08_Design Formula for Sierpinski Gasket

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2. Resonant Frequency of Sierpinski Gasket Monopole Antenna 2.1 Modification of Existing Relation
Figure 1 shows the antenna geometry of a Sierpinski gasket planar monopole. Typically, such antennas exhibit a log-periodic spacing of the resonant frequencies, as well as an increase in the impedance bandwidth at higher bands [3]. It is interesting to note that the band number, n, and the iteration, k, are interchangeable. In other words, band zero and the 0th iteration correspond to the fundamental resonance of the antenna. The first band and the first fractal iteration correspond to the first log-periodic resonant frequency. Therefore, after the first fractal iteration, two resonant frequencies are available: the fundamental and the first log-periodic frequency. This is valid for other higher fractal iterations. In general, the antenna radiates in two distinct bands for a single fractal iteration, three bands for a double fractal iteration, and so on. Puente et al. proposed the following formula [3] for the resonant frequency corresponding to an operating band:
Design Formula for Sierpinski Gasket Pre-Fractal Planar-Monopole Antennas
1 R. K. Mishra , R. Ghatak1 , D. R. Poddar3
'Department of Electronic Science, Berhampur University Bhanja Bihar, Berhampur 760007, Orissa, India E-mail: prof.r. k.mishra@
T
removed or altered without significantly altering the multi-band behavior of a Sierpinski monopole antenna, as mentioned in [5]. Therefore, from an engineering prospective, it is important to determine the resonant frequencies in different Sierpinski modes when other parameters are available. Also, it is equally important to find the side length of the generating triangle when all other parameters are available. We propose an improved closed-form expression for the resonant frequencies, and then use it for finding a closed-form expression for the side length of the generating triangle. fEEE Antennas and Propagation Magazine, Vol. 50, No. 3, June 2008
2Department
of Physics, The University of Burdwan Burdwan, India
3Department
of Electronics and Telecommunication Engineering, Jadavpur University Kolkata, India
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In Equation (4), p =- - 0.230735, and x is one, as explained earlier. The frequencies predicted by this formulation are tabulated in Table 1. However, this will be valid for a Sierpinski gasket under the assumption of perfect magnetic wall boundaries. The following can be noted for the resonant frequency in the of imperfect magnetic walls, which is more realistic: case 1. An equilateral triangle is a zeroth-iteration antenna. For such an antenna, Helszajn and James suggested [6] accounting for an effective value of the side length.
Abstract
An improved formula for determination of resonant frequencies in different Sierpinski modes is developed from other existing formulas, using physical and analytical reasoning. The superiority of the proposed formula over other existing formulas is verified by comparing the results with those from experiments and simulations. A design formula for the side length of the generating Sierpinski triangle is also proposed by algebraic rearrangement of the formula for the resonant frequency. Keywords: Planar antennas; fractal antennas; monopole antennas
Because Equation (1) has a coefficient of 0.26 as compared to 0.245 in Equation (3), the results from these expressions will slightly differ. It is to be noted that in Equation (1), the scale factor is raised to the power of the band number, but in Equation (3), it is raised to the iteration of the Sierpinski gasket geometry. This similarity in form between the expressions of Equations (1) and (3), and the validity of Equations (2) and (1) for the lower and upper bands, implies that a judicious use of Equations (1) and (2) can approximately predict the resonant frequency. Since the conversion of the formn of Equation (2) into Equation (1) occurs in the second band, i.e., for n =I1, it will be appropriate to Ž1 use Equation (1) for n-: and Equation (2) for n = 0. Accordingly, we can use the following expression for the prediction of the resonant frequency:
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Tom Milligan Assoc. Editor, Antenna Designer's Notebook 8204 WNest Polk Place Littleton, CO 80123 USA Tel: +1 (303) 977 7268 Fax: +1(303) 977 8853 E-mail: TMilligan@