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Chapter 6, Solution 1.
(
=+−==
−− t3t3
e6e25
dt
dv
Ci
)
10(1 - 3t)e
-3t
A
p = vi
= 10(1-3t)e
-3t
⋅ 2t e
-3t
= 20t(1 - 3t)e
-6t
W
Chapter 6, Solution 2.
22
11
)120)(40(
2
1
Cv
2
1
w ==
w
2
=
22
1
)80)(40(
2
1
2
=Cv
1
(
)
=−=−=∆
22
21
8012020www 160 kW
Chapter 6, Solution 3.
i = C =
−
=
−
5
160280
10x40
d
t
dv
3
480 mA
Chapter 6, Solution 4.
)0(vidt
C
1
v
t
o
+=
∫
∫
+1tdt4sin6
2
1
=
=
1 - 0.75 cos 4t
Chapter 6, Solution 5.
v =
∫
+
t
o
)0(vidt
C
1
For 0 < t < 1, i = 4t,
∫
−
=
t
o
6
t4
10x20
1
v
dt + 0 = 100t
2
kV
v(1) = 100 kV
For 1 < t < 2, i = 8 - 4t,
∫
+−=
−
t
1
6
)1(vdt)t48(
10x20
1
v
= 100 (4t - t
2
- 3) + 100 kV
Thus v (t) =
<<−−
<<
2t1,kV)2tt4(100
1t0,kVt100
2
2
Chapter 6, Solution 6.
6
10x30
dt
dv
Ci
−
==
x slope of the waveform.
For example, for 0 < t < 2,
3
10x2
10
d
t
dv
−
=
i =
mA150
10x2
10
x10x30
d
t
dv
3
6
==
−
−
C
Thus the current i is sketched below.
t (msec)
150
12
10
2
8
6
4
-150
i(t) (mA)
Chapter 6, Solution 7.
∫∫
+=+=
−
−
t
o
3
3
o
10dt10tx4
10x50
1
)t(vidt
C
1
v
=
=+10
50
t2
2
0.04k
2
+ 10 V
Chapter 6, Solution 8.
(a)
tt
BCeACe
dt
dv
C
600100
600100
−−
−−==
i
(1)
BABCACi 656001002)0(
−
−
=
→
−−== (2)
BAvv +=→=
−+
50)0()0( (3)
Solving (2) and (3) leads to
A=61, B=-11
(b)
J 52500104
2
1
)0(
2
1
32
===
−
xxxCvEnergy
(c ) From (1),
A 4.264.241041160010461100
60010060031003 tttt
eeexxxexxxi
−−−−−−
−−=−−=
Chapter 6, Solution 9.
v(t) =
()
(
)
∫
−−
+=+−
t
o
tt
Vet120dte16
21
1
v(2) = 12(2 + e
-2
) = 25.62 V
p = iv = 12 (t + e
-t
) 6 (1-e
-t
) = 72(t-e
-2t
)
p(2) = 72(2-e
-4
) = 142.68 W
Chapter 6, Solution 10
dt
dv
x
dt
dv
Ci
3
102
−
==
<<
<<
<<
=
s4t316t,-64
s 3t116,
s10,16
µ
µ
µ
tt
v
<<
<<
<<
=
s4t3,16x10-
s 3t10,
s10,1016
6
6
µ
µ
µ
tx
dt
dv
<<
<<
<<
=
s4t3kA, 32-
s 3t10,
s10,kA 32
)(
µ
µ
µ
t
ti
Chapter 6, Solution 11.
v =
∫
+
t
o
)0(vidt
C
1
For 0 < t < 1,
∫
==
−
−
t
o
3
6
t10dt10x40
10x4
1
v kV
v(1) = 10 kV
For 1 < t < 2,
kV10)1(vvdt
C
1
v
t
1
=+=
∫
For 2 < t < 3,
∫
+−=
−
−
t
2
3
6
)2(vdt)10x40(
10x4
1
v
= -10t + 30kV
Thus
v(t) =
<<+−
<<
<<⋅
3t2,kV30t10
2t1,kV10
1t0,kVt10
Chapter 6, Solution 12.
π−π==
−
4sin)(4(60x10x3
dt
dv
Ci
3
t)
=
- 0.7e π sin 4πt A
P = vi = 60(-0.72)π cos 4π t sin 4π t = -21.6π sin 8π t W
W =
∫∫
t dt ππ−=
t
o
8
1
o
8sin6.21pdt
=
π
π
π
8
8
6.
cos
21
8/1
o
= -5.4J
Chapter 6, Solution 13.
Under dc conditions, the circuit becomes that shown below:
i
2
50
Ω
20
Ω
+
−
60V
+
v
1
−
i
1
30 Ω
10
Ω
+
v
2
−
i
2
= 0, i
1
= 60/(30+10+20) = 1A
v
1
= 30i
2
= 30V, v
2
= 60-20i
1
= 40V
Thus, v
1
= 30V, v
2
= 40V
Chapter 6, Solution 14.
(a) C
eq
= 4C = 120 mF
(b)
30
4
C
4
C
1
eq
== C
eq
= 7.5 mF
Chapter 6, Solution 15.
In parallel, as in Fig. (a),
v
1
= v
2
= 100
C
2
+
v
2
−
C
1
+
−
100V
+
v
2
−
C
2
+
−
v
1
C
1
+
v
1
−
+
−
100V
(b)
(a)
w
20
= ==
− 262
100x10x20x
2
1
Cv
2
1
0.1J
w
30
=
=
− 26
100x10x30x
2
1
0.15J
(b) When they are connected in series as in Fig. (b):
,60100x
50
30
V
CC
C
v
21
2
1
==
+
=
v
2
= 40
w
20
=
=
− 26
60x10x30x
2
1
36 mJ
w
30
= =
− 26
4010x30
2
xx
1
24 mJ
Chapter 6, Solution 16
F 2030
80
80
14
µ
=→=
+
+= C
C
Cx
C
eq
Chapter 6, Solution 17.
(a) 4F in series with 12F = 4 x 12/(16) = 3F
3F in parallel with 6F and 3F = 3+6+3 = 12F
4F in series with 12F = 3F
i.e. C
eq
= 3F
(b)
C
eq
= 5 + [6 || (4 + 2)] = 5 + (6 || 6) = 5 + 3 = 8F
(c)
3F in series with 6F = (3 x 6)/9 = 6F
1
3
1
6
1
2
1
C
1
eq
=++=
C
eq
= 1F
Chapter 6, Solution 18.
For the capacitors in parallel
= 15 + 5 + 40 = 60 µF
1
eq
C
Hence
10
1
60
1
30
1
20
1
C
1
eq
=++=
C
eq
= 10 µF
Chapter 6, Solution 19.
We combine 10-, 20-, and 30-
µ
F capacitors in parallel to get 60
µ
F. The 60 -
µ
F
capacitor in series with another 60-
µ
F capacitor gives 30
µ
F.
30 + 50 = 80
µ
F, 80 + 40 = 120
µ
F
The circuit is reduced to that shown below.
12 120
12
80
120-
µ
F capacitor in series with 80
µ
F gives (80x120)/200 = 48
48 + 12 = 60
60-
µ
F capacitor in series with 12
µ
F gives (60x12)/72 = 10
µ
F
Chapter 6, Solution 20.
3 in series with 6 = 6x
3
/(9) = 2
2 in parallel with 2 = 4
4 in series with 4 = (4x4)/8 = 2
The circuit is reduced to that shown below:
20
1
6
2
8
6 in parallel with 2 = 8
8 in series with 8 = 4
4 in parallel with 1 = 5
5 in series with 20 = (5x20)/25 = 4
Thus C
eq
= 4 mF
Chapter 6, Solution 21.
4µF in series with 12µF = (4x12)/16 = 3µF
3µF in parallel with 3µF = 6µF
6µF in series with 6µF = 3µF
3µF in parallel with 2µF = 5µF
5µF in series with 5µF = 2.5µF
Hence C
eq
= 2.5µF
Chapter 6, Solution 22.
Combining the capacitors in parallel, we obtain the equivalent circuit shown below:
a b
40
µ
F
60 µF 30
µ
F
20
µ
F
Combining the capacitors in series gives
C , where
1
eq
10
1
30
1
20
1
60
1
C
1
1
eq
=++= C = 10µF
1
eq
Thus
C
eq
= 10 + 40 = 50 µF
Chapter 6, Solution 23.
(a) 3µF is in series with 6µF 3x6/(9) = 2µF
v
4µF
= 1/2 x 120 = 60V
v
2µF
= 60V
v
6µF
= =(
3
+
)60
36
20V
v
3µF
= 60 - 20 = 40V
(b)
Hence w = 1/2 Cv
2
w
4µF
= 1/2 x 4 x 10
-6
x 3600 = 7.2mJ
w
2µF
= 1/2 x 2 x 10
-6
x 3600 = 3.6mJ
w
6µF
= 1/2 x 6 x 10
-6
x 400 = 1.2mJ
w
3µF
= 1/2 x 3 x 10
-6
x 1600 = 2.4mJ
Chapter 6, Solution 24.
20µF is series with 80µF = 20x80/(100) = 16µF
14µF is parallel with 16µF = 30µF
(a) v
30µF
= 90V
v
60µF
= 30V
v
14µF
= 60V
v
20µF
=
=
+
60x
8020
80
48V
v
80µF
= 60 - 48 = 12V
(b)
Since w =
2
Cv
2
1
w
30µF
= 1/2 x 30 x 10
-6
x 8100 = 121.5mJ
w
60µF
= 1/2 x 60 x 10
-6
x 900 = 27mJ
w
14µF
= 1/2 x 14 x 10
-6
x 3600 = 25.2mJ
w
20µF
= 1/2 x 20 x 10
-6
x (48)
2
= 23.04mJ
w
80µF
= 1/2 x 80 x 10
-6
x 144 = 5.76mJ
Chapter 6, Solution 25.
(a) For the capacitors in series,
Q
1
= Q
2
C
1
v
1
= C
2
v
2
1
2
2
1
C
C
v
v
=
v
s
= v
1
+ v
2
=
2
1
21
22
1
2
v
C
CC
vv
C
C +
=+
s
21
1
2
v
CC
C
+
=v
Similarly,
s
21
2
1
v
CC
C
v
+
=
(b)
For capacitors in parallel
v
1
= v
2
=
2
2
1
1
C
Q
C
Q
=
Q
s
= Q
1
+ Q
2
=
2
2
21
22
2
1
Q
C
CC
QQ
C
C +
=+
or
Q
2
=
21
2
CC
C
+
s
21
1
1
Q
CC
C
Q
+
=
i =
dt
dQ
s
21
1
1
i
CC
C
+
=i ,
s
21
2
2
i
CC
C
+
=i
Chapter 6, Solution 26.
(a)
C
eq
= C
1
+ C
2
+ C
3
= 35µF
(b) Q
1
= C
1
v = 5 x 150µC = 0.75mC
Q
2
= C
2
v = 10 x 150µC = 1.5mC
Q
3
= C
3
v = 20 x 150 = 3mC
(c) w =
J150x35x
2
1
2
22
eq
µ=
vC
1
= 393.8mJ