The document describes circuit analysis techniques for solving differential equations modeling capacitive and inductive circuits driven by voltage or current sources. It presents the differential equations for the basic circuit elements of resistors, capacitors, and inductors. It then shows how to derive solutions for the transient and steady-state responses of circuits using the technique of separating solutions into homogeneous and particular parts.

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. •
•