ความเห็นเพิ่มเติมที่ 2 29 ก.ค. 2554 (22:26) SOLUTION Homework #2
1. (Ref: Ex. 2.3) A hypothetical isotropic antenna is radiating in free-space. At a
distance of 100 m from the antenna, the total electric field (Eθ) is measured to be 5
V/m. Find the
a. Power density (Wrad)
( )
2 2
1 * 5 0.03315
2 2 2 120
r
rad r r
W E H E a a a
η π
= ⎡⎣ × ⎤⎦ = = =
watts/m2
b. Power radiated (Prad)
( )( ) 2
2
0 0
0.03315 sin rad S rad
P W dS r dd
π π
= ∫ = ∫ ∫ θ θ φ
( )
2
2
0 0
0.03315 100 sin d d
π π
= ∫ ∫ θ θ φ
= 4165.75 watts ANS
2. (Ref: Ex. 2.4: c, f) Find the HPBW and FNBW, in radians and degrees, for the
following normalized radiation intensities: (where 0 ≤θ ≤ 900 and 0 ≤φ ≤ 3600 )
a. U (θ ) = cos (2θ )
( ) 0.5 cos (2 ) 1 cos 1 (0.5) 300
h h h 2 U θ = = θ ⇒θ = − =
HPBW = 2(30) = 600
( ) 0 cos (2 ) 1 cos 1 (0) 450
n h h 2 U θ = = θ ⇒θ = − =
FNBW = 2(45) = 900
b. U (θ ) = cos2 (3θ )
( ) 0.5 cos2 (3 ) 1 cos 1 ( 0.5) 150
h h h 3 U θ = = θ ⇒θ = − =
HPBW = 2(15) = 300
( ) 0 cos2 (3 ) 1 cos 1 (0) 300
n h h 3 U θ = = θ ⇒θ = − =
FNBW = 2(30) = 600 ANS
3. (Ref: Ex. 2.7: a) The power radiated by a lossless antenna is 10 watts. The
directional characteristics of the antenna are represented by the radiation intensity
of 2 ( )
0 U = B cos θ watts/unit solid angle ( 0 ≤θ ≤ 900 and 0 ≤φ ≤ 3600 ). Find the
a. Maximum power density (in watts/square meter) at the distance of 1000 m
(assume far-field distance). Specify the angle where this occurs.
2 /2 /2
2
0
0 0 0
sin 2 cos sin radP U dd B d
π π π
= ∫ ∫ θ θ φ = π ∫ θ θ θ
3 / 2
0 0 0 0
0
2 cos 2 1 2 10 15
rad 3 3 3 P B B B B
π θ π
π π
π
⎡ ⎤ ⎡− ⎤ = − ⎢ ⎥ = − ⎢ ⎥ = = ⇒ = ⎣ ⎦ ⎣ ⎦
( ) ( ) 2
2
max 2 2
max max
15 cos 15cos rad
U W U
r r
θ
θ
π
= ⇒ = =
( )
6
2
15 4.7746*10
π 1000
= = − watts/m2 @ θ = 00
6
max 4.7746*10 rad W = − watts/m2 @ θ = 00
b. Exact and approximate beam solid angle A Ω
( )
2 /2
2
0 0
exact cos sin 2
A n 3 U dd
π π π
Ω = ∫ ∫ θ θ θ φ = steradians
0.5 cos2 ( ) cos 1 ( 0.5) 450 h h U = = θ ⇒θ = − = ⇒ HPBW = 2*45 = 900
( ) ( )
2
2 approx 2.4674
A h 2
π
Ω = θ = ⎛⎜ ⎞⎟ =
⎝ ⎠
steradians
c. Exact directivity of the antenna (dimensionless and in dB)
0 ( ) ( )
exact 4 4 6 7.782
exact 2 / 3 A
D
π π
π
= = = =
Ω
dB ANS
4. SCILAB: From an example #2.4 (page 43 in Balaniss textbook, Figure 2.11),
write a program to plot a 3-D picture of an example #2.4 Also write a program to
iteratively compute the HPBW.
//==== "phi" in this program is "THETA" used in lecture.
//==== "theta" in this program is "PHI" used in lecture.
//==== SEE Page 184 in SCILAB book
clear;
phi = linspace(0,%pi/2,31);
theta = linspace(0,2*%pi,61);
[PHI, THETA] = ndgrid(phi,theta);
rho = (cos(PHI).*cos(3*PHI)).^2;
X = rho.*cos(THETA).*sin(PHI);
Y = rho.*sin(THETA).*sin(PHI);
Z = rho.*cos(PHI);
plot3d2(X,Y,Z,alpha=60)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Z
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3 X
-0.3
-0.2
-0.1
0.0
0.1
0.2
Y 0.3
clear;
theta = 0:0.001:1; //== in radian
for i = 1:length(theta)
temp = acos(0.707/cos(3*theta(i)));
err = abs(temp - theta(i));
if (err < 0.01)
break;
end
end
[i temp theta(i) err]
-->252 0.2497667 0.251 0.0012333
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