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CHAP. 6]

6.16

WAVEFORMS AND SIGNALS

121

A radar signal sðtÞ, with amplitude Vm ¼ 100 V, consists of repeated tone bursts. Each tone burst lasts Tb ¼ 50 ms. The bursts are repeated every Ts ¼ 10 ms. Find Seff and the average power in sðtÞ. pffiffiffi Let Veff ¼ Vm 2 be the effective value of the sinusoid within a burst. The energy contained in a single 2 2 burst is Wb ¼ Tb Veff . The energy contained in one period of sðtÞ is Ws ¼ Ts Seff . Since Wb ¼ Ws ¼ W, we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ¼ Ts Seff Seff ¼ ðTb =Ts ÞVeff Seff ¼ Tb =Ts Veff ð40Þ Tb Veff Substituting the values of Tb , Ts , and Veff into (40), we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi Seff ¼ ð50  106 Þ=ð10  103 Þ ð100= 2Þ ¼ 5 V Then W ¼ 102 ð25Þ ¼ 0:25 J.

The average power in sðtÞ is 2 2 =Ts ¼ Seff ¼ 25 W P ¼ W=Ts ¼ Ts Seff

2 2 The average power of sðtÞ ffi is represented by Seff and its peak power by Veff . The ratio of peak power to pffiffiffiffiffiffiffiffiffiffiffiffi average power is Ts =Tb . In this example the average power and the peak power are 25 W and 5000 W, respectively.

6.17

An appliance uses Veff ¼ 120 V at 60 Hz and draws Ieff ¼ 10 A with a phase lag of 608. Express v, i, and p ¼ vi as functions of time and show that power is periodic with a dc value. Find the frequency, and the average, maximum, and minimum values of p. pffiffiffi v ¼ 120 2 cos !t

pffiffiffi i ¼ 10 2 cosð!t  608Þ

p ¼ vi ¼ 2400 cos !t cos ð!t  608Þ ¼ 1200 cos 608 þ 1200 cos ð2!t  608Þ ¼ 600 þ 1200 cos ð2!t  608Þ The power function is periodic. The frequency f ¼ 2  60 ¼ 120 Hz and Pavg ¼ 600 W, pmax ¼ 600 þ 1200 ¼ 1800 W, pmin ¼ 600  1200 ¼ 600 W.

6.18

A narrow pulse is of 1-A amplitude and 1-ms duration enters a 1-mF capacitor at t ¼ 0, as shown in Fig. 6-19. The capacitor is initially uncharged. Find the voltage across the capacitor.

Fig. 6-19 The voltage across the capacitor is 1 VC ¼ C

ðt 1

8 <0 i dt ¼ 106 t : 1V

ðVÞ

for t < 0 for 0 < t < 1 ms (charging period) for t > 1 ms

If the same amount of charge were deposited on the capacitor in zero time, then we would have v ¼ uðtÞ (V) and iðtÞ ¼ 106 ðtÞ (A).

6.19

The narrow pulse is of Problem 6.18 enters a parallel combination of a 1-mF capacitor and a 1-M resistor (Fig. 6-20). Assume the pulse ends at t ¼ 0 and that the capacitor is initially uncharged. Find the voltage across the parallel RC combination.

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