Fractal Monopoles: A Comparative Study

Fractal Monopoles: A Comparative Study Vladimír Hebelka Dept. of Radio Electronics, Brno University of Technology, 612 00 Brno, Czech Republic Email: xhebel02@stud.feec.vutbr.cz Abstract In this paper, we introduce a fractal planar monopole, which consists of self-similar ellipses. These monopoles are compared with the Koch monopole and the Sierpinski gasket as representatives of fractal planar monopoles. All antennas were numerically modeled, fabricated, measured and mutually compared. Comparisons show that elliptic fractal monopoles exhibit a narrower bandwidth, a lower gain and larger dimensions compared to the Koch antenna and the Sierpinski gasket. 2 Fractal Elliptic Monopoles The basic geometry of a fractal elliptic monopole is shown in Figure 1. The antenna is composed of three ellipses (the same geometry) differing in the values of radii (a different dimension). Each ellipse can resonate in a different frequency which helps us to compose a wideband or multiband antenn 1 Introduction Layout of a planar antenna plays an important role in reducing the size, improving the bandwidth and influencing the efficiency of an antenna structure. Exploitation of fractal motives is a frequently used technique of the design of small and efficient antennas. The fractal was defined by Mandelbrot to classify complex geometric structures with a non-integer dimensionality. Fractals have typically a form of self-similar patterns, i.e. patterns of the same shape but different dimensions. Fractal geometry can accurately characterize many non-euclidean features of nature including the length of a coastline, the branching of trees, shapes of mountains, clouds and galaxies. Fractal geometry can find its applications in many areas of science and engineering including the antenna design [5]. In this paper, attention is turned to a critical comparison of different planar fractal monopoles fed by a microstrip transmission line. The antennas were designed for the operation in the first resonance at 7.5 GHz. The designed antennas were simulated in CST Microwave Studio to verify their functionality. The numerical results were verified by measurements. The comparisons comprise a novel elliptic monopole consisting of self-similar ellipses. This elliptic monopole is compared with a planar Koch monopole and Sierpinski one. Monopoles are compared from the viewpoint of impedance and radiation properties. In this paper, all the antennas were designed for the substrate Arlon 25N (the dielectric constant ɛ r = 3.38, and the height of the substrate h = 0.7 mm). Figure 1. Fractal elliptic monopole: the front side (left), the back side (right). The synthesis of the elliptic monopole is shown in Figure 2. During the synthesis, we continuously add shorter but wider ellipses to the bottom of the original ellipse. The dimensions of the designed elliptical monopole are given in Ta 1. The length l 1 of the ellipses is decreased following the ratio 0.65 n, where n is the number of the iteration. The width w 1 of the ellipses is increased following the ratio 1.34 n, where n is the number of the iteration. Computed and measured frequency responses of the return loss of the elliptic fractal monopole consisting of two ellipses (the second-iteration antenna) are depicted in Figure 3. The simulated values of S 11 are lower than 10 db in the interval from 6.60 GHz to 8.76 GHz. The impedance bandwidth is 28.8%, and the computed gain is 3.68 dbi at the frequency 7.5 GHz. 13

is the number of iteration. The synthesized antenna is depicted in Figure 5. -30 db -30 db Figure 4. Directivity patterns of the second-iteration elliptic fractal antenna: E-plane (left), and H-plane (right). Computed and measured frequency responses of the return loss of the modified second-iteration elliptic monopole are depicted in Figure 5. The simulated values of S 11 are lower than 10 db in the interval from 6.49 GHz to 10.09 GHz. The corresponding impedance bandwidth is 48%, and the computed gain of the antenna is G = 3.54 dbi at the frequency 7.5 GHz. Obviously, the modified elliptic fractal antenna exhibits a wider bandwidth than the original antenna, but the gain is maller. The radiation pattern of the modified antenna is practically identical with the pattern of the first antenn Figure 2. Synthesis of the geometry of the elliptic fractal antenna; current distributions for iterations of the elliptic fractal monopole. Figure 4 shows simulated and measured radiation patterns of the second-iteration antenna at the frequency 7.5 GHz. Measured and computed patterns are in good agreement. The differences are probably caused by asymmetric currents excited due to an insufficient symmetrization (a larger reflector should be used). Figure 5. Frequency response of the return loss of the modified second-iteration elliptic fractal antenn In the following section, we are going to compare parameters of elliptic fractal monopoles with the parameters of the Koch monopole and the Sierpinski monopole. 3 Koch Monopole Figure 3. Frequency response of the return loss of the seconditeration elliptic fractal antenn Now, the obtained results are going to be compared with a fractal elliptic antenna which ellipses follow the ratio 0.85 n for the length and the ratio 1.54 n for the width, where n Synthesis of the Koch monopole is depicted in Figure 6 [1], [2]. Zero iteration of the Koch monopole is a straight transmission line of the length l1. In the first iteration, we divide the straight line into three equal parts. In the central part, we create two equal segments of the length l1/3, and form an equilateral triangle. We apply the same procedure on each linear segment in the next iteration step. The iterative transformation does not increase the length of the antenna, but the physical length of the antenna wire 14

is incrased by a factor (4/3) n, where n is the number of the iteration. The length of the zero iteration monopole corresponds to the quarter of the wavelength on the substrate l1 = 7.01 mm. -30 db -30 db Figure 8. Simulated directivity patterns of the Koch monopole: E-plane (left), and H-plane (right). 4 Sierpinski Monopole The Sierpinski gasket is constructed by subtracting a central inverted triangle from the main triangle [3]. After the subtraction, three equal triangles remain on the structure, each one being half the size of the original one. We can apply this process to the remaining triangles to create the next iteration. The synthesis of the Sierpinski gasket is shown in Figure 9. The small gap between the main triangle and smaller triangles is 0.2 mm. Figure 6. Synthesis of the Koch monopole; current distributions for iterations of the Koch monopole. Simulated values of S 11 are lower than 10 db in the interval from 6.64 GHz to 10.06 GHz. The corresponding impedance bandwidth is 45.63%, and the computed gain of the antenna is 4.42 dbi at the frequency 7.5 GHz (see Figure 10). Simulated results clearly show multiband resonances. Figure 7. Frequency response of the return loss of the second iteration of the Koch monopole. Figure 9. Synthesis of the Sierpinski gasket; current distributions for iterations of the Sierpinski gasket. 15

VOL.3, NO.4, DECEMBER 2012 The antennas are matched at frequencies 0.26, (1) where c is the speed of light in vacuum, h is the height of the largest gasket, δ is the log period (δ 2), and n is the number of the iteration. The simulated values of S11 are lower than 10 db in the interval from 6.29 GHz to 11.27 GHz. The corresponding impedance bandwidth is 66.45%. The computed gain of the antenna is 4.26 dbi at the frequency 7.5 GHz. The self-similarities in the antenna geometry cause the multiband behaviour of the antenn Return loss [db] 0-10 -20 Figure 12. Prototypes of studied antennas. -30 5 Summary -40 Measured Simulated -50 0 10 20 30 Frequence [GHz] 40 In the paper, we introduced a planar monopole antenna, which is composed from self-similar ellipses. Properties of this antenna were compared with conventional fractal monopoles the Sierpinski gasket and the Koch monopole. All antennas were designed for the operating frequency 7.5 GHz and 50 Ω microstrip feeding. Dimensions of studied antennas are summarized in Table 1. The comparison shows that the Koch monopole consumes the minimum surface of the substrate. Electrical parameters of studied antennas are summarized in Table 2. Obviously, Elliptic antennas exhibit narrower bandwidth and lower gain compared to the Koch monopole and the Sierpinski gasket. Directivity patterns of all the antennas are similar. 50 Figure 10. Frequency response of the return loss of the second iteration of the Sierpinski monopole Figure 11 shows simulated and measured radiation patterns of the second iteration of the Sierpinski antenna at the frequency 7.5 GHz. Measured and computed radiation patterns are similar. The differences are probably caused by asymmetric currents excited due to an insufficient symmetrization (a larger reflector should be used). Table 2. Electrical parameters of studied antennas. -30 db -30 db Figure 11. Directivity patterns of the Sierpinski gasket antenna: E-plane (left), and H-plane (right). Bandwidth Gain [dbi] Elliptic 1 28.8 % 3.68 Elliptic 2 48.0 % 3.54 Koch 45.6 % 4.42 Sierpinski 66.5 % 4.26 The radiation patterns of the antennas were measured in an anechoic chamber. Comparing simulated and measured patterns, the influence of asymmetric currents is obvious. The simulated and experimental frequency responses of the return loss were in a good agreement. In Table 1, dimensions of all studied antennas are summarized. Photographs of measured prototypes of studied planar monopoles are shown in Figure 12. Acknowledgements Table 1. Dimensions of studied antennas. x y l l1 w1 Elliptic 1 18.00 9.20 9.00 2.16 Elliptic 2 18.00 9.20 9.30 1.84 Koch 15.11 11.56 7.01 0.60 Sierpinski 15.20 9.91 8.60 - The presented research was financially supported by the Czech Science Foundation by the grant 102/12/1274. 16

References [1] BEST, S.R. On the resonant properties of the Koch fractal and other wire monopole antennas. IEEE Antennas and Wireless Propagation Letter, 2002, vol. 1, no. 1, p. 74 76. [2] JAMIL, A., YUSOFF, M.Z., YAHYA, N. Small Koch fractal antennas for wireless local area network. In Proceedings of the International Conference on Communication Systems ICCS 2010. Singapore: National University of Singapore, 2010, p. 104 108. [3] PUENTE-BALIARDA, C., ROMEU, J., POUS, R., CARDAMA, A. On the behaviour of the Sierpinski multiband fractal antenn IEEE Transaction on Antenna and Propagation, 1998, vol. 46, no. 4, p. 517 520. [4] WADELL, Brian C. Transmission line design handbook. Boston: Artech House, 1991, xvi, 513 s. ISBN 08-900-6436-9 [5] FALCONER, K. Fractal geometry: mathematical foundations and applications. New York: Wiley, c1990, xxii, 288 p. ISBN 04-719-2287-0. 17