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The Electrically Small Lim it of Fractal ElementAntennas

Robert G . Hohlfeld and Nathan ohenCen ter fo r Comp u ta t iona l Sc ience ,Boston Universi ty , andFrac ta l An tenn a Sys tems , Inc.

Fractal Antenna Systems, Inc. ,and Metropoli tan College,Boston Universi ty

I. In t roduct ionSmall antenna s pose severe practical restrictions in tradeoffs of directivity,bandwidth, and radiative efficiency. Work by Chu 1948), in particular, prescntedan analytical basis for these restrictions. However, recent work on fractalelements (Cohen and H ohlfeld, 1996) suggests that such apparently complexstructures perform unexpectedly well (Puente, 1997) when their s izes are ljustelectrically small i.e. f r less than a wavelength in size, hut not in the limit ofzero size). Here, we demonstrate that the applicability of Chu s analysis appearsto he overgeneralized when complex (fractal) antennae arc considered. We derivea spherical harmonic expansion of the current distribution of a simple fractal loop,and show that this falls outside of the expansion s truncation assumption of Chu.This affirms that Chu s analysis is valid only for a very specific exam ple of aconventional monopole or dipole which is electrically small. Presently-usedelectrically small guidelines are thus poor at predicting the performance ofcomplex antenna shapes, such as fractal elements in the just electrically smalllimit.

11. Analy t i ca l Approa chThe essential elements of the approach of Chu lies in an expansion, in sphericalharmonic functions, of sou rces and fields within a spherical reference surfa ce ofradius a and the spherical harmonic expansion of an outgoing spherical wavefield. physical solution is identified by imposing matching conditions on thefields at r = a. Chu develops equivalent circuits for the outgoing fields and forthe fields interior to r = and notes that the circuits for the region interior to r = Iare not unique, i e here are m ultiple distinct antenna configurations that yield thesame outgoing wave. The analysis of Chu focuses attention on a tractable class ofantenna s by placing restrictions, notably that the electromagnetic field has only atraveling wave and that the antenna is resonant. These cond itions restrict the classof antenna geometries considered, allowing f or the existence, hut not the example,

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of more complex antenna structures which would not be limited to the findings ofthe analysis.In particular, when Chu s methodology i s applied to a simple current distribution,only a single nonzero term is obtained from a spherical harmonic expansion and,for example, the classic results for gain and radiation resistance of a small currentloop arc immediate. The se results have been noted with respect to fractalantennas see, for example, Puente 1997 , as a first-order guideline inunderstanding performance limitations.Such an application of the classic electrically small limits can be demonstrated tobe inappropriate, even in the context of Chu s methodology. Although thestructure of the outgoing wave may be the same, premature truncation of thespherical harmonic series describing the fields interior and exterior to r = a willlead to misleading analytical results for the gain and efficiency of an antenna. Putsimply, improper assessment of the spherical harmonic expansion of the currentdistribution will lead to a derivation of the Poynting vector, and thus radiativeproperties, which may he substantially inaccurate. It becomes important to showthat an existing complex antenna, in this case a fractal loop, has a many-termedspherical harmonic expansion, to show that there is analytical justification forexcluding such classic electrically small limits.

111. Computation and ResultsWe modeled the lowest resonance of the MI2 antenna (Cohen and Hohlfcld1996 in NEC as a choice of a representative fractal element. T his antennaincorporates a rectangular motif to second iteration, upon a square loop. Theresulting current distribution has two current maxima and minima, much as seenwith a square loop of 1 wavelength circumference. W e do not expect theperformance of MI2 to be optimal as a small antenna, hut it is a well-studiedexample that can illustrate that naive application of sm all antenna limits is invalid.The simulation of MI2 was done at a frequency of 30 MHz and the antenna is0.142h x 0 1341 in size. T hree to five segments were used for each wire in MI2to obtain a high-resolution representation of the current distribution.We determined spherical harmonic coefficients using Mathernatica 4 .0 (Wolfram,1999 by numerical integration of the current distribution, generated by NEC forM12. A high precision numerical integration was performed and standardnumerical tests were made to insure the accuracy and stability of the result withinthe discretization of the current distribution imposed by NEC modeling.Although our current distribution in this example is confined to the plane O rd2,we developed the full spherical harmonic expansion for use in fully three-dimensional problems in future work.

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5 1 15 20Figure I : Magnitude o complex spherical harmonic coeficientsfor the currentdistribution of the A412 antenna modeled in NEC. A representative set of valuesis shown I = 2 open square; = 8 open triangle; = 16 star: 1 = 20, star: 1 =22 box, = 24 diamond The ractal structureo the MI2 antenna generatessignficant spherical harmonic coeficients to much higher order than aconventional antenna.

The mag nitudes of the spherical harmonic coefficients are plotted in Figure 1. Itis clear that significant amplitud es are maintained to very high I and m values,which must be included in a com plete analysis of the small antenna limit of afractal element antenna.We may also address the efficiency of MI2 in qualitative terms. MI2 ischaracterized by two strong current maxima in close proximity, butelectrically distant from each other (Cohen and Hohlfeld, 1996 . This contributesto the generation of strong fields characteris t ic of MI2, and we note is also thetype of electrical configuration explicitly excluded in Chus analysis.

IV. ConclusionsFractal element antennas have significant structure on multiple s ize scales.Therefore, a spherical harmonic expansion of their current distributions, andcorrespondingly their fields, will have significant higher orde r terms, even if theantenna is comparable to or somew hat smaller than a wavelength in size. For thisreason, a naive application of the small antenna limits deriving from Chus workis inappropriate and must be extended by inclusion of the higher order terms.

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While fractal element antennas must still approach zero gain and efficiency in thelimit of zero size , they are not constrained by the performance of simple loopantennas as they approach that limit, as shown by our earlier empirical andmodeling work.With the insight gained from this work on the spherical harmonic expansion offractal element antenna current distributions, we can now proceed to a morefundamental understanding of the operation of fractal element antennas. This willthen lead to understanding of the reasons for high cfficicncy and gain of somefractal antennas and the development of design principles for these antennas.

V. ReferencesChu, L. J., Physical Limitations of Om ni-Directional Antennas, J.Appl . Phys.19 1163-1175.(1948).Cohen, N. L., and Hohlfeld, R. G., Fractal Loops and the Small LoopApproximation, Communicat ions Quar ter ly 6 77-81, (1996).Puente, C., Fractal Antennas,Ph.D. Dissertation at the Dept. of Signal Theoryand Communications, Universitat Politkcnica de Catalunya, June 1997.Wolfram, Stephen, The Mathematica Book, 4tbEdition, Cambridge UniversityPress: Cambridge, (1999).

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