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CHAPTER 6

DEVELOPMENT OF HYBRID FRACTAL TREE

ANTENNA USING MODIFIED KOCH CURVE

An optimized design of printed hybrid fractal tree (PHFT) antenna based on

binary fractal tree geometry is presented and discussed. In this work miniaturization

has been achieved by inserting a modified Koch curve in the conventional fractal

binary tree. The geometrical descriptors of the proposed antenna have been

synthesized using Bacterial Foraging Optimization (BFO) and Particle Swarm

Optimization (PSO) for optimizing the values of electrical parameters within

specifications. A performance comparison has also been done for both of these

computational techniques. Representative results of optimized PHFT antenna for both

simulations and experimental validations are reported in order to access the

effectiveness of the developed approach for reliable implementation in wireless

telemedicine applications.

6.1 Introduction

Integration of various technologies is often used to develop multiple wireless

standards in a single device. Such a task becomes more challenging when a high

degree of miniaturization is also required [13], [122]. Fractal geometries due to their

space filling properties have proved to be very promising tool for miniaturized

multiband antennas [125], [126], [156]. In the last few years, tremendous efforts using

split-ring resonator has been taken to the issue of miniaturization and bandwidth such

106

as, complementary split-ring resonator pair and compact power dividers application

using fractal geometry [171], composite right/left handed transmission line based on

wunderlich shaped extended complementary single split ring resonator pair [20] and

compact balun based on fully artificial fractal shaped composite right/left handed

transmission line [173]. One more class of fractal geometries, the fractal tree has

already been used in antenna designs to produce multi and wideband characteristics or

to achieve miniaturization. The fractal tree includes several families such as the

binary, ternary, three dimensional, etc. [2], [114], [168]. Hybrid fractal trees and

center stubbed fractal trees represent two types of end-load structures that have

proven to be particularly effective in achieving a significant amount of size reduction

[61], [115].

The proposed antenna geometry is based on the hybrid structure obtained by

integrating modified Koch curve and fractal tree, whose geometrical descriptors are

determined by BFO and PSO in order to minimize the linear dimensions of the device

and to obtain the resonant performance characteristics within specifications for

wireless communication and their application in health care industry.

6.2 Design and Structure

Fractals are objects which have a self-similar structure repeated throughout

their geometry. This self-similar structure may be produced by the repeated

application of a generator, and in the case of fractal trees, the generator may be

described as a junction from which several smaller branches, known as child

branches, split from a parent branch [114]. Every branch, with the exception of the

first and final branches, has a generator connected to it at each end: one from which it

is a child and the other to which it is the parent. The proposed model starts from a

107

simple printed monopole as the zeroth

iteration, then applying the fractal tree

generator to the monopole as the first stage of the proposed model and for the second

stage modified Koch is applied to the branches of the fractal tree antenna as shown in

Figure 6.1. This procedure can be repeated to get all the higher iterations of the

structure with the scale factor of 0.5. Now with each successive iteration the length

and strip width of child branch reduces by a factor of 0.5. The proposed structure

provides another way to improve antenna miniaturization techniques that employ

fractal tree geometries as end loads by increasing the density of branches. The

incorporation of modified Koch in fractal tree provides more density to branches.

Fractal hybrid tree structure is appropriate to miniaturization design because of its

space-filling nature. As the iteration increased, the PHFT antenna has the geometric

structure which is easier to realize and can maintain its size reducing nature. The

purpose of designing this new structure (PHFT) is to use it, to tune and control

resonating frequencies.

(a) (b) (c)

Figure 6.1 (a) Ist stage of proposed antenna (b) modified Koch curve (c) 2

nd stage of

proposed model (PHFT).

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The antenna is designed with FR4 substrate having height of 1.6 mm. The

fractal shape allows the PHFT antenna to be effectively reduced in size without

significantly impairing its performance. The proposed geometry is characterized by

Length L, Width W, branch angle 2θ and a very small gap g = 0.01 mm between

ground plane and lower end of radiating element, as shown in Figure 6.2. The

dimensions of the ground plane for the printed geometry is given by, length of ground

plane, Lp and width of the ground plane, Wp. The Lp is kept constant with value of 10

mm whereas the Wp varies according to the branch angle of the structure for

optimizing the structure. The small rectangular partial ground plane provide good

impedance matching.

Figure 6.2 Geometry of proposed PHFT antenna

6.2.1 IFS for PHFT Antenna

An iterative function system (IFS) can be effectively used to generate the

PHFT antenna. A set of affine transformations forms the IFS for modified Koch used

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for its generation. The transformations to obtain the segments of the generator are

[158]:

y

x

y

xW

5

10

05

1

'

'1

(6.1)

5

1

0

05

15

10

'

'2

y

x

y

xW

(6.2)

5

15

1

5

10

05

1

'

'3

y

x

y

xW

(6.3)

5

25

1

10

1

10

3

10

3

10

1

'

'4

y

x

y

xW

(6.4)

5.0

3732.0

10

1

10

3

10

3

10

1

'

'5

y

x

y

xW

(6.5)

5

35

1

5

10

05

1

'

'6

y

x

y

xW

(6.6)

5

45

1

05

15

10

'

'7

y

x

y

xW

(6.7)

5

4

0

5

10

05

1

'

'8

y

x

y

xW

(6.8)

Where W1, W2, W3, W4, W5, W6, W7 and W8 are set of affine linear transformations, and

let A be the initial geometry then the generator is obtained as:

A1 = W1 (A) U W2 (A) U W3 (A) U W4 (A) U W5 (A) U W6 (A) U W7 (A) U W8 (A) (6.9)

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These affine transformations in the generalized case also lead to self-similar fractal

geometry. The fractal similarity dimension is given by the Equation 6.10, where N is

the total number of distinct copies and r is the scale factor of the consecutive iteration

[158].

D = = = 1.293 (6.10)

Fractal dimension is an important characteristic of fractal geometry. This is not a

unique description for the geometry; instead it describes a group of geometries with

similar nature. So a first step in the utilization of fractal properties in antenna design

should include the dimension of the geometry.

6.2.2 Curve Fitting Implementation

The curve fitting method has been used to form a relationship between the

design parameters (2θ) and the corresponding resonant frequency (f) of the proposed

fractal geometry. In case of fractal geometries their resonant properties depend on the

dimensions of the structure. The data sets were generated using EM simulator by

varying the branch angle (2θ) of the antenna and after applying these values,

following equations were obtained that represents the mapping of resonant

frequencies with these design parameters:

f1 = 6.537e-008 (2θ)4-3.091e-005 (2θ)3

+ 0.005597 (2θ)2 - 0.4598 (2θ) + 19.2

(6.11)

f2 = 6.587e-008 (2θ)4 -3.111e-005 (2θ)3

+ 0.005628(2θ)2 – 0.4619 (2θ) + 24.6

(6.12)

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The accuracy of the resonant frequencies calculated using these equations are

99.89%, whereas accuracy using EM simulator is 98.65% in terms of required output.

These accuracies are quite enough for antenna engineering applications.

6.2.3 PSO Implementation

The role of PSO optimization is to find the optimized branch angle (2θ) that

defines the best PHFT antenna for the desirable frequencies of operation. In the PSO

loop, a swarm is initialized with population of random positions and velocities of

antenna parameters (2θ) with their lower and upper bounds in solution space. The

below given Equation 6.13 is taken as a fitness function for PSO to find the designed

parameter of the proposed structure:

Fitness function = (5.2- f1)2

+ (10.6- f2)2 (6.13)

After getting the swarm initialization and a fitness function, the task is to set the

value of the optimization parameters and run the PSO program. The particle position

(SN) and velocity (V

N) was changed according to the Equations 6.14 and 6.15. In the

present work c1 and c2 are set to 2.0 and the inertial weights are varied linearly from

0.9 to 0.4 over iteration, finally w is set at 0.7. The instantaneous frequencies were

developed using curve fitting method. The particles position can be modified

according to the following equations [65]:

SN+1

= SN

+V N+1

(6.14)

VN+1

= w V N

+ c1r1 (Pbest - SN) + c2r2 (gbest - S

N) (6.15)

6.2.4 BFO Implementation

The job of BFO, whose cost function was evaluated using the curve fitting, is

to calculate the optimized values of the design parameter (2θ). In order to start the

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BFO process this parameter was initialized with suitable lower and upper bound that

defines a solution space in which the BFO searches for the optimal design parameter

of the design. The number of steps allowed for swimming Ns, chemotactic loop with

Nc iterations is not very large to avoid trap in a local minima. The probability Ped for a

bacterium is used to disperse to new location. The fitness function, given by Equation

6.13, was developed to find the structure of the PHFT antenna to work at user defined

frequencies. The goal of the algorithm is to find the minimum value for the fitness

function that is the place where maximum number of bacteria is found.

6.3 Results and Discussion

6.3.1 Comparison Between Conventional and PHFT Antenna

The electrical performance of each trial solution is estimated using an

electromagnetic simulator, IE3D which exploits the method of moments to solve the

electric field integral equations. A comparison has been made between PHFT antenna

and conventional fractal tree antenna with similar dimensions (L= 23.7 mm, W= 18.8

mm and branch angle, (2θ =60°). The resonant performance characteristics of both the

antennas are shown in Figure 6.3. It is established that a decrease of the antenna

(PHFT) resonant frequency is obtained, which contributes to 60% of antenna

miniaturization. PHFT can be designed to have low values of reflection by using

generator branching schemes and also the resonant frequency of fractal tree antennas

may be reduced by considering geometries which have a denser configuration of

branches. Fractal trees are space-filling geometries that can be used as antennas to

effectively fit long electrical lengths into small areas. The electrical length of

proposed PHFT has been increased and the input resistance became large enough to

be easily matched to the proposed feeding, while less space is occupied. The

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performance characteristics of conventional tree antenna and PHFT antenna is given

in Table 6.1

Table 6.1 Comparison of conventional tree antenna and proposed PHFT antenna.

Structure Resonan

t

Frequen

cy

(GHz)

Reflectio

n

Coefficie

nt (dB)

Input

Impedan

ce (Ω)

VSW

R

Antenn

a

Efficien

cy

Radiati

on

Efficie

ncy

Gain

(dB)

Conventio

nal Fractal

Tree

5.756

-19.36

49.05

1.241

68.83%

68.79%

2.062

PHFT

Antenna

4.818

-18.68

54.38

1.264

71.4%

75.66%

1.55

9.091

-16.39

41.52

1.357

68.9%

68.85%

1.88

6.3.1.1 VSWR

VSWR is a measure of how well the antenna terminal impedance is matched

to the characteristic impedance of the transmission line of the antenna. It is the ratio of

the maximum to the minimum RF voltage along the transmission line [132]. The

values of VSWR is consider to be good in the range of 1.5 to 2.0, excellent at 1.5 and

it is almost unacceptable at values higher than 2.0. From presented results in Table 6.1

it is observed that conventional and proposed PHFT antenna behaves in similar way

in terms of VSWR for the corresponding resonating frequencies. Figure 6.4 shows the

VSWR of proposed antenna graphically.

6.3.1.2 Input Impedance

The input impedance of the proposed PHFT antenna is shown in Figure 6.5

and Figure 6.6. The presented results indicate that the input impedance values for the

proposed antenna are within acceptable limits and it is well matched. The VSWR

values for PHFT antenna is tabulated in Table 6.1.

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Figure 6.3 Comparison of simulated S11 parameters of conventional antenna and

PHFT antenna

Figure 6.4 VSWR of the proposed PHFT antenna

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Figure 6.5 Real part of input impedance of PHFT antenna

Figure 6.6 Imaginary part of input impedance of PHFT antenna

6.3.1.3 Antenna and radiation efficiency

The antenna efficiency is the ratio of total power radiated by the antenna to the

total power fed to the antenna and this total power fed is the sum of radiated power

116

and power loss. The obtained results shows that besides miniaturization and an

additional band, PHFT antenna provides better radiation and antenna efficiency than

conventional fractal tree antenna at the cost of lower gain values as shown in Figure

6.7 and Figure 6.8.

Figure 6.7 Simulated antenna and radiation efficiency of proposed PHFT antenna.

Figure 6.8 Simulated gain of proposed PHFT antenna

117

6.3.2 Effect of Partial Ground Plane

It is worth mentioning that the structure of the ground plane also affects the

characteristics of the antenna. A parametric study has been done for the proposed

antenna to find the length and width of the partial ground plane. The parametric

variations of the proposed geometry with varying ground plane width are shown in

Figure 6.9. This study shows that by perturbing the ground plane, an excellent

impedance matching response is achieved for a certain ground plane size, beyond

which the response starts degrade. The simulated input impedance graphs for various

ground plane width are shown in Figure 6.10 and Figure 6.11. Notice that for input

impedance analysis, primary resonant frequency for all the structures has been

considered here. The presented results in Table 6.2 reveals that proposed antenna

with partial ground plane length (Lp=10mm) and width (Wp=23 mm), possesses high

impedance matching, better reflection coefficient and wider bandwidth. Also the

removal of partial ground plane augments the size reduction by 43.1% in addition to

the reduction of size due to incorporation of hybrid fractal tree geometry, which

comes at no extra cost or complexity.

Table 6.2 Resonant characteristics of proposed antenna with different ground plane

width

Width

of

Ground

Plane

(Wp)

Resonant

Frequency

(GHz)

Reflection

coefficient

(dB)

Input

Impedance

(ohms)

Bandwidth

(GHz)

Gain

(dB)

11 mm 4.431 -1.94 7.89+ 31.94j - -

15 mm 4.431 -4.10 17.83 + 35.13j - -

19 mm 4.431 -13.18 61.01 + 22.26j 0.03 1.53

23 mm 4.736 -31.61 50.82 + 0.51j 0.51 1.78

27 mm 4.883 -21.84 48.05 - 7.71j 0.45 1.67

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Figure 6.9 Bottom view of proposed geometry with varying ground plane dimensions

(width, Wp)

Figure 6.10 Real part of input impedance of proposed antenna with different ground

plane width.

119

Figure 6.11 Imaginary part of input impedance of proposed antenna with different

ground plane width.

6.3.3 Results of Optimization

The design of the PHFT antenna has been formulated in terms of an

optimization problem by defining and imposing suitable constraints on resonant

frequencies. To obtain a database from simulator for obtaining fitness function, the

branch angle of the fractal tree has been varied from 60 to 180 degree. Using the data,

the equations representing the relationship among different parameters of PHFT

antenna are generated by Curve-fitting method. In order to illustrate the impact and to

increase the confidence in optimization techniques, the proposed antenna was

synthesized using PSO and BFO. The BFO and PSO are quite similar in approach

with subtle differences. PSO has a limitation of being trapped in local minima and it

may converge prematurely. However, BFO can avoid premature convergence by

using its ability to explore and exploit the search space judiciously. The graphical

120

comparison for average best solution using both the techniques is shown in Figure

6.12 and obtained results reveals that BFO outperforms PSO for most of the

iterations. It concludes that the BFO algorithm has an edge over PSO in terms of its

accuracy and robustness. The second iteration of proposed antenna has been

optimized to resonate at user defined frequencies of 5.2 GHz and 10.6 GHz. Figure

6.13 gives the graphical comparison of s-parameters between BFO and PSO. Based

on these studies it is observed that the BFO provides more accurate results in terms of

required resonating frequencies, reflection coefficient and bandwidth than PSO,

which is a primary motive for optimization of the proposed geometry. The various

antenna parameters and their simulated results using both the optimization techniques

have been detailed in Table 6.3. It is interesting to note from Table 6.3, that for most

of the cases the BFO algorithm beats its nearest competitor PSO in a statistically

meaningful way.

Figure 6.12 Average best solutions found using PSO and BFO

121

Figure 6.13 S11 parameter comparisons of PSO and BFO

Table 6.3 Comparison of PSO and BFO results for proposed PHFT antenna.

Param

eters

Length

, L

(mm)

Width,

W (mm)

Branch

Angle

(2θ)

Resonant

Frequenci

es (GHz)

Reflection

Coefficient

(dB)

Bandwi

dth (%)

Radiati

on

Efficien

cy

Comput

ational

time

(secs.)

PSO

Results

22.9 21.9 75.09° 5.31 -25.26 22.33 78.1% 1.5

10.52 -16.68 22.99 71.9%

BFO

Results

22.7 22.2 75.96° 5.27 -27.26 25.04 81.2% 9.1

10.67 -18.66 25.79 77.6%

6.3.4 Experimental Results

The parameters found using developed methodology were used to draw the

structure of the antenna, the structure were then simulated and fabricated for

measurement. The prototype is fabricated using standard printed circuit methods and

the photograph of the fabricated antenna is shown in Figure 6.14. The measured

results obtained using vector network analyzer is shown in Figure 6.15. The simulated

results of resonant characteristics obtained by BFO optimization techniques and

measured results obtained using vector network analyzer, have been plotted

overlapping each other for meaningful comparison as shown in Figure 6.16. It may be

illustrated that the measured results are in good agreement with the simulated results,

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despite a slight frequency shift. This frequency shift is mainly because of the

fabrication imperfections and measurements error are mainly because of spurious

radiations created at the feeding end and the improper coupling of the elements. The

fabricated antenna resonates at 5.24 GHz and 10.69 GHz user defined frequencies

with -26.50 dB and -16.78 dB reflection coefficient respectively. Moreover, it has

been found that fabricated antenna provides wider bandwidth for both the frequency

bands (4.75-5.86 GHz, 10.0-12.22 GHz) covering WLAN, WiMAX and X-band

region for wireless telemedicine applications.

Figure 6.14 Fabricated PHFT antenna

Figure 6.15 Measured results of fabricated PHFT antenna using vector network

analyzer

123

Figure 6.16 Comparisons of S11 parameter of simulated and measured results

6.3.5 Radiation Patterns

Radiation pattern is a graphical representation of the antenna radiation

properties as a function of spherical coordinates. The simulated and measured

radiation characteristics of the optimized PHFT antenna are plotted in Figure 6.17 and

Figure 6.18 respectively.

(a) (b)

Figure 6.17 Simulated radiation patterns of proposed antenna (a) E-plane (b) H-plane

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(a)

(b)

Figure 6.18 Measured radiation patterns of proposed antenna (a) E-plane (b) H-plane

It is observed that the proposed antenna exhibits omnidirectional radiation

patterns at the y-z plane (H-plane) and “8-shape” radiation patterns at the x-z plane (E-

plane). E and H-plane are defined as the plane containing the direction of maximum

radiation and the electric and magnetic field vectors respectively. It is illustrated that

simulated and measured radiation characteristics are in good agreement and the

proposed antenna is linearly co-polarized antenna.

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6.4 Conclusion

In this work comparison between conventional fractal tree antenna and PHFT

antenna is investigated and the study showed a remarkable improvement over

radiation and antenna efficiency with presented structure. A parametric study has been

done for the proposed antenna to find the length and width of the partial ground plane.

The removal of partial ground plane augments the size reduction by 43.1% in addition

to the reduction of size due to incorporation of hybrid fractal tree geometry, which

comes at no extra cost or complexity.

The proposed new geometry has been synthesized by combining two different

fractal shapes in a unique hybrid structure and applying an efficient BFO and PSO

procedure to obtain desired resonant frequencies. The goal of the presented work is to

give a conceptual overview of the BFO technique and introduce it into the fractal

antenna community. This work also provides a critical comparison of BFO and PSO.

The study reveals that BFO provides more accurate results than PSO. The measured

electrical parameters confirm the reliability of the antenna and make it feasible for

wireless telemedicine applications. Representative results of optimized PHFT antenna

for both simulations and experimental validations are reported in order to access the

effectiveness of the developed approach for reliable implementation in wireless

telemedicine applications