Research Article Design of Shaped Beam Planar Arrays of ...

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2013 Article ID 767342 12 pageshttpdxdoiorg1011552013767342

Research ArticleDesign of Shaped Beam Planar Arrays of WaveguideLongitudinal Slots

Giovanni Andrea Casula Giuseppe Mazzarella and Giorgio Montisci

Dipartimento di Ingegneria Elettrica ed Elettronica Universita di Cagliari Piazza drsquoArmi 09123 Cagliari Italy

Correspondence should be addressed to Giuseppe Mazzarella mazzarelladieeunicait

Received 26 October 2012 Revised 10 January 2013 Accepted 21 January 2013

Academic Editor Sembiam R Rengarajan

Copyright copy 2013 Giovanni Andrea Casula et alThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

The Elliottrsquos procedure for the design of a pencil beam waveguide longitudinal slot array has been generalized to encompass thedesign of shaped beam planar slot arrays An extended set of design equations taking into account in an operative way the feedingpart of the array has been devised From this set of equations a general and effective design procedure has been set up shedding lighton the constraints posed by a complex aperture distribution The results of the proposed synthesis procedure have been validatedthrough comparison with a commercial FEM software

1 Introduction

Planar arrays of waveguide slots (Figure 1) have a very longstory since their use dates back at least from the 40s [1] andare still a very popular choice for high-performance antennasystems especially in the higher part of microwave range [2]Therefore their pros and cons from both the mechanical andthe electromagnetic point of view are well known [3] Themain advantages of these antennas are the high efficiencythe polarization purity [2ndash4] a considerable mechanicalstrength a small size and the great accuracy achievableboth in the design and in the realization phase The abovefeatures make the use of these antennas an effective solutionin a wide range of applications [2] such as radar aerospaceand satellite antennas The most common drawbacks ofwaveguide slot arrays are the high realization cost the smalluseful bandwidth and the lack of flexibility since oncethe array is realized its electromagnetic behavior cannot bechanged (though some countermeasures are possible [5 6])

Many slot array configurations are available [2 7ndash15]differing for slot type slot pattern and feeding architecture

The most popular slot array configuration is the resonantarray of longitudinal shunt slots [1] where the slots spacingis half the guided wavelength (at the center or ldquoresonantrdquofrequency) in the slotted waveguide Therefore we considerhere this kind of array which allows quite general feeding

architectures A single array (or subarray) consists of 119872

parallel-slotted waveguides (called radiating waveguides)with a transverse feeding guide as in Figure 1 The feedingand radiating guides are coupled using 119872 series-series slotsSuch slots are still spaced half the guided wavelength in thefeeding waveguide which is therefore the array spacing inthe 119864-plane Since the array bandwidth depends on the arraysize a popular solution is to divide large arrays into subarrayseach one with its own matched input port In this case abeam-forming network (BFN) is required to properly feed thesubarrays A (quadrantal) subarray architecture is requiredalso for monopulse radar antennas

A slot array design splits naturally into an ldquointernalrdquoproblem and an ldquoexternalrdquo one The former is the physicaldesign of the array which realizes an aperture distributionfulfilling the array pattern requirements and whose inputport is well matched The latter is the computation of thisaperture distribution fulfilling the constraints imposed bythe slot array technology

Themost accuratemodel describing the behavior of a res-onant pencil beam and array of longitudinal slots has beenproposed by Elliott in [16] Elliottrsquos work [16] is renownedsince a highly accurate model of a slot array includingmutual coupling is set there for the first time Furthermore itcontains a second less developed though equally importantpoint As a matter of fact the last section of [16] suggests that

2 International Journal of Antennas and Propagation

Feeding waveguide

Input port

Radi

atin

g w

aveg

uide

s119909

119911

Figure 1 Geometry of an array or subarray

the system of nonlinear design equations of a slot array canbe solved in a very effective (and physically meaningful) wayevaluating all mutual couplings on the data of the previousiterative steps since in this way the nonlinear equations foreach slot can be decoupled Actually the decoupling detailsare not described in [16] also because they strongly dependon the array specifications As a matter of fact Elliott himselfpartly developed in [17 eq (56)ndash(58)] the decoupling detailsfor a planar equiphase array but without taking into accountthe feeding networkAfter that useful remarks on large arrayswith a pencil beam are reported in [18] and some results onshaped beam arrays [19] have been reported for the first timein [20 21] where the array is designed using an optimizationtechnique and subsequently in [22] using the Elliottrsquossynthesis procedure Unfortunately the design proceduresdescribed in [20ndash22] have been applied by the authors onlyto linear arrays but a large number of practical applicationscall for planar shaped beam arrays As a matter of fact mostof the antennas used for satellite applications radar systemsaerospace applications and telecommunication systems areusually designed to produce a shaped beam in order toilluminate a selected geographical area with amaximumgainThese antennas require a complex aperture distribution witha phase distribution spanning up to 360∘

The design of a planar array is a completely differentmatter compared to the design of a linear array Actually inthe planar case the radiating slots interact both through theexternal mutual coupling and through the complex feedingnetworkMoreover since the array geometry is not separablethe design of a shaped beam array is significantly differentfrom the design of an array with equiphase slot voltages

To the best of our knowledge a complete synthesisprocedure for a planar waveguide slot array with complexdistribution has not been described in international liter-ature Aim of this paper is to fill this gap providing aneffective and accurate design technique for a shaped beamplanar array which takes into account not only the externalmutual coupling but also the strong slot interaction due tothe feeding network

The first problem to face in the design of a waveguideslot array with a shaped beam pattern is that the excitationphase of each slot cannot span 360∘ A very limited solutionis to realize a shaped beam array with real excitations as

in [23] but the results are not satisfactory On the otherhand array pattern synthesis procedures able to take intoaccount excitations constraints have been proposed [24] andthese procedures could be exploited by suitably limiting theamplitude and phase variation of the slot excitations in orderto get an aperture distribution achievable with a waveguideslot array

In this work we propose an ldquointernalrdquo design procedurefor shaped beam planar arrays of waveguide longitudinalslots by extending Elliottrsquos model and devising from it a newsynthesis procedure This is not straightforward since theoriginal Elliottrsquos equations [17] must be properly modified totake into account a further degree of freedom namely thephase of the slot excitations In Sections 2 and 3 the newderived design equations and the array synthesis procedureare described in detail In Section 4 this procedure hasbeenvalidated and tested in a way independent of the Elliottrsquosmodel using a commercial FEM solver namely HFSS 13 byAnsoft The results obtained with this FEM CAD are in verygood agreement with experimental results as reported inthe open literature for a wide range of applications (see eg[25 26])

A number of shaped beam arrays with different patternshave been designed and two of them are discussed in detailThe analysis performed with HFSS shows that the presentedexamples fulfill the required specifications

2 Design Equations for Slot Arrays

The behavior of a planar slot array is ruled by a set of designequations linking the electrical variables of the array to thegeometrical ones These equations describe

(1) the slot excitation due to the radiating guide [16](2) the external mutual coupling between the slots [16](3) the interaction between the radiating slots due to the

BFN [27ndash29]

The interaction (3) is strongly frequency dependent There-fore since a resonant array is always designed at the centerfrequency only the corresponding equations at this fre-quency will be used in this work

We consider here a planar array composed by119872 radiatingwaveguides each one carrying a possibly different number119873119898

of radiating slots The first radiating slot of the 119898thradiating waveguide is indicated by 119868

119898 and the numbering

proceeds arbitrarily from right to left while 119898

and 119865119898

denote respectively the slot immediately to the right of thefeeding coupling slot and the last slot of the 119898th radiatingwaveguide as shown in Figure 2

This numbering allows to design arrays with regular orirregular aperture shapes in the same way In this referencesystem the array 119864-plane is vertical and the axis of theradiating waveguides is horizontal Each slot is completelycharacterized by its length and its offset with respect to thewaveguide axis which is assumed positive upward

An array of longitudinal slots can also be divided intosubarrays (as in the example in Figure 3) Each subarray ismade of a number of radiating waveguides fed by a feeding

International Journal of Antennas and Propagation 3

119909

119911119898minus1

119898

119898+1

119898minus1 minus 1

119898 minus 1

119898+1 minus 1

119898minus1 + 2

119898minus1 minus 1119898minus1

119898 + 2119898 + 1

119898119898 minus 1

119898+1 + 2 119898+1 + 1119898+1

119898+1 minus 1

119868119898minus1 + 1

119868119898minus1

119868119898 + 1

119868119898

119868119898+1 + 1

119868119898+1

Radiating waveguide 119898 minus 1

Radiating waveguide 119898

Radiating waveguide 119898 + 1

119898minus1 + 1

Figure 2 Radiating slots numbering

waveguide (orthogonal to the radiating ones) through asequence of series-series inclined coupling slot (one for eachradiating waveguide of the subarray) [30] Each feedingwaveguide is then fed at its input node through a series-series inclined coupling slot All the coupling slots have beenchosen resonantTherefore a generic array is composed byNradiating slots119872 radiating waveguides and119876 subarrays andhas consequently 119876 feeding waveguides and 119876 input portsIn the example shown in Figure 3(a) the array is dividedinto 4 subarrays Each subarray is composed by 4 radiatingwaveguides and the design procedure allows each radiatingwaveguide to contain a different number of radiating slotsFigure 3(b) shows the four waveguides each one feeding asubarray and the input port is shown for each subarray

Let 120584119898be the (TE

10fundamental) mode voltage on the

119898th radiating waveguide The array design equations can bewritten taking into account that the mode voltage 119881

119899at the

position of the 119899th radiating slot is different in each radiatingwaveguide and can be written as 119881

119899= (minus1)

119899minus119868119898120584119898 Therefore

the first two sets of design equations for a slot array are [17]

119884119860

119899

119866119877

= 1198951198701119891119899120590119899

119881119878

119899

119881119899

= 1198951198701119891119899120590119899

119881119878

119899

(minus1)119899minus119868119898120584119898

119884119860

119899

119866119877

=21205902

119899

119863119899

(1)

wherein the 119881119878

119899

are the slot excitations required by theaperture distribution and

1198701= minus

2

12057310

sdot120587

119886sdot radic

2

(119896 sdot 119886) (119896 sdot 119887) (2)

120590119899= sin

120587119909119899

119886 (3)

119899=

119884119899

1205902119899

(4)

119891119899= 119891 (119897

119899) =

(1205872119896119897119899) cos120573

10119897119899

(1205872119896119897119899)2

minus (12057310119896)2

(5)

119877119899= 119895120572

119873

sum

119895=1

119895 = 119899

119892119899119895

119881119878

119895

119881119878119899

(6)

119863119899=

2

119899119866119877

+1

1198912119899

119877119899 (7)

wherein 119886 and 119887 are the waveguide transverse dimensions119896 is the wavenumber in free space 119866

119877and 120573

10are the

equivalent admittance and the propagation constant of theTE10

fundamental waveguide mode and 119884119899 119897119899 and 119909

119899are

respectively the self-admittance the length and the offset ofthe 119899th slot of the array

In (6) 119892119899119895is the sum of the external coupling between

the radiating slots [16] and of the internal coupling due to theinteraction between the radiating slots through higher-orderwaveguide modes [31]

At the input node of the 119898th radiating waveguide(Figure 4) the inclined coupling slot feeding the waveguideis modeled (being resonant) as an ideal transformer with acurrent transformation ratio equal to 119862

119898[30 32ndash34]

The input impedance seen at the input of this series-seriestransformer is therefore

119885119898

=1198622

119898

119866119877

119865119898

sum

119899 =119868119898

119884119860

119899

119866119877

(8)

Since the mode voltages 120584119898are not independent for radiating

waveguides fed by the same feeding waveguide we musttake into account the equations of the feeding line Alsothe feeding waveguide is fed by a series-series transformerwith known input current Subsequent equations will bemore clear if we consider each half of the feeding guideas a separate guide With this convention two feedingwaveguides are represented in Figure 5 namely the 119902th and(119902 + 119876)th waveguides The index 119902 assumes therefore thevalues between 1 and 119876 where 119876 represents the number ofthe feeding waveguides of the array (namely the number

4 International Journal of Antennas and Propagation

Subarray 1 Subarray 2

Subarray 4 Subarray 3

(a)

Inputnode 1

Inputnode 4

Inputnode 2

Inputnode 3

Feeding waveguideof subarray 1

Feeding waveguideof subarray 2

Feeding waveguideof subarray 4

Feeding waveguideof subarray 3

(b)

Subarray 1 Subarray 2

Subarray 4 Subarray 3

Inputnode 1

Inputnode 4

Inputnode 2

Inputnode 3

(c)

Figure 3 (a) An example of subarrays structure (b) feedingwaveguides and input ports (c) complete array

of input ports of the array) Let 119868119902be the current flowing

into the last coupling slot of the 119902th feeding waveguide (thefarther from the feeding node) having the same direction of119868119902(see Figure 5) This coupling slot feeds the first radiating

waveguide whichwe denote by 119902The current flowing on the

119884119860119898+1 119881119871119898 119881119877119898

119868119877119898

119862119898

119884119860119898

1198810119898

1198680119898

1205821198924 1205821198924

119868119871119898

Figure 4 Input node of the 119898th radiating waveguide The twoadmittances model the radiating slots

119898

(on the right hand) and119898+1

(on the left hand)

119868119902

119881119902 119882119902119882119902+119876 119881119902+119876

119868119902+119876

A B1205872 1205872

sum119885sum119885

Figure 5 Input node of the feeding waveguide

subsequent coupling slots will be (minus1)119898minus119875119902119868119902 where

119902+ 1 le

119898 le 119902 and

119902represents the coupling slot corresponding

to the last radiating waveguide fed by the 119902th feeding guideThe current 119868

0119898(with 119898 =

119902) flowing on the first radi-

ating waveguide (see Figure 4 for the generic 119898th radiatingguide) fed by the 119902th feeding guide is therefore given by119862119898119868119902 while 119862

119898(minus1)119898minus119875119902119868119902is the current flowing on the other

radiating waveguides with 119902+ 1 le 119898 le

119902

The mode voltage at the slot 119898

(which as shown inFigure 4 is the radiating slot immediately at the right of thecoupling slot feeding the 119898th radiating waveguide) is thengiven by

119881119877119898

= minus119895

119866119877

(minus1198680119898

)

= 119862119898(minus1)119898minus119875119902119868119902

119895

119866119877

(9)

Since (from Figure 2) 119881119873119898

= 119881119877119898

= (minus1)119873119898minus119868119898120584119898 the mode

voltage on the 119898th radiating waveguide can be expressed as

120584119898

= (minus1)119873119898minus119868119898119881119873119898

= (minus1)119873119898minus119868119898119881119877119898

= [119862119898(minus1)119898minus119875119902119868119902

119895

119866119877

] (minus1)119873119898minus119868119898

(10)

Finally as indicated in Figure 5 the input impedance at theport 119860119861 is given by

119866119860119885IN119902

=1

119866119860sum[119902]

119885119898

+1

119866119860sum[119902+119876]

119885119898

119902 = 1 119876

(11)

where the notations [119902] [119902 + 119876] indicate that the sums areextended to all the radiating waveguides fed by the 119902th and

International Journal of Antennas and Propagation 5

(119902 + 119876)th feeding waveguides respectively (see Figure 5) 119866119860

is the equivalent admittance of the TE10

fundamental modein the feeding waveguide

3 Synthesis Procedure

In order to design a slot array we have to solve the nonlinearsystems (1) and (11) which require an iterative solution

The input data of the design procedure are the radiatingslot excitations (namely the 119873 slot voltages 119881

119878

119899

) and theinput impedances 119885

IN119902

at each input node of the array Theprocedure gives as output the lengths and offsets of all theradiating slots

Following Elliott suggestion [16] it is convenient toevaluate the mutual coupling coefficients 119877

119899 given by (6)

using the data of the previous iterative step since smallchanges in offsets and lengths cause only a small changein the mutual coupling With this choice the equations aredecoupled and it is possible to recompute the newparametersof each slot independently of the other slots

A shaped beam array requires a complex aperture dis-tribution therefore the Elliottrsquos design equations (1) must beproperly modified because a further set of requirements thephase of the slot excitations must be taken into account Onthe other hand no further degrees of freedom are availableso a different strategy must be devised in order to extend theElliottrsquos procedure [16] to the shaped beam case

Let

119881119878

119899

=10038161003816100381610038161003816119881119878

119899

10038161003816100381610038161003816119890119895120593

119878

119899 (12)

the slot voltage of the 119899th radiating slot of the arrayA complex slot voltage distribution such as (12) requires

that some other electrical quantities are complex Amongthem there are the feeding currents 119868

119902 We include also a sign

variable into their definition which reads

119868119902=

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816exp (119895120595

119902) 119878119902

119868119902+119876

= minus10038161003816100381610038161003816119868119902+119876

10038161003816100381610038161003816exp (119895120595

119902+119876) 119878119902+119876

(13)

where 119878119902= +minus1 is a sign to be determined and120595

119902is defined

by 120595119902= arc tan(Im[119868

119902]Re[119868

119902]) so that minus1205872 le 120595

119902le 1205872

With this choice in the limit case of a complex distributionbut with all phases of the slot voltages equal to zero (120593119904

119899

= 0)we come back to the equiphase case being 119868

119902= |119868119902|119878119902and

120595119902= 0 without ambiguitySince 119862

119898is a real number we get from (10)

120584119898

= [119862119898(minus1)119898minus119875119902

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816

119895

119866119877

]

times (minus1)119873119898minus119868119898 exp (119895120595

119902) 119878119902

(14)

As a consequence all the mode voltages 120584119898on the radiating

waveguides fed by the same feeding waveguide (identified bythe index 119902) are equiphase

The active admittance using (1) can therefore beexpressed as

119884119860

119899

119866119877

= 1198701119891119899120590119899

119866119877

10038161003816100381610038161003816119881119878

119899

10038161003816100381610038161003816

119862119898

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816

sdot (minus1)119873119898minus119899

sdot (minus1)119898minus119875119902

sdot exp [119895 (120593119878

119899

minus 120595119902)] 119878119902=

21205902

119899

119863119899

(15)

The input impedances 119885IN119902

required at the feeding nodes ofthe array are real numbers in almost all practical applica-tions Therefore it follows from (11) that (sum

[119902]

119885119898)minus1 and

(sum[119902+119876]

119885119898)minus1must have an opposite imaginary part As a

consequence the problem is not determined and the simplerchoice is to require that all the 119885

119898have a real value

Now from (15) it follows that

Im119863119899exp[119895 (120593

119878

119899

minus 120595119902)] = 0 119899 = 1 119873 (16)

as also found in [20 21] for the linear case (120595119902equiv 0)

The left-hand side of (16) depends only on the slot length119897119899 since the offsets are fixed to the values of the previous

iterative step Therefore (16) is the sought equation for thenew value of this length If

119899is the solution of (16) then

120575119899= 119863119899exp[119895 (120593

119878

119899

minus 120595119902)]119897119899=119897119899

119899 = 1 119873 (17)

is real Using (17) in (15) we obtain the following expressionfor the active admittances

119884119860

119899

119866119877

=21205902

119899

120575119899

exp[119895 (120593119878

119899

minus 120595119902)] 119899 = 1 119873 (18)

Then comparing (18) with (15) we get

120590119899= 1198701119891119899

120575119899

2

119866119877

10038161003816100381610038161003816119881119878

119899

10038161003816100381610038161003816

119862119898

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816

(minus1)119873119898minus119899

(minus1)119898minus119875119902119878119902 (19)

Finally by replacing (19) in (15) the active admittance can beexpressed as

119884119860

119899

119866119877

=120575119899

21198702

1

1198912

119899

[

119866119877

10038161003816100381610038161003816119881119878

119899

10038161003816100381610038161003816

119862119898

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816

]

2

exp[119895 (120593119878

119899

minus 120595119902)] (20)

If we put in (8) the active admittances given by (20) the inputimpedance119885

119898seen at the primary of the feeding transformer

can be written as

119885119898

= 1198702

1

119866119877

2

1

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816

2

sdot

119865119898

sum

119899 =119868119898

1205751198991198912

119899

10038161003816100381610038161003816119881119878

119899

10038161003816100381610038161003816

2

times exp[119895 (120593119878

119899

minus 120595119902)] 119898 = 1 119872

(21)

Since the input node is a series one the relation between thecurrents |119868

119902| and |119868

119902+119876| can be expressed as

(minus1)119872119902minus119875119902119878119902

119879119902

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816

=

(minus1)119872119902+119876minus119875119902+119876119878119902+119876

119879119902+119876

10038161003816100381610038161003816119868119902+119876

10038161003816100381610038161003816

119902 = 1 119876 (22)

6 International Journal of Antennas and Propagation

where

119879119902=

1

sum[119902]

1205751198991198912119899

1003816100381610038161003816119881119878

119899

1003816100381610038161003816

2 exp(119895120593119878119899

)

119879119902+119876

=1

sum[119902+119876]

1205751198991198912119899

1003816100381610038161003816119881119878

119899

1003816100381610038161003816

2 exp(119895120593119878119899

)

(23)

Let 119867119902+119876119902

be a real positive parameter defined by

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816= 119867119902+119876

119902

sdot10038161003816100381610038161003816119868119902+119876

10038161003816100381610038161003816 (24)

Using 119867119902+119876

119902

we can write

119879119902119867119902+119876

119902

= 119879119902+119876

(minus1)119872119902minus119875119902(minus1)119872119902+119876minus119875119902+119876119878119902119878119902+119876

(25)

From (25) it follows that 119879119902and 119879

119902+119876must have the same

phase (apart from the sign) once the convergence of theiterative design procedure has been reached

Finally we must fulfill the requirement on the inputimpedance 119885

IN119902

at the secondary of the transformer feedingthe waveguides 119902 and 119902+119876 This impedance is the sum of theinput impedances of the two waveguides

119866119860119885IN119902

= 119866119860119885eq119902

+ 119866119860119885eq119902+119876

(26)

which are (see (11) and (21)) as follows

119866119860119885eq119902

= [

[

1198702

1

119866119860119866119877

2

1

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816

2

timessum

[119902]

1205751198991198912

119899

10038161003816100381610038161003816119881119878

119899

10038161003816100381610038161003816

2

exp[119895 (120593119878

119899

minus 120595119902)]]

]

minus1

119902 = 1 119876

119866119860119885eq119902+119876

= [

[

1198702

1

119866119860119866119877

2

1

10038161003816100381610038161003816119868119902+119876

10038161003816100381610038161003816

2

times sum

[119902+119876]

1205751198991198912

119899

10038161003816100381610038161003816119881119878

119899

10038161003816100381610038161003816

2

exp[119895 (120593119878

119899

minus 120595119902)]]

]

minus1

(27)

Equations (27) can be simplified using (23) as follows

119866119860119885eq119902

=2

1198702

1

119866119860119866119877

10038161003816100381610038161003816119868119902

10038161003816100381610038161003816

2

119879119902exp(119895120595

119902)

119866119860119885eq119902+119876

=2

1198702

1

119866119860119866119877

10038161003816100381610038161003816119868119902+119876

10038161003816100381610038161003816

2

119879119902+119876

exp(119895120595119902+119876

)

(28)

The input impedance 119885IN119902

must have real and positive valueswhile119885eq

119902

and119885eq119902+119876

can be real or complex However we haveenough available degrees of freedom to force both 119885

eq119902

and

119885eq119902+119876

to have real and positive values With this choice wecan fix the phases 120595

119902and 120595

119902+119876as follows

120595119902= minus arg (119879

119902)

120595119902+119876

= minus arg (119879119902+119876

)

(29)

The input impedance 119885IN119902

can be finally expressed as

119866119860119885IN119902

=2

1198702

1

119866119860119866119877

times [10038161003816100381610038161003816119868119902

10038161003816100381610038161003816

2

119879119902exp(119895120595

119902) +

10038161003816100381610038161003816119868119902+119876

10038161003816100381610038161003816

2

119879119902+119876

exp(119895120595119902+119876

)]

=

210038161003816100381610038161003816119868119902

10038161003816100381610038161003816

2

1198702

1

119866119860119866119877

[

[

10038161003816100381610038161003816119879119902

10038161003816100381610038161003816+

10038161003816100381610038161003816119879119902+119876

10038161003816100381610038161003816

(119867119902+119876

119902)2

]

]

(30)

Equation (30) allows to determine |119868119902| from the required

value of 119885IN119902

thus terminating the iterative stepIt is worth noting that in order to avoid convergence

problems the initial values of 120595119902must be properly connected

to the values of the voltage distributionTherefore even in thefirst iterative step the phases of the active admittances mustbe kept relatively small avoiding problems of oscillating ortrapped solutions

The synthesis procedure proposed in this section has nolimitations by itself since it can design the array geometryfor every aperture distribution that a slot array can radiateOn the other hand the excitation phase achievable with alongitudinal radiating slot cannot span the whole 360∘ butis limited to

[minus120593MAX le 120593 le 120593MAX] cup [(180∘

minus 120593MAX)

le 120593 le (180∘

+ 120593MAX)] (31)

where 120593MAX depends slightly on the waveguide dimensionsbut is always not larger than 60∘ However to preventconvergence problems it can be safe to choose a smaller120593MAX for example 50∘

As a consequence an arbitrary voltage distributioncannot always be radiated by a slot array However sincedifferent aperture distributions can radiate essentially equiv-alent shaped patterns this ldquohardwarerdquo limitation can becircumvented using array pattern design techniques whichallow the introduction of appropriate constraints both forvoltage amplitudes and phases (compare [24])The amplitudeconstraints prevent the synthesis procedure to obtain toosmall slot lengths andor offsets while the phase constraintstake into account the limited excitation phase that each slotcan span

4 Results

In order to assess the synthesis procedure a number ofplanar arrays with different size and aperture distributions

International Journal of Antennas and Propagation 7

0

0 025 05 075minus025

minus3

minus6

minus9

minus12

minus15

minus18

minus21

minus24

minus27minus05minus075

0

025

05

075

minus025

minus05

minus075

ltminus30

Figure 6 Contour plot (dB scale) of the 8 times 8 planar array with acircular radiation pattern in the u-v plane

0

05075025 0

0025

05075

5 0 25 00

0

minus5minus10minus15minus20minus25

minus025minus025

minus05minus05

minus075 minus075

0minus25

minus5

minus75

minus10

minus125

minus15

minus175

minus20

minus225

minus25

ltminus275

Figure 7 3D radiation pattern (dB scale) of the 8 times 8 planar arraywith a circular radiation pattern in the u-v plane

have been designed with the procedure of Section 3 In-house softwares have been used to evaluate both the slot self-admittance [35] and the mutual coupling [36 37]

Once the geometry of the designed array has beendetermined an analysis has been performed to checkwhetherthe array requirements are fulfilledThis has been done usingboth [28] and a commercial FEM solver namely HFSS Sincethe former is based on Elliottrsquos model while the latter isindependent of it and is considered essentially equivalent toexperimental verification [25 26] we present here only theHFSS results which fulfill all the requirements and thereforefully assess our procedure It is worth noting that the resultsobtained by the procedure of [28] are equivalent to the onessimulated with HFSS

The architecture of the arrays presented in this sectionis shown in Figure 1 where both the radiating waveguidesand the feeding waveguide are half-heightWR90 waveguides(2286mm times 508mm)with 1mmwall thicknessThe feeding

03

04

05

06

07

08

09

02

01

0 0 025 05 075minus025minus05minus075

0

025

05

075

minus025

minus05

minus075

gt1

Figure 8 Differences between the required and designed radiationpattern (both in dB scale) for the 8 times 8 planar array with a circularradiation pattern in the u-v plane

waveguide has been fed at its side by a waveguide port andthe radiating waveguides have been fed through a series-series inclined resonant coupling slot All the coupling slotshave a length equal to 1707mm a width of 15mm andthe tilt angle with respect to the feeding waveguide axis is45∘ corresponding to a coupling coefficient equal to 1 (see[30 32 33] for details)

Starting from a specified and arbitrary-shaped beampattern we have used the array patterns synthesis proce-dure described in [24] to compute the required excitationsAccording to the considerations made at the end of Section 3about the excitations achievable with a longitudinal radiatingslot appropriate constraints both on the amplitude and onthe phase of the slot excitations are required in order toget an aperture distribution achievable with an array ofslots In particular the maximum phase and the minimumnormalized amplitude of the slot excitations have been setrespectively to 120593MAX = 50

∘ and |119881119878min| = 01

In the first example we present an 8 times 8 planar arrayfed by a single feeding waveguide containing 8 coupling slotsdesigned requiring a circular pattern with a radius equal to025 in the (119906 119907) plane with minus20 dB sidelobes and a rippleof plusmn05 dB The normalized amplitudes and the phases of therequired slot excitations are reported in Tables 1(a) and 1(b)respectively while the corresponding designed slot lengthsand offsets are shown in Tables 1(c) and 1(d)

The contour plot of the simulated (HFSS) far field patternfor the designed array is shown in Figure 6 In Figure 7the 3D far field pattern is depicted and Figure 8 shows thedifference between the pattern obtained using the requiredslot voltages and the designed pattern In the shaped regionthis difference is less than 03 dB Figure 9 shows an enlarge-ment of the shaped region with a ripple of about plusmn05 dBThe frequency response of the array is shown in Figure 10

8 International Journal of Antennas and Propagation

005 015 025

005

01

015

02

025

0

0

minus025minus025

minus02

minus015

minus015

minus01

minus005

minus005

minus08

minus06

minus04

minus02

ltminus1

Figure 9 Ripple of the 8 times 8 planar array with a circular radiationpattern in the u-v plane (expressed in dB)

0

885 89 895 9 905 91 915

Frequency (GHz)

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45

S11

mod

ule (

dB)

10 times 10 arrow8 times 8 circle

Figure 10 Simulated frequency responses (HFSS) of the designedarrays

while Figures 11 and 12 show respectively the 119864-plane andthe 119867-plane far field patterns within the working frequencybandwidth The behaviour of the shaped radiation pattern isstill very good even at the upper and at the lower ends of thebandwidth

The second example is a 10 times 10 planar array fedby a single feeding waveguide containing 8 coupling slotsdesigned requiring an arrow-shaped pattern (see the dashedline in Figure 16 for the required geometry) with minus15 dBsidelobes The normalized amplitudes and the phases of therequired slot excitations are reported in Tables 2(a) and 2(b)and the corresponding designed slot lengths and offsets areshown in Tables 2(c) and 2(d)

Table 1 (a) Required normalized amplitude of slots excitations forthe 8 times 8 planar array with a circular radiation pattern (b) Requiredphase (degrees) of slots excitations for the 8 times 8 planar array with acircular radiation pattern (c) Designed slots lengths (mm) for the8 times 8 planar array with a circular radiation pattern (d) Designedslots offsets (mm) for the 8 times 8 planar array with a circular radiationpattern

(a)

Radiatwaveg

Slot1

Slot2

Slot3

Slot4

Slot5

Slot6

Slot7

Slot8

1 010 010 015 016 016 015 010 0102 010 018 017 010 010 017 018 0103 015 017 019 049 049 019 017 0154 016 010 049 100 100 049 010 0165 016 010 049 100 100 049 010 0166 015 017 019 049 049 019 017 0157 010 018 017 010 010 017 018 0108 010 010 015 016 016 015 010 010

(b)

Radiatwaveg

Slot1

Slot2

Slot3

Slot4

Slot5

Slot6

Slot7

Slot8

1 1299 109 36 00 00 36 109 12992 109 65 193 479 479 193 65 1093 36 193 1299 1394 1394 1299 193 364 00 479 1394 1491 1491 1394 479 005 00 479 1394 1491 1491 1394 479 006 36 193 1299 1394 1394 1299 193 367 109 65 193 479 479 193 65 1098 1299 109 36 00 00 36 109 1299

(c)

Radiatwaveg

Slot1

Slot2

Slot3

Slot4

Slot5

Slot6

Slot7

Slot8

1 1604 1558 1582 1624 1624 1582 1558 16042 1544 1598 1549 1729 729 1549 1598 15443 1623 1482 1617 1627 1627 1617 1482 16234 1592 1347 1685 1616 1616 1685 1347 15925 1589 1317 1633 1634 1634 1633 1317 15896 1574 1524 1671 1616 1616 1671 1524 15747 1628 1560 1563 731 1731 1563 1560 16288 1655 1597 1570 1627 1627 1570 1597 1655

(d)

Radiatwaveg

Slot1

Slot2

Slot3

Slot4

Slot5

Slot6

Slot7

Slot8

1 minus019 minus020 027 minus012 012 minus027 020 0192 minus015 026 minus033 minus021 021 033 minus026 0153 008 minus060 minus013 058 minus058 013 060 minus0084 minus029 086 067 minus105 105 minus067 minus086 0295 024 minus118 minus064 111 minus111 064 118 minus0246 minus020 049 032 minus048 048 minus032 minus049 0207 006 minus025 039 025 minus025 minus039 025 minus0068 030 021 minus025 012 minus012 025 minus021 minus030

The contour plot of the simulated (HFSS) far field patternfor the designed array is shown in Figure 13 In Figure 14

International Journal of Antennas and Propagation 9

Table 2 (a) Required normalized amplitude of slots excitations for the 10 times 10 planar array with an arrow-shaped radiation pattern (b)Required phase (degrees) of slots excitations for the 10 times 10 planar array with an arrow-shaped radiation pattern (c) Designed slots lengths(mm) for the 10 times 10 planar array with an arrow-shaped radiation pattern (d) Designed slots offsets (mm) for the 10 times 10 planar array withan arrow-shaped radiation pattern

(a)

Radiat waveg Slot 1 Slot 2 Slot 3 Slot 4 Slot 5 Slot 6 Slot 7 Slot 8 Slot 9 Slot 101 0229 0224 0206 0143 0100 0100 0110 0211 0238 02062 0299 0289 0237 0100 0100 0100 0190 0301 0336 03063 0249 0235 0102 0100 0198 0323 0438 0467 0390 03064 0122 0100 0240 0428 0592 0696 0723 0634 0460 02825 0237 0285 0525 0764 0926 1000 0969 0791 0549 03106 0237 0285 0525 0764 0925 0999 0968 0790 0549 03107 0120 0100 0241 0429 0591 0695 0721 0633 0459 02818 0249 0233 0102 0100 0202 0326 0439 0468 0390 03079 0299 0287 0234 0100 0100 0100 0190 0301 0336 030510 0228 0223 0207 0144 0100 0100 0112 0212 0239 0206

(b)

Radiat waveg Slot 1 Slot 2 Slot 3 Slot 4 Slot 5 Slot 6 Slot 7 Slot 8 Slot 9 Slot 101 185 000 minus138 minus299 417 506 506 506 506 3762 minus259 minus246 minus280 minus320 505 506 506 506 506 5063 minus359 minus421 minus92 1662 minus179 1434 1306 1306 1306 13064 386 1757 minus1719 minus1787 1736 1642 1553 1490 1452 14835 1306 1469 1715 1776 1766 1718 1671 1659 1679 minus1776 1306 1473 1717 1777 1766 1718 1671 1658 1678 minus1777 385 1765 minus1718 minus1785 1736 1643 1554 1490 1452 14828 minus354 minus417 minus89 1660 minus179 1440 1306 1306 1306 13069 minus256 minus243 minus277 minus318 505 506 506 506 506 50610 183 minus021 minus141 minus300 415 506 506 506 506 386

(c)

Radiat waveg Slot 1 Slot 2 Slot 3 Slot 4 Slot 5 Slot 6 Slot 7 Slot 8 Slot 9 Slot 101 1607 1581 1628 1694 1550 1493 1604 1563 1504 15782 1647 1609 1624 1786 1517 1537 1575 1581 1544 14943 1666 1623 1626 1616 1575 1665 1661 1644 1675 16814 1496 1595 1612 1610 1618 1623 1638 1641 1627 16775 1698 1652 1615 1625 1614 1623 1631 1623 1630 15986 1704 1638 1629 1613 1624 1620 1626 1635 1613 16207 1517 1632 1591 1626 1607 1627 1642 1627 1648 16518 1642 1644 1584 1574 1594 1655 1658 1660 1649 17139 1668 1595 1641 1735 1560 1511 1586 1582 1527 154210 1582 1599 1617 1708 1524 1520 1592 1555 1524 1541

(d)

Radiat waveg Slot 1 Slot 2 Slot 3 Slot 4 Slot 5 Slot 6 Slot 7 Slot 8 Slot 9 Slot 101 019 minus013 021 minus015 010 minus021 011 minus022 040 minus0282 minus032 013 minus014 015 minus019 011 minus021 028 minus042 0693 029 minus014 003 008 minus007 031 minus042 029 minus041 0394 minus034 minus015 016 minus027 038 minus048 055 minus044 027 minus0305 minus034 022 minus036 045 minus066 072 minus069 058 minus038 0306 032 minus023 034 minus049 060 minus074 069 minus056 043 minus0237 029 013 minus018 026 minus039 045 minus060 040 minus031 0308 minus028 019 minus001 minus004 008 minus031 038 minus039 029 minus0459 033 minus010 018 minus010 015 minus015 016 minus030 048 minus05210 minus023 012 minus020 018 minus012 017 minus012 025 minus036 032

10 International Journal of Antennas and Propagation

0

0 20 40 60 80Angel from broadside (degrees)

Far fi

eld

mod

ule (

dB)

minus10

minus15

minus20

minus25

minus30

minus35minus40 minus20minus60minus80

minus5

9GHz905GHz

895GHz

Figure 11 Simulated 119864-plane far field of the 8 times 8 array with acircular radiation pattern

0

0 20 40 60 80Angel from broadside (degrees)

Far fi

eld

mod

ule (

dB)

minus10

minus15

minus20

minus25

minus30

minus35minus40 minus20minus60minus80

minus5

9GHz905GHz

895GHz

Figure 12 Simulated 119867-plane far field of the 8 times 8 array with acircular radiation pattern

the 3D far field pattern is depicted and Figure 15 shows thedifference between the pattern obtained using the requiredslot voltages and the designed one In the shaped region thisdifference is less than 02 dB Figure 16 shows an enlargementof the shaped region Finally the frequency response ofthe array is shown in Figure 10 while Figures 17 and 18show respectively the 119864-plane and the 119867-plane far fieldpatterns in the array frequency bandwidth Also in this casethe behaviour of the shaped radiation pattern remains verysatisfactory even at the upper and at the lower ends of theuseful bandwidth

0

0 025 05 075minus025

minus3

minus6

minus9

minus12

minus15

minus18

minus21

minus24

minus27minus05minus075

0

025

05

075

minus025

minus05

minus075

ltminus30

Figure 13 Contour plot (dB scale) of the 10 times 10 planar array withan arrow-shaped radiation pattern in the u-v plane

05075

025 0

0

0 02505 075

minus025

minus5minus10minus15minus20minus25

minus025minus05 minus05minus075 minus075

0minus25

minus5

minus75

minus10

minus125

minus15

minus175

minus20

minus225

minus25

25 0 0 00 25

ltminus275

Figure 14 3D radiation pattern (dB scale) of the 10times10 planar arraywith an arrow-shaped radiation pattern in the u-v plane

03

04

05

06

07

08

09

02

01

0 0 025 05 075minus025minus05minus075

0

025

05

075

minus025

minus05

minus075

gt1

Figure 15 Differences between the required and designed radiationpattern (both in dB scale) for the 10times10 planar array with an arrow-shaped radiation pattern in the u-v plane

International Journal of Antennas and Propagation 11

005 015 025

005

01

015

02

025

0

minus025minus025

minus02

minus015

minus015

minus01

minus005

minus005

0

minus03

minus06

minus09

minus12

minus15

minus18

minus21

minus24

minus27ltminus3

Figure 16 Ripple of the 10 times 10 planar array with an arrow-shapedradiation pattern in the u-v plane (expressed in dB)

0

0 20 40 60 80Angel from broadside (degrees)

Far fi

eld

mod

ule (

dB)

minus10

minus15

minus20

minus25

minus30minus40 minus20minus60minus80

minus5

9GHz905GHz

895GHz

Figure 17 Simulated 119864-plane far field of the 10 times 10 array with anarrow-shaped radiation pattern

The results of the performed simulations both at thedesign frequency and within the operating frequency band-width of the designed arrays are in very good agreementwith the required specifications and this fully validates theproposed synthesis procedure

5 Conclusion

The use of shaped beam waveguide slot arrays is requiredin various antenna applications such as radar and aerospaceapplications A synthesis procedure for shaped beam planarslot arrays has been presented Starting from the well-knownElliottrsquos model for a (pencil-beam) slot array an extendedset of design equations has been set up to include both the

0

0 20 40 60 80Angel from broadside (degrees)

Far fi

eld

mod

ule (

dB)

minus10

minus15

minus20

minus25

minus30

minus35

minus40 minus20minus60minus80

minus5

9GHz905GHz

895GHz

Figure 18 Simulated 119867-plane far field of the 10 times 10 array with anarrow-shaped radiation pattern

feeding guide interaction between radiating slots and theprovision of complex aperture distribution Then a designprocedure for shaped beam planar arrays has been devisedand assessed through validation against a commercial FEMsoftware

Acknowledgments

The authors would like to thank the associate editor professorSembiamR Rengarajan and the reviewers of this paperTheircomments have added much to the quality of this paperhelping the authors to clarify out thought

References

[1] R J Stegen ldquoSlot radiators and arrays at X-bandrdquo IEEETransactions on Antennas and Propagation vol 1 pp 62ndash641952

[2] R S Elliott Antenna Theory and Design Prentice-Hall NewYork NY USA 1981

[3] S R Rengarajan L G Josefsson and R S Elliott ldquoWaveguide-fed slot antennas and arrays a reviewrdquo Electromagnetics vol 19no 1 pp 3ndash22 1999

[4] G Montisci ldquoDesign of circularly polarized waveguide slotlinear arraysrdquo IEEE Transactions on Antennas and Propagationvol 54 no 10 pp 3025ndash3029 2006

[5] G Montisci and G Mazzarella ldquoFull-wave analysis of awaveguide printed slotrdquo IEEE Transactions on Antennas andPropagation vol 52 no 8 pp 2168ndash2171 2004

[6] G A Casula G Mazzarella and G Montisci ldquoDesign ofslot arrays in waveguide partially filled with dielectric slabrdquoElectronics Letters vol 42 no 13 pp 730ndash731 2006

[7] R C Johnson and H Jasik Antenna Engineering HandbookMcGraw-Hill New York NY USA 2nd edition 1984

12 International Journal of Antennas and Propagation

[8] J Hirokawa M Ando and N Goto ldquoWaveguide-fed parallelplate slot array antennardquo IEEE Transactions on Antennas andPropagation vol 40 no 2 pp 218ndash223 1992

[9] J Hirokawa and M Ando ldquoEfficiency of 76-GHz post-wallwaveguide-fed parallel-plate slot arraysrdquo IEEE Transactions onAntennas and Propagation vol 48 no 11 pp 1742ndash1745 2000

[10] S Park Y Tsunemitsu J Hirokawa and M Ando ldquoCenterfeed single layer slotted waveguide arrayrdquo IEEE Transactions onAntennas and Propagation vol 54 no 5 pp 1474ndash1480 2006

[11] S Costanzo G A Casula A Borgia et al ldquoSynthesis of slotarrays on integrated waveguidesrdquo IEEE Antennas and WirelessPropagation Letters vol 9 pp 962ndash965 2010

[12] G A Casula G Mazzarella and G Montisci ldquoA truncatedwaveguide fed by a microstrip as radiating element for highperformance automotive anti-collision radarsrdquo InternationalJournal of Antennas and Propagation vol 2012 Article ID983281 9 pages 2012

[13] Z Jin G Montisci G Mazzarella M Li H Yang and G ACasula ldquoEffect of a multilayer dielectric cover on the behaviorof waveguide longitudinal slotsrdquo IEEE Antennas and WirelessPropagation Letters vol 11 pp 1190ndash1193 2012

[14] GMontisci GMazzarella andG A Casula ldquoEffective analysisof a waveguide longitudinal slot with cavityrdquo IEEE Transactionson Antennas and Propagation vol 60 pp 3104ndash3110 2012

[15] G A Casula and G Montisci ldquoDesign of dielectric-coveredplanar arrays of longitudinal slotsrdquo IEEE Antennas andWirelessPropagation Letters vol 8 pp 752ndash755 2009

[16] R S Elliott ldquoAn improved design procedure for small arrays ofshunt slotsrdquo IEEE Transactions on Antennas and Propagationvol 31 no 1 pp 48ndash53 1983

[17] R S ElliottThe Design of Waveguide-Fed Slot Arrays edited byY T Lo and S W Lee Van Nostrand Rheinhold New YorkNY USA 1988

[18] G Mazzarella and G Panariello ldquoDesign of slot arrays for SARapplicationsrdquo Alta Frequenza vol 55 no 6 pp 359ndash364 1986

[19] IEEE Standard definitions of terms for antennas IEEE Std 145-1993

[20] R V Gatti and R Sorrentino ldquoSlotted waveguide antennaswith arbitrary radiation patternrdquo in Proceedings of the 34thEuropean Microwave Conference pp 821ndash824 AmsterdamTheNetherlands October 2004

[21] R V Gatti L Marcaccioli and R Sorrentino ldquoDesign of slottedwaveguide arrays with arbitrary complex slot voltage distribu-tionrdquo in IEEEAntennas and Propagation Society Symposium pp3265ndash3268 Monterey Calif USA June 2004

[22] G A Casula G Mazzarella and G Montisci ldquoShaped beamsynthesis technique for linear arrays of waveguide longitudinalslotsrdquo in Proceedings of the IEEE AP-S International SymposiumChicago Ill USA 2012

[23] F Ares R S Elliott and E Moreno ldquoSynthesis of shaped line-source antenna beams using pure real distributionsrdquo ElectronicsLetters vol 30 no 4 pp 280ndash281 1994

[24] G Franceschetti G Mazzarella and G Panariello ldquoArraysynthesis with excitation constraintsrdquo IEE Proceedings H vol135 no 6 pp 400ndash407 1988

[25] K W Leung and L Y Chan ldquoThe probe-fed zonal slot antennacut onto a cylindrical conducting cavityrdquo IEEE Transactions onAntennas and Propagation vol 53 no 12 pp 3949ndash3952 2005

[26] S R Rengarajan M S Zawadzki and R E HodgesldquoWaveguide-slot array antenna designs for low-average-sidelobe specificationsrdquo IEEE Antennas and PropagationMagazine vol 52 no 6 pp 89ndash98 2010

[27] M Hamadallah ldquoFrequency limitations on broad-band perfor-mance of shunt slot arraysrdquo IEEE Transactions on Antennas andPropagation vol 37 no 7 pp 817ndash823 1989

[28] G A Casula and G Mazzarella ldquoA direct computation ofthe frequency response of planar waveguide slot arraysrdquo IEEETransactions on Antennas and Propagation vol 52 no 7 pp1909ndash1912 2004

[29] J C Coetzee J Joubert and D A McNamara ldquoOff-center-frequency analysis of a complete planar slotted-waveguide arrayconsisting of subarraysrdquo IEEE Transactions on Antennas andPropagation vol 48 no 11 pp 1746ndash1755 2000

[30] S R Rengarajan ldquoAnalysis of a centered-inclined waveguideslot couplerrdquo Transactions on Microwave Theory and Tech-niques vol 37 pp 884ndash889 1989

[31] R S Elliott and W R OrsquoLoughlin ldquoThe design of slot arraysincluding internal mutual couplingrdquo IEEE Transactions onAntennas and Propagation vol 34 pp 1149ndash1154 1986

[32] S R Rengarajan and G M Shaw ldquoAccurate characterization ofcoupling junctions in waveguide-fed planar slot arraysrdquo IEEETransactions on Microwave Theory and Techniques vol 42 no12 pp 2239ndash2248 1994

[33] G Mazzarella and G Montisci ldquoWideband equivalent circuitof a centered-inclined waveguide slot couplerrdquo Journal ofElectromagnetic Waves and Applications vol 14 no 1 pp 133ndash151 2000

[34] G A Casula G Mazzarella and G Montisci ldquoEffect of thefeedingT-junctions in the performance of planarwaveguide slotarraysrdquo IEEE Antennas andWireless Propagation Letters vol 11pp 953ndash956 2012

[35] G Mazzarella and G Montisci ldquoRigorous analysis of dielectric-covered narrow longitudinal shunt slots with finite wall thick-nessrdquo Electromagnetics vol 19 no 5 pp 407ndash418 1999

[36] G Mazzarella and G Panariello ldquoOn the evaluation of mutualcoupling between slotsrdquo IEEE Transactions on Antennas andPropagation vol 35 no 11 pp 1289ndash1293 1988

[37] Z Jin G Montisci G A Casula H Yang and J Lu ldquoEfficientevaluation of the external mutual coupling in dielectric coveredwaveguide slot arraysrdquo International Journal of Antennas andPropagation vol 2012 Article ID 491242 7 pages 2012

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