Fractal Antennas. Dr. Ely Levine

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1 Fractal Antennas Dr. Ely Levine

2 contents 1. Fractal Concepts 1.1 Definitions 1.2 Classical Dimension 1.3 Fractal Dimension 1.4 Self Similarity 2. Fractal Antennas 2.1 General 2.2 Fractal Arrays (self-similarity) 2.3 Fractal Radiators (Miniaturization) 3. Conclusions

3 1. Fractal Concepts 1.1 Definitions Fractal (Fractus in Latin) means broken or irregular fragment (Mandelbrot, 1975). A family of complex shape with two features: Non-Integer Dimension and Self Similarity. Founded in many natural structures, built mathematically by successive iterations.

4 1.2 Classical Dimension Number of Dimensions was defined in classical geometry by an integer which reflects the power between a typical length and a typical size. Sphere: multiply the radius by m enlarges the perimeter by m**1 area by m**2 volume by m**3

5 1.2 classical Dimensions (2) Thus: perimeter has D=1 Area has D=2 Volume has D=3 General shape (Euclidean geometry): multiply length by enlarges size by number of Dimensions m m**d D

6 1.3 Fractal Dimensions Koch Snow Flake

7 1.3 Fractal Dimensions (2) Koch Snow Flake Every section contains 4 internal sections Iterative Process Enlarge object by 3 Enlarges length by 4 D = log4/log3 = 1.26

8 1.3 Fractal Dimensions (3) Minkovski Curve (Island) D=log5/log3 = (for IW=0.5)

9 1.3 Fractal Dimensions (4) Shore Line (length depends on scale)

10 1.3 Fractal Dimensions (5) Sierpinski Sponge 3 corridors. If we split into 27 sub-squares we find 7 empty and 20 sponges thus 3**D = 20, D=2.72

11 1.4 Self Similarity Modular design, repeating structures Example: Organizational Chart

12 1.4 Self Similarity (2) Example: Sierpinski Carpet

13 1.4 Self Similarity (3) Example: Cantor Series

14 2. Fractal Antennas 2.1 General Kim and Jaggard, 1986 Lakhatakia, Vardan and Vardan, 1987 Lakhatakia, Holter and Vardan, 1987 Kim, Gerber and Jaggard, 1991 Cohen, 1995, 1996 Werner, Haper and Werner, 1999 Baliardy, Romeu and Cardama, 2000 Gianvittorio and Rahmat Samii, 2002

15 2.1 General (2) Topics Deterministic Fractal Arrays Miniaturization of small antennas Reflection / Diffraction from surfaces (targets) Achievements Scientific interest Limited Performance (2-3 iterations) arrays with small spacing thinned arrays with low sidelobes reduced size elements Still in R&D phase

16 2.2 Fractal Arrays (By Self Similarity) Deterministic Fractal Array P AF p(ψ) = Π GA(δ ψ) p=1 GA(ψ) = array factor of generating sub-array δ = scale of expansion factor p = number of iteration

17 2.2 Fractal Arrays (2) Example (Cantor Array): Array illuminations: V1(x) =101 V2(x) = V3(x) = Array Factors: AF1 (ψ) = cos (ψ) AF2 (ψ) = cos (ψ) cos (3ψ) AF3 (ψ) = cos (ψ) cos (3ψ) cos (9ψ)

18 2.2 Fractal Arrays (3) Example (Cantor Array): Array Factors (p=1,2)

19 2.2 Fractal Arrays (4) Example (Cantor Array): Array Factors (p=3,4)

20 2.2 Fractal Arrays (5) Example (Sierpinski): Filling Factors: P=1 3/4 = 0.75 P=2 9/16 = 0.56 P=3 27/64 = 0.42 P=4 81/256 = 0.32

21 2.2 Fractal Arrays (6) Example (Sierpinski): Spacing (d= λ/2)

22 2.2 Fractal Arrays (7) Example (Sierpinski):

23 2.2 Fractal Arrays (8) Features of Fractal Arrays Algorithms based on the compact product presentation are capable of rapid computations Self Similarity enables frequency independent multi-band behavior. Systematic approach to thinning while keeping low sidelobes

24 2.3 Fractal Radiators (Miniaturization) Miniaturization Small but Long Antennas Loops can be enclosed into finite surface with arbitrary long length Small Antennas that occupies the same volume as their Euclidean counterpart but much longer Novel miniaturization techniques

25 2.3 Fractal Radiators (2) Example: Koch Monopole/Dipole L=h(4/3)**p L= length, h=size, p=iteration

26 2.3 Fractal Radiators (3) Example: Koch Monopole L= 25 cm h=6 cm

27 2.3 Fractal Radiators (4) Example: Koch Monopole (input R)

28 2.3 Fractal Radiators (5) Example: Koch Monopole (input Z)

29 2.3 Fractal Radiators (6) Example: Koch Monopole (experiment) # iteration f(ghz) Rin(ohm)

30 2.3 Fractal Radiators (7) Example: Tree Dipole

31 2.3 Fractal Radiators (8) Tree Dipoles f(mhz)

32 2.3 Fractal Radiators (9) Example: Tree Dipoles (experiment) Koch/tree 3D tree # iteration f(ghz) f(ghz)

33 2.3 Fractal Radiators (10) Example: Loop (Minkovski) Frequency is not proportional to L High cross polarization Narrow bandwidth

34 2.3 Fractal Radiators (11) Example: Microstrip (Koch) L0=19.5 cm, L2=12.0 cm Reduction factor 12/19.5=0.6

35 2.3 Fractal Radiators (12) Microstrip array with close spacing

36 3. Conclusions Arrays Using Self similarity of sub arrays Rapid calculations of patterns Efficient thinning with low sidelobes Multi-frequency (in doubt)

37 3. Conclusions (2) Elements 2-3 iterations at the most Resonant frequency by factor 1.4 Size reduction by factor of 0.7 Q goes up, bandwidth goes down Cross polarization goes up Potential for wide scan phased arrays

Study of Fractal Antennas and Characterization

Study of Fractal Antennas and Characterization Department of Physics M.M.P.G. College, Fatehabad (Haryana) Abstract-Fractal geometry involves a repetitive generating methodology that results in contours