SIMILARITY DIMENSION

AND ITS APPLICATION ON FRACTAL ANTENNA

ByAsst. prof. Nehal Maniar

MKS college

ABSTRACT It is very often impossible to describe nature using only Euclidean geometry, that is in terms of straight lines, circles, cubes and such like. Benoit Mandelbrot realized that it is impossible to describe nature using only Euclidean

geometry , he proposed fractal geometry which uses similarity dimension to describe real object like tree, lightening , coastline ect. In this paper we have tried to explain how similarity dimension(fractal dimension) characterizes the construction of regular fractal by relating similarity dimension and directivity of antenna.

WHAT ARE FRACTALS?A fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is a reduced size of copy of whole. The word fractal was first coined by Benoît Mandelbrot. He suggested that every thing in nature can not be measured with Euclidean geometry that is with integer dimension. He said

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line”.

CLASSIFICATION

Regular fractal

•Mathematically made fractal .•Like Koch curve,serpinski gasket, cantor set ect.

Irregular

fractal

• Statistically similar to mathematical fractals

• Fractal appear in nature like coastline ,lightening, Structure of arteries ect.

FRACTAL GEOMETRYSCALING FACTOR

The concept of dimension is related with scaling.

Consider the line which is divided into N small segment of length ε .Let length of entire line is L and line is of unit length.

The scaling factor is ε/L = ε as L=1.Thus L = N ε = 1

If we divide unit area A into N similar area and unit solid into N similar cube with side , then….

SIMILARITY DIMENSION A = N = 1 and V = N = 1

21 /N 3 31/N We can observe that

exponent of , in each case is a measure of the similarity dimension of an object. in general if is similarity dimension then,

using logarithm we get

sD

1sDN

TRIADIC CANTOR SET Cantor set can be considered simplest fractal object as it contains infinite number of copy of itself or can say cantor set is made up of cantor sets. Let at step k=1 ……

[ ………………….] 0 1/3 2/3 1 So now we have N=2 and ε=1/3.For step k=2…..

[ …… ……………….. ……. ]0 1/9 2/9 1/3 2/3 7/9 8/9 1

Now N=4 = and

In general

22 21/3log log4 2log2 0.6309log1/ log1/(1/9) 2log3s

ND

)log2 log2 0.6309log1/(1/3 log3

c

s ccDc

FRACTAL DIMENSION –KOCH CURVE

1/3

here N =4 and

= 1.261

FRACTAL ANTENNA In modern society people want to be connected via web, phone call or video conference anywhere anytime.

All these technologies work on different frequencies and demands highly efficient multiband antenna which is very compact in size to be fit into small hand set.

Fractal antenna uses fractal geometry for self similar design to maximize the length.

With this theory we can achieve multiple frequency since different part of antenna are similar at different scale.

DIFFERENT FRACTAL ANTENNA

Koch curve antenna

Serpinski gasket antenna

Cantor array antenna

ITERATION FUNCTION SYSTEM Certain fractals can be constructed using iteration ,this process is called iteration function system on elementary part.

Fractals are made of sum up of copies of itself each copy will be smaller copy from previous iteration.

Iteration function system works with series of affine transformation which deals with scaling and translating of an object.

FRACTAL ARRAY Fractal array can be formed recursively through the repetitive application of generating sub array.

A generating array ψ = is phase difference between two elements

=rotation angle of array pattern d = phase shift between two element.is a small element of scale one (say P=1) with scaling factor δ , which will construct larger antenna of higher scale (P>1).

The generating sub array has elements that can be turn off and on to generate certain pattern. An iteration function of this generating sub array can produce the fractal array which will be made of sequence of self-similar sub array.

GA( )p

02 (cos cos )d

0

ITERATION FUNCTION SYSTEM AND CANTOR LINEAR ARRAY

, ,

1 0 1P=1

P=21 0 1 0 0 0 1 0 1101 000 101 000 000 000 101 000 101

P=3

ARRAY FACTOR (SOURCE:FRACTAL ANTENNA ENGINEERING:THE THEORY AND DESIGN OF FRACTAL ARRAY BY DOUGLAS H. WERNER, RANDY L. HAUP AND PINGJUAN L. WERNER )

Array factor for uniform array can be obtained by

for 2N+1 elements (1)

for 2N elements

The array factor for fractal arraycan be

1

12 I cos( )2

2 I cos0 1( ) N

nn

n

NI nn

nA F

( 1)

1( )

PP

PP

AF GA

ARRAY FACTOR (CONT.) Array factor for three element generating sub array 101 can be(from (1), )

For isotropic element total array pattern will be (taking δ=3)

( ) 2cosGA 01, 1, 0nN I I

( 1) ( 1)

1 1( ) (3 )

P PP P

PP P

AF GA GA

DIRECTIVITY The directivity of an antenna is defined as “the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged all directions.”

Radiation intensity ,is surface area per unit solid angle around isotropic radiator. U is radiation intensity (W/ solid angle) is radiation density (W/m2)

Let total power be and be solid element of an element

Directivity will be

2radU r W

2r

radW

04radP Ud U

radP d

0

4rad

U UDU P

MAXIMUM DIRECTIVITY Maximum directivity of cantor array will be at θ = π/2, therefore cosθ = u =0.

Let maximum directivity be

The location of nulls can be found from array factor. For example for given scale P , the nulls of radiation pattern will be

Therefore radiation pattern produced by triadic cantor set has 3P-1 +1 nulls.

(3 1)() (0) 2 , 1,2,3...2P

PP PD u D P

1

1

3( cos ) cos( cos ) 02 2PP

PP

AF

1

13cos( ) 0 (2 1)(1/3)2P

Pku u k

MAXIMUM DIRECTIVITY & FRACTAL DIMENSION The generating sub array of triadic cantor array is special case of uniform cantor arrays, here in sub array we have n=1 (element)turned off or removed and scale factor ,If we take n=2 and, the pattern will be 10101 and likewise n=3 , pattern will be 1010101.The fractal dimension of uniform cantor array can be calculated as

Here we have N= (s+1)/2 and corresponding maximum directivity is

log

log 2

12

s

s

D

1(0) 2P

P PsD D

CONCLUSION Canter array is based on number of designed element and perform better than normal linear array. The fact that fractals are self similar can applied to make small size ,low cost and low weight antenna. Fractal dimension can be used to get maximum array gain and wide band performance. Other fractal geometries like Serpinski gasket, Serpinski carpet , Koch curve etc. also used to design multi band and multifunctional antenna for wireless communication .

BIBLIOGRAPHY 1. Electromagnetic waves and radiating system by E.C.Jordan &K.G.Balmain

2.Antenna Theory-Analysis and Design by C.A.Balanis

3.Chaos and Fractals by Peitgen JurgenSaupe

4. Fractals and chaos by Paul Addition 5. http://www.radio-electronics.com 6.http://www.personal.psu.edu/users/r/l/rlh45/journals

7. http://www.antenna-theory.com 8. rfdesign.com 9. http://www.personal.psu.edu