Zagazig University Faculty of Engineering Electronics & Communications Engineering Dept.
Wideband Microstrip Antennas By
Eng. Hussein Mahmoud Abd El-Salam
A thesis submitted in partial fulfillment
Of
The requirements for the degree of
Master of Science
In
Electronics and Communications Engineering
Under Supervision of
Prof. Kamal H. Awadalla
Electronics and Communications Engineering Department
Faculty of Electronic Engineering – Menoufiya University
Prof. Saber H. Zainud-Deen
Electronics and Communications Engineering Department
Faculty of Electronic Engineering – Menoufiya University
Assoc. Prof. Abdel Hamid A. M. Shaalan
Head of Electronics and Communications Engineering Department
Faculty of Engineering - Zagazig University
2006
Zagazig University Faculty of Engineering Electronics & Communications Engineering Dept.
Wideband Microstrip Antennas By
Eng. Hussein Mahmoud Abd El-Salam
A thesis submitted in partial fulfillment
Of
The requirements for the degree of
Master of Science
In
Electronics and Communications Engineering
Approved By The Examining Committee Signature
Prof. Kamal H. Awadalla ( )
Faculty of Electronic Engineering – Menoufiya University
Prof. Hamdi A. El Mikati ( )
Faculty of Engineering – Mansoura University
Prof. HADIA S. EL HENNAWY ( )
Dean of Faculty of Engineering - Ain Shams University
Assoc. Prof. Abdel Hamid A. M. Shaalan ( )
Head of Electronics and Comm. Eng. Dept.
Faculty of Engineering - Zagazig University
2006
ii
Acknowledgement
I would like to start by thanking ALLAH, without his graciousness the
completion of this work would not have been possible. Allah The Almighty
has entrusted me with the abilities and provided me with the courage to
complete a long journey.
My advisors, Prof. Kamal H. Awadalla, Prof. Saber H. Zainud-Deen
and Prof. Abdel Hamid A. M. Shaalan, deserve a special word of
appreciation. They have always been there with support and guidance
throughout the duration of my graduate studies. My deepest thanks and
feeling of gratitude should go to Eng. Ayman Fekrey and Eng. Hany Zamel
from Electronics Research Institute for their help, software development,
and for all their time and assistance with the construction of antenna
prototypes and many antenna measurements.
My parents, family and friends definitely deserve a special word of
thanks for always being there to support and encourage me. They have
always showed a great deal of interest in my studies.
Finally, I must express my sincere gratitude to my wife, Karima. She knows
more than anyone else about the sacrifices that had to be made. I would like
to thank her for her love, encouragement, patience and understanding
throughout all of my studies. It is much appreciated.
iii
Abstract
Microstrip patch antennas are widely used because of their many
advantages, such as the low profile, light weight, and conformity. However,
patch antennas have a main disadvantage i.e. a narrow bandwidth. Researchers
have made many efforts to overcome this problem and many configurations
have been presented to broaden the bandwidth.
In this thesis the FDTD method with absorbing boundary conditions is
used to characterize several forms of wideband microstrip patch antennas such
as rectangular, circular, and annular ring patch antennas. The time domain
response, the return loss, the input impedance and the radiation patterns of these
patch antennas are obtained.
The C programming language has been used to demonstrate the one, two, and
three-dimensional simulation using the FDTD method and the application of the
perfectly matched layer as the absorbing boundary conditions.
This study has resulted in two original contributions. The first
contribution is the design and fabrication of a new compact wideband
overlapped patches microstrip antenna. In this design the bandwidth of a single
layer microstrip patch antenna is enhanced by using multi-resonance technique
without significantly enlarging the size of the proposed antenna. In this work
the validity of the design concept is demonstrated by two examples with 51.4%
and 56.8% bandwidths. In The first example multiple resonances are achieved
by overlapping three square patches of different dimensions along their
diagonals to form a non-regular single patch, but in the second example a slot is
incorporated into this patch to expand its bandwidth, the second antenna is
designed, fabricated, and measured. These two antennas provide stable far field
Abstract
Zagazig University- Electronics & Comm. Eng. Dept iv
radiation characteristics in the entire operating band, with relatively high gain.
The effects of the substrate thickness and the dielectric constant of the substrate
on the bandwidth have been studied in this work. The feeding technique utilized
in this design is the coaxial probe-feed. The main advantage of this type of
feeding scheme is that the feed can be placed at any desired location inside the
patch in order to match with its input impedance. This feed method is easy to
fabricate and has low spurious radiation.
Another major contribution is the analysis and design of a new
circularly polarized wideband probe-fed microstrip patch antenna with
capacitive feed mechanism. The proposed antenna is designed to achieve three
targets; wide bandwidth up to 27 %, perfect matching at the input (Zin ≈ 50
ohms), and circular polarization at resonance. It is designed to operate at 1.8
GHz. This antenna is applicable to Personal Communication System (PCS)
which uses the frequency range from 1850-1990 MHz. It can be claimed that
this is the first time to realize such microstrip antenna to achieve the three
mentioned targets together.
v
Contents
Acknowledgement ……………………………………...………………..ii
Abstract …………………………………………………...……………..iii
1 Introduction ………………………………………….……………...1
1.1 Background and Motivation …...……………………..….....………..1
1.2 Objective and Scope ……………………………….. …...………..…4
1.3 Original Contribution ………………………………………..….…..5
1.4 Overview of the Thesis …………………………………………..…..7
2 Microstrip Patch Antennas ……..…………….……...………….....10
2.1 Introductory Remarks….…………………………….....……….......10
2.2 Basic Characteristics…………………...…. ………….............….....11
2.3 Advantages and Disadvantages..……………..……………..……….13
2.4 Feed Techniques…...…………………………...…………………...15
2.4.1 Microstrip Line Feed………………………….......…………15
2.4.2 Coaxial Probe Feed ……………………….....…...……........16
2.4.3 Aperture Coupled Feed ……………………...………..…….17
2.4.4 Proximity Coupled Feed……………………………….……18
2.5 Overview of Modelling Techniques…………….……….…..………20
2.5.1 Approximate Methods…………………………………....….20
2.5.1.1 Transmission Line Model…………………………..20
Contents
Zagazig University- Electronics & Comm. Eng. Dept. vi
2.5.1.2 Cavity Model………………...……………………..25
2.5.1.3 The Segmentation Method………………………....28
2.5.2 Full-Wave Methods…………………………………….…...28
2.5.2.1 Method of Moments………………………….…….29
2.5.2.2 The Finite-Element Method…………..……………32
2.5.2.3 Finite-Difference Time-Domain Method……….….34
2.6 Concluding Remarks…………………………………………..…….35
3 Bandwidth Enhancement Techniques…………..………..…….…36
3.1 Introductory Remarks….……………....……………...……….........36
3.2 Bandwidth Definitions…………………………………………....…37
3.2.1 Impedance Bandwidth………………………………….…….37
3.2.2 Pattern Bandwidth ………………………………………...…37
3.2.3 Polarization or Axial Ratio Bandwidth ……………………...38
3.3 Bandwidth Enhancement Techniques………………………………38
3.3.1 Wideband Impedance-Matching Networks…………………..39
3.3.2 Edge-Coupled Patches………………………………………..40
3.3.3 Stacked Patches…………………………………….…………41
3.3.4 Shaped Probes………………………………………….……..43
3.3.5 Capacitive Coupling and Slotted Patches…………………….45
3.3.6 Capacitive Feed Probes……………………………………….47
3.4 New Compact Wideband Overlapped Patches Microstrip
Antennas……………………………………………………….........50
3.5 Concluding Remarks…………………………………………….….52
4 The Finite-Difference Time-Domain Method.................................53
4.1 Introductory Remarks….……………....………................................53
4.2 One-Dimensional Simulation with The FDTD Method…………….54
4.2.1 One-Dimensional Free Space Formulation………………….54
Contents
Zagazig University- Electronics & Comm. Eng. Dept. vii
4.2.2 Stability and The FDTD Method………………………….…59
4.2.3 The Absorbing Boundary Condition In One Dimension…….59
4.2.4 Determining Cell Size…………………………………….…60
4.2.5 Propagation in A Lossless Dielectric Medium………………62
4.2.6 Simulating Different Sources……………………………..…65
4.2.7 Propagation in A Lossy Dielectric Medium…………………66
4.2.8 Calculating The Frequency domain Output…………………69
4.3 Two- Dimensional Simulation with The FDTD Method…………..73
4.3.1 The Perfectly Matched Layer (PML)………………………..77
4.4 Three- Dimensional Simulation with The FDTD Method…………86
4.4.1 Free Space Formulation………………………….. ………...86
4.4.2 The PML in Three Dimensions……………………………...89
4.5 Near-Field to Far-Field Transformation……………………………91
4.6 Concluding Remarks……………………………………………….93
5 FDTD Analysis of Wideband Microstrip antennas........................94
5.1 Introductory Remarks….……………....………................................94
5.2 A Line-Fed Rectangular Patch Antenna………………………...….95
5.2.1 Design Specifications…………………………………….….95
5.2.2 Design Procedure………………………………………..…..96
5.2.3 FDTD Analysis of The Rectangular Microstrip Antenna…...99
5.2.3.1 Frequency-Dependent Parameters…………..............99
5.2.3.2 Numerical Results………………………………….100
5.2.3.3 Radiation Pattern…………………………………..105
5.2.3.4 Other Calculated Parameters…………………...….106
5.3 Wideband E-Shaped Patch Antennas……………………………..109
5.4 Capacitively Probe-Fed wideband Microstrip Antenna………..…113
5.5 Concluding Remarks……………………………………………....120
Contents
Zagazig University- Electronics & Comm. Eng. Dept. viii
6 New Wideband Slotted Overlapped Patches Microstrip
antenna……………………….……………………………………121
6.1 Introductory Remarks….……………....…………………...….......121
6.2 Wideband Overlapped Patches Microstrip antenna (OPMA)…..…123
6.3 New Wideband Slotted overlapped patches microstrip antenna
(SOPMA)…………………………………………………………..132
6.4 Concluding Remarks……………………………………………....136
7 Circularly Polarized Wideband Microstrip Antennas……...…..137
7.1 Introductory Remarks……………………………………………...137
7.2 Dual-Band Circularly Polarized Patch Antenna………..………….138
7.2.1 Antenna Geometry………………………………………….138
7.2.2 Antenna Feed……………………………………………….139
7.2.3 Simulation Results and Discussions………………………..140
7.3 New Circularly Polarized Capacitively Probe-Fed Wideband
Microstrip Antenna………………………………………………...147
7.3.1 Simulation, Analysis, and Discussions…………………….148
7.4 Concluding Remarks……………………………….........................153
8 Conclusions and Future Research………………………………..154
8.1 General Conclusions………………………………………...……..154
8.2 Future Research…………………………………………...……….156
References……………………………………………………................159
Publications ……………………………….…………………………...167
Arabic Summery………………………………….………………………..
1
C H A P T E R 1
Introduction
1.1 BACKGROUND AND MOTIVATION
During recent years, there has been a fast growth in the wireless
communications industry. The deployment of systems such as cellular
telephone networks, wireless local loop networks and wireless local area
networks, is rapidly developing worldwide. As more and more people use
these services, network operators are continuously forced to optimize their
networks so that the maximum amount of capacity, together with quality
coverage, can be obtained from these networks. The field of antenna
engineering is of course central to all wireless technologies and plays a
significant role in the successful deployment and optimization of such
systems. As such, the growing demand for wireless communications, has
stimulated extensive research in order to find new solutions to problems in
antenna engineering.
With the advances in wireless communications technologies and the
associated reproduction of base stations throughout major cities and much
of the countryside, a number of requirements are imposed on the antennas
Chapter 1 Introduction
Zagazig University- Electronics & Comm. Eng. Dept. 2
that are used. From a technological point of view, wireless communications
antennas should be relatively cheap and easy to manufacture, they should be
lightweight and they should be robust. From an environmental point of
view, the antennas should have a minimum impact. As such, these antennas
should have a low profile and should be as compact as possible. This of
course also goes for handset antennas, where the size of such devices is
continuously shrinking.
One type of antenna that fulfills these requirements very well, is the
microstrip antenna. These antennas operate in the microwave frequency
range and are widely used on base stations as well as handsets. They come
in a variety of configurations and have been the topic of what is currently
probably the most active field in antenna research and development. In one
of its most basic forms.
A microstrip antenna is comprised of a metal patch that is supported above
a larger ground plane. It is usually manufactured by printing the patch on a
Fig. 1.1 Cellular base-station antennas. Each antenna array consists of a number of antenna elements
Antenna array
Antenna element
Mast
Chapter 1 Introduction
Zagazig University- Electronics & Comm. Eng. Dept. 3
thin microwave substrate. This configuration is commonly known as the
microstrip patch antenna. Microstrip patches are often used as single-
element antennas, but are also very suitable for use within antenna arrays.
Figure 1.1 shows a typical example of how they can be used in directional
base-station antennas.
Rectangular and circular patches are most common, but any shape that
possesses a reasonably well-defined resonant mode can be used [1]. These
include, for example, annular rings, ellipses and triangles. The patch is a
resonant element and therefore one of its dimensions must be approximately
one half of the guided wavelength in the presence of the dielectric substrate.
There are four fundamental techniques to feed or excite the patch. These are
presented in chapter 2 and include the microstrip-line feed, the probe feed,
the aperture-coupled feed and the proximity-coupled feed.
The main drawback associated with microstrip patch antennas in general is
that they inherently have a very narrow impedance bandwidth (due to their
multilayered configuration, aperture-coupled feeds and proximity-coupled
feeds tend to have a slightly wider bandwidth than probe feeds and
microstrip-line feeds). In most cases, the impedance bandwidth is not wide
enough to handle the requirements of modern wireless communications
systems [2]. The narrow impedance bandwidth of microstrip patch antennas
can be referred to the thin substrates that are normally used to separate the
patch and the ground plane. The general performance trends of a microstrip
patch antenna are illustrated in Figure 1.2. Here, Figure 1.2(a) shows the
typical trend for impedance bandwidth versus substrate thickness, as a
function of the substrate’s dielectric constant, while Figure 1.2(b) shows the
typical trend for surface-wave efficiency versus substrate thickness, also as
a function of the substrate's dielectric constant. From these it can be seen
that, in order to increase the bandwidth, the substrate thickness has to be
increased, while the dielectric constant has to be kept as low as possible.
Chapter 1 Introduction
Zagazig University- Electronics & Comm. Eng. Dept. 4
A low dielectric constant is also required to keep surface-wave losses as
low as possible. Therefore, in order to obtain a wideband microstrip patch
antenna with good surface-wave efficiency, the performance trends of
Figure 1.2 point to a thick substrate with a very low dielectric constant.
1.2 OBJECTIVES AND SCOPE
The specific objectives and scope of the research are described in the
points that follow.
The first objective of this research is the design, analysis and
fabrication of a novel wideband microstrip patch antenna using the
Efficiency
100 %
Bandwidth
Substrate thickness0
εr >> 1
εr > 1
(a)
εr >> 1
εr > 1
εr = 1
0Substrate thickness
(b)
Fig. 1.2 Illustrative performance trends of a microstrip patch antenna. (a) Impedance bandwidth. (b) Surface-wave efficiency.
εr = 1
Chapter 1 Introduction
Zagazig University- Electronics & Comm. Eng. Dept. 5
FDTD method . While still retaining the benefits of low cost, light
weight, low profile, as well as ease of design and manufacture.
The several factors affecting the bandwidth of the microstrip
antenna such as the thickness of the substrate, the dielectric
constant of the substrate and the shape of the patch would be
studied in this thesis.
Another major objective is the design and analysis of a new
circularly polarized wideband probe-fed microstrip patch antenna
with capacitive feed mechanism.
A key objective of this thesis is that, when properly implemented,
FDTD analysis of different shapes of antennas produces results for
near-fields, far-fields, return loss, and input impedance that agree
very well with published experimental data. FDTD method has a
powerful ability to provide, in straight forward manner, results of
antenna structures performance over a wideband of frequency. This
robustness allows the use of the FDTD method to confidently test
proposed for novel antenna designs on the computer before they are
built.
1.3 ORIGINAL CONTRIBUTIONS
This study has resulted in some original contributions. The detailed
contributions are described in the points that follow.
The first contribution which has already been published by the author
[3] is the design and fabrication of a new compact wideband
overlapped patches microstrip antenna. In this design the bandwidth
of a single layer microstrip patch antenna is enhanced by using multi-
resonance technique without significantly enlarging the size of the
proposed antenna. In this work the validity of the design concept is
Chapter 1 Introduction
Zagazig University- Electronics & Comm. Eng. Dept. 6
demonstrated by two examples with 51.4% and 56.8% bandwidths. In
The first example multiple resonances are achieved by overlapping
three square patches of different dimensions along their diagonals to
form a non-regular single patch, but in the second example a slot is
incorporated into this patch to expand its bandwidth, the second
antenna is designed, fabricated, and measured. These two antennas
provide stable far field radiation characteristics in the entire operating
band, with relatively high gain. The effects of the substrate thickness
and the dielectric constant of the substrate on the bandwidth have
been studied in this work. The feeding technique utilized in this
design is the coaxial probe-feed. The main advantage of this type of
feeding scheme is that the feed can be placed at any desired location
inside the patch in order to match with its input impedance. This feed
method is easy to fabricate and has low spurious radiation.
Another major contribution is the design and analysis of a new
circularly polarized wideband probe-fed microstrip patch antenna with
capacitive feed mechanism. The proposed antenna is designed to
achieve three targets; wide bandwidth up to 27 %, perfect matching at
the input (Zin ≈ 50 ohms), and circular polarization at the resonance.
The proposed antenna is designed to operate at 1.8 GHz, so it is
applicable to Personal Communication System (PCS) which uses the
frequency range from 1850-1990 MHz. One can claim that this is the
first time to achieve and realize a microstrip antenna to satisfy the
mentioned three targets together.
1.4 OVERVIEW OF THE THESIS
The thesis consists of 8 chapters including the present one which is
“Introduction”, this chapter presented some background information on
Chapter 1 Introduction
Zagazig University- Electronics & Comm. Eng. Dept. 7
microstrip patch antennas. It was pointed out why the microstrip antennas
are of particular interest for wireless communications systems and what the
challenges are when wideband operation of these antennas is required. With
these in mind, the main objectives and scope of the study were formulated.
In short, it includes the development of a new compact wideband microstrip
antenna, the original contributions that followed, were also summarised.
A short overview of the remaining chapters will now follow.
Chapter 2. “Microstrip Patch Antennas”, which gives an overview of
microstrip patch antennas and reviews the various feeding techniques that
can be used. The analytical and numerical techniques used for the analysis
and design of these antennas are also presented .
Chapter 3. “Bandwidth Enhancement Techniques”, this chapter presents
a study of the different broadbanding techniques of microstrip patch
antennas especially the new trends. The advantages, disadvantages and
requirements of each technique are also discussed. With these in mind, the
new antenna element that forms the basis of this study, will be presented.
Chapter 4. “The Finite-Difference Time-Domain Method”, this chapter
includes advantages of the FDTD method, applications of the FDTD
method, updating equations of the Yee algorithm, stability condition,
absorbing boundary conditions of the FDTD method, and the near-field to
far-field transformation.
This chapter starts off by presenting one-dimensional simulation with the
FDTD method then it presents two-dimensional and three-dimensional
simulation. All the results in this chapter were obtained using the C
programming language and these results have been compared with the
published data and good agreements have been found.
Chapter 1 Introduction
Zagazig University- Electronics & Comm. Eng. Dept. 8
Chapter 5. “FDTD Analysis of Wideband Microstrip antennas”, in this
chapter, the FDTD method will be used to characterize several forms of
wideband microstrip patch antennas such as rectangular, circular, and
annular ring patch antennas. The time domain response, the return loss, the
input impedance and the radiation patterns of these patch antennas are
obtained. The obtained results have been compared to other results
produced using the IE3D software [4] which is based on the method of
moments and good agreements will be shown.
Chapter 6. “New Wideband Slotted Overlapped Patches Microstrip
antenna”, this chapter represents the first contribution that has been
resulted from this study. In this chapter the bandwidth of a single layer
microstrip patch antenna is enhanced by using multi-resonance technique
without significantly enlarging the size of the proposed antenna. In this
work the validity of the design concept is demonstrated by two examples
with 51.4% and 56.8% bandwidths. In The first example multiple
resonances are achieved by overlapping three square patches of different
dimensions along their diagonals to form a non-regular single patch, but in
the second example a slot is incorporated into this complex patch to expand
its bandwidth, the second novel antenna has been designed, fabricated, and
measured.
Chapter 7. “Circularly Polarized Wideband Microstrip Antennas”, this
chapter presents two circularly polarized microstrip antennas.
The first one is the dual-band circularly polarized patch antenna, this patch
has a square shape and it is loaded by four slots close to the radiating edges.
Simulations will be shown and compared with the published data and good
agreements will be shown.
The second one is a new circularly polarized capacitively probe-fed
microstrip antenna, this antenna consists of two small probe-fed rectangular
Chapter 1 Introduction
Zagazig University- Electronics & Comm. Eng. Dept. 9
patches, which are capacitively coupled to the radiating element. The
proposed antenna is designed to achieve three targets; wide bandwidth up to
27 %, perfect matching at the input (Zin ≈ 50 ohms), and circular
polarization at the resonance.
Chapter 8. “ Conclusions and Future Research ”, this chapter contains
general conclusions regarding this study and concludes the thesis with some
recommendations that can be considered for future work.
C H A P T E R 2
Microstrip Patch Antennas
2.1 INTRODUCTORY REMARKS
In high-performance air craft, space craft, satellite and missile applicati-
ons, where size, weight, cost, performance, and ease of installation are
constraints, low profile antennas may be required. Presently there are many
other government and commercial applications, such as mobile radio and
wireless communications, which have similar specifications. To meet these
requirements, microstrip antennas [5] can be used. These antennas are low-
profile, conformal to planar and nonplanar surfaces, simple and inexpensive to
manufacture using modern printed-circuit technology, mechanically robust when
mounted on rigid surfaces, compatible with MMIC (monolithic microwave
integrated circuits) design, and when the particular patch shape and mode are
selected they are very versatile in terms of resonant frequency, polarization,
radiation pattern and impedance. In addition, by adding loads between the patch
and the ground plane, such as pins and varactor diodes, adaptive elements with
variable resonant frequency, impedance, polarization and radiation pattern can
be designed.
10
Chapter 2 Microstrip Patch Antennas
2.2 BASIC CHARACTERISTICS
Microstrip antennas received considerable attention in the 1970s, although
the idea of a microstrip antenna was first proposed by Deschamps in 1953.
Microstrip antennas, as shown in Figure 2.1, consists of a very thin (t << λo
where λo is the free-space wavelength) metallic strip (patch) placed a small
fraction of a wavelength ( h ≤ λo, usually 0.003 λo ≤ h ≤ 0.05 λo ) above a ground
plane. The microstip patch is designed so its pattern maximum is normal to the
patch (broadside radiator). This is accomplished by properly choosing the mode
(Field configuration) of excitation beneath the patch. End-fire radiation can also
be accomplished by good mode selection, the strip (patch) and the ground plane
are separated by a dielectric sheet (referred to the substrate), as shown in Figure
2.1.
Ground plane
Radiating Patch
Dielectric Substrate εr
h
Fig. 2.1 Geometry of microstrip antenna.
There are numerous substrates that can be used for the design of microstrip
antennas, and their dielectric constants are usually in the range of 2.2 ≤ εr ≤ 12.
The ones that are most desirable for antenna performance are thick substrate
with low dielectric constant because they provide better efficiency, large
bandwidth, loosely bound fields for radiation into space, but at the expense of
large element size [6]. Thin substrate with higher dielectric constants is desirable
for MIC (microwave integrated circuits) because they require tightly bound
Zagazig University- Electronics & Comm. Eng. Dept. 11
Chapter 2 Microstrip Patch Antennas
fields to minimize undesired radiation and coupling. Often micostrip antennas
are also referred to as patch antennas. The radiating elements and the feed lines
are usually photoetched on the dielectric substrate.
The radiating patch may be square, rectangular, circular, elliptical, thin strip
(dipole), triangular or any other configuration. These and others are illustrated in
Figure 2.2 and a brief summary of their characteristics follows:
Square and rectangular patches as shown in Figures 2.2(a) and (b) are the first
and probably the most utilized patch conductor geometries. Square patches can
be used to generate circular polarization.
(e) Dipole (a) Square
(i) Star
Fig. 2.2 Representative shapes of microstrip patch elements.
(d) Elliptical
(f) Triangular
(b) Rectangular (c) Circular
(g) Ring sector (h) Annular ring
Circular and elliptical patches as shown in Figures 2.2(c) and (d) are
probably the second most common shape. These patches are slightly smaller
than their rectangular counterpart and as a result have slightly lower gain and
bandwidth. One of the primary reasons the circular geometry was quite
expansively investigated in the past was because of its inherent symmetry.
Microstrip dipoles as shown in Figures 2.2(e) are attractive because they
inherently possess a large bandwidth and occupy less space, which make
them attractive for arrays.
Zagazig University- Electronics & Comm. Eng. Dept. 12
Chapter 2 Microstrip Patch Antennas
Triangular and ring sector patches as shown in Figures 2.2(f) and (g) are
smaller than their rectangular and circular counterparts, although at the
expense of further reduction in bandwidth and gain. Triangular patches also
tend to generate higher cross-polarization levels, because of their lack of
symmetry in the configuration, the bandwidth is typically very narrow.
Annular ring geometries as shown in Figures 2.2(h) are the smallest
conductor shape, once again at the expense of bandwidth and gain. One
problem associated with an annular ring is that it is not a simple process to
excite the lowest order mode and obtain a good impedance match at
resonance.
Star microstrip patches as shown in Figure 2.2(i) have been theoretically
investigated [6] as a radiator of higher-order modes with good symmetry.
2.3 ADVANTAGES AND DISADVANTAGES
Microstrip patch antennas are increasing in popularity for use in wireless
applications due to their low-profile structure. Some of their principal
advantages discussed by J. R. James [6] are given below:
• Light weight and low volume.
• Low profile planar configuration which can be easily made conformal to host
surface.
• Low fabrication cost, hence can be manufactured in large quantities.
• Supports both, linear as well as circular polarization.
• Can be easily integrated with MICs.
• Capable of dual and triple frequency operations.
• Mechanically robust when mounted on rigid surfaces.
Microstrip patch antennas suffer from a number of disadvantages as compared
to conventional antennas. Some of their major disadvantages discussed by J. R.
James [6] are given below:
Zagazig University- Electronics & Comm. Eng. Dept. 13
Chapter 2 Microstrip Patch Antennas
• Narrow bandwidth.
• Low efficiency.
• Low Gain.
• Outsider radiation from feeds and junctions.
• Poor end fire radiator except tapered slot antennas.
• Low power handling capacity.
• Surface wave excitation.
Microstrip patch antennas have a very high antenna quality factor (Q). Q
represents the losses associated with the antenna and a large Q leads to narrow
bandwidth and low efficiency. Q can be reduced by increasing the thickness of
the dielectric substrate. But as the thickness increases, an increasing fraction of
the total power delivered by the source goes into a surface wave. This surface
wave contribution can be counted as an unwanted power loss since it is
ultimately scattered at the dielectric bends and causes degradation of the antenna
characteristics.
However, surface waves can be minimized by use of photonic bandgap
structures as discussed by Qian et al [7]. Other problems such as lower gain and
lower power handling capacity can be overcome by using an array configuration
for the elements.
Zagazig University- Electronics & Comm. Eng. Dept. 14
Chapter 2 Microstrip Patch Antennas
2.4 FEED TECHNIQUES
Microstrip patch antennas can be fed by a variety of methods. These
methods can be classified into two categories contacting and non-contacting. In
the contacting method, the RF power is fed directly to the radiating patch using
a connecting element such as a microstrip line.
In the non-contacting scheme, electromagnetic field coupling is done to transfer
power between the microstrip line and the radiating patch [8]. The four most
popular feed techniques used are the microstrip line, coaxial probe (both
contacting schemes), aperture coupling and proximity coupling (both non-
contacting schemes).
2.4.1 Microstrip Line Feed
In this type of feed technique, a conducting strip is connected directly to
the edge of the microstrip patch as shown in Figure 2.3. The conducting strip is
smaller in width as compared to the patch . This kind of feed arrangement has
the advantage that the feed can be etched on the same substrate to provide a
planar structure.
Microstrip line feed
Radiating Patch
Fig. 2.3 Microstrip Line Feed
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Chapter 2 Microstrip Patch Antennas
The purpose of the inset cut in the patch is to match the impedance of the
feed line to the patch without the need for any additional matching element. This
is achieved by properly controlling the inset position. Hence this is an easy
feeding scheme, since it provides ease of fabrication and simplicity in modeling
as well as impedance matching. However as the thickness of the dielectric
substrate being used, increases, surface waves and spurious feed radiation also
increases, which hampers the bandwidth of the antenna [8]. The feed radiation
also leads to undesired cross polarized radiation.
2.4.2 Coaxial Probe Feed
The coaxial feed or probe feed [9] is a very common technique used for
feeding microstrip patch antennas. As seen from Figure 2.4, the inner conductor
of the coaxial connector extends through the dielectric and is soldered to the
radiating patch, while the outer conductor is connected to the ground plane.
Coaxial Connector
Ground Plane
Substrate εr
Radiating Patch
Fig.2.4 Probe fed Rectangular Microstrip Patch Antenna
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Chapter 2 Microstrip Patch Antennas
The main advantage of this type of feeding scheme is that the feed can be
placed at any desired location inside the patch in order to match with its input
impedance. This feed method is easy to fabricate and has low spurious radiation.
However, its major disadvantage is that it provides narrow bandwidth and is
difficult to model since a hole has to be drilled in the substrate and the connector
protrudes outside the ground plane, thus not making it completely planar for
thick substrates (h > 0.02λo). Also, for thicker substrates, the increased probe
length makes the input impedance more inductive, leading to matching
problems. It is seen above that for a thick dielectric substrate, which provides
broad bandwidth, the microstrip line feed and the coaxial feed suffer from
numerous disadvantages. The non-contacting feed techniques which have been
discussed below, solve these problems.
2.4.3 Aperture Coupled Feed
The aperture coupling of Figure 2.5 is the most difficult of all four to
fabricate and it also has a narrow bandwidth. However, it is somewhat easier to
model and has moderate spurious radiation.
Ground Plane
Substrate 1
Aperture Slot
Substrate 2
Patch
Microstrip Line
Fig. 2.5 Aperture-coupled feed
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Chapter 2 Microstrip Patch Antennas
The aperture coupling consists of two substrates separated by a ground
plane. On the bottom side of the lower substrate there is a microstrip feed line
whose energy is coupled to the patch through a slot on the ground plane
separating the two substrates. The coupling aperture is usually centered under
the patch, leading to lower cross polarization due to symmetry of the
configuration. The amount of coupling from the feed line to the patch is
determined by the shape, size and location of the aperture. Since the ground
plane separates the patch and the feed line, spurious radiation is minimized.
Generally, a high dielectric material is used for the bottom substrate and a thick,
low dielectric constant material is used for the top substrate to optimize radiation
from the patch [10]. The major disadvantage of this feed technique is that it is
difficult to fabricate due to multiple layers, which also increases the antenna
thickness. This feeding scheme also provides narrow bandwidth.
2.4.4 Proximity Coupled Feed
This type of feed technique is also called as the electromagnetic coupling
scheme [11]. As shown in Figure 2.6, two dielectric substrates are used such that
the feed line is between the two substrates and the radiating patch is on top of
the upper substrate.
Substrate 1
Patch
Microstrip Line
Fig. 2.6 Proximity-coupled Feed
Substrate 2
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Chapter 2 Microstrip Patch Antennas
The main advantage of this feed technique is that it eliminates spurious
feed radiation and provides very high bandwidth (as high as 13%) [5], due to
overall increase in the thickness of the microstrip patch antenna. This scheme
also provides choices between two different dielectric media, one for the patch
and one for the feed line to optimize the individual performances.
Matching can be achieved by controlling the length of the feed line and the
width-to-line ratio of the patch [5]. The major disadvantage of this feed scheme
is that it is difficult to fabricate because of the two dielectric layers which need
proper alignment. Also, there is an increase in the overall thickness of the
antenna. Table 2.1 below summarizes the characteristics of the different feed
techniques [6].
Table 2.1 Comparing the different feed techniques
Characteristics
Microstrip Line Feed
Coaxial Feed Aperture coupled
Feed
Proximity Coupled
Feed
Spurious feed radiation
More More Less
Minimum
Reliability Better Poor due to
soldering Good
Good
Ease of fabrication
Easy Soldering and drilling needed
Alignment required
Alignment required
Impedance Matching
Easy Easy Easy Easy
Bandwidth (achieved with
impedance matching)
2-5% 2-5% 2-5%
13%
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Chapter 2 Microstrip Patch Antennas
2.5 OVERVIEW OF MODELLING TECHNIQUES
There are a number of methods that can be used for the analysis of
microstrip patch antennas. Most of these methods fall into one of two broad
categories: approximate methods and full-wave methods [6]. The approximate
methods are based on simplifying assumptions and therefore they have a number
of limitations and are usually less accurate. They are usually used to analyze
single antenna elements as it is very difficult to model coupling between
elements with these methods. However, where applicable, they normally do
provide good physical insight and the computation time is usually very small.
The full-wave methods include all relevant wave mechanisms and rely
heavily upon the use of efficient numerical techniques [12]. When applied
properly, the full-wave methods are reasonably accurate and can be used to
model a wide variety of antenna configurations, including antenna arrays. These
methods tend to be much more complex than the approximate methods and also
provide less physical insight. Very often they also require vast computational
resources and extensive solution times. In the remainder of this section, an
overview of both approximate and full-wave methods will be given.
2.5.1 Approximate Methods
Some of the popular approximate models include the transmission-line
model, the cavity model and the segmentation model. These models usually treat
the microstrip patch as a transmission line or as a cavity resonator.
2.5.1.1 Transmission-Line Model The transmission-line model leads to results that are adequate for most
engineering purposes and entail less computation. Although this method has its
shortcomings, particularly in that it is applicable only to rectangular or square
patch geometries, the model offers a reasonable interpretation of the radiation
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Chapter 2 Microstrip Patch Antennas
mechanism, while simultaneously giving a simple expression of the antenna ’s
characteristics [13].
The basic concept of the transmission-line model is shown in Figure 2.7. This
model is for a rectangular patch fed at the center of the radiating edge. The patch
is characterized as a microstrip transmission-line with a length L, width W, and
GrjBGr jB ZO, ΒgYin
L
Z
X
Y
h h
thickness h. Each radiating edge, with length equal to W, is modeled as a narrow
slot radiating into a half-space. The width of the slot is, for the sake of
convenience, assumed to be equal to the substrate thickness h, As a result, the
rectangular patch antenna can be represented by two admittance connected by an
equivalent microstrip transmission line as shown in the lower half of Figure 2.7,
where the characteristic impedance Zo and the propagation constant βg for the
fundamental mode in the microstrip line are approximated by [14] as:
Fig. 2.7 Plane view of rectangular patch antenna and its equivalent circuit.
W
h
YZ
reff
o
oo
1 (2.1)
reffog K (2.2)
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Chapter 2 Microstrip Patch Antennas
where ηo = The wave impedance in free space.
Ko = The propagation constant in free space.
εreff = Effective dielectric constant.
εreff is related to the intrinsic dielectric constant εr of the substrate as follows
[13]:
2/1
1212
1
2
1
W
hrrreff
(2.3)
Notice that the value of εreff is slightly less then εr because the fringing fields
around the periphery of the patch are not confined in the dielectric substrate but
are also spread in the air
The capacitive component, B, and the conductive component, Gr, which form
each admittance, are related to the fringing field and the radiation loss, and are
respectively approximated by [15] as:
reffo
o
Z
lKB (2.4)
WW
WW
WW
G
oo
ooo
oo
r
2 ,120
235.0 ,60
1
120
35.0 ,90
2
2
2
(2.5)
where ∆L signifies the line extension due to the fringing effect. This value can
be approximated by using the following equation [13]:
8.0258.0
264.03.0412.0
h
Wh
W
hL
reff
reff
(2.6)
The effective length of the patch Leff now becomes:
Leff = L +2 ∆L (2.7)
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Chapter 2 Microstrip Patch Antennas
This effective length is given by [5] as:
reffr
efff
CL
2 (2.8)
where C is the speed of light in free space, and fr is the resonance frequency of
the microstrip antenna.
From the equivalent circuit in Figure 2.7. The input admittance of this patch
antenna can be shown to be the following, if it is regarded as two slot antennas
connected by a transmission line having characteristic admittance and
propagation constant of Yo and βg approximated by Eqs. (2.1) and (2.2):
)tan()(
)tan()(
LjBGjY
LjYjBGYjBGY
gro
gororin
(2.9)
In this case, the resonance condition is given by
ImYin = 0 (2.10)
Where ImYin represents the imaginary part of Yin. From Equation (2.10), the
following condition can be derived:
222
2)tan(
or
og
YBG
BYL
(2.11)
The condition above is used to determine the resonant frequency when the patch
length L is given. The input admittance at resonance can be found by
substituting Eq. (2.11) into (2.9):
Yin = 2 Gr (2.12)
This result is, of course, easily deduced from the equivalent circuit of Figure 2.7.
For an efficient radiation, a practical width that leads to good radiation
efficiencies is given in [8] as:
1
2
2
rrf
CW
(2.13)
In the typical design procedure of a rectangular patch antenna, the thickness and
dielectric constant of the substrate must be known. Once they are given or
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Chapter 2 Microstrip Patch Antennas
Zagazig University- Electronics & Comm. Eng. Dept. 24
determined, a patch antenna that operates at the required resonance frequency
can be designed by following the flow chart shown in Figure 2.8 [8].
εr, h, fr
1
2
2
rrf
CW
reffr
efff
CL
2
8.0258.0
264.03.0412.0
h
Wh
W
hL
reff
reff
2/1
1212
1
2
1
hrr
reff
W
LLL eff 2
END
Fig. 2.8 Flow chart for the design procedure of
a rectangular patch antenna.
Chapter 2 Microstrip Patch Antennas
2.5.١٫2 Cavity Model
Although the transmission-line model discussed in the previous section is
easy to use, it has some inherent disadvantages. Specifically, it is useful for
patches of rectangular design and it ignores field variations along the radiating
edges. These disadvantages can be overcome by using the cavity model. In this
model, the interior region of the dielectric substrate is modeled as a cavity
bounded by electric walls on the top and bottom and magnetic walls on the left
and right. The basis for this assumption is the following observations for thin
substrates ( h << λ).
• Since the substrate is thin, the fields in the interior region do not vary much in
the z direction, i.e. normal to the patch.
• The electric field is z directed only, and the magnetic field has only the transv-
erse components Hx and Hy in the region bounded by the patch metallization
and the ground plane. This observation provides for the electric walls at the
top and the bottom.
Fig. 2.9 Charge distribution and current density creation on the microstrip patch.
Consider Figure 2.9 shown above. When the microstrip patch is energized,
a charge distribution is seen on the upper and lower surfaces of the patch and at
the bottom of the ground plane. This charge distribution is controlled by two
mechanisms-an attractive mechanism and a repulsive mechanism. The attractive
mechanism is between the opposite charges on the bottom side of the patch and
the ground plane, which helps in keeping the charge concentration intact at the
Zagazig University- Electronics & Comm. Eng. Dept. 25
Chapter 2 Microstrip Patch Antennas
bottom of the patch. The repulsive mechanism is between the like charges on the
bottom surface of the patch, which causes pushing of some charges from the
bottom, to the top of the patch. As a result of this charge movement, currents
flow at the top and bottom surface of the patch. The cavity model assumes that
the height to width ratio (i.e. height of substrate and width of the patch) is very
small and as a result of this the attractive mechanism dominates and causes most
of the charge concentration and the current to be below the patch surface. Much
less current would flow on the top surface of the patch and as the height to width
ratio further decreases, the current on the top surface of the patch would be
almost equal to zero, which would not allow the creation of any tangential
magnetic field components to the patch edges. Hence, the four sidewalls could
be modeled as perfectly magnetic conducting surfaces. This implies that the
magnetic fields and the electric field distribution beneath the patch would not be
disturbed. However, in practice, a finite width to height ratio would be there and
this would not make the tangential magnetic fields to be completely zero, but
they being very small, the side walls could be approximated to be perfectly
magnetic conducting surfaces.
Since the walls of the cavity, as well as the material within it are lossless, the
cavity would not radiate and its input impedance would be purely reactive.
Hence, in order to account for radiation and a loss mechanism, one must
introduce a radiation resistance Rr and a loss resistance RL . A lossy cavity
would now represent an antenna and the loss is taken into account by the
effective loss tangent δeff which is given as:
δeff = 1 / QT (2.14)
QT is the total antenna quality factor and has been expressed by [6] in the form:
rcdT QQQQ
1111 (2.15)
• Qd represents the quality factor of the dielectric and is given as :
Zagazig University- Electronics & Comm. Eng. Dept. 26
Chapter 2 Microstrip Patch Antennas
tan
1
d
Trd p
WQ (2.16)
where ωr is the angular resonant frequency.
WT is the total energy stored in the patch at resonance.
Pd is the dielectric loss.
tan δ is the loss tangent of the dielectric.
• Qc represents the quality factor of the conductor and is given as :
h
p
WQ
c
Trc
(2.17)
where Pc is the conductor loss.
∆ is the skin depth of the conductor.
h is the height of the substrate.
• Qr represents the quality factor for radiation and is given as:
r
Trr p
WQ
(2.18)
where Pr is the power radiated from the patch.
Substituting equations (2.15), (2.16), (2.17) and (2.18) in equation (2.14), we get
Tr
reff W
p
h
tan (2.19)
Thus, equation (2.19) describes the total effective loss tangent for the microstrip
patch antenna.
For the TMmn mode, the resonance frequency of a rectangular patch antenna of
length L and width W is given by [13]:
oor
mnmn
kf
2
2/122
L
n
W
mk mn
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Chapter 2 Microstrip Patch Antennas
2.5.1.3 The Segmentation Method
This method is more versatile than both the transmission-line model and
the cavity model, especially in terms of its ability to treat patches with arbitrary
shapes. It is an extension of the cavity model, but instead of treating the patch as
a single cavity, the patch is segmented into sections of regular shapes. The
cavity model is then applied to each section, after which the multiport-
connection method is used to connect the individual sections. This method has
been used, for example, by Palanisamy and Garg [16] for the modelling of a
square-ring patch, while Kumar and Gupta [17] used it for the modelling of
edge-coupled patches. As with the other approximate methods that have been
described, this method also works best for thin, low dielectric-constant
substrates.
2.5.2 Full-Wave Methods
Three very popular full-wave methods that can be used to model
microstrip patch antennas, are the moment method (MoM), the finite-element
method (FEM) and the finite-difference time-domain (FDTD) method. These are
the three major paradigms of full-wave electromagnetic modelling techniques
[2]. Unlike the approximate methods, these methods include all the relevant
wave mechanisms and are potentially very accurate. They all incorporate the
idea of discretising some unknown electromagnetic property. For the MoM, it is
the current density, while for the FEM and FDTD, it is normally the electric
field (also the magnetic field for the FDTD method).
The discretisation process results in the electromagnetic property of interest
being approximated by a set of smaller elements, but of which the complex
amplitudes are initially unknown. The amplitudes are determined by applying
the full-wave method of choice to the total number of elements. Usually, the
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Chapter 2 Microstrip Patch Antennas
approximation becomes more accurate as the number of elements is increased.
Although these methods all share the idea of discretisation, their
implementations are very different and therefore each of the three methods will
now be considered in some more detail.
2.5.2.1 Method of Moments
One of the methods, that provide the full wave analysis for the microstrip
patch antenna, is the Method of Moments [12]. In this method, the surface
currents are used to model the microstrip patch and the volume polarization
currents are used to model the fields in the dielectric slab. It has been shown by
Newman and Tulyathan [18] how an integral equation is obtained for these
unknown currents and using the Method of Moments, these electric field integral
equations are converted into matrix equations which can then be solved by
various techniques of algebra to provide the result. A brief overview of the
Moment Method described by [5] and [18] is given below.
The basic form of the equation to be solved by the Method of Moment is:
F( g) = h (2.20)
where F is a known linear operator, g is an unknown function, and h is the
source or excitation function. The aim here is to find g , when F and h are
known.
The unknown function g can be expanded as a linear combination of N terms to
give:
(2.21) nnn
N
nn gagagagag
......................22111
where an is an unknown constant and gn is a known function usually called a
basis or expansion function. Substituting equation (2.21) in (2.20) and using the
linearity property of the operator F , we can write:
(2.22) hgFa n
N
nn
)(1
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Chapter 2 Microstrip Patch Antennas
The basis functions gn must be selected in such a way, that each F(gn) in the
above equation can be calculated. The unknown constants an cannot be
determined directly because there are N unknowns, but only one equation. One
method of finding these constants is the method of weighted residuals. In this
method, a set of trial solutions is established with one or more variable
parameters. The residuals are a measure of the difference between the trial
solution and the true solution. The variable parameters are selected in a way
which guarantees a best fit of the trial functions based on the minimization of
the residuals. This is done by defining a set of N weighting (or testing) functions
wm = w1,w2,………….,wN in the domain of the operator F . Taking the inner
product of these functions, equation (2.22) becomes:
(2.23)
hwgFwa mn
N
nmn ,)(,
1
where m = 1,2,..........,N.
Writing in Matrix form as shown in [5], we get:
][]][[ mnmn haF (2.24)
where
...
...
...
..............)(,)(,
..............)(,)(,
][2212
2111
gFwgFw
gFwgFw
Fmn
N
n
a
a
a
a
a
.
.3
2
1
hw
hw
hw
hw
h
N
m
,
.
.
,
,
,
3
2
1
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Chapter 2 Microstrip Patch Antennas
The unknown constants an can now be found using algebraic techniques such as
Gaussian elimination. It must be remembered that the weighting functions must
be selected appropriately so that elements of wn are not only linearly
independent but they also minimize the computations required to evaluate the
inner product. One such choice of the weighting functions may be to let the
weighting and the basis function be the same, that is, wn = gn . This is called as
the Galerkin’s Method.
From the antenna theory point of view, we can write the electric field integral
equation as:
E=fe(J) (2.25)
Where E is the known incident electric field.
J is the unknown induced current.
fe is the linear operator.
The first step in the moment method solution process would be to expand J as a
finite sum of basis function given as:
(2.26)
M
iii bJJ
1
where bi is the ith basis function and Ji is an unknown coefficient. The second
step involves the defining of a set of M linearly independent weighting
functions, wj. Taking the inner product on both sides and substituting equation
(2.26) in equation (2.25) we get:
),(,,1
iie
M
ijj bJfwEw (2.27)
where j =1,2,……….M
Writing in Matrix form as,
][]][[ jij EJZ (2.28)
Zagazig University- Electronics & Comm. Eng. Dept. 31
Chapter 2 Microstrip Patch Antennas
Zagazig University- Electronics & Comm. Eng. Dept. 32
)(, iejij bfwZwhere
HwE jj ,
J is the current vector containing the unknown quantities.
The vector E contains the known incident field quantities and the terms of the Z
matrix are functions of geometry. The unknown coefficients of the induced
current are the terms of the J vector. Using any of the algebraic schemes
mentioned earlier, these equations can be solved to give the current and then the
other parameters such as the scattered electric and magnetic fields can be
calculated directly from the induced currents. Thus, the Moment Method has
been briefly explained for use in antenna problems. The Moment Method has
been implemented in some commercial codes. Typical examples of these are
IE3D from Zeland Software, Ensemble from Ansoft and FEKO from EM
Software and Systems. For the modelling of surfaces, IE3D uses basis functions
with both rectangular and triangular support, while Ensemble and FEKO only
use basis functions with triangular support.
.
2.5.2.2 The Finite-Element Method
The FEM is widely used in structural mechanics and thermodynamics. It
was introduced to the electromagnetic community towards the end of the 1960s.
Since then, great progress has been made in terms of its application to
electromagnetic problems [19]. As is the case with the MoM, the FEM is also
mostly applied in the frequency domain. What makes the FEM very attractive, is
its inherent ability to handle inhomogeneous media.
When using the FEM for electromagnetic problems, the electric field is the
unknown variable that has to be solved for. The method is implemented by
discretising the entire volume over which the electric field exists, together with
its bounding surface, into small elements. Triangular elements are typically used
on surfaces, while tetrahedrons can be used for the volumetric elements. Simple
Chapter 2 Microstrip Patch Antennas
linear or higher-order functions on the nodes, along the edges or on the faces of
the elements, are used to model the electric field. For antenna problems, the
volume over which the electric field exists, will have one boundary on the
antenna and another boundary some distance away from the antenna. The latter
boundary is an absorbing boundary, which is needed to truncate the volume.
One viewpoint from which the FEM can be derived, is that of variational
analysis . This method starts with the partial differential equation (PDE) form of
Maxwell's equations and finds a variational functional for which the minimum
(or extremal point) corresponds with the solution of the PDE, subject to the
boundary conditions. An example of such a functional is the energy functional,
which is an expression describing all the energy associated with the
configuration being analyzed, in terms of the electric field. After the boundary
conditions have been enforced, a matrix equation is obtained. This equation can
then be solved to yield the amplitudes that are associated with the functions on
the elements used to model the electric field. The matrix associated with the
FEM, is a sparse matrix due to the fact that every element only interacts with the
elements in its own neighborhood. Other parameters, such as the magnetic field,
induced currents and power loss, can be obtained from the electric field. The
major advantage of the FEM is that the electrical and geometrical properties of
each element can be defined independently. Therefore, very complicated
geometries and inhomogeneous materials can be treated with relative ease. This
implies that the analysis of microstrip antennas with finite ground planes and
layers is also possible. However, the FEM has a few weak points when
compared to methods such as the MoM. The fact that the entire volume between
the antenna surface and the absorbing boundary has to be discretised, makes the
FEM very inefficient for the analysis of highly conducting radiators. Also, for
large three-dimensional structures, the generation of the mesh, into which the
problem is discretised, can become very complex and time-consuming. The
FEM is usually not the preferred method for the analysis of most antenna
Zagazig University- Electronics & Comm. Eng. Dept. 33
Chapter 2 Microstrip Patch Antennas
problems, but is frequently used for the simulation of microwave devices and
eigenvalue problems. An interesting approach is where the FEM is hybridized
with the MoM. These methods are very useful for the analysis of microstrip
antennas inside cavities [20]. Like most other full-wave modelling techniques,
the FEM has been implemented in a few commercial codes. A typical example
is HFSS from Ansoft.
2.5.2.3 Finite-Difference Time-Domain Method
The FDTD method which was introduced by Yee [21] in 1966, is also
very well suited for the analysis of problems that contain inhomogeneous media.
However, unlike the MoM and the FEM, the FDTD method is a time-domain
method and is not restricted to a single frequency at any one time. As compared
to the MoM and the FEM, the FDTD method is much easier to implement as it
makes limited demands on higher mathematics [22].
The FDTD method is also a PDE-based method. However, unlike the FEM, it
does not make use of variational analysis, but directly approximates the space-
and time-differential operators in Maxwell's time-dependant curl equations with
central-difference schemes. This is facilitated by modelling the region of interest
with two spatially interleaved grids of discrete points.
One grid contains the points at which the electric field is evaluated, while the
other grid contains the points at which the magnetic field is evaluated. A time-
stepping procedure is used where the electric and magnetic fields are calculated
alternatively. The field values at the next time step are calculated by using those
at the current and previous time steps. In such a way, the fields are then
effectively propagated throughout the grid. The time stepping is continued until
a steady-state solution is obtained. The source that drives the problem is of
course also some time-dependant function. Frequency-domain results can be
obtained by applying a discrete Fourier transform to the time-domain results.
Zagazig University- Electronics & Comm. Eng. Dept. 34
Chapter 2 Microstrip Patch Antennas
Zagazig University- Electronics & Comm. Eng. Dept. 35
Unlike the MoM and the FEM, no system of linear equations has to be solved
and therefore no matrix has to be stored. As with the FEM, the grid has to be
terminated with an absorbing boundary. The FDTD method has been
implemented in a few commercial codes. Typical examples of these are Fidelity
from Zeland Software and XFDTD from Remcom. For more detailed
descriptions of the FDTD method, chapter 4 in this thesis can be consulted.
2.6 CONCLUDING REMARKS
From the previous discussion on analysis techniques, it is clear that the
approximate methods are inappropriate for the modelling of the new antenna
elements and also for antenna arrays that are based on these elements. This is
partly due to the thick multilayered substrate as well as the fact that accurate
coupling calculations between the various patches are crucial. As for the full-
wave methods, the MoM and FEM method are usually not one of the first
choices when it comes to the modelling of microstrip antennas. These
techniques are much difficult to implement because of their demands on higher
mathematics. The FDTD is far more efficient for such analysis. This method has
a number of attractive features, which include its relatively simple
implementation, its straightforward treatment of inhomogeneous materials, its
ability to generate wideband data from a single run and the fact that no system
of linear equations need to be solved. The analysis of microstrip antennas with
finite ground planes and layers is of course also possible.
Due to the above mentioned reasons, we shall adopt this technique for the
analysis of the new proposed microstrip patch antenna included in this thesis.
36
C H A P T E R 3
Bandwidth Enhancement Techniques
3.1 INTRODUCTORY REMARKS
During recent years, much effort has gone into bandwidth enhancement
techniques for microstrip antennas in general. As such, there is a great amount
of information in the open literature and it covers a very broad range of solutions
that have been proposed thus far [1]. In this chapter, a broad overview will be
given in terms of the various techniques that are currently available to enhance
the bandwidth of patch antennas. The performance, advantages and disadvant-
ages of the most practical approaches will also be discussed. With these in mind,
the new antenna element that forms the basis of this study, will be presented.
In this chapter, Section 3.2 gives an overview of the various definitions
associated with the bandwidth of microstrip patch antenna, Section 3.3 gives an
overview of the bandwidth enhancement techniques especially the new trends,
while Section 3.4 presents the new compact wideband microstrip patch antenna,
employing overlapped patches with coaxial probe feed, this antenna has been
designed, fabricated, and measured.
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 37
3.2 BANDWIDTH DEFINITIONS
Any antenna has a number of associated characteristics, such as input
voltage standing wave ratio (VSWR), beamwidth, sidelobe level, gain, etc. Each
of these characteristics in turn may vary with frequency. If a maximum or
minimum level for any of these is specified, various definitions of bandwidth are
obtained.
3.2.1 Impedance Bandwidth
The impedance variation with frequency of the antenna element results in
a limitation of the frequency range over which the element can be matched to its
feed line.. A more meaningful definition of the fractional bandwidth is over a
band of frequencies where the VSWR at the input terminals is equal to or less
than a desired maximum value (typically less than 2.0), assuming that the
VSWR is unity at the design frequency. This bandwidth is given by K. R.
Carver [23] to be
VSWRQ
VSWR
f
fBW
t
1
Where BW is the impedance bandwidth of the antenna.
Δf is the difference between the two limits of the operating band.
fo is the center frequency (average value of the two limits)
Qt is the total quality factor.
3.2.2 Pattern Bandwidth
The beam-width, side-lobe levels and gain of an antenna all vary with
frequency [24]. If any of these quantities is specified as a minimum or
maximum, the operating frequency range is in turn determined.
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 38
3.2.3 Polarization or Axial Ratio Bandwidth
The polarization properties (linear or circular) of an antenna are usually
preferred to be fixed with frequency. Specifying a maximum cross-polar axial
ratio level can be used to find this bandwidth.
3.3 BANDWIDTH ENHANCEMENT TECHNIQUES
The impedance bandwidth of microstrip patch antennas is usually much
smaller than the pattern bandwidth [25]. This discussion on bandwidth enhance-
ment techniques will therefore focus on input impedance rather than radiation
patterns. There are a number of ways in which the impedance bandwidth of
probe-fed microstrip patch antennas can be enhanced. According to Wong [1],
the various bandwidth-enhancement techniques can be categorized into three
broad approaches:
Impedance matching.
The use of multiple resonances.
The use of lossy materials.
For the purpose of this overview, it has been decided to rather categorize the
different approaches in terms of the antenna structures that are normally used.
These include:
Wideband impedance-matching networks.
Edge-coupled patches.
Stacked patches.
Shaped probes.
Capacitive coupling and slotted patches.
Capacitive feed probes.
Overlapped patches which represents the new antenna element that
forms the basis of this study.
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 39
In terms of Wong 's categories, all these approaches can be identified as making
use of either impedance matching or multiple resonances. In practice, lossy
materials are not frequently used as it limits the radiation efficiency of the
antenna. It will therefore not be considered here.
3.3.1 Wideband Impedance-Matching Networks
One of the most direct ways to improve the impedance bandwidth of
probe-fed microstrip antennas, without altering the antenna element itself, is to
use a reactive matching network that compensates for the rapid frequency
variations of the input impedance. As shown in Figure 3.1, this can typically be
implemented in microstrip form below the ground plane of the antenna element.
The method is not restricted to antenna elements on either thin or a thick
substrates, but the thick substrate will of course add some extra bandwidth.
Pues and Van de Capelle [26] implemented the method by modelling the
antenna as a simple resonant circuit. A procedure, similar to the design of a
bandpass filter, is then used to synthesize the matching network. With this
approach, they have managed to increase the bandwidth from 4.2% to 12% for a
voltage standing-wave ratio (VSWR) of 2:1. Subsequently to that, C.
Nauwelaers [27] used the simplified real frequency technique in order to design
the matching network for a probe-fed microstrip patch antenna. They have
managed to increase the bandwidth of one antenna element from 5.7% to 11.1%
for a VSWR of 1.5:1, and that of another from 9.4% to 16.8% for a VSWR of
2:1. Recently, V. Gupta and S. Sinha [28] have shown how a dielectric-resonator
loaded suspended microstrip patch antenna can increase the bandwidth from
3.2% to 18% for a VSWR of 1.5:1.
The advantages of using impedance-matching networks are that the antenna
elements do not get altered and that the matching network can be placed behind
the antenna's ground plane. As such, the radiation characteristics of the antenna
element stay unchanged, while radiation from the matching network is also
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 40
Ground Plane
Substrates
Matching network below ground plane
minimized. The drawback of this method is that the matching network can
potentially take up space that is very limited when microstrip feed networks are
used to excite the individual elements in an antenna array. Another drawback is
that, for single-element antennas, more than one substrate layer is required to
support the antenna element and the matching network.
3.3.2 Edge-Coupled Patches
The basic idea behind edge-coupled patches, is to increase the impedance
bandwidth of a microstrip patch through the introduction of additional resonant
patches. By doing so, a few closely-spaced resonances can be created. Only one
of the elements is driven directly. The other patches are coupled through
proximity effects. An example of such an arrangement is shown in Figure 3.2.
This approach has been investigated by Kumar and Gupta [17]. The parasitic
patches can be coupled to either the radiating edges, the non-radiating edges or
to both pairs of edges. The approach in [17] uses short transmission lines to
couple the parasitic patches directly to the driven patch. With the edge-coupled
approach, impedance bandwidths of up to 25.8% have been obtained for a
VSWR of 2:1. This was achieved with four parasitic patches coupled to the
driven patch.
Resonant patch
Fig. 3.1 Geometry of a probe-fed microstrip patch antenna with a wideband impedance-matching network.
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 41
The advantages of the edge-coupled approach include the fact that the structure
is coplanar in nature and that it can be fabricated on a single-layer substrate.
However, this approach also has a few drawbacks. Due to the fact that the
different patches radiate with different amplitudes and phases at different
frequencies, the radiation patterns change significantly over the operating
frequencies. The enlarged size of the structure can also be a potential handicap
in many applications. For example, in phased-array applications, the large
separation distances between elements can introduce grating lobes.
3.3.3 Stacked Patches
A very popular technique, which is often used to increase the impedance
bandwidth of microstrip patch antennas, is to stack two or more resonant patches
on top of each other [1]. As with the edge-coupled resonators, this technique
also relies on closely-spaced multiple resonances. However, in this case, the
elements take up less surface area due to the fact that they are not arranged in a
coplanar configuration. Figure 3.3 shows the geometry of such an antenna
element where the bottom patch is driven by a microstrip line and the top patch,
Parasitic patch
ProbeGround Plane
Substrate
Figure 3.2 Geometry of a probe-fed microstrip patch element that is edge-coupled to the parasitic patches.
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 42
which is located on a different substrate layer, is proximity-coupled to the
bottom one. In practice, the patches are usually very close in size so that the
generation of two distinct resonances can be avoided. Different shapes of
patches can be used. These commonly include rectangular patches [29], circular
patches [30] and annular-ring patches [31]. Waterhouse [29] reported a 26% 10
dB return-loss bandwidth for rectangular patches, Mitchell [30] reported a 33%
10 dB return-loss bandwidth for circular patches, while Kokotoff [31] reported a
22% 10 dB return-loss bandwidth for annular-ring patches.
These bandwidths were all obtained for two patches stacked on top of each
other. It is possible to stack more patches, but the performance may not be much
better than with only two patches [1]. Instead of aligning the patches vertically,
some researchers have also used a horizontal offset between the patches [32].
However, due to the structural asymmetry, these configurations exhibit beam
dispersion.
The stacked-patch configuration has a number of advantages over the edge-
coupled configuration. Since it does not increase the surface area of the element,
it can be used in array configurations without the danger of creating grating
lobes. Its radiation patterns and phase centre also remains relatively constant
over the operating frequency band. It has a large number of parameters that can
be used for optimization. However, due to this, the design and optimization
Substrate 2 Bottom patch
Substrate 1
Microstrip Line
Top patch
Ground plane
Fig. 3.3 Geometry of a probe-fed stacked microstrip patch antenna.
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 43
process can also be very complex. Another drawback of stacked patches, is that
it requires more than one substrate layer to support the patches.
3.3.4 Shaped Probes
As was shown in Chapter 1, a thick substrate can be used to enhance the
impedance bandwidth of microstrip patch antennas. However, the input
impedance of probe-fed microstrip patch antennas become more inductive as the
substrate thickness is increased. In order to offset this inductance, some
capacitance is needed in the antenna's feeding structure. One way to implement
such a capacitive feed is to alter the shape of the probe. There are basically two
approaches. In one approach, the probe is connected directly to the patch [33],
while in the other approach, the probe is not connected to the patch at all [34].
The direct feed can be implemented as shown in Figure 3.4(a), where the
feeding structure consists of a stepped probe. The horizontal part of the probe
couples capacitively to the patch. Chen and Chia [33] reported an impedance
bandwidth of 19.5% for a VSWR of 2:1. Another option is to add a stub to one
of the radiating edges of the patch and to feed the stub directly with a probe. For
such an approach, Chen and Chia [35] reported an impedance bandwidth of
25%, once again for a VSWR of 2:1.
The proximity-coupled probe is implemented as shown in Figure 3.4(b), where
the probe is bent into a L-shape. The horizontal part of the probe runs
underneath the patch and also couples capacitively to it. This solution has been
implemented for a variety of patch shapes. Mak et al. reported an impedance
bandwidth of 36% for a rectangular patch in [36] and 42% for a triangular patch
in [37], while Guo et al. reported an impedance bandwidth of 27% for an
annular-ring patch in [38]. These bandwidth figures were all quoted for a VSWR
of 2:1. Instead of a L-shaped probe. A microstrip patch antenna with a shaped
probe, be it directly driven or not, can usually be supported on a single substrate
layer. This makes it extremely suitable for antenna arrays where cost has to be
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 44
Substrate 1patch
minimized. Most of these elements have radiation patterns with a slight squint in
the E-plane and slightly high cross-polarization levels in the H-plane. These are
characteristics of probe-fed microstrip patch antennas on thick substrates. The
stepped probe, though, exhibits somewhat lower cross-polarization levels. The
patches that are directly driven should be more robust that those with the
proximity coupled probes. For the latter ones, care has to be taken with respect
to the proper alignment of the paths and probes. Another advantage of both
approaches is that, since they do not increase the surface area of the element,
they can be used in array configurations without the danger of creating grating
lobes.
Substrate 1
Ground plane
Substrate 2
Fig. 3.4 Geometries of microstrip patch antennas with shaped probes. (a) Stepped probe. (b) L- shaped probe.
Ground plane
Substrate 2
patch
Stepped probe
L - probe
(a)
(b)
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 45
3.3.5 Capacitive Coupling and Slotted Patches
There are two alternative approaches that can also be used to overcome
the inductive nature of the input impedance associated with a probe-fed patch on
a thick substrate. These are capacitive coupling or the use of slots within the
surface of the patch element. Examples of such approaches are shown in Figures
3.5(a) and (b) respectively. It can be argued that these two approaches are
structurally quite similar. The approach in Figure 3.5(a) has a small probe-fed
capacitor patch, which is situated below the resonant patch [39]. The gap
between them acts as a series capacitor. Similarly, the annular slot in Figure
3.5(b) separates the patch into a small probe-fed capacitor patch and a resonant
patch. In this case, the slot also acts as a series capacitor. In principle, both of
these approaches employ some sort of capacitive coupling and are functionally
also, to some degree, equivalent to the L-probe and T -probe as described in
Section 3.3.4.
Liu and kooi [40] combined the capacitively-coupled feed probe with stacked
patches and reported a impedance bandwidth of 25.7% for a VSWR of 2:1. To
achieve this, they used two stacked patches with a small probe-fed patch below
the bottom resonant patch. In another approach, Gonzalez [20] placed a resonant
patch, together with the small probe-fed capacitor patch just below it, into a
metallic cavity. With this configuration, they managed to obtain a impedance
bandwidth of 35.3% for a VSWR of 2:1. Chen and Chia [41] used a small
rectangular probe-fed capacitor patch, located within a notch that was cut into
the surface of the resonant patch. They managed to obtain an impedance
bandwidth of 36% for a VSWR of 2:1.
Some authors also used a rectangular resonant patch with a U-slot in its surface.
The metallic area inside the slot is then driven directly with a probe. Here, Tong
[42] reported a impedance bandwidth of 27% for a VSWR of 2:1, while
Weigand [9] reported an impedance bandwidth of 39 %, also for a VSWR
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 46
of 2:1. In yet another approach, Nie and Chew [43] placed a circular probe-fed
patch within a annular-ring patch, with the circular patch exciting higher-order
modes on the annular-ring patch.
They managed to obtain a 8 dB return-loss bandwidth of 20%. Kokotoff [44]
placed a small shorted circular probe-fed patch within a annular-ring patch, but
with the circular patch exciting the dominant TM11 mode on the annular-ring
patch. They reported a 10 dB return-loss of 6.6%.
The advantage of the approach where the capacitor patch is located below the
resonant patch, is that the cross-polarization levels in the H-plane are lower than
what can be achieved with the approach where the capacitor patch is located
(b)
Ground plane
Substrate 1
Substrate 1
Substrate 2
Fig. 3.5 Geometries of probe-fed microstrip patch antennas where capacitive coupling and slots are used. (a) Capacitive coupling. (b) Annular slot in the surface of the patch.
Ground plane
Substrate 2
Resonant patch
Capacitor patch
probe
(a)
Resonant patch
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 47
within the surface of the resonant patch. However, in order to support the
capacitor patch below the resonant patch, an additional substrate layer might
be required. In contrast, only one substrate layer is required to support the
configuration where the capacitor patch is located inside the surface of the
resonant patch. Furthermore, the capacitor patch below the resonant patch is
prone to alignment errors and can complicate the fabrication process. On the
other hand, when using a capacitor patch within the surface area of a resonant
patch, there can potentially be many design parameters that can complicate the
design of such antenna elements. Here also, an advantage of both approaches is
that, since they do not increase the surface area of the element, they can be used
in array configurations without the danger of creating grating lobes.
3.3.6 Capacitive Feed Probes
There is another approach that can also be used to overcome the inductive
nature of the input impedance associated with a probe-fed patch on a thick
substrate. This is the use of a microstrip patch antenna element with a capacitive
feed probe. Figure 3.6 shows the general geometry of the antenna structure. As
can be seen, it consists of a rectangular resonant patch with a small probe-fed
capacitor patch right next to it. Both patches reside on the same substrate layer.
Both circular and rectangular capacitor patches, as shown in Figures 3.6(a) and
(b) respectively, can be used. G. Mayhew [2] showed that, for a rectangular
resonant patch with a small probe-fed rectangular capacitor patch , a 10 dB
return loss bandwidth of 26.4% could be obtained.
For wideband applications, the two patches can be manufactured on a thin
substrate with a thick low-loss substrate, such as air, right below it. The antenna
element is functionally very similar to most other capacitively-coupled elements.
The gap between the resonant patch and the capacitor patch acts as a series
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 48
thereby offsetting the inductance of the long probe. Once the size of the resonant
patch and the thickness of the substrate have been fixed for a certain operating
frequency and impedance bandwidth, there are basically two parameters that can
be used to control the input impedance of the antenna element. These are the
size of the capacitor patch and the size of the gap between the two patches.
The structural properties of this antenna element can be viewed in
context. First of all, this antenna element can be manufactured on a single
substrate layer due to both the resonant patch and the capacitor patch residing on
the same layer. This is very important for large antenna arrays where lamination
Substrate 1
Substrate 1
Ground plane
Substrate 2
Fig. 3.6 Geometries of the microstrip patch antennas employing capacitive feed probes. (a) Circular capacitor patch. (b) Rectangular capacitor patch.
Ground plane
Substrate 2
Resonant patch
Capacitor patch
probe
(a)
(b)
Resonant patch
probe
Capacitor patch
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 49
can be very expensive. The fact that the capacitor patch is driven directly by a
probe, gives the structure some rigidity. The structure is also less prone to
alignment errors, which can be a factor of merit for antenna elements where the
probe does not make physical contact with any of the resonant patches or where
the capacitor patch is located on a different layer than the resonant patch. The
surface area of the element is
not much larger than that of a resonant patch and therefore it is very suitable for
use within antenna arrays. An advantage that might not be very obvious at first,
is that the antenna element, as opposed to slotted antenna elements, consists of
parts that are regular in shape, it has huge benefits for the analysis of such
antennas, especially for large antenna arrays. Finally, the design of such an
antenna element, as well as tuning of the input impedance, is very
straightforward due to the few parameters that have to be adjusted.
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 50
3.4 NEW COMPACT WIDEBAND OVERLAPPED PATCHES MICROSTRIP ANTENNAS
As was stated earlier on, the basis of this study is a new compact
wideband microstrip patch antenna element [3]. Figure 3.7 shows the general
geometry of the new antenna structure. This antenna has been designed,
fabricated, and measured.
For a conventional rectangular microstrip patch antenna of length L and width
W, the resonance frequency for any TMmn mode is given by James and Hall
[6] to be dependent on the length L, the width W, and the effective dielectric
constant of the substrate. But for the dominant TM10 mode, the resonance
frequency is only dependent on the length L, and the effective dielectric
constant.
W2
W2
W1
W3
W3
W1
W3
W1 W2 W3
W1
S3
S1
S1
S3
Fig. 3.7 Geometry of the multi-resonance wideband patch.
W2
Slot
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 51
Therefore it is clear that the resonance frequency of the rectangular microstrip
patch antenna is a function of its length (L), so if the microstrip patch antenna
has multiple lengths it will be multi-resonance antenna i.e. for every different
length there will be a different resonance frequency, hence the bandwidth of the
microstrip patch antenna can be enhanced. This technique is utilized in the
design of the new microstrip patch antenna.
In this study the bandwidth of a single layer microstrip patch antenna is
enhanced by using multi-resonance technique without significantly enlarging the
size of the proposed antenna. Multiple resonances are achieved by overlapping
three square patches of different dimensions along their diagonals to form a non-
regular single patch as shown in Figure 3.7, a slot is incorporated into this
complex patch to expand its bandwidth.
For handheld wireless systems, a compact single patch on moderately
thick substrate is preferred. For such antenna, achieving more than 25 percent
bandwidth and moderate gain presents a challenge [45]. A 10 dB return loss
bandwidth of 56.8 % has been obtained in this design. This antenna has been
designed, fabricated, and measured. A finite difference time domain (FDTD)
method full wave simulator FIDELITY is used to simulate this antenna. The
obtained results have been compared to the experimental results and good
εr= 2.35
Slot
Probe Feed
h = 3.175 mm
17.1 mm
Y
XGround plane
( Xf , Yf )
Fig. 3.8 The slotted overlapped patches microstrip antenna
Chapter 3 Bandwidth Enhancement Techniques
Zagazig University- Electronics & Comm. Eng. Dept. 52
agreements have been found. This antenna provide stable far field radiation
characteristics in the entire operating band with relatively high gain.
This antenna is fed by a coaxial probe at position ( Xf , Yf ) as shown in Fig.
3.8. The probe feed location and its radius were adjusted in such a way that one
can obtain satisfactory performance. For more detailed descriptions of this new
antenna, chapter 6 in this thesis can be consulted.
3.5 CONCLUDING REMARKS
This chapter presented a broad overview of several approaches that can be
used to enhance the impedance bandwidth of microstrip patch antennas. The
new antenna element, which forms the basis of this study, has also been
introduced, The new antenna element makes use of overlapping three square
patches of different dimensions along their diagonals to form a non-regular
single patch then a slot is incorporated into this complex patch to expand its
bandwidth. This antenna has been designed, fabricated, and measured.
The new antenna has several advantages: This antenna can be easily
fabricated on a single-layer and relatively thin substrate for applications in hand-
held devices. It has been shown that this antenna can easily be used in other
frequency bands with different substrate materials. It achieves 56.8 percent
bandwidth for return loss < -10 dB.
53
C H A P T E R 4
The Finite-Difference Time-Domain Method
4.1 INTRODUCTORY REMARKS
In this chapter, the foundations of the Finite-Difference Time-Domain
(FDTD) electromagnetic field analysis, and the algorithm introduced by Kane
Yee [21] in 1966 to implement the method are outlined. This chapter is
developed to demonstrate how to do three-dimensional electromagnetic
simulation using the finite-difference time-domain (FDTD) method. All the
results in this chapter were obtained using the C programming language and
these results have been compared with the published data and good agreements
have been found.
This chapter is arranged in a graded increase in complexity. Every section
attempts to address an additional level of complexity. The text increases in
complexity in two major ways:
Dimension of Simulation Types of Materials
One-dimensional Free space
Two-dimensional Lossless Dielectric material
Three-dimensional Lossy dielectric material
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 54
4.2 ONE-DIMENSIONAL SIMULATION WITH THE FDTD METHOD
This section is a step-by-step introduction to the FDTD method. It begins
with the simplest possible problem, the simulation of a pulse propagating in free
space in one-dimension. This example is used to illustrate the FDTD
formulation. Subsequent sections lead to more complicated media [46].
4.2.1 One-Dimensional Free Space Formulation
The time-dependent Maxwell’s curl equations in free space are:
1
Ht
E
o
(4.1a)
1
Et
H
o
(4.1b)
Where E : The electric field vector in V/m.
H : The magnetic field vector in A/m.
μo : The magnetic permeability of free space in H/m.
εo : The electric permittivity of free space in F/m.
E and H are vectors in three dimensions, so in general, Eqs. (4.1a) and (4.1b)
represents three equations each. Starting with a simple one-dimensional case
using only Ex and Hy, so Eqs. (4.1a) and (4.1b) become:
1
z
H
t
E y
o
x
(4.2a)
1
z
E
t
Hx
o
y
(4.2b)
These are the equations of a plane wave with the electric field oriented in the x
direction, the magnetic field oriented in the y direction, and traveling in the z
direction.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 55
Taking the central difference approximations for both the temporal and spatial
derivatives gives:
)2/1()2/1(1)()( 2/12/1
z
kHkH
t
kEkEny
ny
o
nx
nx
(4.3a)
)()1(1)2/1()2/1( 2/12/11
z
kEkE
t
kHkH nx
nx
o
ny
ny
(4.3b)
In these two equations, time is specified by the superscript, i.e., “n” actually
means a time t = n. ∆t. It is noticed that every thing is discretized for formulation
into the computer program. The term “n+1” means one time step later. The term
in parentheses represent distance, i.e., “k” actually means the distance z = k . ∆z.
The formulation of Eqs. (4.3a) and (4.3b) assumes that the E and H fields are
interleaved in both space and time. H uses the arguments k+1/2 and k-1/2 to
indicate that the H field values are assumed to be located between the E field
values. This is illustrated in Figure 4.1. Similarly, the n+1/2 or n-1/2 superscript
indicates that it occurs slightly after or before n, respectively.
2/1nxE
K+2+1/2 k-1-1/2 K+1+1/2 K+1/2k-1/2
nyH
k-2 k-1 k K+1 K+2
2/1nxE
k-2 k-1 k K+1 K+2
Fig. 4.1 Interleaving the E and H fields in space and time in the FDTD formulation.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 56
Figure 4.1 shows that the E and H fields are interleaved in space and time in the
FDTD formulation. To calculate Hy(k+1/2), for instance, the neighboring values
of Ex at k and k+1 are needed. Similarly, to calculate Ex(k), the value of Hy at
k-1/2 and k+1/2 are needed.
Eqs. (4.3a) and (4.3b) can be arranged in an iterative algorithm:
)2/1()2/1()()( 2/12/1
kHkHz
tkEkE n
yny
o
nx
nx
(4.4a)
)()1()2/1()2/1( 2/12/11 kEkEz
tkHkH n
xnx
o
ny
ny
(4.4b)
Notice that the calculations are interleaved in both space and time. In
Eq.(4.4a) for example, the new value of Ex is calculated from the previous value
of Ex and the most recent values of Hy. This is the fundamental paradigm of the
finite-difference time-domain (FDTD) method [47].
Eqs. (4.4a) and (4.4b) are very similar, but because εo and μo differ by several
orders of magnitude, Ex and Hy will differ by several orders of magnitude. This
is circumvented by making the following change of variables [48]:
(4.5) ~
EE
This is a system called Gaussian units, which is frequently used by
physicists. The reason for using it here is simplicity in the formulations. The E
field and the H field have the same order of magnitude. This has an advantage in
formulating the perfectly matched layer (PML) which is a crucial part of FDTD
simulation. Substituting Eq. (4.5) into Eqs. (4.4a) and (4.4b) gives:
)2/1()2/1()(~
)(~ 2/12/1 kHkH
z
tkEkE n
yny
oo
nx
nx
(4.6a)
)(~
)1(~
)2/1()2/1( 2/12/11 kEkEz
tkHkH n
xnx
oo
ny
ny
(4.6b)
Once the cell size ∆z is chosen, then the time step ∆t is determined by:
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 57
(4.7) . 2 oc
zt
Where Co is the speed of light in free space. (The reason for this will be
explained later.) Therefore.
(4.8) 5.0.2/1
z
czc
z
t oo
oo
Rewriting Eqs. (4.6a) and (4.6b) in C computer code gives the following:
ex [k] = ex [k] + 0.5 * ( hy [k-1] - hy [k] ) (4.9a)
hy [k] = hy [k] + 0.5 * ( ex [k] - ex [k+1] ) (4.9b)
Note that the n or n+1/2 or n-1/2 in the superscripts is gone. Time is implicit in
the FDTD method. In Eq. (4.9a), the ex on the right side of the equal sign is the
previous value at n-1/2 and the ex on the left side is the new value at n+1/2,
which is being calculated.
Position, however is explicit. The only difference is that k+1/2 and k-1/2 are
rounded off to k and k-1 in order to specify a position in an array in the program.
The first problem to be considered is a simple one-dimensional FDTD
simulation. It generates a Gaussian pulse with unit amplitude in the center of the
problem space as shown in Figure 4.2. This pulse is given by,
2
)(
w
tt o
etpulse .
Where to is the time delay of the pulse.
W is the pulse half width
and the pulse propagate away in both directions as seen in Figure 4.2. The Ex
field is positive in both directions, but Hy is negative in the negative direction.
The following things are worth noting about the program:
1. The Ex and Hy values are calculated by separate loops, and they employ
the interleaving described above.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 58
2. After the Ex values are calculated, the source is calculated. This is done
by simply specifying a value of Ex at the point k = 100, and overriding
what was previously calculated. This is referred to as “hard source”
because a specific value is imposed on the FDTD grid.
0 20 40 60 80 100 120 140 160 180 200-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
FDTD cells
EX
T=100 1D free space
0 20 40 60 80 100 120 140 160 180 200-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
FDTD cells
HY
T=100 1D free space
Fig. 4.2 FDTD simulation of a pulse in free space after 100 time steps. The pulse originated in the center and travels outward.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 59
4.2.2 Stability and The FDTD Method
In this part the determination of the time step is demonstrated. An
electromagnetic wave propagating in free space cannot go faster than the speed
of light . To propagate a distance of one cell requires a minimum time of
∆t = ∆z / co . When dealing with two-dimensional simulation, the propagation
has to be allowed in the diagonal direction, which brings the time requirement to
∆t = ∆z / ( 2 co). Obviously, three-dimensional simulation requires ∆t = ∆z /
( 3 co). This is summarized by the well-known “Courant Condition” [22],[49]:
. ocn
zt
(4.10)
Where n is the dimension of the simulation. Unless otherwise specified. For
simplicity ∆t for any dimension of simulation may be determined by:
. 2 oc
zt
(4.11)
4.2.3 The Absorbing Boundary Condition in One Dimension
Absorbing boundary conditions are necessary to keep outgoing E and H
fields from being reflected back into the problem space. Normally, in calculating
the E field, the surrounding H values are needed; this is a fundamental
assumption of the FDTD method. At the edge of the problem space the value to
one side will be unknown. However, there is an advantage because it is known
that there are no sources outside the problem space. Therefore, the fields at the
edge must be propagating outward. These two facts will be used to estimate the
value at the end by using the value next to it [50].
Suppose it is required to get a boundary condition at the end where k = 0. If a
wave is going toward a boundary in free space, it is traveling at co, the speed of
light. So in one time step of the FDTD algorithm, it travels a distance given by:
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 60
22c.cdistance oo
z
c
zt
o
This equation basically explains that it takes two time steps for a wave front to
cross one cell. So a common sense approach tells us that an acceptable boundary
condition might be:
)1()0( 2 nx
nx EE (4.12)
It is relatively easy to implement this. Simply store a value of Ex (1) or two time
steps, and then put it in Ex (0). Boundary conditions such as these have been
implemented at both ends of the Ex array. Figure 4.3 shows the results of this
simulation. A pulse that originates in the center and propagates outward and is
absorbed without reflecting anything back into the problem space.
4.2.4 Determining Cell Size
Choosing the cell size to be used in an FDTD formulation is similar to
any approximation procedure: enough sampling points must be taken to ensure
that an adequate representation is made. The number of points per wavelength is
dependent on many factors [22], [49]. However, a good rule of thumb is 10
points per wavelength. Experience has shown this to be adequate, with
inaccuracies appearing as soon as the sampling drops below this rate.
Naturally, the worst-case scenario must be used. In general, this will involve
looking at the highest frequencies being simulated and determining the
corresponding wavelength. For instance, suppose a simulation is being run at
400 MHz. In free space, EM energy will propagate at the wavelength
m 75.0sec 104
m/sec 103
MHz 400 1-8
8
oo
c
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 61
Fig. 4.3 Simulation of an FDTD program with absorbing boundary
conditions. Notice that the pulse is absorbed at the edges
without reflecting any thing back.
0 20 40 60 80 100 120 140 160 180 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FDTD cells
T = 225
EX
ABC
Ex
Ex
0 20 40 60 80 100 120 140 160 180 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
EX
T = 100ABC
FDTD cells
0 20 40 60 80 100 120 140 160 180 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
FDTD cells
T=250 ABC
EX
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 62
If the simulation was only in free space one could choose
∆z = λo / 10 = 7.5 cm.
However, for simulating EM propagation in biological tissues, for
instance, it will be necessary to look at the wavelengths in the tissue with the
highest dielectric constant, because this will have the corresponding shortest
wavelength. For instance, muscle has a relative dielectric constant of about 50 at
400 MHz, so
cm 6.10sec 104
m/sec 10424.0
MHz 400
50/1-8
8
oo
c
and one would probably select a cell size of one centimeter.
4.2.5 Propagation in A Lossless Dielectric Medium
In order to simulate a lossless dielectric medium with a conductivity
equals to zero (s/m) and a relative dielectric constant other than one, which
corresponds to free space, the relative dielectric constant of the medium εr have
to be added to Maxwell ’s equations:
1
Ht
E
ro
(4.13a)
1
Et
H
o
(4.13b)
It is assumed that the medium being simulated is nonmagnetic i.e. μ = μo.
Staying with our one-dimensional example and make the change of variable in
Eq. (4.5) .
z
H
t
tE y
oor
x
.
1)(~
z
tE
t
tHx
oo
y
)(~
.1)(
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 63
and then go to the finite difference approximations:
(4.14a) )2/1()2/1(1)(
~)(
~ 2/12/1
z
kHkH
t
kEkEny
ny
oor
nx
nx
(4.14b) )(
~)1(
~1)2/1()2/1( 2/12/11
z
kEkE
t
kHkH nx
nx
oo
ny
ny
From the previous sections
5.01
z
t
oo
so Eq. (4.14) becomes
(4.15a) )2/1()2/1(5.0
)(~
)(~ 2/12/1 kHkHkEkE n
yny
r
nx
nx
(4.15b) )(~
)1(~
5.0)2/1()2/1( 2/12/11 kEkEkHkH nx
nx
ny
ny
Rewriting Eqs. (4.15a) and (4.15b) in C computer code gives the following:
ex [k] = ex [k] + cb [k] * ( hy [k-1] - hy [k] ) (4.16a)
hy [k] = hy [k] + 0.5 * ( ex [k] - ex [k+1] ) (4.16b)
Where
cb [k] = 0.5 / epsilon (4.17)
Over those values of k which specify the dielectric material.
Figure 4.4 shows the results of a program that simulates the interaction of
a pulse traveling in free space until it strikes a dielectric medium which is
located from cell number 100 to cell number 200 and having a relative dielectric
constant of 4.0. The medium is specified by the parameter cb in Eq. (4.17). Note
that a portion of the pulse propagates into the medium and a portion is reflected,
in keeping with basic EM theory.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 64
Fig. 4.4 Simulation of a pulse striking a dielectric material with a εr= 4 and
a conductivity of 0.0 (s/m). The source originates at cell number 5.
Cond. = 0.0
Cond. = 0.0
Cond. = 0.0
Cond. = 0.0
Ex
Ex
Ex
Ex
0 20 40 60 80 100 120 140 160 180 200-0.5
0
0.5
1
FDTD cells
EX
T=320 Eps=4
0 20 40 60 80 100 120 140 160 180 200-0.5
0
0.5
1
FDTD cells
Ex
T=100 Eps=4
0 20 40 60 80 100 120 140 160 180 200-0.5
0
0.5
1
T=220 Eps=4
FDTD cells
0 20 40 60 80 100 120 140 160 180 200-0.5
0
0.5
1
FDTD cells
T=440 Eps=4
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 65
4.2.6 Simulating Different Sources
Up till now, a Gaussian pulse has been used as the source. It is very easy
to switch to a sinusoidal source which can be written in C computer code as:
Source = sin [2 * pi * freq_in * (n*dt) ]
This source originates at cell number 5; the parameter freq_in determines the
frequency of the wave. Figure 4.5 shows the same dielectric medium problem
that was used in the previous section, but with a sinusoidal source. A frequency
of 700 MHz is used. The cell size ∆z may be chosen to be 0.01m then the value
of the time step dt is calculated from Eq. (4.7).
Fig. 4.5 Simulation of a propagating sinusoidal wave of 700 MHz striking
a medium with εr = 4 and a conductivity of 0.0 (s/m).
Cond. = 0.0
Cond. = 0.0
Ex
Ex
0 20 40 60 80 100 120 140 160 180 200-1.5
-1
-0.5
0
0.5
1
1.5
FDTD cells
EX
T=150 Eps=4.0
0 20 40 60 80 100 120 140 160 180 200-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
FDTD cells
T=425 Eps=4.0
EX
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 66
4.2.7 Propagation in a Lossy Dielectric Medium
So far, we have simulated EM propagation in free space or in a simple
media that are specified by the relative dielectric constant εr. However, there are
many media that also have a loss term specified by the conductivity. This loss
term results in the attenuation of the propagation energy.
Once more using the time-dependent Maxwell’s curl equations, but writing them
in a more general form. Which will allow us to simulate propagation in media
that have a specific conductivity:
(4.18a) JHt
E
(4.18b) 1
Et
H
o
J is the current density, which can also be written as:
EJ .
where σ is the conductivity. Putting this into Eq. (4.18a) and dividing thought by
the dielectric constant, it gives
EHt
E
roro
1
Now revert to our simple one-dimensional equation:
)()(1)(
tEz
tH
t
tEx
ro
y
ro
x
and make the change of variable in Eq. (4.5), which gives:
)(~)(
.1)(
~tE
z
tH
t
tEx
ro
y
oor
x
(4.19a)
)(
~1)(
z
tE
t
tHx
oo
y
(4.19b)
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 67
Next take the finite difference approximations for both the temporal and
the spatial derivatives similar to Eq. (4.3a):
)20.4( 2
)(~
)(~
)2/1()2/1(
. 1)(
~)(
~
2/12/1
2/12/1
kEkE
z
kHkH
t
kEkE
nx
nx
ro
ny
ny
oor
nx
nx
Notice that the last term in Eq. (4.19a) is approximated as the average across two
time steps in Eq. (1.20a). From the previous section
5.01
z
t
oo
so Eq. (1.20) becomes
)2/1()2/1(5.0
2
.1)(
~
2
.1)(
~ 2/12/1
kHkH
tkE
tkE
ny
ny
r
ro
nx
ro
nx
or
)2/1()2/1(
2
.1.
5.0
)(~
2
.1
2
.1
)(~ 2/12/1
kHkHt
kEt
t
kE
ny
ny
ror
nx
ro
ronx
Rewriting these equations in C computer code gives the following:
ex [k] = ca [k] * ex [k] + cb [k] * ( hy [k-1] - hy [k] )
hy [k] = hy [k] + 0.5 * ( ex [k] - ex [k+1] )
where
eaf = dt * sigma / (2 * espz * epsilon )
ca [k] = (1.0 - eaf) / (1.0 + eaf )
cb [k] = 0.5 / (epsilon * (1.0 + eaf ))
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 68
0 20 40 60 80 100 120 140 160 180 200-1.5
-1
-0.5
0
0.5
1
1.5
FDTD space
EX
T=500 Eps=4
Cond=.04
Figure 4.6 shows the results of a program that simulates a sinusoidal wave
hitting a lossy medium that has a dielectric constant of 4.0 and a conductivity of
0.04 s/m. The pulse is generated at the far left side and propagate to the right.
Notice that the waveform in the medium is absorbed before it hits the boundary,
so it is not necessary to worry about absorbing boundary conditions.
Figure 4.7 shows the Simulation of a propagating sinusoidal wave of 700
MHz striking a lossy dielectric medium with a dielectric constant of 4.0 and a
conductivity of 1.0 x 106 (s/m). The source originates at cell number 5. Notice
that the waveform in the medium is completely absorbed before it hits the
boundary due to the very large conductivity of the medium.
Fig. 4.6 Simulation of a propagating sinusoidal wave of 700 MHz striking a lossy dielectric medium with εr = 4and a conductivity of 0.04 (s/m). The source originates at cell number 5.
0 20 40 60 80 100 120 140 160 180 200-1.5
-1
-0.5
0
0.5
1
1.5
FDTD space
EX
T=500 Eps=4
Cond.=.04
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 69
4.2.8 Calculating The Frequency domain Output
Up till now, the output of the previous simulations has been the E field
itself, and it has been content to simply watch a pulse or sine wave propagates
through various media. Needless to say, before any such practical application
can be implemented, it will be necessary to quantify the results. Suppose now
that it is required to calculate the E field distribution at every point in a dielectric
medium subject to illumination at various frequencies. One approach would be
to use a sinusoidal source and iterate the FDTD program until it is observed that
a steady state has been reached, and determine the resulting amplitude and phase
at every point of interesting in the medium. This would work, but then this
process must be repeated for every frequency of interest. System theory tells us
Fig. 4.7 Simulation of a propagating sinusoidal wave of 700 MHz striking a lossy dielectric medium with εr = 4 and a
conductivity of 1.0 x106 (s/m). The source originates at cell number 5.
0 20 40 60 80 100 120 140 160 180 200-2
-1.5
-1
-0.5
0
0.5
1
1.5
FDTD space
EX
T=500 Eps=4 Cond=1.0e6
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 70
that the response to every frequency can be obtained if an impulse is used as the
source. one could go back to using the Gaussian pulse. Which, if it is narrow
enough, is a good approximation to an impulse. Then the FDTD program is
iterated until the pulse has died out, and take the Fourier transform of the E field
in the slab. If we have the Fourier transform of the E field at a point, then we
know the amplitude and phase of the E field that would result from illumination
by any sinusoidal source. This, also, has a very serious drawback: the E field for
all time domain data at every point of interest would have to be stored until the
FDTD program is thought iterating so the Fourier transform of the data could be
taken, presumably using a fast Fourier algorithm. This presents a logistical
nightmare.
Here is an alternative. Suppose it is required to calculate the Fourier transform of
the E field E(t) at a frequency f1. This can be done by the equation
.)()( 12
0
11 dtetEfE tfjt
(4.21)
Notice that the lower limit of the integral is 0 because the FDTD program
assumes all causal functions. The upper limit is t , the time at which the FDTD
iteration is halted.
)..()(0
).(21
1
T
n
tnfjetnEfE (4.22)
where T is the number of iterations and ∆t is the time step, so t = T. ∆t.
Equation (4.22) may be divided into its real and imaginary parts
)..2 sin()..(j-
)..2 cos()..()(
10
10
1
ntftnE
ntftnEfE
T
n
T
n
(4.23)
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 71
The above equation may be implemented in computer code as:
Real_pt [m,k] = real_pt [m,k] + ex [k] * cos (2 * pi * freq (m) * dt * n) (4.24a)
imag_pt [m,k] = imag _pt [m,k] + ex [k] * sin (2 * pi * freq (m) * dt * n) (4.24b)
for every point k, in the region interest, it is required only two buffers for every
frequency of interest fm. At any point k, from the real part of E(fi), Real_pt
[m,k], and the imaginary part imag_pt [m,k], one can determine the amplitude
and phase at the frequency fm :
amp [m,k] = sqrt (pow(real_pt[m,k],2.0) + pow(imag_pt[m,k],2.0)) (4.25a)
phase [m,k] = atan2( imag_pt [m,k], real_pt[m,k] ) (4.25b)
Note that there is an amplitude and phase associated with every frequency at
each cell [51],[52]. Figure 4.8 is a simulation of a pulse hitting a dielectric
medium with a dielectric constant of 4, the frequency response at 500 MHz is
also displayed. At T = 200, before the pulse has hit the medium, the frequency
response is 1 through that part of the space where the pulse has traveled. After
400 time steps, the pulse has hit the medium, and some of it has penetrated into
the medium and some of it has been reflected. The amplitude of the transmitted
pulse is determined by [46] as.
667.41
1.2.2
21
1
rr
r
incident
edtransimitt
E
E
Where is the transmission coefficient, εr1 is the dielectric constant of free
space and εr2 is the dielectric constant of the medium.
The value of is the Fourier amplitude in the medium as shown in Figure 4.8.
The Fourier amplitude outside the medium varies between 1- 0.33 and 1+0.333.
This is in keeping with the pattern formed by the standing wave that is created
from a sinusoidal signal, whose reflected wave is interacting with the original
incident wave.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 72
Fig. 4.8 Simulation of a pulse striking a dielectric medium with εr= 4. The top figure is the pulse after 200 time steps. Notice the Fourier amplitude is 1 in the part of the space where the pulse has traveled, but 0 elsewhere. After 400 time steps, the pulse has struck the medium, and part of it has been transmitted and part is reflected. The Fourier amplitude in the medium is 0.667, which is the percentage that has been transmitted.
Ex
Ex
0 20 40 60 80 100 120 140 160 180 200-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
FDTD cells
EX
eps=4 Time Domain T=200
0 20 40 60 80 100 120 140 160 180 2000
0.2
0.4
0.6
0.8
1
1.2
1.4
Amp 2
eps=4 Freq. domain at 500 Mhz
0 20 40 60 80 100 120 140 160 180 200-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
FDTD cells
eps=4 Time domain T=400
EX
FDTD cells
0 20 40 60 80 100 120 140 160 180 2000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
FDTD cells
eps=4 Freq. domain at 500 Mhz
Amp 2
FDTD cells
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 73
4.3 TWO-DIMENSIONAL SIMULATION WITH THE FDTD METHOD
Up to now, the form of Maxwell’s equations that has been used is given in
Eq. (4.1), which uses only the E and H fields. However, a more general form is
Ht
D
(4.26a)
1
Et
H
o
(4.26b)
.. ED ro (4.26c)
where D is the electric flux density. Put these equations in the normalized form
[48], using
~
EE
DDoo
1~
which leads to
Ht
D
oo
.1
~
(4.27a)
Et
H
oo
~1
(4.27b)
~
.~
ED r (4.27c)
When dealing with three dimensional-simulation, there will be six different
fields: Ex, Ey, Ez, Hx, Hy and Hz. In doing two-dimensional simulation, it is
preferred to choose between one of two groups of three vector each: (1) The
transverse magnetic (TM) mode, which is composed of Ez, Hx and Hy or (2) The
transverse electric (EM) mode, which is composed of Ex, Ey and Hz. For
working with the TM mode. Equations (4.27) are now reduced to
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 74
(4.28a)
y
H
x
H
t
D xy
oo
z
1
~
(4.28b)y
E
t
H z
oo
x
~.
1
(4.28c)x
E
t
Hz
oo
y
~.
1
~
.~
zrz ED (4.28d)
As in one-dimensional simulation, it is important that there is a systematic
interleaving of the fields to be calculated. This is illustrated in Figure 4.9.
Putting Eq. (4.28) into the finite difference scheme results in the following
difference equations [21]:
i-1 i i+1 i+2
j-1
j
j+1
Hx Hx Hx Hx
Hx
Hx
X
Y
Hx Hx Hx
Hx Hx Hx
Hy
Hy
Hy Hy Hy
Hy
Hy Hy
Hy
o Ez
Fig. 4.9 Interleaving of the E and H fields for the two-dimensional
TM formulation.
o Ez
o Ez
o Ez
o Ez
o Ez
o Ez
o Ez
o Ez
o Ez
o Ez
o Ez
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 75
y
jiHjiH
x
jiHjiH
t
jiDjiD
nx
nx
oo
ny
ny
oo
nz
nz
)2/1,()2/1,(1
),2/1(),2/1(1),(~
),(~ 2/12/1
(4.29a)
(4.29b)y
jiEjiE
t
jiHjiH nz
nz
oo
nx
nx
),(~
)1,(~
1)2/1,()2/1,( 2/12/11
(4.29c)x
jiEjiE
t
jiHjiH nz
nz
oo
ny
ny
),(
~),1(
~1),2/1(),2/1( 2/12/11
Using the same type of manipulation as in one-dimensional case, including
oc
xt
. 2
For simplicity, it is assumed that ∆x = ∆y, then Eq. (4.29) becomes
)2/1,()2/1,(
),2/1(),2/1(5.0),(
~),(
~ 2/12/1
jiHjiH
jiHjiHjiDjiD
nx
nx
ny
nyn
znz (4.30a)
)1,(~
),(~
5.0)2/1,()2/1,( 2/12/11 jiEjiEjiHjiH nz
nz
nx
nx (4.30 b)
),(~
),1(~
5.0),2/1(),2/1( 2/12/11 jiEjiEjiHjiH nz
nz
ny
ny
(4.30c)
Rewriting Eq. (4.30) in C computer code gives the following:
dz [i] [j] = dz [i] [j] + 0.5 * ( hy [i] [j] - hy [i-1] [j] – hx[i] [j]+hx[i] [j-1] )
hx [i] [j] = hx [i] [j] + 0.5 * ( ez [i] [j] - ez [i] [j+1] )
hy [i] [j] = hy [i] [j] + 0.5 * ( ez [i+1] [j] - ez [i] [j] )
A two-dimensional FDTD simulation that implements the above equations is
shown in Figure 4.10. It has a simple Gaussian pulse source that is generated in
the middle of the problem space and travels outward. The figure demonstrates
the simulation for the first 50 time steps.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 76
010
2030
4050
60
010
2030
4050
600
0.1
0.2
0.3
0.4
0.5
0.6
T = 40
010
2030
4050
60
010
2030
4050
600
0.2
0.4
0.6
0.8
T = 30
010
2030
4050
60
010
2030
4050
600
0.1
0.2
0.3
0.4
0.5
T = 50
010
2030
4050
60
010
2030
4050
600
0.2
0.4
0.6
0.8
1
Cells
Ez
(i,j)
Cells
T = 20
Fig. 4.10 Simulation of a Gaussian pulse initiated in the middle of
the problem space and travels outward.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 77
4.3.1 The Perfectly Matched Layer (PML)
Up to now, the issue of absorbing boundary conditions (ABCs) has been
only briefly mentioned. The size of the area that can be simulated using FDTD is
limited by computer resources. For instance, in two-dimensional simulation of
the previous section, the program contains two-dimensional matrices for the
value of all the fields, dz, ez, hx, and hy. Suppose it is required to simulate a
wave generated from a point source propagating in free space as in Figure 4.10.
As the wave propagates outward, it will eventually come to the edge of the
allowable space, which is dictated by how the matrices have been dimensioned
in the program. If nothing were done to address this, unpredictable reflections
would be generated that would go back inward. There would be no way to
determine which is the real wave and which is the reflected junk. This is the
reason that ABCs have been an issue for as long as FDTD has been used. There
have been numerous approaches to this problem [22],[49].
One of the most flexible and efficient ABCs is the perfectly matched layer
(PML) developed by Berenger [53]. The basic idea is this: if a wave is
propagating in medium A and it impinges upon medium B, the amount of
reflection is dictated by the intrinsic impedances of the two media
BA
AB
(4.31)
which are determined by the dielectric constants ε and permeabilities μ of the
two media
(4.32)
Up to now, it has been assumed that μ was a constant, so when a propagating
pulse went from εr = 1 to εr = 4, as in Figure 4.8, it saw a change in impedance
and reflected a portion of the pulse given by Eq. (4.31). However, if μ changed
with ε so η remained a constant, Γ would be zero and no reflection would occur.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 78
This still doesn’t solve our problem, because the pulse will continue propagating
in the new medium. What it is really needed is a medium that is also lossy so the
pulse will die out before it hits the boundary. This is accomplished by making
both ε and μ of Eq. (4.32) complex, because the imaginary part represents the
part that causes decay.
Let us go back to Eqs. (4.28), but move everything to the Fourier domain . (For
going to the Fourier domain in time, so d / dt becomes jw. This does not affect
the spatial derivatives.)
y
H
x
HcDj xy
oz . (4.33a)
)().()( * wEwwD zrz (4.33b)
orr jw
w
)(* (4.33c)
(4.33d)y
EcjwH z
ox
.
(4.33e)x
EcjwH z
oy
.
Remember that ε and μ have been eliminated from the spatial derivative in
Eqs (4.33a), (4.33b), and (4.33c) for the normalized units . Instead of putting
them back to implement the PML, fictitious dielectric constants and
permeabilities *Fz ,
*Fx and
*Fy are added [54]:
y
H
x
HcyxDj xy
oFzFzz .)().(. ** (4.34a )
)().()( * wEwwD zrz (4.34b )
y
EcyxjwH z
oFxFxx
.)().(. ** (4.34c )
x
EcyxjwH z
oFyFyy
.)().(. ** (4.34d )
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 79
A few things are worth noting: first, the value εF is associated with the flux
density D, not the electric field intensity E; second, two values each of εF have
been added in Eqs. (4.34a), and μF in Eqs. (4.34c) and (4.34d), one for the x
direction and for the y direction; and finally, nothing was added to Eq. (4.34b).
These fictitious values to implement the PML have nothing to do with the real
values of )(* wr which specify the medium.
Sacks, et al. [55] shows that there are two condition to form a PML:
1. The impedance going from the background medium to the PML must be
constant,
yor x mfor *
*
Fm
Fmmo
(4.35)
where ηo is the impedance of the background medium
ηm is the impedance of the PML.
2. In the direction perpendicular to the Boundary (the x direction, for instance),
the relative dielectric constant and relative permeability must be the inverse
of those in the other directions, i.e.,
** 1
FyFx
(4.36a)
** 1
FyFx
(4.36b)
It is assumed that each of these is a complex quantity of the form
yor x mfor * o
DmFmFm jw
(4.37a)
yor x mfor * o
HmFmFm jw
(4.37b)
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 80
The following selection of parameters satisfies Eqs. (4.36a) and (4.36b) [56]:
(4.38a)1 FmFm
(4.38b)o
D
o
Hm
o
Dm
Substituting Eq. (4.38) into (4.37), the value in Eq. (4.35) becomes:
.1/)(1
/)(1*
*
o
o
Fx
Fxmo jwx
jwx
This fulfills the first requirement above. If σ increases gradually as it goes into
the PML, Eqs. (4.34a), (4.34c), and (4.34d) will cause Dz and Hy to be
attenuated.
At the beginning, the PML is implemented only in the x direction. Therefore,
only the x dependent values of *F and
*F are retained in Eq. (4.34)
y
H
x
HcxDj xy
oFzz .)(. *
y
EcxjwH z
oFxx
.)(. *
x
EcxjwH z
oFyy
.)(. * ,
and use the values of Eq. (4.38):
(4.39a)
y
H
x
HcD
jw
xj xy
ozo
D .)(
1
(4.39b)y
EcH
jw
xjw z
oxo
D
.)(
11
(4.39c)x
EcH
jw
xjw z
oyo
D
.
)(1
Note that the permeabilities of Hx in Eq. (4.39b) is the inverse of that of Hy in
Eq. (4.39c) in keeping with Eq. (4.36b). Therefore, this fulfills the second
Requirement for the PML.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 81
Now Eqs. (4.39) have to be put into the FDTD formulation. First, look at the left
side of Eq. (4.39a):
zo
Dzz
o
D Dx
DjDjw
xj
)()(1
Moving to the time domain, and then taking the finite difference approximation,
it yields:
o
Dnz
o
Dnz
nz
nz
o
Dnz
nz
zo
Dz
ti
tjiD
ti
tjiD
jiDjiDi
t
jiDjiDD
i
t
D
.2
).(1
1),(
.2
).(1
1),(
2
),(),()(),(),(~)(
2/12/1
2/12/12/12/1
Putting this equation into Eq. (4.39a) along with the spatial derivatives, it yields:
)2/1,()2/1,(),2/1(),2/1(5.0).(2
),().(3),( 2/12/1
jiHjiHjiHjiHigi
jiDigijiDnx
nx
ny
ny
nz
nz
(4.40)
where once again we have used the fact that
2
1) . 2/(
oo
o cx
cxc
x
t
The new parameters gi2 and gi3 are given by
(4.41a)).2/().(1
1)(2
oD tiigi
(4.41b)).2/().(1
).2/().(1)(3
oD
oD
ti
tiigi
An almost identical treatment of Eq. (4.39c) gives
),(),1(5.0).2/1(2
),2/1().2/1(3),2/1(2/12/1
1
jiEjiEifi
jiHifijiHnz
nz
ny
ny
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 82
where
).2/().2/1(1
1)2/1(2
oD tiifi
).2/().2/1(1
).2/().2/1(1)2/1(3
oD
oD
ti
tiifi
Notice that these parameters are calculated at i+1/2 because of the position of Hy
in the FDTD grid (Fig. 4.9)
Equation (4.39b) will require a somewhat different treatment than the other two.
Start by writing it as
y
E
jw
x
y
EcjwH z
o
Dzox
1)(.
Remember (1/jw) may be regarded as an integration operator over time and jw as
a derivative over time. The spatial derivative will be written as
y
ecurl
y
jiEjiE
y
E nz
nzz
_),()1,(~
2/12/1
Implementation this into an FDTD formulation gives
T
no
Do
nx
nx
y
ecurlt
x
y
ecurlc
t
jiHjiH
0
1 _)(_)2/1,()2/1,(
Note the extra ∆t in front of the summation. This is part of the approximation of
the time domain integral. Finally it yields
)2/1,(2
).(
_.
)2/1,(
)2/1,().(.
_.
)2/1,()2/1,(
2/1
2/1
1
jiItx
ecurly
tcjiH
jiItx
y
ct
ecurly
tcjiHjiH
nHx
o
D
onx
nHx
o
Do
onx
nx
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 83
Eq. (4.39b) is implemented as the following series of equations:
)1,(),(_ 2/12/1 jiEjiEecurl nz
nz
ecurljiIjiI nHx
nHx _)2/1,()2/1,( 2/12/1
)2/1,(.1
_ .5.0)2/1,()2/1,(2/1
1
jiI(i)fi
ecurljiHjiHnHx
nx
nx
With
o
ti(i)fi
2
).(1
In calculating the f and g parameters, it is not necessary to actually vary
conductivities. Instead , an auxiliary parameter is calculated,
o
txn
2
.
that increases as it goes into the PML. The f and g parameters are then
calculated:
(4.42)pmllengthipmllength
iixn _,......,2,1
_*333.0)(
3
)(1 ixn(i)fi
)(1
1)(2
ixnigi
)(1
)(1)(3
ixn
ixnigi
Notice that the quantity in parentheses in Eq. (4.42 ) ranges between 0 and 1.
The factor “0.333” was found empirically to be the largest number that
remained stable. Similarly, the cubic factor in in Eq. (4.42) was found
empirically to be the most effective variation, fi2 and fi3 are different only
because they are computed at the half intervals, i+1/2. The parameters vary in
the following manner:
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 84
fi1(i) from 0 to 0.333
gi2(i) from 1 to 0.750
gi3(i) from 1 to 0.500
Throughout the main problem space, fi1 is zero, while gi2 and gi3 are 1.
Therefore, there is a “seamless” transition from the main part of the program to
the PML (Fig. 4.11).
So far, the implementation of the PML in the x direction has been shown.
Obviously, it must also be done in y direction. Therefore, it is necessary to go
back and add the y dependent terms from Eq. (4.34) that were set aside. So
instead of Eq. (4.34) we have
y
H
x
HcD
jw
y
jw
xj xy
ozo
D
o
D .)(
1)(
1.
(4.43a)
Wave source In vacuum
Decreasing values of fj1; increasing values of fj2, fj3, gj2, and gj3.
Decreasing values of fi1; increasing values of fi2, fi3, gi2, and gi3.
The corners are an overlap of both sets of parameters
Fig. 4.11 Parameters related to the perfectly matched layer (PML)
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 85
y
EcH
jw
y
jw
xjw z
oxo
D
o
D .)(
1)(
11
(4.43b)
x
EcH
jw
y
jw
xjw z
oyo
D
o
D
.)(
1)(
11
(4.43c)
Using the same procedure as before, the following replaces Eq. (4.40):
)2/1,()2/1,(
),2/1(),2/1().5.0).((2).(2
),().(3).(3),( 2/12/1
jiHjiH
jiHjiHjgjigi
jiDjgjigijiD
nx
nx
ny
ny
nz
nz
In the y direction, Hy will require an implementation similar to the one used for
Hx in the x direction giving
),(),1(_ 2/12/1 jiEjiEecurl nz
nz
ecurljiIjiI nHy
nHy _),2/1(),2/1( 2/12/1
),2/1(.1_.5.0.2/12
),2/1(.2/13),2/1(2/1
1
jiI(j)fjecurl)(ifi
jiH)(ifijiHnHy
ny
ny
Finally, the Hx in the x direction becomes
)1,(),(_ 2/12/1 jiEjiEecurl nz
nz
ecurljiIjiI nHx
nHx _)2/1,()2/1,( 2/12/1
)2/1,(.1_.5.0.2/12
)2/1,(.2/13)2/1,(2/1
1
jiI(i)fiecurl)(jfj
jiH)(jfjjiHnHx
nx
nx
Now the full set of parameters associated with the PML are the following:
fi1(i) & fj1(j) from 0 to 0.333
fi2(i), gi2(i), fj2(j), & gj2(j) from 1 to 0.75
fi3(i), gi3(i), fj3(j), & gj3(j) from 1 to 0.50
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 86
It is noticed that the PML can be simply turned off in the main part of the
problem space by setting fi1 and fj1 to zero, and the other parameter to 1. They
are only one-dimensional parameters, so they add very little to the memory
Requirements. However, IHx and IHy are 2D parameters. Whereas memory
requirements are not a main issue while dealing with 2D.
The PML is implemented in a program written in the C programming language.
Figure 4.12 illustrates the effectiveness of an 8 points PML with a sinusoidal
source of the form:
pulse = sin ( 2*pi*1500*1e6*∆t*T )
This sinusoidal source has a frequency of 1500 MHz and is initiated at the center
of the problem space, the space steps used are ∆x = ∆y = 0.01 m, and the total
number of cells are 60 X 60 in the x and y directions respectively. The time step
is calculated using the following formula
ps 16.67 . 2
oc
xt
It is shown in Figure 4.12 that as the wave reaches the perfectly matched layer
(PML) which is eight cells on every side, it is absorbed.
4.4 THREE-DIMENSIONAL SIMULATION WITH THE FDTD
METHOD
In actuality, three-dimensional FDTD simulation is very much like two-
dimensional simulation, it is just harder because all vector fields will be used
and each one is in three dimensions. But the process is straight-forward.
4.4.1 Free Space Formulation
The original FDTD paradigm was described by the Yee cell, (Fig. 4.13),
named, of course, after Kane Yee [21]. It is noticed that the E and H fields are
assumed interleaved around a cell whose origin is at the location i, j, k.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 87
010
2030
4050
60
010
2030
4050
60
-0.4
-0.2
0
0.2T = 60
010
2030
4050
60
010
2030
4050
60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Cells
Ez
(i,j
)
T = 40
Cells
010
2030
4050
60
010
2030
4050
60-0.5
0
0.5 T = 100
Fig. 4.12 Simulation of a sinusoidal source initiated in the middle of the problem space. as the wave reaches the perfectly matched layer (PML) which is eight cells on every side, it is absorbed.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 88
Every E field is located ½ cell width from the origin in the direction of its
orientation; every H field is offset ½ cell in each direction except that of its
orientation. Starting with Maxwell’s equations:
(4.44a)Ht
D
oo
.1
~
)(~
).()(~ * wEwwD r (4.44b)
Et
H
oo
~1
(4.44c)
Once again, the ~ notation will be dropped, but it will always be assumed that
we are referring to the normalized values.
Eqs. (4.44a) and (4.44c) produces six scalar equations:
z
H
y
H
t
D yz
oo
x
1
(4.45a)
x
H
z
H
t
Dzx
oo
y
1
(4.45b)
y
H
x
H
t
D xy
oo
z
1
(4.45c)
Z
X
Y
(i, j, k+1)
(i+1, j, k)
(i, j+1, k)
Ex
(i, j, k)Ey
Ez Hx
Hz
Hy
Fig. 4.13 The Yee cell
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 89
y
E
z
E
t
H zy
oo
x
1
(4.45d)
z
E
x
E
t
Hxz
oo
y
1
(4.45e)
x
E
y
E
t
H yx
oo
z
1
(4.45f)
The first step is to take the finite difference approximations. Using only Eqs.
(4.45c) and (4.45f) as examples:
)2/1,,()2/1,,( 2/12/1 kjiDkjiD nz
nz
(4.46))2/1,,2/1()2/1,,2/1((.
kjiHkjiH
x
t ny
ny
oo
)2/1,2/1,()2/1,2/1,( kjiHkjiH nx
nx
),2/1,2/1(),2/1,2/1(1 kjiHkjiH n
znz
),2/1,(),2/1,1((.
2/12/1 kjiEkjiEx
t ny
ny
oo
(4.47)
),,2/1(),1,2/1( 2/12/1 kjiEkjiE nx
nx
The relation between E and D, corresponding to Eq. (4.44b), is exactly the same
as the one-dimensional or two-dimensional cases, except now there will be three
equations.
4.4.2 The PML in Three Dimensions
The development of the PML for three dimensions closely follows the
two-dimensional version. The only difference is that you deal with three
directions instead of two [54]. For instance, Eq. (4.43a) becomes
(4.48)
y
H
x
HcD
jw
z
jw
y
jw
xj xy
ozo
z
o
y
o
x .)(
1)(
1)(
1.1
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 90
and implementing it will closely follow the two-dimensional development. Start
by rewriting Eq. (4.48) as
y
H
x
H
jw
zcD
jw
y
jw
xj xy
o
zoz
o
y
o
x .)(
1.)(
1)(
1.
(4.49)hcurljw
zchcurlc
o
zoo _.
)(._.
We will define
hcurljw
I DZ _.1
Which is an integration when it goes to the time domain, so Eq. (4.49) becomes
DZ
o
zoz
o
y
o
x Iz
hcurlcDjw
y
jw
xj .
)(_.
)(1
)(1.
The implementation of this into the FDTD parallels that of the two-dimensional
PML, except the right side contains the integration term IDz. Therefore,
following the same math we used in two-dimensional PML, we get
)2/1,2/1,()2/1,2/1,(
)2/1,,2/1()2/1,,2/1(_
kjiHkjiH
kjiHkjiHhcurl
nx
nx
ny
ny
hcurlkjiIkjiI nDZ
nDZ _)2/1,,()2/1,,( 1
)2/1,,().(1_)5.0).((2).(2
)2/1,,()(3).(3)2/1,,( 2/12/1
kjiIkgkhcurljgjigi
kjiDjgjigikjiDnDZ
nz
nz
The one-dimensional g parameters are defined the same as in two-dimensional
PML.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 91
4.5 NEAR-FIELD TO FAR-FIELD TRANSFORMATION
It is not practical to directly calculate far-field data within the FDTD grid
because for most problems the grid space cannot be made large enough to
include the far-field. A. Taflove [22] reported an efficient time-domain near to
far field transformation. This method involves setting up the time dimensional
arrays for the far-field vector potentials. Each array element is determined by
conducting a recursive sum of contributions from the time domain electric and
magnetic current sources just computed via FDTD method on a virtual surface in
a six sided rectangular locus S that completely encloses the structure of interest
in the scattered field zone of the FDTD lattice as shown in Figure 4.14. The
patch electric and magnetic fields exist over only some finite portion S and the
fields elsewhere are zero. The surface electric and magnetic currents densities
respectively are:
EXnM s
ˆ (4.50)
HXnJ s
ˆ (4.51)
Grid boundary (ABC)
Virtual surface
Js Ms
n
Fig. 4.14 Electromagnetic equivalence to transform near-fields to far-fields
SNo sources& zero
fields
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 92
From Figure 4.15, the vector potentials in the far field region are given by:
Nr
esd
R
eJA
jkro
s
jkR
so
44
(4.52)
Lr
esd
R
eMF
jkro
s
jkR
so
44
(4.53)
Where
sdeJNs
rjks cos
sdeMLs
rjks cos
t ˆ poinnobservatioofpositionrrr
t ˆ poinsourceofpositionrrr
rr RRR
rrangle and between :
The electric and magnetic fields due to the vector potentials of Eqs. (4.52) and
(4.53) are given by:
R Equivalent surface S’ (x’,y’,z’)
r r
x z
y
(x,y,z) observation
point
Fig. 4.15 Geometry of a far-field observation point relative to the near-field integration contour and source point.
Chapter 4 The Finite-Difference Time-Domain Method
Zagazig University- Electronics & Comm. Eng. Dept. 93
FXAk
AjEo
1
.1
2 (4.54)
AXFk
FjHo
1
.1
2 (4.55)
From these equations, the theta and phi components of the electric and magnetic
fields can be obtained in the far zone.
4.6 CONCLUDING REMARKS
This chapter has been included to present the foundations of the Finite-
Difference Time-Domain (FDTD) method, and the algorithm introduced by
Kane Yee in 1966 to implement this method.
The C programming language has been used to demonstrate the one, two, and
three-dimensional simulation using the FDTD method and the application of the
perfectly matched layer as the absorbing boundary conditions.
It has been shown that the FDTD method can be used to simulate the
propagation of electromagnetic waves through different mediums specified by
the relative dielectric constant εr, such as free space, lossless dielectric, and
lossy dielectric having finite conductivity. The time domain response for these
simulations has been shown at different time-steps. And the Fourier transform
has been used to calculate the frequency domain output.
94
C H A P T E R 5
FDTD Analysis of Wideband Microstrip antennas
5.1 INTRODUCTORY REMARKS
In this chapter, The transmission-line model described in chapter 2 will
be used to design a rectangular microstrip patch antenna, then this antenna will
be analyzed using the FDTD method and the obtained results will be compared
to other results produced using the IE3D software which is based on the method
of moments and good agreements will be shown.
Next, a single-patch wide-band microstrip antenna will be presented i.e.
the E-shaped patch antenna. Two parallel slots are incorporated into the
rectangular patch of a microstrip antenna to expand it bandwidth. The wide-band
mechanism is explored by investigating the behavior of the currents on the
patch. The obtained impedance bandwidth from this antenna is 26.7 %.
Next, a single-layer capacitive feeding mechanism, consisting of a small
rectangular probe-fed patch, which is capacitively coupled to the radiating
element, will be used to obtain wideband operation for probe-fed microstrip
antennas on thick substrates. The main advantages of this feeding mechanism
are that all the elements reside on a single layer and that it is very easy to fine-
tune the input impedance.
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 95
5.2 A LINE-FED RECTANGULAR PATCH ANTENNA
In this section, the procedure for designing a line-fed rectangular
microstrip patch antenna is explained. Next, a compact rectangular microstrip
patch antenna is designed for use in cellular communications. Finally, the
results obtained from a program which is FDTD based are demonstrated. The
calculated results have been compared to other results produced using IE3D a
commercial simulator [4] based on the method of moment and good agreements
have been found.
5.2.1 Design Specifications
The three essential parameters for the design of a rectangular Microstrip
Patch Antenna are:
• Frequency of operation (fr): The resonant frequency of the antenna
must be selected appropriately to meet the requirements of the
communication system. It should be the center of frequency band
required from such antenna. Let us choose the resonant frequency to be
7.5 GHz [57].
• Dielectric constant of the substrate (εr): The dielectric material selected
for this design is Duroid which has a dielectric constant of 2.2. A
substrate with a low dielectric constant has been selected since it
increases the impedance bandwidth of the antenna. However, this has
detrimental effects on antenna size reduction since the resonant length of
a microstrip antenna is shorter for higher substrate dielectric constant.
• Height of dielectric substrate (h): For the microstrip patch antenna to be
used in hand-held devices, it is essential that the antenna is not bulky.
Hence, the height of the dielectric substrate is selected as .794 mm.
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 96
Hence, the essential parameters for the design are:
• fr = 7.5 GHz.
• εr = 2.2.
• h = 0.794 mm.
5.2.2 Design Procedure
The transmission-line model described in chapter 2 will be used to design
this rectangular microstrip patch antenna which may take the form shown in
Figure 5.1.
Step 1: Calculation of the Width ( W ): The width of the Microstrip patch
antenna is given by equation (2.13) as:
(5.1)1
2
2
rrf
CW
Substituting C = 3X108 m/s, εr = 2.2 and fr = 7.5 GHz, it yields:
W = 16.02 mm
L
WgW
Lg
Fig. 5.1 Top view of Microstrip Patch Antenna.
Radiating Patch
Microstrip line feed
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 97
Step 2: Calculation of Effective dielectric constant ( εreff ): Equation (2.3)
gives the effective dielectric constant as:
2/1
1212
1
2
1
W
hrrreff
(5.2)
Substituting εr = 2.2, W = 16.02 mm and h = 0.794 mm, it yields:
εreff = 2.07
εreff must be in the range ( 1< εreff < εr ). Step 3: Calculation of the Effective length ( Leff ): Equation (2.8) gives the
effective length as:
reffr
efff
CL
2 (5.3)
Substituting εreff = 2.07, C = 3X108 m/s and fr = 7.5 GHz , it yields:
Leff = 14.08 mm
Step 4: Calculation of the length extension ( ∆L ): Equation (2.6) gives the
length extension as:
8.0258.0
264.03.0412.0
hW
hW
hL
reff
reff
(5.4)
Substituting εreff = 2.07, W = 16.02 mm and h = 0.794 mm, it yields:
∆L = 0.421 mm
Step 5: Calculation of actual length of patch ( L ): The actual length is
obtained by re-writing equation (2.7) as:
LLL eff 2
Substituting Leff = 14.08 mm and ∆L = 0.421 mm, it yields:
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 98
L = 13.238 mm.
Step 6: Calculation of the ground plane dimensions (Lg and Wg):
The transmission-line model is applicable to infinite ground planes only.
However, for practical considerations, it is essential to have a finite ground
plane. It has been shown in [1] that similar results for finite and infinite ground
plane can be obtained if the size of the ground plane is greater than the patch
dimensions by approximately six times the substrate thickness all around the
periphery. Hence, for this design, the ground plane dimensions would be given
as:
Lg = 24. 0 mm
Wg = 40.0 mm
For simplicity, the length and the width of the patch and the ground plane have
been rounded off to the following values: L = 13 mm, W = 16 mm, Lg = 24 mm,
Wg = 40 mm (taking into account the length of the microstrip line feed).
Step 7: Specifications of feed type and its location:
A microstrip line feed is to be used in this design. As shown in Figure
5.1. This kind of feed arrangement has the advantage that the feed can be etched
on the same substrate to provide a planar structure.
Hence, for this design, the dimensions of the microstrip line feed would be used
as 2.46 mm X 20 mm [57]. The feed location must be located at that point on
the patch, where the input impedance is 50 ohms for the resonant frequency.
Hence, a trial and error method has been used to locate the feed point. This can
clarify the need for a technique to be developed to find out the appropriate
location of the feed. This is yet a major drawback in the microstrip antenna
design procedure.
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 99
5.2.3 FDTD Analysis of The Rectangular Microstrip Antenna
The 3D finite difference equations and the absorbing boundary
conditions explained in chapter 4 are used to analyze the proposed design
(selected to be the same as in [57]), and to simulate the propagation of a broad-
band Gaussian pulse on the microstrip structure. The essential aspects of the
time domain algorithm are as follows:
Initially (at t = n = 0) all fields are 0.
The following are repeated until the response is ≈ 0:
Gaussian excitation is imposed on port 1.
Hn+1/2 is calculated from FD equations.
En+1 is calculated from FD equations.
Tangential E is set to 0 on conductors.
Save desired field quantities.
n n + l (Increasing the time steps by one and so on).
Compute scattering matrix coefficients from time- domain results.
5.2.3.1 Frequency-Dependent Parameters
In addition to the transient results obtained naturally by the FDTD
method, the frequency-dependent scattering matrix coefficients are easily
calculated.
[V]r = [S] [V]i
where [V]r and [V]i are the reflected and incident voltage vectors, respectively,
and [S] is the scattering matrix. To accomplish this, the vertical electric field in
the substrate at the center of each microstrip port is recorded at every time step.
As in [58], it is assumed that this field value is proportional to the voltage
(which could be easily obtained by numerically integrating the vertical electric
field) when considering propagation of the fundamental mode. To obtain the
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 100
)(
)()(
tVFT
tVFTS
k
jjk
scattering parameter S11(w), the incident and reflected waveforms must be
known. The FDTD simulation calculates the sum of incident and reflected
waveforms. To obtain the incident waveform, the calculation is performed using
only the port 1 microstrip line, which will now be of infinite extent (i.e., from
source to far absorbing wall), and the incident waveform is recorded. This
incident waveform may now be subtracted from the incident plus reflected
waveform to yield the reflected waveform for port 1. The other ports will
register only transmitted waveforms and will not need this computation. The
scattering parameters, Sjk, may then be obtained by simple Fourier transform of
these transient waveforms as:
Note that the reference planes are chosen with enough distance from the circuit
discontinuities to eliminate evanescent waves. These distances are included in
the definition of the circuit so that no phase correction is performed for the
scattering coefficients.
5.2.3.2 Numerical Results
Numerical results have been computed for the line-fed rectangular patch
antenna. This circuit has dimensions on the order of several centimeters, and the
frequency range of interest is from dc to 20 GHz. The operating regions of this
circuit are less than 10 GHz; however, the accuracy of the computed results at
higher frequencies is examined.
The actual dimensions of the microstrip antenna analyzed (according to our
design) are shown in Figure 5.2. The operating resonance which is 7.5 GHz
approximately corresponds to the frequency where L = 13 mm = λ / 2..
Simulation of this circuit involves the straightforward application of the finite-
difference equations, source, and boundary conditions. To model the thickness
of the substrate correctly, ∆Z is chosen so that three nodes exactly match the
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 101
thickness. An additional 13 nodes in the z direction are used to model the free
space above the substrate.
In order to correctly model the dimensions of the antenna, ∆x and ∆y have been
chosen so that an integral number of nodes will exactly fit the rectangular patch.
Unfortunately, this means the port width and placement will be off by a fraction
of the space step. The sizes of the space steps are carefully chosen to minimize
the effect of this error. The space steps used are ∆x = 0.371 mm, ∆y = 0.400
mm, and ∆z = 0.265 mm, and the total mesh dimensions are 65 X 100 X 16 in
the x, y, and z directions respectively. The rectangular antenna patch is thus 35
∆x x 40 ∆y. The length of the microstrip line from the source plane to the edge
of the antenna is 50∆y, and the reference plane for port 1 is 10 ∆y from the edge
of the patch. The microstrip line width is modeled as 7∆x. In choosing the time
step, the smallest dimension ∆z is used to get
spicosecond 441.0.2
oc
zt
The Gaussian half-width is T = 15 ps, and the time delay to is set to be 3T so the
Gaussian will start at approximately 0. The simulation is performed for 8000
time steps. Figure 5.3 shows the Gaussian pulse used in the excitation of the
antenna.
z
y
x
Fig. 5.2 Line-fed rectangular microstrip antenna detail.
L=13.00
W=16.0
0.794 mm 2.46 mm
ε=2.2
2.09 mm
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 102
Incident wave
Reflected wave
Figure 5.4 shows the transient time response of the Ez component at the
reference plane for the microstrip line feeder.
The scattering coefficient results, shown in Figure 5.5, show good agreement
between the calculated data and the simulated data resulted from IE3D- a
commercial simulator based on the method of moment. The operating resonance
at 7.5 GHz is almost exactly shown by both FDTD and IE3D simulation.
Fig. 5.4 The incident and reflected waves at the reference plane on the microstrip line.
0 50 100 150 200 250 300 350 400 450 5000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time Steps
Fig. 5.3 The input Gaussian pulse.
Inci
dent
pul
se
0 1000 2000 3000 4000 5000 6000 7000 8000-0.5
0
0.5
1
Time Steps
Ez
(V/m
)
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 103
Additional resonances are also in relatively good agreement with simulation,
with the shift between both methods increases with the increase in frequency in
a monotonic way.
For a return loss less than –10 dB the frequency band ranges from 7.31 to 7.62
GHz, with a center frequency of 7.465 GHz, the bandwidth (according to the
FDTD results) is calculated as:
% 16.42/)GHz 31.7GHz 62.7(
GHz 31.7GHz 62.7
frequencycener
limitlower -limit upper Bandwidth
The VSWR may be calculated as:
11
11
1
1
S
SVSWR
Figure 5.6 shows that the antenna frequency bandwidth with VSWR<2 covers
the frequency range of 7.33 to 7.63 GHz. This agrees with the less than –10 dB
band of the return loss.
Fig. 5.5 The return loss of the rectangular patch antenna.
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20-50
-40
-30
-20
-10
0
10
Frequency (GHz)
FDTDIE3D
S11
(d
B)
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 104
Input impedance for the antenna may be calculated from the S11(w)
calculation [57] by transforming the reference plane to the edge of the microstrip
antenna,
klj
klj
oin efS
efSZfZ
211
211
)(1
)(1)(
where,
k is the wavenumber on the microstrip
L is the length from the edge of the antenna to the reference plane (10 ∆y)
Zo is the characteristic impedance of the microstrip line.
For simplicity of the Zin calculation, the microstrip line is assumed to have a
constant characteristic impedance of 50 Ω, and an effective permittivity of 1.9 is
used to calculate the wavenumber. Results for the input impedance calculation
near the operating resonance of 7.5 GHz are shown in Figure 5.7.
Fig. 5.6 The voltage standing wave ratio of the rectangular antenna near the operating resonance of 7.5 GHz.
7 7.5 8 8.51
2
6
11
16
20
Frequency (GHz)
IE3D
FDTDV
SW
R
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 105
5.2.3.3 Radiation Pattern
To compute the far-field antenna parameters, such as radiation pattern and
gain. The near-to-far-field transformation principle given in chapter (4) will be
used here.
Since a microstrip patch antenna radiates normal to its patch surface, the
elevation pattern for phi = 0.0 o (H-plane) and phi = 90o (E-plane) degrees
would be important. Figure 5.8 shows the radiation field Pattern of the antenna
under consideration in the (x-z) plane (phi = 0.0 o) and the (y-z) plane (phi = 90o)
at the operating resonance of 7.41 GHz. It is noticed that at the operating
resonance the maximum radiation is obtained in the broadside direction for both
planes. The back-lobe radiation is sufficiently small and is measured to be -40
dB. This low back-lobe radiation is an added advantage for using this antenna in
a cellular phone, since it reduces
Fig. 5.7 The input impedance of the rectangular antenna near the operating resonance of 7.5 GHz.
7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8-50
-40
-30
-20
-10
0
10
20
30
40
50
Frequency (GHZ)
Real
Imag.Z
in (Ω
)
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 106
the amount of electromagnetic radiation which travels towards the users head,
it is noticed that the beam-width in both planes is wide enough.
5.2.3.4 Other Calculated Parameters
Some of the other calculated parameters obtained from IE3D, such as
radiation efficiency, antenna efficiency, directivity, gain, and 3 dB beam-width
for the rectangular patch antenna at 7.5 GHz are demonstrated below.
Radiation efficiency of an antenna is defined as:
Radiation efficiency = Radiated power / Input power
It is to measure how much energy is radiated and how much is lost inside
the antenna from the net input power.
Fig. 5.8 Radiation patterns of the rectangular patch antenna in the
xy plane (phi = 90o) and the xz plane (phi = 0o).
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 107
Input power is the net input power into the antenna. It is defined as:
Input Power = Incident power – Reflected power.
In our case the radiation efficiency was found 88.316 % as shown in
Figure 5.9.
Antenna efficiency is defined as:
Antenna Efficiency = Radiated Power / Incident Power
It considers the reflection as a loss to the antenna. The Antenna efficiency
definition is good for constant wave source which generates a constant
incident wave independent of the reflected wave. In our case the radiation
efficiency was found 86.356 % as shown in Figure 5.9.
Directivity is a measure on how much an antenna concentrates on the
radiation at specific angles. It is defined as:
Directivity = 4 |E(, )|2 / ∫∫|E(, )|2 sin()d d
Where |E(, )| is the relative E-field density at specific angles or the
radiation pattern distribution of the antenna. Note that the directivity of an
antenna is only dependent upon the shape of the pattern or the E(, ) at
all the angles. It is independent of the matching and losses of the antenna.
Its unit is dBi means the dB value compared to an ideally isotropic pattern
or a pattern with constant |E (,)|. In our case the directivity was found
7.762 dBi as shown in Figure 5.10.
Gain is defined as:
Gain = Radiation efficiency * Directivity
In our case the gain was found 7.125 dBi as shown in Figure 5.10.
3 dB beam-width can be calculated from Figure 5.8, In the E-plane, the
3-dB beam-width is 70.12 degrees at 7.41 GHz, but in the H-plane it is
found to be 94.65 degrees at the same frequency. The beam-width in both
planes is wide enough.
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 108
Fig. 5.9 The antenna efficiency and radiation efficiency of the rectangular patch antenna.
Fig. 5.10 The directivity and gain of the rectangular patch antenna.
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 109
5.3 WIDEBAND E-SHAPED PATCH ANTENNAS
This section presents a single-patch wide-band microstrip antenna: the E-
shaped patch antenna. Two parallel slots are incorporated into the patch of a
microstrip antenna to expand it bandwidth. The wide-band mechanism is
explored by investigating the behavior of the currents on the patch. The slot
length, width, and position are optimized to achieve a wide bandwidth [59]. The
antenna geometry is shown in Figure 5.11. The antenna has only one patch,
which is simpler than traditional wide-band microstrip antennas. The patch size
is characterized by (L, W, h) and it is fed by a coaxial probe at position (Xf, Yf).
To expand the antenna bandwidth, two parallel slots are incorporated into this
patch and positioned symmetrically with respect to the feed point. The
topological shape of the patch resembles the letter “E,” hence the name E-
shaped patch antenna. The slot length (Ls), width (Ws), and position (Ps) are
important parameters in controlling the achievable bandwidth.
L
patch
X
Y
Ps
Ws
Ls
Coax.
h
air
Fig. 5.11 Geometry of a wide-band E-shaped patch antenna Consisting of two parallel slots in the patch.
Top View Side View
Slot
Ground Plane
(Xf, Yf)
W
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 110
Figure 5.12 demonstrates the basic idea of the wide-band mechanism of the E-
shaped patch antenna. The ordinary microstrip patch antenna can be modeled as
a simple LC resonant circuit [Fig. 5.12(a)]. Currents flow from the feeding point
to the top and bottom edges. L and C values are determined by these currents
path length. When two slots are incorporated into the patch, the resonant feature
changes, as shown in Fig. 5.12(b). In the middle part of the patch, the current
flows like normal patch. It represents the initial LC circuit and resonates at the
initial frequency.
However, at the edge part of the patch, the current has to flow around the slots
Equivalent Circuit for patch
(a)
Equivalent Circuit for center part, high frequency
Equivalent Circuit for top and bottom parts, low frequency
(b)
Fig. 5.12 Dual resonance: the wide-band mechanism of E-shaped patch antennas. (a) The ordinary microstrip patch antenna. (b) The E-shaped patch antenna.
Patch Shape
J
Patch Shape
J .
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 111
and the length of the current path is increased. This effect can be modeled as an
additional series inductance ∆Ls [60].
So the equivalent circuit of the edge part resonates at a lower frequency.
Therefore, the antenna changes from a single LC resonant circuit to a dual
resonant circuit. These two resonant circuits couple together and form a wide
bandwidth.
As an example, a wide-band E-shaped patch antenna for wireless communi-
cations is characterized in detail [59]. The antenna parameters are listed below
(in millimeters):
(L, W) = (70, 50), h = 15, (Xf, Yf) = (35, 6)
Ls = 40, Ws = 6, Ps = 10.
Figure 5.13 shows the S11 results of this E-shaped patch antenna. The S11 is
calculated using IE3D software and the results have been compared with
measurement results found in [59]. From the figure, one can observe that the E-
shaped patch antenna resonates at 2.04 and 2.46 GHz. These frequencies are
chosen because they are useful frequencies in modern wireless communications.
The E-shaped patch antenna has a wide bandwidth of 26.5%. The simple patch
antenna without slots is also simulated for comparison. It has the same height,
length and width as the E-shaped patch antenna. It is clear that this simple
antenna doesn’t match to 50 Ω.
The variation of the input impedance of the E-shaped patch antenna is shown in
Figure 5.14. It can be observed that the input resistance is compatible with the
50 ohm characteristics of the input feed line (however no perfect matching is
attained). The imaginary part of the input impedance is close to zero during the
operating band of the E-shaped patch antenna.
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 112
Fig. 5.13 S11 of the E-shaped patch antenna (measured and calculated), compared with simple patch antenna without slots.
Fig. 5.14 The input impedance of the E-shaped patch antenna.
1.5 2 2.5 3-25
-20
-15
-10
-5
0
Frequency (GHz)
E-shaped, measured [ 59]
E-shaped, calculated
Simple patch, calculated
S11
(d
B)
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 113
5.4 CAPACITIVELY PROBE-FED WIDEBAND MICROSTRIP ANTENNA
It is well known that one way to increase the bandwidth of microstrip
antennas is to use thick substrates [1]. However, for probe-fed configurations,
this leads to an unavoidable impedance mismatch due to an inductive
component associated with the long probe. In this section, it is shown how a
single-layer capacitive feeding mechanism, consisting of a small rectangular
probe-fed patch, which is capacitively coupled to the radiating element, can be
used to obtain wideband operation for probe-fed microstrip antennas on thick
substrates. The main advantages of this feeding mechanism are that all the
elements reside on a single layer and that it is very easy to fine-tune the input
impedance. Calculated as well as measured results [61] for rectangular, circular
and annular-ring geometries are included. It will also be shown how they
compare to one another.
Figure 5.15 shows the antenna structure of the rectangular, circular and
annular-ring radiating elements, together with the feeding mechanism that
consists of the small rectangular probe-fed patch on the same layer as the
radiating element. The feeding probe is positioned in the center of the small
patch. Both the radiating element and the small patch are supported by a layer of
FR-4, with t = 1.6 mm, εr = 4.4, and loss tangent tan δ = 0.02, which is
suspended in air, at h = 15 mm above a copper ground plane of 150 X 150 mm.
A probe diameter of dp = 0.9 mm is used in all cases.
In order to illustrate the performance of this feeding mechanism, antennas with
rectangular, circular and annular-ring radiating elements were designed [61] to
operate at 1.8 GHz. The annular ring was designed to operate in its TM11 mode.
When operated in the TM11 mode, the annular-ring is smaller than most of its
other counterparts, but unfortunately it also exhibits a very high input impedance
[31]. The capacitive feeding mechanism overcomes these matching problems
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 114
Lp
and at the same time also offers enhanced impedance bandwidth. The IE3D
package from Zeland Software Incorporated, which is based on the method of
moments, was used for the design process and calculated results. The radiating
element sizes can be determined from the models for standard probe-fed
microstrip elements, as the small patch has a negligible effect on the resonant
frequency of the radiating elements. Table 5.1 shows the design parameters and
specifications of the three configurations at a resonant frequency of 1.8 GHz.
Table 5.1 The specifications of the three radiating elements.
Radiating element Design parameters
Rectangular Lp= Wp = 51 mm
Circular R = 32 mm
Annular-ring R1 = 11 mm R2 = 30 mm
y
Lp
w
dp
t
airh
d w
l
Wp
x d
wl
x
y
dl
x
R
R1 R2
zx
Fig. 5.15 Antenna structure for rectangular, circular and annular-ring radiating elements (l = 10mm and w= 5mm).
y
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 115
The input impedance of all three antennas can be controlled by the parameters l,
w, and d as illustrated in Figure 5.16 for the annular-ring (the other elements
behave in a similar way). Each one of these parameters mainly affects either the
resistive or the reactive part of the input impedance. The resistive part of the
input impedance decreases with increasing d, while the reactive part increases
when increasing either l or w. Fine-tuning of the input impedance is therefore
very easy. It is clear that, in order to achieve perfect matching at the input of the
antenna the following parameters should be used:
d = 8 mm, l = 10 mm, w = 5 mm
So these parameters are used to simulate the three configurations, Figure 5.17
shows the return loss of the rectangular, circular, and annular-ring radiating
elements. Table 5.2 also shows very similar calculated and measured 10 dB
return-loss bandwidths for the three antennas.
Table 5.2 calculated and measured 10 dB return-loss bandwidths.
Rectangular Circular Annular-ring
Calculated 25.9 % 25.5 % 25.7 %
Measured [61] 26.4 % 27.9 % 26.1 %
Figure 5.18 shows the variation of the real part of the input impedance of the
rectangular, circular, and annular-ring radiating elements. It is clear that at 1.8
GHz, Rin is very close to 50 ohm. This is of course, the effect of the capacitive
feed mechanism which compensates for the inductive component associated
with the probe.
Figure 5.19 shows the variation of the imaginary part of the input impedance of
the rectangular, circular, and annular-ring radiating elements. It is clear that at
1.8 GHz, Xin is very close to zero ohms. This is of course, due to the same
reason mentioned earlier.
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 116
Fig. 5.16 Effect of parameters variation on the calculated input impedance at 1.8 GHz of the antenna with an annular-ring radiating element.
3 4 5 6 7 8 9 10 11 12 13-20
0
20
40
50
60
80
100
120
RealImaginary
spacing "d" (mm)
Zin
(Ω
)
5 6 7 8 9 10 11 12 13 14 15-30
-20
-10
0
10
20
30
40
50
60
Length of small patch "l " (mm)
Real
Imag.
Zin
(Ω
)
2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5-30
-20
-10
0
10
20
30
40
50
60
Width of small patch "w" (mm)
Real
Imaginary
Zin
(Ω
)
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 117
Fig. 5.17 Return-loss of the rectangular, circular, and annular-ring radiating elements.
Fig. 5.18 Real part of the input impedance of the rectangular, circular, and annular-ring radiating elements.
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 118
Figure 5.20 shows the calculated and measured radiation patterns in the E-plane
(x-z plane) and H-plane (y-z plane) of the circular element. The E-plane pattern
exhibits a slight asymmetry due to the probe. It has been found that this
asymmetry is also dependant on the size of the ground plane. The cross-polar
levels in the H-plane are significantly higher than that in the E-plane. Due to
their low levels, the cross-polar patterns in the E-plane have not been shown.
Once again, the other elements have similar radiation patterns, while Table 5.3
also shows very similar calculated and measured gain values for the three
antennas.
Table 5.3 calculated and measured gain values.
Rectangular Circular Annular-ring
Calculated 7.8 dBi 8.1 dBi 8.0 dBi
Measured [61] 8.2 dBi 8.6 dBi 8.5 dBi
Fig. 5.19 Imaginary part of the input impedance of the rectangular, circular, and annular-ring radiating elements.
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
Zagazig University- Electronics & Comm. Eng. Dept. 119
(b) Fig. 5.20 Co-polar and cross-polar radiation patterns of the antenna
with a circular radiating element. (a) E-plane. (b) H-plane.
(a)
-150-180 -100 -50 0 50 100 150 180.-40
-35
-30
-25
-20
-15
-10
-5
0
Elavation angle (degrees)
Calculted,co-polar.Measured [61], co-polar.
Rel
ativ
e po
we
r (d
B)
-150180 -100. -50 0 50 100 150 180-40
-35
-30
-25
-20
-15
-10
-5
0
Elavation angle (degrees)
Measured[61],cross-polarMeasured[61],co-polarCalculted, co-polarCalculted,cross-polar
Rel
ativ
e po
we
r (d
B)
Chapter 5 FDTD Analysis of Wideband Microstrip antennas
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5.5 CONCLUDING REMARKS
In this chapter, some wideband microstrip antennas have been
investigated in detail. The first one was the conventional rectangular patch
antenna, the transmission-line model has been used for the design process of this
antenna, then the FDTD method has been used to simulate this antenna and the
obtained results have been compared to other results produced using the IE3D
package and good agreements have been found. The time domain response, the
return loss, the input impedance and the radiation patterns of this patch antenna
have been obtained.
Next, the E-shaped patch antenna with wide bandwidth has been
presented. Compared to conventional wide-band microstrip patch antennas, it
has the attractive features of simplicity and small size. The wide-band
mechanism has been explored by investigating the behavior of the currents on
the patch. The obtained impedance bandwidth from this antenna was 26.7 %;
this antenna is applicable to modern wireless communication frequencies of 1.9
to 2.4 GHz.
Next, a single-layer capacitive feeding mechanism for rectangular,
circular and annular-ring probe-fed microstrip antenna elements on thick
substrates has been described. Through calculated and measured results [61],
that agree very well, it has been shown that these elements behave very similarly
and that 10 dB return-loss bandwidths in excess of 25% can be obtained. Given
that this configuration only requires a single substrate layer, the three structures
that have been described in this chapter, prove to be suitable for a wide range of
applications that require wide-band operation at a low cost.
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C H A P T E R 6
New Wideband Slotted Overlapped Patches Microstrip Antenna
6.1 INTRODUCTORY REMARKS
This chapter presents the design, fabrication, measurements, and
simulation of a novel wideband patch antenna. In this chapter the bandwidth of a
single layer microstrip patch antenna is enhanced by using multi-resonance
technique without significantly enlarging the size of the proposed antenna.
There are numerous methods to couple multiple resonances. Examples include
coupled patches [17], patches with slots (e.g. U- and E-shaped) [62], [59],
stacked patches [29], and patches with aperture-coupled feeds [10]. In some
personal wireless communications systems, such as used for triple band option,
an operating bandwidth greater than 30 percent is required. Bandwidth in
excess of 70 percent can be achieved with aperture-coupled stacked patches.
However, such configurations occupy considerable space and are not always
acceptable for integration with other circuitry. For handheld wireless systems, a
compact single patch on moderately thick substrate is preferred. For such
antenna, achieving more than 25 percent bandwidth and moderate gain presents
a challenge [63].
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In [64], the impedance bandwidth of a single-layer microstrip patch
antenna is enhanced by using multi-resonance technique. This microstrip
antenna employs three square patches that are overlapped along their diagonals
to form a non-regular single patch; this antenna has five distinct resonance
frequencies and is designed to operate from 5.09 to 8.61 GHz. It achieves 51.4
percent bandwidth for return loss < -10 dB. It has been noticed that unfortun-
ately the authors failed to give the coordinates and the specifications of the
coaxial probe feed, in this study a trial and error method is used to find out these
missing values. To simulate this antenna The FIDELITY simulator [4] which is
FDTD based is used. This simulator analyzes 3D and multilayer structures of
general shapes. It has been widely used in the design of MICs, RFICs, patch
antennas, wire antennas, and other RF/wireless antennas. It can be used to
calculate and plot the S-parameters, VSWR, current distributions as well as the
radiation patterns. FIDELITY also uses non-uniform meshing. This means that
the sizes of cells in x, y and z directions vary locally according to the
dimensions of objects in a certain area. This helps reduce the computational
domain significantly. For example, usually the dimensions of the feed are much
smaller than the dimensions of the rest of the antenna. With non-uniform
meshing, there is no need to be restricted to the small size cells that are needed
to accurately represent the feed throughout the computational domain. Of
course, due to the small size of the computational domain, a near-to-far field
transformation is used to obtain the far field pattern of the antenna. The obtained
results from FIDELITY have been compared to other results produced using
IE3D a commercial simulator based on the method of moment and good
agreements have been found.
Another antenna is proposed which is in fact a modification of the
overlapped patches microstrip antenna. A slot is incorporated into the complex
patch to expand the antenna bandwidth. It achieves 56.8 percent bandwidth for
return loss < -10 dB. The new proposed antenna is designed, fabricated, and
Chapter 6 New Wideband Slotted Overlapped Patches Microstrip Antenna
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measured. The measured results have been compared to other results produced
using the FIDELITY simulator and good agreements have been found
These two antennas provide stable far field radiation characteristics in the
entire operating band with relatively high gain. The effects of the substrate
thickness and the dielectric constant of the substrate on the bandwidth have
been studied in this work.
6.2 WIDEBAND OVERLAPPED PATCHES MICROSTRIP
ANTENNA (OPMA)
For a conventional rectangular Microstrip patch antenna, the resonance
frequency for any TMmn mode is given by James and Hall [6] as:
2
122
2
W
n
L
mcf
reff
r (6.1)
Where L and W are the length and width of the rectangular patch.
m and n are modes along L and W respectively.
εreff is the effective dielectric constant.
C is the speed of light in free space.
For the dominant TM10 mode, the resonance frequency is given by:
reff
rL
cf
2 (6.2)
From equation (6.2) It is clear that the resonance frequency of the rectangular
microstrip patch antenna is a function of its length (L), So if the microstrip
patch antenna has multiple lengths it will be multi-resonance antenna i.e. for
every different length there will be different resonance frequency, hence the
bandwidth of the microstrip patch antenna can be enhanced. This technique is
utilized in the design of the following microstrip patch antenna.
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Three square patches are overlapped along their diagonals to form a non-regular
single patch as shown in Figure 6.1, The dimensions of the patches are (W1 X
W1), (W2 X W2) and (W3 X W3), respectively. S1 and S3 indicate the
overlapping dimensions of the patches. The structure has five different resonant
lengths as follows: L1, L2, L3, L4 and L5. As an example, an antenna with the
following dimensions was designed: three square patches of dimensions (7.5 X
7.5) mm, (13.5 X 13.5) mm and (7.1 X 7.1) mm with overlapping dimensions
S1=6.4 mm and S3=4.6 mm, a dielectric substrate (Duroid 5875) of relative
permittivity εr =2.35 and thickness h =3.175 mm was used. A copper ground
plane of 37 X 37 mm was used in this design.
This antenna is fed by a coaxial probe at position ( Xf , Yf ) as shown in Figure
6.2. The probe feed location and its radius were adjusted in such a way that one
can obtain satisfactory performance. Using trial and error, it has been found that
at Xf = 4 mm, Yf = 8 mm, and a probe diameter =1.25 mm, the widest bandwidth
of this antenna is obtained.
W1 W2
W3
W3
L5
S1L1
L3
S1 S3
S3
L4L2
W1
W2
Fig. 6.1. Geometry of the multi-resonance wideband patch.
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The FDTD method full wave simulator FIDELITY is used to simulate the
overlapped patches microstrip antenna (OPMA) and the obtained results have
been compared to other results produced using IE3D, a commercial simulator
based on the method of moment and good agreements have been found between
the two generated results as shown in Figure 6.3. For a return loss less than –10
dB the frequency band ranges from 5.09 to 8.61GHz. It has a bandwidth of 51.4
% with the center frequency 6.85 GHz.
Fig. 6.2. The OPMA configuration.
h = 3.175 mm
Substrate (εr = 2.35)
YFeed point Ground plane
(Xf , Yf )Z
Y
X X
Fig. 6.3 The return loss of the OPMA.
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Fig. 6.4. VSWR of the OPMA.
Figure 6.4 shows that the antenna frequency bandwidth with VSWR<2
covers the fequency range of 4.98 to 8.67GHz. This agrees with the less than –
10 dB band of the return loss.
The FIDELITY simulator is then utilized to find out the performance of
this antenna. The gain, the input impedance, and the radiation pattern are
worked out.
Figure 6.5 illustrates the gain of the OPMA against frequency, the gain is
greater than 2 dBi in a frequency range (4.31-7.88 GHz), and the gain variations
are less than about 4 dBi across the operating frequency. Due to the fact that the
radiating apertures of the two edge patches are relatively smaller compared to
those of the main patch, the gain decreases at higher frequencies.
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Fig. 6.5 Gain of the OPMA.
Fig. 6.6 Input Impedance of the OPMA.
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The variation of the input impedance of the OPMA is shown in Figure
6.6. It can be observed that the input resistance is compatible with the 50 ohm
characteristics of the input feed line (however no perfect matching is attained).
The OPMA has five distinct resonance frequencies where the imaginary part of
the input impedance equals zero. The upper four resonances has VSWR < 2, but
the lowest, which is at 4.6 GHz has VSWR close to 3.
Since a microstrip patch antenna radiates normal to its patch surface, the
elevation pattern for = 0 and = 90 degrees would be important. Figure 6.7
shows the gain Pattern of the OPMA in the (x -z) plane ( = 0.0 deg.) at different
frequencies, it is apparent that this antenna provides stable far field radiation
characteristics in the entire operating band with relatively high gain. It is quite
clear that the radiation pattern is not symmetrical because of the asymmetry of
the patch.
It is noticed that at 6.21 GHz the maximum gain is obtained in the broadside
direction and this is measured to be 5.63 dBi for both, = 0 and = 90 degrees.
The back-lobe radiation is sufficiently small and is measured to be -14 dBi for
the above plot. This low back-lobe radiation is an added advantage for using this
antenna in a cellular phone, since it reduces the amount of electromagnetic
radiation which travels towards the users head.
Figure 6.8 shows the Radiation Pattern of the OPMA in the (y -z) plane ( =
90.0 deg.) at different frequencies, it is Clear that this pattern is also not
symmetrical due to the same cause. However, the beamwidth in both planes is
wide enough. (Note that the frequencies at which the radiation patterns are
shown are within the operating frequencies of the OPMA, i.e. S11<-10 dB at
these frequencies. Refer to Figure 6.3 to notice this).
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dBi
Fig. 6.7 Radiation Pattern of the OPMA in the XZ-plane.
Fig. 6.8 Radiation Pattern of the OPMA in the YZ-plane.
dBi
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Fig. 6.9 Effect of the substrate height on the return loss of the OPMA.
Table 6.1 Comparison of different substrate thicknesses.
It is known that the easiest way to incease the bandwidth of a microstrip antenna
is to print the antenna on a thicker substrate. Figure 6.9 and table 6.1 show that
as the thikness of the substrate increases the bandwidth increases. However,
thick substrates tend to trap surface wave modes, especially if the dielectic
constant of the substate is high. In addition, longer coaxial probe feeds will
experience high inductive feed effects. Finally, if the substrate is very thick,
radiating modes higher than the fundamental will be excited. [25] All of these
effects degrade the primary radiator, cause pattern distortion, and detune the
input impedance of the microstrip antenna.
Substrate
thickness
Dielectric
constant Operating Band Bandwidth
2.75 mm 2.35 5.69 – 8.97 GHz 44.7 %
3.0 mm 2.35 5.47 - 8.69 GHz 45.5 %
3.175 mm 2.35 5.09 – 8.61 GHz 51.4 %
3.5 mm 2.35 4.8 – 8.23 GHz 52.6 %
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Another way to increase the bandwidth of a microstrip antenna is to decease the
dielelectic constant of the substarte [25]. Figure 6.10 and table 6.2 show that as
the dielectic constant of the substate decreases the bandwidth increases.
However, this has detrimental effects on antenna size reduction since the
resonant length of a microstrip antenna is shorter for higher substrate dielectric
constant.
In addition, this antenna can easily be used in other frequency bands with
different substrate materials.
Dielectric
constant
Substrate
thickness Operating Band Bandwidth
3.8 3.175 mm 5.43 – 6.75 GHz 21.7 %
2.35 3.175 mm 5.09 – 8.61 GHz 51.4 %
2.2 3.175 mm 5.11 – 8.71 GHz 52.1 %
1.0 3.175 mm 6.90 – 12.07 GHz 54.5 %
Fig. 6.10 Effect of the dielectric constant on the return loss of the OPMA.
Table 6.2 Comparison of different substrate materials.
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6.3 NEW WIDEBAND SLOTTED OVERLAPPED PATCHES MICROSTRIP ANTENNA (SOPMA)
Another antenna is proposed which is in fact a modification of the
overlapped patches microstrip antenna. In the proposed SOPMA a slot is
incorporated into the complex patch to expand its bandwidth. The OPMA
structure used in the previous section is reused here but modified by inserting a
slot. The slot is selected to be 5.1 mm X 0.5 mm and its lower left point is
located at (4.625 mm,5.3 mm ). The new SOPMA structure is shown in Figure
6.11.
The proposed antenna was constructed and studied. Using a Vector
Network Analyzer (Agilent 8719ES) , which covers the frequency range of 50
MHz up to 13.5 GHz and is shown in Figure 6.12. A photo of the proposed
antenna is shown in Figure 6.13.
Figure 6.14 shows very similar measured and calculated return loss S11
for the proposed antenna, it is clear that the SOPMA has 56.8 % bandwidth
compared with 51.4 % of the OPMA i.e. wider bandwidth. Also it is clear that
the value of S11 at resonance is improved by the inserted slot.
Figure 6.15 also shows very similar measured and calculated VSWR for the
proposed antenna, the antenna frequency bandwidth with VSWR<2 covers the
fequency range of 4.78-8.57 GHz with the center frequency 6.675 GHz.
εr= 2.35
Slot
Probe Feed
h = 3.175 mm
17.1 mm
XGround plane
Fig. 6.11 The SOPMA configuration.
Y
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Fig. 6.13 A photo of the slotted overlapped patches microstrip antenna.
(a) Top view, (b) Back view
(b)
Fig. 6.12 A photo of the vector network analyzer used in the measurements.
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4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9-25
-20
-15
-10
-5
0
Frequency (GHz)
S1
1 (
dB
)CalculatedMeasured
4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 91
2
3
4
5
6
7
8
Frequency (GHz)
VS
WR
Calculated
Measured
Fig. 6.14 Measured and calculated return loss for the proposed antenna.
Fig. 6.15 Measured and calculated VSWR for the proposed antenna.
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Zin
(Ω
)
Fig. 6.16 Input Impedance of the SOPMA.
The variation of the input impedance of the SOPMA is shown in Figure 6.16. It
can be observed that this antenna has five resonance frequencies where the
imaginary part of the input impedance equals zero. The upper four resonances
has VSWR < 2, but the lowest, which is at 4.55 GHz has VSWR close to 3.
Figure 6.17 shows that the resonance frequencies of the SOPMA are lower than
the resonance frequencies of the OPMA. This is of course, the effect of the slot
which adds an inductance to the equivalent circuit of the patch, this added
inductance naturally lowers the resonance frequencies as indicated in Figure
6.17.
Fig. 6.17. The imaginary part of the input impedance of both the OPMA and the SOPMA
Chapter 6 New Wideband Slotted Overlapped Patches Microstrip Antenna
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6.4 CONCLUDING REMARKS
In this chapter, two designs for small-size wide-bandwidth microstrip
patch antennas have been presented. The first design employs three square
patches that are overlapped along their diagonals and has been simulated using
two commercial field solvers, the obtained bandwidth was 51.4%. In the second
design a slot is incorporated into the complex patch to expand its bandwidth, the
second design has been fabricted and measured. It achieves 56.8 percent
bandwidth for return loss < -10 dB. Each structure of these designs can be easily
fabricated on a single-layer and relatively thin substrate for applications in hand-
held devices. It has been shown that these antennas can easily be used in other
frequency bands with different substrate materials.
137
C H A P T E R 7
Circularly Polarized Wideband Microstrip Antennas
7.1 INTRODUCTORY REMARKS
This chapter presents two circularly polarized wideband microstrip
antennas.
The first one is the dual-band circularly polarized patch antenna, this patch has
a square shape and it is loaded by four slots close to the radiating edges.
Simulations will be shown and compared with the published data and good
agreements will be shown.
The second one is a new circularly polarized capacitively probe-fed
microstrip antenna; this antenna consists of two small probe-fed rectangular
patches, which are capacitively coupled to the radiating element. The proposed
antenna is designed to achieve three targets; wide bandwidth up to 27 %,
perfect matching at the input (Zin ≈ 50 ohms), and circular polarization at
resonance. It can be claimed that this is the first time to realize such microstrip
antenna to achieve the three mentioned targets together.
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7.2 DUAL-BAND CIRCULARLY POLARIZED PATCH ANTENNA
The multi-frequency antennas are often required in telecommunication
and radar applications. An inherently multi-resonant planar antenna allows to
benefit the well known advantages in size/weight/cost reduction [65].
Typically, a dual-frequency operating mode is achieved by using multi-layer
stacked patches [65]. Single layer structures for linear polarization have been
proposed in [66] and [67]. Both structures are based on a rectangular patch
loaded by two slots. As demonstrated in [66], when these slots are etched close
to the radiating edges, they do not change significantly the first resonant
frequency and the radiation pattern of the patch. Furthermore they introduce
another resonance with a radiation pattern similar to the former. This latter
resonance is strongly dominated by the slot lengths.
In this section the use of slot loaded patches is extended to circular polarization
(CP). This is obtained by properly shaping the patch and by introducing two
other loading slots. Simulations will be shown and compared with the measured
results found in [68].
7.2.1 Antenna Geometry
The geometry of the CP slotted patch is shown in Figure 7.1. The patch
has a square shape and it is loaded by four slots close to the radiating edges.
Both the width of the slots (d) and the distance between the slots and the patch
edges (s) are comparable with the substrate thickness. The dielectric substrate
used is RT-Duroid substrate (thickness t = 1.575 mm, εr = 2.2).
The dimensions of an S-Band CP patch antenna are as follows:
W = L = 40 mm, Ws = Ls = 36 mm, Wp = Lp = 13mm, d = 1mm, s = 1.5mm.
As demonstrated in [66]. the higher order resonant modes of the slotted patch
differ from those excited in a patch without slots. While the TM100 mode
remains unchanged, the TM300 mode in the slotted cavity is strongly modified
Chapter 7 Circular Polarized Wideband Microstrip Antennas
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by the slot loading. As a result, the far field pattern associated to this mode does
not exhibit grating lobes but a rather regular behavior, which is similar to that of
the TM100 mode.
7.2.2 Antenna Feed
The structure may be fed either by coaxial probes or by apertures. The
radiation pattern regularity and the polarization purity are influenced by the
symmetry of the feeds. In order to obtain the highest degree of symmetry either
a cross shaped coupling aperture or four coaxial probes may be used. In the
latter case, the probes have to be fed with phases in rotational sequence [68].
The CP for conventional patches may be obtained by starting from a proper
single feeding point and by detuning the two resonant dimensions. This
arrangement cannot be easily used for our structure because it should require a
simultaneous detuning of both the patch resonant dimensions and the mutual
s
Wp
Ws
Substrate εr
tL
Ls
W
Lp
d
Fig. 7.1 Geometry of the dual-band circularly polarized patch antenna.
Chapter 7 Circular Polarized Wideband Microstrip Antennas
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distance between the loading slots. The use of at least two feeding points is thus
applicable in order to simplify the design of the antenna.
For the considerations previously discussed, a two probe feeding points
configuration was chosen. The two coaxial probes were placed on the two axes
of the patch, at distances Wp and Lp from the edges (Fig. 7.1).
7.2.3 Simulation Results and Discussions
S-Band rectangular patche on RT-Duroid substrate has been simulated
using IE3D software. Perfect agreement is observed between the IE3D
simulation results (Fig. 7.2 and Fig. 7.4) and the measured data [68] shown in
Fig. 7.3 and Fig. 7.5. Please note that the IE3D's results and the measured results
are using different scales.
From an electrical point of view, the structure can be described by 2-port
scattering matrix. Both S11 and S22 perfectly matches each other in magnitude
(Fig. 7.2) and phase (Fig. 7.4) due to the symmetry of the structure. The antenna
has three resonant frequencies at 2.3, 2.73, and 3.24 GHz.
It was claimed in the referenced paper [68] that the resonance at 2.3 GHz and
3.24 GHz are good for circular polarization and the resonance at 2.73 GHz is not
good for circular polarization. The conclusion was based upon the fact that
|S21| and |S12| are almost 0 at 2.3 GHz and 3.24 GHz and are almost 1 at 2.73
GHz.
But there is another conclusion, the circular polarization can be verified by
looking at the radiation patterns (Fig. 7.6), directivity (Fig. 7.7) and efficiency
(Fig. 7.8) of the antenna at the three different frequencies. When the 2-ports are
excited with 90o phase difference, the antenna yields good circular polarized
pattern with cross-polarization below -20 dB from theta = -60o to 60o at 2.3 GHz
and 3.24 GHz. At 2.73 GHz, the pattern is not a circular polarization and the
efficiency is not acceptable.
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Fig. 7.2 Calculated S-parameters for the dual-band CP antenna.
Fig. 7.3 Measured S-parameters from literature [68].
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Fig. 7.4 Calculated phase of S11 and S22 for the dual-band CP patch antenna.
Fig. 7.5 Measured phase of S11 and S22 from literature [68].
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Fig. 7.6 Radiation patterns at the three resonant frequencies.
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Fig. 7.7 Directivity of the dual-band CP patch antenna.
Fig. 7.8 Radiating efficiency of the dual-band CP patch
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Table 7.1 shows the values of the directivity and efficiency at the three
resonant frequencies for the dual-band CP patch antenna.
Table 7.1 The values of the directivity and efficiency at the three resonant frequencies.
Freq. (GHz) Directivity (dBi) Efficiency 2.30 7.5 82.2 %
2.73 4.6 7.0 %
3.24 7.4 71.1 %
It is clear that, at the first and third resonances the directivity and efficiency are
greater than their values at the second resonance, with the efficiency at the
second resonance quite small.
As it is well known, the position of the feeding points severely influences the
matching. For conventional patches, the impedance level increases as the
feeding point moves toward an edge and it is always possible to find an
optimum position for impedance matching. For dual band operations, this
particular feed position has to satisfy the matching requirements at both
frequencies.
Table 7.2 shows the behavior of the impedance matching vs. probe position at
the two frequencies for the first and third resonances. The best trade-off between
the matching at the two frequencies was found at Wp = Lp = 13mm.
Table 7.2 Impedance matching vs. probe positions
|S11| (dB)
Wp = Lp (mm) Freq.= 2.3 GHz Freq.= 3.24 GHz
8 -7.4 -5.6
10.5 -11.1 -7.2
13 -24.3 -12.9
15.5 -7.9 -16.1
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Figure 7.9 shows that the axial ratio at the first and third resonances is less than
3 dB, but at the second resonance, the axial ratio is higher than 3 dB and reaches
about 21 dB. This, also, verifies that the circular polarization occurs at the first
and third resonance.
Fig. 7.9 Axial ratio of the dual-band CP patch antenna..
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7.3 NEW CIRCULARLY POLARIZED CAPACITIVELY PROBE-FED WIDEBAND MICROSTRIP ANTENNA
The capacitively probe-fed microstrip antenna presented in Chapter (5) is
now modified to produce circular polarization (CP). This is obtained by properly
introducing another small rectangular probe-fed patch, which is also capacitively
coupled to the radiating element. The proposed antenna is shown in Figure 7.10.
The proposed antenna is designed to achieve three targets:
Wide bandwidth up to 27 % due to the multilayered substrate structure.
Perfect matching at the input (Zin ≈ 50 ohms) due to the capacitive feed.
Circular polarization at resonance due to the dual feed.
w
l
d
Lp
dp
t
airh
d
w
l
Wp
x
zx
Fig. 7.10 The proposed circularly polarized capacitively probe-fed wideband microstrip antenna .
y
Chapter 7 Circular Polarized Wideband Microstrip Antennas
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The antenna parameters are the same parameters used in Chapter (5). Each
feeding probe is positioned in the center of the small patch. Both the radiating
element and the small patch are supported by a layer of FR-4, with t = 1.6 mm ,
εr = 4.4, and loss tangent tan δ = 0.02, which is suspended in air, at h = 15 mm
above a copper ground plane of 150 X 150 mm. Each probe has a diameter of
dp = 0.9 mm. The other parameters are:
Lp= Wp = 51 mm, d = 8 mm, l = 10 mm, w = 5 mm.
7.3.1 Simulation, Analysis, and Discussions
The Fidelity software which is based on FDTD method is used to simulate
the proposed antenna. FIDELITY uses non-uniform meshing. This means that
the sizes of cells in x, y and z directions vary locally according to the
dimensions of objects in a certain area. This helps reduce the computational
domain significantly.
To model the thickness of the air substrate correctly, ∆Z is chosen so that ten
nodes exactly match the air thickness. For the FR-4 substrate ∆Z is chosen so
that three nodes exactly match the FR-4 thickness.
∆Z = 1.500 mm (in the air substrate)
∆Z = 0.533 mm (in the FR-4 substrate)
An additional 20 nodes of size 0.533 mm in the z direction are used to model
the free space above the substrate.
In order to correctly model the dimensions of the radiating element antenna, ∆x
and ∆y have been chosen so that an integral number of nodes will exactly fit the
square patch. The space steps used are ∆x = ∆y = 0.850 mm,. The radiating
square patch is thus 60 ∆x x 60 ∆y. For the small probe-fed patch, The space
steps used are ∆x = ∆y = 0.50 mm. In choosing the time step, the smallest
dimension of space steps is used to get
Chapter 7 Circular Polarized Wideband Microstrip Antennas
Zagazig University- Electronics & Comm. Eng. Dept. 149
spicosecond 833.0106
5.0
.2
_8
X
mm
c
stepSpacet
o
The patch is excited by a Gaussian pulse of half-width T = 18 ps, and the time
delay to is set to be 3T so the Gaussian will start at approximately 0. The
simulation is performed for 9000 time steps.
Figure 7.11 shows that for a return loss (S11) less than –10 dB the
frequency band ranges from 1.67 to 2.19 GHz. It has a bandwidth of 27 % with
the resonance frequency at 1.78 GHz which is very close to 1.8 GHz used in
modern wireless communications. The S12 at resonance is about -18 dB and it is
an acceptable value to achieve circular polarization.
Figure 7.12 shows the variation of the input impedance of the proposed
antenna. It is clear that at 1.8 GHz, The real part of the input impedance is
exactly 50 ohm, and the imaginary part is very close to zero ohm. This is of
course, the effect of the capacitive feed mechanism which compensates for the
inductive component associated with the probe.
The circular polarization can be verified by looking at the radiation
patterns (Fig. 7.13) of the proposed antenna. When the 2-ports are excited with
90o phase difference, the antenna yields good circular polarized pattern with
cross-polarization below -13 dB from theta = -60o to 60o at 1.78 GHz.
Figure 7.14 shows that the axial ratio is below 3 dB from 1.5 to 2.02 GHz,
, the axial ratio has an acceptable value of 2.2 dB at the resonant frequency of
1.8 GHz.
The effective bandwidth (where both the S11 is less than -10 dB AND the axial
ratio is less than 3 dB), is from 1.67 to 2.02 GHz, this bandwidth equals 18.9 %.
Figure 7.15 shows that the directivity of the proposed circularly polarized
capacitively probe-fed wideband microstrip antenna has an acceptable value of
7.7 dBi at the resonant frequency of 1.8 GHz.
Chapter 7 Circular Polarized Wideband Microstrip Antennas
Zagazig University- Electronics & Comm. Eng. Dept. 150
Figure 7.16 shows that the radiating efficiency of the proposed antenna
has a good value of 93 % at the resonant frequency of 1.8 GHz.
Fig. 7.11 The S-parameters of the proposed circularly polarized capacitively probe-fed wideband microstrip antenna .
Fig. 7.12 The input impedance of the proposed circularly polarized capacitively probe-fed wideband microstrip antenna .
Chapter 7 Circular Polarized Wideband Microstrip Antennas
Zagazig University- Electronics & Comm. Eng. Dept. 151
Fig. 7.13 The radiation pattern of the proposed circularly polarized capacitively probe-fed wideband microstrip antenna .
Fig. 7.14 The axial ratio of the proposed circularly polarized capacitively probe-fed wideband microstrip antenna.
Chapter 7 Circular Polarized Wideband Microstrip Antennas
Zagazig University- Electronics & Comm. Eng. Dept. 152
Fig. 7.15 The directivity of the proposed circularly polarized capacitively probe-fed wideband microstrip antenna .
Fig. 7.16 The radiating efficiency of the proposed circularly polarized capacitively probe-fed wideband microstrip antenna .
Chapter 7 Circular Polarized Wideband Microstrip Antennas
Zagazig University- Electronics & Comm. Eng. Dept. 153
7.4 CONCLUDING REMARKS
It has been shown in this chapter how to use dual-feed for microstrip
patch antenna to obtain circular polarization at specific frequency points.
Two circularly polarized wideband microstrip antennas have been presented.
The first one is the dual-band circularly polarized patch antenna, this patch has
a square shape and it is loaded by four slots close to the radiating edges. The
circular polarization has been verified by two approaches, the first one is the
value of S-parameters at the resonance frequencies. The conclusion was based
upon the fact that when |S21| and |S12| are almost 0, circular polarization occurs,
but when |S21| and |S12| are almost 1, no circular polarization. The second
approach is based on investigating the radiation pattern of the antenna.
The second antenna is a new circularly polarized capacitively probe-fed
microstrip antenna; this antenna consists of two small probe-fed rectangular
patches, which are capacitively coupled to the radiating element. The proposed
antenna is designed to achieve three targets; wide bandwidth up to 27 %,
perfect matching at the input (Zin ≈ 50 ohms), and circular polarization at the
resonance. It can be claimed that this is the first time to realize such microstrip
antenna to achieve the three mentioned targets together.
154
C H A P T E R 8
Conclusions and Future Research
8.1 GENERAL CONCLUSIONS
The principal contributions of this study include the design, fabrication,
and analysis of a new compact wideband overlapped patches microstrip
antenna. In this design the bandwidth of a single layer microstrip patch antenna
is enhanced by using multi-resonance technique without significantly enlarging
the size of the proposed antenna. In this work the validity of the design concept
is demonstrated by two examples with 51.4% and 56.8% bandwidths.
In The first example multiple resonances are achieved by overlapping three
square patches of different dimensions along their diagonals to form a non-
regular single patch, but in the second example a slot is incorporated into this
patch to expand its bandwidth, the second antenna has been designed,
fabricated, and measured. Good agreements have been found between the
calculated results and the experiments.
These two antennas provide stable far field radiation characteristics in the
entire operating band, with relatively high gain. The effects of the substrate
thickness and the dielectric constant of the substrate on the bandwidth have
Chapter 8 Conclusions and Future Research
Zagazig University- Electronics & Comm. Eng. Dept. 155
been studied in this work.
It has been found that, in order to obtain a wideband microstrip patch antenna
with good efficiency, a thick substrate with a very low dielectric constant
should be used.
The feeding technique utilized in this design is the coaxial probe-feed. The
main advantage of this type of feeding scheme is that the feed can be placed at
any desired location inside the patch in order to match with its input impedance.
This feed method is easy to fabricate and has low spurious radiation. Such
antenna configurations are very useful in the wireless communications industry.
To simulate these antennas The FIDELITY simulator which is FDTD based has
been used. The obtained results from FIDELITY have been compared to other
results produced using IE3D a commercial simulator based on the method of
moments and good agreements have been found.
It has been found that, when properly implemented, FDTD analysis of
different shapes of antennas produces results for near-fields, far-fields, return
loss, and input impedance that agree very well with published experimental
data. FDTD method has a powerful ability to provide, in straight forward
manner, results of antenna structures performance over a wideband of
frequency. This robustness allows the use of the FDTD method to confidently
test proposed novel antenna designs on the computer before they are built.
In this thesis the FDTD method has been used to characterize several forms of
wideband microstrip patch antennas such as rectangular, circular, and annular
ring patch antennas. The time domain response, the return loss, the input
impedance and the radiation patterns of these patch antennas are obtained.
Another major contribution is the design and analysis of a new circularly
polarized wideband probe-fed microstrip patch antenna with capacitive feed
mechanism. this antenna consists of two small probe-fed rectangular patches,
which are capacitively coupled to the radiating element. The proposed antenna is
designed to achieve three targets; wide bandwidth up to 27 %, perfect matching
Chapter 8 Conclusions and Future Research
Zagazig University- Electronics & Comm. Eng. Dept. 156
at the input (Zin ≈ 50 ohms), and circular polarization at resonance. This
antenna is designed to operate at 1.8 GHz so it is applicable to Personal
Communication System (PCS) which uses the frequency range from 1850-1990
MHz. One can claim that this is the first time to achieve and realize a microstrip
antenna to satisfy the mentioned three targets together.
It has been found that, the FDTD method has a major disadvantage which
is that it simulates structures in the time-domain. This requires a large memory
storage and large run-times. However, this problem can be reduced by using
modern powerful computers.
It has been found that, the type of feeding and the position of the feeder
affect greatly the value of the resonance frequency. For nonsymmetrical shapes
of patch fed with coaxial probe, different values of resonance frequency can be
obtained through different positions of probe which may be useful to design a
smart antenna by varying the feed point using an appropriately designed feed
network.
8.2 FUTURE RESEARCH
As is the case with all research, there are always more aspects that can be
investigated than what is already been done. Additional work may be suggested
as natural extension of the work reported in this thesis as follows.
Some applications, especially space applications, require the use of
circular polarization. There are various ways to obtain circular
polarization with conventional probe-fed microstrip patch antennas. It
can be done with one probe, or by using sequentially rotated patches.
It should be possible to apply the same ideas to the new circularly
polarized patch antenna presented in this thesis.
Chapter 8 Conclusions and Future Research
Zagazig University- Electronics & Comm. Eng. Dept. 157
Arrays of microstrip patches are important in many applications. They
can be analyzed using FDTD technique, provided suitable computer
resources are available.
Another area for future work is to extend the wideband concept to a
multiband design. It is expected in the future, that a single handset
would serve a number of applications. When the user would be at
home, the handset would operate in the same frequency range as used
by cordless phones and thus would be connected to the local telephone
exchange. When the user would be outside his house, the handset
would connect to the cellular network. On a business trip away from
home, the handset would then connect though the satellite network to
provide service to the user. These different networks would require
that the antenna in the handset is able to operate at separated
frequency bands. The antennas designed in this thesis is uniband
antennas and few are dual-band antennas. Work must be done to
design a multiband microstrip patch antenna which can operate at
several frequencies to serve multiple applications.
The patch antennas, considered in this thesis have a conducting
ground planes and metallization on the top of the substrate. These
electric conductors are assumed to be perfectly conducting and have
zero thickness and are treated by setting the electric field components
that lie on the conductors to zero. It is recommended to extend the
analysis to patch antennas with conductors of finite thickness and
conductivity.
The modeling of wideband antennas often require the analysis to be
performed over a large number of frequency points. It is possible to
reduce the computational time by analyzing the antenna at only a few
selected frequency points and to then use interpolation in some
intelligent way to find the response at other frequency points. Some
Chapter 8 Conclusions and Future Research
Zagazig University- Electronics & Comm. Eng. Dept. 158
people have used the neural network technique to achieve something
like that but in other applications in a different context.
The feed location of the microstrip antenna must be located at that
point on the patch, where the input impedance is 50 ohms for the
resonant frequency. Hence, in this study a trial and error method has
been used to locate the feed point. This can clarify the need for a
technique to be developed to find out the appropriate location of the
feed. This is yet a major drawback in the microstrip antenna design
procedure.
159
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167
Publications
[1] H. M. Abdel-Salam, A. A. Shaalan, S. H. Zainud-Deen, and K. H.
Awadalla, “Compact wideband overlapped patches microstrip antenna,”
Accepted for publication in the 23rd National Radio Science Conf.,
Faculty of Electronic Engineering, Menoufia University, Egypt, March
2006.
1
ملخص الرسالة
, تستخدم الهوائيات الشريطية في نطاق واسع من التطبيقات وذلك لكونها رخيصة الثمن
ولخفة وزنها، وصغر حجمها، وسهولة تصنيعها، , ولسهولة تشكيلها على الجسم الموضوعة عليه
ونظرا . يكروويةفي نطاق الموجات الم وآذلك إلمكانية تجميعها مع الدوائر المتكاملة المستخدمة
لمميزات تلك الهوائيات فقد صار استخدامها شائعا في تطبيقات مختلفة بداية من الطائرات
والصواريخ وسفن الفضاء واالتصاالت المحمولة واالستشعار عن بعد إلى المشعات والمجسات
.المستخدمة في التطبيقات الطبية
هذه الرسالة إلى استخدام فر عيوبه وتهدويعتبر ضيق الحيز الترددي للهوائي الشريطي من أخط
طريقة الفروق المحددة في الحيز الزمني في تحليل الهوائيات الشريطية ذات النطاق الترددي
العريض وتم دراسة عدة أشكال من الهوائيات الشريطية على طبقات من مواد ذات خواص
الدائرية والحلقية والهوائي على موحدة أتجاهيا مثل الهوائيات الشريطية المربعة و المستطيلة و
.Eشكل حرف
آذلك قدمت هذه الرسالة طريقة الطبقات المتوائمة تماما لتحقيق شروط االمتصاص
لحساب المجال في الحيز الزمني والفقد الناتج عن االنعكاس C وتم استخدام لغة البرمجة . الحدية
.اسةومعاوقة الدخل وآذلك مجسم اإلشعاع للهوائيات موضع الدر
وقد تم في هذه الرسالة تصميم وتصنيع و قياس الخواص المختلفة لشكل جديد من الهوائيات
ويعتمد هذا النوع على استخدام مبدأ . الشريطية هو الهوائيات الشريطية المتداخلة صغيرة الحجم
م المحاآي وتم تحليل هذا النوع باستخدا. تعدد الترددات الرنينية في زيادة الحيز الترددي للهوائي
Fidelityطريقة الفروق المحددة في الحيز الزمني ومقارنة النتائج مع النتائج والذي يعتمد على
وتم زيادة الحيز الترددي لهذا النوع الجديد بنسبة . العملية وقد جاءت النتائج متقاربة إلى حد آبير
56.8 .%
يات الشريطية ذات االستقطاب آذلك تم في هذه الرسالة تصميم وتحليل شكل جديد من الهوائ
وتم . الدائري و النطاق الترددي العريض هو الهوائيات الشريطية ذات التغذية الثنائية السعوية
والحصول على مقاومة دخل مقـدارها %. 27ديد بنسبة ـنوع الجـذا الـيز الترددي لهـزيادة الح
رسالةملخص ال
2
آلية الهندسة-جامعة الزقازيق
1.8 ي عند التردد الرنينيطاب دائرـ والحصول أيضا على استق محوري أوم لمجس تغذية50
GHz .
:يمكن تلخيصها على النحو التالي أبواب ثمانيةوتحتوي الرسالة علي
األولالباب :
وآذا التعريفات الخاصة , ويشمل مقدمة تحتوى على نبذة تاريخية عن الهوائيات الشريطية
لشريطية الذي تم تصميمه و آذا الهدف من هذا البحث و توضيح الشكل الجديد من الهوائيات ا, به
.في هذا البحث ، باإلضافة إلى عرض مختصر لكل أبواب الرسالة
:الثاني الباب
يستعرض هذا الباب المواصفات األساسية للهوائي الشريطي، ومزاياه وعيوبه، وطرق
ي التغذية المختلفة والمقارنة بينهم، ويستعرض أيضا بعض طرق التحليل الشائعة للهوائي الشريط
.والمقارنة بينهم
:الثالث الباب
يستعرض هذا الباب الطرق المختلفة وخاصة الحديث منها في توسيع النطاق الترددي
لهوائيات الشريطية، وتشمل هذه الطرق الهوائيات الشريطية ذات الفتحات والهوائيات الشريطية ل
لباب أيضا مبدأ عمل الشكل ويستعرض هذا ا. المتعددة الطبقات والهوائيات الشريطية المتزاوجة
.الجديد من الهوائيات الشريطية الذي تم تصميمه في هذا البحث
: الباب الرابع
يقدم هذا الباب األساس النظري لطريقة الفروق المحددة في الحيز الزمني، مع سرد
شروط االمتصاص الحدية ومنها طريقة الطبقات تطبيقمميزات وتطبيقات هذه الطريقة وطرق
الباب بالتدريج في تناول طريقة الفروق المحددة في الحيز الزمني اآما يتميز هذ .المتوائمة تماما
تم أوال تناول بعد واحد ثم ويتم تناول الطريقة في خطين متوازيين الخط األول هو البعد حيث
من ةومغناطيسي الثاني هو نوع المادة التي يتم انتشار الموجات الكهرطبعدين ثم ثالثة أبعاد والخ
رسالةملخص ال
3
آلية الهندسة-جامعة الزقازيق
ضاء المطلق ثم خالل ـفـ خالل الة تم أوال تناول انتشار الموجات الكهرومغناطيسي حيث. خاللها
σ =constant).(ة ـينـد معـقـ ثم خالل مواد ذات نسبة ف(σ =0)د ـقـفـديمة الـمواد ع
: الباب الخامس
ئي شريطي ذو شكل ويوضح هذا الباب آيفية استخدام نموذج خطوط النقل في تصميم هوا
مستطيل مغذى بخط نقل شريطي ثم استخدام طريقة الفروق المحددة في الحيز الزمني في تحليل
والذي يعتمد على طريقة العزوم وقد IE3Dهذا الهوائي الشريطي ومقارنة النتائج مع برنامج
الناتج عن وتم حساب المجال في الحيز الزمني والفقد . جاءت النتائج متقاربة إلى حد آبير
.االنعكاس ومعاوقة الدخل وآذلك مجسم اإلشعاع للهوائي موضع الدراسة
وتم أيضا تحليل أشكال مختلفة من الهوائيات الشريطية الدائرية والحلقية ذات النطاق الترددي
فتحات لالعريض على طبقات من مواد ذات خواص موحدة أتجاهيا مثل الهوائيات الشريطية ذات ا
والهوائيات الشريطية المتعددة الطبقات وتم زيادة % 24.2حيز الترددي لها بنسبة وتم زيادة ال
%.25.8الحيز الترددي لها بنسبة
:الباب السادس
يقدم هذا الباب تصميم وتصنيع و قياس الخواص المختلفة لشكل جديد من الهوائيات
مد هذا النوع على استخدام مبدأ الشريطية هو الهوائيات الشريطية المتداخلة صغيرة الحجم ويعت
تعدد الترددات الرنينية في زيادة الحيز الترددي للهوائي وتم تحليل هذا النوع باستخدام المحاآي
Fidelityطريقة الفروق المحددة في الحيز الزمني ومقارنة النتائج مع النتائج والذي يعتمد على
وتم زيادة الحيز الترددي لهذا النوع الجديد بنسبة . العملية وقد جاءت النتائج متقاربة إلى حد آبير
56.8.%
:الباب السابع
يقدم هذا الباب تصميم وتحليل شكل جديد من الهوائيات الشريطية ذات االستقطاب الدائري
وتم زيادة الحيز . و النطاق الترددي العريض هو الهوائيات الشريطية ذات التغذية الثنائية السعوية
أوم 50والحصول على مقاومة دخل مقدارها %. 27ا النوع الجديد بنسبة الترددي لهذ
رسالةملخص ال
4
آلية الهندسة-جامعة الزقازيق
وهى أول مرة يمكن GHz 1.8 والحصول أيضا على استقطاب دائري عند التردد الرنيني
. واحد شريطيتحقيق هذه األهداف الثالثة بواسطة هوائي
:الباب الثامن
. لنقاط بحثية مستقبليةفي هذا الباب تم عرض نتائج البحث وآذلك بعض التوصيات
. وفى نهاية الرسالة يوجد قائمة بالمراجع وبيان بالمالحق المتعلقة بالرسالة
جامعة الزقازيق آلية الهندسة االتصاالتو اإللكترونيات قسم هندسة
ذات النطاق الترددى العريضالشريطيةالهوائيات
رسالة مقدمة للحصول على درجة الماجستير في هندسة اإللكترونيات و االتصاالت
جامعة الزقازيق–بكلية الهندسة
مقدمة من مالعبد الس حسين محمود /المهندس
تحت إشراف
مال حسن عوض اهللاآ. د.أ واالتصاالتتهندسة اإللكترونياقسم جامعة المنوفية - االلكترونية بمنوف آلية الهندسة
حلمي زين الدينصابر. د.أ
اإللكترونيات واالتصاالت قسم هندسة جامعة المنوفية - االلكترونية بمنوف آلية الهندسة
عبد الحميد عبد المنعم شعالن. د.م.أ
واالتصاالت تهندسة اإللكترونياقسم رئيس جامعة الزقازيق -آلية الهندسة
2006
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