Department of Electronics and Telecommunications,
Faculty of Information Technology, Mathematics and Electrical Engineering,
Norwegian University of Science and Technology
NTNU
TFE4130 Wave Propagation
Problem Set 11
Problem 1
An AC power supply is attached in the middle of a half-wave dipole antenna (see Fig. 1). The
current distribution along the antenna is given by
πz .
(1)
I(z) = I0 cos
2l
The magnetic field dHφs in point A due to the current element I(z)dz is
dHφs =
I(z)dz
jk
2πf
ω
sin θ · e−jkr , where k =
= .
4π
r
c
c
(2)
Figure 1: Half-wave dipole antenna.
a) Calculate the total magnetic field Hφs in point A from the entire antenna. Do the usual
approximation based on r being much larger than the antenna length 2`.
b) Define the pattern function F (normalized to 1 for maximum field strength) and draw
the radiation pattern (use polar plot for Hφs as a function of θ).
c) The wave impedance in vacuum is ZW = 377 Ω. Use this fact to find the electric field
amplitude Es in point A. What direction does Es have?
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d) Calculate the radiated power per area in point A expressed with Hs or Es . Assume that
r0 = 100 km, θ = π/2, I0 = 10 A and f0 = 100 MHz, and calculate the total power on a
1 m2 surface perpendicular to the direction of propagation.
Useful formulas:
1 jx
e + e−jx ,
2
1 jx
e − e−jx .
sin(x) =
2j
cos(x) =
(3)
(4)
Problem 2
Figure 2: FM-antenna
The FM antenna shown in Fig. 2 radiates at a carrier frequency f0 = 100 MHz and has a
radiation impedance of ZL = 100 Ω. The generator impedance is ZG = 400 Ω and the
generator is situated many wavelengths away from the antenna. The power is fed to the
antenna via a dual transmission line with characteristic impedance Z0 = 400 Ω. A
quarter-wave transformer ensures impedance matching between transmission line and antenna.
a) Give one reason why the characteristic impedance for the transmission line to the
antenna is 400 Ω.
b) Assume that the quarter-wave transformer is removed.
• Find the reflection coefficient Γ at the antenna.
• Find the standing wave ratio S in the transmission line.
• Does the voltage have a maximum or a minimum at the antenna?
c) We wish to make a quarter-wave transformer such that we achieve impedance matching
at f0 = 100 MHz. The relative dielectric constant is εr = 1.2.
• Find the length ` and the characteristic impedance Z00 .
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