Propagation and Antennas

Running head: PROPAGATION AND ANTENNAS 1
Propagation and Antennas
Name
Institution

PROPAGATION AND ANTENNAS 2
Question 1
a0=1 , a1=2 , δ = π
2 , d= λ
4
The expression for an N-element antenna array factor is:
Fa ( θ ) =|∑
i=0
N−1
ai e j ψ 1
e jikdcos ( θ )
|
2
Since it is a two-element array, we substitute N=2 to get:
Fa ( θ ) =|∑
i=0
N−1
ai e j ψ 1
e jikdcos ( θ )
|
2
=|∑
i=0
2−1
ai e jψ1
e jikdcos ( θ )
|
2
=|∑
i=0
1
ai e j ψ1
e jikdcos ( θ )
|
2
¿|a0 e j ψ0
e j (0 )kdcos (θ ) +a1 e j ψ 1
e j(1)kdcos (θ )
|2
¿|a0 +a1 e j δ e j ( 2 π
λ ) dcos ( θ )
|
2
Substituting the values for a0 , a1 , δ ,∧dwe get:
Fa ( θ )=|1+2 e j ( π
2 ) e j ( 2 π
λ )( λ
4 )cos (θ )
|2
=|1+2 e j ( π
2 ) e j ( π
2 )cos (θ )
|2
¿|1+2 e j ( π
2 ) cos ( θ ) + j ( π
2 )|
2
=|1+2 e j ( π
2 ) ( cos ( θ ) +1 )
|
2
¿
|1+2 [ cos ( π
2 ( cos ( θ ) +1 ) ) + jsin ( π
2 ( cos ( θ ) +1 ) ) ]|
2
¿
|1+2 cos ( π
2 ( cos ( θ ) +1 ) )|2
+
|2 sin ( π
2 ( cos ( θ )+ 1 ))|2
¿ 1+4 cos ( π
2 ( cos ( θ ) +1 ) )+ 4 co s2
( π
2 ( cos ( θ ) +1 ) )+4 si n2
( π
2 ( cos ( θ )+1 ) )
¿ 1+4 cos ( π
2 ( cos ( θ ) +1 ) )+ 4 {co s2
( π
2 ( cos (θ ) +1 ) )+ si n2
( π
2 ( cos ( θ )+1 ) ) }
¿ 1+4 cos ( π
2 ( cos ( θ ) +1 ) )+ 4 (1)

PROPAGATION AND ANTENNAS 3
¿ 5+ 4 cos ( π
2 ( cos ( θ ) +1 ) )
Therefore, the array factor Fa ( θ ) =5+ 4 cos ( π
2 ( cos ( θ ) +1 ) )
The plot of the array factor in MATLAB using the code below is shown I figure 1.
%MATLAB code for plotting array factor
clear all;
% Defining theta range
F = zeros(1,360);
for theta=1:360
% change degree to radian
deg2rad(theta) = (theta*pi)/180;
%array factor calculation
F(theta) =abs(5+4*cos(pi/2*(cos(deg2rad(theta))+1)));
end
% plot the array factor
polar(deg2rad,F);
%Title and Axis Labels
title('Array Factor in x-z plane');
xlabel('Theta');
ylabel('F(theta)');

PROPAGATION AND ANTENNAS 4
Figure 1: MATLAB Array Factor Plot
Question 2
Array Factor , Fa ( γ' ) =
si n2
( N γ '
2 )
N2 si n2
( γ '
2 ) , N=number of elements
γ'=kdcos θ−kdcos θ0
but , δ=kdcos θ0 =0 due to equal phase excitation. Hence, γ' =kdcos θ
spacing, d = λ
2 ∧k= 2 π
λ