Physics of Antenna Arrays

Harmonic oscillators
Array of oscillators
Diagram of a linear array

Harmonic oscillators

The basic unit in a transmitting antenna is a isotropic harmonic oscillator, which emits an spherical, monochromatic wave at wavelength lambda

and frequency omega

The oscillator generates an electromagnetic field with electrical component at a certain point

located at a distance

from the antenna in space is given by

$frac{1}{d} mbox{bf Re} left[ z exp left( jmath (omega t - frac{2 pi d}{lambda} right) right],$

where

is a design parameter that allows to scale and change the phase of the electrical field. We refer to this complex number as the weight of the antenna.

Array of oscillators

We now place

such oscillators at the locations $p_k in mathbf{R}^3$ , k=1,ldots,n

. Each oscillator is associated with a complex weight z_k

. Then, the total electrical field received at a point $p in mathbf{R}^3$ is then the sum

$E = mbox{bf Re}left[ exp(jmath omega t) cdot sum_{k=1}^n frac{1}{d_k} z_k exp( frac{-2pi jmath d_k}{lambda} ) right],$

Diagram of a linear array

To simplify, let us assume that the oscillators form a linear array: they are placed on an equidistant grid of points placed on the

-axis, that is, p_k = ke_1

, with

the first unit vector. Let us further assume that the point

under consideration is far away: p = r u

, with $u in mbox{R}^3$ a unit-norm vector that specifies the direction, and the distance to the origin,

, is large.

For a linear array, the electrical field

depends only on the angle between the array and the far away point under consideration, phi

. In fact, a good approximation to

is of the form

$E approx mbox{bf Re} left( frac{1}{r}exp(jmath omega t -frac{2pi r}{lambda}) right) D_z(phi)$

where

denotes the angle between the vector

and

, and the function $D_z : [0,2pi] rightarrow mathbf{C}$ , with complex values

$D_z(phi) := sum_{k=1}^n z_k exp( frac{2pi jmath k cos(phi)}{lambda} )$

is called the diagram of the antenna. We have used the subscript ‘‘

’’ to emphasize that the diagram depends on our choice of the complex weight vector, z=(z_1,ldots,z_n)