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4 1 Introduction

This chapter will focus on array antennas . Arrays are antennascomprised of multiple physically identical individual elements , which are

spatially arranged and excited with respect to each other in order tocreate a specific directional radiation pattern .

To analyze radiation from an array, consider the antenna system depictedin Figure 4.1. The antenna consists of elements, numbered 0 through

1 . All elements are physically identical , but positioned at differentspatial locations and excited by signals that may differ from each other inmagnitude and phase .

Let the position of each element be characterized by a position vector, = 0, , 1, and let the current at the terminals of the individual

elements be denoted by , = 0, , 1. 2

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The electric field ( ) radiated by the array can be found from the far-field vector potential ( ) , which is given by:

where ( ) denotes the total current on all array elements.

(4.1)

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Denoting the current on the individual elements by ( ), = 0, , 1, we have:

and, hence:

(4.2)

(4.3)

Figure 4 2 Arrayreference element and

local coordinate definition 4

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To simplify the integral, we decompose as = + , where is alocal position vector in a coordinate frame attached to element (Fig 4.2).

(4.4)

where ( ) denotes the current on a hypothetical reference element ,positioned at the origin, and excited by a terminal current = 1 .Substituting Eq.(4.4) into (4.3), and rearranging terms, yields:

(4.5)

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Eq.(4.5) indicates that the vector potential associated with radiation by

an array equals the product of two factors : The vector potential associated with the reference element:

A scalar quantity , further termed the array factor , given by:

Using Equations (4.6) and (4.7), we may express the vector potentialassociated with an array also as:

(4.6)

(4.7)

(4.8)6

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The electric field radiated by the array may be obtained as:

The magnetic field can be expressed similarly:

where

(4.9)

(4.10)

(4.11)

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4 2 The Array Factor for Linear Equally Spaced Arrays

Linear arrays consist of individual elements arranged along a straightline . In this and the following sections, we will assume that the arrayconsists of elements which are positioned along the -axis (Fig. 4.3).

If the -coordinates of the array elements are denoted by , = 0, , 1, the expression for the array factor takes the form:

(4.12)

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In many practical applications, the array elements are equally spaced. Ifthe inter-element spacing is denoted by d , the positions of the elements ofa Linear Equally Spaced Array (LESA) are given by:

The array factor of a LESA becomes:

where = . From Eq.(4.14), we conclude that the array factor ofan element LESA can be expressed as a polynomial of degree 1 inthe complex variable . The portion of the unit circle = 1 with phaseangles in between and 0 = is termed " the visible region ".

(4.13)

(4.14)

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(4.15)

Often, we will write Eq.(4.14) in a more suggestive form as:

where denotes the zero of the polynomial (4.14).

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(4.16)

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Solution

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The magnitude of this array factor in the plane is plotted in Figure4.4. The pattern is axisymmetric, i.e. the pattern is identical to thepattern shown here.

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(4.17)

(4.18)

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(4.19)

we obtain:

It is easily shown that the normalized array factor for which

( , ) = 1 is given by:

(4.20)

(4.21)15

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In a Linear CoPhasal Equally Spaced Array (LCPESA) , we chooseelement excitations with the same magnitude, but with a uniform phaseprogression from element to element , as:

The array factor becomes:

where = = ( cos + ). It is now easily verified that the arrayfactor and normalized array factor of the LCPESA are still given by (4.20)

and (4.21), provided that we generalize the definition for

to:

From now on, we will regard the LUEESA as a special case of theLCPESA with = 0 .

(4.22)

(4.23)

= cos + (4.24)

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o Examples

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o Examples

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