of School Electrical and Computer Engineering, Cornell University
ECE 303: Electromagnetic Fields and Waves Fall 2007 Exam 3 November 15, 2007
INSTRUCTIONS:
Only work done on the blue exam booklets will be graded do not attach your own sheets to the exam booklets under any circumstances Every problem must be done in a separate blue booklet so you must have 3 separate blue booklets before starting the exam To get partial credit you must show all the relevant work Correct answers with wrong reasoning will not get points All questions do not carry equal points All questions do not have the same level of difficulty
DO NOT WRITE IN THIS SPACE
1
School of Electrical and Computer Engineering, Cornell University Problem 1 (40 points)
Consider the transmission line circuit shown below: Zs = 50 Vs(t ) Zo = 50 L
z=- Assume that l = 3 cm, v = 3 10 8 m/s, and L = 0.1 nH .
z=0
a) If the source voltage is a step function given by: Vs (t ) = 4u (t ) , find and plot the currents I + (z, t ) , I - (z, t ) , I (z, t ) on the transmission line at time t = 3l 2v . Use the attached sheet to plot your results. Indicate the current values. Partial credit will be given only if proper work is shown. For parts (b) and (c) assume that the source voltage is equal to 4 Volts and it was turned on a very long time ago in the past at time t = - . At time t = 0 , the source voltage is turned off (i.e. it becomes 0 Volts). b) Find the Thevenin equivalent circuit for the circuit on the LEFT of the dashed line, and indicate the values of the Thevenin impedance and the time-dependent Thevenin voltage for times 0 < t < 3l v . You need to make a sketch of the Thevenin voltage for times 0 < t < 3l v . c) Find and plot the voltages V + (z, t ) , V - (z, t ) , V (z, t ) on the transmission line at time t = 3l 2v . Use the attached sheet to plot your results. Indicate the voltage values. Partial credit will be given only if proper work is shown. For part (d) assume that the source impedance Z s is 150 instead of 50 . Also assume that the source voltage is equal to 4 Volts and it was turned on a very long time ago in the past at time t = - . At time t = 0 , the source voltage is turned off (i.e. it becomes 0 Volts). d) Find and plot the voltages V + (z, t ) , V - (z, t ) , V (z, t ) on the transmission line at time t = l 2v . Use the attached sheet to plot your results. Indicate the voltage values. Partial credit will be given only if proper work is shown.
2
School of Electrical and Computer Engineering, Cornell University Problem 2 (30 points)
Consider a Hertzian dipole antenna that is placed on the y-axis closed to an infinite perfect metal plane (that completely occupies the ve side of the y-axis), as shown below. z
y h
Assume that the current phasor of the dipole is given by I = I o e j 3 and the length of the dipole is d.
r r a) Find the expression for the far-field electric field vector E ff (r ) for the radiation emitted by the dipole.
b) Find the expression for the total radiation power emitted by the dipole in terms of the current phasor
I = I o e j 3 of the dipole.
c) Find the expression for the radiation pattern p( , ) for the dipole radiation. d) Sketch the radiation pattern p( , ) in the x-y plane assuming that the distance h equals / 2.
3
School of Electrical and Computer Engineering, Cornell University Problem 3 (30 points)
a) Consider a dielectric stack with thicknesses and refractive indices as shown below:
Ei
n1 = 3 z=0
n2 = 2
n1 = 3
n2 = 2 z
1 / 4
If the incident plane wave is: ^ y E i e - jkz
2 / 2
1 / 4
2 / 2
find the reflection coefficient at the z=0 interface. b) Consider a dielectric slab with thickness and refractive index as below. shown To the right of the slab is an infinite medium of refractive index equal to 2.
Ei
n1 = 3 z=0 slab
T Ei
n2 = 2 z
1 / 8
If the incident plane wave is: ^ y E i e - jkz find the reflection coefficient at the z=0 interface. c) In part (b) if the wave transmitted through the slab, on the right side of the slab, is written as: ^ y TE i e - jkz find the magnitude of the transmission coefficient T.
4
School of Electrical and Computer Engineering, Cornell University
Problem and Part Number: Problem and Part Number:
V+ (z, t )
V- (z, t )
V (z, t )
z = -l
t=
z=0
z = -l
t=
z=0
Problem and Part Number:
Problem and Part Number:
V+ (z, t )
V- (z, t )
V (z, t )
z = -l
t=
z=0
z = -l
t=
z=0
5
School of Electrical and Computer Engineering, Cornell University
Problem and Part Number: Problem and Part Number:
V+ (z, t )
V- (z, t )
V (z, t )
z = -l
t=
z=0
z = -l
t=
z=0
Problem and Part Number:
Problem and Part Number:
V+ (z, t )
V- (z, t )
V (z, t )
z = -l
t=
z=0
z = -l
t=
z=0
6
ECE 303: Electromagnetic Fields and Waves The Complete Smith Chart
0.11
0.12 0.38
0.13 0.37
0.14 0.36
0.1
50
9 0.0
45
0.4
0.39 100
90
0.9
0.15 0.35
80
0.8
1.2
0.7
1.4
0.0
8 0.4 2
0 12
0 .4
55
1
110
40
1.0
70
0.1 6 0.3 4
60
35
1.6
0. 1 7 0.3 3
65 0.5
6
3 0.4 0 13
0.2
1.8
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0.0 5
(+ jX /Z
0.4 5
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0. 0
) /Yo (+jB CE AN T EP SC SU
0.6 60
30
0.1 0.3
2.0
50
8
2
1 0. 9
0. 4
25
4
31 0.
70
14 0
0.4
RE AC TA 75 NC EC OM PO N EN T
4
0.0
6
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0.0 --> WAVELE 0.49 NGTH S TOW ARD 0.0 D <-- 0.49 GEN RD LOA A ERA 0.48 S TOW 7 180 H TO 0.4 170 70 NGT R-- -1 ELE V 0.47 > WA 160 <-- 160 -90 90 -
0.4
15
80
IND UCT IVE
1.0
85
1.2
1.4
1.6
1.8
2.0
3.0
4.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
5.0
10
20
50
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
o) jB/Y E (NC TA EP SC 44 -75 0. SU 40 -1 VE 06 TI 0. C DU IN R - 70 O ), Zo X/
0.48
0.6
-85
8 0.
5.0
1.0
0.2
4 0.0 0 - 15 - 8 0
6
0.4
0.3
5
5
0.0
0.4
1.8
1.6
- 60
0. 7
1.4
0 .8
1.2
- 55
0.9
1.0
0
-5
5 -4
RADIALLY SCALED PARAMETERS
100 40 40 30 0 1 1 0.9
20
10 20 15 2 0.7 0.6 0.8 3 0.5 0.7
5
4 10 4 0.4 0.6 5 0.3
3
2.5 8
2 6 8 9 10 0.1 0.4
1.8 5
1.6 4 12 14
1.4 3 2
1.2 1.1 1 1 30 0 0 0 0 1 1 1
15
TOWARD LOAD --> 10 7 5 1.1 0.1 1.1 0.99 1.2 0.95 0.2 1.3 1.2 0.4 1.4 0.9
4 1.3 1.4 0.6 0.8 1 1.6
3 1.8 1.5 2 2
<-- TOWARD GENERATOR 2 1 3 3 2.5 0.6 0.5 0.4 4 4 3 0.3 5 5 6 4 0.2 5 0.1 10 10 15 10 0 20
1 0.8 0.9
6
7 0.2
20 0.01 0.1
0.05 0.2
1.5
1.6 1.7 1.8 1.9 2 0.8 0.7
0.0 ORIGIN
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
CENTER 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
TR
A
N
SM
0.5
0.3
TR S. RF S. A A W. L. W. TT N P L L EN SM EA O O . . C K S S [ SS C [d B .C O (C dB O ] O EF O ] EF EF F, F, NS F P T. E or P) I
2 0.3
0.6
8 0.1 0 -5 -25
0.5
- 65
3 0.3
2.0
7 0.1
- 30
- 60
4 0.3
6 0.1
-35
-70
0.35
0.15
0.36
0.14 -80
-4 0
0.37
0.13
0.4
0.2
-90
0.12
0.38
0.11 -100
0.39
-12 0 CAP AC ITI VE R EA CT AN CE
0.1
0.4
-110
0.4 1 0.0 9
CO M PO N EN T
4 0.
- 10
10
0.1
0.4
50
20
- 15
4.0
- 20 3.0
1.0
0 .8
0. 8
0.6
0.6
0.4
0.2
0.0 8
0.4 2
(-j
-1
0.0 7 30
0.4
0. 4
0.6
3.0
40
0.2 0.3
20
1 0.2
3
0.3
9 0.2 30
0.8
1.0
4 .0
15
0.22 0.28
5.0
0. 2
10
10
20
0.25 0.26 0.24 0.27 0.23 0.25 0.24 0.26 0.23 0.27 REFLECTION COEFFICIENT IN DE GREE LE OF S ANG ISSION COEFFICIENT IN TRANSM DEGR LE OF EES ANG
0.1
20
50
- 20
0.22
0.28
0.2
1 - 30
0.2 0.3
0.2 9
-4 0
0. 19 0. 31
R BS B ] , P r I SW d S [d EFF , E o S O C O FF . L L. O E RF L. C RF RT N