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==Fractal antennas, frequency invariance, and Maxwell's equations==
==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<ref>Rumsey, V.H. "Frequency Independent Antennas", IRE International Convention Record, Vol. 5, Part 1, pp.114-118, 1957</ref> 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 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<ref>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</ref> that self-similarity was one of the underlying requirements to make antennas invariant. This analysis, based on Maxwell's equations, showed this to be a subset of the more general set of self-similar conditions. Hence 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. Mushiake's earlier work on self complentarity was shown to be limited to impedance smoothness, but not frequency invariance.
A different and also useful attribute of some fractal element antennas is their self-scaling aspect. In 1957, V.H. Rumsey<ref>Rumsey, V.H. "Frequency Independent Antennas", IRE International Convention Record, Vol. 5, Part 1, pp.114-118, 1957</ref> 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 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<ref>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</ref> that self-similarity was one of the underlying requirements to make antennas 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==
==Other uses==

Revision as of 18:39, 21 June 2013

An example of a fractal antenna: a space-filling curve called a Minkowski Island (ref 1)
A planar array 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

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

Antenna elements (as opposed to antenna arrays) made from self-similar shapes were first created by Nathan Cohen [3] then a professor at Boston University, starting in 1988. Cohen's efforts with a variety of fractal antenna designs were first published in 1995(reference 1) (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 (see ref.1), recent independent studies show advantages of the fractal element technology in real-life applications, such as RFID[4] and cell phones.[5]

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."[6][7] However, in the last few years, dozens of studies have shown superior performance with fractals,[8][9] 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[10] 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 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[11] that self-similarity was one of the underlying requirements to make antennas 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][12] 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'.[13]

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.[14][15][16]

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

Notes

  1. ^ a b Nathan Cohen (2002) "Fractal antennas and fractal resonators" U.S. patent 6,452,553
  2. ^ [1]
  3. ^ Nathan Cohen short biography
  4. ^ Ukkonen L, Sydanheimo L, Kivikoski M (26–28 March 2007). "Read Range Performance Comparison of Compact Reader Antennas for a Handheld UHF RFID Reader". IEEE International Conference on RFID, 2007. pp. 63–70. doi:10.1109/RFID.2007.346151. ISBN 1-4244-1013-4. {{cite conference}}: Unknown parameter |booktitle= ignored (|book-title= suggested) (help); Unknown parameter |laysummary= ignored (help)CS1 maint: multiple names: authors list (link)
  5. ^ 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. doi:10.2528/PIERB0903080.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  6. ^ Best,S, (2003). "A Comparison of the Resonant Properties of Small Space-Filling Fractal Antennas" (PDF). IEEE Antennas and Wireless Propagation Letters. 2 (1): 197–200.{{cite journal}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)
  7. ^ Best,S, (2002). "On the Resonant Properties of the Koch Fractal and other Wire Monopole Antennas" (PDF). IEEE Antennas and Wireless Propagation Letters. 1 (1): 74–76.{{cite journal}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)
  8. ^ Singh, Ashutosh K. (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): 51–57. doi:10.2478/s13531-012-0040-2. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
  9. ^ [2]
  10. ^ Rumsey, V.H. "Frequency Independent Antennas", IRE International Convention Record, Vol. 5, Part 1, pp.114-118, 1957
  11. ^ 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
  12. ^ Nathan Cohen (2007) "Fractal antennas and fractal resonators" U.S. patent 7,256,751
  13. ^ Cohen,N.,"Body Sized Wide-Band High Fidelity Invisibility Cloak", FRACTALS, 20,227-232 (2012)
  14. ^ Lancaster, M.; Hong, Jia-Sheng (2001). Microstrip filters for RF/microwave applications. New York: Wiley. pp. 410–1. ISBN 0-471-38877-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
  15. ^ Pourahmadazar, J.; Ghobadi, C.; Nourinia, J.; Shirzad, H. (2010). Mutiband Ring Fractal Monopole Antennas For Mobile Devices. New York: IEEE. pp. 863–866. doi:10.1109/LAWP.2010.2071372.{{cite book}}: CS1 maint: multiple names: authors list (link)
  16. ^ Pourahmadazar, J.; Ghobadi, C.; Nourinia, J.; (2011). Novel Modified Pythagorean Tree Fractal Monopole Antennas for UWB Applications. New York: IEEE. doi:10.1109/LAWP.2011.2154354.{{cite book}}: CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)

References

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