EE 541
University of Southern California Viterbi School of Engineering
J. Choma.
Midterm Examination Solutions
1
Fall Semester, 2010
U
niversity of
S
outhern
C
alifornia
USC Viterbi School of Engineering
Ming Hsieh Department of Electrical Engineering
EE 541:
MIDTERM EXAMINATION
21 October 2010
(SOLUTIONS)
12:30 to 1:55
Problem #1:
Solution
—
—
—
—
—
—
—
—
(25%)
The passive piarchitecture filter shown in Figure (E1) is formed of an ideal induc
tance,
L
, two identical capacitances,
C
, and two resistances,
R
1
and
R
2
.
The filter is to be de
signed so that its transimpedance function,
Z
io
(s) = V
o
/I
i
, provides a Bessel frequency response
featuring a steady state (or group) delay at zero frequency of
T
do
.
L
C
CR
2
R
1
I
i
V
o
Figure (E1)
(a).
How is the zero frequency group delay related to resistances
R
1
and
R
2
and capacitance
C
?
The transimpedance,
Z
io
(s) = V
o
/I
i
, of the subject filter evaluates as the function
12
oi
io
i
23
io
io
RR
1s
RC 1s
RC
VR
o
2
Z
(s)
,
I
L
sL
2
s
L
Cs
L
R+R
⎛⎞
⎜⎟
++
⎝⎠
==
=
+
+
(E11)
where the zero frequency value of the forward transimpedance is
io
io
1
2
R
Z(
0
) RR.
=
±
(E12)
which is obvious by inspection of the given schematic diagram.
Clearly, the filter at hand is a third
order architecture.
For a third order Bessel filter delivering a zero frequency group delay of
T
do
,
the applicable characteristic polynomial, say
D(s)
, is
()
2
do
do
do
21
3
D
(
s
) 1 Ts
Ts
Ts .
51
5
=+
+
+
(E13)
Upon equating like coefficients of complex frequency
s
in
D(s)
and the denominator polynomial on
the far right hand side of (E11), we deduce the constraints,
do
io
L
T2
R
C
,
R +R
(E14)
2
do
5
TL
C
2
=
,
.
(E15)
and
3
2
io
do
T1
5
L
R
C
=
(E16)
If (E16) is divided by (E15), the zero frequency group delay,
T
do
, is found to satisfy
do
io
1
2
T6
R
C
6
R
R
C
.
(E17)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentEE 541
University of Southern California Viterbi School of Engineering
J. Choma.
Midterm Examination Solutions
2
Fall Semester, 2010
(b).
In terms of resistances
R
1
and
R
2
and capacitance
C
, how must inductance
L
be selected to
satisfy the design requirements?
If the delay,
T
do
, from (E17) is substituted into (E14),
io
io
12
L
6R C
2R C
,
R +R
=+
(E18)
whence
L4
R
R
C
=
.
(E19)
Problem #2:
Solution
—
—
—
—
—
—
—
—
—
(35%)
In the filter of Figure (E2), the inductance,
L
, is chosen in accordance with the con
straint,
2
o
L
RC,
=
+
−
L
C
R
o
R
o
R
o
V
s
V
o
Z
in
Figure (E2)
where
R
o
is the resistance terminating the output port of the filter, as well as representing the
Thévenin resistance of the signal source applied to the filter input port.
In addition, note that a
resistance of value
R
o
shunts inductance
L
in the filter.
(a).
Determine the input port scattering parameter,
S
11
, referred to a characteristic impedance of
R
o
.
The input impedance,
Z
in
, of the filter is
()
3
oa
oo
in
2
o
o
2
o
o
Rs
L
R
C
R
Z
o
R
s
L 1s
R
C
1s
R
C
R
C
R
C
R.
This is the end of the preview.
Sign up
to
access the rest of the document.
 '06
 Choma
 Electrical Engineering, Frequency, Input impedance, Lowpass filter, Electrical parameters, Output impedance, Southern California Viterbi

Click to edit the document details