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Vol. 4, No. 5 May 2013Journal of Emerging Trends in Computing and Information Sciences

A Design Methodology for Miniaturized Fractal Slot RFID Antennas Using Particle Swarm Optimization

1,2Department of Electronic and Communications Engineering3Department of Computer Engineering,

E-mail: [email protected]

An optimization-based methodology for the design of miniaturized microstripgives an automated process for generation of antenna geometrical parameters and mainly associated with Particle Swarm Two software programs are used in parallel, and in a coupled synchronized fashion. The first one is the electromagnetic simulMicrowave Studio which gives instantaneous estimation of the antenna performance. The second is MATLABalgorithm in order to optimize the geometry of the antenna after considering two objective functions: return loss and antennamethodology is applied to a third-order Minkowski fractal antenna which is optimizebased on the conventional (non-fractal) antenna, and the sequential one which is based on the optimized previoussimulated results show excellent performance requirementsthan 90% reduction in overall size in comparison with the conventional reference antenna. Keywords: Minkowski fractal, microstrip slot antenna, particle swarm optimizati 1. INTRODUCTION

Recently there is increasing interest in RFID systems operating at microwave frequency (5.8range with higher data transfer rate. In this circumstance, printed microstrip slot antennas are a very attractive choice because of their well-known advantages of low profile, light weight, easy integration with monolithic microwave incircuits, low cost, easy fabrication, and stable radiation patterns [1]-[4]. Therefore, great interests in various wideantennas with different feed methods for applications have been reported in the literature [5]However, miniaturization is still of main concern with the design process of such RFID antennas. The growing need for miniaturization, not only requires small devices, but also small-sized radiators. Fractals, owing to their geometrical properties, can be used successfully in antenna miniaturization and recently some interesting applications have been studied and presented in the scientific literature [10]. Also, the space filling property defined by fractal geometry and applied to a slot antenna element leads to reduce the total area occupied by the antenna [11]-[13].

Usually, antenna miniaturization design involves many

geometrical or material parameters. These parameters may be discrete, and often include constraints in allowable values. Optimizing such antennas to closely approximate desired criterion performance is similar to searching the global solution from a multidimensional solution space. So far, many stochastic evolutionary search techniques, such as Simulated Annealing (SA) and Genetic Algorithms (GAs) have been employed successfully in antenna design [14]. Particle Swarm Optimization (PSO), based on the simulation of a simplified sociological behavior associated with swarm such as bird flocking, and fish schooling, is an alternative optimization

Vol. 4, No. 5 May 2013 ISSN 2079Journal of Emerging Trends in Computing and Information Sciences

©2009-2013 CIS Journal. All rights reserved.

http://www.cisjournal.org

A Design Methodology for Miniaturized Fractal Slot RFID Antennas Using Particle Swarm Optimization

1D. K. Naji, 2J. S. Aziz, 3R. S. Fyath and Communications Engineering, College of Engineering, Alnahrain University, Baghdad, Iraq

Department of Computer Engineering, College of Engineering, Alnahrain University, Baghdad, Iraq

[email protected], [email protected], [email protected] ABSTRACT

based methodology for the design of miniaturized microstrip-fed fractal slot RFID antenna is introduced. This methodology gives an automated process for generation of antenna geometrical parameters and mainly associated with Particle Swarm Two software programs are used in parallel, and in a coupled synchronized fashion. The first one is the electromagnetic simulMicrowave Studio which gives instantaneous estimation of the antenna performance. The second is MATLABalgorithm in order to optimize the geometry of the antenna after considering two objective functions: return loss and antenna

order Minkowski fractal antenna which is optimized using two different approaches, the direct one which is fractal) antenna, and the sequential one which is based on the optimized previous

simulated results show excellent performance requirements with less than 40 of return loss, stable gain and radiation pattern, and more reduction in overall size in comparison with the conventional reference antenna.

Minkowski fractal, microstrip slot antenna, particle swarm optimization, sequential and direct optimization.

Recently there is increasing interest in RFID systems ) to achieve higher range with higher data transfer rate. In this circumstance, printed microstrip slot antennas are a very attractive choice

known advantages of low profile, light weight, easy integration with monolithic microwave integrated circuits, low cost, easy fabrication, and stable radiation

[4]. Therefore, great interests in various wide-slot antennas with different feed methods for 5.8 RFID applications have been reported in the literature [5]-[9]. However, miniaturization is still of main concern with the

The growing need for miniaturization, not only requires small devices, but also

diators. Fractals, owing to their geometrical properties, can be used successfully in antenna miniaturization and recently some interesting applications have been studied

[10]. Also, the space ined by fractal geometry and applied to a

slot antenna element leads to reduce the total area occupied by

design involves many geometrical or material parameters. These parameters may be

d often include constraints in allowable values. Optimizing such antennas to closely approximate desired criterion performance is similar to searching the global solution from a multidimensional solution space. So far, many

echniques, such as Simulated Annealing (SA) and Genetic Algorithms (GAs) have been employed successfully in antenna design [14]. Particle Swarm Optimization (PSO), based on the simulation of a simplified sociological behavior associated with swarm such as bees, bird flocking, and fish schooling, is an alternative optimization

algorithm first proposed by Kennedy and Eberhart in 1995 for solving multidimensional discontinuous problems [15]. The PSO algorithm has been shown to be an effective alternative to other evolutionary algorithms in handling certain kinds of optimization problems. Compared to GA and SA, the PSO algorithm is much easier to implement and apply to design problems with continues or discontinuous parameters. Especially, the PSO, used in conjuelectromagnetic solver, is found to be a revolutionary new approach to antenna design and optimization [16].

It is worthy to note here that, to the author’s knowledge,

there is no published works that usefor miniaturizing fractal RFID antennas. The related published works are concerned with conventional (nonantennas, and use optimization techniques for gain enhancement [17], multiband applications [18, 19] and robust environment and performance characteristics [20, 21]. Recently, Hidetoshi et al. [22] have presented an approach to optimize UHF mender-line RFID antennas for maximizing power and minimization of antenna size.

In this paper, PSO, implemented in MATLAB, is used in

along with CST Microwave Studio (CST MWS), which is an electromagnetic simulator based on finite integration time domain (FITD) [23], to introduce new design methodology for miniaturizing fractal slot antenna. The methodology is applied to Minkowski fractal slot antennThe convergence of the proposed methodology is studied in terms of optimization behavior and success time rate. The originality of the work in this paper resides in analyzing the proposed algorithm in actual real time electromby utilizing both MATLAB (as the optimizer tool) and CST MWS (as the electromagnetic tool).

ISSN 2079-8407 Journal of Emerging Trends in Computing and Information Sciences

477

A Design Methodology for Miniaturized Fractal Slot RFID Antennas

College of Engineering, Alnahrain University, Baghdad, Iraq

College of Engineering, Alnahrain University, Baghdad, Iraq

[email protected]

fed fractal slot RFID antenna is introduced. This methodology gives an automated process for generation of antenna geometrical parameters and mainly associated with Particle Swarm Optimization (PSO). Two software programs are used in parallel, and in a coupled synchronized fashion. The first one is the electromagnetic simulator CST Microwave Studio which gives instantaneous estimation of the antenna performance. The second is MATLAB which is used to run the PSO algorithm in order to optimize the geometry of the antenna after considering two objective functions: return loss and antenna size. The design

d using two different approaches, the direct one which is fractal) antenna, and the sequential one which is based on the optimized previous-order fractal antennas. The

of return loss, stable gain and radiation pattern, and more

on, sequential and direct optimization.

algorithm first proposed by Kennedy and Eberhart in 1995 for solving multidimensional discontinuous problems [15]. The PSO algorithm has been shown to be an effective alternative

her evolutionary algorithms in handling certain kinds of optimization problems. Compared to GA and SA, the PSO algorithm is much easier to implement and apply to design problems with continues or discontinuous parameters. Especially, the PSO, used in conjunction with the numerical electromagnetic solver, is found to be a revolutionary new approach to antenna design and optimization [16].

It is worthy to note here that, to the author’s knowledge, works that use optimization techniques

for miniaturizing fractal RFID antennas. The related published works are concerned with conventional (non-fractal) RFID antennas, and use optimization techniques for gain enhancement [17], multiband applications [18, 19] and robust

ance characteristics [20, 21]. [22] have presented an approach to

line RFID antennas for maximizing power and minimization of antenna size.

In this paper, PSO, implemented in MATLAB, is used in T Microwave Studio (CST MWS), which is an

electromagnetic simulator based on finite integration time domain (FITD) [23], to introduce new design methodology for miniaturizing fractal slot antenna. The methodology is applied to Minkowski fractal slot antenna as an illustrative example. The convergence of the proposed methodology is studied in terms of optimization behavior and success time rate. The originality of the work in this paper resides in analyzing the proposed algorithm in actual real time electromagnetic design by utilizing both MATLAB (as the optimizer tool) and CST MWS (as the electromagnetic tool).


2. PSO ALGORITHM Particle swarm optimization is a population based search

algorithm initialized with a population of random solutions, called particles. Each particle flies through the search space with a velocity that is dynamically adjusted according to its own and its companion’s previous behavior. This velocity consists of three parts, the “social”, the “cognitive”, and the “inertia” parts. The “social” part is the term guiding the particle to the global best position achieved by the whole swarm of particles so far, the “cognitive” term conducts it to the local best position achieved by itself so far, and the “inertia” part is the memory of its previous velocity ∙ . Figure 1 shows the flowchart of a PSO algorithm. During the PSO operation, each candidate solution is represented as a particle with position and velocity represented by and, respectively. Therefore, for dimensional optimization problems, the position and velocity of the ith particle can be represented as ,!, ,", … , ,$% and& where 'is the transpose operator.

Fig. 1: Chart showing the main steps of the PSO algorithm [24].

Define the Solution Space,

Fitness Function and

Population Size

Initialize X, V, P and G

For Each

Iteration

For Each Particle

Evaluate

Fitness

If Fit (X) > Fit (G)

then G = X

If Fit (X) > Fit (P)

then P = X

Solution is Final

G




Particle swarm optimization is a population based search algorithm initialized with a population of random solutions,

. Each particle flies through the search space with a velocity that is dynamically adjusted according to its own and its companion’s previous behavior. This velocity consists of three parts, the “social”, the “cognitive”, and the

“social” part is the term guiding the particle to the global best position achieved by the whole

, the “cognitive” term conducts it to the local best position achieved by itself so

part is the memory of its previous . Figure 1 shows the flowchart of a PSO

algorithm. During the PSO operation, each candidate solution with position and velocity

ively. Therefore, for N-dimensional optimization problems, the position and velocity

th particle can be represented as ,!, ," …, ,$%,

Chart showing the main steps of the PSO algorithm

Each particle of must be kept changed within the allowed solution space, bounded by physically invalid designs that allows an error to encountered by the fitness evaluator (CST MWS) and then an optimization process may be failed. If position of any particle exceed the limit, one of three control approach can be used, which are absorbing wall, reflecting wall, and visible wall. In this paper, an absorbing wall and reflecting wall techniques are used for their effectively pull back any boundary of solution space in that dimensions, by forcing their velocity to zero or changing the sign of their velocity, respectively.

The following formulae demonstrates the updating

process of positions and their velocities matrices and(, respectively, for N geometrical parameters and particles in swarm [25]

() ()*+ , -!.!,-"."/)*+ 0) 0)*+ , ()∆2 In (1) and (2), 2 denotes the current iteration, and

time interval between two consecutive iterations which is assumed to be unity. In (1), 3where 4562 and 4562 are calculated and stored in each iteration. The parameters -! and factors whose values are set in the range [suggested that the best choice is weights for cognition and social parts on average to be 1.0 [25]. In (1), the time varying parameterinertia factor, decreases with iteration number from a maximum value at the first iteration and goes to a minimum at the last iteration. Two statistically independent random variable .! and ." are both uniformly distributed in the interval [0, 1], are introduce to stochastically vary the relative pull of the personal and global best particles.

3. PROPOSED DESIGN METHODOLOGY

PSO algorithm has been tested by different research groups to different benchmark functions and that it is an excellent global optimizer that can be used for different electromagnetic problems especially antenna miniaturization that used in this paper. To calculate the antenna fitness function associatecomprehensive numerical modeling must be carried out to simulate the electromagnetic (EM) performance of the antenna at each stage of optimization. The EM model should be very efficient in both speed of computation and accuracy since the geometry of the fractal antenna is relativthan the conventional counterpart and the dimensions of some structure parameters are much smaller than the operating wavelength. The required EM model features are recovered in this paper by using a commercial EM simulator namely CST MWS. This simulator uses FITD method to assign the EM properties of antennas and has been proven in the literature as a powerful and very accurate tool for this purpose.

Next

Particle

Update Position

Using (1)

Update Velocity

Using (2)


478

must be kept changed within the allowed solution space, bounded by 8,99 , ,9:;< to avoid physically invalid designs that allows an error to encountered by the fitness evaluator (CST MWS) and then an optimization

position of any particle exceed the limit, one of three control approach can be used, which are absorbing wall, reflecting wall, and visible wall. In this paper, an absorbing wall and reflecting wall techniques are used for their effectively pull back any particles that flies outside the boundary of solution space in that dimensions, by forcing their velocity to zero or changing the sign of their velocity,

The following formulae demonstrates the updating process of positions and their velocities matrices = > ?, 0

geometrical parameters and M

3)*+ 0)*+∆2 0)*+∆21

2

denotes the current iteration, and ∆2 is the time interval between two consecutive iterations which is 3 and / are = > ? matrices

are calculated and stored in each and -" are the cognitive and social

factors whose values are set in the range [1.0, 2.0]. It has been suggested that the best choice is -! -" 2.0makes the weights for cognition and social parts on average to be 1.0 [25]. In (1), the time varying parameter, which is called inertia factor, decreases with iteration number from a maximum value at the first iteration and goes to a minimum at the last iteration. Two statistically independent random

are both uniformly distributed in the ], are introduce to stochastically vary the relative

pull of the personal and global best particles.

PROPOSED DESIGN METHODOLOGY

PSO algorithm has been tested by different research groups to different benchmark functions and the results show

it is an excellent global optimizer that can be used for different electromagnetic problems especially antenna miniaturization that used in this paper. To calculate the antenna fitness function associated with the PSO algorithm, a

modeling must be carried out to simulate the electromagnetic (EM) performance of the antenna at each stage of optimization. The EM model should be very efficient in both speed of computation and accuracy since the geometry of the fractal antenna is relatively more complicated than the conventional counterpart and the dimensions of some structure parameters are much smaller than the operating wavelength. The required EM model features are recovered in this paper by using a commercial EM simulator namely CST MWS. This simulator uses FITD method to assign the EM properties of antennas and has been proven in the literature as a powerful and very accurate tool for this purpose.


3.1 PSO-FITD Algorithm

In this work, the fractal RFID tag antenna is optimized using PSO technique while the FITD method is used in parallel with it to compute the EM part of the fitness function (see Fig. 2). The PSO technique runs under MATLAB environment and the FITD method is offered by CST MWS software package. For each generation of the PSO algorithm, the antenna geometrical parameters are updated and mapped to CST MWS to simulate the EM properties of the antenna. According to the EM simulator results that mapped back to MATLAB environment, the fitness function is evaluated by the PSO kernel.

Fig. 2: Flowchart of the PSO/FITD algorithm.

3.2 Optimization model The general nonlinear global optimization problem to be

solved is mathematically defined as: find the set !, ", … , 9 of ? variables that will minimize the function

=ABAA5C6D4E5-22FG 0, HIB 2G5-FB62JIAB26:L H H M,A 1,2, … , where Cis the fitness function, Gconstraint, is the inequality constraint, and vector of design variables. Also, L and Mupper bounds on the ? design variables, respectively.

The goal for the electromagnetic miniaturization design

considered here is to minimize the size of the fractal antenna by altering the geometrical parameters within allowed prescribed ranges while keeping the return loss below a desired threshold valueN!!O at the required resonance frequencyCP. A suitable optimization model is

PSO Antenna

Parameter Fitness Evaluation

MATLAB

Yes No Enough Iteration?

Update Antenna Parameters

Initial Antenna Parameters




In this work, the fractal RFID tag antenna is optimized PSO technique while the FITD method is used in

parallel with it to compute the EM part of the fitness function (see Fig. 2). The PSO technique runs under MATLAB environment and the FITD method is offered by CST MWS

the PSO algorithm, the antenna geometrical parameters are updated and mapped to CST MWS to simulate the EM properties of the antenna. According to the EM simulator results that mapped back to MATLAB environment, the fitness function is evaluated by

Flowchart of the PSO/FITD algorithm.

The general nonlinear global optimization problem to be solved is mathematically defined as: find the set

variables that will minimize the

3I H 034 ,3- , ? is the equality

is the inequality constraint, and is the Mare the lower and design variables, respectively.

The goal for the electromagnetic miniaturization design considered here is to minimize the size of the fractal antenna by altering the geometrical parameters within allowed prescribed ranges while keeping the return loss below a

at the required resonance . A suitable optimization model is

=ABAA5theUitnessfunction

]A2 ^N!!:_ N!!

, a b9b9*!cd 1e

ND4E5-22Fb9 f b9*!cd

IB 2G5-FB62JIAB26:LgG5J5

N!!:_ 20 log jk9:k9: In (4), D refers to the Heaviside step function while

b9and b9*!cd denote, respectively, the area of the nthand optimized B 1th-order fractal antennas. In (5), and k9:_ refer, respectively, to the return loss and the input impedance of the antenna at the resonance frequencykcis the characteristic impedance (

Note that the optimization fitness function (4) consists of

two objective functions which are related to antenna return loss N!! and antenna area

]A2 F4Elmm , F4EnP:

where

F4Elmm ^N!!:_ N!!Oo

and

F4EnP: a b9b9*!cd 1

where F4Elmm and F4EnP: denote the return loss and area objective functions, respectively.

The objective functionF4E

matching at the desired frequency"0" and"N!!O". Its zero value denotes that the goal is achieved, that is to say a return loss of at least desired frequency CPis satisfied. The objective function F4EnP: is used to achieve a minimum area from the optimization process. The range of 0, 1; it is zero if the area of nequal to the optimized area of the counterpart, and to " 1" when the area of the nthfractal antenna is zero (not physically allowed). Thus the value of the total fitness function multiobjective optimization problem ranges between a minimum value of " 1" to a maximum value of

CST MWS

Return Loss r++

Optimal Design


479

!!Oo ∙ D^N!!:_ N!!Oo e 4

H H M , A 1,2, … , ?

j :_ kc:_ , kcj 5

refers to the Heaviside step function while denote, respectively, the area of the nth-order

order fractal antennas. In (5), N!!:_ refer, respectively, to the return loss and the input

impedance of the antenna at the resonance frequencyCP, and is the characteristic impedance (kc 50Ω.

Note that the optimization fitness function (4) consists of two objective functions which are related to antenna return

6I

o ∙ D^N!!:_ N!!Oo64

e6-

denote the return loss and area objective functions, respectively.

F4Elmm represents the amount of matching at the desired frequencyCPand its value between

. Its zero value denotes that the goal is achieved, that is to say a return loss of at least N!!O at the

is satisfied. The objective function is used to achieve a minimum area from the

optimization process. The range of F4EnP: is between it is zero if the area of n-th order fractal antenna is

equal to the optimized area of the B 12G-order when the area of the nth-order

fractal antenna is zero (not physically allowed). Thus the value "]A2"which is considered as

multiobjective optimization problem ranges between a to a maximum value of"N!!O"


3.3. Antenna Design via Miniaturization Methodology (Sequential and Direct) An antenna miniaturization methodology for fractal based

microstrip antenna is investigated in this section. Two optimization approaches, sequential and direct approaches, (SOA) and (DOA), respectively, this work.

In the SOA, the following steps are performed (see Fig. 3)

i) Design a conventional (non-fractal) microstrip antenna using CST MWS. The dimensions of the structure parameters are scanned to achieve the design requirements (N!! H N!!O ) at the desired resonance This antenna will be considered as a reference antenna for the next optimization stage and its area is denoted by

ii) Apply the PSO/FITD algorithm described in subsection 3.2 to the reference antenna. This procedure yields the optimized zero-order fractal antenna (i.e., optimized reference antenna) of areIbtcd. Equation 4 is used here to describe the fitness function after replacingandb9*!cd bybP_ .

iii) Design an optimized first-order fractal antenna usingPSO-FITD algorithm. The input parameters used to start the PSO technique are the values of the structure parameters associated with the optimized version of the previous-order antenna (zero-order fractal antenna). Equation (4) is used to calculated the fitness function after replacing b9 by b! and b9*!cd by btcd .

iv) Repeat step (iii) to design the optimized version of the next-order fractal antennas.

v) Stop when the desired order of the fractal antenna, achieved.

In the direct optimization approach, the desired

fractal antenna (Bu) is optimized directly with respect to the reference antenna using one-shot optimization technique: Hence (4) is used after replacing b9by bbP_ . where b9vwx denotes here an area of the desiredantenna.

4. DESIGN OF MINKOWSKI FRACTAL ANTENNA

To validate the ability of this methodology in antenna design, a third-order Minkowski fractal slot antenna is chosen as an example to miniaturize it at microwave ISM band of5.8. Two optimization approaches, sequential and direct, are applied for this requirement in order to check if one can get the same performance and area for the designed antennas (i.e., a global optimizer approach). The three orders of Minkowski fractal structure is shown in Fig. 4.

The geometry of the conventional microstrip

antenna is illustrated in Fig. 5. A rectangular slot with lengthy and width zis printed at the centre on the ground side of a dielectric substrate of length yand antenna is fed with the main 50Ω stripline followed by




Miniaturization Methodology

An antenna miniaturization methodology for fractal based microstrip antenna is investigated in this section. Two

and direct optimization respectively, are considered in

, the following steps are performed (see Fig. 3)

fractal) microstrip antenna using CST MWS. The dimensions of the structure parameters are scanned to achieve the design requirements

) at the desired resonance frequencyCP. This antenna will be considered as a reference antenna for the next optimization stage and its area is denoted bybP_. Apply the PSO/FITD algorithm described in subsection 3.2 to the reference antenna. This procedure yields the

order fractal antenna (i.e., optimized . Equation 4 is used here to

describe the fitness function after replacingb9 bybt

order fractal antenna using the FITD algorithm. The input parameters used to start

the PSO technique are the values of the structure parameters associated with the optimized version of the

order fractal antenna). e fitness function after

Repeat step (iii) to design the optimized version of the

Stop when the desired order of the fractal antenna, Bu, is

optimization approach, the desired-order ) is optimized directly with respect to the

shot optimization technique: b9vwxand b9*!cd by

denotes here an area of the desired-order

DESIGN OF MINKOWSKI FRACTAL

To validate the ability of this methodology in antenna order Minkowski fractal slot antenna (MFSA)

t at microwave ISM . Two optimization approaches, sequential

and direct, are applied for this requirement in order to check if one can get the same performance and area for the designed antennas (i.e., a global optimizer approach). The three orders

ture is shown in Fig. 4.

The geometry of the conventional microstrip-fed slot antenna is illustrated in Fig. 5. A rectangular slot with

is printed at the centre on the ground and widthz. The

stripline followed by an op-

Fig 3: Flowchart showing the main steps of the proposed Methodology.

-en-circuited microstrip line printed on the other side of the substrate. To reduce costs, the antenna is printed on commercial available FR4 dielectric substrate with a permittivity of 4.3 and loss tangent (achieve very thin antenna requirement for RFID applications, a substrate height G 0.8z! of the main 50Ω-stripline is designed through CST MWS and found to be1.56

Yes

SOA

Design Reference Antenna at Resonance

Frequency CP

Select Optimization Approach, SOA or

DOA

PSO/FIT Algorithm Using eqn. (4)

SOA or DOA

B f Bu

Optimum Designed Order Fractal Antenna at CP with Area b b

Fractal Order, B B

Optimum Designed Bu- order Fractal

Antenna at CP with Area b b9vwx

cd

No

Fractal Order, B


480

Flowchart showing the main steps of the proposed

circuited microstrip line printed on the other side of the substrate. To reduce costs, the antenna is printed on commercial available FR4 dielectric substrate with a permittivity of 4.3 and loss tangent (tan 0.02). In order to achieve very thin antenna requirement for RFID applications, is used. The stripline width

stripline is designed through CST .

Design Reference Antenna at Resonance

Select Optimization

PSO/FIT Algorithm

DOA

Optimum Designed B-Order Fractal Antenna at

b9cd

B , 1

Optimum Designed order Fractal

with Area

PSO/FIT Algorithm

Using eqn. (4)

Fractal Order, B B9

0


Fig. 4: Scheme of the Minkowski fractal slot antenna orders

structure. (a) (MFSA0). (b) (MFSA1)(MFSA3).

Fig. 5: Geometry of the reference microstrip fed(a) Feed side. (b) Ground side. (c) Bottom side.

It is clear from Figs. 4 and 5 that, there are eight geometrical parameters common for the four fractal antennas (MFSA0 – MFSA3). Four of these parameters are related to the ground side (lengthy, widthz, slot lengthwidthz) and the other four parameters are for the feed side (stripline lengthsy! andy" and widthsThe stripline width z"is set to 1.56microstrip feed-line and 0.8-substrate thickness as stated before. This leaves seven common parameters to be adapted during the optimization process.

In this work, five of the common parameters enter the

optimization process as scaling factors with respect to the other two main parameters, ground length This is useful to ensure that physically invalid structures to be not constructed and consequently prevent total failure of the optimization process. These scale factors are

~




Scheme of the Minkowski fractal slot antenna orders (MFSA1). (c) (MFSA2) (d)

Geometry of the reference microstrip fed-slot antenna. Bottom side.

It is clear from Figs. 4 and 5 that, there are eight common for the four fractal antennas

MFSA3). Four of these parameters are related to , slot lengthy, and slot

) and the other four parameters are for the feed side and widthsz! andz"). 56 for a 50Ω-substrate thickness as stated

before. This leaves seven common parameters to be adapted

In this work, five of the common parameters enter the optimization process as scaling factors with respect to the other two main parameters, ground length yand widthz. This is useful to ensure that physically invalid structures to be

nstructed and consequently prevent total failure of the optimization process. These scale factors are

TABLE RANGES OF THE DESIGN PARAMETERS FOR THE

MINIATURIZED FRACTAL

Parameter Ranges

y (mm) 3~40

z (mm) 3~40

x 0.4~0.9

x 0.4~0.9

xm 0.1~0.5

x 0.1~0.4

x y y⁄ ; xm y! z;⁄ x y"⁄

The fractal antennas (MFSA1

structure parameters whose number equal to four times the antenna fractal order. Thus the numbers of additional parameters is 4, 8, and 12 for 1stantennas, respectively, (see Fig. 4). It is clear from the above discussion that the number of geometrical parameters increases with the order of the fractal. Thus the computation time of the optimization technique increases as the order of the fractal antenna increases. To solve this limitation, four scaling factors are introduced here for any fractal order. These four scaling factors are,,follows

y:9 ô9 > yy9 ô9 > z

z:9 ô9 > zz9 ^ o9 > y

where

y:! y:!, y:" y:", …y! y!, y" y",

z:! z:!, z:" z: z! z!, z" z" Table I lists the constraints applied to antenna geometrical

parameters during the optimization process. The design methodology starts with design of the

reference slot antenna which represents the 0thMFSA0. The reference antenna design starts operating frequency CP (5.8permittivity (]4, P 4.3) and the thickness of the substrate (G 0.8) which is very thin suitable for RFID tag applications. For exciting the operating frequency atdimensions of ground and slot can be roughly designed with the following equations [26]


481

TABLE I RANGES OF THE DESIGN PARAMETERS FOR THE

MINIATURIZED FRACTAL-SLOT ANTENNA.

Parameter Ranges

x 0.01~0.1

0.1~0.33

0.2~0.33

0.1~0.33

0.2~0.33

x z z7I⁄ z;⁄ x z" y⁄ 74

The fractal antennas (MFSA1-MFSA3) have additional structure parameters whose number equal to four times the antenna fractal order. Thus the numbers of additional

4, 8, and 12 for 1st-, 2nd-, and 3rd-order fractal antennas, respectively, (see Fig. 4). It is clear from the above discussion that the number of geometrical parameters increases with the order of the fractal. Thus the computation

technique increases as the order of the fractal antenna increases. To solve this limitation, four scaling factors are introduced here for any fractal order. These four

,, and defined as

8I

84

8c

8

B 1, 2, … , Bu

… , y:9vwx y:9vwx9I

… , y9vwx y9vwx94

:", … , z:9vwx z:9vwx9-

", … , z9vwx z9vwx9

lists the constraints applied to antenna geometrical parameters during the optimization process.

The design methodology starts with design of the reference slot antenna which represents the 0th-order structure MFSA0. The reference antenna design starts by selecting the ), substrate with required

) and the thickness of the substrate ) which is very thin suitable for RFID tag

applications. For exciting the operating frequency atCP, the dimensions of ground and slot can be roughly designed with


y z t2 y z t

2P__

P__ P , 12 , P 1

21

1 , 12G z⁄

Here (t - CP⁄ , is the free space wave length, speed of light in vacuum, P__ is the effective relative permittivity of the dielectric substrate, z is the ground width, yis the ground length, z is the ground slot width, ground slot length and z! is the stripline width. Using (10a) and (10b), the reference slot antenna is designed using the following initial parameters

z y 26.0IB z y 14 The designed slot antenna is simulated using CST MWS

by sweeping the ground and slot parameters within the limit

z y t2 1 ∓ 5%,z y t

2__1 ∓ 10%

The simulation takes into account other structure parameters associated with the stripline parameters, strip lengthsy!, y" and strip widthz". Table final antenna geometrical parameters for the reference slot antenna (see Fig. 5).

In PSO technique, the selection of the particle number and

the maximum iteration depends on the dimension of the solution space and the fitness function. The numbparticles should be comparable to the dimension of the solution space to obtain a good convergence. A reasonably large number of iterations are also necessary for the particles to get converged. In this paper, the best number of particles used is chosen between3? to5?, where ?antenna geometrical parameters that are associated with the optimization technique. Thus, a 28-particle-0th-order fractal antenna and 33-particle for antennas MFSA1, MFSA2 and MFSA3. Furthermore, a stop criterion is chosen such that 50 PSO iterations are reached or the fitness function remains unchanged with less than 2% error for at least 20 successive iterations.

TABLE II

OPTIMIZED ANTENNA GEOMETRICAL PARAMETERS TO ACHIEVE CP 5.8FOR THE REFERENCE SLOT

ANTENNA.

Parameter Value (mm)

Parameter

y 27.00 y" z 25.70 z! y 12.00 z" z 13.20 G

y! 8.48




10I

104

z!10-

, is the free space wave length, - is the is the effective relative

is the ground width, is the ground slot width, y is the is the stripline width. Using (10a)

and (10b), the reference slot antenna is designed using the

14.411

The designed slot antenna is simulated using CST MWS by sweeping the ground and slot parameters within the limit

12I

124

The simulation takes into account other structure parameters associated with the stripline parameters, strip

. Table II shows the final antenna geometrical parameters for the reference slot

In PSO technique, the selection of the particle number and the maximum iteration depends on the dimension of the solution space and the fitness function. The number of particles should be comparable to the dimension of the solution space to obtain a good convergence. A reasonably large number of iterations are also necessary for the particles to get converged. In this paper, the best number of particles ?is the number of antenna geometrical parameters that are associated with the

-swarm is used for particle for antennas MFSA1,

Furthermore, a stop criterion is chosen such that 50 PSO iterations are reached or the fitness function remains unchanged with less than 2% error for at least 20

OPTIMIZED ANTENNA GEOMETRICAL PARAMETERS TO FOR THE REFERENCE SLOT

Value (mm) 5.14 1.56 0.64 0.80

5. RESULTS AND DISCUSSION

5.1. PSO Behavior and Optimized Geometrical Parameters

Illustrative results related to the design of a third

Minkowski fractal antenna are given here. The threshold value of N!!, N!!O, used in the fitness function is 6 reveals the progress of the PSO algorithm as a function of iteration number for both sequential and direct approaches. Return loss and area objective functions as well total fitness function are illustrated in this figure. The behavior of the total fitness function is characterized by two representations here. The first representation marks the best value of the total fitness function up to the current iteration. Such representation is useful to record the performance progress that occurs during the history of the optimization process. The best value of the fitness function up to the last iteration gives the required solution. The second representation marks the best fitness function at each PSO iteration and illustrates the instantaneous variation of the fitness funcprocess.

The results in Fig. 6 reveals that the two approaches of

optimization reach nearly the same value of fitness of 0.9 for the 3rd-order fractal antenna (MFSA3). Further, for the initial population, the return loss objective function of the best individual is greater than greater than12. However, the results are improved quickly at iterations12,20,19,12, and 8 to MFSA0, MFSA1, MFSA2, MFSA3 and direct approach MFSA3, respectively.

The optimized antennas are simulated by using a laptop

(hp pavilion dv6,2.54, 4-the sequential approach, the simulation times required to optimize 0th-order, 1st-order, 2ndantennas are 53.50, 65.42, 44.52 and 63.98 hour. In the direct approach, 71.34 hours is required to optimize the 3rdfractal antenna. Thus, the average time of each optimization run is about 60 hours.

It is worth to mention here that one of the main limitations

of the proposed methodology is the relatively long computational time and therefore it requires an efficient PC system to handle it successively without run failure.

The PSO algorithm creates an optimal miniaturized

antenna size in the two optimization approaches. Table summarizes the final optimized geometrical parameters. One can observe from Table III that both sequential and direct optimized 3rd-order fractal antennas have almost the same geometrical and performance parameters. This proves that the proposed methodology has the ability to reach the same optimized dimensional results for the miniaturizei.e., global optimization results.


482

RESULTS AND DISCUSSION

PSO Behavior and Optimized Geometrical

Illustrative results related to the design of a third-order Minkowski fractal antenna are given here. The threshold value

, used in the fitness function is 25 .Figure e progress of the PSO algorithm as a function of

iteration number for both sequential and direct approaches. Return loss and area objective functions as well total fitness function are illustrated in this figure. The behavior of the total

is characterized by two representations here. The first representation marks the best value of the total fitness function up to the current iteration. Such representation is useful to record the performance progress that occurs during

timization process. The best value of the fitness function up to the last iteration gives the required solution. The second representation marks the best fitness function at each PSO iteration and illustrates the instantaneous variation of the fitness function during the optimization

The results in Fig. 6 reveals that the two approaches of optimization reach nearly the same value of fitness of

order fractal antenna (MFSA3). Further, for the initial population, the return loss objective function of the 10 with total fitness values

. However, the results are improved quickly at for sequential approach related

to MFSA0, MFSA1, MFSA2, MFSA3 and direct approach

The optimized antennas are simulated by using a laptop -core CPU and 4 RAM). In

quential approach, the simulation times required to order, 2nd-order, and 3rd-order fractal

antennas are 53.50, 65.42, 44.52 and 63.98 hour. In the direct approach, 71.34 hours is required to optimize the 3rd-order

Thus, the average time of each optimization

It is worth to mention here that one of the main limitations of the proposed methodology is the relatively long computational time and therefore it requires an efficient PC

e it successively without run failure.

The PSO algorithm creates an optimal miniaturized antenna size in the two optimization approaches. Table III summarizes the final optimized geometrical parameters. One

that both sequential and direct order fractal antennas have almost the same

geometrical and performance parameters. This proves that the proposed methodology has the ability to reach the same optimized dimensional results for the miniaturized antenna,


Fig. 6: Variation of return loss and area objectives as well total fitness with PSO iteration number for different fractal order. order (b) 1st-order (c) 2nd-order (d) 3rd-




Variation of return loss and area objectives as well total fitness with PSO iteration number for different fractal order. -order (sequential) and (e) 3rd-order (direct).


483

Variation of return loss and area objectives as well total fitness with PSO iteration number for different fractal order. (a) 0th-


Fig.6: Continued.

OPTIMIZED GEOMETRICAL PARAMETERS OF MINKOWSKI FRACTAL SLOT ANTENNA

RSA 27.000 25.700

SA

MFSA0 21.350 13.500

MFSA1 8.410 13.447

MFSA2 3.560 18.800

MFSA3 3.111 18.656

DA MFSA3 3.229 19.123

Legend: SA= Sequential Approach; DA=Direct Approach;

SIMULATION RESULTS OF MINKOWSKI FRACTAL SLOT ANTENNA.

r++

RSA -53.32

SA

MFSA0 -40.33

MFSA1 -62.12

MFSA2 -51.84

MFSA3 -40.55

DA MFSA3 -40.45

Legend: SA= Sequential Approach; DA=

Ant

enna

T

ype

Opt

imiz

atio

n

App

roac

h

Ant

enna

T

ype

Opt

imiz

atio

n

App

roac

h




TABLE III OPTIMIZED GEOMETRICAL PARAMETERS OF MINKOWSKI FRACTAL SLOT ANTENNA

Antenna Geometrical Parameters

)+ ) ) 0.446 0.516 0.330 0.200 0.025 NA NA

0.623 0.826 0.119 0.325 0.016 NA NA

0.864 0.604 0.204 0.371 0.010 0.305 0.197

0.747 0.820 0.195 0.373 0.082 0.13 0.211

0.820 0.700 0.247 0.339 0.035 0.238 0.218

0.737 0.602 0.328 0.299 0.020 0.209 0.135

Direct Approach; RSA=Reference Slot Antenna; MFSA=Minkowski Fractal Slot Antenna;

TABLE IV SIMULATION RESULTS OF MINKOWSKI FRACTAL SLOT ANTENNA.

Antenna Performance Parameters

/

%

/

¡ /

¡ /

4.57 87.42 0.73 5.43 6.17 694.00

4.33 93.73 0.74 5.52 6.27 288.00

2.39 89.00 0.49 5.56 6.08 113.00

2.07 89.37 1.45 5.37 6.83

2.18 91.01 1.89 5.42 7.31

2.22 90.47 1.69 5.40 7.10

DA= Direct Approach; ¡= -10 dB lower frequency; ¡ = -

¢


484

OPTIMIZED GEOMETRICAL PARAMETERS OF MINKOWSKI FRACTAL SLOT ANTENNA

NA NA NA

NA NA NA

0.197 0.300 0.117

0.211 0.287 0.152

0.218 0.282 0.187

0.135 0.298 0.135

Minkowski Fractal Slot Antenna; NA=Not

SIMULATION RESULTS OF MINKOWSKI FRACTAL SLOT ANTENNA.

£

∆£ %

694.00 -

288.00 58.47

113.00 83.71

67.00 90.35

58.03 91.64

61.74 91.10

-10 dB higher frequency.


5.2. Performance Results of the Optimized MFSA The electromagnetic properties of the optimized antennas

are simulated using CST MWS. Table IVsimulation results, namely, return lossN!!total antenna efficiency., bandwidthreduction∆b. The size reduction is computed as

∆b bP_ bbP_

where b and bP_ is, respectively, the area of the antenna under observation and reference antenna.

Note further that the optimized referen58.5% area reduction with respect to the conventional

reference antenna. Introducing the 2nd-order fractal geometry yields 73.26% area reduction with respect to the optimized reference antenna.

The results in Table IV also highlight the

findings 1) A return loss less than 40 , the gain greater than

and efficiency greater than 87% are obtained for all the designed antennas.

2) Bandwidth enhancement is obtained when the antenna is designed with fractal order more the one. The(sequential), and 3rd (direct) fractals offer1.69 bandwidth, respectively, while the reference antenna gives 0.73 bandwidth.

3) Both sequential and direct optimized 3rdantennas have almost the same performance parameters.

The return losses of the optimized antennas are shown in

Fig. 7. A result related to the reference antenna is also given for comparison purposes. It is clear from this figure and Table 4 that the 10 lower frequency is approximately the same for all antennas as compared with the frequency which increases with increasing of fractal order. Also one can notice that a second resonance frequency appears as the order of the fractal is exceedfractal geometry enhances the antenna bandwidth and opens the possibility to design the structure for two

Figure 8a and 8b show, respectively, the gain and

efficiency for the designed antennas. It is clear from these figures that the third-order fractal has almost similar spectral behavior for both optimization approaches. Also a stable gain of 2 over 5.6– 6.6 is obtained for the thirdfractal in both approaches.

The radiation characteristics of the optimiz

are also studied and the results are depicted inshows the 2D radiation pattern in the elevation direction¥∅ 90°) and¥¨ 90°) planes and azimutal direction∅ 0°) plane at 5.8 for the optimized antennas. It is appears that antenna radiates a nearly omindirectionally ©∅ component in the order of fractal is more than 1, but the radiation patterns show two nulls for ©ª component at 0° and




Performance Results of the Optimized MFSA

The electromagnetic properties of the optimized antennas IV lists some of the !!, antenna gain,

, bandwidthz, and size reduction is computed as

13

is, respectively, the area of the antenna

Note further that the optimized reference antenna offers area reduction with respect to the conventional

order fractal geometry area reduction with respect to the optimized

also highlight the following

, the gain greater than 2 , are obtained for all the

Bandwidth enhancement is obtained when the antenna is designed with fractal order more the one. The 2nd, 3rd (sequential), and 3rd (direct) fractals offer1.46,1.89, and

bandwidth, respectively, while the reference

Both sequential and direct optimized 3rd-order fractal formance parameters.

The return losses of the optimized antennas are shown in Fig. 7. A result related to the reference antenna is also given for comparison purposes. It is clear from this figure and Table

lower frequency is approximately the same for all antennas as compared with the 10 higher frequency which increases with increasing of fractal order. Also one can notice that a second resonance frequency appears as the order of the fractal is exceeds 1. Thus the fractal geometry enhances the antenna bandwidth and opens the possibility to design the structure for two-band operation.

show, respectively, the gain and efficiency for the designed antennas. It is clear from these

order fractal has almost similar spectral behavior for both optimization approaches. Also a stable gain

is obtained for the third-order

The radiation characteristics of the optimized antennas are also studied and the results are depicted in Fig. 9 which

radiation pattern in the elevation ) planes and azimutal

for the optimized antennas. It is appears that antenna radiates a nearly -plane when the order of fractal is more than 1, but the radiation patterns show ¨ 180° for -

Fig. 7: Simulated return losses of the optimized antennas. A result related to the reference antenna is given for comparison purposes.

(a)

(b)

Fig. 8: (a) Gain of the optimized antennas. optimized antennas. A result related to the reference antenna is given for comparison purposes.

-plane and two nulls for ©∅∅ 270° for ¥-planes. Note further that the antennas have relatively strong cross-polarized radiation co-polarized radiation). It is interesting to notice the strong similarity between patterns for at the three planes¥, ¥andmaximum electric field components antenna planes to clarify the similarity between the patterns.

The 3D radiation characteristics of the optimized

antennas are depicted in Figs. pattern are plotted for zero-, 1stantennas, respectively. One can notice from these figures that omnidirectional radiation is achieved when the fractal order increases beyond 1


485

Simulated return losses of the optimized antennas. A result eference antenna is given for comparison purposes.

(b)

Gain of the optimized antennas. (b) Efficiency of the optimized antennas. A result related to the reference antenna is

component at ∅ 90° and planes. Note further that the antennas have

polarized radiation « 18 below polarized radiation). It is interesting to notice the strong

similarity between patterns for ©ªcomponents for all antennas and.Table V shows the

maximum electric field components ©ªand ©∅for the three antenna planes to clarify the similarity between the patterns.

radiation characteristics of the optimized antennas are depicted in Figs. 10a-d where the 3D radiation

, 1st-, 2nd-, and 3rd-order fractal One can notice from these figures that

is achieved when the fractal order


.

Fig. 9: Radiation Patterns for the optimized antennas; (a) Zero

¬

® -plane

-plane

¬∅

¬∅

¬

¬

®-plane

¬∅




Radiation Patterns for the optimized antennas; (a) Zero-order (b) 1st-order (c) 2nd-order (d) 3rd


486

order (d) 3rd-order fractal.


Fig. 10: 3D Radiation Patterns for the three fractal order iteration of Minkowski fractal slot antenna;

(b) 1st-Order (c) 2nd- Order (d) 3rd- Order It is worth to note here the following findings which are

common for all the optimized antennas 1) The electric field components ©∅

respectively, a co-polar component (i.e. maximum) and a cross-polar (i.e. minimum) in ¥ and -

2) The electric field components ©ª respectively, a co-polar component and a crossplane.

3) The ratio of co-polar to cross-polar components is greater than 58.4 for ¥-plane and greater than ¥-plane and more than 22.8 for -

MAXIMUM RADIATION ELECTRIC FIELD VALUES (IN

Radiation plane

1st-order

¬ ¬∅ ¬¯¬°¯

®, ±²° 53.9 71.7 17.8

® , ∅ ±²° 79.1 -200 279.1

, ∅ ²° 56.3 79.1 22.8




3D Radiation Patterns for the three fractal order iteration of Minkowski fractal slot antenna; (a)Order

It is worth to note here the following findings which are

and ©ªrepresent, polar component (i.e. maximum) and a

-plane. and©∅ represent,

polar component and a cross-polar in ¥

polar components is greater plane and greater than 17.8 for

-plane.

Table VI presents a performance comparison between the optimized 3rd-order fractal antenna selected from this work and some RFID antennas reported in the literature for 5.8 GHz applications. It’s worth to emphasize here that the antenna designed in this work is characterized by the smallest area compared with other structures. The area of this antenna corresponds to 0.52 of the area of the antenna reported in Ref. [9] which has the smallest area among previous antennas listed in Table VI. Note further that theoffers the lowest gain and highest bandwidth compared with others.

TABLE V RADIATION ELECTRIC FIELD VALUES (IN ³/) FOR OPTIMIZED ANTENNAS.

2nd-order 3rd-order

¬ ¬∅ ¬¯¬°¯ ¬ ¬∅

¬¯¬°¯

52.4 75.4 23 49.7 76.2 26.5

77.2 -200 277.2 76.8 -73.4 150.2

41.8 77.2 35.4 41.9 76.8 34.9


487

(a) Zero-order

presents a performance comparison between the order fractal antenna selected from this work

and some RFID antennas reported in the literature for 5.8 GHz applications. It’s worth to emphasize here that the antenna

is characterized by the smallest area compared with other structures. The area of this antenna corresponds to 0.52 of the area of the antenna reported in Ref. [9] which has the smallest area among previous antennas

. Note further that the designed antenna offers the lowest gain and highest bandwidth compared with

) FOR OPTIMIZED ANTENNAS.

3rd-order (direct)

¬ ¬∅ ¬¯¬°¯

50.9 76.3 25.4

150.2 76.9 18.5 58.4

41.6 76.9 35.3


PERFORMANCE COMPARISON BETWEEN THE 3RDRFID REPORTED ANTENNAS IN THE LITERATURE AT 5.8 GHZ ISM BAND.

Ref Antenna Structure Aim

This work

Stripline-fed 3rd-order Fractal Slot Antenna

Miniaturization

[9]

CPW-fed dual folded-strip monopole antenna

Miniaturization

[8]

CPW-fed shorted F-shaped monopole antenna

Miniaturization

[5]

CPW-fed Slot Antenna

Broadband

[7]

CPW-fed Folded Slot Antenna

Dual-band(2.45, 5.8) GHz

[6]

CPW-fed Slot Antenna

Miniaturization

[18]

CPW-fed Slot antenna

Tri-Band (0.95, 2.45, 5.8) GHz

6. CONCLUSION An efficient methodology of designing miniaturized

fractal antennas for RFID applications has been proposed. The antenna geometrical parameters are optimized using algorithm which runs on MATLAB environments and synchronously coupled to full wave electromagnetic simulator implemented using CST Microwave Studio software. The used optimization objective functions reflex both return loss and antenna size. The proposed design methodology has been applied to Minkowski fractal slot antenna operated at5.8. The results reveals that more than in overall antenna size can be obtained for fractal order two or more as compared with the conventional reference antenna. Further, excellent performance requirements are obtained with less 40 return loss and more than 2 radiation pattern has been achieved. The 3rdantenna has a just overall size of3(i.e.,0.0217", where is the wavelength at 5.8 GHz).




TABLE VI PERFORMANCE COMPARISON BETWEEN THE 3RD-ORDER FRACTAL ANTENNA REPORTED IN THIS WORK AND

RFID REPORTED ANTENNAS IN THE LITERATURE AT 5.8 GHZ ISM BAND.

Aim Laminate µ° ¶

> £

Miniaturization

FR4 Epoxy

4.4

0.80 3.11 > 18.65

58.0

Miniaturization

FR4 Epoxy

4.4

1.60 14.0 > 8.0

112.0

Miniaturization

FR4 Epoxy

4.4

1.60 13.0 > 16.8

218.4

Broadband

FR4 Epoxy

4.4

1.60

16 > 15 240.0

band (2.45, 5.8) GHz

FR4 Epoxy

4.4

1.00

20 > 31.8 663.0

Miniaturization

FR4 Epoxy

4.3

1.50

30 > 30 900.0

Band (0.95, 2.45, 5.8) GHz

FR4 Epoxy

4.4

1.52

30 > 30 900.0

An efficient methodology of designing miniaturized fractal antennas for RFID applications has been proposed. The antenna geometrical parameters are optimized using PSO algorithm which runs on MATLAB environments and synchronously coupled to full wave electromagnetic simulator implemented using CST Microwave Studio software. The used optimization objective functions reflex both return loss

ed design methodology has been applied to Minkowski fractal slot antenna operated

. The results reveals that more than 90% reduction in overall antenna size can be obtained for fractal order two or more as compared with the conventional reference antenna. Further, excellent performance requirements are obtained with gain with stable radiation pattern has been achieved. The 3rd-order fractal 3.11 > 18.65"

is the wavelength at 5.8 GHz).

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