Page 189 - Schaum's Outline of Theory and Problems of Electric Circuits

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
178

Fig. 8-19 [CHAP. 8

Knowing the end conditions on v C , we can write
þ
v C ¼½v C ð0 Þ v C ð1ފe t= þ v C ð1Þ ¼ 25ð1 e t=10 Þ ðVÞ
wherein t is measured in ms.
The current in the capacitor is given by
dv C t=10
i C ¼ C ¼ 5e ðAÞ
dt
and the current in the parallel 10-
resistor is
v C t=10
i 10
¼ ¼ 2:5ð1 e Þ ðAÞ
10
Hence, i ¼ i C þ i 10
¼ 2:5ð1 þ e t=10 Þ ðAÞ
The problem might also have been solved by assigning mesh currents and solving simultaneous differ-
ential equations.

8.6 For the time functions listed in the first column of Table 8-2, write the corresponding amplitude
and phase angle (cosine-based) and the complex frequency s.

See columns 2 and 3 of the table.
Table 8-2
Time Function A 8 s

iðtÞ¼ 86:6A 86:6 08 A 0
3 3
2 10 t
iðtÞ¼ 15:0e ðAÞ 15:0 08 A 2 10 Np/s
vðtÞ¼ 25:0 cos ð250t 458Þ ðVÞ 25:0 458 V j250 rad/s
vðtÞ¼ 0:50 sin ð250t þ 308Þ ðVÞ 0:50 608 V j250 rad/s
iðtÞ¼ 5:0e 100t sin ð50t þ 908Þ ðAÞ 5:0 08 A 100 j50 s 1
iðtÞ¼ 3 cos 50t þ 4 sin 50t ðAÞ 5 53:138 A j50 rad/s

8.7 For each amplitude and phase angle in the first column and complex frequency s in the second
column in Table 8-3, write the corresponding time function.
See column 3 of the table.
Table 8-3
A 8 s Time Function
10 08 þj120 10 cos 120 t
2 458 j120 2 cos ð120 t þ 458Þ
5 908 2 j50 5e 2t cos ð50t 908Þ
15 08 5000 j1000 15e 5000t cos 1000t
100 308 0 86.6