07-Élec-a7 (version anglaise) -novembre 2011

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07-ÉLEC-A7 ELECTROMAGNETICS – PROPAGATION of 4

ORDRE DES INGÉNIEURS DU QUÉBEC

NOVEMBER 2011 SESSION

Open-book examinationCalculators : only authorized modelsDuration : 3 hours

07-ELEC-A7

ELECTROMAGNETICS - PROPAGATION

Important notice

Hand-in the questionnaire with your exam booklets.Indicate clearly all units.




Question 1 (25 points)

Consider a transmission line having a characteristic impedance of Zo=50 and an unknownpropagation constant ( = + j). A section of this line is terminated by a load having a

normalized impedance of 0.5- j1, as shown in Figure 1.1. The frequency of operation is 1 GHz.

Figure 1.1

a) Place the load impedance on the Smith chart and label it as ‘Point A’. (2 points)b) Give the circuit representation of ZL and compute the values of its elements. (3 points)

For a length L of this line ( L ), the measured normalized input impedance is inin

o

Z z

Z = 2.

c) Placein

z on the Smith and label it as ‘Point B’. (2 points)

d) Using the Smith chart, find the electric size of the line (measured in degrees and ).Illustrate clearly all the steps followed. (5 points)

e) Is the line lossy? If yes, compute the total attenuation in dB for the length L and theattenuation constant in dB/ . (6 points)

f) Match Zin to Zo using a single section of lossless transmission line only. Explain the

matching technique used and give the electric length and characteristic impedance of thetransmission line used. (7 points)


A plane wave traveling in an infinite ideal dielectric medium (o, , = 0) has a magnetic fieldgiven by the following phasor:

/4ˆ ˆ( ) ( ) A/m

j z

o H z H x jy e

a) Find the analytical expression of the magnetic field ( ) E z in phasor notation. (4 points)

b) Find the explicit expressions for ),( t z E

and ),( t z H

. (4 points)c) What is the polarization of this wave? (4 points)

For the remainder of this question, we suppose that H o = 10 A/m, = 40x 108 rad/s and =60 rad/m.

Zo,

L

Zin

ZL




Find:d) The phase velocity of and the wavelength of the wave. (3 points)e) The relative permittivity of the dielectric medium. (3 points)

f) The average power density of the wave: *1Re ( ) ( )

2

E z H z (3 points)

This plane wave is incident normally on a lossless dielectric slab of thickness /4 at thefrequency of operation. The slab is backed by air.g) Compute dielectric constant of slab. (4 points)h) Compute the reflected power density. (5 points)


Consider the parallel plate waveguide of Figure 3.1 where the two conducting planes areseparated by a distance a.

Figure 3.1The waveguide is air-filled and operates in a mode to be determined but having the followingfield component:

2020 cos(2 )

j z

z H j x e A m

a) Is this mode TEM, TE or TM? Justify clearly your answer. (2 points)b) Find the time domain expression of

z H , namely ( , , , ) z z H H x y z t . (3 points)

c) Find the value of a in meters for the two cases where the mode index is m = 1 or m = 2. (5points)

d) Find the operating frequency and cut-off frequency of the mode for m = 1. (5 points)

x

air a

zy





Consider a dipole antenna of length l located at the origin as shown in Figure 4.1.

Figure 4.1.

a) Draw the shape of the radiation pattern for this antenna in the elevation and azimuthplanes for l . (4 points)b) What field components does this antenna ( l ) radiate in the far field and what are

their expressions? (4 points)

Suppose that this antenna has a directivity of 2dBi and a gain of 2dBi.c) What is the antenna efficiency? (5 points)d) What is the total radiated power by the antenna in Watts and in dBm if the power

delivered to its input is 1 Watt? (5 points)

This antenna is used to receive an incoming plane wave whose electric field is given by:

ˆˆ

2ˆˆ( )

k j z yi

o E E z y e

V/m.e) What is the orientation of the dipole that maximizes the received power? (8 points)f) What fraction of the maximum power will be received when the antenna is kept vertical?

(4 points)

r

x

z

yl

07-Élec-a7 (version anglaise) -novembre 2011

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