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The solutions to problem 1.1 to problem 1.4 from the ece360 homework 1, due on september 7, 2007. The problems involve deriving a state-space model and input-output differential equation for an electric circuit, as well as applying the laplace transform to certain functions.
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Problem 1.
o
(^1) v R 2 1 v 2
C 1 C 2
−
R
i
−
v (t) v (t)
Derive a state-space model for the above electric circuit in the form: [ v ˙ 1 v ˙ 2
[ a 11 a 12 a 21 a 22
] [ v 1 v 2
]
[ b 1 b 2
] vi(t)
vo =
[ c 1 c 2
] [ v 1 v 2
]
Problem 1.
Using the results of Problem 1.1, derive the input-output differential equation of the circuit in the form:
d^2 vo dt^2 +^ α
dvo dt +^ βvo^ =^ γvi(t)
Problem 1.
(a) Apply the definition of the Laplace transform and show that:
L
{ e−at
s + a
(b) Use the linearity property of the Laplace transform and show that:
L {cosh at} = s s^2 − a^2 L {sinh at} =
a s^2 − a^2
Problem 1.4 (Problem B-2-4)