problems on plane wave incidence, waveguides and...

Problems on Plane Wave Incidence,

Waveguides and Transmission Lines

ELECTROMAGNETIC ENGINEERING

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 10Vm−1. This wave propagates in air and impinges normally on a dielectric medium

with εr = 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.

z 0

Ei

dielectric

(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 εr = 2.25, µr = 1 and σ = 10−4 Sm−1.

3 At 200 MHz a given medium is characterized by σ = 0, µr = 15(1− j3) and εr = 50(1− j1). Find:

(a) η/η0, λ/λ0 and v/v0, 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 µr = εr = 6 − j6 is located under a perfectly conducting plate.

Assuming that a electromagnetic wave propagating in air with a frequency of 500MHz impinges

normally in this material, find the thickness of the material that reduces the amplitude of the

reflected wave by 35 dB?

Ei

conductor

material

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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 εr2and d such that there are no

reflected wave. Write the answer as a function of εr1, εr3

and frequency f .

εr1

-

-

Medium 2

Medium 3

d

Medium 1

z = −d z = 0

z

εr2

εr3

6 Consider a plane electromagnetic wave that is normally incident on a glass lens of a camera with

εr = 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.

glass air

incidentwave

Figure A

dielectric

glass air

incidentwave

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 (λ0 = 560 nm,

where λ0 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 (λ0 = 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, µr = 1. The incident wave has a frequency of f = 1.5 MHz and

the electric field at the interface z = 0 is given by,

~E(0, t) = 10 sin(2πft)y (V/m).

(a) Determine the magnetic field in the conductor.

(b) Show that the reflection coefficient is approximately given by Γ ≈ (ηr−1)(1−ηr) ≈ −1+2ηr =√

2ωε0/σ(1 + j) − 1, where ηr =√

jωε0/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.

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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 = 105 S/m (graphite), ε = ε0 and µ = µ0,

find the attenuation constant in dB/m.

(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 6π√

2ej45o

. The incident electric field is given by

~E = (x − jy)10e−j20πz (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

~E = xE0e−jπ(4y+3z).

(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 (εr = 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 air-

glass with an angle of incidence of 30. The wave frequency is 600THz (1THz = 1012 Hz), corre-

sponding to green light, and the index of refraction of glass is 1.6. Assuming that the electric field

amplitude of the incident wave is 50Vm−1, determine:

(a) the reflection and transmission coefficients;

(b) the expressions for ~E and ~H in the glass.

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MAP-Tele – 2008/2009

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

~Ei(x, y) = zE0e−j4π(4x+3y) (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

~Hi(x, z) = 0.2e−j(3x+4z)(0.6z − 0.8x) (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 ~Ei;

(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 εr = 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

~Ei(x, y) = 10e−j4π(8x+15y)z (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?

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17 A plane wave propagating in a nonmagnetic medium with dielectric constant εr = 2.25 is charac-

terized by~Ei(x, y) = 10e−j4π(3x+4y)(−0.8x + 0.6y) (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~Ei(x, y) = 10e−j4π(3x+4y)(−0.8x + 0.6y + z) (V/m).

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MAP-Tele – 2008/2009

— Waveguides and Cavities —

1 Show that the TM1 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 parallel-

plate waveguide for the TMn 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 TEn 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 TM1 and TE1

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 TE1, TM1, TE2 and TM2 modes.

(b) Knowing that the guide operates at a frequency of 40 GHz, indicate which modes can propa-

gate.

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 permit-

tivity 2.25. The magnetic field inside the guide is characterized by

Hz = 50 cos (100πy) cos(

3π × 1010t − βz)

(A/m).

Determine:

(a) the mode that is propagating;

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MAP-Tele – 2008/2009

(b) the phase constant (β);

(c) the wave impedance;

(d) ~E and ~H.

9 A waveguide with parallel-plates separated by 1 cm is filled with a dielectric with relative permit-

tivity 4 and operates at 12 GHz. The electric field inside the guide is characterized by

Ez = 40 sin (100πy) 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) ~E and ~H;

(e) ~Sav.

10 A parallel-plate waveguide is filled with two different dielectrics, with permittivities ε1 and ε2.

d 1

2

y

x

d

(a) Starting from the wave equations in each medium, show that for the TE modes the longitudinal

component of the magnetic field satisfies

H0z =

A sin(h1y) + B cos(h1y) medium 1

C sin(h2y) + B cos(h2y) medium 2

where h21 = γ2 + ω2µ0ε1 and h2

2 = γ2 + ω2µ0ε2, and A,B e C are constants.

(b) Determine ~E0 and ~H0.

(c) Using the previous result and appropriate boundary conditions, show that the following con-

dition is satisfied

h1 tan(h2d) + h2 tan(h1d) = 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 εr = 4. Determine the phase constant, the phase velocity and the wave

impedance for the TE10 and TM11 modes when the guide is operating at 15GHz.

12 A rectangular waveguide filled with air has dimensions a = 1 cm and b = 0.6 cm.

(a) Determine the cutoff frequencies of the TE10 and TE20 modes

(b) For a 18GHz 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 15GHz. The

electric field inside the guide is characterized by:

Ez = 20 sin(40πx) sin(50πy)e−jβz V/m.

(a) What mode is propagating inside the guide?

(b) Find β.

(c) Determine the ratio Ey/Ex.

14 A rectangular waveguide with dimensions a = 2.5 cm and b = 1 cm operates at a frequency below

15.1GHz.

(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 εr = 4

and µr = 1?

15 A rectangular waveguide with dimensions a = 1.5 cm, b = 0.8 cm is filled with a dielectric with

µ = µ0 and ε = 4ε0. The magnetic field inside the guide is characterized by

Hx = 2 sin(πx

a

)

cos

(

3πy

b

)

sin(π × 1011t − 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 TE10 mode can propagate and, if so, calculate the phase and group velocities.

(b) Repeat for the TM11 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.

b

d

1

2

y

x a

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(a) Show that the following condition is satisfied for the TEm0 modes with angular frequency ω:

h1 tan(h2d) + h2 tan(h1(a − d)) = 0,

where h21 = γ2 + ω2µε1 and h2

2 = γ2 + ω2µε2, and γ is the propagation constant.

(b) Obtain the cutoff frequency for the TE10 mode by solving the previous equation when γ = 0.

Assume that a = 2.286cm, d = a/2, ε1 = ε0 and εr2 = 2.25.

19 A rectangular waveguide (a = 5 cm; b = 2 cm), filled with air, operates at 1 GHz. For the TM21

mode and assuming that the maximum amplitude of Ez 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

18GHz. The electric inside this guide is characterized by

Ez = 20 sin(50πx) sin(100πy)e−jβz.

(a) What mode is propagating inside the guide? Find its cutoff frequency.

(b) Determine β and vg.

(c) Find the average power propagated inside the guide.

21 A rectangular waveguide will operate in a frequency band of 1.0GHz with central frequency 4.0GHz.

A safety margin of 20% should be considered between the band boundaries and the first and second

mode cutoff frequencies, as shown below.

f (GHz)

2.12c

uf

f = 0.40 =f 12.1 cl ff = 1cf 2cf

(1st mode)

1 GHZ

(2nd mode)

(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 εr = 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 = 2b = 4d.

a = 2b

b = 2d d

x

y z

(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 ∇ × ~E = −jωµ ~H and ∇ × ~H = jωε ~E in cylindrical coordinates,

obtain the expressions of the transverse components (along r and φ) of ~E and ~H as functions of

the longitudinal components (along z).

26 A solution of the following Bessel differential equation

d2R(r)

dr2+

1

r

dR(r)

dr+ R(r) = 0

can be obtained by considering that R(r) is of the form R(r) =∑∞

m=0 amrm. By substituting this

expression in the differential equation, determine the general expression for the coefficients am.

27 Show that the Bessel functions of the first kind satisfy the following conditions.

(a) J ′n(x) = 1

2 [Jn−1(x) − Jn+1(x)].

(b) J−n(x) = (−1)nJn(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 TM01 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 (4ε0, µ0).

(a) Assuming that the guide is operating at 4 GHz, determine which modes are allowed to prop-

agate.

(b) Under the previous conditions, obtain the average Poynting vector of the TE11 mode.

(c) Determine the frequency range [fmin, fmax] 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 10GHz

in the TM010 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 TE211 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 = µ0 and ε = 2.5ε0 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 20GHz.

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 εr = 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 Ez = 10 cos(hdy)e−jβz (V/m).

dielectric

y

z

b air

(a) Find the other components of the electric field in the dielectric.

(b) Knowing that hd = 229.7 m−1, calculate the phase constante β and the decay coefficient ν of

the fields in the air.

(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.

conductor

dielectric

air y

x

b

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 n1 and

thickness b and a conducting plate.

conductor

y

x

b n1

n2

Knowing that the guide is located in an infinite medium with refractive index n2 (n2 < n1),

determine, for the TM modes,

(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 θi, 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, respec-

tively. Determine:

(a) The numerical aperture of the fiber.

(b) The maximum radius that allows propagation in the single-mode regime at a wavelength of

1.3µm.

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44 The refractive indices of the core and cladding of an optical fiber are n1 = 1.5 and n2 = 1.45,

respectively.

(a) Determine the maximum radius that allows propagation in the single-mode regime at a fre-

quency of 2 × 1014 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

50Ω and phase velocity 108 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, µc = µ0 = 4π × 10−7 H/m and

σc = 5.8 × 107 S/m, and for the dielectric εr = 2.6, µ = µ0 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 = µ0,

εr = 2.25 e σ = 10−3 S/m. The conductors are made of copper with µ = µ0 and σc = 5.8×107 S/m

and the line is operating at 1 GHz.

(a) Obtain the line parameters R, L, G e C.

(b) Find α, β, vf and Z0 for this coaxial line.

5 Consider a distortionless transmission line (R/L = G/C) with Z0 = 50Ω, α = 40 × 10−3 Np/m

and vf = 2.5 × 108 m/s.

Obtain the line parameters and the wavelength λ at a frequency of 250 MHz.

6 A transmission line operating at 125 MHz has Z0 = 40Ω, 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 = 30Ω/Km, L = 0.1 H/km, G = 0 and C = 20µF/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 4.5GHz has L = 2.4µH/m and Z0 = 85Ω. Calculate the

phase constant, β, and the phase velocity.

9 Operating at a frequency of 300 MHz, a lossless line with 50Ω and length 2.5 m is connected to a

load ZL = (60 + j20)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 approxi-

mately

(a) Zin = (R + jωl)l, with a short-circuit termination.

(b) Zin = (G − jωC)/((G2 + (ωC)2)l, with an open-circuit termination.

12 A lossless transmission line terminates in a load ZL = (30 − j60)Ω. The wavelength is 5 cm and

the line characteristic impedance is 50Ω. 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 Z0 = 50Ω, determine the load impedance.

14 A lossless transmission line with length l = 0.35λ terminates in a load of impedance ZL = (60 +

j30)Ω. Obtain ΓL, SWR and Zi when Z0 = 100Ω.

ZLZ0Zi

0,35λ

15 A load of 500Ω 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 535ej64

Ω

when the line is open-circuited and is 467.5ej10

Ω when the line is short-circuited.

(a) Determine the characteristic impedance and the propagation constant.

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(b) Find the line parameters R, L, G e C.

(c) Assume that the line is connected to a load ZL = 400Ω. 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, Vg(t) = 5 cos(2π × 109t) (V) and internal impedance Zg = 50Ω, is

connected to a lossless line with 50Ω. The line, 5 cm long, terminates at a load ZL = (100−j100)Ω.

If v = c, determine:

(a) The reflection coefficient at the load.

(b) The impedance at the input of the line, Zi.

(c) The voltage and current at the input of the line, Vi and Ii.

18 A transmission line operating at ω = 106 rad/s has α = 8 dB/m, β = 1 rad/m and Z0 = (60+j40)Ω,

and is 2 meters long. If the line is connected to a generator with Vg = 10ej0

V and Zg = 40Ω,

and terminates in a load (20 + j50)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) = 60e0.0025x cos(108t + 2x) + 12e−0.0025x cos(108t − 2x)

where x is the distance measured from the load. If ZL = 300Ω determine:

(a) The attenuation constant (α), the phase constant (β) and the phase velocity.

(b) The characteristic impedance, Z0, and the current I(x, t).

20 Consider a length l of a transmission line with characteristic impedance Z0 and propagation constant

γ.

Z0, γ V1

I1 I2

V2

+

–

+

–

(a) Determine the parameters A,B,C,D of the relation

[

V1

I1

]

=

[

A B

C D

][

V2

I2

]

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



MAP-Tele – 2008/2009

(b) Show that the previous section of the line can be represented by the equivalent circuit

V1

I1 I2

V2

+

–

Z1/2 Z1/2

Y2

+

–

when

Z1 = 2Z0 tanhγl

2

Y2 =sinh γl

Z0

21 Use the Smith chart to obtain the reflection coefficient corresponding to each of the following load

impedances:

(a) ZL = 3Z0

(b) ZL = (2 − j2)Z0

(c) ZL = −j2Z0

(d) ZL = 0.

22 Use the Smith chart to obtain the normalized load impedances corresponding to the following

reflection coefficients:

(a) ΓL = 0.5

(b) ΓL = 0.5∠60

(c) ΓL = −1

(d) ΓL = 0.3∠ − 30

(e) ΓL = 0

(f) ΓL = j

23 A transmission line of length 30 m with Z0 = 50Ω operates at 2 MHz and terminates at a load

ZL = 60 + j40Ω. If v = 0.6c, determine:

(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 ZL = 100Ω, the measured SWR is 2.5. Use the

Smith chart to determine the two possible values of Z0.

25 A lossless transmission line of length 1.3λ and characteristic impedance 100Ω is terminated by a

load (100 + j50)Ω. Using Smith chart, determine:

(a) The reflection coefficient at the load and the SWR

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MAP-Tele – 2008/2009

(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 50Ω 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:


(b) The load impedance, ZL.

(c) The length and load impedance of a transmission line such that its input impedance is equal

to ZL.

27 A lossless transmission line with 50Ω terminates at a load ZL = (50 + j25)Ω. Use the Smith chart

to calculate:


(b) The SWR.

(c) The impedance at a distance of 0.35λ from the load.

(d) The admittance at a distance 0.35λ 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 2.3λ from the load.

(b) The location of the point nearest to the load where the admittance is Yi = −j0.04 S.

29 A lossless transmission line with 50Ω and length 0.6λ terminates at a load ZL = (50+ j25)Ω. At a

distance 0.3λ from the load a resistance R = 30Ω is inserted, as shown below. Use the Smith chart

to determine the input impedance, Zi, of this circuit.

ZL

Z0 Z0

Zi

0,3λ

R

0,3λ

30 In order to match a lossless transmission line with 50Ω to a load with ZL = (75− j20) (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 300Ω operates at 100 MHz and is termi-

nated by a load ZL = 77.6− j49.4Ω. Design a single stub (terminated in a short-circuit) to match

the line.

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MAP-Tele – 2008/2009

32 In order to match a load with impedance 100Ω to

a coaxial cable with characteristic impedance 50Ω

and vf = 2 × 108, a reactive element is inserted in

parallel with the line, as shown in the figure. The

line is operating at 50 MHz.

RL

Z0 Z0

Zi jX

d

(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 108 m/s.

A

l

d

ZL

B

Z0

(a) Knowing that the line operates at 100 MHz and is matched to its load by a single short-

circuited stub of length l = 0.14 m, which is located at a distance d = 0.46 m from the load

ZL, 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:

L1 = L2 = 0.1 mH/km and C1 = C2 = 2.5 nF/km. The first line, 400 m long, is terminated by

a load of impedance 200Ω, 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 con-

nection to the other two lines?

(b) Consider now that the third line has a characteristic impedance of 100Ω and a phase constant

of π25 rad/m. Design a single stub terminated in a short-circuit that matches this line to its

load.

35 A 9m long lossless transmission line is terminated by a load ZL = 15 + j30Ω. This line has

Z0 = 50Ω and vf = 2c/3, and operates at 100 MHz.

(a) Determine the input impedance of the line.

(b) Find the location of the voltage minimum nearest to the load.

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MAP-Tele – 2008/2009

(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 ZL = 100 + j100Ω to a lossless transmission

line with Z0 = 300Ω. The stubs are separated by 3λ/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 ZL = 100+ j100Ω to a lossless transmission

line with Z0 = 200Ω. The two stubs are separated by 3λ/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 ZL = 30Ω to a lossless transmission

line with Z0 = 100Ω.

39 A lossless transmission line, filled with air and with characteristic impedance Z0 = 50Ω, is termi-

nated by a load of impedance ZL = 40 + j30Ω. 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, 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 Z0 = 50Ω, terminated by a load ZL = 72 + j96Ω, is matched

using a quarter-wave transformer. Use the Smith chart to determine:

(a) The location of the quarter-wave transformer relative to the load, d1.

(b) The characteristic impedance of the matching line.

(c) The SWR of the line with length d1

(d) The SWR in the matching line.

41 Consider a lossless transmission line of length 1 m, with Z0 = 50Ω and v = 2c/3, and terminated

by a load ZL = 25Ω. A step voltage is applied at the line input at t = 0, by a generator with

Vg = 60 V and Rg = 100Ω.

(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 = 25ns.

42 A lossless transmission line with 40 Ω, εr = 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



MAP-Tele – 2008/2009

(a) Sketch the time evolution of the voltage at the load for 0 < t < 5 µs.

(b) Determine the final value of the voltage at the load.

43 A pulse with an amplitude of 15 V and a duration 1µs is applied through a series resistance of

25Ω to the input terminals of a lossless transmission line with 50Ω. 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 8µs. Assume that the dielectric constant in the transmission line is

2.25.

Z0Vg(t)

Vg(t)

Zg

1 µs t0

15 V

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

Z0 = 50Ω, ε = 2.25 and the internal resistance of the generator is Rg = 50Ω. Determine:

(a) The voltage of the generator.

(b) The length of the line.

(c) The impedance of the load.

6 µs t0

5

3

45 A lossless transmission line with parameters C = 100 pF/m and L = 250 nH/m, is terminated by

a resistive load RL. At t = 0 a step voltage is applied to this line by a generator with Vg = 18 V

and Rg. The following graphic shows the time evolution of the voltage at some point A in the line

for t between 0 and 3 µs.

V (V)

t (µs)

12

18

1.5 2.5 3

21



MAP-Tele – 2008/2009

(a) Determine the location of point A relative to the generator and the length of the line.

(b) Determine RL and Rg.

(c) Draw the reflection diagram for t between 0 and 8 µs.

(d) Sketch the time evolution of the voltage at the load for t between 0 and 8 µs.

(e) Show that the final value (t → +∞) of the voltage at the load is RL

RL+RgVg.

22



MAP-Tele – 2008/2009

— Plane wave incidence: Solutions —

1 (a) ~Ei(z) = 10e−j 4π3

z(x + jy) V/m

(b) Γ = −1/3 τ = 2/3

(c) ~Er(z) = − 103 ej 4π

3z(x + jy) V/m

~Et(z) = 203 e−j 8π

3z(x + jy) V/m

~Eair(z) = 203

[

e−j 4π3

z − j sin(

4π3 z

)

]

(x + jy) V/m

(d)Pav,r

Pav,i= 1/9

Pav,t

Pav,i= 8/9

2 (a) ~Ei(z) = 10e−j 4π3

z(x + jy) V/m

(b) Γ = −0.2e−j0.0048

τ = 0.8ej0.0028

(c) ~Er(z) = −2ej(−0.0048+ 4π3

z)(x + jy) V/m~Et(z) = 8e−0.012ze−j(6.283z−0.0028)(x + jy) V/m~Eair(z) =

[

10e−j 4π3

z − 2ej( 4π3

z−0.0048)]

(x + jy) V/m

(d)Pav,r

Pav,i= 0.04

Pav,t

Pav,i= 0.96

3 (a) ZZ0

= 0.797 − j0.188λλ0

= vv0

= 32.8 × 10−3

(b) δ = 4.85 mm

(c) −8.96 dB

(d) Γ = −0.101 − j0.115 = 0.153e−j2.29

4 d = 3.21 cm

5 εr2=

√εr1

εr3, d = nλ2

4 (n odd)

6 (a) 5.07%

(b) εrd = 1.58 , d = 111.3 nm

(c) 0.41%

7 —

8 (a) ~E1(z) = −j20 sin(10πz)x V/m~H1(z) = 1

6π cos(10πz)y A/m~E1(z, t) = 20 sin(10πz) cos(ωt − π

2 )x V/m~H1(z, t) = 1

6π cos(10πz) cos(ωt)y A/m

(b) z = −0.05 m.

(c) α = 2.11 × 105 dB/m

(d) 0.26%

(e) right-hand circularly polarized wave

9 (a) ~Er(z) = 9ej(20πz+3.04)(x − jy) V/m

23



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(b) incident wave: right-hand circular polarization

reflected wave: left-hand circular polarization wave

(c) γ2 = 200π(1 + j) m−1

(d) ~Et(z) = 1.345e−200πze−j(200πz−0.74)(x − jy) V/m

(e) ~Hi(z) = 112π e−j20πz(y + jx) A/m

~Hr(z) = 9120π ej(20πz+3.04)(−y − jx) A/m

~Ht(z) = 0.1285π e−200πze−j(200πz+0.046)(y + jx) A/m

(f) ~Sav, medium 1 = 0.1508π z W/m2

(g) ~Sav, medium 2 = 0.1508π e−400πz z W/m2

10 (a) ani = 0.8y + 0.6z

θi = 53.1o

(b) ~Hi = E0

60π e−jπ(4y+3z)(0.6y − 0.8z) (A/m)

(c) Γ⊥ = ej153o

~Er = E0ej153o

e−jπ(4y−3z)x (V/m)

(d) ~Ear = 0.47E0ej76.5o

e−9.8ze−j4πyx (V/m)

11 (a) right-hand circular polarization

(b) reflected: right-hand elliptic polarization

transmitted: left-hand elliptic polarization

12 (a) Γ⊥ = −0.274

τ⊥ = 0.726

(b) ~E(x, z, t) = 36.3 cos[

12π × 1014t − 6.4π × 106(0.950z + 0.313x)]

y (V/m)~H(x, z, t) = 0.154 cos

[

12π × 1014t − 6.4π × 106(0.950z + 0.313x)]

(0.313z − 0.950x) (A/m)

13 (a) perpendicular polarization

(b) εr1 = 4

θi = 53.1

(c) ~Hi = E0

60π e−j4π(4x+3y)(−0.8y + 0.6x) (A/m)

(d) Γ⊥ = ej1.61

τ⊥ = 1.39ej0.8050

~Er = E0ej1.61e−j4π(4x−3y)z (V/m)

(e) ~Eair = E0

(

e−12πy + ej1.61ej12πz)

e−j16πxz (V/m)

14 (a) f = 2.39 × 108 Hz

θi = 36.9

(b) ~Ei = 24πe−jπ(3x+4z)y (V/m)

(c) ~Er = −24πe−jπ(3x−4z)y (V/m)~Hr = −0.2e−jπ(3x−4z)(0.6z + 0.8x) (A/m)

(d) z = π8 (1 − 2n), n integer

(e) ~Sav, air = 18.1 sin2(4z) x W/m2

15 (a) Γ⊥ = −0.431

(b) 0.464 rad

24



MAP-Tele – 2008/2009

16 (a) n1 = 1.5

(b) θi = 61.9

θt = 41.4

(c) ~Er = −3.6e−j4π(−8x+15y)z (V/m)~Et = 6.4πe−j 90.7π(0.75x+0.66y)z (V/m)

(d) 87.04%

17 (a) f = 2 GHz

(b) ani = 0.6x + 0.8y

anr = −0.6x + 0.8y

ant = 0.8x + 0.6y

(c) ~Er = 0 (V/m)~Et = 7.5e−j 80π

3(0.8x+0.6y)(−0.6x + 0.8y) (V/m)

18 —

25



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— Waveguides and Cavities: Solutions —

1 —

2 Upper plate: ρs = (−1)n γhAne−γz, ~Js = (−1)n jωε

h Ane−γz z

Lower plate: ρs = −γhAne−γz, ~Js = − jωε

h Ane−γz z

3 Upper plate: ~Js = (−1)n+1Bne−γzx

Lower plate: ~Js = Bne−γzx

4 (a) TM1 mode:~E(x, y, z, t) = A1 sin(πy/b) cos(ωt − βz)z + βb/π cos(πy/b) cos(ωt − βz − π/2)y~H(x, y, z, t) = ωεb/πA1 cos(πy/b) cos(ωt − βz + π/2)x

TE1 mode:~E(x, y, z, t) = ωµb/πB1 sin(πy/b) cos(ωt − βz + π/2)x~H(x, y, z, t) = B1 cos(πy/b) cos(ωt − βz)z + βb/π sin(πy/b) cos(ωt − βz + π/2)y

(b) —

5 (a) (fc)TE1= (fc)TM1

= 15 GHz

(fc)TE2= (fc)TM2

= 30 GHz

(b) TEM, TM1, TE1, TM2, TE2.

6 —

7 (a) TE1 and TM1: (fc)1 = 3GHz

TE2 and TM2: (fc)2 = 6GHz




(b) TEn and TMn:

n fc (GHz) vf (m/s) λ (m) β (rad/m) ZTE (Ω) ZTM (Ω)

1 3 3.03 × 108 0.015 131.8π 381.3 372.7

2 6 3.14 × 108 0.016 127.2π 395.2 359.6

3 9 3.36 × 108 0.017 119.0π 422.2 336.7

4 12 3.75 × 108 0.0187 106.7π 471.3 301.6

5 15 4.53 × 108 0.0227 88.19π 570.0 249.4

(c) vf = 3 × 108 m/s, β = 133.3π rad/m, λ = 0.015m, ZTEM = 377Ω

(d) TEn and TMn:

n α (Np/m)

1 14.9π

2 37.7π

3 58.5π

4 78.9π

5 99.1π

8 (a) TE2

(b) β = 111.8π (rad/m)

26



MAP-Tele – 2008/2009

(c) 107.3π (Ω)

(d) ~E = −6000π sin(100πy) sin(3π × 1010t − βz)x (V/m)~H = 25

√5 sin(100πy) sin(3π × 1010t − βz)y + 50 cos(100πy) cos(3π × 1010t − βz)z (A/m)

9 (a) TM1, fc = 7.5 GHz

(b) β = 392.4 rad/m

(c) vf = 1.92 × 108 m/s

(d) ~E = 40 [−j1.25 cos(100πy)y + sin(100πy)z] e−j392.4z (V/m)~H = j0.34 cos(100πy)e−j392.4zx (A/m)

(e) ~Sav = 8.5 cos2(100πy)z (W/m2)

10 —

11 TE10: (fc)10 = 3GHz, β = 615.75 rad/m, vf = 1.53 × 108 m/s, ZTE = 192.34 (Ω)

TM11 : (fc)11 = 8.078 GHz, β = 529.67 (rad/m), vf = 1.78 × 108 m/s, ZTM = 158.9 (Ω).

12 (a) (fc)10 = 15 GHz, (fc)20 = 30 GHz

(b) λ = 1.67 cm, λg = 3.02 cm

13 (a) TM21

(b) β = 241.4 (rad/m)

(c) Ey/Ex = (50/40) tan(40πx) cot(50πy)

14 (a) (fc)01 = 15 GHz, (fc)10 = 6 GHz, (fc)20 = 12 GHz

(b) Modes:

TE10,TE20,TE01,TE11,TM11, TE30,TE21,TM21,TE31,TM31,

TE40,TE41,TM41,TE02,TE50.

15 (a) TM13 or TE13

(b) (fc)13 = 28.57 GHz

(c) γ = jβ, β = 1.72 × 103 (rad/m)

(d) (ZTE)13 = 229.7 (Ω), (ZTM)13 = 154.6 (Ω)

16 (a) (fc)10 = 1.74GHz < f , vf = 3.33 × 108 (m/s) > c, vg = 2.702 × 108 (m/s)

(b) (fc)10 = 3.88GHz < f , vf = 12.3 × 108 (m/s) > c, vg = 7.32 × 107 (m/s)

17 E0 = 63.77 (V/m)

18 (a) —

(b) (fc)TE10= 5.06 GHz

19 —

20 —

21 —

22 TE101, TE011, TE102, TE012, TM110

23 (a) TE101; fTE101= 4.80 GHz

(b) TE101; fTE101= 3.04 GHz

27



MAP-Tele – 2008/2009

24 —

25

H0r = − 1

h2

(

γ∂H0

z

∂r− jωε

r

∂E0z

∂φ

)

H0φ = − 1

h2

(

γ

r

∂H0z

∂φ+ jωε

∂E0z

∂r

)

E0r = − 1

h2

(

γ∂E0

z

∂r+

jωµ

r

∂H0z

∂φ

)

E0φ = − 1

h2

(

γ

r

∂E0z

∂φ− jωµ

∂H0z

∂r

)

26 am = 0, m odd; am = (−1)m/2

2m[(m/2)!]2 , m even

27 —

28 —

29 (a) diameter = 2.2 cm

(b) TM01,TE11,TE21

30 (a) TM01, TE01, TM11, TE11 and TE21

(b) ~Sav = 0.0029E20 (J ′

0(hr))2 z (W/m2)

(c) from 8.79 GHz to 11.48 GHz

31 (a) TM01, TE11 and TE21

(b) for TM01:~Sav = βωε

2h2 E20 (J ′

0(hr))2 z (W/m2) 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 TM010, TE111, TM110, TM011, TE211, TM111, TE011

34 —

35 —

36 dmin = 6.1 mm

37 (a) ~Sav = βωεA2

h2

1

cos2(h1y)z

(b) 0

38 —

39 (a) Inside the dielectric (0 < y < b)~E = jωµ

h1

B sin(h1y)e−jβzx

~H = Be−jβz(

cos(h1y)z + jβh1

sin(h1y)y)

In the air (0 < y < b)~E = − jωµ0

ν B cos(h1b)e−ν(y−b)e−jβzx

~H = B cos(h1b)e−ν(y−b)e−jβz

(

z − jβν y

)

(b) ν = −µ0

µ h1 cot(h1b)

(c) fc = (n−1/2)2b

√µε−µ0ε0

, n = 1, 2, . . .

40 ρS = 0; ~JS = Be−jβzx

41 (a)

√

(

ωc

)2(n2

1 − n22) − h2

1 =(

n2

n1

)2

h1 tan(h1b)

28



MAP-Tele – 2008/2009

(b) fc = (n−1)c

2b√

n2

1−n2

2

, n = 1, 2, . . .

42 εr > 1 + sin2 θi

43 (a) N.A. =√

n21 − n2

2 ; θa = arcsin√

n21 − n2

2

(b) N.A. = 0.986 ; θa = 1.404 rad

44 (a) N.A. = 0.2425

(b) a < 2.05µm

29



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 = 1.0Ω/m, L = 1.67 × 10−7 H/m, G = 0, C = 1.72 × 10−10 F/m

4 (a) R = 0.788Ω/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, Z0 = 27.7 + j0.098Ω, vf = 2 × 108 m/s

5 R = 2Ω/m, L = 200 nH/m, G = 0.8 mS/m, C = 80 pF/m, λ = 1 m

6 (a) R = 0.8Ω/m, L = 38.2 nH/m, G = 0.5 mS/m, C = 23.9 pF/m

(b) l = 172.7 m

7 (a) Z0 = 70.75∠ − 1.37 = 70.73 − j1.69Ω

(b) γ = 2.12 × 10−4 + j8.89 × 10−3 m−1

(c) vf = 7.07 × 105 m/s

(d) 3.68 dB

8 β = 798 rad/m, vf = 3.54 × 107 m/s

9 Zi = 60 + j20Ω

10 λ/4

11 —

12 (a) Γ = 0.632e−j71.6

(b) SWR = 4.43

(c) z′max(1) = 2 cm

(d) z′min(1) = 0.75 cm

13 ZL = 83.24 − j51.27Ω

14 Γ = 0.307e132.5

, SWR = 1.886, Zi = 64.8 − j38.3Ω

15 —

16 —

17 (a) ΓL = 0.620e−j29.74

(b) Zi = 17.855e45.44

= 12.53 − j12.72Ω

(c) Vi = 1.4e−j34.0

, Ii = 78.4e11.4

(mA)

30



MAP-Tele – 2008/2009

18 (a) Zi = 60.25 + j38.79Ω

(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, vf = 0.5 × 108 m/s

(b) Z0 = 200Ω

I(x, t) = 0.3e2.5×10−3x cos(108t + 2x) − 0.06e−2.5×10−3x cos(108t − 2x)A

20 —

21 —

22 —

23 (a) ΓL = 0.352∠56

(b) SWR = 2.088

(c) Zi = 23.97 + j1.35Ω

24 Z0 = 40Ω, Z ′0 = 250Ω

25 —

26 (a) ΓL = 0.6∠ − 135

(b) ZL = 14.25 − j19Ω

(c) lmin = 0.35m, Rmin = 12.5Ω

27 (a) ΓL = 0.24∠75

(b) SWR = 1.65

(c) Z(l = 0.35λ) = 30 − j (Ω)

(d) y(l = 0.35λ) = 1.7/50 + j0.08/50 (S)

(e) 0.105λ

(f) 0.105λ

28 (a) Zin = −j154Ω

(b) 0.074λ

29 Zi = 95 − j70Ω

30 d1 = 0.104λ, l1 = 0.173λ ; d2 = 0.314λ, l2 = 0.327λ

31 —

32 —

33 —

34 —

35 —

36 —

37 (a) lA1= 0.375λ, lB1

= 0.25λ, lA2= 0.125λ, lB2

= 0.074λ

(b) lA1= 0.125λ, lB1

= 0, lA2= 0.375λ, lB2

= 0.324λ

31



MAP-Tele – 2008/2009

38 Z ′0 = 54.77Ω

39 —

40 (a) d1 = 0.053λ

(b) Z ′0 = 106.07

(c) SWR = 4.5

(d) SWR = 2.1

41 (a) Γg = 1/3, ΓL = −1/3, T = 5ns, V +1 = 20V, V −

1 = −6.67V, V +2 = −2.22V, V −

2 = 0.74V

(b) V (0.5, t) = 0, 0 ≤ t < T/2; V (0.5, t) = 20V, T/2 ≤ t < 3T/2

V (0.5, t) = 13.33V, 3T/2 ≤ t < 5T/2, V (0.5, t) = 11.11V, 5T/2 ≤ t < 7T/2

V (0.5, t) = 11.85V, 7T/2 ≤ t < 9T/2

42 —

43 V (200, t) = 0V, 0µs ≤ t < 1µs; V (200, t) = 10V, 1µs ≤ t < 2µs

V (200, t) = 0V, 2µs ≤ t < 3µs; V (200, t) = −10V, 3µs ≤ t < 4µs

V (200, t) = 0V, 4µs ≤ t < 5µs; V (200, t) = 10/3V, 5µs ≤ t < 6µs

V (200, t) = 0V, 6µs ≤ t < 7µs; V (200, t) = −10/3V, 7µs ≤ t < 8µs

44 (a) Vg = 10V

(b) l = 100m

(c) RL = 21.43Ω

45 —

Collection of problems from different sources:

• Problems from the course ”Electrotecnia Teorica”, LEEC-FEUP

• Exams from the course ”Electrotecnia Teorica”, LEEC-FEUP

• D. Cheng, “Field and Wave Electromagnetics”, Addison-Wesley

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problems on plane wave incidence, waveguides and...

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