3203416 . Wideband Microstrip Antennas

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


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 ( )


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



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.



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



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



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



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 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



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



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.



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:



• 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.



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



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



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



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



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%



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



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)



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)



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




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.


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



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 :



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



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



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



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



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)




)(, 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


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



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.




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


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


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.



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.



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



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.



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.



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.



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



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)



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



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



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



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



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.



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



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



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.


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


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


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.



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.



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:



(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.



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.



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:



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



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



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)(



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.



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



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



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)



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 ))



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



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



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)



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.



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



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



(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



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.



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.



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.



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 )



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)



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.



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



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



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:



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)



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



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.



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.



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



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



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.



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



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.



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.


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


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


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.



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



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:



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.



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



)(

)()(

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



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



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

)



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)



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



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

)



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).



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.



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.



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



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 .



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.



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)



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



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



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.



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

)



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.



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.



(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)




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.

Chapter 6 New Wideband Slotted Overlapped Patches Microstrip Antenna

121

Zagazig University- Electronics & Comm. Eng. Dept.

C H A P T E R 6

New Wideband Slotted Overlapped Patches Microstrip Antenna


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].


122


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


123


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.


124


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.


125


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.


126


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.


127


Fig. 6.5 Gain of the OPMA.

Fig. 6.6 Input Impedance of the OPMA.


128


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).


129


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


130


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 %


131


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.


132


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


133


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.


134


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.


135


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


136



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


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.

Chapter 7 Circular Polarized Wideband Microstrip Antennas


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



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.



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.



Fig. 7.2 Calculated S-parameters for the dual-band CP antenna.

Fig. 7.3 Measured S-parameters from literature [68].



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].



Fig. 7.6 Radiation patterns at the three resonant frequencies.



Fig. 7.7 Directivity of the dual-band CP patch antenna.

Fig. 7.8 Radiating efficiency of the dual-band CP patch



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



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..



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



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



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.



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 .



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.



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 .




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


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



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.



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



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


3


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


4


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

3203416 . Wideband Microstrip Antennas

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Transcript of 3203416 . Wideband Microstrip Antennas