Page 196 - Schaum's Outline of Theory and Problems of Electric Circuits

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
CHAP. 8]

Fig. 8-30 Fig. 8-31 185

8.25 Find the natural resonant frequency, j j, of a series RLC circuit with R ¼ 200
, L ¼ 0:1H, C ¼ 5 mF.
Ans: 1000 rad/s

8.26 A voltage of 10 V is applied at t ¼ 0 to a series RLC circuit with R ¼ 5
, L ¼ 0:1H, C ¼ 500 mF. Find the
transient voltage across the resistance. Ans: 3:60e 25t sin 139t ðVÞ

8.27 In the two-mesh circuit shown in Fig. 8-31, the switch is closed at t ¼ 0. Find i 1 and i 2 , for t > 0.
Ans: i 1 ¼ 0:101e 100t þ 9:899e 9950t ðAÞ; i 2 ¼ 5:05e 100t þ 5:00 þ 0:05e 9950t ðAÞ

8.28 A voltage has the s-domain representation 100 308 V. Express the time function for (a) s ¼ 2 Np/s,
t
1
(b) s ¼ 1 þ j5s . Ans: ðaÞ 86:6 e 2t ðVÞ; ðbÞ 100 e cos ð5t þ 308Þ ðVÞ
8.29 Give the complex frequencies associated with the current iðtÞ¼ 5:0 þ 10e 3t cos ð50t þ 908Þ ðAÞ.
Ans: 0; 3 j50 s 1

1
8.30 A phasor current 25 408 A has complex frequency s ¼ 2 þ j3s . What is the magnitude of iðtÞ at
t ¼ 0:2s? Ans: 4:51 A

8.31 Calculate the impedance ZðsÞ for the circuit shown in Fig. 8-32, at (a) s ¼ 0; ðbÞ s ¼ j1 rad/s,
(c) s ¼ j2 rad/s, (d) jsj¼1. Ans: ðaÞ 1
; ðbÞ 1:58 18:438
; ðcÞ 1:84 12:538
; ðdÞ 2

Fig. 8-32 Fig. 8-33

8.32 The voltage source in the s-domain circuit shown in Fig. 8-33 has the time-domain expression
t
v i ðtÞ¼ 10e cos 2t ðVÞ
t
Obtain i o ðtÞ. Ans: 7:07e cos ð2t þ 98:138Þ ðAÞ

8.33 In the time domain, a series circuit of R, L, and C has an applied voltage v i and element voltages v R , v L , and
v C . Obtain the voltage transfer functions (a) V R ðsÞ=V i ðsÞ, (b) V C ðsÞ=V i ðsÞ: