The Sierpinski Antenna
The Sierpinski Antenna
 
 

Introduction

    The Sierpinski Gasket is one of the oldest fractal shapes. It is named after Waclaw Sierpinski, the Polish mathematician that extensively studied it. The fractal form is composed by 3 triangular sets, being each one itself a Sierpinksi Gasket. A classical procedure for constructing it is burning a triangular hole in the central part of a solid triangular shape, and keep iterating the same procedure over each new triangle formed this way. The object has a fractal dimension D=1.585 and a characteristic scale factor d=2 relating the several gaskets within the structure.

Antenna Description

    The antenna was built by printing a 5 iteration, 8.9 cm tall Sierpinski gasket over a Cuclad 250-GT microwave circuit dielectric substrate. The monopole configuration was chosen first for its simplicity and feeding scheme. The antenna was mounted orthogonally to a 80 cm x 80 cm ground plane and fed through an SMA coaxial connector. Owing to the antenna geometry, one expected current flowing from the feeding vertex to the antenna tips and power becoming radiated, i.e., driven out from the antenna, at those fractal iterations that matched the operating wavelength. Since the antenna contained 5 scale-levels with a characteristic scale-factor of 2 relating all of them, it appeared reasonable to assume that the antenna would perform in a similar way at 5 wavelengths (5 bands) spaced also by a factor of 2.

The Sierpinski Dipole Results

    The input parameters (input return-loss, resistance and reactance) versus frequency are shown in the next plot. Frequency axis is plotted in logarithmic scale to emphasize the log-periodic behavior of the antenna. The figure clearly displays 5 bands corresponding to 5 VSWR minimums equally spaced on the log-frequency axis. The band spacing is, as expected, a factor of  2, i.e., the fractal shape characteristic scale factor. There is a shift on the position of the first band due to the truncation effect (the structure lacks of larger iterations to keep symmetry with respect to the other bands).

    A Sierpinski dipole fed by means of a coaxial taper balun was built to measure the radiation patterns. The following plot displays, from left to right and top to down, the full 3D patterns at 2 GHz, 4 GHz, 8 GHz and 16 GHz, i.e., at each of the four upper bands. The most remarkable feature of that plot is the strong similarity among patterns. The behavior is clearly distinct from that of a classical single-band antenna which modifies its radiating properties when changing the operating wavelength.
 
Sierpinski Antenna Input Parameters 
Measured and computed (FDTD, DOTIG4) input parameters of the Sierpinski monopole. From top to down, input return-loss, resistance and reactange vs. log-frequency. Five bands (return-loss minimums) are clearly distinguished. Bands are spaced by a factor of 2, the same scale-factor relating the several fractal iterations.		 Sierpinski Dipole 3D Measured Patterns 
The meaured Sierpinski Dipole radiation patterns at the four upper bands. Compare the results with the same patterns for a linear dipole.

Conclusions

    The Sierpinski antenna is the first reported example of a fractal shape antenna with a multiband behavior. That is, an antenna that keeps a similar behavior (radiation patterns and input parameters) through several bands. The number of bands and their positions are strongly related to the antenna geometry, which demonstrates the tight link between the fractal nature of the antenna and its electromagnetic behavior.

For further reading ...

..check the references below. Get a broader scope of references at the fractal shape antenna historical review www page.
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(c)  by Carles Puente Baliarda,, 1998. Last update 7th April 1998. Send comments to the webmaster.