114
WAVEFORMS AND SIGNALS
[CHAP. 6
Fig. 6-13 EXAMPLE 6.23 The voltage v ¼ V0 e jtj= , > 0, is connected to a capacitor. Find the current i in the capacitor. Sketch v and i for V0 ¼ 10 V, C ¼ 1 mF, and ¼ 1 ms. Using i ¼ C dv=dt, for t < 0;
v ¼ V0 et=
and
i ¼ I0 et=
for t > 0;
v ¼ V0 e t=
and
i ¼ I0 e t=
where I0 ¼ CV0 = . For V0 ¼ 10 V, C ¼ 1 mF, and ¼ 10 3 s, we get I0 ¼ 10 mA. (b), respectively.
Graphs of v and i are shown in Figs. 6-14(a) and
Fig. 6-14
6.11
DAMPED SINUSOIDS A damped sinusoid, with its amplitude decaying exponentially has the form vðtÞ ¼ Ae at cos ð!t þ Þ
ð38Þ
This function will be discussed in more detail in Chapter 8. EXAMPLE 6.24 The current i ¼ I0 e at cos !t passes through a series RL circuit. (a) Find vRL , the voltage across this combination. (b) Compute vRL for I0 ¼ 3 A, a ¼ 2, ! ¼ 40 rad/s, R ¼ 5 and L ¼ 0:1 H. Sketch i as a function of time.