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EE 541
University of Southern California Viterbi School of Engineering
Choma
Solutions, Homework #07
103
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 #07
Fall, 2011
Due: 11/09/2011
Choma
Solutions
−
Problem #37:
When terminated at its output port in a resistance,
R
, the passive network abstracted in
Figure (P37a) is to produce the voltage transfer function,
2
o
2
s
ss
1
V
1
aQ
a
H(s)
,
V2
a1s
s
11
a
a
1
aQ
a
where parameters
“a”
and
“Q”
are positive constants and the transfer function is written in
terms of a normalized impedance of
1 ohm
and a normalized frequency of
1 radian per second
.
In general, a normalized
one-ohm
impedance corresponds to an actual
R-ohm
impedance, while
1 radian per second
reflects an actual frequency of
o
radians -per-second
.
V
o
V
s
V
s
Passive
RLC
Network
R
R
R
Z(s
)
a
Z(s)
b
R
R
R
V
o
(a).
(b).
Figure (P37)
(a).
If the network is realized as the constant resistance tee structure offered in Figure (P37b),
give normalized mathematical expressions for the requisite impedances,
Z
a
(s)
and
Z
b
(s)
.
The given transfer function can be written in the form,
2
2
2
2
2
1+
aQ
a
1
,
sa
1
s
a
1
s
s
Qa
aQ
a
Q
a
1
aQ
a
(P37-1)
which implies that, with
R = 1
Ω
,

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*Sign up*EE 541
University of Southern California Viterbi School of Engineering
Choma
Solutions, Homework #07
104
Fall Semester, 2011
2
a
2
b
sa
1
s
Qa
1
Z (s)
.
Z(
s
)
ss
1+
aQ
a
(P37-2)
(b).
Determine the relationship between parameters
“Q”
and
“a,”
such that
Z
a
(s)
is realizable
as an interconnection of a minimal number of branch elements.
By continued fraction expansion,
2
a
22
2
1
s
1
Z (s)
.
s
1
s
a 1 Q
aQ
a
a Q
a
Q
1
s
s
(P37-3)
Observe that the coefficient of the
(s/a)-term
in this expansion is potentially negative, which would
preclude the realization of
Z
a
(s)
with positive resistances, capacitances, and inductances.
On the
other hand, if this term were to be constrained to zero, the
(s/a)-term
vanishes, thereby giving rise
to the possibility of realizing
Z
a
(s)
with a minimum number of branch elements. Accordingly, set
1
Q.
a1
(P37-4)
(c).
Given the relationship between
“Q”
and
“a”
found in the preceding part of this problem,
draw the normalized realizations of impedances
Z
a
(s)
and
Z
b
(s)
.
Express all normalized
element values exclusively in terms of
“Q
.
”
With
Q
constrained in accordance with the solution to the preceding part of this problem, the ex-
pansion for
Z
a
(s)
becomes
a
2
2
2
11
Z (s)
,
Q
1
s
s
a
Q
a
aa 1
Qs
1
s
s
(P37-5)
which suggests a normalized impedance realization consisting of an inductance of value
1/Q
placed
in shunt with the series combination of a capacitance,
Q/a
2
, and a resistance,
a(a+1)
.
Since
5
2
2
2
QQ

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