The version for printing
2003 Antennas and Propagation short course
The Yagi-Uda antenna.
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.
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.
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.
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
TWO ISOTROPES 1/2 WAVELENGTH APART FED IN PHASE
(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
TWO ISOTROPES 1 WAVELENGTH APART, IN PHASE
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
TWO ISOTROPES 1/2 WAVELENGTH APART ANTIPHASE
FOUR ISOTROPES SPACED 1/4 WAVELENGTH IN PHASE
FOUR ISOTROPES SPACED 1/2 WAVELENGTH, IN PHASE.
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
YAGI-UDA DESIGN, 4 ELEMENTS
Note that this is the H-plane pattern (in the plane at right angles to the rods), as it assumes that each of the elements is omnidirectional. If we wish the E-plane pattern (in the plane of the rods) we must pointwise-multiply this "array pattern" by the "element pattern", which in this case is a simple half-wave-dipole E-plane pattern.
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])^2 + (y-p[i])^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, shading=Z, thickness=2); 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.
Copyright © D.Jefferies 1997, 2001, 2002, 2003.