Fractal Antenna

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FRACTAL ANTENNAS Introduction to fractal technology and presentation of a fractal antenna adaptable to any

transmitting frequency - The “FRACTENT”

By Werner Hödlmayr, DL6NDJ “Prediction is very difficult, especially if it’s about the future.” (Niels Bohr)

his article is an overview about fractal geometry in general, and fractal electrodynamics in particular, with applications of this new frontier of science. As a practical example in section 3, a series of 3 fractal antennas are described, that have

been built and tested around the centre frequency of 145 MHz by using the first and second iteration of a tent transformation. The new science of fractal electrodynamics offers very interesting possibilities for designing small, broadband, and efficient antennas for restricted space. 1. Introduction 1.1 What are Fractals, What is Fractal Geometry? From the very beginning I would like to point out that the subject is not easy and will not be exhausted by any means in this article. Further reading is necessary for those interested in more detail about self-similarity and Chaos in nature. A huge amount of literature concerning this subject is available. The term “Fractal” means linguistically “broken” or “fractured” from the Latin “fractus.” Benoit Mandelbrot, a French mathematician, introduced the term about 20 years ago in his book “The fractal geometry of Nature” [1]. However, many of the fractal functions go back to classic mathematics. Names like G. Cantor (1872), G. Peano (1890), D. Hilbert (1891), Helge von Koch (1904), W. Sierpinski (1916) Gaston Julia (1918) and other personalities played an important role in Mandelbrot’s concepts of a new geometry. Fractals are geometrical shapes, which are self-similar, repeating themselves at different scales. At the beginning, these mathematical shapes were treated only as abstract curiosities until the1990`s. Then different branches of science discovered their practical applicability. Since that time, the number of applications has exploded. Today, we find fractals in the field of image compression algorithms, weather prediction, integrated circuits, filter design and other disciplines including… antenna design! Let us take an example of a fractal shape. The triangle in Fig. 1, called the Sierpinski triangle, is a common self-similar geometrical figure. It also has been used as a very effective antenna in the GHz frequency range. The geometric construction of such a triangle is simple. One starts with the black equilateral shape and takes afterwards, in different steps, the middle of the sides and generates respectively 3, 9, 27, 81, triangles which are self similar and exactly scaled down versions of the initiating shape. The same procedure can be observed in Fig. 2 where a Koch curve is iterated in 3 steps.

Fig. 1 It is interesting to know something about the “Dimension” of such a fractured

structure. The term “Dimension” in mathematics has different meanings. The common definition is the “Topologic Dimension” in which a point has the dimension 0, a line has the

T

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dimension 1, a surface has the dimension 2 and a cube has dimension 3. In other words, one could describe the dimension of an object with a number of parameters or coordinates.

However, describing a fractal shape this way would give us serious trouble because, for example, fractal curves exist with infinite length (dimension 1) which can generate a surface of finite area (dimension 2). So, how can we define a fractal dimension? Mandelbrot gives a suggestive example. Take an arbitrary unit of length x, and see how many times you are using this unit to cover the entire length of the fractured line. Let us say you used it N times, so the total length of your fractal is N*x. In this case the fractal dimension according to Mandelbrot is:

D0x

log N( )log x( )

lim→

:=0x

log N( )log x( )

lim→

The fractal dimension D is called also the “the crippling factor” of a Fractal and can be written in a more simple form like D

ln N( )ln γ( )−

:= where N is the number of the non-

overlapping copies of the whole and γ is the scaling factor of these copies.

Fig. 2

The value of the dimension is independent of the kind of logarithms used. Let us take an example and calculate D for the above Koch curve. If we start from a straight line having the length of 1, then the first “crippling” will have the length 4*1/3, the second iteration will have the length 4*4 and a scaling factor of 3*3 and so on. If we calculate the value for D, we get D

log 4( )log 3( )

:=

An interesting point is the existence of fractal curves that have space-filling (or self-

avoiding) properties. These curves having a dimension of 1 (line) can generate by self-similarity a surface having dimension 2. or a volume with dimension 3. The curves are called after the names of the mathematicians Peano and Hilbert who discovered them. We’ll see later that also some other curves, like the Tent transformation, have similar properties. [See note in section 7] 1.2 Fractal electrodynamics Soon after scientists discovered the practical aspect of fractal geometry, research began in the field of electrodynamics [2, 3, 4.]. To date, most efforts have been concentrated in understanding the physical process and mathematical background of the interaction between electromagnetic waves and fractal structures. This work produced successful devices for synthesis, modelling and optical regimes of microwave and millimetre waves. However, the main applications were for commercial telecommunication devices. These were compact antennas with properties such as broadband capability, frequency-independence, small size, and multi-frequency operability. Other applications included Fractal miniaturization of passive networks and components, fractal filters and resonators. [5] The next section will show the state of the art for fractal antennas. One salient observation is that researchers focused on the frequency ranges that were commercially interesting. This

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concentration left the amateur bands rather “underdeveloped.” This might explain why few Hams are familiar with fractal antennas. 2. Fractal Antennas It is interesting to note that the first antennas using fractal shapes were done many years before Mandelbrot. Among these are very well known antennas like the spiral and the log-periodic. Starting in the 1950s, Victor H. Rumsey did a comprehensive study about new ways of looking to antennas. In his book, Frequency Independent Antennas, he says, “…the impedance and pattern properties of an antenna will be frequency independent if the antenna shape is specified only in terms of angles.” [6].

The upper statement known also as the “Rumsey principle” derives the self-scalability of a frequency-independent antenna from the Maxwell equations which become wavelength-independent under a coordinate transformation like this:

Xxλ

:=

Yyλ

:=

Zzλ

:= where λ is the wavelength.

With this principle and with the new, highly convoluted irregular shapes of fractal functions in mind, we have the tools for the birth of fractal antennas. The new fractal shapes allow graphical scaling to a degree where multi-frequency operation is possible with very small sizes and multiple operating bands. One of the first “tries” was a fractal antenna based on the Sierpinski (triangle). Current variations include the Sierpinski Monopole, the Sierpinski Dipole, the Sierpinski Carpet patch antenna, and others. Remember, this shape has a dimension between line (1) and surface (2) depending on the degree of iteration or “crippling”. The better a curve fills up a surface, the better for a multi-frequency antenna. Specialists Fig. 3 name this property “Lacunarity.”1

Consider the Sierpinski Monopole in Fig. 3.This is a monopole antenna resonant at frequencies of 0.44, 1.75, 3.51, 7.01 and 13.89GHz having an input resistance of 50Ω. One can easily see the 5 resonant frequencies of the structure by looking to the 5 circles marking the respective triangles.

It is even more impressive to see the current distribution in the areas of resonance shown in Fig. 4. Fig. 5, and Fig. 6 show the far field radiation pattern of a Sierpinski structure for the respective stages of iteration from 1 to 4 [7].

1 Lacuna, -ae means gap or missing part. The term used for Fractal Antennas describes a shape having a characteristic “density of gaps” hence, Lacunarity.

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Fig. 4

Fig. 5

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Fig. 6 For such very small antenna structures, we legitimately can ask about their quality factor (their “Q”). Also, how do these antennas behave when related to L.J. Chu and H. Wheeler´s theory about the fundamental limits on the size of any antenna? From their work, the approximate value of Q for a small antenna is defined as:

Q1

k a⋅( )3:=

where k is the wave number and a is the radius of a sphere enclosing the antenna.

This equation is important because it establishes at resonance, a relation between the Bandwidth and Q of a small antenna independently of its shape. In other words, the efficiency of a small antenna depends on how well the radiant is using the spherical volume having radius a.

Fig. 7 Other studies [14] show that Chu´s theory is valid only for simple TE or TM fields and does not apply for compound fields where we have TE+TM in the right phase. This leaves open the possibility to building small compound antennas with a size below the known limit.

Classical antennas are very far from this volume-filling ideal or a TE+TM field situation. In contrast, numerical results obtained by N. Cohen with a Minkowski fractal loop, show a relative high radiation resistance and low Q in spite of antenna’s size [8, 9, 10]. See Fig. 7.This behavior is confirmed also by my fractal antenna experiments presented in the next section of this article.

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Another consequence of a fractal shape is the existence of a multitude of “places” along the antenna wire where the charges are strongly accelerated hence, producing radiation. It is possibly necessary to reconsider the Chu/Wheeler limits in the light of this type of fractal antenna performance.

Another approach by a research group from Pennsylvania State University [11] was to use a Hilbert curve as a fractal antenna for the VHF/UHF band. This curve, in its 3rd iteration (Fig. 8) has very good surface filling properties approaching to a fractal dimension of 2 (D = 1.834). One conclusion of the group was that the higher the order of iteration of a fractal, the lower its resonant frequency. This conclusion may help overcome some of the fundamental

Fig. 8 limitations of antenna engineering, since fractals do not obey Euclidian geometry.

Many more fractal-shaped antennas exist already on the market and are used in cars, mobile phones and data-transfer devices. Here I mentioned only a handful of approaches to this subject.

3. The “FRACTENT” Experiment The temptation to do my own experiments with fractal antennas was too big to resist! With the hope of discovering and learning more about how nature works, I decided to use a fractal curve called the Tent2. According to my knowledge, this curve has not been used before for building antennas. Like all other initiators of fractal functions, the Tent is a very simple transformation of the “self-avoiding” or “space-filling” curve.

The theory about this kind of curve is presented in detail in [12]. Regarding our application, let us have a look at Fig. 9.

2 The function has the shape of a tent.

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Fig. 9 In the first picture we see the tent having the sides equal to the side of the square hence forming an isosceles triangle. The second picture shows a square containing 9 scaled down rectangles whereas step 2 contains 81 and step 3 contains 729 scaled rectangles. All these areas contain a self-similar tent curve scaled, rotated and folded, generating apparently a chaotic shape. It was a challenge to make an antenna of it! I decided to build 3 models starting with a tedious simulation on my NEC WIN+ antenna simulation program around a centre frequency of 145 MHz. Anyone that has used antenna modeling software probably understands why the simulation was tedious. Imagine typing all those wire coordinates! The 3 models are: a. A dipole, comprising two step1 quadrants. b. A loop, comprising four step1 quadrants c. A loop, comprising four step2 quadrants 3.1 The “FRACTENT” Dipole The picture in the Fig. 10 shows the model of a fractal dipole for the 2m band.

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Fig. 10 One can clearly see the step1 of the fractal shape described in Fig. 9. As we see from the diagrams below, the input impedance is quite low (10Ω) and for this reason I was forced to find a solution of impedance matching to 50Ω. In this case, I used a Gamma match that worked just as desired.

Fig. 11

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Fig. 12 Fig. 13

Fig. 14

The size of the dipole is 2 x 15cm.Compare this with a conventional ½ wave dipole that is about

100 cm long! The elementary square, from which the antenna was derived, is 5 *5cm. From the diagrams above, one can see the characteristic dipole current distribution and radiation pattern. The multi-frequency behavior is observed in Fig. 11. The resonant frequencies are 145MHz, 361.5MHz and 560MHz. These have respective input impedances of 10Ω, 20Ω and 50Ω. It is interesting to note that these frequencies are logarithmically spaced, like those of the Sierpinski Monopole in Fig. 3. Fractal structures sometimes allow truncation of the wire without influencing the resonant frequency of the antenna. In the case of this dipole, I found that the last segment of the structure could be removed without altering the 145 MHz resonance of the structure. 3.2 The “FRACTENT” Loop 1(first iteration) This design in Fig. 15 comprises four squares made of polycarbonate transparent material, each containing 9 self-similar copies of the tent transformation. The size of the elementary square is the same as for the dipole design, 5 x 5cm. The four squares have been assembled on a support having oblique slits that allowed minor adjustments of the resonant frequency. A slight compensation of inductivity was necessary in order to get the right impedance match.

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Fig. 15

Fig. 16

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Fig. 17

Fig. 18

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Fig. 19

The “FRACTENT” Loop 1, in comparison with the dipole, has a much larger Bandwidth and the multi-frequency character is more pronounced. The loop area is now 30 cm on a side. A conventional loop with the same dimensions would have an input impedance of about 2 ohms and a matched bandwidth of about 0.2 MHz. 3.3 The “FRACTENT” Loop 2 (Second iteration) This model was a real craftsmanship challenge for me. It required 200 calibrated drilled holes in a fiberglass support, 200 pins fixed in position, 200 simulated wires and a lot of “cut and try” activity for getting the right starting frequency.

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Fig. 20

Fig. 21

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Fig. 22

Fig. 23

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Fig. 24 Fig. 25 I would like to make one observation regarding the simulation of fractal structures with NEC-type modeling software. When simulating a classical (Euclidian) antenna, one normally uses between 10 and 20 segments to simulate a wavelength. A fractal structure is not linear. Consequently, a larger number of segments have to be selected in order to obtain an accurate computation of antenna parameters. The “FRACTENT” Loop2 gave me great satisfaction during the practical tests that I did in my modest home laboratory. Using the substitution method with two Ground Planes and a calibrated attenuator, I measured a maximum gain of 2.0dBd. The radiation pattern is shown in Fig. 26 and the plot of VSWR versus Frequency in Fig. 27. The input impedance was very much higher than the simulated value in free space. However, a professional laboratory test will follow and show the real values for gain and impedance.

Fig. 26

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Fig. 27 4. Conclusions and Outlook I tried in this article to shed some light on a relatively new field of electrodynamics with very interesting potential applications for building small and efficient antennas. The experiments presented with the three 145MHz antennas, will certainly not stop at this stage. The final aim is to develop a small, multi-frequency, antenna for the HF range based on similar fractal functions.

The attentive reader will note that the spacing of resonant frequencies in a fractal structure is logarithmic, whereas most of the HF amateur bands follow a linear sequence. This means that the antenna structure has to have a scaling which follows the same logarithmic law. To build such an antenna is a challenge asking for a professional simulation program (which allows more than 10,000 segments) and very much perseverance. 5. Acknowledgments

I would like to thank Prof.Dr. Heinz-Otto Peitgen for the permission to reproduce Fig. 9 and the editors of IEEE for the permission given to use Fig. 1-6 from [7]. Many thanks are addressed to Bill Miller, KT4YE (GARDS member) for reading very carefully the draft of this article and for making valuable observations and text corrections. –30- 6. References

[1] Mandelbrot, B.B. (1983): The Fractal Geometry of Nature. W.H. Freeman and Company, New York. [2] Jaggard, D.L. (1990): On Fractal Electrodynamics. In Recent Advances in Electromagnetic Theory, H.N. Kritikos and D.L. Jaggard, editors, Springer-Verlag, New York. [3] Jaggard, D.L. (1991): Fractal Electrodynamics and Modeling. In Directions in Electromagnetic Wave Modeling. Plenum Publishing Co., New York, 435-446 [4] Jaggard, D.L. (1995): Fractal Electrodynamics: Wave Interactions with Discretely Self-Similar Structures. In Electromagnetic Symmetry, Taylor and Francis Publishers, Washington, D.C., 231-281.

[5] FRACTUS, The Technology of Nature, www.fractus.com [6] V.H. Rumsey, Frequency Independent Antennas, Academic Press, 1966 [7] Douglas H. Werner, Raj Mittra, Frontiers in Electromagnetics, IEEE Press [8] Cohen, N. (Summer 1995): Fractal Antennas Part 1 In Comm. Quart., 7-22 [9] Cohen, N. (Summer 1996): Fractal Antennas Part 2. In Comm. Quart., 53-66

[10] Cohen, N. R.G. Hohlfeld, (Winter 1996): Fractal Loops and the Small Loop Approximation. Comm. Quart,. 77-81.

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[11] K.J. Vinoy, K.A. Jose, V.K. Varadan, and V.V. Varadan (November 2000): Hilbert Curve Fractal Antenna: A small resonant antenna for VHF / UHF Applications. The Pennsylvania State University. [12] Heinz-Otto Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals, New Frontiers of Science Springer-Verlag New York, 1992

[13] John D. Kraus, Antennas McGraw-HILL International Editions [14] Dale M. Grimes, "Bandwidth and Q of Antennas Radiating TE and TM Modes." IEEE Transactions on electromagnetic compatibility, vol 37. Nr 2. May 1995 7. Note The Tent transformation (or

function) is defined as follows:

f(x) = a*x for x ≤ 0.5

f(x) = a (1-x) for x > 0.5

For example, if we consider the

angle α having 60° (like the initiator

of the “Fractent”)

then, a = 1.73 and f(x) will be

f(x) = 1.73x for x ≤ 0.5

f(x) = 1.73(1-x) for x > 0.5

Fig. 28 [Editor's Note: Some of the difficulties of modeling fractal structures with highly symmetrical properties can be overcome in advanced NEC software that gives access to the entire set of possible geometry commands. A given structure can be repeated, with the one or more new copies rotated and/or moved within the coordinate system--the GM command. As well, a structure may be symmetrically repeated without adding wires or segments--the GX command. Such models need a good volume of comments (the CM command) to ensure that others can read them easily--or that you can read them a year from now. I have constructed--as one example--monopole arrays with over 100 monopoles and two different, closely spaced, wire-grid planes with only 11 geometry entries. As well, in NEC-2, when the segment length drops below about 8 times the wire radius, one should invoke the EK (extended kernel) command. (The revised algorithms of NEC-4 do not require the command.) However, the command is not available on some entry-level software. LBC]

BRIEF BIOGRAPHY OF AUTHOR I was born 1942 and started to build crystal radios and ship models at the age of ten. I obtained my Ham license, DL6NDJ which came quite late in life in 1980. I received my degree of electrical engineering in 1965, but never built power stations or large electric motors. Instead, my attention was focused in a different direction, on the early developing field of Medical engineering and scientific instruments like spectrophotometers and gas chromatographs. This activity was to become my professional career specialty. I am an active member of the GARDS, an International antenna R&D group.

1

1

f(x)

x α

Other hobbies besides experimenting with antennas are, oil painting (3 years school of art), Lyrics, languages (speak five fluently) and building Stirling engine models. I have been retired since the year 2002 and am very happy to be allowed to apply all of my time to these wonderful things!

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