EE 350
PROBLEM SET 3
DUE: 24 Sep 2007
Reading assignment: Lathi Sections 2.5 and 2.6
Exam I
is scheduled for Thursday, September 27 from 8:15 pm to 10:15 pm in room 121 Sparks for all sections. The
exam covers material problem sets 1 through 3, and laboratory#1. The exam is closedbook, but you may bring one
8 1/2 by 11 inch note sheet, Calculators are
not
allowed as graphical/scientific calculators are capable of graphing
functions and solving ODEs: skills that you must be capable of doing by hand.
Sections will meet for Laboratory #1 during the week of September 17.
Problem 11:
(18 points)
Consider the firstorder passive circuit in Figure 1.
Figure 1: Firstorder passive circuit with input voltage
f
(
t
) and output voltage
y
(
t
).
1. (4 points) A standard form for expressing an ODE is
d
n
y
dt
n
+
a
n

1
d
n

1
y
dt
n

1
+
· · ·
+
a
0
y
=
b
m
d
m
f
dt
m
+
b
m

1
d
m

1
f
dt
m

1
+
· · ·
+
b
0
f.
By introducing the derivative operator
D
≡
d/dt
and the polynomials
Q
(
D
)
=
D
n
+
a
n

1
D
n

1
+
· · ·
+
a
1
D
+
a
0
P
(
D
)
=
b
m
D
m
+
b
m

1
D
n

1
+
· · ·
+
b
1
D
+
b
0
,
an ODE may also be represented by the compact form
Q
(
D
)
y
(
t
) =
P
(
D
)
f
(
t
)
.
(1)
For the circuit in Figure 1, carefully derive an expression for the ODE that relates the output voltage
y
(
t
)
to the input voltage
f
(
t
) and express your answer using the form in equation 1. Provide expressions for the
polynomials
Q
(
D
) and
P
(
D
) in terms of the parameters
R
1
,
R
2
, and
C
.
2. (1 point) State the characteristic equation and find its root(s) in terms of the parameters
R
1
,
R
2
, and
C
.
3. (1 point) In general a characteristic root
λ
that maybe complex. If the real part of
λ
is strictly negative, then
the corresponding
natural (or characteristic) mode
e
λt
exponentially decays to zero as
t
increases. In this
case the
time constant
associated with the natural mode is defined as
τ
=
−
1
/
Re(
λ
). The time constant
τ
is
a measure of how quickly a natural mode relaxes towards zero. The smaller
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 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB
 Trigraph, Impulse response

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