Computing The Radiation Characteristics of Linear Arrays(With MATLAB code), Array factor Equation
ARRAY FACTOR EQUATION
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Introduction

Radiation Characteristics of Linear Arrays. Computing The Radiation Characteristics of Linear Arrays(With MATLAB code), Array factor Equation.

The radiation characteristics of a linear array refer to the way in which the array radiates energy into space. The most important radiation characteristics are the directivity, gain, and beamwidth. Directivity is a measure of the focus of the radiation in a particular direction, gain is the increase in power in the desired direction, and beamwidth is the angular width of the main lobe of the radiation pattern. The radiation characteristics of a linear array can be computed using analytical methods, such as the Fourier Transform, or numerical methods, such as the Method of Moments.

The mathematical formula for computing the radiation characteristics of a linear array depends on the type of array and the desired characteristic. Here are a few commonly used formulas:

  1. Directivity: For a uniform linear array of N isotropic point sources, the directivity can be calculated as:

D = (N^2)/2

  1. Gain: The gain of a linear array can be defined as the ratio of the power radiated in the direction of maximum radiation to the power that would be radiated by a single isotropic radiator with the same total input power. It can be calculated as:

G = D * (λ^2) / (4π)

where λ is the wavelength of the radiated energy.

  1. Beamwidth: The beamwidth of a linear array can be defined as the angular width of the main lobe of the radiation pattern. It can be calculated as:

BW = 70° / (N^0.5)

ARRAY FACTOR EQUATION

Software Required

Matlab

Algorithm

➢ START
➢ Assign phi value & declare the step size
➢ Phi=(0:360)./(pi/180)
➢ Read number of source
➢ Read the distance between the source
➢ Read the phase difference between the source
➢ Calculate the psi value
➢ Psi=(2pidcos(phi))delta
➢ Calculate the energy
➢ E=(1/n)(sin(npsi/2)./sin(psi/2))
➢ Polar plot against phi and absolute values of E
➢ stop

Procedure

  1. Define the array parameters: Define the number of elements, inter-element spacing, and complex weights for each element.
  2. Compute the array factor: Use the Array Factor equation to calculate the array factor for a range of angles.
  3. Plot the radiation pattern: Plot the magnitude of the array factor as a function of angle to obtain the radiation pattern of the array.

Matlab Code

clear all;
close all;
clc;
phi = (0:1:360).*(pi/180);
n = input('Enter the number of sources:');
d = input('Enter the spacing between the sources as fraction of wavelength:');
delta = input('Enter the phase differnce between the sources:');
psi = (2*pi*d*cos(phi))+delta;
E = (1/n)*(sin(n*psi/2)./sin(psi/2));
polarplot(phi,abs(E),'Linewidth',3);

%for to get various output plot
%For Plot1: Souces=2, spacing=0.5:0.5, Phase difference=pi/2
%For Plot2: Souces=2, spacing=0.5:0.25, Phase difference=pi/2
%For Plot1: Souces=2, spacing=0.5:0.5, Phase difference=0
%For Plot1: Souces=2, spacing=0.5:0.5, Phase difference=pi

Matlab code output

Conclusion

The array factor for a linear array of N point sources is given by:

AF = ∑_{n=0}^{N-1} a_n e^{jkdcosθ_n}

where a_n is the complex weight or excitation of the n-th element, k is the wave number, d is the inter-element spacing, and θ_n is the angle between the array axis and the direction of interest. The array factor represents the superposition of the individual element patterns and provides a measure of the overall radiation pattern of the array. The magnitude of the array factor is proportional to the directivity of the array, while its phase describes the directionality of the radiation pattern. By plotting the magnitude of the array factor as a function of angle, the radiation pattern of the array can be obtained.

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