ELEC 390
Theory and Applications of Electromagnetics
Spring 2012
Homework Assignment #9 – due in Dana 301 at 5 pm on Tuesday, May 1, 2012
Instructions, notes, and hints:
You may make reasonable assumptions and approximations in order to compensate for missing
information, if any. Provide the details of all solutions, including important intermediate steps.
You will not receive credit if you do not show your work. Unless otherwise indicated, you may
use Matlab to make difficult or time-consuming calculations. If you do, include a print-out of the
m-file or screen display that shows your work. Use SI units, unless others are specified.
Special Note: For all problems that ask for the type of polarization, you must determine the
answer by hand (i.e., show your work). You may use the polarization demo web site to check
your work.
Assignment:
1. Solve Prob. 9.19 in the textbook. The “physical cross-section” of the antenna is the projected
area of the dipole’s wire surface relative to an incoming wave in the direction of maximum
sensitivity. Also, find the radius of the circular area that is equivalent to the dipole’s effective
area, and compare it to the length of the dipole.
2. The WVIA transmitter on the roof of the Dana Engineering Building produces an output
power of around 700 W, which is supplied to an antenna with a gain of around 6 dBi.
Assume that the transmission line between the transmitter and the antenna has 2 dB of loss
and that all receiving stations are in the direction of maximum radiation from the transmitting
antenna. Suppose that a certain FM broadcast receiver tuned to WVIA at 90.5 MHz requires
a received signal strength of –100 dBm (1013 W) to produce mostly noise-free sound. The
radio uses a half-wave dipole antenna oriented so that the signal arrives from a direction that
is within the half-power beamwidth of the dipole. How far can the receiver be from the
WVIA transmitting site and still produce mostly noise-free sound?
3. Solve Prob. 9.23 in the textbook. For part c, the total noise power in the receiver is given by
Equation (9.71). The signal-to-noise ratio is defined in Equation (9.72).
4. Solve Prob. 7.2 in the textbook. “Nonmagnetic” means that  = o. The problem is asking
you to find the full mathematical expressions for the electric and magnetic field vectors (E
and H). The expressions should be in numerical form, not symbolic form.
5. Solve Prob. 7.8 in the textbook with the additional condition that the electric field must point
in the direction 45° above the positive x-axis (i.e., halfway between the positive x-axis and
the positive y-axis) at time t = 0.1 ns at the location (x, y, z) = (0, 0, 2 cm).
(continued on next page)
6. Plot the locus of the electric field vector as a function of time for the wave described by each
of the following phasor expressions. In each case, identify the type of polarization the wave
has. If the polarization is circular or elliptical, specify the sense (left-hand or right-hand). If
linear, specify the angle the electric field makes with the y-axis.
~
a. E  yˆ 0.2e j 0.3 e  j 2.1x  zˆ 0.8e j 0.3 e  j 2.1x V/m
~
b. E  xˆ 5e j 0.3 e  j 0.22 z  yˆ 5e  j 0.7 e  j 0.22 z V/m
7. A transmitter that is part of a communication link generates an EM plane wave that
propagates through air. It can be described mathematically by the expression shown below.
The engineers analyzing the communication link set up a coordinate system in which the
transmitter is located at the origin and the receiving station is located 10 km away along the
x-axis. What type of polarization is present at the receiving site? If the polarization is circular
or elliptical, specify the sense (left-hand or right-hand). If linear, specify the angle the
electric field makes relative to the θ̂ direction.
e  j 2.1R ˆ
e  j 2.1R
~
E  φˆ 7.0e  j 0.2
 θ 3.8e j 0.3
mV/m
R
R
8. Find an expression for the magnetic field associated with the electric field given in Prob. 7.
9. A uniform plane wave propagating in the +z-direction is partially represented by the
mathematical expression shown below. The expression for the y-component (Ey) is missing.
Find the mathematical form for Ey that results in the expression below describing a righthand circularly polarized (RHCP) wave.
~
E  xˆ 0.2e  j 0.1 e  j 4.8 z  yˆ E y mV/m
10. The expression for the electric field in air of an elliptically polarized wave is given below.
The frequency of operation is 100 MHz.
~
E  yˆ 5e  j 0.2 e  jkx  zˆ 2e j 0.3 e  jkx V/m
a. A center-fed dipole is placed in a plane parallel to the yz-plane and connected to a
receiver. What orientation with respect to the y-axis should the dipole have in order to
maximize the received signal power? That is, what angle should the dipole make with the
y-axis?
b. What is the rotational sense of the wave (RH or LH)?
c. What is the corresponding expression for the magnetic field?