Problem 6.26 The electric field radiated by a short dipole antenna is given in
spherical coordinates by
E(R, ;t) =
2 102
sin cos(6 108t 2 R) (V/m).
R
Find H(R, ;t).
Solution: Converting to phasor form, the electric field is given by
e ) = E = 2 10
E(R,
R
Problem 2.17 Using a slotted line, the voltage on a lossless transmission line was
found to have a maximum magnitude of 1.5 V and a minimum magnitude of 0.6 V.
Find the magnitude of the loads reflection coefficient.
Solution: From the definition of the St
Problem 2.29 Show that the input impedance of a quarter-wavelengthlong
lossless line terminated in a short circuit appears as an open circuit.
Solution:
Zin = Z0
For l = 4 , l =
2
ZL + jZ0 tan l
Z0 + jZL tan l
4 = 2 . With ZL = 0, we have
Zin = Z0
jZ0 ta
Problem 2.64 Use CD Module 2.7 to design a quarter-wavelength transformer to
match a load with ZL = (100 j200) to a 50- line.
Solution: Figure P2.64(a) displays the first solution of Module 2.7 where a /4
section of Z02 = 15.5015 is inserted at distance d
Problem 2.25 Apply CD Module 2.4 to generate plots of the voltage standingwave pattern for a 50- line terminated in a load impedance ZL = (100 j50) .
Set Vg = 1 V, Zg = 50 , r = 2.25, l = 40 cm, and f = 1 GHz. Also determine S,
dmax , and dmin .
Solution:
Problem 2.35 For the lossless transmission line circuit shown in Fig. P2.35,
determine the equivalent series lumped-element circuit at 400 MHz at the input to
the line. The line has a characteristic impedance of 50 and the insulating layer has
r = 2.25.
Z
Problem 2.37 A lossless transmission line is terminated in a short circuit. How
long (in wavelengths) should the line be for it to appear as an open circuit at its input
terminals?
sc = jZ tan l. If l = ( /2 + n ), then Z sc = j ().
Solution: From Eq. (2.
Problem 2.40 A 100-MHz FM broadcast station uses a 300- transmission line
between the transmitter and a tower-mounted half-wave dipole antenna. The antenna
impedance is 73 . You are asked to design a quarter-wave transformer to match the
antenna to the li
Problem 2.44 For the circuit shown in Fig. P2.44, calculate the average incident
power, the average reflected power, and the average power transmitted into the infinite
100- line. The /2 line is lossless and the infinitely long line is slightly lossy. (Hi
Problem 5.24 In a certain conducting region, the magnetic field is given in
cylindrical coordinates by
4
H = [1 (1 + 3r)e3r ]
r
Find the current density J.
Solution:
4
1
3r
H = z
r [1 (1 + 3r)e ]
J =
r r
r
1
= z [12e2r (1 + 2r) 12e2r ] = z 24e3r A/m2 .
Problem 2.15 Find and Z0 of a distortionless line whose R = 2 /m and
G = 2 104 S/m.
Solution: From the equations given in Problem 2.13,
= R G = [2 2 104 ]1/2 = 2 102 (Np/m),
r
r
1/2
L
R
2
Z0 =
=
=
= 100 .
C
G
2 104
Problem 2.60 Repeat Problem 2.59 using CD Module 2.6.
Solution: In Module 2.6, after setting Z0 = 75 , and d = 0.6 , cursor is used to
move the load point so that S = 1.8 and simultaneously r = 60 . Once that was
accomplished, the following values were ex
Problem 2.65 Use CD Module 2.7 to design a quarter-wavelength transformer to
match a load with ZL = (50 + j10) to a 100- line.
Solution: Figure P2.65 provides a summary of the two possible solutions, obtained
by going through multiple steps to insert a /4
Problem 1.10 An oscillator that generates a sinusoidal wave on a string completes
20 vibrations in 50 s. The wave peak is observed to travel a distance of 2.8 m along
the string in 5 s. What is the wavelength?
Solution:
T=
50
2.8
= 2.5 s,
up =
= 0.56 m/s,
Problem 2.24 A 50- lossless line terminated in a purely resistive load has a
voltage standing-wave ratio of 3. Find all possible values of ZL .
Solution:
S1 31
=
= 0.5.
S+1 3+1
For a purely resistive load, r = 0 or . For r = 0,
1+
1 + 0.5
ZL = Z0
= 50
= 1
Problem 1.24 If z = 3e j /6 , find the value of ez .
Solution:
z = 3e j /6 = 3 cos /6 + j3 sin /6
= 2.6 + j1.5
ez = e2.6+ j1.5 = e2.6 e j1.5
= e2.6 (cos 1.5 + j sin 1.5)
= 13.46(0.07 + j0.98)
= 0.95 + j13.43.
Problem 1.23 If z = 3 j4, find the value of ez .
Solution:
ez = e3 j4 = e3 e j4 = e3 (cos 4 j sin 4),
4
e3 = 20.09,
and 4 rad = 180 = 229.18 .
Hence, ez = 20.08(cos 229.18 j sin 229.18 ) = 13.13 + j15.20.
Problem 2.10 Use CD Module 2.3 to design a 100- microstrip transmission line.
The substrate thickness is 1.8 mm and its r = 2.3. Select the strip width w, and
determine the guide wavelength at f = 5 GHz. Include a printout of the screen
display.
Solution:
Problem 2.11 A 50- microstrip line uses a 0.6-mm alumina substrate with r = 9.
Use CD Module 2.3 to determine the required strip width w. Include a printout of the
screen display.
Solution: According to the solution provided by CD Module 2.3, the required
Problem 2.46 An antenna with a load impedance
ZL = (75 + j25)
is connected to a transmitter through a 50- lossless transmission line. If under
matched conditions (50- load) the transmitter can deliver 20 W to the load, how
much power can it deliver to th
Problem 2.47 Use the Smith chart to find the reflection coefficient corresponding
to a load impedance of
(a) ZL = 3Z0
(b) ZL = (2 j2)Z0
(c) ZL = j2Z0
(d) ZL = 0 (short circuit)
Solution: Refer to Fig. P2.47.
(a) Point A is zL = 3 + j0. = 0.5e0
(b) Point B
Problem 5.30 In the model of the hydrogen atom proposed by Bohr in 1913, the
electron moves around the nucleus at a speed of 2 106 m/s in a circular orbit of
radius 5 1011 m. What is the magnitude of the magnetic moment generated by the
electrons motion?
Problem 7.5 A wave radiated by a source in air is incident upon a soil surface,
whereupon a part of the wave is transmitted into the soil medium. If the wavelength
of the wave is 60 cm in air and 20 cm in the soil medium, what is the soils relative
permit
Problem 7.9 For a wave characterized by the electric field
E(z,t) = x ax cos( t kz) + y ay cos( t kz + )
identify the polarization state, determine the polarization angles ( , ), and sketch
the locus of E(0,t) for each of the following cases:
(a) ax = 3 V
Problem 7.20 The skin depth of a certain nonmagnetic conducting material is 3 m
at 2 GHz. Determine the phase velocity in the material.
Solution: For a good conductor, = , and for any material s = 1/ . Hence,
up =
2 f
=
= 2 f s = 2 5 109 3 106 = 9.42 104
Problem 7.19 Ignoring reflection at the airsoil boundary, if the amplitude of a
3-GHz incident wave is 10 V/m at the surface of a wet soil medium, at what depth will
it be down to 1 mV/m? Wet soil is characterized by r = 1, r = 9, and = 5 104
S/m.
Solutio
Problem 7.14 Plot the locus of E(0,t) for a plane wave with
E(z,t) = x sin( t + kz) + y 2 cos( t + kz)
Determine the polarization state from your plot.
Solution:
y
t=/2
2
1
z
1
x
t=0
2
Figure P7.14: Locus of E versus time.
E = x sin( t + kz) + y 2 cos( t
Problem 7.34
Repeat Problem 7.33 for a wave traveling in a lossy medium in which
E = x 100e20y cos(2 109t 40y) (V/m)
H = z 0.64e20y cos(2 109t 40y 36.85 )
(A/m)
The box has dimensions a = 1 cm, b = 2 cm, and c = 0.5 cm.
Solution:
(a)
H
S(t) = E
= x 100e20