Problems on Plane Wave Incidence,
Waveguides and Transmission Lines
E LECTROMAGNETIC E NGINEERING
MAP – Tele
Maria Inês Barbosa de Carvalho
November 2008

Faculdade de Engenharia
Electromagnetic Engineering
MAP-Tele – 2008/2009
— Plane wave incidence —
1 The electric field intensity of a left-hand circularly polarized electromagnetic wave of frequency
200 MHz is 10 Vm − 1 . This wave propagates in air and impinges normally on a dielectric medium with ε r
= 4 that is located in the region z ≥ 0. The x component of the incident electric field phasor has a maximum at z = 0 when t = 0.
E i dielectric
0 z
(a) Determine the electric field phasor of the incident wave.
(b) Find the reflection and transmission coefficients.
(c) Obtain the electric field phasors of the reflected and transmitted waves, and of the total field in the region z ≤ 0.
(d) Find the fraction of the incident average power that is reflected by the interface and the one that is transmitted to the dielectric medium.
2 Repeat the previous problem, assuming now that the region z ≥ 0 is filled wiht a poor conductor with ε r
= 2 .
25, µ r
= 1 and σ = 10 − 4 Sm − 1 .
3 At 200 MHz a given medium is characterized by σ = 0, µ r
= 15(1 − j 3) and ε r
= 50(1 − j 1). Find:
(a) η/η
0
, λ/λ
0 and v/v
0
, where η is the intrinsic impedance and v is the phase velocity (the subscript 0 corresponds to values in air);
(b) the skin depth δ ;
(c) the attenuation in dB for a distance of 5 mm;
(d) the reflection coefficient for a wave propagating in air that impinges normally in this medium.
4 A nonconducting material with µ r
= ε r
= 6 − j 6 is located under a perfectly conducting plate.
Assuming that a electromagnetic wave propagating in air with a frequency of 500 MHz impinges normally in this material, find the thickness of the material that reduces the amplitude of the reflected wave by 35 dB?
conductor material
E i
1

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5 The three regions shown in the figure are perfect dielectrics. For a wave propagating in medium 1 and impinging normally in the interface located at z = − d , find ε r
2 reflected wave. Write the answer as a function of ε r
1
, ε r
3 and d and frequency f .
such that there are no z = − d z = 0
Medium 1 ε r 2
Medium 3
z
Medium 2
ε r 1
ε r 3
d
6 Consider a plane electromagnetic wave that is normally incident on a glass lens of a camera with
ε r
= 2 .
5, as illustrated in figure A. Assume that the lens has an infinite thickness in order to neglect the reflections at the glass-air end interface.
air glass air dielectric glass incident wave incident wave
Figure A Figure B
(a) Find the fraction of the incident power that is reflected by the lens.
(b) In order to eliminated the reflections of the visible light corresponding to yellow ( λ
0
= 560 nm, where λ
0 is the vacuum wavelength), a dielectric coating with a thickness of λ/ 4 is applied to the lens (figure B). Find the dielectric constant of this material. What is the thickness of the dielectric layer?
(c) Under the previous conditions, what fraction of the incident blue light ( λ
0
= 475 nm) is reflected by the lens?
7 A plane electromagnetic wave propagating in air along + z ( z < 0) impinges normally at z = 0 on a conductor with σ = 61 .
7 MS/m, µ r
= 1. The incident wave has a frequency of f = 1 .
5 MHz and the electric field at the interface z = 0 is given by,
~
(0 , t ) = 10 sin(2 πf t y (V/m).
(a) Determine the magnetic field in the conductor.
(b) Show that the reflection coefficient is approximately given by Γ ≈ ( η r
− 1)(1 − η r
) ≈ − 1+2 η r
= p
2 ωε
0
/σ (1 + j ) − 1, where η r
= p jωε
0
/σ .
(c) Find the fraction of the incident power that is lost for the conductor after the reflection.
(d) Find the skin depth in the conducting material.
8 A uniform plane wave, linearly polarized along x , with amplitude 10 V/m and frequency 1.5 GHz, is propagating in air and impinges normally on a perfectly conducting surface located at z = 0.
(a) Obtain the phasor and instantaneous expressions for the electric and magnetic fields in air.
2

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(b) Find the location closest to the conducting surface where the magnetic field is always zero.
(c) Assuming now that the conducting surface has σ = 10 5 S/m (graphite), ε = ε
0 find the attenuation constant in dB/m.
and µ = µ
0
,
(d) Under the previous conditions, obtain the fraction of the incident power that is absorbed by the conducting plate.
(e) If the incident wave has left-hand circular polarization, what type of polarization will the reflected wave have? Justify your answer.
9 A plane wave is propagating in air ( z < 0) and impinges normally on a good conductor ( z > 0) that is nonmagnetic and has intrinsic impedance 6 π 2 e j 45 o
. The incident electric field is given by
~
= (ˆ − j ˆ )10 e − j 20 πz
(V / m) .
(a) Obtain the electric field phasor of the reflected wave.
(b) What is the state of polarization of the incident and reflected waves?
(c) Find the propagation constant in the region z > 0.
(d) Determine the electric field phasor in medium 2.
(e) Obtain the magnetic field phasors of the incident, reflected and transmitted waves.
(f ) Obtain the time-average Poynting vector in the regions z < 0 and z > 0 and show that power is conserved during incidence.
10 Assume that the region z > 0 is filled with air, while the region z < 0 is filled with a dielectric with refractive index 2. In the dielectric is propagating a wave characterized by
~
= ˆ E
0 e
− jπ (4 y +3 z )
.
(a) Determine the direction of propagation of the incident wave and the corresponding angle of incidence.
(b) Find the magnetic field phasor of the incident wave.
(c) Obtain the reflection coefficient and the electric field phasor of the reflected wave.
(d) Write the expression for the electric field phasor in air.
11 A left-hand circularly polarized wave impinges at an interface between two different media with and angle of 45 ◦ .
(a) If the two media are air (medium 1) and a perfect conductor (medium 2), determine the state of polarization of the reflected wave.
(b) For the interface air-polystyrene ( ε r
= 2 .
5), determine the state of polarization of the reflected and transmitted waves.
12 A wave propagating in air with perpendicular polarization impinges obliquely at an interface airglass with an angle of incidence of 30 ◦ . The wave frequency is 600 THz (1 THz = 10 12 Hz), corresponding to green light, and the index of refraction of glass is 1 .
6. Assuming that the electric field amplitude of the incident wave is 50 Vm − 1 , determine:
(a) the reflection and transmission coefficients;
(b) the expressions for
~ and
~ in the glass.
3

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Electromagnetic Engineering
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13 In the region defined by y < 0, filled with a nonmagnetic material (medium 1), is propagating a plane wave of frequency 1 .
5 GHz that is characterized by the phasor
E i
( x, y ) = ˆ E
0 e − j 4 π (4 x +3 y )
(V/m) .
This wave impinges obliquely on the interface with the region y > 0, that is filled with air.
(a) What is the state of polarization of this wave relatively to the plane of incidence?
(b) Find the relative permittivity of medium 1 and the angle of incidence.
(c) Obtain the magnetic field phasor of this wave.
(d) Find the reflection and transmission coefficients, and obtain the electric field phasors of the reflected wave and of the wave in the air.
(e) Explain how it is possible to obtain a circularly polarized wave from the incidence of a linearly polarized wave on an interface with a medium with lower refractive index. Justify your answer with the necessary calculations.
14 A plane wave propagating in air is characterized by the following phasor
~ i
( x, z ) = 0 .
2 e
− j (3 x +4 z )
(0 .
z − 0 .
8ˆ ) (A/m) .
This wave impinges on a perfectly conducting surface located at z=0. Determine:
(a) the frequency and the angle of incidence;
(b) the phasor E i
;
(c) the electric and magnetic field phasors of the reflected wave;
(d) the location of the electric field amplitude maxima in the air;
(e) the time-average Poynting vector in the air. What is direction of transport of energy?
15 A plane wave propagating in air (medium 1) with perpendicular polarization is incident with an angle of 30 ◦ on a medium (medium 2) with relative permittivity ε r
= 5.
(a) Obtain the reflection coefficient.
(b) If the wave propagates in medium 2 and impinges on medium 1, find the critical angle.
16 In the region defined by x < 0, filled with a nonmagnetic material (medium 1), is propagating a plane wave of frequency 6 .
8 GHz that is characterized by
~ i
( x, y ) = 10 e − j 4 π (8 x +15 y )
ˆ (V/m) .
This wave impinges obliquely in the region x > 0, filled with a nonmagnetic material with refractive index 2.
(a) Determine the refractive index of medium 1.
(b) Obtain the angles of incidence and transmission.
(c) Obtain the electric field phasors of the reflected and transmitted waves.
(d) What fraction of the incident power is transmitted to medium 2?
4

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17 A plane wave propagating in a nonmagnetic medium with dielectric constant ε r terized by
= 2 .
25 is charac-
E i
( x, y ) = 10 e − j 4 π (3 x +4 y ) ( − 0 .
8ˆ + 0 .
y ) (V/m) .
This wave is obliquely incident on the interface with the region x > 0, filled with a nonmagnetic material with refractive index 2.
(a) What is the frequency of this wave?
(b) What are the direction of propagation of the incident, reflected and transmitted waves?
(c) Obtain the electric field phasors of the reflected and transmitted waves.
18 Repeat the previous problem, assuming now that the electric field phasor of the incident wave is given by
E i
( x, y ) = 10 e − j 4 π (3 x +4 y )
( − 0 .
8ˆ + 0 .
6ˆ + ˆ ) (V/m) .
5

Faculdade de Engenharia
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— Waveguides and Cavities —
1 Show that the TM
1 mode in a parallel-plate waveguide can be seen as a superposition of two plane waves propagating obliquely.
2 Obtain the expressions of the surface charge and surface current densities on the plates of a parallelplate waveguide for the TM n mode. Indicate the direction of the current on the plates.
3 Obtain the expression of the surface current density on the plates of a parallel-plate waveguide for the TE n mode. Indicate the direction of the current on the two plates.
4 Consider a parallel-plate waveguide with plates separated by a distance b .
(a) Write the instantaneous expressions of the electric and magnetic fields for the TM
1 and TE
1 modes.
(b) For the previous modes and t = 0, sketch the electric and magnetic field lines in the yz plane.
5 A parallel-plate waveguide is filled with air and its plates are separated by 1 cm.
(a) Determine the cutoff frequency of the TE
1
, TM
1
, TE
2 and TM
2 modes.
(b) Knowing that the guide operates at a frequency of 40 GHz, indicate which modes can propagate.
6 A waveguide with parallel-plates separated by a distance b is filled with a dielectric with parameters
( ε, µ ).
(a) Sketch the ω − β diagrams for the first three TM and TE modes, and for the TEM mode.
(b) Indicate how it would be possible to obtain the phase and group velocities at a given frequency from a ω − β diagram.
7 A waveguide with parallel-plates separated by 5 cm is filled with air.
(a) Determine the cutoff frequencies for the first five TE and TM modes.
(b) Knowing that the guide operates at 20 GHz, determine the phase velocity, the wavelength, the phase constant and the impedances for the previous modes.
(c) Compare the obtained values with the ones relative to a TEM mode with 20 GHz.
(d) Now assume that the guide operates at 2 GHz. Determine the attenuation constants for the previous modes.
8 A waveguide with parallel-plates separated by 2 cm is filled with a dielectric with relative permittivity 2 .
25. The magnetic field inside the guide is characterized by
H z
= 50 cos (100 πy ) cos 3 π × 10
10 t − βz (A / m) .
Determine:
(a) the mode that is propagating;
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(b) the phase constant ( β );
(c) the wave impedance;
(d)
~ and
~
.
Electromagnetic Engineering
MAP-Tele – 2008/2009
9 A waveguide with parallel-plates separated by 1 cm is filled with a dielectric with relative permittivity 4 and operates at 12 GHz. The electric field inside the guide is characterized by
E z
= 40 sin (100 πy ) e − jβz
(V / m) .
Determine:
(a) the mode that is propagating and its cutoff frequency;
(b) the phase constant ( β );
(c) the phase velocity;
(d) ~ and ~ ;
(e)
~ av
.
10 A parallel-plate waveguide is filled with two different dielectrics, with permittivities ε
1 and ε
2
.
y
2 d x
1 d
(a) Starting from the wave equations in each medium, show that for the TE modes the longitudinal component of the magnetic field satisfies
H
0 z


A sin( h
1 y ) + B cos( h
1 y ) medium 1
=

C sin( h
2 y ) + B cos( h
2 y ) medium 2
(b) where h 2
1
= γ 2 + ω 2 µ
0
ε
1 and h 2
2
= γ 2 + ω 2 µ
0
ε
2
, and A, B e C are constants.
Determine 0 and 0 .
(c) Using the previous result and appropriate boundary conditions, show that the following condition is satisfied h
1 tan( h
2 d ) + h
2 tan( h
1 d ) = 0 .
(d) Obtain the equation for the cutoff frequencies of the different TE modes.
11 Consider a rectangular waveguide with dimensions a = 2 .
5 cm and b = 1 .
0 cm that is filled with a lossless dielectric with ε r
= 4. Determine the phase constant, the phase velocity and the wave impedance for the TE
10 and TM
11 modes when the guide is operating at 15 GHz.
12 A rectangular waveguide filled with air has dimensions a = 1 cm and b = 0 .
6 cm.
(a) Determine the cutoff frequencies of the TE
10 and TE
20 modes
(b) For a 18 GHz operating frequency, determine the guided wavelength for each propagating mode and compare it with the wavelength in free space.
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MAP-Tele – 2008/2009
13 A rectangular waveguide with dimensions 5 cm × 2 cm is filled with air and operates at 15 GHz. The electric field inside the guide is characterized by:
E z
= 20 sin(40 πx ) sin(50 πy ) e − jβz
V / m .
(a) What mode is propagating inside the guide?
(b) Find β .
(c) Determine the ratio E y
/E x
.
14 A rectangular waveguide with dimensions a = 2 .
5 cm and b = 1 cm operates at a frequency below
15 .
1 GHz.
(a) Compute the cutoff frequency of the different TE and TM modes in propagation.
(b) How many TE and TM modes can propagate if the guide is filled with a dielectric with ε r
= 4 and µ r
= 1?
15 A rectangular waveguide with dimensions a = 1 .
5 cm, b = 0 .
8 cm is filled with a dielectric with
µ = µ
0 and ε = 4 ε
0
. The magnetic field inside the guide is characterized by
H x
= 2 sin
πx a cos
3 πy b sin( π × 10
11 t − βz ) (A/m)
Determine:
(a) The mode that is propagating inside the guide.
(b) The cutoff frequency.
(c) The propagation constant.
(d) The wave impedance.
16 A rectangular waveguide filled with air has dimensions a = 8 .
636 cm and b = 4 .
318 cm, and operates at 4 GHz.
(a) Find if the TE
10 mode can propagate and, if so, calculate the phase and group velocities.
(b) Repeat for the TM
11 mode.
17 A rectangular waveguide is filled with air and has dimensions a = 4 cm and b = 2 cm. This guide transports an average power of 2 mW in the dominant mode. If the operating frequency is 10 GHz, determine the maximum amplitude of the electric field inside the guide.
18 A rectangular waveguide is filled with two different dielectrics, as illustrated in the figure.
y
1
2 b x d a
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(a) Show that the following condition is satisfied for the TE m 0 modes with angular frequency ω : h
1 tan( h
2 d ) + h
2 tan( h
1
( a − d )) = 0 , where h 2
1
= γ 2 + ω 2 µε
1 and h 2
2
= γ 2 + ω 2 µε
2
, and γ is the propagation constant.
(b) Obtain the cutoff frequency for the TE
10 mode by solving the previous equation when γ = 0.
Assume that a = 2 .
286cm, d = a/ 2, ε
1
= ε
0 and ε r 2
= 2 .
25.
19 A rectangular waveguide ( a = 5 cm; b = 2 cm), filled with air, operates at 1 GHz. For the TM
21 mode and assuming that the maximum amplitude of E z at z = 0 is 1 kV/m:
(a) Show that there is no propagation at this frequency (evanescent mode);
(b) Determine the distance z such that the amplitude of the longitudinal component of the electric field is 0 .
5% of its value at z=0;
(c) Now assume that the guide is operating at 20 GHz. Determine: i. The phase constant and the wave impedance; ii. The average power transmitted by the guide.
20 A rectangular waveguide with dimensions a = 4 cm and b = 2 cm is filled with air and operates at
18 GHz. The electric inside this guide is characterized by
E z
= 20 sin(50 πx ) sin(100 πy ) e − jβz
.
(a) What mode is propagating inside the guide? Find its cutoff frequency.
(b) Determine β and v g
.
(c) Find the average power propagated inside the guide.
21 A rectangular waveguide will operate in a frequency band of 1 .
0 GHz with central frequency 4 .
0 GHz.
A safety margin of 20% should be considered between the band boundaries and the first and second mode cutoff frequencies, as shown below.
1 GHZ
(1 st
mode) f l
=
1 .
2 f c 1 f
0
=
4 .
0 f u
= f c 2
1 .
2 f c 2
(2 nd
mode)
f (GHz)
(a) Assuming that the guide is filled with air, find the guide dimensions, a and b .
(b) Now assume that this guide (with the previously calculated dimensions a and b ) is filled with a dielectric with r
= 3 .
2. Which modes can propagate in the specified frequency band?
22 A rectangular cavity is filled with air and has dimensions a = 5 cm, b = 4 cm and d = 10 cm. Find the five lowest modes of the cavity.
23 A rectangular cavity has dimensions a = 4 cm, b = 3 cm and d = 5 cm. Find the dominant mode and its resonant frequency when
(a) the cavity is filled with air;
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(b) the cavity is filled with a dielectric with relative permittivity 2 .
5.
24 A rectangular cavity is filled with air and its dimensions satisfy a = 2 b = 4 d .
z y d b = 2d a = 2b x
(a) Find the dimensions of the cavity such that the resonant frequency of the dominant mode is
6 GHz.
(b) Show that in the dominant mode the electric field of a rectangular cavity is parallel to the shortest dimension of the cavity. Assume that the shortest dimension is oriented along z .
25 Starting from the equations ∇ ×
~
= − H and ∇ × obtain the expressions of the transverse components (along
~
= E in cylindrical coordinates, r and φ ) of
~ and
~ as functions of the longitudinal components (along z ).
26 A solution of the following Bessel differential equation d 2 R ( r ) dr 2
+
1 r dR ( r )
+ R ( r ) = 0 dr can be obtained by considering that R ( r ) is of the form R ( r ) = P
∞ m =0 a m r m . By substituting this expression in the differential equation, determine the general expression for the coefficients a m
.
27 Show that the Bessel functions of the first kind satisfy the following conditions.
(a) J 0 n
( x ) =
1
2
[ J n − 1
( x ) − J n +1
( x )].
(b) J
− n
( x ) = ( − 1) n J n
( x ).
Note:
1
( − n )!
= 0 , ∀ n ∈ N .
28 A circular waveguide is filled with air and has a diameter of 90 mm. Determine:
(a) The TE and TM modes that can propagate at an operating frequency of 5 GHz.
(b) The relative velocity ( v/c ) of each mode at a frequency 1 .
1 times their cutoff frequency.
29 A 10GHz signal is transmitted in a circular waveguide filled with air.
(a) Knowing that the lowest cutoff frequency of the guide is 20% lower than the frequency of the transmitted signal, find the diameter of the guide.
(b) If the guide now operates at 15GHz, which modes are allowed to propagate.
30 Consider a circular waveguide filled with air, with a 1 cm radius and operating at 19 GHz.
(a) Which modes are allowed to propagate?
(b) Obtain the expression for the average Poynting vector of the TM
01 mode.
(c) Determine the maximum frequency range of a signal that only propagates in the dominant mode.
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31 A circular waveguide with a 2 cm radius is filled with a dielectric of parameters (4 ε
0
, µ
0
).
(a) Assuming that the guide is operating at 4 GHz, determine which modes are allowed to propagate.
(b) Under the previous conditions, obtain the average Poynting vector of the TE
11 mode.
(c) Determine the frequency range [ f min
, f max
] in which the guide should operate if only the dominant mode is allowed to propagate and the attenuation of all other modes is larger than
200 dB/m.
32 A circular cavity has a length equal to its diameter. Knowing that the cavity oscillates at 10 GHz in the TM
010 mode, determine the cavity length.
33 A circular cavity is filled with air and has a length equal to its radius Determine the seven lowest modes of the cavity.
34 A circular cavity is filled with air and has a length equal to its radius.
(a) Knowing that the TE
211 mode oscillates at 6 GHz determine the cavity dimensions.
(b) Find the lowest resonant frequency of the cavity and the associated mode.
35 Show that the propagation velocity of a electromagnetic wave inside a dielectric waveguide lies between the propagation velocities of a plane wave in the dielectric medium and in the exterior medium of the guide.
36 A dielectric slab with parameters µ = µ
0 and ε = 2 .
5 ε
0 is located in the air. Find the minimum thickness of the slab that allows the propagation of an even TM or TE mode at an operating frequency of 20 GHz.
37 Consider a dielectric slab of thickness b and parameters ( µ, ε ) located in the air.
(a) Determine the average power density propagated in the guide by the TM dominant mode.
(b) Determine the average power transmitted in the transverse direction.
38 Consider a dielectric slab of thickness 2 cm and dielectric constant ε r
= 2 located in the air. An even TM mode propagates in this guide at a frequency of 12 GHz. The longitudinal component of the electric field inside the dielectric is given by E z
= 10 cos( h d y ) e − jβz (V / m).
y
z air dielectric b
(a) Find the other components of the electric field in the dielectric.
(b) Knowing that h d
= 229 the fields in the air.
.
7 m − 1 , calculate the phase constante β and the decay coefficient ν of
(c) Under the previous conditions, obtain the longitudinal component of the electric field in the air.
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39 Consider a planar dielectric waveguide consisting of a dielectric slab of thickness b and parameters
( µ, ε ) and a perfectly conducting plate.
y air b dielectric conductor x
Knowing that the the guide is located in the air, find for the TE modes
(a) the expressions of the electric and magnetic field phasors;
(b) the characteristic equation;
(c) the cutoff frequency of the different modes.
40 Determine, for the waveguide of the previous problem, the expressions for the surface charge and current densities on the conducting plate.
41 Consider a planar dielectric waveguide consisting of a dielectric slab with refractive index n
1 thickness b and a conducting plate.
and y n
2 n
1 b x conductor
Knowing that the guide is located in an infinite medium with refractive index n
2 determine, for the TM modes,
( n
2
< n
1
),
(a) the characteristic equation;
(b) the cutoff frequency of the different modes.
42 A cylindrical rod made of a dielectric transparent material can be used to guide light as a result of the total internal reflection. Find the material minimum dielectric constant that allows a wave, incident at one end with an angle θ i
, to be guided in the rod until it emerges at the other end.
43 Consider an optical fiber with a core and cladding refractive indices equal to 1 .
48 and 1 .
46, respectively. Determine:
(a) The numerical aperture of the fiber.
(b) The maximum radius that allows propagation in the single-mode regime at a wavelength of
1 .
3 µ m.
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44 The refractive indices of the core and cladding of an optical fiber are n
1 respectively.
= 1 .
5 and n
2
= 1 .
45,
(a) Determine the maximum radius that allows propagation in the single-mode regime at a frequency of 2 × 10 14 Hz.
(b) Explain why single-mode fibers are advantageous for long distance transmission of information.
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— Transmission Lines —
1 A transmission line with length l connects a load to a sinusoidal voltage source operating at a frequency f . Assuming that the propagation velocity of the wave in the line is c , in which of the following cases it is reasonable to ignore the presence of the transmission line in the circuit analysis?
(a) l = 20 cm, f = 10 KHz
(b) l = 50 km, f = 60 Hz
(c) l = 20 cm, f = 300 MHz
(d) l = 1 mm, f = 100 GHz
2 Obtain the parameters R, L, C e G of a lossless transmission line with characteristic impedance
50 Ω and phase velocity 10
8 m/s.
3 A parallel-plate transmission line operating at 1 GHz consists of two copper plates of width 1 .
5 cm separated by a dielectric of thickness 0 .
2 cm. For the copper, µ c
= µ
0
= 4 π × 10 − 7 H/m and
σ c
= 5 .
8 × 10 7 S/m, and for the dielectric ε r
= 2 .
6, µ = µ
0 and σ = 0. Obtain the line parameters
( R , L , G e C ).
4 In a coaxial transmission line, the inner conductor diameter is 0 .
5 cm and the exterior conductor diameter is 1 cm. The space between the two conductors is filled with a dielectric with µ = µ
0
,
ε r
= 2 .
25 e σ = 10 − 3
S/m. The conductors are made of copper with µ = µ
0 and σ c
= 5 .
8 × 10
7
S/m and the line is operating at 1 GHz.
(a) Obtain the line parameters R , L , G e C .
(b) Find α , β , v f and Z
0 for this coaxial line.
5 Consider a distortionless transmission line ( R/L = G/C ) with Z
0 and v f
= 2 .
5 × 10 8 m/s.
= 50 Ω, α
Obtain the line parameters and the wavelength λ at a frequency of 250 MHz.
= 40 × 10 − 3 Np/m
6 A transmission line operating at 125 MHz has Z
0
= 40 Ω, α = 0 .
02 Np/m and β = 0 .
75 rad/m.
(a) Obtain the line parameters R , L , G e C .
(b) At what distance is the line voltage attenuated by 30 dB?
7 A phone line has R = 30 Ω / Km, L = 0 .
1 H/km, G = 0 and C = 20 µ F/km. Assuming f = 1 kHz, obtain:
(a) The line characteristic impedance.
(b) The propagation constant.
(c) The phase velocity.
(d) The attenuation in dB after 2 km.
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8 A lossless transmission line operating at 4 .
5 GHz has L = 2 .
4 µ H/m and Z
0 phase constant, β , and the phase velocity.
= 85 Ω. Calculate the
9 Operating at a frequency of 300 MHz, a lossless line with 50 Ω and length 2 .
5 m is connected to a load Z
L
= (60 + j 20) Ω. Assuming v = c , find the input impedance.
10 A lossless transmission line is short-circuited. What length (in wavelengths) should the line have in order to appear as an open circuit at its input terminals.
11 Show that the input impedance of a very short slightly lossy line ( αl 1 and βl 1) is approximately
(a) Z in
= ( R + jωl ) l , with a short-circuit termination.
(b) Z in
= ( G − jωC ) / (( G 2 + ( ωC ) 2 ) l , with an open-circuit termination.
12 A lossless transmission line terminates in a load Z
L
= (30 − j 60) Ω. The wavelength is 5 cm and the line characteristic impedance is 50 Ω. Find:
(a) The relflection coefficient at the load.
(b) The SWR of the line.
(c) The location of the voltage maximum nearest to the load.
(d) The position of the current maximum nearest to the load.
13 In a lossless transmission line, the first voltage minimum is located at 4 cm from the load; the second minimum is located at 14 cm from the load; the voltage SWR is 2 .
5. If the line is lossless and Z
0
= 50 Ω, determine the load impedance.
14 A lossless transmission line with length l = 0 .
35 λ terminates in a load of impedance Z
L
= (60 + j 30) Ω. Obtain Γ
L
, SWR and Z i when Z
0
= 100 Ω.
Z i
Z
0
Z
L
0,35 λ
15 A load of 500 Ω is fed by a line of length 2 km operating at 20 kHz. The amplitude of the voltage at the load is 95% of the amplitude at the input of the line and the phase difference between these voltages is 2 rad. Knowing that this line is matched, find
(a) The propagation constant.
(b) The line parameters.
16 In a phone line of length 5 km operating at 1 kHz it is known that the input impedance is 535 e j 64 ◦ when the line is open-circuited and is 467 .
5 e j 10 ◦
Ω when the line is short-circuited.
Ω
(a) Determine the characteristic impedance and the propagation constant.
15

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Electromagnetic Engineering
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(b) Find the line parameters R , L , G e C .
(c) Assume that the line is connected to a load Z
L
= 400 Ω. Knowing that the current at the load is 0 .
5 A, obtain the voltage and current at the input of the line. Also find the amplitude of the incident voltage wave at the load.
17 A sinusoidal voltage source, V g
( t ) = 5 cos(2 π × 10 9 t ) (V) and internal impedance Z g
= 50 Ω, is connected to a lossless line with 50 Ω. The line, 5 cm long, terminates at a load Z
L
= (100 − j 100) Ω.
If v = c , determine:
(a) The reflection coefficient at the load.
(b) The impedance at the input of the line, Z i
.
(c) The voltage and current at the input of the line, V i and I i
.
18 A transmission line operating at ω = 10 6 rad/s has α = 8 dB/m, β = 1 rad/m and Z
0
= (60+ j 40) Ω, and is 2 meters long. If the line is connected to a generator with V g
= 10 e j 0 ◦ V and Z g
= 40 Ω, and terminates in a load (20 + j 50) Ω, obtain:
(a) The input impedance.
(b) The input current.
(c) The current in the middle point of the line.
19 The voltage in a distortionless transmission line is given by
V ( x, t ) = 60 e 0 .
0025 x cos(10 8 t + 2 x ) + 12 e − 0 .
0025 x cos(10 8 t − 2 x ) where x is the distance measured from the load. If Z
L
= 300 Ω determine:
(a) The attenuation constant ( α ), the phase constant ( β ) and the phase velocity.
(b) The characteristic impedance, Z
0
, and the current I ( x, t ).
20 Consider a length l of a transmission line with characteristic impedance Z
0 and propagation constant
γ .
I
1
I
2
+ +
V
1
Z
0
,
γ
V
2
– –
(a) Determine the parameters A, B, C, D of the relation
"
V
1
I
1
#
=
"
A B
C D
# "
V
2
I
2
# between the voltages and currents in the two terminals of the line. Also show that the following relations are satisfied
A = D
AD − BC = 1
16

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(b) Show that the previous section of the line can be represented by the equivalent circuit
I
1
I
2
+ Z
1
/2 Z
1
/2 +
Y
2
V
1
–
V
2
– when
Z
1
= 2 Z
0 tanh
γl
2
Y
2
= sinh γl
Z
0
21 Use the Smith chart to obtain the reflection coefficient corresponding to each of the following load impedances:
(a) Z
L
= 3 Z
0
(b) Z
L
= (2 − j 2) Z
0
(c) Z
L
= − j 2 Z
0
(d) Z
L
= 0.
22 Use the Smith chart to obtain the normalized load impedances corresponding to the following reflection coefficients:
(a) Γ
L
= 0 .
5
(b) Γ
L
= 0 .
5 ∠ 60 ◦
(c) Γ
L
= − 1
(d) Γ
L
= 0 .
3 ∠ − 30 ◦
(e) Γ
L
= 0
(f ) Γ
L
= j
23 A transmission line of length 30 m with Z
0
Z
L
= 60 + j 40 Ω. If v = 0 .
6 c , determine:
= 50 Ω operates at 2 MHz and terminates at a load
(a) The reflection coefficient.
(b) O SWR.
(c) The input impedance.
Obtain the results analytically and also by using the Smith chart.
24 In a lossless transmission line terminated at a load Z
L
= 100 Ω, the measured SWR is 2 .
5. Use the
Smith chart to determine the two possible values of Z
0
.
25 A lossless transmission line of length 1 .
3 λ and characteristic impedance 100 Ω is terminated by a load (100 + j 50) Ω. Using Smith chart, determine:
(a) The reflection coefficient at the load and the SWR
17

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(b) The line input impedance;
(c) The location of the voltage minimum nearest to the load.
26 The SWR of a lossless transmission line with 50 Ω terminating at a load with unknown impedance is 4 .
0. The distance between consecutive voltage minima is 40 cm and the first minimum is located at 5 cm from the load. Using the Smith chart, determine:
(a) The reflection coefficient.
(b) The load impedance, Z
L
.
(c) The length and load impedance of a transmission line such that its input impedance is equal to Z
L
.
27 A lossless transmission line with 50 Ω terminates at a load Z
L
= (50 + j 25) Ω. Use the Smith chart to calculate:
(a) The reflection coefficient.
(b) The SWR.
(c) The impedance at a distance of 0 .
35 λ from the load.
(d) The admittance at a distance 0 .
35 λ from the load.
(e) The shortest length of the line in order to have a purely resistive input impedance.
(f ) The location of the first voltage maximum.
28 A lossless transmission line terminates in a short-circuit. Use the Smith chart to determine:
(a) The impedance at a distance of 2 .
3 λ from the load.
(b) The location of the point nearest to the load where the admittance is Y i
= − j 0 .
04 S.
29 A lossless transmission line with 50 Ω and length 0 .
6 λ terminates at a load Z
L
= (50 + j 25) Ω. At a distance 0 .
3 λ from the load a resistance R = 30 Ω is inserted, as shown below. Use the Smith chart to determine the input impedance, Z i
, of this circuit.
Z
0
R
Z
0
Z i
Z
L
0,3 λ 0,3 λ
30 In order to match a lossless transmission line with 50 Ω to a load with Z
L
= (75 − j 20) (Ω) a single stub is used. Use the Smith chart to determine the length and the location of the stub.
31 A lossless transmission line with characteristic impedance 300 Ω operates at 100 MHz and is terminated by a load Z
L the line.
= 77 .
6 − j 49 .
4 Ω. Design a single stub (terminated in a short-circuit) to match
18

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32 In order to match a load with impedance 100 Ω to a coaxial cable with characteristic impedance 50 Ω and v f
= 2 × 10 8 , a reactive element is inserted in parallel with the line, as shown in the figure. The line is operating at 50 MHz.
Electromagnetic Engineering
MAP-Tele – 2008/2009
Z i
Z
0 jX d
Z
0
R
L
(a) Using the Smith chart, determine the value of the reactive element to be inserted (inductor or capacitor) and its location relative to the load.
(b) What is the value of the parameter SWR in two sections of the line?
(c) If the cable is 1 m long, obtain the input impedance of the line with and without reactive element.
33 Consider a lossless transmission line with characteristic impedance 50 Ω and phase velocity 10 8 m/s.
d
A B
Z
0 Z
L l
(a) Knowing that the line operates at 100 MHz and is matched to its load by a single shortcircuited stub of length l = 0 .
14 m, which is located at a distance d = 0 .
46 m from the load
Z
L
, obtain the impedance of the load.
(b) Under the previous conditions, the voltage maximum in the section AB of the line is 30 V.
Sketch the variation of the line voltage as a function of the distance to the load, clearly indicating the minimum value of the voltage and the locations of the voltage minima and maxima.
34 Two lossless transmission lines operating at 40 MHz have the following characteristic parameters:
L
1
= L
2
= 0 .
1 mH/km and C
1
= C
2
= 2 .
5 nF/km. The first line, 400 m long, is terminated by a load of impedance 200 Ω, whereas the second one has a length of 501 .
6 m and is open-circuited.
The input terminals of the two lines are connected in parallel and act as a load for a third lossless line.
(a) What is the characteristic impedance of the third line that eliminates reflections in the connection to the other two lines?
(b) Consider now that the third line has a characteristic impedance of 100 Ω and a phase constant of π
25 load.
rad/m. Design a single stub terminated in a short-circuit that matches this line to its
35 A 9 m long lossless transmission line is terminated by a load Z
L
Z
0
= 50 Ω and v f
= 2 c/ 3, and operates at 100 MHz.
= 15 + j 30 Ω. This line has
(a) Determine the input impedance of the line.
(b) Find the location of the voltage minimum nearest to the load.
19

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(c) Match the line to the load using 2 stubs terminated in short-circuit and separated by λ/ 8.
36 A double stub is used to match a load of impedance Z
L
= 100 + j 100 Ω to a lossless transmission line with Z
0
= 300 Ω. The stubs are separated by 3 λ/ 8 and one of the stubs is located at the load.
(a) Obtain the length of the two stubs assuming that they are terminated by an open-circuit.
(b) What is the value of the SWR in the main line and in the section between the two stubs?
(c) In the section between the two stubs, find the location of the voltage maximum nearest to the load.
37 A double stub is used to match a load with impedance Z
L
= 100 + j 100 Ω to a lossless transmission line with Z
0
= 200 Ω. The two stubs are separated by 3 λ/ 8, and one of the stubs is located at the load. Obtain the lengths of the two stubs assuming that:
(a) both stubs are terminated by a short-circuit;
(b) both stubs are terminated by an open-circuit.
38 Desing a quarter-wave adaptor to match a load of impedance Z
L
= 30 Ω to a lossless transmission line with Z
0
= 100 Ω.
39 A lossless transmission line, filled with air and with characteristic impedance Z
0
= 50 Ω, is terminated by a load of impedance Z
L
= 40 + j 30 Ω. A quarter-wave adaptor is going to match this line at a frequency of 3 GHz.
(a) Obtain the length of the matching line in centimeters.
(b) Obtain the characteristic impedance of the matching line, Z 0
0
, and its location relative to the load, d .
(c) Find the values of the parameter SWR in the different sections of the system.
40 A lossless transmission line with Z
0
= 50 Ω, terminated by a load using a quarter-wave transformer. Use the Smith chart to determine:
Z
L
= 72 + j 96 Ω, is matched
(a) The location of the quarter-wave transformer relative to the load, d
1
.
(b) The characteristic impedance of the matching line.
(c) The SWR of the line with length d
1
(d) The SWR in the matching line.
41 Consider a lossless transmission line of length 1 m, with Z
0
= 50 Ω and v = 2 c/ 3, and terminated by a load Z
L
= 25 Ω. A step voltage is applied at the line input at t = 0, by a generator with
V g
= 60 V and R g
= 100 Ω.
(a) Draw V ( z, t ) in a reflection diagram.
(b) Use the diagram to sketch V ( t ) in the middle point of the line for t between t = 0 and t = 25 ns.
42 A lossless transmission line with 40 Ω, ε r
= 2 .
25 and 200 m long, is terminated by an open-circuit.
At t = 0 a voltage is applied at the line input by a generator with 40 V and internal impedance
120 Ω.
20

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(a) Sketch the time evolution of the voltage at the load for 0 < t < 5 µ s.
(b) Determine the final value of the voltage at the load.
43 A pulse with an amplitude of 15 V and a duration 1 µ s is applied through a series resistance of
25 Ω to the input terminals of a lossless transmission line with 50 Ω. The line is 400 m long and is terminated by a short-circuit. Determine the time evolution of the voltage at the middle point of the line for t between 0 and 8 µ s. Assume that the dielectric constant in the transmission line is
2 .
25.
V g
(t)
15 V
Z g
V g
(t) Z
0
0 1
µ s t
44 At the input terminals of a lossless transmission line a step voltage is applied at t = 0. As a result, the following time evolution of the input voltage was observed. The line is characterized by
Z
0
= 50 Ω, ε = 2 .
25 and the internal resistance of the generator is R g
= 50 Ω. Determine:
(a) The voltage of the generator.
(b) The length of the line.
(c) The impedance of the load.
5
3
0 6
µ s t
45 A lossless transmission line with parameters C = 100 pF/m and L = 250 nH/m, is terminated by a resistive load R
L
. At t = 0 a step voltage is applied to this line by a generator with V g
= 18 V and R g
. The following graphic shows the time evolution of the voltage at some point A in the line for t between 0 and 3 µ s.
V (V)
18
12
1
.
5 2
.
5 3 t ( µ s)
21

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Electromagnetic Engineering
MAP-Tele – 2008/2009
(a) Determine the location of point A relative to the generator and the length of the line.
(b) Determine R
L and R g
.
(c) Draw the reflection diagram for t between 0 and 8 µ s.
(d) Sketch the time evolution of the voltage at the load for t between 0 and 8 µ s.
(e) Show that the final value ( t → + ∞ ) of the voltage at the load is
R
L
R
L
+ R g
V g
.
22

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— Plane wave incidence: Solutions —
1 (a) E i
( z ) = 10 e − j 4 π
3 z (ˆ + j y ) V / m
(b) Γ = − 1 / 3
(c)
τ = 2 / 3
E r
( z ) = −
~ t
( z ) =
~ air
( z ) =
20
3
10
3 e
20
3
− e h j j 4 π
3
8 π
3 e − j z z (ˆ
4 π
3
+ j y ) V
4 π
3
/ m
(ˆ + j y ) V / m z − j sin z i
(ˆ + j y ) V / m
(d)
P av , r
P av , i
P av , t
P av , i
= 1 / 9
= 8 / 9
2 (a) E i
( z ) = 10 e − j 4 π
3 z (ˆ + j y ) V / m
(b) Γ = − 0 .
2 e − j 0 .
0048
τ = 0 .
8 e j 0 .
0028
(c)
~ r
( z ) = − 2 e j
(
− 0 .
0048+
4 π
3 z )(ˆ
+ j ˆ ) V / m
E t
( z ) = 8 e − 0 .
012 z h ~ air
( z ) = 10 e − j e
4 π
3
− j (6 .
283 z − 0 .
0028) z − 2 e j
( 4 π
3
(ˆ z − 0 .
0048
+
) i j y
(ˆ
) V
+ j y
/
) m
V / m
(d)
P av , r
P av , i
P av , t
P av , i
= 0 .
04
= 0 .
96
3 (a) Z
Z
0
λ
λ
0
= 0 .
797 − j 0 .
188
= v v
0
= 32 .
8 × 10 − 3
(b) δ = 4 .
85 mm
(c) − 8 .
96 dB
(d) Γ = − 0 .
101 − j 0 .
115 = 0 .
153 e − j 2 .
29
4 d = 3 .
21 cm
5 ε r
2
=
√
ε r
1
ε r
3
, d = n
λ
2
4
( n odd)
6 (a) 5 .
07 %
(b) ε rd
= 1 .
58 , d = 111 .
3 nm
(c) 0 .
41 %
7 —
8 (a) E
1
( z ) = − j 20 sin(10 πz )ˆ V / m
~
1
( z ) =
1
6 π cos(10 πz )ˆ A / m
E
1
( z, t ) = 20 sin(10 πz ) cos( ωt −
~
1
( z, t ) =
1
6 π cos(10 πz ) cos( ωt )ˆ
π
2
)ˆ
A /
V m
/ m
(b) z = − 0 .
05 m.
(c) α = 2 .
11 × 10 5 dB/m
(d) 0 .
26 %
(e) right-hand circularly polarized wave
9 (a) E r
( z ) = 9 e j (20 πz +3 .
04) (ˆ − j y ) V / m
23

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Electromagnetic Engineering
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(b) incident wave: right-hand circular polarization reflected wave: left-hand circular polarization wave
(c) γ
2
= 200 π (1 + j ) m − 1
(d)
~ t
( z ) = 1 .
345 e − 200 πz e − j (200 πz − 0 .
74) (ˆ − j y ) V / m
(e)
(f )
H i
( z ) =
~ r
( z ) =
H t
( z ) =
1
12 π
9
π e − j 20 πz
120 π
0 .
1285 e
(ˆ
− 200 πz e
+ j x ) A / m e j (20 πz +3 .
04) ( − ˆ − j x ) A / m
− j (200 πz +0 .
046) (ˆ + j x ) A / m
~ av , medium 1
=
0 .
1508
π
ˆ W / m 2
(g)
~ av , medium 2
=
0 .
1508
π e − 400 πz ˆ W / m 2
10 (a) ˆ ni
= 0 .
8ˆ
θ i
= 53 .
1 o
+ 0 .
z
(b) H i
= E
0
60 π e − jπ (4 y +3 z ) (0 .
6ˆ − 0 .
z ) (A / m)
(c) Γ
⊥
E r
= e j 153 o
= E
0 e j 153 o e − jπ (4 y − 3 z ) ˆ (V / m)
(d)
~ ar
= 0 .
47 E
0 e j 76 .
5 o e − 9 .
8 z e − j 4 πy ˆ (V / m)
11 (a) right-hand circular polarization
(b) reflected: right-hand elliptic polarization transmitted: left-hand elliptic polarization
12 (a) Γ
⊥
τ
⊥
= − 0 .
274
= 0 .
726
(b)
~
( x, z, t ) = 36 .
3 cos 12 π × 10 14 t − 6 .
4 π × 10 6 (0 .
950 z + 0 .
313 x ) ˆ (V / m)
~ ( x, z, t ) = 0 .
154 cos 12 π × 10 14 t − 6 .
4 π × 10 6 (0 .
950 z + 0 .
313 x ) (0 .
313ˆ − 0 .
x ) (A / m)
13 (a) perpendicular polarization
(b) ε r 1
= 4
θ i
= 53 .
1 ◦
(c) H i
= E
0
60 π e − j 4 π (4 x +3 y ) ( − 0 .
8ˆ + 0 .
x ) (A / m)
(d) Γ
⊥
= e j 1 .
61
τ
⊥
= 1 .
39
~ r
= E
0 e e j 0 .
8050 j 1 .
61 e − j 4 π (4 x − 3 y ) ˆ (V / m)
(e) E air
= E
0 e − 12 πy + e j 1 .
61 e j 12 πz e − j 16 πx ˆ (V / m)
14 (a) f = 2 .
39 × 10 8 Hz
θ i
= 36 .
9 ◦
(b)
~ i
= 24 πe − jπ (3 x +4 z ) ˆ (V / m)
(c) E r
= − 24 πe − jπ (3 x − 4 z ) ˆ (V / m)
~ r
= − 0 .
2 e − jπ (3 x − 4 z ) (0 .
6ˆ + 0 .
x ) (A / m)
(d)
(e) z = π
8
(1 − 2 n ) , n integer
~ av , air
= 18 .
1 sin
2
(4 z ) ˆ W / m 2
15 (a) Γ
⊥
= − 0 .
431
(b) 0 .
464 rad
24

Faculdade de Engenharia
16 (a) n
1
= 1 .
5
(b) θ i
= 61 .
9 ◦
θ t
= 41 .
4 ◦
(c)
~ r
= − 3 .
6 e − j 4 π ( − 8 x +15 y ) ˆ (V / m)
E t
= 6 .
4 πe − j 90 .
7 π (0 .
75 x +0 .
66 y ) ˆ (V / m)
(d) 87 .
04%
17 (a) f = 2 GHz
(b) ˆ ni
= 0 .
6ˆ + 0 .
8ˆ a nr
= − 0 .
6ˆ + 0 .
y
ˆ nt
= 0 .
8ˆ + 0 .
6ˆ
(c) E r
= 0 (V
~ t
= 7 .
5 e − j
/ m)
80 π
3
(0 .
8 x +0 .
6 y ) ( − 0 .
6ˆ + 0 .
8ˆ ) (V / m)
Electromagnetic Engineering
MAP-Tele – 2008/2009
18 —
25

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— Waveguides and Cavities: Solutions —
1 —
2 Upper plate: ρ s
= ( − 1) n γ h
Lower plate: ρ s
= −
γ h
A n e
A n e − γz ,
− γz , J s
=
~ s
= ( − 1) n jωε h
− jωε h
A n e − γz
A n e − γz
3 Upper plate:
Lower plate:
J s
= ( − 1) n +1 B n
~ s
= B n e − γz e − γz
4 (a) TM
1 mode:
~
( x, y, z, t ) = A
1
{ sin( πy/b ) cos( ωt − βz )ˆ + βb/π cos( πy/b ) cos( ωt − βz − π/ 2)ˆ }
~
( x, y, z, t ) = ωεb/πA
1 cos( πy/b ) cos( ωt − βz + π/ 2)ˆ
TE
1 mode:
~
( x, y, z, t ) = ωµb/πB
1 sin( πy/b ) cos( ωt − βz + π/ 2)ˆ
~
( x, y, z, t ) = B
1
{ cos( πy/b ) cos( ωt − βz )ˆ + βb/π sin( πy/b ) cos( ωt − βz + π/ 2)ˆ }
(b) —
5 (a) ( f c
)
TE
1
( f c
)
TE
2
= ( f c
)
TM
1
= ( f c
)
TM
2
= 15 GHz
= 30 GHz
(b) TEM, TM
1
, TE
1
, TM
2
, TE
2
.
6 —
7 (a) TE
1
TE
2
TE
3
TE
4
TE
5 and TM
1
: ( f c
)
1 and TM
2
: ( f c
)
2 and TM
3
: ( f c
)
3 and TM
4
: ( f c
)
4 and TM
5
: ( f c
)
5
= 3 GHz
= 6 GHz
= 9 GHz
= 12 GHz
= 15 GHz
(b) TE n and TM n
:
2
3
4
5 n f c
1
(GHz) v f
3
6
9
12
15
(m / s) λ (m) β (rad / m) Z
TE
3 .
03 × 10 8
3 .
14 × 10 8
3 .
36 × 10 8
3 .
75 × 10 8
4 .
53 × 10 8
0
0
0
0
0
.
.
.
.
.
015
016
017
0187
0227
131
127
119
106
88 .
.
.
.
.
8
2
0
7
19
π
π
π
π
π
381
395
422
471
570
(Ω)
.
.
.
.
.
3
2
2
3
0
Z
TM
372
359
336
301
249 .
.
.
.
.
(Ω)
7
6
7
6
4
(c) v f
= 3 × 10 8 m / s, β = 133 .
3 π rad / m, λ = 0 .
015 m, Z
TEM
= 377 Ω
(d) TE n and TM n
:
4
5
2
3 n α (Np / m)
1 14 .
9 π
37
58
78
99
.
.
.
.
7
5
9
1
π
π
π
π
8 (a) TE
2
(b) β = 111 .
8 π (rad/m)
26

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Electromagnetic Engineering
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(c) 107 .
3 π (Ω)
(d)
~
= − 6000
~ = 25
π sin(100 πy ) sin(3 π ×
5 sin(100 πy ) sin(3 π × 10
10
10 t
10 t
−
−
βz
βz
)ˆ
)ˆ (V / m)
+ 50 cos(100 πy ) cos(3 π × 10 10 t − βz )ˆ (A / m)
9 (a) TM
1
, f c
= 7 .
5 GHz
(b) β = 392 .
4 rad / m
(c) v f
= 1 .
92 × 10 8 m / s
(d)
~
= 40 [ − j 1 .
25 cos(100 πy )ˆ + sin(100 πy )ˆ ] e − j 392 .
4 z
~
= j 0 .
34 cos(100 πy ) e − j 392 .
4 z ˆ (A / m)
(V / m)
(e) S av
= 8 .
5 cos 2 (100 πy )ˆ (W / m 2 )
10 —
11 TE
10
: ( f c
)
10
= 3 GHz, β = 615 .
75 rad/m, v f
= 1 .
53 × 10 8 m/s, Z
TE
= 192 .
34 (Ω)
TM
11
: ( f c
)
11
= 8 .
078 GHz, β = 529 .
67 (rad/m), v f
= 1 .
78 × 10 8 m/s, Z
TM
= 158 .
9 (Ω).
12 (a) ( f c
)
10
= 15 GHz, ( f c
)
20
= 30 GHz
(b) λ = 1 .
67 cm, λ g
= 3 .
02 cm
13 (a) TM
21
(b) β = 241 .
4 (rad/m)
(c) E y
/E x
= (50 / 40) tan(40 πx ) cot(50 πy )
14 (a) ( f c
)
01
= 15 GHz, ( f c
)
10
= 6 GHz, ( f c
)
20
= 12 GHz
(b) Modes:
TE
10
, TE
20
, TE
01
, TE
11
, TM
11
, TE
30
, TE
21
, TM
21
, TE
31
, TM
31
,
TE
40
, TE
41
, TM
41
, TE
02
, TE
50
.
15 (a) TM
13 or TE
13
(b) ( f c
)
13
= 28 .
57 GHz
(c) γ = jβ , β = 1 .
72 × 10 3 (rad/m)
(d) ( Z
TE
)
13
= 229 .
7 (Ω), ( Z
TM
)
13
= 154 .
6 (Ω)
16 (a) ( f c
)
10
= 1 .
74 GHz < f , v f
= 3 .
33 × 10 8 (m/s) > c , v g
= 2 .
702 × 10 8 (m/s)
(b) ( f c
)
10
= 3 .
88 GHz < f , v f
= 12 .
3 × 10 8 (m/s) > c , v g
= 7 .
32 × 10 7 (m/s)
17 E
0
= 63 .
77 (V/m)
18 (a) —
(b) ( f c
)
TE
10
= 5 .
06 GHz
19 —
20 —
21 —
22 TE
101
, TE
011
, TE
102
, TE
012
, TM
110
23 (a) TE
101
; f
TE
101
= 4 .
80 GHz
(b) TE
101
; f
TE
101
= 3 .
04 GHz
27

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Electromagnetic Engineering
MAP-Tele – 2008/2009
24 —
25
H
0 r
= − 1 h 2
E
0 r
= −
1 h 2
γ
∂H 0 z
∂r
γ
∂E 0 z
∂r
− jω r
∂E
∂φ
0 z
+ jωµ r
∂H 0 z
∂φ
H
0
φ
= − 1 h 2
E
0
φ
= −
1 h 2
γ r
γ r
∂H 0 z
∂φ
∂E
∂φ
0 z
+ jω
∂E 0 z
∂r
− jωµ
∂H 0 z
∂r
26 a m
= 0 , m odd; a m
=
( − 1) m/ 2
2 m [( m/ 2)!] 2
, m even
27 —
28 —
29 (a) diameter = 2 .
2 cm
(b) TM
01
, TE
11
, TE
21
30 (a) TM
01
, TE
01
, TM
11
, TE
11 and TE
21
(b) S av
= 0 .
0029 E 2
0
( J 0
0
( hr )) 2 ˆ (W / m 2 )
(c) from 8 .
79 GHz to 11 .
48 GHz
31 (a) TM
01
, TE
11 and TE
21
(b) for TM
01
:
S av
=
βωε
2 h 2
E 2
0
( J 0
0
( hr )) 2 ˆ (W / m 2 ) with β = 116 .
7 rad/m and h = 1202 .
4 m − 1
(c) from 2 .
20 GHz to 2 .
82 GHz
32 d = 23 mm
33 TM
010
, TE
111
, TM
110
, TM
011
, TE
211
, TM
111
, TE
011
34 —
35 —
36 d min
= 6 .
1 mm
37 (a) S av
=
βωεA
2 h 2
1 cos 2 ( h
1 y )ˆ
(b) 0
38 —
39 (a) Inside the dielectric (0 < y < b )
~
= jωµ h
1
B sin( h
1 y ) e − jβz
~
= Be − jβz cos( h
1 y )ˆ + jβ h
1 sin( h
1 y
In the air (0 < y < b )
~ = − jωµ
0
ν
B
~
= B cos( h
1 cos( b ) e h
1 b ) e − ν ( y − b )
− ν ( y − b ) e − jβz e − jβz
ˆ − jβ
ν
)ˆ
(b) ν = −
µ
0
µ h
1 cot( h
1 b )
(c) f c
=
2 b
( n − 1 / 2)
µε − µ
0
ε
0
, n = 1 , 2 , . . .
40 ρ
S
= 0;
~
S
= Be − jβz ˆ
41 (a) q
ω c
2
( n 2
1
− n 2
2
) − h 2
1
= n
2 n
1
2 h
1 tan( h
1 b )
28

Faculdade de Engenharia
(b) f c
=
2 b
√ − n
1)
2
1 c
− n 2
2
, n = 1 , 2 , . . .
42 ε r
> 1 + sin
2
θ i
43 (a) N.A.
= p n 2
1
− n 2
2
;
(b) N.A.
= 0 .
986 ;
θ a
= arcsin
θ a
= 1 .
404 rad p n 2
1
− n 2
2
44 (a) N.A.
= 0 .
2425
(b) a < 2 .
05 µ m
Electromagnetic Engineering
MAP-Tele – 2008/2009
29

Faculdade de Engenharia
Electromagnetic Engineering
MAP-Tele – 2008/2009
— Transmission Lines: Solutions —
1 l/λ ≤ 0 .
01
(a) l/λ = 6 .
67 × 10 − 6 can be neglected
(b) l/λ = 0 .
01 can be neglected
(c) l/λ = 0 .
2 can be neglected
(d) l/λ = 0 .
33 can be neglected
2 —
3 R = 1 .
0 Ω / m, L = 1 .
67 × 10 − 7
H/m, G = 0, C = 1 .
72 × 10 − 10
F/m
4 (a) R = 0 .
788 Ω / m, L = 139 nH/m, G = 9 .
1 mS/m, C = 181 pF/m
(b) α = 0 .
143 Np/m, β = 31 .
5 rad/m, Z
0
= 27 .
7 + j 0 .
098 Ω, v f
= 2 × 10 8 m/s
5 R = 2 Ω / m, L = 200 nH/m, G = 0 .
8 mS/m, C = 80 pF/m, λ = 1 m
6 (a) R = 0 .
8 Ω / m, L = 38 .
2 nH/m, G = 0 .
5 mS/m, C = 23 .
9 pF/m
(b) l = 172 .
7 m
7 (a) Z
0
= 70 .
75 ∠ − 1 .
37 ◦ = 70 .
73 − j 1 .
69 Ω
(b) γ = 2 .
12 × 10 − 4 + j 8 .
89 × 10 − 3 m − 1
(c) v f
= 7 .
07 × 10 5 m/s
(d) 3 .
68 dB
8 β = 798 rad/m, v f
= 3 .
54 × 10 7 m/s
9 Z i
= 60 + j 20 Ω
10 λ/ 4
11 —
12 (a) Γ = 0 .
632 e − j 71 .
6 ◦
(b) SWR = 4 .
43
(c) z 0 max
(1) = 2 cm
(d) z 0 min
(1) = 0 .
75 cm
13 Z
L
= 83 .
24 − j 51 .
27 Ω
14 Γ = 0 .
307 e 132 .
5 ◦
, SWR = 1 .
886, Z i
= 64 .
8 − j 38 .
3 Ω
15 —
16 —
17 (a) Γ
L
= 0 .
620 e − j 29 .
74 ◦
(b) Z i
= 17 .
855 e 45 .
44 ◦
= 12 .
53 − j 12 .
72 Ω
(c) V i
= 1 .
4 e − j 34 .
0 ◦ , I i
= 78 .
4 e 11 .
4 ◦ (mA)
30

Faculdade de Engenharia
Electromagnetic Engineering
MAP-Tele – 2008/2009
18 (a) Z i
= 60 .
25 + j 38 .
79 Ω
(b) I ( z = 0) = 93 .
03 ∠ − 21 .
15 ◦ (mA)
(c) I ( z = l/ 2) = 35 .
10 ∠ 281 ◦ (mA)
19 (a) α = 0 .
0025 Np/m, β = 2 rad/m, v f
= 0 .
5 × 10 8 m/s
(b) Z
0
= 200 Ω
I ( x, t ) = 0 .
3 e 2 .
5 × 10 − 3 x cos(10 8 t + 2 x ) − 0 .
06 e − 2 .
5 × 10 − 3 x cos(10 8 t − 2 x ) A
20 —
21 —
22 —
23 (a) Γ
L
= 0 .
352 ∠ 56 ◦
(b) SWR = 2 .
088
(c) Z i
= 23 .
97 + j 1 .
35 Ω
24 Z
0
= 40 Ω, Z 0
0
= 250 Ω
25 —
26 (a) Γ
L
= 0 .
6 ∠ − 135 ◦
(b) Z
L
= 14 .
25 − j 19 Ω
(c) l min
= 0 .
35 m, R min
= 12 .
5 Ω
27 (a) Γ
L
= 0 .
24 ∠ 75 ◦
(b) SWR = 1 .
65
(c) Z ( l = 0 .
35 λ ) = 30 − j (Ω)
(d) y ( l = 0 .
35 λ ) = 1 .
7 / 50 + j 0 .
08 / 50 (S)
(e) 0 .
105 λ
(f ) 0 .
105 λ
28 (a) Z in
= − j 154 Ω
(b) 0 .
074 λ
29 Z i
= 95 − j 70 Ω
30 d
1
= 0 .
104 λ , l
1
= 0 .
173 λ ; d
2
= 0 .
314 λ , l
2
= 0 .
327 λ
31 —
32 —
33 —
34 —
35 —
36 —
37 (a) l
A
1
= 0 .
375 λ , l
B
1
= 0 .
25 λ , l
A
2
= 0 .
125 λ , l
B
2
= 0 .
074 λ
(b) l
A
1
= 0 .
125 λ , l
B
1
= 0, l
A
2
= 0 .
375 λ , l
B
2
= 0 .
324 λ
31

Faculdade de Engenharia
Electromagnetic Engineering
MAP-Tele – 2008/2009
38 Z 0
0
= 54 .
77 Ω
39 —
40 (a) d
1
= 0 .
053 λ
(b) Z 0
0
= 106 .
07
(c) SWR = 4 .
5
(d) SWR = 2 .
1
41 (a) Γ g
= 1 / 3, Γ
L
= − 1 / 3, T = 5 ns, V
1
+
= 20 V, V
1
− = − 6 .
67 V, V
2
+
= − 2 .
22 V, V
2
− = 0 .
74 V
(b) V (0 .
5 , t ) = 0 , 0 ≤ t < T / 2; V (0 .
5 , t ) = 20 V , T / 2 ≤ t < 3 T / 2
V (0 .
5 , t ) = 13 .
33 V , 3 T / 2 ≤ t < 5 T / 2, V (0 .
5 , t ) = 11 .
11 V , 5 T / 2 ≤ t < 7 T / 2
V (0 .
5 , t ) = 11 .
85 V , 7 T / 2 ≤ t < 9 T / 2
42 —
43 V (200 , t ) = 0 V , 0 µ s ≤ t < 1 µ s; V (200 , t ) = 10 V , 1 µ s ≤ t < 2 µ s
V (200 , t ) = 0 V , 2 µ s ≤ t < 3 µ s; V (200 , t ) = − 10 V , 3 µ s ≤ t < 4 µ s
V (200 , t ) = 0 V , 4 µ s ≤ t < 5 µ s; V (200 , t ) = 10 / 3 V , 5 µ s ≤ t < 6 µ s
V (200 , t ) = 0 V , 6 µ s ≤ t < 7 µ s; V (200 , t ) = − 10 / 3 V , 7 µ s ≤ t < 8 µ s
44 (a) V g
= 10 V
(b) l = 100 m
(c) R
L
= 21 .
43 Ω
45 —
Collection of problems from different sources:
•
•
•
D. Cheng, “Field and Wave Electromagnetics”, Addison-Wesley
32