Fundamentals of Electromagnetic
Fields and Waves: II
Spring 2010, EE 30358, Electrical Engineering, University of Notre Dame
Final Exam
Friday, May 7th, 2010, Time: 10:30 - 12:30.
Notes: There are Four questions in this exam. Answer all. BOX your final answers. Draw
figures and use the provided Smith Charts wherever necessary.
1) Waveguides and Transmission Lines (6 Points):
A transmission line of intrinsic impedance Z0 = 50 Ω is used to excite the TM11 mode of an
air-filled metallic square waveguide. Show that if the dimensions of the square waveguide are
(a, a), there will be no reflection at only one frequency f0 . Find this frequency as a function
of the dimensions of the waveguide ‘a’ and other EMag constants. Can a similar impedance
matching be obtained if the same transmission line is used to excite TE modes?
2) Radiation Efficiency of an Antenna (4 Points):
We have seen that the effective
q resistivity of a non-magnetic metal wire carrying a current of
frequency f is given by ρ = πfσµc 0 , where µ0 is the permeability of free space, and σc is the
dc conductivity of the metal. A center-fed l = 4 cm long dipole antenna made of copper wire
(dc conductivity σc = 5.8 × 107 S/m and radius a = 0.4 mm) is used to radiate EMag waves
at f = 75 MHz. Find the radiation resistance Ra of the antenna, and the radiation efficiency,
Ra
, Rloss being the ac resistive loss of the wire.
which is defined as η = Ra +R
loss
3) Short Antenna (8 Points):
Figure 1: Short Antenna for Problem 3.
In the derivation of a short dipole antenna of length l (l << λ) in class, we assumed a constant
current. However, since the current should go to zero at the ends of the antenna, it is better to
assume a triangular current variation,
1
l
ˆ
I(z)
= I0 (1 − 2 zl ), 0 ≤ z ≤
2
l
z
= I0 (1 + 2 l ), − ≤ z ≤ 0
2
This is shown in Fig 1. Using this current variation, find
a) The far-field electric field E(r, θ, φ),
b) The far-field power density vector Pav (r, θ, φ),
c) The directivity D,
d) The radiation resistance Ra , and
e) Compare the results with those we know for a constant current antenna - do they make sense?
Explain.
4) All together (12 Points):
The now famous λ = 21 cm wavelength EMag radiation (called just the “21 cm line”) which originates from hyperfine atomic transitions in hydrogen is one of the most extensively used signals
for imaging galaxies and their motions, because it passes through most galactic and terrestrial
objects without being absorbed - unless of course you use an antenna to catch them. Let’s say
you are hired to design a setup for efficient detection of the 21 cm line. Since the radiated power
coming from distant galaxies is rather feeble (goes down as 1/r 2 , and r is astronomically large!),
the detection scheme has to coax out the most of the weak received signal. Since antennas are
reciprocal devices (they detect similarly as they radiate), let us look at the radiation properties
alone.
a) Choose a half-wavelength antenna. Find the length and the total input impedance (real +
imaginary) of the antenna.
b) A transmission line of intrinsic impedance Z0 = 50 Ω is used to drive this antenna. Using
the Smith chart, find the VSWR of the signal.
c) Design a scheme to impedance match the transmission line to the antenna so that it can
receive all the signal and not reflect any back.
d) Use what you have learnt on antenna arrays to suggest a method to reconstruct a spatial
image of the 21 cm line signal coming from different directions in space. Fig 2 shows an image
of the Milky Way reconstructed in this fashion (it can be done!).
Figure 2: The Milky Way imaged by antennas detecting the 21 cm line.
–End–
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