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Fractal antenna

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Title: Fractal antenna  
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Subject: Self-complementary antenna, Antenna efficiency, Antenna boresight, Electrical length, Spurious emission
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Fractal antenna

A fractal antenna is an antenna that uses a fractal, self-similar design to maximize the length, or increase the perimeter (on inside sections or the outer structure), of material that can receive or transmit electromagnetic radiation within a given total surface area or volume.

Such fractal antennas are also referred to as multilevel and space filling curves, but the key aspect lies in their repetition of a motif over two or more scale sizes,[1] or "iterations". For this reason, fractal antennas are very compact, multiband or wideband, and have useful applications in cellular telephone and microwave communications.

A good example of a fractal antenna as a spacefilling curve is in the form of a shrunken fractal helix.[2] Here, each line of copper is just a small fraction of a wavelength.

A fractal antenna's response differs markedly from traditional antenna designs, in that it is capable of operating with good-to-excellent performance at many different frequencies simultaneously. Normally standard antennas have to be "cut" for the frequency for which they are to be used—and thus the standard antennas only work well at that frequency.

This makes the fractal antenna an excellent design for wideband and multiband applications. In addition the fractal nature of the antenna shrinks its size, without the use of any components, such as inductors or capacitors.

Log periodic antennas and fractals

An example of a fractal antenna: a space-filling curve called a Minkowski Island[3]

The first fractal "antennas" were, in fact, fractal "arrays", with fractal arrangements of antenna elements, and not recognized initially as having self-similarity as their attribute. Log-periodic antennas are arrays, around since the 1950s (invented by Isbell and DuHamel), that are such fractal arrays. They are a common form used in TV antennas, and are arrow-head in shape.

Fractal element antennas and performance

A planar array fractal antenna

Antenna elements (as opposed to antenna arrays) made from self-similar shapes were first created by Nathan Cohen[4] then a professor at Boston University, starting in 1988.

Cohen's efforts with a variety of fractal antenna designs were first published in 1995[5] (thus the first scientific publication on fractal antennas), and a number of patents have been issued from the 1995 filing priority of invention. Most allusions to fractal antennas make reference to these "fractal element antennas".

Many fractal element antennas use the fractal structure as a virtual combination of capacitors and inductors. This makes the antenna so that it has many different resonances which can be chosen and adjusted by choosing the proper fractal design. This complexity arises because the current on the structure has a complex arrangement caused by the inductance and self capacitance. In general, although their effective electrical length is longer, the fractal element antennas are themselves physically smaller, again due to this reactive loading.

Thus fractal element antennas are shrunken compared to conventional designs, and do not need additional components, assuming the structure happens to have the desired resonant input impedance. In general the fractal dimension of a fractal antenna is a poor predictor of its performance and application. Not all fractal antennas work well for a given application or set of applications. Computer search methods and antenna simulations are commonly used to identify which fractal antenna designs best meet the need of the application.

Although the first validation of the technology was published as early as 1995,[5] recent independent studies show advantages of the fractal element technology in real-life applications, such as RFID[6] and cell phones.[7]

One researcher has stated to the contrary that fractals do not perform any better than "meandering line" (essentially, fractals with only one size scale, repeating in translation) antennas. Specifically quoting researcher Steven Best: "Differing antenna geometries, fractal or otherwise, do not, in a manner different than other geometries, uniquely determine the EM behavior of the antenna."[8][9] However, in the last few years, dozens of studies have shown superior performance with fractals,[10][11] and the below reference of frequency invariance conclusively demonstrates that geometry is a key aspect in uniquely determining the EM behavior of frequency independent antennas.

Fractal antennas, frequency invariance, and Maxwell's equations

A different and also useful attribute of some fractal element antennas is their self-scaling aspect. In 1957, V.H. Rumsey[12] presented results that angle-defined scaling was one of the underlying requirements to make antennas "invariant" (have same radiation properties) at a number, or range of, frequencies. Work by Y. Mushiake in Japan starting in 1948[13] demonstrated similar results of frequency independent antennas having self-complementarity.

It was believed that antennas had to be defined by angles for this to be true, but in 1999 it was discovered[14] that self-similarity was one of the underlying requirements to make antennas frequency and bandwidth invariant. In other words, the self-similar aspect was the underlying requirement, along with origin symmetry, for frequency 'independence'. Angle-defined antennas are self-similar, but other self-similar antennas are frequency independent although not angle-defined.

This analysis, based on Maxwell's equations, showed fractal antennas offer a closed-form and unique insight into a key aspect of electromagnetic phenomena. To wit: the invariance property of Maxwell's equations. This is now known as the HCR Principle. Mushiake's earlier work on self complementarity was shown to be limited to impedance smoothness, as expected from Babinet's Principle, but not frequency invariance.

Other uses

In addition to their use as antennas, fractals have also found application in other antenna system components including loads, counterpoises, and ground planes. Confusion by those who claim "grain of rice"-sized fractal antennas arises, because such fractal structures serve the purpose of loads and counterpoises, rather than bona fide antennas.

Fractal inductors and fractal tuned circuits (fractal resonators) were also discovered and invented simultaneously with fractal element antennas.[1][15] An emerging example of such is in metamaterials. A recent invention demonstrates using close-packed fractal resonators to make the first wideband metamaterial invisibility cloak at microwave frequencies (US patent 8,253,639). Peer reviewed publication may be found in the scholarly journal 'FRACTALS'.[16]

Fractal filters (a type of tuned circuit) are another example where the superiority of the fractal approach for smaller size and better rejection has been proven.[17][18][19]

As fractals can be used as counterpoises, loads, ground planes, and filters, all parts that can be integrated with antennas, they are considered parts of some antenna systems and thus are discussed in the context of fractal antennas.

See also


  1. ^ a b Nathan Cohen (2002) "Fractal antennas and fractal resonators" U.S. Patent 6,452,553
  2. ^
  3. ^ Reproduction of a description of the first fractal element antenna, created in 1988
  4. ^ Nathan Cohen short biography
  5. ^ a b Cohen, N. (Summer 1995). "Fractal Antennas". Communications Quarterly: 9. 
  6. ^ Ukkonen L, Sydanheimo L, Kivikoski M (26–28 March 2007). "IEEE International Conference on RFID, 2007". pp. 63–70.  
  7. ^ N. A. Saidatul, A. A. H. Azremi, R. B. Ahmad, P. J. Soh, and F. Malek (2009). "Multiband Fractal Planar Inverted F Antenna (F-Pifa) for Mobile Phone Application". Progress In Electromagnetics Research B 14: 127–148.  
  8. ^ Best,S, (2003). "A Comparison of the Resonant Properties of Small Space-Filling Fractal Antennas". IEEE Antennas and Wireless Propagation Letters 2 (1): 197–200.  
  9. ^ Best,S, (2002). "On the Resonant Properties of the Koch Fractal and other Wire Monopole Antennas". IEEE Antennas and Wireless Propagation Letters 1 (1): 74–76.  
  10. ^ Singh, Ashutosh K.; Kabeer, Reneez A.; Ali, Z.; Singh, V. K.; Shukla, M. (16 November 2012). "Performance analysis of first iteration koch curve fractal log periodic antenna of varying flare angles". Central European Journal of Engineering 3 (1): 51–7.  
  11. ^
  12. ^ Rumsey, V.H. "Frequency Independent Antennas", IRE International Convention Record, Vol. 5, Part 1, pp.114-118, 1957
  13. ^ Mushiake, Y. (March 1949). "Origination of self-complementary structure and discovery of its constant-impedance property". J. IEE Japan (in Japanese) 69 (3). p. 88. 
  14. ^ Hohlfeld R, Cohen N (1999). "Self-similarity and the geometric requirements for frequency independence in Antennae". Fractals 7 (1): 79–84. doi:10.1142/S0218348X99000098
  15. ^ Nathan Cohen (2007) "Fractal antennas and fractal resonators" U.S. Patent 7,256,751
  16. ^ Cohen,N.,"Body Sized Wide-Band High Fidelity Invisibility Cloak", FRACTALS, 20,227-232 (2012)
  17. ^ Lancaster, M.; Hong, Jia-Sheng (2001). Microstrip filters for RF/microwave applications. New York: Wiley. pp. 410–1.  
  18. ^ Pourahmadazar, J.; Ghobadi, C.; Nourinia, J.; Shirzad, H. (2010). Mutiband Ring Fractal Monopole Antennas For Mobile Devices. New York: IEEE. pp. 863–866.  
  19. ^ Pourahmadazar, J.; Ghobadi, C.; Nourinia, J.; (2011). Novel Modified Pythagorean Tree Fractal Monopole Antennas for UWB Applications. New York: IEEE.  


  • Cohen, N. (Summer 1995). "Fractal Antennas". Communications Quarterly: 9. 
  • US Patents: 6104349; 6127977; 6140975; 6445352; 6452553; 6476766; 6985122; 7019695; 7126537; 7145513; 7190318;7215290; 7256751.
  • Cohen, N.,"NEC Analysis of a Fractalized Monofilar Helix in the Axial Mode", ACES Conference Proceedings, April 1998, p. 1051

External links

  • How to make a fractal antenna for HDTV or DTV
  • Literature Review of the Fractal antennas - November 2014
  • Tri-Band Fractal Antennas for RFID Applications
  • CPW-Fed KOCH SNOWFLAKE Fractal Antenna for UWB Wireless Applications
  • Video of a fractal antenna monopole using fractal metamaterials
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