C i i 0 i c 02 ms 20 v 20 v 25 v 10 k t c e e r υ ω

c. I
i
= 0
i
C
=
/0.2 ms
20 V 20 V 25 V
10 k
t
C
e
E
=
R
υ
− ⎡ ⎤
− −
−
⎣ ⎦
Ω
= 2.5 mA e
− t /0.2 ms
d.
17. υ
C
= V
i
+ ( V
f
− V
i
)(1 − e
− t/RC
) τ = RC = (2 k Ω )(10 μ F)
= 8 + (4 − 8)(1 − e
− t /20 ms
) = 20 ms
= 8 − 4(1 − e
− t /20 ms
)
= 8 − 4 + 4 e
− t /20 ms
= 4 + 4 e
− t /20 ms
υ
C
= 4 V(1 + e
− t /20 ms
)
18. V
i
= 10 V, V
f
= 2 V, τ = RC = (1 k Ω )(1000 μ F) = 1 s
υ
C
= V
i
+ ( V
f
− V
i
)(1 − e
− t / τ
)
= 10 V + (2 V − 10 V)(1 − e
− t
)
= 10 − 8(1 − e
− t
)
= 10 − 8 + 8 e
− t
υ
C
= 2 V+ 8 V e
− t
19. V
i
= 10 V, I
i
= 0 A Using the defined direction of i
C
i
C
=
(10 V 2 V)
1 k
− −
Ω
e
− t / τ
τ = RC = (1 k Ω )(1000 μ F) = 1 s
i
C
=
8 V
1 k
t
e
− −
Ω
and i
C
= − 8mA e
− t

338 CHAPTER 24
20. τ = RC = (5 k Ω )(0.04 μ F) = 0.2 ms (throughout)
υ
C
= E (1 − e
− t / τ
) = 20 V(1 − e
− t /0.2 ms
)
(Starting at t = 0 for each plot)
a. T =
1 1
500 Hz
=
f
= 2 ms
2
T
= 1 ms
5 τ = 1 ms =
2
T
b. T =
1 1
100 Hz
=
f
= 10 ms
2
T
= 5 ms
5 τ = 1 ms =
1
5 2
T
⎛ ⎞
⎜ ⎟
⎝ ⎠
c. T =
1 1
5 Hz
=
f
= 0.2 ms
2
T
= 0.1 ms
5 τ = 1 ms =
10
2
T
⎛ ⎞
⎜ ⎟
⎝ ⎠
21. The mathematical expression for i
C
is the same for each frequency!
τ = RC = (5 k Ω )(0.04 μ F) = 0.2 ms
and i
C
=
/0.2 ms
20 V
5 k
t
e
−
Ω
= 4 mA e
− t /0.2 ms
a. T =
1
= 2 ms, = 1 ms
500 Hz 2
T
5 τ = 5(0.2 ms) = 1 ms =
2
T
b. T =
1
100 Hz
= 10 ms,
2
T
= 5 ms
5 τ = 1 ms =
1
5 2
T
⎛ ⎞
⎜ ⎟
⎝ ⎠
c. T =
1
5000 Hz
= 0.2 ms,
2
T
= 0.1 ms
5 τ = 1 ms = 10
2
T
⎛ ⎞
⎜ ⎟
⎝ ⎠

CHAPTER 24 339
22. τ = 0.2 ms as above
T =
1
500 Hz
= 2 ms
5 τ = 1 ms =
2
T
0 →
2
T
: υ
C
= 20 V(1 − e
− t /0.2 ms
)
2
T
→ T : V
i
= 20 V, V
f
= − 20 V
υ
C
= V
i
+ ( V
f
− V
i
)(1 − e
− t / τ
)
= 20 + ( − 20 − 20)(1 − e
− t /0.2 ms
)
= 20 − 40(1 − e
− t /0.2 ms
)
= 20 − 40 + 40 e
− t /0.2 ms
υ
C
= − 20 V+ 40 V e
− t /0.2 ms
T →
3
2
T : V
i
= − 20 V, V
f
= +20 V
υ
C
= V
i
+ ( V
f
− V
i
)(1 − e
− t / τ
)
= − 20 + (20 − ( − 20))(1 − e
− t / τ
)
= − 20 + 40(1 − e
− t / τ
)
= − 20 + 40 − 40 e
− t / τ
υ
C
= 20 V − 40 V e
− t /0.2 ms
23. υ
C
= V
i
+ ( V
f
− V
i
)(1 − e
− t/RC
)
V
i
= 20 V, V
f
= 20 V
υ
C
= 20 + (20 − 20)(1 − e
− t/RC
)
= 20 V (for 0 →
2
T
)
For
2
T
→ T , υ
i
= 0 V and υ
C
= 20 V e
− t/ τ
τ = RC = 0.2 ms
with
2
T
= 1 ms and 5 τ =
2
T
For T →
3
2
T , υ
i
= 20 V
υ
C
= 20 V(1 − e
− t / τ
)
For
3
2
T → 2 T , υ
i
= 0 V
υ
C
= 20 V e
− t / τ

340 CHAPTER 24
24. τ = RC = 0.2 ms
5 τ = 1 ms =
2
T
V
i
= − 10 V, V
f
= +20 V
0 →
2
T
:
υ
C
= V
i
+ ( V
f
− V
i
)(1 − e
− t / τ
)
= − 10 + (20 − ( − 10))(1 − e
− t / τ
)
= − 10 + 30(1 − e
− t / τ
)
= − 10 + 30 − 30 e
− t / τ
υ
C
= +20 V − 30 V e
− t /0.2 ms
2
T
→ T : V
i
= 20 V, V
f
= 0 V
υ
C
= 20 V e
− t /0.2 ms
25. Z
p
: X
C
=
1 1
2 2 (10 kHz)(3 pF)
=
fC π π
= 5.31 M Ω
Z
p
=
(9 M 0 )(5.31 M 90 )
9 M 5.31 M j
Ω ∠ ° Ω ∠ − °
Ω − Ω
= 4.573 M Ω ∠− 59.5 °
Z
s
: C
T
= 18 pF + 9 pF = 27 pF
X
C
=
1 1
2 2 (10 kHz)(27 pF)
T
=
fC π π
= 0.589 M Ω
Z
s
=
(1M 0 )(0.589 M 90 )
1 M 0.589 M j
Ω ∠ ° Ω ∠ − °
Ω − Ω
= 0.507 M Ω ∠− 59.5 °
V
scope
=
(0.507 M 59.5 )(100 V 0 )
=
(0.257 M 0.437 M ) + (2.324 M 3.939 M )
s i
s p
j j
Ω ∠ − ° ∠ °
+ Ω − Ω Ω − Ω
V Z
Z Z
=
6
6
50.7 10 V 59.5
5.07 10 59.5
× ∠ − °
× ∠ − °
= 10 V ∠ 0 ° =
1
10
(100 V ∠ 0 ° )
=
s p
θ θ
Z Z
= − 59.5 °
26. Z
p
: X
C
=
5
1 1
(10 rad/s)(3 pF)
=
C ω
= 3.333 M Ω
Z
p
=
(9 M 0 )(3.333 M )
9 M 3.333 M j
Ω ∠ ° Ω
Ω − Ω
= 3.126 M Ω ∠− 69.68 °
Z
s
: X
C
=
5
1 1
(10 rad/s)(27 pF)
=
C ω
= 0.370 M Ω
Z
s
=
(1M 0 )(0.370 M 90 )
1 M 0.370 M j
Ω ∠ ° Ω ∠ − °
Ω − Ω
= 0.347 M Ω ∠− 69.68 °
✓ =
p s
θ θ
Z Z

CHAPTER 24 341
V
scope
=
(0.347 M 69.68 )(100 V