a Matlab package for phased array beam shape inspection

16 4 A PHASED ARRAY
∆ m = û · d m of the element from the plane P u normal to û, and is Ψ m = 2π∆ m /λ.
Therefore, the total field at the target is
E = ∑ ∑
E m = E 0 e i2πû· dm λ . (36)
m
m
The sum multiplying the field E 0 in Eq. (36) takes care of the relative phases of the
elements caused by their different positions, and is called the array factor AF. There
may be also inherent phase and amplitude differences between the elements. These can
be accounted for by multiplying the complex exponentials by complex amplitudes b m ,
and these can also be bundled into the array factor, which becomes
AF(û) = ∑ m
b m e i2πû· dm λ . (37)
The sum goes through all the elements comprising the array.
Often in the litterature the exponent in the array factor Eq. (37) is written in terms
of the wave vector k = kû = (2π/λ)û of the plane wave arriving to the target from the
antenna. In this note, we prefer to work with the dimensionless unit vector û instead,
and will write most of our equations in terms of dimensionless quantities only. The basic
reason is that the physical problem at hand, the far field antenna gain pattern, depends
only on dimensionless quantities, like beam directions and the ratio of various lengths
to the wavelength, and is thus inherently dimensionless. We also prefer to work directly
on the discrete problem using the tools of discrete math, instead first moving matters
to the continuous domain and then returning via Dirac delta-functions. Sometimes, our
approach seems to result in slightly more compact equations. 1
In the e3ant package, we assume that the array is rectangular in shape, regularly
spaced, and in the xy-plane, with the longer side along the x-axes. We split the amplitudes
b m = b mxmy
in such away that we can write the array factor as
AF(u x , u y ) =
M x−1,M y−1
∑
m x=0,m y=0
a mxm y
e i(2πuxDx−δx)mx e i(2πuyDy−δy)my . (38)
Here D x = d x /λ and D y = d y /λ specify the array element spacing in the x- and y-
directions, and M x and M y are the number of element rows in the y-and x-directions.
The cartesian x- and y-coordinates u x and u y of û, also called the direction cosines
because u x = cos(û, x), uniquely specify the pointing direction
√
in the upper half sphere,
the z-component is found by the normalization as u z = + 1 − u 2 x − u 2 y. The cartesian
coordinates can be expressed in terms of the usual spherical coordinates φ and θ, where
the azimuth angle φ is measured in the xy-plane counter-clockwise from the positive
x-axis, that is, towards the positive y-axis; and θ is the polar angle, measured from the
positive z-axis, and is the complement of the elevation angle. But note that azimuth in
the EISCAT pointing geometry programs is measured in clockwise direction from the
north, irrespective of how our array is oriented. The spherical-to-cartesian conversion is
u x = sin(θ) cos(φ) (39)
u y = sin(θ) sin(φ) .
The standard Matlab function sph2cart can be used for this transformation.
1 For instance, compare the equations 9.83-9.86 in Kildal’s (factually excellent but typographically
horrible) book Foundations of antennas, to our Eq. (69).