fundamental antennas

8/13/2019 fundamental antennas

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1

2008

First edition

Ahmed M. Alaa


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Contents

Introduction . 8

Chapter 1 : Basic antenna terminology..9

1.1 Radiation pattern

1.2 Directivity

1.3 Gain1.4 Efficiency

1.5 Types of antennas

Chapter 2 : Dipole antenna..34

2.1 Introduction2.2Balanced and

Unbalanced Systems

2.3Image theory

2.4 Monopoles

2.5Disadvantages


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Contents

Chapter 3 : Loop antennas..61

3.1 Introduction

3.2 Design Parameters

3.3 Equivalent Circuits3.4 Loop antenna

Configurations

3.5 Applications in mobile

Communication system

Chapter 4 : Yagi Uda antennas..77

4.1 Introduction

4.2 Components

4.3 Design procedure

4.4 Advantages

4.5 The folded dipole


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Contents

Chapter 5 : Reflectorantennas..92

5.1 Why Reflectors ?

5.2 Types of reflectors

According to geometry5.3 Types of Parabolic

Surfaces

5.4 Methods of feeding

Parabolic reflectors

5.5 Using Image theory

To calculate field

5.6 Using GTD to calculate

The field


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Contents

Chapter 6 : Microstrip antennas..105

6.1 Components

6.2 Types of microstrip

Antennas6.3 Feeding techniques

6.4 Advantages

6.5 Disadvantages

6.6 Techniques to overcome

Disadvantages

6.7 Microstrip arrays

6.8 Feeding of arrays

6.9 Microstrip vs. reflectors.


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Contents

Chapter 7 : Fractal antennas..130

7.1 Definition

7.2 Characteristics

7.3 Types of fractals

7.4 Advantages


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Introduction

This book presents a collection of presentationsI gave And tutorials I made previously for basic

concepts of Antenna design , it shows you aconceptual overview for Each type of antennasand software programs that you Can use to design

them , their advantages , Disadvantages andapplications they are used in without Involvingany complicated equations. The book can be

Considered a quick guide for amateurantenna designers Or readers interested in

understanding how antennasWork with no prerequisites


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Chapter 1

Thomas Edison usedAntennas in 1885 !

Basic antenna terminology


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Basic Antenna

terminologyOutline

1. Radiation Pattern

2. Directivity

3. Gain

4. Efficiency

5. Types of antennas


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1.1 Radiation Pattern

The distribution of power or its

Derivatives ( power density , power

Intensity ) in the space around theAntenna , relative to the maximum

Magnitude , i.e. : Radiation pattern

Is concerned with the proportion

Of magnitudes and not their values..

The pattern varies according to

Different wand u.

An example to a radiation pattern in

Cartesian coordinates


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Radiation Pattern

An example to a radiation pattern in

Polar coordinates

u

w

Azimuth plane

Elevation plane


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Radiation Pattern : Half power beam width

The beam width is the angle included between two angles in which u (u , w)

Is equal to half Umax , where U is the power intensity . The half power beam

Width = u1 - u2 . Where u1 and u2 are the angles where U is half its

Max value , the same for the elevation angle .

The Half power beam widths are :aAzimuth plane beam width

bElevation plane beam width


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Radiation Pattern : Half power beam width

When the patterns mathematical formula is independent on phi , the pattern

Is symmetric about the zaxis , then the Azimuth plane beam width is equal

To the elevation plane beam width .

Calculating Azimuth plane beam width

Putting u = p / 2 , we can calculate

Phi 1 and Phi 2

Putting w = p / 2 , we can calculate

Theta 1 and Theta 2

Calculating elevation plane beam width


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Radiation Pattern : Azimuth plane half power beam width


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Radiation Pattern : Elevation plane half power beam width


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Radiation Pattern : First Null beam width

The beam included by angles where the power is ZERO , usually the first

Nulls bound the major lobe of the radiation pattern , the first null beam width

Is calculated by estimating the angles where the power intensity is

Zero .


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Radiation Pattern : Directive Antennas

Some Applications we need the receiving or transmitting process to be

Directed in a certain direction , the radiation pattern then have a major lobe

With most of the power concentrated in a certain beam .


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Side lobes :lobes

That have lower

Power than major

Lobes ( also calledMinor lobes ) .

Back lobe :The

Lobe directedTo the earth in

3D representation

Major lobes :the

Lobes with highestPower concentration

( usually present in

Directive antennas)

The decart plot of a directive antenna


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The 3D plot of a directive antenna


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Radiation Pattern : Omindirectional Antennas

The decart plot of an

omindirectional antenna

The distribution of power

Around the antenna

Is nearly equal .


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1.2 . Directivity

Directivity : The measure of how much power , power density or power

Intensity is concentrated in a certain beam

D = Umax / Uo

Where Uo is the average power intensity and Umax is maximum intensity

When Umax = Uo , the antenna is omindirectional & D = 1 = 0 dB .


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Directivity

The directivity is usually inversely proportional with the half power beam width

D a ( 1 / HPBW )

U ( u ,w)

u

U ( u ,w)

u

Ideal case D =

Infinity , and HPBW

= 0 .

( a Pulse where ALL

Power isconcentrating

At one point .)

Omindirectional


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1.3 . Gain

Gain : The directivity after considering the antennas efficiency .

G = D *hUsually measured in dB .


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1.4 . Efficiency

The Efficiency of an Antenna is divided into three parts :

aRadiation Efficiency

bMismatch

cPolarization losses .


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Efficiency : Radiation Efficiency

Radiation Efficiency :The efficiency of the antenna itself , regardless of

The antenna system , and the polarization mismatch , it is related to the

Material of the antenna .

Radiation Efficiency

=( Radiated Power )

/ ( Radiated Power +

Lost Power ) .

Sometimes called =

ecd


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Efficiency : Reflection Mismatch

~

Zo

Zin

An equivalent circuit for an

Antenna attached to aGenerator , the input

Impedance of the load

( antenna ) is not equal to

Zin but the transmission

Line transforms it according

To its characteristicImpedance Zo .


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Efficiency : Polarization losses

If the Polarization of the incident wave is not matching with the polarization of

The antenna , losses results in and measured by polarization loss factor

PLF .

Antenna

Polarization Received Signal

Cross -Polar

Component

CoPolar

Component

Lost Component

PLF = Cos


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1.5 . Types of Antennas

1Wire Antennas

3Microstrip Antennas

5Reflector Antennas

2Aperture Antennas

4Array Antennas

6Lens Antennas .


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Chapter 2

C.A.Balanis is one of

The most important

antenna scientists , and

Contributed with a

famous book

Antenna theory.

Dipole antenna


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Dipole Antenna

Outline

1. Introduction

2. Balanced and

Unbalanced Systems

3. Image theory

4. Monopoles

5. Disadvantages

Practical Example


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2.1. Introduction

The dipole antenna is the simplest antenna , despite of not being used

Practically in applications , it is used to test antenna labs ( so it is considered

The reference antenna ) , a dipole antenna consists of 2 wires ( lambda /4 forIts length ) , the two wires are separated by a gap and their terminals are

Connected to the transmitter or the receiver

l/ 4l/ 4

This type of dipoles is called

Half wave length dipole as the

Total length is lambda / 2 .

+

-


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Introduction : dipole configuration


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Introduction :Characteristics

The directivity is nearly equal to 1.6 dimensionless and about 2 -> 2.2 dB ,

The input impedance is usually 73 + 42.5 j ohms and the radiation resistance

Is nearly 73 ohm .


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Introduction : Radiation Pattern I

The radiation pattern for the

Electric field for a folded dipole

antenna


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Introduction : Radiation Pattern II

The radiation pattern of the dipole , all the field is electric as shown .


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The radiation pattern of the dipole , the magnetic field equals zero .

No radiation

Pattern for the

Magnetic field

H !!This means that

A dipole is an

Electric field

Antenna

Introduction : Radiation Pattern III


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Introduction : Radiation Pattern IV

When the length of the dipole exceeds lambda the radiation pattern takes

A new shape due to the appearance of the grating lobes where the major

Lobes divides into multiple lobes .


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A system with two input terminals , a positive and negative terminals , the

Dipole antenna is a balanced system because it has two terminals and this

Is why it is not widely used in applications .

2.2 . Balanced and Unbalanced Systems

Balanced System

Balanced

System+-

2 input terminals


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Balanced and Unbalanced Systems

Unbalanced System

A system with one input terminal , having a single pole and a ground plane, we desire an unbalanced system because when mounting an antenna in a

Device only one input will is used for each component and all components have

A common ground .

UnbalancedSystem1 input terminal


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Balanced and Unbalanced Systems :

Baluns


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2.3 . Image theory

When a single pole is near an infinite plane conductor , virtual sources ( images )

Will be introduced to account for their reflections , the plane conductor can be

Considered a ground and thus we can construct an antenna that have the same

Behavior of a dipole but having a single pole , this type of antennas is called

Monopoles , and have the advantage of being an unbalanced system .

Conductors Fields

Electric conductorPEC

Magnetic Conductors

PMC

Electric field

Magnetic field


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Image theory

s= infinity s= infinity

When electric and magnetic fields are near electric and magnetic fields their

Images are in the following directions :

PEC PMC


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Monopoles


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Monopoles : Coaxial cables ( Coax )

Coaxial cables consists of a central and a ground plane , it is used to connect

The monopole to the load ( ex: a TV ) .

Ground plane

Central cable

Dielectric material


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We benefit from the ground plane of cable by welding it to the ground of monopole

And welding it to the ground of monopoles and welding the central cable to the

Wire ( the monopole ) .

Ground plane

Central cable


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We can even make a monopole from just a coaxial cable !

Central cable

And the pole of

The monopole

Antenna at the

Same time..

Ground plane

Of the monopole

And the ground

Plane of the coax

At the same time..

~

Equivalent to


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Monopoles : Baluns

When we use a dipole instead of a monopole , we should use a balun, which

Is a device that converts a balanced system to an unbalanced system , the

Word balun is the abbreviation of Balanced to Unbalanced converter .

Balanced

SystemBalun


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2.5 . Disadvantages

An Electric field antenna , this means that the magnetic field H is

Zero at near field , this makes dipoles incompatible with portable

Combination .

Dipoles are balanced systems , this makes it difficult to mount themOn any device without the use of baluns .


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Practical Example

Try connecting a terminal of a cable like the one shown in the figure toa port in your TV , the other terminal acts as a monopole ( but with a bad

Performance ) , and you can enjoy watching your TV !!


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Practical Example

When designing your dipole or monopole , you can reduce the length of your

Design by covering it with a dielectric material with permittivity e, the length

Is reduced then by 1 / e

Dielectric coverMaterial

Antenna with

Reduced length .


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Chapter 3

C.A.Balanis is one of

The most important

antenna scientists , and

Contributed with afamous book

Antenna theory.

Loop antenna


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Loop Antennas

Outline

1. Introduction

2. Design Parameters

3. Equivalent Circuits4. Loop antenna

Configurations

5. Applications in mobile

Communication system

Practical Example


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3.1. Introduction

As the dipole is the reference ( conventional ) electric field antenna , loops

Are the reference magnetic field antenna . Loop antennas can take different shapes

Like square , circle , triangle , ellipse or any other closed shape.

In dipoles current

Moves till

discontinuityoccurs

And then radiates

( Electric field ).

When current

Circulates in theLoop it is obvious

That a magnetic

Field is produced.

i

i


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Introduction : Geometry


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Introduction : Radiation Pattern

A small loop is equivalent to an infinitesimal magnetic dipole , whose axis

Perpendicular to the plane of the loop.

The elevation and azimuth

Plane radiation pattern of a

Loop antenna .


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The 3D radiation

Pattern of loop

Antenna , showing

The geometry of

The loop in blue.


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The radiation pattern

Of a loop for magneticField , the dominant

Radiation is magnetic

And this is why

Loops are magnetic

Field antennas .


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Introduction

Types of loops are :

Electrically Small Electrically large

C < l/ 10C : circumference C ~ l


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3.2. Design Parameters

The radiation resistance of loop antennas is very small and sometimes

Less than the loss resistance , this makes them receivers rather than

Transmitters where signal to noise ratio is more important than efficiency .

Methods of increasing radiation resistance :

1Increasing its perimeter (electrically)

2Increasing number of turns

3Inserting a ferrite core with high

Permeability ( ferrite loops ).


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Design Parameters

Design parameters :

1Perimeter of the loop ( circumference).

3 Spacing between turns .

2Increasing number of turns.

4Thickness .

5Presence of a ferrite core .


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Design Parameters

The effect of design parameters on added resistance:

Ron: Normalized

Added resistance.

N : Number of turns

N = 8

N = 7

N = 6

1.0 1.5 2.0 2.5 3.0

Ron

Spacing

We seek a design with the

Minimum spacing and

Maximum turns to satisfy

Maximum radiation resistance.


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Design Parameters

Resistance

Inductance

Capacitance

Reactance

Resonance occurs

When the capacitance

And inductance

Vanishes and

resistance is maximumThis is the

Area we select the

Design within

Impedance

Thickness to circumference ratio


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3.5. Loops in mobile communication

1Loops are alternative to monopoles , the most widely

Used element for hand held portable mobile

Communication.

2Loops are used in portable pagers , but very few in

Transceivers due to high resistance and inductance.

3Loops are very immune to noise , having low noise

To signal ratio makes them suitable for interfering

And fading environment.


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Chapter 4

The Yagi Antenna is a

directional

antenna invented by

Dr. Hidetsugu

Yagi of Tohoku

Imperial

University and his

assistant, Dr. Shintaro Uta.

Yagi antenna


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Yagi Uda

Antennas

Outline

1Introduction

2Components

3Design procedure

4Advantages

5The folded dipole


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4.1 . Introduction

One of the most popular antennas used in home TV is the yagi uda array , it is

A very practical radiator in the HF ( 330 MHz ) , VHF ( 30300 MHz) andUHF ( 3003000 MHz ) ranges .

The Yagiuda antenna is primarily an array of linear dipoles with one elementServing as the feed while the others act as parasitic elements .


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Introduction

This arrangement extends for arrays of loops , an antenna that is very popular

Among ham radio operators is the quad antenna .

~

Driven

Reflectors


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4.2 . Components

The yagi uda antenna consists of a number of linear dipole elements :

-One of which is energized directly by a feed transmission line while the others act

as parasitic radiators whose currents are induced by mutual coupling .

-Parasitic radiators are divided into reflectors and directors.

-The feed element is usually a type of dipoles called a folded dipole used

For operation in the end fire mode .

~Driven

Reflector Directors


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Components : geometry


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Components : 3D display


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4.3 . Design procedure

To achieve the end fire mode the design is characterized by :

Parasitic elements in the direction of the beam are smaller than feed element

( directors )

The driven element is slightly less than l / 2 ( ~ 0.45 l0.49 l )

The directors should be about ( ~ 0.4 l0.45 l ) ; less than the feed element


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Design procedure

The separation between the directors is between 0.3 to 0.4 lambda .

A yagi uda array of 6 lambda total length was found to have an overall gain

Independent on the directors separation

The length of the reflector is somewhat greater than the feed element

The directors are not necessarily of the same length or diameter !


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Design procedure

Most antennas has from 6 to 12 directors .

The separation between the feed element and the reflector is less than that of

The feed and the nearest director ( nearly 0.25 lambda )


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Design procedure

The 3D radiation pattern


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Design procedure

The 2D radiation pattern


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Design procedure

The SWR plot of the yagi uda


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4.4 . Advantages

Light weighted

Simple to build

Low cost .


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4.5 . The folded dipole

~The folded dipole is frequently used as the feeding element

As it has good directional characteristics , it is

Recommended that the width


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Chapter 5

The first cassegrain

Reflector was designed

By Laurent cassegrainIn 1672 .

Reflector antenna


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Reflector

Antennas

Outline

1- Why Reflectors ?

2Types of reflectorsAccording to geometry

3Types of Parabolic

Surfaces

4Methods of feeding

Parabolic reflectors

5Using Image theoryTo calculate field

6Using GTD to calculate

The field


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5.1. Why Reflectors ?

While using aperture antennas we always need to increase the aperture

Area to increase its directivity ,but as this is not practical , instead of using

Large apertures we place a reflecting surface face to face with the aperture( or any other antenna ) , the reflecting surface collimates radiation to

The small aperture and thus we satisfied high directivity with a small

Aperture , and overcame space limitations.

A side view of

An aperture of

A large area

A side view of

An aperture of

A small area

And a reflecting

Surface used.


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5.2. Types according to geometry

Plane reflectors Corner reflectors

Curved reflectors


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Types according to geometry : 90 degree corner

To better collimate the energy in the forward direction , the geometrical shapeOf the plane reflector must be changed to prohibit radiation in the back and

Side directions .

The 90 degreecorner reflector has a unique property , the ray incident on

It reflects exactly in the same direction , so it is not used in military applications

To prevent radars from detecting airplanes positions.


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Types according to geometry

The most important software used for simulating reflector antennas is Grasp.

An example for an

openGL plot for allobjects of a reflector

Antenna using Grasp 9 .


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5.3.Types of parabolic surfaces

Parabolic Cylinder ParabolaHyperbola

Focus is a line Focus is a point


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5.4. Methods of feeding parabolic reflectors

Dual offsetFrontfed reflectors Offset reflectors Cassegrain fed


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Methods of feeding parabolic reflectors

Why we use Offset reflectors ( single and dual ) ?

To avoid blockagecaused by struts , we use half a dish and adjust the

Feeding element in a way that makes the antenna equivalent to a singleReflector .

Why we use cassegrain fed reflectors ?

This increases the focal length and thus increases the directivity .


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5.5.Using Image theory in calculating fields

We use the image theory to find a system of fields but

The GTD is more accurate because here we assume

Virtual sources .

2n : number of images , c = 180 / n .

C = 180 C = 90 C = 60


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Using Image theory in calculating fields

E1

E2

E3

E4

En

Total field : E = E1 + E2 + E3 + E4 + .. En


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5.6 . Using GTD in calculating fields

Using GTD instead of the image theory results in more accuracy

As we dont assume virtual sources . The GTD (geometrical

Theory of diffraction) accounts for reflection and diffraction of

Rays after calculating the reflection and diffraction coefficients .

A satellite dish is a parabolic


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A satellite dish is a parabolic

reflector antenna

Ch t 6


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Chapter 6

Microstrip antennas

Are considered themost practical antennas

For mobile communication !

Microstrip antenna


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Microstrip

Antennas Outline

1- Components

2- Types of microstrip

Antennas

3- Feeding techniques4- Advantages

5- Disadvantages

6- Techniques to overcome

Disadvantages

7- Microstrip arrays

8- Feeding of arrays

9- Microstrip vs. reflectors.

6 1 C


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6.1. Components

A microstrip antenna consists of :

Patch ( radiating

Element )Feed

Dielectric

Ground planecopper

The patch ( radiating element ) may be circular , rectangular or any other shape .


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6 3 F di h i


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6.3.Feeding techniques

Direct feeding by coaxial

Feed line ( probe )

Microstrip line

Feed

Feeding by coupling

Aperture

coupled

feed

Proximity

coupled

feed

Feeding techniques : Direct feed by


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Feeding techniques : Direct feed bycoaxial fees line

The inner ( central ) of the coax is attached to the patch while

The outer ground is welded to the ground of the microstrip

( like the monopole ) .

Patch

Coaxial

Equivalent circuit

F di h i f d l


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Feeding techniques : Microstrip feed line

It is a conducting strip of much smaller width compared to the

Patch , it is easy to fabricate and simple to match ..

F di t h i f di b li


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Feeding techniques : feeding by coupling

Aperture

coupled

feed

Proximity

coupled

feed

The most difficult to fabricate

And has a narrow band ,

Depends on two substrates and

A ground with a slot .

Has a band width of 13% ,

however it is difficult to fabricate.

6 4 Ad t


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6.4 . Advantages

1High accuracy in manufacturing , the design is executed by

Photo etching

2Easy to integrate with other devices

3An array of microstrip antennas can be used to form a

Pattern that is difficult to synthesize using a single element.

4 We can obtain high directivity using microstrip arrays

Ad t


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Advantages

5 Have a main radiating edge , this makes it useful for mobile

Phones to avoid radiation inside the device .

6Small sized applicable for handheld portable communication

7 Smart antennas when combined with phase shifters .

6 5 Di d nt


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6.5 . Disadvantages

4An array suffers presence of feed network decreasing

Efficiency , also microstrip antennas are relatively expensive .

1 Narrow band width ( 1% ) , while mobiles need ( 8% )

2 Low efficiency , especially for short circuited microstripantenna

3 Some feeding techniques like aperture and proximity

Coupling are difficult to fabricate

6.6 . Techniques for overcoming


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q g

disadvantages

Conventional techniques Non conventional techniques

1- Decreasing dielectric

Constant

2- Increasing thickness

3- Increasing width .

1- Aligned parasitic elements

2- Using stacked parasitic

Elements.


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Techniques for overcoming


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q g

disadvantages : Stacked parasitic elements

Rather than aligning them ,We can even combine the two

Methods and modulate the

Patchs shape to yield widest

Band width .

6 7 Microstrip Arrays


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6.7 . Microstrip Arrays

2 ^ n

2 ^ n

Feed

Network

Microstrip Arrays


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Microstrip Arrays

The optimum spacing is 0.8lo , length must be = lambda / 2 .

Advantages of microstrip arrays

1Used to synthesize a required pattern difficult to achieve with

A single element.

3 Increases directivity .

2 Used to scan the beam of an antenna system

Microstrip Arrays


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Microstrip Arrays

Disadvantages of microstrip arrays

1 Narrow bandwidth ( 1 % ) .

2 Low efficiency

3 If the separation is more than lambda , grating lobes appear

4 Feed network decreases efficiency .

6 8 Feeding of arrays


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6.8 . Feeding of arrays

A microstrip antenna uses feed network which may be either :

2Corporate feed .

1Series feed

Sometimes feed networks are synthesized with the antenna !

Feeding of arrays : Series feed


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Feeding of arrays : Series feed

Series feed

Feeding of arrays : Corporate feed


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Feeding of arrays : Corporate feed

Corporate feed

6 9 Microstrip vs Reflectors


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6.9 . Microstrip vs. Reflectors

Preferred for low directivityapplications

Performed for high directivity

applications as the effect of blockage

Is less

Lower efficiency Higher efficiency

Suffers low efficiency caused by

Feed network for arraysSuffers blockage caused by fixation

Struts

Microstrip Antennas Reflector Antennas

Microstrip vs Reflectors


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Microstrip vs. Reflectors

Smart antennas , uses electronicscanning when combined with phase

Shifters

Uses mechanical scanning .

More accurate manufacturing by

photo etching

Less accuracy , sometimes parabolic

Surfaces are rough

Feeding is by coupling or coax feed

LinesUses other antenna ( dipole ,

monopole , apertures , ..etc) as

A feed

Microstrip Antennas Reflector Antennas

Flat plane Microstrip Antenna


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Flat plane Microstrip Antenna

Chapter 7


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Chapter 7

Fractal antennas are

Very compact as they

Utilize the same

Physical area of classicAntennas but with an

Electrically large length !

Fractal antenna

Fractal


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Fractal

Antennas

Outline

1Definition

2Characteristics

3Types of fractals

4Advantages

7 1 - Definition


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7.1 Definition

A fractal antenna is an antenna that uses a fractal, self-similar design

to maximize the length, or increase the perimeter

(on inside sections or the outer structure), of material that

can receive or transmit electromagnetic signals within a giventotal surface area or volume. [ source : wikipedia ]

A fractal is : a recursively generated geometry that has fractionalDimensions.

Definition : fractal generation


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Definition : fractal generation

Some software productscan generate fractals

And fractal maps , the

Opposite figure shows

A koch loop after several

Iterations .

7.2 Characteristics


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7.2 Characteristics

A fractal antenna's response differs markely from traditional antenna designs, in

that it is capable of operating with good-to-excellent performance at many

different frequencies simultaneously. Normally standard antennas have to be "cut" for

the frequency for which they are to be usedand thus the standard

antennas only work well at that frequency. This makes the fractal antenna an

excellent design for wideband and multiband applications.

Characteristics


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Characteristics

Fractal antennas satisfies the requirements of wireless communication

Systems :

1Wideband

2Multiband

3Low profile

4Small antenna

Characteristics


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Characteristics

The band width of an antenna can be improved as the geometry of the

The antenna best utilizes the available planar area of a circle of radius r

That encloses the antenna .

Fractal antennas utilizes the available space in a sphere of radius r in an

Efficient way

The quality factor Q is inversely proportional with the band width.

Characteristics


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Characteristics

The concept of fractals is frequently used in electromagnetism , and also used

To represent nature .

A Fern fractal

Represents a plant



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Fractals may be:

Deterministic Random

-Von Koch snowflake- Sierpinski gaskets

- Minkowski island

Types of fractals : Koch loop


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Fractals that begin with a basic geometry (initiator) and uses a recursive

Algorithm t produce copies of themselves .

Initiator Generator



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yp p

Iterations 2 31

Types of fractals : Minkowski island


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yp

A Minkowski islandA Minkowski island after more iterations

As plotted by the directx display of 4nec2

Software ( by Arie voor )

Types of fractals : Sierpinski gaskets


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yp p g

Determined by the nodes of a Pascal triangle which are numbered by

the excitation coefficients of the binomial array decided by J.S.stone

( 1 + x ) ^ ( m1 ) = 1 + ( m -1 ) * x + ( ( m1 ) ( m2 ) ( x ^ 2 ) ) / 2!+ ( ( m1 ) ( m2 ) ( m3 ) ( x ^ 3 ) ) / 3! +.

1 element

2M + 1 = 1

M = 0

A1 = 1

2 elements

2M = 2

M = 1

A1 = 1 , A2 = 1

3 elements

2M +1 = 3



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yp p g

The Pascal triangle



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yp p g

If the nodes with numbers divisible by a prime number p ( p = 2 , 3 , 5 , )

is deleted the result is a sierpinski gasket of mod-p

Types of fractals : Random fractals


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yp

7.4Advantages


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g

Fractal antennas results in more compact antennas , but can resonate

And has input resistance that are much greater than classic geometriesOf loops and dipoles

The first resonance for a linear dipole occurs at lambda / 2 overall length

Which can be physically large for some frequencies

Advantages


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The higher iterative geometries , the lower resonant frequencies because

Its overall length becomes electrically large .

Documents

fundamental antennas