Synthesis of a Three-Dimensional Triband (L1-L2 GPS and Wi-Fi) Prefractal Tree Antenna

UNIVERSITY

OF TRENTO

DEPARTMENT OF INFORMATION AND COMMUNICATION TECHNOLOGY

38050 Povo – Trento (Italy), Via Sommarive 14

http://www.dit.unitn.it

S

YNTHESIS OF A

T

HREE

-D

IMENSIONAL

T

RIBAND

(L1-L2

GPS

AND

W

I

-F

I

)

P

REFRACTAL

T

REE

A

NTENNA

R. Azaro, E. Zeni, T. Gazzini, R. Dallapiccola, A. Massa

November 2006

and Wi-Fi) Prefra tal Tree Antenna

Renzo Azaro, Member, IEEE, Edoardo Zeni, Tommaso Gazzini, Ramona Dallapi ola,

Andrea Massa, Member, IEEE

Department of Information and Communi ation Te hnologies,

University of Trento,Via Sommarive 14,38050 Trento - Italy

Tel. +39 0461 882057,Fax +39 0461 882093

E-mail: andrea.massaing.unitn.it, {renzo.azaro,edoardo.zeni}dit.unitn.it

and Wi-Fi) Prefra tal Tree Antenna

Renzo Azaro, Member, IEEE, Edoardo Zeni, Tommaso Gazzini, Ramona Dallapi ola,

Andrea Massa, Member, IEEE

Abstra t

In this letter, the synthesis of a 3D wire antenna with a tree-like shape and

op-erating at three dierent frequen y bands (

L1 − L2 GP S

bands and

W i

− F i

) is

presented. The geometri al parameters of theantennaaredetermined bymeans of

a parti leswarm-based optimization pro edureexploitingthedegrees of freedom of

thereferen etreegeometry for ttingtheele tri al proje t onstraints. Theresults

of thenumeri al simulation areshownand ompared withthose fromexperimental

measurements for assessingthe ee tiveness ofthesynthesispro edure.

Key Words - Antennas Synthesis, Fra tal Antennas, Tree-like Antennas, Multiband

Be ause of the growing demand of miniaturized and multiband antennas for ele troni

devi es employing multiple wireless standards, several resear hes have been devoted at

analyzingthe ele trodynami behaviorof fra taland pre-fra talgeometries[1℄[2℄[3℄. As

a matter of fa t, fra tal and pre-fra tal geometries present very interesting features in

terms of multiband behaviorand overall size.

Besides planargeometries [4℄[5℄[6℄, easilybuilt on diele tri substrates by meansof

stan-dard and low- ostphotolitographi te hniques, three-dimensional fra taltree-likeshapes

have been investigated [7℄. As a matter of fa t, despite their omplex realization, these

stru tures present veryinteresting ele trodynami properties thanksto the large number

ofdegrees offreedominthe hoi eofthevaluesoftheirdes riptivegeometri parameters.

In [7℄, Petko and Werner showed that fra tal tree geometries an be fruitfully employed

as end-loads for the miniaturizationof onventional dipoles or monopoleantennas.

However, although a multi-band behavior an be realized, the spe tral distribution of

multiple resonant frequen ies is stri tly related to the geometri al parameters of the

self-similar stru ture with xed relationships among the frequen ies whatever the

fra -tal shape. Therefore, the allo ationof several working frequen y bands inthe frequen y

spe trum turns out to be a omplex task with  onstrained hoi es. In order to

over- ome su h a limitation, the insertion of rea tive loads in the fra tal tree stru ture has

beenproposed in[7℄atthe ost of anin reased omplexityin the onstru tion pro ess of

the devi e.

Anotheree tivemethodologyforobtainingamultibandbehavior,without onstraintson

resonan efrequen ies,isbasedontheintrodu tionofperturbationsbothinsizeandlength

of the fra tal geometry[8℄[9℄[10℄[11℄. A ording tothis guideline, this paperpresents the

design of a three-dimensional pre-fra tal tree antenna operating at three dierent

fre-quen y bands(

GP S L

1

-

L

2

bandsand

W i

− F i

). The synthesis of the pre-fra tal

geome-try is arried out through a numeri aloptimizationpro edurebased on aparti leswarm

optimizer(

P SO

)[12℄[13℄[14℄, whi hoptimizesthe geometri alparameters des ribing the

tree geometryavoiding the insertionof any lumped load toobtaina multibandbehavior.

three-and measurementson anexperimental prototype.

2 Triband Pre-Fra tal Tree Antenna Design and

Opti-mization

The synthesis of the three-dimensional geometry of the multiband prefra tal tree has

been formulated as an optimization problem by xing suitable onstraints in terms of

gain values and impedan e mat hing at the input port (Return Loss values) in both

L

1

and

L

2

frequen y bands of the

GP S

system ( entered respe tively at

f

1

= 1575.42

MHz

and at

f

2

= 1227.60

MHz) as well as in the

W i

− F i

band ( entered at

f

3

= 2440.00

MHz).

As far as the impedan e mat hing is on erned, a Return Loss lower than

−10 dB

is

required. Moreover, the radiation hara teristi s of the three-dimensional antenna are

required to satisfy the following onstraints. Con erning

L

1

and

L

2

GP S

bands, an

hemispheri al overage isneeded withgainvalues suitablefor ommer ial

GP S

re eivers.

More in detail, gain values greater than

G

min

= 3.0 dBi

at

θ

= 0

o

and greater than

G

min

= −4.0 dBi

at

θ

= 70

o

by assuming the presen e of a lownoise

30 dB

preamplier

stage at the input port of the

GP S

re eiver be ause of the weakness of the

GP S

signals

at the earth surfa e. Furthermore, taking into a ount the performan es required by a

ommer ial mobile transmitter/re eiver in the

W i

− F i

band, the following onstraints

have been dened: gain values greater than

G

min

= −6.0 dBi

at

θ

= 70

o

and an average

gain greater than

G

min av

= −10.0 dBi

in the angular range

70

o

_{< θ <}

₉₀

o

. Finally, in

order to minimizethe overall size of the antenna, the

3D

stru ture is required tobelong

to a volume having dimensions

λ

0

4

×

λ

0

4

×

λ

0

4

, being

λ

0

4

∼

= 6 cm

at the lowest resonan e

frequen y (

f

2

= 1227.60 MHz

).

A ording to [7℄[15℄, the self similar stru ture of the prefra tal tree has been derived by

repeatedlyapplyingthe so- alledHut hinsonoperatorup tothestage

i

= 3

. Asshown in

Fig. 1, startingfrom atrunk of length

l

0

lyingalong the

z

-axis ina artesian oordinate

Ex ept the trunk and the nal bran hes, every bran h is onne ted to a jun tion at its

ownterminations. Thus, itisatthe sametime hildandparent ofotherbran hes. These

stru tures an be grouped in dierent lasses depending on: (a) the numberof bran hes

R

that lead oa jun tion,(b) thebentangle

ξ

b

with respe t tothe parent dire tion,and

( )the angle

ξ

s

that separates the onse utive bran hes leadingo the samejun tion. In

this paper, in order to simplify the building pro edure, a four bran hes (

R

= 4

) fra tal

tree hara terized by

ξ

b

= 90

o

and

ξ

s

= 90

o

has been onsidered. Therefore, the resulting

fra tal shape is omposed by bran hes lying alongthe axesof the oordinatesystem and

the antenna stru ture is uniquely determined by: (i) the trunk length

l

0

and (ii) the

lengths

l

i,j

of the bran hes,

i

being the iteration index and

j

(

j

= 1, ... , R

i

) denotes the

j

-th bran h generated at the

i

-th stage (

i

= 1, ..., I

) as pi torially shown in Fig. 1(b)

(

i

= 1

), Fig. 1( ) (

i

= 2

), and Fig. 1(d) (

i

= 3

). Summarizing, the

I

-th order prefra tal

tree is onstituted by

L

=

P

I

i=1

R

(I−i+1)

+ 1

segments (the bran hes and the trunk) of

ir ular ross-se tion

r

= 0.5 mm

in radius.

Under these assumptions, the des riptive parameters of the pre-fra tal tree (trunked at

I

= 3

fortuningthreedierentresonan efrequen ies)havebeenoptimizedbyminimizing

the following ost fun tion

ℑ γ
=

M −1

X

m=0

N −1

X

n=0

T −1

X

t=0



max



0,

G

min

{t∆θ, n∆φ, m∆f } − Φ {t∆θ, n∆φ, m∆f }

G

min



(1)

+

V −1

X

v=0



max



0,

Ψ {v∆f } − V SW R

max

V SW R

max



where

γ

= {l

0

, l

i,j

, i

= 1, ... I = 3, j = 1, ... , R

i

_}

,

∆f

is the sampling frequen y step in

the

L1

,

L2

and

W i

− F i

bands,

∆θ

and

∆φ

are the sampling angular steps of the gain

fun tion,

Φ

γ = Φ {t∆θ, n∆φ, m∆f}

is the gain fun tion of the radiating stru ture

γ

omputed at(

θ

= t∆θ

,

φ

= n∆φ

,

f

= m∆f )

,and

Ψ

γ = Ψ {v∆f}

isthe

V SW R

value

at

f

= m∆f

. Moreover, sin e besides ele tri al onstraints some other onditions have

been stated for limitingthe physi alsize of the antenna, a set of geometri al onstraints

have been dened toavoidthe synthesis of unfeasible stru tures.

(

M oM

) ele tromagneti simulator[17℄ forminimizing(1), thus determiningthe antenna

stru ture. More in detail,starting fromthe trialsolutions

γ

(k)

p

(

p

∈ [1, P ]

and

k

∈ [1, K]

beingtheindexoftheparti leinthesolutionsswarmandtheiterationindex,respe tively)

iteratively generatedby the

P SO

, the prefra talgenerator determines the orresponding

antenna stru ture. Then, the

V SW R

and gain values are omputed by means of the

M oM

simulator, whi h takes into a ount the presen e of a referen e ground plane of

innite extent. The iterative pro ess ontinues until

k

= K

or

ℑ

opt

_{≤ η}

,

η

being the

onvergen e threshold and

ℑ

opt

_{= min}

k

n

min

p

h

ℑ



γ

(k)

_p

io

.

3 Numeri al Analysis and Experimental Validation

Con erning the

P SO

implementation, the following setup has been used: a swarm of

P

= 8

trial solutions, a threshold

η

= 10

−3

, and a maximum number of iterationsequal

to

K

= 2000

. The remaining

P SO

parameters have been set taking into a ount the

guidelines proposed in [13℄ and a ording tothe rules dened in [16℄.

As an illustrative example of the optimization pro ess, Fig. 2 shows the evolution of

the behaviors of the

V SW R

in orresponden e with the optimal solution determined at

su essive iterations of the optimization pro ess. As it an be observed, starting from a

ompletely mismat hed onguration (

k

= 0

), the solution improves (

k

= 10, k = 50

)

until an antenna design is found (

k

= k

conv

) that ts the ele tri al spe i ations. As

far as the geometri onstraints are on erned, the synthesized stru ture turns out to be

adequatesin eitsmaximumdimensionsareequalto

3.28 [mm]

alongthe

x

-axis,

5.17 [mm]

along the

y

-axis,and

3.82 [mm]

alongthe

z

-axis.

From the omputationalpointofview, Figure3showsthe plotoftheoptimalvalueofthe

ost fun tion

ℑ

opt

versus the iterationnumber

k

. As it an be noti ed, the optimization

required

k

conv

= 75

with a

CP U

-time at ea h iteration of about

5 sec

(

P entium IV

,

1800 MHz

,

512 MB

RAM).

Thanks to the solution estimated at the end of the

P SO

-based optimization, a

proto-type of the three-dimensional antenna has been built (Fig. 4). In order to arry out

on-90 × 140 [cm

2

_]

and the Return Loss values have been measured with a s alar network

analyzerby pla ingthe measurementsetupinsideanane hoi hamber. Figure5shows a

omparison between omputed and measured values that points out asatisfa tory

agree-ment amongsimulations and experiments.

A similar omparative evaluation has also been performed for the radiation properties

of the antenna. More in detail, Figure 6 shows measured and simulated gain fun tions

in the horizontal plane [

θ

= 0

o

- Fig. 6(a)℄ and in a verti al plane [

φ

= 0

o

- Fig. 6(b)℄.

As required, the antenna prototypeallows anhemispheri al overage inthe

L

1

-

L

2

GP S

bands and, onsidering the

30 dB

preampli ation stage at the input port, the gain

val-ues turn out to be ompliant with the requirements (

Φ

GP S

{θ = 0

o

_{} > G}

min

{θ = 0

o

} =

3 dBi

and

Φ

GP S

{θ = 70

o

_{} > G}

min

{θ = 70

o

} = −4 dBi

). Similar on lusions hold true

for the

W i

− F i

frequen y band (

Φ

W i−F i

{θ = 70

o

} > G

min

{θ = 70

o

} = −6 dBi

and

Φ

Av GP S

{70

o

< θ <

90

o

} > G

min Av

{70

o

< θ <

90

o

} = −3 dBi

).

4 Con lusions

Thedesignandtheoptimizationofathree-dimensionaltribandprefra taltreeantennahas

beendes ribed. Theantenna stru turehas been synthesized byoptimizingthelengths of

thebran hesofaprefra taltreewithasuitableparti leswarm-basedalgorithminorderto

omply withgeometri alandele tri alrequirements. Aprototypeofthe tribandantenna

has been built and an exhaustive set of omparisons between measured and simulated

antenna parameters has been arriedout.

A knowledgments

This work has been supported in Italy by the Progettazione di un Livello Fisi o

'In-telligente' per Reti Mobili ad Elevata Ri ongurabilità, Progetto di Ri er a di Interesse

[1℄ D. H.Wernerand R.Mittra,Frontiersin ele tromagneti s.Pis ataway: IEEEPress,

2000.

[2℄ J. Gianvittorio and Y. Rahmat-Samii, "Fra tals antennas: a novel antenna

minia-turization te hnique, andappli ations,"IEEE Antennas Propagat. Mag.,vol.44,pp.

20-36, Feb. 2002.

[3℄ D. H. Werner, R. L. Haupt, and P. L. Werner, Fra tal antenna engineering: the

theory and design of fra tal antenna arrays, IEEE Antennas Propagat. Mag., vol.

41, pp. 37-59, O t. 1999.

[4℄ M. Sindou, G. Ablart, and C. Sourdois, "Multiband and wideband properties of

printed fra tal bran hed antennas," Ele tron. Lett., vol.35, no. 3, pp. 181-182,Feb.

1999.

[5℄ C. Puente, J. Claret, F. Sangues, J. Romeu; M.Q. Lopez-Salvans, and R. Pous,

Multibandpropertiesof afra taltree antennagenerated by ele tro hemi al

deposi-tion, Ele tron. Lett., vol.32,no. 25,pp. 2298-2299, De . 1996.

[6℄ D.H.Werner,A.R.Bretones,andB.R.Long,Radiation hara teristi softhin-wire

ternary fra tal trees, Ele tron. Lett., vol.35,no. 8,pp. 609-610, De .1999.

[7℄ J. S. Petko and D. H. Werner, Miniature re ongurable three-dimensional fra tal

tree antennas, IEEE Trans.AntennasPropagat.,vol.52,no. 8,pp. 1945-1956,Aug.

2004.

[8℄ C.Puente,J.Romeu,R.Pous,X.Gar ia,andF.Benitez,Fra talmultibandantenna

based on the Sierpinskigasket, Ele tron. Lett., vol. 32,pp. 1-2, Jan. 1996.

[9℄ M. Fernandez Pantoja,F. Gar ia Ruiz, A.Rubio Bretones, R.Gomez Martin,J. M.

Gonzales-Arbesù, J.Romeu,and J.M.Rius,GAdesignofwirepre-fra talantennas

and omparisonwithothereu lideangeometries, IEEEAntennasWirelessPropagat.

Gomez Martin, J.M. Gonzales-Arbesù, J. Romeu,J. M. Rius, P. L.Werner,and D.

H. Werner,GAdesignof smallthin-wire antennas: omparisonwithSierpinskitype

prefra tal antennas, IEEE Trans.Antennas Propagat., vol.54, pp. 1879-1882, Jun.

2006.

[11℄ R. Azaro, F. De Natale, M. Donelli, E.Zeni, and A. Massa, Synthesis of a

prefra -tal dual-band monopolar antenna for GPS appli ations, IEEE Antennas Wireless

Propagat. Lett., vol.5, pp. 361-364,2006.

[12℄ J. Kennedy,R.C. Eberhart,and Y.Shi, Swarm Intelligen e.San Fran is o: Morgan

Kaufmann Publishers, 2001.

[13℄ J. Robinson and Y. Rahmat-Samii,"Parti le swarm optimization in

ele tromagnet-i s," IEEE Trans. Antennas Propagat., vol. 52,pp. 397-407,Feb. 2004.

[14℄ M.DonelliandA. Massa,"A omputationalapproa hbasedonaparti leswarm

op-timizerformi rowaveimagingoftwo-dimensionaldiele tri s atterers,"IEEETrans.

Mi rowave Theory Te hn., vol.53, pp. 1761-1776, May 2005.

[15℄ D. H. Werner, P. L. Werner, and D. H. Chur h, Geneti ally engineered multiband

fra tal antennas, Ele tron. Lett., vol. 37,pp. 1150-1151, Sep. 2001.

[16℄ R. Azaro, F. De Natale, M. Donelli, A. Massa, and E. Zeni, "Optimized design of

a multi-fun tion/multi-bandantenna for automotive res ue systems," IEEE Trans.

Antennas Propagat., vol.54, no. 2,pp. 392-400,Feb. 2006.

[17℄ R.F.Harrington, FieldComputationbyMomentMethods.Malabar,Florida: Robert

•

Figure 1. Geometryofthe Tri-BandPrefra talTreeAntenna: (a)

i

= 0

,(b)

i

= 1

, ( )

i

= 2

, and (d)

i

= I = 3

.

•

Figure 2. SimulatedReturnLossattheinputportofthe Tri-BandPrefra talTree

Antenna at dierent iterations of the optimization pro ess: (a)

k

= 0

, (b)

k

= 10

,

( )

k

= 50

,and (d)

k

= k

conv

.

•

Figure 3. Behaviorof the ost fun tionversus the iterationnumber.

•

Figure 4. Photograph of the prototype of the Tri-BandPrefra tal Tree Antenna.

•

Figure 5. Comparison between measured and simulated ReturnLoss values.

•

Figure 6. Computedandmeasuredvaluesofthegainfun tioninthe(a)horizontal

plane (

θ

= 0

o

)and inthe (b) verti alplane (

φ

= 0

o

x

y

z

l

₀

x

y

z

l

_1,j

(a) (b)

x

y

z

2,j

l

x

y

z

l

_3,j

( ) (d)

-25

-20

-15

-10

-5

0

3.0

2.5

2.0

1.5

1.0

Return Loss

Frequency [GHz]

Iteration 0

Iteration 10

Iteration 50

Iteration k

_Conv

10

-3

10

-2

10

-1

10

0

10

1

10

2

10

3

10

20

30

40

50

60

70

80

90

100

Iteration number k

Φ

tot

0

-5

-10

-15

-20

-25

1.0

1.5

2.0

2.5

3.0

Return Loss

Frequency [GHz]

Measured

Simulated

0

-10

-20

-30

Gain [dBi]

0

30

60

90

120

150

180

210

240

270

300

330

Measured (1227 MHz)

Measured (1575 MHz)

Measured (2440 MHz)

Simulated (1227 MHz)

Simulated (1575 MHz)

Simulated (2440 MHz)

(a)

10

0

-10

-20

-30

Gain [dBi]

90

60

30

0

-30

-60

-90

-120

-150

180

150

120

Simulated (1227 MHz)

Simulated (1575 MHz)

Simulated (2440 MHz)

Measured (1227 MHz)

Measured (1575 MHz)

Measured (2440 MHz)

(b)