ECE3085  Homework #1
Due: Jan. 4, 2009
Problem 1.
(8 pts)
Write the dynamic equations describing the circuit in Figure 1(a), where
L
= 1
H
,
R
= 2Ω
, and
C
= 1
F
, with the initial conditions
y
(0) = 2
V
and
˙
y
(0) = 1
V
. In particular,
(a) Write them as a second order differential equation where
V
out
is the measurement
y
and
V
in
is the input
u
.
(b) Transform the differential equation into the Laplace domain.
(c) Assuming a zero input, solve the differential equation for
y
(
t
)
using Laplace transform methods.
(d) Verify your answer numerically, by modifying the differential equation from Homework 1, Problem 6b (as I did
in the solutions to the last homework).
Problem 2.
(6 pts)
Using Matlab’s
roots
command determine the poles, associated modes, and stability type for
the following transfer functions:
(a)
G
(
s
) =
1
s
3
+2
s
2
+21
s

58
,
(b)
G
(
s
) =
s

1
s
3
+11
s
2
+39
s
+29
, and
(c)
G
(
s
) =
1
s
4
+5
s
3
+11
s
2
+22
s
+10
.
Problem 3.
(10 pts)
You are given the circuit shown in figure 1(b).
(a) Using the impedance method and the voltage division rule, determine the transfer function
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 Spring '08
 CITRIN
 Electromagnet, Laplace, DC Motor, Vout Vin

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