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6.013/ESD.013J Electromagnetics and Applications, Fall 2005
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Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.013 Electromagnetics and Applications
Problem Set #7
Fall 11/2/05
_____________________________________________________________________________________
Suggested Reading Assignment: Sections 5.2, 10.6.4
Issued: 10/25/05
Due:
Problem 7.1
An unusual type of distributed system is formed by series capacitors and shunt inductors.
(a) What are the governing partial differential equations relating the voltage and current?
Hint:
Review Lecture 10, pp. 2-3 (Section I.C.)
(b) What is the dispersion relation between ω and k
for signals of the form e j
( t kz
?
(c) What are the group ( d
ω dk
) and phase velocities ( ω k
) of the waves? Why are such systems called “backward wave”?
(d) A voltage
V
0 cos ω t
is applied at z = -l
with the voltage and current distributions along the line? z = 0
end short circuited. What are the
(e) What are the resonant frequencies of the system?
Problem 8.5 in
Electromagnetic Field Theory: A Problem Solving Approach,
by Markus Zahn, 1987. Used with permission.
1

Problem 7.2
For the transmission line shown, the length of the line is ¼ wavelength frequency ω of the voltage source.
( λ 4 ) at the driving
25(1 + j
)
(a) Find the values of lumped reactive admittance Y = jB and non-zero source resistance
R s that maximizes the power delivered by the source.
(Hint:
Do not use the Smith chart.)
(b) If the lumped reactive admittance Y=jB is made from a short circuited transmission line of length l
and characteristic impedance
Z i.e., l
= a
λ , what is a
?
0
= 50 Ω , what is l
in terms of wavelength λ ,
(c) What is the time-average power dissipated in the load?
(d) The driving frequency of the voltage source is now doubled. What is the transmission line length in terms of wavelengths λ at the frequency 2 ω ? Repeat (a) to (c).
Adapted from Problem 8.19 in
Electromagnetic Field Theory: A Problem Solving Approach
, by Markus Zahn, 1987.
Used with permission.
Problem 7.3
(a) Find the time-average power delivered by the source for the transmission line system shown below when the switch is open or closed.
(Hint:
Do not use the Smith chart.)
25
100
(b) For each switch position, what is the time average power dissipated in the load resistor
R
L
?
Adapted from Problem 8.20 in
Electromagnetic Field Theory: A Problem Solving Approach
, by Markus Zahn, 1987.
Used with permission.
2

Problem 7.4
A 100-ohm TEM transmission line operating at frequency f is terminated with a load consisting of a
100-ohm resistor in series with an inductor having a reactance of 100j, as illustrated. Additional details and values are shown in the figures.
�
L 100 100j ohms a) In terms of the complex reflection coefficient reflected?
�
L of the load, what fraction A of the power incident upon the load is b) For this load what is the numerical value of the complex reflection coefficient �
L
= a + jb? c) At what fraction of a wavelength q = D/ Γ (and in terms of � , see figure), is the distance D from the load of the first point where Z(z) is purely real?*
Z o
= 100 � q wavelengths 0
� degrees jX n
= j
Load
R n
= 1
R n
= K d) To match this load a quarter-wave transformer is inserted at the first point where Z(z) is purely real. In terms of K, what should be the characteristic impedance Z wave transformer (see figure)?
T of the quartere) What is K? jX n
= 0 f) Find another set of values for q, K and Z
T that allow this load to be matched with a quarter-wave transformer with Z real.
T
*Hint for part (c): Use Smith Chart given below .
Z o
= 100 �
Smith chart jX n
= -j toward generator
Z
0
= 100 �
�
L
Z
T
Γ/ 4 q wavelengths
Load
100 + 100j ohms
Image by MIT OpenCourseWare.
3

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G
N
E
L
E
V
A
W
H
T
T
S
O
W
A
R
D
E
G
0.49
N
E
R
A
T
O
R
—
>
0.47
0.48
170
160
0.1
0.46
85
0.1
0.4
0.09
0.41
110
55
0.08
0.06
0.44
0.07
0.43
130
0.2
0.05
150
0.45
75
0.3
80
D
U
C
T
IV
E
R
E
A
C
T
A
N
C
E
140
70
C
O
M
PO
N
EN
T
0.4
(+ j
X
/Z o)
, O
R
IN
65
0.42
0.5
CA
PA
CI
TI
120
0.6
60
VE
S U
S C
E P
0.7
T A
N C
E ( + j B /
Y o )
0.8
0.11
0.39
100
50
0.12
0.38
90
0.13
0.37
45
0.4
0.2
0.2
0.4
0.6
0.6
0.8
0.8
1.2
0.14
0.36
80
40
1.0
1.0
1.4
0.15
0.35
70
35
1.6
0.16
0.34
60
0.17
0.33
1.8
30
2.0
0.32
50
0.18
25 0.31
0.19
40
20
3.0
0.3
0.2
4.0
15
30
0.29
0.21
0.28
0.22
5.0
20
10
A
N
L
G
10
T
F
O
E
20
A
R
N
M
A
G
N
E
L
F
O
E
R
0.27
0.26
0.23
0.24
50
RESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)
0.2
0.49
-170
0.1
0.48
0.47
H
G
N
E
V
A
W
—
<
-160
0.46
0.04
Y
0.45
-150
0.2
-80
0.05
E
C
N
A
T
P
E
SC
0.3
SU
E
V
T
C
U
D
-75
-140
IN
O
R
0.4 o
Z
/ X j
-70
0.06
0.44
-130
0.07
0.43
N
T
N
E
-65
0.5
P
O
M
C
O
C
E
N
0.08
-120
0.42
A
C
T
0.6
A
-60
R
E
0.09
0.41
E
V I
T I
C
0.7
A P
A
C
0.1
-110
0.4
-55
0.11
0.8
-100
-50
0.9
0.12
-90
-45
0.2
0.4
0.39
0.38
0.4
0.37
0.13
0.6
0.6
0.8
0.8
1.2
-40
-80
0.14
0.36
1.0
1.0
0.15
1.4
-35
-70
0.35
1.6
0.16
1.8
0.34
-60
-30
0.17
0.33
2.0
-25
-50
0.18
0.32
-20
-40
3.0
0.19
0.31
0.2
50
20
0.24 0.26
5.0
4.0
-15
-30
0.21
0.29
10
S
-10
-20
0.22
0.28
0.23
0.27
0.3
∞
100 40
∞
40 30
0
20
1
10
20
2
1 0.9 0.8 0.7 0.6
1 0.9 0.8
0.0
ORIGIN
0.1 0.2
15
3
0.5
0.7
5 4 3
4
0.4
0.6
5
0.3
10
6
0.5
2.5
7
8
8
0.2
0.4
0.3 0.4 0.5 0.6
2 1.8
1.6
6
9 10
5 4
12 14
1.4
3
0.1
0.3
0.05
0.2
RADIALLY SCALED PARAMETERS
2
1.2 1.1 1
20
0.01
0.1
1 1 1
30
∞
0
0 0
0 1
15
TOWARD LOAD —>
10
1.1
0.99
0.1
1.2
7
1.1
0.2
0.95
1.3
5 4
0.8
3
1.2 1.3 1.4
0.4 0.6
0.8
1
1.6 1.8 2
1.5
1.4
0.9
1.5
0.7
2
<— TOWARD GENERATOR
2 1
3
3
4
4
5
5 6
10 20
∞
10 15
∞
1.6 1.7 1.8 1.9 2
0.6
2.5
0.5
3 4 5 10
∞
0.4 0.3 0.2 0.1 0
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
ATTEN. [dB]
S.W. LOSS COEFF
RFL. LOSS [dB]
S.W. PEAK (CONST. P)
TRANSM. COEFF, P
TRANSM. COEFF, E or I