Spiral Array Architecture, Design, Synthesis and Application

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Published in IET Microwaves, Antennas & Propagation

Received on 10th September 2009

Revised on 7th November 2010

doi: 10.1049/iet-map.2010.0301

ISSN 1751-8725

Spiral array architecture, design, synthesis andapplicationA. Jafargholi M. Kamyab M. Veysi

Electronic Engineering Department, K.N. Toosi University of Technology, PO Box 16315-1355, Tehran, Iran

E-mail: [email protected]

Abstract: This study introduces a new array architecture, in which antenna elements are arranged in a spiral curve. The spiral

array enhances ultra-wideband (UWB) pattern characteristics compared to alternative array geometries of similar elements,without requiring a complex feed network for frequency change compensation. A number of examples are illustrated todemonstrate the array capability in UWB array designs. It is revealed that for the same number of elements and curvaturelengths, a spiral array has a wider radiation bandwidth than the corresponding linear and circular arrays. In addition, it is alsodemonstrated that the spiral architecture discussed here can be best suited for small antenna array applications. The arrayfactor and the bandwidth of the spiral array are calculated theoretically. The simulation results are found to be in goodagreement with the theoretical calculations.

1 Introduction

Array elements, feed networks and array architecture are three

main factors determining the array performance [1]. In the pastseveral decades, the ultra-wideband (UWB) antenna elementshave been developed with input impedance that remainsrelatively constant within a reasonably wide frequency range[28]. As an important parameter in the array design, the feednetwork generally depends on the array elements andarchitecture. The selection of the array architecture for therealisation of a desired array antenna is determined byrequirements on the array bandwidth, pattern, size and scanrange [1]. In addition, broad radiation and input impedance

bandwidths are two major characteristics of the antenna arraysfor UWB applications. Since the input impedance bandwidthdepends on the characteristics of the element used in the array,the radiation bandwidth is the limiting factor. In other words,the radiation bandwidth of the array limits its practical bandwidth.

Since antenna arrays play a significant role both in directionfinding and in increasing the capacity of the systems, the use ofsuitable array architecture is becoming increasingly important.From an architecture point of view, there are still manytheoretical as well as practical open issues despite the basicconfigurations found in the literature [1]. The broadbandfrequency characteristics can be obtained based on thefundamental theory introduced by Rumsey in the fall of1954 [9]. As a result, we speculate that the spiral arrayarchitecture, in which antenna elements are arranged in aspiral curve, may be frequency independent. The simulationresults confirmed this claim is provided later. Consequently,

this paper is mainly focused on the radiation bandwidthenhancement and on the basic design principle for the spiralarray architecture. The proposed spiral architecture exhibitsgreat potential for applications such as radio direction

finding and UWB systems. The combination of similarUWB antenna elements to form a spiral array is aninteresting way of improving the directional properties of the

UWB antennas without altering their size. Furthermore, the proposed array offers more degrees of freedom thanconventional arrays, namely, circular and linear arrays.

In UWB systems, one is interested in having an antenna orantenna array which maintains a substantially stable radiation

pattern over the entire frequency range of interest to keepUWB pulse distortion as small as possible. However, the

bandwidth of conventional arrays (linear and circular arrays) islimited because the radiation pattern scans with frequency andthus moves off the target. In order to avoid this beam squint

problem, wide band arrays are required. A number ofexamples are presented to demonstrate the capability of thespiral array in UWB array antenna designs, all of which aredesigned based on array architecture rather than elements orfeed network. However, the total radiation pattern of theantenna array is equal to the product of the active element andarray radiation vectors, referred to as pattern multiplication.Therefore deployed UWB antenna elements should also havea relatively stable radiation pattern over the frequency range ofinterest. To this aim, a miniaturised UWB monopole antennawith stable radiation pattern is proposed, which is simple tomanufacture and is well suited to being incorporated into aspiral array. The simulation was performed with CSTMICROWAVE STUDIO based on the finite integration method.

2 Array characterisation

A three-dimensional array with an arbitrary geometry isshown in Fig. 1. For obvious reasons, the coordinate systemof choice is spherical where the location of the mth elementis characterised by rm = (rm, um, fm). The unit vector

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pointing from the incident wave source to the origin cantherefore be represented as k= (1, u, w). Throughout thiscontribution, the incident wave source is assumed to belocated at the far field. In order to calculate the array factor,it is necessary to find the phase angle of the received planewave from the source at each of the elements with respectto the origin given by

jm = bkrm= b[xm sin(u)cos(w)+ym sin(u) sin(w)+zm cos(u)] (1)

Here, element locations are designated as (xm, ym, zm,1 m M) and M is the number of array elements. Andthus the array factor is calculated from

AF(u, w) =Mm=1

Imej(jm+dm) (2)

In the above equation, Im and dm denote the magnitude andphase of the weighting of the mth element, respectively. For

the spiral topology, the position vector of the mth elementcan be expressed by

rm = reafm = r0 eafm cos(fm) x+ r0 eafm sin(fm)y (3)

Here, a is the spiral constant, which specifies the increasing

rate of the spiral radius proportionate to angle fm, and xand yare the unit vectors along the x- andy-axis, respectively.

Two possible types of spiral array architectures are depictedin Fig. 2. The parameters of the spiral array are labelled inFig. 2 where r0 is the distance of the first element from theorigin and fm (m2 1)p/b denotes the angle of the mthelement relative to the x-axis. It is indispensable to point outthatb (b . 0) is one of the important parameters of the spiral

array, which identifies locations of the array elements. It alsoincreases the available degrees of freedom in theoptimisation of the array performance.

Substituting (3) into (1) reduces it to

jm =2p

lr0 e

afm sin ucos(w fm) (4)

And thus the resultant array factor can be expressed by

AF(u, f) =Mm=1

Imej((2p/l)r0e

afm sin ucos(wfm)+dm) (5)

The spiral array bandwidth can be also calculated using the perturbation theory [Appendix]. In order to gain furtherinsight into the spiral array bandwidth, (6) introduces aspiral-to-linear bandwidth ratio (SLBWR)

SLBWR= (1/2)M(M+ 1)(eap/b 1)

[e(ap/b)M 1] for r0 = dLinear(6)

where dLinear is the element separation for the linear array. Forthe contracting spiral array whose radius decreases as wincreases (i.e. a , 0), SLBWR for the large value of M

(M b) can be expressed by

SLBWR|M1 1

2M2(1 eap/b) (7)

The effects of the spiral array parameters, namely, number of

Fig. 1 Schematic of an arbitrary three-dimensional array

Fig. 2 Two possible array architectures, where the element locations are specified by (3)

a Contracting spiral, a 20.125, M 5, b 1.75, f0 5 GHz, r0 0.04 mb Expanding spiral, a 0.1, M 7, b 3, f0 7 GHz, r0 0.01 m

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array elementsM, spiral constanta andb on the array bandwidthare investigated in order to obtain some engineering guidelinesfor spiral array designs. The SLBWRs of five- and seven-elements spiral arrays for different values of b are shown inFig. 3. As can be seen, the SLBWR decreases as the spiralconstant (a) increases. It can also be noticed that the curves ofthe contracting spiral (a , 0) have steeper slopes as comparedto those of the expanding spiral (a. 0). In contrast to the

contracting spiral, the SLBWR of the expanding spiralincreases with the value of b. Fig. 4 shows the SLBWR of thecontracting and expanding spirals with the number ofelements as a parameter. As can be seen, increasing thenumber of array elements increases the SLBWR in both cases(a 20.125 and 0.1).

Based on the uniform amplitude and progressive phaseexcitation method, for the main lobe pointing to (u0, f0),the phase dm of the radiating elements can be expressed by[1, 10]

dm = 2p

lr0 e

afm sin u0 cos(f0 fm) (8)

By using (3) and (8), the element locations and excitationscan be easily determined so that the main lobe points to(u0, f0) (90, 0). To demonstrate the capability of thespiral array to generate the UWB radiation, two examplesare studied here. The first one is a five-element contractingspiral array whose parameters are labelled in Fig. 2a. Aseven-element expanding spiral array, whose parameters arelabelled in Fig. 2b, is also studied here. The excitation

currents and locations of the elements, of both cases, arelisted in Table 1. The simple monopole antennas are usedas array elements. The simulated radiation patterns of the

previously mentioned spiral arrays are shown in Figs. 5 and6 and compared to the analytical results. It should be

pointed out that the isolated element pattern is considered inthe analytic calculations.

As can be seen, the agreement between the simulation andanalytical results is reasonable. Small difference between thesimulation and analytical results is attributed to the existenceof the mutual coupling between the array elements, which isnot considered in the analytic calculations. In the simulations,the mutual coupling between the array elements plays an

Fig. 3 SLBWR against spiral constant a, with the value of b as a parameter

a M 5b M 7

Fig. 4 SLBWR against number of array elements M, with the value of b as a parameter

a a 0.1b a 20.125



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important role in determining of the array characteristics.Here, since the element separation is small, the mutualcoupling effect is considerable. The obvious difference

between the analytical and simulation results at lowfrequencies is due to the fact that the element separation isrelatively smaller in terms of wavelength at the lowerfrequency. For the expanding spiral array (design II), since

the element separation is smaller than that of the contractingspiral (design I), this difference is clearer (see Fig. 6a). Asa result, the use of active element pattern could bring theanalytical and simulation results into better agreement. On

the other hand, the front-to-back ratios of these arraydesigns are relatively low. To eradicate this problem, thegenetic algorithm (GA) is applied to the array design toimprove the antenna efficiency. The improved radiation

patterns of the expanding spiral array (design II) at differentfrequencies are also shown in Fig. 6. The effect of GA onthe contracting spiral array (design I) has been investigated

latter when we are trying to practically realise the spiralarray antenna. The excitation currents of the antennaelements obtained from GA are also listed in Table 2.Fig. 7 shows the frequency response of the antenna

Table 1 Excitation currents and locations of array elements

Elem. no. 1 2 3 4 5 6 7

design I location

(x, y) (cm)

(4.00, 0) (20.71, 3.12) (22.30,21.18) (1.27, 1.59) (1.016, 21.27)

current

(complex)

20.50+ 0.866i 0.735+ 0.677i 20.7436+ 0.668i 0.236 0.9716i 0.485 0.8745i

design II location

(x, y) (cm)

(1.00, 0) (0.55, 0.961) (20.616, 1.067) (21.369,0) (20.760, 21.316) (0.844, 21.462) (1.874, 0)

current

(complex)

0.104 0.994i 0.6860.727i 0.618+ 0.785i 20.4227+ 0.906i 0.44+ 0.8976i 0.32720.9449i 20.92353834i

Fig. 5 Normalised radiation pattern of the contracting spiral array antenna (design I), atu 908

a 4 GHzb 5 GHzc 6 GHz

Fig. 6 Normalised radiation pattern of the expanding spiral array antenna (design II), at u 908:

a 4 GHzb 5 GHzc 6 GHz



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directivity in the main beam direction, (u0, f0) (90,0), in both cases. The 3 dB directivity bandwidth of thecontracting and expanding spiral arrays are 5:1 (rangingfrom 1.5 to 7.5 GHz) and 4.375:1 (ranging from 2.4 to10.5 GHz), respectively. It is worthwhile to point out herethat the expanding spiral array (design II) has more stableand better characteristics than the contracting spiral array(design I). However, since the space between the antennasin the expanding spiral array described in Fig. 2b is verysmall, its practical realisation is very difficult to achieve.Consequently, the contracting spiral array described inFig. 2a is selected for practical realisation.

3 Comparison of spiral, circular and lineararray architectures

Various array architectures have been proposed and investigated

in the literature [1, 1014]. Besides the spiral array architectureproposed in this paper, the linear and circular array architecturesare popularly used in antenna engineering. Fig. 8 comparesthe geometries of the spiral, linear and circular arrays. Tomake a fair comparison, the number of elements and curvaturelengths are the same in all cases. The total length of an

M-element linear array with an element separation of dLinearis L (M2 1)dLinear whereas the circumference of an

M-element circular array with a radius of r0,circular isCc 2pr0,Circular. In addition, for an M-element spiral array,the array circumference can be expressed by

CS =

r0 1+a2

a(ea

fM

1) (9)

To have a fair comparison, the element separation forlinear array and the radius of the circular array are selected

to satisfy (10).

r0,Circular= CS1

2p

dLinear= CS1

(M 1)(10)

The parameters of the spiral array are the same as those givenin the caption ofFig. 2a whereas the parameters of the circularand linear arrays are labelled in Fig. 8. For all arraygeometries, simple monopole antennas are used as arrayelements. Directivities of the arrays with differentgeometries in the main beam direction are plotted inFig. 9a. As revealed in the figure, the spiral arrayarchitecture provides a wider radiation bandwidth. The 3 dBradiation bandwidth of the spiral array architecture isapproximately 4.1 and 1.9 times wider than that of thelinear and circular arrays, respectively. Further insight issought through the investigation of the front-to-back ratio ofcircular and spiral arrays, as shown in Fig. 9b. As revealedin the figure, the spiral array provides larger FBR over theentire frequency band of interest compared to the circulararray. The cross-polarisation level (with respect to the 0 dBco-polarisation level) comparison of the arrays withdifferent geometries is shown in Fig. 9c. The fact that thecircular array is a symmetrical arrangement causes theradiation pattern to experience lower cross-polarisationlevels as compared to the spiral and linear arrays. However,the cross- polarisation level of the spiral array geometry issignificantly lower than that of the linear array because of

its more symmetrical configuration. Finally, it should benoted that the spiral arrangement allows the beam to besteered in any direction in the azimuth plane, similar to thelinear and circular arrangements [1].

Fig. 7 Frequency response of the simulated directivity in the main

lobe direction, for the spiral array

Fig. 8 Geometries of the spiral, circular and linear arrays:

dlinear 4.7 cm and r0,circular 3.04 cm

Table 2 Excitation currents of array elements obtained from genetic algorithm

Elem. no. 1 2 3 4 5 6 7

current (complex)

design I

20.012

+ 0.0206i20.07

0.0646i

0.112

0.1005i

20.071

+ 0.29443i20.0575

+ 0.1037i

current (complex)

design II

20.0274

+ 0.261i20.1224

+ 0.1296i0.2136

+ 0.2713i0.0453

0.09716i

0.1322

+ 0.2693i20.05317

+ 0.1535i20.1394

0.0541i



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4 Practical realisation of the spiral arrays

Since the main concept outlined in this paper is focused on theradiation patterns and on the basic design principles for thespiral array, the input impedance characteristics which dependon the characteristics of the UWB-elements used in the arraywere not fully considered in the previous sections, similar tothe procedure used by other authors [1]. Consequently, asimple monopole antenna was used in the basic description ofthe spiral arrays. However, in practice, the mutual coupling

between the array elements plays an important role indetermining the array characteristics. In the spiral arrayantennas, since the element spacing sometimes becomes small,the UWB antenna elements need to be small enough so thatthey can be arranged in a spiral curve with minimum mutualcoupling. In addition, it is expected that spiral array gain will

be degraded by rotation of the element radiation pattern withinthe antenna impedance bandwidth. To eradicate these

problems, a miniaturised UWB antenna element with stableradiation pattern within the impedance bandwidth is proposed

in this section. To this aim, a square planar metal-platemonopole antenna is selected to provide wideband frequencycharacteristics. The main reason is that the radiation patterns ofthe square planar metal-plate monopole antennas are usually

less degraded within the antenna impedance bandwidth. Theidea comes from the fact that location and number of feedingstrips significantly affect the current distribution on the planar

Fig. 9 Comparison between different array geometries (without GA):

a Directivity [in the main lobe (u 908, w 08) direction]b FBRc Cross-polarisation level

Fig. 10 Omnidirectional UWB antenna: W 10.5 mm, W1

7.4 mm, L 1 mm, H 20 mm, SL 6.2 mm, G1 0.2 mm,

G2 0.1 mm, G3 0.8 mm and WG 20 mm



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monopole antennas which in turn affect the antenna impedancebandwidth [15]. The geometry of the proposed planar monopoleantenna is shown in Fig. 10. The antenna is fed through theground plane by a coaxial cable. In the feeding point, thefeeding strip is divided into two parts, as shown in Fig. 10.The antenna parameters are also labelled in Fig. 10. Fig. 11ashows the voltage standing wave ratio (VSWR) of the

proposed miniaturised UWB monopole antenna. As can beseen, the antenna VSWR is acceptably small (VSWR 3) in

the frequency range from 2.8 to 8.5 GHz. The frequencyresponse of the antenna directivity is also plotted in Fig. 11b.To investigate the practical realisation of the spiral arrayarchitecture, the proposed miniaturised UWB antenna elementsare arranged in a spiral curve (design I). The geometry of theresultant spiral array is shown in Fig. 12. The excitation currentsof the antenna elements can also be found in Table 2. TheVSWRs of the array elements are shown in Fig. 13a. As can beseen, all the array elements have a VSWR lower than 2.5 inthe frequency range from 2.5 to 7.5 GHz. Fig. 13b shows themutual coupling between the array elements. As can be seen,the mutual couplings between the array elements are acceptableover the entire frequency band of interest (always below29 dB).

The CST-simulated radiation patterns of the proposed spiralarray at different frequencies are shown in Fig. 14. The mainlobe of the spiral array composed of proposed UWBmonopole antennas and backed by an infinite ground planealways points to the u 908 direction, the same as a verticalmonopole backed by an infinite ground plane. But the beamdirection begins to move towards a high-elevation angle asthe ground plane size reduces and thus the main beam

Fig. 11 CST simulated results of the proposed planar monopole antenna

a VSWRb Directivity of UWB antenna element

Fig. 12 Simulated model of the contracting spiral array

Fig. 13 CST simulated results of the proposed spiral array shown in Fig. 12

a VSWR of the array elementsb Mutual couplings between the array elements



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direction depends on the ground plane size. In the case at hand,the ground plane size is 20 cm 20 cm. As frequencyincreases and wavelength decreases, the ground plane sizeincreases as compared to the wavelength and thus the beamdirection moves towards a low-elevation angle, as revealed

in Fig. 14. The frequency response of the directivity (in thebeam direction (u 358)) of the proposed spiral array shownin Fig. 12 is plotted in Fig. 15. The directivity of the samespiral array composed of ideal monopole antennas backed by

an infinite ground plane [in the main beam direction(u 908)] is also plotted for comparison.

5 Conclusion

In this paper, a novel planar array for UWB applications has been introduced and thoroughly investigated. A faircomparison between the spiral array architecture and itsconventional counterparts, namely, circular and linear arrayarchitectures, exhibits the ability of the proposed architectureto enhance the radiation bandwidth of an array. Several spiralarray antennas have been designed and simulated to confirmthe theoretical calculations. However, the UWB spiral arraysgenerally require a miniaturised UWB antenna with stableradiation pattern. To eradicate this problem, a novelminiaturised UWB monopole antenna with stable radiation

pattern within the antenna impedance bandwidth has beenproposed. It was revealed that the proposed antenna is a goodcandidate for spiral array antennas so that the mutualcouplings between the array elements are not that significant.

6 References

1 Allen, B., Dohler, M., Okon, E.E., Malik, W.Q., Brown, A.K., Edwards,D.J.: Ultra-wideband antennas and propagation for communications,radar and imaging (John Wiley, New York, 2007)

2 Li, P., Liang, J., Chen, X.: Study of printed elliptical/circular slotantennas for ultra wideband applications, IEEE Trans. AntennasPropag., 2006, 54, (6)

3 Ammann, M.J., Chen, Z.N.: A wide-band shorted planar monopolewith bevel, IEEE Trans. Antennas Propag., 2003, 51, (4), pp. 901903

4 Agrawall, N.P., Kumar, G., Ray, K.P.: Wide-band planar monopoleantennas, IEEE Trans. Antennas Propag., 1998, 462, pp. 2942955 Liang, J., Chiau, C., Chen, X., Parini, C.G.: Printed circular disc

monopole antenna for ultra wideband applications, IEE Electron.Lett., 2004, 40, (20), pp. 12461248

Fig. 14 Radiation patterns of the spiral array antenna shown in Fig. 13, atw 0 plane, for different frequencies

Fig. 15 Frequency response of the directivity, in the u 358

direction, for the proposed spiral array shown in Fig. 12

As a reference, the directivity of the same spiral array composed of idealmonopole antennas backed by an infinite ground plane, in the u 908direction, is also depicted in this figure



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6 Wu, X.H., Chen, Z.N.: Comparison of planar dipoles in UWBapplications, IEEE Trans. Antennas Propag., 2005, 53, pp. 19731983

7 Eldek, A.A.: Ultrawideband double rhombus antenna with stableradiation patterns for phased array applications, IEEE Trans.Antennas Propag., 2007, 55, (1), pp. 8491

8 Wang, F.J., Zhang, J.-S.: Wideband printed dipole antenna for multiplewireless services, J. Electromagn. Waves Appl., 2007, 21, (11),pp. 14691477

9 Dyson, J.D.: The equiangular spiral antenna, IRE Trans. AntennasPropag., 1959, 7, (2), pp. 181187

10 Balanis, C.A.: Antenna theory (John Wiley & Sons, Inc., 1997)11 Kraus, J.D.: Antennas (McGraw-Hill, 1988)12 Ng, B.P., Er, M.H., Kot, C.: Linear array geometry synthesis with

minimum sidelobe level and null control, IET Microw. AntennasPropag., 1994, 141, (3), pp. 162166

13 Khdier, M.M., Christodoulou, C.G.: Linear array geometry synthesis withminimum side lobe level and null control using particle swarmoptimization,IEEE Trans. Antennas Propag., 2005, 53, (8), pp. 26742679

14 Mahmoud, K.R., El-Adawy, M., Ibrahem, S.M.M.: A comparison between circular and hexagonal array geometries for smart antennasystems using particle swarm optimization algorithm, ProgressElectromagn. Res., 2007, PIER 72, pp. 7590

15 Wong, K.L., Wu, C.H., Su, S.W.: Ultrawide-Band square planar metal- plate monopole antenna with a trident-shaped feeding strip, IEEETrans. Antennas Propag., 2005, 53, (4), pp. 12621269

7 Appendix

7.1 Bandwidth calculation of the spiral arrayantenna

To reduce the equations complexity, the first-element locationis expressed in terms of the array centre frequency ( f0) so that

r0 = dspirl0 = dspirc

f0(11)

where dspir is a constant andl0 is the free space wavelength atf0. And thus

r0l= dspir

c

f0/

c

f r0

l= dspir

f

f0(12)

Substituting (r0/l) into (5), the array factor can be reorganisedin terms of frequency ( f) and angle (w) as

AF(w, f) =Mm=1

Im ej(2pdspir(f/f0) e

afm sin ucos (wfm)+dm) (13)

And thus, forw f0, one can write

AF(f0, f) =M

m=1Im e

j(2pdspir(f/f0) eafm sin u cos(f0

fm)

+dm)

(14)

Considering the fact that the array factor around the main lobeis a smooth function, the Taylor expansion of (14), about the

point f f0 results in

AF(f0, f)=Mm=1

Im ej(2pdspire

afm sinucos(f0fm)+dm)

+j 2pdspirf0

(ff0)sinuMm=1

Im

eafm cos(f0fm)

ej(2pdspir eafm

sin ucos(f0fm)+dm)(15)

In the above equation, higher-order basis functions areneglected for simplicity. And thus

|AF(f0, f)| Mm=1

|Im| +2pdspir

f0|ff0|

Mm=1

|Im eafm | (16)

For uniform feed distribution, Im I0, some manipulationgives

|AF(f0, f)| 1+2pdspir

Mf0|ff0|

Mm=1

|eafm | (17)

Assume that the normalised main beam is defined by|AF(f0, f)| 1 1, where 1 is a small positive value. Thus

1 1|10 |AF(f0, f)| 1+2pd

spirMf0

|ff0|Mm=1

eafm

(18)

And thus one can easily write

|f0 f|f0

j2pdspir

Mm=1

eafm

1(19)

where j is a small positive value. Finally, the fractional

bandwidth of the spiral array can be expressed as

BWfract.Spiral =2|f0 f|

f0 jpdspir

Mm=1

eafm

1(20)

By applying the above method to the uniform linear array(ULA), we have

BWfract.ULA =2|f0 f|

f0 jpdlin

M

m

=1

m

1(21)

where dlin dlinear/l0 and dLinear is the element separation.After some manipulations, a spiral-to-linear bandwidth ratio(SLBWR) can be defined as below

SLBWR= (1/2)M(M+ 1)[e(ap/b)M 1]/(eap/b 1)

= (1/2)M(M+ 1)(eap/b 1)

[e(ap/b)M 1] , for r0 = dLinear

SLBWR=

Cs

2r0

M(M+ 1)(M 1)

(eap/b 1)

[e(ap/b)M 1],

for the same array lengths (22)



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Spiral Array Architecture, Design, Synthesis and Application