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ECE360: Homework 1 - Circuit State-Space Model & Input-Output Equation, Assignments of Electrical and Electronics Engineering

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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Pre 2010

Uploaded on 08/19/2009

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ECE360 HOMEWORK #1 DUE: FRIDAY, SEPTEMBER 7, 2007
Problem 1.1
o
1R2
v1v2
C1C2
+
R
i
+
v (t)v (t)
Derive a state-space model for the above electric circuit in the form:
"˙v1
˙v2#="a11 a12
a21 a22 #" v1
v2#+"b1
b2#vi(t)
vo=hc1c2i"v1
v2#+ [d]vi(t)
Problem 1.2
Using the results of Problem 1.1, derive the input-output differential equation of the circuit in the
form:
d2vo
dt2+αdvo
dt +βvo=γvi(t)
Problem 1.3
(a) Apply the definition of the Laplace transform and show that:
Lneato=1
s+a
(b) Use the linearity property of the Laplace transform and show that:
L {cosh at}=s
s2a2
L {sinh at}=a
s2a2
Problem 1.4 (Problem B-2-4)

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ECE360 HOMEWORK #1 DUE: FRIDAY, SEPTEMBER 7, 2007

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

]

  • [d] vi(t)

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)