Fractal Dimension versus Quality Factor: Modelling with NEC. José M. González and Jordi Romeu. Total number of pages: 17

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1 FRACTALCOMS Exploring the limits of Fractal Electrodynamics for the future telecommunication technologies IST Fractal Dimension versus uality Factor: Modelling with NEC Author(s): Participant(s): Workpackage and task: WP1, T1.1 Security: Nature: Abstract: José M. González and Jordi Romeu UPC, EPFL and ROME Public Report Version and date: 1.1, Total number of pages: 17 This report of activities is aimed to check through computer simulations the hypothesis that monopoles with high fractal dimension have lower. For carrying out the simulations the classical freeware Numerical Electromagnetics Code (NEC) has been used. Proprietary software FIESTA (Fast Integral Equation Software for scatterers and Antennas in 3D) has been used too to correlate some results. Keyword list: antenna, fractal, Koch, Miniaturization, Peano, prefractal, quality factor, Sierpinski - 1 -

2 RELATED WP AND TASKS (FROM THE PROJECT DESCRIPTION) WP1. Theory of fractal electrodynamics. Task 1.1: Understanding fractal electrodynamics phenomena 1 EXTENDED ABSTRACT This report of activities is aimed to check through computer simulations the hypothesis that monopoles with high fractal dimension have lower. For carrying out the simulations the classical freeware Numerical Electromagnetics Code (NEC) has been used. Proprietary software FIESTA (Fast Integral Equation Software for scatterers and Antennas in 3D) has been used too to correlate some results. Although the use of fractal monopoles is technologically not feasible, the investigations have been carried out using prefractals. This was not a limitation, fortunately, because the benefit of miniaturization using fractals could be achieved in a few iterations, after which the increase in complexity of the curves do not afford a meaningful increase in the performance of the antennas [Gianvittorio, 2]. The main results are: Computer simulations agree with the fact that greater fractal dimension means lower. Simulations show that at a given frequency additional degree of iteration do not change the behavior of the structure. Topology is also important to reduce the quality factor of a monopole. The higher the number of loops for the same fractal dimension, the smaller the quality factor. Prefractal monopoles are longer than a monopole of the same size, having a small radiation resistance and greater ohmic losses. Consequently they are less efficient than monopoles, and not become suitable for the majority of practical applications. On the other hand greater miniaturization is reached with these structures. Future work aims to: Research on new structures expected to have, as Peano monopoles, the maximum rates of miniaturization (Hilbert monopoles and Sierpinski Carpet monopoles). Using 3D geometries for reaching greater miniaturization ratios. Fractal apertures as structures that increase the radiation resistance. Increasing the pre-fractal number of iterations using FIESTA

3 2 INTRODUCTION A trend in the scientific effort of researchers points towards the miniaturization of communication systems. Antennas are part of these systems and their miniaturization is a fundamental. The fundamental limitation for antenna miniaturization is well known [Chu, 1948] [McLean, 1996] but it is not yet reached. In fact, Hansen [Hansen, 1981] writes about the fundamental limitation that...has only been approached but never equaled, much less exceeded. Miniaturised antennas in modern communication systems means the use of electrically small antennas (always compared with wavelength) and in this case their bandwidth or their quality factor always changes with size. The quality factor is the ratio of the radiated energy from the antenna to its stored energy. For antennas with >>1, the quality factor is the inverse of the fractional bandwidth. The fundamental limit of the quality factor for a linearly polarized small antenna has been determined by McLean and is (1) 1 1 = + (1) 3 3 kr k r being k the wave number associated with the electromagnetic field and a the radius of the smallest sphere that closes the antenna. From equation (1) it is concluded that the smaller the antenna, the higher is its quality factor. Hansen abstracts Chu results writing that... it relates the lowest achievable to the maximum dimension of an electrically small antenna, and this result is independent of the art that is used to construct the antenna within the hypothetical sphere. This hypothetical sphere is the one that has radius a and closes the antenna. In this sense, the art of filling the space surrounding the antenna could help in improving its radiation capability. Mathematical geometries that efficiently fill the space are fractals, so it is worthwhile considering them as good candidates for efficient miniaturized antennas. The objective is, then, the design of a miniaturized (small) antenna with a low quality factor. Although the results presented on this document are referred to wire antennas, characteristics and trends are not limited to these kinds of antennas. Wire antennas have been used for their easy modelling and fabrication. 3 FIRST STEP TO MINIATURIZATION: THE KOCH POLE A first attempt to increase the space-filling efficiency of an antenna is the use of a Koch monopole [Puente, 1998][Puente-Baliarda, 2]. It is constructed from a Koch curve adding a ground plane at one end. The Koch curve is generated after an iterative procedure that maintains the size of the curve growing its length by a factor of 4/3 at each iteration. Figure 1 shows the evolution on complexity of the fractal Koch monopole and figure 2 the computed improvement on its quality factor as the number of iterations increase from (linear monopole) to 3 ( monopole). Evolution of its input resistance and reactance are shown on figures 3a and 3b. As the iteration number - 3 -

4 increases the resonant frequency of the antenna becomes smaller, down the small antenna limit (ka<1), approaching to a limit. At few iterations there is no meaningful variation on impedance and matching. Even though a true fractal is not feasible due to technological limitations, few iterations are needed to reach a behavior where changes are not distinguished. The resulting geometry is a prefractal. K K1 K2 K3 Koch Fractal Figure 1. Construction of the Koch monopole. These simulations have been carried out with a 6 cm high monopole, wire radius of.12 mm and neglecting the ohmic losses of copper. 1 4 K- K-1 K-2 K Fundamental limit 1 1 Figure 2. uality factor of Koch monopoles of several iterations

5 35 2 Input Resistance (Ω) K-4 K-2 K-1 K- Input Reactance (Ω) K- K-4 K-2 K-1 (a) f (GHz) -2 (b) f (GHz) Figure 3. a) Input resistance and b) reactance of the Koch monopole. 4 LOWERING : THE GENERALIZED KOCH POLE The ability of the Koch monopole for making good use of its surrounding space has its origin in the fractal dimension of the structure, which is This fact suggested the use of other Koch monopoles, that we call Generalized Koch monopoles, to verify the straight dependence of fractal dimension and quality factor. These Generalized Koch (GK) monopoles are constructed in the same way that the Koch monopole, using the IFS (Iterated Function System) method, and have different fractal dimensions according to the variant selected. Variant n needs 2n+4 transformations each one with an scale factor of 1/(n+3). The higher the variant, the higher is the number of zigzags that the monopole has. Figure 4 shows the first iteration of some variants. The GK monopoles so constructed are the initiators of the infinite iterative procedure that generates the fractal. The fractal dimension of the GKn monopole (Generalized Koch monopole, variant n) is (2). ( 2n + 4) ( n + 3) log D = (2) log Figure 5 shows the evolution of the fractal dimension of the GK monopoles versus variant number. Maximum dimension for a GK curve is and is found for variant n=1. From variant n=7 on, fractal dimension is below GK, the conventional Koch monopole. Comparing the quality factors of simulated GK, GK1 and GK7 for the same iteration we should verify the relation between quality factor and fractal dimension. In figures 6a are shown the evolution of with for GK-1, GK1-1 and GK7-1, and in figure 6b the evolution of for GK-2, and GK7-2. These are the first and the second iteration of the Generalized Koch monopoles of variations (conventional Koch), 1 and 7. For the frequency ranges where these monopoles are considered small antennas, it was expected that the best happened for GK1 due to its greater fractal dimension. s for GK and GK7 were expected to be similar

6 n+1 h GK h/3 GK1 h/4 GK7 h/1 3 1 GKn 2 h/(n+3) Figure 4. Construction of Generalized Koch monopoles. Fractal Dimension Variant (n) Figure 5. Fractal Dimension of GK monopoles vs. variants GK-2 GK GK-1 GK1-1 GK GK-2 GK7-2 (a) 5 Fundamental limit Fundamental limit Figure 6. a) uality factor of the first iteration of Generalized Koch monopoles of variant (conventional Koch), 1 and 7. b) uality factor for the second iteration of the same structures. (b) - 6 -

7 As expected, the increasing number of iterations in the prefractals shows a decreasing value of the quality factor for the same variant of the Generalized Koch. But in any case the quality factor of GK7 should be worst than GK1 and figure 6a shows better values. Although this should be accomplished for the fractal, technological limitations make only feasible working with prefractals. Hence, for prefractals not only the fractal dimension of the fractal with infinite intricacy is important. Maybe the number of bendings of the antenna affects its radiation, and as the number of bendings is increased so does the radiation. To tackle this question some other variants of Generalized Koch monopoles of different orders and several depths of iteration have been simulated. All of them have been designed having similar number of bendings. Table 1 summarizes for the simulated structures, the fractal dimension, the number of bendings and the wire length needed to fabricate the monopoles. Monopole Iteration Fractal Wire length Bendings Dimension (m) GK GK GK GK Table 1. Simulated GK monopoles. Figure 7 shows the simulated quality factor for these structures. GK7-2 has lower fractional bandwidth than GK-4, and the behavior of GK1-3 and GK2-3 is almost the same, having both of them the best quality factor achievable with any of the Generalized Koch monopoles. As expected fractal dimension of the fractal orders the quality factors of the monopoles but just when the prefractal has reached a degree of detail not resolved for the used wavelength. On the other hand, increasing the number of iterations the resonant frequency is reduced. Fractal dimension has the same effect over the resonant frequency. Though both cases increase the wire length their influence on the reduction of the resonant frequency is not of the same intensity. Table 2 abstracts the first resonant frequency f and wire length for the Generalized Koch monopoles variants, 1, 2 and 7 at resonance. uality factor at =.8 is shown too

8 GK-4 GK1-3 GK2-3 GK Fundamental limit Figure 7. uality factors of GK-4, GK1-3, GK2-3 and GK7-2. Monopole Iteration Heigth First resonance f at Wire length at λ (m) (m) (MHz) =.8 GK GK GK GK Table 2. First resonance, electric wire length at resonance and at =.8 for several GK monopoles. 5 INCREASING FRACTAL DIMENSION TO REACH GREATER MINIATURIZATION Greater miniaturization factors could be achieved using fractal curves (monopoles) that better fill the space that surrounds the antenna. Curves whose fractal dimension is greater than the Koch monopole could be obtained from the Sierpinski gasket, with fractal dimension , or the Peano curve (D=2). The first iterations of a Sierpinski Arrowhead curve and a Peano curve are shown on figure 8a and 8b. The quality factor for several fractal monopoles with fractal dimensions ranging from 1.2 til 2 are shown on figure 9. All are represented on the range of below their first resonant frequency, where all behave as small monopoles (figure 1a). All of the antennas are inscribed in a 6 cm radius circle. As expected, increasing fractal dimension means decreasing the quality factor. Hence, the highest miniaturization factor is reached - 8 -

9 using the Peano Monopole, although it has the typical (for small antennas) inconvenience of its low radiation resistance (figures 1b and 1c). 1 st iteration 2 nd iteration 1 st iteration 2 nd iteration (a) 3 rd iteration 6 th iteration (b) 3 rd iteration Figure 8. a) Sierpinski Arrowhead curves and b) Peano curves Fundamental limit Figure 9. factors of Koch monopole, Generalized Koch monopole, Sierpinski Arrowhead monopole, Peano monopole and a conventional λ/4 monopole

10 -1 (a) Γ (db) f (MHz) Input Resistance (Ω) (b) f (MHz) Input Reactance (Ω) (c) f (MHz) Figure 1. a) Matching frequencies at 5 W of fractal monopoles of different fractal dimensions; b) input resistance and c)input reactance. On figures 11a and 11b the computed quality factors of Sierpinski Arrowhead Monopoles (from first to fifth iteration) and Peano Monopoles (from first to second iteration) are represented. Like other fractal curves, increasing the iteration or the fractal dimension means reducing the quality factor. On figure 11b the computed quality factor for the second iteration of the Peano monopole shows the effect of the resonance near =

11 SA-1 SA-2 SA-3 SA-5 K Fundamental limit (a) 1 5 P-1 K (b) Figure 11. a) uality factors of Sierpinski Arrowhead monopoles from iteration 1 to iteration 5. b) s of Peano monopoles from first to second iteration

12 6 AN EXAMPLE OF MINIATURIZATION: RESONANT ANTENNA To truly verify the degree of miniaturization feasible with fractal structures an example is proposed. Figure 12 shows a scaled representation of several fractal monopoles all of them resonant at 8 MHz. Table 3 summarizes the height of the antennas and their wire length. From figure 12 is really assesed the increase on the miniaturization factor as the fractal dimension of the curve increases. Their quality factors, plotted on figure 13c, were computed on the range where the antennas behave as small antennas. Monopoles built with curves of high fractal dimension have lower quality factors when working as small antennas than a λ/4 monopole but their radiation resistance and bandwidth (figures 13a and 13b) make them not appealing for practical applications. Fractal Height Monopole Iteration Dimension (cm) Wire length at λ R in (Ω) k h 8 MHz λ/ Koch Generalized Koch Variant 1 Sierpinski Arrowhead Peano Table 3. Height and wire length for several fractal monopoles tuned at 8 MHz. Fractal dimension, radianlength and at resonance are shown too

13 λ/4 9 cm Figure 12. Scaled representation of several fractal monopoles resonant at 8 MHz. Input Resistance (Ω) (a) f (MHz) Input Reactance (Ω) (b) f (MHz) (c) 1 1 Fundamental limit Figure 13. a) Input resistance; b) input reactance and c) for several fractal monopoles whose resonant frequency is 8 MHz

14 7 THE INFLUENCE OF TOPOLOGY Fractals are not only characterized by their fractal dimension, but also for their topology. As volumes in a 3D space, and surfaces in a 2D space, quite different topologies could be found for the same fractal dimension. Nevertheless from the point of view of the quality factor only fractal dimension seems to be important. Let us verify this assumption comparing several prefractal monopoles circumscribed in a circle of the same radius (.15 m in the example) and with the same fractal dimension: Sierpinski Arrowhead monopole (SA) Delta-wired Sierpinski monopole (DWS) Y-wired Sierpinski monopole (YWS) Heart-like Sierpinski monopole (HLS) Koch-Sierpinski monopole (K1S) Though quite different in appearance (see figure 14), an infinite iteration of the IFS creates the same fractal, the Sierpinski gasket monopole with fractal dimension m Monopole Sierpinski Arrowhead Monopole Delta-wired Sierpinski Monopole Y-wired Sierpinski Monopole Heart-like Sierpinski Monopole Koch- Sierpinski Monopole Figure 14. Fractal monopoles based on the Sierpinski gasket. The quality factors for these monopoles were computed and are shown on figure 15. All of the fractal monopoles have the same fractal dimension and as a consequence according with the initial hypothesis their factor should be the same. Nevertheless slight differences are shown according to the topology. As the number of closed paths (loops) on the monopole increases the factor is reduced. In this sense, the Sierpinski Arrowhead monopole is the one with the greater factor, as it is constructed with an open wire. The Sierpinski monopole manufactured with a Koch island of first iteration has the greater number of loops and as a consecuence has the smaller factor. Figures 16 a) and b) show input resistance and input reactance for the monopoles compared with a monopole. All of the Sierpinski monopoles have the same first resonant frequency (there are small differences due to variations on the perimeter of the antennas) and behave as multiband antennas

15 SA-3 DWS-3 YWS-3 HLS-3 K1S Figure 15. Computed factors for the Sierpinski curve-based monopoles. Input Resistance (Ω) SA-3 DWS-3 YWS-3 HLS-3 K1S-3 Input Reactance (Ω) SA-3 DWS-3 YWS-3 HLS-3 K1S f (MHz) f (MHz) Figure 16. a) Computed input resistance and b) input reactance of the Sierpinski curvebased monopoles

16 8 REFERENCES [Chu, 1948] Chu, L.J.: Physical limitations of omni-directional antennas, Journal of Applied Physics, 1948, 19, pp [Gianvittorio, 22] Gianvittorio, J.P.; Rahmat-Samii, Y.: Fractal Antennas: a novel antenna miniaturization tecnique, and applications, IEEE Antennas and Propagation Magazine, Feb. 22, Vol. 44, No. 1, pp [Hansen, 1981] Hansen, R.C.: Fundamental limitations in antennas, Proc. IEEE, Feb. 1981, 69, pp [McLean, 1996] McLean, J. S.: A re-examination of the fundamental limits on the radiation of electrically small antennas, IEEE Transactions on Antennas and Propagation, May 1996, 44, (5), pp [Puente, 1998] Puente, C.; Romeu, J.; Pous, R.; Ramis, J.; Hijazo, A.: Small but long Koch fractal monopole, Electronics Letters, 8 Jan. 1998, 34, (1), pp [Puente-Baliarda, 2] Puente-Baliarda, C.; Romeu, J.; Cardama, A.: The Koch monopole: a small fractal antenna, IEEE Transactions on Antennas and Propagation, November 2, 48, (11), pp

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