Page 157 - Schaum's Outline of Theory and Problems of Electric Circuits

[CHAP. 7
FIRST-ORDER CIRCUITS
146
7.2 In Problem 7.1, obtain the power and energy in the resistor, and compare the latter with the initial
energy stored in the capacitor.
p R ¼ v R i ¼ 25e 125t ðWÞ
ð ð
t t
w R ¼ p R dt ¼ 25e 125t dt ¼ 0:20ð1 e 125t Þ ðJÞ
0 0
The initial stored energy is
2
6
2
1
1
W 0 ¼ CV 0 ¼ ð40 10 Þð100Þ J ¼ 0:20 ¼ w R ð1Þ
2 2
In other words, all the stored energy in the capacitor is eventually delivered to the resistor, where it is
converted into heat.
7.3 An RC transient identical to that in Problems 7.1 and 7.2 has a power transient
t=0:00001
p ¼ 360e ðWÞ
R
Obtain the initial charge Q 0 ,if R ¼ 10
.
2
p R ¼ P 0 e 2t=RC or ¼ 10 5 or C ¼ 2 mF
RC
ð t
t=0:00001
w R ¼ p R dt ¼ 3:6ð1 e Þ ðmJÞ
0
2
Then, w R ð1Þ ¼ 3:6mJ ¼ Q 0 =2C, from which Q 0 ¼ 120 mC.

7.4 The switch in the RL circuit shown in Fig. 7-21 is moved from position 1 to position 2 at t ¼ 0.
Obtain v R and v L with polarities as indicated.

Fig. 7-21

The constant-current source drives a current through the inductance in the same direction as that of the
transient current i. Then, for t > 0,
i ¼ I 0 e Rt=L ¼ 2e 25t ðAÞ
v R ¼ Ri ¼ 200e 25t ðVÞ
v L ¼ v R ¼ 200e 25t ðVÞ

7.5 For the transient of Problem 7.4 obtain p R and p L .

p R ¼ v R i ¼ 400e 50t ðWÞ
p L ¼ v L i ¼ 400e 50t ðWÞ

Negative power for the inductance is consistent with the fact that energy is leaving the element. And, since
this energy is being transferred to the resistance, p R is positive.