ECE416 Project Report:Design and Implementation of Compact Microstrip

Fractal Antennas

by

Paul Simedrea

Department of Electrical and Computer EngineeringFaculty Advisor: Dr. Serguei Primak

Electrical/Computer Engineering Project ReportSubmitted in partial fulfillment of the requirements

for the degree of Bachelor of Engineering Science

The University Of Western OntarioLondon, Ontario

March 29, 2004

c© Paul Simedrea, 2004All rights reserved

Abstract

The project undertaken was to construct antennas using fractal patterns in order to

obtain desired performance properties such as compact size and multi-band behaviour.

One of the prevailing trends in modern wireless mobile devices is a continuing decrease

in physical size. In addition, as integration of multiple wireless technologies becomes

possible, the wireless device will operate at multiple frequency bands. A reduction

in physical size and multi-band capability are thus important design requirements

for antennas in future wireless devices. The use of fractal patterns in antenna de-

sign provides a simple and efficient method for obtaining the desired compactness

and multi-band properties. Two proof-of-concept fractal antennas were designed and

built. A compact, multi-band antenna based on the Sierpinski gasket fractal was de-

signed to operate at 2.4 and 5.0 GHz and a compact antenna based on the Koch fractal

was designed to operate at 900 MHz. The design process consisted of initial theo-

retical calculations follwed by extensive numerical simulations, the results of which

were used as guidelines for the physical antenna implementation. Numerical electro-

magnetics simulations were performed using the software packages Sonnet and NEC2,

based on the Method-of-Moments (MoM) simulation technique. The two antennas

were implemented as microstrip patches on printed circuit boards (PCB) and were

found to perform in agreement with design expectations. The design methodology

used, simulation and test results, as well as design recommendations are presented.

Keywords: fractal antenna, koch monopole, sierpinski monopole, microstrip patch.

i

Acknowledgements

Special thanks to my friends and family who have been of great support throughout

my undergraduate years. Especially, I would like to thank my colleagues: Mr. Chris

Snow, Mr. Carlos O’Donell and Mr. Derek Hunter for their assistance during this

project. Much credit for the ideas presented in this project report is due to Dr.

Serguei Primak, my project advisor. Working with him for the past two years has

been a unforgettable learning experience. In addition, a special thanks to the staff of

the ECE Electronics Shop staff who did a marvelous job at producing the antenna

prototype PCB boards whose design is presented in this report.

ii

Contents

1 Introduction 11.1 Brief Technical Overview . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Design Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Outline of the Report . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Literature Review 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Fractal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Iterated Function Systems . . . . . . . . . . . . . . . . . . . . 42.2.2 The Sierpinski Gasket Fractal . . . . . . . . . . . . . . . . . . 52.2.3 The Koch Fractal . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Microstrip Patch Antennas . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Fractal Microstrip Antennas . . . . . . . . . . . . . . . . . . . . . . . 8

2.4.1 The Sierpinski Gasket Monopole . . . . . . . . . . . . . . . . . 82.4.2 The Koch Fractal Monopole . . . . . . . . . . . . . . . . . . . 8

2.5 Antenna Performance Simulators . . . . . . . . . . . . . . . . . . . . 9

3 Statement of Problem and Methodology of Solution 103.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Solution Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Theoretical Development . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3.1 Performance Requirements . . . . . . . . . . . . . . . . . . . . 113.3.2 Fractal Antenna Patterns . . . . . . . . . . . . . . . . . . . . 123.3.3 Implementation Considerations . . . . . . . . . . . . . . . . . 12

3.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.1 Software Simulators . . . . . . . . . . . . . . . . . . . . . . . . 13

3.5 Physical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 143.6 Experimental Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.6.1 Antenna Resonance Testing . . . . . . . . . . . . . . . . . . . 153.6.2 Radiation Performance Testing . . . . . . . . . . . . . . . . . 15

4 Design of the Sierpinski Gasket Monopole 164.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Selection of Operating Frequency . . . . . . . . . . . . . . . . . . . . 16

iii

4.3 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 Hardware Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4.1 Dielectric Substrate . . . . . . . . . . . . . . . . . . . . . . . . 174.4.2 Connectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.3 Ground Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.4 Signal Feed System . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 Software Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.6 Final Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Design of the Koch Fractal Monopole 215.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Selection of Operating Frequency . . . . . . . . . . . . . . . . . . . . 215.3 Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4 Hardware Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.4.1 Dielectric Substrate . . . . . . . . . . . . . . . . . . . . . . . . 225.4.2 Physical Design . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.5 Software Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Final Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Results and Discussion 256.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.2.1 Sierpinski Gasket Monopole Results . . . . . . . . . . . . . . . 256.2.2 Koch Fractal Monopole Performance . . . . . . . . . . . . . . 28

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.3.1 Sierpinski Gasket Monopole . . . . . . . . . . . . . . . . . . . 316.3.2 Koch Fractal Monopole . . . . . . . . . . . . . . . . . . . . . . 32

7 Conclusions 347.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A Source Code Listings 36

B Antenna Schematics 37

C Antenna Illustrations 40

D Selected Product Data Sheets 42

iv

List of Figures

2.1 A 4-iteration Sierpinski gasket. . . . . . . . . . . . . . . . . . . . . . 52.2 Iterations of the Koch fractal. . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Experimental setup for antenna reflection coefficient measurements. . 14

4.1 Signal feeding techniques for microstrip patches. . . . . . . . . . . . . 194.2 General depiction of the Sierpinski monopole. . . . . . . . . . . . . . 20

5.1 Sketch of the 900 MHz Koch monopole. . . . . . . . . . . . . . . . . . 23

6.1 Reflection coefficient of a 4-iteration Sierpinski monopole. . . . . . . . 266.2 Simulated charge density for the Sierpinski monopole. . . . . . . . . . 286.3 Koch monopole radiation pattern. . . . . . . . . . . . . . . . . . . . . 296.4 Reflection coefficient of the 900 MHz Koch monopole . . . . . . . . . 30

B.1 Detailed schematic of the 2.4/5.0 GHz Sierpinski monopole . . . . . . 38B.2 Detailed schematic of the 900 MHz Koch monopole . . . . . . . . . . 39

C.1 Photo of the Sierpinski Monopole. . . . . . . . . . . . . . . . . . . . . 41C.2 Photo of the Koch Monopole. . . . . . . . . . . . . . . . . . . . . . . 41

D.1 Datasheet for FR-4 (page 1 of 2) . . . . . . . . . . . . . . . . . . . . 43D.2 Datasheet for FR-4 (page 2 of 3) . . . . . . . . . . . . . . . . . . . . 44D.3 Datasheet for FR-4 (page 3 of 3) . . . . . . . . . . . . . . . . . . . . 45D.4 Datasheet for an SMA flange connector (page 1 of 3). . . . . . . . . . 46D.5 Datasheet for an SMA flange connector (page 2 of 3). . . . . . . . . . 47D.6 Datasheet for an SMA flange connector (page 3 of 3). . . . . . . . . . 48

v

List of Tables

4.1 Sierpinski gasket monopole geometry parameters. . . . . . . . . . . . 17

5.1 900 MHz Koch monopole design parameters. . . . . . . . . . . . . . . 24

6.1 Sierpinski gasket monopole performance parameters. . . . . . . . . . 286.2 Koch monopole performance parameters. . . . . . . . . . . . . . . . . 31

vi

Chapter 1

Introduction

1.1 Brief Technical Overview

The goal of this project is to design proof-of-concept antennas which have properties

beneficial to modern wireless communication receivers.

Although fractal geometry has been known to mathematics for a century, fractal

antenna engineering research is a relatively very recent development because consid-

erable computing speed is required to complete their design. It has been discovered

that fractal shapes radiate electromagnetic energy well and have several properties

that are advantageous over traditional antenna types. They can be used as multi -band

antennas, which can radiate signals at multiple frequency bands. Another desireable

property is that they are compact, meaning that they can occupy a portion of space

more efficiently than other antenna types.

Such antennas could be used to improve the functionality of modern wireless

communication receivers such as cellular handsets.

1.2 Motivation

As already mentioned in the previous section, fractal antenna technology could be

applied to cellular handsets. Because fractal antennas are more compact, they would

more easily fit in the receiver package. Currently, many cellular handsets use quarter-

wavelength monopoles which are essentially sections of radiating wires cut to a deter-

mined length. Although simple, they have excellent radiation properties. However,

for systems operating at 900 MHz such as GSM, the length of these monopoles is often

1

longer than the handset itself, posing a nuissance to the user. It would be highly bene-

ficial to design an antenna with similar radiation properties as the quarter-wavelength

monopole while retaining its radiation properties.

Other prevailing trends in wireless communications technology could also benefit.

More and more systems are introduced which integrate many technologies. They are

often required to operate at multiple frequency bands and so they require antenna

systems which accomodate that requirement. Examples of systems using a multi-band

antenna are varieties of common wireless networking cards used in laptop computers.

These can communicate on 802.11b networks at 2.4 GHz and 802.11g networks at 5

GHz.

Using fractal patterns in antenna design can help solve the problems mentioned

above.

1.3 Design Methodology

Since the purpose of the project is to design fractal antennas, it is useful to refine the

goals in order to obtain a design methodology:

• Given the academic nature of the design project, the monetary budget is limited.

The design must thus employ technology that is low in cost. This is a good

design rule in any case, as cheaper is always better.

• To show the potential for application to common technology, the antennas must

be designed for operating at useful frequency bands. In addition, an effort

should be made to design for radiation properties similar to those of common

systems.

• Most importantly, the advantages of fractal antennas must be proven. It must

be shown that the antenna designs are more compact than traditional antennas

or that they have multi-band behaviour.

1.4 Outline of the Report

A review of current research literature on fractal antenna technology is given in Chap-

ter 2. The literature review examines some of the relevant facets of fractal theory,

2

potential technologies for physical construction, and reviews the concepts behind ap-

plying fractal theory to antenna research.

Chapter 3 elaborates on the design methodology mentioned in the previous sec-

tion. Design requirements are presented there and a general strategy for design is

discussed. Chapters 4 and 5 then detail the actual design of the antennas that was

performed and Chapter 6 presents and discusses the complete design’s simulated and

experimental test results. Finally, some recommendations on further work as well as

a concluding statement are given in Chapter 7.

3

Chapter 2

Literature Review

2.1 Introduction

This chapter presents a literature review of the theory of fractal antennas. An in-

troduction to fractal theory is first discussed by giving examples of two fractals of

interest: the Sierpinski Gasket and the Koch fractal. Microstrip patch antenna tech-

nology is then presented to outline its application to fractal antennas. Finally, a

review of current research work on specific fractal antennas and numerical simulation

techniques is given.

2.2 Fractal Theory

Fractals are a class of shapes which have no characteristic size [1]. Each fractal is

composed of multiple iterations of a single elementary shape. The iterations can

continue infinitely, thus forming a shape within a finite boundary but of infinite

length or area. This compactness property is highly desirable in mobile wireless

communication applications because smaller receivers could be produced.

2.2.1 Iterated Function Systems

Many useful fractals can be generated by Iterated Function Systems (IFS). An ex-

tended discussion of IFS is found in [2, 3]. Briefly, IFS work by applying a series of

affine transformations w to an elementary shape A over many iterations. The affine

4

Figure 2.1: A 4-iteration Sierpinski gasket.

transformation w(x, y), comprising rotation, scaling and translation, is given by:

w(x, y) =

a b e

c d f

0 0 1

·

x

y

1

(2.1)

The set of affine transforms W (A), known as the Hutchinson operator is given by:

W (A) =

N⋃

n=1

wn(A) = w1(A) ∪ w2(A) ∪ w3(A) . . . ∪ wN(A) (2.2)

The fractal can then be generated by applying the operator W to the previous geom-

etry for k iterations. Thus:

A1 = W (A0), A2 = W (A1), . . . Ak+1 = W (Ak) (2.3)

2.2.2 The Sierpinski Gasket Fractal

The Sierpinski Gasket fractal is generated by the IFS method. As depicted in Figure

2.1, a triangular elementary shape is iteratively scaled, rotated and translated, then

removed from the original shape in order to generate a fractal. It is interesting to

note that after infinite iterations of the fractal, the entire shape has an infinite area

but is bounded by a finite perimeter.

5

2.2.3 The Koch Fractal

The Koch fractal curve is one of the most well-known fractal shapes. It consists

of repeated application of the series of IFS affine transformations [2] given in (2.4).

Multiple iterations of the Koch fractal are shown in Figure 2.2. To form the first

iteration (n = 1 in Figure 2.2), the affine transform w1 scales a straight line to one-

third of its original length. The transform w2 scales to one-third and rotates by 60.

The third transform, w3 is similar to w2 but rotating by −60. Finally the fourth

transform, w4, is simply another scaling to one-third and a translation. It can be

seen in Figure 2.2 how this set of transforms are applied to each previous iteration to

obtain the next.

w1 =

1

30 0

0 1

30

0 0 1

w2 =

1

3cos 60 −1

3sin 60 1

3

1

3sin 60 1

3cos 60 0

0 0 1

w3 =

1

3cos 60 1

3sin 60 1

3

−1

3sin 60 1

3cos 60

√3

2

0 0 1

w4 =

1

30 0

1

3

2

30

0 0 1

(2.4)

An important characteristic of the Koch fractal worthy of note is that the un-

folded length of the fractal approaches infinity as the number of iterations approach

infinity. However, the area which bounds the fractal remains constant [2]. This

property can be used to minimize the space use of a simple wire monopole or dipole

antenna [4, 5]. The length ln of the Koch fractal at each iteration n increases expo-

nentially with respect to the n = 0 length l0, as given by:

ln = l0 ·(

4

3

)n

(2.5)

2.3 Microstrip Patch Antennas

The microstrip patch antenna has become one of the most versatile radiating ele-

ment solutions for a large variety of systems [6]. Microstrip patches are essentially

conductive radiating elements etched on a thin dielectric layer. A ground plane is

etched on the opposing side of the dielectric layer from the radiating element. Mi-

6

n=0

n=1

n=2

n=3

Figure 2.2: Iterations of the Koch fractal.

crostrip patches are resonant-type antennas, meaning that in order for the antenna

to radiate, one of the dimensions of the radiating patch must be approximately half

the wavelength of the electrical excitation signal being fed to the patch.

An advantage of microstrip and printed antennas is that they can be easily and

inexpensively built using commonly available printed circuit board (PCB) technology.

Microstrip patches can thus also be produced at small size and profile, allowing easy

integration into the skins of various systems such as airplanes or cellular handsets [6].

Because they are resonant-type antennas, microstrip patches are efficient radiators.

The efficiency of microstrip patches is generally between 95% and 99% [6].

Microstrip patches also have some disadvantages, most important of which is

their limited impedance bandwidth. Because microstrip patches are resonators, its

impedance will inherently be largely real over only a small percentage around the

operating frequency for which it was designed. This characteristic prevents the use of

microstrip patches to a number of very wide-band applications. However impedance

bandwidths of up to 10% can commonly be achieved using microstrip patches, a value

sufficient for many applications.

7

2.4 Fractal Microstrip Antennas

2.4.1 The Sierpinski Gasket Monopole

The Sierpinski gasket monopole antenna was first presented by Puente et. all. It is

discussed in detail in [7, 8, 9, 10] . The Sierpinski gasket was chosen by researchers

due to its resemblance to the well-known, triangular (bow-tie) monopole antenna [10].

In addition, the triangular structure presents a simple way of feeding current to the

antenna. Current is fed through a connector at the bottom tip of the monopole.

With the exception of an iterative transmission line model presented in [9], the

theoretical principles governing operation of the Sierpinski gasket are not discussed in

great quantitative detail in literature. A qualitative analysis described in [10] provides

some insight, however. The current fed to an antenna will generally concentrate

over a region in size comparable to the current’s wavelength. The introduction of

boundaries at such locations in the Sierpinski gasket allows radiation to occur. Thus,

each resonant frequency is directly related to the dimensions of each sub-gasket.

Consequently, the 4-iteration gasket in Figure 2.1 has four resonant frequencies.

The resonant frequency band spacing is related to the similarity factor δ of the

gasket [10]. The gasket shown in Figure 2.1 has δ = 2, meaning a ratio of 2 would be

seen between adjacent bands in the corresponding monopole. A general relationship

[10] between the resonant frequency of the monopole and gasket parameters is given

by:

fn = kc

hcos(α/2)δn (2.6)

where c is the speed of light, h is the height of the monopole, α is the flare angle and

n is the resonant band number. Thus, the resonant frequency band locations can be

adjusted by varying α and δ.

2.4.2 The Koch Fractal Monopole

The Koch fractal monopole was introduced as a means to reduce the size of the

traditional quarter-wave (λ/4) straight-wire monopole [4, 5]. Size reduction is most

important for wireless communication systems operating at relatively low frequencies.

GSM, for instance, operates at 900 MHz. If a λ/4 monopole were used in a GSM

handset, its height would be h = 8.33 cm. For smaller handsets such an antenna may

be too long and a smaller, lower gain, (h λ) monopole would be used instead.

8

The space-filling properties of the Koch fractal, as shown by (2.5), can be used

to create a smaller λ/4 monopole while preserving the radiation characteristics of a

straight-wire monopole [5]. The rough, uneven shape of the fractal facilitates radia-

tion. Although (2.5) suggests that it is theoretically possible to fit an infinitely long

fractal curve in an arbitrarily small space, it is discussed in [5] that the electrical

length of a Koch monopole does not increase at the same pace as its physical length.

Thus, increasing the number of iterations to minimize the size of the monopole will

eventually no longer increase its electrical length. However, the greater limitation on

the physical length is the minimum physical trace thickness that can be produced

by the PCB printing process. The pattern can simply not be photo-etched correctly

using a line thickness relatively large compared to its length. Even if very small thick-

nesses were possible, resistive losses due to reduced conductor area would adversely

affect radiation efficiency [5, 6].

Since the Koch fractal has very complex geometry, it can most reliably be im-

plemented by using printed antenna techniques similar to those used in microstrip

patches. Thus, instead of using a wire to form the fractal, the pattern is printed on

a dielectric substrate and mounted on a reflective ground plane. At relatively low

frequencies dielectric losses are marginal and thus special dielectrics are not needed

to maintain high antenna efficiency [5, 6]. Materials commonly in use for PCB man-

ufacturing, such as FR-4 (see datasheet in Appendix D), can be used.

2.5 Antenna Performance Simulators

Several numerical techniques exist which predict the performance of free-space and mi-

crostrip antennas. All numerical techniques involve solving discrete forms of Maxwell’s

equations for time-varying fields in 1D, 2D or 3D space. The most popular techniques

in use are Method-of-Moments (MoM) and Finite Difference Time Domain (FDTD).

An excellent treatise on both methods is given in [11].

9

Chapter 3

Statement of Problem andMethodology of Solution

3.1 Problem Statement

The purpose of this project is to design fractal antennas which have desired multi-

band resonance and compact profile. The designs of a multi-band Sierpinski Gasket

monopole and of a compact Koch monopole antenna are undertaken to achieve this

goal.

3.2 Solution Overview

Since using fractals as an approach to antenna design is a relatively new development

in the field of antenna research, the Sierpinski and Koch microstrip antennas were

selected for this project. They are simple to design and their radiation properties are

far better documented in research literature than those of other types of fractals.

Before undertaking the design of both antennas, it is prudent to establish a

general design plan and to determine the constraints imposed on that plan. The

design plan includes the following phases:

1. Theoretical development: rough calculations of antenna parameters as well

as develop a general idea of the physical implementation of the antenna.

2. Numerical simulation: perform software simulations in order to verify the

theoretical design and adjust any parameters to predict the desired antenna

performance.

10

3. Physical implementation: undertake physical construction of the antennas

based on simulation-confirmed parameters.

4. Experimental testing: measure antenna performance is measured and com-

pare with the simulated results. If necessary, the antennas’ parameters are

adjusted and another implementation iteration and testing is undertaken.

Details of these design phases are discussed in the remaining sections of this

chapter.

3.3 Theoretical Development

3.3.1 Performance Requirements

Considerations for the following performance characteristics should be taken into

account before proceeding with a design:

• Resonant Frequency: Narrow band at design frequency, as is inherent of

resonant antennas. The Sierpinski Gasket monopole would exhibit a number of

resonant bands, as controlled by the design. The Koch monopole would have

only one resonant band of interest.

• Impedance Bandwidth: As discussed in Section 2.3, resonant-type anten-

nas generally have low impedance bandwidth. The input impedance remains

approximately constant in a narrow band of frequencies, generally beteween

5—10% of the centre frequency.

• Input Impedance: Should be resistive and large in value, although the first

resonance usually has low impedance and a noticeable reactive component [12].

• Radiation Pattern: As both antennas are most suitable for use in low power

applications such as mobile receivers, their radiation patterns should be suitable

for such use. This should be achieveable since both the Sierpinski and Koch

antennas are monopoles, inherently having a very uniform radiation pattern in

all directions [12, 13].

11

• Radiation Efficiency: Microstrip antennas have high radiation efficiency, be-

tween 90—99.9%. The thickness and relative dielectric constant εr of the dielec-

tric substrate used affect radiation efficiency. For higher efficiency, substrates

with low dielectric constants should be used [6].

• Space Efficiency (Compactness): Due to the space-filling properties of both

the Sierpinski and Koch fractals, the antenna designs should exhibit efficient

space usage. The Koch fractal should exhibit a much reduced profile when

compared with the traditional straight wire, λ/4 monopole. Although the aim of

the Sierpinski monopole design is multi-band operation, the antenna is expected

to be compact, since the fractal structure of the antenna represents smaller

antennas bounded by larger ones.

• Gain: The gain of the fractal antennas should be comparable to that of tradi-

tional antennas with similar radiation properties. This would allow the fractal

antennas to be used in similar applications.

3.3.2 Fractal Antenna Patterns

After establishing the required resonant frequencies, initial calculations of the geo-

metrical properties of the fractal radiators can be undertaken. The fractal pattern

features must be comparable in size to the wavelength of the input signal. As a

start, the height of each triangular structure in the Sierpinski gasket pattern should

be approximately λ/2. The resonant frequency spacing can then be approximated by

varying the parameters in equation (2.6). The Koch monopole pattern can be gen-

erated using the IFS transformation parameters given in (2.4) and scaled to a height

of approximately λ/4.

3.3.3 Implementation Considerations

Further adjustments to the geometry of the fractal are necessary due to the presence

of the dielectric substrate, which increases the electrical length of the pattern. This

is performed with the aid of numerical simulations. Caution must be used when

selecting the dielectric substrate, however: the thickness of the dielectric and its

permittivity constant is a significant factor in radiation losses. The radiation efficiency

of microstrip patches are discussed at length in [6].

12

Techniques of feeding the signal to the antenna must also be considered. The

position of the feeding point and connector on the radiating element may affect input

impedance or could introduce spurious radiation. Many connector types are avail-

able, with varying frequency responses. Popular connector types for high-frequency

applications include BNC, Type-N, SMA and SMB.

3.4 Numerical Simulation

As mentioned in Section 2.5, because of their geometrical complexity, the radiation

properties of fractal antennas are very difficult to predict using symbolic calculations.

A software package which uses either the FDTD or MoM technique must be used to

simulate performance. This step is also intended to minimize trial and error in the

physical design and testing phases.

3.4.1 Software Simulators

There are a number of commonly available software packages which allow the sim-

ulation of antenna parameters. Some of the best known are SONNET1, XFDTD2,

HFSS3 and various packages based on the NEC24 code. XFDTD and HFSS are ex-

cellent professional design tools which offer a great deal of simulation flexibility and

analysis options. Unfortunately, evaluation or academic versions of these programs

are not offered and their price would well exceed the budget allowed for this project.

SONNET, however, is offered as a feature-limited evaluation package. It uses the

MoM technique to simulate 2D surfaces including traces on dielectric layers, which

is essential for microstrip antenna modeling. The software is user-friendly and with

some effort it can be used to model realistic structures despite the feature limitations.

Software based on the NEC2 code is freely available. NEC2 uses 1D MoM, which al-

lows modeling of wire structures. This is ideal for modeling free-space antennas such

as arrays of dipoles. Although not as user-friendly as SONNET, NEC2 is more flexible

and offers more analysis options.

1Sonnet USA: http://www.sonnetusa.com2Remcom Inc.: http://www.remcominc.com3Ansoft Corporation: http://www.ansoft.com/products/hf/hfss/4NEC2: http://www.nec2.org

13

Antenna

Network Analyzer

Figure 3.1: Experimental setup for antenna reflection coefficient measurements.

3.5 Physical Implementation

The physical implementation of the antenna is the simplest phase of the design pro-

cess. Since both antennas are to be implemented as printed microstrip patches or

lines, this step involves the plotting of a PCB layout using a CAD software package.

Several PCB design software packages are available to download as shareware

from the Internet, such as Eagle5 and CirCAD6. The PCB design can be submitted to

the ECE Department’s Electronics Shop for production. The final step before testing

is to attach a connector for the coaxial cable which feeds an excitation signal to the

antenna.

3.6 Experimental Testing

The experimental testing of the antennas proceeds in two separate phases. The initial

tests can confirm resonance at the intended operating frequencies and gather informa-

tion on antenna input impedance. A second, more elaborate testing procedure could

be used to measure the radiation characteristics of the antennas. Since the antennas

are to operate in the gigahertz range, it is crucial to note that measurements will in-

evitably be error-prone due to interference of electromagnetic fields with surrounding

objects.

5CadSoft Online: http://www.cadsoftusa.com/6Holophase: http://www.holophase.com

14

3.6.1 Antenna Resonance Testing

To measure the location of the resonant bands of an antenna, a simple setup as

described by Figure 3.1 can be used. Simply, the antenna is connected by a cable

to one port of a network analyzer. Measuring the power of the reflected signal while

varying the frequency of the input signal gives a good indication at what frequency

the antenna radiates. At the resonant frequencies, the reflected signal’s power should

be much smaller than that of the input signal. In this way the input impedance and

impedance bandwidth can also be measured. Measurements should be conducted

in an environment isolated from external radiation because antennas are inherently

susceptible to interference.

3.6.2 Radiation Performance Testing

Although conceptually simple, a much more elaborate experiment is required to mea-

sure the far-field radiation characteristics of an antenna. A function generator is

connected using a coaxial cable to the test antenna which then radiates the signal. A

reference antenna with a well-known radiation pattern and gain is fed the same signal

and transmits a calibration signal. The test antenna can be rotated about its axis

for far-field radiation pattern measurements. Another antenna with well-known radi-

ation characteristics is used at the receiver end, some distance from the transmitter.

The signal received is measured using a spectral analyzer to determine the received

power. This experiment is discussed in more detail in [12].

The complexity of the far-field experiment is caused by the amount of calibration

required due to power losses in the antennas and transmission lines. In addition,

radiated fields must be uniform. The contribution of fields from wall reflections are

another complication and source of measurement errors. Since such great precision

is required, this experiment would be very time consuming and thus not feasible for

this project. The far-field radiation properties will thus only be simulated in software

for the scope of this project.

15

Chapter 4

Design of the Sierpinski GasketMonopole

4.1 Introduction

This chapter presents the design of the Sierpinski gasket monopole. The purpose

of building this monopole is to produce an antenna that can operate at multiple

frequencies while retaining the compactness property.

4.2 Selection of Operating Frequency

The goal of the Sierpinski monopole design is to achieve multi-band resonance. To

achieve that goal, it is useful to design the antenna to operate at a set of frequency

bands commonly used by wireless devices. Two bands of interest are the 2.4 GHz

and 5.0 GHz unlicensed bands which, respectively, are used for applications such as

802.11a/b and 802.11g wireless networking. Although lower frequency bands are also

useful for many applications, microstrip patch technology is most advantageous at

higher frequencies where required trace element sizes are physically smaller. Choosing

these bands also facilitates the design and analysis of the proof-of-concept antenna,

as discussed in the next section.

4.3 Fractal Geometry

The calculation of the initial physical parameters of the Sierpinski gasket fractal

pattern required for the monopole are presented here. Final geometry parameters are

16

given in Table 4.1. The initial estimation of the fractal geometry is obtained by using

(2.6), presented again below:

fn = kc

hcos(α/2)δn

We begin by calculating the similarity factor, δ, from the ratio of the resonant

frequencies desired: f2 = 2.4 GHz and f3 = 5.0 GHz. We obtain δ = 2.08 ≈ 2,

which will allow for a very simple and symmetric fractal pattern. Each triangular

structure of the gasket is twice as large as its sub-structure and thus it is very simple

to manually define the pattern in the software simulator. Since the height of the

triangular structure resonating at f2 is h2 = λ2/2 = 3.05 cm, the height of the

monopole is calculated to be h = 2h2 = 6.1 cm.

The number of iterations needed to generate the required fractal is nmax = 4. It

should be noted that although the complete fractal structure will resonate at two other

frequencies, f1 and f4, they are simply included in the pattern to provide continuity

so that truncation effects do not affect the resonant bands of interest. Because of

this, it is mentioned in [10] that f1 ≈ f2/3.5.

The triangle flare angle, α, was chosen to be 60 as a starting point. Since the

similarity factor δ = 2, all triangular elements should be equilateral. The constant k

is given in [10] as ∼ 0.15, however it is dependent on the dielectric substrate type and

thickness used. It is only used as a first guess for this design and final parameters are

fully confirmed through simulation.

Geometry Parameter Value

Similarity factor, δ: 2Height, h: 6.1 cmFlare angle, α: 60

Max. iterations, nmax: 4

Table 4.1: Sierpinski gasket monopole geometry parameters.

4.4 Hardware Design

4.4.1 Dielectric Substrate

Many different dielectric types may be used for this design, but the only available

material available for PCB printing at the ECE department’s Electronics Shop is

17

G10-FR4. The dielectric substrate G10-FR4 is essentially an epoxy and glass fabric

laminate, with a dielectric constant εr = 4.8 at low frequencies. A data sheet is

provided in Appendix D. The thickness of the dielectric available is 1.6 mm. Thus,

there is no flexibility in the type and thickness of material we can use. However,

an examination the efficiency plots in [6], p.33, reveals that FR4 at the available

thickness should still have an efficiency of approximately 95% at 2.4 GHz and 90% at

5.0 GHz. These values are acceptable for the purposes of this proof-of-concept design.

4.4.2 Connectors

To connect the antenna to transmission lines for testing, a BNC connector was first

considered. Some testing using an HP7853D network analyzer concluded that BNC

connectors do not have very good frequency response in the gigahertz range. Thus,

50Ω SMA connectors were chosen as an alternative. SMA connectors have excellent

frequency response up to 18 GHz (see data sheet in Appendix D), although adapters

are required to connect with test equipment and transmission lines which generally

employ Type-N connectors. Type-N connectors were considered since they also have

good frequency response, but they are too large to properly attach to an antenna of

the size required in this design.

4.4.3 Ground Plane

A ground plane of size 10.5x10.5 cm was printed on the opposite side of the substrate

from the fractal pattern, as is required in microstrip patch design. Generally, larger

ground planes allow better radiation performance, but a trade-off must be made

against the final size of the antenna allowed. In addition, larger areas of PCBs cost

more to produce.

4.4.4 Signal Feed System

There are a number of techniques to feed a signal to a microstrip patch [6,14], although

the most advantageous for this design are edge and probe feeding. A depiction of each

technique is included in Figure 4.1.

18

Feed line

Patch

(a) Edge feed system

Patch

Dielectricsubstrate

Probe feed

Connector

Top

Bottom

(b) Probe feed system

Figure 4.1: Signal feeding techniques for microstrip patches.

Edge Feeding

The edge feeding technique shown in Figure 4.1(a) simply consists of printing a trace

from and edge of the patch to the edge of the dielectric substrate. This makes

the attachment of a connector very simple and can change the impedance of the

antenna, allowing some matching to be performed. However, it has the disadvantage

of introducing spurious radiation, since the trace will inevitably radiate.

Probe Feeding

Probe feeding, depicted in Figure 4.1(b) consists of drilling a hole through the ground

plane and dielectric substrate then connecting the transmission line or connector

directly to the patch. Since the signal source is essentially behind the ground plane,

this eliminates spurious radiation. In addition, it allows the signal to be fed at any

location on the radiating patch. The disadvantage of this technique is that it is more

difficult to implement since care must be made that the hole through the dielectric is

drilled properly and that the signal probe does not short-circuit to the ground plane.

To minimize chances of error in measurement, the probe feeding technique is

chosen. An SMA PCB-mount connector is thus connected through the dielectric

substrate to a corner of the triangular fractal.

19

Figure 4.2: General depiction of the Sierpinski monopole.

4.5 Software Simulation

The program SONNET was used to simulate the performance of the Sierpinski gasket

monopole. Since SONNET can model the radiation of 2D structures on dielectric

substrates, it was ideal for this design. The fractal structure designed in Section 4.3

was drawn using the simulator’s editor and then discretized into a rectangular mesh.

The reflection coefficient variation with input signal frequency was simulated in order

to determine the resonant frequencies of the antenna. The geometry was adjusted as

necessary in order to obtain a simulated resonance at the desired frequencies of 2.4

and 5.0 GHz, thus producing a good design prototype.

4.6 Final Design

A general depiction of the final design of the Sierpinski gasket monopole is shown

in Figure 4.2. More detailed diagrams can be found in Appendix B. Images of the

physical implementation are given in Appendix C.

20

Chapter 5

Design of the Koch FractalMonopole

5.1 Introduction

This chapter presents the design of the Koch fractal monopole. The purpose of

building this monopole is to produce a more space-efficient quarter-wave monopole

design while maintaining the radiation properties of the traditional quarter-wave,

straight-wire monopole.

5.2 Selection of Operating Frequency

The operating frequency of 900 MHz was chosen for the design of the Koch fractal

monopole. This frequency band is used for cellular wireless telephony through the

GSM system. Also, the space-filling properties of the Koch monopole are more ad-

vantageous at this comparatively lower band since at frequencies like 2.4 and 5 GHz

the wavelength is small enough to produce relatively small antennas [5]. Dielectric

losses are small at this frequency, thus it is possible to use a dielectric substrate which

reduces electrical length without incurring noticeable radiation losses.

5.3 Fractal Geometry

The geometric complexity required that the pattern be automatically generated.

The geometry of the Koch fractal used in this design was generated using the

Matlab script kochgen.m listed in Appendix A. The IFS algorithm described in

21

Section 2.2.1 was implemented using the set of transforms specific to the Koch fractal

given by (2.4). The script outputs the line vertex coordinates of Koch fractal of any

given iteration. A scaling factor can also be given as a parameter to the script in

order to generate coordinates for any physical size needed.

A 3-iteration Koch fractal was generated to provide maximum height reduction.

Although a 4- or higher-iteration fractal would be a further improvement, it is dis-

cussed in Section 2.4.2 that this would cause problems in printing the pattern on the

dielectric substrate and that only very small increments in electrical lengths would

be added.

The Koch fractal was then scaled to have an equivalent unfolded length identical

to the height of the straight-wire λ/4 monopole. At 900 MHz, a straight-wire λ/4

monopole has a height of h = 8.33 cm in free space. According to expression (2.5),

the equivalent Koch monopole would have a height of only 3.51 cm. However it was

determined in simulations that the electrical length was longer. This height is further

reduced by the high dielectric constant of the substrate, as described in the next

section.

5.4 Hardware Design

5.4.1 Dielectric Substrate

As with the Sierpinski monopole design, the only available dielectric substrate is G10-

FR4, with a relative dielectric constant εr = 4.8 and a thickness of 1.6 mm. However,

at a frequency of 900 MHz, radiation losses are expected to be negligible [6]. When

electromagnetic waves propagate through a dielectric, they travel at a speed given

by:

v =1√µε

, (5.1)

where ε = εrε0. Since λ = v/f , the wavelength inside of the dielectric can then be

expressed by:

λ =1√

µεrε0f(5.2)

This relationship is used to determine the required height of the monopole in

22

order to radiate at 900 MHz while comlpletely immersed in the FR-4 dielectric. This

scaling effect is modeled using software simulations. Since the fractal pattern is essen-

tially printed at the boundary between air and the dielectric substrate, the calculation

of the required physical height of the Koch fractal is not as simple as indicated by

(5.2). Some numerical simulations are required to simulate the effects on electrical

length.

Figure 5.1: Sketch of the 900 MHz Koch monopole.

5.4.2 Physical Design

A metallic ground plane is mounted perpendicular to the antenna. The ground plane

is required to generate an “image” of the monopole so that an equivalent dipole

is produced. Thus the perpendicular ground plane is what allows the monopole

to function like a dipole antenna by simulating a perfectly conducting ground. To

minimize diffractive effects and produce good reflection, a steel sheet of approximately

0.5 mm thick is used. The dimensions of the sheet are 40x40 cm.

23

The monopole is fed at one end through a 50Ω SMA connector, which has

good frequency response. The connector was secured using two washers, allowing

the monopole to stand rigidly and perpendicular onto the ground plane.

5.5 Software Simulation

Because of the 16MB mesh size limitation, the design of the Koch monopole could

not be completed only using SONNET. The maximum mesh size allowed is simply

too small to simulate any pattern more complex than the first iteration of the Koch

fractal. Thus, NEC2 was used to adjust any design parameters. However, SONNET

was used to find an equivalent εr due to the dielectric boundary effect. A straight

wire monopole which resonates at 900 MHz was simulated in SONNET on an FR-4

dielectric sheet. A 900 MHz straight wire monopole was then simulated in NEC2

exhibiting identical characteristics to the SONNET simulation. By using expression

(5.2), an equivalent εr ≈ 2.2 was obtained. This allowed a further minimization of the

Koch monopole height to h = 4.1 cm, a significant improvement over the free-space

height of 8.33 cm.

5.6 Final Design

A sketch of the final design is shown in Figure 5.1. Final physical design parame-

ters are given in Table 5.1. More detailed schematics as well as illustrations of the

constructed Koch monopole are given in Appendices B and C, respectively.

Design Parameter Value

Height, h: 4.1 cmTrace line thickness: 0.3 mmMax. iterations: 3Ground plane size: 40x40 cmDielectric type: FR-4

Table 5.1: 900 MHz Koch monopole design parameters.

24

Chapter 6

Results and Discussion

6.1 Introduction

The Sierpinski gasket and Koch fractal monopoles discussed in Chapters 4 and 5,

respectively, have been built and tested. The experimental and simulation results

for each of the antennas are presented in Section 6.2, then analyzed and discussed in

Section 6.3.

6.2 Results

Software simulations and experimental tests were used in order to evaluate the per-

formance of the antenna designs. Experimental results are compared with simulation

performance estimates in order to verify that the designs perform as intended.

6.2.1 Sierpinski Gasket Monopole Results

Reflection Coefficient

Determining the value of the input reflection coefficient of the antenna is necessary

to determine the location of the resonant bands. The input reflection coefficient, Γin,

is obtained from the expression:

Γin =Zin − Z0

Zin + Z0

, (6.1)

where Zin is the input impedance of the antenna and Z0 is the characteristic impedance

used in the transmission line, used here as a reference. The absolute value of the re-

25

flection coefficient can also be expressed as the ratio of the reflected power from the

antenna input, Pref and the power delivered to the antenna, Pin, as given in expres-

sion (6.2). It is clear that the reflection coefficient will be low at frequencies where

the reflected power is small, indicating that power was radiated. In addition, from

examining (6.1), we can obtain the input impedance of the antenna and hence how

well we can match it to the transmission line.

|Γin| =Pref

Pin

(6.2)

The reflection coefficient of the Sierpinski Gasket monopole was simulated using

SONNET. A simulated sinusoidal signal was fed through the corner of the monopole,

as was discussed in Chapter 4. The port parameter S11, which is equivalent to the

reflection coefficient was then calculated as a function of frequency and plotted, as

shown in Figure 6.1. The frequency range used spans from 0 to 6 GHz to match the

capabilities of the available experimental hardware.

0 1000 2000 3000 4000 5000 6000−30

−25

−20

−15

−10

−5

0

5

S11

vs. Frequency for a Dual−Band Sierpinski (2.4/5.0 GHz) Monopole

Frequency (MHz)

S11

(dB

)

MeasurementSonnet simulation

Figure 6.1: Reflection coefficient for a 4-iteration Sierpinski monopole. (Z0 =50Ω)

26

The physical measurement of the reflection coefficient was performed in a shielded

chamber using a network analyzer. The Sierpinski gasket was connected directly to

one port of a 6 GHz HP8753D network without using any transmission lines, to

avoid introducing experimental errors from losses in the coaxial lines. An SMA to

Type-N adapter was used, however, because only a 50Ω Type-N calibration kit for

the HP8753D analyzer was available. Using a reference characteristic impedance of

Z0 = 50Ω, the variation of reflection coefficient with frequency was measured in the

range of 0—6 GHz. It can be seen from the deep valleys in plot of the data in Figure

6.1 that the monopole has resonant frequencies at both 2.4 and 5.0 GHz.

From examining Figure 6.1, we can see that both the simulation and measured

values are approximately in agreement, although some noticeable discrepancies occur

above 3 GHz. The simulation curve deviates from the measurement curve, although

still indicates radiation at around 5 GHz. This deviation was caused by the memory

limitation of 16 MB in SONNET. As a result, a comparably coarse mesh had to be

used to simulate the structure. The accuracy of the simulation is directly related to

the mesh element size (smaller is better, resulting in a larger mesh) [11].

Another interesting feature in Figure 6.1 to note is the location of the first res-

onant frequency band, f1 (see Section 4.3). The simulation shows this location quite

accurately at 700 MHz, although the measurement curve shows a number of reso-

nances in the region of the expected f1.

Input Impedance

The input impedance of the Sierpinski gasket monopole is approximately Zin = 50Ω

at both the 2.4 and 5.0 GHz resonant bands, as indicated by the very low values

of the measured and simulated reflection coefficients. This property appears to be

inherent of the antenna structure [10] and can be useful in matching, as discussed in

Section 6.3.

Impedance bandwidths of 8.3% and 4.4% were calculated at the 2.4 and 5.0 GHz

bands, respectively. This is similar to what is expected of microstrip patches.

Simulated Charge Density

In order to verify visually that the antennas function as expected, we can simulate

and plot the charge density on the Sierpinski monopole’s surface [10]. Since charge

density indicates the presence of surface currents, we can easily see what regions of

27

(a) 2.4 GHz (b) 5.0 GHz

Figure 6.2: Simulated charge density for the Sierpinski monopole.

the fractals radiate more than others. A plot of the surface charge density is shown

in Figure 6.2. In Figure 6.2(a) we can see that most of the charge is distributed over

the second fractal iteration at the resonant frequency f2 = 2.4 GHz. As expected,

examining Figure 6.2(b) indicates that most of the charge is distributed over the third

iteration at the next resonant frequency f3 = 5.0 GHz.

Performance Parameter Value

First resonant frequency: 2.4 GHzSecond resonant frequency: 5.0 GHzImpedance Bandwidth: 8.3% (2.4 GHz) and 4.4% (5.0 GHz)Input Impedance 50 Ω

Table 6.1: Sierpinski gasket monopole performance parameters.

Antenna Performance Parameters

The measured antenna performance parameters are summarized in Table 6.1.

6.2.2 Koch Fractal Monopole Performance

Reflection Coefficient

The reflection coefficient variation with frequency was measured and simulated using

the same procedures as outlined in Section 6.2.1. NEC2, instead of SONNET, was used

28

XY

Z

5.0 dBi

Figure 6.3: Koch monopole radiation pattern.

to simulate the antenna parameters. The results of the simulation and experiment

are given in Figure 6.4. As can be seen, the experiment and simulation values are

highly in agreement. The Koch monopole is observed to resonate at 900 MHz and to

have harmonic resonances at 2.6, 4 and 5.7 GHz. A close-up of the 900 MHz band is

shown in Figure 6.4(b).

The impedance bandwidth of the Koch monopole at 900 MHz was calculated

to be approximately 14.4%, which is higher than the expected maximum of 10% for

microstrip antennas.

Radiation Patterns

Although not measured due to the complexity of the experiment required, the radi-

ation patterns of the Koch monopole were only simulated using NEC2. A 900 MHz

sinusoidal signal source was simulated at one end of the antenna, where the feed point

is located in the physical design. The Koch fractal structure was discretized to () el-

ements in the NEC2 simulation for high accuracy. An infinite, perfectly conducting

ground plane was introduced in order to simulate the ground plane used in the design.

The polar plot of the variation of gain in 3D is given in Figure 6.3. The radiation

pattern is very uniform in all directions. It is consistent with the classic doughnut

29

0 1000 2000 3000 4000 5000 6000−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

S11

vs. Frequency for a 3−Iteration, 900 MHz Koch Monopole

S11

(dB

)

Frequency (MHz)

NEC2 simulationMeasurement

(a) S11 in the 0 to 6 GHz region

600 700 800 900 1000 1100 1200 1300

−8

−7

−6

−5

−4

−3

−2

−1

0

1

S11

vs. Frequency for a 3−Iteration, 900 MHz Koch Monopole

S11

(dB

)

Frequency (MHz)

NEC2 simulationMeasurement

(b) S11 in the 900 MHz region

Figure 6.4: Reflection coefficient measurements for the 3-iteration 900 MHz Kochmonopole. (Z0 = 50Ω)

30

Performance Parameter Value

Resonant frequency: 915 MHzImpedance bandwidth: 14.4%Input impedance (simulated): 22.5 + j12.6 ΩGain in horizontal plane (simulated): 5.0 dBiRadiation efficiency (simulated): 98.8 %Front-to-back ratio (simulated): 1Compactness: 4.1/8.33 = 49 %

Table 6.2: Koch monopole performance parameters.

shape characteristic of the straight wire λ/4 monopole, and consequently that of the

λ/2 dipole.

It is noted that the maximum directivity gain of the Koch monopole is 5.0 dBm,

in the horizontal plane. This is comparable to the simulation of a straight-wire

monopole which resulted in a gain of 5.18 dBm in the same direction.

Antenna Performance Parameters

The antenna performance parameters are summarized in Table 6.2. Simulated and

measured parameters are included.

6.3 Discussion

Since every design has benefits and disadvantages, it is useful to discuss the most rel-

evant points of each. The advantages and disadvantages are evaluated in the sections

below.

6.3.1 Sierpinski Gasket Monopole

Benefits

The largest advantage of this design is that by using the self-similar structure of the

Sierpinski fractal, we can essentially create two antennas which operate at a desired

pair of frequencies in the same space as one. This multi-band behaviour and efficient

space usage is beneficial in many applications where space is at a premium, such as

receivers for wireless networking in laptop computers or wireless PDAs.

The design has good impedance bandwidth, and its impedance is 50Ω at both

resonant frequencies. This is quite fortunate since no matching networks would be

31

required in order to connect this antenna to a standard 50Ω transmission line. In

addition, since the impedance is identical at both resonant bands, the rather difficult

problem of matching the antennas at two different frequencies is eliminated.

Disadvantages

Because continuity is required in the fractal structure to obtain the desired location

and separation of the design resonant frequencies, a superfluous larger iteration of

the fractal structure is used. The height of the Sierpinski monopole thus is twice that

of the iteration which produces its first resonant band of interest at 2.4 GHz. This

extra iteration increases the physical profile of the antenna.

The attentive reader will have noticed that 2.4 and 5.0 GHz are nearly harmonics

of each other. A dual-band antenna is not necessary to operate at these frequencies,

since a simple monopole or half-wave dipole could be used. While this is true, the

purpose of this design was to prove that the concept can be used to formulate useful

antennas. It is simple to see how the design procedure outlined in this report could be

used to generate antennas with band-spacings different than 2, where the harmonics

of a monopole would not exist.

6.3.2 Koch Fractal Monopole

Benefits

The greatest advantage of the Koch monopole design is its initial purpose: compact-

ness. A size reduction of 51% was achieved over the straight-wire, λ/4, free-space

monopole. This is highly significant for applications such as GSM cellular phones

which regularly employ λ/4 monopoles. Since it is half the size of the traditional

monopole, it could easily be completely integrated within the case of the phone,

eliminating the protruding monopoles commonly seen on many cellular phones.

The Koch monopole design has excellent impedance bandwidth, allowing some

flexibility in the types of applications where it could be used. Since the radiation

pattern is highly uniform and identical to that of a traditional λ/4 monopole, it could

be used in nearly any type of wireless communications receiver. The very similar

gain to the traditional λ/4 monopole is another benefit of the design. Thus the Koch

monopole presents an excellent, compact solution to the traditional straight-wire

monopole.

32

Disadvantages

A noticeable disadvantage of the Koch monopole is the requirement for a reflective

ground plane. In real-world applications the Earth ground is used to provide re-

flections is used while the antenna’s ground terminal is connected to the system’s

ground. This was not done in the proof-of-concept design presented in this report

because measurement accuracy and consistency were required which would not be

possible without the metallic ground plane used.

33

Chapter 7

Conclusions

7.1 Future Work

Since the area of fractal antenna engineering research is still in its infancy, there are

many possibilities for future work on this topic. The Sierpinski and Koch fractals

were chosen for this project because they are the best documented fractal antenna

types in current research. However, many possible fractal structures exist which may

undoubtedly have desireable radiation properties. Thus, a possible avenue for future

work is to investigate other types of fractals for antenna applications. A novel and

intriguing development is the use of fractal patterns for antenna arrays [15].

More immediate future goals related to the design presented in this report in-

clude the full measurements of the radiation properties of the two antenna prototypes

built. Given proper time and resources, the radiation pattern measurement technique

described in Section 3.6.2 should be used to completely verify the performance of the

antenna prototypes.

Finally in the distant future, the successful and fully evaluated prototypes should

be integrated with existing communication systems. The Koch fractal monopole, for

instance, offers great promise for application in GSM cell phones. Further simulations,

measurements as well as modification of the design would naturally be required

7.2 Conclusions

The compact, multi-band Sierpinski gasket monopole and the compact Koch fractal

monopole designs presented in this report are an excellent alternative to traditional

34

antenna systems in mobile wireless receivers. The Sierpinski gasket monopole per-

forms adequately at both the 2.4 and 5.0 GHz frequency bands and demonstrates

space-efficiency through its self-similar fractal structure. The Koch monopole ex-

hibits excellent performance at 900 MHz and has radiation properties nearly identical

to that of traditional, straight-wire monopoles at that frequency.

The goals of this design project were thus successfully accomplished.

35

Appendix A

Source Code Listings

Listing A.1: kochgen.m1 f u n c t i on k = kochgen ( s t a r t l e n g t h , max i t e r )2

3 % kochgen .m − g en e r a t e s c o o r d i n a t e s f o r a Koch f r a c t a l cu rve4 %5 % k = kochgen ( s t a r t l e n g t h , max i t e r )6 %7 % INPUT − s t a r t l e n g t h : l e n g t h to s c a l e the f r a c t a l to8 % − max i t e r : number o f i t e r a t i o n s9 % OUTPUT − Koch cu rve c o o r d i n a t e s

10 %11 % Author : Paul S imedrea (15/01/2004)12

13 %wn = [ a b e ; c d f ; 0 0 1 ]14

15 w1 = [ 1 / 3 0 0 ; 0 1 / 3 0 ; 0 0 1 ] ;16 w2 = [1/3∗ cos ( p i /3) −1/3∗ s i n ( p i / 3 ) 1 / 3 ; . . .17 1/3∗ s i n ( p i /3) 1/3∗ cos ( p i / 3 ) 0 ; 0 0 1 ] ;18 w3 = [1/3∗ cos ( p i /3) 1/3∗ s i n ( p i / 3 ) 1 / 2 ; . . .19 −1/3∗ s i n ( p i /3) 1/3∗ cos ( p i /3) 1/6∗ s q r t ( 3 ) ; 0 0 1 ] ;20 w4 = [ 1 / 3 0 2 / 3 ; 0 1 / 3 0 ; 0 0 1 ] ;21

22

23 %coo r d i n a t e space i s i n cm24 v1 = [ 0 1 ; . . .25 0 0 ; . . .26 1 1 ] ;27

28 %gene r a t e f r a c t a l to s p e c i f i e d number o f i t e r a t i o n s29 f o r i = 1 : max i t e r30 v1a = w1∗v1 ;31 v2a = w2∗v1 ;32 v3a = w3∗v1 ;33 v4a = w4∗v1 ;34

35 v t = [ v1a v2a v3a v4a ] ;36 v1 = vt ;37 end

38 v t = s t a r t l e n g t h ∗ v t ( 1 : 2 , : ) ;39 p l o t ( v t ( 1 , : ) , v t ( 2 , : ) )40

41 r e t u r n

36

Appendix B

Antenna Schematics

37

105

mm

105mm

60,39 mm

1,6

mm

2.4

/5.0

GH

z S

ierp

insk

i Mo

no

po

le

Figure B.1: Detailed schematic of the 2.4/5.0 GHz Sierpinski monopole

38

A

DETAIL

A

400mm

400

mm

41mm

30mm

900MHzKochFractalMonopole

Figure B.2: Detailed schematic of the 900 MHz Koch monopole

39

Appendix C

Antenna Illustrations

40

Figure C.1: Photo of the Sierpinski Monopole.

Figure C.2: Photo of the Koch Monopole.

41

Appendix D

Selected Product Data Sheets

42

Figure D.1: Datasheet for FR-4 (page 1 of 2)http://www.injectorall.com/techsheetFR4.htm

43

Figure D.2: Datasheet for FR-4 (page 2 of 3)http://www.injectorall.com/techsheetFR4.htm

44

Figure D.3: Datasheet for FR-4 (page 3 of 3)http://www.injectorall.com/techsheetFR4.htm

45

Figure D.4: Datasheet for an SMA flange connector (page 1 of 3).

46

Figure D.5: Datasheet for an SMA flange connector (page 2 of 3).

47

Figure D.6: Datasheet for an SMA flange connector (page 3 of 3).

48

Bibliography

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[3] H. O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers of

Science. New York: Springer-Verlag, 1992.

[4] C. Puente, J. Romeu, R. Pous, J. Ramis, and A. Hijazo, “Small but long Kochfractal monopole,” Electronics Letters, vol. 34, pp. 9–10, Jan. 1998.

[5] C. P. Baliarda, J. Romeu, and A. Cardama, “The Koch monopole: A small fractalantenna,” IEEE Trans. Antennas and Propagation, vol. 48, pp. 1773–1781, Nov.2000.

[6] R. B. Waterhouse, Microstrip Patch Antennas. Boston: Kluwer Academic Pub-lishers, 2003.

[7] C. Puente, J. Romeu, R. Pous, X. Garcia, and F. Benitez, “Fractal multibandantenna based on the Sierpinski gasket,” Electronics Letters, vol. 32, pp. 1–2,Jan. 1996.

[8] J. Romeu and J. Soler, “Generalized Sierpinski fractal multiband antenna,” IEEE

Trans. Antennas and Propagation, vol. 49, pp. 1237–1239, Aug. 2001.

[9] C. P. Baliarda, C. B. Borau, M. N. Rodero, and J. R. Robert, “An iterativemodel for fractal antennas: Application to the Sierpinski gasket antenna,” IEEE

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magnetics. New York: IEEE Press, 1998.

49

[12] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Wiley, Sec-ond ed., 1997.

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[16] S. Ramo, J. R. Whinnery, and T. V. Duzer, Fields and Waves In Communication

Electronics. New York: Wiley, 1965.

[17] L. C. Godara, Handbook Of Antennas In Wireless Communications. New York:CRC Press, 2002.

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50

Vita

NAME: Paul SimedreaPLACE OF BIRTH: Timisoara, RomaniaYEAR OF BIRTH: 1980

SECONDARY EDUCATION: Sir Frederick Banting S.S. (1994-1999)HONOURS AND AWARDS: First Prize, Electrical Engineering Design Day 2004

2003/04 IEEE Inc. Award2002/03 Hydro One ScholarshipNSERC Undergraduate Research Award 2003SharcNET Student Fellowship 2002Dean’s Honour List (2000, 2001, 2002, 2003)UWO Entrance Scholarship (1999)

51