Polar form for a Loop Antenna with Uniform Current: Theory and Matlab code
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  • MATLAB program to compute the: (a) Maximum directivity (dimensionless and in dB),
  • Radiation resistance (Rr),
  • Normalized current distribution,
  • Directivity pattern (in dB) in polar form,
  • Normalized far-field amplitude pattern (H theta, in dB) in polar form for a loop antenna with uniform current.

Introduction

The polar form for a loop antenna with uniform current can be determined using the far-field approximation and the formula for the electric field radiated by a circular loop antenna.

In the far-field approximation, the radiated electric field is proportional to the derivative of the current with respect to time, and is given by:

E_theta = k * I * R * sin(theta) / (4 * pi * R)

where I is the current flowing in the loop, R is the radius of the loop, theta is the angle between the direction of observation and the plane of the loop, and k is the wave number.

For a loop antenna with uniform current, the radiated electric field in polar coordinates can be written as:

E_theta = k * I * R^2 / (2 * R) * sin(theta)

This expression represents the polar form for the electric field radiated by a loop antenna with uniform current, and can be used to compute the radiation characteristics of the antenna, such as the directivity, gain, and beamwidth.

Software Required

Matlab Software or online matlab

Algorithm

➢ START 
➢ Initialize eta and theta
➢ Eta=(180*pi)
➢ Theta=(1:180)*(pi/180)
➢ Declare step-size
➢ Read radius of loop
➢ Calculate F and U
o F=besselj(1,(2*pi*A*sin(theta)))
o U=A^2*(2*pi)^2*F.^2*eta/8
➢ Calculate radiated power
o Prad=sum(u.*sin(theta)*dth*2*pi)
➢ Calculate directivity
o D=(A*pi*max(u))/prad
➢ Converting in dB
o D_db=10*log10(D)
➢ Calculating radiating resistance
o Rr=2*prad
➢ Initialize IO and r and lambda values
o IO=1, r=10, lambda=1
➢ Calculate k value
o K=(2*pi)/lambda
➢ Calculate H-theta,HdB and theta
➢ Polar plot normalized for field elevation pattern for loop with set up 
rlim(f) constant & current 
➢ sto

Matlab Code

clear all;close all; clc;
format long;

%-Definition of constants and initialization---%

%Free space impedance:
eta = 120*pi;
% Angle vector:
theta=(1:180)*(pi/180);
%Integration step size
dth = theta(2) - theta(1);

%Reading the radius of the loop
A = input ('\nSpecify radius of loop in wavelengths:');

%-------------------------------------------------------
%diation intensity calculation:
%A^2*omega*2 X mu^2 |I0|^2*J1^2(k*A*sin(theta))
%U
%8*eta
%Using omega = 2*pi*f and f = C/lambda,
%C 1/sqrt(mu * epsilon) and eta = sqrt{mu/epsilon) = 120*pi 
%Simplified radiation intensity is given by:
% A2*(2*pi*2*eta |I0|^2*J1^2*(k*A*sin(theta))
%----------------------------------------------------------

F = besselj(1,(2.0*pi*A*sin(theta)));
U = A^2*(2*pi)^2*F.^2*eta/8;
%Radiated power:
Prad = sum(2*pi*U.*sin (theta) *dth);
%Directivity:
D = (4.0*pi*max(U))/Prad;
D_dB = 10*log10(D);
%Padiation resistance:
Rr = 2.0*Prad;

%Calculation of eievation pattern:
% -A*k*I0*exp(-jkr)
% H_theta = ----------------------*J1*(k*A*sin(theta))
% 2*r
% Normalized Peak Current
I0 = 1;
%Distance r in neters:
r=10;

% Wavellength in meters:
lambda=1; 

%Wave nunber:
k=(2*pi)/lambda;

H_theta=-A*k*I0*exp(-1j*k*r)*besselj(1,k*A*sin(theta))/(2*r);
HdB= 20*log10(abs(H_theta)/max(abs(H_theta)));
HdB = [HdB fliplr(HdB)];
theta = [theta, theta+pi];
figure(1),polarplot(theta,HdB);
title ('Normalized far field elevation pattern for loop with constant current');
rlim([-40 01]);

%Printing the values:
fprintf('\nMaximum radiated power: %f watts\n',Prad);
fprintf(' \nMaximum directivity: %f (dimensionless) \n',D);
fprintf (' \nMaximum directivity: %f (dBi) \n' ,D_dB);
fprintf('\nRadiation resistance: %f ohms\n',Rr);


Code Output


Conclusion

In conclusion, the polar form for a loop antenna with uniform current is a useful representation of the radiated electric field from the antenna. It provides information about the distribution of radiated power in different directions and can be used to evaluate the directivity, gain, and beamwidth of the antenna.

The polar form can be derived using the far-field approximation and the formula for the electric field radiated by a circular loop antenna. It takes into account the current flowing in the loop, the radius of the loop, and the angle between the direction of observation and the plane of the loop.

The polar form is an important tool for the analysis and design of loop antennas, and is widely used in various applications, such as radio communication, navigation, and sensing systems. Understanding the polar form and its application is essential for the optimization and performance improvement of loop antennas.

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