VHF COM r-. l lT'\"TCATTO~S 4/2001

AI/gel Vilaseca, HB9SL V

The Fractal Antenna

A Revolutionary idea

Fractals are mathematical tuncnons.This concept is normally connectedwith computer-gen erat ed image s,Fractals became extremely popular inthe eighti es for two reasons: the fir-stfactor was the metcurtc dcvctupment ofmicro-informatics and the rapid in­crease in th e com p utationa l powe r ofmt cro-cnmputcrs! T he sccon u conu-lb­lIting factor was the work of the world­r ennwn cd French ma th ematician ,Benuit .\ l a ndclh rol. wh o had made hisnam e working in the field of fractals.

I.Order and cha os

Fractals were or igmailly just a math­ematical con cept . Sinn : then. they ha vebecome a tool which allows splendidIlTlagt:s to be generated by computers. Butthars nOI all they also offer a generalconcept of the universe. a unifying princi­plc of science. thanks to Benoit Mandel­bro t and num erou s other researchers.

Frac tals are linked to chaos theory . Tn19R8. a book was publi shed called"CHAOS" by James Gle ick , and becamevery popular. Tt includes several astonish­ing propositions which have since be­come common know ledge : like the oneabout the butterfly in China wh ich fl apsits wings and, through a chain of chaotic

events , subsequentl y causes a tornado inthe Caribbean,

Fractals combine order and disorder in aunique way ! Ifwe look attentiv ely at onecit' the illust rations ill Fig. J above, wecan sec that they arc disordered within asmall frame and yet ordered on a largescale. A fracta l can be chaotic , as in thesediagrams. or de terministic. "Determini s­tic" here means that they are composedon the bas is of a single patt ern (forexample a tr iang le or a rectang le), whichis called the generator , and that they arcbased on successively changing scales(wh ich art: ca lled "repetitions"] . T henum ber ofrepetitions can he infinite . Youneed at least two o f them to be able tospeak o f self-similarity. Examples offractals art: the Koc h curve , Fig. 2 Oil thele ft, the Sicrpinski triangle . or the "cy lin­der head gas ket" devel oped by Mandel­brot . Fig . 2 below .

The term "fractal" was coin ed by BenoitMandclbrot. It means that it can heproven .that fracta l images have ll O com­plete d imens ions i.e. they arc incomplete.This doc s not happen in Eu clidian geom­etry! In classical geometry, a point isdefin ed as having no dimensions. A linehas one dim ension, a plane has two andfinally a volume has three dimensions . Sofar so good. It can be demons trat ed tha t agiven fractal. for exampleJog 4 / Jog 3 = 1.2618 dimensions

or again

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log 2 f log 3 --' 0.6309 dimensions! Thispoint can he understood intuitively. If. forexample. we take a Sieminski triangle,this is an area. so it lias two dimensions.For the first repet ition. •1 triunulc isremoved from the who le and hen: aga inwe lind onec more three smaller trinn­gtcs. b UI these also st ill hnve two dimcn­sions. If we repeal this procedure illstages , the areas become smal ler andsmaller. Aller an infinite number of rep­ctittons. the trian gle will be math: lip ofan innum erab le number or areas whichwill he infinitely small. TIll' entire areawill he approach ing zero. hut ncvcnhc­less the Sicrpinski triangle appl'ars topossess an area. It can be mnthcnuuicallydemonstrated that the numb er ufdimcn­sions is less than two.

2.Fro m theory to pract ise

powerful antenna s (with regard to thesize . radiatio n efficicncv . antenna cffi­cicncy and band widlli Of slight sidelobes). There arc some ante nnas whichcan just not be lIesignc.1without fractals .

Fracta l antenna:'> arc a new area of devol­opment only -I or 5 ycurs old: for thisreason, there nrc \ cry lew descriptions 10be found . ri g. ,~ shows two industrialdc\cloPIIK't1IS which have already beenproduced.

This area of dcvcloprncur looks verypromi sing aga inst the background of anenormous expansion in wireless commu­uication . The developers arc wo rkingaway at it intensively uud protecting theirown developments through putcnts. How­O:\Cf. liulc by little it is. becoming po"..i­blc 10 gl'l at the secrets of even th is newtype oj' antenna . I his article is already abeginning,

The subject of this artic le is the upplica­ttou nf the theor y of fractals to the designof antennas. The objective is nor simp lyto discuss a modern subjec t hUI 10 dcm­onstrare that there really is some potentialhere. Resea rchers have already invcsu­gated this path extensively and haveestablished that the theo ry of fractalsmakes it possib le to create 1l111d t more

3.Characteristics of a frac tal

To obtain some idea of what a fracta l is,try and answer the following question:how lone is the coast of Corsica? Thatseems k; be an ..:asy quest ion. All youneed 10 do is find a map of Cors ica andpick up a ruler' - right? Wrong!

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Fi~ 2 : l<:xlulIJllcs of the Koch Curve, tilt' xpier pin sk l Trillllglt· an d III(' CylinderHead G as ket rtcveluped by Mnn rtcfb r ut

You take a large-scale map and a pai r ofcornpusscs. Working from the scale ofthemap . scr the points, c.g. 10 50 kill andthen measure, approximately, the lengthof the Corsican coast line. Now repeal theprnt:ess once more, but this time set thecompass po ints tn 5 kill. Since the coast­line is very Jagged and consis ts of nu mer­ous promontories and bays. the va lue1!OW obta ined will be greater than the firstresu lt. [f Vi e usc more and more detailedmaps and smaller ant! smaller measure­ment steps , the value at each stage will begreater than the preceding value. Wcwould thus finally arrive. in theo ry , at an

infinite length.

It could be said that th is theore tica lcontour ofCorsica is a fracta l diagram: itgives us a coastline with all infinitenumber o f indentations. T hey all look thesame, irrespect ive of whether they extend1 km. or 10 or 100 km . lucidcntall y, thesame also applies ttl a cloud. There arcalso computer programs whic h app lyfrnctnl theory to the synth etic generationof landscap es. And take it fro m me, I C.111

confidently assert that these syntheticlandscapes (fig. 4) look more " realistic"than reality!

Fi~ 3 : [ X311l1lIl'S o f industr ial devetopm cnts IIr Fractal Antennas, from Si~ t t'mas

Ra dianres S.A.

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Fi~"' : A Fracu lVanMr lilJpe

Lundveapc, produced wlrh soh wa rc catted "Bryce" [ro m H..

And so we come III il characteristic whichloan he extremely valuable for an antennait alw'lys behaves the same way withwaves of varying wave lengths, irrespcc­live of whether the wuvclc nuth is 10 emJ m or [0 m. In other words~a wide ban,;antenna.

If we usc a dipole which has a trad alform instead of {he standard rectilinearform. we can establish that it is made upof a large number of sections. each ufdifferent length. and that each will radiatein u certain part Ill' the spectrum as wellas possible (Fig. 5).

On the othe r hand, everyo ne knows thatthe shortest dis tance between two poi ntsis a straigh t line . Hut it has now likewisebeen recognised that the longest distanceis the fractal linc! In th is way, we obtaincurved radiating clements which result invr ry much more compact antennas withless losses and a higher degree of effi­cicncy. Fig. 6

Another J....l'isible way of lI ~i n~ the theoryof Fractals is the log pe riodic antenna: itis well-known for its extensive bandwidth. In II log period ic ntucuua. thetheory of fractals is applied, not just tothe individual clements 11\11 10 the entirearchitecture of the antenna. Although ithad already been invented many years

Fi~ 5: A Fra ctal dipole dev elopedmi Jl~ lh c- Koch curve by ~a t ha n

Cohen - .'l ll l-t.

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Fig 6: The development of the Fracta l form of a n~cta n g l t" T he r ight hand sideslum's the iteration process.

<I\!O. befo re there was anv talk or Fractals.it can be recognised th:lt it has on e ortheir essential characteristics : self-simi­lar ity . T his is one charac teristic propertywhich immediate ly stri kes any one "....husees a tructal image . What this observersees is one clement repea ted infinitely,from the tiniest detail right up \0 theoveral l VIew . Like the mannclade jar onwhich there is a label with a photo of alittle girl holding a marmclade jar, 0 11

winch there is a pho to of a little g irl ­etc ..

It can be demonstrated math ematically

Fig 7: A Fr act al a nte nna developedfor a mobile phone by Nnthau Cohen ­NlIR

that. for an antenna to have good broad­band functioning, it m ust have a poin t o rsymmet ry and it must be self- similar. 1.1..: .,its appearance must al ways be the same,irre spective o f the scale . III other words,it must obey the law of fractals.

Var ious wel l-known fracta l represents­tions, such as the Sieminsk i triangle. havebeen put forward as an tenna s. The com­mon po int of all these struc tures IS thengood behaviour III re lation \0 their rcac­tivity, which usually means we call dov,.. ithuut any matching. nus implies asimplification of des ign, greater reliab il­ity and lower losses.

It is possi ble to create radiating fractalstructure s for the UHF range as printedc ircuits . In the ultra-s hort wave range aswell. thank s to the good ratio of thewave length to the mec hanical length withthis approach, printed circu it structurescan he use d (Fig. 7) .

Th ere are structures drafted in one piece ,b ut there are also an tennas cons isting or ametallic plan e. in which holes hav e beenstamped in acc orda nce with fractal disrri­bution .

The concept of fractal antennas can beapp lied to indiv idual antennas or to an­tenna arrays. In the ease of arrays, the

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Fi~ 101: I{il:,h l - a Koch cu rve, Left - A Sierplus ki fri :III/..:l l' developed as a Ilr infl'llclrcult h~' T he Car les Puente Pfll)fl'c hnic at Barcelona Univ ersit y

distribution of elements call he regu lar. asill ce rtain radar antennas, or irregular, a ...in certain radio telesco pe ar rays, winchwere built by linking together alreadyexisting instruments, which arc d istrib­uted all over the world.

A fractal array can combine the robust­ness of an irregular array with the effi­cicncy or a regula r array, and this withonly a qua rter o f the elements.

In the case of individual antennas we C3 11.

as an exa mple, take the Koch curve or theSicrpin...ki tri angle (Fig. R).

These representations, e.g. o f the coiledshape, combine inductances and capaci­tances, which removes tile need for anymatching circuit and extends the trans­mission hand. in that it improv es the gain.Antennas have already been put on themarket which can advantageo usly replacethe rubber antenna.. of porta ble eq uip­mcnr. Some have bee n created usingsmall printed circuits which are intc­grated into the housing of <J mobilephone.

It is true that fractal antennas allowminiuturi..arion, but we should not gobeyond a reduct ion in size hy a factor of2 to 4. Otherwise there is a danger ofsacrificing the yield. And, just as with allothe r antennas, it will naturally not bepossible to have smal l d imensions, large

band widths and high gain simuhanc­ouslv . But fmctal antennas call comecloser to this than other tYre~ and thor­oughly success ful compromises can beobtained between these three chnmctcr iv­tics.

Any type o r antenna can be Iracrulisc d:monopole . dipole. helix, primed anten­nas, etc .. The clements are termed inaccord ance with a fractal represcnnuion.OJ" gaps arc introduced into the antenna.

In the case of fl at antennas such asprinted circuits. which arc ninde up ofseveral clements. the coupling betweenthem is generally the main factor whichrestr icts their power i.c. gain and theirimpedance matching. This problem isavoid ed ill the case of the fractal ,1l1 IC ll­nas.

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Expcr lmcutnl Ira ctali satinn Of :'l

loop

Nathan Cohen. N lI R, had the followingexperience: he manufactured a frame an­tcnna with all edge length of J5 cent ime­trcs. in the form of a rect ang ular printedcircuit. l ie then manufactured three otherfrac talised arucnn as with the same dimcu-

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Fig I): Experiment a l re sult s 011 a Quad Loop by Nathan Cohen ~ ,\ IIW.

SiO IlS, hut each time adding a repet ition inaccordance wit h the Koch curve , Natu­rally the Sil l' of the loop went u p eac htime.

Compare this with the problem of theCors ica n coast which we mentioned car­licr. lie then measured the reflectedpow er (proportional to the SW RJ of eac hor these antennas. This to ld him that thefr equencies at which the SW R V'iUS lowest(which means th at the impedance herewas approximately 50 Ohms) kept chang­ing, and th at th e more peaks there wereon the frequency ax is the lower thesewe re (Fig , 9).

For an tenn as with three repetitious. forexample, we have a usable peak at 0 11C­

third of the fre quency of the non-fractal-

ised loop, It should be borne ill m ind that1110 peaks arc no t harmonics o r the startfrequency, II can he determ ined subse ­qu ent ly th at the peaks als o correspond Iozones in which the impedanc e is real i.c .not rea ctive, If the rad iation re sistance isat 50 Ohms and is not reactive, thatmea ns that no addi tional match ing isnecessary ,

5.Fractalisation of a dipole

In th is example we are not deal ing wirhexperiments bu t w ith the resu lts whichwere calc ulated by Nathan Cohen , ~ l lR,

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fiil.: 10: FructalDipulc-, hy x arhan Cohen - ~IIR

with the- help of hNEe-r software (Fig.10 ).

I lore. instead of a loop. we take a dipoledimensioned for (i:'i \ 1H7,. to which weapply two repetitions of tile Koch paucmin succession . Ilut this lime we sweep amuch bigger runge. going right up to 5(ITT!. With a rectilinear dipole we nowmeasure a host of points with a low SWRfor all harmonic s of (is Mllz.

On the other hand. the Iractuliscd dipolesdemo nstra te ex pansi on and low ertroughs. The optimal impedance i.e.• forwhich the trough is weakest lies around350 Ohms.

Thus we can sec that fractalisation can beused [0 create antenna.. with an extremely

wide trunsruission band.

6.Practical Applica tions

A practical application of the fractaltheory has been put forward by NathanCohen, .'J1IR: a quad with two eleme ntsfor the lu-m-Band. f ig. 11

It measures 1.5 m x 1.5 m without anyefficienc y losses as against the unfractaf­ised version. It does not need a matchingcircuit. and i t ~ impedance, Z. is 50 Ohms.It can thus be cncrgiscd d irectly through acoaxial cable. Fig 12.

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Fig 11: A TwoElement Qua d forth e 10m ba nd bvNa tha n Cohen .,"J\"'IIR

Using thi s anten na, radio contact couldbe made with European locations at IWatt an d with the Pacific reeicn at 2Watts . A furthe r version was "measuredfor the 2-m band, wi th a ga in of 4 d ll anda frout-to-bac k rat io of at least 15 dlt .The 2-m ve rsion has a transmission handwidth of 500 kTI/ with an SWR below 2.

Fig. 14 shows a general view o f theantennas. The two clem ent s ha ve thesame dime nsions. Thev can be manufnc­tured using copper wire (I. S mm . ormore). The simple st me thod is to lise atemp late a ll a wooden hoard on whichnai ls are knocked in at the bend ingpoints, I to 2(i, in accordance w ith thopatte rn, ami the cop pe r wire can th us bevery precisely bent. Each of th e fo ursides of an d ement is hcnt on the tern ­plate III success ion III this way .

Fig . 13 shows a quad clement for 10 mwi th corresponding dimensional sp ccifi­cat ions . The framewo rk of the qua d caneas ily be manufactured us ing I've tub­ing, The same materia l can be used forthe transverse struts. The typical impcd­ancc of the antenna can he increased ifthe len gth of the m iddle transverse strutsis increa sed.

Instead of the coax cable, a coil w ith two

turns is connected to tile distnhuti nspoint o r the re flector, so that its trc­quc ncy of reson ance call be decreased byctpp. (iOn kllz. TIl(: distance between theact ive cle ment and the reflector is 170em.

Like all frac tal ante nnas , this antenna isalso resona nt in more than one band.:"J I IR de tected reso nan ces at 52. 97, 125and 14 1 ~ l H z . At 125 \ f llz. the fron t­to-hack ratio is In dn and the gain IS

likew ise 10 dlj. which 1S ;1 lot for anant enna whic h is co ntained III a cube witha waveleng th or onl y 0.6.

Fi~ 12: The measu red and th eoreticalSWR for th e Two E lement Q uad

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0.'. T wo frltdal antenn as for the7(1-clII hand

their dimensions were matched. IIt: Si l11U­

fated the properties of two quad an tennaswith two, then with three loops (Fig 1~and 16), Unfortunately he gives no detailsof the precise dimensions of his antennas.

Another pract ica l implementation for theAmerican 70-cm band is proposed IlyRichard Kutter of the Univcrs ity orDay­101 1. To dcrnonct ratc the validity or theconcept {If the fractal nntennn, he firstexamined a dipol e and an antenna proveda temp ting con tro l. He then comparedthem with two fractaliscd loops, one withone repetition, the other with 1\','0. Theradiation and SWR simulations were car­r ied nut in the 70-cm band using :\HNI­I'\I~C Pro. To ensure that the fmctalisedantennas remained in the desired band.

6.2. Fr actul autcunav for micrnwa vcs

The higher the frequen cy involved, thefewer details we ar c given by authors . Itis, in fac t, in the Hinges of mobiletelephony uud microwaves that the nppli­cations are mos t interesting from thecommercial p oint o f view...

Here are some examples of ult ra-highIh..xlucncy antennas. on winc h numerous

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Fi:.: 15: Simulation nf a Two Element Quad fur the 70 em hand

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Fi~ 16: S illlulatiun of a Two Element Q uat! for th e 70 em h:1I1(1

Fi:;: 17: T ht'Sil' rp in<;kiT r hll11:ll' used atmlcrewav cfn 'clul ' lldl'<;

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Fi:.: IN: Exa mr ll'\ of a nten na .. scale d for ,arion.. microwave fr equencies

Fig 19: A wid eh und a ntennadt"dllpt"l! by Xath an Cuhen - Yl f R

Fi:.: 211 : A fractal vcrs fon of the ht'li\a nt enna b~ :"Iat ha n Cohe n - :"IIU

225

teem s arc current ly 'II work all ove r tbcworld. Gut. who knows? Per haps, onceagain. another unporlant discovery willemerge from the amateur radio commu­nity?

6.3. T he Slcrplusk l tr iangle

One structure uflon used in ultra-highfreque ncy engineering is the Sicrpinskitriangle. l t is powered through one of thecorners , in that it is connected up to theintern al cond uctor or a coaxial line. TheSCrlTnm g is connected to the earth sur­face, which can be both a closed surfaceand also a fractal figure. The beha viourof such an antenna can be analv xcd withthe help of software based 0 1; the mo­ments method, such as. fur example.Emxigln. The program has already beenintroduced here. The said meth od makesit possi ble to calculate the intensity of thecurrents which prevail in the conductors.

The poi nt of opt imal matching for patch

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Fig 21 : l\1 ~

Fav ouri te. a bea ma nte nnadevelop ed by Th eCades PuentePol ytechnic atBarcelonaUn iver sity

antenna s is generall y empirically deter­ruin ed However. a start has been madeon using mathematical models 10 developsimulation s, which wil l ccrtainlv make itpossible to create more powerful simul n­lions in the coming years.

7.Literature

r11 "CHAOS" hy CJ leick, 19:\7,ISO:'\' 0 7493 B600 1

[2J httr :iifradcnna.colll"

[31 http-Z'www.crhc.uiuc .cduc- kuucrthc­sis/

[4] htlp :/I\Y\\"w.amlab.ee.llcla.edu<johng/fractal.html

[51 http ://www-tsc. upc .o.:s/ed 'research linc-/an tcnnas/fractals 'frac tal antennas.htlm