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ECE 450 Fall 2005 FINAL EXAM

December 12, 2005

This is a closed book/notes exam. You are allowed to use only:

a) Three sheets of notes (written on both sides); b) Calculator. c) The Smith Charts provided, in case you choose to use them as a

graphical aid for answering transmission line-related questions. Perfect score is 350. Show all your work for full credit. Type and sign your name below and indicate the section you are in. Also, type your name in the space indicated at the top of each page. NAME: ____SOLUTIONS_________ SIGNATURE:_____________________ SECTION: ____________________ Problem

1 Problem

2 Problem

3 Problem

4 Problem

5 Problem

6 TOTAL

Useful constants: 12 70 08.854 10 (F/m); 4 10 (H/m).

= =

3

PROBLEM 1 [50 points] NAME_________________ A time-harmonic, uniform electromagnetic plane wave is propagating in a non-magnetic

0( ) = medium (Medium 1). The electric field vector of the wave is given by

( )7 ( , ) 10 cos 2 10 3 (V/m)z xz t e t z = E a a) Calculate the attenuation constant, phase constant and wavelength in Medium 1. (15

points)

( ) ( )0

1

( , ) exp cosAttenuation constant: (Nepers/m).Phase constant: 3 (m ).

2 2Wavelength: m.3

xz t E z t z

=

=

=

= =

E a

b) Calculate the intrinsic impedance of the wave in Medium 1 and the time-average power density. (15 points)

( )

( )

( ) ( ) ( )

0 7 70

0

22

(2 10 )(4 10 )(1 3)

2.4 0.8 7.94767 exp 18.435 ( )

Time-average power density: 1 1 | ( ) | ( ) Re cos 5.96831exp 2 (W/m )2 2 | |z z z

jj jj

j jj j j

j j

zz z

= + = = + += + = +

= + =

= = = EP a E H a a

4

c) Calculate the time-average power dissipated inside the volume in Medium 1 bounded by the planes 0, 1 m, 0, 1 m, 0, ,px x y y z z d= = = = = = where pd is the penetration depth in the medium. (10 points)

( )( )2 1 1 m ( 0) ( ) 5.96831 1 exp 2 (W) 5.160587 (W)

dissipated z p z

dissipated

P z z d

P

= = = =

=

P a P a

d) At (2 / ) mz = , the wave encounters a planar interface between Medium 1 and air. Calculate the magnitude of the transmitted electric field in the air region. (10 points)

0

The incident electric field at 2 / (m) is (in phasor form)2 2( 2 / ) ( 2 / ) 10exp exp 3 1.353353exp( 6) (V/m).

The transmission coeffficient at the planar interface is:2

inc

z

E z E z j j

=

= = = = =

=+ ( )0

trans

240 1.96;7.94767exp 18.435 120

Hence, |E | | ( 2 / ) | 2.65364 (V/m)inc

j

E z

= +

= =

5

PROBLEM 2 [75 points] NAME_________________

a) The total, time-average power radiated in air by a Hertzian dipole driven by a cur-rent 10 0 (A)AI = is 50 W. The operating frequency is 100 MHz. Calculate the physical length of the Hertzian dipole. (15 points).

22 22

3 (m)

1 1 802 21 0.1067644 (m)80

tot rad A A

cf

P R I I

= =

= =

= =

b) If instead of the Hertzian dipole, a half-wave dipole is used, what should be the cur-rent magnitude 0| |I at the feed of the dipole so that the total, time-average radiated power remains 50 W? (10 points).

20 0

73 ( )1 | | | | 1.17 (A)2

rad

tot rad

R

P R I I

= =

6

c) You are asked to design an antenna system that radiates a circularly-polarized wave along specific directions in the far-field region. The system will consist of two Hertzian dipoles of the same length, both placed at the origin of the reference coor-dinate system, with the dipole in Part a) above oriented along the z axis. What should be the current phasor, BI , of the second dipole and its orientation in order for the system of the two dipoles to produce the circularly-polarized wave in the far-field region in the direction 45 , 90 = = ? (25 points).

0

0

Dipole A: ( 45 , 90 )4

With dipole B oriented along the axis, its far-field is given by

( 45 , 90 ) sin 454

Hence, the total electric field is:

j r

A A

j r

B B

ej Ir

x

ej Ir

= = =

= = =

E a

E a

E 02 ( 45 , 90 )

4 2

For the field to be circularly polarized, 2 .

j r

A B

B A

ej I Ir

I j I

= = = +

=

a a

d) Consider, next, a different antenna system, consisting once again of two Hertzian dipoles of the same length, arranged in the array configuration depicted in the figure

below. The operating frequency is 100 MHz, 0.75 md = and 2j

A BI I e

= .

Find the array factor for the array and sketch, i) the group pattern on the x-y plane, and ii) the resultant pattern on the x-y plane. (25 points)

3 (m); 0.75 (m) ; .4 2

cosArray factor: 2cos 2cos cos .2 4 4

c df

d

= = = = =

+ = +

x

z Air

dy

AIBI

7

Group Pattern: ( ) 2 cos cos4 4

On the plane the pattern looks as follows:

G

xy

= +

Since on the xy plane the z-directed Hertzian dipole has an isotropic radiation pattern, the resultant pattern is the same with the group pattern depicted above.

x

y

2

2

2

8

PROBLEM 3 [50 points] NAME_________________ A time-harmonic, uniform plane is incident obliquely onto an air-dielectric interface as shown in the figure below. The electric field of the incident wave is perpendicularly po-larized and its phasor form at the origin is 1.0 (V/m)y=E a . The operating frequency is 1011 Hz.

a) Calculate the apparent phase constants and the apparent wavelengths of the inci-

dent wave along x and z. (10 points)

1

Since the incident wave is propagating in a denser medium,we must check whether the angle of incidence is greater or equal

1to the critical angle. sin 41.81 .2.25

; hence, total internal refle

c

i c

= =

>

10 0

10 0

ction occurs.22.25 sin 45 500 2 (m ) 2 2 (mm)

22.25 cos 45 500 2 (m ) 2 2 (mm)

x xx

z zz

= = = =

= = = =

b) Calculate the components of the wave vector of the transmitted wave into the air

region. (10 points)

-1

2 2 2 2 2 10 0

Since the apparent phase velocities of the incident and transmitted waves

along the interface are the same, we have = =500 2 (m ).1000Also, it is, (m ).3 2

tz z

tx tz tx tz j

+ = = =

z

x Air

Dielectric ( = 2.250, = 0)i = 45o

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c) Calculate the phasor form for the total electric field at the interface. (8 points) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

18.4

2

0, 0, 0, 1 0, ;

alternatively, 0, 0, 0, .Since the angle of incidence is larger than the critical angle it is,

2cos 1.89736661cos sin

2.25

inc ref inc

trans inc

ji

i i

x z x z x z x z

x z x z x z

ej

= = = + = = + =

= = = = =

= =

E E E E

E E E

( ) ( )

35

18.435Hence, 0, 1.8973666 expjy zx z e j z= = E a

d) Calculate the maximum and minimum values of the magnitude of the total electric field in the dielectric. Also, calculate the distance between a maximum and its ad-jacent minimum. (12 points)

The magnitude of the reflection coefficient is one. Hence, the maximumand minimum values of the total electric field in the dielectric are, respectively,2 (V/m) and 0. The distance between a maximum and its adjacent minimum is one quarter of the

2apparent wavelength along ; hence, it is (mm).2

x

e) A dielectric slab waveguide is formed by introducing a second, planar air-dielectric interface at a distance d below the interface depicted in the figure for this problem. Calculate d such that only the TE10 mode propagates at the operat-ing frequency of 1011 Hz in the resulting slab waveguide. (10 points)

Since the cutoff frequency of the TE00 is 0, it is not possible to have a dielectric slab waveguide in which only the TE10 mode propagates at a finite frequency.

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PROBLEM 4 [60 points] NAME_________________ A. The characteristic impedance for a general lossy line is given by

0R j LZG j C

+=

+

which is in general a complex quantity. However, there is a special condition for which a particular combination of non-zero R, G, L and C parameters gives the usual

0 RealLZC

= =

a) Find the mathematical relationship between R, G, L and C necessary for this spe-

cial condition to be fulfilled and derive the corresponding real part and imaginary part of the propagation constant j = + . (Hint: this transmission line is called distortionless because the results for and do not depend of frequency). (20 points)

( )( ) ( )( )( )( )

( ) ( ) ( ) ( )

{ }

0

0

or / / distortionless condition

/ /

/ /

/ / since / /

Re or

R j L L R j L LZ RC j LC GL j LCG j C C G j C C

RC GL R L G C

j R j L G j C GL C j L RC L j C

LC G C j R L j

LC R L j LC G C j G C j R L j

C RRL Z

+ += = = + = +

+ +

= =

= + = + + = + +

= + +

= + = + + = +

= = =

{ }

0 G

Im

L G ZC

j LC

= =

= =

11

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B. Consider the solution for a single stub matching problem illustrated on the Smith chart below. The transmission line is lossless with characteristic impedance Z0 = 100. Make sure any work done on the chart is clearly indicated and report numerical an-swers and comments under each question on the next page.

yR

(a) zR

(b) y(dstub)

(a) R

(c)

(d)

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a) Determine the actual load impedance ZR and the load reflection coefficient R. (6 points)

R 0

R

2.5 from Smith chartZ 2.5 250 actual load impedance

0.43 from Smith chart

RzZ

== =

=

b) Determine the actual line admittance at the location of the stub, before the stub is

inserted. (6 points)

0

( ) 1.0 0.95 from Smith chart( ) ( ) / 0.01 0.0095 [S] actual admittance

STUB

STUB STUB

y d jY d y d Z j

= += = +

c) Determine the length of a short circuited stub necessary to fulfill this impedance

matching design, using the Smith chart. (12 points)

( ) 0.950.1295 from Smith chartstub

y stub jL

= =

d) Determine the line location where the stub is inserted in terms of wavelength. (6 points)

1.6stubd =

e) At which line locations could one insert a quarter wavelength transformer for an

alternative impedance matching design, instead of using a stub line? Determine the characteristic impedance of the quarter wavelength transformer that could be in-serted at the location closest to the load.(10 points)

A quarter wavelength transformer could be inserted at any location with real line impedance, either at locations of maximum or minimum of the standing wave pat-tern. Since ZR = 250 > Z0, there is a maximum of standing wave pattern at the load location and a transformer can be inserted there, with characteristic impedance

/ 4 0 100 250 25000 158.114RZ Z Z = = = =

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PROBLEM 5 [60 points] NAME_________________ A technician is working inside a long service tunnel with rectangular cross-section of di-mensions a = 2.5m and b = 1.2m. Because of the metallic armature embedded in the walls, the tunnel behaves approximately like an ideal rectangular wave guide filled with air. The technician carries a consumer radio that can receive the following frequency bands

AM broadcast band: 530 kHz 1.7 MHz FM broadcast band: 88.0 MHz 108.0 MHz NOAA weather radio: 162.4 165.55 MHz

a) Determine the bandwith for monomode propagation inside the wave guide struc-

ture formed by the tunnel. (15 points)

Cutoff frequencies for propagation modes are obtained from 2 2

10

20

01

2

( ) 60.0MHz fundamental mode( ) 120.0MHz 2nd mode( ) 125.0MHz 3rd mode

pc

c

c

c

v m nfa b

f TEf TEf TE

= +

==

=

b) Discuss briefly the behavior of each frequency ranges listed above, inside the wave guide structure formed by the tunnel. (20 points)

AM Band It is well below the cutoff frequency of the fundamental mode, TE10. Therefore, it can only be present as evanescent wave with a very strong attenua-tion factor, and cannot propagate inside the wave guide. FM band It falls within the range of the monomode bandwidth and can propa-gate as TE10 mode inside the wave guide. NOAA band it is higher than the cutoff frequency of several modes, and can propagate in multimode conditions inside the wave guide.

f 10( )cf TE 20( )cf TE

monomode bandwidthBW = 60 MHz

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c) How many modes can propagate in the wave guide structure at the maximum op-eration frequency of 165.55 MHz? (15 points)

In addition to TE10, TE20, and TE01, also the TE11 and TM11 modes, both with cut-off frequency of about 138.5 MHz, can propagate. The next modes have cutoff frequencies higher than the operation frequency specified and cannot propagate. For instance, TE21 and TM21 have cutoff fre-quency of about 173.0 MHz, and TE30 has cutoff frequency of about 179.9 MHz.

d) Determine the group velocity of the TE01 mode at the maximum operating fre-

quency of 165.55 MHz. (10 points)

( ) 2 2601 8 801 6

125 10( ) 1 3 10 1 1.966 10 m/s165.55 10

cg p

f TEv TE v

f

= = =

16

PROBLEM 6 [55 points] NAME_________________

a) You need to realize a reactive impedance Z = j 50. All you have is a length

of transmission line terminated by a short circuit. What additional informa-tion do you need and how would you go about realizing the wanted imped-ance? (8 points)

One needs to know the characteristic impedance of the line as well as the wavelength (or equivalently frequency and permittivity of the line insulator) since the input impedance of the short circuited line is given by

( )0 02tan tan 50Z j Z L j Z L j

= = =

From this relation one can obtain the length L of the line necessary to realize the wanted reactive impedance.

b) A lossless transmission line with characteristic impedance Z0 = 100 is ter-

minated by a load impedance ZR = 300. What percentage of the incident power is reflected by the load? (8 points)

The load reflection coefficient is

0

0

300 100 0.5300 100

RR

R

Z ZZ Z

= = =

+

The reflected time-average power is proportional to |R|2 = 0.25, therefore, 25% of the time-average power is reflected by the load.

c) What is the polarization of an electromagnetic plane wave, characterized by the electric field phasor E = 10 ay + j10 az? (8 points)

The polarization is circular, since the two components of the electric field phasor have same magnitude and /2 phase difference, since j = exp(j/2). The electric field is parallel to the {y,z} plane, therefore propagation is either in the positive or negative xdirection. For positive x propagation, z - y = + /2 and the polarization is left-handed, for negative x propagation it is right-handed.

d) A material medium has intrinsic impedance = 50 44.8. Classify the material. (7 points)

Since = 44.8 45 the material is a good conductor.

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e) Consider a planar interface between two dielectric media, with permittivity 1 = 250 in medium 1 and 2 = 40 in medium 2. An electromagnetic plane wave is incident obliquely from medium 1. Determine the Brewster angle for total transmission and the critical angle for total reflection. (7 points)

f) For the same system in e) above, consider the two possible cases for oblique incidence, with a plane wave propagating either from medium 1 or from me-dium 2. For each case, indicate for which wave polarization one may have to-tal transmission at the Brewster angle or total reflection at the critical angle. (7 points)

Case 1) The Brewster angle occurs only for parallel polarization and the critical angle oc-curs for both perpendicular and parallel polari-zation, since 1>2. Case 2) The Brewster angle occurs only for parallel polarization, but since 1>2 there can be no critical angle (total reflection) in this di-rection of propagation.

g) Common features but also important differences characterize electromagnetic

plane waves and the far-field approximation of waves radiated by a dipole an-tenna. For each item in the list below, mark PW if the feature is specific for a uniform plane wave, mark DW if the feature is specific for a far-field dipole wave, or mark BOTH if it is a common feature. (10 points)

1. Electric field, magnetic field and propagation vector are perpen-dicular to each other __ PW __ DW Both

2. Surfaces of constant phase are spheres __ PW DW __ Both 3. Field intensities are uniform on surfaces of constant phase PW __ DW __ Both 4. Field intensities decrease as the inverse of the distance __ PW DW __ Both

5. The impedance of the medium is the proportionality constant be-tween electric field and magnetic field __ PW __ DW Both

Medium 1 = 25 0

Medium 2 = 4 0

Medium 1 1 = 25 0

Medium 2 2 = 4 0

Case 1)

Case 2)

-1 12 2

1 1 2

-1 2

1

Brewster angle

tan sin 21.8

Critical angle

sin 23.578

= = +

=

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