EE-301 END-SEMESTER EXAMINATION

Instructions: There are NINE questions in this paper and the Maximum score is 100. This is a closed

book examination.

Useful Relations:

a) Voltage reflection coefficient at the load end:

.0

0

ZZ

ZZ

L

L

L

b) At any point on the line, the ratio of the phasor voltage to phasor current is given in the

impedance transformation. For a lossless line we have:

C

LZLC 0,

2

r

c) Maxwell's equations in differential and integral form:

t

DJH

t

BE

B

D

0

SS

S

enc

SdDt

SdJldH

SdBt

ldE

SdB

QSdD

0

d) Maxwell’s equations in phasor form:

DjJH

BjE

B

D

0

e) The magnetic flux is given by:

S

SdB

,

f) Auxiliary equations of applied fields inside media:

HHB

EED

rm

re

00

00

)1(

)1(

f) In a medium with finite conductance , relative permittivity r with the conduction

current density J

, we have:

0

,

jEJ rr

g) The current continuity equation relating the charge density and the current density J

:

Jt

h) For propagating waves in a media, the propagation constant and impedance is given by:

Z,

i) Energy carried by electric and magnetic fields in instantaneous and phasor form:

HBWDEW

HBWEDW

ME

ME

~~

2

1~,

~~

2

1~

2

1,

2

1

j) Poynting theorem in instantaneous form:

SdPSdHEJEW

tW

tV

ME

k) Poynting vector in instantaneous and phasor form:

HEPHEP~~~

,

l) Average power in phasor form:

PHEPav

~Re

2

1~~Re

2

1

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QUESTIONS

1) A coaxial cable consists of an inner conductor of radius a and an outer conductor of

radius b . The space between the conductors is filled with a dielectric of permittivity and

permeability . The length of the cable is L . The inner conductor has a surface charge

density of S .

a) Find the electric field in the region between the conductors. (3 Marks)

b) Find the capacitance of the structure using the definition VQC / (4 Marks)

c) Assuming that a current I flows through the inner conductor, find the magnetic field H

between the conductors. (3 Marks)

d) Find the inductance of the structure using the definition IL / (4 Marks)

e) A voltage source, v , is connected to a pure resistor R by the same coaxial cable as

shown below. Find the Poynting vector inside the dielectric and show that it leads to the

same instantaneous power in the resistor as methods used in circuit analysis. (6 Marks)

(Total: 20 Marks)

2) A coaxial cable with inner radius 5 mm and outer radius 6 mm and length 500 mm is filled

with a dielectric of relative permittivity 7.6r , and a voltage tV 377sin240 is applied

between the outer and inner conductors. The voltage is applied between the conductors by

a signal generator connected via perfectly conducting wires. a) Determine the displacement

current between the conductors and, b) compare it with the conduction current through the

wires. c) How and why are they related? d) What happens if the conducting wires are not

perfect and had a finite conductivity?

(8 Marks)

3) A transmission line of length L is connected in between an ideal dc voltage source (no

input impedance) with voltage 0V and a matched load 0ZZL . At 0t the voltage source

is turned on. Assume that the signal velocity along the line is v .

a) Sketch the voltage across the load LV as a function of time. (2 Marks)

b) Now assume that the load is not matched such that 0ZRZ LL . Explain qualitatively

what happens at the load end and at the generator end with time. (4 Marks)

c) Sketch the voltage across the load LV as a function of time. (6 Marks)

d) What happens to the voltage across the load LV after in steady state? (3 Marks)

(Total: 15 Marks)

4) The complex electric field of a uniform plane wave is given by

yjj ezejxE ˆ)1(ˆ210 4/2

(a) Find the polarization of the wave (linear, circular or elliptical) (2 Marks)

(b) Determine the sense of rotation (clockwise or anticlockwise) (2 Marks)

(c) Sketch the figure the electric field traces as a function of t (2 Marks)

(d) Determine the time average power flow density of the wave. (2 Marks)

(Total: 8 Marks)

5) A uniform plane wave with electric field in the plane of incidence, whose incident electric

field has an x component with an amplitude of mV /10 3 , is traveling in a free-space

medium and is normally incident upon a lossy flat earth (conductor) as shown in the figure

below.

Assume that the constitutive parameters of the earth are 0202 ,9 and

mS /10 1

2

with the frequency being 1 MHz .

a) Can we regard the earth as a good conductor? Why? (2 Marks)

b) Derive an expression for and calculate the intrinsic impedance of the earth. (3 Marks)

c) Calculate the total electric field magnitude on the air side of the interface (3 Marks)

d) Calculate the skin depth of the earth and determine the variation of the conduction

current density in the earth (4 Marks)

(Total: 12 Marks)

6) In a certain cross section of a rectangular waveguide the instantaneous components of

the electric field are given by

0,sincos,cossin

zxy E

b

y

a

xBE

b

y

a

xAE

Identify the mode of operation and calculate the cut-off frequency (5 Marks)

7) The time harmonic complex fields inside a source-free conducting pipe of rectangular

cross section (waveguide shown below) filled with free space are given by:

byaxea

xEyE

zj z

0,0,sinˆ

0

Where,

2

02

1

az

, 0E is a constant and 000 . For a section of waveguide

of length l along the z axis, determine

a) The corresponding complex magnetic field. (3 Marks)

b) The supplied complex power. (3 Marks)

c) The exiting complex power. (3 Marks)

d) The dissipated real power. (3 Marks)

e) Verify the conservation of energy equation in integral form is satisfied for this set of fields

inside this section of the waveguide. (4 Marks)

(Total: 16 Marks)

8) Consider a typical antenna set up with a transmitter and a receiver in its far field.

a) Draw the equivalent circuit of a transmitting antenna. Define various power metrics

involved. (2 Marks)

b) Assume that a maximum power of maxP is radiated along the main direction. Define the

directivity, gain and efficiency of the antenna. (3 Marks)

c) Derive a relationship between the three quantities defined above. (3 Marks)

d) Draw the equivalent circuit of the receiving section. How is the open circuit voltage

estimated? (2 Marks)

(Total: 10 Marks)

9) The radiation pattern function of a finite length dipole antenna of length L is given by:

sin

coscoscos)(

LLF

Sketch the radiation patterns for 2

3,

and 2 dipole antennas. (6 Marks)

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