SQUARE LOOP ANTENNA MINIATURIZATION USING FRACTAL GEOMETRY

Hassan M. EIkamchouchi* Dept. of EE, Faculty of Engineering, Alexandria University. email: helkamchouchi@ieee.org

Mona N. Abd El-SaIam Communications Sector, Telecom. Egypt. email: mnabil@ieee.org

Abshact: Antenna design is a very tricky problem. Common designs are sensitive to only a narrow range of frequencies, and are not efjicient if they are smaller than a quarter ofthe wavelength. This is a problem for small, portable antennas, such as those on cellular phones. Fractal antenna designs can overcome some ofthese problems. Experiments have shown that antennas built with only a small number of iterations of a fractal process can exhibit sensitiviry at severalfrequencies. As the number of iterations increases, the lowestfrequency of the antenna gets lower, and additional higher frequencies are added. Also, fractal antennas can operate efticiently at one-quarter the size of more traditional designs. Properly harnessed, these features represent real advantages. Fractals already are being used for compact, multifrequency antennas in cellular phones and military communications hardware. Fractal Antennas are now used in cellular phone antennafitting inside the bo& ofthe phone, and the multifrequency aspect ofthe ontenna will allow GPS to be incorporated in the phone. Other applicalions include compoct, multrfrequency wireless LAN and maritime antennas. In this paper we present a miniaturization model of a square loop antenna using a non-return to zero pulse as a generator instead of the square pulse in Minkowski fractal model that allows an increase in the total electric length without occupying more space. This led to an easily matched compact loop antenna, but with a small sacrfice of the gain.

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I. Introduction

Fractal geometries [ I ] are generated in an iterative fashion, leading to self-similar structures. In a fractal no matter how closely the structure is studied, there never comes a point where the fundamental building blocks can be observed. The reason for this intricacy is that the fundamental building blocks of fractals are scaled versions of the fractal shape. This can be compared to not being possible to see the ending reflection when standing between two mirrors. Closer inspection only reveals another mirror with an infinite number of mirrors

' reflected inside. [2]

11. Proposed geometry description

The proposed model is an iterative model to a normal square loop with a generator of the shape shown in figure I , by replacing each straight wire in the square with another wire of the generator shape, we get a new shape as shown in figure 6, a second iteration figure 8 will also results in a similar overall shape of the first iteration but with more details. Successive iterations can be repeated for eternity but practical manufacturing will prohibit the miniaturization process to a certain level.

I F Indentation width

Fig. I Generator for the proposed square loop frarfalization.

111. Mathematical Modeling

The numerical simulations of the antenna system are carried out via the method of moments. Numerical modeling commercial software (NEC) [5] is used in all simulations. The modeling process is simply done by dividing all straight wires into short segments where the current in one segment is considered constant along the length of the short segment, but a

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segment length is restricted to the diameter of the wire. All conductors are assumed perfect. In NEC, the basis and weight functions are different, weight function is being chosen as a set of delta functions with a set of points on the conducting surface. Short straight segments model wires in NEC with the current on each segment represented by three terms a constant, a sine, and a cosine. This expansion provides rapid solution convergence. I t has the added advantage that the fields of sinusoidal currents are easily evaluated in closed form. The amplitudes of the constant, sine, cosine terms are related such that their sum satisfies physical conditions on the local behavior of current and charge at the segment ends. [6]

IV. Numerical Analysis

In order to clearly describe the benefits of the proposed geometry, the first and second iteration of Minkowski fractal geometly [7] are compared to the obtained results of the proposed shape. In Minkowski model a retum to zero (RZ) pulse shape is used as the generator signal, while in our proposed model a non-retum to zero ( N U ) pulse is used. In RZ pulse the initial segment is divided to three segments of equal length and the indentation width must be less than one third of the initial wire length as it will intersect with the one from the adjacent side. On the other hand the N U pulse the initial segment is divided to four equal length segments and the indentation width can be one and half times of the new segment length and the final length of the first iteration can be 3 % of the original length. A method of moments based software is used where all wires were divided into small segments to maintain a constant current over the segment length and delta gap voltage source is imposed over the comer of the shape.

V. Numerical Results

As fractal geometry is introduced to overcome the low values of the input impedance of small loops antennas at high frequencies, the analysis is carried out on a 6cm side length square loop and a wire diameter of 0.001cm. A voltage source of 1 volt and a frequency range from 300 MHz - 6 GHz in free space.

The major withdrawal of the fractal technique is the gain loss, the total gain in the three planes XZ, ZY and XY of the comer-fed square loop, Minkowski fractal, and the proposed fractal s for the Is' and 2"d iterations are shown in figures 2 and 3 respectively. It can be noted that IdB loss in gain compared to the original square loop, but this loss can be tolerated depending on the application that is used to.

The 3D radiation pattems are calculated at a frequency of Z.IGHz, for the comer-fed square loop antenna figure 4 -all figures are 90'rotated to view the antenna- shows that we got the typical radiation of an eight figure in the horizontal plane and an omni directional one in the vertical plane. As expected due to fractalization the direction of the radiation has been reversed and we got an eight figure in the vertical plane for both the first iteration of fractalization in Minkowski fractal figure 5 , and the proposed fractal figure 6. The second iteration proves a consistency of the radiation pattern shape of Minkowski fractal loop figure 7 and the field of the proposed fractal figure 8 but with broader beam.

The current distribution on the wire segments is shown in figure 9 at a frequency of 2.1GHz, the periodicity of the distribution is expected in all cases due to the similarity of the shapes, the current amplitude didn't exhibit an extreme change for all cases, except the rapid increase in the value for the I" proposed fractal loop which is expected due to the resonance and low resistance at 2.1GHz.

The frequency response of the input impedance of both Minkowski fractal and the proposed fractal for a frequency ranging from 300 MHz - 6 GHz as small loops usually used for wireless applications. Figure I O shows the input resistance response, the proposed fractal loop as expected has a higher value at higher frequencies due to the increase in the total electric length for both the I " and Znd iterations. Figure 11 shows the imaginary part of the input impedance, at higher frequencies the values of the reactance are inductive and can easily he compensated by capacitive loads.

VI. Conclusions

Fractals are space-filling geometries that can be used as antennas to effectively fit long electrical lengths into small areas. This concept has been applied to square loop wire

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antenna, the electrical length has been increased and the input resistance became large enough to be easily matched to commercial coaxial feedings. Fractal antennas perform much like square loop antennas, but in a much smaller package. The fractal shape allows the square loop antenna to be effectively reduced in size without significantly impairing performance. Gain loss can be tolerated in a certain degree for the sake of the increase in the input resistance. The NRZ pulse has increased the electrical length of the original loop by 4 times in two iterations, the diameter of 6cm loop is 24cm, and the diameter of 24cm fractal is 96cm occupying the same area, while maintaining almost the same characteristics of the previously introduced generators.

References

[I] Hohlfeld, R., and Cohen, N., “Self-similarity and the geometric requirements for frequency independence in antennae,“ Fractals 7 (1 999), 79-84 httd/classes.vale.edu/math I90a/Fractals/references/ReiHC.html [2] Paul Bourke “An Introduction to Fractals” May 1991 htto://astronomv.swin.edu.au/-~bourke/f~ctal~f~cintro/ [3] Douglas H. Werner, Randy L. Haupt, and Pingjuan L. Werner, “Fractal Antenna Engineering: The Theory and Design of Fractal Antenna Arrays,” IEEE Anfennas and Propagarion Magazine 41,s 1999, pp. 37-59. [4] John Gianvittorio, “Fractal Antennas: Design, characterization an Applications,” Master’s Thesis, University of California, LA 2000. [ 5 ] NEC-WIN Professional vl.la, Nittany-Scientific Inc. 1997 [6] G. J. Burke and A. J. Poggio “Numerical Electromagnetic Code (NEC)- Program description”, January1981 Lawrence Livermore Laboratory. [7] N. Cohen, “Fractal Antenna Applications in wireless Telecommunications”. Professional Program Proceedings, pp. 43-9. Electronic Industries Forum of New England May 1997.

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