1
School of Electrical and Computer Engineering, Cornell University
ECE 303: Electromagnetic Fields and Waves
Fall 2005
Homework 12
Due on Dec. 02, 2005 by 5:00 PM
Reading Assignments:
i) Review the lecture notes.
ii) Review sections 9.19.5, 9.7, 9.8 of the paperback book
Electromagnetic Waves
.
Problem 12.1: (Receiving and transmitting antenna system)
Consider the following setup for a mobile wireless receiver
Assume that:
GHz
0
.
1
cm
15
km
1
150
30
2
1
2
1
=
=
=
=
=
=
f
d
d
L
o
o
θ
The antennas are always assumed to be matched to their respective circuits (But I can give you the
unmatched case in the final exam – be sure to ask your recitation instructors how to handle that case).
a) Find the ratio
in
out
P
P
assuming that the antennas do not dissipate power (i.e.
0
=
diss
R
).
b) Find the ratio
in
out
P
P
assuming that each antenna has a 25
Ω
impedance contribution due to internal
dissipation (i.e.
Ω
=
25
diss
R
).
P
in
d
1
P
out
d
2
1
2
L
Base station
Mobile receiver
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Problem 12.2: (Nturn small wire loop antenna)
Consider the
N
turn small wire loop antenna shown below. In this problem you will verify the antenna
theorem:
()
()
φ
θ
π
λ
,
4
,
2
G
A
=
for the small wire loop antenna in the same way as was done in the lecture notes for a shortdipole
antenna. Since the expression for gain has already been derived in the lecture notes, you will need to
calculate the antenna effective area directly and then verify the above relation. Consider plane wave
radiation with Poynting vector magnitude
P
incident from the
direction. You will need to find the
power received
P
rec
by a matched circuit connected to the antenna.
a) Find the magnitude 
H
 of the Hfield for the incident plane wave given its Poynting vector magnitude
P
.
b) Using Faraday’s law, find the voltage phasor
V
generated across the opencircuited ends of the loop
when a plane wave with magnetic field phasor
H
passes through the loop. Assume that the plane wave is
incident from the
direction and the Hfield is polarized in the
ˆ
direction (as shown in the figure
above).
Hint:
don’t forget the angular dependence here. Note that when angle
is 0degrees no magnetic
flux passes through the loop and when the angle
is 90degrees maximum magnetic flux passes through
the loop.
c) From you result in part (b), find the maximum power
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 Fall '06
 RANA
 Electromagnet, Wavelength, Electromagnetic spectrum, plane wave, antenna array, Incident plane wave

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