EE 541
University of Southern California Viterbi School of Engineering
Choma
Solutions, Homework #08
119
Fall Semester, 2011
U
niversity of
S
outhern
C
alifornia
USC Viterbi School of Engineering
Ming Hsieh Department of Electrical Engineering
EE 541:
Solutions, Homework #08
Fall, 2011
Due: 11/22/2011
Choma
Solutions
−
Problem #42:
The distributed line in Figure (P42) is lossless and is characterized by a characte
ristic impedance of
Z
o
.
It operates at a frequency for which its physical length is
N
wavelengths.
Z
o
Z
in
Figure (P42)
(a).
In terms of
Z
o
and the sine and cosine of appropriate functions of
N
, derive an expression
for the indicated impedance,
Z
in
.
The schematic diagram of the network at hand is redrawn as Figure (P42.1) to highlight the I/O
port currents,
I
1
and
I
2
, as well as the I/O port voltages,
V
1
and
V
2
.
The interrelationships among
these electrical port variables are arguably best modeled through use of the yparameters for the
distributed network.
In particular,
Z
o
Z
in
I
1
I
2
I
2
I
V
1
V
2
V
Figure (P42.1)
ir
11
ri
22
YY
IV
V
,
V
(P421)
where use has been made of the fact that the port voltages,
V
1
and
V
2
, in Figure (P42.1) are identi
cally equal to the indicated voltage,
V
.
In (P421), a lossless line gives
i
oo
Y,
Zj
2
π
Nj
Z
2
π
N
tanh
tan
(P422)
and
r
Y.
2
π
Z
2
π
N
sinh
sin
(P423)
Since current
I
is the sum of the port currents,
I
1
and
I
2
,
12
o
21
1
III
2
Y
Y
V
V
.
jZ
2
π
N2
π
N
tan
sin
(P424)
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View Full DocumentEE 541
University of Southern California Viterbi School of Engineering
Choma
Solutions, Homework #08
120
Fall Semester, 2011
In view of the fact that
Z
in
is simply
V/I
,
o
o
in
Z
Z
j2
π
N
j
V
2
2
Z
.
11
I2
π
N1
2
π
N2
π
N
sin
cos
tan
sin
(P425)
(b).
Evaluate
Z
in
for line lengths of
¼
and
½
wavelengths.
For
N = ¼
,
Z
in
= –jZ
o
/2
, while for
N = ½
,
Z
in
= 0
.
Solutions
−
Problem #43:
The quarter wavelength transmission line in the schematic diagram of Figure (P43)
has a real characteristic impedance of
Z
o
.
The circuit is to be designed as a resonant structure
having a radial tuned frequency of
o
.
R
l
C
l
C
s
Z
o
¼
V
o
V
i
V
s
R
s
Z(
j )
in
V
x
x
Figure (P43)
(a).
In terms of
o
, load resistance
R
l
, signal source resistance
R
s
, and load capacitance
C
l
, give
an expression for capacitance
C
s
, such that the indicated input impedance,
Z
in
(j
ω
)
, matches
the source resistance at frequency
o
.
From the Class Lecture Aids, the impedance,
Z
x
(j
ω
)
, seen looking into the input port of a quarter
wavelength line terminated in an impedance,
Z
l
(j
ω
)
, is
22
2
2
oo
o
x
ll
o l
l
ZZ
Z
j
ω
)1
j
ω
RC
j
ω
ZC.
j
ω
)R
R
(P431)
It follows that the given network can be supplanted by the lumped equivalent model offered in Fig
ure (P43.1).
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 Choma
 Electrical Engineering, Frequency, Trigraph, Input impedance, Southern California Viterbi, California Viterbi School

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