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ECE 233
FINAL
2011-2012 FALL
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1- f (t )   e  2t  e 8t . Draw f(t) using analytic techniques (indicate important points in
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your plot) (10 points)
2- There are two capacitor (not connected to each other initially when t=0-). The initial energy
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of the first capacitor is Einitial_ C1  C1V12 (0  ) and the initial energy of the second capacitor
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is Einitial_ C2  C 2V22 (0  ) . When t=0 the capacitors are connected in series and total energy
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of the system becomes E final . Show that the inequality E final  Einitial_ C1  Einitial_ C2 always
holds. (10 points)
3- The state equation for the state variable x(t) for a circuit is given by
d 2x
dx
 2  x  Sin (t ) . (5 points)
2
dt
dt
a) What is the characteristic equation for the circuit?
b) What are the natural frequencies of the circuit?
c) What is the structure of homogeneous solution for the state variable x(t)?
d 2x
4- The state equation for the state variable x(t) for a circuit is given by
 2 x  e t . (5
2
dt
points)
a) What is the characteristic equation for the circuit?
b) What are the natural frequencies of the circuit?
c) What is the structure of the homogeneous solution for the state variable x(t)?
5- The state equation for the state variable x(t) for a circuit is given by
d 2x
dx
1
 2  2x  .
2
dt
t
dt
(5 points)
a) What is the characteristic equation for the circuit?
b) What are the natural frequencies of the circuit?
c) What is the structure of homogeneous solution for the state variable x(t)?
d 2x
6- The state equation for the state variable x(t) for a circuit is given by
 x  4Sin (t ) .
dt 2
Find the particular solution x p (t ) . (13 points)
Hint: The structure of particular solution is x p (t )  t  A1Cos(t )  A2 Sin (t ) 
7- Find the second order differential equations governing IL(t) (the inductor current) and
VC(t) (the capacitor voltage) for a series RLC (series connected capacitor inductor and
resistor) circuit whish is fed by a series connected voltage source of VS(t). (17 points)
8- For the parallel RLC circuit below the initial conditions for the system variables are as
follows: VC(0)= 5 Volt, IL(0)=0 Amper. When 0  t  ln( 2) switch S is open. When
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t  ln( 2) switch S is closed. The circuit parameters are given as R=1 Ω, C  Farad, L1 
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Henry, L2  Henry. Find IL(t) and VC(t) when 0  t  ln( 2) and when t  ln( 2) .
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s
IL
L1
L2
C
+
VC
R
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