted by HFSS: fr = 195.3292Mhz, is 1.9208MHz lower
than that value, i.e., 0.98% lower. Nonetheless, because this offset is consistent, the
circuit model of the Z antenna shown in Fig. 4.3, where the Adev, Bdev, Cdev, and
Ddev represent the antenna block without the lumped element, can be used to find
what kind of internal matching network is required to represent the antenna block
shown in Fig. 4.3.
In Fig 4.3, the Adev, Bdev, Cdev, and Ddev terms are the elements of the ABCD
105
195.23 195.232 195.234 195.236 195.238 195.24 195.242 195.244 195.246 195.248−60
−50
−40
−30
−20
−10
0
Freq(MHz)
S11
(dB
)
Figure 4.2: The HFSS-predicted S11 values of the Z antenna
−vs
+ Zs [
Adev Bdev
Cdev Ddev
]
L
Figure 4.3: Z Antenna Circuit Model
matrix which represents the antenna block and the L is the lumped element inductor.
The proposed internal matching network would replace the block between the dashed
lines in Fig 4.3. The resulting circuit model is shown in Fig 4.6, where ANET ,
BNET , CNET , and DNET are the ABCD matrix parameters of the internal matching
network. Then, to obtain a low return loss S11, the antenna input impedance must be
made equal or nearly equal to the source impedance Zs. Based on this consideration,
the internal matching network was designed such that
1 Zs
0 1
=
Adev Bdev
Cdev Ddev
ANET BNET
CNET DNET
, (4.1)
i.e., the antenna structure combined with the internal matching network was de-
signed to have the source resistance value. The ABCD matrix of the internal
matching network can then be determined analytically as
106
Figure 4.4: ANSOFT Designer Circuit Model for the Z Antenna
197.05 197.1 197.15 197.2 197.25 197.3 197.35−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
Freq(MHz)
S11
(dB
)
Figure 4.5: ANSOFT Designer predicted S11 values for the Z antenna loaded witha passive inductor
ANET BNET
CNET DNET
=
1 Zs
0 1
Adev Bdev
Cdev Ddev
−1
. (4.2)
To calculate the ABCD matrix of the internal matching network, the S or Z pa-
rameters were obtained for the frequencies in the interval of interest from the HFSS
simulations which included the lumped port element. These values were then con-
verted to the requisite antenna ABCD parameters. The internal matching network
(IMN) block was calculated from (4.2) in MatLab. The results were then recon-
verted to the S-parameter form and incorporated into the Designer model as the
N -port element shown in Fig. 4.7. The Designer predicted S11 values for the IMN-
based Z antenna system are shown in Fig 4.8. The return loss is below −30dB for
over a 20% fractional bandwidth. These results clearly demonstrate that an appro-
priately designed IMN can lead to a matched electrically small antenna over very
107
−vs
+ Zs [
Adev Bdev
Cdev Ddev
] [
ANET BNET
CNET DNET
]
Figure 4.6: Antenna circuit model with NET load
broad frequency range.
Figure 4.7: ANSOFT Designer circuit representation of the IMN-based Z antenna
A real IMN circuit could be developed from these ABCD matrix results. How-
ever, it could be very challenging because there are four independent variables. Note
that in Fig 4.6, the matching network is connected to the antenna on one port and
is shorted on the other port. According to the ABCD matrix definition,
VNET,1
INET,1
=
ANET BNET
CNET DNET
VNET,2
INET,2
, (4.3)
where VNET,1 and INET,1 are the voltage and current at the left port of the matching
network in Fig 4.6 and VNET,2 and INET,2 are the voltage and current at irs right
port. Since VNET,2 = 0, this model yields the relation
ZNET =VNET,1
INET,1=
BNET
DNET
. (4.4)
Utilizing this relation, the circuit model given in Fig 4.6 with the shorted matching
108
180 185 190 195 200 205 210 215 220−90
−80
−70
−60
−50
−40
−30
Freq(MHz)
S11
(dB
)
Figure 4.8: ANSOFT Designer predicted S11 vales for the IMN-based Z antenna
−vs
+ Zs [
Adev Bdev
Cdev Ddev
] ZNET
Figure 4.9: Antenna circuit model with equivalent load
network can be simplified to the circuit model in Fig 4.9. Both HFSS and Designer
co-simulations were preformed for the simplified circuit model; the same S11 values
shown in Fig 4.8 were obtained. The calculated ZNET values in the whole sweep
range are found to be pure imaginary values, whose imaginary part is always positive
and changing with frequency. Consequently, one finds that the IMN can be imple-
mented by a frequency dependent inductor, i.e., to have the Z antenna resonate at
different frequencies and, hence, to expand its very limited bandwidth, an inductor
with frequency dependent values is needed.
4.3 Inductor Versus Resonant Frequency
The Z antenna results were used to establish a relation between its inductor value
and its resonant frequency fr. For such an antenna structure, one finds that its
resonant frequency
109
fr =1
2π
1√
LeffCeff
, (4.5)
where Leff and Ceff are, respectively, its effective inductance and capacitance. Ac-
cording to this relation, if Ceff remains the same, then the effective inductance must
satisfy the relation
Leff =a
fr2 , (4.6)
where a−1 = 4π2Ceff is a constant. Thus, the Z antenna will be resonant at fr if
its effective inductance Leff satisfies (4.6). It should be noted that this effective
inductance is composed of the inductance of the lumped element, L, and of all of
the radiating elements, L0. For a fixed geometry, Ceff does not change its value.
Moreover, it is found that the lumped element inductance L is much larger than L0,
which means Leff ∼ L. Consequently, the resonance frequency of the Z antenna can
be controlled simply by changing the value of the lumped element inductor. These
properties are also true for the electrically small stub and canopy antennas to be
discussed below.
Satisfaction of the he Leff − fr relation (4.6) was readily demonstrated with
the set of discrete HFSS simulation results shown in Fig. 4.10. The frequency was
swept from 60MHz to 1.0GHz. In this sweep the inductor values were varied and
only small adjustments to the height of the monopole antenna were made to bring
the Z antenna radiation resistance back into match with the source. Comparing
these discrete results with those given by the analytical expression (4.6), one finds
very good agreement.
4.4 Bandwidth Enhancement for Metamaterial-inspired ESAs
Having established that the Z antenna can be predictably tuned by varying its
lumped element value and the monopole height, we investigated its achievable band-
width by only varying the inductor value. Moreover, because the stub and canopy
110
0 1000 2000 3000 4000 5000 6000 7000 80000
200
400
600
800
1000
1200
Inductor ( nH )
Res
onan
t fre
quen
cy (
MH
z)
ActualEstimated
Figure 4.10: Resonant frequency of the Z antenna as its lumped element inductorvalue is varied
antennas are also realized as lumped element controlled resonant near-field para-
sitics, they were also included in our studies. We have found that when all of these
resonant near field parasitic antennas are designed with passive inductors, their
bandwidths are restricted by the Chu-lower bound. However, when active inductors
are included, significant enhancements of their bandwidths can be realized.
4.4.1 Z antenna
The Z antenna was first studied to show its behavior as the value of the lumped
element inductor was varied. Because the Z antenna in Fig. 4.1 has a very small
ka value and a high Q ratio, its bandwidth was found to be very limited. As will
be discussed below, this made it difficult to enhance its value even with an active
element. Consequently, a larger ka value Z antenna was designed. In particular, this
Z antenna had ka = 0.266, fr = 877.715MHz, QRatio = 11.2, and FBW10dB = 0.1%
for an 100nH inductor. Twenty four different inductor values were considered in the
neighborhood of this original value. The HFSS simulations predicted the resonance
frequencies shown in Fig. 4.12. The variation of these resonant frequency values
as a function of the inductance was curve fit with a minimum mean square error
(MMSE) approach. It was found that the frequency dependent inductor value can
111
be expressed by the relation:
L =a1f 2
+ a0, (4.7)
where a1 = 8.113 × 107 and a0 = −5.299. The units of the inductance, L, and the
frequency, f , are, respectively, nH and MHz, For the corresponding metric units,
respectively, H and Hz, this relation becomes
L =a1 × 103
f 2+ a0 × 10−9. (4.8)
The curve fitting results shown in (4.7) and (4.8) are consistent with the relation
(4.6) when L0 = a0×10−9, i.e., note that a0 is negative and recall that Leff = L+L0.
Figure 4.11: Z Antenna with ka = 0.266
The frequency dependent inductor L values predicted by (4.7) or (4.8) cannot be
generated by a simple circuit element. In particular, we note that these values have
a non-Foster reactance behavior, i.e., the inductance is decreasing quickly enough
with increasing frequency so that ∂ω(ωL) < 0. It is straightforward to show that
an equivalent circuit can be synthesized with active elements to provide the same
112
830 840 850 860 870 880 890 900 910 920 93090
95
100
105
110
115
Freq(MHz)
Indu
ctor
(nH
)
Curve FittingHFSS
Figure 4.12: Results obtained by curve fitting of the inductor values
impedance values. The frequency dependent impedance, ZL, corresponding to the
inductance L can be written in the form:
ZL = j ω L = j (2πf) L = j 2πfa1 × 103
f 2+ j 2πfa0 × 10−9
=1
j 2πf(− 14π2a1×103
)+ j 2πfa0 × 10−9 =
1
jωCneg
+ jωLneg,
where we have introduced the equivalent capacitor and inductor terms, Cequ and
Lequ, respectively. The series circuit shown in Fig. 4.13, which consists of this equiv-
alent capacitor and inductor, produces the desired frequency dependent impedance.
According to (4.9), the component values in Fig. 4.13 that reproduce the curve fit
in Fig. 4.12 are
Cequ = − 1
4× π2a1 × 103= −0.31222 pF
Lequ = a0 × 10−9 = −5.299 nH.
The predicted negative values for this negative capacitor and inductor circuit
can be realized with a negative impedance converter circuit [47]. In particular,
113
Lequ
Cequ
Figure 4.13: Negative lumped element circuit model
the negative impedance converter (NIC) element shown in Fig. 4.14 produces the
following relation between the input impedance and the desired load:
Zin = −k ZL (4.9)
and k is a positive constant. A typical NIC circuit [47] that produces the desired
values is shown in Fig. 4.15, i.e., its input impedance is defined as
Zin = −R2
R1ZL. (4.10)
ZLNICZin
Figure 4.14: Circuit with negative impedance converter that is equivalent to thenegative element circuit
In our set of 24 fine resolution HFSS simulations about the original resonance
frequency, we also had to examine the overall efficiency behavior of the Z antenna.
In particular, we considered only the lumped inductor values that maintained the
overall efficiency level with no changes in the monopole height or any other design
parameter for this specific set of nearby resonant frequencies. We note that like
114
R2R1
ZL
Zin
Figure 4.15: Floating negative impedance converter circuit
Ceff , the built in inductance L0 of the Z antenna is also constant since the antenna
structure was maintained without any changes. The derivative with respect to the
inductance value of (4.6) shows that the rate of change of the resonant frequency
with respect to the inductor value is given by the expression
∂fr∂Leff
= − 1
4π√
LeffCeffLeff
= − 1
2Leff
fr, (4.11)
which can also be re-written approximately (i.e., recall that Leff ∼ L since L0 L)
as
∆frfr
= −1
2
∆Leff
Leff
∼ −1
2
∆L
L, (4.12)
According to (4.12), it can then be concluded that to obtain a 10% bandwidth, the
change of the inductor value must be approximately 20%, which is confirmed by the
results given in Fig. 4.12.
The L-fr relation provides a guideline to determine the ka values in the de-
sign and implementation of the Z antenna to allow for adequate variation in the
parameters so that an actual implementation of the active circuit design might be
realized. Because of the nature of the curve fitting, there are always errors between
the values specified by the resulting curve and the exact (as specified by the HFSS
simulations) values. A meaningful curve fit should provide inductor values close
enough to the exact values that one could fulfill some additional practical criteria,
e.g., S11 < −10dB at every frequency in the range of interest.
115
Assume the inductor value is L1 at frequency f1. Then assume that the curve
fitting yields an inductor value L1+δL, which corresponds to the resonant frequency
f1 + δf . Assume that the resulting Z antenna has a 10dB bandwidth ∆f . If
δf ≤ ∆f/2 at frequency f1, then one will have S11 ≤ −10dB. Consequently, the
bandwidth ∆f must be broad enough to accommodate the maximum error in the
curve fitting. It thus requires that the ka value should be large enough to insure
this requisite bandwidth.
In an actual implementation the actual errors in the inductor value may be much
larger simply because of the manufacturer’s tolerance values (manufacturing imper-
fections) of the components in Fig. 4.15. The Z antenna has a 10dB bandwidth
of 0.1% for ka = 0.266. Neglecting the curve fitting error, the circuit in Fig. 4.15
must then be constructed with inductor values having a 0.1% accuracy. Such com-
ponents would most likely be extremely expensive since generally a 1% tolerance is
considered to be very good. Note that the bigger the error in the inductor value,
the broader the bandwidth has to be to guarantee the overall performance of the
antenna system; and, therefore, the need for larger ka values and/or lower Qratio
values when this occurs.
4.4.2 Stub Antenna
In the bandwidth enhancement process for the active IMN-based Z antenna, it was
emphasized that it is necessary to minimize the error between the curve fit values
and the actual inductor values, which yielded the resonant, matched conditions, so
that the S11 values resulting from the curve-fit inductor values fall below the 10dB
bandwidth criterion. For the Z antenna, this criterion was met by increasing the Z
antenna size to a larger value: ka = 0.266. This led to a broader 10dB bandwidth
and, hence, a larger fractional bandwidth limit. The curve fitting errors associated
with defining the active inductor can then be accommodated by the design. In the
Z antenna, the meander line, i.e., the “Z” portion of the parasitic element, was
designed originally in [21] to provide additional inductance to the system, as well
as to enhance the radiation mechanism. However, in our active inductor design
116
studies, it was found that the meander line inductance is actually negligible when
compared to the lumped element inductance. On the other hand, the complexity
of the meander line itself caused some difficulties in the convergence of the HFSS
simulations, and thus produced numerical sensitivities in their predicted values. In
addition, it was recognized that a structure, which has a more complex design, will
generally lead to non-trivial fabrication sensitivities. Consequently, we felt that the
curve fitting errors associated with parasitic elements whose designs were simpler,
would be lower and would thus more readily lead to a successful active element
design. We thus decided to investigate whether the stub antenna introduced in [22],
which has a simpler near-field parasitic element and can be designed to have a lower
Q-ratio value, would lead to improved curve fits and, hence, to lower curve fitting
error values.
The stub antenna is a metamaterial-inspired ESA which was first introduced in
[22] and whose Q ratio behavior was further studied in [45]. Although a stub antenna
in [45] with Qratio = 4.94 was introduced, its ka = 0.0375 leads to a bandwidth which
is too narrow for our purposes. We thus designed the stub antenna shown in Fig.
4.16. It has a coax-fed monopole whose radius and height are, respectively, 0.5mm
and 9.2mm, and a parasitic whose radius and height are, respectively, 1.205mm and
17.35mm, and whose center is located 10mm from the center of the monopole. The
length of the inductor and the conductor of the parasitic are, respectively, 3.35mm
and 14mm. This stub antenna has ka = 0.1092, the radius a being measured from
the center of the parasitic; it resonates at 299.6839Mhz; and it has Qratio = 11.03
when the lumped element inductor L = 570nH . A discrete set of 20 additional HFSS
simulations, symmetrically located about the center resonant frequency, based on
6nH increments of the inductor value at the center frequency (i.e, approximately 1%
of L = 570nH) were then run. The resulting HFSS-predicted inductor-frequency
sweep and the corresponding curve fit results for this antenna are labeled as Ant1
in Fig. 4.17. The associated curve fitting error percentage and the corresponding
fractional bandwidth limiting values are given in Fig. 4.18. Although this one-
stub antenna has a Q ratio similar to the Z antenna, one observes from Fig. 4.18
117
that it has, as expected, a lower error level and, hence, a further separation from the
limiting fractional bandwidth values. One finds that a 10% bandwidth enhancement
can be achieved for this ka = 0.1092 stub antenna. To lower the Q ratio, the radius
of the parasitic element shown in Fig. 4.16 was increased from 1.205mm to 3mm
and the inductor value was decreased to L = 282nH . This thicker parasitic element
one-stub antenna has ka = 0.1094 and resonates at 300.3901MHz. This means
Qratio = 8.2. A similar set of HFSS simulations were run based on 3nH increments
of the L = 282nH inductor value. The results for this antenna are labeled as Ant2
in Figs. 4.17 and 4.18. One observes that with a lower Q-ratio, the curve fitting
errors have been decreased while the fractional bandwidth error limits have been
increased. These results imply that it will be easier to design and achieve an active
inductor element version of the lower Q-ratio passive antenna. To emphasize this
point further, the four parasitic element stub antenna version of the Qratio = 11.03
one-stub antenna was obtained. It is shown in Fig. 4.19. From [45] it was known
that this four-stub antenna has a lower Q ratio than the one-stub case. In particular,
setting each inductor to L = 780nH , adjusting the copper portion of the parasitic so
that its overall height was 13.30mm, and decreasing the monopole height to 7.3mm,
the resonant frequency was fr = 299.3409MHz giving ka = 0.1090, the a now
being taken with respect to the center of the monopole. Thus, Qratio = 6.48 for
the four-stub antenna. Another similar set of HFSS simulations were run based on
1% increments of the L = 780nH inductor value. The results for this antenna are
labeled as Ant3 in Figs. 4.17 and 4.18. The fractional bandwidth criterion values
are increased further while the curve fitting errors are decreased further. It is now
clear that for the same electrical size and a similar parasitic structure, a lower Q
value results in a larger tolerance between the curve fitting errors and the limiting
FBW10dB values. This means that there is a smaller accuracy requirement for the
active internal matching network implementation. We note that in all three stub
antenna cases with ka ≈ 0.11, one finds that their active versions can have more
than a 10% fractional bandwidth.
118
Figure 4.16: One-leg stub antenna
285 290 295 300 305 310 315 320200
300
400
500
600
700
800
900
Freq(MHz)
Indu
ctor
(nH
)
HFSS Value:Ant1Curve Fitting:Ant1HFSS Value:Ant2Curve Fitting:Ant2HFSS Value:Ant3Curve Fitting:Ant3
Figure 4.17: Inductor-frequency (L-F) sweep for the stub antenna cases
119
285 290 295 300 305 310 315 3200
0.005
0.01
0.015
Freq(MHz)
Per
cent
age
Error:Ant1BW
10dB:Ant1
Error:Ant2BW
10dB:Ant2
Error:Ant3BW
10dB:Ant3
Figure 4.18: Comparison of the curve fitting errors and the FBW10dB values for thestub antenna cases
Figure 4.19: Four-leg stub antenna
4.4.3 Canopy Antenna
To achieve yet a smaller ka-valued antenna that achieves more than a 10% fractional
bandwidth, the one-, two-, and four-leg canopy antennas introduced in [45] were
considered. All of these antennas have the same, even lower Q ratio: Qratio = 1.75.
The curve fitting procedure was performed explicitly for the one-leg canopy antenna
120
shown in Fig. 4.20 and for the four-leg canopy antenna shown in Fig. 4.21. For both
cases, each leg was treated as an ideal inductor; the canopy was treated as copper
whose thickness coincided with the diameter of the inductor. For an outer radius
a = 7.417575345mm, an inductor L = 408nH , a 0.2mm shell thickness, a 4.4mm
inductor height, a 0.5mmmonopole radius, and a 1.98mmmonopole height, the one-
leg canopy antenna has a resonance frequency, fr,center = 297.4MHz, and, hence,
has ka = 0.0467. With a passive inductor, its fractional bandwidth is 0.0133%. We
take this resonance frequency as the center frequency of the active inductor sweep.
The results for a discrete set of 20 more HFSS simulations symmetrically located
about the center resonant frequency, based on 4nH increments of the inductor value
at the center frequency (i.e., approximately 1% of L = 408nH) are shown in Fig.
4.22. The curve fit developed from these HFSS simulation results is also shown in
Fig. 4.22. The derived constants for the curve fit (4.7) are a1 = 3.6702 × 107 and
a0 = −5.4525. The errors in the resonance frequencies as calculated with the curve
fit are shown in Fig. 4.23 along with the corresponding HFSS-predicted limiting
FBW10dB values. One observes that the curve fitting resonance frequency errors are
even more separated from their limiting FBW10dB values than they were for the stub
antenna cases, thus ensuring that the active one-leg ka = 0.0467 canopy antenna
would be resonantly well-matched to the source over more than a 10% bandwidth.
The same procedure was also performed for the four-leg canopy antenna. It
should be noted that although the canopy antenna with four inductors also has
Qratio = 1.75, its actual bandwidth is slightly narrower than that of the one-leg
canopy antenna because its radiation efficiency is a little higher. However, because
of the symmetry of the four-leg canopy antenna, the HFSS simulations could be
run with two perfect H symmetry planes, which reduced these problems to a quar-
ter of their original sizes and, hence, allowed us to enhance the discretization used
to further reduce their numerical errors. Except for two changes, all of the con-
stituent elements remained the same. With the inductor value now L = 1600nH
and the height of the monopole now 1.88mm, the HFSS-predicted value for the
center frequency was fr,center = 300.0567MHz; the ka value was thus ka = 0.0466.
121
The active inductor sweep was again taken to be a discrete set of 20 more HFSS
simulations symmetrically located about the center resonant frequency with 16nH
(1%) increments of the inductor value, which now of course, was L = 1600nH . The
results for this active inductor sweep are also shown in Fig. 4.22; the corresponding
curve fit results are also shown. The derived constants for the curve fit (4.7) are
a1 = 1.4454 × 108 and a0 = −5.3682. The curve fitting errors in the resonance
frequencies are shown in Fig. 4.23 along with the corresponding HFSS-predicted
FBW10dB limiting values. As anticipated, because of the smaller modeling errors,
one does find a further separation between the curve fitting errors and the corre-
sponding limiting FBW10dB values in Fig. 4.23. These results demonstrate that the
active four-leg ka = 0.0466 canopy antenna would also be resonantly well-matched
to the source over more than a 10% bandwidth.
Figure 4.20: One-leg canopy antenna
4.5 Conclusions
In this chapter, the use of an active internal matching network for several near-field
resonant parasitic element antennas was considered. Simulations of the electrically
122
Figure 4.21: Four-leg canopy antenna
280 285 290 295 300 305 310 315 320200
400
600
800
1000
1200
1400
1600
1800
Freq(MHz)
Indu
ctor
(nH
)
HFSS Value:One LegCurve Fitting:One LegHFSS Value:Four LegCurve Fitting:Four Leg
Figure 4.22: L-F sweep for the one- and four-leg canopy antennas
123
280 285 290 295 300 305 310 315 3200
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
−3
Freq(MHz)
Per
cent
age
Error:One LegBW
10dB:One Leg
Error:Four LegBW
10dB:Four Leg
Figure 4.23: Comparison of the curve fitting errors and the FBW10dB values for theone- and four-leg canopy antennas
small Z antenna led to the development of its circuit model representation. An
internal matching network version of this model was then proposed and validated.
This result revealed that the requisite IMN was simply a frequency independent
inductor and that with such an element, the electrically small Z antenna could
be resonantly matched to the source over a specified frequency range. A relation
between the required inductor value and the resonance frequency at which matching
was maintained was developed. It was found that it had a non-Foster behavior which
could be realized with a negative impedance converter. A curve-fitting procedure
based on a set of HFSS simulations in which the inductor value was varied and the
frequency at which resonant matching was obtained then led to the definition of the
inductor values that needed to be obtained with the active inductor. Comparisons
of the errors between the resonance frequencies predicted by the HFSS simulations
based on these curve fitting results and the original discrete set of values and the
corresponding 10dB fractional bandwidths obtained from the latter defined whether
the IMN-based (active inductor) near-field resonant parasitic antenna would work
or not.
This active internal matching network procedure was then applied to three spe-
cific ESAs: the Z, stub and canopy antennas. It was demonstrated that more
124
than a 10% fractional bandwidth could be realized with a ka = 0.266 Z antenna,
a ka ≈ 0.11 stub antenna, and a ka = 0.0467 canopy antenna. By considering
these passive antenna systems with ever decreasing Q-ratio values and with a cor-
responding ever increasing accuracy of their HFSS simulations, it was shown that
the separation between the curve fitting errors and the 10dB fractional bandwidth
limits could be increased. Consequently, the even lower ka valued active versions
of these systems surpass the desired bandwidth goals. While theoretical implemen-
tations of the requisite NIC circuit realizations of the active inductors considered
here were presented, we are now concentrating on fabricating actual prototypes to
validate these designs. We hope to present these results in the future.
125
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
The research efforts described in this dissertation were focused on using metama-
terial (MTM) concepts to achieve antenna performance improvements. The MTM
concepts involved in this dissertation were of two types: one is the transmission line
based MTM and one is the volumetric, inclusion-based MTM.
The first part of the dissertation described the introduction of TL MTM phase
shifters to correct the frequency dispersion associated with an LPDA antenna. The
frequency dispersion phenomenon was studied with a differentiated Gaussian pulse
because its frequency spectrum, while being very broad band, had zero DC compo-
nents. Because it radiates the differentiated Gaussian pulse without any frequency
dispersion, an infinitesimal dipole antenna was introduced as the reference antenna.
The concept of a received signal’s fidelity was introduced to measure the perfor-
mance of an LPDA antenna. This fidelity (FD) figure of merit was defined in terms
of a likelihood measure between the received waveform radiated by the reference
infinitesimal dipole and the LPDA antenna under consideration. It is calculated as
the maximum value of the correlation between these two waveforms normalized by
their energies. The fidelity term, FD, was also introduced to accommodate the π
phase difference associated with negative and positive values of the waveforms. The
LPDA, as a wide bandwidth, multi-element antenna, radiates different frequency
components from different sets of its elements. This behavior results in different
time delays in the feeding and wave traveling times, which directly impacts the fre-
quency components received at the same observation point from these different sets
of radiating elements. The low frequency components experience more time delays
in both the feeding and wave traveling times and, consequently, exhibit the most
frequency dispersion. Since the frequency dispersion is caused by different time
delays for different frequency components, we proposed the use of left-handed and
126
right-handed phase shifters to provide the phase adjustments required to remove
these delays for the resonance frequencies associated with the LPDA. In particular,
it was shown that the lower frequencies needed the most compensation to achieve
high fidelity values. In contrast to the normal phase shifters which consist of a serial
inductor and a shunt capacitor, which provide a phase delay, the left handed phased
shifters are composed of a serial capacitor and a shunt inductor and produce a phase
advance. Both of these phase shifters are of compact size and were applied to the
LPDA antenna without changing its log-periodic geometry. Moreover, because all of
the elements radiate primarily along the endfire direction of the array, the direction
of interest to improve the performance of the LPDA in generating pulses is primarily
in this main beam direction.
The software and simulation tools used for this LPDA portion of the dissertation
included MATLAB and HFSS. Since HFSS is a frequency domain, finite element
method (FEM), Fourier transforms were employed to bridge the frequency domain
simulation tools with the desired time domain outcomes, i.e., a high FD or FD
calculated for the time domain waveforms under consideration. The basic analysis
model adopted was a frequency domain circuit model. Again, Fourier transforms
were needed for the circuit model analysis. The efficacy of the phase adjustments
required to improve the fidelity of the output waveforms was demonstrated by the
optimum solution whose current phase is a linear function of frequency and whose
current magnitude remains the same. For the optimum solution, the fidelity is
over 97%. The phase shifter values are determined by matching the phases at
the resonant frequencies of the dipoles to those of the optimum solution. After
introducing these phase adjustments, the fidelity of the LPDA antenna was shown
to be significantly improved. In addition to the time domain fidelity, the properties
of the modified LPDA antenna were also checked. It was shown that the return
loss of the modified LPDA was still under 10dB in the frequency range of interest.
On the other hand, the gain was shown to become only slightly lower in the low
frequency range. The lower gain in this low frequency range was determined to be
due to the lower current magnitudes on the longer radiation elements, an effect was
127
caused by the phase shifters. Moreover, the phase shifters also led to a change in
some of the antenna patterns associated with the individual radiators. These effects
were found to be associated with the fact that the phase shifters were designed
specifically for the dominant resonant frequencies of the elements in the array. The
current phase was adjusted to be as close to the value associated with the optimum
solution. However, for the other frequencies, the current phase is actually far from
the optimum solution. These two defects, the current amplitudes and the narrow
bandwidth phase adjustment, could be improved by applying DNG phase shifters.
Both the left handed and right handed phase shifters cause extra group delays,
which in turn causes changes in the performance of the antenna. With the DNG
phase shifters, both the phase delay and the group delays can be made positive or
negative and, hence, a better solution over a wider bandwidth could be achieved.
On the other hand, the DNG phase shifter would require an active amplifier to
overcome the resistance loss. Nonetheless, this is possible and is currently under
investigation. By adjusting the amplifier gain, the overall gain of the modified
LPDA could be improved further.
The remainder of this dissertation was concerned with metamaterial-inspired
electrically small antennas. In Chapter 3, three types of metamaterial-inspired elec-
trically small antennas were introduced, the Z, stub, and canopy antennas. To ease
the fabrication process, the Z antenna was redesigned and fabricated using Rogers
Duroid. Inductors from COILCRAFT were adopted in the fabrication. In the mea-
surements, the conductive loss of the inductors and the ground plane size were taken
into account and several different Z antennas were measured. For each case, com-
parisons between the HFSS predicted results and the measured results show very
good agreement, including the S11 values and the total radiated power. These mea-
surement results demonstrated that our ’internal matching network’ design concept
was a viable one. For the purposes of comparison, external matching networks such
as the MFJ tuner, a single and adouble stub tuner, were applied to the electrically
small monopole to match it to a 50Ω source. The measurement results shows that
the Z antenna has more than a 20dB radiated power improvement compared to
128
any of the external matching cases. In this dissertation, one of the research inter-
ests was to design MTM-inspired ESAs with Qratio approaching unity relative to
the Chu limit value, while maintaining nearly complete matching to the source and
a high radiation efficiency. Guided by the internal matching network concept of
the Z antenna, the stub antennas were proposed. Their Qratio values with a single
parasitic element were examined. Stub antennas with multiple parasitic element
were then proposed to more efficiently utilize the minimum enclosing sphere. The
simulation results showed that by using multiple parasitic elements, the Qratio could
be decreased from 5+ to 2+. The canopy antenna was then proposed to achieve
even lower Qratio values. A canopy antenna with ka ∼ 0.046, and Qratio = 1.75
was introduced. Fields and current distributions were provided to explain the ra-
diation mechanism of the canopy antenna and comparisons were made between the
canopy antenna and several other low Qratio ESAs. For a better understanding of
the radiation mechanism of the canopy antenna, two circuit models were developed
to explore the possible ways to bring the Qratio close the the Thal limiting Q value
of 1.5. Filling the interior of the minimum enclosing sphere of the canopy antenna
with a metamaterial having a non-zero permittivity that was smaller than one was
explored to achieve Q values below that limit. While theoretically possible, the dis-
persion associated with a real metamaterial was found to bring the Qratio back over
the Thal limit. Further investigations to reduce the passive Q value with related
configurations should be investigated in the future.
Despite having several successful ESA designs that approach the Chu and Thal
limits, their bandwidths are not of much practical use because those designs were
very electrically small. The concept of an active internal matching network idea was
then developed to demonstrate that highly electrically small ESAs can be designed
to have large bandwidths. Their compact size and nearly complete matching to the
source offer great flexibilities in antenna system designs and applications.
The analysis of the internal matching networks also introduced another interest-
ing and useful simulation technique for applications to these metamaterial-inspired
ESAs. Using Ansoft HFSS-Designer co-simulations, the Z antenna without the
129
lumped element was treated as a two-port component whose one port is connected
to the 50Ω source and whose other port is connected to the IMN. For such a two-port
component, an IMN was determined that could achieve nearly complete impedance
matching for the port connected to the source. This IMN was demonstrated to be a
frequency dependent inductor. Over a 20% fractional bandwidth was demonstrated
with Ansoft Designer simulations. An analytical expression for the frequency depen-
dent inductor was determined by curve fitting the results from a set of distinct fre-
quency HFSS simulations. The inductor value was specified and the HFSS-predicted
resonance frequency was then obtained. A curve fit of all of the results was obtained
over a specified frequency range around the original resonance frequency. An ana-
lytical relation between the inductor and the resonance frequency was established in
this manner. The effects on the inductor values and the resulting resonance frequen-
cies from errors in the modeling and, hence, on our ability to provide a design that
will operate well over a large frequency band with fixed components, were deter-
mined. With the analytical model of the IMN, circuit implementation designs were
shown to require a negative capacitor and a negative inductor. It was also shown
that these circuit elements could be realized with well-known negative impedance
converter designs. Active internal matching networks were designed and simulated
for the Z, stub and canopy antennas. A canopy antenna with ka = 0.047 and more
than 10% fractional bandwidth was demonstrated.
Currently, multiple input multiple output (MIMO) communication systems are
of great research interest because of the promised process gain. Multiple, well sep-
arated receiving and transmitting antennas with good performances are needed in
MIMO systems to fully benefit from diversity gain. The types of ESAs introduced
in this dissertation are very good candidates for MIMO applications because their
mutual coupling is much weaker that resonant half-wavelength antennas because
of their compact size, which allows one to position each ESA with a much smaller
distance of separation. The compact ESAs and smaller separation distances can be
utilized to design compact MIMO antenna systems. The ESAs can also be used to
design compact antenna arrays for applications such as high gain or beam steering
130
capability. In our undergoing work, it has been shown that with less than λ/10 sepa-
ration, the mutual coupling term, (S12, between two identical canopy antennas with
ka ∼ 0.047 is about −20dB. The internal matching concept can be also adopted to
design multi-band antennas by introducing parasitic elements with different reac-
tances in the very near field of the driven elements. Because of the small coupling
between the parasitic elements at each band, only the corresponding parasitic el-
ement radiates. We have successfully designed a dual-band Z antennas without
increasing ka value. Further investigations in these ultra-dense array configurations
is highly recommended.
131
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