Page 190 - Schaum's Outline of Theory and Problems of Electric Circuits

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
CHAP. 8]
p ffiffiffi 179
8.8 An amplitude and phase angle of 10 2 458 V has an associated complex frequency
1
s ¼ 50 þ j 100 s . Find the voltage at t ¼ 10 ms.
p ffiffiffi
vðtÞ¼ 10 2e 50t cos ð100t þ 458Þ ðVÞ
At t ¼ 10 2 s, 100t ¼ 1 rad ¼ 57:38, and so
p ffiffiffi
v ¼ 10 2e 0:5 cos 102:38 ¼ 1:83 V

8.9 A passive network contains resistors, a 70-mH inductor, and a 25-mF capacitor. Obtain the
respective s-domain impedances for a driving voltage (a) v ¼ 100 sin ð300t þ 458ÞðVÞ,
100t
(b) v ¼ 100e cos 300t ðVÞ.
(a) Resistance is independent of frequency. At s ¼ j300 rad/s, the impedance of the inductor is
3
sL ¼ð j300Þð70 10 Þ¼ j21
and that of the capacitor is
1
¼ j133:3
sC
1
(b)At s ¼ 100 þ j300 s ,
3
sL ¼ð 100 þ j300Þð70 10 Þ¼ 7 þ j21
1 1
¼ ¼ 40 j120
6
sC ð 100 þ j300Þð25 10 Þ

8.10 For the circuit shown in Fig. 8-20, obtain v at t ¼ 0:1 s for source current ðaÞ i ¼ 10 cos 2t (A),
t
(b) i ¼ 10e cos 2t (A).
2ðs þ 2Þ s þ 3
Z in ðsÞ¼ 2 þ ¼ð4Þ
s þ 4 s þ 4
(a)At s ¼ j2 rad/s, Z in ð j2Þ¼ 3:22 7:138
. Then,

V ¼ IZ in ¼ð10 08Þð3:22 7:138Þ¼ 32:2 7:138 V or v ¼ 32:2 cos ð2t þ 7:138Þ ðVÞ
and vð0:1Þ¼ 32:2 cos ð18:598Þ¼ 30:5V.
1
(b)At s ¼ 1 þ j2s , Z in ð 1 þ j2Þ¼ 3:14 11:318
. Then
t
V ¼ IZ in ¼ 31:4 11:318 V or v ¼ 31:4e cos ð2t þ 11:318Þ ðVÞ
and vð0:1Þ¼ 31:4e 0:1 cos 22:778 ¼ 26:2V.

Fig. 8-20 Fig. 8-21

8.11 Obtain the impedance Z in ðsÞ for the circuit shown in Fig. 8-21 at (a) s ¼ 0, (b) s ¼ j4 rad/s,
(c) jsj¼ 1.