antenna array analysis and synthesis

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20

Antenna ArrayAnalysis and Synthesis

The topics in this chapter include:

� Retarded potentials

� Array structures

� Weighting functions

� Actual source antennas

� Equivalent source antenna

� Linear array analysis

� Different forms of linear arrays

� Linear array synthesis

� Schelkunoff unit circle

� Sum and difference patterns

� Dolph-Chebyshev synthesis of sumpatterns

� Taylor synthesis of sum patterns

� Bayliss synthesis of difference patterns

� Planar array synthesis

20–1 Introduction

This chapter deals with some basic aspects of array antenna analysis and synthesis. Theapproach adopted involves systematic building up of concepts ultimately leading to antennadesign. Since the matter involves many aspects only the most essential ones are brieflydescribed. In order to understand the array analysis, first the concepts relating to retardedpotentials are introduced, followed by introduction toweighting functions for Actual SourceAntennas and Equivalent Source Antennas. These weighting functions are then related toarray factors, which are taken to be the basis for antenna analysis and subsequently forsynthesis. Initially the analysis problem is directly addressed in terms of array factor. Laterthe application of Schelkunoff Unit Circle, which is related to the array factor, has beenexplored. Similarly there are many approaches to synthesize an antenna array. Other aspectsof antennas that need to be addressed, viz. sum and difference patterns, beam width andlocation of the main lobe, location of nulls, number, levels and locations of side lobes,directivity of antenna, etc. Since all the above aspects cannot be addressed in a singleintroductory chapter, the matter is covered only to the extent of introducing the subject. Fordetailed study reader is advised to go through the references given at the end of the chapter.

863

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864 Chapter 20 Antenna Array Analysis and Synthesis

20–2 Retarded Potentials

The analysis of all antenna problems normally revolves around certain field quantities. Themost important among them ismagnetic vector potential denoted by ‘A’, which is definedin terms of Magnetic flux density by:

B ≡ ∇ × A (1)

The quantity A in itself is defined in terms of a source (current density J Amp/m2) by:

A =∫v

µJdv4πR

Since, Idl = kds = Jdv, where I is the current in Amps, k is the linear current density inAmp/m, dl, ds and dv are the elemental values of length, surface and volume respectively,A can also be written as:

A =∫v

µJdv

4πR=

∫s

µkds

4πR=

∫l

µIdl

4πR(2)

Parameter A satisfies steady magnetic filed Laws (i.e. Biot–savart Law and Ampere’sCircuital Law)

If the field source is time varying, (2) can be written as

A =∫v

µ[J]dv4πR

(3)

Where [J] indicates that every t appearing in the expression for J has been replaced bya retarded time t ′ where,

t ′ = t − R

v(4)

In view of (1)A can be related to other field quantities (viz.H, E, D) and Poynting vectorP by using the Maxwell’s equations and other relations:

H = B/µ (5)

∇ × H = ε∂E/∂t, (for free space) σ = 0, or J = 0) (6)

D = εE (7)

|E / H| = η =√(µ/ε) (8)

and

P = E × H (9)

In all above equations µ and ε are the conventional media parameters, v is the velocityof wave propagation and R is the distance from source to the point at which field is to beestimated

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20–2 Retarded Potentials 865

A also satisfies the equation:

∇2A = −µJ + µε∂2A∂t2

(10)

If field is time invariant (10) reduces to common form of Poisson’s equation:

∇2A = −µJ (11)

If J is also zero, it reduces to Laplace’s equation:

∇2A = 0 (12)

In view of Helmholtz Theorem, “any vector field due to a finite source is specifieduniquely if both the curl and the divergence of the field are specified”. Since the curl of Ahas already been specified by (1), vector A can be specified uniquely only if its divergenceis also specified. The problem is resolved by using Lorentz Gauge condition, which relateselectric scalar potential � with ∇· A by:

∇ · A = −µε∂�∂t

(13)

A is further related to E and � throught the equation

E = −∇�− ∂A

∂t(14)

∇2� = −ρv

ε+ µε

∂2�

∂t2(15)

Since A and � are related by (13) (provided both are time variant), both or any of thesequantities may be used for the study of antenna behaviour. The magnetic vector potential‘A’ however, finds more favour with antenna theory since it more conveniently leads to theother field quantities.

Normally field problems are solved in terms of electric currents and charges and all theabove equations play their appropriate role. However, in many applications use of fictitiousmagnetic currents and charges is also useful and all the above equations need to be re-writtenin terms of such fictitious sources. The two sets of some of the above equations are producedbelow:

1. If only Electric currents (J ) and Electric charges (ρ) exists

∇ × He = J + ε∂Ee/∂t (16)

∇ × Ee = −µ∂He/∂t (17)

Be = ∇ × A (18)

Ee = −∇�+ ∂A/∂t (19)

2. If only Magnetic currents (Jm) and Magnetic charges (ρm) exists

∇ × Hm = εEm (20)

∇ × Em = −µHm − Jm (21)

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LINEAR ARRAY (1D)

PLANAR ARRAY (2D)FS ARRAY (3D)

Figure 20–1

Dm = ∇ × F (22)

Hm = −∇ + ∂F/∂t (23)

In above equations prefix e and m refer to fields due to electric and magnetic sources.

3. It is often required to solve problems having both electric and magnetic distributions.This is the case when both electric and magnetic sources simultaneously exist. Two ofthe equations to be satisfied by E and H (without prefixes) can be written as:

E = −∇�+ ∂A/∂t − (1/ε)∇ × F (24)

H = −∇ + ∂F/∂t − (1/µ)∇ × A (25)

Similarly other equations can be accordingly modified.

In view of the assumption of sinusoidal time variation, all field quantities can be char-acterized by ejωt . Further, all spatial variations are characterized by e−γR , where γ isthe propagation constant. The parameter γ normally consists two factors namely attenu-ation constant α and phase shift constant β (sometimes also referred as wave number k,k = β = ω/v = 2π/λ) and is written as γ = α + jβ. In case of perfect dielectric α = 0.For free space α is usually neglected and spatial variation is characterized only be e−jβR .

20–3 Array Structures

An array may be defined as a collection of elements. It may be composed of large numberof elements, which may run from few to thousands and are limited by practical constraints.The elements in an array may be Dipoles, Polyrods, Helix, Spirals, Log Periodic Struc-tures, Slots or Horns. Dipoles and Slots generate broad band pattern, and are widely usedin simple structures. Polyrods, Helices, Spirals and Log Periodic Structures have moredirectivity. Slots and Horns are used at UHF and Microwave frequencies and are easy toconstruct. The purpose of using array antennas is to achieve more directivity and to generate

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20–3 Array Structures 867

( ) Cylindricala ( ) Sphericalb

Figure 20–2Elements forming Curved Planes

steereable beams. Arrays may be made to generate many search beams, tracking beams orsimultaneously both.

Arrays may be linear (one-dimensional), planar (two-dimensional) or frequency sen-sitive (FS) (three-dimensional) configurations. Figures 20–1 and 20–2 illustrate some suchconfigurations. In linear arrays the elements are aligned along a line. Planar arrays maybe Flat (square or rectangular), Curved (circular, semi-cylindrical/cylindrical or semi-spherical/spherical) in shape. The elements may be arranged into different types of grids asshown in Fig. 20–3. Rectangular grids, square grids and circular grids are, however, morecommonly used. Rectangular grids may be used with rectangular or circular boundaries.Similarly circular grids may also be used with either type of boundaries. However, rect-angular grids are more commonly used with rectangular boundaries and circular grid withcircular boundaries. These are illustrated in Fig. 20–4.

Rectangular planar arrays generate Fan-shaped beams. Square and Circular planararrays generate Pencil shaped beams. Planar arrays have lower side lobes (SL) than curved,due to superior illumination. Planar arrays require less maintenance. Planar arrays can scanangle (off broadside) limited to 60◦−70◦, thus several faces are needed for full hemisphericalcoverage

In arrays the elements are normally equispaced. Non-equispaced structures are sometimes also used. Element spacing governs lobe grating. There are certain advantages ofnon-equispaced structures, e.g. these require less number of elements for comparable beamwidth.

The major problems with arrays include mutual coupling between elements resultingin changed radiation pattern and radiation resistance. Large coupling results in poor pattern,raised side-lobes and mismatching with transmitting or receiving circuits.

Each element in an array acts as a source or radiator. As long as the element size isso small that it can be regarded as a point source the problem of evaluation of distancesbetween different elements or an element and the field point remains simple. The momentsize increases, the radiators will be regarded as collection of point sources, each pointsource having its different geometrical location. Besides, if the shape of the radiator is alsoarbitrary the problem further complicates. In such a situation the current distribution on these

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Square grid square cell Square grid square cell Square grid rectanglar cell

Triangular grid hexagonal cell

Triangular grid square cell

Triangular grid rectanglar cell

Figure 20–3

Rectangular BoundaryRectangular Grid(commonly used)

( )a

Rectangular BoundaryCircular Grid

Elements arranged in different forms

Circular BoundaryRectangular Grid

Circular BoundaryCircular Grid

(commonly used)( )d( )c( )b

Figure 20–4

elements may or may not be accurately known. If the actual current distribution on radiatorsis accurately known or assumed to be so, the antennas composed of such radiators can beclassified as Actual Source Antennas. This class of radiators include dipoles and helices.Alternatively, if the assumption for current distribution used in actual source antennas doesnot hold well but for which the close-in fields can be accurately described, for estimationof field distribution actual sources can be replaced by equivalent sources. Such antennasare termed as Equivalent Source Antennas. Radiators for this case may include slots and

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20–4 Weighting Functions 869

horns. These (equivalent) sources must be so chosen that they produce the same fields at allpoints exterior to the surfaces as the actual antennas does.

Thus for the purpose of analysis, antenna arrays can be broadly classified into(l) Actual Source Antennas and(2)Equivalent Source Antennas. Themethod of analysis for these is briefly described

in the following section.

20–4 Weighting Functions

20–4a Case - I: Actual Source Antennas

Consider Fig. 20–5, which shows an antenna array containing a number of elements num-bered 1, 2, . . . , i, j, . . .. Each of these radiators is composed of a large number of pointsources A,B,C, . . . , P ,Q . . .. These point sources confined to some finite volume V areassumed to be oscillating harmonically with time at an angular frequency ω.

Also assume the origin to lie within the volume V . The volume V that contains all thesources is dimensionally much smaller in comparison to the distance between any source-point to the field point. There are no sources left outside this volume or lying on any relatedsurface. These time varying sources with current density J (ξ, ς, ζ, t) or charge densityρ(ξ, ς, ζ, t) are assumed to exists in (or surrounded by) free space, which is characterizedby µ0 and ε0. These sources are assumed to be arbitrarily located at (ξ, ς, ζ ) in rectangularcoordinate system. Figure 20–6 illustrates the coordinate system, location of volume Vtherein, location of point source (at ξ, ς, ζ ) and the field point (at x, y, z). The magneticvector potential A or electric scalar potential � at arbitrary location of field point (x, y, z)can be written as:

A(x, y, z, t) =∫v

µ0J (ξ, ς, ζ )

4πRej(ωt−βR)dv (1)

�(x, y, z, t) =∫v

ρ(ξ, ς, ζ )

4πε0Rej(ωt−βR)dv (2)

In terms of location ξ, ς, ζ and x, y, z the distance R can be given by

R = [(x − ξ)2 + (y − ς)2 + (z − ζ )2]1/2 (3)

B BP P

A AQ Q

Point radiators Point radiators

Volumewith

Surface

V

S

Volumewith

Surface

V

SLocalorigin

Localorigin

Element 1 Element 2

B P

A Q

Point radiators

Volumewith

Surface

V

SLocalorigin

Element i

B P

A Q

Point radiators

Volumewith

Surface

V

SLocalorigin

Element j

CC C

Figure 20–5

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Figure 20–6

( , , )ξ ς ζ

VS

θ

R

r

( , , )x y z

Location ofpoint sourceor radiator ina volume

havingsurface

V

S

φ

y

zField point(point at whichField is to beevaluated)

x

Further in view of the correspondence between cartisian and spherical coordinates:x = r sin θ cosφ, y = r sin θ sin φ and z = r cos θ , and the binomial expansion:

R = r − (ξ sin θ cosφ + ς sin θ sin φ + ζ cos θ)+ 0(r−1)

= r − �, on neglecting the last termWhere, � = ξ sin θ cosφ + ς sin θ sin φ + ζ cos θ

Thus A of (1) can now be written as

A(x, y, z) = µ0ej (ωt−βr)

4πr

∫v

J (ξ, η, ζ )ejβ�dv (4)

Equation (4) is written as a product of two terms. The first one of these terms representsan outgoing spherical wave and is given by:

µ0 × ej (ωt−βr)

4πr(5)

The second term, called directional weighting function0(θ, φ) is given by:

0(θ, φ) =∫v

J (ξ, η, ζ )ejβ�dξdηdζ (6)

The average power with complex field quantities E and H (both represented in sphericalcoordinates containing r, θ and φ components, i.e. E = Erar + Eθaθ + Eφaφ and H =Hrar +Hθaθ +Hφaφ) can be obtained by using complex Poynting vector P(θ, φ):

P(θ, φ) = 1

2Re[E ×H ∗] =

[β2η

(4πr)2

](1/2)[0θ0

∗θ +0φ0

∗φ ] (7)

The radiation pattern associated with Eθ is called the θ -polarized or vertically polarizedpattern. The radiation pattern associated with Eφ is called the φ-polarized or horizontally

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20–4 Weighting Functions 871

polarized pattern. Thus

Pr,θ (θ, φ) = (1/2)

[β2η

(4πr)2

][0θ(θ, φ)]

2 (8)

Pr,φ(θ, φ) = (1/2)

[β2η

(4πr)2

][0φ(θ, φ)]

2 (9)

where,

0θ(θ, φ) =∫v

[cos θ cosφJx(ξ, ς, ζ )+ cos θ sin φJy(ξ, ς, ζ )

− sin θJz(ζ, ς, ζ )]ejβ�dξdςdζ (10)

0φ(θ, φ) =∫v

[− sin φJx(ξ, ς, ζ )+ cosφJy(ξ, ς, ζ )]ejβ�dξdςdζ (11)

Equations (10 and 11) form the basis of pattern analysis and synthesis of antennas. Forantenna analysis, if the actual current distribution is known, the weighting functions0θ and0φ can be determined from (10) and (11) and the power radiation patterns can be obtainedfrom (8) and (9). For synthesis if the desired patterns are specified, (10) and 11) can be usedto find the current distributions.

20–4b Case - II: Equivalent Source Antennas

In this case electric sources are to be supplemented by fictitiousmagnetic sources for faithfulreproduction of external fields. If real electric sources (ρ andJ ) create electromagnetic fieldsE1, B1 and the fictitious magnetic sources (ρm and Jm) create electromagnetic fields E2,B2, all of these fields must satisfy Maxwell’s equations and conform to the same boundaryconditions, since the second set of sources is replacing the first set. Away from these sources,no distinction in characteristic features of fields can be made in order to determine the typeof source, which has resulted in this field.

In case of actual source antennas the magnetic vector potential A and electric scalarpotential � can be defined and related expressions can be derived. In this case we defineelectric vector potential F and magnetic scalar potential �m. The solutions are obviouslysimilar. If the electric and magnetic sources are confined to reside in surfaces, then linearcurrent densitiesK Amp/m andKm magnetic Amp/m replaces J and Jm. Similarly the realcharge densities ρs C/m2 and ρsm magnetic C/m2 replaces ρ and ρm. The potential functionscan now be written as:

A(x, y, z, t) = µ0

∫s

K(ξ, ς, ζ )

4πRej(ωt−βR)ds (12)

F(x, y, z, t) = µ0

∫s

Km(ξ, ς, ζ )

4πRej(ωt−βR)ds (13)

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�(x, y, z, t) =∫s

ρs(ξ, ς, ζ )

4ε0πRej(ωt−βR)ds (14)

�m(x, y, z, t) =∫s

ρsm(ξ, ς, ζ )

4ε0πRej(ωt−βR)ds (15)

As before, the far field form of vector potential functions A and F (12) and (13)/scalarpotential functions � and �m (14) and (15) can be written as the product of outgoingspherical wave factor (same as (5) with the directional weighting functions (similar to (6)).In this case weighting functions [0(θ, φ) for A and ϑ(θ, φ) for F] for vector potentialfunctions are:

0(θ, φ) =∫v

K(ξ, η, ζ )ejβ�dξdηdζ (16)

ϑ(θ, φ) =∫v

Km(ξ, η, ζ )ejβ�dξdηdζ (17)

Once vectors A and F are known the field quantities E and H can be readily obtainedby using the defining equations of Section (20–2). The complex Poynting vector givingthe average power density can also be evaluated in terms of A and F, as before. Besides,the transverse components of 0 and ∂ can be obtained by expanding (16) and (l7). Theresulting equations are:

0θ(θ, φ) =∫s

[cos θ cosφKx(ξ, ς, ζ )+ cos θ sin φKy(ξ, ς, ζ )

− sin θKz(ξ, ς, ζ )]ejβ�ds (18)

0φ(θ, φ) =∫s

[− sin φKx(ξ, ς, ζ )+ cosφKy(ξ, ς, ζ )]ejβ�ds (19)

ϑθ(θ, φ) =∫s

[cos θ cosφKxm(ξ, ς, ζ )+ cos θ sin φKym(ξ, ς, ζ )

− sin θkzm(ξ, ς, ζ )]ejβ�ds (20)

ϑφ(θ, φ) =∫s

[− sin φKxm(ξ, ς, ζ )+ cosφKym(ξ, ς, ζ )]ejβ�ds (21)

20–5 Linear Array Analysis

The matter embodied in this section encompasses the most general cases of antennas em-ployed in practical applications. In such antennas the elements (normally less than halfwavelength in their maximum dimensions) in an array are of the same type, equispacecd

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20–5 Linear Array Analysis 873

and similarly oriented. The relative physical positioning of the elements and their relativeelectrical excitations (in terms of current magnitudes and phases) are the parameters thatnormally govern the shape of radiation pattern. The analysis of the effects on radiationpattern caused by varying excitation reveals its ability:

1. to form a radiation pattern with main beam and side lobes.

2. to control angular placement of the main beam.

3. to select beam sharpness by choosing the length of the array.

4. to create a difference pattern.

5. to produce a shaped beam pattern without nulls.

As discussed earlier a current density distribution J (ξ, ς, ζ )ejωt contained in a finitevolume V causes a far field pattern (in θ and φ planes) and can be represented by weightingfunctions given by (l0) and (11) and (18)–(21) in the previous section. After evaluation ofvarious field expressions for components of E in spherical coordinates it can be shown thatEθ/0φ = Eφ/0θ . The weighting function0θ(θ, φ) can be viewed as vertically polarizedcomponent of the far field and0φ(θ, φ) as horizontally polarized component. It can also bestated that if magnetic currents are introduced as secondary sources ϑθ(θ, φ) and ϑφ(θ, φ)will have the similar expressions if Jm replaces J . When only that part of field due tomagnetic sources is being considered then Eϑ/ϑφ = Eφ/ϑθ . Thus ϑφ(θ, φ) can be viewedas vertically polarized component of the far field and ϑθ(θ, φ) as horizontally polarizedcomponent. In view of the above, the weighting function concept applies equally well foractual source and equivalent source antenna arrays.

Assume that an array contains 2N + 1 arbitrarily shaped, identical, similarly oriented,discrete radiating elements. Since, these elements are not point sources, these may beassumed to be composed of large number of point sources, as shown in Fig. 20–5. A pointPi(xi, yi, zi) in the i th element and a pointPj (xi, yi, zi) in the j th element occupies the sameposition vis-a-vis the current distribution. This collection of reference points describes therelative positions of different (N + 1) elements. Similarly another set of pointsQi(ξ, ς, ζ )

andQj(ξ, ς, ζ ) in the i th and j th elemnents can also be defined.It is convenient to establish local coordinate system at each of these reference points.

In view of Fig. 20–7

ξi = ξ − xi, ςi = ς − yi, ζi = ζ − zi (1)

LetJx(ξ, ς, ζ ) = Jx(xi+ξi, yi+ςi,+zi+ζi) andJx(ξ, ς, ζ ) = Jx(xj+ξj , yj+ςj+zj+ζj )be the current densities in i th and j th elements, which happened to be the same as per ourinitial assumption. Since all elements are identical and similarly oriented, if Ii is the complextotal current at the terminals of the i th radiator and Ij is the complex total current at theterminals of the j th radiator, it leads to the following:

Jx(xi + ξi, yi + ςi, zi + ζi)/Jx(xj + ξj , yj + ςj , zj + ζj ) = Ii/Ij (2)

provided ξi = ξj , ςi = ςj , ζi = ζj and Jx belongs to corresponding physical points inthe two elements. The terminals referred to above may be the cross-section of a waveguidefeeding a slot, or a junction of coaxial cable and a helix along with its ground plane.

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0

X, ξ

( + + )ξ ς ζ2 2 2

( + + )x y z2 2 2

y, ς

P

|( – ) + ( – ) + ( – )|x y zξ ς ζ2 2 2

ςiPi

ξi

z, ζ ζi ( + + )ξ ς ζi2 2 2

i i

( + + )ξ ς ζi2 2 2

i i( + + )ξ ς ζ

i2 2 2

i i

i th element

ζi

ζi

Local originof elementsith

Local originof elementsith

Pi Pj

ξi ξi

ζi ζi

Qi Qi

ith elements jth elements

(a)

( + + )x y zi i i2 2 2

(b)

Figure 20–7

From (2) and (22-4-11) the weighting function may be written as

0φ(θ, φ) =N∑i

∫Vi

[− sin φJx(xi + ξi, yi + ςi, zi + ζi)+ cosφJy(xi + ξi, yi

+ςi, zi + ζi)]ejβ�dξidςidζi (3)

= 0φ,a(θ, φ)0ϕ,e(θ, φ) (4)

If the origin of principal coordinates is fixed at a point P0(x0, y0, z0) in the above equation,it will merely result in phase change in array factor and can lead to:

0φ,a(θ, φ) =N∑i=0

Ii

I0ejβ(xi sin θ cosφ+yi sin θ sin φ+zi cos θ) (5)

0ϕ,e(θ, φ) =∫V0

[− sin φJx(ξ0, ς0, ζ0)+ cosφJy(ξ0, ς0, ζ0)]

·ejβ(ξ0 sin θ cosφ+ς0 sin θ sin φ+ζ0 cos θ)dξ0dς0dζ0 (6)

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An expression for0θ(θ, φ) can also be obtained in a similar manner as

0θ(θ, φ) = 0θ,a(θ, φ)0θ,e(θ, φ) (7)

where,

0θ,a(θ, φ) =N∑i=0

Ii

I0ejβ(xi sin θ cosφ+yi sin θ sin φ+zi cos θ) (8)

0θ,e(θ, φ) =∫V0

[cos θ cosφJx(ξ0, ς0, ζ0)+ cos θ sin φJy(ξ0, ς0, ζ0)

− sin θJz(ξ0, ς0, ζ0)] · ejβ(ξ0 sin θ cosφ+ς0 sin θ sin φ+ζ0 cos θ)dξ0dς0dζ0 (9)

The expressions of0φ,e(θ, φ) and0θ,e(θ, φ) involve the current distributions in onlyoneof the elements. Thepatterns thus obtained are called element patterns or element factors.These expressions are not only identical but involve relative current levels in differentelements and their relative placements. Since, subscripts θ and φ in (5) and (6) spell onlythe orientation (polarization) of field, and henceforth can be dropped. The resulting notationsretain no vector characteristics and can be named as array patterns or array factors. Forconsideration of polarization one can revert to the original expressions of (5) and (6) andevaluate0φ(θ, φ) and0θ(θ, φ) from (4) and (7).

20–5a Different Forms of Linear Arrays

Now let ri be the distance from P0(x0, y0, z0) to Pi(xi, yi, zi) and the line connecting P0and Pi have direction cosines cosαi, cosβi, cos γi . If all the elements lie along a commonline, then α = β = γ for every Pi . The antenna thus formed is termed a linear array andits array factor can be written as:

0a(θ, φ) =N∑n=0

In

I0ejβrn(cosα sin θ cosφ+cosβ sin θ sin φ+cos γ cos θ) (10)

Equation (10) can be taken as the starting point to study different forms of linear arrays,and the modified versions of (10) can be accordingly obtained for the following cases.

1. If array elements are equispaced and odd (2N + 1) in number the zeroth element canbe taken in the mid of array. If now rn is replaced by nd , where d is the commonspacing between elements, (10) results in a general form of array factor expressionfor a uniformly spaced linear array.

0a(θ, φ) =N∑n=0

In

I0eβnd(cosα sin θ cosφ+cosβ sin θ sin φ+cos γ cos θ) (11)

If all elements lie along one of the axis, z-axis (say), i.e. at zi = 0, ±d, ±2d,· · · ±Nd, cos α = cos β = 0, cos γ = 1 and the resulting pattern will be

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φ-independent or rotationally symmetric. Equation (11) reduces to:

0a(θ) =N∑n=0

In

I0ejβnd cos θ (12)

(a) If all currents are equal and in phase the distribution results in uniformly excitedbroadside array, with an array factor:

0a(θ) =N∑n=0

ejβnd cos θ (13)

Equation (13) represents a sum of phasors with common unit magnitude,possessing phase angles which depend onQ, but for a given value ofQ areprogressive multiples of the basic angle ψ , where ψ is given by:

ψ = βd cos θ = (2πd/λ) cos θ (14)

Equation (14) is same as Equation (4-13-2) with φ replaced by θ and δ taken to bezero. The term δ is referred as αz in the subsequent text. If ψ = 0, it requiresθ = (2k + 1)π/2, for k = 0, 1, 2, . . . and the radiation from such a configurationof elements is perpendicular to the line in which elements are arranged, and ismaximum at θ = π/2, 3π/2, . . .. Such an array was identified as broadside arrayin Case 1 of Section (4–13).

(b) Alternatively, if the phase change between adjacent elements is required to makethe field a maximum in the direction of array (θ = 0) the array is termed as endfire array. For this case ψ and θ are zero and δ = −(2πd/λ). This is described inCase 2 and 3 of Section (4–13).

(c) If the currents are unequal and tapered (i.e. maximum in the center and minimumat the end elements of the array the distribution results in broadside array withtapered excitation.

(d) If the phase angles (δ or α) are progressively increasing, In = I0e−jnα the

distribution results in a scanned beam array and the array factor for this case canbe written as:

0a(θ) =N∑

n=−Nejn(βd cos θ−αz) (15)

(e) A combination of any of the above cases is also possible resulting in (say)tapered scanned beam array.

2. If the number of elements in an equispaced linear array is even (say 2N ) the arrayelements can lie along z-axis at ±d/2, ±3d/2, . . . the array factor can be written as

0a(θ) =N∑

n=−N

I−NI1

e−j [(2N−1)/2]βd cos θ + · · · + I−1I1

e−j (1/2)βd cos θ + ej (1/2)βd cos θ

+IN

I1ej [(2N−1)/2]βd cos θ (16)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


A number of different types of arrays can be derived from (16) in the similar manneras obtained for the case of odd elements. A detailed discussion relating to differenttypes of arrays is included in Chapter 4.

20–5b Schelkunoff Unit Circle

To understand different arrays afresh consider a linear array containing N + 1 elements(where N may be even or odd integer) lying along z-axis. The array factor of (15) can berewritten in the modified form as under.

0a(θ) =N∑n=0

In

I0ejn(βd cos θ−αz) (17)

Since, the ratio of currents will be a complex quantity, (17) may be further modifiedby substituting:

w = ejψ (18)

where as before,

ψ = βd cos θ − αz (19)

Equation (19) and (4-13-2) are same except containing different symbols to representquantities. The array factor now transforms into:

0a(w) =N∑n=0

In

I0ejnψ =

N∑n=0

In

I0wn = IN

I0

N∑n=0

In

INwn (20)

On expanding the summation term and writing the expanded terms in reverse order,the modulus of modified form of array factor becomes:

|0a(w)| =∣∣∣∣INI0

∣∣∣∣ •∣∣∣∣wN + IN−1

INwN−1 + · · · + I0

INw0

∣∣∣∣ =∣∣∣∣INI0

∣∣∣∣ • |f (w)| (21)

If the summation of (21) is replaced by a product termN∏n=1

(w − wn), |f (w)| on ex-pansion can be written as:

|f (w)| = |w − w1| • |w − w2| • |w − w3| • · · · • |w − wN | (22)

Thus in view of the above |f (w)| cannot only be expressed as a polynomial in w ofdegree one less than the number of elements in the equispaced array, it can also be used assubstitute to the array factor (as the two differ only by a multiplying factor). The roots of thepolynomial are linked to the array excitation, since current ratios comprise the coefficientsof the polynomials.

The following fundamental theorems (due to Schelkunoff) in fact lay the foundationto the understanding of antenna arrays.

Theorem I Every linear array with the commensurable separations between the elementscan be represented by a polynomial, and every polynomial can be interpreted as a lineararray.

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


Since the product of two polynomials is again a polynomial, a corollary to theorem Iis:

Theorem II There exists a linear array with a space factor equal to the product of thespace factors of two linear arrays.

Theorem III The space factor of a linear array of n apparent elements is the product of(n− 1) virtual couplets with their null points at the zeroes of E.

The space factor of an array is defined as the radiation pattern of a similar array ofnon-directive or isotropic elements. Thus the term space factor, array pattern or array factorcan be used to spell the same meaning.

The parameter E referred to in Theorem III is the electric field intensity, which canbe obtained after knowing the magnetic potential A from the basic equations of Section20–2. A in itself can be obtained in terms of array factor, which in turn is expressed as apolynomial. Thus the product of (22) will result in a polynomial equation of the followingform. This equation for f (w) can represent A or E.

|f (w)| = |a0w0 + a1w1 + a2w

2 + · · · + am−2wm−2 + am−1wm−1|= |a0 + a1w + a2w

2 + · · · + am−2wm−2 + am−1wm−1|Alternatively,

|f (w)| = |wm + a1wm−1 + a2w

m−2 + · · · + 1|From (18) and (19), it is evident that w is a complex quantity and shall represent a

complex plane. Angle θ on the contrary represents real space. The roots of |f (w)| thus liein a complexw plane and are related, to the field pattern in real θ space. Since θ varies from0 to π,w (in view of (18) and (19)) will vary from some initial value ψi = kd −αz to somefinal value ψf = −kd − αz. Equation (18) if plotted, results in a circle as shown in Fig.20–8(a). This figure illustrates real and imaginary axes of w, arbitrary location of ψ andcorresponding ejψ , increasing direction of θ , arbitrary locations of θ = 0 and θ = π , andthe corresponding positions ofψi andψf (which in turn correspond to first and last roots of(22) on the periphery of a unit circle. From (22) it can be concluded that if the rootsw1,w2,w3, . . . wm (within the range of w) are placed on the unit circle then for |f (wm)| = 0, anarray pattern containing N nulls will result. If all the roots are placed off the unit circle orare outside the range of w the resulting radiation pattern will be free from nulls. This circleis known as Schelkunoff unit circle. Figure 20–8(b) illustrates placement of four rootsw1, w2, w3, and w4 on the unit circle. These roots belong to a five-element array problem.If the excitation currents in amplitude and phases are same (i.e. αz = 0) (19) reduces toψ = βd cos θ and (20) becomes:

0a(w) = f (w) =N∑n=0

wn = 1− wN+1

1− w(23)

In view of DeMoviver’s theorem the roots of f (w) can be evaluated from

wm = ej2πm/(N+1) For m = 1, 2, . . . , N (24)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


θ increasing

θ = 0 Ψi

Ψ

w e= jΨ

Im Im

θ π= Ψf

Ψ αi z= –k dΨ αf z= – –k d

Re

w1

d1

w

d2w2

d3 d4

w3

w4

( )a ( )b

Re

Figure 20–8

Figure 20–9

d

Ψ

Ψ �

( , )x y� �w �

yIm

Rex

( , )x yw

Complex- planew

Equation (24) is illustrated by Fig. 20–8(b). Since number of roots have to be one lessthan number of elements in an array, the four roots obtained from (24) are±2π/5 = ±72◦

and±4π/5 = ±144◦. Thevalue of |f (w)| is obtainedbymultiplyingdistancesd1,d2,d3 andd4 shown in Fig. 20–8(b). Asw moves along the unit circle these distances change and thustheir product also changes. Whenever w coincides with one of the roots the correspondingdistance d = 0 and hence |f (wm)| = 0, resulting in a null in the array factor.

Fig. 20–9 shows a circle with location of a root w′ at an angle ψ ′, w is located at anarbitrary angle ψ . To estimate the distance d between w and w′ the following procedurecan be followed.

If the real and imaginary axes of the complexw-plane lie along x and y axes, the coor-dinates ofw andw′ can be given as (x, y) and (x ′, y ′). In fact x, y, x ′ and y ′ can themselvesbe written as x = r cosψ = cosψ , y = r sinψ = sinψ , x ′ = r cosψ ′ = cosψ ′ andy ′ = r sinψ ′ = sinψ ′ (since r = 1 for assumed unit circle). Thus d .

√[cosψ ′ − cosψ)2+

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


(sinψ ′ − sinψ)2]. Thus any change in ψ will result a change in the value of d . If thereare more roots located on the unit circle distances d1, d2 . . . can be evaluated by using theequation dm =

√[(cosψ ′

m − cosψ)2 + (sinψ ′m − sinψ)2] for m = 1, 2, . . .. As w moves

along the unit circle ψ changes all the distances change and hence their product will alsochange.

20–5c Different Forms of Linear Arrays

To obtain different forms of arrays the following steps may be followed:

1. Select the number of elements (N + 1) and spacing between elements in terms ofwavelength λ(d = λ/2, 2λ/2, 3λ/2 . . .)

2. From Eqn. (wm = ej2πm/(N+1)) calculate the root positions for m = 1, 2, . . . , N .

3. Place these roots w1, w2 . . . on the unit circle

4. Obtain wi and wf from Eqn.(18).

5. Start moving clockwise from wi until wf is reached. On such movement w will havemany discrete locations. Measure distance from w to w1, w2 . . . and list these as d1,d2, . . .. Evaluate the product of these distances for every discrete location of w. Eachsuch product shall give value of |f (w)|.

6. Whenever w will coincide w1, w2, . . . one of the distances d1, d2, . . .. Will be zero,indicating a null at that location.

7. Maximum values of product will give, maximas either of main lobe or that of sidelobes. These maximum values need not be equal.

8. Since w represents a complex plane and is related to θ which represents real space,the values of θ corresponding to each discrete value of w can be obtained.

9. Plot the product of distances in the real space to get the radiation pattern.

Knowledge of the lobe heights (maximas) and the null positions is all that is needed toproduce a radiation pattern.

Adopting the above procedure for the problem of five element array, if d = λ/2, thenfrom equation ψ = βd cos θ , ψi = π and ψf = −π (since θ varies from 0 to π andβ = 2π/λ or βd = π) all the four roots are within the range of w (i.e. 0 to 2π ). Fig. 20–10 reveals the formation of radiation pattern and its relation to Fig. 20–8(b). As w movesclockwise from ejπ to e−jπ through ej0, the product of distances (d1, d2, etc.) first gives ahalf side lobe, followed by a null at w2, a side lobe, a null at w1, a main lobe, a null at w4,a side lobe, a null at w3, and thereafter another half side lobe. The nulls in real space canbe determined from the relation:

ψm = 2πm/(N + 1) = kd cos θm

If d = λ then ψI = 2π and ψf = −2π , w ranges two full revolutions around theunit circle. In such a case there will be extra main beams at θ = 0 and θ = π . A suitablerestriction is to be imposed on ψi to avoid the extra lobes. The reader is advised to see thereferences at the end of chapter for this and more involved cases.

Kraus-38096 kra21032_ch20 May 4, 2006 12:4

20–6 Linear Array Synthesis 881

Figure 20–10

ejπ

e–jπ

θ

W2

W1

W4

W3

Z

e– 0j

X Y- Plane

20–6 Linear Array Synthesis

In synthesis of antennas the requisite current distribution in an equispaced array is to beobtained to satisfy a specified desired array pattern. In case of linear arrays this desiredpattern has to be a function of θ (or φ) alone, so that the array factor can be expressed in theform of0a(θ) (or ϑa(θ)). For planar arrays the desired pattern will be a function of both, θand φ. These functions [0a(θ) or ϑa(θ)] may encompass a variety of array configurationsincluding

(a) sum patterns with: (i) uniform side lobes, (ii) symmetrically tapered side lobes, (iii)asymmetric side lobes

(b) difference patterns with same topographies of side lobes,(c) patterns with neither nulls nor side lobes.Since all these aspects cannot be covered in an introductory chapter the discussion

in this section is confined only to very essential features and techniques of synthesis. Insynthesis the term sum and difference patterns forms the basis for selection of the techniquesemployed. The meaning of sum and difference patterns is explained as follows.

20–6a Sum and Difference Patterns:

If an antenna is illuminated by two offset feeds, two offset radiation patterns are generated.Such an arrangement is frequently used in mono-pulse tracking radars, in which a parabolicreflector is fed by two offset horns. If tracking is to be accomplished in (say) azimuth onlytwo horns are to be placed side-by-side as shown in Fig. 20–11a. In case the tracking is tobe accomplished in both, azimuth and elevation, four offset horns arranged in Fig. 20–11bare to be used. If in a two-horn case, the two offset beams generated Fig. 20–11c are of thesame polarity they get summed, the beam shown by, Fig. 20–11d will result Alternativelyif beams generated are of opposite polarity Fig. 20–11e the difference of two beams willresult as in Fig. 20–11f . The generation of same or opposite polarity beams will dependon the relative polarity of feeds, i.e. the two feeds may be in phase or out of phase. The tworadiation patterns, so obtained, are termed as sum pattern and difference pattern.

These patterns find extensive use in many practical applications. Tracking of RadarsSatellites and Space vehicles are some such applications. In Fig. 20–11 signals to be summed

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


A B

BA

DC

+ +++ +– +

(a)

(b)

(c) (d) (e) (f)

Sum Pattern

Sum = +

Case (a)

or

+ + +

Case (b)

A B

A B C D

Difference Pattern

– for (a)or

( + ) – ( + ) for elevation Case (b)or

Difference –

( + C) – ( + ) for azimuth Case (b)

A B

A B C D

A B DFeed Horn

Arrangement

Figure 20–11

or differenced in case of azimuth or elevation are also mentioned. The creation of sum ordifference pattern is not confined to reflectors or lenses; antenna arrays are equally suitablefor such ventures. Thus synthesis requires understanding of the meanings of these patterns,since generation of these patterns is related to the number of elements in an array. If thisnumber is odd, an array is not suitable to generate a difference pattern. On the other hand, anarray with even number of elements is equally suitable for creation of sum and of differencepatterns. Further, the description given above is not confined to transmitting antennas; theapplication of sum and difference pattern is equally valid for receiving antennas. In this casethe incoming signal are added or subtracted to get the sum or difference patterns. Normallyin monopulse tracking, sum pattern is used for transmission. In case of reception, it is usedto obtain range information whereas one or two difference patterns are used to track thetarget in azimuth or elevation or in both.

To produce sum and difference patterns with main beam of sum pattern pointing atan angle θ0, and the twin main beams of the difference pattern straddling θ0, both patternsexhibiting a symmetrical side lobe structure with 2N equispaced elements in an array, thearray factor for separation d can be written as:

0a(θ) =−1∑

n=−N

In

I1ej [(2n−1)/2]βd(cos θ−cos0) +

N∑n=1

In

I1ej [(2n−1)/2]βd(cosφ−cos0) (1)

If all current amplitudes are real, and In = I−n, Eqn (1) for sum pattern becomes:

S(θ) = 2N∑n=1

In

I1cos[(2n− 1)(πd/λ)(cos θ − cos θ0)] (2)

Similarly for difference pattern In = −I−n, and Eqn (1) becomes:

D(θ) = 2jN∑n=1

In

I1sin[(2n− 1)(πd/λ)(cos θ − cos θ0)] (3)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


The sum pattern from an array with 2N + 1 (odd number) elements can be given bymodifying Eqn. (20–5–11) as:

s(θ) = 1+ 2N∑n=1

In

I1cos[2n(πd/λ)(cos θ − cos θ0)] (4)

Amajor class of synthesis problems can be circumvented by (1)–(4). To obtain specifiedside lobe topography, the required current distribution, In/I1 or In/I0 for sum or differencepatterns can be obtained from these equations.

If the Schelkunoff unit circle is correlated to the design of equispaced linear arrays,excited to give an array factor with main beam accompanied by side lobes the followingcan be concluded.

1. For 2N + 1 elements, if 2N roots (in complex conjugate pairs) are placed on the unitcircle, a symmetrical sum pattern results. The adjustment of the position of these rootpairs can alter the side lobe heights. Further by clustering the root pairs closer toψ = π can also reduce the levels of side lobes. This option will however, broaden themain beam.

2. For 2N elements, if 2N − 1 roots are placed on the unit circle with one root atψ = −π and the remaining inN − 1 complex conjugate pairs, a symmetrical sumpattern will result. The side lobe heights again can be altered by adjusting root pairpositions. As before the level of side lobes can also be reduced by clustering the rootpairs closer to ψ = π but at the expense of main lobe widening.

3. Occurrence of roots in complex conjugate pairs, results in pure real coefficients ofpolynomial f (w). These coefficients appear in symmetrical pairs and hence thecurrent distribution in an array is symmetrical in amplitude.

20–6b Dolph-Chebyshev Synthesis of Sum Pattern

Dolph addressed the problem of proper root positioning or proper excitation to give a sumpattern with uniform side lobes of specified levels by exploring the properties of Chebyshevpolynomials. These polynomials denoted by Tm(u), with index m and a variable argumentu, are the solutions of the differential equation:

i − u2d2Tm

du2− u

dTm

du+m2Tm = 0 (5)

If m(= 2N) is an even integer:

T2N(u) =N∑n=0

(−1)N−n N

N + n(N+n2N )(2u)2n (6)

If m(= 2N − 1) is an odd integer:

T2N−1(u) =N∑n=1

(−1)N−n 2N − 1

2(N + n− 1)(N+n−12n−1 )(2u)2n−1 (7)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


Figure 20–12

m = 6

m = 7

+1 u

( , )u b0

( , )u b0

T um( )

T um( )

+1

–1

0–1

+1 u

0

–1

+1

The term of the form

(r

s

)stands for binomial coefficient

r!

s!(r − s)!. Equation (6) and

(7) can also be written as

Tm(u) = cos(m cos−1 u) for − 1 ≤ u ≤ 1

= cosh(m cosh−1 u) for ≤ u ≥ 1 (8)

The plots ofTm(u) are shown in Fig. 20–12 for even and odd indices. These figures illustrate:

1. Tm(u) oscillates cosinusoidally (for even m) and sinusoidally (for odd m). Also for|u| < 1, Tm(u) exhibits symmetry and anti-symmetry about u = 0 for respectivevalues of even and odd m.

2. For |u| > 1, Tm(u) rises hyperbolically, in continuity, in the respective directions.Larger the value of m steeper is the slope.

The task involved now is to make a correspondence between variable u to the real anglevariable θ . Such correspondence will correlate Tm(u) to 0a(θ). This was developed byconsidering basic equation for array factor, with the assumption of a uniform progressivephase and symmetrical amplitude distribution. As a result (4) and (2) for odd and evennumber of elements can be rewritten in the form as given below:

S

(ψ

2

)= 1+ 2

N∑n=1

In

I0cos 2n

ψ

2(9)

S

(ψ

2

)= 2

N∑n=1

In

I1cos(2n− 1)

ψ

2(10)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


where,

ψ = βd(cos θ − cos θ0) (11)

In view of the transformation

u = u0 cos(ψ/2) (12)

As θ ranges from 0 to θ0 to π , ψ changes from ψ1 = βd(1 − cos θ0) to 0 to ψf =−βd(1+ cos θ0), u will assume values from ui = u0 cos(ψi/2) to u0 to uf = u0 cos(ψf /2).

If u0 is chosen to be Tm(u0) = b with 20 log10 b as the assumed side lobe level, thena pattern consisting of a main beam at the relative lobe level b and a family of side lobes,all at the same (unit) height, will result. The implementation of the Dolph design procedurefor a simple case of uniform array can thus be summarized as below.

1. Select the number of elements by which an array is to be implemented. This gives thedegree of Chebyshev polynomial m, which is one less than the number of elements

2. Select the desired side lobe level (SLL) in dB’s (i.e. SLL = 20 log10 b).

3. Find u0 from equation Tm(u0) = b.cosh(m cosh−1 u0) = b . . . for|u| ≥ 1or u0 = cos((cosh−1b)/m)

4. Determine the roots of Tm(u) by using relation

Tm(u) = cos(m cos−1 u) . . . For − 1 ≤ u ≤ 1

Get the roots from cos(m cos−1 u) = 0 as up = ± cos[{cos−1[(2p − 1)π/2]}/m]5. Evaluate up for p = 1, 2, 3, . . .

6. From (12), obtain ψ = ψp for different values of up7. If ψp is known, values of wp (from w = ejψ) can be evaluated. This will lead to the

known value of f (w). From equation ψ = βd cos θ − αz, θp can be evaluated

8. Since |f (w)| = |w − w1||w − w2||w − w3| . . . the product will result in apolynomial equation:

f (w) = wm + a1wm−1 + a2w

m−2 + . . .+ 1

The coefficients a1, a2, . . .will represent the relative magnitude of currents in differentelements of array. These coefficients will be symmetrical about a central element, i.e. thecurrent in first and last, second and last but one and so on will be equal.

It is possible to determine the Dolph current distribution for the general case withoutgoing through the intermediate steps of finding up, ψp andwp and the multiplication to getf (w). The relative current distribution for an array of known side lobe level, after evaluatingu0 can be calculated as below:

1. For any array with 2N + 1 elements:

In =N∑p=n

(−1)N−p N

N + p

N+pC2p

2pCp−nu2p0 (13)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


2. For any array with 2N elements:

In =N∑p=n

(−1)N−p 2N − 1

2(N + p − 1)N+p−1C2p−1 2p−1Cp−nu

2p−10 (14)

These relations are obtained by inserting (12) in (6) and (7) and using the relations

cos2nψ

2= 1

22n

n∑q=0

εq(2nn−q

)cos

2qψ

2(15)

cos2n−1ψ

2= 1

22n−2

n∑q=1

εq(2n−1n−q

)cos(2q − 1)

ψ

2(16)

where εq = 1 if q = 0, otherwise εq = 2

20–6c Taylor Synthesis of Sum Patterns

When a current distribution is to be obtained for producing a radiation pattern with a singlemain beam of a prescribed width and specific scan position, along with a family of sidelobes having a common specified level, the problem is similar to that of a discrete lineararray. The synthesis procedure must permit determination of the line source distributioncorresponding a specific pattern. Dolph method leads to the solution of such a problemwithdiscrete sources. The method is also applicable for the cases where a source configurationcan be viewed as a continuous line source. A horn of aperture size a × b, with a >> λ andb << λ belongs to this category.

The Taylor distributions can be sampled and applied to the design of discrete array.This alternative can also be extended to planar apertures. To this extent it is a competitorto the Dolph method. However, Taylor method offers an added advantage, since this can beextended to deal with circular planar apertures. Such an extension is not possible in case ofDolph.

For a continuous line source of small cross-section s stretching along (say) z-axis froma to +a (20–4–10) and (20–4–11) can take the form

0θ(θ) =a∫

−as[cos θ cosφJx(ζ )+ cos θ sin φJy(ζ )− sin θJz(ζ )]e

jβ�dζ (17)

0φ(θ) =a∫

−as[− sin φJx(ζ )+ cosφJy(ζ )]e

jβ�dζ (18)

where � = ζ cos θ . If the direction of current density is assumed to be the same in everyaperture element dζ , that is:

J (ζ ) = (C1ax + C2ay + C3az)g(ζ ) (19)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


with C1, C2, and C3 as constants and ax , ay , and az are respective unit vectors.

0θ(θ) = (C1 cos θ cosφ + C2 cos θ sin φ − C3 sin θ)s

a∫−a

g(ζ )ejβ� cos θdζ (20)

0φ(θ) = (−C1 sin φ + C2 cosφ)s

a∫−a

g(ζ )ejβ� cos θdζ (21)

The multiplying factors before integrals are called the element factors for0θ and0φ .The integrals (fortunately same) give the array factors for the line source. The general arrayfactor can be written as

S(θ) =a∫

−ag(ζ )ejβ� cos θdζ (22)

Thus for a given desired sum pattern the aperture distribution g(ζ ) is to be obtained.There can be many approaches to obtain the array factors g(ζ ). It can be an assumed

value, e.g. g(ζ ) =Ge−jbζ withG and b having constant values. Its substitution in (22) givesthe relation for sum pattern for a uniformly excited discrete array for L = 2a, provided thelimit d/λ → 0

S(θ) = 2Ga[{sin βa(cos θ − b/β)}/{βa(cos θ − b/β)}] (23)

The substitution u = (2a/λ) (cos θ − b/β) in (23) leads to a ‘universal power pattern’shown by Fig. 20–9, which is obtained by implementing the equation:

f (u) = 20 log10

∣∣∣∣S(θ0)S(θ)

∣∣∣∣ = 20 log10

∣∣∣∣ sin πuπu

∣∣∣∣ (24)

in which θ0 = cos−1 b/β is the poynting angle of main beam.A plot of f (u) results in a sum pattern with symmetrical side lobes where the field

heights are inversely proportional to u. The closest pair of side lobes is down by 13.5 dB.Since θ changes in real space from 0 to θ0 toπ , the value of u changes from (2a/λ)(1−b/β)

to 0 to −(2a/λ) (1 + b/β). The number of side lobes will depend on the aperture length(2a/λ).

An approximate universal pattern can be constructed by selecting an integer n′ suchthat |u| ≥ n′. The nulls of new pattern have to occur at an integral value of u. In order todepress the intervening side lobes a little the next pair of nulls in the vicinity of main beamhave to occur at u = ±un−1, where |un−1| > n′ − 1. Also the penultimate (last but one) pairof nulls needs to be shifted to u = ±un−2, where |un−2| > n′ − 2 and so on. Such a patterncan be expressed by the function:

s(u) = sin πu

πu

n−1∏n=1

(1− u2/u2n)

n−1∏n=1

(1− u2/n2)

(25)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


0

–10

–20

–30

–40

–50

dB

–15 –10 –5 0 5 10 15

2aλ

u = (cos – cos )θ θ0

sin πuπu

Figure 20–13

Thus the innermost n′ − 1 pairs of nulls are removed from the original sin πu/πupattern and replaced by new pairs at modified positions ± un. Taylor obtained the followingrelation for determining the new null positions:

un = n′[A2 + (n− 1/2)2

A2 + (n′ − 1/2)2]1/2 (26)

From side lobe level = 20 log10 b and b = coshπA gives value of A.The aperture distribution g(ζ ) given by (22) to satisfy pattern given by (25) and (26)

let g(ζ ) = h(ζ, )e−pζ , where h(ζ ) can be represented by:

h(ζ ) =∞∑

m= 0

Bm cos(mπζ/a) (27)

Equation (22) becomes

S(u) =∞∑

m= 0

Bm

a∫−a

cos(mπζ/a) ejuπζ/adζ (28)

Discarding of odd integrand results

S(u) =∞∑

m= 0

Bm

a∫−a

cos(mπζ/a) cos(uπζ/a)dζ (29)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


Thus integral is zero unless u is an integer and equals m for such limiting case,

S(0) = 2aB0 and S(m) = aBm, for m = 1, 2, . . . (30)

From (25) S(m) = 0 for m ≥ n′, thus this Fourier series truncates. The continuousaperture distribution is obtained to be:

g(ζ ) = e−jpζ

2a[S(0)+ 2

n−1∑m= 1

S(m) cos(mπζ/a)dζ ] (31)

Modified Taylor patterns If equal importance is attached to all the directions Taylorpatterns may not be optimum in some applications. It is often desirable to shape the pat-terns so as to allow high side lobe level in unimportant directions and low side lobe levelin regions adjacent to the main beam or in the desired direction. For such cases Taylordistribution obviously needs modification, which can be achieved by using perturbationprocedure resulting in new form of equations.

20–6d Bayliss Synthesis of Difference Patterns

The Taylor line source distribution g(ζ ) given by (31) can be seen as a product of a uniformprogressive phase term e−jpζ and a pure real amplitude distribution function:

h(ζ ) = 1

2a[S(0)+ 2

n−1∑m= 1

S(m) cos(mπζ/a)] (32)

It is the first term, which determines direction of the main beam. The expression forh(ζ ) contains all even Fourier series terms that are non zero at the end points. In the caseof uniform distribution h(ζ ) is constant and corresponds to only first term of this Fourierseries. It results in the pattern of Fig. 20–13 and can be modified by root location. If a linesource distribution is obtained, which can be represented by a Fourier series containing allodd terms that are non zero at the end points, (32) can be written as:

h(ζ ) =∞∑

m= 0

Bm sin[(m+ 1/2)(πζ/a)] (33)

If only the zeroth term of this Fourier series are taken a generic difference patternshould result from the aperture distribution:

g(ζ ) = sin

(πζ

2a

)e−jpζ (34)

When (34) is used in the array factor common to (20) and (21), it leads to

D(θ) =a∫

−ag(ζ )ejβζ cos θ dζ

= j

a∫−a

[cos{(u− 1/2)(πζ/a)} − cos{(u+ 1/2)(πζ/a)}]dζ (35)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


–10

0

–20

–30

–40

–50–7.5 –5 –2.5 0 2.5 5 7.5

2aλ

u = (cos – cos )θ θ0

D u( ) =πu

u u

cos

( – 1/2) ( + 1/2)

πu

dB

Figure 20–14

where u is given by (26). The integration and simplification of (26) ultimately leads to:

D(u) = πu cosπu

(u− 1/2)(u+ 1/2)(36)

The plot of difference pattern obtained from (36) is shown in Fig. 20–14. The nulls inview of (36) will occur at u = 0 and cos πu or when:

u = ±(n+ 1/2), for → n = 1, 2, . . . (37)

If the lobe pairs in the vicinity of main lobes are to be depressed (36) is to be modified:

D(u) = πu cosπu

n−1∏n=1

(1− u2/u2n)

n−1∏n=0

[1− {u/(n+ 1/2)}2](38)

The synthesis problem thus boils down to finding new null locations ±un. In view ofa parametric study with the aid of computer Bayliss obtained relations given by Equation(39) for root replacement:

un = 0 for n = 0 (a)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4

20–7 Planar Arrays 891

un = (n′ + 1/2)

(ζ 2n

A2 + n′2

)1/2

for n = 1, 2, 3, 4 (b)

un = (n′ + 1/2)

(A2 + n2

A2 + n′2

)1/2...

for n = 5, 6 (c) (39)

where parameterA and ζ 2n are related to side lobe level (SLL) and their appropriated valuesare given in Table 20–1.

Table 20–1

(SLL) 15 20 25 30 35 40A 1.0079 1.2247 1.4355 1.6413 1.8431 2.0415ζ1 1.5124 1.6962 1.8826 2.0708 2.2602 2.4504ζ2 2.2561 2.3698 2.4943 2.6275 2.7675 2.9123ζ3 3.1693 3.2473 3.3351 3.4314 3.5352 3.6452ζ4 4.1264 4.1854 4.2527 4.3276 4.4093 4.4973

Bayliss difference pattern can be obtained on multiplying (33) by e−jpζ , inserting theresult in (35) and using (39). The resulting equation is:

D(u) = 2j∑m

Bm

a∫0

sin[(m+ 1/2)(πζ/a)] sin(uπζ/a)dζ (40)

If u is midway between two integers (say n + 1/2) the integral is zero unless n = m,for which: D(m+ 1/2) = jaBm. Further D(m+ 1/2) = 0 for m ≥ n′, the Fourier seriestruncates. The aperture distribution to produce Bayliss pattern is finally obtained to be:

g(ζ ) = e−jpζn−1∑m=0

D(m+ 1/2) sin[(m+ 1/2)πζ/2] (41)

20–7 Planar Arrays

In order to understand a planar array it is necessary to understand its structure vis-a-vis thenature of plane, distribution and spacing of elements. The planar array may be of square,rectangular or circular shape. The elements used to shape a plane may be in the form ofa square grid, a rectangular grid or a circular grid. Rectangular grids are normally usedwith rectangular boundaries and circular grids with circular boundaries. In these grids theelements may be equispaced or non-equispaced. The spacings along the two axes of a planemay be equal or unequal. The plane of an array may be divided into symmetrical quadrantsfor an appropriate mode of excitation to generate sum and/or difference patterns.

If the aperture distribution in a planar array is such that its resulting pattern is the productof patterns of two orthogonal linear arrays, such a distribution is termed as a separableaperture distribution. If such a separation is not realizable the distribution may be termedas inseparable aperture distribution. The synthesis in separable case is simply the extension

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


Figure 20–15

�dy

dx

x

y

�

r

P r( , , )� �

z

of the techniques developed for linear arrays. In non-separable cases new techniques need tobe developed to tackle the problems. In the following text only the separable distributions arebeing discussed in brief. The issue of non-separable distributions and detailed descriptionof separable cases are thoroughly discussed by Elliot [1]. The two array configurations,which have been discussed include:

1. Arrays with Rectangular Boundary

2. Arrays with Circular Boundary

20–7a Arrays with Rectangular Boundary

Figure 20–15 illustrates a rectangular planar array in which the elements are arranged inthe form of a rectangular grid. The number of elements along x and y-axes are assumedto be 2Nx + 1 and 2Ny + 1 and their separations as dx and dy respectively. An mnth,element in an array is located at ξm = mdx and ζn = n dy where −Nx ≤ m ≤ Nx and−Ny ≤ n ≤ Ny . The current of this mnth element can be designated by Imn. The arrayfactor given by (20–5–8) can be modified to:

0a(θ, φ) =Nx∑

m=−Nx

Ny∑n=−Ny

Imn

I00ejβ sin θ(mdx cosφ+ndy sin φ) (1)

Equation (1) is equally valid if the representative current is magnetic, for which0a(θ, φ) will be replaced by ϑa(θ, φ).

If each row has the same current, even though the current levels in different rows aredifferent, i.e. if Imn/Im0 = I0n/I00, the current distribution is said to be separable and (1)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


can be written as:

0a(θ, φ) = 0x(θ, φ)0y(θ, φ) (2)

where,

0x(θ, φ) =Nx∑

m=−Nx

Imejmβ sin θ cosφdx (3)

0y(θ, φ) =Ny∑

n=−Ny

Inejmβ sin θ sin φdy (4)

and

Im = Im0/I00, In = I0n/I00 (5)

are the normalized current distributions in a row of elements parallel to x axis and y axisrespectively.

Equation (2) indicates the principle of multiplication of patterns of two linear arrayslying along x-axis (3) and y-axis (4). If Imn and I00 differ in phase by the factor e−j (mαx+nαy)

0a(θ, φ) =Nx∑

−Nx

Imejm(βdx sin θ cosφ−αx) ·

Ny∑−Ny

Inejn(βdy sin θ cosφ−αdy) (6)

The amplitude distribution Im and In are pure real and have uniform phase progressionαx and αy in x and y directions. lf the distributions are also symmetric, the factors0x and0y represent patterns consisting conical main beams with side lobes, which are rotationallysymmetric about x and y axes respectively. The main beam of 0x makes an angle θx withpositive x axis and satisfies the relation:

βdx cos θx − αx = βdx sin θ cosφ − αx = 0

cos θx = αx

βdx− sin θ cosφ (7)

Similarly for 0y :

cos θy = αy

βdy− sin θ cosφ (8)

Equation (2) spells that planar array factor is the product of two linear array factors. Itfurther reveals that the two conical main beams and associated side lobes of each conicalpattern intersect to form a planar radiation pattern of a planar antenna. A criterion needs to bedeveloped to ensure an intersection. For given spacings dx and dy and given inter-elementphase shift αx and αy the equations, which provide unique pointing direction (θ0, φ0) inz>0 plane are:

tan θ0 = (αydx)/(αxdy) (9)

sin2 θ0 = (αx/βdx)2 + (αy/βdy)

2 (10)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


Figure 20–16

a

ϕdρ

ρ y

dϕ

θ

r

P r( , , )θ φz

x

Equation (10) indicates the situation of no intersection. If sin2 θ0 = 1, (10) reduces toan elliptic relation:

(αx/βdx)2 + (αy/βdy)

2 = 1 (11)

which limits the ranges of ax (or αy) for specified values of βdx and βdy.Since, the desired pattern has to be a function of both, θ andφ, the problemboils down to

finding the proper current distribution across a finitewidth aperture so as to produce a desiredradiation pattern. There are several techniques of synthesis like Fourier Integral method,Woodward technique that synthesizes pattern-requiring null filling, Dolph technique usingChebyshev polynomials to deduce discrete current distributions that yield sum patterns withuniform side lobes and the Taylor procedure, which accomplishes basically the same goalbut for continuous line sources and modifies the Dolph procedure.

It is also possible to determine theDolph current distribution for general (separable) casewithout going through the intermediate steps as was done in Section (20–6). In equations(20–6–13) and (14), current In were estimated since it was a one-dimensional problem.In planar array (involving two dimensions) Im can also be similarly evaluated by makingsuitable changes in (20–6–13) and (20–6–14).

Arrays with Circular Boundary: If a planar aperture with circular boundary of radiusa, contains a linear (unidirectional) current density distribution (20–4–18) and (l9) or 20–4–2) and (22) can be written as:

0θ(θ, φ) = cos θ sin φ∫s

ky(ξ, ζ )ejβ�dξdζ (12)

0θ(θ, φ) = cosφ∫s

ky(ξ, ζ )ejβ�dξdζ (13)

where,

� = ξ sin θ cosϕ + ζ sin θ sin ϕ (14)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


The integral in (12) and (13) can be taken to be the array factors for linearly polarized planaraperture distribution. In view of Fig. (20–16),

ξ = ρ cosϕ, ς = ρ sin ϕ (15)

If ky(ξ, ς) is designated as k(ρ, ϕ), the common integral becomes:

F(θ, φ) =a∫0

2π∫0

k(ρ, ϕ)ejβρ(sin θ cos(φ−ϕ)ρdρdϕ (16)

The aperture distribution can be represented by Fourier Series

k(ρ, ϕ) =∞∑

n=−∞kn(ρ)e

jnϕ (17)

Also in view of Bessel expansion

ejβρ sin θ cos(φ − ϕ) =∞∑

m=−∞jmJm(kρ sin θ)e

jm(φ−ϕ) (18)

Thus (16) becomes

F(θ, φ) =∞∑

m=−∞

∞∑n=−∞

2π∫0

a∫0

kn(ρ)(j)mJm(kρ sin θ)e

jmφej (n−m)ϕρdρdϕ (19)

The ϕ integrals of (19) has a non zero value only when m = n, which leads to:

F(θ, φ) = 2π∞∑

n=−∞jnejnφ

a∫0

kn(ρ)Jn(kρ sin θ)ρdρ (20)

For getting a φ independent pattern n should be restricted to zero value. This restrictionleads to:

F(θ) = 2π

a∫0

k0(ρ)J0(kρ sin θ)ρdρ (21)

On substituting u = 2a sin θ/λ, p = πρ/a, g0(p) = 2a2k0(ρ)/π , F(θ) transforms to

F(u) =π∫0

pg0(ρ)J0(up)dρ (22)

For a uniformly excited circular aperture, g0(p) = 1 is the representative conditionand (22) results in a sum pattern given by:

S(u) = J1(πu)/πu (23)

The pattern obtained from Eqn. (23) is illustrated by (Fig. 20–17) which contains mainbeam plus a family of side lobes that decay in height as the side lobe position becomesmore remote from the main beam. Since this pattern is rotationally symmetric the main

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


�

Figure 20–17

lobe is a pencil beam, surrounded by ring-side-lobes, the number of which in the visibleregion depends on the aperture size. Since u = 2a sin θ/λ, the range of u corresponds to0 < u < 2a/λ of visible space.

The main problem now is to get the function that modifies the side lobe structure sothat near in lobes are at controlled height. Taylor tackled the problem by moving innermostside lobes to achieve the desired side lobe level for interleaving side lobes.

If the roots of J1(u) are defined by

J1(πγln) = 0, n = 1, 2, . . . (24)

Then a modification of (24) can be written as

S(u) = [J1(πu)/(πu)]∏

(1− u2/u2n)/L(1− u2/γ 2ln) (25)

Equation (25) accomplishes the purpose of removing the first n1 − 1 root pairs of (24)and replacing them by n − 1 root pairs of new positions un. Taylor obtained the new root

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


Figure 20–18

position at

U 2n = γ 2ln{A2 + (n− 0.5)2}/{A2 + (n1 − 0.5)2} (26)

where,−20 log10(cos hπA) is the desired side lobe level. A typical circular Taylor patternis shown in Fig. 20–18, with n = 6 and desired side lobe level of−15 dB. The near in sidelobes are seen to be somewhat drooping.

To get the aperature distribution g0(p) to produce this type of pattern g0(p) can beexpressed in a series form

g0(p) =∑

BmJ0(γ1mρ) (27)

Thus (23) becomes

S(u) =∑

Bm

∫ρJ0(γ1mρ)J0(uρ)dρ (28)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4


0 n/4 n/2 3 /4n n0

1.0

Figure 20–19

In (28) each mth term will contribute toward S(u). The contribution of kth term S(γ1k)

towards sum can be written as

S(γ1k) = Bk

∫ρJ 2

0(γ1kρ)dρ (29)

From which

Bk = (2/π2)S(γ1k)/J0(γ1kπ) (30)

Since, S(γ1k) = 0 for k > n1, the series truncates and the aperature distribution is givenby:

g0(p) = (2/π2)∑

S(γ1k)J0(γ1mρ)/J20(γ1kπ) (31)

where, S(γ1k) can be computed from (27)The aperture distribution corresponding to the pattern of Fig. 20–18 is shown in Fig. 20–

19.Discretization of a circular planar aperture by employing a grid system comprising

concentric circles is a more natural arrangement, though it may not necessarily lead to apractical system from feeding aspect. The radii of the family of concentric circles may begiven by

ρm = (2m− 1)d/2 (32)

Kraus-38096 kra21032_ch20 May 4, 2006 12:4

References 899

where d is the inter radial spacing. If d is also the spacing between edges and elementsalong any circle, then

2πρm = (2m− 1)πd = Nmd (33)

in which Nm is the number of elements on the mth circle. The values of Nm satisfying (33)need not be integers. In such cases these can be rounded to the nearest integer. The sumpattern can be obtained to be

S(θ, φ) = 4∑ ∑

Imn cos(kξmn sin θ cosφ) cos(kςnm sin θ sin φ) (34)

References

Elliot, R. S: Antenna Theory and Design, Prentice Hall, New Delhi, 1985.

Jordon, E. C. and Balmain, K. G.: Electromagnetic Waves and Radiating Systems, Prentice Hall Ltd, New Delhi,1987.

Jesic, H.:. Antenna Engineering Hand book, McGraw-Hill Book Co., 1981.

Weeks, W. L. Antenna Engineering", Tata McGraw-Hill, 1974.

antenna array analysis and synthesis

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