L11 Antenna Array Part 2

8/12/2019 L11 Antenna Array Part 2

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Dr. Mohamed Ouda

Electrical Engineering Department

Islamic University of Gaza

2013

Antenna Theory

EELE 5445Lecture 11: Antenna Array


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Advantages of linear array with nonuniformamplitudeThe most often met BSAs, classified according to the type of their excitationamplitude, are:

a) the uniform BSArelatively high directivity, but the side-lobe levels

are high;

b) DolphTschebyscheff (Chebyshev) BSAfor a given number ofelements maximum directivity is next after that of the uniform BSA; side-

lobe levels are the lowest in comparison with the other two types of arrays

for a given directivity;

c) binomial BSAdoes not have good directivity but has very low side-lobe levels (when d =/2, there are no side lobes at all).

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Array factor AF)

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Array factor AF) of a linear array withnonuniform amplitude distribution

Let us consider a linear array with an even number (2M) of elements, locatedsymmetrically along the z-axis, with excitation, which is also symmetrical

with respect to z=0. For a broadside array (=0),

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If the linear array consists of an odd number (2M+1) of elements,

located symmetrically along the z-axis, the array factor is

The normalizedAF derived can be written in the form

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Notice that as the current amplitude is tapered more toward the edges

of the array, the side lobes tend to decrease and the beamwidth tendsto increase.


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Binomial broadside array The binomial BSA was investigated and proposed by J. S. Stone to synthesize

patterns without side lobes.

First, consider a 2element array (along the z-axis).

The elements of the array are identical and their excitations

are the same. The array factor is of the form

If the spacing is d /2 and =0 (broad-side maximum), the array pattern|AF| has no side lobes at all.

Second, consider a 2element array whose elements are

identical and the same as the array given above.

The distance between the two arrays is again d.

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This new array has anAF of the form

AF = (1+ Z)(1+ Z) =1+ 2Z + Z2.

Since (1+ Z) has no side lobes, (1+ Z)2does not have side lobes either.

Continuing the process for an N-element array produces

AF = (1+ Z)N-1

If d /2, the aboveAFdoes not have side lobes regardless of the number

of elements N. The excitation amplitude distribution can be obtained easily

by the expansion of the binome. Making use of Pascals triangle,

the relative excitation amplitudes at each element of an (N+1)-element

array can be determined. 12


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The excitation distribution as given by the binomial expansion gives

the relative values of the amplitudes.

It is immediately seen that there is too wide variation of the

amplitude, which is a disadvantage of the BAs.

The overall efficiency of such an antenna would be low. Besides, the

BA has relatively wide beam. Its HPBW is the largest as compared

to this of the uniform BSA or the DolphChebyshev array.

An approximate closed-form expression for the HPBW of a BA with

d =/2 is

where L=(N-1)d is the arrays length. The AFs of 10-element

broadside binomial arrays (N=10) are given below.

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The directivity of a broadside BA with spacing d =/2 can be

calculated as

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DolphChebyshev array DCA) Dolph proposed (in 1946) a method to design arrays with any desired

sidelobe levels and any HPBWs. This method is based on the approximation of the pattern of the array by a

Chebyshev polynomial of order m, high enough to meet the requirement

for the side-lobe levels.

A DCA with no side lobes (sidelobe level of -dB) reduces to thebinomial design.

4.1. Chebyshev polynomials

The Chebyshev polynomial of order m is defined by

(1)

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A Chebyshev polynomial Tm(z) of any order m can be derived via a recursion

formula, provided Tm1(z) and Tm2(z) are known:

(2)

Thus

If |z|

1, then the Chebyshev polynomials are related to the cosine functions.

We can always expand the function cos(mx) as a polynomial of cos(x) of

order m, e.g.,

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Similar relations hold for the hyperbolic cosine function.

Comparing the trigonometric relation with the expression for T2(z) above,

we see that the Chebyshev argument zis related to the cosine argument x

by Then

Properties of the Chebyshev polynomials:

1) All polynomials of any order mpass through the point (1,1).

2) Within the range -1z1, the polynomials have values within [1,1].3) All nulls occur within -1 z 1.

4) The maxima and minima in the z[-1,1] range have values +1 and1,respectively.

5) The higher the order of the polynomial, the steeper the slope for |z|>1.

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Chebyshev array design

The main goal is to approximate the desired AF with a Chebyshev

polynomial such that the side-lobe level meets the requirements, and themain beam width is as small as possible.

An array of Nelements has an AF approximated with a Chebyshev

polynomial of order m, which is always M=N-1

where N=2M , if Nis even; and N=2M +1, if N is odd. In general, for a given side-lobe level, the higher the order m of the

polynomial, the narrower the beamwidth. However, for m > 10, the

difference is not substantial.

The AF of an N-element array is identical with a Chebyshev polynomial if

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Here, u=(d /)cos. Let the side-lobe level be

Then, the maximum of TN-1is fixed at an argument z0(|z0|>1), where

The previous equation corresponds to maxAF(u)=AFmax(u0) .

Obviously, z0must satisfy the condition:|z0|>1, where TN-1>1. The maxima of |TN-1(z)| for |z|>1are equal to unity and they correspond

to the side lobes of theAF. Thus,AF(u)has side-lobe levels equal b to R0.

TheAFis a polynomial of cosu, and the TN-1(z)is a polynomial of z where

the limits for z are

Since

the relation between z and cosu must be normalized as

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Design of a DCA of elementsgeneral procedure:

1) Expand the AF as given by (1) or (2) by replacing each cos(mu)

term (m =1,2,...,M ) with the power series of cosu .

2) Determine z0such that TN-1(z0)=R0(voltage ratio).

3) Substitute cosu =z/z0 in the AF as found in step 1.4) Equate the AF found in Step 3 to TN-1(z0) and determine the

coefficients for each power of z.

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Maximum affordable d for Chebyshev arrays

This restriction arises from the requirement for a single major lobe

For a given array, when varies from 0oto 180o, the argument z assumes

values

The extreme value of z to the left on the abscissa corresponds to the end-fire

directions of the AF. This value must not go beyond z=-1.

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Otherwise, minor lobes of levels higher than 1 (higher than R0) will

appear. Therefore, the previous inequality must hold for =0oor

=180o:

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Planar arrays

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Planar arrays provide

directional beams, symmetrical patterns

with low side lobes,

much higher directivity

than that of theirindividual element.

In principle, they can

point the main beam

toward any direction.

Applications trackingradars, remote sensing,

communications, etc.


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The AF of a linear array of M elements along the x-axis is

where sincos=cos xis the directional cosine with respect to the x-axis (x isthe angle between r and the xaxis).

It is assumed that all elements are equispaced with an interval of dx and a

progressive shift x. Im1denotes the excitation amplitude of the element at the

point with coordinates: x =(m-1)dx,y=0. In the figure above, this is the element of the m-th row and the 1stcolumn of

the array matrix.

If N such arrays are placed at even intervals along they direction, a

rectangular array is formed.

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We assume again that they are equispaced at a distance dy and there is a

progressive phase shift yalong each row.

We also assume that the normalized current distribution along each of

the x-directed arrays is the same but the absolute values correspond to afactor of I1n(n=1,...,N). Then, the AF of the entire MxN array is

Or Where

In the array factors above

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The pattern of a rectangular array is the product of the array factors of the

linear arrays in the x and y directions.

In the case of a uniform planar (rectangular) array, Im1= I1n= I0for all m

and n, i.e., all elements have the same excitation amplitudes. Thus,

The normalized array factor is obtained as

The major lobe (principal maximum) and grating lobes of the terms

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l d l h h


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are located at angles such that

The principal maximum corresponds to m=0, n=0.

In general, xand yare independent from each other. But, if it is requiredthat the main beams of SxMand SyNintersect (which is usually the case),

then the common main beam is in the direction:

If the principal maximum is specified by (0,0), then the progressive

phases xand ymust satisfy

When xand y are specified, the direction of the main beam can be

found by simultaneously solving the previous equations

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To avoid grating lobes, the spacing between the elements must be less

than (dx


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L11 Antenna Array Part 2