38

CHAPTER 3

MINKOWSKI FRACTAL ANTENNA FOR

DUAL BAND WIRELESS APPLICATIONS

3.1 INTRODUCTION

The explosive growth in wireless broadband services demands new

standards to provide high degree of mobility and enhanced data transmission

(Yin and Alamouti 2006). Among the emerging standards, WiMAX or the

IEEE 802.16 standard is one of the most promising broadband wireless

technologies nowadays. It is characterized by its high data rates, large

coverage area, and flexible design.

An important amendment to IEEE 802.16 is the IEEE 802.16e

standard which adds a capability for full mobility support to WiMAX

(Hoadley and Javed 2005). An important requirement of WiMAX wireless

systems is the significant increase in data throughput and link range without

additional bandwidth or transmit power, with a combination of higher spectral

efficiency and link reliability or diversity (Jenkins 2008). The system

throughput is significantly improved with the support of multiple-input-

multiple-output (MIMO) antenna technology as well as Beamforming and

Advanced Antenna Systems (AAS), which are referred to as "smart" antenna

technologies (Pedersen et al 2003).

Multiple-input-multiple-output (MIMO) antenna systems have been

reported for wireless applications to improve the capacity of the radio

communication (Naguib et al 1994, Konanur et al 2005, Sayeed and

39

Raghavan 2007, Zhang and Chen 2008, Chen et al 2008). This technology

brings potential benefits in terms of coverage, self installation, low power

consumption, frequency re-use and bandwidth efficiency. The performance

aspects of MIMO system shows a significant increase in throughput using an

antenna array compared to single antenna system (Guterman et al 2004).

However, MIMO techniques create an additional difficulty of housing multiband antennas with low level of mutual coupling between the

antenna elements (Salvekar et al 2004). Multiple antennas in a portable device

require that the spacing between antennas be small and compact. With small

antenna element spacing, the mutual coupling can be significant, and this is

accounted for the design of MIMO antenna systems (Wennstrom 2002).

Many attempts to design small and compact antennas in the past

have been endeavoured through slots, shorting and folded geometries, but it

was not until the introduction of fractals in antenna engineering that this could be done in a most efficient and sophisticated way (Wang and Lee 2004,

Khodaei et al 2008, Gianvittorio and Rahmat Samii 2002, Gianvittorio 2003).

Fractal antennas have facilitated miniaturization as they are electrically small

and at the same time self-resonant radiators. Minkowski fractal geometry has

received lot of attention in respect of reduction in the size of the conventional loop antenna leading to compactness and miniaturization (Cohen 1995). Tight

packing of Minkowski Island loop antenna elements in the array and reduced

mutual coupling was achieved without affecting the bandwidth (Gianvittorio

and Rahmat Samii 2002).

Recent studies have shown that the application of printed Koch

curve fractal shapes in MIMO antenna design allows reduced mutual coupling

between antenna elements in (MIMO) system for multiband applications

besides providing miniaturization (Guterman et al 2004). However this requires PIFA configuration and the use of slot to achieve multiband

behaviour.

40

This chapter evaluates the performance of fractal array based on

Minkowski geometry for WiMAX dual band operation with MIMO

application. The generation of Minkowski fractal geometry is explained in

section 3.1. The structure has been used in microstrip configuration. The

numerical analysis of Minkowski patch antenna is presented in section 3.3.

The performance evaluation of Minkowski fractal antenna and the results are

discussed in section 3.4. The Minkowski fractal based two element array for

WiMAX application is presented in section 3.5. The simulated results are

experimentally validated in section 3.6. A brief summary of the work

proposed is presented in section 3.7.

3.2 MINKOWSKI FRACTAL GEOMETRY

Minkowski fractals were first introduced by Hermann Minkowski

in the form of representation and definition of geometries in the year 1885

(Strobl 1985, Schwermer 1991).

The basic square patch geometry is compressed five smaller

squares to form the Minkowski fractal geometry as shown in Figure 3.1 This

is explained in terms of affine transformation here.

W

w1(x,y)=(a1x+b1y+e1, c1x+d1y+f1)

w2(x,y)= (a2x+b2y+e2, c2x+d2y+f2)

w3(x,y)= (a3x+b3y+e3, c3x+d3y+f3)

w4(x,y)= (a4x+b4y+e4, c4x+d4y+f4)

w5(x,y)= (a5x+b5y+e5, c5x+d5y+f5)

w1

w2

w5

Figure 3.1 Generation of Minkowski fractal geometry

41

Iterated function systems (IFS) represent an extremely versatile

method for conveniently generating a wide variety of useful fractal structures

(Michael Barnsley 1993). These iterated function systems are based on the

application of a series of affine transformations, w, defined by

x a b x ew y c d y f

(3.1)

or, equivalently, by

( , ) ( , )w x y ax by e cx dy f (3.2)

where a, b, c, d, e, and f are real numbers. In affine transformation, w, is

represented by six parameters

a b ec d f

(3.3)

such that a, b, c, and d control rotation and scaling, while e and f control

linear translation.

Let w1, w2, ..., wN be a set of affine linear transformations, and A be

the initial geometry, then a new geometry is produced by applying the set of

transformations to the original geometry A, and is represented as

1

( ) ( )N

nn

W A w A

(3.4)

where W is known as the Hutchinson operator (Peitgen 1992).

The IFS coefficient values to generate Minkowski fractal geometry

as shown in Figure 3.7 are tabulated in Table 3.1. The successive fractal

42

geometry can be obtained by repeatedly applying the previous geometry. The

fractal similarity dimension of this geometry is given by

log5 1.465log3

D (3.5)

This implies that there are five copies of squares are generated in

successive iteration having dimension scaled down to one third.

Table 3.1 IFS Transformation coefficients for the Minkowski fractal

geometry

wi ai bi ci di ei fi

w1 1/3 0 1/3 0 0 0

w2 1/3 0 1/3 0 1/6 0

w3 1/3 0 1/3 0 0 1/6

w4 1/3 0 1/3 0 1/6 1/6

w5 1/3 0 1/3 0 1/3 1/3

The iterative procedure is continued to get the successive stages of

Minkowski fractal patch antenna is shown in Figure 3.2. The starting

geometry of the Minkowski fractal antenna is the initiator square patch, and

the successive iterations of fractal antenna is obtained by replacing each of the

four straight sides of the starting structure with the generator with indentation

as shown in the Figure 3.2. The indentation width S, can vary from 0 to 1.

43

Figure 3.2 Generation of Minkowski fractal patch antenna (a) Initiator patch (b) First iterated Minkowski fractal patch (c) Second iterated Minkowski fractal patch (d) Third iterated Minkowski fractal patch

3.3 SBTD ANALYSIS OF MINKOWSKI FRACTAL ANTENNA

The governing equations for a coaxial fed Minkowski fractal antenna shown in Figure 3.2 are the following updated equations as explained

in Chapter 2.

2

1/2 , 1/2, 1/2 1/2 , 1/2, 1/2 , 1 2, 13, 1 2, 1 2

1 .X X yK m n k m n i k m n i

im n

tH H a Ez

2 2

, 1, , 1,3 3

1 1. .z zi k m i n i k m i n

i i

a E a Ey y

(3.6)

2

1/2 1 2, , 1/2 1/2 1 2, , 1/2 1, 1 2,31 2, , 1 2

1 .y y zK m n k m n i k i m n

im n

tH H a Ex

2 2

1 2, , 1 1 2, , 13 3

1 1. .x xi k m n i i k m n i

i i

a E a Ez z

(3.7)

44

2

1 , , 1 2 , , 1 2 1 2 1/2 , , 1/23, , 1/2

1 .z z yK m n k m n i k i m n

im n

tE E a Hx

2 2

1 2 , 1/2 , 1/2 1 2 , 1/2 , 1/23 3

1 1. .x xi k m i n i k m i n

i ia H a H

y y

(3.8)

where the coefficient {ai}, dxx

xSxQa ii

2/1 (3.9)

A Gaussian pulse of 2

0

.exp.)(

TtAtf is used as excitation field.

3.3.1 Antenna Structure Discretization

Figure 3.3 shows the top view and cross section of the second

iterated Minkowski fractal antenna.

Figure 3.3 Top view and the cross section of the second iterated

Minkowski microstrip fractal patch antenna

45

3.3.2 Boundary Conditions

(i) on the ground plane

Since the entire antenna lay on the ground plane, a perfect

electric conductor boundary condition is chosen on the ground

plane of the antenna, as the entire antenna lies on the ground

plane. That is Ex=Ey=0, and Ez exists in the substrate

(ii) on the metallized antenna, it is assumed that Ex=Ey=0 for k=3

where k is the representation of the cell for the metal antenna

and Ez exists in the free space above the antenna

(iii) on the feed line conductor, the respective cells have Ex=Ey=0

and Ez exists in the substrate around the conductor

3.3.3 Results and Discussion

A Minkowski fractal antenna having a dimension of 28.6mm

28.6mm and the dielectric substrate having εr =4.4 and thickness 1.6mm is

chosen for analysis. This antenna is analysed using SBTD technique with

mesh specification of 100 100 8 Yee cells. The cell size

x = y =0.286mm, z =0.2mm is chosen to accommodate the single element

antenna structure conveniently. The time step selected is 0.88 ps to ensure

courant stability condition. The coding for SBTD is developed in MATLAB.

The reflected voltage of the Minkowski fractal antenna at the input port is

presented in Figure 3.4.

46

Figure 3.4 Reflected Voltage at the input port of the antenna

The simulated input characteristic of the antenna are obtained using

Empire 3D EM simulator. To validate the analysis of the Minkowski fractal

antenna, the antenna is fabricated as shown in Figure 3.5 and the

measurements are made using vector network analyzer.

Figure 3.5 Prototype of the second iteration Minkowski fractal antenna

47

Figure 3.6 shows the comparison of SBTD with the FDTD,

simulated and measured return loss. The results shows the resonance

behaviour of the second iterated Minkowski fractal antenna operating at

1.8GHz and it shows good agreement with the measured results.

Figure 3.6 Return loss (S11) of the Second iteration Minkowski antenna

The SBTD technique used smaller number of meshes for the

computation to get the same accuracy. The measured result shows a small

degree of variation from the simulated and the numerical results. This is due

to the mesh truncation at the edges of the fractal antenna during the EM

simulation. The performance of SBTD technique is also compared with

FDTD method in respect of CPU time and memory requirement. SBTD

technique still gives good performance with coarse mesh having accurate

result. Compared to conventional FDTD method, the SBTD technique

reduces by a factor of 3.5 and 1.3 of the computer memory and computation

speed respectively.

48

3.4 PERFORMANCE EVALUATION OF MINKOWSKI

FRACTAL ANTENNA

3.4.1 Preamble

In order to study the effectiveness of miniaturization in Minkowski

fractal antenna, the antenna at different stages of fractal iterations are studied

through simulation software ADS 2002C. Then the influence of parameter

such as indentation of the Minkowski fractal antenna on the frequency

response is studied in order to make it suitable for dual band operation.

3.4.2 Influence of Minkowski Fractal Iteration

Figure 3.7 shows the first three stages of Minkowski fractal antenna.

These structures are simulated, fabricated and tested. The antenna has been

fabricated at different stages of fractal iterations and the measured results are

compared with that of simulated values. Figure 3.8 shows the prototype of

Minkowski fractal patch antenna at different stages of fractal iteration.

Figure 3.7 First three stages of Minkowski fractal antenna

49

Figure 3.8 Prototype of Minkowski microstrip fractal antenna in

different iteration

Figure 3.9 presents the return loss of simple microstrip antenna (K0)

resonating at 2.41GHz

2.25 2.3 2.35 2.4 2.45 2.5-30

-25

-20

-15

-10

-5

0

Frequency (GHz)

Retu

rn lo

ss (d

B)

Simulated

Measured

Figure 3.9 Comparison of simulated and measured return loss of

simple microstrip patch (K0) antenna

50

The first iteration of the fractal patch resonates at 2.2GHz and

hence it provides 12 % miniaturization as shown in Figure 3.10. The second

iteration of the fractal patch further lowers the resonance to 16 % as shown in

Figure 3.11. For this fractal, it is seen that the second iteration of the fractal

actually has a lower resonant frequency than the first iteration. This is because

the current does not follow the straight path, but rather ease around the edges

of the patch increased electrical length and hence the miniaturization of

antenna.

Figure 3.12 shows the return loss of third iterated Minkowski

fractal antenna. Table 3.3 shows the resonant frequencies at first and second

iteration and the frequency reduction factor. As iteration increases, there

exists a shift in resonance frequency towards lower value. This is due to the

increase in the electrical dimension of the Minkowski fractal antenna.

Figure 3.10 Comparison of simulated and measured return loss of first

iteration Minkowski fractal (K1) antenna

51

Figure 3.11 Comparison of simulated and measured return loss of

second iteration Minkowski fractal (K2) antenna

Figure 3.12 Comparison of return loss of Minkowski microstrip fractal

antenna at different stages of iteration

52

Table 3.2 Resonant frequencies and scale factor between adjacent

bands in Minkowski fractal antenna

n fn(GHz) S11 (dB) fn/fn+1

0 2.4 -25 -

1 2.1 -12 1.14

2 1.7 -34 1.24

Figure 3.13 shows the radiation patterns of the Minkowski fractal

patch antenna at different stages and first iterated Minkowski antenna in the

parallel and perpendicular plane respectively. The antenna shows bidirectional

radiation in horizontal plane and uniform radiation in the direction

perpendicular to the plane of antenna. There is a slight reduction in the

measured value at null direction which may be due to the cable loss during the

measurement.

Figure 3.13 Radiation patterns of Square patch antenna (KO), First

iterated Minkowski fractal patch (K1) and Second iterated

Minkowski fractal patch (K2)

53

3.4.3 Significance of Indentation Width

The slots between fractal segments have indentation having a width

SV and SH along x and y directions as shown in Figure 3.14. The influence of

indentation width is studied so that the Minkowski fractal antenna can be

customized for the required specifications.

Figure 3.14 Minkowski fractal antenna showing Indentation widths

The effect of varying the input indentation is shown in the plot of

the input characteristics of the antenna shown in Figures 3.15 and 3.16. Three

indentation sizes for the first iteration of the antenna have been simulated. It

can be seen that the indentations miniaturize the patch and that an increase in

the indentation, increases the amount of miniaturization. The patches have a

dimension of 28.6 mm by 28.6 mm and are printed on a 1.6mm-thick FR4

substrate which has a dielectric constant of 4.4. The square patch is resonant

at 2.41 GHz and the perimeter is 114.4 mm, which is 2λ at the resonant

frequency. The patch that uses the first iteration of Minkowski fractal antenna

indented by 5.4 mm is resonant at 2.1 GHz. The perimeter of this patch has

increased to 158.4 mm, which is 3.385λ at this frequency. The patch that uses

6.2 mm for the indentation width is resonant at 1.95 GHz and the perimeter is

161.3 mm which is 4.25λ. It can be seen that the effect of increasing the

54

indentation has diminishing returns, in the sense that the miniaturization

tapers off as the indentation is increased.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Iteration n

Reso

nant

freq

uenc

y (G

Hz)

sv=5.4mmsv=4.8mmsv=3.7mm

Figure 3.15 Resonant frequency of a first iterated Minkowski fractal

patch for varying indentation widths, SV and SH.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Iteration n

Res

onan

t fre

quen

cy (G

Hz)

Sh=9.7Sh=8.7Sh=7.8Sh=6.5

Figure 3.16 Resonant frequency of a first iterated Minkowski fractal

patch for varying indentation widths, SV and SH

55

Table 3.3 shows the fractal dimension and the perimeter at

resonance for various indentation width and the generating iterations.

Table 3.3 Minkowski fractal antenna fractal dimension and respective

indentation width scaling factor

Indentation width scaling

factor

Fractal iterations

n

Fractal dimension

(λ)

Perimeter at resonance

(λ) 0 0 1 2

0.53 1 1.08 2.14 0.53 2 1.08 2.84 0.64 1 1.14 2.36 0.64 2 1.14 2.53 0.72 1 1.29 2.86 0.72 2 1.29 3.03 0.85 1 1.36 3.133 0.85 2 1.36 3.324 0.98 1 1.42 3.385 0.98 2 1.42 4.251

3.5 CUSTOMIZATION OF MINKOWSKI FRACTAL ANTENNA

FOR DUAL BAND WIRELESS APPLICATION

The demand for miniaturized dual band antenna and compact array

for WiMAX MIMO antenna system have necessitated the design of

Minkowski fractal antenna array. Single Input Single Output (SISO) and

multiple input multiple output (MIMO) antennas using Minkowski fractal

patch is designed by customizing the values of the Iterative Function

Coefficients (IFS) coefficients. The optimized IFS coefficients of the first

56

iteration Minkowski fractal antenna to meet the frequency of resonance for

dual band applications are shown in Table 3.4.

Table 3.4 IFS transformation coefficients for the optimized

Minkowski fractal geometry

wi ai bi ci di ei fi

w1 0.287 0 0.239 0 0 0

w2 0.259 0 0.239 0 0.605 0

w3 0.279 0 0.256 0 0 0.609

w4 0.253 0 0.253 0 0.609 0.609

w5 0.253 0 0.209 0 0.326 0.275

Figures 3.17 and 3.18 show the layout and the prototype of

proposed optimized Minkowski fractal antenna and its suitability for dual

band operation. Microstrip line is used to feed the antenna in order to have

easier integration with the subsequent RF systems and circuitry. The input

characteristic is measured and the results are compared with that of simulated

value as shown in Figure 3.19.

Figure 3.17 Layout of first iterated Minkowski microstrip fractal patch

antenna

57

Figure 3.18 Prototype of the Minkowski antenna

2 3 4 5 6 7-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (GHz)

Ret

urn

loss

(dB

)

SimulatedSBTDMeasured

Figure 3.19 Return loss of optimized Minkowski fractal antenna

It is seen from the investigation on Minkowski fractal antenna, the

non uniform change in IFS coefficients has significant change in the resonant

frequency has resulted in dual band operation. Larger the indentation widths

SH and SV, lower the resonant frequency and hence further miniaturization of

square patch antenna. Proper selection of these values has resulted required

resonance.

58

Figure 3.20 shows the two element Minkowski fractal antenna

configuration.

Figure 3.20 Three dimensional view of 2 element Minkowski microstrip

fractal array

The array antenna is analysed using SBTD method and its

resonance behaviour is obtained. The array antenna is fabricated and tested in

an anechoic chamber. The photograph of two elements Minkowski array is

presented in Figure 3.21.

Figure 3.21 Prototype of 2 element Minkowski microstrip fractal array

59

The return loss obtained using SBTD method, simulated results

obtained using Empire software and the measured results obtained in an

anechoic chamber are shown in Figure 3.22, and they are all in agreement

with each other. The antenna is matched (S11< -10dB) at the WiMAX

frequencies 2.5 GHz and 5.8 GHz respectively. The coupling coefficient S21

at 2.5GHz and 5.8GHz frequencies obtained are -20db and -24db respectively,

showing less coupling between the fractal patches. This is presented in

Figure 3.23. It is observed that the proposed Minkowski fractal array provides

dual band operation at WiMAX frequencies 2.5 GHz and 5.8 GHz due to the

resonant behaviour of the fractal elements. This in turn has resulted in

reduced mutual coupling.

1 2 3 4 5 6 7-25

-20

-15

-10

-5

0

Frequency (GHz)

Ret

urn

loss

(dB

)

SimulatedSBTDMeasured

Figure 3.22 Return loss of the Minkowski microstrip fractal array at

both the input ports 1 and 2

60

Figure 3.23 Coupling coefficients between the elements of Minkowski

microstrip fractal array

The radiation pattern at 2.5GHz and 5.8GHz frequency are plotted in Figure 3.24.

2.5GHz

5.8GHz

Figure 3.24 Radiation pattern of Minkowski microstrip fractal antenna

array in H and E plane for 2.5GHz and 5.8GHz

61

In the lower frequency band, the antenna exhibits max gain of

8.84db and in the high frequency band, the antenna demonstrates max gain of

7.39db which is acceptable for WiMAX application.

3.6 SUMMARY

A Minkowski fractal geometry based microstrip patch antenna is

investigated for miniaturization. The influence of fractal indentation width is

also studied. A compact two element Minkowski fractal antenna array is

customized for Multiple-Input Multiple-Output (MIMO) supported WiMAX

dual band application which shows reduced mutual coupling. The

experimental results are validated using SBTD based numerical results and

the simulated results obtained using ADS and Empire XCcel simulator.