# Antenna arrays

The version for viewing

Antennas notes.

## Array definitions.

An array of antenna elements is a spatially extended collection of N similar radiators or elements, where N is a countable number bigger than 1, and the term "similar radiators" means that all the elements have the same polar radiation patterns, orientated in the same direction in 3-d space. The elements don't have to be spaced on a regular grid, neither do they have to have the same terminal voltages, but it is assumed that they are all fed with the same frequency and that one can define a fixed amplitude and phase angle for the drive voltage of each element.

## Element pattern, Array pattern.

The polar radiation pattern of a single element is called the "element pattern". It is possible for the array to be built recursively; for example the element may itself be an array, as would be the case if we had an array of Yagi-Uda antennas. A Yagi-Uda antenna may be thought of as an array of dipoles with different amplitudes and phases of the dipole currents.

The array pattern is the polar radiation pattern which would result if the elements were replaced by isotropic radiators, having the same amplitude and phase of excitation as the actual elements, and spaced at points on a grid corresponding to the far field phase centres of the elements.

## Pattern multiplication.

If we assume that all the polar radiation patterns of the elements taken individually are identical (within a certain tolerance) and that the patterns are all aligned in the same direction in azimuth and elevation, then the total array antenna pattern is got by multiplying the array pattern by the element pattern. It does not matter if we consider the patterns in question to be both patterns of radiated power, or both patterns of amplitudes having modulus and phase angle, provided the resulting pattern is interpreted as being of the same type as the original patterns.

## Calculation of array patterns

The radiated field strength at a certain point in space, assumed to be in the far field, is calculated by adding the contributions of each element to the total radiated fields. The field strengths fall off as 1/r where r is the distance from the isotrope to the field point. We must take into account any phase angle of the isotrope excitation, and also the phase delay which is due to the time it takes the signal to get from the source to the field point. This phase delay is expressed as 2 Pi radians times (r/lambda) where lambda is the free space wavelength of the radiation. Contours of equal field strength may be interpreted as an amplitude polar radiation pattern. Contours of the squared modulus of the field strength may be interpreted as a power polar radiation pattern.

Here is an example of a power polar radiation pattern for two isotropes spaced 1/4 wavelength apart along the x axis (horizontally on your screen or paper) and fed with equal amplitudes and phases......-->

TWO ISOTROPES 1/4 WAVELENGTH APART FED IN PHASE

If we increase the spacing to 1/2 wavelength, but still keep the excitation in phase and equal amplitudes, we see deep nulls developing.......-->

TWO ISOTROPES 1/2 WAVELENGTH APART FED IN PHASE

If we restore the original 1/4 wavelength spacing and feed the isotropes in phase quadrature (that is, there is Pi/2 phase shift between the excitations), we see a single lobe develop. This pattern is close to a Cardioid......-->

(To make the lobe face the other direction we would have to reverse the relative phase, and feed the second isotrope at -Pi/2 phase angle.)

TWO ISOTROPES 1/4 WAVELENGTH APART 90 DEGREES PHASE SHIFT

Returning now to feeding the two isotropes in phase, as we increase the spacing we see more sub-lobes or sidelobes develop. Here is a picture for spacing of a whole wavelength....-->

TWO ISOTROPES 1 WAVELENGTH APART, IN PHASE

And if we increase the spacing to six whole wavelengths, there are large numbers of lobes developing......-->

This is a method used to make sharp beams for radio telescopes using "Very Long Baseline Interferometry". In this method two elements are spaced thousands of kilometres apart, which makes the individual lobes very narrow indeed.

TWO ISOTROPES 6 WAVELENGTHS APART IN PHASE

If we now take two isotropes spaced by a half a wavelength and feed the elements in antiphase (Pi phase difference) we see the pattern with horizontal lobes rather than vertical lobes....-->

TWO ISOTROPES 1/2 WAVELENGTH APART ANTIPHASE

Now we turn to arrays of four elements, just to illustrate the ideas further. Equally excited (same amplitudes and phases) and spaced along the x axis at intervals of 1/4 wavelength, we see......-->

FOUR ISOTROPES SPACED 1/4 WAVELENGTH IN PHASE

and again, if we increase the spacing to 1/2 wavelength we get more sidelobes....-->

FOUR ISOTROPES SPACED 1/2 WAVELENGTH, IN PHASE.

To see the pattern multiplication, we add a row of 4 extra isotropes, spaced by 1/4 wavelength and fed in phase quadrature, below the original row of 4 sources. This has the effect of multiplying the pattern seen above by the cardioid pattern seen earlier. Of course, the cardioid has the main lobe facing upwards because the quadrature array is spaced out along the y axis.......-->

PATTERN MULTIPLICATION. 8 ISOTROPES 4x2.
1/2 WAVELENGTH SPACING FOR 4 ELEMENTS ALONG X (IN PHASE)
1/4 WAVELENGTH SPACING TO 4 ELEMENTS ALONG Y (IN QUADRATURE)

```

x    x    x    x

x    x    x    x
```

Finally, let us take a guess at the amplitudes and phases of the elements in a 4-element Yagi-Uda antenna. The pattern we arrive at looks like this........

YAGI-UDA DESIGN, 4 ELEMENTS

and for completeness, the Maple program used to calculate this pattern is given below....you may wish to extract this from the web page and use it yourselves suitably modified.

### XMAPLE program for 4 element Yagi-Uda approximation

```
Thanks to Jonathan Deane for the Maple programming
>>>>>>>>>>>>>>>>>>>>>>>>>>cut here>>>>>>>>>>>>>>>>>>>>>>>>>>>>
with(plots):

# ------------------
# Default parameters
# ------------------

NS := 4;		# No. of sources
GRD := [100,100];  	# Grid size
Cutoff := .07;		# Maximum amplitude allowed
Xrange := -50..250;	# x-range for plotting
Yrange := -150..150;	# y-range for plotting

#positions of elements p in wavelengths
p := [[-.2,0],[0,0],[.31,0],[.62,0]];

#amplitudes of element drives A
A := [1.3,1.5,1.0,0.7];

#phases of element drives phi in radians
phi := [-3*Pi/5,0,Pi*(1-.31),2*Pi*(1-.31)];

# ------------------
# Plotting procedure
# ------------------

#to invoke type    plot_amp(p,A,phi);

plot_amp := proc(p, A, phi)
local	i, r, amp;
amp := 0;
for i from 1 to NS do
r := sqrt( (x-p[i][1])^2 + (y-p[i][2])^2 );
amp := amp + (A[i]/r) * exp(-I*(r*2*Pi + phi[i]));
od;
amp := min(evalc(abs(amp)), Cutoff);
contourplot(amp^2, x=Xrange, y=Yrange, grid=GRD, axes=normal,
end:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>end program>>>>>>>>>>>>>>>>>>>>>>>>>>>

```

For commercial software for a PC platform, you could try http://www.nittany-scientific.com/ although as I have not tried out these packages myself I cannot comment on their ease of use or accuracy. At this site there are many useful onward web links.

Good plotting!