Sierpinski Fractal Loop Antenna

Performance of Sierpinski Fractal Equitriangular Loop Antenna Rabindra K. Mishra1, Deepak R. Poddar2, Rowdra Ghatak3, 1 Sambalpur University, Jyoti Vihar, Burla, Orissa – 768018 (India) 2Jadavpur University, Kolkata, W. Bengal (India) 3 NIST, Palur Hills, Berhampur, Orissa – 761008(India) E-mail:{[email protected]; [email protected]; [email protected] }

Abstract Fractals are a whole new class of geometry that is attracting In this paper Sierpinski loop antenna have been analyzed. the antenna engineers to redesign some of the existing The increased usages of the wireless systems have forced structures for obtaining multiband operation. Fractal the antennae designers to try out new shapes to have geometry is a mathematical language used to describe multiband characteristics. A step towards it is to incorporate complex shapes and has self-similar characteristics. This the fractal geometry in the antenna designing. The starting property helps to generate these shapes with repeated point is the equitriangular loop antennae, which has been iterations. Unlike Euclidian geometry, the dimensions of modified to generate Sierpinski loop fractal antenna. This fractal shapes are not integers. The “scale symmetry” has has resulted in a more emphasized multiband operation, the implication that objects look the same on many different which is of great engineering importance so far as using the scales of observation. The Sierpinski fractal is shown in antennae for various wireless systems is concerned. figure 1. The Sierpinski fractal starts from an equilateral triangle. The transformation is accomplished by dividing the triangle in to four smaller equilateral triangles each having 1. Introduction half the area of the original triangle and then the middle triangle is removed. The process is repeated for the three Within the wireless domain there is need for multiple outer triangles, each time deleting the middle triangle. As resonance radiators. For achieving this design goal we have we keep applying the iterations then the area will approach used the fractal geometry for defining the antenna shapes. zero and will become so after an infinite iterations. Benoit B. Mandelbrot first introduced this branch of The dimensions of such fractal shapes can be calculated by mathematics in the year 1975 [1]. Sierpinski fractal Loop box dimension estimation method [6]. According to this antennas studied here can be broadly classified in to two after first iteration there are three triangles and is half the major categories namely the electrically small and the log 3 electrically large loops. The fractal loop antenna has been original size, so the dimension is which is compared with an equitriangular antenna because of the log 2 inherent triangular shape of the fractal loop under approximately 1.58. The fractal dimension measures the consideration. Sierpinski planar antennas have been quality of a fractal’s geometrical features such as analyzed and compared with the Euclidian counterpart [2], augmentation, detail and intricacy. [3], [4]. The planar structures have been analyzed in detail whereas very preliminary results are available [5] for the non-Euclidian loops. The Sierpinski loop with perimeters 2, 5 and 0.5 is analyzed using method of moment based WIPL-D software. L Iterations up to second stage are reported here for the above- mentioned structures. The fractal geometry emphasizes multiband operations in each of the iteration, reported in this L/2 paper.

2. Fractals Figure 1. Sierpinski Fractal Loop1st iteration

1 coordinate points were then used to generate the fractal structure, which was analysed, by using a MoM based 3. Sierpinski Loop Antenna Design software WIPL-D. Sierpinski fractal is a well-known curve. It starts from a 4. Current Distribution equilateral triangle. In the next step the mid point of the sides are found. These points are then connected by means The current distributions for curved wires have been given of wires. The radius of the wire has been chosen to be a in [8] which explain the relation between current gauge no. nine wire i.e. 1.4478 mm. The perimeters of distribution and the corner of a curved wire. The current 666mm, 1665 mm and 166.5 mm corresponding to 2, 5 distribution gets perturbed greatly if a current null coincides and 0.5 are considered. The first two cases come under the with the corner, while it remains same if maxima fall on a electrically large loop antennas and the last one is an corner. The current distribution in a Sierpinski loop antenna electrically small loop. The structures are shown in figure2. after first iteration is shown in figure 3.

Y Figure 3. Current distribution in a Sierpinski triangle loop antenna after 1st iteration for top feed.

By symmetry the total horizontal current is nullified. So the radiation is in the x-y plane. The corresponding time varying fields are cancelled due to the current in opposite directions. (c) (d) Figure 2. Sierpinski loop antenna (a) top driven 1st iteration (b) 2nd iteration top driven 5. Results and Discussions (c) 1st iteration base driven (d) 2nd iteration loop driven The equitriangular loop antenna for top driven is resonant at 1.45 GHz and the bottom driven case is resonant at the same All structures are analyzed for both top and base driven frequency with difference in the return loss values which – cases. The loop antenna in assumed to be on the y-z plane. 18.34 dB in the former case and –9.20 dB in the later case as shown in the figure 4. This shows that the top driven loop Polygonal loop antennas have been studied a long before in antenna for the triangular loop performs better as per the the 1980’s [7], which concluded that the top driven one findings reported in [7]. performs better than the bottom driven one. It was reported that the design criteria for a triangular loop antenna is the The first iterated Sierpinski equitriangular loop antenna, as flare angle, which in this case is 60. This work also shown in figure 5, is resonant at two frequencies namely at investigates the effects of feeding on S-parameters of the 1.5 GHz and 3 GHz with the S11 values being –38.17 dB antenna. and –17.5 dB respectively for the base driven case. For the top driven case the resonant frequency is same but the S11 During the initial stages of design the wavelength was values are –19dB and –17 dB respectively. chosen to be 333 mm, which corresponds to a frequency of 900 MHz. Then the perimeter of the equilateral triangle and For the second iterated Sierpinski loop antenna there are the Sierpinski triangle was taken to be twice and five times three resonances for both top and base driven cases as that of the wavelength. Another structure was also tried out shown in figure 6. These are 2.9 GHz, 3.0 GHz and 6.0 which was half of the wavelength. Then the coordinates of GHz. The corresponding return loss values are –14 dB, the antenna were determined using an algorithm for the -13.32 dB and –20 dB for the top driven case. For the base generation of the Sierpinski triangular loop. These driven case the values are –21 dB, -20 dB and -13.32 dB respectively. The S11 value for the third frequency is same 2 for the third resonant frequency. The above results pertain to the perimeter of 666 mm. For the perimeter of 1665 mm the equitriangular loop is resonant at 0.6 GHz with a return loss value of –13.61 dB for the top driven case and –8.0 dB for the base driven case as shown in figure 7. The equitriangular loop, the first iterated Sierpinski and the second iteration Sierpinski loop is compared for the top driven case as shown in figure 8. It is found that the equitriangular loop is resonant at 0.6 GHz and the first iterated Sierpinski is resonant at 0.6 GHz and 1.1 GHz with return loss being-16.04 dB and –14.5 dB. The second iteration Sieprinski loop is resonant at three distinct frequencies. The first two occurring at 1.2 GHz and 2.4 GHz respectively and the third is at 3.5 GHz with S11 Figure 5. Comparison of S11parameter for Sierpinski loop values being –17 dB, -14.5 dB and –9.44 dB respectively. antenna, 1st iteration, for top and bottom feeds. In figure 9 the same structures as mentioned in the above paragraph are compared for base driven conditions. The first iterated Sierpinski loop is resonant at 0.6 and 1.1 GHz with S11 values of –12.5 dB and –15.12 dB respectively. The second iterated Siepinski loop antenna is resonant at 1.2 GHz and 2.4 GHz with the return loss values of –24.10 dB and –16.0 dB. The third resonance occurs at 3.5 GHz with S11 value being –11.0 dB. This shows that the Siepinski antennas emphasizes on the multiband nature of the fractal antenna.

For the Sierpinski loop antenna with perimeter of 166.5mm the resonance occurs at 6.6 GHz for the top driven case with return loss values of –44.98 dB and the base driven antenna is resonant at the same frequency with the S11 value of –12 dB as shown in figure 10. The top driven frequency has a Figure 6. Comparison of S11parameter for Sierpinski sharp defined frequency where as the base driven antenna equitriangular loop antenna for top and bottom driven nd has much wider 3dB bandwidth. designs for 2 iteration.

Figure 4 Comparison of S11parameter for equitriangular loop antenna for top and bottom feeds. Figure 7. Comparison of S11parameter for 2nd iteration quitriangular loop antenna for bottom and top feed designs for the loop perimeter being 5 times the wavelength.

3 design procedure has been adopted based upon mapping the simple loop antenna in fractal domain, which has resulted in the multiband character. It is seen that the top driven structures be it a simple equitriangular loop or Sierpinski loop the performance is better as compared to the base driven ones. 6. Conclusion

Sierpinski loop antennas for first and second iteration have been investigated for different perimeters. They have been in turn compared with the corresponding zeroeth iteration structure that is the equitriangular loop. It is seen that the Figure 8. Comparison of S11parameter for 2nd iteration first iteration is resonant at two resonant frequencies and the Equitriangular, Sierpinski 1st iteration and 2nd iteration loop second at three different resonant frequencies. The antenna for top feed designs for the loop perimeter being 5 Sierpinski loop antenna for all the iterations tried out is times the wavelength. better than the equitriangular loop antenna for both the top and base driven cases. The multiband characteristics of Sierpinski loop antenna emphasize its simultaneous usage in various wireless systems. 7.Acknowledgement

We are grateful to Electronics and Electrical Communication Engineering Department of IIT Kharagpur for providing us WIPL-D software, under QIP programme of the department, for carrying out the work.

Figure 9. Comparison of S11parameter for Equitriangular, References Sierpinski equitriangular loop antenna for bottom driven designs of 1st 2nd iteration. [1] B. B. Mandelbrot, The Fractal Geometry of Nature, San Francisco, CA: Freeman, 1983. [2] C.Puente-Baliarda, J.Romeu, R.Pous, A.Cardama, “On the Behavior of the Sierpinski Multiband Fractal Antenna”, IEEE Trans. on Antennas & Propagation, vol.46, no.4, pp.517 - 524, April 1998. [3] J. Parron, J. M.Rius, and J.Romeu, “ Analysis of a Sierpinski Fractal Patch Antenna Using the Concept of Macro Basis Function,” IEEE International Symposium on Antennas and Propagation Digest, Vol.3, Boston, Massachusetts, pp 616 -619, July 2001. [4] Jaume Anguera, Enrique Martínez, Carles Puente, Carmen Borja, and Jordi Soler, “Broad-Band Dual- Frequency Microstrip Patch Antenna With Modified Sierpinski Fractal Geometry”, IEEE Transactions On Antennas And Propagation, Vol. 52, No. 1, January 2004. st Figure 10. Comparison of S11parameter for Sierpinski 1 [5] Ahmad Karami and Ibrahim Karami, “ Fractal wired iteration antenna for top feed and bottom feed designs for Loop Triangle Antenna”, proceedings of IEEE-ICECS the loop perimeter being 0.5 times the wavelength. 2003. [6] H.O.Peitgen, H.Jurgens and D.Saupe, Chaos and Here the Sierpinski loop antenna has been compared with a Fractals: New Frontiers of Science. New York, Springer Euclidian loop antenna and it is seen that the Sieprinksi loop Verlag, Inc., 1992. is resonant at different frequencies in accordance with the [7] Takehiko Tsukiji and Shigehumi Tou, “ On Polygonal Loop size of the loop. We see that the structure results in Antennas,” IEEE Transactions On Antennas And enhancement of effective electrical length and a special Propagation, Vol.AP-28, pp 571-575, No.4, July 1980.

4 [8] P.A.Kennedy,“Loop Antenna Measurement,” IRE Transactions Antennas Propagation, Vol.AP-4, pp 610, Oct 1956.