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3. •
•

4. •

5. •
•
RC
dv(t)
dt
+ v(t) = vs
(t)
L
R
di(t)
dt
+ i(t) = is
(t)

6. RC
dv(t)
dt
+ v(t) = vs (t)
L
R
di(t)
dt
+ i(t) = is
(t)
dy(t)
dt
+ y(t) = x(t)
y(t) = unknown variable =
v(t) for the capacitive case
i(t) for the inductive case
x(t) = forcing function =
vS
(t) for the capacitive case
iS
(t) for the inductive case
= time constant =
RC for the capacitive case
L /R for the inductive case

7. •
•

8. •
•
–
–
dy(t)
dt
+ y(t) = x(t)
dy*
(t)
dt
+ y*
(t) = 0
dy** (t)
dt
+ y**
(t) = x(t)
y(t) = y*
(t) + y**
(t)

9. •
•
•
dy*
(t)
dt
+ y*
(t) = 0

10. •
y*
(t) yHA
(t) yN
(t)
dy(t)
dt
+ y(t) = x(t)
dy(t)
dt
+ y(t) = 0

11. •
•
y**
(t) yP
(t) yF
(t)
y(t) = yHA
(t) + yP
(t) yN
(t) + yF
(t)

12. •
•
•
is=I
v = L
diL
dt
= L
d(constant)
dt
= 0

13. •
•
is=I
L
di(t)
dt
+ Ri(t) = 0 for t 0

14. L
di(t)
dt
+ Ri(t) = 0 L
di
dt
dt + R i dt = 0
Ldi = R i dt
di
i
=
R
L
dt
di
ii (0)
i (t )
=
R
L
dt
0
t
ln
i(t)
i(0)
=
R
L
t i(t) = i(0)e ( R /L )t
with t 0
i(0) = i(0 ) = I0
i(t) = I0
e ( R /L )t
with t 0

15. i(t) = I0
e ( R /L )t
with t 0
v(t) = R i(t) = RI0
e ( R /L )t
with t 0

16. v(0 ) = L
diL
dt t=0
= L
dI0
dt
= 0 with t < 0
v(0+) = L
di
dt t=0
= L
d
dt
I0 e ( R /L )t
( )t=0
= RI0 with t 0
v(t) = R i(t) = RI0
e ( R /L )t
with t 0
v(0+) = R iR (0+) = RI0
Is=I
v(0 ) = R iR
(0 ) = R 0 = 0

17. •
p(t) = i(t)v(t) = I0
e( R /L )t
I0
R e( R /L )t
= I0
2
R e 2( R /L )t
for t 0
w(t) = p(t)dt
0
t
= I0
2
R e( 2 R /L )t
dt
0
t
= I0
2
R
e( 2 R /L )t
2R /L 0
t
=
I0
2
R
2R /L
e( 2 R /L )t
[ ]0
t
=
=
1
2
LI0
2
1 e( 2 R /L )t
( ) for t 0

18. •
i I0
=
I0
t
di
dt t=0
=
d
dt
Io
e ( R /L )t
( )t=0
= I0
R
L
=
I0

19. •
i(t) = I0
e t /
with t 0
=
L
R

20. •
•
•
i = C
dvC
dt
= C
d(constant)
dt
= 0

21. •
•
v(0 ) = V0
C
dv
dt
+
v
R
= 0 for t 0

22. •
v(0 ) = V0
= v(0+)
C
dv
dt
+
v
R
= 0 for t 0
v(t) = v(0+)e
t
RC
= V0
e
t
RC
for t 0

23. v(t) = v(0+)e
t
RC
= V0
e
t
RC
for t 0
i(t) =
v(t)
R
=
V0
R
e
t
RC
for t 0
i(0 ) = 0; i(0+) =
V0
R

24. •
p(t) = i(t)v(t) =
V0
2
R
e
2t
RC
for t 0
w(t) = p(t)dt
0
t
=
1
2
CV0
2
1 e
2t
RC
( ) for t 0

25. •
= RC

26. dy(t)
dt
+ y(t) = 0
dy(t)
dt
+ y(t) = 0
dy(t)
dt
= y(t)
y(t) = A est

27. •
•
dy(t)
dt
+ y(t) = 0 ( s +1)Aest
= 0 s +1 = 0
Aest
0
y(t) = Aest
s +1 = 0 s =
1
y(t) = Aest
t 0 y(0) = A

28. • y(t) = y(0)e t /

29. y( ) = y(0)e 1
=
1
e
y(0) 0.37y(0)
dy(t)
dt t=o
=
1
y(0)e t /
t=0
=
y(0)

31. •
•
•

32. •
•
•

33. •
•

34. •
dy(t)
dt
+ y(t) = x(t)
1
et /
et / dy(t)
dt
+
et /
y(t) =
et /
x(t)
d
dt
et /
y(t)( ) =
et /
x(t)
d
dt
et /
y(t)( )0
t
dt =
et /
x(t)
0
t
dt

35. d
dt
et /
y(t)( ) =
et /
x(t) d
dt
et /
y(t)( )0
t
dt =
et /
x(t)
0
t
dt
d et /
y(t)( )0
t
=
1
x(t)et /
0
t
dt et /
y(t) e0
y(0) =
1
x(t)et /
0
t
dt
y(t)et /
y(0) =
1
x(t)et /
0
t
dt y(t) = y(0)e t /
+
1
e t /
x(t)et /
0
t
dt
•
•
•

36. ycomplete = ynatural + y forced
ycomplete
= y(t)
ynatural = y(0)e t /
y forced =
1
e t /
x(t)et /
0
t
•
•

37. •
•
x(t) = XS
(with Xs
being a constant)
x(t) =
0 for t < 0
XS
for t 0
= XS
u(t)
x(t)
Xs

38. •

40. •
dy(t)
dt
+ y(t) = XS for t 0
y forced
=
1
e t /
XS
et /
0
t
dt =
XS
e t /
et /
0
t
dt =
XS
e t /
et /
[ ]0
t
for t 0
y forced
= XS
1 e t /
( ) for t 0
ynatural
= y(0)e t /
for t 0

41. •
•
•
y(t) = ynatural
+ y forced
= y(0)e t /
+ XS
(1 e t /
) =
= y(0)e t /
+ XS
XS
e t /
for t 0
y( ) = y(0)e + XS XSe = XS
y(t) = y(0)e t /
XSe t /
+ XS = [y(0) y( )]e t /
+ y( ) for t 0

42. •
y(t) = [y(0) y( )]e t /
+ y( ) for t 0 (with y( ) = XS )
t 0

43. •
•
y(t) = [y(0) y( )]e t /
+ y( ) for t 0

44. •
y(t) = [y(0) y( )]e t /
+ y( ) for t 0 (with y( ) = XS )
ytransient (t) = [y(0) y( )]e t /
for t 0 (with y( ) = XS )
y(t ) = [y(0) y( )]e /
+ y( ) for t 0 (with y( ) = XS )

45. •
•
y(t) = ytransient
+ ysteady state
for t 0
ytransient
= [y(0) y( )]e t /
ysteady state
= y( )
for t 0
ytransient = y(0)e t /
y( )e t /
for t 0
ysteady state
= y( ) = XS

46. VS
= Ri(t) + L
di(t)
dt
for t 0 VS
u(t) = Ri(t) + L
di(t)
dt
i(t) =
VS
R
+ I0
VS
R
e t /
for t 0

47. i(t) =
VS
R
+ I0
VS
R
e t /
for t 0

48. i(t) =
VS
R
+ I0
VS
R
e t /
for t 0
v(t) = L
di(t)
dt
= (VS
I0
R)e t /
for t 0

49. C
dvC
dt
+
vC
R
= IS
for t 0 C
dvC
dt
+
vC
R
= IS
u(t)
vC
(t) = IS
R + (V0
IS
R)e t /
for t 0
iC
(t) = C
dvC
dt
= (IS
V0
R
)e t /
for t 0

50. vC
(t) = IS
R + (V0
IS
R)e t /
for t 0
iC
(t) = C
dvC
dt
= (IS
V0
R
)e t /
for t 0

51. •
•
•

52. •
•

53. •
•

54. •
•
•
•

55. •

56. iaux
=
vaux
10K
7i + i
i =
vaux
20K
iaux
=
vaux
10K
6
vaux
20K
= vaux
(0.1 10 3
0.3 10 3
)
vaux
iaux
= 5K

57. •

58. iC
= iR
i
vO
(t) = Ri(t)
i(t) = C
dvI
(t)
dt
vO (t) = RC
dvI (t)
dt
•
•

59. •
•

60. •
•
iC
= iR
i
i(t) =
vI
(t)
R
i(t) = C
dvO
(t)
dt
vO (t) =
1
RC
vI (t)dt + vO (0)
0
t

61. •
•
•
•
•
vO
(t) =
VI
RC
t + vO
(0)
dvO
(t)
dt
=
VI
RC

62. •
•

63. •
•
vO (t) =
1
RC
vI (t)dt + vO (0)
0
t

64. •
•