U
niversity of
S
outhern
C
alifornia
USC Viterbi School of Engineering
Ming Hsieh Department of Electrical Engineering
EE 541:
Solutions, Homework #07
Fall, 2010
Due: 11/16/2010
Choma
Solutions
−
Problem #31:
A particular type of multi-loop feedback amplifier is designed to deliver a driving
point input impedance,
Z
in
(s)
, that is identical to the Thévenin equivalent resistance,
R
, of the
signal source,
V
s
.
Although the design achieves
Z
in
(s) = R
at low signal frequencies, the input
impedance degrades at high signal frequencies in accordance with the model shown in Figure
(P31a).
Note therein that
Z
in
(s)
matches the signal source resistance at low frequencies but de-
grades toward zero at high frequencies.
Compensate for this input impedance response by intro-
ducing an appropriate impedance,
Z(s)
, in series with the signal source and the input port, as
suggested in Figure (P31b). It is to be understood that the objective of
Z(s)
is to ensure that the
indicated impedance,
Z
t
(s)
, is identical to
R
for all signal frequencies.
The
Z(s)
adopted for this
compensation scheme should exploit interconnections of only passive, lumped elements.
Show
a schematic diagram of
Z(s)
, with all element values thereof expressed in terms of model resis-
tances
R
1
and
R
and capacitances
C
1
and
C
2
.
R
1
Z (s)
in
R
C
2
V
i
C
1
R
V
s
+
−
(a).
R
1
Z (s)
in
R
C
2
V
i
C
1
R
V
s
+
−
(b).
Z(s)
Z (s)
t
V
t
Figure (P31)
(a).
Design the network shown in Figure (P31b) if
R = 50
Ω
,
C
1
= 20 fF
,
C
2
= 3 fF
, and
R
1
=
75
.
The natural candidate for the required impedance,
Z(s)
, can be found in the Class Lecture Aids.
In
particular, the compensation takes on the form shown in Figure (P31.1).
In this diagram,
()
1
21
11
b
2
12
1
1
1
1
2
1
R
sC
sC
1
sRC
1
Z
.
sC
sC
1
sR C
sC
sC
s R C C
R
sC
sC
⎛⎞
+
⎜⎟
++
⎝⎠
==
=
(P31-1)
Since constant input resistance at the input port requires
Z
a
Z
b
= R
2
,