Design of Sierpinski Gasket Monopole Fractal Antenna for Multiband Applications

DOI : 10.17577/IJERTCONV1IS06080

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Design of Sierpinski Gasket Monopole Fractal Antenna for Multiband Applications

Design of Sierpinski Gasket Monopole Fractal Antenna for Multiband Applications


Department of ECE Department of ECE Anna University PSN college of Engg & Tech. Coimbatore Tirunelveli

Abstract In this paper, a compact design and construction of microstrip feed Sierpinski gasket monople fractal antenna for multiband applications is presented. Self-similarity and space filling property of fractal technology is exploited in the antenna design to reduce the physical size, increase bandwidth and gain. The proposed antenna is designed on FR4 substrate with dielectric constant of 4.6 and the multiband behavior is analyzed through five fractal iterations. This antenna has a size of 82.43mm×37mm and is simulated using CST Microwave Studio suite 2009. The simulated results show that the return loss is less than -10 dB and covers more than 10 bands in the frequency range of 0.6 GHz – 16 GHz for the purpose of investigation possibilities.

structures can give rise to miniaturized wideband antennas having radiation pattern similar to larger antennas.

Multiplication of an antenna size by a factor generally decreases the operating frequency of the antenna by same factor, so antenna geometries and its dimensions are the

main factors determining their operating frequencies [4].

In this paper, Sierpinski Gasket Monopole fractal antenna (SGMFA) for multiband applications is proposed. The structure is designed and feed is through a microstrip line with 50 ohms microstrip line [7, 8].


    Keywords Multiband, Sierpinski gasket, Microstrip feed.


The dramatic development of a variety of wireless applications have remarkably increase the demand of multiband antennas with smaller dimensions than conventional possible. This has initiated antenna research in various directions, one of which is fractal shaped antenna element [1, 2]. First Mandelbrot proposed fractal geometries in 1951[3], which were extensively used in various engineering fields. Fractals are class of shapes which has no characteristic shapes. Fractal is a fragmented geometric shape that can be sub-divided into parts each of which is a reduced copy of the whole. By using fractal, we can describe any real world objects such as clouds, mountains, turbulences and coastlines.

Each fractal is composed of multiple iterations of single elementary shape. The iteration can continue infinitely, thus forming a shape within a finite boundary but of infinite length or area. This shows that fractal shapes are compact, meaning that they can occupy a portion space more efficiently than other antennas. Since the fractal structures are generated by a recursive process, they can produce a very long length or a wide surface area in a limited space. Consequently, fractal

Department of ECE Thiagarajar college of Engg Madurai

  1. Theory

    Mandelbrot offered the following definition: A fractal is by definition a set for which the Hausdorff dimension strictly exceeds the topological dimension which he later retracted and replaced with: A fractal is a shape made of parts similar to the whole in some way. So, possibly the simplest way to define a fractal is as an object that appears self-similar under varying degrees of magnification, and in effect, possessing symmetry across scale, with each small part of the object replicating the structure of the whole.

    A fractal antenna is an antenna that uses a fractal, self-similar design to maximize the length, or increase the perimeter of material that can receive or transmit electromagnetic radiation within a given total surface area or volume.

    Some properties which most fractals have:

    1. Fractals have detail at arbitrarily small scales;

    2. Fractals are usually defined by simple recursive processes;

    3. Fractals are too irregular to be described in traditional geometric language;

    4. Fractals have some sort of self-similarity;

    5. Fractals have fractal dimension.

  2. Principles

    Given a curve, we can transform it into parts ( actually represents the number of segments), where the whole is times the length of each of the parts. The fractal dimension is therefore


    The th iteration of a fractal structure takes more space than a basic segment, but it occupies less space than a filled two- dimensional area.

    Considering the properties of fractal geometry, it is interesting to see what benefits are derived when such structure is applied to the antenna field [4]. Fractals are abstract objects that cannot be physically implemented. Nevertheless, some related geometries can be used to approach an ideal fractal that are useful in constructing antennas. Usually, these geometries are called pre-fractals or truncated fractals. In other cases, other geometries such as multi-triangular or multilevel configurations can be used to build antennas that might approach fractal shapes and extract some of the advantages that can theoretically be obtained from the mathematical abstractions. In general, the term fractal antenna technology is used to describe those antenna engineering techniques that are based on such mathematical concepts that enable one to obtain a new generation of antennas with some features that were often thought impossible in the mid 1980s.

    After all the work carried out thus far, one can summarize the benefits of fractal technology in the following way:

    1. Self-similarity is useful in designing multiband behavior.

    2. Fractal dimension is useful to design electrically small antennas, such as the Hilbert, Minkowski, and Koch monopoles or loops, and fractal-shaped microstrip patch antennas.

    3. Mass fractals and boundary fractals are useful in obtaining high-directivity elements, under-sampled arrays, and low side-lobe arrays.

While Euclidean geometries are limited to points, lines, sheets, and volumes, fractals include the geometries that fall somewhere between these distinctions. Therefore, a fractal can be a line that approaches a sheet. The line can meander in such a way as to effectively almost fill the entire sheet. These space- filling properties lead to curves that are electrically very long, but fit into a compact physical space. This property can lead to the miniaturization of antenna elements. In the previous section, it was mentioned that pre-fractals drop the complexity in the geometry of a fractal that is not distinguishable for a particular application. For antennas, this can mean that the in- tricacies that are much smaller than a wavelength in the band of usable frequencies can be dropped out [5].

Fractal geometry has self-similarity, which can create effective antennas of different scale. This can lead to multiband characteristics of antennas, allowing for similar performance at various frequencies [6]. Traditional wideband antennas (spiral and log-periodic) and arrays can be reanalyzed in view of

fractal geometry to shed new light on their operating principles. That is, a number of new configurations can be used for the antenna elements with good multiband characteristics.


    One of the fractal structures was discovered in 1916, by a Polish mathematician Waclaw Sierpinski [6]. The Sierpinski sieve of triangles is characterized by a certain multiand behavior owing to its self-similar shape, where a monopole antenna based on the Sierpinski gasket has been shown to be an excellent candidate for multiband applications. Generation of the fractal, namely the four steps of conventional Sierpinski triangle iterations, is shown in Fig. 3. The middle triangles are removed from the antenna, leaving three (first step) or nine (second step) equally sized triangles, which are one-half or one-third the height of the original triangle. From an antenna engineering point of view, a useful interpretation of fig.1 is that the dark triangular areas represent a metallic conductor, whereas the white triangular areas represent regions where metal has been removed.

    Fig. 1.The first four stages in the construction of a conventional Sierpinski gasket fractal.

    1. Iterated Function System Used to Generate Fractal Structures

      The iterated function system represents a versatile method for convenient generation of a wide variety of useful fractal structures [6] and is based on application of a series of affine transformations, , defined as

      w (2)

      or as the equivalent


      where a, b, c, d, e, and f are real numbers.

      Hence, the affine transformation, w, is represented by six



      where , , , and control rotation and scaling, while and control linear translation.

      Now, let us consider as a set of affine linear transformations, and as the initial geometry. The new geometry, formed after applying these transformations, and aggregating the results of w1(A),w2(A),,wN(A), can be represented by

      Where W(A) is known as the Hutchinson operator.


      miniaturized to suite different wireless applications. Carles Puente first describes the multiband behavior of Sierpinski Gasket monopole geometry. The geometry of Sierpinski Gasket Antenna up to 5th iteration is presented in Fig. 2.

      The fractal geometry can be obtained by repeatedly applying to each subsequent geometry. For example, if the set represents the initial geometry, then we will have


      An iterated function system generates a sequence that converges to a final image, , in such a way that


      This image is called the attractor of the iterated function system and represents a ÒfixedpointÓ of .

      The iterated function system for generating the Sierpinski tri- angle gasket can be described in the following sequence:


      In this case, the initial set, , is a triangle. Three affine linear transformations are applied to and the results of these trans- formations are then combined together

      Fig. 2. Schematic of proposed Sierpinski Gasket Antenna


      0.6 GHz to 16 GHz

      Substrate Thickness (h)


      Substrate Dielectric Constant (r)


      Loss Tangent ()


      Sierpinski gasket fractal antenna

      Length (L) = 26.73 mm

      Width (W) = 37 mm

      Feed Line

      Length(l) = 55.7 mm

      Width (w) = 1.5 mm

      Ground Plane

      Length (L1) = 55.7 mm

      Width (w1) = 37 mm

      Table1: Design Considerations


      to form the first iteration of a Sierpinski gasket, denoted by (Fig. 1). The second iteration of Sierpinski gasket, , can now be obtained by applying the same three affine transformations to . Higher order versions of Sierpinski gasket are generated by repeating this iterative process until the desired resolution is achieved. After the th iteration the gasket will include triangles, each being a small-scale copy of the whole structure. Iterated function system has proven to be a very powerful design tool for fractal antenna engineers. This is primarily because they provide a general framework for the description, classification, and manipulation of fractals.


      Self-similarity is a property common to many fractals, but in order to become a useful radiator it is necessary for the fractal antenna to fulfill the specific requirements for the desired frequencies.

      The Sierpinski gasket in a monopole configuration has good matching to at the resonant frequencies, a log-period band spacing of 2, and a fairly invariant radiation pattern in all bands, which is very similar to the pattern of a monopole.

    2. Design Procedure

    In order to use same antenna for different applications required the antenna to be a multiband antenna and

    The antenna system consists of two metallic layers with the antenna printed on the top side, over the ground plane at the bottom of the PCB using the above design considerations. In this work, microstrip feeding technique is used. The location of microstrip feed to the monopole is adjusted to match with its input impedance (usually 50 ohm). A commercial software CST (Computer Simulation Technology) microwave studio has been used to model and simulate the proposed antenna.


    The fig. 3. shows the simulation results of fourth iteration. The sierpinski gasket monopole antenna resonates at the three frequency bands one is at 2 GHz with a return loss of -12.3 dB and second at 2.4 GHz with a return loss of -26.1dB and third at 5 GHz with a return loss of -10.2 dB.

    Fig. 3. Simulated return loss

    Different iteration will produce a different amount of Return Loss, dB and bandwidth. The simulated return loss upto fourth iteration are combined and the results are compared. The return loss of fourth iteration shows a good performance compared to other iterations.

    Fig. 4. Combined result: Simulated return loss

    The radiation pattern is a graphical depiction of the relative field strength transmitted from or received by the antenna. Antenna radiation patterns are taken at one frequency, one polarization, and one plane cut. The measured and computed gain patterns of the E and E components of the electric field at the three principal plane cuts ( = 90°, = 0°, = 90°) for the first three operating bands are depicted in Fig. 5. The components are normalized with respect to the maximum total electric field value and expressed in dB. Fig. 5 . shows the radiation pattern of sierpinski gasket fractal antenna (a) At 2 GHz (b) At 2.45 GHz (c) At 5 GHz.

    Fig. 5. Radiation Pattern of 2, 2.45, 5GHz


From result and discussion, it can be concluded that the self similarity in the structure for the 4th iteration Sierpinski Gasket Monopole Fractal antenna is observed to possess multiband behavior. This multiband Sierpinski gasket monopole antenna can cover more than 10 bands in the frequency band of 0.6 GHz-16 GHz. Furthermore, as the no. of iterations increases number of resonant frequencies which gives a multiband performance to the designed antenna structures. The simulated results indicate that the antennas exhibit a good return loss, and multiband frequencies suitable for multiband applications.


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