Assignment title: Information
Swinburne Higher Education Division: Exam Paper
HET316 Electromagnetic Waves, 2013
Page 2 of 12
1. A thin walled conducting cylinder of circular cross section and radius
a cm 1 runs parallel to a conducting ground plane. The centre of the cylinder
is a height h cm 5.1 above the ground plane. The ground plane is at a
potential of 0V and the cylinder is at a potential V.
The electric potential ((x,y)) in the region above the ground plane and
outside the conducting cylinder is given by:
0 2 2 2 byxbyxVyx )(ln)(ln),( 2
where ahb 22 and
0 ln 2 bahbah )( 2
V
V
(a). Use the result above to prove that, in the region between the cylinder and the
ground plane, the y component of the electric field is given by:
0 2 22 2
)()(
2
byx
by
byx
by
VE
y
(b). (i). In the region between the cylinder and the ground plane, where do you
expect the magnitude of the electric field to be a maximum? (A mathematical
proof is not required).
(ii). On the basis of your answer above, and given that the region between
the cylinder and the ground plane is filled with air, and that the breakdown
field for air is 6 V/m 103 , find the upper limit of the potential difference that
can be maintained between the cylinder and the ground plane.
Question 1 continues on the next page.
h
a
L
z
y x
Swinburne Higher Education Division: Exam Paper
HET316 Electromagnetic Waves, 2013
Page 3 of 12
(c). (i). What is the electric field in the region below the ground plane
(assuming that this region is a good conductor)? Do you expect to find any
free charge in the region below the ground plane? Explain.
(ii). At a point r, on the surface of a conducting body, the surface charge
density is given by S 0En rr )()( , where En is the component of the
electric field normal to the surface at the point r. Use this result to show that
the charge per unit length stored on the ground plane is given by
4 V00
QL
.
(Hint: you may assume that the field at the ground plane is given by the field
at y = 0; also the following integral may be of use:
xb 22
bdx
.)
(iii). Calculate the capacitance per unit length of the cylinder and ground
plane system.
(10 + 12 + 14 = 36 marks)
2.(a). In a conducting region at high frequencies the electric field decreases
approximately exponentially with distance from the surface of the region. For
a conductor filling the half-space x 0 the electric field is of the form:
E ),,,( z ˆ 0 tjx xeeEtzyx 0 ;
where E0 and are (possibly complex) constants and is the frequency of the
signal in rad/s.
(i). Use Faraday's law in its differential form to find an expression for the
magnetic B field in the medium.
(ii). Then use the Ampere-Maxwell law in differential form to find an
expression for the current density, J, in the conducting medium. (Assume a
non-magnetic medium with permittivity .)
(iii). For an Ohmic conductor EJ , where is the conductivity of the
medium. Use this result, together with you answers for parts (i) and (ii) of this
question, to find an expression for the constant in terms of the constants
0 and ,,
Question 2 continues on the next page
Swinburne Higher Education Division: Exam Paper
HET316 Electromagnetic Waves, 2013
Page 4 of 12
(b). In the context of wave propagation in general, explain briefly what you
understand by the following terms:
(i). Standing wave.
(ii). Linear medium.
(iii). Dispersive medium.
(iv). Phase velocity.
(v). Group velocity.
(vi). Evanescent wave.
(c). A wave travels in 1D in a medium with a dispersion relation of the form:
222
0
2
c
where is the frequency (in rad/s) is the propagation constant and 0 and c
are constants. Prove that for this medium the phase (
v p ) and group ( vg )
velocities satisfy the relation
2
cvv
gp .
(18 + 12 + 6 = 36 marks)
3.(a). A lossy transmission line has the following parameter values:
nF/m 10
H/m 25
/m 2
S/m 40.0
G R L C
The signal frequency is 1 MHz and the line is terminated with a 50 load.
Calculate the characteristic impedance of the line, the voltage reflection
coefficient at the load and the voltage standing wave ratio in the vicinity of the
load.
(b). A 75 lossless transmission line is terminated with an unknown load ZL. The
VSWR measured on the terminated line is 4 and the first and second minima
of the standing wave pattern are found at distances of 3 cm and 13 cm from
the load respectively.
(i). Find the load impedance, ZL.
(ii). Suggest a position (as a fraction of the wavelength ) for a lossless /4
transformer needed to match the load to the 75 line and find the required
characteristic impedance of the /4 transformer.
(16 + 20 = 36 marks)
Swinburne Higher Education Division: Exam Paper
HET316 Electromagnetic Waves, 2013
Page 5 of 12
4.(a). A monochromatic TEM plane wave travelling in a non-conducting, nonmagnetic medium of relative permittivity r 50 , is incident normally on the
plane surface of a similar medium of relative permittivity. r 30 . Estimate
the fraction of the incident power which is transmitted into the second
medium.
(b). A coaxial transmission line can support a TEM wave at all frequencies. In
addition it can support TE and TM waveguide modes with dispersion relations
of the form:
2222
c
c
where is the propagation constant and c 1 is the speed of EM
radiation in the medium filling the region between the inner and outer
conductors. For the lowest waveguide mode (TE11), the constant
ba
c
c
2
,
where a and b are the radii of the inner and outer conductors respectively.
(i). Explain briefly why, in principle, the existence of TE and TM modes
may limit the usefulness of the coaxial line at higher frequencies.
(ii). Estimate the lowest frequency at which this may be a problem for a
high power, air filled coaxial line for which a = 9.08 mm, b = 23.2 mm,
0 and 0 .
(iii). For radiation with frequency equal to 99% of the frequency you
calculated in part (ii) of this question, estimate the penetration distance for the
(evanescent) lowest waveguide mode, (i.e. the distance for the fields to fall to
1/e of their original value).
(iv). What would be the lowest cutoff frequency for an air-filled circular
waveguide with radius equal to that of the outer conductor of the coaxial line
described in part (ii) of this question.
(10 + 6 + 6 + 8 + 6 = 36 marks)
Swinburne Higher Education Division: Exam Paper
HET316 Electromagnetic Waves, 2013
Page 6 of 12
5.(a). Explain briefly what you understand by the term Radio Horizon and explain
why the distance to the radio horizon usually exceeds that to the geometrical
horizon.
(b). Two dish antennas, operating at a frequency of 600 MHz, are separated by 50
km. A large hill of height h m 200 is situated at the mid-point of the direct
line of sight between the antennas. Estimate the height of antenna towers
needed to avoid significant attenuation of the signal due to the hill. (You may
ignore the curvature of the Earth.)
(c). In order to detect small space debris and near Earth objects such as asteroids,
the 70 m diameter antenna at the Goldstone solar system radar has a 500kW
transmitter operating at 8.56 GHz. This system can detect objects out to a
distance of 10 m 105.1 from the Earth. Assuming a radar cross-section of
about 1 m2 for the distant object, estimate the power detected for a radar signal
returned from a target at the limiting distance.
(d). Explain briefly how the position of a GPS receiver can be determined by
measuring ranges to 3 satellites. Explain how these ranges are measured and
why the range to a 4th satellite is also required.
(6 + 12 + 12 + 6 = 36 marks)
_____________________________________________________________________
END OF EXAM QUESTIONS
Swinburne Higher Education Division: Exam Paper
HET316 Electromagnetic Waves, 2013
Page 7 of 12
List of major expressions and values of constants
1. Material Properties:
Conductivity of copper : 582 10 . 7 1 -1m
Relative permeability of copper: 1
Relative permittivity of copper: 1
Conductivity of aluminium: 1055.3 m1-17
Relative permeability of aluminium: 1
Relative permittivity of aluminium: 1
2, Constants:
Permeability of free space: 0 Hm 104 -17
Permittivity of free space: 0 -12 Fm 10 8.85418782 -1
Speed of light in free space c s m 92458 299 -1
Impedance of free space 376.73
0 0
0
Radius of Earth: 6370 km
Effective Radius of Earth 8470 km (Normal atmosphere)
3. Electric Fields
Point charge 2
0
ˆ
4 r
Q a r
E
in spherical polar coordinates
Line charge
r
a rL
E
ˆ
2
0
in cylindrical polar coordinates
4. Maxwell's Equations
t
t
D
JH
B
E
D B
0
C S
C S
S S
d
dt
d
d
d
dt
d
d
d
Qd
JlB AE
ABlE
AB
AE
000
0
.
0.
1
.
0
0
r
r
ED
HB
5. Potentials
t
V
E A 2
1
12
r r
dVV lE
AB
Swinburne Higher Education Division: Exam Paper
HET316 Electromagnetic Waves, 2013
Page 8 of 12
6. Transmission Lines
line lossless afor
)tan(
)tan(
Z(x)
; 1 line lossless afor
2
ohms
0
0
0
0
xjZZ
xjZZ
Z
LC
vfv
jZY
CjG
LjR
Z
L
L
p p
Reflection coefficients :
for voltage :
0 0
ZZ
ZZ
L L
v
for current
0
0
ZZ
ZZ
L
L
i
Transmission coefficients :
for voltage :
0
2
ZZ
Z
t
L
L
v
for current
0
2 0
ZZ
Z
t
L
i
Voltage Standing Wave Ratio
v v
VSWR
1 1
Standing wave minimum
1
4
min
x
7. Plane wave propagation:
Lossless media Conducting media
1
2 2
v p
2
if then )1( and
if
2
1
2
2 2
j j
j
j
j
j
E field reflection coefficient :
12
12
v
E field transmission coefficient :
12
2 2
Wave impedance :
EH
Swinburne Higher Education Division: Exam Paper
HET316 Electromagnetic Waves, 2013
Page 9 of 12
8. Conversion
mdB mnepers )/(69.8)(
9. Waveguides:
c
c
22
,
1
c
v p
vg pg cvv 2
2
Rs
Parallel plates:
ca
n
c
,
Rectangular :
2 2
b
n
a
m
c
c
(TE and TM modes)
2
2
1
21
c
s ab c
Rb
(TE10 mode)
Circular :
r
cs
mn
c (TE modes)
r
ct
mn
c (TM modes)
2
2 222
1
c
s c mn msm
Rr
(TEmn modes)
smn
n \ m 0 1 2
1 3.832 1.841 3.054
2 7.016 5.331 6.706
3 10.173 8.536 9.969
tmn
n \ m 0 1 2
1 2.405 3.832 5.136
2 5.520 7.016 8.417
3 8.654 10.173 11.620
Swinburne Higher Education Division: Exam Paper
HET316 Electromagnetic Waves, 2013
Page 10 of 12
10. Propagation:
Far field zone: r 2 D2
2
4 r
P
GP
tt
; t PGEIRP
G 4 A
2
P
Z
1
0
2
E
2
4
r
GPGP
r rt
A dB ( ) . log log ( ) ( ) 92 45 20 20 10 f d G dB G dB 10 t r
s Rh 2
Width of first Fresnel zone at midpoint = d
11. Radar:
43
2
det
)4( r
GG
PP rt
2 d sin
12. Shielding:
j
j
2
21
2
1 0
2
EE
21
1
1 0
2
HH
dB 69.8
d
A , dB
4
log20
2
1
10
R
, B e 20 1 log10 2d dB.
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0 1.0 1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.6
1.6
1.6
1.8
1.8
1.8
2.0
2.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.0
1.0
20
-20
30
-30
40
-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
± 180
-90 90
85
-85
80
-80
75
-75
70
-70
65
-65
60
-60
55
-55
50
-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.19
0.19
0.2
0.2
0.21
0.21
0.22
0.22
0.23
0.23
0.24 0.24
0.25
0.25
0.27 0.26 0.26
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49 0.49
0.0
0.0
ANGLE OF TRANSMISSION COEFFICIE
NT IN DEGREES
ANGLE OF REFLECTION COEFFICIENT I
N DEGREES
—> WAVELENGTHS TOWARD GENERATOR
—>
<— WAVELENGTHS TOWARD LOAD <
—
INDUCTIVE REACTANCE CO
MPONENT (+jX/Zo), OR CAPACITI
VE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE CO
MPONENT (-jX/Zo), OR INDUCTIVE S
USCEPTANCE (-jB/Yo)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR
2040100 10 45 2.53 1.11.21.41.61.82
SWR
∞ 1
3040 20 15 810 123456
dBS
∞ 1
1015 7 45 3 2 1 ATTEN. [dB]
1 1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 ∞ S.W. LOSS COEFF
RTN. LOSS [dB] 0 1 2 3 4 5 6 7 8 9 10 12 14 20 30 ∞
RFL. COEFF, P 0.40.50.60.70.80.91 0.3 0.2 0.1 0.05 0.01 0 0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15
RFL. LOSS [dB]
0 ∞
0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 ∞ S.W. PEAK (CONST. P)
RFL. COEFF, E or I 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. COEFF, P
CENTER
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
TRANSM. COEFF, E or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1.0 1.0 1.0
1.2
1.2
1.2
1.4
1.4
1.4
1.6
1.6
1.6
1.8
1.8
1.8
2.0
2.0
2.0
3.0
3.0
3.0
4.0
4.0
4.0
5.0
5.0
5.0
10
10
10
20
20
20
50
50
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1.0
1.0
1.0
1.0
20
-20
30
-30
40
-40
50
-50
60
-60
70
-70
80
-80
90
-90
100
-100
110
-110
120
-120
130
-130
140
-140
150
-150
160
-160
170
-170
± 180
-90 90
85
-85
80
-80
75
-75
70
-70
65
-65
60
-60
55
-55
50
-50
45
-45
40
-40
35
-35
30
-30
25
-25
20
-20
15
-15
10
-10
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.19
0.19
0.2
0.2
0.21
0.21
0.22
0.22
0.23
0.23
0.24 0.24
0.25
0.25
0.27 0.26 0.26
0.27
0.28
0.28
0.29
0.29
0.3
0.3
0.31
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.35
0.35
0.36
0.36
0.37
0.37
0.38
0.38
0.39
0.39
0.4
0.4
0.41
0.41
0.42
0.42
0.43
0.43
0.44
0.44
0.45
0.45
0.46
0.46
0.47
0.47
0.48
0.48
0.49 0.49
0.0
0.0
ANGLE OF TRANSMISSION COEFFICIE
NT IN DEGREES
ANGLE OF REFLECTION COEFFICIENT I
N DEGREES
—> WAVELENGTHS TOWARD GENERATOR
—>
<— WAVELENGTHS TOWARD LOAD <
—
INDUCTIVE REACTANCE CO
MPONENT (+jX/Zo), OR CAPACITI
VE SUSCEPTANCE (+jB/Yo)
CAPACITIVE REACTANCE CO
MPONENT (-jX/Zo), OR INDUCTIVE S
USCEPTANCE (-jB/Yo)
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
RADIALLY SCALED PARAMETERS
TOWARD LOAD —> <— TOWARD GENERATOR
2040100 10 45 2.53 1.11.21.41.61.82
SWR
∞ 1
3040 20 15 810 123456
dBS
∞ 1
1015 7 45 3 2 1 ATTEN. [dB]
1 1.1 1.2 1.3 1.4 1.6 1.8 2 3 4 5 10 20 ∞ S.W. LOSS COEFF
RTN. LOSS [dB] 0 1 2 3 4 5 6 7 8 9 10 12 14 20 30 ∞
RFL. COEFF, P 0.40.50.60.70.80.91 0.3 0.2 0.1 0.05 0.01 0 0.1 0.2 0.4 0.6 0.8 1 1.5 2 3 4 5 6 10 15
RFL. LOSS [dB]
0 ∞
0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.5 3 4 5 10 ∞ S.W. PEAK (CONST. P)
RFL. COEFF, E or I 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 0.99 0.95 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 TRANSM. COEFF, P
CENTER
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
TRANSM. COEFF, E or I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ORIGIN