Sample Problems EE 204 Electromagnetics II

Sample Problems
EE 204 Electromagnetics II
1.
The arrangement shown in the figure below, represent an elementary AC generator,
where a square loop of wire, 0.6 m on the side rotates with angular velocity of
ω = 60π rad sec around the x-axis in a magnetic field B = 0.059 a z Web m 2 .
(a)
Calculate the induced emf across the slip ring (i.e. across the load).
(b)
Sketch the emf waveform for one revolution as the loop rotates in a clockwise
direction from position A, B, C , D, A as shown.
(c)
What is the effective (rms ) value of the current and voltage across the load of 2 Ω ?
(d)
How much average power is delivered to the load?
(e)
Sketch the instantaneous current and power in the diagram of part (b) and show the
average power line in your sketch.
1
2.
A circular disk of radius r shown in the Figure,
rotates at angular velocity ω rad . sec . in a
uniform magnetic flux density B = Bo a z . Sliding
contacts connects a voltmeter to the disk.
(a)
3.
(b)
Find the induced voltage between the center
and the rim of the disk.
What should be the reading of the voltmeter for
Bo = 0.05 web . m 2 , r = 30 cm. and
ω = 80π rad . sec .
(c)
Find the direction of current following through the voltmeter.
ω
A conductor 2 cm in length placed parallel to the Z − axis at radius 50 cm on a nonmagnetic circular frame and rotates at 1200 rpm as shown below.
a)
If the conductor is rotating in a radial magnetic field B = 2.5ar web m 2 , find the
induced voltage across the conductor and show its polarity.
b)
Repeat (a) if the conductor lies along the radial direction parallel to the plane of
rotating circle.
2
4.
5.
Two transceivers are located inside a river
communicating with each other. The
parameters of the river water are shown in
the figure below. If the magnitude of the
conduction current density is twice that of
displacement current density, find the
following:
a)
The transmission frequency in the river
b)
The attenuation of the transmitted signal
c)
The distance at which the amplitude of
the electric field becomes one-fifth of its
value at the transmitter.
d)
If the transmission frequency is reduced
by one-tenth of that in (a), then what
would be the distance at which the
amplitude of the electric field still be
one-fifth of its value at the transmitter?
µr = 1
ε r = 81
σ = 7.482 × 10 −3
A student would like to kill insects located in a
sandy earth at depth of 10 m . He transmitted
high intensity electric field given by:
E = E x ax = 375 cos(wt − β Z ) ax,
w = 2π × 10 8 rad . s
a)
Find the reflected and transmitted
electric field.
b)
It is required an average power of 40 W m 2
to kill the insects, would the insects be
killed? Show your work and confirm.
c)
Find the average energy density at the insects.
d)
What is the electric and magnetic field at the insects?
3
Z
ε r = 12
σ = 10 −3 mhos / m
6-A.
A lossless transmission line of characteristic impedance Z o is terminated by an antenna
of resistive impedance Z L as shown. Calculate, Γ, S and sketch the voltage and current
distribution along the line for the following cases.
6-B.
The antenna load Z L of part (c ) was not equal
to the characteristic impedance Z o of the
transmission line. Therefore the antenna was not
matched to the transmission line and reflections
occurred with weak signal reception at the
antenna. However, to avoid this, a student
suggested to insert a section of a new
transmission line of length l between the
antenna and the original transmission line as
shown below. What should be the length and the
characteristic impedance of the new line?
4
7.
A submarine equipped with a half-wave dipole antenna located just below the water
surface communicating with a ship on top of it at 1 KHz. As the submarine dives
downward, the received signal continues to suffer attenuation. However, the receiver in
the submarine is capable to receive signal as the attenuation increases up to 24 dB beyond
which communications will be interrupted.
a. Calculate the signal loss in dB m.
b. How far can the submarine dive down deeper, before communication with the ship
will be interrupted?
c. What is the length of the dipole on the submarine?
Sea parameters are: [ ε r = 81, σ = 4 mho m, µ r = 1 ]
8.
A vertically polarized wave was transmitted to an identical Horn antenna as shown
below. Find:
a. The minimum distance between horn antenna, in order to receive far field power?
b. The maximum effective aperture of the horn.
c. The maximum power received (in dB ): if both antennas have 100% efficiencies and
both vertically polarized.
d. The power received if the receiving antenna is rotated by 45° around its axis.
e. The power received if the receiving antenna is horizontally polarized (rotated
by 90° ).
RX
TX
5
9.
In the figure shown below, a student #1 using half-wave dipole antenna communicating
at 200 MHz with student #2 located across the street with a short dipole antenna of
length 0.02 λ with radiation intensity given by:
U (θ . ϕ ) = Ao sin 2 θ
=0
0 ≤ θ ≤ π and
----- elsewhere
0 ≤ φ ≤ 2π
If the radiated output power for each antenna is 5w,
Calculate the following:
a) The currents required through each antenna.
b) The high frequency loss resistance Rt for each antenna.
c) The radiation efficiency ξ for each antenna assuming that each dipole is perfectly
matched to the transmitters and each is made of copper rod, with diameter of
2 × 10 −3 λ
(σ = 5.7 × 10 7 mhos m).
d) The directive gain G d (θ , ϕ ) for antenna #2.
e) The maximum received power by each antenna (assuming both are vertically
polarized).
f) A third student (student #3), started to intercept the signal transmitted by the short
dipole #2, using identical antenna as shown below. How much power this student
will receive?
6
10.
A microwave receiving station of parabolic antenna is located at a distance of 20 km from
an identical transmitting antenna of output power 10W and antenna gain 45.8 dB. If both
antennas are vertically polarized with efficiency of 60%, calculate the following:
a) The minimum distance required between the antennas in order to receive far field
power.
b) The diameter of the transmitting antenna.
c) The maximum effective aperture of the receiving antenna.
d) The power Pr received by the receiving antenna.
e) Repeat part (d ) if the transmit antenna changes polarization to:
(i)
Horizontal polarization
(ii)
45° from vertical
12 GHz
20 Km.
Gr = 45.8 dB
Gt = 45.8 dB
Pr ?
11.
An engineer working in the Arabsat Satellite Complex in Riyadh, would like to design a
law loss waveguide to route the satellite signal received at C − band (4 GHz ) , to the
control room 100 meter away from the earth station feed-horn. He used copper conductor
(σ = 5.8 × 10 7 mhos m) with air dielectric so that only TE10 mode should propagate
down the guide with transmission frequency 20% above cutoff frequency as safety factor.
Calculate the following:
a) The dimension of the waveguide which will permit only TE10 to propagate but all
other modes including TE 01 will be blocked down at least by 20% below cut-off.
b) The attenuation due to copper losses in Nep. m
c) The total loss suffered by the received signal at the control room in dB.
d) The group velocity U g along the guide.
e) The cut-off wavelength λc of the guide.
f) The characteristic impedance of the waveguide.
7
3
5
100m
15m
100m
100 m
4
12.
2
1
A student with a short dipole antenna of length 0.02 λ at the center of a circle at
location #1 , as shown in the figure above, is communicating at 100 MHz with three
students using identical antennas in location 2, 3, & 4 each at 100 m away. Each dipole
antenna has a radiated output power of 4.19W with radiation intensity given by;
U (θ , φ ) = 0.5 sin 2 θ
=0
a)
b)
c)
d)
0≤θ ≤π
elsewhere
Find the directive gain Gd (θ ,φ ) for each dipole antenna.
Find the power received by each antenna.
A fifth student located in the fifth floor 15 m high, tried to intercept the signal as
shown in the figure. How much power he will receive if his antenna is:
(i)
the identical short dipole antenna
(ii)
half-wave dipole antenna
Calculate the required currents passing through.
(i)
the short dipole antenna
(ii)
the half-wave dipole antenna assuming same output radiated power of 4.19W
for communicating.
(Assume 100% efficiencies for all antennas)
8