EE 541
University of Southern California Viterbi School of Engineering
Choma
Solutions, Homework #02
17
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 #02
Fall, 2011
Due: 09/08/2011
Choma
Solutions
−
Problem #06:
Consider the linear, bilateral, and symmetric two-port network abstracted in Figure (P6a).
Observe that the load
(R
l
)
terminating the output port of the network is purely resistive, as is the
intrinsic impedance
(R
s
)
associated with the applied signal source.
Linear, Bilateral,
Symmetric
Two-Port Network
I
2
V
2
V
1
R
l
R
s
V
s
I
1
Z
in
Figure (P6a)
(a).
In terms of the open circuit impedance parameters,
z
ij
, of the subject network, show that
maximum power transfer between the applied signal source and the network input port re-
quires
2
11
s
11
l
12
zR
z
.
If a general, linear two-port network is modeled by its open circuit impedance parameters,
z
ij
, the
terminal volt-ampere characteristics of the network abide by
11
1
1
1
2
2
Vz
I
z
I
,
(P6-1)
and
22
1
1
2
2
2
I
z
I
.
(P6-2)
Since the terminating load resistance,
R
l
, in the subject linear network subjugates the output port
current,
I
2
, and the output port voltage,
V
2
, to the Ohm’s constraint,
V
2
=
−
R
l
I
2
, (P6-2) delivers
1
1
2
2
2
l
2
I
z
I
R
I
,
(P6-3)
which, in turn, produces
21 1
2
22
l
zI
I.
(P6-4)
Note then that
z
21
is a measure of the degree to which input port current,
I
1
is transferred to the out-
put port as a current,
I
2
, since the input -to- output (I/O) current ratio
I
2
/I
1
, is clearly proportional to
the transimpedance parameter,
z
21
.
If we now insert (P6-4) into (P6-1), we arrive at a driving point
input impedance,
Z
in
, of
2
2
1
in
11
12
2
l
z
Z
z.
Iz
R
(P6-5)
But in a bilateral network,
z
12
= z
21
, while a symmetric two-port exhibits
z
22
= z
11
.
Thus, (P6-5) for

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University of Southern California Viterbi School of Engineering
Choma
Solutions, Homework #02
18
Fall Semester, 2011
a symmetric, bilateral, linear two-port network is
2
12
in
11
11
l
z
Z
z.
zR
(P6-6)
Now, maximum signal power transfer occurs at the network input port when the driving point input
impedance of the subject network is a conjugate match to the source impedance.
In this case, the
source impedance is a purely real resistance,
R
s
, which means that
Z
in
= R
s
is required to effect
maximum power transfer between the applied signal source and the network input port.
Setting
Z
in
= R
s
in (P6-6), we therefore conclude that the applicable design requirement is
2
11
s
11
l
12
z
.
(P6-7)
(b).
What network condition must be satisfied if the indicated input impedance,
Z
in
is designed
to be
KR
l
, where
K
is a positive, frequency invariant constant?

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