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UNIVERSITY OF NAIROBI FACULTY OF ENGINEERING DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING PROJECT INDEX: PRJ 18 PATCH ANTENNA ARRAY FOR THE 2.4 GHz ISM BAND By Waihenya Peter Ndung’u F17/28805/2009 Supervisor: Dr. Wlifred N. Mwema Examiner: Prof. H Ouma A Project submitted in partial fulfillment of the requirements for the award of the degree Of

Transcript of eie.uonbi.ac.keeie.uonbi.ac.ke/sites/default/files/cae/engineering/eie... · Web viewAntenna Theory...

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UNIVERSITY OF NAIROBI

FACULTY OF ENGINEERING

DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING

PROJECT INDEX: PRJ 18

PATCH ANTENNA ARRAY FOR THE 2.4 GHz ISM BAND

By

Waihenya Peter Ndung’u

F17/28805/2009

Supervisor: Dr. Wlifred N. Mwema

Examiner: Prof. H Ouma

A Project submitted in partial fulfillment of the requirements for the award of the degree

Of

Bachelor of Science in ELECTRICAL AND INFORMATION ENGINEERING of the Univeristy of Nairobi

Submitted on: April 24th, 2015

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DECLARATION OF ORIGINALITY

Declaration Form for Students UNIVERSITY OF NAIROBI Declaration of Originality Form This form must be completed and signed for all works submitted to the University for examination.

Name of Student: WAIHENYA PETER NDUNG’U

Registration Number: F17/28805/2009

College: COLLEGE OF ARCHITECTURE AND ENGINEERING

Faculty/School/Institute: ENGINEERING

Department: ELECTRICAL AND INFORMATION ENGINEERING

Course Name: FINAL YEAR PROJECT

Title of the work: DESIGN OF A PATCH ANTENNA ARRAY FOR THE 2.4GHz ISM BAND

DECLARATION

1. I understand what Plagiarism is and I am aware of the University’s policy in this regard 2. I declare that this project is my original work and has not been submitted elsewhere for examination, award of a degree or publication. Where other people’s work or my own work has been used, this has properly been acknowledged and referenced in accordance with the University of Nairobi’s requirements. 3. I have not sought or used the services of any professional agencies to produce this work 4. I have not allowed, and shall not allow anyone to copy my work with the intention of passing it off as his/her own work 5. I understand that any false claim in respect of this work shall result in disciplinary action, in accordance with University Plagiarism Policy.

Signature

Date 24/04/2015

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DEDICATION

This project is dedicated to my loving parents Dr. and Mrs. Waihenya for their utmost support and encouragement throughout my education life and especially when I was undertaking my BSc. Electrical and Electronics Engineering degree.

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AKNOWLEDGEMENT

I would like to express my heartfelt gratitude to Dr Wilfred Mwema of the department of Electrical and Information Engineering, University of Nairobi for his critical guidance as my project supervisor. I would also like to sincerely thank my family and classmates for their support. May God bless you.

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ABSTRACT

In this study, a coaxial fed patch antenna array for application in the 2.4GHz ISM band was implemented using the Ansoft HFSS software. Standard formulas were used to calculate different parameters of the antenna. These were just used as a basis of design as some parameters varied considerably during simulation. A good extent of the antenna design was hence done through trial and error. The proposed antenna was designed to work at 2.44GHz frequency band. A fractional bandwidth of 2.62%, which was not close to the desired 10% and a reflection coefficient of -18.2131dB were attained. This may have been brought about by poor impedance matching and a high level of spurious feed radiation and surface waves. A way of improving the bandwidth would have been to use proximity coupling feeding method which offers the highest bandwidth (as high as 13%) and is somewhat easy to model and has low spurious radiation. However, its fabrication would have been more difficult. A directivity of 8.53dB was achieved. This was a fairly high though directivity increase could have been studied through use of different substrate material and thickness.

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LIST OF FIGURES

Figure 2.1 Antenna measurement co-ordinate system………………………………………………………………….……...12Figure 2.2 Microstrip antenna and coordinate system…………………………………………………………………..…….16Figure 2.3 Microstrip line and its electric field lines, and effective dielectric constant…………………...……..17Figure 2.4 Physical and effective lengths of rectangular microstrip patch…………………………………..…………18Figure 2.5 Feed arrangements for microstrip patch arrays…………………………………………………………..……….20Figure 3.1 Configuration for compensated right-angled bends……………………………………………………..……….23Figure 3.2 Characteristics of the step width junction discontinuity………………………………………………….…...24Figure 3.3 T-junction discontinuity compensation and minimization of the effect……………………………..….24Figure 3.4 4 element patch antenna HFSS model………………………………………………………………………………....27Figure 3.5 4-element patch antenna PCB layout with dimensions……………………………………………………......27Figure 3.6 Implemented 4-element patch antenna array……………………………………………………………….……..28Figure 3.7 ground plane of the patch array…………………………………………………………………………………….……..29Figure 3.8 N male to sma female cable………………………………………………………………………………………….…......29Figure 4.1 Return loss S11 obtained for the patch array…………………………………………………………………….…..31Figure 4.2 Simulated E-Plane (phi=90 °, theta varying) ………………………………………………………………………....32Figure 4.3 Simulated H-plane (theta=90 °, phi varying)………………………………………………………………………....32Figure 4.4 3D radiation pattern…………………………………………………………………………………………………………....33Figure 4.5 E-Plane and H-Plane patterns in rectangular coordinates …………………………………………………....34Figure 4.6 VSWR plot…………………………………………………………………………………………………………………………....35Figure 4.7 Smith chart of the proposed patch antenna………………………………………………………………………....36Figure A1.4 Rectangular microstrip patch and its equivalent circuit transmission-line model………………..40Figure A1.5 Recessed microstrip- line feed……………………………………………………………………………………………43

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LIST OF TABLES

Table 4.1 Variation of antenna parameters with changes in dimensions……………………………………………....30Table 4.2 Variation of resonance frequency with changes in patch feed length………………………………….…34Table 4.3 HFSS Antenna Parameters in HFSS…………………………………………………………………………………………37

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TABLE OF CONTENTS

DECLARATION OF ORIGINALITY……………………………………………………………………………………………………………….2

DEDICATION…………………………………………………………………………………………………………………………………………....3

AKNOWLEDGEMENT………………………………………………………………………………………………………………………………..4

ABSTRACT…………………………………………………………………………………………………………………………………………..……5

LIST OF FIGURES………………………………………………………………………………………………………………………………………6

LIST OF TABLES………………………………………………………………………………………………………………………………………..7

CHAPTER 1: INTRODUCTION…………………………………………………………………………………………………………………10

CHAPTER 2: LITERATURE REVIEW…………………………………………………………………………………………………………11

Fundamental Specifications of Antennas ……………………………………………………………………………………………..11

Microstrip Antennas………………………………………………………………………………………………………………………………14

Basic Characteristics…………………………………………………………………………………………………………………..………….15

Transmission Line Model Analysis for a Rectangular Patch…………………………………………………………………….16

Arrays and Feed Networks………………………………………………………………………………………………………………....…19

CHAPTER 3: THE DESIGN METHODOLOGY………………………………………………………………………………………….…21

Design Procedure………………………………………………………………………………………………………………………………..…21

Ground Plane……………………………………………………………………………………………………………………………………..….22

Microstrip Discontinuities………………………………………………………………………………………………………………………23

Main Beam Direction…………………………………..…………………………………………………………………………………………25

Matching of Microstrip Lines to the Source……………………………………………………..…………………………………….25

Quarter Wave Transformer……………………………………………………………………………………………………………………25

Simulation…………………………………………………………………………………………….……………………………………………….27

Fabrication………………………………………………………………………………………….…………………………………………………28

CHAPTER 4: HFSS SIMULATION RESULTS AND ANALYSIS……………….……………………………………………………..30

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Variation of Patch Length and Width……………………………………………………………………………………………………..30

Reflection Coefficient……………………………………………………………………………………………………………………………..31

Radiation Pattern…………………………………………………………………………………………………………………………………..32

Inset Feed Position…………………………………………………………………………………………………………………………………34

VSWR Plot………………………………………………………………………………………………………………………………………………35

Smith Chart……………………………………………………………………………………………………………………………………………36

Ground Plane………………………………………………………………………………………………………………………………………...36

H plane Inter-Element Separation….……………………………………………………………………………………………………...37

E Plane Inter-Element Separation………………………………..…………………………………………………………………………37

CHAPTER 5: CONCLUSION…………………………………………….……………………………………………………………………….39

APPENDICES…………………………………………………………………………………………………………….……………………………40

Appendix A……………………………………………………………………………………………………….……………………………………40

Conductance…………………………………………………………………………………………………….………….………………………..40Resonant Input Resistance……………………………………………………………………………………………………..………………41

Appendix B…………………………………………………………………………………………………………………………………………….45

Matlab Code for calculation of the insed feed position where the input impedance is 50 Ohms…………….45Matlab code for calculation for the width of the 50 ohm line……………………………………………………………..….45

REFERENCES………………………………………………………………………………………………………………………………………….47

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CHAPTER 1: INTRODUCTION

An antenna is a transducer between a guided wave and a radiated wave, or vice versa. The structure that "guides" the energy to the antenna is most evident as a coaxial cable attached to the antenna. A patch antenna is a type of radio antenna with a low profile, which can be mounted on a flat surface. It consists of a flat sheet of metal, usually copper, mounted on a larger sheet of metal called a ground plane. A patch array antenna is, in general, some arrangement of multiple patch antennas that are all driven by the same source. Frequently, this arrangement consists of patches arranged in orderly rows and columns (a rectangular array). The reason for these types of arrangements is higher gain. Higher gain commonly implies a narrower beamwidth and that is, indeed, the case with patch arrays.

This report presents the design and analysis of patch network antenna array for the 2.4GHz ISM band which is largely license exempt and can be accessed freely for example bluetooth. The antenna will be designed with an aim of achieving high directivity and at least a 10% fractional bandwidth. The antenna will have a center frequency of 2.44 which is almost the same as the given ISM band center frequency. It was so chosen so as to have a bandwidth whose range is falls within the 2.4 Ghz band. The work presented here is the continuation or enhancement of the 2013 final year patch antenna array project where a basic 4 element patch antenna array was designed without much emphasis on the gain, directivity or bandwidth.

The report consists of five chapters. After the introduction, the necessary theoretical background is presented in the second chapter. Then a chapter describing the design and all the steps and choices made for the patch antenna array follows. An Analysis of the simulated results together with discussions is done in chapter four. The conclusion, which includes a short summary of the design achievements, is presented in chapter five.

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CHAPTER 2: LITERATURE REVIEW

An antenna is generally a bidirectional device, that is, the power through the antenna can flow in both directions, coupling electromagnetic energy from the transmitter to free space and from free space to the receiver, and hence it works as a transmitting as well as a receiving device. Transmission lines are used to transfer electromagnetic energy from one point to another within a circuit and this mode of energy transfer is generally known as guided wave propagation. An antenna can be thought of as a mode transformer which transforms a guided-wave field distribution into a radiated-wave field distribution. It can also be thought of as a mode transformer which transforms a radiated-wave field distribution into a guided-wave field distribution (since the two waves may have different impedances, it may also be thought of as an impedance transformer) [8].

Fundamental Specifications of Antennas

Lobes

Any given antenna pattern has portions of the pattern that are called lobes. A lobe can be a main lobe, a side lobe or a back lobe and these descriptions refer to that portion of the antenna pattern in which the lobe appears. In general, a lobe is any part of the pattern that is surrounded by regions of weaker radiation. So a lobe is any part of the pattern that sticks out [15].

Radiation Pattern

Radiation pattern is graphical representation of the relative field strength transmitted from or received by the antenna. It is measurement of radiation around the antenna. Antenna radiation patterns are taken at one frequency, one polarization and one plane cut. The patterns are usually presented in polar or rectilinear form with a dB strength scale. It is important to state that an antenna radiates energy in all directions, at least to some extent, so the antenna pattern is actually three-dimensional. It is common, however, to describe this 3D pattern with two planar patterns, called the principal plane patterns. These principal plane patterns can be obtained by making two slices through the 3D pattern through the maximum value of the pattern or by direct measurement. It is these principal plane patterns that are commonly referred to as the antenna patterns [14, 15].

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Azimuth and Elevation Plane (E and H plane)

Characterizing an antenna's radiation properties with two principal plane patterns works quite well for antennas that have well-behaved patterns, that is, not much information is lost when only two planes are shown. Figure 2.1 shows a possible coordinate system used for making such antenna measurements [15].

Figure 2.1 Antenna measurement co-ordinate system

The term azimuth is commonly found in reference to "the horizon" or "the horizontal" whereas the term elevation commonly refers to "the vertical". When used to describe antenna patterns, these terms assume that the antenna is mounted (or measured) in the orientation in which it will be used. In Figure 2.1, the xy-plane (θ=90 °) is the azimuth plane (E-plane). The azimuth plane pattern is measured when the measurement is made traversing the entire xy-plane around the antenna under test. The elevation plane (H-plane) is then a plane orthogonal to the xy-plane, say the yz-plane (Φ=90 °). The elevation plane pattern is made traversing the entire yz-plane around the antenna under test [15].

The Poynting vector describes both the direction of propagation and the power density of the electromagnetic wave. It is found from the vector cross product of the electric and magnetic fields and is denoted S:

S=E×H ¿w /m2(2−1)

Root mean square (RMS) values are used to express the magnitude of the fields.H ¿ is the complex conjugate of the magnetic field phasor. The magnetic field is proportional to the

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electric field in the far field. The constant of proportion is η, the impedance of free space (η =376.73):

|S|=S=¿E ¿2

ηw/m2(2−2)

Because the Poynting vector is the vector product of the two fields, it is orthogonal to both fields and the triplet defines a right-handed coordinate system: (E, H, S) [6].

Return Loss

Return loss is a measure of the reflected energy from a transmitted signal. It is a logarithmic ratio measured in dB (decibel) that compares the power reflected by the antenna to the power that is fed into the antenna from the transmission line. The larger the value of return loss the less is the energy reflected. For good impedance matching resonant frequency must lie below −10dB . [14] .

Bandwidth

Bandwidth is defined as the range between upper cut-off frequency ( f U) at -10 dB and lower cut-off (f L¿ frequency at -10 dB. Bandwidth indicates range of frequency for which an antenna provides satisfactory operation [14].

3-dB Beamwidth

Also known as the Half Power Beamwidth (HPBW) is typically defined for each of the principle planes. The 3-dB beamwidth in each plane is defined as the angle between the points in the main lobe that are down from the maximum gain by 3dB. This is the point where the magnitude of the radiation pattern decreases by 50% (or -3 dB) from the peak of the main beam [14, 15].

VSWR

VSWR stands for Voltage Standing Wave Ratio. The parameter VSWR is a measure that numerically describes how well the antenna is impedance matched to the radio or transmission line it is connected to. The smaller the VSWR the better the antenna matched to the transmission line and the more the power delivered to the antenna. For the perfect matching VSWR = 1, there is no reflection and return loss. In the real system it is very hard to achieve a perfect match, so it is defined that having VSWR < 2 is still good matching system [14].

Directivity

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Directivity of an antenna is a measure of the concentration of the radiated power in a particular direction [14]. If the antenna had 100% radiation efficiency, all directivity would be converted to gain. Typical half wave patches have efficiencies well above 90% [13].

Antenna Gain

Gain is a measure of the ability of the antenna to direct the input power into radiation in a particular direction and is measured at the peak radiation intensity [6].It is standard practice to use an isotropic radiator as the reference antenna in this definition. An isotropic radiator is a hypothetical lossless antenna that radiates its energy equally in all directions. This means that the gain of an isotropic radiator is G=1 (or 0 dB). It is customary to use the unit dBi (decibels relative to an isotropic radiator) for gain with respect to an isotropic radiator [15].

Polarization

The Polarization of an antenna is the polarization of the wave radiated by the antenna in the far field [8]. Polarization is a property of waves that can oscillate with more than one direction [16].The plane in which the electric field varies is also known as the polarization plane. For optimum system performance, transmit and receive antennas must have the same polarization [13].

Front-to-back ratio

The front-to-back (F/B) ratio is used a figure of merit that attempts to describe the level of radiation from the back of a directional antenna. Basically, it is the ratio of the peak gain in the forward direction to the gain 180-degrees behind the peak. On a dB scale, it is just the difference between the peak gain in the forward direction and the gain 180-degrees behind the peak [15].

Microstrip Antennas

Microstrip antennas are also referred to as patch antennas. They are low profile, conformable to planar and non-planar surfaces, simple and inexpensive to manufacture using modern printed-circuit technology, mechanically robust when mounted on rigid surfaces, compatible with MMIC designs and when the particular patch shape and mode are selected, they are very versatile in terms of resonant frequency, polarization, pattern and impedance [1].

Major operational disadvantages of microstrip antennas are their low efficiency, low power, high Q (sometimes in excess of 100), poor polarization purity, poor scan performance, spurious

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feed radiation and very narrow frequency bandwidth, which is typically only a fraction of a percent or at most a few percent. There are methods, however, such as increasing the height of the substrate that can be used to extend the efficiency (to as large as 90 percent if surface waves are not included) and bandwidth (up to about 35 percent). However, as the height increases, surface waves are introduced which usually are not desirable because they extract power from the total available for direct radiation (space waves). The surface waves travel within the substrate and they are scattered at bends and surface discontinuities, such as the truncation of the dielectric and ground plane, and degrade the antenna pattern and polarization characteristics [1].

Basic Characteristics

Microstrip antennas, as shown in Figure 2.2, consist of a very thin (t ≪ λ0, whereλ0 is the free-space wavelength) metallic strip (patch) placed a small fraction of a wavelength (h≪ λ0, usually 0.003λ0 ≤ h≤0.05λ0) above a ground plane. The microstrip 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 judicious mode selection. For a rectangular patch, the length L of the element is usually λ0/3 <L<λ0/2. The strip (patch) and the ground plane are separated by a dielectric sheet (referred to as the substrate). 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≤ε r ≤12. The ones that are most desirable for good antenna performance are thick substrates whose dielectric constant is in the lower end of the range because they provide better efficiency, larger bandwidth, loosely bound fields for radiation into space, but at the expense of larger element size [1].

The radiating elements and the feed lines are usually photo-etched on the dielectric substrate. The radiating patch may be square, rectangular, thin strip (dipole), circular, elliptical, triangular, or any other configuration. Square, rectangular, dipole (strip), and circular are the most common because of ease of analysis and fabrication, and their attractive radiation characteristics, especially low cross-polarization radiation [1].

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Figure 2.2 Microstrip antenna and coordinate system

There are many configurations that can be used to feed microstrip antennas. The four most popular methods are the microstrip line, coaxial probe, aperture coupling, and proximity coupling. The microstrip-line feed is easy to fabricate, simple to match by controlling the inset position and rather simple to model. However as the substrate thickness increases, surface waves and spurious feed radiation increase, which for practical designs limit the bandwidth [1].

There are various methods of analysis for microstrip antennas with the most popular models being the transmission-line, cavity, and full wave models (which include primarily integral equations/Moment Method). The transmission-line model is the easiest of all, it gives good physical insight, but is less accurate and it is more difficult to model coupling [1].

Transmission–Line Model Analysis for a Rectangular Patch

Fringing Effects

Because the dimensions of the patch are finite along the length and width, the fields at the edges of the patch undergo fringing. This is illustrated along the length in Figures 2.2(a, b) for the two radiating slots of the microstrip antenna. The same applies along the width. The amount of fringing is a function of the dimensions of the patch and the height of the substrate. For the principal E-plane (xy-plane) fringing is a function of the ratio of the length of the patch L to the height h of the substrate (L/h) and the dielectric constant ε r of the substrate. Since for microstrip antennas L/h≫1, fringing is reduced; however, it must be taken into account

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because it influences the resonant frequency of the antenna. The same applies for the width. For a microstrip line shown in Figure 2.3(a), typical electric field lines are shown in Figure 2.3(b). This is a nonhomogeneous line of two dielectrics; typically the substrate and air. As can be seen, most of the electric field lines reside in the substrate and parts of some lines exist in air. As W/h≫1 and ε r≫ 1, the electric field lines concentrate mostly in the substrate. Fringing in this case makes the microstrip line look wider electrically compared to its physical dimensions. Since some of the waves travel in the substrate and some in air, an effective dielectric constant ε reff is introduced to account for fringing and the wave propagation in the line [1].

Figure 2.3 Microstrip line and its electric field lines, and effective dielectric constant

The effective dielectric constant is defined as the dielectric constant of the uniform dielectric material so that the line of Figure 2.3(c) has identical electrical characteristics, particularly propagation constant, as the actual line of Figure 2.2(a).

Effective Length, Resonant Frequency, and Effective Width

Because of the fringing effects, electrically the patch of the microstrip antenna looks greater than its physical dimensions. For the principal E-plane (xy plane), this is demonstrated in Figure 2.4(a) where the dimensions of the patch along its length have been extended on each end by a distance ∆ L, which is a function of the effective dielectric constant ε reff and the width-to height ratio (w/h) [1].

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Figure 2.4 Physical and effective lengths of rectangular microstrip patch

Since the length of the patch has been extended by ∆ L on each side, the effective length of the patch is now (L=λ /2 for for dominant TM 010 mode with no fringing)

Leff=L+2∆L(2−4 )

For the dominant TM 010 mode, the resonant frequency of the microstrip antenna is a function of its length. Usually given by

¿¿

where v0 is the speed of light in free-space. Since (2-5) does not account for fringing, it must be modified to include edge effects and should be computed using

¿¿

where

q=¿¿

The q factor is referred to as the fringe factor(length reduction factor). As the substrate height increases, fringing also increases and leads to larger separation between the radiating edges and lower resonant frequencies [1].

Arrays and Feed Networks

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Usually the radiation pattern of a single element is relatively wide, and each element provides low values of directivity (gain). In many applications it is necessary to design antennas with very directive characteristics (very high gains) to meet the demands of long distance communication. This can only be accomplished by increasing the electrical size of the antenna. Enlarging the dimensions of single elements often leads to more directive characteristics. Another way to enlarge the dimensions of the antenna, without necessarily increasing the size of the individual elements, is to form an assembly of radiating elements in an electrical and geometrical configuration. This new antenna, formed by multielements which are driven by the same source, is referred to as an array. In most cases, the elements of an array are identical. This is not necessary, but it is often convenient, simpler, and more practical [1].l

The total field of the array is determined by the vector addition of the fields radiated by the individual elements. This assumes that the current in each element is the same as that of the isolated element (neglecting coupling). This is usually not the case and depends on the separation between the elements. To provide very directive patterns, it is necessary that the fields from the elements of the array interfere constructively (add) in the desired directions and interfere destructively (cancel each other) in the remaining space. Ideally this can be accomplished, but practically it is only approached. In an array of identical elements, there are at least five controls that can be used to shape the overall pattern of the antenna [1]. These are:

1. The geometrical configuration of the overall array (linear, circular, rectangular, spherical, etc.)

2. The relative displacement between the elements3. The excitation amplitude of the individual elements 4. The excitation phase of the individual elements 5. The relative pattern of the individual elements

Arrays are very versatile and are used, among other things, to synthesize a required pattern that cannot be achieved with a single element. In addition, they are used to scan the beam of an antenna system, increase the directivity, and perform various other functions which would be difficult with any one single element. The elements can be fed by a single line or by multiple lines in a feed network arrangement. The first is referred to as a series-feed network while the second is referred to as a corporate-feed network. The corporate-feed network is used to provide power splits of 2n (i.e., n=2, 4, 8, 16, 32, etc.). This is accomplished by using either tapered lines, or using quarter-wavelength impedance transformers [1].

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Figure 2.5 Feed arrangements for microstrip patch arrays

Corporate-fed arrays are general and versatile. With this method the designer has more control of the feed of each element (amplitude and phase) and it is ideal for scanning phased arrays, multi-beam arrays, or shaped-beam arrays. The phase of each element can be controlled using phase shifters while the amplitude can be adjusted using either amplifiers or attenuators [1].

Those who have been designing and testing microstrip arrays indicate that radiation from the feed line, using either a series or corporate-feed network, is a serious problem that limits the cross-polarization and side lobe level of the arrays [38]. Both cross-polarization and side lobe levels can be improved by isolating the feed network from the radiating face of the array. This can be accomplished using either probe feeds or aperture coupling [1].

In microstrip arrays, as in any other array, mutual coupling between elements can introduce scan-blindness which limits, for a certain maximum reflection coefficient, the angular volume over which the arrays can be scanned. For microstrip antennas, this scan limitation is strongly influenced by surface waves within the substrate. This scan angular volume can be extended by eliminating surface waves. One way to do this is to use cavities in conjunction with microstrip elements. It has been shown that the presence of cavities, either circular or rectangular, can have a pronounced enhancement in the E-plane scan volume, especially for thicker substrates. The H-plane scan volume is not strongly enhanced. However the shape of the cavity, circular or rectangular, does not strongly influence the results [1].

CHAPTER 3: THE DESIGN METHODOLOGY

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A rectangular patch was chosen as the basis of the design because of its ease of fabrication and analysis. The microstrip line was used as the feeding method as it is easy to fabricate, simple to match by controlling the inset feed position and rather simple to model. The antenna was designed to work in the 2.4GHz ISM band which has a frequency range of 2.4-2.5GHz, a center frequency of 2.450GHz, a bandwidth of 100MHz and is freely available worldwide. Some applications in the 2.4GHz ISM band include the home microwave oven, sulphur lamps, communication applications such as wireless LANs, bluetooth and radio control equipment such as low power remote control of toys [3].

Design Procedure

The FR4 Glass Epoxy, whose loss tangent is 0.002, was chosen as the dielectric material substrate.

To commence the design procedure assumes, specific information had to be included: dielectric constant of the substrate (ε r ¿, the resonant frequency ( f ¿¿ r )¿ and the height of the substrate, h.

ε r=4.3 , f r=2.44GHz ,h=1.6mm

For an efficient radiator, the practical width that leads to good radiation efficiencies is

W= 12 f r√μo εo √ 2

εr+1=

vo2 f r √ 2

ε r+1(3−1)

¿37.58mm where vois the free-space velocity of light.

The initial values (at low frequencies) of the effective dielectric constant are referred to as the static values, and they were calculated as

W /h>1

εreff=¿

ϵr+12 +

εr−12 ¿ ¿

¿3.99

A very popular and practical approximate relation was then used to find the normalized extension of the length as

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∆ Lh

=0.412(εreff +0.3)(

Wh

+0.264 )

(εreff−0.258)(Wh

+0.8)(3−3)

∆ L=0.741mm

The actual length of the patch was determined by solving L as,

L= 12 f r√ε reff √μo εo

−2∆ L(3−4)

¿29.15mm

For efficient transfer of power from a transmission line to the patch antenna, the input impedance of the patch antenna needed to be matched to the characteristic impedance of the transmission line. It was observed that impedance seen by a transmission line attached to the radiating edge was very high, and also the impedance (ratio of voltage to current) decreased as one moved towards the center of the patch. Therefore, depending on the characteristic impedance of the transmission line, an appropriate point on the patch was chosen through calculation as the feed point [8].

In order to access the appropriate impedance point on the patch, a recess was created in the patch. The recess or inset feed was used to improve the impedance matching between the patch and the feed line. The inset feed position, where the input impedance was 50 ohms and the lengths and widths for the microstrip feeds were calculated using the matlab code in appendix B. A FDTD-Finite Difference Time Domain- analysis shows that the inset disturbs the transmission line or cavity model and increases the impedance variation with distance compared to a coaxial probe feed given a patch resonant length L and feed position yo from the center. Transmission line analysis method was applied as it gives a good insight. However, it is more difficult to model coupling as well as less accurate [1, 6, 8].

Ground Plane

As part of the antenna, the ground plane should be infinite in size as for a monopole antenna but in reality this is not easy to apply besides a small size of ground plane is desired. In practice, it has been found that the microstrip impedance with finite ground plane width (Zo) is practically equal to the impedance value with infinite width ground plane (Zi) , if the ground widthW g is at least greater than 3*W. The radiation of a microstrip antenna is generated by the

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fringing field between the patch and the ground plane, the minimum size of the ground plane is therefore related to the thickness of the dielectric substrate.[9, 11].

The size of the ground plane was chosen as 114mmlengthby 148mmwidth.

Microstrip Discontinuities

Surface waves are electromagnetic waves that propagate on the dielectric interface layer of the microstrip. The propagation modes of surface waves are practically TE and TM. Surface waves are generally at any discontinuity of the microstrip. Once generated, they travel and radiate, coupling with other microstrip of the circuit, decreasing isolation between different networks and signal attenuation. Surface waves are a cause of crosstalk, coupling, and attenuation in a multi-microstrip circuit. For this reason surface waves are always an undesired phenomenon [9].

A discontinuity in a microstrip is caused by an abrupt change in geometry of the strip conductor, and electric and magnetic field distributions are modified near the discontinuity. The altered electric field distribution gives rise to a change in capacitance, and the changed magnetic field distribution to a change in inductance.

a) Bends Four 90° bends were encountered in the design. This brought about excess capacitance at the square corners making the characteristic impedance value to be lower than that of the uniform connecting lines. A bend of this angle doesn’t work well above a few GHz due to a high VSWR. The same holds true for bends with angles greater than 90°Compensation for the microstrip corner bend was made by the use of decreased capacitance technique. Since experiments on various bends have proven that a decrease in the input reflection coefficients can be achieved if the corner is chamfered (mitered), the following configuration was applied

Figure 3.1 Configuration for compensated right-angled bends; W is the width of the line

Therefore,1.8×2.62=4.716mm

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b) Step Width Junction This discontinuity was found at the λ /4Transformers. The effect of the fringing capacitance associated with the wider line of the step discontinuity is similar to an increase in the length of that line

Figure 3.2 Characteristics of the step width junction discontinuity

In terms of distributed elements, the discontinuity capacitance C has the effect of an increase in length of the wide line w1, and an equal decrease in length of the narrow line w2. To compensate for the excess capacitance, the wider line w1 was made to be electrically longer by a length of 9.26mm.

c) T-Junction These discontinuities were found in the patch antenna array as branch –lines. The T-Junctions were easily compensated for by simply adjusting the lengths of the different lines. The offset in the main line is usually very small, and the main effect is on the length of the stub

Figure 3.3 T-junction discontinuity compensation and minimization of the effect

w1=12=2.62mm;0.7w1=0.7×2.62=1.834mm

Main Beam Direction :

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For the 4-element array of figure 3.4, the main beam was directed broadside to the array by ensuring there was no input phase difference from element to element. To implement an even number of in-phase patch elements, the feed network needed to be carefully designed. The distance from the 50-ohm SMA source to each patch element needed to be identical or multiples of λ. Unequal line lengths would have produced phase shifts, which would yield fixed beams that would be scanned away from the broadside. A quarter-wave transformer was used to match the 100-ohm line to a 50-ohm line. The 100-ohm microstrip line was fed using a 50-ohm SMA. In the design of an effective in-phase radiator, the distance between the patch elements needed to be optimized to yield a peak gain. The antenna-array chapter in Antenna Theory by Balanis provided insight on the optimum antenna separation distance. The author identified a separation distance of λ /2 as providing the optimal gain. In the design, this separation was used as 31.33mm [2, 4].

Matching of Microstrip Lines to the Source

The characteristic impedance of a transmission line of the microstrip feed patch was designed with respect to the source impedance. The characteristic impedance Zo of the transmission line from the source with respect to the source impedance Z s was

Zo=n .Zs(3−5)

Zo=2×50=100Ohms

Where the factor n was the number of twigs emanating from the node connected to the source. The inner conductor of the coax was soldered to the 100-ohm microstrip line, and the outer conductor connected to the ground plane. Since the coax fed two 100-ohm microstrip lines in parallel, no mismatch occurred at this input as the parallel combination of the two microstrip lines was equal to 50-ohm [4, 10].

Quarter-wave Transformer:

For the input impedance of a transmission line of length L with a characteristic impedance Zo and connected to a load with impedance ZA :

Z¿ (−L )=Zo [ ZA+ j Zo tan (βL)Zo+ j Z A tan (βL) ](3−6)

When the length of the transformer is a quarter wavelength;

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Z¿ (L= λ4 )= Zo

2

Z A(3−7)

The above states that by using a quarter-wavelength of a transmission line, the impedance of the load ZA can be transformed by the above equation.

Hence by using a transmission line with a characteristic impedance of 50-ohms, the 50 ohm inset feed line was matched to

Zo=√50∗50 (3−8 )

¿50ohms

Where Zo = Characteristic impedance of the quarter-wavelength transformer

This ensured that no power would be reflected back to the SMA feed point as it tried to deliver power to the antenna [5].

The length of the quarter wavelength transformer was calculated as

L= λ4=

λ04 √εreff

(3−9)

¿15.39mm

Where λ = Effective wavelength

λo = Free space wavelength

Simulation

The antenna array was designed using the Ansoft HFSS 13.0 software. HFSS is a 3D full wave electromagnetic field simulator. It uses the finite element method together with adaptive meshing to solve the wave equations. If a 3D model has been made, HFSS sets up the mesh automatically. HFSS computes S-parameters, can calculate and plot both the near and far field radiation and compute important antenna parameters such as gain and radiation efficiency. This software was used to vary the sizes of the patches, microstrip feed lines and ground plane in order to come up with the desired results [12]. Figure 3.4 illustrates the HFSS antenna model.

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Figure 3.4 4 element patch antenna HFSS model

Figure 3.5 4-element patch antenna PCB layout with dimensions

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E-Plane

H-Plane

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Fabrication

As per the HFSS designs, masks for fabrication of the microstrip antenna and the ground plane were designed using AutoCAD. The mask images were then transferred to transparent films before being photoengraved to a double sided PCB by exposure to UV light for 60 seconds. The PCB was then suspended in Sodium Hydroxide developer for a minute to develop photoresist. It was washed after which chemical etching done using a solution of iron chloride to create the patch antenna. The etched copper pattern was rinsed in water and again exposed to UV light for a minute. It was immersed in the developer to remove the photoresist and finally cleaned with water. After air drying, an RF RP-SMA connector (through-hole, Jack (male pin) right angle PCB mount connector) with solder post was soldered at the center of the PCB from the backside. An RG58/U cable was used to connect to the SMA connector. The other end of the cable was terminated to an N male connector. This was to be used as the connector to a spectrum analyzer [7].

Figure 3.6 Implemented 4-element patch antenna array

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Figure 3.7 ground plane of the patch array

Figure 3.8 N male to sma female cable

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CHAPTER 4: HFSS SIMULATION RESULTS AND ANALYSIS

Variation of Patch Length and Width

Dimensions calculated in the design procedure were used to create the 4 element array patch antenna. The antenna, however, did not produce acceptable results. In order to shift the S11 minima towards the desired center frequency of 2.4GHz, the length and width of the patch were shortened as follows

Table 4.1 Variation of antenna parameters with changes in dimensions

Length(mm) Width(mm) Resonance Frequency(GHz) Peak Directivity(dB)38.47 29.85 2.27 7.3434.47 29.85 2.33 7.7830.47 28.66 2.45 8.3030.47 26.85 2.56 7.8430.47 23.85 2.87 7.34

A length of 30.47mm and width of 28.66mm were selected as the S11 minima operated at the center frequency. It was observed that a decrease in width increased the resonance frequency. This is due to the increase in ∆ L and ε reff . The input impedance at resonance also increased because the radiation from the radiating edges decreases, which increases the radiation resistance. The bandwidth of the antenna decreases. There is a decrease in the directivity, efficiency, and hence gain, resulting from a decrease in the effective aperture of the antenna. Effective aperture (also known as effective area) is the area over which the antenna collects energy from the incident wave and delivers it to the receiver load [8, 17].

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Reflection Coefficient and Bandwidth

Figure 4.1 shows the reflection coefficient [S11] of the proposed antenna in dB. S11 gives the reflection coefficient at the inset feed position where the input to the microstrip patch antenna was applied. It should be less than -10dB for an acceptable operation. It shows that the proposed antenna had a frequency of resonance of 2.44GHz [18].

Figure 4.1 Return loss S11 obtained for the patch array

The simulated impedance bandwidth of about 63.3MHz (2.4721-2.4088 GHz) was achieved at −10dB reflection coefficient (VSWR≤2). The reflection coefficient value that was achieved at this resonant frequency was equal to -18.2131 dB. This reflection coefficient value suggested that there was good matching at the frequency point below the -10dB region [18].

The fractional bandwidth achieved for the antenna was

BW=f U−f L

f C×100%=2.4721−2.4088

2.44045=2.62%(4−1)

where

f C=f U+ f L

2(4−2)

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f C, f U and f L are the center, upper and lower cutoff frequencies respectively.

Radiation Pattern

Figure 4.2 Simulated E-Plane (phi=90 °, theta varying)

Figure 4.3 Simulated H-plane (theta=90 °, phi varying)

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The radiation patterns in the E-plane and H-plane of the patch antenna array at 2.44GHz for yo=10.545mm are shown in Figure 4.2 and Figure 4.3 above. They are also referred to as the azimuth plane and elevation plane pattern respectively. The coplanar components in the E and H planes are Eθ in the Φ=0 ° and EΦ in Φ=90 ° planes.

Figure 4.4 shows the simulated 3-D radiation pattern with gain of 5.2235 dB for proposed antenna configuration at 2.44GHz. .

Figure 4.4 3D radiation pattern

The strongest energy was radiated outward, in the yz-plane, at the widths of the patch elements and at an angle of 36 °. It was observed that the antenna had an azimuth plane beamwidth of about 57 ° and an elevation plane beamwidth of 41 ° as indicated on the patterns in figures 4.2 and figure 4.3 by the blue lines. These lines were drawn where the gain was down from the peak by -3dB. The beamwidths were the total angular width between the two 3dB points on the curves. [15].

The azimuth and elevation patterns were derived by simply slicing through the 3D radiation pattern. For the azimuth plane pattern, slicing was done through the xz plane aty=0, while for the elevation plane the slicing was done through the yz plane atx=0.

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Figure 4.5 E-Plane and H-Plane patterns in rectangular coordinates

The Figure 4.5 shows that the antenna had two main lobes which were 180 ° out of phase with each other. It was used to determine the half-power beamwidths for the radiation patterns as the peaks and 3 dB points below them could easily be picked.

Inset Feed Position

Initially, the length of the inset feed position was calculated as yo=14.15mm from the edge of the antenna. The slot width was chosen as 3.62mm which was 1mm greater than that of the microstrip feed. An increase in the width of the slot brought about an increase in the resonance frequency. The microstrip feed going into the patch element was 15.67mm in length which is equal to λ /4 wavelength. The resulting resonance frequency was below the desired value hence the length had to be increased as shown below

Table 4.2 Variation of resonance frequency with changes in patch feed length

Length of Feed(mm) Resonance Frequency(GHz)15.67 1.7718.67 2.2620.67 2.2925.67 3.29

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The feed length of 18.67mm was chosen for analysis as it was closer to the center frequency and also not too long.

Changing of the inset feed position yo affected the resonance frequency of the patch antenna. The longer the length, the lesser the resonance frequency became and vice versa. Lesser directivity, gain as well as magnitude of the S11 parameter were realized when a longer length was used.

VSWR Plot

Figure 4.6 shows the VSWR (Voltage Standing Wave Ratio) plot for the designed antenna. The value of the VSWR should lie between 1 and 2. SWR is used as an efficiency measure for transmission lines, electrical cables that conduct radio frequency signals, used for purposes such as connecting radio transmitters and receivers with their antennas, and distributing cable television signals [18].

Figure 4.6 VSWR plot

Here the value for the proposed microstrip patch antenna was 1.2801 at the resonating frequency of 2.44GHz.

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Smith Chart

Figure 4.7 Smith chart of the proposed patch antenna

The smith chart is a graphical representation of the normalized characteristic impedance. It provides the information about the impedance match of the radiating patch. The smith chart for the designed patch antenna array showed an input impedance of 51.73+12.47i ohms at resonant frequency 2.44GHz. The magnitude of the input impedance was 53.21 which showed that accurate machine was not achieved. This was due to shifting of the inset feed position away from the center of the patch element which was done in order to improve the directivity, gain and return coefficient of the antenna.

Ground Plane

For a finite ground plane, the resonance frequency of the antenna was almost the same but the input impedance was slightly higher than that of the infinite ground. It was observed that an increase in the dimensions of the ground plane increased the resonance frequency and magnitude of the S11 parameter. There was an increase in the directivity and hence gain.

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H Plane Inter-element Separation

From the variation of the spacing between the patch elements in the H-plane, it was observed that as the spacing was increased, the magnitude of the return loss S11 as well as the directivity of the antenna decreased. This meant that the gain decreased. The resonance frequency, radiated power and efficiency however increased. A separation of λ /2 was chosen for the simulation as it gave the optimal gain.

E plane Inter-element separation

An increase in the E-plane separation gave similar results to that of the H-plane. However, a separation of λ /2 could not be achieved because of the orientation of the patch elements as well as the different lengths of the microstrip feeds.

Table 4.2 HFSS Antenna Parameters in HFSS

The table 4.2 shows a summary of the antenna parameters from the HFSS software. The software did not give the antenna parameters summary in decibels as shown. The directivity D

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and efficiency 𝜂 were 7.1273 and 46.7%, which gave a gain G (=𝜂D) of the antenna as 3.32. The front to back ratio was 242.9, implying that there was a difference of about 23.85 dB between the peak gain in the forward direction and the gain 180-degrees behind the peak. This is evidence of presence of back lobes from the radiation [15].

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CHAPTER 5: CONCLUSION

A 4 element, microstrip fed patch antenna array of rectangular shaped radiating elements was successfully designed and implemented using the FR4 Epoxy Glass substrate. Through analysis with the Ansoft HFSS simulation software, it was observed that the antenna worked in the 2.4GHz ISM band by having a resonance frequency of 2.44GHz, and had a fractional bandwidth of 2.26% and a directivity of 8.53dB . The patch antenna array was coaxially fed through a 50 ohm cable with a 50 ohm sma-connector. Impedance matching was done well though not accurately. The maximum achievable gain by the antenna was 5.2235 dB.

Time did not allow for more analysis to be done through different design simulations and testing of the prototype. This was caused by the lack of the necessary testing equipment and environment like the spectrum analyzer and anechoic chamber which is a room designed to completely absorb reflections of either sound or electromagnetic wave [1].

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APPENDICES

Appendix A

Conductance

Each radiating slot is represented by a parallel equivalent admittance Y (with conductance G and susceptance B). This is shown in FigureA1. 4.

Figure A1.4 Rectangular microstrip patch and its equivalent circuit transmission-line model

The slots are labeled as #1 and #2. The equivalent admittance of slot #1, based on an infinitely wide, infinite slot, is given by

Y 1=G1+ jB1(A1)

Where for a slot of finite width W

G1=W120 λo

¿

B1=W120 λo

¿

Since slot #2 is identical to slot #1, its equivalent admittance is

Y 2=Y 1 ,G2=G1 ,B2=B1(A4 )

In general, the conductance is defined as

G1=2Prad

¿vo ¿2 (A5)

The radiated power is written as

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Prad=¿vo ¿

2

2 πηo∫0

π

¿¿

Therefore, the conductance of (1-10) can be expressed as

G1=I1

120π2(A7)

Where

I 1=∫0

π

¿¿¿

¿−2+cos (X )+X S i (X )+ sin (X )X

(A8)

X=koW (A 9)

Asymptotic values of (1-12) and (1-12a) are

G1={ 190 (Wλ0 )2

W ≪λo

1120 (Wλo )W ≫ λo

(A10)

Resonant Input Resistance

The total admittance at slot #1 (input impedance) is obtained by transferring the admittance of slot #2 from the output terminals to the input terminals using the admittance transformation equation of transmission lines. Ideally the two slots should be separated by λ/2 where λ is the wavelength in the dielectric (substrate). However, because of fringing the length of the patch is electrically longer than the actual length. Therefore the actual separation of the two slots is slightly less than λ/2. If the reduction in length is properly chosen (typically 0.48 λ<L<0.49 λ), the transformed admittance of slot #2 becomes

~Y 2=G2+ j B2=G1− j B1(A11)

Or

G2=G1(A12)

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B2=−B1(A13)

Therefore the total resonant input admittance is real and is given by

Y ¿=Y 1+Y 2=2G1(A 14)

Since the total input admittance is real, the resonant input impedance is also real, or

Z¿=1Y ¿

=R¿=12G1

(A15)

The resonant input resistance, as given by (A15), does not take into account mutual effects between the slots. This can be accomplished by modifying (A15) to

R¿=1

2(G1±G12)(A 16)

Where the plus (+) sign is used for modes with odd (antisymmetric) resonant voltage distribution beneath the patch and between the slots while the minus (-) sign is used for modes with even (symmetric) resonant voltage distribution. The mutual conductance is defined, in terms of the far-zone fields, as

G12=1

¿V 0¿2ℜ∬s E1×H 2

¿ . d s(A17)

Where E1 is the electric field radiated by slot #1, H 2 is the magnetic field radiated by slot #2, V o is the voltage across the slot, and the integration is done over a sphere of large radius. It can be shown that G12 can be calculated using

G12=1

120π2∫0

π

¿¿¿

Where Jo is the Bessel function of the first kind of order zero.

As shown by above the input resistance is not strongly dependent upon the substrate height h. In fact for very small values of h, such thatk oh≪1, the input resistance is not dependent on h. Modal-expansion analysis also reveals that the input resistance is not strongly influenced by the substrate height h. It is apparent that the resonant input resistance can be decreased by increasing the width W of the patch. This is acceptable as long as the ratio of W/L does not exceed 2 because the aperture efficiency of a single patch begins to drop, as W/L increases beyond 2. The resonant input resistance, as calculated by (1-17), is referenced at slot #1. However, it has been shown that the resonant input resistance can be changed by using an inset feed, recessed a distance yo from slot #1, as shown in Figure A1.5.

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Figure A1.5 Recessed microstrip- line feed

This technique can be used effectively to match the patch antenna using a microstrip-line feed whose characteristic impedance is given by

Zc={ 60√ εreff

ln [ 8hW o+W o

4h ] , W o

h ≤1(A19−A)

120 π

√ εreff [W o

h+1.393+0.667 ln (

W o

h+1.444)]

,W o

h>1(A 19−B)

Where W ois the width of the microstrip line, as shown in Figure A1. 5. Using modal-expansion analysis, the input resistance for the inset feed is given approximately by

R¿ ( y= yo )= 12(G1±G12) [cos2(πL yo)+G1

2+B12

Y c2 sin2( πL yo)−B1

Y csin( 2πL yo)](A 20)

Where Y c=1/Zc. Since for most practical microstrips G1/Y c≪1 and B1/Y c≪1, (1-20) reduces to

R¿ ( y= yo )= 12 (G1±G12 )

cos2( πL yo)¿ R¿ ( y=0)cos

2( πL yo)(A 21)The values obtained using (A20) agree fairly well with experimental data. However, the inset feed introduces a physical notch, which in turn introduces a junction capacitance. The physical notch and its corresponding junction capacitance influence slightly the resonance frequency, which typically may vary by about 1%. It is apparent from (A20A) that the maximum value

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occurs at the edge of the slot ( yo=0) where the voltage is maximum and the current is minimum; typical values are in the 150–300 ohms. The minimum value (zero) occurs at the center of the patch (yo=L/2) where the voltage is zero and the current is maximum. As the inset feed point moves from the edge toward the center of the patch the resonant input impedance decreases monotonically and reaches zero at the center. When the value of the

inset feed point approaches the center of the patch ( y0=L /2), the cos2(π yo

L) function varies

very rapidly; therefore the input resistance also changes rapidly with the position of the feed point. To maintain very accurate values, a close tolerance must be preserved.

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Appendix B

Matlab Code for calculation of the insed feed position where the input impedance is 50 Ohms

≫er=4.3; %dielectric constantf=2.44e9; %frequency in Hzh=1.6; %substrate thickness in mm

la=(3e9/f)*1000;k=(2*pi)/la;w=30.47; %width of the patch in mml=29.28; %length of the patch in mm

x=k*w;i1=-2+cos(x)+(x*sinint(x))+(sin(x)/x);g1=i1/(120*pi*pi); %conductance

%jb=besselj(0,(k.*l.*sin(th)));

a=@(th)(((sin((x./2).*cos(th))./cos(th)).^2).*(besselj(0,k.*l.*sin(th)))).*(sin(th)).^3;a1=quad(a,0,pi);g12=a1/(120*pi*pi); %in siemensr_in=1/(2*(g1+g12));inset=(l/pi)*(acos(sqrt(50/r_in))) %inset feed point distance in mm

inset =

14.2961

Matlab code for calculation of antenna dimensions

er=4.2; %er=input('Enter the di-electric constant:') f=2.44*10^9; %f=input('Enter the frequency (GHz):') h=0.16*10; %h=input('Enter the substrate thickness (in mil)')

wid=(3e8/(sqrt((er+1)/2)*2*f))*1000 %width of patch in mm

e_eff=((er+1)/2)+ (((er-1)/2)* (1+((12*h)/wid))^-0.5) %Effective dielectric constant

l_eff=(3e8/(2*f*sqrt(e_eff)))*1000 %Effective Length

del_l=(((e_eff+0.3)*((wid/h)+0.264))/((e_eff-0.258)*((wid/h)+0.8)))*(0.412*h) %Normalized extension of length

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L=l_eff-(2*del_l) %Actual length of Patch

wid = 38.1254

e_eff = 3.9048

l_eff = 31.1100

del_l = 0.7435

L = 29.6230

Matlab code for calculation for the width of the 50 ohm line

>> solve [x/1.6+0.667*ln((x/1.6)+1.444)]=2.3868 ans = 2.6182207203339794813467175866412

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REFERENCES

[1] Constantine A. Balanis Antenna Theory, Analysis and Design, 3rd Edition, John Wiley & Sons, Inc., 2005

[2] John Hermann, Zach Dyals, and Angel Arcia, “Analysis and Design of ISM-Band Patch Antenna Array”, ECE 4370: 5.8GHz High-Directivity Antenna, 2012, Group 3, pp. 1-13

[3] http://en.m.wikipedia.org/wiki/ISM_band

[4] Design of Microstrip Patch Antenna using FR4 substrate http://just.edu.jo/nihad/files/mat/591/report.pdf

[5] http://www.antenna-theory.com/tutorial/txline/transmission5.php

[6] Thomas A. Milligan Modern Antenna Design, 2nd Edition, John Wiley & Sons, Inc., 2005

[7] Deepak Sood,Gurpal Singh, Chander Charu Tripathi, Suresh Chander Sood & Pawan Joshi, “Design, fabrication and characterization of microstrip square patch antenna array for X-band applications”, Indian Journal of Pure & Applied Physics, 2008, Vol. 46, pp. 593-597

[8] A.R. Harish, M.Sachidananda, Antennas and Wave Propagation, 4th Edition, Oxford University Press, 2007

[9] Iulan Rosu, “Microstrip, Stripline, and CPW Design”, YO3DAC/ VA3IUL, http://www.qsl.net/va3iul

[10] John R. Ojha and Marc Peters, “Patch Antennas and Microstrip Lines, Micrrowave and Millimeter Wave Technologies Modern UWB antennas and equipment”, Igor Mini(Ed.), 2010, ISBN 978-953-7619-67-1, In-Tech DOI: 10.5772/9016.

[11] Yi Huang, Kevin Boyle, Antennas From Theory to Practice, John Wiley and Sons, Ltd, Publication, 2008

[12] Fredrick Gulbrandsen, “Design of an X-band Phased Array Patch Antenna”, 2013, Norwegian University of Science and Technology, Department of Electronics and Telecommunications http://www.diva-portal.org/smash/get/diva2:646810/FULLTEXT01.pdf

[13] D. Orban and G.J.K.Moernaut, “The Basics of Patch Antennas, Updated”, RF Globalnet Newsletter, 2009

http://rfglobalnet.com/doc/technical-article-the-basics-of-patch-antenna-0001

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[14] Ruchi Kadwane, Vinaya Gohokar, “Design and Characteristics Investigation of Multiband Microstrip Patch Antenna for Wireless Application”, International Journal of Emerging Engineering Reserch and Technology, 2014, Vol. 2, Issue 3, pp. 61-66

[15] http://www.cisco.com/c/en/us/products/collateral/wireless/aironet-antennas-accessories/prod_white_paper0900aecd806a1a3e.html

[16] http://en.m.wikipedia.org/wiki/Polarization_(waves)

[17] Girish Kumar, K.P. Ray, Broadband Microstrip Antennas, Artech House, Inc.,2003

[18] Jaswinder Kaur, Rajesh Khanna, “Co-axial Fed Rectangular Microstrip Patch Antenna for 5.2 Ghz WLAN A pplication”, Universal Journal of Electrical and Electronic Engineering, 2013, Vol. 1, Issue 3, pp 94-98

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